POWER SYSTEM STABILITY AND CONTROL

•

P. KUNDUR Vice-President, Power Engineering Powertech Labs Inc., Surrey, British Columbia

Formerly Manager Analytical Methods and Specialized Studies Department Power System Planning Division, Ontario Hydro, Toronto, Ontario

and

Adjunct Professor Department of Electrical and Computer Engineering

U I I I V t M b l i y Ul I U I U I U U , I U I U M I U , W l l l d l l U

Edited by Neal J. Balu Mark G. Lauby Power System Planning and Operations Electrical Systems Division Electric Power Research Institute 3412 Hillview Avenue Palo Alto, California

Program

-McGraw-Hill, Inc. New York San Francisco Washington, D.C. Auckland Bogota Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto

Contents

xix

xxi

FOREWORD

PREFACE

PART I GENERAL BACKGROUND

1 GENERAL CHARACTERISTICS OF MODERN POWER SYSTEMS 3

1.1 Evolution of electric power systems 3 1.2 Structure of the power system 5 1.3 Power system control 8 1.4 Design and operating criteria for stability 13 References 16

2 INTRODUCTION TO THE POWER SYSTEM STABILITY PROBLEM 17

2.1 Basic concepts and definitions 17 2.1.1 Rotor angle stability 18 2.1.2 Voltage stability and voltage collapse 27 2.1.3 Mid-term and long-term stability 33

2.2 Classification of stability 34 2.3 Historical review of stability problems 37 References 40

VII

viii Contents

PART II EQUIPMENT CHARACTERISTICS AND MODELLING

3 SYNCHRONOUS MACHINE THEORY AND MODELLING 45

3.1 Physical description 46 3.1.1 Armature and field structure 46 3.1.2 Machines with multiple pole pairs 49 3.1.3 MMF waveforms 49 3.1.4 Direct and quadrature axes 53

3.2 Mathematical description of a synchronous machine 54 3.2.1 Review of magnetic circuit equations 56 3.2.2 Basic equations of a synchronous machine 59

3.3 The dqO transformation 67 3.4 Per unit representation 75

3.4.1 Per unit system for the stator quantities 75 3.4.2 Per unit stator voltage equations 76 3.4.3 Per unit rotor voltage equations 77 3.4.4 Stator flux linkage equations 78 3.4.5 Rotor flux linkage equations 78 3.4.6 Per unit system for the rotor 79 3.4.7 Per unit power and torque 83 3.4.8 Alternative per unit systems and transformations 83 3.4.9 Summary of per unit equations 84

3.5 Equivalent circuits for direct and quadrature axes 88 3.6 Steady-state analysis 93

3.6.1 Voltage, current, and flux linkage relationships 93 3.6.2 Phasor representation 95 3.6.3 Rotor angle 98 3.6.4 Steady-state equivalent circuit 99 3.6.5 Procedure for computing steady-state values 100

3.7 Electrical transient performance characteristics 105 3.7.1 Short-circuit current in a simple RL circuit 105 3.7.2 Three-phase short-circuit at the terminals of

a synchronous machine 107 3.7.3 Elimination of dc offset in short-circuit current 108

3.8 Magnetic saturation 110 3.8.1 Open-circuit and short-circuit characteristics 110 3.8.2 Representation of saturation instability studies 112 3.8.3 Improved modelling of saturation 117

3.9 Equations of motion 128

Contents ix

3.9.1 3.9.2 3.9.3 3.9.4 3.9.5

References

Review of mechanics of motion Swing equation Mechanical starting time Calculation of inertia constant Representation in system studies

128 128 132 132 135 136

4 SYNCHRONOUS MACHINE PARAMETERS 139

4.1 Operational parameters 139 4.2 Standard parameters 144 4.3 Frequency-response characteristics 159 4.4 Determination of synchronous machine parameters 161 References 166

5 SYNCHRONOUS MACHINE REPRESENTATION IN STABILITY STUDIES 169

5.1 Simplifications essential for large-scale studies 169 5.1.1 Neglect of stator p\\i terms 170 5.1.2 Neglecting the effect of speed variations on stator voltages 174

5.2 Simplified model with amortisseurs neglected 179 5.3 Constant flux linkage model 184

5.3.1 Classical model 184 5.3.2 Constant flux linkage model including the effects of

subtransient circuits 188 5.3.3 Summary of simple models for different time frames 190

