Model Based Fault Locating on Distribution Feeders.

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ABSTRACT

SMITH, JUSTIN WADE. Model Based Fault Locating on Distribution Feeders. (Under thedirection of Dr. Mesut E Baran.)

As electric power systems grow in complexity and size, ensuring system reliability and con-tinuity has become a major concern. Failure to locate faults quickly can result in prolongedoutage times, customer safety concerns and lost revenues. Over the past few decades distri-bution networks have evolved into large and complex networks capable of carrying thousandsof customers. As of late, utilities have displayed particular interest in distribution fault lo-cation technology to reduce such widespread impacts. This thesis aims to develop a moderndistribution fault locating algorithm.

The proposed fault locating algorithm is a model based algorithm that uses a short circuitmodel of the feeder to locate the fault. In the short circuit model, a fault is placed at everynode and the observed fault current at the substation is recorded in tabular format. During anactual fault condition, the algorithm compares the recorded fault current at the substation tothe short circuit modelling data. This comparison allows the fault locator to identify the exactnode in the network that is faulty.

In many cases, the short circuit model will indicate that there are several unique nodes withthe same fault current magnitude. This problem is overcome with a proposed sub-algorithmcalled the localization algorithm. This algorithm uses protective device data and load ow datato determine the protective device that has interrupted the fault. By knowing the protectivedevice that has interrupted the fault, the fault can be localized to a specic region of the

network.To test the fault locating algorithm, four different test cases are proposed. Three test cases

are performed using a Progress Energy Carolinas feeder model. Detailed fuse, recloser andsubstation breaker models are inserted into the feeder model to test popular protective schemessuch as fuse saving and fuse blowing. Finally, the Stewart Street 12.47kV feeder is tested usingDEW models provided by Allegheny Power. The Stewart Street feeder is much larger than theProgress Energy Carolinas feeder model and contains hundreds of nodes and tapped loads.

The proposed localization sub-algorithm directly relies on the accuracy of the load owsolution. In the nal test case, we assume the load to be a gaussian random variable by which

the known load is perturbed about a mean load operation point. This approach introduces loaduncertainty and measures the fault locating algorithms sensitivity to load variability.


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Model Based Fault Locating on Distribution Feeders

byJustin Wade Smith

A thesis submitted to the Graduate Faculty of North Carolina State University

in partial fulllment of therequirements for the Degree of

Master of Science

Electrical Engineering

Raleigh, North Carolina

2013

APPROVED BY:

Dr. Subhashish Bhattacharya Dr. Srdjan Lukic

Dr. Mesut E BaranChair of Advisory Committee


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DEDICATION

To my parents Jeffrey W. Smith and Cindy A. Smith. Without you, none of this would havebeen possible. You gave me strength when I had nothing left and encouragement when I

couldnt carry on. This one is for you.

To my grandfather Harold W. Smith who never got to see my thesis completed.

To my friend Asa Gray, who encouraged me. His hard work and dedication showed me thatthe best things in life are the hardest to get.

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BIOGRAPHY

Justin Wade Smith was born in Salisbury, North Carolina, United States of America. Justinattended high school at Northwood High in Chatham County, North Carolina and graduated

in 2004. He received a Bachelor of Science in Electrical Engineering from North Carolina StateUniversity in 2009. Upon completion of his bachelors degree he joined the graduate school atNCSU working towards his Master of Science degree in Electrical Engineering with an emphasison power system protection and fault locating under the direction of Dr. Mesut Baran withthe NSF FREEDM Systems Center.

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ACKNOWLEDGEMENTS

I would like to thank my advisor Dr. Mesut E. Baran, and my other committee members, Dr.Subhashish Bhattacharya and Dr. Srdjan Lukic.

I would also like to acknowledge Larry Alesi at Schweitzer Engineering Laboratories(SEL) forhis guidance. Much of my knowledge in the eld of power system protection can be attributedto the mentorship and dedication of Mr. Alesi. He is true mentor and friend.

I would like to thank Bette Gray of Rodanthe, N.C. for helping me chase my dreams. My familycould not have been blessed with a greater friend.

I would like to thank my friends Nick Parks and Jon McDonald. Their support and friendship

led me to success.

I would also like to thank my friends at Progress Energy Carolinas: Ronald Chip Moore,Juan Yancey and Lawrence Roberts.

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TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Challenges of Distribution Fault Locating . . . . . . . . . . . . . . . . . . . . . . 11.3 Proposed Solution: Model Based Algorithm . . . . . . . . . . . . . . . . . . . . . 21.4 Glossary of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Chapter 2 Modern Distribution Fault Locating Algorithms . . . . . . . . . . . . 52.1 Introduction to Distribution Fault Locating Algorithms . . . . . . . . . . . . . . 52.2 Technique with Two-port Network Section Representation(Das Method) . . . . . 5

2.2.1 Overview of Das Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.3 Fault Detection and Classication . . . . . . . . . . . . . . . . . . . . . . 62.2.4 Developing an Equivalent Radial Network . . . . . . . . . . . . . . . . . . 102.2.5 Load Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.6 Estimating nodal pre-fault voltages and currents . . . . . . . . . . . . . . 122.2.7 Estimating Voltages and Currents at the Remote End and at the Fault . 152.2.8 Calculating the Distance to the Fault: Single Line to Ground . . . . . . . 182.2.9 Assessment of the Das Algorithm: Advantages and Disadvantages . . . . 19

2.3 Girgis Apparent Impedance Method . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.1 Overview of Girgis Method . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.2 Direct Determination of Distance to the Fault . . . . . . . . . . . . . . . . 212.3.3 Assessment of the Girgis Algorithm: Advantages and Disadvantages . . . 26

2.4 Fault Locating using Digital Fault Recorder Data(Saha Algorithm) . . . . . . . . 272.4.1 Overview of Saha Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.2 Fault Loop Impedance Determination . . . . . . . . . . . . . . . . . . . . 272.4.3 Determination of the Faulty Node . . . . . . . . . . . . . . . . . . . . . . 302.4.4 Distance to the Fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.5 Assessment of the Saha Algorithm: Advantages and Disadvantages . . . . 38

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Chapter 3 Model Based Fault Locating . . . . . . . . . . . . . . . . . . . . . . . . . 403.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Sampled Data Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.3 Steady State Fault Current Extraction . . . . . . . . . . . . . . . . . . . . . . . . 433.4 Fault Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.1 Sliding Fault Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.4.2 Selecting Possible Fault Locations from Fault Tables . . . . . . . . . . . . 473.4.3 Fault Identication and Fault Table Selection . . . . . . . . . . . . . . . . 47

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3.4.4 Calculation of Fault Currents in the Fault Table: Fault Resistance . . . . 493.5 Localization Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5.1 Localization Using Protective Devices . . . . . . . . . . . . . . . . . . . . 503.5.2 Zones of Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.5.3 Localization using Fuse Characteristics . . . . . . . . . . . . . . . . . . . . 53

3.5.4 Estimating Fault Current Through the Fuse . . . . . . . . . . . . . . . . . 563.6 Localization using Load-Flow Rejection . . . . . . . . . . . . . . . . . . . . . . . 583.6.1 Calculating Best Matched Device from Load Flow Rejection . . . . . . . . 59

3.7 Combining Localization Data for Best Match . . . . . . . . . . . . . . . . . . . . 613.7.1 Final Ranking of Each Possibility . . . . . . . . . . . . . . . . . . . . . . . 62

3.8 Localization Flow Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Chapter 4 Fault Locator Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . 654.1 Introduction to the Notional Feeder, Stewart Street Feeder and Test Cases . . . . 654.2 Notional Feeder Fault Tables and Short Circuit Data . . . . . . . . . . . . . . . . 664.3 Test Case 1: Notional Feeder Testing With No Localization . . . . . . . . . . . . 67

4.3.1 Introduction: Test Conditions and Procedure . . . . . . . . . . . . . . . . 674.3.2 FLA Testing for Line-to-Ground Faults . . . . . . . . . . . . . . . . . . . 674.3.3 FLA Testing for Line-to-Line Faults . . . . . . . . . . . . . . . . . . . . . 684.3.4 Test Case 1 Results: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4 Test Case 2: Localization using Load Flow and Protective Devices . . . . . . . . 704.4.1 Introduction: Test Conditions and Procedures . . . . . . . . . . . . . . . 714.4.2 FLA Testing for Line-to-Ground Faults . . . . . . . . . . . . . . . . . . . 714.4.3 A-Ground Fault at Node 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 724.4.4 A-Ground Fault at Node 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4.5 Limitations of the Localization Algorithm: A-Ground Fault at Node 16 . 77

4.5 Test Case 3: Fault Locating on Fuse Saving or Fuse Blowing Schemes . . . . . . 794.5.1 Introduction: Notional Feeder with Fuse Blowing Coordination . . . . . . 79

4.5.2 Introduction: Fuse Blowing Scheme Test Conditions . . . . . . . . . . . . 804.5.3 FLA Testing for Line-to-Ground Faults: Fuse Blowing . . . . . . . . . . . 804.5.4 Line-to-Ground Fault at Node 17 . . . . . . . . . . . . . . . . . . . . . . . 804.5.5 Line-to-Ground Fault at Node 6 . . . . . . . . . . . . . . . . . . . . . . . 834.5.6 Introduction: Notional Feeder with Fuse Saving Coordination . . . . . . . 854.5.7 Introduction: Fuse Saving Scheme Test Conditions . . . . . . . . . . . . . 854.5.8 FLA Testing for Line-to-Ground Faults: Fuse Saving . . . . . . . . . . . . 854.5.9 Line-to-Ground Fault at Node 10 . . . . . . . . . . . . . . . . . . . . . . . 864.5.10 Line-to-Ground Fault at Node 6 . . . . . . . . . . . . . . . . . . . . . . . 884.5.11 Limitations of the FLA on Fuse Blowing or Fuse Saving Coordinated

