link-springer-com-443.webvpn.jmu.edu.cn€¦ · The Power Electronics and Power Systems Series...

Power Electronics and Power Systems

Arun G. PhadkeJames S. Thorp

Synchronized Phasor Measurements and Their Applications Second Edition

Power Electronics and Power Systems

Series editors

Joe H. Chow, Rensselaer Polytechnic Institute, Troy, New York, USAAlex M. Stankovic, Tufts University, Medford, Massachusetts, USADavid Hill, The University of Hong Kong, Sydney, New South Wales, Australia

The Power Electronics and Power Systems Series encompasses power electronics,electric power restructuring, and holistic coverage of power systems. The Seriescomprises advanced textbooks, state-of-the-art titles, research monographs, profes-sional books, and reference works related to the areas of electric power transmissionand distribution, energy markets and regulation, electronic devices, electricmachines and drives, computational techniques, and power converters andinverters. The Series features leading international scholars and researchers withinauthored books and edited compilations. All titles are peer reviewed prior topublication to ensure the highest quality content. To inquire about contributing tothe Power Electronics and Power Systems Series, please contact Dr. Joe Chow,Administrative Dean of the College of Engineering and Professor of Electrical,Computer and Systems Engineering, Rensselaer Polytechnic Institute, JonssonEngineering Center, Office 7012, 110 8th Street, Troy, NY USA, 518-276-6374,[email protected].

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Arun G. Phadke • James S. Thorp

Synchronized PhasorMeasurements and TheirApplicationsSecond Edition

123

Arun G. PhadkeDepartment of Electrical and ComputerEngineering

Virginia TechBlacksburg, VAUSA

James S. ThorpDepartment of Electrical and ComputerEngineering

Virginia TechBlacksburg, VAUSA

ISSN 2196-3185 ISSN 2196-3193 (electronic)Power Electronics and Power SystemsISBN 978-3-319-50582-4 ISBN 978-3-319-50584-8 (eBook)DOI 10.1007/978-3-319-50584-8

Library of Congress Control Number: 2016959259

© Springer International Publishing AG 2008, 2017This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made.

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Preface to the Second Edition

The first edition of this book was published in the beginning of 2008. Theseintervening years have seen a steadily growing adaption of this technology toimprove the monitoring, protection, and control of the power systems around theworld. Many countries are busily installing phasor measurement units (PMUs) andwide area measurement systems (WAMS) on their networks. In the USA, activitiesrelated to this technology continue to be supported by various electric utilities,government agencies—particularly the US Department of Energy—and many otherindustry groups. Although the number of substations equipped by PMUs andWAMS remains small compared to the size of the US power grid, an excellent starthas been made, and in general, individual US utilities continue their efforts toimprove the coverage provided by WAMS.

In our view, the new frontier in this area is the development of new applicationsof PMUs and WAMS. Work on new applications continues in many organizationsaround the world. Postmortem analysis after major system disturbances is a veryuseful application which assists in determining the causes and timelines of eventswhich contributed to the system disturbance. To this end, obtaining and keepingrecords of such events are important, requiring selection of important data fromWAMS which will validate system models used in simulations of the postmortemanalysis. At present, there is no convenient tool which will examine all the saveddata and determine which data segments are of most interest. We expect applica-tions to achieve such a data selection task to be one of the outcomes of the presentresearch.

State estimation using PMU measurements alone or in conjunction with SCADAis being developed in many organizations. A new development in this regard is toperform state estimation in phase coordinates, rather than using positive sequenceestimation. Protection and control using WAMS is also being developed by manyorganizations, and some of those research results are reported in this book.

We have clarified many sections in this book which we hope will make our ideasclearer.

A particularly beneficial feature of this revision is that we have been able to drawupon the latest work done by our colleagues. Instead of paraphrasing their work

v

ourselves, we invited them to make direct contributions to this book. Three col-leagues and friends have made such contributions, which have been attributed tothem at appropriate places in this book. Dr. Anamitra Pal, a former graduate studentat Virginia Tech, has contributed to several sections in this book, and in addition, hecontributed Chap. 9, which deals with the advanced developments in the area ofcontrol using wide area measurement systems. Kenneth E. Martin has been leadingthe effort to create PMUs and WAMS standards in the Power System RelayingCommittee of the Power and Energy Society, as well as the parallel effort in IEC tomake the standards more precise and relevant to the evolving technology. He hascontributed the new Sect. 5.6, which deals with the present state of these standards.We recognize that standards are a moving target, and they will surely be modified inthe coming years as newer developments in this technology emerge. The work onthe calibration of instrument transformers and estimating the transmission networkparameters was originally developed by Dr. Zhongyu Wu while she was a Ph.D.student at Virginia Tech. She has continued to work in this area and has contributedto Sect. 7.6 bringing together the latest developments in this field.

We are grateful to Dr. Pal, Mr. Martin, and Dr. Wu for their important contri-butions to our book. We believe this has added greatly to the overall coverage of thesubject.

Finally, we wish to thank many colleagues throughout the world who have usedour book and have corresponded with us on questions of mutual interest. We hopethis new edition of this book will continue to be of interest to students, researchers,and industry practitioners.

Wilsonville, Oregon Arun G. PhadkeBlacksburg, Virginia James S. ThorpJanuary 2017

vi Preface to the Second Edition

Preface to the First Edition

Synchronized phasor measurements have become the measurement technique ofchoice for electric power systems. They provide positive sequence voltage andcurrent measurements synchronized to within a microsecond. This has been madepossible by the availability of Global Positioning System and the sampled dataprocessing techniques developed for computer-relaying applications. In addition topositive sequence voltages and currents, these systems also measure local frequencyand rate of change of frequency and may be customized to measure harmonics,negative and zero-sequence quantities, as well as individual phase voltages andcurrents. At present, there are about two dozen commercial manufacturers of phasormeasurement units (PMUs), and industry standards developed in the Power SystemRelaying Committee of IEEE have made possible the interoperability of units fromdifferent manufacturers.

Recent spate of spectacular blackouts on power systems throughout the worldhas provided an added impetus to wide-scale deployment of PMUs. Positivesequence measurements provide the most direct access to the state of the powersystem at any given instant. Many applications of these measurements have beendiscussed in the technical literature, and no doubt many more applications will bedeveloped in the coming years.

The authors have been associated with this technology since its birth, and theyand their colleagues and students have produced a rich body of literature on thesubject of phasor measurement technology and its applications. Other researchersaround the world have also made significant contributions to the field. Our aim inwriting this book is to present to the interested reader a coherent account of thedevelopment of the technology and of the emerging applications of these mea-surements. It is our hope that this book will help power system engineers under-stand the basics of synchronized phasor measurement systems. This technology isbound to inaugurate an era of improved monitoring, protection and control of powersystems.

Blacksburg, Virginia Arun G. PhadkeJanuary 2008 James S. Thorp

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Contents

Part I Phasor Measurement Techniques

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Historical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Phasor Representation of Sinusoids . . . . . . . . . . . . . . . . . . . . . . 51.3 Fourier Series and Fourier Transform . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Sampled Data and Aliasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . . . . . . . . . 16

1.5.1 DFT and Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . 201.5.2 DFT and Phasor Representation . . . . . . . . . . . . . . . . . . 22

1.6 Leakage Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Phasor Estimation of Nominal Frequency Inputs . . . . . . . . . . . . . . . 292.1 Phasors of Nominal Frequency Signals. . . . . . . . . . . . . . . . . . . . 292.2 Formulas for Updating Phasors. . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.1 Non-recursive Updates . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.2 Recursive Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Effect of Signal, Noise, and Window Length . . . . . . . . . . . . . . . 342.3.1 Errors in Sampling Times . . . . . . . . . . . . . . . . . . . . . . . 37

2.4 Phasor Estimation with Fractional Cycle Data Window . . . . . . . 372.5 Quality of Phasor Estimate and Transient Monitor . . . . . . . . . . . 392.6 DC Offset in Input Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.7 Non-DFT Estimators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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3 Phasor Estimation at Off-Nominal Frequency Inputs . . . . . . . . . . . . 473.1 Types of Frequency Excursions Found in Power Systems . . . . . 473.2 DFT Estimate at Off-Nominal Frequency with a Nominal

Frequency Clock. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.1 Input Signal at Off-Nominal Frequency . . . . . . . . . . . . . 48

3.3 Post Processing for Off-Nominal Frequency Estimates . . . . . . . . 563.3.1 A Simple Averaging Digital Filter for 2f0 . . . . . . . . . . . 563.3.2 A Re-sampling Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Phasor Estimates of Pure Positive Sequence Signals. . . . . . . . . . 583.4.1 Symmetrical Components . . . . . . . . . . . . . . . . . . . . . . . 58

3.5 Estimates of Unbalanced Input Signals . . . . . . . . . . . . . . . . . . . . 623.5.1 Unbalanced Inputs at Off-Nominal Frequency . . . . . . . . 623.5.2 A Nomogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.6 Sampling Clocks Locked to the Power Frequency . . . . . . . . . . . 693.7 Non-DFT Type Phasor Estimators . . . . . . . . . . . . . . . . . . . . . . . 71References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Frequency Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.1 Historical Overview of Frequency Measurement . . . . . . . . . . . . 734.2 Frequency Estimates from Balanced Three-Phase Inputs . . . . . . 744.3 Frequency Estimates from Unbalanced Inputs . . . . . . . . . . . . . . 784.4 Nonlinear Frequency Estimators . . . . . . . . . . . . . . . . . . . . . . . . . 784.5 Other Techniques for Frequency Measurements . . . . . . . . . . . . . 81References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Phasor Measurement Units and Phasor Data Concentrators . . . . . . 835.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.2 A Generic PMU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3 The Global Positioning System. . . . . . . . . . . . . . . . . . . . . . . . . . 855.4 Hierarchy for Phasor Measurement Systems . . . . . . . . . . . . . . . . 865.5 Communication Options for PMUs. . . . . . . . . . . . . . . . . . . . . . . 885.6 Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.6.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.6.2 Synchrophasor Measurement . . . . . . . . . . . . . . . . . . . . . 945.6.3 Synchrophasor Communication . . . . . . . . . . . . . . . . . . . 1035.6.4 PDC Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Transient Response of Phasor Measurement Units . . . . . . . . . . . . . . 1116.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.2 Nature of Transients in Power Systems . . . . . . . . . . . . . . . . . . . 112

6.2.1 Electromagnetic Transients . . . . . . . . . . . . . . . . . . . . . . 1126.2.2 Electromechanical Transients. . . . . . . . . . . . . . . . . . . . . 113

6.3 Transient Response of Instrument Transformers . . . . . . . . . . . . . 1176.3.1 Voltage Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.3.2 Current Transformers. . . . . . . . . . . . . . . . . . . . . . . . . . . 118

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6.4 Transient Response of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.4.1 Surge Suppression Filters . . . . . . . . . . . . . . . . . . . . . . . 1196.4.2 Anti-aliasing Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.5 Transient Response During Electromagnetic Transients . . . . . . . 1216.6 Transient Response During Power Swings . . . . . . . . . . . . . . . . . 122

6.6.1 Amplitude Modulation. . . . . . . . . . . . . . . . . . . . . . . . . . 1236.6.2 Frequency Modulation. . . . . . . . . . . . . . . . . . . . . . . . . . 1256.6.3 Simultaneous Amplitude

and Frequency Modulation . . . . . . . . . . . . . . . . . . . . . . 1266.6.4 Aliasing Considerations in Phasor Reporting Rates . . . . 128

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Part II Phasor Measurement Applications

7 State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.1 History-Operator’s Load Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.2 Weighted Least Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.2.1 Least Square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.2.2 Linear Weighted Least Squares . . . . . . . . . . . . . . . . . . . 1357.2.3 Condition Numbers, Leverage,

and LAV in Linear Least Squares . . . . . . . . . . . . . . . . . 1377.2.4 Nonlinear Weighted Least Squares . . . . . . . . . . . . . . . . 140

7.3 Static State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417.4 Bad Data Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1457.5 State Estimation with Phasor Measurements . . . . . . . . . . . . . . . . 147

7.5.1 Linear State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 1507.5.2 An Alternative for Including Phasor Measurements. . . . 1537.5.3 Incomplete Observability Estimators . . . . . . . . . . . . . . . 1537.5.4 Partitioned State Estimation. . . . . . . . . . . . . . . . . . . . . . 159

7.6 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.6.1 Calibration with Positive Sequence Measurements . . . . 1657.6.2 Calibration with Phase Measurements . . . . . . . . . . . . . . 1687.6.3 Simultaneous Calibration of Line Parameters

and Transducers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1747.7 Dynamic Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8 Control with Phasor Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1858.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1858.2 Linear Optimal Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1858.3 Linear Optimal Control Applied to the Nonlinear Problem. . . . . 1878.4 Coordinated Control of Oscillations . . . . . . . . . . . . . . . . . . . . . . 193

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8.5 Polytopic Control Using LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . 1978.5.1 Phasor Measurement-Based Adaptive Control . . . . . . . . 2058.5.2 Future Research on Phasor-Based Controls . . . . . . . . . . 207

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

9 Phasor Measurement-Enabled Decision Making . . . . . . . . . . . . . . . . 211Anamitra Pal9.1 Discrete Event Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2119.2 Decision Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

9.2.1 Classification and Regression Tree (CART) . . . . . . . . . 2149.2.2 Fisher’s Linear Discriminant Applied to

Synchrophasor Data (FLDSD) Technique . . . . . . . . . . . 2159.2.3 Applications of FLDSD in Power Systems . . . . . . . . . . 218

9.3 Synchrophasor Data Conditioning and Validation. . . . . . . . . . . . 2229.3.1 Three Sample-Based Quadratic

Prediction Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 2239.3.2 A Methodology for Performing Synchrophasor

Data Conditioning and Validation . . . . . . . . . . . . . . . . . 2279.3.3 Alternate Approaches for Addressing Data Quality

Issues in a LSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

10 Protection Systems with Phasor Inputs . . . . . . . . . . . . . . . . . . . . . . . 24510.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24510.2 Differential Protection of Transmission Lines . . . . . . . . . . . . . . . 24510.3 Distance Relaying of Multiterminal Transmission Lines . . . . . . . 24810.4 Adaptive Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

10.4.1 Adaptive Out-of-Step Protection . . . . . . . . . . . . . . . . . . 25010.4.2 Security Versus Dependability. . . . . . . . . . . . . . . . . . . . 25310.4.3 Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25510.4.4 Adaptive System Restoration. . . . . . . . . . . . . . . . . . . . . 255

10.5 Control of Backup Relay Performance . . . . . . . . . . . . . . . . . . . . 25710.5.1 Hidden Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

10.6 Intelligent Islanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26110.7 Supervisory Load Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

11 Electromechanical Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . 26511.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26511.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26711.3 Electromechanical Telegrapher’s Equation . . . . . . . . . . . . . . . . . 27111.4 Continuum Voltage Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . 27311.5 Effects on Protection Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 275

11.5.1 Overcurrent Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27611.5.2 Impedance Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

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11.5.3 Out-of-Step Relays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27711.5.4 Load Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

11.6 Dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27911.7 Parameter Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Contents xiii

Part IPhasor Measurement Techniques

Chapter 1Introduction

1.1 Historical Overview

Phase angles of voltage phasors of power network buses have always been ofspecial interest to power system engineers. It is well known that active (real) powerflow in a power line is very nearly proportional to the sine of the angle differencebetween voltages at the two terminals of the line. As many of the planning andoperational considerations in a power network are directly concerned with the flowof real power, measuring angle differences across transmission have been of con-cern for many years. The earliest modern application involving direct measurementof phase angle differences was reported in three papers in the early 1980s [1–3].These systems used LORAN-C, GOES satellite transmissions, and the HBG radiotransmissions (in Europe) in order to obtain synchronization of reference time atdifferent locations in a power system. The next available positive-goingzero-crossing of a phase voltage was used to estimate the local phase angle withrespect to the time reference. Using the difference of measured angles on a commonreference at two locations, the phase angle difference between voltages at two buseswas established. Measurement accuracies achieved in these systems were of theorder of 40 μs. Single-phase voltage angles were measured, and of course, noattempt was made to measure the prevailing voltage phasor magnitude. Neither wasany account taken of the harmonics contained in the voltage waveform. Thesemethods of measuring phase angle differences are not suitable for generalization forwide-area phasor measurement systems and remain one-of-a-kind systems whichare no longer in use.

The modern era of phasor measurement technology has its genesis in researchconducted on computer relaying of transmission lines. Early work on transmissionline relaying with microprocessor-based relays showed that the available computerpower in those days (1970s) was barely sufficient to manage the calculations neededto perform all the transmission line relaying functions.

© Springer International Publishing AG 2017A.G. Phadke and J.S. Thorp, Synchronized Phasor Measurementsand Their Applications, Power Electronics and Power Systems,DOI 10.1007/978-3-319-50584-8_1

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A significant portion of the computations was dedicated to solving six fault loopequations at each sample time in order to determine whether any one of the tentypes of faults possible on a three-phase transmission line is present. The search formethods which would eliminate the need to solve the six equations finally yielded anew relaying technique which was based on symmetrical component analysis ofline voltages and currents. Using symmetrical components, and certain quantitiesderived from them, it was possible to perform all fault calculations with a singleequation. In a paper published in 1977 [4], this new symmetrical component-basedalgorithm for protecting a transmission line was described. As a part of this theory,efficient algorithms for computing symmetrical components of three-phase voltagesand currents were described, and the calculation of positive sequence voltages andcurrents using the algorithms of that paper gave an impetus for the development ofmodern phasor measurement systems. It was soon recognized that the positivesequence measurement (a part of the symmetrical component calculation) is of greatvalue in its own right. Positive sequence voltages of a network constitute the statevector of a power system, and it is of fundamental importance in all of the powersystem analysis. The first paper to identify the importance of positive sequencevoltage and current phasor measurements and some of the uses of these measure-ments was published in 1983 [5], and this last paper can be viewed as the startingpoint of modern synchronized phasor measurement technology. The GlobalPositioning System (GPS) [6] was beginning to be fully deployed around that time.It became clear that this system offered the most effective way of synchronizingpower system measurements over great distances. The first prototypes of themodern phasor measurement units (PMUs) using GPS were built at Virginia Techin the early 1980s, and two of these prototypes are shown in Fig. 1.1. The prototypePMU built at Virginia Tech was deployed at a few substations of the BonnevillePower Administration, the American Electric Power Service Corporation, and theNew York Power Authority. The first commercial manufacture of PMUs withVirginia Tech collaboration was started by Macrodyne in 1991 [7]. At present, anumber of manufacturers offer PMUs as a commercial product, and deployment ofPMUs on power systems is being carried out in earnest in many countries aroundthe world. IEEE published a standard in 1991 [8, 9], governing the format of datafiles created and transmitted by the PMU. A revised version of the standard wasissued in 2005.

Concurrently with the development of PMUs as measurement tools, researchwas ongoing on the applications of the measurements provided by the PMUs. Theseapplications will be discussed in greater detail in later chapters of this book. It canbe said now that finally the technology of synchronized phasor measurements hascome of age, and most modern power systems around the world are in the processof installing wide-area measurement systems consisting of the phasor measurementunits.

4 1 Introduction

1.2 Phasor Representation of Sinusoids

Consider a pure sinusoidal quantity given by

xðtÞ ¼ Xm cosðxtþ/Þ ð1:1Þ

ω being the frequency of the signal in radians per second, and ϕ being the phaseangle in radians. Xm is the peak amplitude of the signal. The root-mean-square(RMS) value of the input signal is (Xm/√2). Recall that RMS quantities are par-ticularly useful in calculating active power and reactive power in an AC circuit.

Equation (1.1) can also be written as

xðtÞ ¼ Re Xmejðxtþ/Þn o

¼ Re ejðxtÞn o

Xmej/h i

It is customary to suppress the term ej(ωt) in the expression above, with theunderstanding that the frequency is ω. The sinusoid of Eq. (1.1) is represented by acomplex number X known as its phasor representation:

xðtÞ $ X ¼ Xm=ffiffiffi2

p� �ej/ ¼ ðXm=

ffiffiffi2

pÞ cos/þ j sin/½ � ð1:2Þ

(a)

GPSreceiver

PMU

Signalcondition-ing

(b)

UserInter-face

Fig. 1.1 The first phasor measurement units built at the power systems research laboratory atVirginia Tech. The GPS receiver clock was external to the PMU, and with the small number ofGPS satellites deployed at that time, the clock had to be equipped with a precision internaloscillator which maintained accurate time in the absence of visible satellites

1.2 Phasor Representation of Sinusoids 5

A sinusoid and its phasor representation are shown in Fig. 1.2.It was stated earlier that the phasor representation is only possible for a pure

sinusoid. In practice, a waveform is often corrupted with other signals of differentfrequencies. It then becomes necessary to extract a single frequency component ofthe signal (usually the principal frequency of interest in an analysis) and thenrepresent it by a phasor. Extracting a single frequency component is often done witha Fourier transform calculation. In sampled data systems, this becomes the discreteFourier transform (DFT) or the fast Fourier transform (FFT). These transforms arereviewed in the next section. The phasor definition also implies that the signal isunchanging for all time. However, in all practical cases, it is only possible toconsider a portion of time span over which the phasor representation is considered.This time span, also known as the ‘data window’ is very important in phasorestimation of practical waveforms. It will be considered in greater detail in latersections.

1.3 Fourier Series and Fourier Transform

1.3.1 Fourier Series

Let x(t) be a periodic function of t, with a period equal to T. Then, x(t + kT) = x(t) for all integer values of k. A periodic function can be expressed as a Fourierseries:

φ

Real

Imag

inar

yφ

t=0

(a) (b)

PhasorXm

Fig. 1.2 A sinusoid (a) and its representation as a phasor (b). The phase angle of the phasor isarbitrary, as it depends upon the choice of the axis t = 0. Note that the length of the phasor is equalto the RMS value of the sinusoid

6 1 Introduction

xðtÞ ¼ a02

þX1k¼1

ak cos2pktT

� �þ

X1k¼1

bk sin2pktT

� �ð1:3Þ

where the constants ak and bk are given by

ak ¼ 2T

Zþ T=2

�T=2

xðtÞ cos 2pktT

� �dt; k ¼ 0; 1; 2; . . .

bk ¼ 2T

Zþ T=2

�T=2

xðtÞ sin 2pktT

� �dt; k ¼ 1; 2; . . .

ð1:4Þ

The Fourier series can also be written in the exponential form

xðtÞ ¼X1k¼�1

ak ej2pktT ð1:5Þ

With

ak ¼ 1T

Zþ T=2

�T=2

xðtÞe�j2pktT dt; k ¼ 0;�1;�2; . . . ð1:6Þ

Note that the summation in Eq. (1.5) goes from −∞ to +∞, while the sum-mations in Eq. (1.3) go from 1 to +∞. The change in summation limits isaccomplished by noting that the cosine and sine functions are even and oddfunctions of k, and thus, expanding the summation limits to (−∞ to +∞) andremoving the factor 2 in front of the integrals for ak and bk lead to the desiredexponential form of the Fourier series.

Example 1.1 Consider a periodic square wave signal with a period T as shown inFig. 1.3. This is an even function of time. The Fourier coefficients (in exponentialform) are given by

ak ¼ 1T

Zþ T=4

�T=4

e�j2pktT dt; k ¼ 0;�1;�2; . . .

¼ 1pk

sinkp2

� �

1.3 Fourier Series and Fourier Transform 7

Hence,

α0 = 1/2,α1 = 1/π, α−1 = 1/π,α3 = −1/3π, α−3 = −1/3π, andα5 = 1/5π, α−5 = 1/5π, etc., and all even coefficients are zero.

Thus, the Fourier series of the square wave signal is

xðtÞ ¼ 12þ 2

pcos

2ptT

� �� 13cos

6ptT

� �þ 1

5cos

10ptT

� ��

� �

The sum of first seven terms of the series is shown in Fig. 1.4.

1.3.2 Fourier Transform

There are several excellent textbooks devoted to the subject of Fourier transforms[10, 11]. The reader should consult those books for a more complete account of theFourier transform theory. Here, we present only those topics which are of directinterest for phasor estimation in power system applications.

The Fourier transform of a continuous time function x(t) satisfying certain in-tegrability conditions [10] is given by

Xðf Þ ¼Zþ1

�1xðtÞ e�j2pftdt ð1:7Þ

1

T/2- T/2

Fig. 1.3 A square wavefunction with a period T, withduty cycle equal to half, withthe t = 0 axis so chosen thatthe function is an evenfunction

T/2-T/2

1

Fig. 1.4 A square wave approximated by 7 terms of the Fourier series. With more terms, thewaveform approaches the square shape. The oscillations are known as the Gibbs phenomenon andare inescapable when step functions are approximated by the Fourier series

8 1 Introduction

and the inverse Fourier transform recovers the time function from its Fouriertransform:

xðtÞ ¼Zþ1

�1Xðf Þ ej2pftdf ð1:8Þ

An important function frequently used in calculations using sampled data is theimpulse function δ(t) defined by

xðt0Þ ¼Zþ1

�1dðt � t0ÞxðtÞdt ð1:9Þ

The impulse function (also known as a distribution or a Dirac delta function) is asampling function in the sense that when the integration in Eq. (1.9) is performed,the result is the sampled value of the function x(t) at t = t0. The integrals of the typeshown in Eq. (1.9) are known as convolutions. Thus, the sampling process atuniform intervals ΔT apart can be considered to be a convolution of the input signaland a string of impulse functions δ(t − kΔT) where k ranges from −∞ to +∞.

The convolutions of two time functions and their Fourier transforms have aconvenient relationship. Consider the convolution z(t) of two time functions x(t) andy(t)

zðtÞ ¼Zþ1

�1xðsÞyðs� tÞds � xðtÞ � yðtÞ ð1:10Þ

The important result regarding convolutions is the following property:

Property 1 The Fourier transform of a convolution is equal to the product of theFourier transform of the functions being convolved, or:

If sðtÞ ¼ xðtÞ � yðtÞ; then Sðf Þ ¼ Xðf Þ:Yðf Þ

and similarly, the inverse Fourier transform of a convolution of two Fouriertransforms is a product of the corresponding inverse Fourier transforms:

If Zðf Þ ¼ Xðf Þ � Yðf Þ; then zðtÞ ¼ xðtÞ � yðtÞNext, we illustrate the second of the above two statements. Consider the func-

tions x(t) = cos(ω0t) and y(t) = sin(ω0t), with ω0 = 2πf0. The Fourier transforms ofx(t) and y(t) are as follows:


Xðf Þ ¼Zþ1

�1cosð2pf 0tÞe�j2pftdt ¼

Zþ1

�1

e�j2pðf�f 0Þt þ e�j2pðf þ f 0Þt

2dt

¼ 12dðf � f 0Þþ dðf þ f 0Þ½ �

and similarily

Yðf Þ ¼ j2dðf þ f 0Þ � dðf � f 0Þ½ �

The Fourier transforms of a pure cosine wave of unit amplitude are a pair of realimpulse functions in frequency domain located at ±f0 and that of a pure sine waveof unit amplitude is a pair of imaginary impulse functions of opposite signs at ±f0.

The convolution of the two Fourier transforms determined above in the fre-quency domain is as follows:

Sðf Þ ¼Zþ1

�1

12dð/� f 0Þþ dð/þ f 0Þ½ � j

2dðf þ f 0 � /Þ � dðf � f 0 � /Þ½ �d/

¼ j4

Zþ1

�1dð/� f 0Þdðf þ f 0 � /Þþ dð/þ f 0Þdðf � f 0 � /Þd/½

�dð/� f 0Þdðf � f 0 � /Þ � dð/þ f 0Þdðf � f 0 � /Þ�d/

Using the sampling property of the integrals involving impulse functions

Sðf Þ ¼ j4dðf � 2f 0Þ � dðf � 2f 0Þ½ �

The inverse Fourier transform of S(f) is clearly

SðtÞ ¼ 12sinð4pftÞ ¼ sinð2pftÞ cosð2pftÞ ¼ xðtÞ:yðtÞ

This property of convolutions will be used in discussing the sampling processand the DFT. Some other properties of the Fourier transform which are particularlyuseful in our development are stated next with accompanying examples.

Property 2 The Fourier transform of an even function is an even function offrequency. If the even function is real, the Fourier transform is also real and even.

Consider an even function of time, x(t), so that x(−t) = x(t). Let x(t) be complex,x(t) = r(t) + js(t). The Fourier transform X(f) of this function is given by

10 1 Introduction

Xðf Þ ¼Zþ1

�1xðtÞe�j2pftdt ¼

Zþ1

�1rðtÞe�j2pftdt

Zþ1

�1sðtÞe�j2pftdt

¼Zþ1

�1rðtÞ cosð2pftÞdtþ j

Zþ1

�1rðtÞ sinð2pftÞdt

þ jZþ1

�1sðtÞ cosð2pftÞdt �

Zþ1

�1sðtÞ sinð2pftÞdt

The second and fourth integrals are zero, since the integrands are odd functionsof time. Thus,

Xðf Þ ¼Zþ1


Zþ1

�1sðtÞ cosð2pftÞdt

Since cos(2πft) = cos(−2πft), it follows that X(f) = X(−f).Also, if x(t) is real = r(t), the Fourier transform of x(t) is R(f), which is real and

even.

Property 3 The Fourier transform of an odd function is an odd function of fre-quency. If the odd function is real, the Fourier transform is imaginary and odd.

Consider an odd function of time, x(t), so that x(−t) = −x(t). Let x(t) be complex,x(t) = r(t) + js(t). The Fourier transform X(f) of this function is given by

Xðf Þ ¼Zþ1


Zþ1

�1rðtÞe�j2pftdtþ j

Zþ1

�1sðtÞe�j2pftdt

¼Zþ1


Zþ1

�1rðtÞ sinð2pftÞdtþ j

Zþ1

�1sðtÞ cosð2pftÞdt

�Zþ1


The first and third integrals are zero, since the integrands are odd functions oftime. Thus,

Xðf Þ ¼ jZþ1

�1rðtÞ sinð2pftÞdt �

Zþ1



Since sin(2πft) = −sin (−2πft), it follows that X(f) = −X(−f).Also, if x(t) is real = r(t), the Fourier transform of x(t) is jR(f), which is imag-

inary and odd.

Property 4 The Fourier transform of a real function has an even real part and anodd imaginary part.

Consider a real function of time x(t) = r(t) + j0. The Fourier transform is givenby

Xðf Þ ¼Zþ1


Zþ1

�1rðtÞ sinð2pftÞdt ¼ R1ðf Þþ jR2ðf Þ

Since the cosine and sine functions are, respectively, even and odd functions offrequency, it is clear that R1(f) is an even function, and R2(f) is an odd function offrequency.

Property 5 The Fourier transform of a periodic function is a series of impulsefunctions of frequency.

If x(t) is a periodic function of t, with a period equal to T, it can be expressed asan exponential Fourier series given by Eqs. (1.5) and (1.6):

xðtÞ ¼X1k¼�1

akej2pktT

with

ak ¼ 1T

ZþT=2

�T=2

xðtÞe�j2pktT dt; k ¼ 0;�1;�2; . . .

The Fourier transform of x(t) expressed in the exponential form is given by

Xðf Þ ¼Zþ1


Zþ1

�1

X1k¼�1

akej2pktT

" #e�j2pftdt ¼

X1k¼�1

Zþ1

�1ake

j2pktT e�j2pftdt

¼X1k¼�1

Zþ1

�1ake

�j2pt kT�ff gdt

where the order of the summation and integration has been reversed (assuming thatthis is permissible). Setting f0 = 1/T, the fundamental frequency of the periodicsignal, the integral of the exponential term in the last form is the impulse functionδ(kf0 − f), and thus, the Fourier transform of periodic x(t) is

12 1 Introduction

Xðf Þ ¼X1k¼�1

akd f � kT

� �; with

ak ¼ 1T

ZþT=2

�T=2

xðtÞe�j2pktT dt; k ¼ 0;�1;�2; . . .

These are a series of impulses located at multiples of the fundamental frequencyf0 of the periodic signal with impulse magnitudes being equal to amplitude of eachfrequency component in the input signal.

Property 6 The Fourier transform of a series of impulses is a series of impulsefunctions in the frequency domain.

Consider the function

xðtÞ ¼X1k¼�1

dðt � kTÞ

This is a periodic function with period T. Hence, its Fourier transform (byProperty 5 above) is

Xðf Þ ¼X1k¼�1

ak d f � kT

� �; with

ak ¼ 1T

Zþ T=2

�T=2

dðtÞe�j2pktT dt; k ¼ 0;�1;�2; � � �

Since the delta function in the integrand produces a sample of the exponent att = 0, ak is equal to 1/T for all k, and the Fourier transform of x(t) becomes

Xðf Þ ¼ 1T

X1k¼�1

d f � kT

� �

which is a pulse train in the frequency domain at intervals kf0 and a magnitude of 1/T.

Example 1.2 Consider a rectangular input signal as shown in Fig. 1.5. This is aneven function of time.

The Fourier transform of this time function is given by

Xðf Þ ¼Zþ1


Zt1 þT0

t1

e�j2pftdt ¼ ej2pf t1 þ T02ð ÞT0

sin 2pf T02

2pf T0

2


The first term in the Fourier transform is a phase shift factor and has beenomitted from the plot in Fig. 1.5b for convenience. If the rectangular wave iscentered at the origin, t1 = −T0/2, and the phase shift factor vanishes. This is also inkeeping with property 1.2 of the Fourier transform given above, which states thatthe Fourier transform of a real even function must be real and even function offrequency.

1.4 Sampled Data and Aliasing

Sampled data from input signals are the starting point of digital signal processing.The computation of phasors of voltages and currents begins with the samples of thewaveform taken at uniform intervals kΔT (k = 0, ±1, ±2, ±3, ±4,…}. Consider aninput signal x(t) which is being sampled, yielding sampled data x(kΔT). We mayview the sampled data as a time function x′(t) consisting of uniformly spacedimpulses, each with a magnitude x(kΔT)

x0ðtÞ ¼X1k¼�1

xðkDtÞ dðt � kDTÞ ð1:11Þ

It is interesting to determine the Fourier transform of the sampled data functiongiven by Eq. (1.11). Note that the sampled data function is a product of the functionx(t) and the sampling function δ(t − kΔT), the product being interpreted in the senseof Eq. (1.9). Hence, the Fourier transform X′(f) of x′(t) is the convolution of theFourier transforms of x(t) and of the unit impulse train. By property 1.6 of Sect. 1.3,the Fourier transform of the impulse train is

Dðf Þ ¼ 1DT

X1k¼�1

d f � kDT

� �ð1:12Þ

(b)

T0

1/T0 3/T0

2/T0

(a)

1

+T0/2-T0/2

Fig. 1.5 a A rectangular function of time with the t = 0 axis so chosen that the function is an evenfunction. The duration of the signal is 2T0. b Fourier transform of the function

14 1 Introduction

Hence, the Fourier transform of the sampled data function is the convolution ofΔ(f) and X(f)

X 0ðf Þ ¼ 1DT

Zþ1

�1Xð/Þ

X1k¼�1

d f � kDT

� /

� �d/

¼ 1DT

X1k¼�1

Zþ1

�1Xð/Þd f � k

DT� /

� �d/

¼ 1DT

X1k¼�1

X f � kDT

� �ð1:13Þ

Once again, the order of summation and integration has been reversed (it beingassumed that this is permissible), and the integral is evaluated by the use of thesampling property of the impulse function.

The relationship between the Fourier transforms of x(t) and x′(t) is as shown inFig. 1.6. The Fourier transform of x(t) is shown to be band-limited, meaning that ithas no components beyond a cutoff frequency fc. The sampled data have a Fouriertransform, which consist of an infinite train of the Fourier transforms of x(t) cen-tered at frequency intervals of (k/ΔT) for all k. Recall that the sampling interval isΔT, so that the sampling frequency fs = (1/ΔT).

If the cutoff frequency fc is greater than one-half of the sampling frequency fs, theFourier transform of the sampled data will be as shown in Fig. 1.7. In this case, thespectrum of the sampled data is different from that of the input signal in the regionwhere the neighboring spectra overlap as shown by the shaded region in Fig. 1.7.This implies that frequency components estimated from the sampled data in thisregion will be in error, due to a phenomenon known as aliasing.

X(f)

f

f

f

Δ(f)1/ΔT

fs =1/ΔT

fs

X’(f)

fc

fc

Fig. 1.6 Fourier transform of the sampled data function as a convolution of the transforms X(f) and Δ(f). The sampling frequency is fs, and X(f) is band-limited between ±fc

1.4 Sampled Data and Aliasing 15

It is clear from the above discussion that in order to avoid errors due to aliasing,the bandwidth of the input signal must be less than half of the sampling frequencyutilized in obtaining the sampled data. This requirement is known as the Nyquistcriterion.

In order to avoid aliasing errors, it is customary in all sampled data systems usedin phasor estimation to use anti-aliasing filters which band-limit the input signals tobelow half of the sampling frequency chosen. Note that the signal input cutofffrequency must be less than one-half of the sampling frequency. In practice, thesignal is usually band-limited to a value much smaller than the one required formeeting the Nyquist criterion. Anti-aliasing filters are generally passive low-passR-C filters [12], although active filters may also be used for obtaining a sharp cutoffcharacteristic. In addition to passive anti-aliasing filters, digital filters may also beused in special cases (e.g., with oversampling and decimation). All anti-aliasingfilters introduce frequency-dependent phase shift in the input signal which must becompensated for determining the phasor representation of the input signal. Thiswill be discussed further in Chap. 5 where the ‘synchrophasor’ standard isdescribed.

1.5 Discrete Fourier Transform (DFT)

Discrete Fourier transform (DFT) is a method of calculating the Fourier transformof a small number of samples taken from an input signal x(t). The Fourier transformis calculated at discrete steps in the frequency domain, just as the input signal issampled at discrete instants in the time domain. Consider the process of selecting N

f

f

Δ(f)

X(f)

1/ΔT

fs =1/ΔT

fs

f

X’(f)

fcfc

Fig. 1.7 Fourier transform of the sampled data function when the input signal is band-limited to afrequency greater than half of the sampling frequency. The estimate of frequencies from sampleddata in the shaded region will be in error because of aliasing

16 1 Introduction

http://dx.doi.org/10.1007/978-3-319-50584-8_5

samples: x(kΔT) with {k = 0, 1, 2, …, N − 1}, ΔT being the sampling interval.This is equivalent to multiplying the sampled data train by a windowing function w(t), which is a rectangular function of time with unit magnitude and a span of NΔT.With the choice of samples ranging from 0 to N − 1, it is clear that the windowingfunction can be viewed as starting at –ΔT/2 and ending at (N − 1/2)ΔT. Thefunction x(t), the sampling function Δ(t), and the windowing function w(t) alongwith their Fourier transforms are shown in Fig. 1.8.

Consider the collection of signal samples which fall in the data window: x(kΔT) with {k = 0, 1, 2, …, N − 1}. These samples can be viewed as beingobtained by the multiplication of the signal x(t), the sampling function δ(t), and thewindowing function ω(t):

yðtÞ ¼ xðtÞdðtÞwðtÞ ¼XN�1

k¼0

xðkDTÞdðt � kDTÞ ð1:14Þ

where once again the multiplication with the delta function is to be understood inthe sense of the integral in Eq. (1.9). The Fourier transform of the sampled win-dowed function y(t) is then the convolution of Fourier transforms of the threefunctions.

f

f

Δ(f )

X(f)

1/ΔT

fs =1/ΔT

fc

1T0

1/T0 3/T0

2/T0

t

x(t)

t

δ(t)

ΔT

T0 – ΔT/2– ΔT/2

W(f)

Fig. 1.8 Time functions and Fourier transforms x(t), δ(t), and ω(t). Note that once again, thephase shift factor from Ω(f) has been omitted

1.5 Discrete Fourier Transform (DFT) 17

The Fourier transform of y(t) is to be sampled in the frequency domain in orderto obtain the DFT of y(t). The discrete steps in the frequency domain are multiplesof 1/T0, where T0 is the span of the windowing function. The frequency samplingfunction Φ(f) is given by

Uðf Þ ¼X1

n¼�1d f � n

T0

� �ð1:15Þ

and its inverse Fourier transform (by property 1.6 of Fourier transforms) is

/ðtÞ ¼ T0

X1n¼�1

dðt � nT0Þ ð1:16Þ

In order to obtain the samples in the frequency domain, we must multiply theFourier transform Y(f) with F(f). To obtain the corresponding time-domain functionx′(t), we will require a convolution in the time domain of y(t) and ϕ(t):

x0ðtÞ ¼ yðtÞ � /ðtÞ

x0ðtÞ ¼ yðtÞ � /ðtÞ ¼XN�1

k¼0

xðkDTÞdðt � kDTÞ" #

� T0

X1n¼�1

dðt � nT0Þ" #

¼ T0

X1n¼�1

XN�1

k¼0

xðkDTÞdðt � kDT � nT0Þ" # ð1:17Þ

This function is periodic with a period T0. The functions x(t), y(t), and x′(t) areshown in Fig. 1.9. The windowing function limits the data to samples 0 throughN − 1, and the sampling in frequency domain transforms the original N samples intime domain to an infinite train of N samples with a period T0 as shown in Fig. 1.9c.Note that although the original function x(t) was not periodic, the function x′(t) is,and we may consider this function to be an approximation of x(t).

The Fourier transform of the periodic function x′(t) is a sequence of impulsefunctions in frequency domain by property 1.5 of the Fourier transform. Thus,

X 0ðf Þ ¼X1

n¼�1and f � n

T0

� �; with

an ¼ 1T0

ZT0�T0=2

�T0=2

x0ðtÞ e�j2pntT0 dt; n ¼ 0;�1;�2; . . .

ð1:18Þ

Substituting for x′(t) in the above expression for αn,

18 1 Introduction

an ¼ 1T0

ZT0�T0=2

�T0=2

T0

X1m¼�1

XN�1

k¼0

xðkDTÞdðt � kDT � mT0Þ" #( )

e�j2pntT0 dt;

n ¼ 0;�1;�2; . . .

ð1:19Þ

The index m designates the train of periods as shown in Fig. 1.9c. Since thelimits on the integration span one period only, we may remove the summation onm, and set m = 0, thus using only the samples over the period shown in bold inFig. 1.9c. Equation (1.15) then becomes

an ¼ZT0�T0=2

�T0=2

XN�1

k¼0

xðkDTÞdðt � kDTÞ" #

e�j2pntT0 dt; or

an ¼XN�1

k¼0

ZT0�T0=2

�T0=2

xðkDTÞdðt � kDTÞe�j2pntT0 dt;

¼XN�1

k¼0

xðkDTÞe�j2pnknDTT0 n ¼ 0;�1;�2; . . .

ð1:20Þ

Since there are N samples in the data window T0, NΔT = T0. And therefore

t t

y(t)

ΔT

x(t)

(a)(b)

T0 – ΔT/2– ΔT/2

(c)

Fig. 1.9 a The input function x(t), its samples (b), and c the Fourier transform of the windowedfunction x′(t)


an ¼XN�1

k¼0

xðkDTÞe�j2pknN ; with n ¼ 0;�1;�2; . . . ð1:21Þ

Although the index n goes over all positive and negative integers, it should benoted that there are only N distinct coefficients αn. Thus, αN+1 is the same as α1, andthe Fourier transform X′(f) has only N distinct values corresponding to frequenciesf = n/T0, with n ranging from 0 through N − 1:

X 0 nT0

� �¼

XN�1

k¼0

xðkDTÞe�j2pknN ; with n ¼ 0; 1; 2; . . .;N � 1 ð1:22Þ

Equation (1.22) is the definition of the DFT of N input samples taken at intervalsof ΔT. The DFT is symmetric about N/2, and the components beyond N/2 simplybelong to negative frequency. Thus, the DFT does not calculate frequency com-ponents beyond N/(2T0), which also happens to be the Nyquist limit to avoidaliasing errors.

Also note that any real function of time can be written as a sum of a real and anodd function. Consequently, by properties 2 and 3 above, any real function of timewill have real parts of the DFT as even functions of frequency and the imaginaryparts of the DFT will be odd functions of frequency.

1.5.1 DFT and Fourier Series

The Fourier series coefficients of a periodic signal can be obtained from the DFT ofits sampled data by dividing the DFT by N, the number of samples in the datawindow. Thus, the Fourier series for a function x(t) can be expressed by theformula:

xðtÞ ¼X1k¼�1

akej2pktT ¼

X1k¼�1

1N

XN�1

n¼0

xðkDTÞe�j2pknN

" #ej2pktT ð1:23Þ

As there are only N components in the DFT, the summation on k in Eq. (1.23) isfrom {k = 0, …, N − 1}.

Example 1.3 Consider a periodic function x(t) = 1 + cos 2πf0t + sin 2πf0t. Thefunction is already expressed in terms of its Fourier series, with a0 = 2, a1 = 1, andb1 = 1. The signal is sampled 16 times in one period of the fundamental frequency.The sampled data, the DFT, and the DFT divided by 16 (N, the number of samples)are shown in the following table. (Table 1.1)

20 1 Introduction

The last column contains the Fourier series coefficients. Note that the DCcomponent a0 appears in the 0th position, while the fundamental frequency com-ponent appears in the 2nd and 15th position. The cosine term being an evenfunction produces real parts which are even functions of frequency (0.5 at ±f0),while the sine term is an odd function of time and produces odd functions offrequency (±j0.5 at ± f0). The coefficient a1 is obtained by adding the real partscorresponding to f0 and −f0 in the (DFT/16) column, while the coefficient b1 isobtained by subtracting the imaginary part of the −f0 term from the imaginary partof the f0 term:

a0 = 2X0 = 2a1 = Real (X1 + XN−1) = 1b1 = Imaginary (X1 – XN−1) = 1

From the above example, it is clear that for real functions x(t), the Fourier seriescoefficients of a periodic function can be obtained from the DFT of its sampled databy the following formulas:

a0 = 2 � X0

ak = 2 � Real (Xk)bk = 2 � Imaginary (Xk) for k = 1, 2, …, N/2 − 1.

Table 1.1 Sampled data and Fourier transform of the periodic function t = 1 + cos 2πf0t + sin2πf0t

Sample No. x(t) Frequency DFT X = DFT/16

0 2.0000 0 16.0000 1.000

1 2.3066 f0 8.0000 + j8.0000 0.5000 + j0.5000

2 2.4142 2f0 0.0000 − j0.0000 0.0000 + j0.0000

3 2.3066 3f0 0.0000 − j0.0000 0.0000 + j0.0000

4 2.0000 4f0 0.0000 + j0.0000 0.0000 + j0.0000

5 1.5412 5f0 −0.0000 + j0.0000 0.0000 + j0.0000

6 1.0000 6f0 0.0000 + j0.0000 0.0000 + j0.0000

7 0.4588 7f0 0.0000 − j0.0000 0.0000 + j0.0000

8 0.0000 – −0.0000 0.0000 + j0.0000

9 −0.3066 −7f0 0.0000 + j0.0000 0.0000 + j0.0000

10 −0.4142 −6f0 0.0000 − j0.0000 0.0000 + j0.0000

11 −0.3066 −5f0 −0.0000 − j0.0000 0.0000 + j0.0000

12 −0.0000 −4f0 0.0000 + j0.0000 0.0000 + j0.0000

13 0.4588 −3f0 0.0000 + j0.0000 0.0000 + j0.0000

14 1.0000 −2f0 0.0000 + j0.0000 0.0000 + j0.0000

15 1.5412 −f0 8.0000 − j8.0000 0.5000 − j0.5000


1.5.2 DFT and Phasor Representation

A sinusoid x(t) with frequency kf0 with a Fourier series

xðtÞ ¼ ak cosð2pkf 0tÞþ bk sinð2kpf 0tÞ

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2k þ b2k

q� �cosð2pkf 0tþ/Þ where/ ¼ arctan

�bkak

� �ð1:24Þ

has a phasor representation (see Sect. 1.2)

Xk ¼ 1ffiffiffi2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2k þ b2k

q� �ej/ ð1:25Þ

where the square root of 2 in the denominator is to obtain the rms value of thesinusoid. The phasor in complex form becomes

Xk ¼ 1ffiffiffi2

p ðak � jbkÞ ð1:26Þ

Using the relationship of the Fourier series coefficients with the DFT, the phasorrepresentation of the kth harmonic component is given by

Xk ¼ 1ffiffiffi2

p 2N

XN�1

n¼0

xðnDTÞe�j2pknN

¼ffiffiffi2

p

N

XN�1

n¼0

xðnDTÞ cos2pknN

� �� j sin

2pknN

� �� ð1:27Þ

Using the notation x(nΔT) = xn, and 2π/N = θ (θ is the sampling angle measuredin terms of the period of the fundamental frequency component)

Xk ¼ffiffiffi2

p

N

XN�1

n¼0

xn cosðknhÞ � j sinðknhÞf g ð1:28Þ

If we define the cosine and sine sums as follows:

Xkc ¼ffiffiffi2

p

N

XN�1

n¼0

xn cosðknhÞ ð1:29Þ

Xks ¼ffiffiffi2

p

N

XN�1

n¼0

xn sinðknhÞ ð1:30Þ

22 1 Introduction

Then, the phasor Xk is given by

Xk ¼ Xkc � jXks ð1:31Þ

Equations (1.29) through (1.31) will be used to represent the phasor in most ofthe computations in the rest of our discussion.

Example 1.4 Consider a signal consisting of a DC component and 60, 120, and300 Hz components:

x tð Þ ¼ 0:5þ cosð120ptþ p=4Þþ 0:2 cosð240ptþ p=8Þþ 0:3 cos 600ptð Þ

Note that the signal is real, but not an even or odd function of time, and hence,by property 1.4 above, the real part of the Fourier transform will be even, and theimaginary part will be odd functions of frequency.

The signal is sampled at 1440 Hz, and the following 24 samples are obtainedover a window of 16.66 ms, which corresponds to one period of the 60 Hz signal.There will be 24 frequency samples of the DFT. They are calculated in Table 1.2)

Table 1.2 Spectrum created by the discrete Fourier transform

Sample No. x(k) Frequency DFT DFT/24

0 1.6919 0 12.0000 + j0.0000 0.5000 + j0.0000

1 1.1994 f0 8.4853 – j8.4853 0.3535 − j0.3535

2 0.5251 2f0 2.2173 − j0.9184 0.0924 − j0.0383

3 0.2113 3f0 0.0000 − j0.0000 0.0000 − j0.0000

4 0.2325 4f0 0.0000 − j0.0000 0.0000 − j0.0000

5 0.0915 5f0 3.6000 − j0.0000 0.1500 − j0.0000

6 −0.3919 6f0 −0.0000 − j0.0000 −0.0000 − j0.0000

7 −0.7776 7f0 0.0000 − j0.0000 0.0000 − j0.0000

8 −0.6420 8f0 −0.0000 − j0.0000 −0.0000 − j0.0000

9 −0.2113 9f0 −0.0000 − j0.0000 −0.0000 − j0.0000

10 −0.0474 10f0 −0.0000 + j0.0000 −0.0000 + j0.0000

11 −0.2454 11f0 0.0000 − j0.0000 0.0000 − j0.0000

12 −0.3223 – 0.0000 + j0.0000 0.0000 + j0.0000

13 0.0441 −11f0 0.0000 + j0.0000 0.0000 + j0.0000

14 0.5271 −10f0 −0.0000 − j0.0000 −0.0000 − j0.0000

15 0.6356 −9f0 −0.0000 + j0.0000 −0.0000 + j0.0000

16 0.4501 −8f0 −0.0000 + j0.0000 −0.0000 + j0.0000

17 0.5119 −7f0 0.0000 + j0.0000 0.0000 + j0.0000

18 1.0223 −6f0 −0.0000 + j0.0000 −0.0000 + j0.0000

19 1.5341 −5f0 3.6000 + j0.0000 0.1500 + j0.0000

20 1.5898 −4f0 0.0000 + j0.0000 0.0000 + j0.0000

21 1.3644 −3f0 0.0000 + j0.0000 0.0000 + j0.0000

22 1.3648 −2f0 2.2173 + j0.9184 0.0924 + j0.0383

23 1.6420 −f0 8.4853 + j8.4853 0.3535 + j0.3535


The Fourier series coefficients are as follows

a0 = 1.0a1 = 0.707b1 = −0.707a2 = 0.1848b2 = −0.0766a5 = 0.3b5 = 0.000

leading to the Fourier series

x tð Þ ¼ 0:5þ 0:707 cos 120ptð Þ � 0:707 sin 120ptð Þþ 0:1848 cos 240ptð Þ� 0:0766 sin 240ptð Þþ 0:3 cos 600ptð Þ

which agrees with the expression for the input signal.

1.6 Leakage Phenomena

The calculation of the DFT implies truncation of the sampled data outside the datawindow. As shown in Fig. 1.9c, the effect of sampling and windowing is to create aperiodic function which replicates the samples of the original function in repeatingdata windows. In general, this new function has discontinuities at the windowboundaries, and these discontinuities lead to a spurious spectrum which is a con-tinuous function of frequency. The side lobes of the Fourier transform of thewindowing function are superimposed on the spectrum of the original signal in thedata window and lead to errors in the Fourier transform calculated from the sampleddata. This phenomenon is known as the leakage effect.

Example 1.5 Consider an input signal with a frequency 60.05 Hz being sampled at1440 Hz, x(t) = cos(120.1πt), and xk = cos (120.1kπt/1440), k = 0, 1, …, 23(Table 1.3).

The Fourier series coefficient for the fundamental frequency is as follows:

a1 = 2 � Real X1(f)b1 = 2 � Imaginary X1(f)

The phasor in polar coordinates is found to be (1.0004/√2)∠0.1499°. The truevalue of the phasor as seen from the expression for x(t) is of course (1.0/√2)∠0°.The computation error is due to the leakage effect. In a later chapter, we willconsider the off-nominal frequency phasor estimation in greater detail and offeralternate methods of eliminating the small error introduced by the leakage effect.

Figure 1.5 shows the Fourier transform of a single square wave which is repe-ated in Fig. 1.10a. Fourier transform of the square wave shown in Fig. 1.5b isrepeated in Fig. 1.10b. The Fourier transform of the square windowing function

24 1 Introduction

used in calculating the DFT has side lobes as shown in Fig. 1.10b, which areresponsible for the leakage effect. It is possible to use other types of windowingfunctions which produce side lobes which are smaller than those produced by thesquare wave. A popular windowing function which has this property is the Hanningfunction, given by

hðtÞ ¼ 0:5 1þ cos2ptT0

� �for � T0=2� t� T0=2 ð1:32Þ

The Fourier transform of the Hanning function is

Hðf Þ ¼ T0=2

ð1=T20 � f 2Þ

sinðpfT0ÞðpfT0Þ

ð1:33Þ

The Hanning function, its Fourier transform, and the Fourier transform of thesquare window function are shown in Fig. 1.10c, d.

Table 1.3 Leakage effect in DFT calculations

Sample No. x(k) Frequency DFT DFT/24

0 1.0000 0 0.0199 + j0.0000 0.0008 + j0.0000

1 0.9659 f0 12.0048 − j0.0314 0.5002 − jj0.0013

2 0.8658 2f0 −0.0068 + j0.0000 −0.0003 + j0.0000

3 0.7066 3f0 −0.0026 + j0.0000 −0.0001 + j0.0000

4 0.4992 4f0 −0.0014 + j0.0000 −0.0001 + j0.0000

5 0.2578 5f0 −0.0010 + j0.0000 −0.0000 + j0.0000

6 −0.0013 6f0 −0.0007 + j0.0000 −0.0000 + j0.0000

7 −0.2603 7f0 −0.0005 + j0.0000 −0.0000 + j0.0000

8 −0.5015 8f0 −0.0005 + j0.0000 −0.0000 + j0.0000

9 −0.7085 9f0 −0.0004 + j0.0000 −0.0000 + j0.0000

10 −0.8671 10f0 −0.0004 + j0.0000 −0.0000 + j0.0000

11 −0.9665 11f0 −0.0003 + j0.0000 −0.0000 + j0.0000

12 −1.0000 – −0.0003 + j0.0000 −0.0000 − j0.0000

13 −0.9652 −11f0 −0.0003 − j0.0000 −0.0000 − j0.0000

14 −0.8645 −10f0 −0.0004 − j0.0000 −0.0000 − j0.0000

15 −0.7048 −9f0 −0.0004 − j0.0000 −0.0000 − j0.0000

16 −0.4970 −8f0 −0.0005 − j0.0000 −0.0000 − j0.0000

17 −0.2552 −7f0 −0.0005 − j0.0000 −0.0000 − j0.0000

18 0.0039 −6f0 −0.0007 − j0.0000 −0.0000 − j0.0000

19 0.2628 −5f0 −0.0010 − j0.0000 −0.0000 − j0.0000

20 0.5038 −4f0 −0.0014 − j0.0000 −0.0001 − j0.0000

21 0.7103 −3f0 −0.0026 − j0.0000 −0.0001 − j0.0000

22 0.8684 −2f0 −0.0068 − j0.0000 −0.0003 − j0.0000

23 0.9672 −f0 12.0048 + j0.0314 0.5002 + j0.0013

1.6 Leakage Phenomena 25

It is shown in Fig. 1.10 that the Hanning windowing function has side lobeswhich are much smaller than those of the square windowing function. Thus, usingHanning function in calculating a DFT leads to much smaller leakage effect, andconsequently, the errors due to this effect are much reduced.

Another function known as the Hamming function is sometimes used as awindowing function. This function is quite similar to the Hanning function and isgiven by [13]

hðtÞ ¼ 0:54þ 0:46 cos2ptT0

� �for � T0=2� t� T0=2 ð1:34Þ

Other windowing functions could be used to meet specific requirementsregarding the leakage effect.

It is worth pointing out that in power system work, the principal contributor tothe leakage effect is the off-nominal frequency input signals, when the samplingfrequency is based upon the nominal power system frequency. For example, onemay use a sampling frequency of 1440 Hz, which corresponds to 24 samples perperiod of the system nominal frequency of 60 Hz, while the actual power systemfrequency may be different from 60 Hz. Power system frequency never deviatesfrom the nominal value by more than a few millihertz. In such a case, the dis-continuity at the window boundary is quite small, and the leakage effect even withthe square windowing function is quite small. It is therefore a common practice inpower system work to use the square windowing function because of its simplicity.

(a) (b)

(c) (d)

1

1

T0/2- T0/2

T0/2- T0/2

f0 = 1/T0- f0

- f0f0

T0

T0/2

Fig. 1.10 a The rectangular window function and b its Fourier transform. The Hanning function(c), and its Fourier Transform (d)

26 1 Introduction

References

1. Missout, G., & Girard, P., (1980). Measurement of bus voltage angle between Montreal andSept-Iles. IEEE Transactions on PAS, 99(2), 536–539.

2. Missout, G., Beland, J., & Bedard, G. (1981). Dynamic measurement of the absolute voltageangle on long transmission lines. IEEE Transactions on PAS, 100(11), 4428–4434.

3. Bonanomi, P. (1981). Phase angle measurements with synchronized clocks—Principles andapplications. IEEE Transactions on PAS, 100(11), 5036–5043.

4. Phadke, A. G., Hlibka, T., & Ibrahim, M. (1977). Fundamental basis for distance relayingwith symmetrical components. IEEE Transactions on PAS, 96(2), 635–646.

5. Phadke, A. G., Thorp, J. S., & Adamiak, M. G. (1983). A new measurement technique fortracking voltage phasors, local system frequency, and rate of change of frequency. IEEETransactions on PAS, 102(5), 1025–1038.

6. There is great wealth of information about the GPS system available in various technicalpublications. A highly readable account for the layman is available at the web-site http://wikipedia.com. There the interested reader will also find links to other source material.

7. Macrodyne Model. (1690). PMU disturbance recorder. Macrodyne Inc. 4 Chelsea Place,Clifton Park, NY, 12065.

8. IEEE Standard for Synchrophasors for Power Systems, C37.118-2005. Sponsored by thePower System Relaying Committee of the Power Engineering Society, pp. 56–57.

9. IEEE Standard for Synchrophasors for Power Systems, IEEE 1344-1995. Sponsored by thePower System Relaying Committee of the Power Engineering Society.

10. Papoulis, A. (1962). The fourier integral and its applications. New York: McGraw-Hill.11. Brigham, E. O. (1974). The fast fourier transform. Englewood Cliffs: Prentice Hall.12. Phadke, A. G., & Thorp, J. S. (1994). Computer relaying for power systems. Research Studies

Press. Reprinted August 1994.13. Walker, J. S. (1996). Fast fourier transforms (2nd ed.). New York: CRC Press.

References 27

http://wikipedia.com


Chapter 2Phasor Estimation of NominalFrequency Inputs

2.1 Phasors of Nominal Frequency Signals

Consider a constant input signal x(t) at the nominal frequency of the power systemf0, which is sampled at a sampling frequency Nf0. The sampling angle θ is equal to2π/N, and the phasor estimation is performed using Eqs. (1.25)–(1.27).

xðtÞ ¼ Xm cosð2pf0tþ/Þ ð2:1Þ

The N data samples of this input xn:{n = 0, 1, 2, …, N − 1) are

xn ¼ Xm cosðnhþ/Þ ð2:2Þ

Since the principal interest in phasor measurements is to calculate the funda-mental frequency component, we will set k = 1 in Eqs. (1.25)–(1.27) to produce thefundamental frequency phasor obtained from the sample set xn. The superscript(N − 1) is used to identify the phasor as having the (N − 1)st sample as the lastsample used in the phasor estimation.

XN�1c ¼

ffiffiffi2

p

N

XN�1

n¼0

xn cosðnhÞ ¼ffiffiffi2

p

N

XN�1

n¼0

Xm cosðnhþ/Þ cosðnhÞ

¼ffiffiffi2

p

NXm

XN�1

n¼0

cosð/Þ cos2ðnhÞ � 12sinð/Þ sinð2nhÞ

� �¼ Xmffiffiffi

2p cosð/Þ ð2:3Þ

It is to be noted that the summation of the sin(2nθ) term over one period isidentically equal to zero, and that the average of the cos2(nθ) term over a period isequal to 1/2.


29

http://dx.doi.org/10.1007/978-3-319-50584-8_1

http://dx.doi.org/10.1007/978-3-319-50584-8_1

http://dx.doi.org/10.1007/978-3-319-50584-8_1

http://dx.doi.org/10.1007/978-3-319-50584-8_1

The sine sum is calculated in a similar fashion:

XN�1s ¼

ffiffiffi2

p

N

XN�1

n¼0

xn sinðnhÞ ¼ffiffiffi2

p

N

XN�1

n¼0

Xm cosðnhþ/Þ sinðnhÞ

¼ffiffiffi2

p

NXm

XN�1

n¼0

12cosð/Þ sinð2nhÞ � sinð/Þ sin2ðnhÞ

� �

¼ � Xmffiffiffi2

p sinð/Þ ð2:4Þ

The phasor XN−1 is given by

XN�1 ¼ XN�1c � jXN�1

s ¼ Xmffiffiffi2

p cosð/Þþ j sinð/Þ½ � ¼ Xmffiffiffi2

p ej/ ð2:5Þ

It is to be understood that Eq. 2.5 gives the fundamental frequency phasorestimate, even though the subscript k = 1 has been dropped for the sake of sim-plicity. The result obtained in Eq. (2.5) conforms with the phasor definition given inChap. 1, and the phase angle ϕ of the phasor is the angle between the time when thefirst sample is taken (corresponding to n = 0) and the peak of the input signal.

