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POWER SYSTEMDYNAMICSStability and ControlSecond EditionJan MachowskiWarsaw University of Technology, PolandJanusz W. BialekThe University of Edinburgh, UKJames R. BumbyDurham University, UK
POWER SYSTEMDYNAMICSStability and ControlSecond EditionJan MachowskiWarsaw University of Technology, PolandJanusz W. BialekThe University of Edinburgh, UKJames R. BumbyDurham University, UK
This edition ﬁrst published 2008C 2008 John Wiley & Sons, Ltd.Registered ofﬁceJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United KingdomFor details of our global editorial ofﬁces, for customer services and for information about how to apply forpermission to reuse the copyright material in this book please see our website at www.wiley.com.The right of the author to be identiﬁed as the author of this work has been asserted in accordance with theCopyright, Designs and Patents Act 1988.All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted,in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except aspermitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may notbe available in electronic books.Designations used by companies to distinguish their products are often claimed as trademarks. All brandnames and product names used in this book are trade names, service marks, trademarks or registeredtrademarks of their respective owners. The publisher is not associated with any product or vendor mentionedin this book. This publication is designed to provide accurate and authoritative information in regard to thesubject matter covered. It is sold on the understanding that the publisher is not engaged in renderingprofessional services. If professional advice or other expert assistance is required, the services of a competentprofessional should be sought.Library of Congress Cataloging-in-Publication DataMachowski, Jan. Power system dynamics: stability and control / Jan Machowski, Janusz W. Bialek,James R. Bumby. – 2nd ed. p. cm. Rev. ed. of: Power system dynamics and stability / Jan Machowski, Janusz W. Bialek,James R. Bumby. 1997. Includes bibliographical references and index. ISBN 978-0-470-72558-0 (cloth) 1. Electric power system stability. 2. Electric power systems–Control. I. Bialek, JanuszW. II. Bumby, J. R. (James Richard) III. Title. TK1010.M33 2008 621.319 1–dc22 2008032220A catalogue record for this book is available from the British Library.ISBN 978-0-470-72558-0Typeset in 9/11pt Times New Roman by Aptara Inc., New Delhi, India.Printed in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire
ContentsAbout the Authors xiiiPreface xvAcknowledgements xixList of Symbols xxiPART I INTRODUCTION TO POWER SYSTEMS1 Introduction 31.1 Stability and Control of a Dynamic System 31.2 Classiﬁcation of Power System Dynamics 51.3 Two Pairs of Important Quantities: Reactive Power/Voltage and Real Power/Frequency 71.4 Stability of a Power System 91.5 Security of a Power System 91.6 Brief Historical Overview 122 Power System Components 152.1 Introduction 15 2.1.1 Reliability of Supply 15 2.1.2 Supplying Electrical Energy of Good Quality 16 2.1.3 Economic Generation and Transmission 16 2.1.4 Environmental Issues 162.2 Structure of the Electrical Power System 16 2.2.1 Generation 18 2.2.2 Transmission 18 2.2.3 Distribution 19 2.2.4 Demand 192.3 Generating Units 19 2.3.1 Synchronous Generators 20 2.3.2 Exciters and Automatic Voltage Regulators 21 2.3.3 Turbines and their Governing Systems 252.4 Substations 352.5 Transmission and Distribution Network 35 2.5.1 Overhead Lines and Underground Cables 35 2.5.2 Transformers 36 2.5.3 Shunt and Series Elements 41 2.5.4 FACTS Devices 43