Feeders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.6 Notional Feeder Test Results Summary . . . . . . . . . . . . . . . . . . . . . . . . 914.6.1 Test Case 1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.6.2 Test Case 2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.6.3 Test Case 3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.7 Test Case 4: Fault Locating on Large Scale Feeders . . . . . . . . . . . . . . . . . 934.7.1 Introduction: Stewart Street 12.47kV Feeder . . . . . . . . . . . . . . . . 93

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4.7.2 Introduction: Test Conditions and Procedures . . . . . . . . . . . . . . . 944.7.3 Fault Tables for the Stewart Street Feeder . . . . . . . . . . . . . . . . . . 954.7.4 Load Flow Analysis on the Stewart Street 12.47kV Feeder . . . . . . . . . 954.7.5 MATLAB Fault Modelling of the Stewart Street 12.47kV Feeder . . . . . 964.7.6 Fault at Pole P4622 on the Stewart Street 12.47kV Feeder . . . . . . . . . 98

4.7.7 FLA Test Results for P4622 Fault . . . . . . . . . . . . . . . . . . . . . . 1034.7.8 Fault at Pole P4266 on the Stewart Street 12.47kV Feeder . . . . . . . . . 1054.7.9 FLA Test Results for P4266 Fault . . . . . . . . . . . . . . . . . . . . . . 1094.7.10 FLA Test Results Summary for Stewart Street Feeder . . . . . . . . . . . 111

4.8 Sensitivity Analysis: System Load Perturbations . . . . . . . . . . . . . . . . . . 1124.8.1 Introduction: Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . 1124.8.2 System Load Perturbations: Test Conditions and Procedures . . . . . . . 1124.8.3 System Load Perturbations on Test Case 1-No Localization . . . . . . . . 1144.8.4 System Load Perturbations on Test Case 2 . . . . . . . . . . . . . . . . . 1164.8.5 System Load Perturbations on Test Case 3-Fuse Saving . . . . . . . . . . 1184.8.6 System Load Perturbations on Test Case 3-Fuse Blowing . . . . . . . . . 120

Chapter 5 FLA Testing Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Appendix A Notional Feeder Load Flow Data . . . . . . . . . . . . . . . . . . . . . . 127

A.1 Notional Feeder Real Power Flow Data . . . . . . . . . . . . . . . . . . . . . 127A.2 Notional Feeder Reactive Power Flow Data . . . . . . . . . . . . . . . . . . 129A.3 12kV Capacitor Bank Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 130A.4 Transformer Bank Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130A.5 Source Impedance Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130A.6 Line Impedance Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

A.6.1 Positive Sequence Line Impedance Data . . . . . . . . . . . . . . . . 131A.6.2 Negative Sequence Line Impedance Data . . . . . . . . . . . . . . . . 132A.6.3 Zero Sequence Line Impedance Data . . . . . . . . . . . . . . . . . . 133A.6.4 Positive and Zero Sequence Shunt Capacitance Data . . . . . . . . . 134

A.7 Relay Settings and Fuse Characteristics . . . . . . . . . . . . . . . . . . . . 135A.8 Fuse Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Appendix B Modelling of the Notional Feeder using MATLAB . . . . . . . . . . . . 138B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138B.2 Modelling of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138B.3 Modelling of System Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

B.3.1 Load Modelling Under Fault Conditions . . . . . . . . . . . . . . . . 142B.4 Modelling of Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143B.5 Modelling of Capacitor Banks . . . . . . . . . . . . . . . . . . . . . . . . . . 143B.6 Modelling of Multi-Winding Transformers . . . . . . . . . . . . . . . . . . . 144B.7 Modelling of Feeder Transformer . . . . . . . . . . . . . . . . . . . . . . . . 145

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Appendix C Distribution Fault Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 147C.1 System Fault Currents and Voltages via Numerically Computed Thevenin

Equivalents and Sensitivity Matrices . . . . . . . . . . . . . . . . . . . . . . 147C.1.1 Thevanins Theorem and Superposition Principle . . . . . . . . . . . 148C.1.2 Pre-Fault and Faulted Systems in DEW . . . . . . . . . . . . . . . . 151

C.1.3 Forming the Phase Thevanin Matrix . . . . . . . . . . . . . . . . . . 155C.1.4 System Fault Characteristics . . . . . . . . . . . . . . . . . . . . . . 158C.1.5 3-Phase Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160C.1.6 Phase-to-Phase-Ground Faults . . . . . . . . . . . . . . . . . . . . . 162C.1.7 Phase-to-Phase Faults . . . . . . . . . . . . . . . . . . . . . . . . . . 164C.1.8 Single Line-to-Ground Faults . . . . . . . . . . . . . . . . . . . . . . 166

C.2 Verication of DEW Fault Current Calculation: Example Feeder . . . . . . 167C.2.1 Validation with MATLAB Simulink Model . . . . . . . . . . . . . . 173

Appendix D Supplemental FLA Simulation Results for Stewart Street 12.47kV Feeder174D.1 Pole P4622 Supplemental Recorded Results . . . . . . . . . . . . . . . . . . 174D.2 Pole P4266 Supplemental Recorded Results . . . . . . . . . . . . . . . . . . 178

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LIST OF TABLES

Table 3.1 Fault Table for Figure 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Table 3.2 Increased Resolution Fault Table for Figure 3.7 . . . . . . . . . . . . . . . 47

Table 4.1 Notional Feeder Fault Table for SLG Faults on A Phase. . . . . . . . . . . 66Table 4.2 Line-to-Ground Simulation Results for Test Case 1 . . . . . . . . . . . . . 69Table 4.3 Phase-to-Phase Fault Simulation Results for Test Case 1 . . . . . . . . . . 69Table 4.4 Percent Mismatch Table for Protective Device Localization. . . . . . . . . 74Table 4.5 Percent Mismatch Table for Load Rejection Localization . . . . . . . . . . 74Table 4.6 Percent Mismatch Table for Protective Device Localization . . . . . . . . 76Table 4.7 Percent Mismatch Table for Load Rejection Localization . . . . . . . . . . 77Table 4.8 Percent Mismatch Table for Protective Device Localization . . . . . . . . 82Table 4.9 Percent Mismatch Table for Load Rejection Localization . . . . . . . . . . 82Table 4.10 Percent Mismatch Table for Protective Device Localization . . . . . . . . 84

Table 4.11 Percent Mismatch Table for Load Rejection Localization . . . . . . . . . . 84Table 4.12 Percent Mismatch Table for Protective Device Localization . . . . . . . . 87Table 4.13 Percent Mismatch Table for Load Rejection Localization . . . . . . . . . . 87Table 4.14 Percent Mismatch Table for Protective Device Localization . . . . . . . . 89Table 4.15 Percent Mismatch Table for Load Rejection Localization . . . . . . . . . . 89Table 4.16 Substation Bus Load Flow Solution for Stewart Street Feeder. . . . . . . . 96Table 4.17 Pre-Fault Load Flow Values . . . . . . . . . . . . . . . . . . . . . . . . . . 98Table 4.18 Fault Analysis Calculated Parameters . . . . . . . . . . . . . . . . . . . . 100Table 4.19 MATLAB Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 102Table 4.20 Pre-Fault DEW Load Flow Values . . . . . . . . . . . . . . . . . . . . . . 107Table 4.21 Success-Failure Rate on Node 10, Test Case 1(No Localization) . . . . . . 114Table 4.22 Success-Failure Rate on Node 17, Test Case 1(No Localization) . . . . . . 115Table 4.23 Success-Failure Rate on Node 4, Test Case 1(No Localization) . . . . . . . 115Table 4.24 Success-Failure Rate on Load Perturbation Test for Node 10 Fault . . . . 117Table 4.25 Success-Failure Rate on Load Perturbation Test for Node 17 Fault . . . . 117Table 4.26 Success-Failure Rate on Load Perturbation Test for Node 10 Fault with

Fuse Saving Feeder Coordination. . . . . . . . . . . . . . . . . . . . . . . . 118Table 4.27 Success-Failure Rate on Load Perturbation Test for Node 17 Fault with

Fuse Saving Feeder Coordination. . . . . . . . . . . . . . . . . . . . . . . . 119Table 4.28 Success-Failure Rate on Load Perturbation Test for Node 10 Fault with

Fuse Blowing Feeder Coordination. . . . . . . . . . . . . . . . . . . . . . . 120Table 4.29 Success-Failure Rate on Load Perturbation Test for Node 17 Fault with

Fuse Blowing Feeder Coordination. . . . . . . . . . . . . . . . . . . . . . . 121

Table A.1 Relay Settings for Substation Breaker A and Substation Breaker B . . . . 135Table A.2 Relay Settings for Recloser A and Recloser B . . . . . . . . . . . . . . . . 135

Table C.1 Fault types and their frequency of occurrence[23]. . . . . . . . . . . . . . . 159Table C.2 L-G Voltage Results of DEW vs. MATLAB Powerow Script . . . . . . . 168

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Table C.3 Current Results of DEW vs. MATLAB Script . . . . . . . . . . . . . . . . 169Table C.4 MATLAB Simulink Fault Current Results . . . . . . . . . . . . . . . . . . 173