2.2 Formulas for Updating Phasors

2.2.1 Non-recursive Updates

Considering that the phasor calculation is a continuous process, it is necessary toconsider algorithms which will update the phasor estimate as the newer datasamples are acquired. When the Nth sample is acquired after the previous set ofsamples has led to the phasor estimate given by Eq. (2.5), the simplest procedurewould be to repeat the calculations implied in Eqs. (2.3)–(2.5) for the new datawindow which begins at n = 1 and ends at n = N.

XN�1 ¼ffiffiffi2

p

N

XN�1

n¼0

xn ðcosðnhÞ � j sinðnhÞ½ �

XN ¼ffiffiffi2

p

N

XN�1

n¼0

xnþ 1 ðcosðnhÞ � j sinðnhÞ½ �ð2:6Þ

The two windows are shown in Fig. 2.1. Phasor 1 is the result of phasor esti-mation over window 1, while phasor 2 is calculated with the data in window 2. Thefirst sample in window 1 is lagging the peak of the sinusoid by an angle ϕ, while the

30 2 Phasor Estimation of Nominal Frequency Inputs

http://dx.doi.org/10.1007/978-3-319-50584-8_1

first sample of window 2 (n = 1) lags the peak by an angle (ϕ + θ), θ being theangle between samples.

It should be clear from Fig. 2.1 that in general, the phasor obtained from aconstant sinusoid of nominal power system frequency by this technique will have aconstant magnitude and will rotate in the counter-clockwise direction by angle θ asthe data window advances by one sample. Since the phasor calculations are per-formed fresh for each window without using any data from the earlier estimates,this algorithm is known as a non-recursive algorithm. Non-recursive algorithms arenumerically stable but are somewhat wasteful of computation effort as will be seenin the following.

Figure 2.2 is another view of the non-recursive phasor estimation process. Asnewer samples are obtained, the table of sine and cosine multipliers is moved downto match the new data window. In this figure, the multipliers are viewed as samplesof unit magnitude sine and cosine waves at the nominal power system frequency.The new data window has N − 1 samples in common with the old data window. Inactual computation, these are simply stored as tables of sine and cosine, which areused repeatedly on each window as needed.

2.2.2 Recursive Updates

The formulas for calculating the (N − 1)th and (N)th phasors by the non-recursivealgorithm are

phasor 1

(a) (b)

φ

φ + θ

θ

φ

n = N -1

n = Nn = 0

n = 1

window 1window 2

phasor 2

Fig. 2.1 Update of phasor estimates with N sample windows. Phasor 1 is calculated with samplesn = 0, …, N − 1, while phasor 2 is calculated with samples n = 1, 2, … N. θ is the angle betweensuccessive samples based on the period of the fundamental frequency

2.2 Formulas for Updating Phasors 31

XN�1 ¼ffiffiffi2

p

N

XN�1

n¼0

xne�jnh

XN ¼ffiffiffi2

p

N

XN�1

n¼0

xnþ 1e�jnh

ð2:7Þ

The multipliers for a given sample are different in the two computations. Forexample, the multiplier for (n = 2) sample in the first sum is e−j2θ, while themultiplier for the same sample in the second sum is e−jθ.

It should be noted that samples xn: {n = 1, 2, …, N − 1) are common to bothwindows. The second window has no x0, so that it begins with x1, and it ends withxN, which did not exist in the first window. If one could arrange to keep themultipliers for the common samples the same in the two windows, one would saveconsiderable computations in calculating XN. If we multiply both sides of thesecond equation in (2.6) by e−jθ, we obtain the following result:

X_N

¼ e�jhXN ¼ffiffiffi2

p

N

XN�1

n¼0

xnþ 1e�jðnþ 1Þh

¼ XN�1 þffiffiffi2

p

NðxN � x0Þe�jð0Þh ð2:8Þ

where use has been made of the fact that e−j(0)θ = e−jNθ, since N samples spanexactly one period of the fundamental frequency. The phasor defined by Eq. (2.7)

phasor 1

phasor 2New sample, window 2

Samples of unit magnitude sine and cosine functions

t

φ

φ1

φ+θ

φ2 = φ1 + kθ

Inputsignal

Fig. 2.2 Non-recursive phasor estimation. There are 12 samples per cycle of the power frequencyin this example. Fresh calculations are made for each new window as new samples are obtained.The phasor for a constant input signal rotates in the counter-clockwise direction by the samplingangle, in this 30°


differs from the non-recursive estimate by an angular retardation of θ. Theadvantage of using this alternative definition for the phasor from the new datawindow is that (N − 1) the multiplications by the Fourier coefficients in the newwindow are the same as those used in the first window. Only a recursive update onthe old phasor needs to be made to determine the value of the new phasor. Thisalgorithm is known as the recursive algorithm for estimating phasors. In general,when the last sample in the data window is (N + r), the recursive phasor estimate isgiven by

X_N þ r

¼ e�jhXNþ r�1 þffiffiffi2

p

NðxNþ r � xrÞe�jrh

¼ X_Nþ r�1

þffiffiffi2

p

NðxN þ r � xrÞe�jrh ð2:9Þ

When the input signal is a constant sinusoid, xN+r is the same as xr, and thesecond term in Eq. (2.8) disappears. The phasor estimate with data from the newwindow is the same as the phasor estimate with data from the old window when theinput signal is a constant sinusoid. In general, the recursive algorithm is numericallyunstable. Consider the effect of an error in the estimate from one window—forexample, caused by a round-off error. This error is always present in all the phasorestimates from then on. This property of the recursive phasor algorithms must bekept in mind when practical implementation of these algorithms is performed [1].Nevertheless, because of the great computational efficiency of the recursive algo-rithm, it is usually the algorithm of choice in many applications.

Unless stated otherwise explicitly, we will assume that only the recursive formof the phasor estimation algorithm is in use (Fig. 2.3).

t

φ

φ1

φ+θ

φ2 = φ1phasor 1 & 2

New sample, window 2

Samples of unit magnitude sine and cosine functions

Inputsignal

New samples, sine and cosine

Fig. 2.3 Recursive phasor estimation. There are 12 samples per cycle of the power frequency inthis example. Fresh calculations are made for each new window as new samples are obtained. Newsine and cosine multipliers are used on the new sample. The phasor for a constant input signalremains stationary

2.2 Formulas for Updating Phasors 33

Example 2.1 Consider the 60 Hz signal

xðtÞ ¼ 100 cosð120ptþ p=4Þ

sampled at the rate of 12 samples per cycle, i.e., at a sampling frequency of 720 Hz.The first 18 samples, and the non-recursive and recursive phasor estimates obtainedusing Eqs. (2.6) and (2.8) beginning with sample no. 12 (at which time the first datawindow is completely filled) are shown in the following (Table 2.1).

As expected, the non-recursive phasor estimates produce a constant magnitudeof 100/√2 with an initial angle of π/4 (45°), and then for each successive estimate,the angle increases by 30°.

2.3 Effect of Signal, Noise, and Window Length

The input signals are rarely free from noise. A spurious frequency componentwhich is not a harmonic of the fundamental frequency signal may be considered tobe noise. One may also have induced electrical noise picked up in the wiring of theinput signal. Leakage effect caused by the windowing function has already beendiscussed in Chap. 1, and it too contributes to an error in phasor estimation andshould therefore be considered as a type of noise in the input.

Table 2.1 Phasor estimates of sampled data

Sample No. Sample xn Non-recursive phasor estimate Recursive phasor estimate

0 70.7107

1 25.8819

2 −25.8819

3 −70.7107

4 −96.5926

5 −96.5926

6 −70.7107

7 −25.8819

8 25.8819

9 70.7107

10 96.5926

11 96.5926

12 70.7107 70.701∠45° 70.701∠45°

13 25.8819 70.701∠75° 70.701∠45°

14 −25.8819 70.701∠105° 70.701∠45°

15 −70.7107 70.701∠135° 70.701∠45°

16 −96.5926 70.701∠165° 70.701∠45°

17 −96.5926 70.701∠195° 70.701∠45°


http://dx.doi.org/10.1007/978-3-319-50584-8_1

As an approximation, we will consider the noise in the input signal to be a zeromean, Gaussian noise process. This should be a good approximation for the elec-trical noise picked up in the wiring and signal conditioning circuits. The other twosources of noise, viz. non-harmonic frequency components and leakage phenomenaneed further consideration. A phasor measurement system may be placed in anarbitrarily selected substation and will be exposed to input signals generated by thepower system which is likely to change states all the time. Each of the powersystem states may lead to different non-harmonic frequencies and leakage effects,and the entire ensemble of conditions to which the phasor measurement system isexposed may also be considered to be a pseudo-random Gaussian noise process.

Consider a set of noisy measurement samples

xn ¼ Xm cosðnhþ/Þþ en; n ¼ 0; 1; 2; . . .;N � 1f g ð2:10Þ

where εn is a zero-mean Gaussian noise process with a variance of σ2. If we set(Xm/√2) cos (ϕ) = Xr and (Xm/√2) sin (ϕ) = Xi, the phasor representing the sinusoidis X = Xr + jXi. We may pose the phasor estimation problem as one of the findings—the unknown phasor estimate from the sampled data through a set of N overdeter-mined equations:

x0x1x2�

xN�1

266664

377775 ¼

ffiffiffi2

pcosð0Þ � sinð0ÞcosðhÞ � sinðhÞcosð2hÞ � sinð2hÞ

� �cos½ðN � 1Þh� � sin½ðN � 1ÞhÞ

266664

377775

Xr

Xi

� �þ

e1e2e3�

eN�1

266664

377775 ð2:11Þ

or, in matrix notation

½x� ¼ ½S�½X� þ ½e� ð2:12Þ

Assuming that the covariance matrix W of the error vector is σ2 multiplied by aunit matrix

W½ � ¼ r2½1� ð2:13Þ

the weighted least squares solution of Eq. (2.11) provides the estimate for thephasor

½X_ � ¼ ½STW�1S��1STW�1½x� ð2:14Þ

Using (2.13) for W, and calculating {STS]−1 for the S in Eq. (2.11):

½X_ � ¼ ½STW�1S��1½STW�1�½x� ¼ ½STS��1½ST �½x� ¼ 1N½ST �½x� ð2:15Þ

2.3 Effect of Signal, Noise, and Window Length 35

Since the noise is a zero-mean process, the estimate given by (2.15) is unbiased,and the expected value of the estimate is equal to the true value of the phasor. If X isthe true value of the phasor, the covariance matrix of the error in the phasor estimateis

E ½X_ � X�½X_ � X�Th i

¼ ½STW�1S��1 ð2:16Þ

Substituting for [W] from Eq. (2.13), the covariance of the error in phasorestimate is (σ2/N). The standard deviations of error in real and imaginary parts ofthe phasor estimate are (σ/√N). We may thus conclude that higher sampling rateswill produce improvement in phasor estimates in inverse proportion of the squareroot of the number of samples per cycle. Alternatively, if longer data windows areused (multiple of cycles), then once again the errors in phasor estimate go down asthe square root of the number of cycles used. Thus, a four cycle phasor estimate istwice as accurate as a one cycle estimate in with noisy input.

Example 2.2 Consider a 60 Hz sinusoid

xðtÞ ¼ 100 cosð120ptþ p=4Þþ eðtÞ

in a noisy environment, with the Gaussian noise ε having a zero mean and astandard deviation of 1. The source of noise could be electromagnetic interference,quantization errors, or harmonic and non-harmonic components in the input signal.If the phasing of the harmonic and non-harmonic signals is random, the noise modelmay be approximated by a zero-mean Gaussian characteristic.

The signal is sampled at six different sampling rates: 8, 16, 32, 64, 128, and 256times per cycle. The signal samples are created with appropriately modeled noiseinput for 1000 cycles, and 1000 estimates of the phasor value are calculated. Thestandard deviations of the errors in the 1000 phasor estimates as well as its theo-retical value (σ/√N) are given in Table 2.2 and are also shown in Fig. 2.4.

These results show very good agreement with the expected results. As men-tioned earlier, increasing the data window size at a fixed sampling rate, rather thanthe number of samples in the same data window will produce similar results.

Table 2.2 Phasor estimation of a noisy signal

No. of samplesper cycle N

Standard deviation ofinput noise

Standard deviation of phasorestimate error (volts)

σ/√N

8 1 0.3636 0.3536

16 1 0.2601 0.2500

32 1 0.1794 0.1768

64 1 0.1231 0.1250

128 1 0.0880 0.0884

256 1 0.0626 0.0625


2.3.1 Errors in Sampling Times

Another possible source of error in input data samples is the error in timing of thesampling pulses. One possible source of errors is that the sampling clock is not preciselyat a multiple of the power system frequency. This case will be dealt with in the nextchapter, when we consider the phasor estimation problem at off-nominal frequencies.

In this section, we will consider the case where the sample times are corrupted bya Gaussian random noise with standard deviation varying by up to 10% of thesampling interval. Large errors of this type should not exist in modern measuringsystems. Nevertheless, when sampling pulses are generated with the help of softwareclocks, it is possible to encounter random errors in sampling times. The followingnumerical example considers errors of this type. The errors are truncated at 3 timesthe standard deviation in order to eliminate impossibly large sampling clock errors.

Example 2.3 Consider a 60 Hz sinusoid

xðtÞ ¼ 100 cosð120ptþ p=4Þ

which is sampled at tn = nΔT + ε, where the Gaussian noise ε has a zero mean anda standard deviation of bΔT, with the parameter b varying between 0.0 and 0.10.The signal is sampled at a sampling rate of 32 times per cycle. The signal samplesare created with sampling time errors for 1000 cycles, and 1000 estimates of thephasor value are calculated. The standard deviations of the errors in the 1000 phasorestimates are given in Table 2.3 and are also shown in Fig. 2.5.

2.4 Phasor Estimation with Fractional Cycle DataWindow

The weighted least squares solution technique developed in Sect. 2.2 is a conve-nient vehicle for calculating phasors from fractional cycle data windows. It shouldbe remembered that fractional cycle phasor estimates are necessary in developing

2 4 8 160.05

0.15

0.25

0.35

σ/√N

standard deviation of phasor error(volts)

√N

Fig. 2.4 Standard deviationof phasor error due tozero-mean Gaussian noise inthe input. The result of 1000phasor estimates at eachsampling rate is shown by thesolid line, and the theoreticalvalue of the standarddeviation (σ/√N) is shown bythe dotted line

2.3 Effect of Signal, Noise, and Window Length 37

high speed relaying applications, and not particularly useful in wide area phasormeasurement applications whereby measurement times of a few cycles areacceptable. Nevertheless, it is instructive to include a discussion of fractional cyclephasor estimation.

Consider the use of M samples of a sinusoid for estimating phasors, the sinusoidhaving been sampled at a sampling rate of N samples per cycle. M < N produces afractional cycle phasor estimation algorithm.

As before, the input is set of noisy measurement samples

xn ¼ Xm cosðnhþ/Þþ en; fn ¼ 0; 1; 2; . . .;M � 1Þ ð2:17Þ

where εn is a zero-mean Gaussian noise process with a variance of σ2. The samplingangle θ is equal to 2π/N.

x0x1x2�

xM�1

266664

377775 ¼

ffiffiffi2


� �cos½ðM � 1Þh� � sin½ðM � 1ÞhÞ

266664

377775

Xr

Xi

� �þ

e1e2e3�

eM�1

266664

377775 ð2:18Þ

Table 2.3 Errors in phasor estimation due to noisy inputs

Coefficient ‘b’ of input noise describedabove

Standard deviation of phasor estimate error(volts)

0.00 0.0000

0.02 0.0705

0.04 0.1373

0.06 0.2111

0.08 0.2809

0.10 0.3432

0.060 0.02 0.04 0.08 0.100

0.1

0.2

0.3

‘b’

Stan

dard

dev

iatio

n of

err

ors i

n ph

asor

est

imat

e (v

olts

)Fig. 2.5 Standard deviationof phasor error due tozero-mean Gaussian noise insampling clock pulses. Theresult of 1000 phasorestimates at different values of‘b’ coefficients are shown


or, in matrix notation

x½ � ¼ S½ � X½ � þ ½e� ð2:19Þ

As before, the weighted least squares solution of Eq. (2.17) provides the esti-mate for the phasor

½X_ � ¼ ½STW�1S��1STW�1½x� ð2:20Þ

Using (2.13) for W, and calculating {STS]−1 for the S in Eq. (2.11):

½X_� ¼ ½STW�1S��1½STW�1�½x� ¼ ½STS��1½ST �½x� ð2:21Þ

Unlike in the case of the full cycle phasor estimation, [STS]−1 is no longer asimple matrix:

½STS� ¼ 2

PM�1

n¼0cos2ðnhÞ PM�1

n¼0cosðnhÞ sinðnhÞ

PM�1

n¼0cosðnhÞ sinðnhÞ PM�1

n¼0sin2ðnhÞ

2664

3775 ð2:22Þ

It can be shown that for a half-cycle estimation, with M = N/2, the least squaressolution is very similar to the DFT estimator.

2.5 Quality of Phasor Estimate and Transient Monitor

Phasor estimates obtained from a data window represent the fundamental frequencycomponent of the input confined to the data window. When a fault occurs on thepower system, there is a series of data windows which contain pre- and post-faultdata. This is illustrated in Fig. 2.6 for an assumed voltage waveform during a fault.

It should be clear that although a phasor estimate will be available for all datawindows (including the ones that are shaded in Fig. 2.6), only phasors whichbelong entirely to the pre- or post-fault periods are of interest. The phasors com-puted for the shaded windows of Fig. 2.6 do not represent any meaningful systemstate, and a technique is needed to detect the occurrence of mixed states within adata window.

A technique known as ‘Transient Monitor’ [2] provides a measure to indicate a‘quality’ of the estimate and can also be used to detect the condition when a datawindow contains mixed state waveforms. Consider the process of computing thedata samples ðx_nÞ in a window from the estimated phasor which has been estimatedfrom a sample set (xn):

2.4 Phasor Estimation with Fractional Cycle Data Window 39

½x_n� ¼ffiffiffi2


� �cos½ðN � 1Þh� � sin½ðN � 1Þh�

266664

377775 X

_

r

X_

i

" #ð2:23Þ

Substituting for the phasor estimate from Eq. (2.15)

x_n

h i¼

ffiffiffi2


� �cos½ðN � 1Þh� � sin½ðN � 1Þh�

26666664

37777775

�ffiffiffi2

p

N

cosð0Þ cosðhÞ cosð2hÞ � cos½ðN � 1ÞhÞ� sinð0Þ � sinðhÞ � sinð2hÞ � � sin½ðN � 1ÞhÞ

� �xn½ � ð2:24Þ

Multiplying the matrices and simplifying

x_n

h i¼ 2

N

1 cosðhÞ cosð2hÞ � cos½ðN � 1ÞhÞcosðhÞ 1 cosðhÞ � cosð0Þcosð2hÞ cosðhÞ 1 � cosðhÞ

� � � 1 �cos½ðN � 1ÞhÞ cosð0Þ cosðhÞ � 1

266664

377775½xn� ð2:25Þ

Pre-faultPost-fault

Windows with all pre-fault data

Windows with all post-fault data

Fig. 2.6 Transition from pre-fault to post-fault waveforms. The shaded windows contain mixedwaveform data


where use has been made of the fact that N = 2π. The difference between the inputdata and the re-computed sample data from the phasor estimate is the error ofestimation [tn]:

tn½ � ¼ ½xn � x_n�

¼

1� 2N � 2

N cosðhÞ � 2N cosð2hÞ � � 2

N cos½ðN � 1ÞhÞ� 2

N cosðhÞ 1� 2N � 2

N cosð3hÞ � � 2N cosð0Þ

� 2N cosð2hÞ � 2

N cosð3hÞ 1� 2N � � 2

N cosðhÞ� � � 1� 2

N �� 2

N cos½ðN � 1ÞhÞ � 2N cosð0Þ � 2

N cosðhÞ � 1� 2N

26666664

37777775xn½ �

ð2:26Þ

If the input signal is a pure sinusoid at fundamental frequency, all entries of [tn]will be identically equal to zero. However, when the input signal is noisy orcontains a composite window of two different sinusoids, [tn] is not zero, and onemay use the sum (Tn) of the absolute values of its elements as a measure of the errorof estimation.

Tn ¼XN�1

k¼0

tkj j ð2:27Þ

This sum has been referred as a ‘Transient Monitor’ and can be used as ameasure of the ‘quality’ of the phasor estimate.

Example 2.4 Consider a composite 60 Hz voltage waveform samples described by

xn ¼ 100 cosðnhþ p=4Þ; for n ¼ 0; 1; 2; . . .; 35

xn ¼ 50 cosðnhþ p=8Þ; for n ¼ 36; 37; 38; . . .; 71

with a sampling rate of 24 samples per cycle; thus, θ = π/12.The data samples, recursive phasor estimates, and the function Tn are shown in

Table 2.4.Note that in the interest of saving space, several rows which do not show

interesting transitions have been omitted. The phasor estimates and the transientmonitor are plotted in Fig. 2.7.

Note that the phasor estimate remains stationary at (50 + j50) and(32.6641 + j13.5299), while the input signal is 70.7∠45° and 50∠22.5°, respec-tively, and the transition from one value to another takes 24 samples, the width ofphasor estimation window. The transient monitor provides a good indication of thequality of the phasor estimate, it being high during the transition period when thephasor estimate is unreliable.

2.5 Quality of Phasor Estimate and Transient Monitor 41

2.6 DC Offset in Input Signals

Fault currents in a power system often have an exponentially decaying dc com-ponent, which is generally known as the dc offset. Occasionally, voltage waveformsmay also have a dc offset due to capacitive voltage transformer transients. In bothcases, the dc offsets decay to negligible values in a few cycles. If the phasorestimate is performed while a dc offset is present in a waveform, one is likely to getsignificant errors in phasor estimate while the dc offset is nonzero. The transientmonitor described in Sect. 2.4 can be used to alert the user that the phasor estimatethus obtained is unreliable.

Table 2.4 Transient Monitorfor a transient signal

SampleNo.

Samplevalue

Phasor Tn

1 70.7107 0

2 50.0000 0

23 96.5926 0

24 86.6025 50.0 + j50.0 0.0000

36 −86.6025 50.0 + j50.0 0.0000

37 −46.1940 48.5553 + j50.0 51.4677

38 −39.6677 47.9672 + j50.1576 67.5480

39 −30.4381 48.1998 + j50.0233 68.1205

40 −19.1342 48.9970 + j49.2261 80.9808

41 −6.5263 49.9518 + j47.5723 129.1521

42 6.5263 50.6149 + j45.0978 195.2854

43 19.1342 50.6149 + j42.0587 267.1946

44 30.4381 49.7583 + j38.8619 331.2446

45 39.6677 48.0811 + j35.9570 382.1787

46 46.1940 45.8392 + j33.7151 413.6845

47 49.5722 43.4397 + j32.3297 430.7721

48 49.5722 41.3320 + j31.7650 436.9438

49 46.1940 39.8874 + j31.7650 432.8577

50 39.6677 39.2993 + j31.9225 426.1713

51 30.4381 39.5318 + j31.7883 431.3592

52 19.1342 40.3290 + j30.9910 433.7538

53 6.5263 41.2839 + j29.3372 424.5001

54 −6.5263 41.9469 + j26.8628 405.5234

55 −19.1342 41.9469 + j23.8236 367.7588

56 −30.4381 41.0903 + j20.6269 314.0214

57 −39.6677 39.4132 + j17.7219 243.3327

58 −46.1940 37.1713 + j15.4800 162.9157

59 −49.5722 34.7718 + j14.0947 77.7374

60 −49.5722 32.6641 + j13.5299 0.0000

72 49.5722 32.6641 + j13.5299 0.0000


Since many phasor applications are dedicated to relatively slow phenomena, it isnot essential that dc offset be handled in any special way; one only needs to be alertto the estimate quality indicated by the transient monitor. However, in computerrelaying applications, powerful techniques have been developed to remove dcoffsets before phasors are estimated, and in very specific applications of phasorswhich require very high speed of response, it may be necessary to employ algo-rithms which will remove the dc offset from the signals. This section provides abrief summary of the available techniques for this purpose. We will consider onlythe dc offset in current waveforms when a fault occurs. Similar techniques areapplicable to the handling of dc offset in the voltage waveforms as well.

The earliest technique used in relays for removing the dc offset from faultcurrents is the one of using a ‘mimic’ circuit in the secondary winding of a currenttransformer [3]. Figure 2.8 shows the primary fault circuit, and the current trans-former secondary winding with a burden (r + jωl) such that the ratio R/L is matchedexactly by the burden ratio r/l. In this case, the dc offset in the current is not presentin the voltage e2(t) across the burden, and the burden voltage can be used as a signalwhich is proportional to the current and is free from the dc offset. The primary faultcircuit and the CT secondary burden are both primarily inductive in nature, andhence, the mimic circuit acts as a differentiator. Thus, it has the property ofamplifying any high frequency noise that may be present in the current. However, itshould be remembered that the primary fault current is itself is produced by theR–L circuit, and thus has attenuated the high frequency noise that may be present inthe voltage signal. Thus, although the mimic circuit is a differentiator, the noisecontent of the voltage across it is similar to that in the primary voltage.

For computer relays, there is a least squares solution technique available foreliminating the dc offset, which is free from the noise amplification properties of themimic circuit. Consider a fault current i(t), containing a dc offset is given by

iðtÞ ¼ A cosðxtÞþB sinðxtÞ � Ce�t=T for t� 0

¼ A� C t ¼ 0� ð2:28Þ

20 4000

20

40

(a) (b)

0

450

1 24 36 60 72

Fig. 2.7 Result of phasor estimation in a data stream with mixed input signals. a Phasorestimates. The transition from a solid phasor at 50 + j50 to a new phasor of 32.6641 + j13.5299 isshown by open circles. b The transient monitor Tn. Note that it is high during the transition fromone phasor value to another. When the input signal is a pure sinusoid, the Tn becomes zero

2.6 DC Offset in Input Signals 43

This expression assumes that the current just before the occurrence of fault is(A − C), and that the dc offset decays with a time constant T, which for the circuitof Fig. 2.8 is equal to L/R seconds.

Consider a sample set of M data points obtained from this current waveform

in ¼ A cosðnhÞþB sinðnhÞ � Crn; for fn ¼ 0; 1; 2; . . .;M � 1Þ ð2:29Þ

where θ is the sampling angle equal to 2π/N, N being the number of samples percycle of the nominal frequency, and r is the decrement factor for the decaying dccomponent in one sample time

r ¼ e�DT=T ð2:30Þ

If we now assume that the decrement factor r is known, the only unknowns inEq. (2.30) are A, B, and C. Taking the overdetermined set of M data points,

i0i1

iM�1

2664

3775 ¼

1 0 �1cos h sin h �r

cosðM � 1Þh sinðM � 1Þh �rM�1

2664

3775

ABC

24

35 ð2:31Þ

As usual, the above equation can be solved for A, B, and C, and then by adding(Cr−n) to each sample of the current, the dc offset can be removed from thewaveform.

It is possible that the value of the time constant T is not known exactly and anapproximate value must be used for ‘r’. The algorithm is tolerant of reasonableerrors in the value of ‘r’, as is seen by the following numerical example.

Example 2.5 Consider a fault current waveform with full dc offset given by

iðtÞ ¼ 100 cosð120ptÞþ 100 sinð120ptÞ�100e�t=0:05

The current is zero before the occurrence of the fault. The dc offset decay timeconstant is 50 ms.

RL

l

r

Faulte(t)

i(t)

i2 (t)e2(t)

Fig. 2.8 Fault circuit and themimic burden in CTsecondary to eliminate the dcoffset in fault current


The true value of the phasor must be (100/√2)(1 − j1). The dc offset is removedby applying Eq. (2.31) for a window of one cycle at a time. It is assumed that anerror of ±10% is made in the decrement factor ‘r’ used in Eq. (2.31). The resultingerror in (√2 × phasor) is shown in Fig. 2.9.

It can be seen from Fig. 2.9 that the errors of estimation of phasors are less than0.2% even though the time constant errors are of the order of 10%.

The least square solution described above is some times described as a ‘digitalmimic’ procedure. However, it must be pointed out that this process is not adifferentiator, and consequently, there is no amplification of noise in the currentsignal in this process.

2.7 Non-DFT Estimators

A number of papers dealing with the problem of computing phasors from sampleddata have been published over the last several years. Several papers considervariations on the Fourier transform method, with special emphasis on the problemof dealing with off-nominal frequency signals. We will consider such signals in thenext chapter, where the performance of the fixed frequency Fourier transform onoff-nominal frequency input signals will be discussed. Among the variations of thebasic Fourier technique, least squares methods, Kalman filter methods, and Pronymethods have been discussed. As the main thrust of these variations is to deal withoff-nominal frequency signals, we will defer their discussion to a later chapter.

References

1. Phadke, A. G., & Thorp, J. S. (1994). Computer relaying for power systems (pp. 127–129).New York: Research Studies Press Ltd., Wiley.

2. ibid. pp. 151–152.3. Mason, C. R. (1956). Art and science of protective relaying. New York: Wiley.

-0.15 0 0.15-0.15

0

0.15

(100 – j100)

assumed ‘T’ = 45 ms

assumed ‘T’ = 55 ms

Fig. 2.9 Errors in phasorestimate caused by errors intime constant estimate. Twocycles worth of data areshown. The solid arrowsshow the direction ofincreasing time

2.6 DC Offset in Input Signals 45

Chapter 3Phasor Estimation at Off-NominalFrequency Inputs

3.1 Types of Frequency Excursions Found in PowerSystems

Phasors are a steady state concept. In reality, a power system is never in a steadystate. Voltage and current signals have constantly changing fundamental frequency(albeit in a relatively narrow range around the nominal frequency) due to changes inload and generation imbalances and due to the interactions between real powerdemand on the network, inertias of large generators, and the operation of automaticspeed controls with which most generators are equipped. In addition, when faultsand other switching events take place, there are very rapid changes in voltage andcurrent waveforms, and depending upon the definition of frequency, one wouldhave to accept that power system waveforms under these conditions contain a verywide band of frequencies ranging from DC to hundreds of kilohertz. The consid-eration of various interpretations of frequency which are of interest in power systemengineering will be considered in Chaps. 4 and 6.

In this chapter, we will focus on the changes in power system frequency due toresponses to load generation imbalances and when the power system is in aquasi-steady state and is operating with a frequency which may be different from itsnominal value. It will be assumed that power system voltages and currents arebalanced, and the frequency changes are only due to speed changes of the rotors ofpower system generators. As these speed changes are slow (as compared to thenominal power system frequency), one may consider the progress of such speedchanges as a sequence of quasi-steady states when the waveforms are observed overa small window—for example over one period of the power frequency.

Most integrated power systems operate in a relatively narrow band of frequencywithin 0.5 Hz from its nominal value. Under exceptional circumstances—forexample when small islands of generators and load are isolated from the rest of thenetwork—the frequency excursions may be as large as ±10 Hz. However, thepower system operation at such extreme excursions are usually controlled and


47

http://dx.doi.org/10.1007/978-3-319-50584-8_4

http://dx.doi.org/10.1007/978-3-319-50584-8_6

brought back to normal values by available control actions. Where the islands areprimarily powered by hydro-electric generators, the system may operate at largefrequency deviations for extended periods.

3.2 DFT Estimate at Off-Nominal Frequencywith a Nominal Frequency Clock

It is assumed that the sampling clock is a fixed frequency clock with sampling rateswhich are multiples of the nominal power system frequency. Recursive phasorcalculation formulas were developed in Chap. 2, Eqs. (2.8) and (2.9) and are usedto consider phasor estimation when the power system frequency differs from thenominal.

Equation (2.9) is reproduced here for ready reference as Eq. (3.1).

X_Nþ r

¼ e�jhXNþ r�1 þffiffiffi2

p

NðxNþ r � xrÞe�jrh

¼ X_Nþ r�1

þffiffiffi2

p

NðxNþ r � xrÞe�jrh ð3:1Þ

It should be clear that if the input signal is a constant sinusoid of nominal powersystem frequency xN+r = xr, and consequently Eq. (3.1) confirms that the resultingphasor would also remain constant [1].

3.2.1 Input Signal at Off-Nominal Frequency

Now assume that the input signal is at a frequency

x ¼ x0 þDx ð3:2Þ

where ω0 is the nominal power system frequency. For a 60 Hz system, ω0 is120π rad/s.

The input signal is once again assumed to be


The corresponding phasor representation is ðXm=ffiffiffi2

p Þðej/Þ, which is the sameformula as given in Eq. (1.2). In fact the definition of a phasor representation of asinusoid is independent of the frequency of the signal. We would get this phasorwith the phasor computation formulas given above if the Fourier coefficients werealso from sines and cosines of the same period. However, when fixed frequencyclocks related to the nominal power system frequency are used, the resulting

48 3 Phasor Estimation at Off-Nominal Frequency Inputs

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http://dx.doi.org/10.1007/978-3-319-50584-8_2

http://dx.doi.org/10.1007/978-3-319-50584-8_1

phasors would be different from the true value given above. These considerationsare illustrated in Fig. 3.1.

Figure 3.1 shows the off-nominal frequency being much higher than the nominalfrequency. As mentioned before, in normal power systems the deviation in fre-quency would certainly be much smaller. However, the figure illustrates the conceptinvolved. If the phasor calculation is performed with off-nominal sine and cosinewaves, the result will be the correct phasor for the given sinusoid. However, usingthe nominal frequency, sines and cosines for phasor calculation will introduce anerror in phasor estimation. Clearly, the error made in phasor estimation will dependupon the difference between the nominal and actual frequency.

The input signal can be expressed as follows:

xðtÞ ¼ Xm cosðxtþ/Þ ¼ffiffiffi2

pRe½ðXm=

ffiffiffi2

pÞðej/ÞðejxtÞ�

¼ffiffiffi2

pRe Xejxt

� � ð3:4Þ

where X is the correct value of the phasor at the off-nominal frequency and thefunction ‘Re’ is the real value function. Expressing the real value as the average of acomplex number and its complex conjugate

xðtÞ ¼ ðffiffiffi2

p=2Þ Xejxt þX�e�jxt

� � ð3:5Þ

The kth sample of the signal represented by Eq. (3.5) is given by

xðtÞ ¼ ð1=ffiffiffi2

pÞ XejxkDt þX�e�jxkDt� � ð3:6Þ

The phasor representation of x(t), i.e., X′ (which is different from X unless thesystem frequency is equal to the nominal value ω0), is calculated using Eq. (3.1)

φ

t=0off-nominal sine

off-nominal cosine

nominal sine nominal cosinex

off-nominal signal

Fig. 3.1 Phasor calculation at off-nominal frequency signals with sampling clock synchronizedwith the nominal power system frequency

3.2 DFT Estimate at Off-Nominal Frequency … 49

with xr as the first sample. Note that Eq. (3.1) uses sine and cosine terms at thenominal power system frequency ω0. Thus, X 0

r is given by

X 0r ¼

ffiffiffi2

p

N

XrþN�1

k¼r

xk e�jkx0Dt

¼ 1N

XrþN�1

k¼r

XejkxDt þX�e�jkxDt� �

e�jkx0Dt ð3:7Þ

Making use of the identity

ejx � 1 ¼ ejx = 2ðejx = 2 � e�jx = 2Þ¼ 2jejx = 2 sin ðx = 2Þ

the two summations in Eq. (4.9), which are geometric series, can be expressed inclosed-form as [2]

X 0r ¼ Xejrðx�x0ÞDt sin Nðx�x0ÞDt

2

N sin ðx�x0ÞDt2

( )ejðN�1Þðx�x0ÞDt

2

þX�e�jrðxþx0ÞDt sin Nðxþx0ÞDt2

N sin ðxþx0ÞDt2

( )e�jðN�1Þðxþx0ÞDt

2 ð3:8Þ

or

X 0r ¼ PXejrðx�x0ÞDt þQX�e�jrðxþx0ÞDt ð3:9Þ

where P and Q are coefficients in Eq. (3.9) which are independent of ‘r’:

P ¼ sin Nðx�x0ÞDt2

N sin ðx�x0ÞDt2

( )ejðN�1Þðx�x0ÞDt

2 ð3:10Þ

Q ¼ sin Nðxþx0ÞDt2

N sin ðxþx0ÞDt2

( )e�jðN�1Þðxþx0ÞDt

2 ð3:11Þ

A Qualitative Graphical Representation

The phasor estimate at off-nominal frequency is given by Eq. (3.9). It should benoted that for all practical power system frequencies, ω − ω0 is likely to be verysmall, and hence (ω + ω0 = 2ω0 + Δω) is very nearly equal to 2ω0. A qualitativerepresentation of Eq. (3.9) is shown in Fig. 3.2.

In Fig. 3.2, the phasors X and X* are attenuated by complex gains P and Q asshown in Fig. 3.2a. PX rotates in the anticlockwise direction at an angular speed of


http://dx.doi.org/10.1007/978-3-319-50584-8_4

(ω − ω0) = Δω. The phasor QX* rotates in the clockwise direction at a speed(ω + ω0), which is approximately equal to 2ω0. Figure 3.2b shows the resultantphasor, which is made of the two components. The resultant phasor thus has amagnitude and phase angle variation at a frequency 2ω0 (approximately) super-imposed on a monotonically rotating component at Δω. The qualitative variation ofthe magnitude and phase angle of the estimate of an off-nominal input signal isshown in Fig. 3.3.

Note that in Fig. 3.3, the effect of the Q � XQ term has been exaggerated in orderto illustrate the behavior of the estimate. As will be seen in the next chapter, theactual effect is quite small when practical frequency excursions are considered.

The constants P and Q are complex numbers, and their values depend upon thedeviation between the nominal frequency and the actual signal frequency. Thisdependence is illustrated in Figs. 3.4 and 3.5 for a nominal frequency of 60 Hz, afrequency deviation in the range of ±5 Hz, and a sampling rate of 24 samples percycle.

Note that the maximum attenuation occurs at a deviation of 5 Hz from nominalfrequency, being around about 98.8%. For a 2 Hz deviation, the attenuation is at

( 0)

-( 0)

PX

( 0)

PX

QX*

X

X*

(a) (b)

X

-( 0)

QX*

Fig. 3.2 Qualitative illustration of phasor estimates at off-nominal frequency

(a) (b) t t

∠Xr΄ Δωt

2nd harmonic

|X||PX|

≈ 2nd harmonic

r΄│

Fig. 3.3 Magnitude and angle variation with time of phasor estimate of an off-nominal signal


99.8%, which for practical cases can be completely disregarded. The phase angleerror corresponds to about 3° per Hz deviation, varying linearly in the ±5 Hz range.Remembering that the factor P affects the principal term of the quantity beingmeasured, the effect of this factor can often be neglected. For the sake of com-pleteness, the data plotted in Fig. 3.4 is also provided in Table 3.1.

The effect of sampling rate on the attenuation and phase shift is relatively minor.For example, for a +2.0 Hz deviation, by varying the sampling rate from 12 to 120samples per cycle, the effect on P is as shown in Table 3.2.

As can be seen from Table 3.2, the sampling rate affects the attenuation and thephase shift only slightly.

The effect of frequency deviation on the magnitude and phase angle of theattenuation factor Q is shown in Fig. 3.5 for frequency excursions in the range of±5 Hz.

Note that at the nominal frequency, the magnitude of Q is 0. It increases almostlinearly as a function of the frequency deviation, being about 0.008 per unit per Hz.Note also that at negative frequency deviations, the multiplier is also negative, so inthat sense what is plotted is not the absolute value of Q. The phase angle of Q is 15°at the nominal frequency, and the phase angle varies linearly with respect to fre-quency deviation. For the sake of completeness, the data plotted in Fig. 3.5 is alsoprovided in Table 3.3.

The effect of sampling rate on the attenuation and phase shift of Q is alsorelatively minor. For example, for a +2.0 Hz deviation, by varying the samplingrate from 12 to 120 samples per cycle, the effect on Q is as shown in Table 3.4.

As can be seen, the sampling rate does not affect the attenuation, but does affectthe phase shift significantly.

-5 -4 -2 0 2 4 5 0.988

0.992

0.996

1

0

1015

-5 -10-15

5

Mag

nitu

de

Frequency deviation

Phas

e Sh

ift –

degr

ees

(dot

ted)

Fig. 3.4 Factor P as afunction of frequencydeviation

-0.05-0.04

-0.02

0.02

0.040.05

Frequency deviation

Mag

nitu

de

-5 -4 -2 0 2 4 5

10

15

2530

0.00

20

5

0

Phas

e Sh

ift –

degr

ees

(dot

ted)

Fig. 3.5 Factor Q as afunction of frequencydeviation


Example 3.1 Numerical example: single phase off-nominal frequency signal.As an illustration of the application of the algorithms derived above, consider an

example of a sinusoid having an rms value of 100 at a frequency of 60.5 Hz. Let theassumed phase angle of the phasor be π/4, so that the correct phasor representationof this signal is X = 100 ejπ/4. Assume that this input signal is sampled at a fre-quency of 24 times the nominal frequency, or at 1440 Hz for a 60 Hz system. From

Table 3.1 Magnitude andphase angle of P

Δf |P| ∠P (°)

−5 0.9886 −14.37

−4.5 0.9908 −12.94

−4 0.9927 −11.5

−3.5 0.9944 −10.06

−3 0.9959 −8.62

−2.5 0.9972 −7.19

−2 0.9982 −5.75

−1.5 0.9990 −4.31

−1 0.9995 −2.87

−0.5 0.9999 −1.44

0 1.0000 0

0.5 0.9999 1.44

1 0.9995 2.87

1.5 0.9990 4.31

2 0.9982 5.75

2.5 0.9972 7.19

3 0.9959 8.62

3.5 0.9944 10.06

4 0.9927 11.5

4.5 0.9908 12.94

5 0.9886 14.37

Table 3.2 Effect of thesampling rate on P for afrequency of 62 Hz

Sampling rate |P| ∠P (°)

12 0.9982 5.5

24 0.9982 5.75

36 0.9982 5.83

48 0.9982 5.87

60 0.9982 5.9

72 0.9982 5.92

84 0.9982 5.93

96 0.9982 5.94

108 0.9982 5.94

120 0.9982 5.95


Tables 3.1 and 3.3, the coefficients P and Q corresponding to this case(Δf = +0.5 Hz) are P = [email protected]° and [email protected]°, respectively.

The input signal is sampled at 1440 Hz, and 200 samples are processed by theFourier formula Eq. (3.1). Table 3.5 lists first 25 samples of the signal:

The estimated phasor magnitude and angle calculated by the recursion formulafor 160 samples of the input signal are shown in Figs. 3.6 and 3.7. Note that thesecond harmonic ripple anticipated in the theory is evident in both figures. The

Table 3.3 Magnitude andphase angle of Q

Δf |Q| ∠Q (°)

−5 −0.0434 29.37

−4.5 −0.0390 27.94

−4 −0.0346 26.5

−3.5 −0.0302 25.06

−3 −0.0258 23.62

−2.5 −0.0215 22.19

−2 −0.0171 20.75

−1.5 −0.0128 19.31

−1 −0.0085 17.87

−0.5 −0.0042 16.44

0 0 15

0.5 0.0042 13.56

1 0.0084 12.12

1.5 0.0125 10.69

2 0.0166 9.25

2.5 0.0206 7.81

3 0.0246 6.37

3.5 0.0285 4.94

4 0.0324 3.5

4.5 0.0363 2.06

5 0.0400 0.62

Table 3.4 Effect of thesampling rate on Q for afrequency of 62 Hz

Sampling rate |Q| ∠Q (°)

12 0.0172 24.5

24 0.0166 9.25

36 0.0164 4.17

48 0.0164 1.62

60 0.0164 0.1

72 0.0164 −0.92

84 0.0164 −1.64

96 0.0164 −2.19

108 0.0164 −2.61

120 0.0164 −2.95


amplitude of the ripple in the magnitude is about 0.42 (zero-to-peak of the variationin Fig. 3.6), which is identical to the predicted value for |Q| of 0.0042 for a fre-quency deviation of 0.5 Hz as seen from Table 3.3.

The estimated angle shown in Fig. 3.7 contains an average slope correspondingto Δωt. Here also the second harmonic ripple in the angle estimate is evident.

Table 3.5 First 25 samplesof the 60.5 Hz signal 100ejπ/4

Sample No. xk Sample No. xk1 100.0000 14 −67.2091

2 70.4433 15 −32.4138

3 36.0062 16 4.6272

4 −0.9256 17 41.3476

5 −37.7932 18 75.2034

6 −72.0424 19 103.8489

7 −101.3004 20 125.2995

8 −123.5400 21 138.0691

9 −137.2205 22 141.2730

10 −141.3941 23 134.6891

11 −135.7716 24 118.7737

12 −120.7424 25 94.6294

13 −97.3480

20 60 100 140

99.2

99.6

100

100.4

100.8

Sample number

|X΄|

Fig. 3.6 Magnitude ofphasor estimate of a signal at60.5 Hz

20 60 100 140

50

60

70

∠X΄ i

n de

gree

s

Sample number

Fig. 3.7 Angle of phasorestimate of a signal at 60.5 Hz


3.3 Post Processing for Off-Nominal Frequency Estimates

3.3.1 A Simple Averaging Digital Filter for 2f0

A very effective filter to correct for the errors introduced by the factor Q is toaverage three successive values of the estimate such that their relative phase anglesare 60° and 120° at the nominal fundamental frequency, which would correspond to120° and 240° for the second harmonic ripple. The result of applying such a filter tothe data in Figs. 3.6 and 3.7 is shown in Figs. 3.8 and 3.9, respectively. As can beseen, the second harmonic variation has been practically eliminated, and theremaining errors in the estimation are negligible.

Note that even after the single phase signals were filtered with the three-pointalgorithm, there was a small amount of residual second harmonic ripple. This is tobe expected, as the ripple in the single phase estimation process is at 2ω0 + Δωrather than at 2ω0, hence the three-point averaging does not eliminate the rippleexactly.

20 60 100 140

99.2

99.6

100

100.4

100.8

|X΄|

Sample number

Fig. 3.8 Magnitude ofphasor estimate of a signal at60.5 Hz using three-pointaveraging

20 60 100 140

50

60

70

Sample number

∠X΄ i

n de

gree

s

Fig. 3.9 Angle of phasorestimate of a signal at 60.5 Hzusing three-point averaging


3.3.2 A Re-sampling Filter

Another very effective filter is the re-sampling filter. Using the Fourier method tocalculate the phasor using Eq. (3.9), the signal frequency is estimated by taking thederivative of the phasor angle (see Chap. 4). With this estimated frequency, thesamples of the original signal at the estimated frequency are calculated using aninterpolation formula so that the new sampling rate corresponds to the estimatedfrequency. Assuming that the input signal is a sinusoid, the interpolation formulacan be derived as shown below.

Consider the input signal at frequency ω and a sampling clock corresponding toa frequency N times the nominal power system frequency ω0. Using the notation ofSect. (3.2), the sampling interval is 2π/N rad of the nominal frequency. Considerthe samples corresponding to the sampling index ‘k’ and ‘k + 1’ as shown inFig. 3.10 obtained at a sampling rate of N samples per cycle at the nominal fre-quency ω0. It is required that sample number ‘m’ corresponding to the sampling rateof N samples per cycle of the estimated frequency ω be calculated using aninterpolation formula.

Assuming that the input signal is given by


The samples corresponding to ‘k’ and ‘k + 1’ are given by

xk ¼ Xm cos½khþ/�; xkþ 1 ¼ Xm cos½ðkþ 1Þhþ/� ð3:13Þ

Sampling pulses at k k+1

n

θ

γ off-nominal signal

Sampling pulses at estimated Nω

Fig. 3.10 Re-sampling process applied to an off-nominal signal

3.3 Post Processing for Off-Nominal Frequency Estimates 57

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It is required that the sample xn corresponding to a sampling pulse generated bythe sampling clock corresponding to the frequency ω

xn ¼ Xm cos½khþ cþ/� ð3:14Þ

where the angles θ and γ are expressed based on the estimated frequency ω. Usingtrigonometric identities, it can be shown that

xn ¼ xk sinðh� cÞf g= sin hþ xkþ 1 sin cf g= sin h ð3:15Þ

Phasor estimation is then performed using the re-sampled data. These re-sampleddata phasors have very little errors of estimation.

Example 3.2 Phasor estimation with re-sampled data.Consider the data in Table 3.5, which is obtained from a signal at 60.5 Hz. The

data is taken at 60 × 24 samples per second. The re-sampled data obtained bysinusoidal interpolation as in Eq. (3.15) are given in Table 3.6.

3.4 Phasor Estimates of Pure Positive Sequence Signals

3.4.1 Symmetrical Components

The symmetrical components of three-phase voltages and currents are defined bythe formula

Table 3.6 First 25 samples of the re-sampled 60.5 Hz signal 100ejπ/4

SampleNo.

xk γ Re-sampleddata

SampleNo.

xk γ Re-sampleddata

1 100.0000 100.0000 14 −67.2091 0.2356 −70.7107

2 70.4433 0.2618 70.7107 15 −32.4138 0.2334 −36.6025

3 36.0062 0.2596 36.6025 16 4.6272 0.2313 −0.0000

4 −0.9256 0.2574 0.0000 17 41.3476 0.2291 36.6025

5 −37.7932 0.2553 −36.6025 18 75.2034 0.2269 70.7107

6 −72.0424 0.2531 −70.7107 19 103.8489 0.2247 100.0000

7 −101.3004 0.2509 −100.0000 20 125.2995 0.2225 122.4745

8 −123.5400 0.2487 −122.4745 21 138.0691 0.2203 136.6025

9 −137.2205 0.2465 −136.6025 22 141.2730 0.2182 141.4214

10 −141.3941 0.2443 −141.4214 23 134.6891 0.2160 136.6025

11 −135.7716 0.2422 −136.6025 24 118.7737 0.2138 122.4745

12 −120.7424 0.2400 −122.4745 25 94.6294 0.2116 100.0000

13 −97.3480 0.2378 −100.0000


X0

X1

X2

24

35 ¼ 1

3

1 1 11 a a2

1 a2 a

24

35 Xa

Xb

Xc

24

35 ð3:16Þ

where the phase quantity phasors are used to calculate the symmetrical components.Note that we are considering phasors calculated according to Eq. (3.7), i.e., at thewindow starting at sample number r, and with Fourier coefficients corresponding tonominal power system frequency. Using Eq. (3.7) to represent each of the phasors(adding the appropriate phase identifier as a subscript), Eq. (3.8) becomes:

X 0r0

X 0r1

X 0r2

24

35 ¼ 1

3

1 1 11 a a2

1 a2 a

24

35 PXaejrðx�x0ÞDt þQX�

ae�jrðxþx0ÞDt

PXbejrðx�x0ÞDt þQX�be

�jrðxþx0ÞDt

PXcejrðx�x0ÞDt þQX�c e

�jrðxþx0ÞDt

24

35 ð3:17Þ

or

X 0r0

X 0r1

X 0r2

24

35 ¼ 1

3

PðXa þXb þXcÞejrðx�x0ÞDt þQðX�a þX�

b þX�c Þe�jrðxþx0ÞDt

PðXa þ aXb þ a2XcÞejrðx�x0ÞDt þQðX�a þ aX�

b þ a2X�c Þe�jrðxþx0ÞDt

PðXa þ a2Xb þ aXcÞejrðx�x0ÞDt þQðX�a þ a2X�

b þ aX�c Þe�jrðxþx0ÞDt

24

35

ð3:18Þ

Example 3.3 Numerical example: balanced three-phase off-nominal frequencysignal.

Now, consider a three-phase balanced source of frequency 60.5 Hz

Xa ¼ 100ejp=4

Xb ¼ 100ejðp=4�2p=3Þ

Xc ¼ 100ejðp=4þ 2p=3Þð3:19Þ

This corresponds to the symmetrical components

X0 ¼ 0

X1 ¼ 100ejp=4

X2 ¼ 0

ð3:20Þ

The first 25 samples of these signals taken at a sampling rate of 1440 Hz aregiven in Table 3.7.

A plot of the three phasor magnitudes is shown in Fig. 3.11.It can be seen that the magnitude of errors in each of the phasors is identical to

that seen in Sect. 3.3. The magnitudes of positive and negative sequence compo-nents obtained from these signals are shown in Figs. 3.12 and 3.13. The angle ofthe positive sequence component is shown in Fig. 3.14. The negative sequence

3.4 Phasor Estimates of Pure Positive Sequence Signals 59

Table 3.7 First 25 samplesof the positive sequence60.5 Hz signal 100ejπ/4

Sample No. xka xkb xkc1 100.0000 36.6025 −136.6025

2 70.4433 70.9777 −141.4210

3 36.0062 100.4354 −136.4415

4 −0.9256 122.9347 −122.0091

5 −37.7932 136.9168 −99.1235

6 −72.0424 141.4129 −69.3705

7 −101.3004 136.1117 −34.8113

8 −123.5400 121.3804 2.1597

9 −137.2205 98.2395 38.9810

10 −141.3941 68.2924 73.1017

11 −135.7716 33.6139 102.1577

12 −120.7424 −3.3935 124.1360

13 −97.3480 −40.1658 137.5139

14 −67.2091 −74.1553 141.3645

15 −32.4138 −103.0072 135.4210

16 4.6272 −124.7225 120.0953

17 41.3476 −137.7967 96.4491

18 75.2034 −141.3241 66.1207

19 103.8489 −135.0602 31.2113

20 125.2995 −119.4391 −5.8605

21 138.0691 −95.5429 −42.5262

22 141.2730 −65.0273 −76.2457

23 134.6891 −30.0065 −104.6827

24 118.7737 7.0933 −125.8670

25 94.6294 43.7016 −138.3310

20 60 100 140

99.2

99.6

100

100.4

100.8 |X΄a| |X΄b| |X΄c|

|X΄|

Sample number

Fig. 3.11 Phasor magnitudesof individual phase quantitiesin balanced 60.5 Hz signals


20 60 100 140

99.2

99.6

100

100.4

100.8

|X1΄|

Sample number

Fig. 3.12 Positive sequence voltage magnitude estimate for a 60.5 Hz balanced signal

20 60 100 140

0.411

0.413

0.415

0.417

0.419

Sample number

|X2΄|

Fig. 3.13 Negative sequence voltage magnitude estimate for a 60.5 Hz balanced signal

20 60 100 140

50

60

70

∠X

1in

deg

rees

Sample number

Fig. 3.14 Positive sequence voltage angle estimate for a 60.5 Hz balanced signal

3.4 Phasor Estimates of Pure Positive Sequence Signals 61

component angle is not shown in a plot, since it is not very instructive, it rotates at(approximately) 2ω0. The zero sequence component is identically zero.

Figures 3.12 and 3.13 show that the positive and negative sequence componentshave no second harmonic ripple as in case of a single phase input. The magnitudesare constant at 99.9886 and 0.4197, respectively. This agrees with the estimates of|P| and |Q| (0.9999 and 0.0042 from Tables 5.1 and 5.3, respectively) at 0.5 Hzdeviation from the nominal frequency.

3.5 Estimates of Unbalanced Input Signals

Most phasor measurement applications call for measurements under normal systemconditions. This usually implies balanced three-phase voltages and currents.However, it is common to have some degree of unbalance in the power system dueto unbalanced loads and un-transposed transmission lines. Estimates of suchunbalances (negative and zero sequence) range between 0 and 10% of the positivesequence component. The effect of such unbalances on positive sequence mea-surement at off-nominal conditions is considered in this section.

3.5.1 Unbalanced Inputs at Off-Nominal Frequency

When the off-nominal three-phase signals are unbalanced, (either or both) theirnegative and zero sequence components are nonzero. The effect of unbalances inthe input signals can be best studied by considering the estimated sequence com-ponents in the presence of unbalances. The phase components in terms of theirsymmetrical components are as follows:

Xa ¼ X0 þ X1 þ X2f gXb ¼ X0 þ a2X1 þ aX2

� �Xc ¼ X0 þ aX1 þ a2X2

� � ð3:21Þ

The symmetrical components in Eq. (3.21) are true symmetrical components ofthe input signals.

Substituting Eq. (3.16) into Eq. (3.18) leads to symmetrical component esti-mated with nominal frequency phasor computation process:


http://dx.doi.org/10.1007/978-3-319-50584-8_5

http://dx.doi.org/10.1007/978-3-319-50584-8_5

X 0r0

X 0r1

X 0r2

264

375 ¼ 1

3

Pð3X0Þejrðx�x0ÞDt þQð3X�0Þe�jrðx�x0ÞDt



264

375

¼ Pejrðx�x0ÞDtX0

X1

X2

264

375þQe�jrðx�x0ÞDt

X�0

X�2

X�1

264

375 ð3:22Þ

In obtaining Eq. (3.22), use has been made of the following identities:

a� ¼ a2

a2� ¼ a

1þ aþ a2 ¼ 0

ð3:23Þ

Equation (3.22) displays an interesting result: at off-nominal frequencies, thepositive and negative sequence components of the input signals create false neg-ative and positive sequence components, respectively, which introduce errors in theestimate of the positive and negative components. Zero sequence component alonemakes an error contribution to the zero sequence estimate. These error contributions—because of the multiplier Q—vanish as the frequency approaches the nominalfrequency (i.e., ω → ω0).

Example 3.4 Numerical example: unbalanced three-phase off-nominal frequencysignal.

Now, consider an unbalanced input at 60.5 Hz. The three-phase inputs areassumed to be made up of the following symmetrical components;

X0 ¼ 10ejp=4

X1 ¼ 100ejp=4

X2 ¼ 20ejp=4ð3:24Þ

The corresponding phase quantities are given by

Xa ¼ 130ej0:7854

Xb ¼ 85:44e�j1:4105

Xc ¼ 85:44ej2:9813ð3:25Þ

As before, these signals are sampled at 1440 Hz, and the phasor representationof their symmetrical components was estimated. The first 25 samples of these inputquantities are given in Table 3.8.

The magnitude and angle of the symmetrical components calculated from thesephasors are shown in Figs. 3.15, 3.16, 3.17, 3.18, 3.19, and 3.20.

3.5 Estimates of Unbalanced Input Signals 63

Table 3.8 First 25 samplesof the unbalanced 60.5 Hzsignals

Sample No. xka xkb xkc1 130.0000 19.2858 −119.2814

2 91.5763 49.7413 −120.1814

3 46.8080 76.7506 −112.7550

4 −1.2033 98.4425 −97.5167

5 −49.1312 113.3140 −75.5221

6 −93.6552 120.3349 −48.2953

7 −131.6905 119.0187 −17.7224

8 −160.6020 109.4566 14.0783

9 −178.3867 92.3111 44.9037

10 −183.8123 68.7701 72.6180

11 −176.5030 40.4646 95.3012

12 −156.9652 9.3556 111.3817

13 −126.5524 −22.4016 119.7454

14 −87.3719 −52.6068 119.8129

15 −42.1380 −79.1673 111.5795

16 6.0153 −100.2428 95.6156

17 53.7519 −114.3733 73.0272

18 97.7644 −120.5798 45.3794

19 135.0036 −118.4322 14.5876

20 162.8894 −108.0794 −17.2149

21 179.4898 −90.2386 −47.8247

22 183.6548 −66.1459 −75.1211

23 175.0958 −37.4704 −97.2130

24 154.4058 −6.1989 −112.5697

25 123.0182 25.5021 −120.1273

20 60 100 140

9.92

9.96

10.0

10.04

10.08

Sample number

|X0 |

Fig. 3.15 Zero sequence voltage magnitude estimate for a 60.5 Hz unbalanced signal


Note the presence of the second harmonic ripple in all the signals. The governingequation is (3.22). The zero sequence result (Fig. 3.15) shows a steady level of 10,with a superposition of a peak-to-peak second harmonic component of 0.042, whichcorresponds to QX�

0 , |Q| being 0.0042 for a frequency deviation of 0.5 Hz.Similarly, Fig. 3.17 shows a second harmonic of 0.084 peak-to-peak in the positivesequence estimate, which agrees with the expected value of QX�

2 with X2 magnitudebeing 20. Similarly, Fig. 3.19 shows the negative sequence component estimate of

20 60 100 140

50

60

70

∠X

0΄ in

degr

ees

Sample number

Fig. 3.16 Zero sequence voltage angle estimate for a 60.5 Hz unbalanced signal

20 60 100 14099.8

99.9

100

100.1

Sample number

|X1΄|

Fig. 3.17 Positive sequence voltage magnitude estimate for a 60.5 Hz unbalanced signal

20 60 100 140

50

60

70

∠X

1΄in

deg

rees

Sample number

Fig. 3.18 Positive sequence voltage angle estimate for a 60.5 Hz unbalanced signal


a steady level of 20, with a superimposed second harmonic component of 0.42peak-to-peak corresponding to QX�

1 , X1 having a magnitude of 100.All the second harmonic components can be eliminated by the three-point filter

algorithm discussed in Sect. 3.3.1. The results are similar to those shown inFigs. 3.8 and 3.9.

3.5.2 A Nomogram

We now summarize the result of analysis of unbalances at off-nominal frequency.The factor Q defined by Eq. (3.11) and plotted in Fig. 3.5 provides the magnitudeof the (ω + ω0) component in the output phasor when a single phase phasor atoff-nominal frequency is estimated, or when the positive sequence phasor from anunbalanced three-phase source at off-nominal frequency is estimated. Note that inthis discussion, it is assumed that the off-nominal frequency phasors being mea-sured are constant. Equations (3.9) and a part of Eq. (3.22) are reproduced here asEqs. (3.26) and (3.27) for ready reference:

20 60 100 140

19.2

19.6

20.0

20.4

20.8

Sample number

|X2΄|

Fig. 3.19 Negative sequence voltage magnitude estimate for a 60.5 Hz unbalanced signal

20 60 100 140

50

60

70

Sample number

∠X

1΄in

de g

rees

Fig. 3.20 Negative sequence voltage angle estimate for a 60.5 Hz unbalanced signal


X 0r ¼ PXejrðx�x0ÞDt þQX�e�jrðx�x0ÞDt ð3:26Þ

X 0r1 ¼ PXejrðx�x0ÞDt þQX�

2e�jrðx�x0ÞDt ð3:27Þ

The two equations have similar forms. While the (ω + ω0) term in (3.26) has amagnitude proportional to QX*, the corresponding term in Eq. (3.27) is proportionalto QX2

*. We may thus present the (ω + ω0) term for a single phase phasor mea-surement as an error term which is a function of Δω, while the corresponding termfor positive sequence measurement will depend upon Δω as well as upon thenegative sequence component. Normalizing the two equations,

X 0r ¼ X Pejrðx�x0ÞDt þQ

X�

X

� �e�jrðx�x0ÞDt

� �ð3:28Þ

X 0r1 ¼ X1 Pejrðx�x0ÞDt þQ

X�2

X1

� �e�jrðx�x0ÞDt

� �ð3:29Þ

As noted previously, P is almost equal to 1.0 for all practical frequencyexcursions.

If the phase angle of the single phase phasor X in Eq. (3.28) is θ while the phaseangles of positive and negative sequence phasors X1 and X2 are θ1 and θ2,respectively,

X�

X

� �¼ e�2jh ð3:30Þ

X�2

X1

� �¼ k21e

�jðh2 þ h1Þ ð3:31Þ

where k21 is the ratio of magnitudes of the negative sequence to positive sequencecomponent in the quantities being measured (k21 represents the per unit negativesequence component in the inputs). Thus, Eqs. (3.28) and (3.29) become

X 0r ¼ X Pejrðx�x0ÞDt þQe�jrðx�x0ÞDt�2jh

h ið3:32Þ

X 0r1 ¼ X1 Pejrðx�x0ÞDt þQk21e

�jrðx�x0ÞDt�jðh2 þ h1Þh i

ð3:33Þ

The magnitude of the per unit (ω + ω0) component in Eq. (3.32) is simply |Q|,whereas that in Eq. (3.33) is |Qk21|. The single phase measurement error componentat (ω + ω0) can be obtained from a plot of |Q| versus Δω as shown in Fig. 3.21.