vi Contents2.6 Protection 54 2.6.1 Protection of Transmission Lines 54 2.6.2 Protection of Transformers 56 2.6.3 Protection of Busbars 57 2.6.4 Protection of Generating Units 572.7 Wide Area Measurement Systems 58 2.7.1 WAMS and WAMPAC Based on GPS Signal 58 2.7.2 Phasors 59 2.7.3 Phasor Measurement Unit 61 2.7.4 Structures of WAMS and WAMPAC 623 The Power System in the Steady State 653.1 Transmission Lines 65 3.1.1 Line Equations and the π -Equivalent Circuit 66 3.1.2 Performance of the Transmission Line 67 3.1.3 Underground Cables 723.2 Transformers 72 3.2.1 Equivalent Circuit 72 3.2.2 Off-Nominal Transformation Ratio 743.3 Synchronous Generators 76 3.3.1 Round-Rotor Machines 76 3.3.2 Salient-Pole Machines 83 3.3.3 Synchronous Generator as a Power Source 89 3.3.4 Reactive Power Capability Curve of a Round-Rotor Generator 91 3.3.5 Voltage–Reactive Power Capability Characteristic V(Q) 95 3.3.6 Including the Equivalent Network Impedance 1003.4 Power System Loads 104 3.4.1 Lighting and Heating 105 3.4.2 Induction Motors 106 3.4.3 Static Characteristics of the Load 110 3.4.4 Load Models 1113.5 Network Equations 1133.6 Power Flows in Transmission Networks 118 3.6.1 Control of Power Flows 118 3.6.2 Calculation of Power Flows 122PART II INTRODUCTION TO POWER SYSTEM DYNAMICS4 Electromagnetic Phenomena 1274.1 Fundamentals 1274.2 Three-Phase Short Circuit on a Synchronous Generator 129 4.2.1 Three-Phase Short Circuit with the Generator on No Load and Winding Resistance Neglected 129 4.2.2 Including the Effect of Winding Resistance 133 4.2.3 Armature Flux Paths and the Equivalent Reactances 134 4.2.4 Generator Electromotive Forces and Equivalent Circuits 140 4.2.5 Short-Circuit Currents with the Generator Initially on No Load 146 4.2.6 Short-Circuit Currents in the Loaded Generator 149 4.2.7 Subtransient Torque 150
Contents vii4.3 Phase-to-Phase Short Circuit 152 4.3.1 Short-Circuit Current and Flux with Winding Resistance Neglected 153 4.3.2 Inﬂuence of the Subtransient Saliency 156 4.3.3 Positive- and Negative-Sequence Reactances 159 4.3.4 Inﬂuence of Winding Resistance 160 4.3.5 Subtransient Torque 1624.4 Synchronization 163 4.4.1 Currents and Torques 1644.5 Short-Circuit in a Network and its Clearing 1665 Electromechanical Dynamics – Small Disturbances 1695.1 Swing Equation 1695.2 Damping Power 172 5.2.1 Damping Power at Large Speed Deviations 1755.3 Equilibrium Points 1765.4 Steady-State Stability of Unregulated System 177 5.4.1 Pull-Out Power 177 5.4.2 Transient Power–Angle Characteristics 179 5.4.3 Rotor Swings and Equal Area Criterion 184 5.4.4 Effect of Damper Windings 186 5.4.5 Effect of Rotor Flux Linkage Variation 187 5.4.6 Analysis of Rotor Swings Around the Equilibrium Point 191 5.4.7 Mechanical Analogues of the Generator–Inﬁnite Busbar System 1955.5 Steady-State Stability of the Regulated System 196 5.5.1 Steady-State Power–Angle Characteristic of Regulated Generator 196 5.5.2 Transient Power–Angle Characteristic of the Regulated Generator 200 5.5.3 Effect of Rotor Flux Linkage Variation 202 5.5.4 Effect of AVR Action on the Damper Windings 205 5.5.5 Compensating the Negative Damping Components 2066 Electromechanical Dynamics – Large Disturbances 2076.1 Transient Stability 207 6.1.1 Fault Cleared Without a Change in the Equivalent Network Impedance 207 6.1.2 Short-Circuit Cleared with/without Auto-Reclosing 212 6.1.3 Power Swings 215 6.1.4 Effect of Flux Decrement 215 6.1.5 Effect of the AVR 2166.2 Swings in Multi-Machine Systems 2206.3 Direct Method for Stability Assessment 222 6.3.1 Mathematical Background 223 6.3.2 Energy-Type Lyapunov Function 225 6.3.3 Transient Stability Area 227 6.3.4 Equal Area Criterion 228 6.3.5 Lyapunov Direct Method for a Multi-Machine System 2306.4 Synchronization 2376.5 Asynchronous Operation and Resynchronization 239 6.5.1 Transition to Asynchronous Operation 240 6.5.2 Asynchronous Operation 241 6.5.3 Possibility of Resynchronization 242