Table D.1 Other Likely Fault Possibilities for P4622 . . . . . . . . . . . . . . . . . . 177Table D.2 Other Likely Fault Possibilities for P4266 . . . . . . . . . . . . . . . . . . 180

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LIST OF FIGURES

Figure 1.1 12kV Radial Feeder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Figure 1.2 FLA High Level Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Figure 2.1 Radial Distribution Feeder[25]. . . . . . . . . . . . . . . . . . . . . . . . . 7Figure 2.2 Flow Chart for Determining fault type[25]. . . . . . . . . . . . . . . . . . 8Figure 2.3 Line section Bus M and Bus R. . . . . . . . . . . . . . . . . . . . . . . . 9Figure 2.4 Radial Distribution Feeder[25]. . . . . . . . . . . . . . . . . . . . . . . . . 10Figure 2.5 Voltage and Current relationship between M and R[25]. . . . . . . . . . . 13Figure 2.6 Consolidated loads at the remote end, Node N[25]. . . . . . . . . . . . . . 15Figure 2.7 Fault between Nodes x and x + 1(= y)[25]. . . . . . . . . . . . . . . . . . 16Figure 2.8 Fault locator load model as constant impedance with system load model

as constant power [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Figure 2.9 Simple Feeder with no laterals or taps . . . . . . . . . . . . . . . . . . . . 22

Figure 2.10 Single line-to-ground fault sequence networks. . . . . . . . . . . . . . . . 23Figure 2.11 B-to-C Fault . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2.12 Sequence Network Diagram for a Phase-to-Phase Fault. . . . . . . . . . . 28Figure 2.13 Simple Substation with Two Parallel Feeders during Pre-Fault Conditions. 31Figure 2.14 Simple Substation with Two Parallel Feeders during Fault conditions. . . 32Figure 2.15 Cascaded Line Sections of Distribution Feeder. . . . . . . . . . . . . . . . 33Figure 2.16 Faulted Distribution Feeder[1]. . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 2.17 Circuit Representation of Network for a Fault located between 1 and k. . 35Figure 2.18 Circuit Representation of Network for a Fault located between k and k +1. 36Figure 2.19 Tapped Load inserted between node 2 and node k. . . . . . . . . . . . . . 39

Figure 3.1 High-Level Overview of FLA. . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 3.2 Handling of Raw Data Passed to the Fault Locator. . . . . . . . . . . . . 42Figure 3.3 Sampling of Fault Current Recorded at 64 Samples Per Cycle . . . . . . 42Figure 3.4 Bus Voltage During a Ground Fault. . . . . . . . . . . . . . . . . . . . . . 43Figure 3.5 RMS value of the 60Hz Fundamental during Line-to-Ground Fault. . . . 44Figure 3.6 Algorithm for detecting steady state fault current. . . . . . . . . . . . . . 45Figure 3.7 Generic 6 Node Feeder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 3.8 Phase-to-Phase evolving into a 3-Phase fault[13]. . . . . . . . . . . . . . . 50Figure 3.9 Generic 6 Node Feeder with Fuses and Overcurrent Relay. . . . . . . . . 51Figure 3.10 Operating Times for a 8620 A Fault Through Current. . . . . . . . . . . 52Figure 3.11 Observed Current during Fault Conditions for Figure 3.9. . . . . . . . . . 53Figure 3.12 Zones of Protection for Figure 3.9. . . . . . . . . . . . . . . . . . . . . . . 54

Figure 3.13 FLA Fuse Localization Algorithm. . . . . . . . . . . . . . . . . . . . . . . 54Figure 3.14 Superimposed Currents Through Faulted Fuse. . . . . . . . . . . . . . . . 56Figure 3.15 Fault Current Estimation Algorithm in FLA. . . . . . . . . . . . . . . . . 58Figure 3.16 Current Rejection after Recloser Lockout. . . . . . . . . . . . . . . . . . . 59Figure 3.17 Load Flow Rejection Algorithm in the FLA. . . . . . . . . . . . . . . . . 60

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Figure 3.18 FLA Localization Algorithm Flow Chart. . . . . . . . . . . . . . . . . . . 64

Figure 4.1 Simulation Testing Procedure using MATLAB Simulink. . . . . . . . . . 67Figure 4.2 Test Case 1 Notional Feeder One-Line Model. . . . . . . . . . . . . . . . 68Figure 4.3 Test Case 2 Testing Procedure. . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 4.4 Notional Feeder One-Line Model with added Reclosers and Breakers. . . 72Figure 4.5 Test Case 2: Substation Phase A RMS current during a fault at Node 17. 73Figure 4.6 Test Case 2: Substation Phase A RMS current during a fault at Node 7. 75Figure 4.7 Fault Locator Ranking of Fault at Node 16. . . . . . . . . . . . . . . . . 78Figure 4.8 Fuse Blowing Scheme on Notional Feeder. . . . . . . . . . . . . . . . . . . 79Figure 4.9 Fault Locator Observed Fault Current for Fault at Node 17. . . . . . . . 81Figure 4.10 Fault Locator Observed Fault Current for a fault at Node 6. . . . . . . . 83Figure 4.11 Fault Locator Observed Fault Current for a Fault at Node 10. . . . . . . 86Figure 4.12 Fault Locator Observed Fault Current for a Fault at Node 6. . . . . . . . 88Figure 4.13 Fault Locator Observed Fault Current for Node 8 Fault(Fuse Blowing

Coordination). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Figure 4.14 Stewart Street 12.47kV Feeder Modelled in DEW. . . . . . . . . . . . . . 93Figure 4.15 Test Procedure for Stewart Street Feeder using DEW. . . . . . . . . . . . 94Figure 4.16 Reected Fault Current at the Substation Bus for a Fault at Pole P4622. 95Figure 4.17 Pre-Fault Short Circuit Model. . . . . . . . . . . . . . . . . . . . . . . . . 97Figure 4.18 Faulty Short Circuit Model. . . . . . . . . . . . . . . . . . . . . . . . . . 97Figure 4.19 Post-Fault Short Circuit Model. . . . . . . . . . . . . . . . . . . . . . . . 98Figure 4.20 P4622 Node Location in Stewart Street 12.47kV Feeder Modelled in DEW. 99Figure 4.21 Reected Fault Current at the Substation due to the fault itself. . . . . . 100Figure 4.22 Observed Fault Current at FUSE654421506 during loaded conditions. . . 101Figure 4.23 Observed Fault Current at Substation during loaded conditions. . . . . . 101Figure 4.24 Possible Fault Locations in Stewart Street 12.47kV Feeder for a fault at

P4622. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Figure 4.25 Best Matched Locations for a fault at P4622 as calculated by the FLA. . 104Figure 4.26 P4266 Node Location in Stewart Street 12.47kV Feeder Modelled in DEW.105Figure 4.27 Upstream Fuse(FUSE1052482746) Load Flow Solution. . . . . . . . . . . 106Figure 4.28 Pre-Fault Voltage at P4266. . . . . . . . . . . . . . . . . . . . . . . . . . 106Figure 4.29 Reected Fault Current at the Substation due to the fault itself. . . . . . 107Figure 4.30 Observed Fault Current at upstream fuse(FUSE1052482746) during loaded

conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Figure 4.31 Possible Fault Location in Stewart Street 12.47kV Feeder for a fault at

P4266. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Figure 4.32 Best Matched Locations for a fault at P4266 as calculated by the FLA. . 110Figure 4.33 Normally Distributed Per-Unit Load Power. . . . . . . . . . . . . . . . . 113

Figure A.1 SandC T-Speed Fuse Minumum Melt Characteristics. . . . . . . . . . . . 136Figure A.2 SandC T-Speed Fuse Total Clearing Time Characteristics. . . . . . . . . 137

Figure B.1 Nominal PI Line Model[28]. . . . . . . . . . . . . . . . . . . . . . . . . . 139Figure B.2 Line Model Block in MATLAB Simulink[28]. . . . . . . . . . . . . . . . . 140

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Figure B.3 3-Phase Dynamic Load Representing Exponential Load Functions. . . . . 143Figure B.4 System Thevanin Equivalent. . . . . . . . . . . . . . . . . . . . . . . . . . 143Figure B.5 Three Phase MATLAB Simulink Source. . . . . . . . . . . . . . . . . . . 144Figure B.6 12kV Capacitor Bank. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Figure B.7 Three Winding Transformer. . . . . . . . . . . . . . . . . . . . . . . . . . 145

Figure B.8 MATLAB Multi-Winding Transformer Block. . . . . . . . . . . . . . . . 145Figure B.9 Per Phase Representation of 3-Phase Feeder Transformer. . . . . . . . . . 146Figure B.10 MATLAB Three Phase Transformer. . . . . . . . . . . . . . . . . . . . . 146

Figure C.1 System Thevanin Equivalent Component Model. . . . . . . . . . . . . . . 148Figure C.2 Power System containing a source, load buses and bus F . . . . . . . . . . 149Figure C.3 Power System before a fault occurrence at bus F. . . . . . . . . . . . . . 149Figure C.4 Open switch replaced by voltage source. . . . . . . . . . . . . . . . . . . . 150Figure C.5 Closed Switch Replaced by Two Sources. . . . . . . . . . . . . . . . . . . 150Figure C.6 Circuit during pre-fault conditions. . . . . . . . . . . . . . . . . . . . . . 151Figure C.7 Calculation of currents due to the fault itself. . . . . . . . . . . . . . . . . 151Figure C.8 Converting the network to a thevanin equivalent. . . . . . . . . . . . . . . 152Figure C.9 Pre-Fault system showing several loads and bus N(future faulted bus). . 152Figure C.10 Faulted System with Fault applied at Node N. . . . . . . . . . . . . . . . 153Figure C.11 Pre-Fault system with scaled loads and test load inserted at the faulted

bus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Figure C.12 Test load inserted on Phase A of the faulted node. . . . . . . . . . . . . . 156Figure C.13 Test load inserted on Phase B of the faulted node. . . . . . . . . . . . . . 157Figure C.14 Test load inserted on Phase C of the faulted node. . . . . . . . . . . . . . 158Figure C.15 Fault Root Causes and Percent Occurrence[7]. . . . . . . . . . . . . . . . 158Figure C.16 Model of a 3 Phase Fault. . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Figure C.17 Model of a Phase-Phase-Ground Fault. . . . . . . . . . . . . . . . . . . . 162Figure C.18 Model of a Phase-Phase Fault. . . . . . . . . . . . . . . . . . . . . . . . . 164