In case of an unbalanced three-phase input, the contribution at (ω + ω0) is afunction of two variables: Q and k21. Thus, the result must be presented as a curved


plane in a three dimensional plot or as a family of curves in a two dimensional plot.These results are shown in Figs. 3.22 and 3.23, respectively.

For representation of the data on a surface in three dimensions, we use theexpression for Q given in Eq. (3.11), and instead of (ω + ω0), we use the equivalentform (2ω0 + Δω) = (240π + 2πΔf). Thus,

Fig. 3.21 Per unit error contribution at frequency (ω + ω0) when measuring a single phasequantity with a frequency deviation of Δf

-5

0

5

00.05

0.10.15

0.2-0.01

-0.005

0

0.005

0.01

Frequency deviation Δf Per unit negative sequence

Com

pone

nt o

f 0)

Fig. 3.22 Per unit phasor estimate error at frequency (ω + ω0) when measuring an unbalancedthree-phase quantity with a frequency deviation of Δf. The per unit value of the negative sequencecomponent in terms of positive sequence component is k21. The figure is a curved surface in3-dimensions representing Eq. (3.34)


k21Q ¼ k21sin Nð240pþ 2pDf ÞDt

2

N sin ð240pþ 2pDf ÞDt2

( )ð3:34Þ

Further, we use the same data sampling rate as was used in earlier discussions:N = 24, and Δt = 1/1440 s. For this sampling rate and letting Δf vary between±5 Hz, leads to the surface shown in Fig. 3.22.

As noted previously, the factor Q changes sign with Δf. In order to simplifyFig. 3.23, the dotted lines, which belong to negative Δf have been plotted with theirsigns reversed. The surface in Fig. 3.22 does show the correct sign change in Q asΔf becomes negative.

3.6 Sampling Clocks Locked to the Power Frequency

It has been assumed thus far that the sampling process is keyed to the nominalfrequency of the power system regardless of the prevailing power system fre-quency. As the power system is rarely at the nominal frequency, error terms appearin the phasor estimation process. The error terms were represented by the factorQ in the previous sections and are responsible for the second harmonic ripple in thephasor magnitude and angle estimates. It has already been demonstrated above thatthe error terms are very small for normally expected frequency excursions in apower system, and furthermore the errors can be eliminated with the help ofappropriate filtering techniques.

0.00

0.004

0.008

Frequency deviation (Hz)

Com

pone

nt a

t (ω

−ω0)

0` 1 4 5 32

0.002

0.006

Solid line (ω ≥ω0) Dotted line (ω≤ω0) k21 = 0.2

k21 = 0.15

k21 = 0.1

k21 = 0.05

k21 = 0.0

Fig. 3.23 Per unit contribution to phasor estimate at frequency (ω + ω0) when measuring anunbalanced three-phase quantity with a frequency deviation of Δf. The per unit value of thenegative sequence component in terms of positive sequence component is k21


Another option, although not as frequently employed in PMU technology, is totrack the power system frequency and alter the sampling clock to match the periodof the prevailing power system frequency [3]. A generic flowchart of such a schemeis shown in Fig. 3.24. The frequency of the power system is estimated by mea-suring the zero crossing intervals of the voltage signals, or by using one of thefrequency estimation techniques described in Chap. 4 (Alternatively, phase-lockedloops may be used to track the power system frequency, and then sampling clockpulses generated at the desired sampling rate). In any case, as the voltage wave-forms may undergo step changes due to switching operations, the frequencytracking system must be designed with some care. In addition, if the voltage signalused for frequency measurement should be lost (due to a fault, or due to a blownfuse in the voltage transformer circuit), an assured fall-back position must beavailable to the frequency tracking process.

If the sampling rate is matched to the power system frequency, there is no errorin phasor estimation. The estimated phasor will be correct, and no ripple corre-sponding to the Q factor is observed. It is of course obvious that if the frequencymeasurement is in error, then the phasor estimate will also be in error.

An issue with this technique of phasor estimation is the correlation between themeasured phasor and the time tag with which the measurement must be associated.As will be seen in the discussion of the IEEE standard which defines the require-ments for PMUs, the time tags of phasor measurements must coincide with the GPSsecond marker and with multiples of nominal periods of power system frequencymeasured from the GPS signal. For example, if the time tag rate of 30 per second isselected for a 60 Hz system (i.e., once every two cycles of the nominal powersystem frequency), the allowable time tags for the PMU are as shown Fig. 3.25a.These time tag instants are of course related to the GPS clock and are not related tothe sampling clock generated from the power system frequency measurement. If thefirst sample of the data window from which the phasor is estimated is as shown in

FrequencyEstimator

Samplingclock

A/DConverter

SampledData

AnalogInputs

Fig. 3.24 Sampling clocksynchronized to the powersystem frequency. Thesampling rate (samples percycle) is multiplied by thepower system frequency todetermine the sampling clockfrequency


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Fig. 3.25b, the estimated angle of the phasor will be θ—being the angle betweenthe first sample and the peak of the sinusoid. It then becomes necessary that theinterval between the first sample of the data window and the time tag be determinedso that the correct angle corresponding to the time tag (ϕ) be reported as the angle ofthe phasor.

3.7 Non-DFT Type Phasor Estimators

There are a number of alternative algorithms described in the literature [4–7] forestimating phasors from sampled data. Assuming that the sinusoid under consid-eration is given by

xðtÞ ¼ X1 cosxtþX2 sinxt ð3:35Þ

where X1, X2 and ω are all treated as unknown. Taking sufficient samples of thesampled data, one could now formulate the estimation problem as a non-linearWLS problem. In addition to finding the phasors (through X1 and X2), the procedurewould also determine the power system frequency. One could also include the DCoffset in this formulation and estimate the DC component at the same time [4].

Other techniques cited in the literature include neural networks [8], KalmanFilter [9], and wavelets [10]. The interested reader will find explanations of thesemethods in the cited references. For this book, we concentrate on the DFT-basedtechniques described in this chapter, as they provide simple, elegant, and accurateestimation of the parameters of interest.

θ

Time-tag instantaccording to theStandard

φ

First sample

GPS t=0

2T 4T

(a) (b)

Fig. 3.25 a Phasor time tags at multiples of fundamental frequency period T. Example here showsreporting rate of once every two cycles. b The phasor angle estimated by the variable frequencyclock is θ, as determined by the waveform and the instant when the first sample is taken. The phaseangle which must be reported in the output is ϕ

3.6 Sampling Clocks Locked to the Power Frequency 71

References

1. Phadke, A. G., Thorp, J. S., & Adamiak, M. G. (1983). A new measurement technique fortracking voltage phasors, local system frequency, and rate of change of frequency. IEEETransactions on Power Apparatus and Systems, PAS-102(5), 1025–1038.

2. Phadke, A. G., & Thorp, J. S. (1991). Improved control and protection of power systemsthrough synchronized phasor measurements. Control and Dynamic Systems, 43 (AcademicPress, Inc.).

3. Benmouyal, G. (1991). Design of a combined digital global differential and volt/hertz relayfor step transformer. IEEE Transactions on Power Delivery, 6(3), 1000–1007.

4. Terzija, V. V., Djuric, M. B., & Kovacevic, B. D. (1994). Voltage phasor and local systemfrequency estimation using Newton type algorithm. IEEE Transactions on Power Delivery, 9(3), 1368–1374.

5. Sidhu, T. S., & Sachdev, M. S. (1996). An iterative DSP technique for tracking power systemfrequency and voltage phasors. In Canadian Conference on Electrical and ComputerEngineering (Vol. 1, pp. 115–118), May 26–29, 1996.

6. Kamwa, I., & Grondin, R. (1991). Fast adaptive schemes for tracking voltage phasor and localfrequency in power transmission and distribution systems. In Transmission and DistributionConference, 1991, Proceedings of the 1991 IEEE Power Engineering Society (pp. 930–936),September 22–27, 1991.

7. Yang, J.-Z., & Liu, C.-W. (2000). A precise calculation of power system frequency andphasor. IEEE Transactions on Power Delivery, 15(2), 494–499.

8. Dash, P. K., Panda, S. K., Mishra, B., & Swain, D. P. (1997). Fast estimation of voltage andcurrent phasors in power networks using an adaptive neural network. IEEE Transactions onPower Systems, 12(4), 1494–1499.

9. Girgis, A. A., & Brown, R. G. (1981). Application of Kalman filtering in computer relaying.IEEE Transactions on Power Apparatus and Systems, PAS-100.

10. Chi-kong, W., Ieng-tak, L., Chu-san, L., Jing-tao, W., & Ying-duo, H. (2001). A novelalgorithm for phasor calculation based on wavelet analysis. Power engineering societysummer meeting, 2003, IEEE (Vol. 3, pp. 1500–1503), July 15–19, 2001.


Chapter 4Frequency Estimation

4.1 Historical Overview of Frequency Measurement

Power system frequency measurement has been in use since the advent of alter-nating current generators and systems. The speed of rotation of generator rotors isdirectly related to the frequency of the voltages they generate. The Watt-typefly-ball governor of steam turbines (Fig. 4.1) is essentially a frequency-measuringdevice which is used in a feedback control system to keep the machine speed withina limited range around the nominal value. However, this measurement is availableonly at the generating stations, and there is a need for measuring frequency ofpower system at network buses away from the generating stations.

The earliest frequency measurement for power frequency voltages was per-formed by mechanical devices which employed mechanical resonators (similar totuning forks) tuned to a range of frequencies around the nominal power frequency[1]. Such a frequency meter of mid-1950s vintage is shown in Fig. 4.2a. Anotherfrequency-measuring instrument of about the same period is a resonance-typedevice, whereby tuned resonant circuits at different frequencies are energized by thesecondary voltage obtained from a voltage transformer, and the circuit that is inresonance provides the frequency measurement (Fig. 4.2b) [1]. Typical resolutionof these meters was of the order of 0.25 Hz.

The next advance in frequency measurement came with the introduction ofprecise time measurement techniques. By measuring the time interval betweenconsecutive zero crossings of the voltage waveform, the frequency of the voltagecould be determined. Clearly, the accuracy of such a measurement depends on theprecision of time measurement, as well as on the accuracy with which the zerocrossing of the waveform could be determined. This latter measurement is affectedby the presence of noise in the measurement, varying harmonic frequencies andlevels, and the performance of the zero-crossing detector circuits.

Synchronized phasor measurements offer an opportunity for measuring powersystem frequency which eliminates many of these error sources. It should be noted


73

that the frequency measurement on a power system is primarily dedicated to esti-mating rotor speed(s) of connected generators. As such, the positive sequencevoltage measurement is an ideal vehicle for frequency measurement. In addition,phasors reflect the fundamental frequency components of the voltages, and har-monics do not affect frequency measurement based upon phasors. Techniques formeasuring frequency from phasors are described in the following sections.

4.2 Frequency Estimates from BalancedThree-Phase Inputs

Frequency and rate of change of frequency can be estimated from the phase anglesof phasor estimates [2]. It was pointed out in Chap. 3 that positive sequence phasorestimated from balanced inputs at off-nominal frequencies has a minor attenuation

HP LP

Generator

Watt typespeedgovernor

SteamValve

Reset control

SpeedSensor

Fig. 4.1 Mechanical speed sensing used in a Watt-type speed governor of a steam turbine

Fig. 4.2 a A mechanical resonance-type frequency meter. b An electrical resonance-typefrequency meter. These instruments are for a 50 Hz power system

74 4 Frequency Estimation

http://dx.doi.org/10.1007/978-3-319-50584-8_3

in phasor magnitude, and both the magnitude and phase angle estimates are freefrom a ripple of approximate second harmonic. Setting the negative sequencecomponent of the input X2 = 0 in Eq. (3.22), the estimate of the positive sequencevoltage is given by

X 0r1 ¼ PX1e

jrðx�x0ÞDt ð4:1Þ

The magnitude of P is the attenuation factor, and phase angle of P is a constantoffset in the measured phase angles. The angle of the phasor X 0

r1 advances at eachsample time by ðx� x0ÞDt where ω is the signal frequency, ω0 is the nominalsystem frequency, and Δt is the sampling interval.

It should be clear from Eq. (4.1) that the first and second derivatives of phaseangle of the phasor estimate would provide an estimate of Dx ¼ ðx� x0Þ and therate of change of frequency. Since there are errors of estimation in phasor calcu-lation, it is desirable to use a weighted least squares approach over a reasonable datawindow for calculating the derivatives of the phase angle.

Assume that the positive sequence phasors are estimated over one period of thenominal frequency and that the phasors calculated with several consecutive datawindows over a span of 3–6 cycles are used for frequency and rate of change offrequency estimation.

Let ½/k�fk ¼ 0; 1; . . .;N � 1g be the vector of ‘N’ samples of the phase angles ofthe positive sequence measurement. The vector ½/k� is assumed to be monotonicallychanging over the window of ‘N’ samples. As the phase angles of the phasorestimate may be restricted to a range of 0–2π, it may be necessary to adjust theangles to make them monotonic over the entire spanning period by correcting anyoffsets of 2π radians which may exist. This is illustrated in Fig. 4.3.

If the frequency deviation from the nominal value and the rate of change offrequency at t ¼ 0 are Dx and x0, respectively, the frequency at any time ‘t’ isgiven by

xðtÞ ¼ ðx0 þDx þ tx0Þ ð4:2Þ

(b)

Ang

le2π

Time

Ang

le

Time

2π

(a)

0 0

Fig. 4.3 a Phasor estimates produce angles which are restricted to a range of 0–2π. b Forestimating the frequency and rate of change of frequency, the offset of 2π in the phase angleestimates is removed

4.2 Frequency Estimates from Balanced Three-Phase Inputs 75

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The phase angle is the integral of the frequency:

/ðtÞ ¼Z

xdt ¼Z

ðx0 þ tDxþ tx0Þdt ¼ /0 þ tx0 þ tDxþ 12t2x0 ð4:3Þ

/0 is the initial value of the angle. Assuming that the recursive algorithm is usedfor estimating the phasors, the term tx0 is suppressed from the estimated phaseangles (see Sect. 2.2.2). Thus, the phase angle as a function of time becomes

/ðtÞ ¼ /0 þ tDxþ 12t2x0 ð4:4Þ

If ϕ(t) is assumed to be a second-degree polynomial of time

/ðtÞ ¼ a0 þ a1tþ a2t2 ð4:5Þ

it follows that at t ¼ 0,

Dx ¼ a1x0 ¼ 2a2

ð4:6Þ

Or, in terms of Hz and Hz/sec,

Df0 ¼ a1=ð2pÞ and f 0 ¼ a2=ðpÞ ð4:7Þ

The vector of ‘N’ angle measurements is given by

/0/1/2

..

.

/N�1

2666664

3777775 ¼

1 0 01 Dt Dt2

1 2Dt 22Dt2

..

. ... ..

.

1 ðN � 1ÞDt ðN � 1Þ2Dt2

266664

377775

a0a1a2

24

35 ð4:8Þ

In matrix notation,

½/� ¼ ½B�½A� ð4:9Þ

where [B] is the coefficient matrix in Eq. (4.8). The unknown vector [A] is cal-culated by the weighted least squares (WLS) technique (4.9):

½A� ¼ ½BTB��1BT½/� ¼ ½G�½/� ð4:10Þ


http://dx.doi.org/10.1007/978-3-319-50584-8_2

where

½G� ¼ ½BTB��1BT ð4:11Þ

The matrix [G] is precalculated and stored for use in real time. It has ‘N’ rowsand 3 columns. In real time, [G] is multiplied by ½/] to obtain the vector [A], andfrom that, the frequency and rate of change of frequency at any time t (which is amultiple of Dt) can be calculated. This time is usually associated with the time tagfor which the measurement is posted.

Example 4.1 Numerical example of frequency and rate of change of frequencyestimation.

Consider an input with a frequency of 60.5 Hz and a rate of change of frequencyof 1 Hz per second. The polynomial for phase angles is given by

/ðtÞ ¼ /0 þ 2p� 0:5� tþð1=2Þt2 � 1� 2p

The initial angle ϕ0 is assumed to be 0.1 rad. Assuming that the phasors arecalculated at a sampling rate of 24 samples per cycle of the nominal power systemfrequency, the time step is Δt = (1/1440) s. Phase angles over a span of 4 cycles aretabulated below with and without a Gaussian random noise with zero mean and astandard deviation of 0.01 rad (Table 4.1).

The estimated frequency and rate of change of frequency using the weightedleast squares formulation are found to be Δf = 0.5000 Hz and f′(0) = 1.0000 Hz/s

Table 4.1 Partial list of 96phase angle samples with andwithout noise

Sample no. Phase angleswithout noise

Phase angleswith noise

1 0.1000 0.1007

2 0.1022 0.1017

3 0.1044 0.1062

4 0.1066 0.1077

5 0.1088 0.1075

6 0.1109 0.1103

… … …

88 0.3013 0.3015

89 0.3037 0.3033

90 0.3062 0.3056

91 0.3086 0.3087

92 0.3111 0.3123

93 0.3135 0.3110

94 0.3160 0.3166

95 0.3185 0.3175

96 0.3209 0.3219

4.2 Frequency Estimates from Balanced Three-Phase Inputs 77

with no noise in the phase angle measurement and Δf = 0.4968 Hz andf′(0) = 1.0550 Hz/s with the noise, respectively.

The estimates of Δf and f′ for different amounts of random noise in phase anglemeasurements are shown in Table 4.2.

The results in Table 4.2 are for 1000 Monte Carlo trials with the specifiedstandard deviation of the noise. It is clear that the rate of change of frequency ismore sensitive to the amount of noise in the input. Also, the estimates are essen-tially zero-mean processes.

4.3 Frequency Estimates from Unbalanced Inputs

The effect of unbalance in input signals has been analyzed in Sect. 3.5. Equation (3.22)provides the formula for the estimate of the positive sequence component when there isa negative sequence component present in the input signal:

X 0r1 ¼ PX1e

jrðx�x0ÞDt þQX�2e

jrðx�x0ÞDt ð4:12Þ

where Q is given by Eq. (3.11) and X2 is the negative sequence component in theinput signals. The effect of the second term of Eq. (4.12) is to produce a ripple inthe angle estimate of the positive sequence component. This ripple can be elimi-nated by one of the filtering techniques described in Sect. 3.3. When the ripple inangle is eliminated, the frequency and rate of change of frequency can be estimatedas in Sect. 4.2. The error performance of the estimates is then identical to thatcorresponding to the balanced input signals.

4.4 Nonlinear Frequency Estimators

It is possible to formulate the frequency and rate of change of frequency estimationproblem as a nonlinear estimation problem from the input signal waveform [3, 4].Consider a single-phase input having a frequency deviation of Δω and a rate ofchange of frequency ω′ (as in Eq. 4.4)

Table 4.2 Effect of random noise on frequency and rate of change of frequency estimation

σ of randomnoise in radians

Mean offrequencyestimate in Hz

σ offrequencyestimate in Hz

Mean of rate ofchange of frequencyestimate in Hz/s

σ of rate of changeof frequencyestimate in Hz

0.0001 0.5000 0.0003 1.0000 0.0096

0.0005 0.5001 0.0017 0.9979 0.0498

0.0010 0.5000 0.0033 0.9986 0.0982

0.0050 0.5000 0.0167 1.0038 0.4888

0.0100 0.4998 0.0331 1.0033 0.9660


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http://dx.doi.org/10.1007/978-3-319-50584-8_3

xðtÞ ¼ X cos /0 þ tDxþ 12t2x0

� �ð4:13Þ

‘N’ samples of this signal at a sampling interval of Δt are {xk, k = 0, 1, …,N − 1}. It is assumed that there are four unknowns in the data samples:

z ¼X/0Dxx0

2664

3775 ð4:14Þ

The function x(t) is a nonlinear function of the four unknowns, and if ‘N’ isgreater than four, a nonlinear weighted least squares iterative technique can be usedto solve the four unknowns.

Assuming reasonable initial values of the four unknowns: [z0], the initial esti-mates of the function x(t) are [x0]. Using the first-order terms of Taylor series torepresent the nonlinear function around [z0]

½x� x0� ¼ @x@X

@x@/0

@x@Dx

@x@x0

h iz¼z0

Dz½ �: ð4:15Þ

where the partial derivatives are columns of ‘N’ rows evaluated at the assumedvalue of the unknown vector [z0]. Representing the matrix of partial derivatives bythe Jacobian matrix [J], the weighted least squares solution for [Δz] is

½Dz� ¼ ½JTJ��1½x� x0� ð4:16Þ

The four partial derivatives in Eq. (4.15) are obtained by differentiating theexpression for x(t):

J1 ¼ @x@X

¼ cos /0 þ tDxþ 12t2x0

� �

J2 ¼ @x@u0

¼ �X sin /0 þ tDxþ 12t2x0

� �

J3 ¼ @x@Dx

¼ �Xt sin /0 þ tDxþ 12t2x0

� �

J4 ¼ @x@x0 ¼ �X

t2

2sin /0 þ tDxþ 1

2t2x0

� �ð4:17Þ

Having calculated the corrections [Δz] in Eq. (4.16), they are added to [z0] toproduce the answer at the end of the first iteration. The process is repeated until theresidual [x − x0] becomes smaller than a suitable tolerance.

4.4 Nonlinear Frequency Estimators 79

Example 4.2 Numerical example of nonlinear frequency and rate of change offrequency estimation.

Consider a single-phase input with an amplitude of 1.1, a frequency 60.5 Hz att = 0, a rate of change of frequency of 1 Hz per second, and a phase angle ϕ0 of π/8.The initial values for starting the iteration are assumed to be

X ¼ 1:0

/0 ¼ 0

Dx ¼ 0

x0 ¼ 0

The sampling rate is assumed to be 1440 Hz, and 96 samples of the input signalare used to estimate the signal parameters.

Table 4.3 lists the first 10 values of the input signal, the estimated signal with thevector [z0], and the first 10 entries of the Jacobian matrix.

The corrections vectors at the end of the first iteration are

DðXÞ ¼ �0:0365

Dð/0Þ ¼ 0:4072

DðDxÞ ¼ 4:2767

Dðx0Þ ¼ �32:2073

At the end of four iterations, correct values for the unknowns are obtained.It is necessary to consider the effect of noise in the sampled data on the per-

formance of the nonlinear frequency estimator. Zero-mean normally distributedrandom noise was added to the sampled data of the above example, and the effecton the results obtained after 5 iterations evaluated. 1000 Monte Carlo trials producethe result shown in Table 4.4. It can be seen that the mean of the parameterestimation is very close to the true value, although the standard deviation of the

Table 4.3 First 10 values of [x], [x0], and [J] at the beginning of the iteration

[x] [x0] [J1] [J2] [J3] [J4] × 104

1 1.0163 1.0000 1.0000 0 0 0

2 0.8712 0.9659 0.9659 −0.2588 −0.0002 −0.0006

3 0.6658 0.8660 0.8660 −0.5000 −0.0007 −0.0048

4 0.4143 0.7071 0.7071 −0.7071 −0.0015 −0.0153

5 0.1340 0.5000 0.5000 −0.8660 −0.0024 −0.0334

6 −0.1555 0.2588 0.2588 −0.9659 −0.0034 −0.0582

7 −0.4343 0.0000 0.0000 −1.0000 −0.0042 −0.0868

8 −0.6830 −0.2588 −0.2588 −0.9659 −0.0047 −0.1141

9 −0.8843 −0.5000 −0.5000 −0.8660 −0.0048 −0.1336

10 −1.0244 −0.7071 −0.7071 −0.7071 −0.0044 −0.1381


estimation increases very rapidly as the size of the noise added increases. The rateof change of frequency is practically unusable when the noise exceeds 1% of thesignal peak value. The amplitude and phase angle estimates are quite good even forvery large sample errors.

4.5 Other Techniques for Frequency Measurements

A number of other techniques for measuring power system frequency have beenpublished in the technical literature [5–8]. These references are provided for theinterested reader as a sample of what is available and is by no means a completelisting of papers dealing with frequency measurement. In general, the fasterapproaches (measurements made within one or two periods of the power frequencysignal) tend to have greater errors than those using longer data windows. It is wellto keep in mind that a traditional use of frequency measurement is in underfre-quency load shedding. Relays used for that purpose tend to have operating times ofthe order of 5–6 cycles of the nominal power frequency. This is probably a goodsize for a data window to be used in frequency estimation.

One should not have excessively long data windows for frequency estimation inorder to improve the accuracy of the estimate. During transient stability swings, thefrequency of the power system may change rapidly. Thus, a long window mayinclude significantly different frequencies over the window span, and once again,the frequency estimation may be in error. We will consider the effect of changingfrequency due to transients in Chap. 6.

Table 4.4 Effect of sample noise on the estimation of signal parameters

r samplenoise

MeanX

Mean/0

MeanDx

Meanx0

rX r/0 rDx rDx0

0.0 1.1000 0.3927 0.5000 1.0000 0 0 0 0

0.01 1.1000 0.3929 0.4982 1.0511 0.0014 0.0037 0.0430 1.3061

0.02 1.1000 0.3924 0.5011 0.9927 0.0028 0.0076 0.0878 2.6582

0.03 1.1001 0.3923 0.5028 0.8986 0.0043 0.0112 0.1323 4.0170

0.04 1.1001 0.3926 0.4992 1.0135 0.0056 0.0144 0.1620 4.9061

0.05 1.1001 0.3928 0.5045 0.8704 0.0072 0.0187 0.2153 6.4638

0.06 1.1007 0.3930 0.5068 0.7559 0.0086 0.0223 0.2543 7.7155

0.07 1.1004 0.3930 0.5028 0.8911 0.0099 0.0269 0.3029 9.0593

0.08 1.1010 0.3910 0.5185 0.4604 0.0119 0.0293 0.3424 10.4239

0.09 1.1003 0.3942 0.4814 1.6546 0.0129 0.0337 0.3877 11.7737

0.10 1.1006 0.3943 0.4957 1.0495 0.0143 0.0376 0.4451 13.5048

The signals are the same as for Example 4.1

4.4 Nonlinear Frequency Estimators 81

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References

1. Lythall, R. T. (1953). The J. & P. Switchgear book (5th ed., pp. 441–442). Charlton, London,S.E. 7: Johnson & Phillips Ltd.

2. Phadke, A. G., Thorp, J. S., & Adamiak, M. G. (1983). A new measurement technique fortracking voltage phasors, local system frequency, and rate of change of frequency. IEEETransactions on Power Apparatus and Systems, 102(5), 1025–1038.

3. Sachdev, M. S., & Giray, M. M. (1985). A least error squares technique for determiningpower system frequency. IEEE Transactions on Power Apparatus and Systems, PAS-104(2),437–444.

4. Terzija, V. V., Djuric, M. B., & Kovacevic, B. D. (1994). Voltage phasor and local systemfrequency estimation using newton-type algorithms. IEEE Transactions on Power Delivery,9(3), 1368–1374.

5. Sidhu, T. S., & Sachdev, M. S. (1998). An iterative technique for fast and accuratemeasurement of power system frequency. IEEE Transactions on Power Delivery, 13(1),109–115.

6. Girgis, A. A., & Hwang, T. L. D. (1984). Optimal estimation of voltage phasors and frequencydeviation using linear and non-linear Kalman filtering. IEEE Transactions on Power Apparatusand Systems, 103(10), 2943–2949.

7. Moore, P. J., Carranza, R. D., & Johns, A. T. (1994). A new numeric technique for high-speedevaluation of power system frequency. IEEE Proceedings-Generation, Transmission andDistribution, 141(5), 529–536.

8. Hart, D., Novosel, D., Hu, Y., Smith, B., & Egolf, M. (1996). A new frequency tracking andphasor estimation algorithm for generator protection. Paper No. 96, SM 376-4-PWRD, 1996,IEEE-PES Summer Meeting, Denver, July 28–August 1, 1996.


Chapter 5Phasor Measurement Units and PhasorData Concentrators

5.1 Introduction

The history of phasor measurement unit (PMU) evolution was discussed in Chap. 1.In this chapter, we will consider certain practical implementation aspects of thePMUs and the architecture of the data collection and management system necessaryfor efficient utilization of the data provided by the PMUs. One of the mostimportant features of the PMU technology is that the measurements aretime-stamped with high precision at the source, so that the data transmission speedis no longer a critical parameter in making use of this data. All PMU measurementswith the same time stamp are used to infer the state of the power system at theinstant defined by the time stamp. It is clear that PMU data could arrive at a centrallocation at different times depending upon the propagation delays of the commu-nication channel in use. The time tags associated with the phasor data provide anindexing tool which helps create a coherent picture of the power system out of suchdata. The Global Positioning System (GPS) has become the method of choice forproviding the time tags to the PMU measurements and will be described briefly inthe following sections. Other aspects of the overall PMU data collection systemsuch as phasor data concentrators and communication systems will also be con-sidered in this chapter.

The industry standards which define file structures for compliant PMUs havebeen very important to ensure inter-operability of PMUs made by different man-ufacturers and will be considered in Sect. 5.6.

Kenneth E. Martin contributed to Sect. 5.6.


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5.2 A Generic PMU

The phasor measurement units manufactured by different manufacturers differ fromeach other in many important aspects. It is therefore difficult to discuss the PMUhardware configuration in a way which is universally applicable. However, it ispossible to discuss a generic PMU, which will capture the essence of its principalcomponents.

Figure 5.1 is based upon the configuration of the first PMUs built at VirginiaTech (and shown in Fig. 1.1). Remember that PMUs evolved out of the develop-ment of the Symmetrical Component Distance Relay. Consequently, the structureshown in Fig. 5.1 parallels that of a computer relay. The analog inputs are currentsand voltages obtained from the secondary windings of the current and voltagetransformers. All three phase currents and voltages are used so that positivesequence measurement can be carried out. In contrast to a relay, a PMU may havecurrents in several feeders originating in the substation and voltages belonging tovarious buses in the substation.

The current and voltage signals are converted to voltages with appropriate shuntsor instrument transformers (typically within the range of ±10 V) so that they arematched with the requirements of the analog-to-digital converters. The samplingrate chosen for the sampling process dictates the frequency response of theanti-aliasing filters. In most cases, these are analog-type filters with a cutoff fre-quency less than half of the sampling frequency in order to satisfy the Nyquistcriterion. As in many relay designs [1], one may use a high sampling rate (calledoversampling) with corresponding high cutoff frequency of the analog anti-aliasingfilters. This step is then followed by a digital ‘decimation filter’ which converts thesampled data to a lower sampling rate, thus providing a ‘digital anti-aliasing filter’concatenated with the analog anti-aliasing filters. The advantage of such a scheme isthat the effective anti-aliasing filters made up of an analog frontend and a digitaldecimation filter are far more stable as far as aging, and temperature variations areconcerned. This ensures that all the analog signals have the same phase shift andattenuation, thus assuring that the phase angle differences and relative magnitudesof the different signals are unchanged.

Anti-aliasingfilters

A/D conv.

GPSreceiver

Phase-lockedoscillator

AnalogInputs

Phasormicro-

Modem

Second Of Century Counter

One pulse per second

processor

Fig. 5.1 Major elements ofthe modern phasormeasurement unit. Allelements of the PMU with theexception of the GPS receiverare to be found in computerrelays as well

84 5 Phasor Measurement Units and Phasor Data Concentrators

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As an added benefit of the oversampling technique, if there is a possibility ofstoring raw data from samples of the analog signals, they can be of great utility ashigh bandwidth Digital Fault Recorders.

The sampling clock is phase locked with the GPS clock pulse (to be described inthe following section). Sampling rates have been going up steadily over the years—starting with a rate of 12 samples per cycle of the nominal power frequency in thefirst PMUs, to as high as 96 or 128 samples per cycle in more modern devices, asfaster analog-to-digital converters, and microprocessors have become common-place. Even higher sampling rates are certainly likely in the future leading to moreaccurate phasor estimates, since higher sampling rates do lead to improved esti-mation accuracy [1].

The microprocessor calculates positive sequence estimates of all the current andvoltage signals using the techniques described in Chaps. 2–4 earlier. Certain otherestimates of interest are frequency and rate of change of frequency measuredlocally, and these also are included in the output of the PMU. The time stamp iscreated from two of the signals derived from the GPS receiver. This will be con-sidered in greater detail in the next section. For the moment, it is sufficient to saythat the time stamp identifies the identity of the UTC second (Universal TimeCoordinated) and the instant defining the boundary of one of the power frequencyperiods as defined in the IEEE standard to be considered in Sect. 5.6 below.

Finally, the principal output of the PMU is the time-stamped measurement to betransferred over the communication links through suitable modems to a higher levelin the measurement system hierarchy. It is the specification of these output filestructures which is the subject of the industry standard for PMUs to be consideredin Sect. 5.6.

5.3 The Global Positioning System

The Global Positioning System (GPS) was initiated with the launch of the firstBlock I satellites in 1978 by US Department of Defense [2]. By 1994 the completeconstellation of 24 modern satellites was put in place. (In 2007, there are 30 activesatellites in orbit, the extra satellites providing for greater accuracy in estimation ofspatial coordinates of the receivers. Block I and II satellites have been retired.)These are arranged in 6 orbital planes displaced from each other by 60° and havingan inclination of about 55° with respect to the equatorial plane (see Fig. 5.2). Thesatellites have an orbital radius of 16,500 miles and go around the earth twiceduring one day. They are so arranged that at least six satellites are visible at mostlocations on earth, and often as many as 10 satellites may be available for viewing.The most common use of the GPS is in determining the coordinates of the receiver,although for the PMUs the signal which is most important is the one pulse persecond. This pulse as received by any receiver on earth is coincident with all otherreceived pulses to within 1 μs. In practice, much better accuracies of synchro-nization—of the order of a few hundred nanoseconds—have been realized.

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The GPS satellites keep accurate clocks which provide the one pulse-per-secondsignal. The time they keep is known as the GPS time which does not take intoaccount the earth’s rotation. Corrections to the GPS time are made in the GPSreceivers to account for this difference (leap-second correction) so that the receiversprovide UTC clock time. The identity of the pulse is defined by the number ofseconds since the time that the clocks began to count (January 6, 1980). It should benoted that the PMU Standard (see Sect. 5.6) uses UNIX time base with asecond-of-century (SOC) counter which began counting at midnight on January 1,1970.

At present, there are a number of GPS-like systems being deployed by othernations, with similar goals. It is expected that the GPS will remain the principalsource of synchronization for PMUs for the foreseeable future.

5.4 Hierarchy for Phasor Measurement Systems

The phasor measurement units are installed in power system substations. Theselection of substations where these installations take place depends upon the use tobe made of the measurements they provide. The optimal placement of PMUs willbe considered in some of the following chapters which discuss some of theapplications of phasor measurements.

In most applications, the phasor data is used at locations remote from the PMUs.Thus, an architecture involving PMUs, communication links, and data

Fig. 5.2 Representation ofthe GPS satellite disposition.There are 4 satellites in eachof the 6 orbits, which orbitaround the earth with a periodof half a day


concentrators must exist in order to realize the full benefit of the PMU measurementsystem. A generally accepted architecture of such a system is shown in Fig. 5.3.

In Fig. 5.3, the PMUs are situated in power system substations and providemeasurements of time-stamped positive sequence voltages and currents of allmonitored buses and feeders (as well as frequency and rate of change of frequency).The measurements are stored in local data storage devices, which can be accessedfrom remote locations for postmortem or diagnostic purposes. The local storagecapacity is necessarily limited, and the stored data belonging to an interesting powersystem event must be flagged for permanent storage so that it is not overwrittenwhen the local storage capacity is exhausted. The phasor data is also available forthe real-time applications in a steady stream as soon as the measurements are made.There may well be some local application tasks which require PMU data, in whichcase it can be made available locally for such tasks. However, the main use of thereal-time data is at a higher level where data from several PMUs is available.

The devices at next level of the hierarchy are commonly known as phasor dataconcentrators (PDCs). Typical function of the PDCs is to gather data from severalPMUs, reject bad data, align the time stamps, and create a coherent record ofsimultaneously recorded data from a wider part of the power system. There are localstorage facilities in the PDCs, as well as application functions which need the PMUdata available at the PDC. This can be made available by the PDCs to the localapplications in real time. (Clearly, the communication and data management delaysat the PDCs will create greater latency in the real-time data, but all practicalexperience shows that this is not unmanageable. The question of data latency willbe further considered when applications are discussed in later chapters.)

One may view the first hierarchical level of PDCs as being regional in their datagathering capability. On a systemwide scale, one must contemplate another level ofthe hierarchy (super data concentrator in Fig. 5.3). The functions at this level aresimilar to those at the PDC levels—i.e., there is facility for data storage of dataaligned with time tags (at a somewhat increased data latency), as well as a steady

Super Data Concentrator

Data Concentrator

PMU PMU PMU PMU PMU PMU

Data Concentrator

PMUs located in substations

Data storage

Applications

Fig. 5.3 Hierarchy of the phasor measurement systems and levels of phasor data concentrators

5.4 Hierarchy for Phasor Measurement Systems 87

stream of near real-time data for applications which require data over the entiresystem.

Figure 5.3 shows the communication links to be bidirectional. Indeed, most ofthe data flow is upward in the hierarchy, although there are some tasks, whichrequire communication capability in the reverse direction. Usually these are com-mands for configuring the downstream components, requesting data in a particularform. In general, the capacity for downward communication is not as demanding asone in the upward direction. These issues will be considered in Sect. 5.6 where theprevailing industry standard for PMUs [3] will be discussed.

5.5 Communication Options for PMUs

Communication facilities are essential for applications requiring phasor data atremote locations. Two aspects of data transfer are significant in any communicationtask [4, 5]. Channel capacity is the measure of the data rate (in kilobits per secondor megabits per second) that can be sustained on the available data link. The secondaspect is the latency, defined as the time lag between the time at which the data iscreated, and when it is available for the desired application. The data volumecreated by the PMUs is quite modest, so that channel capacity is rarely a limitingfactor in most applications. On the other hand, some applications may requirerelatively small latency—in particular applications for real-time control of powersystems. At the other extreme are postmortem analysis applications, which requirePMU data to help analyze the power system performance during major distur-bances. These applications are not affected by large delays in transferring the data.Several applications of PMU data will be considered in the following chapters.

The communication options available for PMU data transfer may be classifiedaccording to the physical medium used for communication [6]. Leased telephonecircuits were among the first communication media used for these purposes.Switched telephone circuits can be used when data transfer latency is not ofimportance. More common electric utility communication media such as PowerLine Carrier and microwave links have also been used and continue to be used inmany current applications. Of course, the medium of choice now is fiber-optic linkswhich have unsurpassed channel capacity, high data transfer rates, and immunity toelectromagnetic interference. Figure 5.4 [6] shows typical construction of afiber-optic cable commonly used in electric utility industry. The most populardeployment of fiber-optic cables is in the ground wires of transmission lines. Theground wires may carry multiple fibers which may be used for other communica-tion, protection, and control applications for power system operation and man-agement. Other configurations of fiber-optic links may involve separate towers forthe fiber cable in the electric utility right-of-way, wrapping the fiber cable aroundthe phase conductors, or direct burying the fiber cable in the ground (see Fig. 5.5).

The technology of fiber optics is changing very rapidly [7]. Fibers may be singlemode (meaning that the entire fiber cross section is homogeneous material) or


multi-mode with graded index or step-index change in the refractive index of thefiber and the cladding material (see Fig. 5.6). Multimode fibers tend to have greaterloss per km because of the partial loss of energy due to refraction at the boundarybetween the fiber core and the cladding. Single-mode fibers propagate opticalwaves along the axis of the fiber and have minimal loss during transmission. Thewavelength of light used in these systems ranges from 900 to 1800 nm. Typical lossin the fibers may range from 0.5 (single-mode fibers) to 4 db/km (multimode fibers).In addition, loss in the connectors and repeaters must also be taken into account.

Galvanized steel rods

Plastic tubes

Plastic jacket Aluminum alloy tube

Kevlar strength members

Fibers

Fig. 5.4 Typical fiber optic cable construction. Such cables are in wide use on electric utilitysystems

Fiber bundle wrapped aroundPhase conductor

(a)

(b)

(c)

Fiber bundle in ground wire

Fiber bundle direct-buried

Fiber bundle on separate towers

Power line

TransmissionTower

Fig. 5.5 Arrangement of fiber-optic bundle commonly employed by electric utilities. a The fiberis in the ground wire. b The fiber bundle is strung on separate towers on transmission lineright-of-way. c The fiber-optic cable is directly buried

5.5 Communication Options for PMUs 89

Depending upon the length of transmission path and allowed transmission lossbudget, an appropriate type of fiber is selected for a given application.

One may also classify the communication facilities based upon the communi-cation protocols in use. Here also the field is rapidly changing, and it is onlypossible to mention a few of the available protocols which have been used in phasormeasurement applications. The IEEE standard applicable to PMU technology [3]discusses the general requirements for communications with PMUs in Annex I ofthe standard. When serial communication over an RS-232 is used, the entire datastream from the PMU (as defined in the PMU Standard and discussed more fully inthe next section) is to be mapped in proper order on to the serial communicationport. The communication system may apply any protocol, encryption, or change theordering of the data, as long as it is restored to its original format at the receiver end.

Phasor Measurement Unit messages may also be mapped in their entity into TCP(Transmission Control Protocol) [8] or UDP (User Datagram Protocol) [9] and willbe accessed by using standard Internet Protocol functions. The Internet Protocol(IP) may be carried over Ethernet or other available transport means.

In recent years, IEC Standard 61850 has been introduced to facilitate electricutility substation automation including protection and control [10]. In its presentversion, this standard has not been identified as being useable for PMUs. It may bethat advances in the state of the art in PMU technology and substation communi-cation technology will lead to the acceptance of IEC 61850 by the PMUcommunity.

Single mode fiber

Graded index fiber

Step index fiber

~ 200 µm cladding

~ 50-100 µmcore

~ 10 µm

Index ofRefraction

Inputpulse

Outputpulse

Fig. 5.6 Types of fibers used in fiber-optic communication, their relative dimensions, and modesof transmission


5.6 Standards

Kenneth E. Martin

5.6.1 History

The first PMU developmentThe first PMU was built in 1987 at Virginia Tech. It was built using a standardVME chassis with a Motorola 68000 CPU, an A/D digitizing board, a serialcommunication board, and an analog signal interface built by the research team (seeFig. 1.1) It used a commercial GPS receiver to provide a synchronized onepulse-per-second (PPS) signal to synchronize the measurements and a 720 PPS tosynchronize the sampling of analog signals. Communication was through a RS232serial port operating at 4800 bytes per second (BPS). This speed allowed sending 12measurements per second in a binary format with a Binary-Coded Decimal(BCD) time tag.

The first commercial PMU, the Macrodyne 1690 (Fig. 5.7), was introduced in1990 and used the same measurement technique as used in the Virginia Tech unit,but incorporated a GPS receiver internally. This PMU sent data at varioususer-selected rates up to 30/s. Two serial ports provided data output through either adirect connection or a modem that adapted the signal to a utility communicationsystem. This PMU had the capability for configuration settings by both the frontpanel and via the serial interface.

These two PMUs formed the basis of understanding for development of the firstsynchrophasor standard. The measurement was at a significantly higher speed thanmost existing measurements. It captured records of power system dynamics more

Fig. 5.7 First commercial PMU, Macrodyne 1690, manufactured by Macrodyne Inc.

5.6 Standards 91

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accurately and with more detail than before. This was demonstrated by some of theearly measurements, both in laboratory tests and in system events (see Figs. 5.8 and5.9).

The First Standard, IEEE 1344Since the whole technique was new, the developers felt it was a good time todevelop a standard so that the evolving PMU technology would be compatible.A project to create a standard for synchrophasor measurements was initiated in1992 in the IEEE Power System Relay Committee (PSRC) of the Power and

Time (Seconds) Time (Seconds)

280

380

Phasor Remote Unit Test on Digital Test SystemAt 24 Jan. ‘92, 16:08:12

300

340

380

Vol

tage

(K

V)

Vol

tage

(K

V)

-30 0 30 60 -0.4 0.0 0.5 1.0

Fig. 5.8 Laboratory test of the first PMU showing the response to voltage-step change andsinusoidal amplitude modulation at 0.3 and 3.0 Hz, recorded by the author at BPA

Phasor Measurement System DataAt 7 Dec. ‘90, 10:12:31

Phasor Measurement System DataAt 7 Dec. ‘90, 10:12:31

-30 0 30 60 90 120

Time (Seconds)-30 0 30 60 90 120

Time (Seconds)

Pha

se D

iffer

ence

(Deg

rees

)

Pow

er ( M

ega w

atts

)

30

800

900

1000

1100

1200

1300

1400

1500

Fig. 5.9 Recording of a generator drop event using the first PMU, recorded by the author at BPA


Energy Society of IEEE. At that time, the whole measurement process was not fullyexplored. Consequently, the effects of off-nominal frequency, interference, andrapidly changing system dynamics were not considered in this first standard. Thatstandard, IEEE 1344, focused on the time synchronization of the measurement. Itidentified timing sources and set requirements for synchronizing the measurementand sampling of the waveform. Time of day (seconds through century) was alsorequired through inputs such as a high-precision IRIG-B or 1 PPS plus through aserial port. The phase angle was fixed to a cosine at nominal frequency, which is thenormal reference in mathematical analysis. This standard specified messaging forreporting the measurements. The PMU sent the measurements in a binary formatdata message. The contents, scaling, and names were described in a configurationmessage. A header message provided additional readable information.

The Second Standard, C37.118-2005Phasor measurement system deployments increased in the late 1990’s after somemajor power system incidents. In deploying these systems, it was found that theIEEE 1344 standard lacked some features that were needed for wide-area deploy-ment including better identification of bad measurements, ability to aggregate datafrom many PMUs, and more measurement identifiers. In the new standard, testconditions were specified and the resulting measurements were compared with theexpected errorless result. This comparison includes both phase and magnitude ofthe phasor. These tests were specified for the steady-state measurement only.

This standard was a great success. It brought attention to basic PMU measure-ment problems and provided a benchmark to vendors as a design goal. Mostwide-area synchrophasor systems (outside of China which has a similar but dif-ferent standard) in operation in the 2010s follow this standard. The communicationsformed a baseline for further IEEE and IEC developments.

The Third Standard RevisionThe IEEE and the International Electrotechnical Commission (IEC) had discussedmaking C37.118 a joint standard, since the IEC did not have a synchrophasorstandard. However, the IEC standard series separates communication and mea-surement aspects into different Technical Committees (TC) and standards.The IEEE decided it would be best to split the C37.118 standard into two standards—C37.118.1 for measurements and C37.118.2 for communications. The mea-surement standard, C37.118.1, extended performance requirements with smallchanges in the existing steady-state limits and new tests for dynamic measurements.Test conditions included modulated signals, frequency ramps, and step changes.Requirements were established for the frequency and rate of change of frequency(ROCOF) measurements, since these measurements were established and were usedextensively. Measurement classes were specified, and many other provisions werestrengthened.

The communication standard, C37.118.2, was extended to support operation inlarger grids. However in support of the many existing systems in operation ordevelopment, changes were both limited and made backward compatible. The IEC61850 WG developed TR 90-5 to extend that standard for operation with

5.6 Standards 93

synchrophasor systems. Both of these standards will probably be further developedto support synchrophasor communications as the technology advances.

5.6.2 Synchrophasor Measurement

The standard specifies measurement performance under both steady-state anddynamic conditions. Performance is evaluated with a number of tests that charac-terize measurement capability and represent signals that are found in power sys-tems. The standard defines criteria for evaluating the measurement of these signals.Originally the synchrophasor standard only addressed synchrophasor estimationeven though both frequency and rate of change of frequency (ROCOF) werereported along with synchrophasors. Since these measurements are widely used andtheir derivation is closely related to synchrophasor estimation, the StandardsWorking Group decided they should be evaluated with criteria similar to those ofsynchrophasors.

The original formulation of synchrophasors was based on the phasor equivalentof a cosine wave with the time referenced to UTC. This worked well forsteady-state measurements but did not provide a reference for measurement underdynamic signal changes. Chapter 4 has additional discussions and explanations asto how the frequency and rate of change of frequency are estimated.

Measurement EvaluationA key element of the synchrophasor standards is the evaluation of measurementperformance. In order to evaluate the measurement results, Total Vector Error(TVE) was introduced. A synchrophasor measurement can be expressed in rect-angular coordinates as XrðnÞþ jXiðnÞ where Xr and Xi are the real and imaginarycomponents of the estimate, respectively. Similarly if the ideal or reference value isXr(n) + jXi(n) taken at the same instant t(n), the TVE is defined by:

TVE nð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðXrðnÞ � Xr nð ÞÞ2 þðXiðnÞ � XiðnÞÞ2

ðXrðnÞÞ2 þðXiðnÞÞ2

sð5:1Þ

Total Vector Error is a scalar quantity. While the error mechanisms for magnitudeand phase angle are somewhat different, the effect of small magnitude or phaseangle errors on many applications is similar. This approach simplified evaluationand setting limits and has worked reasonably well in practice.

Frequency and ROCOF limits are also simply scalar quantities:

FrequencyMeasurement Error : FE ¼ fmeasured � fref ð5:2Þ


http://dx.doi.org/10.1007/978-3-319-50584-8_4

ROCOFError : RFE ¼ ðdf =dtÞmeasured � ðdf =dtÞref ð5:3Þ

Note that the formulation in C37.118.1 defined these error quantities as mag-nitude errors (absolute value), but FE and RFE are signed errors (the evaluation isspecified in absolute value). For reporting it is often useful to know whether theestimate is high or low. Defining the evaluation as a signed number encourages testlaboratories to report signed errors which is useful to both manufacturers and users.

Performance EvaluationThe most important function of a measurement standard is establishing methods toevaluate how well a measuring device performs. In this case, we need to evaluatehow well a PMU estimates synchrophasors, frequency, and ROCOF. The standardspecifies tests that characterize the PMU output under a variety of conditions. Thetests specify a signal to be measured and the expected results (reference). Theevaluation is the difference between the reference value and the PMU estimate. Thetests are based on power system operating conditions including steady-state on andoff-nominal frequency, interference such as from introduced noise or harmonics,and dynamic changes such as loss of generation or load, switching, and systeminteractions that can produce oscillations of machine rotors. The following sectionsdescribe the testing conditions, the test evaluation, and the tests.

The standard does not specify any environmental conditions other than standardlaboratory conditions (temperature *23 °C and humidity <90%). It directs users tospecify environmental requirements appropriate to their installation and referencesstandards that list guidelines. Measurement reporting is also a critical part of asynchrophasor system. Estimation algorithms are generally simpler and moreaccurate when they operate at a rate synchronous to the nominal power systemfrequency. Data is much easier to manage in a system if all data are at the same rate.This is particularly important with synchrophasors since phasor angle between twoPMUs can be precisely compared only when the measurements are at the same time.Because of these factors, standard reporting rates that are a multiple or submultiplesof the nominal frequency are specified. They are 10, 12, 15, 20, 30, 60, and 120/s for60 Hz and 10, 25, 50, and 100/s for 50 Hz systems. Other rates can be used, butthese rates must be provided and used to demonstrate compliance with the standard.

The PMU is also required to create a flag to indicate whether any errors occurredduring the synchrophasor estimation process. This flag includes errors such asmemory check failure, A/D calibration failure, and calculation overflow. Thisrequirement captures the functionality of error reporting that was provided in theC37.118 standard by the STATUS word in the communication protocol.

Compliance VerificationIt is more beneficial to the industry to base the standard on performance criteria toencourage industry to develop better estimation algorithms. One factor that sig-nificantly affects performance is filtering. Less filtering allows faster response butallows noise and interference to degrade the measurement. Conversely more fil-tering provides a more precise response but is slower. The user community has

5.6 Standards 95

different applications that benefit from different approaches. Consequently, the WGestablished classes of performance. The P class performance criterion allows forfaster response but reduced protection from interference and aliasing. M classrequires better measurement over a wider range, but allows longer response time.More classes may be established as the need arises.

Reporting rate is the other major factor affecting performance. A slowerreporting rate causes more delay in responding, narrows the measurement band-width, and is more subject to aliasing. Consequently, all the performance require-ments are specified in terms of the particular test, the performance class, and thereporting rate.

Compliance verification consists of various tests and evaluation of how the PMUresponds to the test. Each test is described with specific test signals and a testapplication description. The standard stops short of detailed test procedures sincethese can vary depending on the equipment being used. The reference phasor,frequency, and ROCOF signal formulas are given for evaluating the PMU mea-surements. Error limits are given for each test and based on performance class andreporting rate.

Test signals can be provided as analog waveforms or digital sample streams. Thelimits given in C37.118.1 and 60255-118-1 are based on analog waveforms withthe PMU doing the signal conditioning and A/D conversion. The standard requiresthat these waveforms must be low noise, undistorted signals so error allowance isonly allowed for the frontend errors such as input protection, scaling, and A/Dquantizing. When digitized signals are used, testing will normally be performed bya test device that produces digital samples by formula, so frontend errors areeliminated. Under these circumstances, testing the limits should be tightened upsince no allowance for PMU frontend errors is needed. Changes for using digitalinput streams are discussed in Annex E of 60255-118-1, but actual requirements arenot specified in this standard.

Tests are specified in terms of nominal input level as stated by the manufacturer.These may vary among manufacturers and the particular intended installation. Theuser needs to be sure the PMU will match the intended installation, just the same aswith environmental requirements.

All tests are specified for balanced 3-phase input to support evaluation of pos-itive sequence. However, if single-phase measurements are produced, they must beevaluated using the same tests and error limits. Phasors, frequency, and ROCOF areall evaluated using the same tests.

Steady-State Measurement EvaluationThe first need is to assure that the measurements are comparable and accurate over areasonable range of operation in steady state. These tests include the following:

1. Varying signal frequency from nominal value by ±2/5 Hz

2. Voltage magnitude from 10/80 to 120% of nominal value

3. Current magnitude from 10 to 200% of nominal value(continued)


(continued)

4. Harmonic interference rejection 1/10% of the main signa

5. Out-of-band (subharmonic) interference rejection 0/10% of the main signa

Note Dual number requirements (x/y) indicate the exact number depends on class and reportingrate. User must consult the standard for details

The first 3 tests assure that phasor, frequency, and ROCOF measurements areaccurate over a range of magnitude and frequency that might be expected in apower system. Note that when the frequency is off nominal, the phase angle rotates,so the frequency variation test effectively tests phase angle.

Tests 4 and 5 assure that harmonics and other interfering signals will not degradethe measurement. The harmonic test requires testing with harmonics of the fun-damental (nominal) from the 2nd to the 50th. The test for out-of-band (OOB)-interfering signals looks for rejection of signals that are out of the passband but arenot harmonics of the fundamental signal. The passband of the measurement islimited to signals within the Nyquist frequency for the given reporting rate (e.g., areporting rate of 30/s has a Nyquist frequency of 15 Hz, so OOB signal includesany frequency ≥15 Hz). The effective measurement passband will be narrower dueto filtering, but the Nyquist frequency sets a hard limit for all devices. This testprobes the frequency range outside the Nyquist limits from DC to the secondharmonic with a single-frequency sine wave. This shows susceptibility to inter-ference and noise that can alias into the passband. All harmonic and OOB testsrequire TVE ≤ 1% except the M class OOB which is TVE ≤ 1.3%. The FE limitvaries from 0.005 to 0.025 Hz and RFE from 0.1 to 0.4 Hz/s except wheresuspended.

No limits are set for ROCOF under OOB and harmonic interference. Tests usingsimple calculation methods have shown the error for these interference conditionscan be very large. This is largely due to the fact that any noise tends to get highlyamplified since ROCOF is the second derivative of phase (except when the poly-nomial fit method is used to determine the frequency and the rate of change offrequency, as explained in Chap. 4). Limits that allow for these high errors are notvery meaningful for the rest of the range. So rather than establishing high errorallowances, it was felt that it was better to wait for research to come up with betterestimation methods that would allow better measurements before setting limits.Therefore, C37.118.1 and 60255-118-1 do not have requirements for ROCOFperformance under interference conditions.

Measurement Evaluation Under Dynamic Operating ConditionsSynchrophasor measurement under dynamic operating conditions is the mostimportant aspect of synchrophasors, but is difficult to evaluate. It requires gener-ating test signals with well-defined waveforms that have dynamics that are preciselytime aligned. Test equipment that can generate accurate amplitudes along with timealignment was not available when PMUs were first introduced, so accurate dynamictesting was not possible. However, test equipment with the required precision hasbeen developed and these tests can now be performed with good precision.

5.6 Standards 97

http://dx.doi.org/10.1007/978-3-319-50584-8_4

Measurement evaluation can be considered from both instrumentation and powersystem’s point of view. Instrumentation evaluation involves determining theaccuracy at steady state, the measurement bandwidth, settling time, and measure-ment noise. Other aspects such as stability (aging) and environmental operatingconditions may be considered as well. From the power system’s viewpoint, themeasurements should be able to track dynamic system changes accurately andreliably. Thus, the tests need to characterize the measurement capability with testsignals that also reasonably represent those that would be found in a power system.Of the many different tests considered for dynamic performance verification,modulations, frequency ramps, and steps were chosen. These tests determineinstrumentation capability, look at aspects of implementation that can causeproblems with synchrophasors, and replicate power system phenomena. These testscan also be generated and evaluated reliably.

To simplify testing, all tests are performed with balanced 3-phase input signals.Both single-phase and positive sequence phasors can be evaluated, and must meetthe same requirements. Testing has shown that PMUs accurately calculate sym-metrical components with both amplitude and phase angle imbalances, so testingthis conversion is not required. However, unbalanced phases in power systems are areality and can introduce some additional measurement problems. For example,harmonic leakage from off-nominal system frequency can be canceled by positivesequence if the system is balanced. If the system is not balanced, this interferencemust be removed by filtering. Since it is impossible to choose a level of imbalancethat is right for every situation, testing for measurement performance underunbalanced conditions is left to users who have special concerns with it.

The a-phase test signal formulas for modulation, frequency ramp, and stepchange are as follows:

Xa ¼ Xm½1þ kx cosðxtÞ fcos� ½x0tþ ka cosðxt � pÞ�g ð5:4Þ

Xa ¼ Xm cos½x0tþ pRf t2� ð5:5Þ

Xa ¼ Xm 1þ kxuðtÞ½ �fcos ½x0tþ kau tð Þ�g ð5:6Þ

where Xm is the amplitude of the input signal, ω0 is the nominal power systemfrequency, ω is the modulation frequency, Rf is the frequency ramp rate, u(t) is theunit step, and kx and ka are the amplitude and phase angle modulation indexes(12) or the unit step sizes (14). B and C phase signals are similar except witha ±2/3π phase offset.

Measurement bandwidth determines the range of frequency over which themeasurement responds. Modulation bandwidth is from DC (steady state) to thepoint that the measurement has reduced response, typically when the response hasrolled off 3 dB (≈30% amplitude reduction). Bandwidth can be determined bymodulating a signal parameter and increasing the modulation frequency until themeasurement of that parameter has rolled off to the 3 dB limit. Phasors include bothphase angle and amplitude measurement, so the bandwidth of both can be


determined. They will likely be similar, but not necessarily the same. Frequencyand ROCOF should respond to angle modulation but not to amplitude modulation.Oscillations, both forced and modal, are observed in power systems as modulatedsignals.

The modulation test applies sinusoidal modulation to the amplitude and phase ofthe signal (12). Bandwidth is determined by increasing the modulation frequencyfrom 0.1 Hz to a frequency where the measurement response drops by 3 dB. Thiscan be determined by finding the point where the TVE increases to 3%. (TVE isbased on the overall signal. The modulation is 10% of the overall signal amplitude.The 3 dB drop will be in the modulated signal. 3 dB ≈ 30%, so the test looks for30% drop in the 10% modulating signal which is measured as 30% × 10% = 3%TVE on the overall signal.)

The test requirement is that the bandwidth is a minimum of 2/5 Hz, dependingon the class and reporting rate. This is checked by varying frequency from 0.1 to2/5 Hz and checking that the TVE <3% over the interval. This test does notdetermine the full bandwidth. The procedure given in 60255-118-1 Annex I extendsthe frequency range to determine the actual bandwidth and allows certifying thePMU for the full bandwidth. Here the minimum bandwidth assessment is done forboth amplitude and phase by setting the modulation indices kx and ka alternately to10% and 0.

Figures 5.10, 5.11, 5.12, and 5.13 illustrate the modulation test as simulated in amodeling environment (MATLAB). Only the amplitude modulation is shown, sincelow-level phase modulation is very difficult to see in a plotted signal. In thisexample, the nominal power system frequency is 50 Hz and it is sampled at 16samples/cycle (800 samples/second). The reporting rate is set at 25 frames/second,so the Nyquist rate is 12.5 Hz and the filtering is designed to reduce the response byat least 20 dB by that frequency. Figure 5.10 shows the modulated 50 Hz signal.The modulation appears as an envelope on waveform with frequencies of 0.2 Hz

0 2 4 6 8 10 12 14 16 18 20

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nitu

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e)

Signal MagnitudeFig. 5.10 10% amplitudemodulation of the 50 Hz ACwaveform. Modulationfrequency of 0.2 Hz (0–5 s),0.5 Hz (5–7 s), and 1–13 Hz(1 s at each frequency) isvisible

5.6 Standards 99

(0–5 s), 0.5 Hz (5–7 s), and 1–13 Hz (1 s each beyond 7 s). Figure 5.11 shows thereference or theoretical phasor for that modulation. Note that the modulation andtheoretical phasor amplitude is constant over the entire testing range. Figure 5.12shows the signal as measured by the PMU. The response remains very flat until themodulation frequency reaches 6 Hz where it rolls off rapidly. Figure 5.13 shows theresulting TVE. The TVE is very small until the modulation frequency reaches 6 Hzwhere it begins to rise rapidly, though it does not exceed the 3% TVE limit until thefrequency reaches 8 Hz. This example uses the reference algorithm from thestandard which is designed to meet the standard’s requirements.

The frequency ramp test determines how well the PMU tracks changing valueswhile frequency moves over the required PMU operating range of 2/5 Hz,depending on class and reporting rate. It demonstrates capability of keeping

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Reference Phasor MagnitudeFig. 5.11 Reference phasorbased on the waveformformulas, which follows theenvelope of the modulatedsignal. This is the value themeasurement should produce

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PMU Measurement Phasor MagnitudeFig. 5.12 Phasor magnitudeas measured by the PMU.PMU is set for M class with25/s reporting, so responserolls off quickly after 12 s.The modulation frequency is6 Hz from 12 to 13 Hz andincreases by 1 Hz over each1 s interval thereafter


measurements time aligned over large frequency excursions. Frequency tracking isbasic to many of the PMU estimation processes, such as sample rate and theestimation window, so it is important that it works well. In a power system, loss of agenerator or a load will cause a ramp in frequency. This test assures that the PMUwill make good measurements when one of these events occurs.