viii Contents 6.6 Out-of-Step Protection Systems 244 6.6.1 Impedance Loci During Power Swings 245 6.6.2 Power Swing Blocking 248 6.6.3 Pole-Slip Protection of Synchronous Generator 249 6.6.4 Out-of-Step Tripping in a Network 251 6.6.5 Example of a Blackout 253 6.7 Torsional Oscillations in the Drive Shaft 253 6.7.1 The Torsional Natural Frequencies of the Turbine–Generator Rotor 253 6.7.2 Effect of System Faults 259 6.7.3 Subsynchronous Resonance 2617 Wind Power 265 7.1 Wind Turbines 265 7.1.1 Generator Systems 269 7.2 Induction Machine Equivalent Circuit 274 7.3 Induction Generator Coupled to the Grid 277 7.4 Induction Generators with Slightly Increased Speed Range via External Rotor Resistance 280 7.5 Induction Generators with Signiﬁcantly Increased Speed Range: DFIGs 282 7.5.1 Operation with the Injected Voltage in Phase with the Rotor Current 284 7.5.2 Operation with the Injected Voltage out of Phase with the Rotor Current 286 7.5.3 The DFIG as a Synchronous Generator 287 7.5.4 Control Strategy for a DFIG 289 7.6 Fully Rated Converter Systems: Wide Speed Control 290 7.6.1 Machine-Side Inverter 291 7.6.2 Grid-Side Inverter 292 7.7 Peak Power Tracking of Variable Speed Wind Turbines 293 7.8 Connections of Wind Farms 294 7.9 Fault Behaviour of Induction Generators 294 7.9.1 Fixed-Speed Induction Generators 294 7.9.2 Variable-Speed Induction Generators 2967.10 Inﬂuence of Wind Generators on Power System Stability 2968 Voltage Stability 299 8.1 Network Feasibility 299 8.1.1 Ideally Stiff Load 300 8.1.2 Inﬂuence of the Load Characteristics 303 8.2 Stability Criteria 305 8.2.1 The d Q/dV Criterion 305 8.2.2 The dE/dV Criterion 308 8.2.3 The dQG /dQL Criterion 309 8.3 Critical Load Demand and Voltage Collapse 310 8.3.1 Effects of Increasing Demand 311 8.3.2 Effect of Network Outages 314 8.3.3 Inﬂuence of the Shape of the Load Characteristics 315 8.3.4 Inﬂuence of the Voltage Control 317 8.4 Static Analysis 318 8.4.1 Voltage Stability and Load Flow 318 8.4.2 Voltage Stability Indices 320
Contents ix 8.5 Dynamic Analysis 321 8.5.1 The Dynamics of Voltage Collapse 321 8.5.2 Examples of Power System Blackouts 323 8.5.3 Computer Simulation of Voltage Collapse 326 8.6 Prevention of Voltage Collapse 327 8.7 Self-Excitation of a Generator Operating on a Capacitive Load 329 8.7.1 Parametric Resonance in RLC Circuits 329 8.7.2 Self-Excitation of a Generator with Open-Circuited Field Winding 330 8.7.3 Self-Excitation of a Generator with Closed Field Winding 332 8.7.4 Practical Possibility of Self-Excitation 3349 Frequency Stability and Control 335 9.1 Automatic Generation Control 336 9.1.1 Generation Characteristic 336 9.1.2 Primary Control 339 9.1.3 Secondary Control 341 9.1.4 Tertiary Control 345 9.1.5 AGC as a Multi-Level Control 346 9.1.6 Defence Plan Against Frequency Instability 347 9.1.7 Quality Assessment of Frequency Control 349 9.2 Stage I – Rotor Swings in the Generators 350 9.3 Stage II – Frequency Drop 353 9.4 Stage III – Primary Control 354 9.4.1 The Importance of the Spinning Reserve 356 9.4.2 Frequency Collapse 358 9.4.3 Underfrequency Load Shedding 360 9.5 Stage IV – Secondary Control 360 9.5.1 Islanded Systems 361 9.5.2 Interconnected Systems and Tie-Line Oscillations 364 9.6 FACTS Devices in Tie-Lines 370 9.6.1 Incremental Model of a Multi-Machine System 371 9.6.2 State-Variable Control Based on Lyapunov Method 375 9.6.3 Example of Simulation Results 378 9.6.4 Coordination Between AGC and Series FACTS Devices in Tie-Lines 37910 Stability Enhancement 38310.1 Power System Stabilizers 383 10.1.1 PSS Applied to the Excitation System 384 10.1.2 PSS Applied to the Turbine Governor 38710.2 Fast Valving 38710.3 Braking Resistors 39110.4 Generator Tripping 392 10.4.1 Preventive Tripping 393 10.4.2 Restitutive Tripping 39410.5 Shunt FACTS Devices 395 10.5.1 Power–Angle Characteristic 395 10.5.2 State-Variable Control 397 10.5.3 Control Based on Local Measurements 400 10.5.4 Examples of Controllable Shunt Elements 404 10.5.5 Generalization to Multi-Machine Systems 406 10.5.6 Example of Simulation Results 414
x Contents10.6 Series Compensators 416 10.6.1 State-Variable Control 417 10.6.2 Interpretation Using the Equal Area Criterion 419 10.6.3 Control Strategy Based on the Squared Current 420 10.6.4 Control Based on Other Local Measurements 421 10.6.5 Simulation Results 42310.7 Uniﬁed Power Flow Controller 423 10.7.1 Power–Angle Characteristic 424 10.7.2 State-Variable Control 426 10.7.3 Control Based on Local Measurements 428 10.7.4 Examples of Simulation Results 429PART III ADVANCED TOPICS IN POWER SYSTEM DYNAMICS11 Advanced Power System Modelling 43311.1 Synchronous Generator 433 11.1.1 Assumptions 434 11.1.2 The Flux Linkage