Figure C.19 Model of a Phase-to-Ground Fault. . . . . . . . . . . . . . . . . . . . . . 167Figure C.20 Simple Radial Feeder with three load buses. . . . . . . . . . . . . . . . . 168Figure C.21 DEW Event Report showing the current seen at the substation for a

bolted 3-Phase fault at Bus 1. . . . . . . . . . . . . . . . . . . . . . . . . . 172

Figure D.1 Worst Matched Locations for a fault at P4622 as calculated by the FLA. 175Figure D.2 Other Likely Fault Locations for a fault at P4622 as calculated by the

FLA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Figure D.3 Worst Matched Locations for a fault at P4266 as calculated by the FLA. 178Figure D.4 Other Likely Fault Locations for a fault at P4266 as calculated by the

FLA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

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Chapter 1

Introduction

1.1 Introduction

Modern utilities are required to transmit and distribute electric power over vast regions depend-ably while keeping costs low for its customers. As electric power systems grow in complexityand size, ensuring system reliability and continuity has become a major concern. Failure tolocate faults quickly can result in prolonged outage times, customer safety concerns and lostrevenues. Over the past few decades, distribution networks have evolved into large and complexnetworks capable of carrying thousands of customers. As of late, utilities have displayed partic-ular interest in distribution fault location technology to reduce such widespread impacts. Withthe addition of fault locators, utilities are able to reduce the search radius substantially. Thisimprovement has allowed utilities to expedite restoration time and reduce economic impact.

1.2 Challenges of Distribution Fault Locating

Currently most of the research being performed in the eld of fault location has been fo-cused on transmission networks[1]. Transmission networks are generally very simple, homo-geneous throughout and contain few tap lines or branches. The simplicity of transmissionsystem topology greatly reduces the complexity of the algorithm. Many transmission faultlocating algorithms have been shown to be accurate using basic fault locating techniques andmethods([10],[30]). Unlike transmission networks, distribution systems contain various conduc-tor sizes in addition to many load taps, laterals, and branches. Fault locating on such complextopologies presents many challenges not present in transmission systems. An example of amodern radial distribution feeder is shown in Figure 1.1.

One of the challenges of fault locating on distribution networks is the presence of signicantsystem loading. Unlike transmission networks, distribution systems contain many tapped loads

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Figure 1.1: 12kV Radial Feeder.

between the fault locator and the fault itself. Modern distribution networks make it infeasible

to place measurement devices at every branch on the network. This implies that system loadconditions throughout the network remain an unknown quantity. Therefore, distribution faultlocation algorithms must be robust enough to handle load uncertainty.

Many modern distribution fault locating algorithms use apparent impedance at the substa-tion to calculate the location of the fault. In many distribution networks there exist multiplenodes that have the same apparent impedance during fault conditions. This problem results inmultiple calculated possibilities throughout the network. As a result, fault locating algorithmsmust be able to localize and rank these possibilities from most to least likely.

1.3 Proposed Solution: Model Based AlgorithmThe proposed fault locating algorithm(FLA) is a model based algorithm(MBA) that uses ashort circuit model of the feeder to locate the fault. In the short circuit model, a fault isplaced at every node and the observed fault current at the substation is recorded in tabular

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format(fault tables). During an actual fault condition, the algorithm compares the recordedfault current at the substation to the short circuit modelling data. This comparison allows thefault locator to identify the exact node in the network that is faulty. A high level overview of the FLA is shown in Figure 1.2.

Figure 1.2: FLA High Level Overview.

In many cases, the short circuit model will indicate that there are several unique nodes withthe same fault current magnitude. This problem is overcome with a sub-algorithm in the FLAcalled the Localization Algorithm. This sub-algorithm uses protective device data and loadow data to determine the protective device that has interrupted the fault. By knowing thedevice that interrupted the fault, we can de-rank all other possibilities as a poor choices.

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1.4 Glossary of Terms

MBA : Model Based Algorithm, which is the proposed algorithm in this paper. Short circuitmodels are used to locate the faulty node.

FLA : Fault Locating Algorithm

IEEE : Institute of Electrical and Electronics Engineers

NF : Notional Feeder, which refers to the test network built by Progress Energy Carolinas inMATLAB Simulink.

PEC : Progress Energy Carolinas, referring to the utility based in Raleigh, NC.

P.U. : A per-unit quantity, expressed as a quantity on a dened system base unit.

DFT : Discrete Fourier Transform, referring to the family of techniques based on signal decom-position into sinusoids.

DFR : Digital Fault Recorder.

RMS : Root Mean Square, referring to the process of calculating the quadratic mean, repre-senting the measure of the magnitude of a varying function[12].

EPRI : Electric Power Research Institute

SandC : Refereeing to the fuse and protective device manufacturer.

DEW : Referring to the fault analysis and load ow software package. The Stewart Street12.47kV feeder was built by Allegheny Power and tested in the software package.

ASPEN One-Liner : Software fault analysis and coordination tool used to check protectivedevice coordination.

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Chapter 2

Modern Distribution Fault LocatingAlgorithms

2.1 Introduction to Distribution Fault Locating Algorithms

Fault locating on transmission and distribution networks requires the algorithm to quicklyand accurately calculate the location of the fault. Transmission systems by comparison tendto be much simpler, containing few lateral taps and homogeneous conductor sizing. Experi-mental data has shown that modern fault locating algorithms on transmission system to bevery accurate([10],[30]). Distribution systems present a unique challenge for fault locating dueto lateral taps(single and multiphase laterals), complex topologies, load uncertainty and non-homogeneous nature of the system. This set of challenges makes distribution fault location

distinctly different from transmission[25].The algorithms presented in this section attempt to confront many of the challenges of

distribution fault locating. Each algorithm is assessed and derived in full detail to show itsstrengths and weaknesses. At the end of each algorithm derivation, a section is included toreview the advantages and disadvantages of each.

2.2 Technique with Two-port Network Section Representation(DasMethod)

The following method developed by Ratan Das([25]), uses fundamental voltages and currentsavailable via measurement devices at the substation. The authors method addresses manyof the known problems with distribution fault location including: non-homogeneous cabling,system loading, fault impedance and multi-estimate conversion. In the following sections, adetailed assessment of the method is performed to illustrate advantages and dis-advantage of

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the Das method.

2.2.1 Overview of Das Method

The alogrithm proposed by Das uses fundamental voltages and currents at the substation bus

provided by utility measurement equipment. The location of the fault is estimated by computingthe apparent reactance from the fundamental voltage and current phasors at Bus M in Figure2.1. Once all possible fault locations have been identied, estimates of the voltages and currentsat the fault and remote end are calculated. The nal step of the algorithm is to estimate thedistance to the fault from the beginning of the line segment. The Das fault locating algorithmcan be decomposed into seven major steps.

Data Acquisition.

Preliminary estimate of the faulted line section.

Modication of the radial model(Equivalent Model Development).

Load Modelling.

Estimation of voltage and currents at the fault and at the remote end.

Estimating the distance to the fault from the line origination node.

Converting multiple fault locations into a single estimate.

2.2.2 Data Acquisition

After a fault is detected, the fundamental frequency voltage and currents at the substationbus(Node M) are saved. The data saved includes the pre-fault(fault has not occurred) andfault(fault has occurred) voltage and current phasors. Once a fault has been detected, pre-faultvoltage and current are saved 1 cycle before the fault occurrence. Fault data is saved 3 cyclesafter the fault occurrence to minimize infeed by motors.

2.2.3 Fault Detection and Classication

The rst step in many fault locating algorithms is detecting that a fault has occurred. TheDas algorithm uses current thresholds or pickup values to declare that a fault has occurred.After the fault has been detected, the algorithm attempts to classify the fault condition as oneof the following types: 3-phase, Line-to-Line, Line-to-Ground or Line-to-Line-to-Ground. Theow chart in Figure 2.2 is used to determine the fault type and the faulty phase A, B or C.

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It must also be noted that zero sequence currents are used to discriminate between phase-to-phase faults and phase-to-phase-to-ground faults. If the zero sequence current threshold isexceeded, this indicates the presence of a ground fault.

Figure 2.1: Radial Distribution Feeder[25].

Estimating the Faulted Line Section

After the fault has occurred, a preliminary estimate of the faulted line section is made. Tomake this estimate, detailed knowledge of the line parameters in needed along with the faultvoltages and currents. We rst consider a Phase A-to-Ground fault F between nodes x andx + 1 = y in Figure 2.1. If load is neglected, the apparent impedance from node M to the faultis dened as:

Z m = V amI am

(2.1)

We can dene the apparent reactance as:

X m = ImV amI am

(2.2)

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Figure 2.2: Flow Chart for Determining fault type[25].