The ramp test starts at a constant off-nominal frequency and ramps throughnominal to another equally off-nominal frequency as shown in Fig. 5.14. Themeasurements are compared by time tag to the ramped signal to assure theyaccurately track the signal. Equation 5.6 defines the ramped signal; the ramp rate Rf

0 2 4 6 8 10 12 14 16 18

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TVE PMU PhasorFig. 5.13 TVE remains verylow until the modulationfrequency reaches 6 Hz at12 s and rises steadilythereafter

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Frequency ramps-- positive (blue) & negative (green)Fig. 5.14 Ramp test starts atan offset frequency andcontinues through nominal toa higher or lower frequency ofequal amount. A testwith ±5 Hz frequency rangeis shown

5.6 Standards 101

is positive for a positive ramp and negative for a negative ramp. The ramp limits are2/5 Hz above and below the nominal system frequency, depending on the class andreporting rate. The ramp rate used in the compliance test is ±1 Hz/s. This valuewas chosen as one that reasonably represents the maximum ramp speed that wouldbe seen in a medium-to-large power system. Smaller systems could have fasterrates, and in the presence of an electronically generated power source, the frequencychange could be nearly instantaneous. The 1 Hz/s rate is satisfactory for mostsystems, but, with the rapid growth of renewables, may have to be reconsidered.Reports at the end of the ramp are excluded to prevent nonlinear and end of rangeeffects that result from the testing environment from degrading the evaluation.

The step test creates a step change in phase or magnitude of the AC signal. Thesteps are small (≈10%) to allow the PMU to operate through the change in a linearoperating region. The response time is the time from when the measurement leavesthe starting steady-state measurement to when it settles into the steady state fol-lowing the step. The standard also requires reported values to stay reasonably closeto what they should be through the step. To that end, overshoot and undershoot islimited to 10% of the step size, and the point where the measurement makes the steptransition must be close to the actual time it occurs. Switching in a power systemcreates step changes, so they are common.

The step test applies a step in amplitude or phase of the signal (Fig. 5.15). Theresponse time is evaluated as the time when the measurement leaves the steady-statevalue until it re-enters and stays at the steady state. The delay time is the differencebetween the input step time and the time where the stepped parameter (amplitude orphase) crosses the midpoint of the transition (50% response) between the startingand final values. Overshoot and undershoot are the amounts the response is over orunder the starting or final value. These are limited to 10% of the step magnitude toprevent erroneous evaluation. These parameters are illustrated in Fig. 5.15.

A latency test requires that the PMU send the measurements within a few sampleperiods of the measurement to assure that the PMU does not add unnecessary delay.

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

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Fig. 5.15 Step test showing response of PMUs, with a 10% magnitude step at t = 0


5.6.3 Synchrophasor Communication

When synchrophasors were introduced, existing communication protocols were tooslow or had too much overhead for anticipated real-time installations. So the firststandard, IEEE 1344, included a basic communication protocol which was con-tinued into the IEEE C37.118 standard series.

IEEE Standard C37.118 [11–13]This protocol is a messaging system rather than a complete communication pro-tocol. The message types, the message contents, and message exchange are spec-ified. The choice of underlying communication systems such as RS-232 serial orInternet Protocol over Ethernet is left to the implementer. The C37.118 standardonly requires that the messages be mapped in their entirety to the underlyingsystem. This assures that PMUs and other devices that use this protocol will dealwith standardized messages.

The messaging is set up in a way that keeps implementation simple and con-sistent. All messages (also called frames) follow the same overall format with aheader for message parsing and identification and a termination CRC(CHK) following the information (Fig. 5.16). The header information includes thefollowing:

• SYNC word—16-bit word led by a sync byte and followed by frame type andversion identification.

• FRAMESIZE word—16-bit unsigned integer of number of bytes in frame. Thislimits the maximum frame size to 65,535 bytes.

• IDCODE—16-bit word identification word. The IDCODE from each PMU isretained in the configuration message and can be used to identify the particularsource PMU. Every message also has an IDCODE which a receiving device canuse to sort data and reject unwanted messages. This provides some additionalsecurity and helps overcome routing problems.

• SOC—32-bit second of century. This is the count of seconds since 1/1/1970. Itprovides the basic time tag for the message.

• FRACSEC—32-bit fraction of second. This is the fraction of second in a 24-bitunsigned integer plus an 8-bit time-quality flag.

There are four message types specified: command, configuration, data, andheader.

Fig. 5.16 Communication frame structure in Standard C37.118

5.6 Standards 103

1. The command message is sent from the device that will be receiving data to thedevice supplying data. Commands include turn-on/turn-off data and sendconfiguration/header information. A block of commands are reserved for futurestandards assignment, and another block is assigned to users for dedicatedapplications. The frame can be extended allowing users to send special infor-mation or instructions using this messaging.

2. Configuration messages provide meta-data for the measured data. This includesthe station name (substation where PMU is located), signal names and scaling,data formats (e.g., floating point/integer), timescale for fractional time, nominalsystem frequency, and the data reporting rate. The first configuration message isConfig 1 which describes the full PMU measurement capability. This allows areceiving device to determine whether more measurements can be added.Config 2 is identical to Config 1 except that it describes the contents of an actualdata stream output. The data stream and associated Config 2 message shouldhave the same IDCODE to confirm their association. Config 3 also identifies thecontents of a particular data stream, but includes much more informationincluding longer names, more scaling, and PMU information. Config 3 can beextended for larger data sets.

3. The data message contains the measured data which includes phasors, fre-quency, ROCOF, analog, and digital. Analog data are intended as sampledwaveforms of any type that the user wishes to include. Digital data are Booleanindications that are mapped into 16-bit words. All data except digital can berepresented in 32-bit IEEE floating point or 16-bit integer formats. Phasors canbe in polar or rectangular format. Every message includes a 16-bit STATUSword from the PMU that indicates any time, measurement, or PMU errors aswell as processing and trigger information. This word is carried through with themeasurements so that the users can be advised if the data may be degraded.

4. The header message is a free-form, human readable message. It allows thevendor or user to add specific information about the data or equipment that canbe retrieved remotely by a receiving device. The header message is sent onrequest by command.

New messages can be defined using reserved message type and version numberdesignations. This allows upgrading the protocol in revisions of the standard withnew messages and still maintaining backward compatibility. The protocol wasupgraded with the Config 3 message in 2011, and new equipment successfullyoperates with 2005 version equipment.

Typically synchrophasor data is reported from a PMU in a substation to a phasordata concentrator (PDC) in a control center. From there, it is sent to applications orsent on to a higher level control center such as an ISO where it combined with datafrom other utilities (Fig. 5.17). This protocol supports multilevel data aggregation.The data from each PMU is imbedded along with its STATUS word in the mes-sages sent from the PDC (Fig. 5.18). The configuration message includes theoriginal station name and IDCODE for data identification as well as scaling,naming, and other signal information. With higher level aggregation, the messages


can get very large particularly the configuration messages since they have names foreach signal. Config 3 allows going past the 64 KB message limitation by allowingthe message to be extended into as many as 65,534 frames, each 64 KB. If data setsget larger than this, different techniques are required.

As mentioned before, this messaging protocol can be carried over any kind ofunderlying communication system. Internet Protocol (IP) over Ethernet has becomea de facto industry standard for substations and control centers. Consequently,standard methods for using C37.118 protocol over IP/Ethernet have developed. It isinteresting that these methods are so common that virtually every vendors’implementations have worked together even without a published standard. It is atestament to the simplicity of the C37.118 methods that they can be implemented soconsistently.

The basic operation of synchrophasor systems under C37.118 is client–server.The data source—either a PMU or a PDC—is the server and the client—a PDC or adata application—is the client that will receive the data. A PDC requests data fromPMUs and is therefore a client, and also provides data to other devices and istherefore also a server. In this scheme, the client initiates making a connection andcontrols the data flow with commands.

The operating modes are commanded or spontaneous. In commanded operation,the server only sends data, configuration, and header data on command.

Fig. 5.17 Hierarchy in synchrophasor data systems

Fig. 5.18 Data message aggregation

5.6 Standards 105

Spontaneous operation is when the server sends data based on a setting or commandthat is external to the synchrophasor data system. This method is not described inthe standard, but is commonly used anyway. It is useful where security or firewallconcerns block incoming data including commands. Since the receiving deviceneeds configuration information, it can be sent automatically on a periodic basis orprovided offline.

The actual connections can be TCP, UDP, or TCP/UDP. A straight TCP con-nection is the simplest. The client requests and establishes a TCP connection withthe server. Over that connection, commands, data, and metadata are exchanged. Theadvantage of this type of connection is that it is simple and the status of theconnection is known to both ends, so communication problems can be readilyassessed. The disadvantage is that TCP has more overhead than UDP and canbackup if there are lost message issues.

A UDP–UDP connection sends messages both ways by UDP. The messages arecompact and there is no backup problem for lost messages. However, lost messagesand lost connections are not detected by the interface, so diagnosing problems canbe more difficult than with TCP. This method also requires externally determiningor establishing the destination ports.

TCP/UDP was developed to resolve the problems listed above. The initialconnection is done with TCP as described, but the data is sent with UDP. The morereliable TCP is used for commands and configuration. This still requires estab-lishing a destination port for the UDP data stream, but minimizes traffic andeliminates data backups.

Both UDP methods can use multicast to send data to multiple destinations withminimal traffic. It is particularly useful with spontaneous data transmission. Therehave been concerns about using multicast since it allows routing to many locationswhich are not necessarily known by the data manager. However as higher ratemeasurements are implemented, the data traffic load can become significant.Multicast can reduce the load and can be managed to maintain security.

C37.118 over Ethernet and its extensions are widely used worldwide. It hasproven to be simple to implement and easy to use. It has been applied to every stageof communication from the PMU to the large-scale applications using data fromhundreds of PMUs. Since it can be used with any underlying communicationprotocol, it is easy to adapt to new requirements.

IEC Standard 61850 TR 90-5IEC 61850 is the IEC protocol for substation communication. It was originallyestablished for substation automation, protection, and control within the substation,but has been extended for protection and control applications. Technical report TR90-5 was developed to extend 61850 for synchrophasor communications based onsystems that had been implemented or anticipated at the time. The TR is the defacto standard to guide development until the contents can be incorporated into thenext published version of the standard.

TR 90-5 described a spectrum of use cases to establish performance parameters.These cases covered use of synchrophasor data ranging from support of event


analysis using recorded data to high-speed controls. Subsequent developmentincluded modeling aspects, data definitions, data profiles, and communicationsecurity based on these use cases.

61850 uses logical nodes (LN) to model functional elements and the data thatthese elements will use or provide. The communication interface to the IEDs thatimplement these functions will expose these data objects. The model that an IEDimplements can be described by the system configuration language (SCL). Forsynchrophasors, existing LNs for measurements (MMXU and MSQI) can be usedto represent the PMU functional elements. PDCs can be considered proxy servers;however, users may need to develop additional LNs for PDCs and applications assynchrophasor systems expand.

61850 was originally designed for use in a substation, so many message typeswere sent directly over Ethernet. This is fast and simple, but has no method forsending on to other systems. Synchrophasor communication focuses on wide-areadata exchange which requires routing messages over networks. Since existingsample values were not routable, TR 90-5 defines R-SV, which maps sample valuesto UDP/IP a routable protocol. Another profile maps data to TCP for users that needthat mode, though UDP is the preferred transport mode.

TR 90-5 also defines many other details for carrying synchrophasor data usingthe 61850 standard. It defines application profiles for the data. The control blockprofiles are detailed. Some new calculation math was needed for M and P classes,and a data object was defined for ROCOF. An important addition was optionalend-to-end security which can cover the PMU to the end application. It coversauthentication, integrity, and confidentiality. It uses symmetric keys for speed indata encryption and asymmetric encryption for security in key distribution. Thesesecurity methods are published in IEC 62351. This provides a level of security thathas not been developed for IEEE C37.118.2.

IEC 61850 is a complete communication system that extends from the messagesdown to the low levels of communication on the wire. It chiefly uses theManufacturing Messaging System (MMS) for communication of profiles and othermetadata, and IP for data communication. But it is flexible, so other methods can beadapted. It is intended to bring seamless communication between IEDs in thesubstation and on into the control center. While there is a significant implemen-tation in the substation, the wide-area implementation is new and not extensivelydeveloped. That development will happen as users find advantages over the existingprotocols in use.

Other StandardsProfiles in other protocols have been developed for carrying synchrophasor data.OPC (OLE for process control) has been used to transfer synchrophasor data at thenative 30/s rate from a PDC to an application. It worked well with a small data set,but expansion was questionable due to high CPU usage. Variations of OPC, such asOPC-DA, may be capable of operating with larger data sets and higher rates due togreater efficiency.

5.6 Standards 107

Both DNP3 and ICCP have been used to convey synchrophasor data. These areEMS-oriented protocols and have been used for downsampled phasor data, withsampling at rates around 1/s or 1/2s. These protocols are designed to be used withtraditional EMS systems and are not expected to run at very high rates. Other datatransfer-oriented protocols such as FMS, Modbus, and PROFIBUS could be usedbut no implementations have been published.

Standards OutlookStandards are essential for coordinating technology development. Properly handledstandards encourage technology development by assuring that products will operatecorrectly in their intended environment. Inventors create new technology withassurance that there will be a market their products will fit into. Developers refer tostandards to find the expected performance requirements. End users can use stan-dards for their purchase specifications, knowing the product will meet a set ofrequirements that has been established by subject matter experts. They are reassuredthat the products they buy will meet some basic interoperability conditions.

The synchrophasor standards have followed the general development pattern.The first standard was based on a new technology without deployment experience.Subsequent revisions brought back operational experience to strengthen weak areasand resolve unanticipated problems. Revisions of this standard have somewhatpaced technology development by requiring improvements in performance. TheWorking Group has tried to assure that such developments were possible throughresearch and testing before publishing new requirements. While this has beenchallenging, it has encouraged improvements that support this continuing tech-nology development that is required to keep pace with growing deployments. Therapid expansion of synchrophasor systems will no doubt bring back new per-spectives which will require further standard revisions. In a technological world,standards are always a work in progress.

5.6.4 PDC Files

The phasor data concentrator (PDC) and the super phasor data concentrator (SPDC)in Fig. 5.3 are important elements of the overall PMU system organization. Theirprincipal functions are to collate data from different PMUs with identical time tags,to create archival files of data for future retrieval and use, and to make data streamavailable to application tasks with appropriate speed and latency. Yet, there are noindustry standards for the PDC data files. However, it is generally understood thatPDCs will have file structures similar to those of PMUs. There are no commerciallyavailable PDCs at this time. Most existing PDCs have been custom built byresearchers or manufacturers of PMUs. As wider implementation of PMU tech-nology takes place, the industry will no doubt work toward creating standards forthese important components of the overall PMU infrastructure.


References

1. Phadke, A. G., & Thorp, J. S. (1988). Computer Relaying (book). New York: ResearchStudies Press, Wiley.

2. There is great wealth of information about the GPS system available in various technicalpublications. A highly readable account for the layman is available at the web-site http://wikipedia.com. There the interested reader will also find links to other source material.

3. IEEE Standard for Synchrophasors for Power Systems, C37.118-2005. Sponsored by thePower System Relaying Committee of the Power Engineering Society (pp. 56–57).

4. Khatib, A.-R. A. (2002). Internet-based wide area measurement applications in deregulatedpower systems, Ph.D. dissertation Virginia Tech, July 2002.

5. Snyder, A. F., et al. (2000). Delayed-Input wide-area stability control with synchronizedphasor measurements and linear matrix inequalities. In Power engineering society summermeeting (Vol. 2, pp. 1009–1014). July 16–20, 2000.

6. Horowitz, S. H., & Phadke, A. G. (2004). Power system relaying (2nd ed.), Chapter 6.Research Studies Press Ltd., Reprinted May 2004.

7. Hecht, J. (2002). Understanding fiber optics (4th ed.). Upper Saddle River, NJ, USA:Prentice-Hall. ISBN 0-13-027828-9

8. Postel, J. (Ed.). (1981). Internet protocol—DARPA internet program protocol specification,RFC 791, USC/Information Sciences Institute, September 1981.

9. Postel, J. (1980). Internet protocol, RFC 760. USC/Information Sciences Institute, January1980.

10. Apostolov, A. (2006). Communications in IEC 61850 based substation automation systems.In Power systems conference: Advanced metering, protection, control, communication, anddistributed resources (pp. 51–56), 2006. PS ’06, March 14–17.

11. IEEE Standard for Synchrophasors for Power Systems, IEEE 1344-1995. Sponsored by thePower System Relaying Committee of the Power Engineering Society.

12. Depablos, J., Centeno, V., Phadke, A. G., & Ingram, M. (2004). Comparative testing ofsynchronized phasor measurement units. In Power engineering society general meeting,IEEE, (vol. 1, pp. 948–954), June 6–10, 2004.

13. IEEE Standard Common Format for Transient Data Exchange (COMTRADE) for PowerSystems, IEEE C37.111-1991, Sponsored by the Power System Relaying Committee of thePower Engineering Society.

References 109



Chapter 6Transient Response of PhasorMeasurement Units

6.1 Introduction

As has been mentioned before, phasor is a steady-state concept. In its classicalinterpretation, it is a complex number representation of a pure sinusoidal waveformof single frequency although the frequency is not explicitly displayed in the phasorrepresentation. In power systems, usually the implied frequency is the nominalfrequency of the power system. However, it is well known that the power system israrely operating precisely at the nominal system frequency. We have considered theeffect of steady-state off-nominal frequency signals on the phasor measurementprocess in Chap. 3. It is also well known that the power system voltages andcurrents have various harmonic, non-harmonic, and transient components. Theharmonic and non-harmonic (steady-state) components are filtered to variousdegrees by appropriate analog or digital filters.

There are transient phenomena occurring on power systems due to a variety ofcauses which produce transient components in current and voltage waveforms.A PMU calculates phasors from sampled data continuously, and it is certain thatsome of these phasor estimates will involve sampled data containing transientcomponents. The subject of this chapter is to investigate the nature of PMUresponse to various power system transients.

To consider the transient response of a PMU, we must consider the chain ofcomponents in the signal path from the power system up to the phasor outputdelivered by the PMU. Principal elements of this chain are shown in Fig. 6.1.Power system transients result from faults, switching operations, and relativemovement of large generator rotors. These sources of transients are representedsymbolically in Fig. 6.1. In the next section, we will consider the nature of tran-sients generated by each of these phenomena. The voltages and currents of thepower system are converted to lower level signals by current and voltage trans-formers (instrument transformers). The signals are then processed by analog anddigital filters serving the purpose of surge suppression, anti-aliasing filtering, and


111

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decimation filtering as appropriate. The filtered signals are then sampled beforephasor computation is performed in the PMU processor. Each of the componentsidentified in Fig. 6.1 affects the transient waveforms, and therefore, the final phasoroutput is produced by the PMU. The following sections will consider these phe-nomena in some detail.

6.2 Nature of Transients in Power Systems

For the purpose of the present discussion, transients in power systems may beclassified into two categories: electromagnetic transients and electromechanicaltransients.

6.2.1 Electromagnetic Transients

Switching operations and faults produce step changes in the voltage and currentwaveforms. On long transmission lines and in transformer and reactor windings,there may be multiple reflections of generated transients. Resonances in the powernetwork create additional frequencies in the waveforms during these phenomena.Another source of electromagnetic transients is lightning. Such transients may beclassified as electrical or electromagnetic transients [1, 2]. The effect of thesetransients is to introduce high-frequency components in the signals. These tran-sients dissipate within a short time and the waveforms then return to aquasi-steady-state condition. The frequencies of signals produced by electromag-netic transients can be summarized as shown in Fig. 6.2.

Power System

Instrument Transformers

Analog and digital filters

Sampled Data

PhasorCalculation

PMU

Xr+ jXi

Fig. 6.1 Generation and passage of transient phenomena from power system to the output of thePMU. Principal components of the chain which affect transient response of the PMU are illustrated

112 6 Transient Response of Phasor Measurement Units

The harmonic phenomena indicated in Fig. 6.2 are typically produced by powerelectronic devices. It should be noted that the harmonic frequencies are multiples ofthe prevailing network frequency, which may be different from the nominal powerfrequency. (It should be remembered that phasor estimate performed with samplingrates keyed to nominal power frequency do not eliminate harmonics of off-nominalpower system frequency.)

Network resonances are caused by shunt and series capacitors, and line chargingcapacitances interacting with various reactances in the network. Winding resonancephenomena are characteristic of transformer, generator, and reactor windings.Faults may produce some very high-frequency components, especially if arcing isinvolved in the fault.

With the exception of phenomena at the low end of the spectrum (100–1000 Hz), the electromagnetic transients are severely attenuated by the filtersemployed in PMUs. Step changes induced in voltages and currents by the faults,and the low-frequency oscillations induced by electromechanical phenomena dis-cussed next are the phenomena of concern in phasor estimation.

6.2.2 Electromechanical Transients

Electromechanical transients are created by the movement of rotors of large gen-erators (and motors) connected to power networks. Normally, all rotors are oper-ating at synchronous speed, with rotor angles at relative positions which arerequired to meet the load demand on the network. When disturbances such as faultsor line outages take place, there is an imbalance between the mechanical andelectrical powers of the machines, and rotor angles begin to deviate from theirsteady-state values. The rotor movement causes the frequency of the generatedvoltage to deviate from its nominal value. The movement of rotors of differentmachines occurs at different rates depending upon their moment of inertias, and thepower imbalance between the inputs and outputs. In general, all generators operateat speeds which may differ from synchronous speed by different (although small)amounts.

Frequency (Hz)106105104103

Lightning, Traveling waves

Winding resonances, Switching surges

Faults

NetworkResonances

Harmonics

Fig. 6.2 Frequencies ofvarious types ofelectromagnetic transients.Lightning phenomena havesteep wave fronts, which maybe interpreted as frequenciesas shown above

6.2 Nature of Transients in Power Systems 113

Consider the power system shown in Fig. 6.3 with ‘n’ generators and ‘m’ buses.Each machine is operating at a different speed, and its output is represented by thecurrent of the appropriate frequency and the generator internal voltage bus isidentified as the source of the current. The power system is shown to have onlythree generators, but in general, there will be a much larger number of machinescontributing current injections to the network.

For the present discussion, we may consider the loads to be represented byimpedances. In this case, the above network with ‘n’ generator buses and a totalof ‘m’ buses can be represented by a bus impedance matrix [ZB] of dimensionm × m. The voltage at any bus ‘i’ is given by

Ei ¼Xmk¼1

ZikIk ð6:1Þ

Equation (6.1) clearly shows that the voltage at any network bus is obtained by asuperposition of voltages of different frequencies. The contribution of generatorswhich are near bus ‘i’ will be dominant in this superposition.

Superposition of voltages with unequal but close frequencies leads to resultantvoltages which show magnitude and phase angle modulation [3]. For example,consider the signal x(t) at a bus obtained by superposition of two signals withfrequencies (ω) and (ω + Δω) with amplitudes X1 and X2, respectively. It isassumed that X2 is much smaller than X1, implying that the generator which pro-duced X1 is closer to the bus at which x(t) is being measured. Without loss ofgenerality, we may assume the superimposing signals are sine waves and that theirphase angles at t = 0 are zero.

xðtÞ ¼ X1 sinðxtÞþX2 sinðxþDxÞt ð6:2Þ

I1(ω1)

In(ωn)

I2(ω2)

Fig. 6.3 Power system withgenerators contributingcurrent injections into thenetwork at unequalfrequencies


Using a trigonometric identity, Eq. (6.2) can be written as

xðtÞ ¼ X1 sinðxtÞþX2 sinðxþDxÞt¼ X3 sinðxtþ/Þ

where

X3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX21 þX2

2 þ 2X1X2 cosðDxtÞq

and

tanð/Þ ¼ X2 sinðDxtÞX1 þX2 cosðDxtÞ

ð6:3Þ

If X2 is much smaller than X1, the expressions for X3 and ϕ can be simplified:

X3 ffi X1 þX2 cosðDxtÞ ð6:4Þ

and

tanð/Þ ffi X2

X1sinðDxtÞ � 1

2X2

X1

� �2

sinð2DxtÞ ffi 0 ð6:5Þ

It is clear from Eq. (6.4) that the effect of this type of superposition is tomodulate the amplitude of the signal with a frequency equal to the differencebetween the two frequencies. The phase angle is also modulated, but for smallvalues of X2, this effect is negligible.

It is also interesting to note that a pure amplitude modulation of a sinusoid isequivalent to a superposition of three sinusoids of center frequency and two sidebands:

X1 þX2 cosðDxtÞf g sinðxtÞ¼ X1 sinðxtÞþ 1

2X2 sinðxþDxÞtþ 1

2X2 sinðx� DxÞt ð6:6Þ

Equation (6.6) is an approximation for signal of Eq. (6.2).

Example 6.1 Consider the superposition of two signals:

xðtÞ ¼ 100 sinð120ptÞþ 10 sinð100ptÞ

The resulting signal is shown in Fig. 6.4a.The amplitude modulation is clearly visible in Fig. 6.4 and can be approximated

by Eq. (6.6):

xðtÞ ffi 100½1þ 5 cosð20ptÞ� sinð120ptÞ

The plot of the approximation is shown in Fig. 6.4b and is very close to that inFig. 6.4a.

6.2 Nature of Transients in Power Systems 115

When several signals of different frequencies are superimposed to form x(t), theresulting expressions are more complex but essentially produce amplitude andphase modulation of the center frequency. This is illustrated in the next example.

Example 6.2 Consider the superposition of three signals:

xðtÞ ¼ 100 sinð120ptÞþ 5 sinð110ptÞþ 10 sinð115ptÞ

Figure 6.5 is representative of voltage and current waveforms on power net-works following large disturbances. The response of PMUs to transients of thisnature is one of the most important considerations in determining their ability to beeffective in providing feedback to controllers and special protection functions.

0 0.04 0.08 0.12 0.16 0.2-100

0

100

(a)

(b)

Time in seconds

0 0.04 0.08 0.12 0.16 0.2-100

0

100

Time in seconds

Fig. 6.4 Superposition oftwo sinusoids approximatedby amplitude modulation.a Superposition andb amplitude modulation

0 0.2 0.4 0.6 0.8 1

-100

0

100

Time in seconds

Fig. 6.5 Superposition ofthree signals representinggenerators’ contributions atdiffering frequencies andmagnitudes


6.3 Transient Response of Instrument Transformers

The response of these transformers can be considered under two categories: fre-quency response and response to step functions. The frequency response is ofinterest in order to assess how the higher frequencies generated by power systemtransients will be transmitted to subsequent stages of Fig. 6.1 by instrumenttransformers. As mentioned before, the filtering performed in succeeding stages islikely to suppress most high-frequency components. The step-function response isof particular interest as fault and switching generated transients do contain stepchanges in voltages and currents. Saturation effects in current transformers andsubsidence effects in CVTs are of particular interest, as they affect the fundamentalfrequency estimation performed by PMUs.

6.3.1 Voltage Transformers

Frequency ResponsePower systems use two types of voltage transformers: potential transformers (PTs)or capacitive voltage transformers (CVTs) [4]. Potential transformers are similar topower transformers, with primary and secondary windings on magnetic cores. ThePTs have essentially flat response to frequencies which are passed through by thelow-pass filters (surge suppression) and anti-aliasing filters. Figure 6.6 is adaptedfrom [5] and is representative of frequency response characteristics of several PTs[6]. For the purposes of PMU measurements, we may assume that the PTs have flatresponse in the frequency range of interest.

CVTs have resonances at lower frequencies relative to the PTs. Resonancefrequencies encountered in CVT responses depend heavily upon the values of thecapacitors, details of the ferro-resonance suppression circuits, and other systemcomponents. A representative CVT frequency dependence response curve is shownin Fig. 6.7 [7].

Step-Function ResponseFor all practical purposes, the step-function response of a potential transformer maybe assumed to be transparent. The capacitive voltage transformers have subsidence

1.0

2.0

3.0

4.0

0.00 10 20 30 40 50

Freq

uenc

y de

pend

ent

trans

form

er ra

tio

Harmonic order

Fig. 6.6 Frequency responsecharacteristics of severalpotential transformers.Adapted from Ref. [5]

6.3 Transient Response of Instrument Transformers 117

transient response which depends upon the elements of the voltage transformercircuit, and also upon at what point on the voltage waveform the step function hasoccurred [4]. Figure 6.8a, b are representative of CVT subsidence transients.

The subsidence transients are usually associated with faults on the network, andthe phasor estimates performed with fault data are usually of no interest in PMUapplications. The ‘transient monitor’ function discussed in Sect. 6.5 below isresponsible for identifying the presence of fault data in the data window of PMUestimation and could be used to flag the phasor estimate as being unusable.

6.3.2 Current Transformers

Frequency ResponseMost current transformers in use in substations are magnetic core multiwindingtransformers. Frequency response of these transformers is flat for up to high

Gai

n db

10 100 1000 10000

Frequency, Hz

600

Fig. 6.7 Frequency response characteristics of several capacitive voltage transformers. Adaptedfrom Ref. [7]

(a) (b)

Primary voltagePrimary voltage

Subsidence transient Subsidence transient

Fig. 6.8 Subsidence transient of a CVT. a Step function occurring at voltage maximum andb step function occurring at voltage zero


frequencies of the order of 50 kHz [8] and will no longer be considered here as thebandwidth of the anti-aliasing filters is much lower than these frequencies.

Step-Function ResponseThe principal concern in current transformer response to faults is the possibility ofsaturation of the magnetic core. When the core saturates, the secondary current hasseverely distorted waveforms depending upon the CT burden, the remanence in thecore, and the amount of DC offset in the fault current. A representative CT outputwaveform due to heavy saturation is shown in Fig. 6.9.

The CT saturation transients are usually associated with fault currents with DCoffsets, and the phasor estimates of waveforms during fault conditions are usually ofno interest in PMU applications. The ‘transient monitor’ function could be used toidentify the presence of fault data in the data window of PMU estimation and couldbe used to flag the phasor estimate as being unusable.

6.4 Transient Response of Filters

6.4.1 Surge Suppression Filters

Surge suppression filters are designed to block destructive transients created byswitching and arcing events in the substations. The frequency of such signals is inthe megahertz range [4]. Thus, the filters may have cutoff frequencies of the order of100 s kHz. It is therefore not necessary to consider the effect of these filters onPMU performance.

6.4.2 Anti-aliasing Filters

Frequency ResponseAnti-aliasing filters used in PMUs have a cutoff frequency which is less than halfthe sampling frequency. If the signals are sampled at 1000 Hz, the corresponding

Primary current

Secondary current

Fig. 6.9 CT saturation.Primary current has a largeDC offset, leading to coresaturation which may persistfor considerable time

6.3 Transient Response of Instrument Transformers 119

anti-aliasing filter will have a cutoff frequency of about 400 Hz. The design ofanti-aliasing filters may vary considerably depending upon the preference of thePMU manufacturer. We may consider a generic anti-aliasing filter made up of twostages of R–C circuits as shown in Fig. 6.10 [9].

The output of the anti-aliasing filters has a phase lag depending upon the fre-quency of the input signal. The application functions using PMU data are usuallyinterested in signals with frequencies very close to the nominal power frequency.The phase lag at the output of a two-stage R–C filter with (R1 = 1260 Ω,R2 = 2520 Ω, and C1 = C2 = 0.1 μF) for frequencies in the range of ±5 Hz aroundthe nominal frequency is shown in Fig. 6.11. This figure corresponds to ananti-aliasing filter designed to match a sampling rate of 12 times the power systemfrequency of 60 Hz. It is a simple matter to determine this characteristic for anyother filter design. Recall that this phase shift must be compensated in the output ofthe PMU in accordance with the requirements of the PMU standard discussed inChap. 5.

Step-Function ResponseThe anti-aliasing filters respond to step-function inputs caused by faults as shown inFig. 6.12. The filter as in Sect. 6.4.2.1 shown in Fig. 6.10 is used to produce thisfigure.

R1 R2

(a) (b)

C2C1

Gai

n

f c

1.0

0.010

Fig. 6.10 A two stage R–C anti-aliasing filter. a Filter circuit and b Frequency response

55 56 57 58 59 60 61 62 63 64 65-11.8

-11

-10

Frequency, Hz

Phas

e la

g, d

egre

es

Fig. 6.11 Phase lagproduced by a two-stage R–Canti-aliasing filter. Thesampling rate is assumed tobe 720 Hz, and the filtercutoff frequency is 330 Hz


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Section 6.5 discusses the response of PMUs to transients caused by faults andother switching events. It will be shown there that phasor estimates produced byPMUs during faults are meant to be discarded, as they are not representative of thepower system state.

6.5 Transient Response During ElectromagneticTransients

As mentioned in Sect. 6.4, high-frequency transients will be removed from PMUinputs by the filtering stage. What remains important for our consideration is theeffect of step changes in voltages and currents brought about by faults andswitching operations. The step changes are further modified by the filtering stage,and the phase angle of the fundamental frequency signal is shifted (lag). This isillustrated in Fig. 6.12 for a voltage waveform following a fault which reduces thefundamental frequency voltage to a low value.

Using the recursive phasor estimation process with one cycle data window, thephasor estimate of the pre-fault waveform would be obtained in data windowswhich contain only the pre-fault data. This is illustrated by data windows 1, 2, and 3in Fig. 6.13a. (Also see Sect. 2.4). The corresponding phasor is X1 in Fig. 6.13b.When the data window is fully occupied by post-fault data as with windows N andN + 1 in Fig. 6.13a, the phasor estimate becomes X2 for all succeeding windows.However, while the windows contain both the pre- and post-fault data as withwindows 4, 5, 6, …, the phasor estimate travels along a trajectory from X1 to X2 asshown in Fig. 6.13b. These phasor values are not representing the state of the powersystem and must be discarded in application of the phasor data.

It is important that the transitional phasors be recognized and flagged in order toavoid their use in applications. A possible technique for accomplishing this wasdescribed in Sect. 2.4 with the use of a ‘transient monitor’ function which is ameasure of the quality of phasor estimates [9]. This function estimates the differ-ence between the input data samples and the data samples which correspond to theestimated phasor waveform (see Fig. 6.14).

Time, msec.0 100 200

-100

0

100

200

volta

ge

Fig. 6.12 Response of atwo-stage R–C anti-aliasingfilter of Fig. 6.10 to a stepchange in voltage waveform

6.4 Transient Response of Filters 121

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Equations (2.26) and (2.27) provide formulas for computing the ‘transientmonitor,’ and Example 2.4 shows the calculation for a transitional waveform.

The transient monitor (or a similar ‘quality’ function) should be estimated at thesame time that the phasor is estimated. When this function is outside acceptablebounds, it would be an indication that the phasor estimate may be for a transitionalsignal, or a signal with excessive noise. At present, the industry standard does notcall for such a specification to be included in the PMU output. However, it would bea desirable feature which may be provided by manufacturers of PMUs and even-tually adopted in a future standard revision.

6.6 Transient Response During Power Swings

Section 6.2.2 considered the effect of superposition of signals with differing fre-quencies in power system voltages and currents during transient stability swings. Itshould be remembered that these swings are relatively slow phenomena, withoscillation frequencies in the range of 0.1–10 Hz. It has been shown there thatsuperposition effects can be approximated by frequency and amplitude modulation

InputOutput

1 23 4 56 N

N+1 1 2 3

N N+1

4

56

…

…

(a) (b)

X1

X2

Fig. 6.13 Response of PMU to step changes in input signals due to faults or switching operations

Input samplesEstimated phassor samples

Input waveform

Estimated phasor waveform

Data window

Fig. 6.14 Input data samplesand samples corresponding toestimated phasor. Thedifference between the two isused to define a ‘transientmonitor’ function


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of voltages and currents. The two modulations are likely to be present simultane-ously, although it is possible to have amplitude or frequency modulation by itselfunder certain system conditions. The performance of the synchronized phasormeasurement systems for modulated signals is considered in this section. We willconsider these modulation phenomena in sequence. In each case, the effect ofoff-nominal frequency operation will be discussed.

6.6.1 Amplitude Modulation

Assume that the power system is operating close to nominal frequency (i.e., within±5 Hz of the nominal frequency) and that a transient stability swing has beeninitiated. The corresponding voltage and current signals may be assumed to beamplitude modulated with a frequency ωm. It is expected that system frequency ωwill be close to the nominal power frequency of 120π (Δω varying between −10πand +10π), while ωm will vary between 0 and 20π (corresponding to 0–10 Hz).

xðtÞ ¼ffiffiffi2

p½1þ 0:2 sinðxmtÞ� cosðxtÞ ð6:7Þ

A balanced three-phase input with amplitude modulation as above and atoff-nominal frequency will produce a positive sequence measurement with just theamplitude modulation.

PMUs estimate positive sequence phasors continuously, i.e., a new phasor isestimated whenever a new data sample is obtained. The data window is equal to oneperiod of the nominal system frequency. Under these conditions, it is clear that thecalculated phasors will follow the amplitude modulation perfectly, as long as themodulation frequencies are low.

Example 6.3 A balanced three-phase input with a system frequency excursion ofΔω = 10π (5 Hz) and an amplitude modulation frequency of 2 Hz is given inEq. (6.8).

XaðtÞ ¼ffiffiffi2

p½1þ 0:2 sinð4ptÞ� cosð130ptÞ

XbðtÞ ¼ffiffiffi2

p½1þ 0:2 sinð4ptÞ� cosð130pt � 2p=3Þ

XcðtÞ ¼ffiffiffi2

p½1þ 0:2 sinð4ptÞ� cosð130ptþ 2p=3Þ

ð6:8Þ

The result of estimating positive sequence phasor with a sampling rate of1440 Hz produces a positive sequence phasor shown in Fig. 6.15 when theamplitude modulation frequency ωm is set equal to 1, 2, 3, 5, and 10 Hz. It is shownin this figure that balanced three-phase inputs reproduce amplitude modulationfaithfully, and no additional filtering is necessary.

Now consider the case of a single-phase input with amplitude modulation and atoff-nominal frequency. It can be expected that in this case the phasor estimate will

6.6 Transient Response During Power Swings 123

show a signal at 2(ω + ω0) (see Sect. 3.2), which must be filtered in order toeliminate the possibility of aliasing errors when the phasor output is to be sampled.Figure 6.16 shows the result of estimating the phasor when the input is as given byEq. (6.7) with ω = 65 Hz, and ωm = 5 Hz.

Using the three-point filtering technique of Sect. 3.3.1, the frequency componentat 2(ω + ω0) is eliminated for all practical purposes. The result of this filter appliedto the phasor estimates in Figs. 6.16 and 6.17. It is shown in these figures that asbefore, the three-point filter removes the 2(ω + ω0) component completely.

0 400 800 14000

0.2

0.6

1

Estim

ated

pha

sor m

agni

tude

Sample number

Fig. 6.15 Balanced three-phase inputs at 65 Hz. Positive sequence magnitude estimate foramplitude modulation at 1, 2, 3, 5, and 10 Hz

0 100 200 3000

0.2

0.6

1

Estim

ated

pha

sor m

agni

tude

Sample number

Fig. 6.16 Single-phase inputat 65 Hz. Magnitude estimatefor amplitude modulation at5 Hz

Sample number0 100 200 300

0

0.2

0.6

1

Estim

ated

pha

sor m

agni

tudeFig. 6.17 Single-phase input

at 65 Hz. Three-pointaveraging filter. Magnitudeestimate for amplitudemodulation at 5 Hz


http://dx.doi.org/10.1007/978-3-319-50584-8_3

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It should also be noted that in the presence of unbalances, similar remarks aboutfiltering apply for the positive sequence estimate. When positive sequence com-ponent is estimated, the 2(ω + ω0) component is further multiplied by the per-unitvalue of the negative sequence component, and the three-point filter will removethis component from the positive sequence estimate.

6.6.2 Frequency Modulation

Consider a frequency modulation of the type

x ¼ x0 þA sinðx1tÞ ð6:9Þ

so that the frequency of the input signal varies sinusoidally between (ω0 + A) and(ω0 − A) with a frequency ω1. The corresponding phase function ϕ is the integral ofthe frequency ω:

/ ¼ x0t � Ax1

� �cosðx1tÞ ð6:10Þ

Balanced three-phase inputs with this phase function are as follows:


pcosð/Þ ¼

ffiffiffi2

pcos x0t � A

x1

� �cosðx1tÞ

� �


pcosð/Þ ¼

ffiffiffi2

pcos x0t � A

x1

� �cosðx1tÞ � 2p

3

� �


pcosð/Þ ¼

ffiffiffi2

pcos x0t � A

x1

� �cosðx1tÞþ 2p

3

� � ð6:11Þ

Example 6.4 As an example, consider the case of a 60 Hz nominal frequency inputvarying between 61 and 59 Hz at a frequency of 1 Hz. Since we are considering asingle-phase input, we should expect a second harmonic ripple in the estimatedphasor angle and magnitude. The result of estimating phasors from a single phaseinput is shown in Fig. 6.18.

Note that as the frequency excursions reach extreme values (61 and 59 Hz), theamount of signals at 2(ω + ω0) are at a maximum. Note also the 1 Hz modulationfrequency reproduced in the envelope of the 2(ω + ω0) signal. As before, athree-point averaging filter removes this component almost perfectly. The result isshown in Fig. 6.19.

A positive sequence voltage estimated from a balanced three-phase input iscompletely free of these 2(ω + ω0) components. This is illustrated in Fig. 6.20.Although barely noticeable, the amplitude of the estimated phasor does differ from1.0 as determined by the factor P discussed in Chaps. 4 and 5.


http://dx.doi.org/10.1007/978-3-319-50584-8_4

http://dx.doi.org/10.1007/978-3-319-50584-8_5

6.6.3 Simultaneous Amplitude and Frequency Modulation

During stability swings, both magnitude and frequency of the three-phase voltagesand currents in a power system could be modulated. Equation (6.12) representssuch input quantities. Note that the frequency of the amplitude modulation andfrequency modulation is equal, viz. ω1.


p½1þ 0:2 sinðx1tÞ� cos x0t � A

x1

� �cosðx1tÞ

� �



x1

� �cosðx1tÞ � 2p

3

� �



x1

� �cosðx1tÞþ 2p

3

� � ð6:12Þ

Both the amplitude and frequency modulations arise out of the superposition ofthe various generator currents. It is therefore natural that both modulation fre-quencies be the same.

0 200 400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

Output sample number.Es

timat

ed P

haso

r Mag

nitu

de

Fig. 6.18 Phasor estimatesfor single-phase input withfrequency modulation

0 200 400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

Output sample number.

Estim

ated

Pha

sor M

agni

tude

Fig. 6.19 Phasor estimatesfor single-phase input withfrequency modulation. Samecase as in Fig. 6.18. Effect ofa three-point averaging filter


Example 6.5 The phasor estimate of a single-phase input having a 20% amplitudemodulation and a 1 Hz frequency excursion around 60 Hz with both the frequencyand amplitude modulation frequency of 2 Hz is shown in Fig. 6.21. As is shownthis figure, the net outcome is a superposition of results in Sects. 6.6.1 and 6.6.2.

The three-point averaging applied to this estimate effectively eliminates theripple at 2(ω + ω0), as shown in Fig. 6.22.

As before, balanced three-phase inputs produce positive sequence phasor esti-mates which have no ripple at 2(ω + ω0) as shown in Fig. 6.23. The amplitudemodulation in this case was 0.2 per unit zero-to-peak. The output shown inFig. 6.23 (although not discernable from this figure because of the scale used) has amagnitude swing of 0.1995, which is 99.75% of the amplitude of input modulation.

0 200 400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1


timat

ed P

haso

r Mag

nitu

deFig. 6.20 Phasor estimatesfor positive sequencecomponent from balancedthree-phase inputs withfrequency modulation.Frequency excursion andmodulation frequency same asin Fig. 6.19

0 200 400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

1.2


Estim

ated

Pha

sor M

agni

tudeFig. 6.21 Phasor estimates

for single-phase input withfrequency and amplitudemodulation. Modulationfrequency is 2 Hz. Frequencyexcursions of ±1 Hz around60 Hz

0 200 400 600 800 1000 1200 14000

0.20.40.60.8

11.2


Estim

ated

Pha

sor

Mag

nitu

de

Fig. 6.22 Phasor estimatesfor single-phase input withfrequency and amplitudemodulation. Result ofapplying three-point filteralgorithm. Modulationfrequency is 2 Hz. Frequencyexcursions of ±1 Hz around60 Hz


The slight drop in measured amplitude modulation is the result of the factor P atoff-nominal frequency as explained in Chaps. 4, 5 and 6.

When three-phase inputs are unbalanced, the positive sequence estimate willonce again have a ripple at 2(ω + ω0) depending upon the amount of the negativesequence component present in the input. This ripple can also be eliminated byfiltering, the three-point filter being the most efficient one in this regard.

6.6.4 Aliasing Considerations in Phasor Reporting Rates

It is clear from the foregoing discussion that electromechanical transients whichreflect the movement of machine rotors are reproduced faithfully by DFT-basedphasor estimators. However, since the phasor estimates are reported to higher levelsof the hierarchy at a rate which is much lower than that utilized in the figures of thissection (once every sample), it becomes necessary to consider the effect of thisreporting rate on the frequency response observable at the higher levels of hierar-chy. Consider a phasor reporting rate of 30 Hz, i.e., once every two cycles for a60 Hz power system. At this reporting rate, the phasors will be able to reproducecorrectly oscillation frequencies lower than 15 Hz without causing errors due toaliasing. If the power system signals happen to have frequencies higher than 15 Hz,they must be removed by appropriate filtering before they are forwarded to appli-cations at the upper hierarchical levels. Table 6.1 summarizes the filteringrequirements for various phasor reporting rates.

0 200 400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

1.2


timat

ed P

haso

r Mag

nitu

deFig. 6.23 Phasor estimatesfor balanced three-phase inputwith frequency and amplitudemodulation. Modulationfrequency is 2 Hz. Frequencyexcursions of ±1 Hz around60 Hz

Table 6.1 Filtering requirements for various phasor reporting rates

Phasor reporting rate (Hz) Cutoff frequency of phasor processing filters (Hz)

60 30

30 15

20 10

15 7.5

10 5


http://dx.doi.org/10.1007/978-3-319-50584-8_4

http://dx.doi.org/10.1007/978-3-319-50584-8_5

http://dx.doi.org/10.1007/978-3-319-50584-8_6

It should be noted that most electromechanical oscillation frequencies observedin large power networks are well above the lowest cutoff frequency in Table 6.1.Nevertheless, it is possible that there are small signals at higher frequencies causedby resonances or by machines of low inertia. It is therefore necessary to includesuch a filter in all phasor measurement and application systems.

Any of the standard filter design techniques could be used to accomplish thistask. A very simple averaging filter with appropriate averaging window may beused as illustrated in the following example.

Example 6.6 Consider a phasor estimate in a 60 Hz power system with 5 and20 Hz modulating frequencies. The sampling rate is assumed to be 1440 Hz (i.e.,24 samples per cycle), and the modulating 5 and 20 Hz components are 10 and20%, respectively, of the fundamental frequency. Considering only the magnitudemodulation here, the phasor magnitude of the kth sample may be represented by

Xkj j ¼ 1:0þ 0:1 cosk

24� 12

� �þ 0:2 cos

k24� 3

� �

The phasors are to be reported at a rate of 20 Hz, so that all frequency com-ponents 10 Hz and above must be eliminated by the filter. A 10 Hz signal will havea period of 144 samples (1440/10). Thus, we may construct an averaging filter witha width of 144 samples. Thus, a new phasor magnitude is created by averagingsuccessive 144 samples of the original data set (see Fig. 6.24).

It should be noted that the result of averaging is available at the end of theaveraging period, which produces the plot with the thin solid line. On the otherhand, by assigning the average to the middle of the averaging window, the boldsolid line is produced, which has the correct phase information about the modu-lating signal at 5 Hz.

Signals with magnitude and phase angle modulation are treated similarly. Formost practical purposes, a simple averaging filter (the so-called box-car filter) issufficiently accurate to meet the Nyquist criterion associated with the phasorreporting rate.

0 200 400 600 800 1000 1200 14000

0.2

0.4

0.6

0.8

1

1.2

Estim

ated

Pha

sor M

agni

tude


Output of averaging filter tagged at window end

Input phasor magnitude

Output of averaging filter tagged at window center

Fig. 6.24 Averaging filterfor eliminating frequenciesabove the Nyquist rate


In summary, the present industry standard (C37.118) for PMUs does not specifythe requirements for transient response of PMUs. However, it is to be expected thatfuture revision of the standard will deal with this requirement in order to achieveinteroperability between PMUs manufactured by different manufacturers. Theprincipal task of PMUs is to measure positive sequence voltages and currents in thepower network. By definition, phasors and positive sequence are steady-stateconcepts. Thus, as discussed in this chapter, phasors estimated during fault andother switching events should be flagged as not representing any state of the net-work. The swings of machine rotors are treated as a sequence of steady-stateconditions, and PMUs are shown to measure the rotor swings accurately for allpractical power systems. Finally, the issue of phasor reporting rate and possiblealiasing effects due to the presence of higher swing frequencies in the network hasalso been addressed. It is shown that a simple averaging filter does an effective jobof taking care of the aliasing problem, when the window of the averaging filter isadjusted to the phasor reporting rate.

References

1. Greenwood, A. (1971). Electrical transients in power systems. New York: Wiley Interscience.2. Chowdhuri, P. (1996). Electromagnetic transients in power systems. New York: Research

Studies Press Ltd.: Wiley.3. Phadke, A. G., Kasztenny, B. (2008). Synchronized phasor and frequency measurement under

transient conditions. Accepted for publication in IEEE Transactions on Power Delivery.4. Horowitz, S. H., & Phadke, A. G. (2008) Power system relaying, 4th printing. New York:

Wiley, Second Edition.5. Seljeseth, H. et al. (1998) Voltage transformer frequency response. Measuring harmonics in

Norwegian 300 kV and 132 kV power systems. In 8th international conference on harmonicsand quality of power, ICHQP ’98, IEEE/PES and NTUA, Athens, Greece, October 14–16,1998.

6. Meliopoulos, A. P., Sakis, A. P., et al. (1993). Transmission level instrument transformers andtransient event recorders characterization for harmonic measurements. IEEE Transactions onPower Delivery, 8(3), July 1993.

7. Working Group C-5 of IEEE Power System Relaying Committee. (2000). Mathematicalmodels for current, voltage, and coupling capacitor voltage transformers. IEEE Transactions onPower Delivery, 15(1), January 2000.

8. Samesima, M. I., et al. (1991). Frequency response analysis and modeling of measurementtransformers under distorted current and voltage supply. IEEE Transactions on PowerDelivery, 6(4), 1762–1768.

9. Phadke, A. G., & Thorp, J. S. (2000). Computer relaying for power systems. Research StudiesPress, Reprinted May, 2000, p. 150.


Part IIPhasor Measurement Applications

Chapter 7State Estimation

7.1 History-Operator’s Load Flow

Before the advent of state estimation, the power system operator had responsibilityfor many real-time control center functions including scheduling generation andinterchange, monitoring outages and scheduling alternatives, supervising scheduledoutages, scheduling frequency and time corrections, coordinating bias settings, andemergency restoration of the system. All of this was done with operating guidesproduced by the planning department after running a large number of load flows. Asis always the case, the actual events faced by the operator were occasionallyunexpected and had not been included in the planning cases. The solution was tosupply the operator with a load flow program installed in the control center. Theoperator could manually enter data describing the current situation and get a loadflow corresponding to the real world. Unexpectedly, the operator’s load flow didnot work well. The problems were caused by insufficient data, non-uniform data,and errors in the data and in the model.

It was important that the operator’s load flow be accurate so that the outage reliefwould be accurate. It was also recognized that the planning load flow was not exactlywhat the operator needed. What was required in the control center was a process ofusing a large number of imprecise measurements to estimate the existing state of thesystem. Early-state estimation algorithms [1] used measurements of line flows, bothreal and reactive power, to estimate the bus voltage angles and magnitudes. Thecomplex bus voltages are the state of the system since given an accurate model of thenetwork the bus voltages determine the complex power flows in the lines and all thecomplex power injections. Unfortunately, prior to synchronized phasor measure-ments, the state could not be measured directly but only inferred from the unsyn-chronized power flow measurements. This fact and the process of getting largenumbers of measurements into the control center forced the first state estimators to

Dr. Zhongyu Wu has contributed to Sects. 7.6.2 and 7.6.3.


133

make compromises which persist today and have an influence on how phasormeasurements must be integrated into existing state estimation algorithms.

7.2 Weighted Least Square

7.2.1 Least Square

The terms least squares and weighted least squares have to do with the criterion forselecting a solution to the overdefined equations produced when more measure-ments than states are represented in the estimation problem. Suppose the equationsare linear and in the form

y ¼ Ax ð7:1Þ

where x is the state, y the measurements, and A a matrix with more rows thancolumns. In general, Eq. (7.1) does not have a solution and should be written as

y ¼ Axþ e ð7:2Þ

where the vector ε represents errors in the measurements. The ‘least squares’solution is based on assuming the errors are independent and identically distributedwith mean 0 and variance 1. That is with the unit matrix denoted by I

Efeg ¼ 0; EfeeTÞ ¼ I ð7:3Þ

The optimization problem is to find the estimate, x which minimizes

E y� Axð ÞT y� Axð Þn o

¼ yTy� 2yTAx� xTATAx ð7:4Þ

The solution is

x ¼ ATA� ��1

ATy ð7:5Þ

Equation (7.5) is what MATLAB produces for the \ operation. That is, x ¼ Any.Example 7.1

Let A ¼

7 2 13 5 01 4 6�1 3 42 �3 5

266664

377775 y ¼

15011216

266664

377775

134 7 State Estimation

Then,

ATA ¼64 24 1924 63 2319 23 78

24

35 ATy ¼

14632169

24

35

and

x ¼2:0744�0:99611:9551

24

35 y� Ax ¼

0:5165�1:24281:1794�0:7578�0:9123

266664

377775

7.2.2 Linear Weighted Least Squares

Least SquaresSuppose there was more information about the errors ε in Eq. (7.2) in the form of acovariance matrix

E eeT� � ¼ W ð7:6Þ

Even in the simple case where W is diagonal, Eq. (7.6) allows different errors to beweighted differently. The objective is to minimize

E y� Axð ÞTW�1ðy� AxÞn o¼ yTW�1y� 2yTW�1Ax� xTATW�1Ax ð7:7Þ

which has the solution

x ¼ ATW�1A� ��1

ATW�1y ð7:8Þ

An understanding of weighted least squares can be obtained by considering thediagonal version of the matrix W. If W is diagonal, then the objective function issimply

E y� Axð ÞTW�1 y� Axð Þn o

¼X yi � yið Þ2

Wiið7:9Þ

7.2 Weighted Least Square 135

where

y ¼ Ax ~y ¼ y� y ð7:10Þ

The variable ~y, the difference between the actual measurement and the estimatedmeasurement in Eq. (7.9) or Eq. (7.4), is called the measurement residual. Whenthe measurement errors are identically distributed, we are minimizing the sum of thesquares of these residuals. When the measurement errors are independent but ofdifferent sizes, we are dividing the residuals by the measurement variances tonormalize things. That is, if a measurement error is large, then a larger residual isaccepted, while a small measurement error demands a smaller residual. The mean of~y is zero, and the covariance of ~y is

Ef~yg ¼ 0 Covð~yÞ ¼ A ATW�1A� ��1

AT ð7:11Þ

A tall A matrix can cause numerical difficulties. State estimation frequentlyinvolves thousands of measurements. A could have thousands of rows, and in thatcase, the QR algorithm can be employed. If we write

W�1 ¼ MTM; MA ¼ QR0

� �ð7:12aÞ

where Q is orthogonal, i.e., QQT ¼ I and R is upper triangular then

ðATW�1AÞ�1ATW�1 ¼ R�1 I 0½ �QTM ð7:12bÞExample 7.2 Suppose everything is the same as in Example 7.1 except

W ¼

1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 100 00 0 0 0 100

266664

377775

That is, the last two measurements are ten times larger than the first three.

x ¼2:1762�1:29462:3277

24

35 ~y ¼

0:0280�0:05570:0360�1:2508�3:8746

266664

377775

The residuals corresponding to the better measurements are considerably smallerthan the residuals for the poor measurements as would be expected. The covariancematrix for the measurement residual also shows the effect of the unequal mea-surement error variances. The first 3 × 3 block of covariance matrix is almost a unitmatrix, while the last two residuals are strongly correlated with the first three.


Covð~yÞ ¼

0:9925 0:0123 �0:0033 �0:3076 0:79640:0123 0:9784 0:0089 0:1486 �1:3378�0:0033 0:0089 0:9902 0:7136 0:6733�0:3076 0:1486 0:7186 0:6298 0:02270:7964 �1:3378 0:6733 0:0227 3:2611

266664

377775

7.2.3 Condition Numbers, Leverage, and LAV in LinearLeast Squares

Condition NumbersEven when there is insufficient information to supply a W matrix, there is fre-quently evidence that caution is require in using least squares. One is the conditionnumber jðAÞ of the matrix A which is a measure of how sensitive the solution tothe least squares problem is to errors in y. The condition number of A is given by

jðAÞ ¼ Ak k2 ATA� ��1

A��

2ð7:13Þ

A large condition number means that even small errors in y can cause large errors inx. A very large condition number indicates near singularity of A. The 2 norms ofthe matrix in Eq (7.13) are the largest singular values of the matrices, i.e.,Ak k2 = the square root of the largest eigenvalue of A * A where * denoted con-

jugate transpose. It is well to check condition numbers whenever using leastsquares.

LeverageThe matrix H in Eq. (7.14) is called the hat matrix or influence matrix and gives theestimate of y in Eq. (7.15) from Eq. (7.10), i.e.,

y ¼ Hy for H ¼ A ATA� ��1

AT ð7:14Þ

The diagonal entries of H are called leverages. If A is n × m, n > m, and rank m,then the trace of the matrix H is m and the average diagonal entry, hii, is m/n. Thiscan be shown by examining the trace of H and using properties of the trace. H isn × n with positive diagonal entries, but the sum of the diagonal entries is onlym. As a rule of thumb, if hii is greater than 2m/n (twice the average), then it ispossible that the ith measurement is exerting influence over the answer. The actualeffects depend on y as shown in the next example.

Tr A ATA� ��1

AT

¼ Tr AAT ATA� ��1

¼ Tr Imð Þ ¼ m ð7:15Þ


Example 7.3

Given A ¼

1 0:251 0:51 0:751 11 1:251 1:51 1:751 21 2:251 2:51 3

266666666666666664

377777777777777775

and y ¼

0:24270:54560:35870:97981:6511:53461:76031:81652:24232:5581�2:8934

266666666666666664

377777777777777775

The threshold is 4/11 = 0.3636, and the only diagonal entry above the threshold isH(11,11) = 3.797. The best least squares fit to all the data (red circles) is shown inFig. 7.1a and is obviously effected by the 11th data point. A change in the value ofy(11) is shown in green in Fig. 7.1b along with the result of the complete removalof y(11) shown in blue.

L1 NormThe bad data point in the previous example can also be addressed by using othererror criteria. Clearly, the biggest absolute error dominates a sum of squared errors.It would be even worse if we only considered the biggest error. Alternately,summing only the absolute value of the errors would seem to be a way to mitigatethe leverage points. The norm formed as the sum of the absolute errors is referred toas the L1 norm and is denoted by xk k1 for a scalar and for a vector xk k1¼

Pi xij j.

The problem can be recast as

minx

Ax� yk k1 ð7:16Þ

The problem can then be converted to a linear programming problem as

H ¼A �I

�A I

0 �I

264

375; fT ¼ 0 1 T

h i; b ¼

z

�z

0

264

375; y ¼ x

s

� �

min f Ty

Hy� b

ð7:17Þ

where s is the vector of residuals. It is also necessary to add constraints to theproblem to produce Eq. (7.18)


A �IA I0 �II 0�I 0

266664

377775

xs

� ��

z�z0xmax

�xmin

266664

377775 ð7:18Þ

Using xmin = −2 xmax = 2, the solution to the previous example using the L1norm is shown in Fig. 7.2.

0 2 4 6 8 10 12-3

-2

-1

0

1

2

3

0 2 4 6 8 10 12-3

-2

-1

0

1

2

3

(a)

(b)

Fig. 7.1 Best least squares fitwith a leverage point (a) andremoved (b)


7.2.4 Nonlinear Weighted Least Squares

If the measurements are a nonlinear function of the state,

z ¼ hðxÞþ e; EðeÞ ¼ 0; E eeT� � ¼ W ð7:19Þ

Then, the task is to find x to minimize

JðxÞ ¼ z� h xð Þ½ �TW�1 z� h xð Þ½ � ð7:20Þ

Equation (7.14) must be minimized recursively by linearizing hðxÞ about xk, andthe value of x at the last iteration,

hðxÞ ¼ h xk� �þH x� xk

� � ð7:21Þ

where H is a matrix of first partial derivatives of the elements of h with respect tothe components of x evaluated at xk. If

D xð Þ ¼ x� xk; Dz ¼ z� h xk� � ð7:22Þ

Then, one step in the iteration is given by the solution of Eq. (7.18) which is anincremental version of Eq. (7.8)

HTW�1HDx ¼ HTW�1Dz ð7:23Þ

Dx ¼ HTW�1H� ��1

HTW�1Dz ð7:24Þ

The covariance of the resulting estimate is given by the matrix H xð ÞW�1H xð Þ� ��1.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

L1

L2

data

Fig. 7.2 L1 and L2 solutionsto the example


Example 7.3

Let hðxÞ ¼ x2 � 4x + 43x� 1

� �z ¼ 4

1

� �H ¼ 2x� 4

3

� �w ¼ 1 0

0 1

� �

From Eq. (7.18), the iteration is

xkþ 1 ¼ xk þ 1

2xk � 4ð Þ2 þ 92xk �4 3

� � 4� xk� �2 þ 4xk � 41� 3xk þ 1

� �

Starting at x0 ¼ 4, the values are given in Table 7.1 (Fig. 7.3).

7.3 Static State Estimation

The power system application of the preceding begun after the 1965 NortheastBlackout involved analog measurements and less sophisticated communicationsystems than available today. These facts forced an approximation on the problemfrom the outset. The measurements from a supervisory control and data acquisition(SCADA) system composed of remote terminal units (RTUs) in the substationswere obtained in sequence by polling. The data scan took long enough that thesystem was actually in a slightly different state when the scan was complete than ithad been at the beginning. The approximation was to assume the system did notchange during the scan—that the system was static. Perhaps, it is better to view theresulting estimate of the state to be the state of a hypothetical system which couldsimultaneously support the complete set of measurements. Of course, depending onhow long the scan takes and what changes in load and generation take place duringthe scan, the hypothetical system may not exist or may be quite different from thereal system. While scans have become quicker, the introduction of phasor mea-surements forces reconsideration of the static assumption.

The state of the system is the collection of system bus voltages which arecomplex numbers. Conventional static state estimation algorithms use the magni-tude and angle of these voltages as the states and measurements of real and reactivepower flows and injections along with some voltage magnitude measurements. The

Table 7.1 First 7 iterationsfor Example 7.4

k xk

1 2.80

2 1.6042

3 0.4416

4 0.2861

5 0.2756

6 0.2746

7 0.2745


use of synchronized phasor measurements can make it attractive to also consider theproblem in rectangular coordinates with real and imaginary parts of voltages andcurrents.

Elaborate error models were developed [2] for the analog measurements that haddependence on the size of the measurement and the full scale of the meter. Digitaltechnology makes these models less appropriate. The common elements in mea-surements, both old and new, are the current transformers, CTs, and voltagetransformers, PTs. The current transformers, in particular, have a fixed bias that isnot modeled with the measurement error ε described in the weighted least squaressection.