Equations in the Stator Reference Frame 434 11.1.3 The Flux Linkage Equations in the Rotor Reference Frame 436 11.1.4 Voltage Equations 440 11.1.5 Generator Reactances in Terms of Circuit Quantities 443 11.1.6 Synchronous Generator Equations 446 11.1.7 Synchronous Generator Models 453 11.1.8 Saturation Effects 45811.2 Excitation Systems 462 11.2.1 Transducer and Comparator Model 462 11.2.2 Exciters and Regulators 463 11.2.3 Power System Stabilizer (PSS) 47011.3 Turbines and Turbine Governors 470 11.3.1 Steam Turbines 471 11.3.2 Hydraulic Turbines 476 11.3.3 Wind Turbines 48111.4 Dynamic Load Models 48511.5 FACTS Devices 488 11.5.1 Shunt FACTS Devices 488 11.5.2 Series FACTS Devices 48812 Steady-State Stability of Multi-Machine System 49112.1 Mathematical Background 491 12.1.1 Eigenvalues and Eigenvectors 491 12.1.2 Diagonalization of a Square Real Matrix 496 12.1.3 Solution of Matrix Differential Equations 500 12.1.4 Modal and Sensitivity Analysis 509 12.1.5 Modal Form of the State Equation with Inputs 512 12.1.6 Nonlinear System 51312.2 Steady-State Stability of Unregulated System 514 12.2.1 State-Space Equation 515 12.2.2 Simpliﬁed Steady-State Stability Conditions 517 12.2.3 Including the Voltage Characteristics of the Loads 521 12.2.4 Transfer Capability of the Network 522
Contents xi12.3 Steady-State Stability of the Regulated System 523 12.3.1 Generator and Network 523 12.3.2 Including Excitation System Model and Voltage Control 525 12.3.3 Linear State Equation of the System 528 12.3.4 Examples 52813 Power System Dynamic Simulation 53513.1 Numerical Integration Methods 53613.2 The Partitioned Solution 541 13.2.1 Partial Matrix Inversion 543 13.2.2 Matrix Factorization 547 13.2.3 Newton’s Method 548 13.2.4 Ways of Avoiding Iterations and Multiple Network Solutions 55113.3 The Simultaneous Solution Methods 55313.4 Comparison Between the Methods 55414 Power System Model Reduction – Equivalents 55714.1 Types of Equivalents 55714.2 Network Transformation 559 14.2.1 Elimination of Nodes 559 14.2.2 Aggregation of Nodes Using Dimo’s Method 562 14.2.3 Aggregation of Nodes Using Zhukov’s Method 563 14.2.4 Coherency 56514.3 Aggregation of Generating Units 56714.4 Equivalent Model of External Subsystem 56814.5 Coherency Recognition 56914.6 Properties of Coherency-Based Equivalents 573 14.6.1 Electrical Interpretation of Zhukov’s Aggregation 573 14.6.2 Incremental Equivalent Model 575 14.6.3 Modal Interpretation of Exact Coherency 579 14.6.4 Eigenvalues and Eigenvectors of the Equivalent Model 582 14.6.5 Equilibrium Points of the Equivalent Model 589Appendix 593References 613Index 623
About the Authors Professor Jan Machowski received his MSc and PhD degrees in Elec- trical Engineering from Warsaw University of Technology in 1974 and 1979, respectively. After obtaining ﬁeld experience in the Dispatching Centre and several power plants, he joined the Electrical Faculty of Warsaw University of Technology where presently he is employed as a Professor and Director of the Power Engineering Institute. His areas of interest are electrical power systems, power system protection and control. In 1989–93 Professor Machowski was a Visiting Professor at Kaiser- slautern University in Germany where he carried out two research projects on power swing blocking algorithms for distance protection and optimal control of FACTS devices. Professor Machowski is the co-author of three books published inPolish: Power System Stability (WNT, 1989), Short Circuits in Power Systems (WNT, 2002) andPower System Control and Stability (WPW, 2007). He is also a co-author of Power System Dynamicsand Stability published by John Wiley & Sons, Ltd (1997). Professor Machowski is the author and co-author of 42 papers published in English in interna-tional fora. He has carried out many projects on electrical power systems, power system stabilityand power system protection commissioned by the Polish Power Grid Company, Electric PowerResearch Institute in the United States, Electrinstitut Milan Vidmar in Slovenia and Ministry ofScience and Higher Education of Poland. Professor