5.4 Reactive capability limits 191 5.4.1 Reactive capability curves 191 5.4.2 V curves and compounding curves 196

References 198

6 AC TRANSMISSION 199

6.1 Transmission lines 200 Electrical characteristics 200 Performance equations 201 Natural or surge impedance loading 205 Equivalent circuit of a transmission line 206 Typical parameters 209

6.1.1 6.1.2 6.1.3 6.1.4 6.1.5

X Contents

6.2

6.3 6.4

6.1.6 6.1.7 6.1.8 6.1.9 6.1.10 6.1.11 6.1.12

Performance requirements of power transmission lines Voltage and current profile under no-load Voltage-power characteristics Power transfer and stability considerations Effect of line loss on V-P and Q-P characteristics Thermal limits Loadability characteristics

Transformers 6.2.1 Representation of two-winding transformers 6.2.2 Representation of three-winding transformers 6.2.3 Phase-shifting transformers Transfer of power between active sources Power-flow analysis 6.4.1 Network equations 6.4.2 Gauss-Seidel method 6.4.3 Newton-Raphson (N-R) method 6.4.4 Fast decoupled load-flow (FDLF) methods 6.4.5 Comparison of the power-flow solution methods 6.4.6 Sparsity-oriented triangular factorization 6.4.7 Network reduction

References

7 POWER SYSTEM LOADS

211

211

216

221

225

226

228

231

232

240

245

250

255

257

259

260

264

267

268

268

269

271

7.1 Basic load-modelling concepts 7.1.1 Static load models 7.1.2 Dynamic load models

7.2 Modelling of induction motors 7.2.1 Equations of an induction machine 7.2.2 Steady-state characteristics 7.2.3 Alternative rotor constructions 7.2.4 Representation of saturation 7.2.5 Per unit representation 7.2.6 Representation in stability studies

7.3 Synchronous motor model 7.4 Acquisition of load-model parameters

7.4.1 Measurement-based approach 7.4.2 Component-based approach 7.4.3 Sample load characteristics

References

271

272

274 279 279 287 293

296

297

300

306

306

306 308 310 312

Contents xi

8 EXCITATION SYSTEMS 315

8.1 Excitation system requirements 8.2 Elements of an excitation system 8.3 Types of excitation systems

8.3.1 DC excitation systems 8.3.2 AC excitation systems 8.3.3 Static excitation systems 8.3.4 Recent developments and future trends

8.4 Dynamic performance measures 8.4.1 Large-signal performance measures 8.4.2 Small-signal performance measures

8.5 Control and protective functions 8.5.1 AC and DC regulators 8.5.2 Excitation system stabilizing circuits 8.5.3 Power system stabilizer (PSS) 8.5.4 Load compensation 8.5.5 Underexcitation limiter 8.5.6 Overexcitation limiter 8.5.7 Volts-per-hertz limiter and protection 8.5.8 Field-shorting circuits

8.6 Modelling of excitation systems 8.6.1 Per unit system 8.6.2 Modelling of excitation system components 8.6.3 Modelling of complete excitation systems 8.6.4 Field testing for model development and verification

References

9 PRIME MOVERS AND ENERGY SUPPLY SYSTEMS 377

9.1 Hydraulic turbines and governing systems 377 9.1.1 Hydraulic turbine transfer function 379 9.1.2 Nonlinear turbine model assuming inelastic water column 387 9.1.3 Governors for hydraulic turbines 394 9.1.4 Detailed hydraulic system model 404 9.1.5 Guidelines for modelling hydraulic turbines 417

9.2 Steam turbines and governing systems 418 9.2.1 Modelling of steam turbines 422 9.2.2 Steam turbine controls 432 9.2.3 Steam turbine off-frequency capability 444

tion

315 317 318 319 320 323 326 327 327 330 333 333 334 335 335 337 337 339 340 341 342 347 362 372 373

XII Contents

449 449 455 459 460

9.3 Thermal energy systems 9.3.1 Fossil-fuelled energy systems 9.3.2 Nuclear-based energy systems 9.3.3 Modelling of thermal energy systems

References

10 HIGH-VOLTAGE DIRECT-CURRENT TRANSMISSION 463

10.1 HVDC system configurations and components 464 10.1.1 Classification of HVDC links 464 10.1.2 Components of HVDC transmission system 467

10.2 Converter theory and performance equations 468 10.2.1 Valve characteristics 469 10.2.2 Converter circuits 470 10.2.3 Converter transformer rating 492 10.2.4 Multiple-bridge converters 493