In polar coordinates, the angles θp and magnitudes Vp are the states where

Ep ¼ Vpejhp ð7:25Þ

The complex flows and injections are nonlinear. For example, the real power flowfrom bus p to bus q with a series impedance zpq between p and q and a shuntadmittance at bus p of yp is

ppq ¼ V2p Ypq cos bpq

� �þ yp cos ap

� �� VpVq cos hp � hp � bpq

� � ð7:26Þ

where αp and βpq are the angles in Eq. (7.21)

1zpq

¼ Ypq e�jbpq yp ¼ yp

ejap ð7:27Þ

0 1 2 3 40

20

40

60

80

100

x

J(x)

Fig. 7.3 Objective functionand the first four iterations


In general,

z ¼ hðV; hÞþ e ð7:28Þ

With assumptions as in Sect. 7.2 that the measurement errors have zero mean andare independent,

E ef g ¼ 0; E eeT� � ¼ W; wij ¼ 0; wii ¼ r2i ð7:29Þ

The estimate is formed by minimizing

JðV; hÞ ¼ z� hðV; hÞ½ �TW�1 z� hðV; hÞ½ �

¼Xmi¼1

zi � hiðV ; hÞð Þ2r2i

ð7:30Þ

Following the steps in Eqs. (7.16)–(7.18),

hðV; hÞ ¼ h Vk; hk� �þH V �Vk

h �hk

� �ð7:31Þ

where H is a matrix of first partial derivatives of the elements of h with respect tothe components of x evaluated at xk. If

DVDh

� �¼ V �Vk

h �hk

� �; Dz ¼ z� h Vk; hk

� � ð7:32Þ

Then, one step in the iteration is given by

HTW�1HDV

Dh

� �¼ HTW�1Dz

GDV

Dh

� �¼ HTW�1Dz

ð7:33Þ

The gain matrix G in Eq. (7.33) is large and sparse, and Eq. (7.33) is solved withGaussian elimination. Note that H is much like the load flow Jacobian in terms ofsparsity. With organization into active and reactive power, the equivalent of the fastdecoupled load flow can be used to simplify Eq. (7.33). First order the measure-ments into real or active power (sub A) and reactive power (sub R)

z ¼ zAzR

� �; zA ¼ Pkm

Pk

� �; zR ¼

QkmQkVk

24

35 ð7:34Þ

7.3 Static State Estimation 143

And then, write the states as angles followed by voltage magnitudes to form

GAA 00 GRR

� �DhDv

� �¼ HTW�1 DzA

DzR

� �ð7:35Þ

The off-diagonal blocks in Eq. (7.35) are zero under the same assumptions that areused in the fast decoupled load flow, viz. that angle differences are small, thatvoltage magnitudes are near one, and that the X/R of the lines is large. An evenstronger assumption can be made and G computed with angles set equal to zero andvoltage magnitudes set equal to one. Note in this case, G does not have to berecomputed between iterations. The price is inevitably that more iterations arerequired.

Example 7.4 A thirty-bus system is shown in Fig. 7.4. Figure 7.5 shows the errorsin bus voltage angles and magnitudes for a specific case. All real and imaginaryflows and injections are measured with a random error with a sigma of 1% of themagnitude of the complex power, and all voltage magnitudes are measured with asigma of 1% per unit. The results correspond to a specific set of random errorsadded to a load flow solution.

2

7

5

28

3

4

6

89

11

24

10

3022

21

12

14

23

16

26

27

13

2925

20

18

19

15

1

17

Fig. 7.4 Thirty-bus system for Example 7.4


The data for the thirty-bus system along with the state estimation program usedhere is included in a suite of free software available at the Matpower Web sitehttp://www.pserc.cornell.edu/matpower. The example has bus numbering specificto bus 1 being the swing bus connected to buses 2 and 3 as in Fig. 7.4.

7.4 Bad Data Detection

One of the most important functions of a state estimator is to identify and reject baddata [3]. Bad data can arise from problems in the measuring unit or in the com-munication of that data. If it is caused by an uncalibrated measuring instrument, itprobably is modest in size and may even fit within the model of the random errorsin Eq. (7.28). A communication error, however, might produce an immense error.The estimator could be seriously damaged by one huge measurement error. Thesolution, of course, is to eliminate measurements that have such large errors beforeperforming the calculations. This is possible because of the ability to compute themeasurement residuals.

Fig. 7.5 Bus voltage angle and magnitude errors for the 30-bus system in Fig. 7.4

7.3 Static State Estimation 145

http://www.pserc.cornell.edu/matpower

~z ¼ z� h V; h h i

ð7:36Þ

Equation (7.17) gave the covariance of ~y in the linear case. Recognizing the con-nection between A in the linear case and H in the nonlinear case, the covariance of~y is given by

R ¼ Cov ~yð Þ ¼ H HTW�1H� ��1

HT ð7:37Þ

If we normalize the vector of residuals by their covariance matrix

c ¼ ~yTR�1~y ð7:38Þ

we obtain a v2 (chi-squared) random variable. It has E{c} = m, the number ofmeasurements, and is concentrated around its mean. The probability density form = 40 is shown in Fig. 7.6.

The density becomes more concentrated as the number of degrees of freedom(m) increases so that with some confidence, the quantity c can be used to determineif some data fits the model. For m = 40 in the figure, if c was greater than 60 or lessthan 20, it would be reasonable to say something was wrong. For any m, upper andlower bounds on c can be set to initiate further tests of individual residuals.

The individual residuals can be normalized by their variance. One procedurelargest, the normalized residual LNR test, is as follows:

(i) Normalize the residuals by the measurement variances

~zin ¼ zi � hi bV ; h h i=ri ð7:39Þ

where the subscript n indicates a normalized quantity,(ii) Rank the normalized residuals,

Fig. 7.6 Chi-squared density with 40 degrees of freedom


(iii) Eliminate the measurements with residuals above some threshold or simplythe largest,

(iv) Repeat the estimation problem without the measurements in (iii),(v) Check c again, and(vi) Return to (i) if c is too large.

The m functions used in Example 7.4 also use LNR and rejects data with anormalized residual greater than 2.5. In fact, the case shown in Fig. 7.5 rejected asingle measurement.

To show the effect of interacting bad data, consider the system in Fig. 7.7.The affected section is redrawn in Fig. 7.7. If the measurement of the real flow

from bus 3 to 4 is zero (the actual flow is 12.56 MW), the bad data detectionidentifies it and rejects it. The results of the estimate after rejecting the bad data areshown in Fig. 7.8.

If interacting bad data is added in the form of another zero measurement of realflow from bus 3 to bus 4, then the bad data rejection ranks the flow from bus 2 tobus 1 as the first bad data rejects it and then rejects the reactive injection at bus 7 inthe next iteration. The resulting errors are shown in Fig. 7.9. Larger errors in bothvoltage angle and magnitude in the first eight buses are visible.

The issue recognized in [4] is that bad data can reinforce itself and force theLNR procedure to eliminate good data. If the bad data is statistically independent,then the interaction is unlikely.

7.5 State Estimation with Phasor Measurements

The addition of even a few direct measurements of angle to the previous formu-lation has a number of advantages and creates a symmetry in the problem statement.Eq. (7.34) becomes Eq. (7.40) when angle measurements are included.

1

3

2

57

0

4

Fig. 7.7 A portion of thesystem in Fig. 7.4 with onebad measurement

7.4 Bad Data Detection 147

z ¼ zAzR

� �, zA ¼

Pkm

Pk

hk

24

35; zR ¼

QkmQkVk

24

35 ð7:40Þ

The matrix H is modified in an obvious manner, but otherwise, the previousdevelopment applies. The Matpower m files from Example 7.5 have the anglemeasurements included. If the structure of Example 7.5 is maintained and a com-plete set of phasor angle measurements are added with measurement error variancesof 0.02°, the performance shown in Fig. 7.8 results (Fig. 7.10).

0 5 10 15 20 25 30-0.5

0

0.5Voltage Angle Error (deg)

0 5 10 15 20 25 30-2

-1

0

1x 10-3 Voltage Magnitude Error (p.u.)

Fig. 7.8 Errors with one bad measurement of the real power flow from buses 3 to 4 removed

Fig. 7.9 Errors with interacting data. The measured flow from 1 to 2 was incorrectly identified asbad data along with the reactive injection at bus 7


Note the angle errors are smaller as would be expected. A complete set of anglemeasurements would be quite expensive and is used only as an example. We willconsider the selection of the location for a few phasor measurements in the sequel.One issue that must be remembered is that phasor measurements have universaltime as a reference, that is, the sampling instants determine the reference for thePMU data. The conventional state estimator has a particular bus as a reference. Ifthe angle measurements are added without considering the different references, thealgorithm is liable not to converge. The solution is to obtain a common reference.An obvious approach is to measure the angle of the bus that is the reference for theconventional estimator with a PMU.

In addition to measuring bus voltages, PMUs can measure the currents in linesconnected to the bus. The addition of this data further complicates the formulationbecause it creates a tension between rectangular and polar coordinates. The pre-ceding is a polar formulation with the PMU measurement modeled as a measure-ment of voltage angle. The actual measurement is inherently one of the real andimaginary parts of the bus voltage and line currents. In the next section, a linear,rectangular estimator will be formulated using only these linear PMU measure-ments. However, integrating line current measurements into a conventional esti-mator with the systems state expressed in polar coordinates means expressing theline currents as nonlinear functions of the magnitude and angle of the bus voltagesor arguing that the PMU measures the magnitude and angle of the line currents. Ofcourse, the angle and magnitude can be computed from the rectangular parts, butthe issue is the covariance of the measurement errors and the resulting covariance ofthe error is the estimates.

0 5 10 15 20 25 30-0.02

-0.01

0

0.01Voltage Angle Error (deg)

0 5 10 15 20 25 30-4

-2

0

2x 10-4 Voltage Magnitude Error (p.u.)

Fig. 7.10 Errors with a complete set of phasor measurements of angles with a sigma of 0.02°

7.5 State Estimation with Phasor Measurements 149

7.5.1 Linear State Estimation

If an estimate could be formed with only PMU data, then the issues of data scan andtime skew could be eliminated. The PMU data would be time-tagged and the staticassumption removed. We could obtain an estimate of a dynamic system at aninstant in time. The estimate might be obtained a small time after the measurementsbecause of communication delays, but it would be an estimate of the state of thesystem at the instant the measurements were made. There are several issues thatmust be addressed. One is the need for redundancy to eliminate bad data, and theother is how many PMUs are required. At one extreme, if there was a PMU at everybus, we would be measuring the state not estimating it. The loss of a measurementin such a case would only mean we lost information about the bus in question butstill had knowledge of all other buses.

The first observation is that a PMU in a substation could easily have access toline currents in addition to the bus voltage. Sampling both voltages and currents atthe same sampling instants would mean that all phasors would be on the samereference. With a model of the transmission line, the knowledge of the line currentcan be used to compute the voltage at the other end of the line. Measuring linecurrents can extend the voltage measurements to buses where no PMU is installed.With a large number of PMUs, the redundancy issue is addressed. On the otherhand, the smallest number of PMUs needed to indirectly measure all the busvoltages and the optimum PMU location to achieve this has been a subject of anumber of papers [5–7].

To begin the linear formulation, consider the pi equivalent shown in Fig. 7.11[8–10].

Ep

Eq

IpqIqp

2664

3775 ¼

1 00 1

ypq þ yp0 �ypq�ypq ypq þ yq0

2664

3775 Ep

Eq

� �ð7:41Þ

A current measurement–bus incidence matrix is defined in a manner similar to theelement–bus incidence matrix. It has as many rows as measurements of currents andas many columns as there are buses (excluding ground). Two other matrices areneeded as shown in Fig. 7.12. If m is the number of current measurements, n thenumber of lines measured, p the number of buses with voltage measurements, andq the number of buses in the system, then A is an m × q incidence matrix and y isan m × m diagonal matrix of admittances.


pqypq

yp0 yq0

EpEq

IpqIqp

Fig. 7.11 Pi equivalent for a transmission line

1

23 4

0

I1 I2I3

I4

I5

I6y1

y2

y3

y4

Fig. 7.12 An example with six (m) current measurements on four (n) lines, three (p) voltagemeasurements and four (q) and buses

PMU1

PMU2

Fig. 7.13 Two PMUs usedto observe a nine-bus system.The larger circles representPMUs, while the smallercircles are shaded to indicatewhich PMU is responsible forthe current measurement

The matrices are

A ¼

1 �1 0 0�1 1 0 00 1 0 �10 1 �1 00 �1 0 10 0 �1 1

26666664

37777775; and y

y1 0 0 0 0 00 y1 0 0 0 00 0 y3 0 0 00 0 0 y2 0 00 0 0 0 y3 00 0 0 0 0 y4

26666664

37777775

ð7:42Þ


With the shunt branches for each pi section denoted by the subscript 0,

yS ¼

y10 0 0 00 y10 0 00 y30 0 00 y20 0 00 0 0 y300 0 0 y40

26666664

37777775

ð7:43Þ

Then, the measurement vector composed of p voltages and m currents can bewritten as

z ¼ IIyAþ ys

� �Eb½ � ¼ BEB ð7:44Þ

where II is a unit matrix from which rows corresponding to missing bus voltagesare removed. For the example in Fig. 7.12,

yAþ yS ¼

y1 þ y10 �y1 0 0�y1 y1 þ y10 0 00 y3 þ y30 0 �y30 y2 þ y20 �y2 00 �y3 0 y3 þ y300 0 �y4 y4 þ y40

26666664

37777775

ð7:45Þ

or

E1

E2

E4

I1I2I3I4I5I6

26666666666664

37777777777775¼

1 0 0 00 1 0 00 0 0 1


26666666666664

37777777777775

E1

E2

E3

E4

2664

3775 ð7:46Þ

The equations are linear, and Eq. (7.46) is in the form z = B EB

x ¼ BTW�1B� ��1

BTW�1z ¼ Mz ð7:47Þ

Unlike the earlier state estimator, this equation is linear, and hence, no iterationsare needed. As soon as the measurements are obtained, the estimate is obtained bymatrix multiplication. The matrix M that converts the measurements to the stateestimate is constant as long as the bus structure does not change. It can be computed


off-line and stored for real-time use. Under certain conditions of measurementconfiguration, the matrix M becomes real, simplifying the computations evenfurther [8–10].

7.5.2 An Alternative for Including Phasor Measurements

An alternate procedure for incorporating the phasor measurements into a conven-tional estimator is presented in [11]. If the traditional state estimator is in place, thenrather than the substantial changes required for in Sect. 7.5.1, it is possible toconsider the phasor measurements sequentially with the traditional SCADA scan.That is, form the conventional estimate, take it and its covariance matrix, and thenimagine the phasor measurements as in Sect. 7.5.1 as an addition. We can imaginecombining the two with a measurement equation of the form.

E 1ð Þ

S2

� �¼ I

H2

� �E½ � Cov E 1ð Þ

S2

� �� ¼ HT

1W�11 H1 00 W2

� �ð7:48Þ

where Eð1Þ is the estimate from the conventional estimator and S2 is the residualfrom the linear measurements. The old covariance matrix is in polar coordinates,while the new must be converted from rectangular to polar. It can be shown that thesolution to Eq. (7.48) is the same as the solution of

S1S2

� �¼ H1

H2

� �E½ � Cov

S1S2

� �� ¼ W1 0

0 W2

� �ð7:49Þ

which is a nonlinear hybrid estimate handling both the traditional SCADA mea-surements and the phasor measurements.

7.5.3 Incomplete Observability Estimators

One of the disadvantages of traditional state estimators is that at the very minimum,a complete tree of the network must be monitored in order to obtain a state estimate.The phasor-based estimators have the advantage that each measurement can standon its own, and a relatively small number of measurements can be used directly ifthe application requirements could be met. For example, consider the problem ofcontrolling oscillations between two systems separated by great distance. In thiscase, only two measurements would be sufficient to provide a useful feedbacksignal.


But in terms of a state estimator application using only PMUs, the obviousquestion is how many PMUs need to be installed in order to measure the state of thesystem using line currents as discussed in Sect. 7.5.1. Given the number of linesconnecting to each node in a power system is approximately 3, it is clear that aPMU is not necessary at every bus.

The light gray buses in Fig. 7.14 are unobservable to a depth of 1 in the sensethat they are only one bus away from an observable bus. It is possible to imaginehaving so few PMUs that depths of unobservability of 2 or 3 or more wereachieved. Algorithms to find PMU placements to minimize the number of PMUsfor a given depth have been developed [5]. The complete observability case hasbeen approached in a number of ways with a consensus that PMUs are required atapproximately one-third of the buses to obtain complete observability. Someexamples are given in Table 7.2 (Fig. 7.13).

Incidence MatricesThe techniques used in [5] involve enumerating trees and can becometime-consuming as system size increases. The number of possible trees for a1500-bus system is overwhelming. The idea of searching a tree is appealing whenattempting to determine PMU locations which have a certain depth of observability.If only complete observability is of interest, the techniques in [6, 7] are moreefficient. These calculations involve integer programming and the network inci-dence matrix. The incidence matrix approach can also be extended to degrees ofobservability. Consider the network graph shown in Fig. 7.13. The incidencematrix is a square matrix with the dimension of the number of buses. There is a oneon each diagonal and a one in the ij the position if bus i is connected to bus j. Thematrix for the network is given by Eq. (7.44). If we imagine placing a PMU at bus3, for example, we would learn the voltages at buses 2, 3, 4, and 6 which happen tobe the nonzero entries in column 3 of A.

PMU

Indirectlyobserved

Unobserved

Fig. 7.14 Unobservable buses


A ¼

1 1 0 0 0 1 0 01 1 1 0 0 0 1 00 1 1 1 0 1 0 00 0 1 1 1 0 0 00 0 0 1 1 0 0 11 0 1 0 0 1 0 00 1 0 0 0 0 1 10 0 0 0 1 0 1 1

266666666664

377777777775

ð7:50Þ

In [6], the problem of locating the minimum number of PMUs to completelyobserve the network is stated as an integer programming problem in the form.

min fTx

subject toAx� 0; xi ¼ 1 or 0

fT ¼ 1 1 1 � � � 1½ �ð7:51Þ

The form in Eq. (7.45) is the simplest binary integer programming problem.Equality constraints can be added, and conventional injection and flow measure-ments can be accommodated [7]. The depth of observability calculation, however,was not considered in the approach. It is surprisingly easy to include by consideringthe effect of taking powers of the A matrix. In [12], it is stated that

Theorem The ij entry in the nth power of the incidence matrix for any graph ordiagraph is exactly the number of different paths of length n, beginning at vertex iand ending at vertex j.

The proof is by induction and is easy to see in our example. The signum functiony = sgn(x) of A2 is the incidence matrix of another graph which has branches addedto the graph in Fig. 7.15 as shown in Fig. 7.16. The network associated with A4 forthis example has all nodes connected to all other nodes. In a ‘small-world’ context,the example has four degrees of separation, i.e., every node can be reached fromany other node by going through 4 branches.

Table 7.2 Observability results for a few systems [5]

Test system Size (buses/lines) Complete observability Depth 1 Depth 2 Depth 3

IEEE 14 bus (14, 20) 3 2 2 1

IEEE 30 bus (30, 41) 7 4 3 2

IEEE 57 bus (57, 80) 11 9 8 7

System α (270, 326) 90 62 56 45

System β (444, 574) 121 97 83 68


A2 ¼

3 2 2 0 0 1 1 02 4 2 1 0 2 2 12 2 4 2 1 2 1 00 1 2 3 2 1 0 10 0 1 2 3 0 1 22 2 2 1 0 3 0 01 2 1 0 1 0 3 20 1 0 1 2 0 2 3

266666666664

377777777775

ð7:52Þ

The 1–5 path is such an example. In terms of depth of unobservability, a singlePMU at nodes 2, or 3, or 4, or 7 gives a depth of unobservability of 3. The numberof paths is not important in placing PMUs, so the sgn(An) plays the role of A in theinteger programming formulation for the depth of observability problem.

A common problem is to determine the optimum-phased deployment of PMUs.There is usually a limited annual budget to put a certain number in PMUs in placeeach year. The ultimate goal is to have a completely observable system of mea-surements at the end of a multiyear time window. It would be best if the PMUsinstalled in the first year represented a good choice given the small numberinvolved. For example, a set of PMUs that gave some degree of unobservability ineach year with a progression to complete observability at the end of period would

1 2 3 4 5

6

7 8

Fig. 7.15 A network graph

1 2 3 4 5

6

7 8

Fig. 7.16 Graph of sgn(A * A)


be ideal. The incidence matrix approach offers a convenient solution to this prob-lem. Suppose x0 is a solution to the complete observability calculation in Eq. (7.51)and consider

min fTx1subject to sgnðAAÞ½ �x1 � 0;

f � x0ð ÞTx1 ¼ 0

fT ¼ 1 1 1 � � � 1½ �

ð7:53Þ

The use of sgn(AA) assures the solution has depth of observability of one, andthe equality constraint says x1 must be chosen from the locations in x0. For a largesystem, a great deal of computation is involved in solving for x0, while x1 is muchquicker and succeeding solutions are quicker yet. The general form is in Eq. (7.54)

min fTxn

subject to sgn Anþ 1� �� xn � 0

f � xn�1ð ÞTxn ¼ 0

ð7:54Þ

For the 57-bus system studied in [5] and [6], there are 15 ‘zero injection buses’(buses with no generation or load). The different treatments of these buses produceslight variation in the results. The technique in [5] produced the numbers inTable 7.2. If the zero injection buses are simply reduced by network equivalencing,a 42-bus network is created. The 42-bus network is more ‘connected’ than theoriginal because the elimination of a bus that has connections to both buses p and qproduces a new line from p to q. The elimination of 15 buses creates a number ofadditional lines. The repeated application of Eq. (7.54) produces a nested solutionfor the reduced network, but if the reduced network is too different from the originalnetwork, there are still unanswered questions for the original network.

A reasonable approach is to limit the network reductions to obvious situations.An example system has 1443 buses and 1929 branches. There are 104 buses thathave three or fewer branches connected to them. If they are eliminated by networkreduction, the number of branches grows to 2178 and the nested solutions areshown in Table 7.3. If only the five buses with only two branches are removed, thenumber of branches grows to 1940 and the computation takes considerably longer.Again, the results are in Table 7.3.

Table 7.3 Observability results of two versions of the 1246-bus system

# Connections tobuses removed

(Buses/lines) Depth 0 Depth 1 Depth 2 Depth 3 Depth 4 Depth 5

1 or 2 or 3 (1339/2178) 433 218 131 74 54 43

1 or 2 (1458/1940) 445 248 135 83 61 42


7.5.3.1 Criticality and Redundancy in PMU Placement

The concept of optimal PMU placement has been a highly researched topic sincePMUs came to be used commercially. Primarily, there have been two methodsfollowed by power engineers for addressing this issue [13]:

1. Development of a prioritized list of placement sites based essentially onobservability;

2. Placement of PMUs to correctly represent critical dynamics of the system.

The first method is concerned with state estimation and so does not take intoaccount transient and dynamic stability of the system. The second approach doesnot consider complete observability as one of its priorities. The net outcome is thatfor the same system, different ‘optimal’ PMU placement sets are created dependingon the methodology followed. Since it is not possible for any utility to implementall of the schemes that are proposed, either transient/dynamic stability, or observ-ability, or both are compromised. One way to reconcile these two approaches is byinitially placing PMUs on the important buses of the network identified on the basisof system studies/topologies and then computing for the optimal number of PMUsrequired for complete system observability. Called the critical bus-based binaryinteger optimization (CBBBIO) technique for PMU placement [14], the criticalbuses comprise of

1. High-voltage buses,2. High-connectivity buses,3. Buses relevant to transient/dynamic stability perspective,4. Small-signal control buses, etc.

It is a common practice to place PMUs initially on the high-voltage networkbecause from a protection perspective, it is important to monitor this network in realtime. High connectivity is system specific and in practice is indicative of substationswhich have good communication facilities and/or of areas where expansions/installations can be easily carried out. The buses relevant to the transient/dynamicstability of the system are selected based on their relevance in preventing voltagecollapses and minimizing impacts of faults and/or for their participation towarddamping interarea oscillations. Potential small-signal control buses are the buseswhere controllers are placed. This includes locations of flexible AC transmissionsystems (FACTS) devices, energy storage devices (ESDs), HVDC terminals, etc.

Another issue that needs to be addressed when computing the optimal placementof PMUs in a network is the question of redundancy. The norm in the industry is tohave an N − 1 contingency criterion in the system. An N − 1 contingency meansthat even if one PMU or transmission line goes down, the system remainsobservable. But studies have shown that for PMU-based observability under N − 1contingency criterion, a very large number of PMUs will be required [15]. Theconcept of criticality can alleviate this concern because if the critical buses areselected carefully, then by providing redundancy to their measurements alone, the


system can be made secure with much fewer PMUs. A modification of theobservability equation given in Eqs. (7.47)–(7.48) can handle criticality andredundancy as shown below.

If xinit be the set of critical buses where a value of one indicates that the index isa critical bus while a value of zero indicates otherwise, then for complete observ-ability, the desired equation will be

min gTx

subject toAx� f

xTinitxn ¼ nnz xinitð Þð7:54aÞ

In Eq. (7.54a), g ¼ f � wxinit where w is a scalar greater than 1 and nnzðyÞ countsthe number of nonzero elements in y. For computing higher depths of unobserv-ability, the constraints become

sgn Anþ 1� �� xn � f

f � xn�1ð ÞTxn ¼ 0ð7:54bÞ

Table 7.4 gives a comparison of the number of PMUs required at different depths ofunobservability without and with the consideration of high-voltage buses beingcritical buses. In Table 7.4, the traditional method used for computing the numberof buses on which PMUs must be placed for different depths of unobservability isbased on the algorithm developed in Sect. 7.5.3.

Table 7.5 compares the total number of PMUs required for complete observ-ability of the IEEE 118-bus system using the CBBBIO technique and other similaralgorithms. For the CBBBIO technique, redundancy is provided to the measure-ments of only the critical buses which were the high-voltage buses and thehigh-connectivity buses (buses with 8 or more connections). From the table, itbecomes clear that an intelligent choice of critical buses can provide an optimalcost–benefit ratio with respect to PMU placement.

A 283-bus network of the Central American Power Transmission Network and a750-bus North and East Indian Power Transmission Network were also used forcomparison purposes. The details of the systems can be found in [16]. Based on thecriteria set previously, 24 buses were identified to be critical for the CentralAmerican system, whereas 65 buses were found to be critical for the Indian system.The result obtained for these two systems is given in Table 7.6.

7.5.4 Partitioned State Estimation

Early-state estimators were restricted to the highest voltage levels and did notextend to voltages as high as 138 kV in some utilities. The advent of PMU tech-nology made it possible to imagine using the existing conventional estimator


supplemented by PMU measurements in portions of the lower voltage system.Again, because a few PMU measurements are useful, it is not necessary to fullymeasure the low-voltage network. A second more recent problem of the same typeis the seams issue. Two adjoining ISOs with large and elaborate state estimationprograms representing immense investment in people and systems would like tocombine their state estimates. Typically, the people who would actually have to doit have the most reservations. Given the size of an ISO estimator, some reservationsare justified. Combining two 30,000-bus estimators that probably work differentlyand their own quirks is a daunting prospect. Both problems involve the concept ofjoining state estimators.

Table 7.4 Comparison of CBBBIO technique with the traditional method when onlyhigh-voltage buses are considered critical (from [14])

Depths ofunobservability

IEEE 118-bus system IEEE 300-bus system

Notconsideringany bus ascritical

Considering thehighest voltagebuses as critical

Notconsideringany bus ascritical

Considering thehighest voltagebuses as critical

0 32 39 87 96

1 16 23 47 55

2 8 17 34 41

3 7 13 19 29

4 4 13 14 24

5 3 12 9 20

6 2 11 8 18

7 2 7 16

8 2 5 15

9 2 3 15

10 1 3 15

11 3 14

12 2

13 2

14 2

15 2

16 1

Table 7.5 Minimum number of PMUs for complete observability under N − 1 criterion, in theabsence of conventional measurements for IEEE 118-bus system (from [14])

Primary andbackup method[18]

Integer linearprogramming [15]

Local redundancymethod [24]

Proposedapproach

Numberof PMUs

65 64 61 39


Some have suggested that the only optimum answer to these problems is to startover and pool the models and data and buy more computers. If a simpler solutionthat saves money, time, and frustration can be found, it certainly seems worthconsidering. Consider the system in Fig. 7.17. The boundary buses in Fig. 7.17could be the high sides of transformers connecting high- and low-voltage subsys-tems or the buses at the end of the tie-lines between ISOs. An attractive approach tothe problem is to let each system use their existing state estimator but recognize thatthe two estimators do have different references. By including the boundary buses inboth systems, we can estimate the difference between the two references. If NB isthe number of boundary buses and sub 1 and 2 denote estimates from each side,then ϕ is the estimated difference between the two references

/ ¼ 1NB

XNBi¼1

h1i � h2;i

ð7:55Þ

Example 7.6 The 30-bus system is divided into two subsystems as shown inFig. 7.16. Two estimators are formed using the tie-line flows as sources or loads.The reference for system 1 is bus 1 as before, but the reference for system 2 is bus23. The voltage errors are shown in Fig. 7.19. The diamond shows the correctederrors in system 2 after the reference is estimated.

Table 7.6 Number of PMUs required at different depths of unobservability when redundancy ofcritical buses is considered (from [14])

Depths ofunobservability

Central American transmissionsystem (283 buses)

North and East Indian transmissionsystem (750 buses)

Notconsideringany buses ascritical

Consideringcritical buseswith redundancy

Notconsideringany buses ascritical

Consideringcritical buseswith redundancy

0 87 101 161 208

1 61 67 80 131

2 37 47 41 92

3 24 37 20 77

4 13 33 9 74

5 11 32 7

6 9 30 3

7 8 30 3

8 6 29 2

9 5 1

10 5

11 5

12 5

13 4

14 1


Nonlinear Residual Minimization MethodA study of alternative approaches to integrating adjoining estimators is in [17]where the solution that combines small computational burden, accuracy, androbustness was chosen as the nonlinear residual minimization. Let the subscriptC denote calculated quantities given and N be the total number of tie-lines betweenthe two systems. The calculated quantities for a line from node a to node b aregiven by Eqs. (7.56) and (7.57). The method (NLRM) in which the angle difference

System 2

System 1

Boundary Buses

Fig. 7.17 Two systems with boundary buses

Fig. 7.18 Thirty-bus system divided into two subsystems


between systems, ϕ, was chosen to minimize J(ϕ) in Eq. (7.58). In Eq. (7.58), thesubscript m denotes measured quantities

PC ¼ Vaj j2Zab

cos hzð Þ � Vaj j Vbj jZab

cos hz þ dab � /ð Þ ð7:56Þ

QC ¼ Vaj j2Zab

sin hzð Þþ Vaj j2Bab

2� Vaj j Vbj j

Zabsin hz þ dab � /ð Þ ð7:57Þ

J uð Þ ¼

Pm1�PC1rP1

Qm1�QC1rQ1

Pm2�PC2rP2

Qm2�QC2rQ2

..

.

PmN�PCNrPj

QmN�QCNrQN

��

��

ð7:58Þ

Using Eq. (7.52) with g ¼ hz þ dab

cos g� /ð Þ ¼ cos g cos/þ sin g sin/

sin g� /ð Þ ¼ sin g cos/� cos g sin/ð7:59Þ

Fig. 7.19 Voltage errors before and after correction


Then, the objective function is given by Eq. (7.60)

Jð/Þ ¼ Xþ Y cos /þZ sin /k k2 ð7:60Þ

Since X, Y, and Z are constants, Eq. (7.54) is equivalent to Eq. (7.55) where A, B,C, D, and E are constants

Jð/Þ ¼ A cos /þB sin /þC cos / sin /þD cos2 /þE sin2 / ð7:61Þ

The minimization of Eq. (7.61) involves taking the first derivative with respect to/, equating it to zero and using Newton’s method to solve the resulting scalarnonlinear equation.

7.6 Calibration

Calibration as an issue in state estimation predates phasor measurements and is stillan issue with phasor measurements. In the previous sections, we have modeled themeasurement error as a simple additive error. Our numerical examples have usedrather small standard deviations expressed as a percent of the measured quantity;that is, if we measured a flow of 10 MWs, we added a random error with a standarddeviation of say 1% of the 10 MW to both the real and imaginary measurements. Itis assumed in our previous development that the error has zero mean and if theestimation process is repeated every few minutes the error is a particular mea-surement will be independent from measurement to measurement and have thesame statistical description. Two measurements of a complex flow are as shown inFig. 7.18a. It is assumed that one measurement is approximately half the other inmagnitude with a slightly different angle. A small circle of error about the correctvalue with a radius proportional to the size of the measurement is drawn to rep-resent the error. It is a circle in the complex plane because we are assuming thesame size error is added to the real and imaginary measurements. In Fig. 7.18, thesolid lines are the true values and the dashed lines are the hypothetical measure-ments (Fig. 7.19).

The problem lies with the transducers, the current, and the potential trans-formers. They contribute to a systematic error as shown in Fig. 7.18b. If the truecurrent is I, the measured value is γ I where γ is always the same complex number:complex because there are both a magnitude and a phase shift error. While γ is notknown when a single estimate is formed, it has been suggested that over time, theseunknown constants could be learned, i.e., that the measurements could be cali-brated. The circles in Fig. 7.20 were large enough for early analog measurementsthat many solved the problem by just increasing the standard deviation to take thesystematic errors into consideration.


7.6.1 Calibration with Positive Sequence Measurements

The current and voltage transducers exist on individual phases of a three-phasesystem. Each of these transducers has an unknown ratio correction factor (RCF).The estimation of these RCFs must be based on phasor measurements on individualphases. However, some of the earlier research [18] was conducted using positivesequence measurements for the calibration. This approach assumed that althoughthe actual RCFs are in phase quantities, there is an equivalent RCF which may beassociated with the positive sequence measurements.

Consider phasors Xa,b,c belonging to phases, X being either the current or thevoltage phasor. Additional subscript ‘m’ indicates the measured values of the pha-sors, while a symbol without the ‘m’ subscript denotes the actual value of themeasured signal (which is of course unknown). The relationship between themeasured and true phasors is shown in Eq. (7.62) where γa,b,c represent the RCFs ofthe corresponding instrument transformers:

Xm;a ¼ caXa

Xm;b ¼ cbXb

Xm;c ¼ ccXc

ð7:62Þ

Recall that the positive sequence phasor X1 and its measured value Xm,1 aregiven by (α being the usual 120° phase shift operator)

X1 ¼ Xa þ aXb þ a2Xc� �

=3

Xm;1 ¼ Xm;a þ aXm;b þ a2Xm:c� �

=3ð7:63Þ

If the power system may be assumed to have balanced voltages and currents, thephase quantities can be expressed in terms of the positive sequence quantity:

Imaginary

Real RealReal(a) Electronic error model (b) CT and PT error (c) Combination

Imaginary

Fig. 7.20 Error models for the calibration problem

7.6 Calibration 165

Xa ¼ X1

Xb ¼ a2X1

Xc ¼ aX1

ð7:64Þ

so that Eq. (7.64) becomes

X1 ¼ Xa

Xm;1 ¼ caX1 þ cbX1 þ ccX1ð Þ=3 ¼ X1 ca þ cb þ ccð Þ=3 ð7:65Þ

The effective RCF for the positive sequence measurement becomes

cm ¼ ca þ cb þ ccð Þ=3 ð7:66Þ

The only approximation used here is that the phasors represent a balanced system.As long as the estimation is done with normal loading conditions and not withessentially unbalanced conditions resulting from unbalanced faults, this should besufficiently accurate. Small unbalances produce second-order corrections inEq. (7.66), but are not significant. In the rest of this section, all the measurementsand phasors are positive sequence quantities, and the RCFs for each measurementare for positive sequence phasors as given in Eq. (7.66).

Mathematically, the model has changed from Eq. (7.22) to

zi ¼ cihiðV ; hÞþ e ð7:67Þ

and the objective is to estimate the γ’s . Since every CT and VT contribute a γ, it isclear that in the worst case, the γ’s cannot be estimated from a single set ofmeasurements. The fact that the γ’s are constant must be exploited to calibrate themeasurements over time. More complicated models than Eq. (7.66) are alsorequired. Calibration of a power flow measurement, for example, would require aquadratic model since both a CT and a VT are involved. If the number of mea-surements to be calibrated is small enough, the unknown γ’s can be added to thestate, the Jacobian adjusted accordingly, and the procedure in Sect. 7.3 followedwith an augmented state. However, most calibration techniques involve the use ofmultiple load flow scans and assume that the RCFs of CTs and VTs are constantwhile the data is collected. In [15], it is reported that data was used over a period ofseveral days to perform a calibration.

Another approach is to calibrate CTs and VTs separately for different systemconditions [18]. For a lightly loaded system, currents are small and systematic CTRCFs make little contribution. If some voltage measurements are known to be good(using precision voltage transformers), then unbiased current measurements and afew unbiased voltage measurements are sufficient to estimate the system state andthe γ’s for the remaining voltages. A linear estimator using only PMU measure-ments is considered in [18]. One approach is to augment system state to include thecomplex bus voltages and the γ’s for the uncalibrated voltage measurements.


If there are Nb buses and NL lines, then a complete set of linear measurementswould include Nb complex voltages and 2NL complex current measurements. If Nt

of the voltages are already calibrated, then there are (Nb − Nt) γ’s and Nb complexvoltages to estimate with Nb + 2NL measurements. The problem is nonlinearbecause γ’s appear as a multiplier of the currents and voltages. Using the system inSect. 7.5.1 as an example, if we assume the voltage at bus one is free of RCF errorbut E2 and E4 have RCF errors, then Eq. (7.46) can be written as

E1

E2

E4

I1I2I3I4I5I6

26666666666664

37777777777775¼

1 0 0 00 c2 0 00 0 0 c4


26666666666664

37777777777775

E1

E2

E3

E4

2664

3775 ð7:68Þ

The nonlinear estimation problem converges in two iterations with the matrixM in the form of Eq. 7.69. The β and δ in Eq. (7.69) express a dependence of γ2 andγ4 on the residuals in E2 and E4. The estimation of voltage γ’s under light load doesnot require accumulation of many data points. One light load measurement issufficient to calibrate the voltage measurements given that some number of voltages(at least one) are already calibrated. Clearly, more measurements with a lightlyloaded case will improve the estimate of the voltage RCFs. The measurement vectorz is the same as in the lightly loaded case.

Given the voltage multipliers have been estimated, the heavy load case can beexamined to estimate the CT RCFs. The issue that must be addressed is that thenumber of measurements is not really adequate to estimate the CT

M ¼

x b 0 x x x x x xx 0 d x x x x x xx 0 0 x x x x x xx 0 0 x x x x x xx 0 0 x x x x x xx 0 0 x x x x x xx 0 0 x x x x x x

2666666664

3777777775

ð7:69Þ

In the example, if γ2 and γ4 are assumed to be known (from the light loadestimation process) but γ’s are introduced for the current measurements, thenEq. (7.68) becomes Eq. (7.70) for the heavily loaded case.

7.6 Calibration 167

z ¼

1 0 0 00 c2 0 00 0 0 c4

c1 y1 þ y10ð Þ �c1 y1ð Þ 0 0�c2 y1ð Þ c2 y1 þ y10ð Þ 0 0

0 c3 y3 þ y30ð Þ 0 �c3 y3ð Þ0 c4 y2 þ y20ð Þ �c4 y2ð Þ 00 �c5 y3ð Þ 0 c5 y3 þ y30ð Þ0 0 �c6 y4ð Þ c6 y4 þ y40ð Þ

26666666666664

37777777777775E ð7:70Þ

Now, there are 10 states to be estimated: the six current RCFs and the four voltages.If a second set of measurements are taken later and combined with the first 9measurements, then we would have 18 measurements and 14 unknowns (4 voltagesfrom the first measurement, 4 more voltages from the second, and 6 unknown γ’s).

The Jacobian for two measurements is given in Eq. (7.71) where

H ¼N 0 0CY 0 diag YEð1Þ� �0 N 00 CY diag YEð2Þ� �

2664

3775 for x ¼

Eð1Þ

Eð2Þ

c

24

35 ð7:71Þ

C is a diagonal matrix of the current RCFs, Y is the matrix Eq. (7.45), E(1) and E(2)

are voltage estimates corresponding to the two load flow scans, and N is thediagonal matrix in Eq. (7.72).

N ¼1 0 00 c2 00 0 c4

24

35 ð7:72Þ

7.6.2 Calibration with Phase Measurements

Zhongyu Wu

Most utilities in the world are installing PMUs to obtain the full observation ofhigh-voltage power systems. Reference [19] presented a fully observed 500-kVsubsystem that has PMU metering voltages and currents at both ends of eachtransmission line. Full PMU observation is required to calibrate three-phase voltageand current transformers. It requires that at least one set of three-phase voltagetransformers should be precalibrated and have negligible ratio correction factors(RCF). PMUs utilize signals from three-phase current and voltage transformers (CTand VT) as inputs to obtain three-phase measurements. Calibration of phasormeasurements is carried out to estimate the systematic errors introduced by CT andVT. The relationship between the true and measured three-phase phasors is shown


in Eq. (7.73) where γabc represents the RCF vector in three phases of the corre-sponding instrument transformers, diag(γabc) is a square matrix with RCF vector onits main diagonal, and εabc represents the vector of random noise in three phases(X representing voltages or currents):

Xabc ¼ diag cabcð ÞXm;abc þ eabc ð7:73Þ

Ratio correction factor of individual phase current and voltage transformers isassumed to be constant over a period during which multiple load flow scans aremade. Three-phase pi equivalent model of a transmission line is shown in Fig. 7.19.The parameters of a π-type transmission line model are given in Eq. (7.74), whereZabcpq is the symmetric three-phase impedance matrix of the line pq, Yabc

pq is the

symmetric three-phase admittance matrix of the line pq, and Babcp0 is the symmetric

three-phase susceptance matrix at bus p end of the line. The three-phase suscep-tance at the two ends of a line is assumed to be the same, that is, Babc

p0 ¼ Babcq0

(Fig. 7.21).

Zabcpq ¼

Zapq Zab

pq Zacpq

Zabpq Zb

pq Zbcpq

Zacpq Zbc

pq Zcpq

2664

3775; Yabc

pq ¼ Zabcpq

�1¼

Yapq Yab

pq Yacpq

Yabpq Yb

pq Ybcpq

Yacpq Ybc

pq Ycpq

2664

3775

Babcp0 ¼

Bap0 þBab

p0 þBacp0 Bab

p0 Bacp0

Babp0 Bb

p0 þBabp0 þBbc

p0 Bbcp0

Bacp0 Bbc

p0 Bcp0 þBac

p0 þBbcp0

2664

3775

ð7:74Þ

Fig. 7.21 Pi equivalent for a three-phase transmission line

7.6 Calibration 169

The relationship between true current and voltage measurements for the systemmodel of Fig. 7.19 is shown in Eq. (7.75), where Eabc

p is the true three-phase

voltage vector at bus p, Eabcq is the true three-phase voltage vector at bus q, Iabcpq is

the true three-phase current vector at bus p, and Iabcqp is the true three-phase currentvector at bus q.

0 ¼ Yabcpq þBabc

p

Eabcp � Yabc

pq Eabcq � Iabcpq

0 ¼ �Yabcpq Eabc

p þ Yabcpq þBabc

p

Eabcq � Iabcqp

ð7:75Þ

Considering the error model in Eq. (7.73), the relationship between measuredvalues of three-phase voltage and current phasor measurements is given byEq. (7.76), where cabcv;p is the vector of three-phase RCFs of voltage measurements at

bus p and cabci;pq is the vector of three-phase RCFs of current measurements. All RCFs

cabc� �

are unknowns needed to be estimated in this section.

0 ¼

Yabcpq þBabc

p �Yabcpq . . . �I3�3

..

. ...

. . . ...

�Yabcpq Yabc

pq þBabcp . . . �I3�3

..

. ...

. . . ...

2666664

3777775 diag Eabc

m . . .Iabcm

� ��

cabce;p

cabce;q

..

.

cabci;pq

..

.

cabci;qp

..

.

26666666666664

37777777777775þ e

ð7:76Þ

In a system of n buses and m branches, Eq. (7.76) represents 6m algebraicequations with [3n + 6m] unknowns: 3n for RCFs of each voltage: cabce and 6m forRCFs of each line current measured at both ends: cabci .

Ratio correction factors are assumed to be unchanging over the period of mul-tiple load scans. A total of ‘s’ load scans provide 6ms algebraic equations with3n + 6m unknowns, and Eq. (7.76) becomes Eq. (7.77). AY + B is a ð6mÞ � ð3nÞmatrix, diagðEÞ is the ð3nÞ � ð3nÞ diagonal matrix with 3n voltage measurementson its main diagonal, diagðIÞ is a ð6mÞ � ð6mÞ diagonal matrix with 6m currentmeasurements on its main diagonal, and subscripts ‘1; 2; . . .; s’ indicate a matrixwith ‘s’ sets of blocks AY þBð Þdiag Eð Þ � diagðIÞ½ � in a row. The size of first matrixon the right-hand side in Eq. (7.77){[(AY + B)diag(E) − diag(I)]} is ð6msÞ�ð3nþ 6mÞ. The second matrix on the right-hand side {γ} is an unknown matrixcontaining the RCFs of 3n + 6m measurements.


0 ¼ AY þBð ÞdiagðEÞ � diagðIÞ½ �1;2;...s� c½ �e ð7:77Þ

To estimate RCF vector with ‘s’ sets of load scenarios, at least one set ofthree-phase VT needs to be a high-precision (precalibrated) devices with RCFsequal to 1. It is assumed that the voltage transformers installed at bus p arehigh-precision devices: cabcv;p ¼ 1; 1; 1½ � þ eabc. With these known RCFs, thenumber of unknowns reduces to 3n + 6m − 3, and Eq. (7.77) changes to Eq. (7.78)

Z ¼ H � cþ e ð7:78Þ

The matrix H of size 6msð Þ � 3nþ 6m� 3ð Þ; is the first matrix in Eq. (7.79). Itexcludes RCFs of voltage transformers at bus ‘p’, and the unknown γ is of size3nþ 6m� 3ð Þ � 1 of unknown RCF vector of n − 1 voltage transformers and m-branch flows. Z is a 6msð Þ � 1 matrix in Eq. (7.79), where the subscript ‘1, …, s’means the measurements at different load scenarios.

Z ¼

� Yabcpq þBabc

p

Eabcp;1

Yabcpq Eabc

p;1Zeros 6m� 6ð Þ

..

.

� Yabcpq þBabc

p

Eabcp;s

Yabcpq Eabc

p;sZeros 6m� 6ð Þ

2666666666664

3777777777775

ð7:79Þ

c is calculated by linear least squares solution of Eq. (7.80).

c ¼ HTH� ��1

HT � Z ð7:80Þ

Accuracy of the result is impacted by two factors: (1) errors in the precalibratedvoltage measurements at bus p and (2) the random noise in PMU measurements. Toreduce the error induced by precalibrated measurements, more than one set ofprecise voltage transformers may be used to produce more reliable Z matrix.Utilizing more load scans would also help alleviate the effect of random noise. Thisis illustrated by Example 7.7.

Example 7.7 Consider a three bus, two branch power system. Buses are 1, 2, and 3,and lines are 1–2 and 2–3. The assumed line parameters of the system are givenbelow:

Z12 ¼0:001þ j0:0i 0:0012þ j0:006 0:001þ j0:005

0:0012þ j0:006 0:0012þ j0:01i 0:002þ j0:0010:001þ j0:005 0:002þ j0:001 0:0015þ j0:015

24

35

7.6 Calibration 171

Z23 ¼0:002þ j0:02 0:001þ j0:003 0:001þ j0:0050:001þ j0:003 0:0022þ j0:022 0:002þ j0:0020:001þ j0:005 0:002þ j0:002 0:0025þ j0:025

24

35

B1 ¼j0:5 �j0:05 �j0:01

�j0:05 j0:48 �j0:015�0:01i �0:015i 0:55i

24

35

B2 ¼j0:2 �j0:01 �j0:007

�j0:01 j0:18 �j0:004�j0:007 �j0:004 j0:22

24

35

Ratio correction factors are randomly generated with magnitude in the range of[0.94, 1.06] and angle in the range of [−4°, 4°]. The chosen RCFs of three-phasevoltage measurements at buses 1, 2, and 3 are as follows:

ct e;1 ¼ 1; 1; 1½ �; These voltage transformers are assumed to be high-precisiondevices.

ct e;2 ¼ 0:98� j0:05; 1:03þ j0:005; 0:97þ j0:01½ �ct e;3 ¼ 1:024þ j0:002; 0:99� j0:001; 0:96� j0:003½ �

The chosen RCFs of three-phase current measurements at each end of the twolines are as follows:

ct i;12 ¼ 1:01� j0:05; 0:96þ j0:002; 0:996� j0:003½ �;ct i;23 ¼ 1:012� j0:002; 1:016þ j0:001; 0:99� j0:0011½ �cti;21 ¼ 0:98þ j0:1; 0:99þ j0:02; 1:02� j0:001½ �cti;32 ¼ 0:998� j0:001; 0:981� j0:009; 1:012� j0:001½ �

Four cases are compared to study the impact of different magnitudes of randomnoise in PMU measurements.

Case 1: In this case, 58 load scans are used; random noise eabc in Eq. (7.77) isassumed to be in the range of [−1%, 1%] of the measurements. The high-precisionvoltage transformer at bus 1 is assumed to have random errors in the range of[−0.1%, 0.1%].Case 2: Six load scans are used; random noises eabc in Eq. (7.77) are in the range of[−1%, 1%] of the measurements. The high-precision voltage transformer at bus 1 isassumed to have random errors in the range of [−0.1%, 0.1%].Case 3: Fifty-eight load scans are used; random noises eabc in Eq. (7.77) are in therange of [−1%, 1%] of the measurements. The high-precision voltage transformer atbus 1 is assumed to have random errors in the range of [−0.001%, 0.001%].


Case 4: Fifty-eight load scans are used; random noises eabc in Eq. (7.65) are in therange of [−0.1%, 0.1%] of the measurements. The high-precision voltage trans-former at bus 1 is assumed to have random errors in the range of [−0.001%,0.001%].

MATLAB simulation results for the 4 cases:

1. The differences between estimated RCFs and true RCFs of voltage transformers(magnitude in %, angle in degree) are as follows:

ct � cestj j Case 1 Case 2 Case 3 Case 4

Magn (%) Angle Magn Angle Magn Angle Magn Angle

Max 0.13 0.039 0.831 0.107 0.134 0.0391 0.013 0.0026

Mean 0.064 0.013 0.419 0.036 0.066 0.012 0.007 9.8e−4

Min 0.029 0.002 0.227 0.001 0.03 3.5e−4 0.004 9.5e−5

2. The differences between estimated RCFs and true RCFs of current transformersare as follows:

ct � cestj j Case 1 Case 2 Case 3 Case 4


Max 0.847 1.614 2.024 4.7534 0.7715 1.6143 0.0652 0.1582

Mean 0.284 0.801 1.074 3.4471 0.2854 0.7794 0.0254 0.0777

Min 0.0292 0.475 0.042 2.7982 0.0718 0.3656 0.0039 0.0364

3. The differences between estimated RCFs of three-phase voltage transformersand true RCFs of voltage transformers are as follows:

Et�EestEt

Case 1 Case 2 Case 3 Case 4


Max 1.079 0.041 1.318 0.1071 1.084 0.041 0.108 0.003

Mean 0.477 0.0130 0.48 0.038 0.477 0.012 0.048 9.88e−4

Min 0.003 2.6e−5 0.018 5.2e−4 0.007 7.4e−6 7.65e−4 6.98e−6

4. The differences between estimated RCFs of three-phase current transformersand true RCFs of current transformers are as follows:

It�IestIt

Case 1 Case 2 Case 3 Case 4


Max 1.823 1.615 2.681 4.754 1.749 1.615 0.163 0.583

Mean 0.578 0.801 1.186 3.448 0.566 0.779 0.055 0.078

Min 0.007 0.474 0.092 2.798 0.004 0.363 2.65e−4 0.036

7.6 Calibration 173

Case 1 and Case 2 have the same random noise of phase transformer RCFs andthe same errors of precalibrated voltage transformers at bus 1. However, case 1 uses58 load scans, while Case 2 only uses 6 load scans. The results of Case 1 are moreaccurate than those of Case 2 in all the four tables above. It demonstrates that moreload scans can lower random noise impacts on calibration accuracy.

Random noises in PMU measurements are consistent in Case 1 and Case 3, aswell as the number of load scans. The only difference is that the errors in precal-ibrated RCFs of voltage transformers at bus 1 in Case 3 are 100 times smaller thanthose in Case 1. However, the accuracy levels of results in Case 1 and Case 3 areonly slightly different, and this indicates that the impacts of errors in precalibratedvoltage measurements are minimal when the random noise is the dominant effect.

On the other hand, when the precalibrated voltage measurements have the sameerrors, random measurement noise in Case 4 is 10 times smaller than that in Case 3.The results in the four tables above prove that random measurement noise has directimpact on the accuracy of the calibration results.

7.6.3 Simultaneous Calibration of Line Parametersand Transducers

Zhongyu Wu

Most of the calibration research [19–21] is based on the assumption that eithersystem model of line parameters is accurate to estimate CT and VT errors, or CTand VT measurements are accurate enough to estimate line parameters.Section 7.6.2 introduces a linear estimation methodology to calibrate phase mea-surements, assuming the values of three-phase impedance and susceptance areknown. In reality, transmission line parameters may also have errors. Traditionaltransmission line parameters are determined by the geometry of the transmissionline conductors, their electrical characteristics, and transmission tower configura-tions. It is recognized that line parameters may be in error because of changingweather conditions or simply due to data entry errors. The errors in line parametersaffect the accuracy of CT and VT calibration, state estimation, power systemmonitoring, and all applications requiring line parameters as input.

This section introduces a methodology to calibrate phase measurements whentaking system line parameters as unknowns, in other words to calibrate lineparameters, CTs, and VTs simultaneously. Similarly, the methodology also requiresthat:

1. The power system has sufficient number of PMUs to provide completeobservability and that the transmission line currents are measured at each end ofall the transmission line.


2. At least one substation has its three-phase voltage transformers, current trans-formers, and connected PMU precalibrated, providing negligible RCFs in cor-responding CTs and VTs: Xabc ¼ diag cabcð ÞXm;abc þ e with cabc ¼ ½1; 1; 1�.A number of sets of measurements under multiple load scans are needed to

support enough redundancy for estimation. It is assumed that (1) RCFs are constantduring the period of multiple load scans collected and (2) line parameters are alsoconstant over that period.

Equation (7.81) shows the relationship after replacing the currents and voltagesin Eq. (7.75) with the error models in Eq. (7.73). The goal of this section is tocalculate the values of Zabc; Yabc; cabce and cabci .

0 ¼ Iþ Zabcpq Yabc

q

cabcep Eabc

mp � cabceq Eabcmq � Zabc

pq cabcipq Iabcmpq

0 ¼ �cabcep Eabcmp þ Iþ Zabc

pq Yabcq

cabceq Eabc

mq � Zabcpq cabciqp I

abcmqp

8<: ð7:81Þ

To solve for all these unknowns, the principal goal is to procure extra rela-tionship among the unknowns. Multiply the first equation in Eq. (7.81) by

Iþ Zabcpq Yabc

q

and get Eq. (7.82).

0 ¼ Iþ Zabcpq Yabc

q

2cabcep Eabc

mp � Iþ Zabcpq Yabc

q

cabceq Eabc

mq

� Iþ Zabcpq Yabc

q

Zabcpq cabcipq I

abcmpq

0 ¼ �cabcep Eabcmp þ Iþ Zabc

pq Yabcq

cabceq Eabc

mq � Zabcpq cabciqp I

abcmqp

8>>><>>>:

ð7:82Þ

Substitute elements in Eq. (7.82) using symbols X1, X2, X3, X4, and X6 as definedin Eqs. (7.83) and (7.84) leading to Eq. (7.86)

X6 ¼ Iþ Zabcpq Yabc

q

ð7:83Þ

X1 ¼ X6cabceq

X2 ¼ X26c

abcep

X3 ¼ X6Zabcpq cabcipq

X4 ¼ Zabcpq cabciqp

8>>><>>>:

ð7:84Þ

0 ¼ X2Vabcp � X1Vabc

q � X3Iabcpq

0 ¼ �cabcep Vabcp þX1Vabc

q � X4Iabcqp

(ð7:85Þ

Equation (7.85) may be represented as a nonlinear function f1 as shown inEq. (7.86)

7.6 Calibration 175

f1 X1;X2;X3;X4; cabcep

¼ 0 ð7:86Þ

The first two expressions in Eq. (7.84) may be rewritten as

X�11 ¼ cabceq

�1X�16

X2 cabcep

�1¼ X2

6

By premultiplying the second equation with the first,

X�11 X2c

abc�1ep ¼ cabc�1

eq X6

The right-hand expression can be written explicitly as in Eq. (7.87):

cabc�1eq X6 ¼

X16

caeq

X26

caeq

X36

caeqX46

cbeq

X56

cbeq

X66

cbeqX76

cceq

X86

cceq

X96

cceq

26664

37775 ¼

X11

caeqð Þ^2X21

caeqcbeq

X31

caeqcceq

X41

caeqcbeq

X51

cbeqð Þ^2X61

cbeqcceq

X71

caeqcceq

X81

cbeqcceq

X91

cceqð Þ^2

266664

377775 ð7:87Þ

Using the following expressions, the left-hand term in Eq. (7.87) can berewritten as Eq. (7.88).

N ¼ X11 X5

1X91 � X6

1X81

� �þX21 X6

1X71 � X4

1X91

� �þX31 X4

1X81 � X5

1X71

� �T1 ¼ X1

2 X51X

91 � X6

1X81

� �þX42 X3

1X81 � X2

1X91

� �þX72 X2

1X61 � X3

1X51

� �T2 ¼ X2

2 X51X

91 � X6

1X81

� �þX52 X3

1X81 � X2

1X91

� �þX82 X2

1X61 � X3

1X51

� �T3 ¼ X3

2 X51X

91 � X6

1X81

� �þX62 X3

1X81 � X2

1X91

� �þX92 X2

1X61 � X3

1X51

� �T4 ¼ X1

2 X61X

71 � X4

1X91

� �þX42 X1

1X91 � X3

1X71

� �þX72 X2

1X61 � X3

1X51

� �T5 ¼ X2

2 X61X

71 � X4

1X91

� �þX52 X1

1X91 � X3

1X71

� �þX82 X2

1X61 � X3

1X51

� �T6 ¼ X3

2 X61X

71 � X4

1X91

� �þX62 X1

1X91 � X3

1X71

� �þX92 X2

1X61 � X3

1X51

� �T7 ¼ X1

2 X41X

81 � X5

1X71

� �þX42 X2

1X71 � X1

1X81

� �þX72 X1

1X51 � X2

1X41

� �T8 ¼ X2

2 X41X

81 � X5

1X71

� �þX52 X2

1X71 � X1

1X81

� �þX82 X1

1X51 � X2

1X41

� �T9 ¼ X3

2 X41X

81 � X5

1X71

� �þX62 X2

1X71 � X1

1X81

� �þX92 X1

1X51 � X2

1X41

� �

X�11 X2c

abc�1ep ¼

T1Ncaep

T2Ncbep

T3Nccep

T4Ncaep

T5Ncbep

T6Nccep

T7Ncaep

T8Ncbep

T9Nccep

2664

3775 ð7:88Þ


Equations (7.87) and (7.88) lead to the functional expression

f2 X1;X2; cabcep ; cabceq

¼ 0 ð7:89Þ

The details of f2 are shown in Eq. (7.90). From the physical perspective, it isclear that the values of RCFs must be nonnegative. Consequently, the negativesquare roots are ignored in the first three equations in Eq. (7.90).

0 ¼ caeq �ffiffiffiffiffiffiffiffiffiffiffiNcaepX

11

T1

q0 ¼ cbeq �

ffiffiffiffiffiffiffiffiffiffiffiNcbepX

51

T5

q0 ¼ cceq �

ffiffiffiffiffiffiffiffiffiffiffiNccepX

91

T9

q0 ¼ X2

1Ncbep � T2 caeqc

beq

0 ¼ X31Nc

cep � T3 caeqc

ceq

0 ¼ X41Nc

aep � T4 caeqc

beq

0 ¼ X61Nc

cep � T6Kb

vqKcvq

0 ¼ X71Nc

aep � T7Ka

vqKcvq

0 ¼ X81Nc

bep � T8Kb

vqKcvq

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

ð7:90Þ

From the expressions in Eq. (7.84), additional relationships lead to Eqs. (7.91)–(7.93).

f3 X1;X6; cabceq

¼ X1 � X6c

abceq ¼ 0 ð7:91Þ

f4 X3;X6; Zabcpq ; cabcipq

¼ X3 � X6Z

abcpq cabcipq ¼ 0 ð7:92Þ

f5 X4; Zabcpq ; cabciqp

¼ X4 � Zabc

pq cabciqp ¼ 0 ð7:93Þ

Equation (7.94) represents the sum of all currents (k branches) connected to bus q,which by Kirchhoff’s law must add to 0.

f6 cabciq1 ; . . .; cabciqn

¼ cabciq1 I

abcq1 þ � � � þ cabciqk I

abcqk ¼ 0 ð7:94Þ

Nonlinear Eq. (7.95) is finally obtained from Eqs. (7.87), (7.88), and (7.91)through (7.93). In an n-bus m-branch system, Eq. (7.95) contains 57m + 3n − 6unknowns: Each branch has five 9-element matrixes (X1 to X4, and X6), one6-element symmetric matrix of three-phase impedances, and 6 RCFs of three-phasecurrent measurements in two directions; each bus has 3 RCFs of three-phasevoltage measurements and excludes the 6 RCFs of precalibrated voltage mea-surements and current measurements.

7.6 Calibration 177

f ðxÞ ¼ f1; f2; f3; f4; f5; f6½ � ð7:95Þ

Multiple load scans provide redundant measurements for estimation. With ‘s’sets of load scans, f1–f5 contain 6msþ 36m algebraic equations. When ‘t’ buseshave two or more branches connected, f6 provides additional 3ts algebraic equa-tions. Gauss–Newton method with correction is applied to solve the nonlinearestimation having 6msþ 36mþ 3ts equations with 57mþ 3n� 6 unknowns.

Gauss–Newton Method with CorrectionGiven 6msþ 45mþ 3ts nonlinear equations with 57mþ 3n� 6 unknowns, the goalis to find x to minimize the residual of the equations in a least squares sense shownin Eq. (7.96).

12

fxk k22! min; which is@

@x12

fxk k22� �

¼ 0 ð7:96ÞLinearizing fx at point xk with a correction step of D,

f xð Þ � f xkð Þþ @f xkð Þ@x

Dþ 12DT @

2f xkð Þ@x@x

D ð7:97Þ

Second-order terms in Eq. (7.97) may be assumed to be small.

f xð Þ � f xkð Þþ @f xkð Þ@x

D ð7:98Þ

Jacobian matrix J of partial derivatives in Eq. (7.98) is a 6msþ 36mþ 3tsð Þ �57mþ 3n� 6ð Þ sparse matrix.