Let us dene the line segment between Bus M and Bus R as the line segment in Figure 2.3.And the sequence impedance matrix for the line segment between Bus M and Bus R:

Z 012 =Z 00 Z 01 Z 02Z 10 Z 11 Z 12Z 20 Z 21 Z 22

(2.3)

Assuming de-coupled terms(equal self and mutual impedances):

Z 012 =Z 0 0 00 Z + 00 0 Z

(2.4)

Where Z 00 = Z 0, Z 11 = Z + and Z 22 = Z . We also assume that the negative and positivesequence impedances are equal for this line segment: Z + = Z

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Figure 2.3: Line section Bus M and Bus R.

We can transform the sequence impedance matrix to the phase impedance matrix using thefollowing transform:

Z abc = 13

T Z 0 0 00 Z + 00 0 Z +

3T 1 (2.5)

Where T is:

T =1 1 11 2 1 2

(2.6)

Where 2 = 1 120. This yields the following result:

Z abc =

2Z (1)3 +

Z (0)3

Z (0)3 Z

(1)

3Z (0)

3 Z (1)

3Z (0)

3 Z (1)

32Z (1)

3 + Z (0)

3Z (0)

3 Z (1)

3Z (0)

3 Z (1)

3Z (0)

3 Z (1)

32Z (1)

3 + Z (0)

3

=Z aa Z ab Z acZ ba Z bb Z bcZ ca Z cb Z cc

(2.7)

We can now dene the self impedance of phase A, Z aa as:

Z aa = 2Z (1)

3 +

Z (0)

3 (2.8)

The author denes the modied reactance as:

Im (Z aa ) = X aa = 2X (1)

3 +

X (0)

3 (2.9)

If the modied reactace is less the apparent reactance than the fault must be located beyond

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the remote node. To illustrate this point, if:

X aa < X m : Fault is Beyond Node RX aa > X m : Fault is Between Node M and Node R

If the fault is beyond the remote node of the rst line section, the next section is added tothe rst to obtain the total modied reactance dened by the following general summation:

X total =N

n =1

2X (1)n3

+ X (0)n

3 (2.10)

X total < X m : Fault is Beyond Current Line SectionX total > X m : Fault is Located on Current Line Section

This general procedure is continued into X total > X m , indicating the faulted line section.

Given the large size of distribution feeders, there is likely more than one possible fault loca-tion. If this occurs, all possible fault locations are recorded and analysed individually. In latersections, ranking of these possibilities from most likely to lease likely is discussed.

2.2.4 Developing an Equivalent Radial Network

Once all possible fault locations have been established, the radial feeder model with lateralsis converted to a network without laterals. All lateral loads between Bus M and the faultare consolidated at the tap origination. To illustrate this point we refer back to our radialdistribution feeder shown in Figure 2.1 with lateral taps K and L. To eliminate laterals K andL, they are lumped with all other loads connected to x 1. The nal result is shown in Figure2.4.

Figure 2.4: Radial Distribution Feeder[25].

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2.2.5 Load Modelling

System loading on distribution networks can introduce large errors when calculating distanceto the fault along a faulted line segment. To mitigate this, the effects of loads are taken intoaccount in the Das algorithm. It is assumed that system load is dependent on voltage. We

begin our analysis of system loading by considering the following static exponential load modelfor real and reactive power consumed by the load:

P (V, p) = P 0V V 0

p(2.11)

Q(V, q) = Q0V V 0

q(2.12)

The above equations represent the consumed load at various voltages based on a nominalpower and voltage. The terms p and q terms represent the real and reactive power sensitivity

to changes in voltage. To illustrate the sensitivity terms, we begin by dening a relativesensitivity function[31]:

S F =F F 0 0 x 0

(2.13)

S F = F x 0

0F 0

(2.14)

For Active Power:

S P V =P

P 0V V 0 V 0

(2.15)

S P V = P V V 0

V 0P 0

(2.16)

For Reactive Power:

S QV =QQ 0V V 0 V 0

(2.17)

S Q

V = QV V 0

V 0Q0 (2.18)

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If we take a derivative of the load model in Equation 2.11 active power with respect tovoltage we get:

P V

= pP 0V 0

V V 0

p 1(2.19)

Evaluating V = V 0 we get:

P V V 0

= pP 0V 0

(2.20)

Solving for p:

p = S P V =

P P 0V V 0

(2.21)

We can also do the same for q:

q = S QV =

QQ 0V V 0

(2.22)

We can see intuitively that the coefficients p and q represent the load sensitivity to voltage.Therefore, the apparent power absorbed by the load is:

S (V, p, q) = P 0V V 0

p

+ jQ 0V V 0

q

(2.23)

Using Equation 2.23 formulation, a voltage dependent impedance value can be derived torepresent the load at various voltages. The values of G0 and B0 are calculated at nominal

voltage and corresponding value of Y load can be calculated at any voltage. This relationship isextremely important when calculating load currents under fault conditions:

Y load (V, p, q) = G0V V 0

p 2+ jB 0

V V 0

q 2 (2.24)

2.2.6 Estimating nodal pre-fault voltages and currents

A vital part of the Das algorithm is the compensation for system loading. The algorithm com-pensates for system load by calculating the nominal load impedance and applying it to Equation

2.24. After the nominal pre-fault load impedance is calculated, the appropriate impedance atany voltage can be calculated. With this result we can easily calculate the load impedance un-der faulted conditions and compensate for the effects of load. To calculate the load impedanceunder pre-fault conditions, the following parameters are needed: pre-fault load voltage, pre-fault power factor(assumed to be known), percent of the total load at the load node(assumed

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to be known). The pre-fault voltage is solved for by using a two-port network representation of each distribution line. Using the two-port model, we can begin at the measurement node andcalculate the corresponding current injection and voltage at each node.

Before calculation of pre-fault values, the algorithm suggested by Das estimates the loadsat all nodes up to the fault using a database. The following formulation is used:

S load =Connected Load at Node X

Total Connected Load Total Pre-Fault Load (2.25)

The total pre-fault load is measured before the fault occurred by using the voltage and currentphasors measured by the fault locator. It can be see from the above equation that the measuredpre-fault load is apportioned to each node based on the percent loading of the node.

After the apparent power is calculated at each node(Equation 2.25), the load admittancecan be calculated:

Y 0 = V 20S load

1(2.26)

The above equation illustrates a very important point of this algorithm, the load impedancecannot be calculated until the pre-fault load voltage V 0 is solved for1. Beginning at the measurednode, we can calculate the load voltage at each node using the following two-port networkrelationship:

Figure 2.5: Voltage and Current relationship between M and R[25].

1 It is assumed that p and q are known from the power database

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The two port equation for Figure 2.5:

V rI rm

= Dmr BmrC mr Amr

V mI mr

(2.27)

Where Dmr

, C mr

, Bmr

and Amr

are as follows:

Dmr = cosh ( mr Lmr ) (2.28)

C mr = sinh ( mr Lmr )

Z smr(2.29)

Bmr = Z mr sinh ( mr Lmr ) (2.30)

Amr = cosh ( mr Lmr ) (2.31)The terms mr and Z smr are the propagation constant and characteristic impedance of the

line, respectively. We can can calculate each of these terms by using the resistance, reactance,conductance and suseptance per unit length of the line.

mr = (r mr + jx mr ) (gmr + jbmr ) (2.32)Z smr = (r mr + jx mr )(gmr + jbmr ) (2.33)

r mr resistance per unit length

xmr reactance per unit length

gmr conductance per unit length

bmr

suseptance per unit length

For short cable lengths in distribution systems the following approximation can be made:

Amr = Dmr = 1

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Bmr = Z smr mr Lmr

C mr = mr Lmr

Z smrResulting in a two-port model for a short distribution line:

V rI rm

= 1 BmrC mr 1

V mI mr

(2.34)

All pre-fault voltages and currents are solved for up until the faulty line section(node x)using the two-port in Equation 2.34. Once the faulty line section has been reached, all loadsbeyond the fault are consolidated at the farthest remote node (node N). Using another two-portequation similar to Equation 2.34, we can solve for the remote end voltage and current. Usinga cascaded two-port model, we can form a cascaded line section equivalent for all nodes fromx to N . This forms a equivalent two port model between the beginning of the faulted linesection(Node x) to the remote node(Node N ).

V n

I n =

De BeC e Ae

V xI xf

(2.35)

Where D e , C e , B e and Ae are cascaded line section equivalent constants from node x + 1 toN .

Figure 2.6: Consolidated loads at the remote end, Node N[25].

2.2.7 Estimating Voltages and Currents at the Remote End and at the FaultThe next step requires the voltages and currents at the fault to be calculated. When the faultoccurs, the fault locator at Node M records the currents and voltages for later use. Since wehave a two-port model representation of each line up until the fault, we can easily calculate

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the voltage at the beginning of the faulted line section(Node x). Also, as expected, the loadsare present during the fault and must be accounted for. Recall that Equation 2.24 allows us tocalculate the load current at any voltage, even during fault conditions.

Beginning at Node M , the two-port network model and Equation 2.24 can be used to calculatecurrents and voltages present at Node R. Using the two-port network model for R to x 1 andthe load model for Node x 1 we can calculate the voltages and currents at node x 1. Thisprocess is completed until you reach the beginning terminal of the faulted line section, Node x.

Figure 2.7: Fault between Nodes x and x + 1(= y)[25].

After the values of V x and I xf have been solve for, we can begin the process of calculatingthe distance to the fault from Node x. The fault is considered to be s length from Node x and1 s from Node x + 1(= y). Therefore, we break the two-port model of the line segment x tox + 1 into two two-port models: one from x to F and the other from F to x + 1(= y).