10.3 Abnormal operation 498 10.3.1 Arc-back (backfire) 498 10.3.2 Commutation failure 499

10.4 Control of HVDC systems 500 10.4.1 Basic principles of control 500 10.4.2 Control implementation 514 10.4.3 Converter firing-control systems 516 10.4.4 Valve blocking and bypassing 520 10.4.5 Starting, stopping, and power-flow reversal 521 10.4.6 Controls for enhancement of ac system performance 523

10.5 Harmonics and filters 524 10.5.1 AC side harmonics 524 10.5.2 DC side harmonics 527

10.6 Influence of ac system strength on ac/dc system interaction 528 10.6.1 Short-circuit ratio 528 10.6.2 Reactive power and ac system strength 529 10.6.3 Problems with low ESCR systems 530 10.6.4 Solutions to problems associated with weak systems 531 10.6.5 Effective inertia constant 532 10.6.6 Forced commutation 532

10.7 Responses to dc and ac system faults 533 10.7.1 DC line faults 534 10.7.2 Converter faults 535 10.7.3 AC system faults 535

Contents л и i

538 539 540 544 -/"TT 544 564 566 577

10.8 Multiterminal HVDC systems 10.8.1 MTDC network configurations 10.8.2 Control of MTDC systems

10.9 Modelling of HVDC systems 10.9.1 Representation for power-flow solution 10.9.2 Per unit system for dc quantities 10.9.3 Representation for stability studies

References

11 CONTROL OF ACTIVE POWER AND REACTIVE POWER 581

11.1 Active power and frequency control 581 11.1.1 Fundamentals of speed governing 582 11.1.2 Control of generating unit power output 592 11.1.3 Composite regulating characteristic of power systems 595 11.1.4 Response rates of turbine-governing systems 598 11.1.5 Fundamentals of automatic generation control 601 11.1.6 Implementation of AGC 617 11.1.7 Underfrequency load shedding 623

11.2 Reactive power and voltage control 627 11.2.1 Production and absorption of reactive power 627 11.2.2 Methods of voltage control 628 11.2.3 Shunt reactors 629 11.2.4 Shunt capacitors 631 11.2.5 Series capacitors 633 11.2.6 Synchronous condensers 638 11.2.7 Static var systems 639 11.2.8 Principles of transmission system compensation 654 11.2.9 Modelling of reactive compensating devices 672 11.2.10 Application of tap-changing transformers to

transmission systems 678 11.2.11 Distribution system voltage regulation 679 11.2.12 Modelling of transformer ULTC control systems 684

11.3 Power-flow analysis procedures 687 11.3.1 Prefault power flows 687 11.3.2 Postfault power flows 688

References 691

XIV Contents

PART III SYSTEM STABILITY: physical aspects, analysis, and improvement

12 SMALL-SIGNAL STABILITY 699

12.1

12.2

12.3

12.4 12.5 12.6 12.7 12.8 12.9

Fundamental concepts of stability of dynamic systems 12.1.1 12.1.2 12.1.3 12.1.4

State-space representation Stability of a dynamic system Linearization Analysis of stability

Eigenproperties of the state matrix 12.2.1 12.2.2 12.2.3 12.2.4 12.2.5 12.2.6 12.2.7 12.2.8 12.2.9

Eigenvalues Eigenvectors Modal matrices Free motion of a dynamic system Mode shape, sensitivity, and participation factor Controllability and observability The concept of complex frequency

700 700 702 703 706 707 707 707 708 709 714 716 717

Relationship between eigenproperties and transfer functions 719 Computation of eigenvalues

Small-signal stability of a single-machine infinite bus system 12.3.1 12.3.2

Generator represented by the classical model Effects of synchronous machine field circuit dynamics

Effects of excitation system Power system stabilizer System state matrix with amortisseurs Small-signal stability of multimachine systems Special techniques for analysis of very large systems Characteristics of small-signal stability problems

References

726 727 728 737 758 766 782 792 799 817 822

13 TRANSIENT STABILITY 827

13.1 An elementary view of transient stability 827 13.2 Numerical integration methods 836

13.2.1 Euler method 836 13.2.2 Modified Euler method 838 13.2.3 Runge-Kutta (R-K) methods 838 13.2.4 Numerical stability of explicit integration methods 841 13.2.5 Implicit integration methods 842

Contents xv

13.3 Simulation of power system dynamic response 848 13.3.1 Structure of the power system model 848 13.3.2 Synchronous machine representation 849 13.3.3 Excitation system representation 855 13.3.4 Transmission network and load representation 858 13.3.5 Overall system equations 859 13.3.6 Solution of overall system equations 861