J ¼ @f xkð Þ@x

¼

@f1@x1

@f1@x2

@f1@x3

@f1@x4

@f1@x6

@f1@cabcep

0 0 0 0@f2@x1

@f2@x2

0 0 0 @f2@cabcep

@f2@cabceq

0 0 0@f3@x1

0 0 0 @f3@x6

0 @f3@cabceq

0 0 0

0 0 @f4@x3

0 @f4@x6

0 0 @f4@cabcipq

0 @f4@z

0 0 0 @f5@x4

0 0 0 0 @f5@cabciqp

@f5@z

0 0 0 0 0 0 0 @f6@cabcinq

@f6@cabciqn

0

2666666666664

3777777777775

ð7:99Þ

When matrix J xkð ÞTJðxkÞ is non-singular at xk, matrix J xkð ÞTJðxkÞ is positivedefinite and can be inverted. However, if at point xk the Jacobian is not rankdeficient, which means the locally optimal minimizing direction is not the best forglobal optimization, we need to introduce a Gauss–Newton correction to make theminimizing direction adjusted between the negative gradient and the local optimaldirection. In Eq. (7.100), k is the Gauss–Newton correction factor. When k 0,the minimizing direction is along the negative gradient direction.


J xkð ÞTJ xkð Þþ kI

D ¼ �J xkð ÞTf xkð Þ ð7:100Þ

We choose the diagonal of second derivative of the function shown inEq. (7.101).

kI ¼0 0 00 @2f2 xkð Þ

@x6@x60

0 0 0

24

35 ð7:101Þ

Here, @2f2 xkð Þ@x6@x6

¼ �diag 2cabceq

Based on the non-singular matrix J xkð ÞTJ xkð Þþ kI

, Newton iteration is

applied to solve for the unknowns in two steps:

1. Estimate the delta of unknowns in Eq. (7.88), while the initial values for allunknowns are all unit matrix;

DðiÞ ¼ J x ið Þ T

J x ið Þ

þ kI

� ��1

J x ið Þ T

f � f x i�1ð Þ

ð7:102Þ

2. Terminate the iteration when the mismatch is smaller than a preset tolerance;superscript ið Þ indicates the ith iteration.

DðiÞ\e; x iþ 1ð Þ ¼ x ið Þ þDðiÞ

Example 7.8 Calibrate phase measurements and line parameters simultaneously inthe 3-bus 2-branch system used in Example 7.7. The precalibrated three-phasevoltage measurements are at bus 1, and the precalibrated three-phase currentmeasurements are at bus 1 in line 1–2.

Ratio correction factors are randomly generated with magnitude in the range of[0.94, 1.06] and angles in the range of [−4°, 4°]. The RCFs of three-phase voltagemeasurements at bus 1 to bus 3 are as follows:

ct e;1 ¼ 1; 1; 1½ �; since voltage measurements at bus 1 are precalibrated.

ct e;2 ¼ 0:98� i0:05; 1:03þ j0:005; 0:97þ j0:01½ �ct e;3 ¼ 1:024þ j0:002; 0:99� j0:001; 0:96� j0:003½ �

The RCFs of three-phase current measurements, respectively, from bus 1 to bus2, from bus 2 to bus 3, from bus 2 to bus 1, and from bus 3 to bus 2 are as follows:

7.6 Calibration 179

ct i;12 ¼ 1; 1; 1½ �; since the current transformers on this line are assumed to beaccurate.

ct i;23 ¼ 1:012� j0:002; 1:016þ j0:001; 0:99� j0:0011½ �cti;21 ¼ 0:98þ j0:1; 0:99þ j0:02; 1:02� j0:001½ �cti;32 ¼ 0:998� j0:001; 0:981� j0:009; 1:012� j0:001½ �

Fifty-eight sets of measurements under load scans are used to estimate the lineparameters of two branches and to estimate the RCFs of measurements. In Example7.6, it was shown that the larger number of load scans helps to lower the impact ofmeasurement noise. Therefore, comparison of different load scans is skipped in thisexample. Three cases are applied to study the impacts caused by random noise andmeasurement errors at precalibrated bus.

Case 1: Random noises εabc in Eq. (7.76) are in the range of [−1%, 1%] of themeasurements and are in the range of [−0.1%, 0.1%] for voltage and currentmeasurements at bus 1, which are assumed to be precalibrated measurements.Case 2: Random noises εabc in Eq. (7.76) are in the range of [−0.1%, 0.1%] of themeasurements and are in the range of [−0.01%, 0.01%] for voltage and currentmeasurements at bus 1, which are assumed to be precalibrated measurements.Case 3: Random noises εabc in Eq. (7.76) are in the range of [−0.01%, 0.01%] ofthe measurements and are in the range of [−0.01%, 0.01%] for voltage and currentmeasurements at bus 1, which are assumed to be precalibrated measurements.

Results

1. The estimated impedance matrix of branch 12 and branch 23

Z12 True Case 1 Case 2 Case 3

Za12 0.001 + j0.01 0.0008 + j0.01 0.001 + j0.0101 0.001 + j0.01

Zab12 0.0012 + j0.006 0.0013 +j0.0063 0.0012 + j0.0061 0.0012 + j0.006

Zac12 0.001 + j0.005 0.0009 + j0.0046 0.0011 + j0.005 0.001 + j0.005

Zb12 0.0012 + j0.012 0.0017 + j0.0121 0.0013 + j0.012 0.0012 + j0.012

Zbc12 0.002 + j0.001 0.0016 + j0.0007 0.0021 + j0.0009 0.002 + j0.001

Zc12 0.0015 + j0.015 0.0007 + j0.0156 0.0015 + j0.0151 0.0015 + j0.0151

Z23 True Case 1 Case 2 Case 3

Za12 0.002 + j0.02 0.0002 + j0.0196 0.0018 + j0.02 0.002 + j0.02

Zab12 0.001 + j0.003 0.0005 + j0.0032 0.001 + j0.0031 0.001 + j0.003

Zac12 0.001 + j0.005 0.0004 + j0.0052 0.0009 + j0.005 0.001 + j0.005

Zb12 0.0021 + j0.021 0.0023 + j0.0214 0.0021 + j0.0211 0.0021 + j0.021

Zbc12 0.002 + j0.002 0.0022 + j0.0024 0.002 + j0.0021 0.002 + j0.002

Zc12 0.0025 + j0.025 0.002 + j0.0257 0.0025 + j0.0251 0.0025 + j0.0251


2. The differences between estimated RCFs and true RCFs of voltage measure-ments (magnitude n%, angle n degrees)

ct � cestj j Case 1 Case 2 Case 3

Magn (%) Angle Magn Angle Magn Angle

Max 1.2525 0.8130 0.0141 0.0064 0.0095 0.006

Mean 0.5793 0.3721 0.0076 0.0033 0.0049 0.0023

Min 0.3166 0.0882 0.0021 1.95e−5 5.97e−4 2.55e−4

3. The differences between estimated RCFs and true RCFs of currentmeasurements

ct � cestj j Case 1 Case 2 Case 3


Max 2.9782 1.5825 0.1368 0.1443 0.053 0.0307

Mean 0.729 0.3067 0.0443 0.0311 0.0124 0.0088

Min 0.0153 0.0045 6.03e−4 5.56e−4 4.34e−5 5.54e−5

4. The differences between estimated three-phase voltage measurements and truevoltage measurements, except the precalibrated voltage measurements at bus 1

Et�EestEt

Case 1 Case 2 Case 3


Max 2.0122 0.8135 0.1059 0.0067 0.0189 0.006

Mean 0.6736 0.3721 0.0475 0.0033 0.0067 0.0023

Min 5.29e−4 0.0837 2.46e−5 1.39e−5 9.51e−5 2.55e−4

5. The differences between estimated three-phase current measurements and truecurrent measurements

It�IestIt

Case 1 Case 2 Case 3


Max 3.8775 1.5829 0.2352 0.1443 0.0619 0.0307

Mean 0.8972 0.3067 0.07 0.0311 0.0151 0.0088

Min 0.0043 0.0045 2.35e−4 5.51e−4 3.34e−5 5.51e−5

From these results, it is clear that errors of measurement noise are main deter-minants in the accuracy of results. Case 1 has the largest measurement noise andhas the worst accuracy in results, while Case 3 has the smallest measurement noiseand has the best accuracy. Comparing the results of Case 2 and Case 3, which have

7.6 Calibration 181

the same precalibrated measurement error but different random measurement noise,Case 3 has much more accurate results than Case 2. This indicates that errors inprecalibrated measurements have limited impact on results.

7.7 Dynamic Estimators

Before the introduction of phasor measurements, the idea of tracking the state of thepower system was introduced [22]. It was assumed that state of the system could bemodeled as Eq. (7.62) where x(k) is the state at the kth time step, Δt is the time step,r is a maximum rate of change vector, z(k) is the measurement, and v(k) is themeasurement error.

x kþ 1ð Þ ¼ x kð Þþ Dtð ÞrzðkÞ ¼ HxðkÞþ vðkÞ ð7:103Þ

If the term (Δt)r is denoted w(k) and w(k) and v(k) are modeled as zero-mean,independent, Gaussian processes, then the problem is essentially a Kalman filteringproblem [23]. The model incorporates a random evolution of the state through w(k) but loses power system details in the form of the measurements and the linearmodel. Phasor measurements with time tags could produce a modern equivalent ofthe tracking estimator in [24]. PMU measurements could be integrated with theexisting data scans. It is important to model the system dynamics that are beingtracked. There are certainly situations in which the system responds in a roughlypredictable manner. The morning load pickup, for example, is quite dependable asare other daily events. An estimator that assumed that the changes in load werelinear in time with unknown rates of change (which could themselves be estimated)would seem to be a considerable improvement over the static assumption. This is, ineffect, a matter of placing the phasor measurements correctly in time within theSCADA data window. The problem of skew of PMU measurement integrated withconventional data is difficult. If the data scan for conventional measurements takesT seconds and the PMU data can be located anywhere in the T-second interval,should the phasor measurements be distributed in the window or concentrated atone point? And if at one point, should that point be at the beginning, the middle, orthe end of the observation window?

The preceding could ultimately lead to an estimator based on more PMU datathat could follow the state through transient swings. With PMU measurementsavailable as often as every two cycles, transient swings with one-second periodscould be tracked. There would be some delay but a true dynamic estimate that was afew hundred millisecond delayed would still be desirable.

It is also possible to consider the use of real-time phasor measurements in theestimation of parameters or validation of models of components [23]. The use ofphasors and local frequency in estimating machine parameters and internal states


can also be approached as a Kalman filtering problem [25]. At the level of vali-dating system models, one could replace parts of the system with observed phasorvoltage and currents as time-dependent sources and then improve the model ofother parts of the system.

References

1. Allemong, J. J., et al. (1982). A fast and reliable state estimation algorithm for AEP’s newcontrol center. IEEE Transactions on Power Apparatus and Systems, 101(4), 933–944.

2. Dopazo, J. F., et al. (1975). Implementation of the AEP real-time monitoring system. IEEETransactions on Power Apparatus and Systems, 95(5), 1618–1529.

3. Handshin, E., et al. (1975). Bad data analysis for power system static state estimation. IEEETransactions on Power Apparatus and Systems, 94(2), 329–337.

4. Monticelli, A., Wu, F. F., & Yen, M. (1971). Multiple bad data identification for stateestimation using combinatorial optimization. IEEE PAS-90, 2718–2725.

5. Nuki, R. F., & Phadke, A. G. (2005). Phasor measurement placement techniques for completeand incomplete observability. IEEE Transactions on Power Delivery, 20(4), 2381–2388.

6. Gou, B., & Abur, A. (2001). An improved measurement placement algorithm for networkobservability. IEEE Trans on Power Systems, 16(4), 819–824.

7. Abur, A. (2005). Optimal placement of phasor measurements units for state estimation.PSERC Publication. October 06–58, 2005.

8. Phadke, A. G., & Thorp, J. S. (1988). Computer relaying for power systems. Somerset,England: Research Studies Press.

9. Thorp, J. S., Phadke, A. G., & Karimi, K. J. (1985). Real-time voltage phasor measurementsfor static state estimation. IEEE Transactions on Power Apparatus and Systems, 104(11),3098–3107.

10. Phadke, A. G., Thorp, J. S., & Karimi, K. J. (1986). State estimation with phasormeasurements. IEEE Transactions on Power Systems, 1(1), 233–241.

11. Zhou, M., et al. (2006). An alternative for Including Phasor measurements in State Estimation.IEEE Transactions on Power Systems, 21(4), 1930–1937.

12. Korevaar, N. (2002). Incidence is no coincidence. University of Utah Math Circle, October2002.

13. Heydt, G. T., Liu, C. C., Phadke, A. G., & Vittal, V. (2001). Solution for the crisis in electricpower supply. IEEE Computer Applications in Power, 14(3), 22–30.

14. Pal, A., Sanchez, G. A., Centeno, V. A., & Thorp, J. S. (2014). A PMU placement schemeensuring real-time monitoring of critical buses of the network. IEEE Transactions on PowerDelivery, 29(2), 510–517.

15. Dua, D., Dambhare, S. S., Gajbhiye, R. K., & Soman, S. A. (2008). Optimal multistagescheduling of PMU placement: An ILP approach. IEEE Transactions on Power Delivery, 23(4), 1812–1820.

16. Pal, A. (2014). PMU-based applications for improved monitoring and protection of powersystems. Ph.D. dissertation, Electrical and Computer Engineering Department, VirginiaPolytechnic Institute and State University, Blacksburg, April 2014.

17. Jeffers, R. (2007). Wide area state estimation techniques using phasor measurement data.Virginia Tech Report prepared for Tennessee Valley Authority, March 2007.

18. Xu, B., Yoon, Y. J., & Abur, A. (2005). Optimal placement and utilization of phasormeasurements for state estimation. In Proceedings of 15th Power Systems ComputationConference (PSCC), Liege, Belgium (pp. 1–6), August 22–26, 2005.

7.7 Dynamic Estimators 183

19. Wu, Z., Sun, R., & Phadke, A. G. (2012). Three-phase calibration of instrument transformerswith synchronized phasor measurements. In IEEE PES Innovative Smart Grid TechnologiesConference.

20. Zhou, M. (2008). Phasor measurement unit calibration and applications in state estimation.Ph.D. dissertation, Virginia Tech, March 2008.

21. Wu, Z. (2012). Synchronized phasor measurement applications in three-phase powersystems. Ph.D. dissertation, Virginia Tech.

22. Debs, A. S., & Larson, R. E. (1970). A dynamic estimator for tracking the state of a powersystem. IEEE Transactions on Power Apparatus Systems, PAS-893(7), 1670–1678.

23. Gelb, A. (1974). Applied optimal filtering. Massachusetts: MIT Press.24. Abbasy, N. H., & Ismail, H. M. (2009). A unified approach for the optimal PMU location for

power system state estimation. IEEE Transactions on Power Systems, 24(2), 806–813.25. Pilay, P., Phadke, A. G., Linders, D. K., & Thorp, J. S. (1987). State estimation for a

synchronous machine: Observer and Kalman filter approach. In Princeton Conference.


Chapter 8Control with Phasor Feedback

8.1 Introduction

Prior to the introduction of real-time phasor measurements, power system controlwas essentially used by local signals. Feedback control with such locally availablemeasurements is widely used in controlling machines. In other situations, controlaction was taken on the basis of a mathematical model of the system without actualmeasurement of the system. The advent of phasor measurements allows the con-sideration of control based on the measured value of remote quantities. It isexpected that such control will be less dependent on the model of the system beingcontrolled. The fact that most such phenomena are relatively slow is an encouragingfactor for deploying PMUs. The latency of the phasor measurement process is notimportant when the process frequencies are in 0.2–2.0 Hz range. The phasor datawould be time-tagged so that control would be based on the actual state of thesystem a short time earlier. The frequencies are representatives of the electrome-chanical oscillations, transient stability, and certain overload phenomena. The fre-quency of measurements is expected to be of the order 15–30 Hz, which is certainlysufficient to handle the control task.

Studies of control of HVDC systems, excitation control, power system stabi-lizers, and FACTS control will be described in the next few sections. All of theseapplications share common features. The actual processes are inherently nonlinearbecause they involve real power, and there are never enough measurements tototally describe the dynamical system in the same detail as a typical aerospaceapplication. The next section provides a framework to investigate these problems.

8.2 Linear Optimal Control

A general control design used in this section is shown in Fig. 8.1.


185

The control law will minimize the difference between the output desired tra-jectory y(t) and the desired trajectory yd(t). The problem is well studied for linearsystems [1]. For the system in Eq. (8.1)

_xðtÞ ¼ AðtÞxðtÞþBðtÞuðtÞyðtÞ ¼ CðtÞxðtÞ ð8:1Þ

The finite interval problem is to minimize J by choice of u(t), where J is inEq. (8.2)

J ¼ 12

Ztft0

yðtÞ � ydðtÞ½ �TQ yðtÞ � ydðtÞ½ � þ uTðtÞR uðtÞn o

dt

þ 12yðtfÞ � ydðtfÞ½ �TH yðtfÞ � ydðtfÞ½ � ð8:2Þ

The performance index has weighting throughout the interval on the error inachieving the desired output trajectory. The error is weighted by the matrix Q, andthe control effort required is weighted by the matrix R. Typically, Q and R arediagonal with weights that take different units and maximum values of the variablesinto consideration. For example, if the states included the altitude of an aircraft infeet and the angle of attack in radians, then very different diagonal weights arerequired to keep the feet variables from swamping out the radian variable terms.The solution is given by Eq. (8.3) where the feedback gain matrix K(t) and forcingterm g(t) are given in Eq. (8.4).

u�ðtÞ ¼ R�1BT gðtÞ �KðtÞxðtÞ½ � ð8:3Þ

Nondynamicsystem:x(t) state

y(t)=C(t)x(t) Output Vector

u(t) control vector

-linear

desired trajectory

actual trajectory

tfto

y(t)

Fig. 8.1 Controller design

186 8 Control with Phasor Feedback

_KðtÞ ¼ �ATK�KAþKBR�1BTK� CTQC

with KðtfÞ ¼ CTðtfÞHCðtfÞ_gðtÞ ¼ �½AT �KBR�1BT�gðtÞ � CTQyd

with gðtfÞ ¼ CtðtfÞHydðtfÞ

ð8:4Þ

The matrix K in Eq. (8.4) satisfies the so-called Riccati equation [1]. It involvesconsiderable computation and is frequently approximated by its steady-state solu-tion which can be obtained with only algebraic manipulations. The steady state canbe reached if the time interval tf–t0 is long enough. Note that the control u* appliedto the original state equation results in a closed-loop system

_xðtÞ ¼ ½AðtÞ � BðtÞR�1BTKðtÞ�xðtÞþBðtÞR�1BTgðtÞyðtÞ ¼ CðtÞxðtÞ ð8:5Þ

The A matrix from Eq. (8.1) has now been transformed to the closed-loop plantwith “A” matrix as A-BR−1BT K. The transpose of that “A” matrix appears in thedifferential Eq. for g(t). So the g(t) equation in Eq. (8.4) is the adjoint of theclosed-loop system. It is also important to recognize that both equations in Eq. (8.4)are solved backward in time, i.e., the terminal conditions are given rather than initialconditions. These points are important to understand the nature of the approxi-mation as discussed in the next section.

8.3 Linear Optimal Control Applied to the NonlinearProblem

Given phasor measurements of the system, it is possible to measure the differencebetween the states of system we are actually controlling and the state of a model. Ifthe actual system is given by Eq. (8.6), then we can have a simpler model in mind(a linearized or reduced-order model)

_xðtÞ ¼ FðxðtÞ; uðtÞ; tÞyðtÞ ¼ CðtÞxðtÞ ð8:6Þ

_xðtÞ ¼ AðtÞxðtÞþBðtÞuðtÞþ fðtÞwhere fðtÞ ¼ FðxðtÞ; uðtÞ; tÞ � AðtÞxðt) ð8:7Þ

The term f(t) in Eq. (8.7) is the difference between the derivative of the state inthe actual system and in the linear model. Assuming f(t) is known for the timebeing, the solution to the optimal control problem is given by Eqs. (8.3) and (8.4)

8.2 Linear Optimal Control 187

with one small addition. The differential equation for g has an additional term at theend, depending on f as given in Eq. (8.8).

_gðtÞ ¼ �½AT �KBR�1BT�gðtÞ � CTQyd þKf

with gðtfÞ ¼ CTðtfÞHydðtfÞ ð8:8Þ

It is convenient that the Riccati equation is not affected so it can be computedoff-line and stored. Only Eq. (8.8) must be solved in order to determine the control.The problem is that Eq. (8.8) is solved backward in time, and we can compute f(t) forward in time as shown in Fig. 8.2. The solution is to predict f(t) based on thedata that has been stored since the optimization began. The prediction is worst at thebeginning of the interval and improves as time increases. The next few sectionsgive some applications of this approximate control design to some systems.

Example 8.1 HVDC SystemIn [2], the predictive control is applied to a HVDC system as shown in Fig. 8.3.The modeling involved combining the network with the HVDC model, generatormodels, and the exciter models to write the state equations. It was found that thesteady-state value of the Riccati equation was acceptable and greatly reducedthe computational burden. Even a steady-state value of g(t) was used by finding thesteady-state solution of Eq. (8.8) given by Eq. (8.9) (note this value g gives _g ¼ 0:)

g ¼ ½AT �KBR�1BT��1Kf ð8:9Þ

The prediction of f(t) was performed with a straight line, obtained by computingthe moving average of the previous values of f(t). The system is shown in Fig. 8.3.It has two generators, three transformers with two of them having the off-nominalturns ratios, a two-terminal HVDC link, a load bus, and an infinite bus.A three-phase fault is applied as shown for 3 cycles, and the line is removed to clearthe fault. The performance of the controller is shown in Fig. 8.4. The solid curvesare the two rotor angles with a constant current and constant voltage on the HVDC.The dashed curves are the rotor angles obtained with the steady-state Riccatiequation and the steady-state g equation (Eq. 8.9).

f(t)

t

Observation window

Actual f(t)

Predicted f(t)

t0tf

Fig. 8.2 Predicting f(t)


Example 8.2 Excitation ControlA centralized excitation controller can be designed using the same technique. If allthe real-time phasor data were brought to a central location as in [3] and all thecontrol signals for the generator excitation and governor systems computed usingEq. (8.3), then a centralized excitation controller could be designed (Fig. 8.5).

The controller state variables correspond to the incremental changes frompre-fault values. The IEEE Type 1 exciter model with an applied auxiliary inputsignal was used [4]. The desired trajectory corresponds to a desired post-faultequilibrium and to a known state of the power system immediately after the fault.Four machine angles from the 39 bus New England system are shown in Fig. 8.6with the angle of machine 10 as the reference with and without feedback control.Again as in the HVDC example, steady-state values are used for the Riccatiequation and the equilibrium g equation.

Example 8.3 FACTS ControllerA three-machine system with a thyristor controlled series capacitor (TCSC) isshown in Fig. 8.7 [5]. The incremental linear system was obtained by linearizing

1640 MW

820 MW

(200+j20) MVA

590 MVAR

680 MVA3 phase faultcleared in 3 cycles

G 1

G2

Fig. 8.3 HVDC system

δ2

δ1

Constant current, constant voltage control

Optimal Controller5 sec

time

time 5 sec

Fig. 8.4 Performance of theHVDC controller

8.3 Linear Optimal Control Applied to the Nonlinear Problem 189

G3

G2

G1

G8G10

G9

G7

G6

G4G5

Fig. 8.5 Ten-machine system used for a centralized excitation controller

Without Feedback Control

With Optimal Feedback Control

δ1

t (seconds)

δ23

5

Fig. 8.6 Performance of theexcitation control

G2 G3

TCSCPhasorfeedback

G1

Fig. 8.7 FACTS examplesystem


around the initial operating point. IEEE Type I exciters were used for the machines.Sensitivity analysis was used to locate the TCSC for maximum effect on eigenvaluelocation. The reactive loading levels were increased until there was a pair ofunstable modes as shown in Table 8.1.

The phasor measurements control the compensating reactance of the transmis-sion line with TCSC as shown in Fig. 8.8. A PID controller with an input of theangle difference δ1 − δ2 produced an output signal to control the reactance as shownin Fig. 8.8.

The performance of the optimal control solution is shown in Fig. 8.9 for a stepchange in one generator power.

Example 8.4 Power System StabilizerThree control schemes were tested on the four-machine system in Fig. 8.10:Automatic Voltage Regulator on all four machines, a conventional power systemstabilizer on one machine as shown, and phasor feedback on the same machine.

Table 8.1 Eigenvalues forthe system in Fig. 8.7

−60.22 −2.5341

−26.5 0.0021 ± j0.3043

−16.57 −1.5769

−9.21 −0.6859

−5.52454 ± j3.8478 −0.38 ± j0.0494

−7.3475 −1.3394

−2.7664 ± j0.9461 −0.1747

−2.00

2

without TCSC

with optimally controlled TCSC

Rotor angle difference 12

Fig. 8.9 FACTS controllerperformance

Fig. 8.8 Thyristor controlledseries capacitor (TCSC)control

8.3 Linear Optimal Control Applied to the Nonlinear Problem 191

Modal analysis of the system shows that at tie flow of 158 MW, there is an unstableinter-area mode. The modes are shown in Table 8.2. The generators were modeledfollowing the two-axis method [4, 6] and the detailed models used for the governor,turbine, and constant gain excitation systems.

The inter-area mode is stable for tie flow of 50 MW but becomes unstable forlarger flows between the two areas. A comparison of the three control schemes isgiven in Fig. 8.11. In a separate study, the amount of latency that could be toleratedin the phasor measurements for the power system stabilizer was determined to be asmuch as 150 ms, depending on the frequency of the oscillation.

AVR AVR

100ms fault cleared

AVR

PSS

PhasorFeedback

AVRFig. 8.10 Example systemfor AVR PSS and phasorfeedback comparison

Table 8.2 Eigenvalues for example with 158 MW tie flow

Mode Frequency Damping ratio Mode type

−0.05977 ± j7.0365 1.1199 0.0849 Local area 1

−0.6060 ± j7.247 1.1534 0.0833 Local area 2

0.0296 ± j4.1784 0.665 −0.0071 Inter-area

1. AVR aloneMW tie flow

Time

1. AVR alone2. PSS on one machine3. Phasor feed -

back on one machine

Fig. 8.11 Comparison of AVRs, a single PSS and a phasor-based PSS


8.4 Coordinated Control of Oscillations

The power system stabilizers of the preceding example are typically used to controlinter-area oscillations. These oscillations are low-frequency small signal oscillationsthat seem to be growing in number. A single 0.7 Hz oscillation in the WECC hasbeen replaced by as many as five frequencies with some as low as 0.2 Hz.Stabilizers are tuned to damp a specific mode and when installed are effective. Thedifficulty is that as the system changes, the stabilizer is not quite as effective. It isalso conjectured that the stabilizers interact with each other to produce new modes.Given the evolving nature of the frequencies and occurrences of the modes, itwould be best if some strategy could be devised to provide damping for all modesrather than designing specific controllers aimed precisely at presumed modes.Existing approaches have been shown to lack robustness. A parallel to this problemexists in structures both tall buildings and large space structures. In both cases, it isdesirable to damp vibrations without knowing precisely what form the vibrationswill take. Earthquakes and unusual winds for tall buildings and unpredicted dis-turbances on the space station are examples.

A common solution to the structural engineering problem is the use of so-calledcollocated control [7]. It seems that phasor measurements can provide a similarsolution to low-frequency inter-area oscillations in power systems. The basic idea inthe structures problem is to formulate the problem in modal form as in Eq. (8.10)

€gþD _gþK2g ¼ Bu; u ¼ �Fy; y ¼ BTg ð8:10Þ

where g is the vector of modal coordinates, u is the vector of control inputs,and y is the vector of measurements. The matrix K is a diagonal matrixK ¼ diagfx2

1 � � �x2ng with x1\x2\ � � �xn. We assume the damping is propor-

tional to frequency, D ¼ 2aK, where a ¼ diagfa1 � � � ang with ai\1. F is anon-negative definite matrix to be determined. The open loop eigenvalues of thesystem are �aixi � jxi

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� a2i

p. We assume the first K modes are the critical

low-frequency modes for which we wish to provide additional damping. The termcollocated refers to the matrix B appearing with both u and y in Eq. (8.10). Itproduces a convenient form for the eigenvalues of the closed-loop system ofEq. (8.10) given in Eq. (8.11)

€gþDðFÞ _gþK2g ¼ 0; DðFÞ ¼ 2aKþBFBT ð8:11Þ

The collocated form guarantees that the damping added by the feedback does noharm even if the system model changes. The term BFBT is non-negative definiteand behaves like a multidimensional resistive network.

An optimal F can be considered by mapping the complex plane in which theeigenvalues of Eq. (8.11) reside. If �k is a mapping from k, then the eigen values ofEq. (8.11) are given by

8.4 Coordinated Control of Oscillations 193

�k ¼ r � z0 þ krþ z0 � k

k ¼ rð�k� 1Þ�kþ 1

¼ z0 ð8:12Þ

where r and z0 are shown in Fig. 8.12.Equation (8.11) can be mapped using Eq. (8.12) so that if the eigenvalues of the

�k system are in the left half of the �k plane, then the eigenvalues of the k system arein region R1. If A is a non-negative definite matrix, we write A � 0. A sufficientcondition that the F matrix results in shifting the eigenvalues of the closed-loopsystem in the region R1 is that the two matrices in Eq 8.13 be non-negative definite.

�DðF; r; z0Þ ¼ �ðr2 � z20ÞIþ z0DðFÞþK2 � 0

�KðF; r; z0Þ ¼ �ðr � z0Þ2I� ðr � z0ÞDðFÞþK2 � 0ð8:13Þ

The two matrices are obtained by mapping 8.11 into the �k plane. Such an F iscalled feasible. The system described by Eq. (8.11) is stable by its very structure.The problem is that any low frequency eigenvalue may have very small real parts. Ifwe could find a matrix F so that the eigenvalues λ of Eq. (8.12) were in the regionR1 in Fig. 8.12, we would have guaranteed damping of low-frequency modes.A direct test for eigenvalues λ in to be in R1 in Fig. 8.12 is quite difficult, but a testfor the eigenvalues, �k in Fig. 8.12 to be in the left half plane, is simple. Hence, themapping in Eq. (8.12) is used. Recognizing that Eq. (8.13) is in the form:

Gþ z0BFBT � 0 gii ¼ u2i � r2 gij ¼ 0

H� ðr � z0ÞBFBT � 0 hii ¼ v2i hij ¼ 0ð8:14Þ

where ui and vi are scalars depending on the geometry as shown in Fig. 8.13, andG and H are diagonal matrices. Conditions are given in [7], but the numerical test issimply to apply the QR decomposition to the matrix B, i.e., if

Rez0

r

Im

r2-z02

λ plane

Region R1λ plane

Region R2

Re

Im

Fig. 8.12 λ and �k planes


TTB ¼ R0

� �ð8:15Þ

where T is orthogonal and R is upper triangular. Then, Eq. (8.14) is equivalent to

TTGTþ z0R

0

� �F

R

0

� �T¼ TTGTþ z0

RFRT 0

0 0

" #� 0

TTHT� ðr � z0ÞR

0

� �F

R

0

� �T¼ TTHTþðr � z0Þ RFRT 0

0 0

" #� 0 ð8:16Þ

The fact that G and H are diagonal, combined with the assumption that there areno open loop eigenvalues on the circle in Fig. 8.11, allows one remaining trans-formation. If there are no eigenvalues on the circle, then M22 in Eq. (8.17) isinvertible

TTGT ¼ M11 M12

MT12 M22

� �; U ¼ I 0

�M�122 M

T12 I

� �;

UTTTGTU ¼ M11 �M12M�122 M

T12 0

0 M22

" # ð8:17Þ

ui

Re

vi

2ii 1 α−ω

iiωα

z0

r-zo

Im

Fig. 8.13 Values of ui and vi for Eq. 8.14

8.4 Coordinated Control of Oscillations 195

If TTHT = N, then the final conditions are that

M22 � 0

� 1r � z0

R�1 N11 � N12N�122 N

T12

� �R�T

�F

� 1z0R�1 M11 �M12M�1

22 MT12

� �R�T ð8:18Þ

If there are sufficient well-placed measurements and there is one feasible matrixF, then F is not unique. The choice of optimal F as the one with minimumFrobenius norm is suggested in [7]. The Frobenius norm of F is given byEq. (8.19)

Fk k ¼ minF

Xmi¼1

Xmj¼1

f 2ij

!ð8:19Þ

In [7], it is shown that if the problem is one of minimizing the norm inEq. (8.19) subject to the constraints of Eq. (8.18) then one must compute the Schurfactorization of the matrix in Eq. (8.18)

1z0R�1 M11 �M12M�1

22 MT12

� �R�T ¼ SDST ð8:20Þ

In Eq. (8.20), S is orthonormal and D is diagonal. Replace the negative entrieson the diagonal of D with zero and call the result �D. If

SDST � 1

~r � z0R�1 N11 � N12N�1

22 NT12

� �R�T

then F ¼ SDST ð8:21Þ

The minimum norm F obtained from Eq. (8.21) may, in fact, not be feasibleunder some situations. Hence, it is worth examining the closed-loop eigenvaluesbefore using it. The conditions in Eq. (8.18) provide an interval in which F must lie.The matrix in Eq. (8.18) is a more reliable choice for F.

Example 8.5 The following MATLAB example uses a 12-dimensional system witheigenvalues given in Table 8.3. The radius of the circle in Fig. 8.3 is r = 5, and thecenter is at z0 = 4. The B matrix is also shown in Table 8.3. Using F = N1, theclosed-loop eigenvalues are shown in Fig. 8.14.


8.5 Polytopic Control Using LMIs

One way to implement a coordinated control for providing requisite damping over awide range of operating points of a power system is to develop a linear matricinequality (LMI)-based polytopic controller.

An LMI is any constraint of the form:

AðpÞ :¼ A0 þ p1A1 þ p2A2 þ � � � þ pnAn\0 ð8:22Þ

where p ¼ p1; p2; . . . pn is an unknown vector comprising of the optimizationvariables; A0;A1; . . .An are known symmetric matrices, and <0 implies “negativedefinite,” i.e., the largest eigenvalue of AðpÞ is negative. The LMI constraint defined

Table 8.3 Data for MATLAB example

Eigenvalues B Closed loop λs

−0.1 ± j1 3.6946 0.1730 0.1365 −9.3842

−0.2 ± j2 0.6213 1.9797 0.0118 −4.5731

−0.3 ± j3 0.7948 0.2714 2.8939 −0.6803 ± j5.7594

−0.4 ± j4 0.9568 0.2523 0.1991 −0.6741 ± j4.7032

−0.5 ± j5 0.5226 0.9757 0.2987 −0.4494 ± 3.9404

−0.6 ± j6 0.8801 0.7373 0.6614 Four at −1.000

-6 -4 -2 0 2 4

-6

-4

-2

0

2

4

6Open loop λs

Closed loop λ s

Fig. 8.14 Opened- and closed-loop eigenvalues

8.5 Polytopic Control Using LMIs 197

in Eq. (8.22) is a convex constraint on p as A qð Þ\0 and A rð Þ\0 will implyA qþ r

2

� �\0. Moreover, a system of LMI constraints can be thought of as a single

LMI problem whose block-diagonal matrix has the system matrices of thoseindividual systems as its diagonal elements. More details about LMIs can be foundin [8].

A polytopic system is the one whose system matrix varies within a fixedpolytope of matrices. The polytopic H1=H2 formulation is posed with state spacemodels of the form:

_x ¼ AxþB1fþB2u

z1 ¼ C1xþD11fþD12u

z2 ¼ C2xþD22u

y ¼ CyxþDy1fþDy2u

ð8:23Þ

where x is the state of the system, u is the control, f is a disturbance, z1 and z2 arefor the H1=H2 optimizations, and y is the output. The cases that would be used totest the robustness of the controller for a power system application such as tie-lineoutages, load changes, and changes in line flows all have their own version of thesystem denoted by Eq. (8.23). These systems are then the vertices of the polytope.The kth such system denoted by Sk is of the form:

Sk ¼Ak Bk1 Bk2

C1 Dk11 Dk12

Ck2 0 Dk22

Cky Dky1 Dky2

2664

3775 ð8:24Þ

The convex combination of the systems is given by

S S1; S2; . . . Skf g ¼Xki¼1

aiSi:Xi

ai ¼ 1; ai � 0

( )ð8:25Þ

The non-negative numbers a1; a2; . . .; ak are called the polytopic coordinates ofS. Many of the convex combinations in Eq. (8.25) have simple explanations. If S1 isthe base case and S2 represents the outage of a tie-line, then aS1 þ 1� að ÞS2 can bethought of as continuously increasing the impedance of the tie-line until it is open.The advantage of the polytopic approach is that the span of the model uncertainty isconsidered within the design itself.

The objective of a robust H1=H2 control for damping oscillations is the designof a controller u ¼ Kx such that the resulting closed-loop system satisfies thefollowing constraints:

1. The H1 norm of the closed-loop transfer function Tfz1 does not exceed thespecified maximum bound c1, where c1 [ 0. This guarantees that the


closed-loop system is stable and robust. Mathematically, this is denoted byEqs. (8.26a) and (8.26b).

Tfz1\c1 ð8:26aÞ

In terms of system matrices, Tfz1 is defined as

Tfz1 ¼ C1 þD12Kð Þ sI� A� B2Kð Þ�1B1 þD11 ð8:26bÞ

2. The H2 norm of closed-loop transfer function Tfz2 does not exceed the specifiedmaximum bound c2, where c2 [ 0. This guarantees that the closed-loop systemwill have good performance. Mathematically, this is denoted by Eq. (8.27a) and(8.27b).

Tfz2\c2 ð8:27aÞ

In terms of system matrices, Tfz2 is defined as

Tfz2 ¼ C2 þD22Kð Þ sI� A� B2Kð Þ�1B1 ð8:27bÞ

3. The closed-loop system must be D-stable [8]. An example of a D-stable regionis shown in Eq. (8.28). Graphically, it corresponds to the shaded region asshown in Fig. 8.15.

D ¼ s 2 C:Lþ sMþ�sMT\0 ð8:28Þ

In Eq. (8.28), L ¼2l0 0 00 0 00 0 0

24

35 and M ¼

1 0 00 cos h � sin h0 sin h cos h

24

35. The vari-

ables l0 and h are shown in Fig. 8.15.4. It minimizes the H1=H2 trade-off criterion of the form: q T2

fz1�� þ r T2

fz2

�� .This ensures optimal performance while taking into account both disturbancerejection aspects H1ð Þ as well as linear quadratic Gaussian (LQG) aspects H2ð Þ.

ol

θ

Fig. 8.15 Example of aD-stable region (From [22])


Thus, for a system Si : Ai;Bi1;Bi2;Ci1;Ci2;Di11;Di12;Di22f g, the LMIs corre-sponding to the constraints given in Eqs. (8.26a)–(8.28) can be expressed asEqs. (8.29)–(8.31), respectively [9].

AiXþXATi þBi2YþYTBT

i2 Bi1 XCTi1 þYTDT

i12BTi1 �I DT

i11Ci1XþDi12Y Di11 �c2I

24

35\0 ð8:29Þ

Q Ci2XþDi22YXCT

i2 þYTDTi22 X

� �[ 0 ð8:30Þ

LMð Þþ M AiXþBi2Yð Þð ÞþMT AiXþBi2Yð ÞT\0 ð8:31Þ

In Eqs. (8.29)–(8.31), X is the Lyapunov matrix, Q is a positive definite matrix,and is the Kronecker product. Furthermore,

Y ¼ KX

c2\c21Trace Qð Þ\ c2ð Þ2

ð8:32Þ

An example of an LMI control design using a 3-dimensional polytope is shownin Fig. 8.16. The vertices P, Q, and R denote known operating conditions, while theoperating points (a, b, c, d, and e) inside the polytope denote a convex combinationof the known operating points. The convexity property of a polytope ensures that acontrol devised for the vertices will work for any point that lies inside it [10].Therefore, a controller that is able to provide requisite damping for the operatingpoints, P, Q, and R, will also be able to provide suitable damping for operatingpoints a-e.

LMIs have been used to design PSS-based controls in [11], while it has beenused to integrate the control effects of DC lines, SVCs, and energy storage devicesin [12]. Although LMIs gave good results for small-sized systems, its directapplication to bigger systems was limited by its computational complexity [13].Selective modal analysis (SMA) [14] was found to provide a viable solution to thisproblem by reducing the size of the system to the relevant modes of oscillations.

SMA works on the system matrix A of Eq. (8.23). If A ¼ A1 A2

A3 A4

� �, where A1

comprises of the angles and speeds of the generators and A2, A3 and A4 haveinformation about the remaining machine states and the states associated with thecontrols, then SMA follows an iterative procedure to come up with a reduced-ordermatrix that has similar dynamic response as A. The block diagram of SMA is shownin Fig. 8.17. The block X denotes the “more relevant” portion of the system whosedynamics is of interest. The block Y denotes the “less relevant” part of the systemwhich has been collapsed in such a manner that it does not interfere with thedynamics of the portion of interest. In SMA, block X contains the matrix A1, while


block Y contains the matrices A2, A3 and A4. The constituent elements of A1, A2, A3,and A4 are shown in Eq. (8.33).

A1 ¼0 x0I

A21 A22

� �; A2 ¼

0 0

A23 A24

� �;

A3 ¼A31 A32

A41 þB21 A42 þB22

� �; A4 ¼

A33 A34

A43 A44

� � ð8:33Þ

The transfer function of the system in the feedback path denoted by the blockY is

HðsÞ ¼ A2ðsI � A4Þ�1A3 ð8:34Þ

The above equation can be thought of as a combination of two equations asshown below:

Fig. 8.16 LMI control implemented using a 3-dimensional polytope

Fig. 8.17 Block diagram ofSMA (From [13])


HðsÞ ¼ H1ðsÞþH2ðsÞ

H1ðsÞ ¼ A2 sI � A4ð Þ�1 A31 A32

A41 A42

� �

H2ðsÞ ¼ A2 sI � A4ð Þ�1 0 0

B21 B22

� � ð8:35Þ

Being an iterative process, SMA begins with the eigenvalues and eigenvectors ofA1. Using H1ðsÞ to illustrate, if the left eigenvalues and eigenvectors of A1 are diand vi, then wi, and the ith column of the matrix W is formed by

wi ¼ A2 diI � A4ð Þ�1 A31 A32

A41 A42

� �vi ð8:36Þ

A first correction to A1 is then formed by

A1next ¼ A1last þM1 whereM1 ¼ WV�1 ð8:37Þ

In Eq. (8.37), V�1 is the inverse of the matrix of left eigenvectors. The process isrepeated with the eigenvalues and eigenvectors of A1 þM1. A few iterations give agood estimate to the eigenvalues of the A matrix. Given the presence of B inEq. (8.35), the right eigenvectors of A1 are more appropriate for H2. The result is areduced-order system of the form in Eq. (8.38), whereM is an approximation to H1,N is an approximation to H2, and K is the gain matrix obtained as the output of theLMI control.

_x1 ¼ A1 þMþNKð Þx1 ð8:38Þ

Thus, Eq. (8.38) is an approximation to the closed-loop eigenvalues using thefixed matrices M, N and K.

Example 8.6 The modified 4-machine, 2-area test system is shown in Fig. 8.18.The system consists of two similar areas connected by two long, but weak tie-lines.Each area consists of two identical generating units coupled with one another. Theloads are located at buses 4 and 14 with the generator G4 at bus 12 acting as theswing machine. The system has been modeled to operate under heavy stress byincreasing the flow through the inter-connecting transmission lines. There are threedifferent electromechanical modes of oscillations present in this system: oneinter-area mode and two local modes. The control action is provided by the DC line(shown in red).

The A matrix for this system had a dimension of 19 19. The complexeigenvalues of the system were found to be −0.10146 ± i2.9729,−0.12644 ± i8.8531, and −0.14495 ± i8.1709. On performing three iterations ofSMA, it was observed that the eigenvalues of the reduced-order system perfectlymatched those of the original system. Similarly, a linear quadratic regulator


(LQR) feedback control that was developed based on the reduced-order system wasfound to be effective for the original system as seen in Fig. 8.19. In the figure, thered dots and green circles represent the eigenvalue movements for the reducedsystem and full system, respectively, as the LQR gain is increased. From the figure,it becomes clear that as the gain of the control is increased to its maximum value,the eigenvalues move further towards the left indicating that the stability of thesystem has improved. This example demonstrates that a control that is developed

Fig. 8.18 Modified 4-machine, 2-area system

Fig. 8.19 Transition ofeigenvalues with an increasein gain of an LQR control


using the reduced-order system can be successfully transferred back to the fullsystem.

The set of contingency cases that were used to develop the polytopic controllerfor this system is shown in Table 8.4. These cases formed the vertices of thepolytope. The objective was to ensure 5% damping for all operating points that areformed by a convex combination of the contingency cases. The iterations of SMAwere performed for the system matrices corresponding to each of those vertices.SMA was found to converge within three iterations. The results obtained onapplying the SMA-LMI-based polytopic control are shown in Fig. 8.20. The solidline indicates 5% damping boundary. Since all the closed-loop poles lie on the leftof the boundary, they have the requisite amount of damping. A comparison of theLMI-based polytopic control without and with the combination of SMA is shown inTable 8.5.

Table 8.4 Case Bank for theModified 4-machine—2-areasystem

Case No. Case details

1 No contingency case

2 Loads decreased to 90% of their original value

3 Loads increased to 102% of their original value

4 Flow in line 3–101 increased by 200 MW

5 Flow in line 3–101 decreased by 100 MW

Fig. 8.20 Output of the SMA-LMI-based polytopic control for convex combination of the fivecases of the modified 4-machine—2-area system


8.5.1 Phasor Measurement-Based Adaptive Control

The phasor measurement-based coordinated control developed in Sect. 8.5 wasused for damping inter-area oscillations in a 16-machine, 68-bus system in [15], anda 4000+ bus model of the WECC system in [12]. In order to apply the LMI-basedcontrol to such large systems, SMA was extended further to select only thosemachines of the system which had a higher participation factor to the modes ofinterest [13]. However, it was soon realized that a single polytope is not able todamp a large number of operating conditions [16]. The formulation of an adaptivecontrol was found to be the solution to this problem. An “adaptive” controller is onethat modifies its output based on current system conditions. Therefore, an adaptivecontrol methodology could choose the most suitable controller amongst multiplecontrollers that were available, for optimally damping the oscillations that werepresent. An adaptive control methodology that uses multiple polytopes is developedas follows.

The state-space model of the system that is used here is given by Eq. (8.23).Similarly, the polytopic combination of the systems is given by Eq. (8.25) wherethe non-negative quantity a represents the polytopic coordinates. Let n polytopes bedefined for a particular system. Compute Ui, where 1� i� n, as the discretizedmatrix corresponding to the time invariant closed-loop state matrix Ai. Then, aconvex combination of the variables, denoted by x, is given by,

x(mÞ ¼Xni¼1

Uixiðm� 1Þai ð8:39Þ

In Eq. (8.39), m is the number of samples taken, and the elements of x are anglesand speeds of the generators of the system. Equation (8.39) can be solved recur-sively to find the value of a as shown in Eq. (8.40).

ai ¼ Uixðm� 1Þð ÞT Uixðm� 1Þð Þ� �1

Uixðm� 1Þð ÞTxðmÞ ð8:40Þ

The rationale behind Eq. (8.40) is that if the oscillation currently present in thesystem can be damped by the control provided by the ith polytope, then a will

Table 8.5 Comparison ofLMI optimization without andwith SMA for the modified4-machine—2-area system

LMI control SMA-LMI control

System size (A=A1) 19 19 6 6

Size of individual system 24 22 11 9

Size of polytopic system 24 116 11 51

Size of closed-loop system 24 111 11 46

CPU timea (s) 29:286383 1:209831aThe computations were performed on an Intel (R) Core™ i5processor having a speed of 2.40 GHz and an installed memory(RAM) of 5.86 GB


converge to a fixed value for that polytope and diverge for all other polytopes [16].This methodology is analogous to a gain scheduling algorithm, except that itschedules the control, instead of the gain. In a gain scheduling algorithm, a non-linear system is controlled with a set of linear controllers that satisfactorily stabi-lizes the system at various operating points. Observer variables determine theoperating region of the system and then apply the appropriate linear controller.Similarly, this methodology determines what control needs to be used to maintainstability.

The adaptive control methodology is a promising solution for accommodatingwide range of operating conditions of a large-scale power system network [9, 17].The multiple model method especially provides a “well-behaved” solution foradaptively damping oscillations through the observable and measureable schedul-ing variables [18, 19]. These schemes involve (a) linearization of the system modelfor an appropriate set of operating conditions, (b) designing a set of robust oroptimal controllers for the individual conditions, and (c) providing a linear com-bination of the controllers as the final solution. However, due to the inherentnonlinearity of the power system, there is no guarantee that a linear combination ofthe controllers will always show good performance [20]. To avoid the linearcombination of controllers, a dual Youla parameterization-based adaptive controlmethod has also been proposed [21]. But since it was based on the polynomials ofthe transfer functions, the multiplication and division of higher-order polynomialsbrought about the “curse of dimensionality” especially when applied to large sys-tems. Also, the complex iterative process of the loop transfer recovery associatedwith the Youla method limited its practical use.

One way to circumvent this problem and implement an adaptive control in amultiple-input multiple-output (MIMO) system is by using phasor measurementsthat are independent of each other. This situation is illustrated in Fig. 8.21, wherep1, p2 and p3 denote the parameters, while y1 and y2 denote the measurements. Thered square represents the initial equilibrium point, and the blue circles representother equilibrium points (different from the initial equilibrium point). Figure 8.21adepicts the parameter space, while Fig. 8.21b depicts the measurement space. Thechanges in the operating point in the measurement space are caused by the changesthat occur in the parameter space due to changing operating conditions.

When a disturbance occurs in a power system, the current operating point willdeviate from its initial value (the original equilibrium point). This deviation in theparameter space will be reflected by a change in the values of the measurements inthe measurement space. As seen in Fig. 8.21, because of the disturbance, the systemmoved from the red square equilibrium point to the equilibrium point denoted bythe blue circle on the top left of the measurement space. Depending on the con-tingency, the system might have stabilized at a different equilibrium point or evenreturned back to its original equilibrium point. However, the trajectory that thecurrent point follows to reach its final destination from its initial position becomes ameasure of the domain-of-attraction of the different equilibrium points. In [22], thetrajectory was used to train a tree in CART for identifying the most suitableequilibrium points for different disturbances.


8.5.2 Future Research on Phasor-Based Controls

Future research is mainly directed towards making phasor data-driven linear ornonlinear controls that also take into account the time delays that occur in theimplementation of the control. In [23] and [24], researchers analyzed the effect ofcommunication delays on the performance of the control. Reference [23] observedthat a decentralized communication architecture that involves multiple data routinghubs are better suited for control applications which require fast control actions. In[24], the authors defined “equivalent time delay (ETD)” as a suitable criterion forcomparing different wide area control system designs. ETD is defined as the timedelay caused by the information and communication technology (ICT) system inproviding remote input signal to a feedback controller that results in an equaldamping performance for the system with remote input and local input signals. InJanuary 2013, an abrupt change in time synchronization led to a variation in thevalue of area control error (ACE) and finally resulted in a loss of 540 MW load[25]. Such time synchronization deviations can also be caused due to cyberattacks.Therefore, the causes of deviation in the time synchronization of phasor mea-surements and methods that can counter it need to be explored in more detail.

Most of the controls proposed so far have linearized the system in order toimplement the control mechanisms. However, the modern power system is highlynonlinear especially during stressed operating conditions, when the need for controlis paramount. Therefore, developing nonlinear controllers that are robust and cancoordinate different types of controllers are the need-of-the-hour. References [26,27] lay the foundation for such research by showing how wide area measurementscan be used for controlling PSSs and SVCs, in order to damp low-frequencyoscillations in a Central American system and an Australian system, respectively.

0.51

1.52

2.53

3.5

1

2

3

0.5

1

1.5

2

p1p2

p3

y1

y2

disturbance

Equilibrium piontInitial Equilibrium point

(a) parameter space (b) measurement space

Fig. 8.21 Mapping of parameter space onto measurement space (From [22])


References

1. Stengel, R. F. (1986). Stochastic optimal control: Theory and application. New York: Wiley.2. Rostamkolai, N., Phadke, A. G., Thorp, J. S., & Long, W. F. (1988). Measurement based

optimal control of high voltage AC/DC systems. IEEE Transaction on Power Systems, 3(3),1139–1145.

3. Manansala, E. C., & Phadke, A. G. (1991). An optimal centralized controller with nonlinearvoltage control. Electric Machines and Power Systems, 19, 139–156.

4. Kundur, P. (1994). Power system stability and control, Example 12.6, p. 813 McGraw-Hill.5. Smith, M. A. (1994). Improved dynamic stability using FACTS devices with phasor

measurement feedback, MS Thesis, Virginia Tech, 1994.6. Mili, L. Baldwin, T., Phadke, A. G. (1991). Phasor measurements for voltage and transient

stability monitoring and control.Workshop on Application of advanced mathematics to PowerSystems, San Francisco, September 4–6, 1991.

7. Liu., J., Thorp, J.S., & Chiang, H.-D. (1992). Modal control of large flexible space structuresusing collocated actuators and sensors. IEEE Transaction on Automatic Control, 37, 143–47,January, 1992.

8. Boyd, S., El-Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities insystems and control theory. Philadelphia: SIAM books.

9. Majumder, R., Chaudhuri, B., & Pal, B. C. (2005). A probabilistic approach to model-basedadaptive control for damping of interarea oscillations. IEEE Transaction on Power Systems,20(1), 367–374.

10. Pal, A. (2012). Coordinated control of inter-area oscillations using SMA and LMI. M.S.Thesis, Virginia Tech, Blacksburg, May, 2012.

11. Jabr, R. A., Pal, B. C., & Martins, N. (2010). A sequential conic programming approach forthe coordinated and robust design of power system stabilizers. IEEE Transaction on PowerSystems, 25(3), 1627–1637.

12. Pal, A., Thorp, J. S., Veda, S. S., & Centeno, V. A. (2013). Applying a robust controltechnique to damp low frequency oscillations in the WECC. International Journal ofElectrical Power and Energy Systems, 44(1), 638–645.

13. Pal, A., & Thorp, J. S. (2012). Co-ordinated control of inter-area oscillations using SMA andLMI. In Proceedings of the IEEE Power Energy Society Conference Innovative Smart GridTechnology, Washington D.C., January 16–20, pp 1–6

14. Verghese, G. C., Perez-Arriaga, I. J., & Schweppe, F. C. (1982). Selective modal analysiswith applications to electric power systems, Part II: The dynamic stability problem. IEEETrans on PAS, 101(9), 3126–3134.

15. Ma, J., Garlapati, S., & Thorp, J. (2011). Robust WAMS based control of inter areaoscillations. Electric Power Components and Systems, 39(9), 850–862.

16. Vance, K., Pal, A., & Thorp, J. S. (2012). A robust control technique for damping inter-areaoscillations. In Proceedings of the IEEE Power and Energy Conference at Illinois (PECI),Champaign, IL, February 24–25, 2012, pp. 1–8.

17. Fang, D. Z., Yang, X., Chung, T. S., & Wong, K. P. (2004). Adaptive fuzzy logic SVCdamping controller using strategy of oscillation energy descent. IEEE Transaction on PowerSystems, 19(3), 1414–1421.

18. Chaudhuri, B., Majumder, R., & Pal, B. C. (2004). Application of multiple model adaptivecontrol strategy for robust damping of interarea oscillations in power system. IEEETransaction on Control System Technol, 12(5), 727–736.

19. Ma, J., Wang, T., Wang, Z., & Thorp, J. S. (2013). Adaptive damping control of inter-areaoscillations based on federated Kalman filter using wide area signals. IEEE Transaction onPower Systems, 28(2), 1627–1635.

20. Bendtsen, J. D., Stoustrup, J., & Trangbaek, K. (2003). Multi-dimensional gain schedulingwith application to power plant control. In Proceedings of the 42nd IEEE ConferenceDecision Control (vol. 6, pp 6553–6558), Maui, HI, USA, December 9–12, 2003.


21. Ma, J., Wang, T., Wang, S., Gao, X., Zhu, X., Wang, Z., et al. (2014). Application of dualYoula parameterization based adaptive wide-area damping control for power systemoscillations. IEEE Transaction on Power Systems, 29(4), 1602–1610.

22. Wang, T., Pal, A., Thorp, J. S., Wang, Z., Liu, J., & Yang, Y. (2015). Multi-polytope basedadaptive robust damping control in power systems using CART. IEEE Transaction on PowerSystems, 30(4), 2063–2072.

23. Wang, Y., Yemula, P., & Bose, A. (2015). Decentralized communication and control systemsfor power system operation. IEEE Transaction on Smart Grid, 6(2), 885–893.

24. Anh, N. T., Vanfretti, L., Driesen, J., & Hertem, D. V. (2015). A quantitative method todetermine ICT delay requirements for wide-area power system damping controllers. IEEETransaction on Power Systems, 30(4), 2023–2030.

25. Bi, T., Guo, J., Xu, K., Zhang, L., & Yang, Q. (2016). The impact of time synchronizationdeviation on the performance of synchrophasor measurements and wide area damping control.IEEE Trans on Smart Grid, PP(99), 1–8.

26. Sánchez-Ayala, G., Centeno, V., & Thorp, J. (2016). Gain scheduling with classification treesfor robust centralized control of PSSs. IEEE Transaction on Power Systems, 31(3),1933–1942.

27. Vahidnia, A., Ledwich, G., & Palmer, E. W. (2016). Transient stability improvement throughwide-area controlled SVCs. IEEE Transaction on Power Systems, 31(4), 3082–3089.

References 209

Chapter 9Phasor Measurement-Enabled DecisionMaking

Anamitra Pal

9.1 Discrete Event Control

The controls developed in the previous chapter were continuous feedback controls,i.e., the control, u(t), depends on the state, x(t), at each instant of time. Ifx(t) changes, then u(t) changes with a small delay induced by communicationlatency (actually u(t) = f(x(t − Δt)). The power system, however, has other types ofcontrol which can be characterized as discrete in their dependence on state. Thesecontrols are typically stability controls and are characterized by a specific actiontaken when the state exceeds a limit. Examples are use of dynamic brake,high-speed switching of series and shunt capacitors, high-speed generator trippingtriggered by the loss of the DC line, and under frequency and voltage load shed-ding. The control action responds to the state but not continuously. They are a formof discrete-event control where state space has been partitioned by some process. Inmost early stability controls of this type, many off-line simulations were performedin order to develop the rules for application of the control.

An early phasor measurement application was of this form [1]. The attempt wasto control the power flow on the Intermountain and Pacific Intertie HVDC lines in adiscrete form in response to a collection of approximately 20 phasor measurementswhich were to be communicated to the Sylmar substation (the southern terminal ofboth DC lines). The control action was limited to ±500 MW ramps on each line.The measurement locations are shown in Fig. 9.1. The locations were chosen inpart to attempt to get appropriate coverage of dynamic events but also in collab-oration with the participating utilities in terms of accessibility of the PMUs andavailability of communication channels. The power system model was a 176-bussystem with 29 generators that included the two DC lines.

The challenge, from a control perspective, is to find a technique that willdetermine from the limited measurements which of the nine possibilities inTable 9.1 is the correct choice. In attempting to select an approach, it must berecognized that the off-line calculations of the classical approach are still


211

Fig. 9.1 The WECC system for DC line control

Table 9.1 Control options Intermountain Pacific Intertie

0 0

0 +500 MW

0 −500 MW

+500 MW 0

+500 MW +500 MW

+500 MW −500 MW

−500 MW 0

−500 MW +500 MW

−500 MW −500 MW

212 9 Phasor Measurement-Enabled Decision Making

appropriate; that is, the map from phasor measurements can be constructed fromoff-line simulations.

9.2 Decision Trees

Consider the Xs and Os in the sequence of drawings in Fig. 9.2. The horizontal andvertical lines partition the (x, y) plane into regions that are either Xs or Os. Thehorizontal line at y = 1.5 has all Xs below, the vertical line at x = −0.5 has all Xs tothe left.

The partitioning is shown in Fig. 9.2a–c with the decision tree produced by theprocess shown in Fig. 9.2c. Such trees can be constructed by software packages forlarge databases [2]. A useful property of the decision tree training process is that ifthe process is given more data than is required to do the classification it will selectthe necessary inputs and discard the others. As finding the optimum PMU locations

Fig. 9.2 a Initial partitioning. b Final partitioning. c Recursive partitioning and the final decisiontree

9.1 Discrete Event Control 213

is an issue in almost all applications, the use of decision trees has the advantage ofhelping solve the placement problem automatically.

The training data for the decision tree training involved thousands of four-secondETMSP-transient stability runs [1]. Three-phase faults on all buses and transmissionlines with fault durations from 0 to 10 s were used to produce the training cases.The intended use of the tree logic is that the phasor measurements will be presentedto the tree which will be able to decide which of the eight control actions to take.The first test is to see whether the tree can successfully predict that an event will beunstable. The tree was 95% accurate in predicting stability/instability with the errorsbeing on cases that were on the boundary between stability and instability. The treetraining takes place off-line and is time-consuming, but the response of a trainedtree is essentially limited by the delay in the arrival of the PMU data. Thisamounted to approximately 250 ms in the WECC application.

The fundamental requirement on the control is that it has a positive effect, i.e.,the control stabilized some unstable events but did not destabilize events that wouldbe stable without control.

If post-event generator angles for the first T seconds are denoted by diðtÞ,consider the objective function

F ¼ZT0

Xi

MiðdiðtÞ � dcoaÞ2dt ð9:1Þ

where the Ms are machine inertias and δcoa is the center of angle. The performanceindex F strongly penalizes diverging generator angles of large machines and pro-vides the possibility of selecting control options that minimize F over a largenumber of initiating events. With a large number of machines and control options,the computation is substantial but done off-line. Although the decision tree wastrained as a stabilizing control (Eq. 9.1) and was not designed to control islanding,the resulting control would have prevented the December 14, 1995, event in whichthe WECC separated into five islands [3].

9.2.1 Classification and Regression Tree (CART)

Classification and regression tree (CART) is a binary decision tree that is con-structed by splitting the parent node and subsequent nodes into two child nodesrepeatedly, beginning with the root node that contains the whole learning sample.The logic is based on choosing the best split among all possible splits at each parentnode so that the child nodes are purest. The CART algorithm initially grows adecision tree as large as possible and then selectively prunes it upward. Costcomplexity criterion is used in the pruning process. The objective is to attain aminimum-sized tree with minimized cost complexity [2, 4]. Being a nonparametric


decision tree learning technique, CART has been used in a variety of power systemapplications [5, 6].

Classification and regression tree (CART) data is in the form of an array withrows being the events/outcomes and columns being the measurements. In its sim-plest form, CART picks one measurement at a time for performing the splits. Butsynchrophasor data is a complex quantity that can be expressed in polar (magnitudeand angle) or Cartesian (real and imaginary) forms. If CART picks one column, ituses either the real or the imaginary part (or magnitude or angle) of the measurement,but not both. Thus, it is not able to address the complete phasor in a single split.Moreover, although CART allows splitting on linear combinations (LCs) of attri-butes, they are chosen as ‘p chooses d’ [7]. Hence, for performing the split, CART isnot particularly likely to select a linear combination that includes both the real andthe imaginary parts (or magnitude and angle components) of the same phasor.