V f I fx

= 1 sB xysC xy 1 V xI xf

(2.36)

V x+1(= y)I fn

= 1 (1 s)Bxy

(1 s)C xy 1 V f I fn

(2.37)

With the above equations, the current owing through the fault I f is still unknown, alongwith the remote end voltage and current. We can relate the currents at the remote node N andthe I fn current by:

V n

I n = De

Be

C e Ae V x+1(= y)

I fn (2.38)

Where De , C e , Be and Ae are cascaded line section equivalent constants from node x+1(= y)to N . If we substitute Equation 2.37 into 2.38, we get:

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V n

I n =

De BeC e Ae

1 (1 s)Bxy(1 s)C xy 1

V f I fn

(2.39)

The above equation represents a very important relationship between currents owingaround the fault and the remote end load voltage and current. If we simplify 2.39, we get:

V n

I n =

K a + sK b K c + sK dK e + sK f K g + sK h

V f I fn

(2.40)

We can substitute I fn = I fx I f to eliminate I fn : V n

I n =


V f

I fx I f

K c + sK dK g + sK h

(2.41)

And substituting 2.36:

V n

I n =


1 sB xysC xy 1

V xI xf I f


(2.42)

We can substitute I n = Y n V n and re-arrange:

V n

V n Y n+ I f


= K a + sK b K c + sK dK e + sK f K g + sK h

1 sB xysC xy 1

V xI xf

(2.43)

Reducing:

1 K c + sK d

Y n K g + sK h V nI f

= K a + sK b K c + sK dK e + sK f K g + sK h

1 sB xysC xy 1

V xI xf

(2.44)Solving for V n and I f while neglecting second order terms:

V nI f

= 1

K v + sK w K m + sK n sK p

K q + sK r K v + sK u

V xI xf

(2.45)

Equation 2.45 represents the relationship between currents and voltages at the beginning of the faulted line and the voltages and currents for the remote end load under fault conditions.Equation 2.45 broken into equation form:

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V n = 1

K v + sK w[V x (K m + sK n ) + sK pI xf ] (2.46)

I f = 1

K v + sK w[V x (K q + sK r ) + ( K v + sK u ) I xf ] (2.47)

2.2.8 Calculating the Distance to the Fault: Single Line to Ground

For a single line to ground resistive fault, the fault voltage is described as:

V f = I f R f (2.48)

The fault voltage and current can be broken into corresponding sequence components:

V f I f

=V (0)f + V

(1)f + V

(2)f

I (0)f

+ I (1)f

+ I (2)f

= R f (2.49)

Taking the imaginary parts of both sides:

ImV (0)f + V

(1)f + V

(2)f

I (0)f + I (1)f + I

(2)f

= 0 (2.50)

Referring back to Equation 2.37 and Equation 2.47, these equations can also be broken intosequence components:

V (0)f = V (0)

x

sB (0)xy I

(0)xf (2.51)

V (1)f = V (1)

x sB (1)xy I (1)xf (2.52)

V (2)f = V (2)

x sB (2)xy I (2)xf (2.53)

I (0)f = 1

K (0)v + sK (0)w

V (0)x K (0)q + sK

(0)r + K

(0)v + sK

(0)u I

(0)xf (2.54)

I (1)f = 1

K (1)v + sK (1)w

V (1)x K (1)q + sK

(1)r + K

(1)v + sK

(1)u I

(1)xf (2.55)

I (2)f = 1

K (2)v + sK (2)wV (2)x K

(2)q + sK

(2)r + K

(2)v + sK

(2)u I

(2)xf (2.56)

After substituting Equations 2.54-2.57 and Equations 2.51-2.53 into Equation 2.49 whileneglecting higher order terms we obtain the solution for the distance to the fault:

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s = K AR K CI K AI K CR

(K CR K BI K CI K BR ) + ( K DR K AI K DI K AR ) (2.57)

2.2.9 Assessment of the Das Algorithm: Advantages and Disadvantages

The method proposed by Ratan Das, is a excellent fault location method with many advan-tages over other competing algorithms. In this section, a discussion of the advantages anddisadvantages of the Das algorithm are compared to other apparent impedance techniques usedin distribution fault location.

Fault Resistance

An excellent attribute to the Das algorithm is its ability to compensate for fault resistance. Theauthor choose a 23 mile long, 25kV radial feeder to preform tests. The tests performed weresingle line-to-ground faults with fault resistances varying from 5 to 50. The Das algorithmshows that for a SLG fault of 5, the error is less than 1.7%. For a 50 fault, the error wasshown to be less than 2.2% [1].

However, it can be easily shown that the Das algorithm does not work for a bolted fault( R f = 0)using Equation 2.49. Das does note this drawback: ...it is practically impossible to have afault with exactly zero resistance.[25]

Load Compensation

One of the major problems of distribution fault location is compensating for the effects of loads.The Das algorithm does this by developing a voltage dependent load model that is calculatedunder pre-fault conditions. In order to develop the load model, information about the load istaken from a load database. This implies that the utility using the fault locator must havemonitoring equipment at the load or available load ow study data. The drawback of using aload ow table is common for fault locators without communication systems such as SCADAto report real time load data.

In [8], it was shown that inaccurate load studies can lead to signicant errors in faultlocation with the Das algorithm. In Figure 2.8, a radial distribution feeder was modelled withconstant power loads. The load were assumed to be constant impedance loads by the faultlocator, resulting in large errors. The author of [8] concluded that the performance of the Dasalgorithm is not guaranteed without load behaviour studies preformed.

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for a utility. Also, careful attention must be given to the placement of the FCIs in order tomaximize their effectiveness.

2.3 Girgis Apparent Impedance Method

The Girgis Apparent Impedance method was developed by Adly Girgis and uses symmetri-cal components to determine a distance to the fault from the measurement point(usually thesubstation)[26]. The proposed algorithm addresses some of the issues with fault location suchas fault impedance, and fault localization. However, this algorithm does not address issues suchas: unbalanced mutual coupling and non-homogeneous lines.

2.3.1 Overview of Girgis Method

The Girgis method uses fundamental frequency voltage and current phasors available at thesubstation bus to determine the distance to the fault. Before the distance to the fault canbe calculated, the fault type is needed: Line-to-Ground, Line-to-Line, Line-to-Line-to-Ground,or 3-phase. To classify the fault type, changes in current magnitude are observed, indicatingthe faulted phase(s). Once the fault type has been determined, the algorithm uses symmetricalcomponents to determine the distance to the fault. If the solution yields multiple fault locationsthe algorithm will use localization techniques to determine the most likely candidate. Therefore,we can break the Girgis algorithm into three easy steps: Fault Classication, Solution andLocalization.

2.3.2 Direct Determination of Distance to the Fault

To begin our analysis of the Grigis algorithm we consider a simple feeder with no laterals ortaps in Figure 2.9. A single line-to-ground fault is placed d distance from the measurementbus. We will begin our analysis be assuming the system to be unloaded, and the only currentowing at the time of the fault is due to the fault itself.

I (0)faI (1)faI (2)fa

= 13

1 1 11 2

1 2

I fa00

(2.58)

Evaluating Equation 2.58 above yields:

I (0)fa = I (1)fa = I

(2)f a =

I f a3

(2.59)

The above equation directly implies that the positive( I (1)fa ), negative( I (2)fa ) and zero sequence( I

(0)fa )

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currents are equal during the fault. Using this implication, we must connect the positive, neg-ative and zero sequence networks in series as shown in Figure 2.10. The positive, negativeand zero sequence impedance of the line are represented in the gure as Z (1)L , Z

(2)L , and Z

(0)L

respectively. The source voltage is also broken into sequence components V ( i)s .

Figure 2.9: Simple Feeder with no laterals or taps

To calculate the distance d to the fault, we begin with a simple KVL loop around the

positive, negative and zero sequence parts of Figure 2.10 to solve for sequence voltage at thefault:

V (1)f = V (1)

s DI (1)fa Z

(1)L (2.60)

V (2)f = V (2)

s DI (2)fa Z

(2)L (2.61)

V (0)f = V (0)

s DI (0)fa Z

(0)L (2.62)

V (0)f + V (1)

f + V (2)

f = 3I (0)f a Rf (2.63)

Summing equations 2.60, 2.61 and 2.62.

V (0)f + V (1)

f + V (2)

f = V (0)

s + V (1)

s + V (2)

s D I (1)f a Z

(1)L + I

(2)fa Z

(2)L + I

(0)fa Z

(0)L (2.64)

We can substitute: V (0)f + V (1)

f + V (2)

f = V f and V (0)

s + V (1)s + V (2)s = V s .

V f = V s D I (1)f a Z

(1)L + I

(2)fa Z

(2)L + I

(0)fa Z

(0)L (2.65)

The Girgis method assumes that Z (2)L = Z (1)L .

V f = V s D Z (1)L I

(1)f a + I

(2)fa + I

(0)f a Z

(0)L (2.66)

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Figure 2.10: Single line-to-ground fault sequence networks.