13.4 Analysis of unbalanced faults 872 13.4.1 Introduction to symmetrical components 872 13.4.2 Sequence impedances of synchronous machines 877 13.4.3 Sequence impedances of transmission lines 884 13.4.4 Sequence impedances of transformers 884 13.4.5 Simulation of different types of faults 885 13.4.6 Representation of open-conductor conditions 898

13.5 Performance of protective relaying 903 13.5.1 Transmission line protection 903 13.5.2 Fault-clearing times 911 13.5.3 Relaying quantities during swings 914 13.5.4 Evaluation of distance relay performance during swings 919 13.5.5 Prevention of tripping during transient conditions 920 13.5.6 Automatic line reclosing 922 13.5.7 Generator out-of-step protection 923 13.5.8 Loss-of-excitation protection 927

13.6 Case study of transient stability of a large system 934 13.7 Direct method of transient stability analysis 941

13.7.1 Description of the transient energy function approach 941 13.7.2 Analysis of practical power systems 945 13.7.3 Limitations of the direct methods 954

References 954

14 VOLTAGE STABILITY 959

14.1 Basic concepts related to voltage stability 960 14.1.1 Transmission system characteristics 960 14.1.2 Generator characteristics 967 14.1.3 Load characteristics 968 14.1.4 Characteristics of reactive compensating devices 969

14.2 Voltage collapse 973 14.2.1 Typical scenario of voltage collapse 974 14.2.2 General characterization based on actual incidents 975

xvi

14.2.3 Classification of voltage stability 14.3 Voltage stability analysis

14.3.1 Modelling requirements 14.3.2 Dynamic analysis 14.3.3 Static analysis 14.3.4 Determination of shortest distance to instability 14.3.5 The continuation power-flow analysis

14.4 Prevention of voltage collapse 14.4.1 System design measures 14.4.2 System-operating measures

References

15 SUBSYNCHRONOUS OSCILLATIONS

15.1 Turbine-generator torsional characteristics 15.1.1 Shaft system model 15.1.2 Torsional natural frequencies and mode shapes

15.2 Torsional interaction with power system controls 15.2.1 Interaction with generator excitation controls 15.2.2 Interaction with speed governors 15.2.3 Interaction with nearby dc converters

15.3 Subsynchronous resonance 15.3.1 Characteristics of series capacitor-compensated

transmission systems 15.3.2 Self-excitation due to induction generator effect 15.3.3 Torsional interaction resulting in SSR 15.3.4 Analytical methods 15.3.5 Countermeasures to SSR problems

15.4 Impact of network-switching disturbances 15.5 Torsional interaction between closely coupled units 15.6 Hydro generator torsional characteristics References

976 977 978 978 990 1007

1012

1019

1019

1021

1022

1025

1026

1026

1034

1041

1041

1047

1047

1050

1050

1052

1053

1053

1060

1061

1065

1067

1068

16 MID-TERM AND LONG-TERM STABILITY

16.1 Nature of system response to severe upsets 16.2 Distinction between mid-term and long-term stability 16.3 Power plant response during severe upsets

16.3.1 Thermal power plants 16.3.2 Hydro power plants

1073

1073

1078

1079

1079

1081

Contents xvii

16.4 Simulation of long-term dynamic response 1085 16.4.1 Purpose of long-term dynamic simulations 1085 16.4.2 Modelling requirements 1085 16.4.3 Numerical integration techniques 1087

16.5 Case studies of severe system upsets 1088 16.5.1 Case study involving an overgenerated island 1088 16.5.2 Case study involving an undergenerated island 1092

References 1099

17 METHODS OF IMPROVING STABILITY 1103

17.1 Transient stability enhancement 1104 17.1.1 High-speed fault clearing 1104 17.1.2 Reduction of transmission system reactance 1104 17.1.3 Regulated shunt compensation 1105 17.1.4 Dynamic braking 1106 17.1.5 Reactor switching 1106 17.1.6 Independent-pole operation of circuit breakers 1107 17.1.7 Single-pole switching 1107 17.1.8 Steam turbine fast-valving 1110 17.1.9 Generator tripping 1118 17.1.10 Controlled system separation and load shedding 1120 17.1.11 High-speed excitation systems 1121 17.1.12 Discontinuous excitation control 1124 17.1.13 Control of HVDC transmission links 1125

17.2 Small-signal stability enhancement 1127 17.2.1 Power system stabilizers 1128 17.2.2 Supplementary control of static var compensators 1142 17.2.3 Supplementary control of HVDC transmission links 1151

References 1161

INDEX 1167