The problem of splitting on the real or imaginary part of a complex quantity for adecision tree built using real-time PMU data was first studied in [8]. Since complexsynchrophasor measurements always have a reference angle associated with them,[8] proposed rotating along a reference as a potential solution to the problem. Toillustrate, assume a variety of load flow simulations have been performed for asystem of N buses. For the load flow data that is generated, the utility swing bus,say bus n, is the reference. Therefore, the angle of bus n will be zero for all thecases. Now, if the utility places the reference PMU at bus n, then its angle will beused as the reference for all synchrophasor measurements and no change will bemade to the load flow data. However, if the utility decides to place the referencePMU at some other bus, say bus b, where b 2 N and b 6¼ n, then in order to use thenew bus b as the reference, the angle of bus b has to be subtracted from all the otherangles of the data set. Thus, bus b will now be at an angle of zero and all the angleswill be referenced to it (instead of bus n). Graphically, this can be thought of asrotating all the data points by the angle difference between bus n and bus b.

The logic developed in [8], although novel, could not resolve all the problemsassociated with decision making using synchrophasor data. There was no guaranteethat there would always be a reference bus that gave optimal splits in all situations.Similarly, the transfer of the reference from one bus to the other was a verycumbersome process especially when applied to large systems. Finally, it could nothandle data having many attributes (such as trajectories of complex data), nor couldit perform classification in the presence of multiple classes (such as classifyingdifferent types of faults).

9.2.2 Fisher’s Linear Discriminant Appliedto Synchrophasor Data (FLDSD) Technique

Let there be two classes of data sets denoted by qb and qr that need to be separated.Let~p denote a set of observations for each sample of an event that falls within one

9.2 Decision Trees 215

of the two classes. The objective of the classification problem is to find a goodpredictor for the class q of any sample given only an observation ~p. According toFisher’s linear discriminant (FLD), an optimum split can be performed by addingthe covariance of the two data sets and using it on both the data [9]. Let the twoclasses of observations have centroids~lqb and~lqr and experimental covariance Cqb

and Cqr . Then, FLD defines a performance index q which maximizes the projectedclass differences relative to the sum of the projected within-class variability.Mathematically, this is stated as follows:

Maximize q: q ¼~mT ~lqb �~lqr

� �� 2

~mT Cqb þCqr

� �~m

ð9:2Þ

In Eq. 9.2, the vector ~m is the normal to the discriminant hyperplane. On solvingEq. 9.2, it is realized that the maximum separation occurs when

~m ¼ Cqb þCqr

� ��1~lqb �~lqr

� �ð9:3Þ

Now, if ~lqbn ¼ Cqb þCqr

� ��1~lqb and ~lqrn

¼ Cqb þCqr

� ��1~lqr , then any vector

perpendicular to ~m and passing through its midpoint is given by

12

~lqbn�~lqrn

� �þ~wa ð9:4Þ

In Eq. 9.4, α is the optimizing variable. For finding the optimal hyperplane, onehas to minimize the following:

~p� 12

~lqbn�~lqrn

� ��~wa

��2

ð9:5Þ

On solving Eq. 9.5, using weighted least squares (WLS) algorithm, the estimateof α comes out to be the following:

a ¼ ~wT~w� ��1

~wT� �

~p� 12

~lqbn�~lqrn

� �� ð9:6Þ

Using this value of a; we obtain the splitting variable D as follows:

D ¼ I �~w ~wT~w� ��1

~wTh i

~p� 12

~lqbn�~lqrn

� �� ð9:7Þ

From Eq. 9.7, it can be inferred that the original multivariate data can bereplaced in CART by the single variable D: It is also easy to show that thehyperplane perpendicularly bisecting the line joining the two centroids in the new


coordinate system is equivalent to a rotation of the hyperplane perpendicularlybisecting the line joining the two centroids in the original coordinate system,thereby making it no longer perpendicular. This is the geometric interpretation ofFisher’s linear discriminant (FLD). As such, this technique is called FLDSD—Fisher’s linear discriminant applied to synchrophasor data [10].

Example 9.1 Performing splits on complex dataFigure 9.3a denotes a randomly generated set of 40 data points. Without any loss ofgenerality, it can be assumed that the x-axis depicts the real voltages in p.u. whereasthe y-axis depicts the imaginary component of the voltages also in p.u., whenreferenced to the swing bus of an arbitrary power system. The blue circles can thencorrespond to the stable voltages while the red circles will denote the unstablevoltages. The dotted line shows the best splits obtained when this data is fed intoCART. From the figure, it becomes clear that two blue circles have been mis-classified by the first split. Subsequent splits separate the two blue circles from thered circles.

Changing the swing bus for this system will translate to rotating Fig. 9.3a by theangle differences between the original swing bus and the newer swing bus. Twosuch rotations are shown in Fig. 9.3b, c, respectively. In Fig. 9.3b, a single split isbarely able to separate the two distributions, whereas in Fig. 9.3c it is realized that

Fig. 9.3 a Split using traditional CART algorithm. b Split after rotating along arbitrary reference(phase shift of pi/20 radians). c Split after rotating along arbitrary reference (phase shift of—pi/90radians). d Split obtained using FLDSD


an inappropriate choice of reference worsens the performance considerably. Fromthe figures, it becomes clear that rotating along a reference may not always givegood results. The MATLAB results are shown in Fig. 9.3d. The black stars denotethe centroids of the two distributions. From the figure, it is realized that thehyperplane created using FDLSD is able to very easily separate the two groups ofdata.

The FLDSD technique can also be used to separate multiple (more than two)classes, by taking two classes at a time. In order to do so, distances of the datapoints from all the hyperplanes must be initially computed. Taking two classes at atime, Eq. 9.7 can be used to compute for the distances. For n-class distribution, thenumber of hyperplanes formed is as follows:

Number of Hyperplanes ¼ n� n� 1ð Þ2

ð9:8Þ

The distances to the hyperplanes are then fed into CART for selecting theoptimal distance variables required for performing the split. Since all the inputvariables have single attributes, CART can directly select the distance variable thatwill result in the best possible split. By selecting two classes at a time, and pro-ceeding until all the class combinations have been covered, the data can be suitableclassified.

9.2.3 Applications of FLDSD in Power Systems

In Sect. 9.2.2, an algorithm called FLDSD was developed for making decisionsinvolving complex synchrophasor data. CART, which is a nonparametric decisiontree learning technique, was used in the development of that algorithm. FLDSDshowed how the CART logic can be used to make decisions while consideringmeasurements having multiple attributes without needing a reference. Since it isonly the inputs to CART which were modified, and not the logic on which CARToperates, the FLDSD-with-CART approach can be readily applied to any engi-neering problem that involves decision making based on the multivariate data. Inthis section, its uses in the development of an adaptive protection scheme and inidentifying different events captured by a linear state estimator (LSE) are demon-strated. For knowing about other applications of FLDSD in power systems, [10–12]are suggested for further reading.

It has been suggested that the likelihood of hidden failures and potential cas-cading events can be significantly reduced by adjusting the security/dependabilitybalance of protection systems in accordance with the prevailing system conditions.PMUs placed at critical locations in the network provide information regarding thestate of the power system. Using that information, the system can be classified assafe or stressed. A safe system is biased toward dependability whereas a stressedsystem is biased toward security. Dependability is ‘the degree of certainty that a


relay or relay system will operate correctly’; that is, it is a measure of the certaintythat the relays will operate correctly for all the faults for which they are designed tooperate. On the other hand, security ‘relates to the degree of certainty that a relay orrelay system will not operate incorrectly.’ Although, traditionally, protection sys-tems are biased toward dependability, under the stressed system conditions, afavorable bias toward security is more beneficial [13, 14].

A decision tree-based adaptive voting scheme was proposed in [6] to alter thesecurity-dependability balance in accordance with the current state of the system.The voting scheme consists of a set of three independent and redundant relays.Based on the PMU measurements, if the system state is initially found to be ‘safe,’then in the case of a fault, voting is disabled and any of the three relays can trip theline (bias toward dependability). However, if the PMU measurements indicate thatthe system is in a ‘stressed’ condition, then the voting scheme is enabled and theline will trip only if two or more relays see the fault (bias toward security). The busvoltage angles and the currents flowing in the lines were used for decision makingwhile the nodes of the decision tree identified the locations of the PMUs.

This adaptive voting scheme was applied to the heavy winter (HW) and heavysummer (HS) models of a 4000-bus system of California. The total number oftraining cases in the HW case was 4150, whereas in the HS case it was 11,367. Thetotal number of ‘out-of-sample’ cases was 660 for the HW case and 1155 for the HScase. The details behind the creation of the cases can be found in [15]. The resultsobtained using the traditional CART approach are shown in Fig. 9.4a, b. The term‘red’ indicates a ‘stressed’ system, while the term ‘blue’ indicates a ‘safe’ system. Inorder to apply the FLDSD technique to this scheme, the currents were expressed asa complex number, and each complex number was treated as a single entity. Theresulting tree that was obtained is shown in Fig. 9.4c, d. Table 9.2 compares theresults obtained using the FLDSD-with-CART technique with those obtained usingthe traditional CART approach. From the figures and the table, it is realized that byusing the FLDSD approach, a smaller tree (indicating the requirement of lessernumber of PMUs) is able to provide higher classification accuracy. Table 9.3compares the performance of the decision trees obtained using the FLDSD tech-nique with that obtained by rotating along a reference as was proposed in [8]. FromTables 9.2 and 9.3, it becomes clear that the FLDSD technique is more accurate inmaking decisions involving complex synchrophasor data. There are applicationswhere only the voltage angle, for example, is relevant for making decisions.However, relaying in particular deals with complex quantities and the FLDSDtechnique developed here is necessary for making an optimal split [16].

A PMU-only LSE has been developed for the 500-kV network of DominionVirginia Power, a power utility of the USA. The LSE generates an output phasor atthe rate of 30 samples per second [17]. Since FERC: CEII (Federal EnergyRegulatory Commission: Critical Energy Infrastructure Information) preventspublishing results obtained from actual systems, the performance of the LSE wastested on the IEEE-118-bus system. A typical output of the LSE for a three-phasefault on line 26–30 of this system followed by an unsuccessful high-speed reclose isshown in Fig. 9.5. The figure shows the trajectories of the eleven complex positive


sequence voltages for one second. The display has been split into four windows forclarity. The measurements start from the black circles and end on the black squares.The fault point is indicated by a plus sign whereas the reclose operation is indicatedby a diamond. From Fig. 9.5, it can be realized that unless a label is provided to the

Fig. 9.4 a Decision tree for the heavy winter case obtained using traditional CART algorithm.b Decision tree for the heavy summer case obtained using traditional CART algorithm. c Decisiontree for the heavy winter case obtained using FLDSD technique (from Pal et al. [16]). d Decisiontree for the heavy summer case obtained using FLDSD technique (from Pal et al. [16])

Table 9.2 Comparing size of decision trees obtained using traditional CART and the FLDSDtechnique. (from Pal et al. [16])

Scenario Using traditional CART approach Using FLDSD-with-CARTapproach

#Nodes Misclassification rate (%) #Nodes Misclassification rate (%)

Heavy winter 4 1.00 3 0.14

Heavy summer 5 1.00 3 0.20

Table 9.3 Comparing performance of decision trees obtained using rotation of referenceapproach and the FLDSD technique

Scenario #Cases Using a reference forrotation

Using FLDSD technique

#Errors Accuracy (%) #Errors Accuracy (%)

Heavy winter 4810 48 99.00 26 99.46

Heavy summer 12,522 172 98.63 78 99.38

From Pal et al. [16]


plots that the estimator generates, it will be very difficult to identify the event thathas occurred. The FLDSD algorithm provides a solution to this problem by treatingthe trajectory of complex numbers as a single attribute which can then be used forclassifying different events.

For the ten 345-kV lines of the IEEE 118-bus system, ten parallel decision treesare created (one tree per line). Time-tagged breaker statuses identify the relevanttree in the case of an event. The types of events that were identified were singleline-to-ground (SLG) faults and three-phase-to-ground (TPG) faults. For both thefaults, three post-event scenarios (called classes) are considered: no reclose (NR),successful high-speed reclose (SHSR), and unsuccessful high-speed reclose(USHSR). Finally, Zone II (Z2) operations following the occurrence of a SLG or aTPG are also considered. A total of 6674 cases are created for the ten high-voltagelines. The details behind the creation of the cases can be found in [18].

Taking 345-kV line between buses 38 and 65 as an example, the complexvoltages of the two buses were obtained for different classes of events. Since bothbuses 38 and 65 were connected to other 345-kV buses, Zone II operation could bedetected on either end. This resulted in a ten-class classification—three classes forSLG fault, three classes for TPG fault, two classes for Z2 operation on bus 38 end,and two classes for Z2 operation on bus 65 end. A second’s worth of data startingfrom the time of the fault was used for classification purposes. In order to perform a

ten-class classification using FLDSD, distances to 45 ¼ 10� 10�1ð Þ2

� �hyperplanes

were computed and set as inputs to CART. The resulting decision tree is shown inFig. 9.6, where ‘di−j’ denotes the distance of the individual points from thehyperplane separating classes i and j. From the figure, it is observed that for the line38–65, CART chose the distances d1–2, d1–5, d1–6, d1–10, d2–8, and d3–6 as the

Fig. 9.5 345-kV-bus voltage trajectories of IEEE 118-bus system after a three-phase fault on line26–30 followed by an unsuccessful high-speed reclose


optimal variables for performing the splits. Given an actual event, the distancesfrom each data point to these six hyperplanes need to be computed and the treefollowed to the respective terminal node to identify the event that has beencaptured.

PMU data has been used for making critical decisions for quite some time now,but the true potential of these devices has not yet been realized. When placed at abus, they provide real-time measurements of voltages and currents connecting thatbus with the rest of the system. But, being complex quantities, these measurementshave not been addressed completely for decision-making purposes until thedevelopment of the FLDSD technique. Using the FLDSD-with-CART techniquecomplex numbers as well as trajectories of complex numbers can be represented bya single variable. This variable can then be used to perform optimal splits in CART.By utilizing both the real and the imaginary components of the synchrophasor datasimultaneously, a new understanding of PMU data is obtained. The FLDSDtechnique will not only be useful for solving synchrophasor-based problems, but itwill also be useful in other engineering applications that involve decision makingbased on the multivariate data.

9.3 Synchrophasor Data Conditioning and Validation

All energy management systems (EMS) depend on algorithms that process rawpower system data for computing the states of the system. The traditional nonlinearstate estimation techniques rely on supervisory control and data acquisition(SCADA) measurements for providing this raw data. The state estimator results inturn become the basis for other network applications such as real-time contingency

Fig. 9.6 Decision Tree generated for identifying a fault on line 38–65 of IEEE 118-bus systemusing FLDSD technique


analysis and system reliability studies. As the electric utility industry becomes moreand more familiar with synchrophasor technology, the transition of state estimationfrom a traditional nonlinear formulation to one which is purely phasor-based andlinear becomes more and more realistic.

A purely PMU-based state estimator has considerable advantages over a purelySCADA-based or a mixed (SCADA-and-PMU based) state estimator. Firstly, beinglinear, the PMU-only state estimator does not require any iteration. Secondly, it isfree from the data scan that is required in conventional estimators. Thirdly, despiteits formulation as a state estimation problem, the time-tagged data produces anestimate at such a fast enough rate that it can be considered to be truly dynamic.However, similar to the conventional state estimators, a PMU-only state estimatoralso depends on a consistent, reliable stream of input data. Due to the streamingnature of the phasor data, downstream applications which use this data are vul-nerable to network congestion, configuration errors, and equipment failures. Someof the practical data quality issues associated with PMU data are highlighted in [19].Therefore, a computationally simple and efficient methodology for cleaning syn-chrophasor data is required.

9.3.1 Three Sample-Based Quadratic Prediction Algorithm

In order to detect bad data and switching operations that affect network topology, atechnique to predict the next value of every voltage estimate from previous esti-mates of the same voltage was developed in [20]. It is realized that linear changes inload at constant power factor resulted in a quadratic change in the complex voltage.Mathematically, this is stated as follows:

vk tð Þ ¼ ak þ bktþ ckt2 ð9:8Þ

In Eq. 9.8, ak; bk; ck are complex quantities. Equation 9.8 is a polynomial ofdegree m� 1 as shown in Eq. 9.9a and which can be interpolated and expressed inthe form shown in Eq. 9.9b.

y tð Þ ¼ am�1tm�1 þ am�2tm�2 þ am�3tm�3 þ . . .þ a1tþ a0 ð9:9aÞ

y 1ð Þy 2ð Þ...

y m� 1ð Þy mð Þ

266664

377775 ¼

1 1 12 . . . 1m�1

1 2 22 . . . 2m�1

1 ... ..

. . .. ..

.

1 m� 1ð Þ m� 1ð Þ2 . . . m� 1ð Þm�1

1 m m2 . . . mm�1

266664

377775

a0a1...

am�1

am

2666664

3777775 ð9:9bÞ

In Eqs. 9.9a–b, a0; a1; . . .; am�1 are constant coefficients. Equation 9.9bcan be written as y ¼ Va; where V is the Vandermonde matrix. Now, if

9.3 Synchrophasor Data Conditioning and Validation 223

b ¼ b1 b2 . . . bm�1 bm½ � denotes the first row of V�1, then on pre-multiplying both sides of y ¼ Va with b, Eq. 9.10a is obtained in which y 0ð Þ equalsa0, in accordance with Eq. 9.9a [21].

b1y 1ð Þþ b2y 2ð Þþ . . .þ bm�1y m� 1ð Þþ bmy mð Þ ð9:10aÞ

Finally, Eq. 9.10a can be rearranged and written as Eq. 9.10b.

y mð Þ ¼ � bm�1

bmy m� 1ð Þ � bm�2

bmy m� 2ð Þ � . . .� b1

bmy 1ð Þþ y 0ð Þ ð9:10bÞ

The rows of the Pascal’s triangle (shown in Fig. 9.7) indicate the coefficients ofthe model with alternate signs. From a comparison of Eqs. 9.8 and 9.9a, it isrealized that m ¼ 3, and therefore, the third row of the Pascal’s triangle gives thedesired coefficients. As such, the new voltage can be predicted from the estimates ofthe three previous voltages using Eq. 9.11.

y nð Þ ¼ 3y n� 1ð Þ � 3y n� 2ð Þþ y n� 3ð Þ ð9:11Þ

When Eq. 9.11 is used in conjunction with the fact that the measurements aremade at a very high speed (e.g., 30 times a second), it can be realized that the loadchanges occurring at such high rates can be approximated to be linear. This, in turn,implies that the three sample-based quadratic prediction algorithm can be used topredict future voltages irrespective of any change occurring in system topology[22]. Equation 9.11 can also be expressed in the Kalman filter notation as shown inEq. 9.12. This equation will be used in Sect. 9.3.2 for developing a methodologyfor cleaning synchrophasor data.

y kþ 1jkð Þ ¼ 3y kjkð Þ � 3y k � 1jk � 1ð Þþ y k � 2jk � 2ð Þ ð9:12Þ

Fig. 9.7 Pascal’s triangle


The discovery of this ‘quadratic’ dependence of the estimate of the future stateon the estimates of the three previous states gives one the ability to detect datainconsistencies. By using the three prior estimates of a particular measurement, thenext value of that measurement can be predicted. Next, by comparing this estimatewith the actual measurement, an observation residual can be computed. Finally, bycomparing the observation residual with a pre-defined threshold, discrepancies inthe incoming data can be detected.

Table 9.4 Complex voltages for one second at 30 samples per second for four buses of IEEE118-bus system

10 30 64 68

Mag. Angle Mag. Angle Mag. Angle Mag. Angle

1.0498 0.5682 0.9852 0.2723 0.9833 0.3630 1.0031 0.4183

1.0498 0.5669 0.9852 0.2710 0.9833 0.3617 1.0031 0.4169

1.0498 0.5656 0.9852 0.2697 0.9833 0.3603 1.0031 0.4156

1.0498 0.5643 0.9851 0.2684 0.9833 0.3589 1.0031 0.4142

1.0498 0.5630 0.9851 0.2670 0.9833 0.3575 1.0031 0.4129

1.0498 0.5616 0.9851 0.2657 0.9833 0.3561 1.0031 0.4115

1.0498 0.5603 0.9851 0.2643 0.9833 0.3546 1.0031 0.4101

1.0498 0.5589 0.9851 0.2629 0.9833 0.3532 1.0031 0.4086

1.0498 0.5575 0.9851 0.2615 0.9832 0.3517 1.0030 0.4072

1.0498 0.5561 0.9851 0.2601 0.9832 0.3502 1.0030 0.4058

1.0498 0.5547 0.9851 0.2587 0.9832 0.3487 1.0030 0.4043

1.0498 0.5533 0.9851 0.2572 0.9832 0.3472 1.0030 0.4028

1.0498 0.5518 0.9851 0.2558 0.9832 0.3457 1.0030 0.4013

1.0498 0.5504 0.9851 0.2543 0.9832 0.3442 1.0030 0.3998

1.0498 0.5489 0.9851 0.2528 0.9832 0.3426 1.0030 0.3983

1.0497 0.5474 0.9851 0.2513 0.9832 0.3411 1.0030 0.3968

1.0497 0.5460 0.9851 0.2498 0.9832 0.3395 1.0030 0.3952

1.0497 0.5444 0.9851 0.2483 0.9832 0.3379 1.0030 0.3937

1.0497 0.5429 0.9851 0.2467 0.9832 0.3363 1.0030 0.3921

1.0497 0.5414 0.9851 0.2452 0.9832 0.3347 1.0030 0.3905

1.0497 0.5398 0.9851 0.2436 0.9832 0.3331 1.0030 0.3889

1.0497 0.5383 0.9851 0.2420 0.9832 0.3314 1.0030 0.3873

1.0497 0.5367 0.9851 0.2404 0.9832 0.3298 1.0030 0.3857

1.0497 0.5351 0.9851 0.2388 0.9832 0.3281 1.0030 0.3840

1.0497 0.5335 0.9851 0.2372 0.9832 0.3264 1.0030 0.3824

1.0497 0.5319 0.9851 0.2356 0.9832 0.3247 1.0030 0.3807

1.0497 0.5303 0.9851 0.2339 0.9832 0.3230 1.0030 0.3790

1.0497 0.5286 0.9851 0.2322 0.9832 0.3212 1.0030 0.3773

1.0497 0.5269 0.9851 0.2306 0.9832 0.3195 1.0030 0.3756

1.0497 0.5253 0.9851 0.2289 0.9832 0.3177 1.0030 0.3739


Example 9.2 Performance of three sample-based quadratic prediction algorithmThe IEEE 118-bus system has 11 high-voltage (HV) buses: 8, 9, 10, 26, 30, 38, 63,64, 65, 68, and 81. For illustration purposes, complex voltages in p.u. of four of thosebuses (10, 30, 64, and 68) for one second are shown in Table 9.4. The data wascreated by increasing the load and generation at constant power factor so as to mimicthe morning load pickup. The increase rate was set at 60% over a period of 1 h.A second’s worth of data in the midst of the simulation was picked for analysis. Theoutput rate was set at 30 samples per second. The numerical values given in Table 9.4were obtained with MATLAB. Figure 9.8 shows the maximum errors that wereobtained for the different buses over the length of the simulation. From the figure, itbecomes clear that for all the four buses, the maximum errors in the prediction wereof the order of 10−4. It must also be pointed out here that the prediction started fromthe fourth sample as three data points are needed to make the first prediction.

Fig. 9.8 Maximum error inpredicting future voltagebased on previous threesamples of the same voltage


9.3.2 A Methodology for Performing Synchrophasor DataConditioning and Validation

A technique was developed in Sect. 9.3.1 which predicted the next value of eachvoltage estimate from a three-sample history of previous estimates of the samevoltage. Figure 9.9 shows the application of the three sample-based quadraticprediction algorithm on a portion of field data. The figure shows samples andestimates of a complex voltage at 30 samples per second during a period of alow-frequency oscillation at off-nominal frequency. The green line is the actualPMU data while the red circles are the estimates. The oscillation starts from the topright and moves to the bottom left. Since the estimate matches very well with theactual data, this algorithm can be used to detect bad data by using an observationresidual. It can also be used to smooth data by using subsequent measurements toobtain a better estimate which can be thought of as a technique for supplyingmissing data. Thus, it provides a simple and elegant solution to the synchrophasordata quality problem. Moreover, even if a LSE is not desired, this data conditioningand validation methodology can be used independently for detecting bad data andfinding the best estimate.

The block diagram describing the data conditioning and validation process isshown in Fig. 9.10. This is designed so as to fit the linear state estimation for-mulation described in [17]. The raw data is initially passed through the datamonitoring block. It is responsible for validating the quality of the incoming dataand providing information when data cleaning is required. The cleaning block is

Fig. 9.9 Performance of the three sample-based quadratic prediction algorithm on realsynchrophasor data


responsible for conditioning the data when it is possible to do so. The LSE usesknowledge obtained from both of these blocks for performing bad data detection aswell as for providing a best estimate when there is measurement redundancy.

Two simple techniques for validating the quality of the incoming data evenbefore it is received by the LSE are (a) plausibility checks and (b) signal-to-noiseratio (SNR). Plausibility checks comprise of a set of online filters which performsanity checks on the incoming data. In addition to preventing unworthy data frombeing consumed by the application, plausibility checks can also make engineersaware of data quality problems that may require manual intervention. Differenttypes of plausibility checks that would cause measurements to be eliminated beforestate estimation is performed are as follows:

• In-service buses having zero, near-zero, or negative voltage magnitude mea-surement readings,

• For three-phase systems, phasor groups with improper phase relationships,• Bad, missing, or repeated time stamps,• Frequency excursion of 0.1 Hz or more from the average nominal value

(60 Hz),• Rate of change of frequency excursion of 0.03 or more from the average

nominal value (zero), and• Measurements which have a C37.118 status word showing the DataValid bit or

the PMUSync bit or the PMUError bit asserted.

Evaluating the signal-to-noise ratio (SNR) of a signal is an efficient method tomonitor the quality of that signal. Since reconstructing the original sinusoid wouldbe difficult, the assumption made is that the components of the phasor (magnitudeand unwrapped referenced phase angle) are DC signals. Under such conditions,SNR is the mean of the signal divided by the standard deviation of the signal takenover a moving window as described in Eq. 9.13.

SNRDC in dBð Þ ¼ 10 loglr

� �¼ 10 log

meanstd: dev:

� �ð9:13Þ

Signal-to-noise ratio evaluation not only indicates loose connections or potentialhardware problems but also helps diagnose equipment issues much more accurately

Fig. 9.10 Data conditioning and validation module


than the raw voltage measurement. An example of this is shown in Fig. 9.11a, b.Figure 9.11a depicts the SNR magnitude plot of a potential transformer (PT) of a500-kV bus few days prior to a fault. Figure 9.11b depicts the SNR angle plot forthe same PT. It is clear from the figures that both the magnitude and the angle plotsshow a clear indication of a potential problem (identified by the wider spread of theC-phase data in comparison with the other phases). Moreover, since these types ofdevices fail slowly, by regularly monitoring the SNR, signs of failure can beidentified days in advance. For instance, the SNR magnitude and angle plot shownin Fig. 9.11 captured the faulty C-phase data three days before the PT actuallyfailed. Also, since SNR is a relative measure whereas the raw voltage measurementis an absolute one, SNR becomes a suitable candidate for setting alarm limits.

A key assumption made during the computation of the SNR is thequasi-steady-state operating condition of the power system. The size of the movingwindow used to calculate the mean and standard deviation components is criticalfor establishing a baseline criterion for alarming as it dictates the sensitivity of thecalculation. Therefore, an intelligent alarming scheme would be necessary to pre-vent misinterpretation of the SNR during oscillations and when discrete changesoccur in the network. The former can be avoided by using an oscillation detection

Fig. 9.11 a SNR of phase magnitude prior to C-phase PT failure (from Jones et al. [26]). b SNRof phase angle prior to C-phase PT failure (from Jones et al. [26])


algorithm while the latter can be circumvented by alarming only when the SNR ishigher than a pre-defined threshold for a certain period of time.

If the data monitoring block finds issues with the incoming data, then that data ispassed through the data cleaning block which tries to eliminate the issues. Twotechniques for cleaning the synchrophasor data are developed based on the threesample-based quadratic prediction algorithm proposed in Sect. 9.3.1. The first oneis Kalman filter-based filtering and the second one is Kalman filter-basedsmoothing. The classical model of a Kalman filter is given in Eq. 9.14, wherethe symbols have their usual meanings [23].

x kþ 1ð Þ ¼ U kþ 1; kð Þx kð ÞþC kþ 1; kð Þw kð Þz kþ 1ð Þ ¼ H kþ 1ð Þx kþ 1ð Þþ v kþ 1ð Þ ð9:14Þ

Equation 9.14 can be solved recursively as shown in Eqs. 9.15a–e.

x kþ 1jkþ 1ð Þ ¼ U kþ 1; kð Þx kjkð ÞþK kþ 1ð Þ z kþ 1ð Þ � z kþ 1jkð Þð Þ ð9:15aÞ

z kþ 1jkð Þ ¼ H kþ 1ð Þx kþ 1jkð Þ ð9:15bÞ

In Eq. 9.15a, K kþ 1ð Þ is the gain of the Kalman filter and it can be recursivelycalculated from the following equations:

P kþ 1jkð Þ ¼ U kþ 1; kð ÞP kjkð ÞUT kþ 1; kð ÞþC kþ 1; kð ÞQ kð ÞCT kþ 1; kð Þð9:15cÞ

K kþ 1ð Þ ¼ P kþ 1jkð ÞHT kþ 1ð Þ H kþ 1ð ÞP kþ 1jkð ÞHT kþ 1ð ÞþR kþ 1ð Þ �1

ð9:15dÞ

P kþ 1jkþ 1ð Þ ¼ I�K kþ 1ð ÞH kþ 1ð Þ½ �P kþ 1jkð Þ ð9:15eÞ

In Eqs. 9.15c–d, P is the error covariance matrix, Q is the process noisecovariance matrix, and R is the measurement noise covariance matrix. However,when applied to the synchrophasor data conditioning problem, the Kalman filtermodel can be simplified further. Due to the nature of the three sample-basedquadratic prediction algorithm, adjacent state vectors share two of the three statevariables in common, yielding an augmented state vector. This can be imagined as amoving window that contains three snapshots of the system at any given time andwhich moves forward only one snapshot at a time. Therefore, for predicting the nextstate based on Eq. 9.12, x kjkð Þ and x kþ 1jkð Þ can be expressed as follows:

x kjkð Þ ¼x kjkð Þ

x k � 1jk � 1ð Þx k � 2jk � 2ð Þ

24

35 ð9:16aÞ


x kþ 1jkð Þ ¼x kþ 1jkð Þx kjkð Þ

x k � 1jk � 1ð Þ

24

35 ð9:16bÞ

Since the estimate of the future state depends on the three previous state esti-mates, for filtering purposes, it makes sense to depict x kjkð Þ and x kþ 1jkð Þ as 3� 1matrices. Now, it is known that U kþ 1; kð Þ relates the kþ 1 state to the k state, thatis,

x kþ 1jkð Þ ¼ U kþ 1; kð Þx kjkð Þ ð9:16cÞ

Therefore, based on Eqs. 9.12 and 9.16a–c, U kþ 1; kð Þ can be formulated as aconstant as shown in Eq. 9.16d.

U kþ 1; kð Þ ¼3 �3 11 0 00 1 0

24

35 ð9:16dÞ

It is to be noted here that in Eqs. 9.16a–b, the states are the complex voltagemeasurements. Therefore, without any loss of generality, Eqs. 9.17a–b can bewritten.

z kþ 1ð Þ ¼ x kþ 1ð Þþ v kþ 1ð Þ ð9:17aÞ

z kþ 1jkð Þ ¼ x kþ 1jkð Þ ð9:17bÞ

Substituting Eqs. 9.17a–b in Eq. 9.15b, the value of H kþ 1ð Þ comes out to be asfollows:

H kþ 1ð Þ ¼ 1 0 0½ � ð9:17cÞ

Thus, on substituting Eqs. 9.16d and 9.17c in RHS of 9.15a–b, respectively, asimplified model of the filtering technique will be developed as shown inEqs. 9.18a–b.

x kþ 1jkþ 1ð Þ ¼3 �3 11 0 00 1 0

24

35x kjkð ÞþK kþ 1ð Þ z kþ 1ð Þ � z kþ 1jkð Þð Þ ð9:18aÞ

z kþ 1jkð Þ ¼ 1 0 0½ �x kþ 1jkð Þ ð9:18bÞ

Example 9.3 Performance of the Kalman filter developed on the basis of the threesample-based quadratic prediction algorithm


The simulation setup used for Example 9.2 is repeated, except that Gaussian errorsrandomly picked from a normal distribution having zero mean and standard devi-ations of 0.02% for magnitudes and 0.104° for angles are introduced in the complexvoltage measurements. The new data is shown in Table 9.5. MATLAB resultsdescribe the performance of the Kalman filter developed on the basis of the threesample-based quadratic prediction algorithm on this data set. The numerical valuesgiven in Table 9.5 are obtained with MATLAB. Figure 9.12 shows the maximumerrors that were obtained for the different buses over the length of the simulation.From the figure, it becomes clear that for all the four buses, the maximum errors inthe prediction were of the order of 10−3.

The three sample-based quadratic prediction algorithm that had been used forfiltering purposes also applies to the smoothing process. Kalman filter-basedsmoothing estimates the previous states of the system using current measurements.Mathematically, this means solving for x kjjð Þ where j[ k [23]. Three types ofsmoothing techniques are as follows: fixed-interval smoothing, fixed-pointsmoothing, and fixed-lag smoothing. A fixed-interval smoother is primarily meantfor post-experiment analysis and is therefore not suitable for an online application,as is needed here. The fixed-point smoother is also not appropriate for an onlineapplication as it is concerned with the optimal estimate of the state at a singleinstant of time. The fixed-lag smoother is ideal for online applications as it allowsthe point of interest to move forward by recursively removing the oldest mea-surement and covariance information and including the latest. The model of thefixed-lag smoother has a discrete-time state equation in the form of a recursiveKalman filter with an augmented state vector, an associated augmented dynamicalsystem, and an augmented measurement equation as shown in Eq. 9.19. Moredetails about it can be found in [24].

x kþ 1jkþ 1ð Þx kjkþ 1ð Þ

x k � 1jkþ 1ð Þ...

x kþ 1� Njkþ 1ð Þ

266666664

377777775¼

U kþ 1; kð ÞI0

..

.

0

0

0I

..

.

0

0

00

..

.

. . .

. . . 0

. . . 0

. . . 0

. .. ..

.

I 0

26666664

37777775

x kjkð Þx k � 1jkð Þx k � 2jkð Þ

..

.

x k � Njkð Þ

266666664

377777775

þ

K kþ 1ð ÞK1 kþ 1ð ÞK2 kþ 1ð Þ

..

.

KN kþ 1ð Þ

266666664

377777775

z kþ 1ð Þ �H kþ 1ð Þx kþ 1jkð Þð Þ ð9:19Þ

In Eq. 9.19, N is the window length (or lag) and K kþ 1ð Þ,K1 kþ 1ð Þ. . .KN kþ 1ð Þ are the gain matrices. A judicious choice of window lengthis important because although a higher value of N would introduce more delay


between the received measurement and the smoothed state estimate, it will alsoimprove the quality of the estimate by a significant amount. The reason for thisbeing that an estimate obtained using x kþ 1� Njkþ 1ð Þ will be intuitively betterthan one obtained using x kþ 1� Njkþ 1� Nð Þ: A suitable window length wasN ¼ 3: The smoothing technique works in the same way as the filtering techniquewith regards to time and measurement updates. Also, for both the filtering as well asthe smoothing techniques, the initial conditions for all of the estimates and

Table 9.5 Complex voltages for one second at 30 samples per second for four buses of IEEE118-bus system with Gaussian errors embedded in them

10 30 64 68


1.0489 0.5673 0.9835 0.2704 0.9832 0.3624 1.0060 0.4186

1.0530 0.5648 0.9836 0.2715 0.9832 0.3642 1.0021 0.4165

1.0514 0.5659 0.9858 0.2692 0.9836 0.3623 1.0043 0.4163

1.0484 0.5647 0.9841 0.2663 0.9864 0.3588 1.0049 0.4170

1.0499 0.5627 0.9830 0.2677 0.9821 0.3609 1.0030 0.4122

1.0519 0.5643 0.9882 0.2626 0.9813 0.3586 1.0022 0.4141

1.0500 0.5611 0.9877 0.2629 0.9823 0.3544 1.0026 0.4102

1.0497 0.5553 0.9829 0.2629 0.9862 0.3526 1.0040 0.4099

1.0521 0.5556 0.9851 0.2627 0.9823 0.3502 1.0029 0.4063

1.0507 0.5537 0.9887 0.2575 0.9811 0.3522 1.0045 0.4028

1.0508 0.5537 0.9843 0.2599 0.9838 0.3508 1.0039 0.4084

1.0497 0.5500 0.9845 0.2583 0.9835 0.3492 1.0020 0.3999

1.0479 0.5503 0.9837 0.2549 0.9819 0.3462 1.0025 0.4000

1.0515 0.5484 0.9849 0.2503 0.9825 0.3435 1.0047 0.3984

1.0508 0.5486 0.9853 0.2506 0.9853 0.3432 0.9989 0.3978

1.0453 0.5501 0.9815 0.2506 0.9848 0.3434 1.0031 0.3972

1.0539 0.5496 0.9827 0.2492 0.9835 0.3427 1.0040 0.3930

1.0491 0.5422 0.9855 0.2501 0.9809 0.3382 1.0050 0.3961

1.0486 0.5420 0.9872 0.2463 0.9817 0.3338 0.9998 0.3906

1.0511 0.5410 0.9817 0.2484 0.9815 0.3357 1.0017 0.3934

1.0488 0.5372 0.9855 0.2434 0.9804 0.3333 1.0043 0.3867

1.0509 0.5406 0.9845 0.2418 0.9831 0.3319 1.0059 0.3894

1.0486 0.5375 0.9882 0.2408 0.9850 0.3295 1.0052 0.3870

1.0511 0.5359 0.9847 0.2403 0.9821 0.3308 1.0072 0.3843

1.0464 0.5340 0.9855 0.2352 0.9872 0.3301 1.0034 0.3819

1.0486 0.5345 0.9846 0.2355 0.9820 0.3240 1.0019 0.3793

1.0445 0.5328 0.9842 0.2365 0.9850 0.3228 1.0052 0.3795

1.0516 0.5267 0.9878 0.2330 0.9831 0.3220 1.0061 0.3788

1.0480 0.5266 0.9874 0.2305 0.9826 0.3216 1.0061 0.3775

1.0502 0.5267 0.9844 0.2267 0.9825 0.3197 1.0060 0.3743


covariances can be zero without requiring an additional Kalman filter to providethis information as long as it is acknowledged that the values for these will not becorrect until the first window has been completely filled with data. Thus, in essence,the fixed-lag smoothing algorithm is an extension of the filtering technique.Together, the methodology for conditioning and validating synchrophasor data isexpected to address the issues of loss of data from one or several PMUs, loss ofsignals/corrupt signals in one or more PMUs, and stale (non-refreshing) data.

Example 9.4 Performance of the Kalman filter-based smoother developed on thebasis of the three sample-based quadratic prediction algorithmThe simulation setup used for Example 9.3 is repeated here with the additionalcomplexity that some of the data are missing, repeated, or corrupt. The new data isshown in Table 9.6. The locations of the bad data are highlighted in red.

Fig. 9.12 Maximum error inpredicting future voltage thathas noise using a Kalmanfilter developed based on thethree sample-based quadraticprediction algorithm


The MATLAB function BMFLS.m [25] was used to implement the Kalmanfilter-based fixed-lag smoother. The length of the lag window was set at 3. Theintegration of the data with the MATLAB function is left as an exercise. The resultsare shown in Fig. 9.13a–d.

The synchrophasor conditioning and validation scheme based on the threesample-based quadratic prediction algorithm and Kalman filtering/smoothing isable to address data quality issues up to a penetration factor of 20% [26]. For higherpercentages of bad data, the chances of the algorithm diverging increases. This is

Table 9.6 Complex voltages for one second at 30 samples per second for four buses of IEEE118-bus system with embedded Gaussian errors and bad data

10 30 64 68


1.0489 0.5673 0.9835 0.2704 0.9832 0.3624 1.0060 0.4186

1.0530 0.5648 0.9836 0.2715 0.9832 0.3642 1.0021 0.4165

1.0514 0.5659 0.9858 0.2692 0.9836 0.3623 1.0043 0.4163

1.0484 0.5647 0.9841 0.2663 0.9864 0.3588 1.0049 0.4170

1.0499 0.5627 0.9830 0.2677 0.9821 0.3609 1.0030 0.4122

1.0519 0.5643 0.9882 0.2626 0.9813 0.3586 1.0022 0.4141

1.0500 0.5611 0.9877 0.2629 0.9823 0.3544 1.0026 0.4102

1.0497 0.5553 0.9829 0.2629 0.9862 0.3526 1.0040 0.4099

1.0521 0.5556 0.9851 0.2627 0.9823 0.3502 1.0029 0.4063

1.0507 0.5537 0.9887 0.2575 0.9811 0.3522 1.0045 0.4028

0.0000 0.0000 0.9843 0.2599 0.9838 0.3508 1.0039 0.4084

0.0000 0.0000 0.9845 0.2583 0.9835 0.3492 1.0020 0.3999

1.0479 0.5503 0.9837 0.2549 0.9819 0.3462 0.0000 0.0000

1.0515 0.5484 0.9837 0.2549 0.9825 0.3435 1.0047 0.3984

1.0508 0.5486 0.9837 0.2549 0.0000 0.0000 0.9989 0.3978

1.0453 0.5501 0.9837 0.2549 0.0000 0.0000 1.0031 0.3972

1.0539 0.5496 0.9827 0.2492 0.0000 0.0000 1.0040 0.3930

0.0000 0.0000 0.9855 0.2501 0.9809 0.3382 1.0050 0.3961

1.0486 0.5420 0.9872 0.2463 0.9817 0.3338 0.9998 0.3906

1.0511 0.5410 0.9817 0.2484 0.9815 0.3357 1.0017 0.3934

1.0488 0.5372 0.9855 0.2434 0.9804 0.3333 1.0043 0.3867

1.0509 0.5406 0.9845 0.2418 0.9831 0.3319 0.9855 0.2434

1.0486 0.5375 0.9882 0.2408 0.9850 0.3295 0.9855 0.2434

1.0511 0.5359 0.9847 0.2403 0.9821 0.3308 1.0072 0.3843

1.0464 0.5340 0.9855 0.2352 0.9872 0.3301 1.0034 0.3819

1.0486 0.5345 0.9846 0.2355 0.9820 0.3240 1.0019 0.3793

1.0445 0.5328 0.9842 0.2365 0.9850 0.3228 1.0052 0.3795

1.0516 0.5267 0.9878 0.2330 0.9831 0.3220 1.0061 0.3788

1.0480 0.5266 0.9874 0.2305 0.9826 0.3216 1.0061 0.3775

1.0502 0.5267 0.9844 0.2267 0.9825 0.3197 1.0060 0.3743


because if the observation residual remains high, the measurement gets replaced bythe optimal predicted estimate. Under such circumstances, even if after some time,the incoming data becomes good, the error in the optimal prediction would havecompounded so many times that it would not be able to track the raw measurementsanymore [27].

In order to prevent the algorithm from becoming numerically unstable anddiverging, a reset functionality can be built in that activates when the smoothingwindow is completely filled with estimated data. A pseudo-code for implementingthe algorithm’s reset function is shown in Fig. 9.14, where NSubOptimalDataPoints

denotes the number of estimates. If the number of successive bad measurementsreceived by the algorithm equals or exceeds the smoothing window length, then the

Fig. 9.13 a Comparison of filtered and smoothed estimates for Bus 10 of IEEE 118-bus system.b Comparison of filtered and smoothed estimates for Bus 30 of IEEE 118-bus system.c Comparison of filtered and smoothed estimates for Bus 64 of IEEE 118-bus system.d Comparison of filtered and smoothed estimates for Bus 68 of IEEE 118-bus system


algorithm will reset itself. It will start operating normally (afresh) once thesmoothing window gets filled with raw data (and not estimates). Thus, for thequadratic prediction algorithm, there will be a delay of at least three frames.However, proper selection of initial conditions can help the algorithm track thesynchrophasor stream faster. For example, the steady-state error covariance matrixand Kalman filter gains can be saved and used to reinitialize the algorithm whenrequired. Another advantage of the reset functionality is that by using it, contin-gencies or discrete changes in the system can be properly conditioned. Since thethree sample-based quadratic prediction algorithm cannot account for step changes,it takes several samples until the window has moved past the step change, before itcan properly track the stream again. However, by resetting the algorithm at the righttime, a discrete network change can be immediately acknowledged [26].

Fig. 9.13 (continued)


The relevance of the quadratic prediction model-based Kalman filter is profoundfor the synchrophasor data conditioning and validation application. The idea oftracking the state of the power system with a Kalman filter like process wasoriginally suggested in [28]. The difficulty in the use of such a filter for thisapplication is that the number of measurements made using PMUs is inadequate toproduce a successful estimate. As an example, Dominion Virginia Power(DVP) has approximately 4000 buses. Therefore, the traditional Kalman filter willrequire a state equation for the states of all the 4000 buses. However, DVP hasplaced PMUs on only their high-voltage network (500-kV buses) which numberabout 30. In such a scenario, a traditional dynamic/tracking estimator will not workbecause all the states of the system will not be ‘observable.’ The proposed approachconsiders each state individually. Therefore, the predicted value of a state is basedon the previous predicted values of the same state. Hence, this methodology isindependent of the network model/size. This also implies that the process andmeasurement noise associated with this methodology can be scalar quantities.Considerably, the synchrophasor data conditioning and validation methodologydescribed above provides a keener insight into the workings of a PMU-only LSE.

9.3.3 Alternate Approaches for Addressing Data QualityIssues in a LSE

Other formulations have also been proposed to address the synchrophasor dataquality issue in the context of a LSE. A two-level LSE was proposed in [29, 30] in

Fig. 9.14 A pseudo-code to implement the data conditioning algorithm’s reset functionality (fromJones et al. [26])


which the data and the calculations are distributed among the substations and thecontrol center. In that formulation, the states are inferred from the measurements inthe weighted least squares sense, with the usual chi-square test modified for the LSEequations to detect bad data. If the measurement residual vector is e and theweighted sum of its square is f , then f is a random variable having a chi-squaredistribution.

f ¼XNm

i¼1

bei2r2i

ð9:20Þ

In Eq. 9.20, Nm is the number of measurements, r2i is the covariance factors ofthe errors. The expected value of f is equal to the number of degrees of freedomwhich is then used to determine whether bad data is present or not. Identificationand rejection of bad data is carried out based on the largest normalized residualmethod [30, 31].

A LSE framework was proposed in [32] which required PMUs to be placed onsome (not necessarily all) of the HV buses of the system. Its solution being used tosupplement the conventional static state estimator. The framework used polarcoordinates for expressing the phasor quantities. This required the framework to besolved as a nonlinear weighted least squares problem which would be solved in aniterative fashion. The LSE framework improved data consistency by identifyingvoltage phase angle biases and current scaling errors in the PMU data as well as lineparameters and transformer tap ratios using an augmented state vector approach[19]. In this formulation, the residual error vector e; state vector x, and the aug-mented parameter vector a are given by the following:

e ¼eVeheIed

2664

3775; x ¼ V

h

� �; a ¼

/cXBa

266664

377775 ð9:21Þ

In Eq. 9.21, eV and eh are the errors in the voltage magnitudes and angles,respectively, with the phase angle biases associated with the angular components, eIand ed are the errors in the current magnitudes and angles, respectively, with thescaling correction factors associated with the magnitude components, / is thevoltage phase angle bias vector, c is the current scaling correction vector, X and Bare the line parameters (resistance is neglected), and a are the transformer tap ratios.The Gauss–Newton iterative method is used to solve the state estimation problem.In every iteration, the states and parameters are updated by solving for the incre-ment given by the weighted least squares problem:


WHDxDa

� �¼ We ð9:22Þ

In Eq. 9.22, H is the measurement Jacobian matrix and W is a diagonal matrixconsisting of all weights Wi. H is computed as shown below.

H ¼ @e@x

� @e@a�¼

@eV@V 00 @eh

@h@eI@V@ed@V

@eI@h@ed@h

26664

0@eh@/0@ed@/

00@eI@c0

00@eI@X@ed@X

00@eI@B@ed@B

00@eI@a@ed@a

37775 ð9:23Þ

The conditions to be satisfied for a unique solution to the linear state estimationformulation given above are as follows [19]:

1. The measurement Jacobian matrix H has full rank, that is,2NV þ 2NI � 2NB þNa, where NV and NI are the number of unique voltage andcurrent phasors, NB is the number of buses, and Na is the total number ofparameters to be estimated, such as voltage angle biases, current scaling factors,line parameters, and tap ratios.

2. Voltage phase angle bias correction can be imposed if a bus voltage phasor canbe calculated by multiple independent but redundant PMU measurements.

3. Current scaling factor correction can be imposed if the current magnitude can becalculated using multiple independent phasor measurements.

4. Line parameters or tap ratios can be estimated if along with a current phasormeasurement, the bus voltage phasors at both ends of the line can be computedusing other PMU data, independent of the current measurement.

In [33], a set of sanity checks, similar to the plausibility checks described in thedata monitoring block of Fig. 9.10, were suggested so that only good data mayenter the LSE. The bad data detection algorithm of [33] detects unreasonable dataand channel dropouts. It does so by initially excluding measurements containingNaNs, empty cells, or very small values (especially of currents). Next, it eliminatesvoltages that lie outside the range of 0:5p:u:\ Vmeasj j\1:5p:u: and currents thathave magnitudes of 10 Amps or less. The eliminated phasor measurements aretreated as missing data. Other bad data problems (such as erroneous measurements)are detected using the least absolute value method. Finally, the measurements arepassed through a data dropout processor that replaces missing data with the lastreliable measurement. A more elegant methodology for recovering missing syn-chrophasor data (called erasures) by using low-rank matrix completion methods isproposed in [34]. The framework characterizes the temporal and the channel cor-relations in PMU erasures and provides theoretical guarantees of a matrix com-pletion method that recovers correlated erasures. Its extension to an online,real-time PMU erasure estimation, and event detection is also demonstrated.

The formulations described so far addressed synchrophasor data quality issuesfaced at the HV levels of the transmission network. With the development of


μ-PMUs in the pipeline [35], it is possible that in the near future LSEs will beintroduced in active distribution networks [36]. A methodology for eliminating baddata in such a system was proposed in [37]. It combined an autoregressive inte-grated moving average (ARIMA) (0, 1, 0) process model with a discrete Kalmanfilter for detecting anomalies in the data. Then, a Boolean logic routine based on thenetwork topology is identified whether the anomaly was a result of bad data or ifthey were an outcome of a fault. Finally, the bad data was replaced by theirpredicted estimates and the noise covariance was updated to account for theanomaly. The immunity of this methodology in the presence of continuous bad dataand a judicious tuning of the noise covariance are topics of research that can beexplored in the future.

References

1. Rovnyak, S., Taylor, C. W., Mechenbier, J. R., & Thorp, J. S. (1995). Plans to demonstratedecision tree control using phasor measurements for HVDC fast power changes. InConference on Fault and Disturbance Analysis and Precise Measurements in Power Systems,Arlington, VA, November 9, 1995.

2. http://www.salford-systems.com/cart.php3. Rovnyak, S., Taylor, C. W., & Thorp, J. S. (1995). Real-time transient stability prediction—

Possibilities for on-line automatic database generation and classifier training. In Second IFACSymposium on Control of Power Plants and Power Systems, Cancun, Mexico, December 7,1995.

4. Breiman, L., Friedman, J. H., Olshen, R., & Stone, C. J. (1984). Classification and regressiontree. Pacific California: Wadsworth & Brooks/Cole Advanced Books & Software.

5. Swarnkar, A., & Niazi, K. R. (2005). CART for online security evaluation and preventivecontrol of power systems. In Proceedings of the 5th WSEAS/IASME International Conferenceon Electric Power Systems, High Voltages, Electric Machines, Tenerife, Spain, 16–18December 2005, pp. 378–383.

6. Bernabeu, E. E., Thorp, J. S., & Centeno, V. A. (2012). Methodology for asecurity/dependability adaptive protection scheme based on data mining. IEEETransactions on Power Delivery, 27(1), 104–111.

7. Murthy, S. K., Kasif, S., & Salzberg, S. (1994). A system for induction of oblique decisiontrees. J. Artificial Intelligence Research, 2(1), 1–32.

8. Garlapati, S., & Thorp, J. S. (2011). Choice of reference in CART applications using PMUdata. In Proceedings of the 17th Power Systems Computation Conference (PSCC),Stockholm, Sweden, 22–26 August 2011, pp. 1–7.

9. Fisher, R. (1936). The use of multiple measurements in taxonomic problems. Annals ofEugenics, 7(2), 179–188.

10. Li, M., Pal, A., Phadke, A. G., & Thorp, J. S. (2013). Transient stability prediction based onapparent impedance trajectory recorded by PMUs. International Journal of Electrical Powerand Energy Systems, 54, 498–504.

11. Gao, F., Thorp, J. S., Gao, S., Pal, A., & Vance, K. A. (2015). A voltage phasor based faultclassification method for PMU only state estimator output. Electric Power Components andSystems, 43(1), 22–31.

12. Wang, T., Pal, A., Thorp, J. S., Wang, Z., Liu, J., & Yang, Y. (2015). Multi-polytope basedadaptive robust damping control in power systems using CART. IEEE Transactions on PowerSystems, 30(4), 2063–2072.


http://www.salford-systems.com/cart.php

13. IEEE Standard Definitions for Power Switchgear. In IEEE Std. C37.100-1992, 1992,pp. 1–82.

14. Horowitz, S. H., & Phadke, A. G. (1995). Power system relaying (2nd ed.). New York: Wiley.15. Bernabeu, E. E. (2009).Methodology for a security-dependability adaptive protection scheme

based on data mining. Ph. D. Dissertation, Virginia Tech.16. Pal, A., Thorp, J. S., Khan, T., & Young, S. S. (2013). Classification trees for complex

synchrophasor data. Electric Power Components and Systems, 41(14), 1381–1396.17. Jones, K. D., Thorp, J. S., & Gardner, R. M. (2013). Three-phase linear state estimation using

phasor measurements. In Proceedings of the IEEE Power Engineering Society GeneralMeeting, Vancouver, BC, Canada, 21–25 July 2013, pp. 1–5.

18. Pal, A. (2014). PMU-based applications for improved monitoring and protection of powersystems. Ph. D. Dissertation, Virginia Tech.

19. Ghiocel, S. G., Chow, J. H., Stefopoulos, G., Fardanesh, B., Maragal, D., & Blanchard, B.(2014). Phasor-measurement-based state estimation for synchrophasor data quality improve-ment and power transfer interface monitoring. IEEE Trans on Power Systems, 29(2), 881–888.

20. Gao, F., Thorp, J. S., Pal, A., & Gao, S. (2012). Dynamic state prediction based onAuto-Regressive (AR) model using PMU data. In Proceedings of the IEEE Power and EnergyConference at Illinois (PECI), Champaign, IL, 24–25 February 2012, pp. 1–5.

21. Eisinbergt, A., & Pugliese, P. (1994). Exact inversion of a class of Vandermonde matrices. InProceedings of the Fifth SIAM Conference on Applied Linear Algebra, June 1994, pp. 239–243.

22. Pal, A. (2015). Effect of different load models on the three-sample based quadratic predictionalgorithm. In Proceedings of the IEEE Power Engineering Society Conference InnovativeSmart Grid Technologies, Washington D.C, 18–20 February 2015, pp 1–5.

23. Meditch, J. S. (1969). Stochastic optimal linear estimation and control. NY: McGraw-HillBook Company.

24. Moore, J. B. (1973). Discrete-time fixed-lag smoothing algorithms. IFAC-Automatica, 9,163–173.

25. Grewal, M. S., & Andrews, A. P. (2015). Optimal smoothers. In Kalman Filtering: Theoryand Practice with MATLAB, Chapter 6, Fourth Edition, John Wiley & Sons-IEEE Press,pp. 239–279.

26. Jones, K. D., Pal, A., & Thorp, J. S. (2015). Methodology for performing synchrophasor dataconditioning and validation. IEEE Transactions on Power Systems, 30(3), 1121–1130.

27. Jones, K. D. (2013). Synchrophasor-only dynamic state estimation & data conditioning.Ph.D. Dissertation, Virginia Tech.

28. Debs, A. S., & Larson, R. (1970). A dynamic estimator for tracking the state of a powersystem. IEEE Transactions PAS, 89(7), 1670–1678.

29. Yang, T., Sun, H., & Bose, A. (2011). Transition to a two-level linear state estimator–Part I:Architecture. IEEE Transactions on Power Systems, 26(1), 46–53.

30. Yang, T., Sun, H., & Bose, A. (2011). Transition to a two-level linear state estimator–Part II:Algorithm. IEEE Transactions on Power Systems, 26(1), 54–62.

31. Zhang, L., Bose, A., Jampala, A., Madani, V., & Giri, J. (2016). Design, testing, andimplementation of a linear state estimator in a real power system, to appear in IEEE Trans onSmart Grid, pp. 1–8.

32. Vanfretti, L., Chow, J. H., Sarawgi, S., & Fardanesh, B. (2011). A phasor-data-based stateestimator incorporating phase bias correction. IEEE Transactions on Power Systems, 26(1),111–119.

33. Fernandes, E. R., Ghiocel, S. G., Chow, J. H., Ilse, D. E., Tran, D. D., & Zhang, Q. (2016).Application of a phasor-only state estimator to a large power system using real PMU data, toappear in IEEE Trans on Power Systems, pp. 1–9.

34. Gao, P., Wang, M., Ghiocel, S. G., Chow, J. H., Fardanesh, B., & Stefopoulos, G. (2016).Missing data recovery by exploiting low-dimensionality in power system synchrophasormeasurements. IEEE Transactions on Power Systems, 31(2), 1006–1013.


35. Meier, A. V., Culler, D., McEachern, A., & Arghandeh, R. (2014). Micro-synchrophasors fordistribution systems. In Proceedings of the IEEE Power Engineering Society ConferenceInnovative Smart Grid Technologies, Washington D.C., 19–22 February 2014, pp. 1–5.

36. Sarri, S., Pignati, M., Romano, P., Zanni, L., & Paolone, M. (2015). A hardware-in-the-looptest platform for the performance assessment of a PMU-based real-time state estimator foractive distribution networks. In Proceedings of the 2015 IEEE Eindhoven PowerTech,Eindhoven, Netherlands, June 29-July 2 2015, pp. 1–6.

37. Pignati, M., Zanni, L., Sarri, S., Cherkaoui, R., Le Boudec, J. Y., & Paolone, M. (2014).A pre-estimation filtering process of bad data for linear power systems state estimators usingPMUs. In Proceedings of the Power Systems Computation Conference (PSCC), Wroclaw,Poland, 18–22 August 2014, pp. 1–8.

References 243

Chapter 10Protection Systems with Phasor Inputs

10.1 Introduction

Synchronized phasor measurements have offered solutions to a number of vexingprotection problems. These include the protection of series compensated lines,protection of multiterminal lines, and the inability to satisfactorily set out-of-steprelays. In many situations, the reliable measurement of a remote voltage or currenton the same reference as local variables has made a substantial improvementpossible. In some examples, communication of such measurements from one end ofa protected line to the other is all that is required, while in others, communicationacross large distances is necessary.

Phasor measurements are particularly effective in improving the protectionfunctions which have relatively slow response times. For such protection functions,the latency of remote measurements is not a significant issue. For example, backupprotection functions of distance relays and protection functions concerned withmanaging angular or voltage stability of networks can benefit from remote mea-surements with propagation delays with latencies of up to several hundred mil-liseconds. Examples of applications of this nature will be found in Sect. 9.4.

The next two sections will consider improved line protection using phasormeasurements from the remote ends of the line. The following section involvesadaptive protection in which the phasor measurements assist in ‘making adjust-ments automatically in various protection functions in order to make them moreattuned to prevailing system conditions’ [1, 2].

10.2 Differential Protection of Transmission Lines

Differential protection of buses, transformers, and generators is a well-establishedprotection principle that has no direct counterpart in protection of long transmissionlines. Pilot relays use communicated information from remote locations. True


245

http://dx.doi.org/10.1007/978-3-319-50584-8_9

differential protection was not possible before synchronized phasor measurements.Communication over twisted pair of wires up to five miles is described in [2]. Theadvantages of differential protection are important for series compensated lines andtapped lines. There are a number of forms of current differentials for line protection.In the first form, the currents are combined using a communication channel andcompared. In the second form, the currents are sampled and the samples commu-nicated over a wide-band channel, and in the third form, phasors are computed fromthe samples and the phasor values communicated. The first is shown in Fig. 10.1.

The dashed dual slope shown in Fig. 10.1 is used for high current conditionswhere CT accuracy and saturation are more likely. Transmission lines equippedwith series compensation, FACTS devices, or multiterminal lines present protectionproblems which call for differential protection. To date, such transmission lineproblems are solved with ‘differential like’ schemes such as phase comparison. Theeasy availability of synchronized measurements using GPS technology and theimprovement in communication technology make it possible to consider true dif-ferential protection of transmission lines and cables.

Differential protection can be based on computed phasors or on samplesalthough it can be argued that significant shunt elements in the transmission linemake phasors the preferred solution. In either case, it is necessary to synchronizethe sampling and time-tag the result. Phasors can be computed from fractional cycledata windows as in impedance relaying although full cycle windows offer bettersecurity.

If Ii is the current phasor at terminal i (reference direction is positive when thecurrent is flowing into the zone of protection), the differential currents may bedefined as

Idj j ¼Xi

Ii

�� ð10:1Þ

Operate

Protected Element

I1 I2operate

restraint restraint

I1+I2

Restraint

I1-I2

Fig. 10.1 Basic currentdifferential

246 10 Protection Systems with Phasor Inputs

A single restraining current may be constructed by averaging the magnitudes of allterminal currents or taking the maximum of all the terminal currents as the restraint.Alternately, one restraining current for each pair of terminals may be constructed inorder to maintain uniform sensitivity when one of the terminals of a multiterminalline is out of service. This is equivalent to the use of multiple restraints for mul-tiwinding transformers.

If a two terminal line is modeled with the exact π equivalent [3], then the phasorcurrents and voltages are shown in Fig. 10.2.

The impedances Zc1 and Zc2 are the impedances of the possible series capacitornetworks or FACTS devices; Z and Ys are the exact π impedance and admittance,respectively. The relay measures I1, V1, I2, and V2, and then, the differential currentsIx and Iy can be obtained from Eqs. 10.2 and 10.3. Under no fault conditions usingKirchhoff’s current law, Ix = Iy. When a fault occurs, the 60 Hz exact π is no longervalid because the currents and voltages are no longer pure fundamental frequencysignals. A percentage differential characteristic as shown in Fig. 9.1 based on Ix andIy on a per-unit basis, with a modest slope, is capable of sensing faults within thezone defined by the terminal where Ix and Iy are measured.

V3 ¼ V1 � I1Zc1V4 ¼ V2 � I2Zc2

ð10:2Þ

Is1 ¼ V3Ys Is2 ¼ V4YsIx ¼ I1 � Is1 Iy ¼ I2 � Is2

ð10:3Þ

The preceding discussion is for lines of any length because of the exact πequivalent but has the disadvantage of requiring voltage measurements. In [4], anapproximation to the charging current is proposed which does not require voltagemeasurement. The assumption is that each end uses data communicated from theother end to perform the current differential calculation.