Rearranging yeilds:

V f = V s DZ (1)L I

(1)fa + I

(2)fa + I

(0)f a

Z (0)LZ (1)L

(2.67)

Substituting I f a I (0)f a = I

(1)f a + I

(2)fa :

V f = V s DZ (1)L I fa I

(0)fa + I

(0)fa

Z (0)LZ (1)L

(2.68)

If we simplify:

V f = V s DZ (1)

L I f a + I (0)

f a

Z (0)L

Z (1)L

Z (1)L (2.69)

Solving for V s :

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V s = V f + DZ (1)L I f a + I

(0)f a

Z (0)L Z (1)L

Z (1)L (2.70)

Substituting Equation 2.63:

V s = 3I (0)fa Rf + DZ

(1)L I fa + I

(0)fa

Z (0)L Z (1)L

Z (1)L (2.71)

Let k = Z (0)L Z

(1)L

Z (1)L

V s = 3I (0)fa Rf + DZ

(1)L I fa + I

(0)fa k (2.72)

Equation 2.72 can be divided by I fa + I (0)fa k which gives:

V sI fa + I

(0)fa k

= 3I (0)f a Rf

I fa + I (0)fa k

+ DZ (1)L (2.73)

The above equation can be used to calculate the positive sequence reactance to the fault.This is accomplished by taking the imaginary part of both sides of Equation 2.73. Assuming

I f and I fa + I (0)fa k are in phase, the imaginary part of

3I (0)fa R f

I fa + I (0)fa k

would equal 0. This yeilds:

Im V s

I fa + I (0)fa k

= DX (1)L (2.74)

The above solution is called the positive sequence reactance method[9]. The obvious drawbackof this solution is the assumption that I f and I fa + I

(0)fa k are in phase. The Girgis method offers

a different solution which does not make this assumption.

To begin, we dene the apparent impedance seen at the measurement bus as Z app = V sI fa + I

(0)fa k

.

Z app =3I (0)f a Rf

I f a + I (0)f a k

+ DZ (1)L (2.75)

We then substitute I comp = 3I (0)f a , where I comp is the compensating current fed into thefault.

Z app = I comp R f I f a + I

(0)f a k

+ DZ (1)L (2.76)

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Next, the compensating current I comp and the selected current I fa + I (0)f a k are broken into

real and imaginary parts:

I comp = I d + jI q (2.77)

I f a + I (0)f a k = I s 1 + jI s 2 (2.78)

The positive sequence impedance of the line, Z (1)L is broken into real and imaginary partsas well:

Z (1)L = R(1)L + jX

(1)L (2.79)

Substituting 2.77,2.78, and 2.79 into 2.76:

Z app = D R(1)L + jX

(1)L +

R f (I d + jI q)I s 1 + jI s 2 (2.80)

In the above equation, the fault resistance and distance to the fault are unknown. To solvefor the distance to the fault, the apparent impedance is broken in real and imaginary parts.This yields two equations and two unknowns.

Re (Z app ) = DR(1)L + R f

I dI s 1 + I qI s 2I 2s 1 + I 2s 2

N(2.81)

Im (X app ) = DX (1)L + R f

I qI s 1 I dI s 2I

2s 1 + I

2s 2

M(2.82)

Substituting M and N for the terms above:

Re (Z app ) = DR(1)L + NR f (2.83)

Im (X app ) = DX (1)L + MR f (2.84)

Solving for D:

D = Rapp M X app N R (1)L M X

(1)L N

(2.85)

The above equation represent the direct determination of the distance to the fault from themeasurement point.

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2.3.3 Assessment of the Girgis Algorithm: Advantages and Disadvantages

The Girgis algorithm is a very simple algorithm that can be easily implemented by a utility forbasic fault locating. However, the method does have many negative attributes that make it anunacceptable choice for many modern utilities. The following conditions were not considered by

the Girgis algorithm: non-homogenous lines, mutual coupling and system loading conditions.

System Loading

The Girgis algorithm considers the feeder to be unloaded at the time of the fault occurrence.This assumption causes large errors in heavily loaded feeders. In the event that the fault wereto occur at the remote end of a long feeder, it is often the case that the current seen at thesubstation during the fault is not much greater than the load current. In this case, the loadcurrent presents a major issue resulting in degraded accuracy at the remote end of the network.

Non-Homogeneous Lines and Mutual Coupling

The direct determination of the distance to the fault assumes equal mutual coupling and self impedances of the line. In the event that the we have equal self impedance and unequal mutualimpedance or visa-versa this results in coupling between sequence components 1. In practice,equal self impedance and mutual coupling terms are rarely the case.

The Girgis algorithm also assumes the feeder conductors to be homogeneous. Many feedersare composed of many different types and sizes of conductor, resulting in a non-homogeneoussystem. This is a major disadvantage of this algorithm.

Localization of Multiple Fault Possibilities

The Girgis algorithm, much like many impedance based algorithms can return multiple faultpossibilities. For example, if the fault is found to be 1 mile from the measurement point,there may be multiple locations that are 1 mile from the measurement point. The author doesrecognize this as a limitation and presents a excellent solution. When multiple fault locationsare found, the operating characteristics of protective devices are used to eliminate possibilities.This will be discussed in a later chapter.

1 The 3-phase system is no longer decoupled.

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2.4 Fault Locating using Digital Fault Recorder Data(Saha Al-gorithm)

2.4.1 Overview of Saha Method

The algorithm proposed by Mourari Saha uses fundamental voltages and current available atthe substation before and after the fault[1]. The algorithm proposed by Saha can be brokeninto two major steps: calculation of the fault loop impedance and calculation of impedancealong the feeder. By comparing the measured impedance with the calculated feeder impedance,an indication of the fault location can be obtained[1].

2.4.2 Fault Loop Impedance Determination

The algorithm suggested by Saha requires that the positive sequence fault loop impedance becalculated from the available voltages and currents at the substation. This impedance is later

used to determine the faulty node in the network. To begin our analysis we consider a Phase-to-Phase fault involving phases B and C at some node in the network. A phase-to-phase faultis shown in Figure 2.11 at any arbitrary point in the distribution network. The parametersI f a , I fb and I fc are the fault currents measured at the substation.

Assuming the system to be unloaded at the time of fault we can easily show that for a B-to-C fault that I fc = I fb and I fa = 0. We can now transform the fault currents owing in thenetwork into their respective symmetrical components using the transform in Equation 2.86.

I (0)fa

I (1)faI (2)fa

= 13

1 1 11 2

1 2

0I fb

I fb (2.86)

Reducing Equation 2.86 results in the following:

I (1)f a = I (2)fa (2.87)

The above equation forms the foundation of solving for the positive sequence fault loopimpedance. Using Equation 2.87 we can form its circuit representation shown in Figure 2.12.If we preform a simple KVL loop on Figure 2.12, we obtain:

V (1)f a I (1)f a Z

(1)kk Z f I

(1)f a + I

(2)fa Z

(2)kk V

(2)fa = 0 (2.88)

Where the positive sequence measured voltage is V (1)f a and the negative sequence measured

voltage is V (2)f a . The thevanin positive and negative sequence impedances looking into the

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system at some arbitrary bus k is dened as Z (1)kk and Z (2)kk .

Figure 2.11: B-to-C Fault

Figure 2.12: Sequence Network Diagram for a Phase-to-Phase Fault.

Using Equation 2.87, we can eliminate all negative sequence currents:

V (1)f a I (1)f a Z

(1)kk Z f I

(1)f a I

(1)fa Z

(2)kk V

(2)fa = 0 (2.89)

If we solve Equation 2.89 for the fault loop impedance we get:

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V (1)fa V (2)

f a

I (1)f a= Z (1)kk + Z f + Z

(2)kk (2.90)

We can break down Equation 2.90 in much simpler terms by analysing the numerator

portion of the equation. The pre-fault measured voltages can be broken into their correspondingsequence components by using a equation similar to Equation 2.86:

V af V bf V cf

=1 1 11 2 1 2

V (0)af V (1)af V (2)af

(2.91)

If we extract V bf and V cf we get the following equations:

V bf = V (0)

af + V (1)

af 2 + V (2)af (2.92)

V cf = V (0)af + V (1)af + V (2)af 2 (2.93)

Subtracting Equation 2.92 from 2.93 yeilds:

V bf V cf = V (1)

af 2 + V

(2)af 2 (2.94)

Substituting = 1 120:

V bf V cf = V (1)

af 3 j ) + V (2)

af 3 j ) (2.95)

We can now solve for V (2)af

V (1)af

:

V (2)af V (1)

af = V bf V cf 3 j (2.96)

If we substitute 2.96 into 2.90 we get:

V bf V cf 3 jI (1)f a

= Z (1)kk + Z f + Z (2)kk (2.97)

Using a the symmetrical component transform similar to Equation 2.86, we can show that

the positive sequence fault current yields:

I (1)f a = 13

I fb 2I fb (2.98)

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Reducing yields:

I (1)fa = 33

jI fb (2.99)

Substituting Equation 2.99 into Equation 2.97:

V bf V cf 3 j 33 jI fb

= Z (1)kk + Z f + Z (2)kk (2.100)

Reducing Equation 2.100 yields:

V bf V cf I fb

= Z (1)kk + Z f + Z (2)kk (2.101)

If we assume that the positive and negative sequence impedances are equal Z (1)kk = Z (2)kk :

V bf

V cf I fb = 2 Z

(1)kk + Z f (2.102)

During fault conditions 2 I fb = I bf I cf :V bf V cf I bf I cf

= Z (1)kk + Z f (2.103)

Assuming the fault to be bolted Z f = 0:

V bf V cf I bf I cf

= Z (1)kk (2.104)

The above equation represents the positive sequence impedance to the fault. We can usethe available voltages V bf and V cf and currents I fb and I fc to calculate the positive sequencefault loop impedance. However, with the fault resistance being unknown, we must assume that the fault is bolted Z f = 0.