The best synchronization is obviously obtained with GPS. Pre-fault load currentscan also be used for synchronizing. Data communication over a dedicated fiberchannel, while expensive, provides the best performance. A frequency-shift powerline carrier-voice grade channel operating at 64 kbps can also be used. The

Ix Iy

Is1 Is2

+

-

V1

+

-

V2

Z

Ys Ys

Zs1 Zs2Zc1 Zc2

+ +

- -

V3 V4

I1 I2Ix Iy

Is1 Is2

+

-

V1

+

-

V2

Z

Ys Ys

Zs1 Zs2Zc1 Zc2

+ +

- -

V3 V4

I1 I2

Fig. 10.2 Exact π for the protected line

10.2 Differential Protection of Transmission Lines 247

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reliability of current differential schemes can be improved by adding redundantchannels.

10.3 Distance Relaying of Multiterminal TransmissionLines

Occasionally, lines are tapped without the benefit of a high side breaker as shown inFig. 10.3. If there is no communication from terminal C, then the first zone of therelay at A must always under-reach terminals B and C. That is, zone 1 of the relayat A must be set with no infeed. If all the three lines are 10 Ω (secondary) and zone1 is 85% of the line length, then the relay at A has a 17 Ω zone 1 setting. In fact,only 70% of the line from the tap to B is in zone 1 for the relay at A. If themaximum infeed is such that IC = IA, then the impedance seen from A is 17 Ωwhen the fault is at 35% of the line from T to B. The relay reach is reduced, but thealternative of setting zone 1–85% with infeed is that the relay could see faultsbeyond B without infeed. If there is no source at C, then the load current IC is in theopposite direction but is small compared to the fault current. The reach of the firstzone at A would be extended by a very small amount by a modest outfeed (coveredeasily by the 85% setting).

The second zone at A must always overreach the complete lines AB and AC,however. This implies that zone 2 must be set with the infeed present. Imagine thesecond zone at A is set at little greater than 1 + k of the line from A to B with themaximum infeed IC = IA present. The multiplicative effect of the double currentflowing from the point T to second zone setting point as shown in Fig. 10.4 willgive an impedance at A of 30 + 40 kΩ. For example, if zone 2 is conventionally150% (k = 1/2), the zone 2 impedance at A is 50 Ω and could easily overreach zone1. The compromise forced by the tapped line is that zone 2 must be held back.A value of k of 0.1 will give Z2 = 34 Ω which could be difficult for a short linefrom B to D.

Relay

B

C

I V

X2

X3

I I

I

A A

C

+ I C

faultX1

Relay at A does notmonitor Ic

A

T

Fig. 10.3 A tapped transmission line


Depending upon the status of breakers at B and C, the settings of Table 10.1 or10.2 are selected for relay at A as shown in Fig. 10.5.

The use of communication to signal the status of the breaker at C predates phasormeasurements [5]. Adaptive schemes for setting zone 2 which do not use phasormeasurements have also been reported [6]. The number of taps on a single line hasincreased over time, and the protection of a five terminal line, for example, is farfrom simple. Schemes involving phasor measurement have been proposed whichamount to differential relaying similar to bus protection.

Software agents (a software agent is a computer program that takes independentaction based on events in the surrounding environment [7]) have also been pro-posed to deal with these issues [8]. No practical agent-based systems have yet beenreported in the literature.

10 ohms

AB

10 ohms

10 ohmsX

II

2I 2I

10 L ohms

C

D

Fig. 10.4 Second zone setting for a tapped line

Table 10.1 Relay setting distances

Type Distance

Under-reaching 0.7 of the line length

Overreaching 1.5 of the line length

Zone 2 1.2 of the line length

Zone 3 The entire line plus 1.2 of the length of the longest line. The remote busbehind

Setting Table I

Setting Table II

R

BreakersB and CStatus

AFig. 10.5 Changing settingadaptively

10.3 Distance Relaying of Multiterminal Transmission Lines 249

10.4 Adaptive Protection

Conventional protective systems respond to faults or abnormal events in a fixed,predetermined manner. This predetermined manner, embodied in the characteristicsof the relays, is based upon certain assumptions made about the power system.‘adaptive relaying’ accepts that relays may need to change their characteristics tosuit prevailing power system conditions. With the advent of digital relays, theconcept of responding to system changes has taken on a new dimension. Digitalrelays have two important characteristics that make them vital to the adaptiverelaying concept. Their functions are determined through software, and they have acommunication capability. This allows the software to be altered in response tohigher-level supervisory software, under commands from a remote control center orin response to remote measurements.

Adaptive relaying with digital relays was introduced on a major scale in 1987[1, 2]. One of the driving forces that led to the introduction of adaptive relaying wasthe change in the power industry wherein the margins of operation were beingreduced due to environmental and economic restraints and the emphasis on opera-tion for economic advantage. Consequently, the philosophy governing the tradi-tional protection and control performance and design have been challenged [9].Adaptive protection is a protection philosophy which permits and seeks to makeadjustments automatically in various protection functions in order to make themmore attuned to prevailing system conditions. In 1993, a Working Group of the IEEEPower System Relaying Committee issued a report [10] with the results of a surveyof relay engineers in North America questioning their acceptance of 16 specificadaptive functions and soliciting their suggestions for additional adaptive ideas.Examples include adapting transformer protection to the tap changer position,adaptive reclosing, and adapting relay characteristics to changes in load. It can beargued that adaptive relaying schemes address existing relaying deficiencies, makingfalse trips less likely, and improving the speed and dependability of the protectionsystem. The result is an improvement in the reliability of the bulk power system and,in some cases, an increase in allowable power transfer limits. The adaptive relayingapplications of interest in this chapter are those involving phasor measurements.

10.4.1 Adaptive Out-of-Step Protection

It is recognized that a group of generators going out of step with the rest of thepower system is often a precursor of a complete system collapse. Whether anelectromechanical transient will lead to stable or unstable condition has to bedetermined reliably before appropriate control action could be taken to bring thepower system to a viable steady state. Out-of-step relays are designed to performthis detection and also to take appropriate tripping and blocking decisions.


Traditional out-of-step relays use impedance relay zones to determine whether ornot an electromechanical swing will lead to instability. A brief description of theserelays and the procedure for determining their settings are provided here. In order todetermine the settings of these relays, it is necessary to run a large number oftransient stability simulations for various loading conditions and credible contin-gencies. Using the apparent impedance trajectories observed at locations near theelectrical center of the system during these simulation studies, two zones of animpedance relay are set, so that the inner zone is not penetrated by any stable swing.This is illustrated in Fig. 10.6 (which uses reactance type of relay characteristics).

The outer zone is shown by a dashed line, and the inner zone is shown by adouble line. Note that all the stable swing trajectories (shown by dotted lines)remain outside the inner zone, while all the unstable swing trajectories penetrate theouter as well as the inner zone. Although only two impedance characteristics areshown for stable and unstable cases, in reality a large number of such impedanceloci must be examined. The time duration for which the unstable swings dwellbetween the outer and inner zones is identified as T1 and T2 for the two unstablecharacteristics shown in the figure. The largest of these dwell times (with an addedmargin) is chosen as the timer setting for the out-of-step relay. If an actual observedimpedance locus penetrates the outer zone, but does not penetrate the inner zonebefore the timer expires, the swing is declared to be a stable swing. If it penetratesthe outer zone and then the inner zone before the timer runs out, it is an unstableswing. Stable swings do not require any control action, whereas unstable swingsusually lead to out-of-step blocking and tripping actions at predetermined locations.

Problems with Traditional Out-of-Step RelaysTraditional out-of-step relays are found to be unsatisfactory in highly intercon-nected power networks. This is because the conditions assumed when the relaycharacteristics are determined become out-of-date rather quickly, and in reality, theelectromechanical swings that do occur are quite different from those studied whenthe relays are set. The result is that traditional out-of-step relays often misoperate:

R

X

Inner

Outer zone

Stable

Unstable swingUnstable swing

Stable

T1T2

Fig. 10.6 Traditionalout-of-step relay parametersusing reactance-type relaysand timers

10.4 Adaptive Protection 251

They fail to determine correctly whether or not an evolving electromechanicalswing is stable or unstable. Consequently, their control actions also are oftenerroneous, exacerbating the evolving cascading phenomena and perhaps leading toan even greater catastrophe. Wide-area measurements of positive sequence voltagesat networks (and hence swing angles) provide a direct path to determine stabilityusing real-time data instead of using pre-calculated relay settings. This problem isvery difficult to solve in a completely general case. However, progress could bemade toward an out-of-step relay which adapts itself to changing system conditions.Angular swings could be observed directly, and the time series expansions could beused to predict the outcome of an evolving swing. It is highly desirable to developthis technique initially for known points of separation in the system. This is oftenknown from past experience, and use should be made of this information. In time,as experience with this first version of the adaptive out-of-step relay is gained, morecomplex system structures with unknown paths of separation could be tackled.

It should be noted that a related approach was developed for a field trial at theFlorida–Georgia interface [11–14] where the interface was modeled as atwo-machine system. The machines shown in Fig. 10.6 are equivalents of theeastern interconnection on the left and Florida on the right with the four buses beingphysical buses in the interconnection (Fig. 10.7).

The equation of motion of the angle difference between the two rotors of the twomachines is given by Eq. (10.4) where δ = δ1 − δ2. M1 and M2 are the two rotorinertias, and the remaining terms in Eq. (10.4) are obtained from the equivalentsystem [15]. As the system undergoes changes due to a fault and its clearing, theparameters of the differential equation Pc and Pmax change and the classical equalarea criterion can be used to determine stability. That is, the area A1 must besmaller than the area A2 for stability. The issue in adaptive out of step is todetermine the new parameter values Pc and Pmax from real-time measurements.A least squares estimate of Pm [15] from samples of δ is used in [11]. The estimateis obtained from five or six consecutive measurements of δ (Fig. 10.8).

Md2ddt2

¼ Pm þfPc � Pmax sinðd� cÞg ð10:4Þ

PMU

Relay

PMU

Relay

Fig. 10.7 Reduced Florida–Georgia system


Determination of Coherent Groups of MachineAn algorithm can be developed for determining the principal coherent groups ofmachines as the electromechanical swings begin to evolve. Algorithms for inferringrotor angles from observed bus angles are needed. Criteria for judging coherencybetween machines and groups of machines will be developed. It is expected thatcenters of angles for each coherent group will be used in determining theout-of-step condition.

Predicting the Out-of-Step Condition from Real-Time DataIt is of course possible to determine whether or not a swing is unstable by waitinglong enough and observing the actual swing. However, in order to take appropriatecontrol action, it is essential that a reliable prediction algorithm be developed whichprovided the stable–unstable classification of an evolving swing in a reasonabletime. In the Florida–Georgia experiment, a period of observation of actual angularswings for a maximum of 250 ms was used to obtain a reliable prediction of theoutcome. Assuming that the normal periods of power system swings on a largeinterconnected power system are of the order of a few seconds, this target isreasonable. Experiments were conducted on the test system to determine what is theminimum observation period needed to predict the swing outcome with a chosendegree of confidence. With the observed swing evolution, a time series approxi-mation to the swings will be made in order to provide the predicted regions of theswings [11–14].

10.4.2 Security Versus Dependability

The existing protection systems are designed to be dependable at the cost ofsomewhat reduced security. This is a desirable bias when the power system is in a‘normal’ state, meaning that there is a sufficient operational margin in generationand transmission capability. The consequence of not tripping in primary protectiontime when a fault occurs in such cases is catastrophic, in that transient instabilityand system collapse are likely to result. However, when the power system is in a

A2Pre-fault

Pred

ictio

nIn

terv

al

Inte

rval

Post-faultA1

Obs

erva

tion

Fig. 10.8 Equal areacriterion


stressed state, this is an unacceptable bias. Under stressed system conditions, a falsetrip (insecure operation of the protection system) is likely to cause greater damageto the system. It is then desirable to alter the bias of the protection system in favorof increased security with a slightly increased possibility that the primary protectionwould not work as designed in case of a fault.

It should be recognized that a relay has two failure modes. It can trip when itshould not trip (a false trip), or it can fail to trip when it should trip. The two typesof reliability have been designated as ‘security’ and ‘dependability’ by protectionengineers. Dependability is defined as the measure of the certainty that the relayswill operate correctly for all faults for which they are designed to operate, whilesecurity is the measure of the certainty that the relays will not operate incorrectly.The existing protection systems with their multiple zones of protection andredundant systems are biased toward dependability; i.e., a fault is always cleared bysome relays. There are typically multiple primary protection systems often relyingon different principles (one might depend on communications, while another usesonly local information) and multiple backup systems that trip (with some timedelay) if all primary systems fail to trip. The result is a system that virtually alwaysclears the fault but as a consequence permits larger numbers of false trips. Highdependability is recognized as being a desirable protection principle when thepower system is in a normal ‘healthy’ state, and high-speed fault clearing is highlydesirable in order to avoid instabilities in the network. The consequent price paid inoccasional false trip is an acceptable risk under ‘system normal’ conditions.However, when the system is highly stressed, false trips exacerbate disturbancesand lead to cascading events.

An attractive solution is to ‘adapt’ the security–dependability balance inresponse to changing system conditions as determined by real-time phasor mea-surements. The concept of ‘adaptive relaying’ accepts that relays may need tochange their characteristics to suit the prevailing power system conditions. Theability to change a relay characteristic or setting, on the fly, as it were, raised seriousquestions about reliability and responsibility. Adaptive relaying with digital relayswas introduced on a major scale in 1987 [1, 2]. One of the driving forces that led tothe introduction of adaptive relaying was the change in the power industry whereinthe margins of operation were being reduced due to environmental and economicconstraints and the emphasis on operation for economic advantage. With threeprimary digital protection systems, it is possible to implement an adaptive security–dependability scheme by using voting logic (see Fig. 10.9). The conventionalarrangement is that if any of the three relays sees a fault, then the breaker is tripped.A more secure decision would be made by requiring that two of the three relays seea fault before the trip signal is sent to the breaker. The benefit is in avoidingcascading and creating a more reliable system. The price paid for this increasedsecurity under ‘stressed’ system conditions is that there is a somewhat reduceddependability, which is acceptable. The advantage of the adaptive voting scheme isthat the actual relays are not modified, but only the tripping logic responds tosystem conditions.


10.4.3 Transformer

Adaptive transformer protection gained immediate acceptance since it required onlylocal information as opposed to adaptive schemes for other equipment. The slope ofthe characteristic in Fig. 10.1 is chosen to account for a variety of problemsincluding CT ratio mismatches, saturation in CTs and transformers, and off-nominalturns ratios in tap-changing transformers. Slopes as great as 40% exist in sometransformer relays. The price for a large slope is that part winding faults may fall onthe wrong side of the tripping characteristic. Recognizing that in a digital trans-former relay, the sum and the difference in the currents shown in Fig. 9.1 areformed from samples, an adaptive solution to the off-nominal turns ratio is tomonitor the tap changer and modify the trip and restraint currents appropriately.Similarly, a record of the currents in non-fault conditions can be used to determinethe actual CT ratios.

10.4.4 Adaptive System Restoration

It must be accepted that some blackouts are unavoidable. It then becomes essentialthat strategies for restoring power after a blackout with minimum delays and atminimum cost should be put in place. Quick restoration of power is of paramountimportance as it can significantly minimize user inconvenience due to poweroutages. Although pre-calculated restoration strategies obtained from planning-type

System State Assessment

PMU da-PMU da-

Critical System Locations

Supervisory signals

Relay 1

Relay 2

Relay 3

OR

VOTE

AND

Supervisory signal

Breaker trip signal

See detail below

Fig. 10.9 Adjustment ofdependability–securitybalance under stressed systemconditions


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simulation studies are available at present, they are often inadequate because theactual system state is quite different from the one assumed in the planning-typestudies. Real-time wide-area measurements provide an excellent opportunity todetermine a restoration strategy which takes into account the prevailing state of thepower system [16]. Although automated restoration procedures are possible toimplement, for various practical reasons it is desirable to use a cooperativerestoration technique, whereby the computer program suggests a restoration planfor any islands that may have been created and blacked-out and for reconnecting theislands after they have been energized. The operator implements the suggestedrestoration plan based upon the step-by-step procedure provided by the computerprogram.

An example of how real-time data provided by PMUs may have helped inrestoration in the European blackout in 2003 [17]. Review of the sequence of eventsshowed that phase angle information was not known when operators wereattempting to restore the initial line outage. Another example is a recent disturbance

Fig. 10.10 PMU measurements from three areas during reclosing attempts, UCTE disturbanceNovember 4, 2006


in Europe on November 4, 2006 [18]. Figure 9.10 shows PMU readings forreclosing attempts between two areas, including the final successful reclosingbetween those two areas and eventually with the third area. Restoration could onlybe achieved after the phase angles of separated areas became acceptable. In theabsence of real-time data, seven unsuccessful attempts for restoration were made,ultimately leading to success when the angles were favorable. As Fig. 9.10 shows,had real-time data been available to the system operators, the unsuccessful attemptsfor restoration could have been avoided.

A recent study used artificial neural networks (ANN) to accomplish systemrestoration [16]. In this proposed technique, the ANNs are trained for determiningisland boundaries for restoration in each viable island and then determining asequence of switching operations which would lead to restoration taking intoaccount cold-load pickup and control of overvoltages due to light load conditions.

If this approach is followed, then wide-area measurements could be used todetermine prevailing angle differences across breakers used to close tie linesbetween islands, and closing only when the angle differences are within acceptablebounds. If the angles exceed acceptable limits, a generation-load reschedulingwithin the islands would be implemented which would bring the angles withinlimits. This type of restoration scheme formalizes the lessons learned fromuncontrolled restoration attempts as in the example of Fig. 10.10 which may lead tomultiple failures on restoration attempts.

10.5 Control of Backup Relay Performance

It is well known that some backup zones of distance relays are prone to tripping dueto load encroachment during power system disturbances (see Fig. 10.11). This hasled to a call for abandoning the use of backup zones, in particular zone 3 of distancerelays which is used to protect downstream circuits in case their protection systemsfail to remove a fault on those circuits [19]. However, it has also been argued thatthis measure is too drastic and should not be applied as a blanket policy. The remotebackup policy is designed to cover certain contingencies [20] for which no otherprotection is available. Under these circumstances, it becomes necessary to considerways in which the loadability limits imposed by the remote backup zones can becircumvented [21].

Wide-area measurements offer a possibility for restraining the remote backuprelays in the event that the loading is being interpreted by the relay as a fault [22].Consider the conditions illustrated in Fig. 10.11. Zone 3 of relay A is assumed to bepicked up. If a significant negative sequence current is present (indicating anunbalanced fault), the zone 3 pickup is appropriate, and no further action is nec-essary. However, if the currents in the line are balanced, either a three-phase faulton the neighboring circuits or a possible loadability violation may be inferred. ThePMUs at the buses corresponding to the terminals of lines which are to bebacked-up by relay A may then determine whether any of them see a zone 1


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three-phase fault. This can be readily determined by taking the ratio of the positivesequence voltage and current in those line terminals. If none of the PMUs indicatesthat a zone 1 three-phase fault exists, then the zone-3 pickup of relay A must be dueto loadability limit violation. If tripping on this condition by relay A is to beavoided, it would then be possible to block its operation by supervisory control ofits output (Figs. 10.12 and 10.13).

Zone 3

Loadability limit

Fig. 10.11 Loadability limitimposed by a zone 3 setting ofa distance relay. Theillustration shows a mhocharacteristic, which iscommonly used in manyrelays. As the load increasedalong the bold arrow, itwould enter the tripping zoneof the relay and cause aninappropriate trip

Zone 3

PMU PMU PMU

Zone-1picked up ?

N N N

All No?BLOCK

Zone-1picked up ?

Zone-1picked up ?

PMU

Zone-3picked up

Fig. 10.12 Hidden failuremonitoring and control

CB

CBA

B

Fig. 10.13 Reverse local busvulnerability region


10.5.1 Hidden Failures

In examining the data made available by the North American Electric ReliabilityCouncil (NERC) [22], the extent of relay involvement in major disturbancesbecomes obvious. The mechanism has been referred to as ‘hidden failures’ in theprotection system [23]. It is not that relays initiate major disturbances, but they tendto be involved in the spreading of what might have been a more localized event.The NERC data is an annual record of approximately 10 major disturbancesmeasured by size and duration. Over a considerable period, relays have played arole in about 2/3 of these events.

There are thousands of relays that operate correctly in any major event, but ifthere is some defect in a relay, the stressed system conditions that exist in a majordisturbance can cause an incorrect relay operation. The fact that the defect is notnoticed until conditions around it are unusual prompts the term ‘hidden.’Maintenance can be a source of these hidden failures. This is true in power systems,the Internet, and in large chemical plants. The entire process has been described asthe ‘curse of robustness.’ Large complex system is designed to keep working whenmost of the elements are healthy. But under exceptional stress, all the defectiveelements give way and the disturbance cascades to an exceptional extent [23].

Some defects in relays would cause the relay to misoperate immediately and donot qualify as ‘hidden.’ The exact mechanism of hidden failures in a number ofcommonly used protection schemes has been tabulated [24–28]. Using this analysis,it is possible to define the ‘regions of vulnerability’ for hidden failures. The regionof vulnerability for a given relay is the region in the power system where a fault willexpose a hidden failure. To illustrate the concept, we will assume the reach settingsfor relays that operate for faults within a certain distance of their location shown inTable 10.1. An example of the region of vulnerability is shown in Fig. 9.13 for avariety of pilot schemes listed in Table 10.2. The region is the local reverse busregion where faults behind bus A will result in the circuit breaker at B trippingincorrectly for a fault in the shaded region. The mechanism for the failure is shownin Table 10.2.

The hidden failure probability is small and certainly is a function of loading andsystem conditions. The mechanism of hidden failure can be studied by consideringa sample path made up of a sequence of hidden failure trips and correct trips due tooverloads with resulting load and generation shedding. The probabilities are small

Table 10.2 Hidden failures in pilot relays in Fig. 10.1

Relay Hidden failure mode

Directional comparison blocking Fault detector at a cannotpick up—transmitter fails to transmit

Directional comparison unblocking DA continuously picks up

Permissive overreaching transfer trip DA continuously picks up

Permissive under-reaching transfer trip Transmitter continuously transmits

10.5 Control of Backup Relay Performance 259

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enough that multiple hidden failure trips at one branch almost never happen. Thesample path is approximately one-dimensional like a crack as opposed to a forestfire [29]. The hidden failure mechanism itself is sufficient to produce power lawbehavior. The simulation presented in [29] produced the characteristic plot of thelog of the size of the disturbance versus the log of the relative frequency of thedisturbance shown in Fig. 10.14. If two quantities are related by y = xα, the log–logplot similar to Fig. 10.14 is a straight line with slope α, i.e., logy = α logx.

There are a number of papers and approaches to investigating this power lawmechanism [30]. The approach in [29] has been dubbed the hidden failure model.

One approach to reducing hidden failures or reducing their impact is to upgradethe relays at strategic locations determined with engineering judgment or simulation

-16 -14 -12 -10 -8 -6 -4 -2 0-2

0

2

4

6

8

10

12

14

16

Log(load loss)

Log(relative frequency)

Fig. 10.14 Power law relationship from the hidden failure model

PMU

PMU

PMU

PMU

PMU

Islandformation

logic

Coherencydetection

Supervisorycontrol

Fig. 10.15 Islanding basedon coherency detection


techniques. The upgrade is done with an eye toward hidden failures with anemphasis on self-monitoring and checking. The voting technique shown inSect. 10.4.2 is also a possible tool (Fig. 10.15).

10.6 Intelligent Islanding

Separation of a power system into islands is a measure of last resort when the powersystem is stressed (thermal limits, phase angles across system centers beyondacceptable limits, voltage or frequency excursions beyond planned thresholds), andfurther disturbance propagation resulting in system separation is unavoidable. Inprinciple, each resulting island should have a balance between generation and loadin the island. In practice, this may not be the case, and consequently, load orgeneration shedding may be required to bring about a balance and may return theisland to stable operation at normal frequency and voltage profile.

System separation into islands is accomplished using system integrity protectionscheme (SIPS) also known as remedial action schemes (RAS) or system protectionscheme (SPS) [31]. These schemes are designed based on extensive planningstudies covering various reasonable loading levels, topology, planned andunplanned outages, etc. In many practical situations, the prevailing system condi-tions are quite different from those upon which the SIPS settings are based.Consequently, the performance of these systems may not be optimal for the pre-vailing system state.

Wide-area real-time synchronized data provides important information on pre-vailing system conditions to improve the match between SIPS and the actual systemstate. These measurements may be used to either supplement or replace thepre-calculated scenarios and improve the planned system separation in two keyareas:

(a) Using real-time data provided by the PMUs to more accurately determinewhether a power system is heading to an unstable state and whether a networkseparation is necessary to avoid a blackout.

(b) Determine optimal islanding boundaries according to the prevailing systemconditions. For example, establish which groups of generators will separatedue to loss of synchronism and how to optimally balance load and generationin each island formed by coherent generator groups and loads.

Assuming that the PMU measurements are to be added to the existing SIPS plansto improve and speed up instability detection, technical and computationalrequirements are well within the scope of present technology. The implementationwould require a certain number of PMU measurements from optimally placedlocations and dedicated fiber optic channels with data latency of the order of 50 ms.The needed coherency detection algorithms and self-sufficient island identificationalgorithms would have to be developed to suit a specific power system.

10.5 Control of Backup Relay Performance 261

In the absence of prior experience with the prevailing power system state,PMU-based SIPS would lead to islanding operations that are more appropriate forthe existing system state. Wide-area measurement-based decisions made basedupon real-time data should result in islanding operations which would form sus-tainable islands and better prospects for service restoration.

10.7 Supervisory Load Shedding

Under-frequency load shedding and restoration are used in most power systems tomanage frequency excursions when islands of mismatched generation and load areformed. The frequency measurements are performed locally in distribution sub-stations, and pre-assigned feeders tripped when frequency passes through presettrigger points [32]. More recently, with the threat of voltage collapse in many powersystems, voltage-controlled load shedding has also been implemented [33, 34].Voltage collapse is a localized phenomenon, although several other factors (such asreactive power margins in generators) should also be taken into account. This isachieved by using SIPS, with information brought from remote sites.

SIPS system is a form of wide-area measurement-based protection systems. It ispossible to formulate a strategy which would address the issue of load sheddingbefore the frequency begins to decay, or before the voltage begins a dive towardinstability. A conceptual view of such a scheme is shown in Fig. 10.16. One couldconsider measuring a real-time area control error (ACE) by determining the devi-ation of tie-line power flows from their pre-disturbance schedule. While the tie linesremain connected, there is no frequency decay so that the ACE measures directly ashortfall of generation in the network. This shortfall can be weighed against

ΔT ΔQδ V and V′

Load shed? Load shed?

IntegratedLoad shedCommand

PMU PMU

PMU

PMU

PMUPMU

PMU PMU

PMU

PMU

Fig. 10.16 Conceptualintegrated load sheddingbased upon wide-areameasurement systems


predetermined allowable margins to retain stability and line loadings below theircapability. (The margin calculations could be made adaptive to prevailing systemconditions by using trained ANN which would produce a decision as to whether ornot load shedding is to be invoked. The ANN training could be performed using tieflow deviations and bus phase angles at critical locations.) Should the thresholds bebreached, one could initiate load shedding under centrally directed supervisorycontrol. The required wide-area measurements for such a load shedding schemewould be tie-line flows and phase angles at key network buses.

Similar strategy could also be employed for load shedding for voltage control.Wide-area measurements of voltages at key network buses would be collected at acentral location, along with available amounts of reactive support in reactive powersources. Based upon this information, and using rate of change of voltage magnitudesat key buses, it would be possible to determine margin to voltage collapse and throughit the need for load shedding. The advantage of such a scheme would be to considerthe voltage problem in its entirety for the power system and determine appropriateamounts of load to be shed in a coordinated fashion. In fact, the load shedding forreal-power limitations (angular instability or overloads) and reactive power limita-tions (voltage profile violations or insufficient margins to voltage collapse) could beunified under an integrated load shedding control as shown in Fig. 10.16.

References

1. Horowitz, S. H., Phadke, A. G., & Thorp, J. S. (1988). Adaptive transmission systemrelaying. IEEE Transactions on Power Delivery, 3(4), 1436–1445.

2. Rockefeller, G. D., Wagner, C. L., Linders, J. R., Hicks, K. L., & Rizi, D. T. (1988). Adaptivetransmission relaying concepts for improved performance. IEEE Transactions on PowerDelivery, 3(4), 1446–1458.

3. Stevenson, W. D. (1980). Elements of power system analysis. New York: McGraw-Hill.4. ABB Application manual, Line differential protection IED RED 670 ANSI.5. Hedding, R. A., Mekic, F. (2007). Advanced multi-terminal line current differential relaying

and applications. Protective Relay Engineers, 60th Annual Conference, March 2007,pp. 102–109.

6. Sidhu, T. S., Baltazar, D. S., Palomino, R. M., & Sachdev, M. S. (2004). A new approach forcalculating zone-2 setting of distance relays and its use in an adaptive protection system. IEEETransactions on Power Delivery, 10(1), 70–77.

7. Genesereth, M., & Ketchpel, S. (1994). Software agents. Communications of the ACM 37(7),48–52, 147.

8. Coury, D., Thorp, J., Hopkinson, K., & Birman, K. (2001). Improving the protection of EHVteed feeders using local agents. Developments in Power System Protection, ConferencePublication, No. 479, IEE.

9. Thorp, J. S., Phadke, A. G., Horowitz, S. H., & Begovic, M. M. (1988). “Some applications ofphasor measurements to adaptive protection. IEEE Transactions on PAS, 3(2), 791–798.

10. Thorp, J. S., et al. (1993). Feasibility of adaptive protection and control. IEEE Transactionson Power Delivery, 8(3), 975–983.

11. Centeno, V., Phadke, A. G., Edris, A., Benton, J., Gaudi, M., & Michel, G. (1997). Anadaptive out-of-step relay [for power system protection]. IEEE Transactions on PowerDelivery, 12(1), 61–71.

10.7 Supervisory Load Shedding 263

12. Centeno, V., Phadke, A. G., Edris, A. (1997). Adaptive out-of-step relay with phasormeasurement, developments in power system protection. Sixth International Conference on(Conf. Pub l. No. 434), March 25–27, pp. 210–213.

13. Centeno, V., Phadke, A. G., Edris, A., Benton, J., & Michel, G. (1997). An adaptiveout-of-step relay. IEEE Power Engineering Review, 17(1), 39–40.

14. Centeno, V., de la Ree, J., Phadke, A. G., Michel, G., Murphy, R. J., & Burnett, R. O., Jr.(1993). Adaptive out-of-step relaying using phasor measurement techniques. ComputerApplications in Power, IEEE, 6(4), 12–17.

15. Phadke, A. G., & Thorp, J. S. (1988). Computer relaying for power systems. Somerset,England: Research Studies Press.

16. Bretas, A. S., & Phadke, A. G. (2003). Artificial neural networks in power system restoration.IEEE Transactions On Power Delivery, 18(4), 1181.

17. 1186System Disturbance on November 4, 2006, UCTE, February 2007, available at www.ucte.com

18. Developments in UCTE and Switzerland, W. Sattinger, WAMC course at ETH Zürich, 30August–1 September 2005.

19. Horowitz, S. H., & Phadke, A. G. (2003). Boosting immunity to blackouts. Power andEnergy Magazine, IEEE, 1(5), 47–53.

20. Horowitz, S. H., & Phadke, A. G. (2006). Third Zone Revisited. IEEE Transactions on PowerDelivery, 21(1), 23–29.

21. Phadke, A. G., Novosel, D., & Horowitz, S. H. (2007). Wide area measurement applicationsin functionally integrated power systems. CIGRE B-5 Colloquium, Madrid, Spain.

22. http://www.nerc.com/*dawg/dawg-disturbancereports.html.23. Taylor, C. W. (1999). Improving grid behavior. IEEE Spectrum, 36(6), 40–45.24. Tamronglak, S., Horowitz, S. H., Phadke, A. G., & Thorp, J. S. (1996). Anatomy of power

system blackouts: preventive relaying strategies. Power Delivery, IEEE Transactions on, 11(2),708–715.

25. Elizondo, D. C., & De La Ree, J. (2004). Analysis of hidden failures of protection schemes inlarge interconnected power systems. Power Engineering Society General Meeting, IEEE, 1,107–114.

26. De La Ree, J., & Elizondo, D. C. (2004). A methodology to assess the impact of hiddenfailures in protection schemes. Power Systems Conference and Exposition, IEEE PES, 3,1782–1783.

27. Phadke, A. G., & Thorp, J. S. (1996). Expose hidden failures to prevent cascading outages inpower systems. Computer Applications in Power, IEEE, 9(3), 20–23.

28. Elizondo, D. C., De La Ree, J., Phadke, A. G., & Horowitz, S. (2001). Hidden failures inprotection systems and their impact on wide-area disturbances. Power Engineering SocietyWinter Meeting, IEEE, 2, 710–714.

29. Wang, H., & Thorp, J. S. (2001). Optimal locations for protection system enhancement: Asimulation of cascading outages. IEEE Transactions on Power Delivery, 16(4), 528–533.

30. Dobson, I., Chen, J., Thorp, J. S., Carreras, B. A., & Newman, D. E. (2002). Examiningcriticality of blackouts in power system models with cascading events. Proceedings of the35th Annual Hawaii International Conference on System Sciences, January 2002.

31. Horowitz, Stanley H., & Phadke, Arun G. (2008). Computer Relaying for Power Systems (3rded.). RSP: John Wiley & Sons. (book).

32. Westinghouse. (1976). Applied protective relaying. Westinghouse Electric Corporation,Newark, N.J. (Chapter 19).

33. Begovic, M., Novosel, D., Karlsson, D., Henville, C., & Michel, G. (2005). Wide-areaprotection and emergency control. Proceedings of the IEEE, 93(5), 876–891.

34. Madani, V., Novosel, D., Apostolov, A., Corsi, S. (2004). Innovative solutions for preventingwide area disturbance propagation. IREP Symposium for Bulk Power Systems Dynamics andControl VI, Cortina d’Ampezzo, Italy, August 2004.


http://www.ucte.com

http://www.ucte.com

http://www.nerc.com/%7edawg/dawg-disturbancereports.html

Chapter 11Electromechanical Wave Propagation

11.1 Introduction

Several different events connected with the early application of phasor measure-ments prompted consideration of the propagation of transient events in powersystems. The first is typical of what is shown in Fig. 11.1

It is shown that there seems to be a delay between the frequencies at differentpoints in the system. Similar effects were noticed in a variety of experiments [1, 2].The event described in [1] was a load rejection test in Texas monitored with PMUs.The authors state ‘Note the delay detecting the transient between the point closest tothe plant—Venus and the furthest—Robinson. There is nearly a half second delaybetween the onset of the frequency disturbance near the plant and its appearance ata similar level at a remote site. The propagation phenomenon is not clear. It is notelectrical in nature because of the time lag. It appears to be related to the localizedelectrical inertia in the system.’ A similar staged event in July, 1995, showed adelay of approximately a second between PMU’s in Florida and New York [3].

The second motivation was a desire to display the phasor measurementsobtained in the study described in 8.5.1 for the WECC 1994 disturbance. The plotshown in Fig. 11.2 was produced by locating the phasor measurements geo-graphically on a map of the WECC, making the z variable at that point the anglemeasurement, and then fitting a smooth surface to those points. By sampling thephase angles in time, a movie of the surface can be obtained. Constant contour linesare shown below on a map showing state borders and the location of the two DClines. Since to a first approximation, power flows down hill in angle transmissionlines should be constructed perpendicular to the constant contour lines. Of course,the contour map changes in response to system conditions.

The movies produced show wavelike motion of the surface when simulated ormeasured phasor quantities are used to draw the surfaces. There have been earlyattempts to describe this behavior even before phasor measurement data wasavailable. Over a 30-year period [4–7], a number of authors have derived a wave


265

http://dx.doi.org/10.1007/978-3-319-50584-8_8

equation for an idealized continuum power system model. In [4–7], linearizedswing equations are written with the DC power flow assumptions for a uniform,isotropic, and lossless network. A different approach in [7] produced an ellipticpartial differential equation for a continuum load flow. In [7], the load flow equa-tions are written under the DC load flow assumptions, but the uniformity and

Deg

rees

60

Grand Coulee

Malin

Seconds

Seconds

20

-200

0

50 60-400

Vincent

59.95

60.00

50

Malin

Vincent

Freq

uenc

y(a)

(b)

Fig. 11.1 Phasormeasurements in threelocations for an event in theWECC

Fig. 11.2 Phase angle as afunction of location

266 11 Electromechanical Wave Propagation

isotropy constraints are relaxed. In [8–10], a combination of the two approachesproduces a different model.

11.2 The Model

Consider the element at point (x, y) shown in Fig. 10.3 with Fig. 11.3

Ri + jXi p.u. line impedance1/(Rs + jXs) p.u. generator admittanceGs + jBs p.u. shunt admittance|V| voltage magnitudeϕ(x, y) internal voltage phase anglesδ(x, y) external voltage phase anglesM generator inertiaD generator dampingP mechanical power injectionθ angle of the branch with respect to the x–y-axis

The power flow at the external node (x, y) must satisfy the load flow equation inEq. (11.1):

P i+

P i-

V(x,y)

θi

branch ‘i’

Δ

(x,y)x - axis

y -ax

P

Δ

Fig. 11.3 Incremental power system model

11.1 Introduction 267

http://dx.doi.org/10.1007/978-3-319-50584-8_10

XNi¼1

DRi V2��

D2ðR2i þX2

i Þ1� cosðdðx; yÞ � dðxþDxi; yþDyiÞÞ½ �

þ DXi V2��

D2ðR2i þX2

i Þsinðdðx; yÞ � dðxþDxi; yþDyiÞÞ½ �

¼ DRs V2��

D2ðR2s þX2

s Þ1� cosðdðx; yÞ � uðx; yÞÞ½ �

þ DXs V2��

D2ðR2s þX2

s Þsinðdðx; yÞ � uðx; yÞÞ½ � ð11:1Þ

And the power flow at the internal node (x, y) must satisfy the swing equation inEq. (11.2). The next step is to write Taylor series expansions ofdðx� Dxi; y� DyiÞ, to use trigonometric identities for small angles, and to take thelimit as D ! 0. Two coupled partial differential equations are resulted: One is theswing equation in Eq. (11.3) where virtually everything is a function of x and y, i.e.,m(x, y), d(x, y), pm(x, y), gint(x, y), bint(x, y), ϕ(x, y), and δ(x, y); the other equation isthe continuum load flow equation in Eq. (10.4). The conductance G and the sus-ceptance B are 2-by-2 tensor fields, functions of (x, y) which capture thenon-uniformity and anisotropy in the network:

M@2/@t2

þD@/@t

¼ P� DXs V2��

D2ðR2s þX2

s Þ1� cosð/ðx; yÞ � dðx; yÞÞ½ �

DXs V2��

D2ðR2s þX2

s Þsinðuðx; yÞ � dðx; yÞÞ½ � ð11:2Þ

m@2/@t2

þ d@/@t

¼ pm � gint 1� cosð/� dÞ�½ � � bint sinð/� dÞ ð11:3Þ

�r � BðrdÞ½ � þrd � G � rd ¼ gint cosðd� /Þ � 1½ � � bint sinðd� /Þ½ � � gsð11:4Þ

Equations (11.3) and (11.4) are coupled by the angle difference d� /.Equation (11.4) is a continuum load flow with no time dependence. In principle,given a /ðx; yÞ, Eq. (11.4) can be solved for dðx; yÞ. Alternately, given /ðx; yÞ,Eq. (11.3) can be solved for dðx; y; tÞ. That is Eq. (11.3) is a swing equation foreach (x, y).

The phase angle gradient field rd is imaged to the power flow field P through

P ¼ �BðrdÞ ð11:5Þ

Several things can be noticed. The first is that the power injected at a point(x, y) in the system is


http://dx.doi.org/10.1007/978-3-319-50584-8_10

pðx; yÞ ¼ �r � BðrdÞ� �þrd �G � rd ð11:6Þ

The nonlinearity is in the second term and is due to the electrical losses. Thelossless models in [4–7] had linear wave behavior because they had no losses. Itmust be observed that Eqs. (11.3) and (11.4) describe an electromechanical system,not an electromagnetic system. It may be disquieting that linear electrical lossesproduce a nonlinear effect in the electromechanical system. Further, the net powerloss in a region R in the system is given by

PlossR ¼

ZZRðrd �G � rdÞdxdy ð11:7Þ

A linear analogy can be made between the electromechanical model and theelectromagnetic wave propagation model. It is summarized in Table 11.1.

Example 11.1 A Ring System As an example, consider a ring system made up of 64identical generators connected in a ring with identical transmission lines connectingthe generator. If the equilibrium is chosen as a 2π increase in angle in the coun-terclockwise direction around the ring, then power flows in the same direction. Thediscrete model has 64 differential and algebraic equations in the form of Eq. (11.8),or the continuum form is Eq. (11.9) (Fig. 11.4):

Mid2/i

dt2¼ Pm

i � bint sinð/i � diÞ½ �2� cosðdi � diþ 1Þ � cosðdi � di�1Þ½ �þ b sinðdi � diþ 1Þþ sinðdi � di�1Þ½ � ¼ bint sinð/i � diÞ½ � ð11:8Þ

mðxÞ @2u@t2

¼ 2p64

� �2

�bint sinð/ðxÞ � dðxÞÞ½ � ð11:9Þ

@d@x

� �2

�b@2d@x2

¼ bint sinð/ðxÞ � dðxÞÞ½ � ð11:10Þ

Table 11.1 Correspondence between linear electromagnetic and electromechanical systems

Electromagnetism Electromechanical

Quantity Relationship Quantity Relationship

Electric potential E ¼ �rV Voltage phase angle U ¼ �rd

Electric field intensity Phase angle gradient field

Permittivity tensor D ¼ eE Susceptance tensor P ¼ BUElectric flux density q ¼ �rD Power flow density p ¼ �r � PCharge density Power injection density

q ¼ �r � ðerVÞ p ¼ �r � ðBrdÞ

11.2 The Model 269

The model has no source admittance and hence no dispersion. The 64 anglesfrom the discrete model are shown with δ1 at the bottom and δ64 at thetop. Figure 11.5 shows an initial perturbation from the equilibrium about a quarterof the way around the ring propagating in both directions. Remarkably, it grows inthe counterclockwise direction but decays in the clockwise direction.

Example 11.2 A more realistic power flow is around both sides of the ring in thesame direction. Let the equilibrium angles be

dek ¼ kp=32; k ¼ 1; 2; . . .; 32

dek ¼ ð64� kÞp=32; k ¼ 33; 34; . . .; 64

with d ¼ 12e�0:1ðx�15:5Þ2

ð11:11Þ

Then, the waves are as shown in Fig. 11.6.Vertical cross sections of the phase angle for Example 11.2 are shown in

Fig. 11.7. Each of the eight figures is the angle at a fixed instant in time plotted

64 12

63

32 313132

1

Fig. 11.4 Uniform powerflow in a counterclockwisedirection

2π

Time

δn(t ) n=1,2 … 64

0

Fig. 11.5 64 angles fromExample 10.1 versus time


versus position around the ring δ(θ, t), where θ is the position on the ring as shownfor the first cross section. The pulse separates into two pulses in the second crosssection, spreads in the next two cross sections, crosses, and returns to the center inthe last two cross sections.

11.3 Electromechanical Telegrapher’s Equation

If we open up the loop in Example 11.1, we have a line rather than a ring. Thesystem such as that shown in Fig. 11.8 is produced.

If we assume small internal impedances and R ≪ X, the equations become

0

0

π

πδk(t) k=1,2,…32

δk(t) k=33,34 …64

Fig. 11.6 Phase angles versus time for Example 11.2

1 2

3 4

2 3 4 spreading

6

78

5

5 crossing 7, 8 recombining

6

78

5

5 crossing 7, 8 recombining

δ(θ,o)

−π πθ

Fig. 11.7 Vertical cross sections of phase angle

11.2 The Model 271

mðxÞ @x@t

¼ � @p@x

@p@t

¼ �bðxÞ @x@x

ð11:12Þ

where

P ¼ �b@d@x

x ¼ @d@t

ð11:13Þ

The result is a form of the Telegrapher’s equation in angle and power rather thanvoltage and currents.

mðxÞ @2d@t2

¼ pmðxÞþ bðxÞ @2d

@x2ð11:14Þ

which has

velocity ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibðxÞ=mðxÞ

p; Z0 ¼ 1=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibðxÞmðxÞ

pð11:15Þ

In other words, the one-dimensional line has a characteristic velocity of prop-agation and a characteristic ‘impedance’. The impedance is not in ohms, since theratio is angle divided by power rather than volts divided by amps. The velocity ofpropagation measured in the power system is far less than the velocity of light andvaries in different parts of the country. Speeds of hundreds of miles a second to athousand miles per second have been observed in the FNET system [11], where themeasured waves are in frequency rather than in angle. They are, of course, adifferent manifestation of the same waves (Fig. 11.9).

Determining the onset of a wave can be complicated with electromechanicalwaves just as in electromagnetic waves. The use of discriminant functions as intraveling wave relays [12] is possible:

Df ¼ xþ Z0p ¼ 2xþ

Dr ¼ x� Z0p ¼ 2x� ð11:16Þ

Typical plots of angular velocity versus position along a uniform line are shownin Fig. 11.9 at a sequence of times. Although the waves begin crisply, they become

P1 P2 P3 PN

………

Fig. 11.8 A one-dimensionalline


distorted as time goes on. The forward and reverse discriminant functions areshown in Fig. 11.10.

11.4 Continuum Voltage Magnitude

The development of Sect. 11.2 assumed constant voltages. The load flow equationcan be rederived with the external bus voltage magnitude depending on (x, y),that is,

0 20 40 60-5

0

5x 10-4 0 Seconds

0 20 40 60-5

0

5x 10-4 3.7 Seconds

0 20 40 60-5

0

5x 10-4 14.5 Seconds

0 20 40 60-5

0

5x10-4 22.9 Seconds

0 20 40 60-5

0

5x10-4 41 Seconds

0 40 60-5

0

5x 10-4 49.4 Seconds

Fig. 11.9 Time snapshots of angular velocity versus position

0 10 20 30 40 50 60-2

0

2

4

6

8

10x 10-4

Forward DiscriminantReverse Discriminant

Time (seconds)

Fig. 11.10 Forward andreverse discriminate functionsfor the example

11.3 Electromechanical Telegrapher’s Equation 273

Eðx; yÞejdðx;yÞ ð11:17Þ

The resulting two equations are a real power equation:

pðx; yÞ ¼ �r � E2ðBrdÞ � r � EðGrEÞþrE �G � rEþðErdÞ �G � ðErdÞ ð11:18Þ

And a reactive power equation is

qðx; yÞ ¼ �r � EðBrEÞ � r � E2ðGrdÞþrE � B � rEþðErdÞ � B � ðErdÞ ð11:19Þ

A striking simplification is possible if we assume the R/X ratio of the lines isconstant, q ¼ R=X, and use a transformation of real and reactive power introducedin [13]. Equation (11.20) is a rotation of the real and reactive powers so that thenew ‘real power’ is rotated by the line angle as shown in Fig. 11.11

~pðx; yÞ~qðx; yÞ

� ¼ 1

1þ q21 �qq 1

� pðx; yÞqðx; yÞ

� ð11:20Þ

In the new coordinate system, the two partial differential equations become

~pðx; yÞ ¼ �r � E2ðBrdÞ~qðx; yÞ ¼ �r � EðBrEÞþrE � B � rEþðErdÞ � B � ðErdÞ ð11:21Þ

The angle dependence in the second equation can be removed, and a singleequation in the voltage magnitude is written as in Eq. (11.22):

qq

R p

X

~

p~

Fig. 11.11 Rotation of realand reactive powers


bE3 @2E@x2

þ @2E@y2

� �þ bE2~qðx; yÞ ¼ b

4

Zx

0

~pð x_; yÞd x_ þ

Zx

0

~pðx; y_Þd y_

0@

1A ð11:22Þ

Example 11.3 The one-dimensional line shown in Fig. 10.7 with 40 sections hasp(x) and q(x) as shown in Fig. 11.12. The continuous solution obtained fromEq. (11.22) along with the discrete solution is shown in Fig. 11.1 (Fig. 11.13).

11.5 Effects on Protection Systems

One of the motivations for considering electromechanical waves in power system isthe concern about these disturbances on protection systems. It is safe to say thatrelay systems were not designed with the motion of waves in angle and frequency

0 5 10 15 20 25 30 35 40-0.06-0.02

00.02

0 5 10 15 20 25 30 35 40-0.010

0.01

0.02

P

Q

Fig. 11.12 Real and reactive powers for the one-dimensional line

Phase Angle vs. Position Phase Angle vs. Position

Discrete System Continuum System5 10 20 25 30

0

0 35-0.6

-0.4

-0.2

0

0.2

0 5 10 150

0

20 25 30 35 40

0.5

1.0

1.5

5 10 15 20 25 30 35 40

-0.4

-0.2

0

0.2Phase angle

355 1510 20 3025

15-0.6

Voltage Magnitude vs. PositionVoltage Magnitude vs. Position

0

0.5

1

1.5

Fig. 11.13 Discrete and continuous solutions for the voltage on the one-dimensional line

11.4 Continuum Voltage Magnitude 275

http://dx.doi.org/10.1007/978-3-319-50584-8_10

propagating through the system at speeds of hundreds a miles a second. Toinvestigate these effects, the 64-machine system was investigated with each lineprotected with an overcurrent relay, a distance relay, an out-of-step relay, and a loadshedding relay [14]. In theory and as observed on the system, electromechanicalwaves travel at speeds slow enough to make it possible to communicate the exis-tence of the wave to remote locations before the wave arrives. The diagram shownin Fig. 11.14 is a simple model of a monitoring system for such a scheme.

11.5.1 Overcurrent Relays

The overcurrent relays are set to pickup at twice maximum load. The wave prop-agation is generated by applying a pulse with a 0.5 rad peak value to the 16thmachine. In addition, at the 16th machine, two other lines show possible over-current violations. The current wave forms for lines at buses 5 and 36 are shown inFig. 11.15.

11.5.2 Impedance Relays

The distance relays were set for zone 1 at 90% of the line length, zone 2 at 150% ofthe line length, and zone 3 at 150% of the next line length. A similar wave wasinitiated with a 1.5 rad peak at machine 16. Zone 1 of the relay on the adjacent line

No

specified

No

Disturbanceat sourcesystem

High speedbroadcast of

event start and its shape

Receptionat a System

Center

Keyfacility?

Supervision needed?

Yes

Fall -Back topre-selectedresponse for

duration

Yes

Fig. 11.14 Wavepropagation monitoring andcontrol system


on bus 15 was entered by the apparent impedance from the wave as shown inFig. 11.16.

11.5.3 Out-of-Step Relays

The inner zone of the relay is set at j0.2 pu, and the outer zone chosen as j0.25 pu.The timer setting to determine whether the trajectory is a fault or an unstable swingis 0.02 s. A second timer used to distinguish between a stable or unstable swing issat at 0.16 s. A Gaussian disturbance with peak value of 2.5 rad is applied to the16th machine at t = 5 s in order to generate the disturbances.

When the disturbance propagates through the ring system, out-of-step relayslocated at the 14th bus through the 18th bus are entered as shown in Fig. 11.17, and

Fig. 11.15 Overcurrent relay pickup setting and transmission line current

Zone 3

Zone 2

Zone 3

Zone 3

Zone 2

Zone 3

Fig. 11.16 Partial locus ofapparent impedancemovement at bus 15

11.5 Effects on Protection Systems 277

some relays are in danger of tripping due to the wave propagation. Since wavepropagation is a transient phenomenon, it is desirable to block the tripping of theout-of-step relays for the duration of the wave propagation.

11.5.4 Load Shedding

For load shedding, the pickup setting was set at 59.8 Hz where the time delay is tenor more cycles. A Gaussian disturbance with peak value of two radians is applied tothe 16th machine in order to generate the disturbances. Frequencies detected at the15th, 16th, and 17th buses are below 59.8 Hz. The frequency at bus 16 is shown inFig. 11.18.

Fig. 11.17 Partial locus ofapparent impedancemovement at the 16th bus(t1 = 0.16 s, t2 = 0.07 s,t3 = 0.12 s, t4 = 0.11 s, andt5 = 0.12 s)

Fig. 11.18 Frequency at the16th bus (t1 = 0.72 s andt2 = 0.22 s)


11.6 Dispersion

The term rd � G � rd in Eq. (11.6) represents the dispersion in the waves. With theangle written as in Eq. (11.23),

dðx; tÞ ¼ ekx�wt ð11:23Þ

where k is the wave number, and there is a negligible dispersion when

k �ffiffiffiffiffiffiffiffib=b

pand significant dispersion when k �

ffiffiffiffiffiffiffiffib=b

pð11:24Þ

Most of the figures have had little dispersion for clarity, but Fig. 11.19 shows awave with dispersion.

Distortion in the waveforms is also found when the system is non-uniform. Forexample, when the inertias in the 64-machine ring system have a random com-ponent rather than being identical, the cross sections are distorted as in Fig. 11.19.

11.7 Parameter Distribution

There is certainly no expectation that control center software will be called upon tosolve partial differential equations. The aim of this chapter was to establish thatelectromechanical disturbances spread in the power system at speeds much less thatthe speed of light because of the effect of machine inertias. This observation opensthe door to new concepts in adaptive protection and control and the possible pre-vention of blackouts. It is also a view of the system that might be appropriate for aregional coordinator rather than the operators in the ISO control center. In August2003, a view such as Fig. 11.2 of Ohio might have been very useful in New Yorkbefore the disturbance in Ohio spread to the Northeast US. Given this motivation,this section addresses the problem of smoothing the power system data so a picturelike Fig. 11.2 has some degree of accuracy.

Discrete System Continuum System

(source susceptance‘bint’ ≈ line susceptance‘bi’ )

Fig. 11.19 Time snapshots of phase angle versus position

11.6 Dispersion 279

Consider the one-line diagram with a geographical base (where is bus 221 on amap?), the line data, the generator data, and load data, how do we create a con-tinuum model? How do we convert a detailed model with thousands of point andlines into a smooth continuum such as the view of the power system from the spacestation? Consider a density function with the following properties:

f ðxÞ� 0;ZX

f ðsÞds ¼ 1 @mf ðxÞ=@xm exists ð11:25Þ

(It is not required, but a Gaussian density would be a good candidate). Imagineconvolving the data at a point (inertia at a bus, a load in megawatts) with thedensity. The inertia would become distributed in a Gaussian mound around theoriginal point. The width of the distribution determines the accuracy andsmoothness of the model. A very narrow distribution preserves everything but doesnot gain much. A very broad density will smooth out the system loosing precisedetails but give a satellite view of the system. A symbolic version is shown inFig. 11.20.

Figure 11.21 shows a surface similar to Fig. 11.2 for a typical load flow solutionfor the IEEE 118 bus system with the line structure below. The surface is relativelysmooth as would be expected. It is possible a regional reliability coordinator or anoperator might see relatively similar surfaces on a daily or hourly basis so thatunusual system conditions, when they occurred, would stand out. The possibilitythat surfaces like Fig. 11.2 might become the ‘face’ of the system.

CONVOLUTIONWITH IDENTITY

FUNCTION

Gaussian Function

LINE PARAMETERS(ux, uy, vx, vy, t, w)

NODE PARAMETERS(m, d, p, a b, g)

CONVOLUTIONWITH IDENTITY

FUNCTION

Gaussian Function

LINE PARAMETERS(ux, uy, vx, vy, t, w)

NODE PARAMETERS(m, d, p, a b, g)

Fig. 11.20 Convolution with a density function


References

1. Faulk, D., & Murphy, R. J. (1994). Comanche Peak Unit No 2 100 percent load rejection test—Underfrequency and voltage phasors measured across TU electric system. In ProtectiveRelay Conference, Texas A&M, March 1994.

2. Murphy, R. J. (1995). Power disturbance monitoring. In Western Protective RelayingConference.

3. Murphy, R. J. Personal Communication.4. Semlyen, A. (1974). Analysis of disturbance propagation in power systems based on a

homogeneous dynamic model. IEEE Transactions on Power Apparatus and Systems, 93,676–684.

5. Cresap, R. L., & Hauer, J. F. (1981). Emergence of a new swing mode in the western powersystem. IEEE Transactions on Power Apparatus and Systems, PAS-100(4), 2037–2045.

6. Grobovoy, A., & Lizalek, N. (2002). Assessment of power system properties by waveapproach and structure analysis. In Fifth International Conference on Power SystemManagement and Control, April 2002.

7. Dersin, P., & Lewis A. H. (1984). Aggregate feasibility sets for large power networks.In Proceedings of 9th Triennial World Congress IFAC, Budapest, Hungary (Vol. 4,pp. 2163–2168), July 1984.

8. Thorp, J. S., Seyler, C. E., & Phadke, A. G. (1998). Electromechanical wave propagation inlarge electric power systems. IEEE Transactions on Circuits and Systems, 45(6), 614–622.

Fig. 11.21 Phase angle plot for the continuum 118 bus system with the line structure below [15]

References 281

9. Thorp, J. S., Seyler, C. E., Parashar, M., & Phadke, A. G. (1998). The large scale electricpower system as a distributed continuum. Power Engineering Letters, IEEE PowerEngineering Review, 49–50.

10. Parashar, M., Thorp, J. S., & Seyler, C. E. (2004). Continuum modeling of electromechanicaldynamics in large-scale power systems. IEEE Transactions on Circuits and Systems, 51,1851–1858.

11. Zhong, Z., et al. (2005). Power system frequency monitoring network (FNET) implemen-tation. IEEE Transactions on Power Systems, 20(4), 1914–1920.

12. Dommel, H. W., & Michels, J. M. (1978). High speed relaying using traveling wave transientanalysis. IEEE paper No. A78-214-9.

13. Haque, M. H. (1993). Novel decoupled load flow method. IEEE Proceedings-C, 140(1),199–205.

14. Huang, L., Parashar, M., Phadke, A. G., & Thorp, J. S. (2005). Impact of electromechanicalwave propagation on power-system reliability. In Proceedings of 39th CIGRE Conference,Paris, France, August 2005.

15. Parashar, M. (2003). Continuum modeling of electromechanical dynamics in power systems.Ph.D. dissertation, Cornell University.


Index

AAdaptive Relaying, 250Adaptive voting scheme, 254Adjoint, 187Aliasing, 14Amplitude modulation, 115Anti-aliasing filters, 119Area Control Error, 262Artificial Neural Networks, 257Automatic Voltage Regulator, 191

BBackup zones, 257Bad data, 145Bus impedance matrix, 114

CCalibration, 95, 164–166, 168, 174Channel capacity, 88Characteristic “impedance”, 272Characteristic velocity, 272Chi squared, 146Coherency, 253Collocated control, 193Command files, 103–106, 262Complete observability, 154Configuration file, 84, 88, 91, 93, 103–107,

153, 223Continuum power system model, 266Convolution, 9Cosine wave, 10Coupled partial differential equations, 268Curse of robustness, 259CVT, 117

DData files, 4, 108Data window, 6

DC Offset, 42Decision tree

recursive partitioning, 214Dedicated fiber channel, 247Dependability, 254Depths of unobservability, 154Differential protection, 245Digital Fault Recorders, 85Dirac delta function, 9Discriminate function, 272Dispersion, 279Distance relays, 276

EElectromagnetic transients, 112Electromechanical system, 269Electromechanical transients, 112Equal-area criterion, 252ETSMP, 214Even function, 10Exact-π, 247Excitation controller, 189Exciter model, 189Exponential form, 7

FFACTS, 185, 246Fast Fourier Transform (FFT), 6Feedback gain matrix, 186Fiber-optic cable, 88Florida-Georgia, 252Fourier coefficients, 7Fourier series, 6, 8Fourier Transform, 6, 8FRACSEC, 103Fractional cycle data window, 37Frequency, 5Frequency disturbance, 265

© Springer International Publishing AG 2017A.G. Phadke and J.S. Thorp, Synchronized Phasor Measurementsand Their Applications, Power Electronics and Power Systems,DOI 10.1007/978-3-319-50584-8

283

Frequency load-shedding, 262Frequency meter, 73Frequency Modulation, 125Frobenius norm, 196

GGlobal Positioning System, 4GOES, 3GPS, 4Ground wire, 88

HHanning function, 25Header file, 93, 103, 104, 105Hidden failure, 258HVDC, 185

control, 188

IIEC 61850, 90IEEE 118 bus system, 280IEEE Power System Relaying

Committee, 250Incidence matrix

bus–bus, 154current-measurement–bus, 150

Infeed, 248Instrument transformers, 117Integrability, 8Interacting bad data, 147Inter-area oscillations, 193IP, 90Islands, 261

JJoining state estimators, 160

KKalman filtering, 182

LLargest normalized residual, 146Leakage phenomena, 24Lightning, 112Linear State Estimation, 150Loadability limits, 257Load shedding, 278LORAN-C, 3

MMatpower website, 145Measurement accuracies, 3Measurement residual, 136

Microprocessor based relay, 3Mimic, 43

NNegative sequence, 62Newton’s method, 164Nominal frequency, 29Nomogram, 66Non-DFT estimators, 45Nonlinear Residual Minimization, 162Non-recursive, 33Nyquist Criterion, 16

OOdd function, 11Off-nominal Frequency, 24, 26, 45, 48–51, 53,

56, 59, 62, 63, 66, 93, 95, 101, 111, 123,128, 227

Operator’s load flow, 133Out-of-step relays, 250Overcurrent relays, 276

PPDC, 87Percentage differential characteristic, 247Periodic function, 12Phase angles, 3Phase-locked, 85Phasor, 5Phasor Data Concentrators, 87Pilot relays, 245PMU, 4Positive sequence, 4, 59Post processing, 56Power-law, 260Power Line Carrier, 88Power Swings, 122Power System Relaying Committee, 250PT, 117

QQR algorithm, 136Quality of Phasor, 39

RRecursive, 31Remote terminal units, 141Re-sampling filter, 57Restoration, 255Restraining current, 247Riccati equation, 187Ring system, 269, 277, 279Root mean square (RMS), 5

284 Index

SSampled data, 14SCADA, 141Security, 254Security—dependability balance, 254Series compensation, 246Signal Noise, 34Single-mode fiber, 89Single phase, 3Skew, 182SOC, 86, 103Software agent, 249Static state estimation, 141Steam turbine, 73Step function response, 117Superposition, 114Symmetrical component, 4, 58Synchrophasor, 16, 91–95, 97, 103–108, 215,

217–219, 222–225, 227, 230, 234, 235,237, 238, 240

System Integrity Protection Scheme (SIPS),261, 262

TTCP, 90Telegraphers equation

electromechanical, 272

Thirty bus system, 145Thyristor controlled series capacitor, 189Time-series approximation, 253Transient Monitor, 39Transient Response, 111, 112,

117–119, 121, 122, 130

UUDP, 90Unbalanced input, 62US Department of Defense, 85UTC, 85

VVirginia Tech, 4

WWatt, 73WECC, 193Weighted least squares, 76, 134Wide area measurement, 4, 252Windowing function, 17

ZZero-crossing, 3Zero sequence, 62

Index 285

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