2.4.3 Determination of the Faulty Node

After the positive sequence fault loop impedance has been found, we then search for the faultynode. To begin our search for the faulty node we consider a simple substation with two feeders.

Let us assume that the only available measurements are the bus voltage and supply current

from the source: V and I. The pre-fault positive sequence impedance of the fault feeder isdened as Z (1)k . The parallel connected feeder positive sequence impedance is dened as Z

(1)lk .

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Figure 2.13: Simple Substation with Two Parallel Feeders during Pre-Fault Conditions.

During the pre-fault conditions, we can dene the pre-fault positive sequence impedanceseen by the fault locator as Z (1) pre . This is represented in equation form as:

Z (1) pre = Z (1)k Z

(1)lk

Z (1)k + Z (1)lk

(2.105)

During fault conditions, as shown in Figure 2.14, we represent the positive sequence impedance

seen by the fault locator as:

Z (1)f = V b V cI b I c

=Z (1)fk Z

(1)lk

Z (1)fk + Z (1)lk

(2.106)

Our objective is to calculate the positive sequence loop impedance of the faulty feeder Z (1)fk .

Solving Equation 2.106 for Z (1)fk we get:

Z (1)fk =Z (1)f Z

(1)lk

Z (1)lk

Z (1)f (2.107)

If we solve Equation 2.105 for Z (1)lk we get:

Z (1)lk = Z (1) pre Z (1)kZ (1)k Z

(1) pre

(2.108)

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Figure 2.14: Simple Substation with Two Parallel Feeders during Fault conditions.

Substituting Equation 2.108 into Equation 2.107:

Z (1)fk = Z (1)k Z

(1) pre

Z (1) pre Z (1)f

Z (1)k Z (1)

pre

Z (1)k

kzk

(2.109)

Consequently, kzk can also be related to Z (1)

pre and Z (1)lk :

kzk = Z (1) preZ (1)lk

(2.110)

Substituting:

Z (1)fk = Z (1)k Z

(1) pre

Z (1) pre Z (1)f

Z (1)preZ (1)lk

(2.111)

Z (1)fk = Z (1)k Z (1) pre

Z (1) pre Z (1)f kzk

(2.112)

We previously dened Z f as:

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Z (1)f = V a V bI a I b

= V I

(2.113)

If we substitute Equation 2.113 into 2.117:

Z (1)fk = V I kzkV Z (1)pre

(2.114)

The above equation represents the positive sequence loop impedance of the faulty feederduring fault conditions. It is assumed that the impedance of the adjacent parallel feeders donot change during fault conditions(this may not always be the case).

Now that the positive sequence loop impedance of the faulty feeder has been calculated, wecan begin to search for the faulty line. The algorithm suggested by Saha sweeps the networkfor the faulty node until a specic set of criterion exists. One the criterion has been met, thefaulty line section is found.

Figure 2.15: Cascaded Line Sections of Distribution Feeder.

We will begin our search for the faulty node at some node i1 in the network. We will representthe impedance of the cable between i 1 and i by Z

(1)si 1. The load at the remote node i is

represented by Z (1) pi . By using network reduction, we can develop a relationship between theimpedance seen looking into the system at i

1 and i(Which is Z (1)f i and Z

(1)f i

1 respectively).

Z (1)f i 1 =Z (1)f i Z

(1) pi

Z (1)f i + Z (1)

pi

+ Z (1)si 1 (2.115)

Solving for Z (1)f i we get:

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Z (1)f i =Z (1) pi Z

(1)f i 1 Z

(1)si 1

Z (1) pi Z (1)f i 1 + Z

(1)si 1

(2.116)

Let us assume the substation bus to be i 1; assuming the load impedances to be known,we can now traverse each line section and calculate the impedance seen looking into the systemfrom the remote bus. As expected |Z

(1)f i | < |Z

(1)f i 1| as we approach the faulty line. If we were

traverse down the feeder and the value of Im (Z (1)f i ) would be become less than zero, we wouldknow the fault exists between i 1 and i.

To begin our search for our faulty line section, we always begin at the substation and workoutwards. In this particaluar scenario, the substation may represent i1. As we move outwardon the feeder, the value of I m (Z (1)f i ) decreases. When the value of Im (Z

(1)f i ) becomes negative,

the faulty line section has been found[32]. This part of the algorithm only identies that theline section between i 1 and i is faulty. The distance from i 1 to the fault will be coveredin the next section.

2.4.4 Distance to the Fault

After we have identied the fault line section( i and i 1) we must now determine the distancefrom node i 1 to the fault(distance down the line to the fault). Using a change in notation,we will call the faulted line k and k + 1 in the following example( k = i 1 and k + 1 = i).

Figure 2.16: Faulted Distribution Feeder[1].

Using Figure 2.16, we will place a the fault between nodes k and k + 1. Our objective is tocalculate the distance from node k to the fault.

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Let us begin by assuming that the positive sequence loop impedance from node 1 to k, Z (1)1kis known from pre-calculated tabulated values. Also, we will assume that the impedance of theshunt load at node 2 Z (1)

l2 is also known. Any impedance beyond node k is lumped together as

a shunt impedance called Z (1)f . The circuit representation of the faulty network seen by thefault locator is represented by Figure 2.17.

Figure 2.17: Circuit Representation of Network for a Fault located between 1 and k.

We will begin our derivation by assuming that the impedance seen by the fault locator is denedas Z (1)1f .

Z (1)1f = V I

(2.117)

In the above gure, the variable m is used as a percentile over the total length of the cablefrom node 1 to node k. In this example the load is 1 m distance from the measurementpoint and m from the origination terminal of the faulted line section(node k). The equationrepresentation of impedance seen by the fault locator described by Figure 2.17:

Z (1)1f = V I

=mZ (1)1k + Z

(1)f Z

(1)l2

mZ (1)1k + Z (1)f + Z

(1)l2

+ (1 m) Z (1)1k (2.118)

If we solve for the unknown variable, Z (1)f we get:

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Z (1)f =Z (1)1k Z

(1)1f mZ

(1)1k + Z

(1)l2 mZ

(1)1k

2

mZ (1)1k Z (1)l2 Z

(1)1k Z

(1)1f

(2.119)

Previously we dened Z (1)f

as the lumped impedance representative of all elements beyond

k. In order to solve for the distance to the fault from k, we must dene Z (1)f in much moredetail.

We can represent Z (1)f as the circuit shown in Figure 2.18. All elements beyond node kare represented, including the fault resistance. In Figure 2.18, x represents the distance to thefault from node k and Z (1)L represents the positive sequence impedance of the line from nodek to k + 1. Also since a shunt load is present at node k, this is represented by Z (1)lk . We alsomust represent all elements beyond node k + 1; this positive sequence thevanin impedance isrepresented by Z (1)k+1 .

Figure 2.18: Circuit Representation of Network for a Fault located between k and k + 1.

Using network reduction we can reduce Figure 2.18 to the following equation:

Z (1)f =

Z (1)lk xZ (1)L + Rf (1x )Z

(1)L + Z

(1)k +1

R f +(1 x )Z (1)L + Z

(1)k +1

Z (1)lk + xZ (1)L +

R f (1x )Z (1)L + Z

(1)k +1

R f +(1 x )Z (1)L + Z

(1)k +1

(2.120)

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If we set Equation 2.119 and Equation 2.120 equal and solving for Rf :

R f = x2A xB + C (2.121)Where A,B and C are:

A =Z (1)L

2Z (1)lk Z

(1)f

Z (1)lk Z (1)f Z

(1)L + Z

(1)k+1 Z

(1)f Z

(1)lk

(2.122)

B =Z (1)L Z

(1)lk Z

(1)f Z

(1)L + Z

(1)k+1 Z

(1)f Z

(1)lk + 2 Z

(1)f Z

(1)lk

Z (1)lk

Z (1)f Z

(1)L + Z

(1)k+1

Z (1)f Z

(1)lk

(2.123)

C = Z (1)f Z

(1)lk Z

(1)L + Z

(1)k+1

Z (1)lk Z (1)f Z

(1)L + Z

(1)k+1 Z

(1)f Z

(1)lk

(2.124)

Taking the imaginary part of the quadratic function for the fault resistance R f :

Im (R f ) = x2Im (A) + xIm (B ) + Im (C ) = 0 (2.125)Where the imaginary parts of A,B and C are:

Im (A) = A i

Im (B ) = B i

Im (C ) = C i

(2.126)

The resultant roots 1 and 2 of Equation 2.125 are:

1 = B i + B2i 4AiC i2Ai (2.127)

x = 2 =B i B 2i 4Ai C i2Ai (2.128)

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Since 1 is considered a invalid solution, the distance to the fault from k is x = 2.

2.4.5 Assessment of the Saha Algorithm: Advantages and Disadvantages

The algorithm suggested by Saha has many advantages and addresses many of the issues that

other algorithms fail to consider. The algorithm compensates for system loading, fault resis-tance, non-homogeneous feeder conductors and measurement issues(only available current is themain transformer current). However, there are several key disadvantages of the Saha algorithm.One of which is that the positive sequence fault loop impedance is used to calculate the faultyline section assuming unloaded conditions and a bolted fault. The algorithm also presents adirect determination for distance to the fault, but this solution is highly dependent on topologyof the system. Also, the author does not address localization of multiple fault possibilities if there exists two possible paths on a feeder where Im (Z (1)f i ) < 0.

Problems with Determining the Faulty Line Section

The Saha algorithm uses the positive sequence fault loop impedance

Model Based Fault Locating on Distribution Feeders.

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Transcript of Model Based Fault Locating on Distribution Feeders.