Janusz Bialek received his MEng and PhD degrees in Elec- trical Engineering from Warsaw University of Technology in 1977 and 1981, respectively. From 1981 to 1989 he was a lecturer with War- saw University of Technology. In 1989 he moved to the University of Durham, United Kingdom, and since 2003 he has been at the Univer- sity of Edinburgh where he currently holds the Bert Whittington Chair of Electrical Engineering. His main research interests are in sustain- able energy systems, security of supply, liberalization of the electricity supply industry and power system dynamics and control. Professor Bialek has co-authored two books and over 100 research papers. He has been a consultant to the Department of Trade and Industry (DTI) of the UK government, Scottish Executive, Elexon,Polish Power Grid Company, Scottish Power, Enron and Electrical Power Research Institute (EPRI).He was the Principal Investigator of a number of major research grants funded by the Engineeringand Physical Sciences Research Council and the DTI. Professor Bialek is a member of the Advisory Board of Electricity Policy Research Group,Cambridge University, a member of the Dispute Resolution Panel for the Single Electricity MarketOperator, Ireland, and Honorary Professor of Heriot-Watt University, Scotland.
xiv About the Authors Dr Jim Bumby received his BSc and PhD degrees in Engineering from Durham University, United Kingdom, in 1970 and 1974, respectively. From 1973 to 1978 he worked for the International Research and De- velopment Company, Newcastle-upon-Tyne, on superconducting ma- chines, hybrid vehicles and sea-wave energy. Since 1978 he has worked in the School of Engineering at Durham University where he is cur- rently Reader in Electrical Engineering. He has worked in the area of electrical machines and systems for over 30 years, ﬁrst in industry and then in academia. Dr Bumby is the author or co-author of over 100 technical papers and two books in the general area of electrical machines/power systems and control. He has also written numerous technical reports for industrialclients. These papers and books have led to the award of a number of national and internationalprizes including the Institute of Measurement and Control prize for the best transactions paper in1988 for work on hybrid electric vehicles and the IEE Power Division Premium in 1997 for workon direct drive permanent magnet generators for wind turbine applications. His current researchinterests are in novel generator technologies and their associated control for new and renewableenergy systems.
PrefaceIn 1997 the authors of this book, J. Machowski, J.W. Bialek and J.R. Bumby, published a bookentitled Power System Dynamics and Stability. That book was well received by readers who toldus that it was used regularly as a standard reference text both in academia and in industry. Some10 years after publication of that book we started work on a second edition. However, we quicklyrealized that the developments in the power systems industry over the intervening years required alarge amount of new material. Consequently the book has been expanded by about a third and theword Control in the new title, Power System Dynamics: Stability and Control, reﬂects the fact thata large part of the new material concerns power system control: ﬂexible AC transmission systems(FACTS), wide area measurement systems (WAMS), frequency control, voltage control, etc. Thenew title also reﬂects a slight shift in focus from solely describing power system dynamics to themeans of dealing with them. For example, we believe that the new WAMS technology is likely torevolutionize power system control. One of the main obstacles to a wider embrace of WAMS bypower system operators is an acknowledged lack of algorithms which could be utilized to controla system in real time. This book tries to ﬁll this gap by developing a number of algorithms forWAMS-based real-time control. The second reason for adding so much new material is the unprecedented change that has beensweeping the power systems industry since the 1990s. In particular the rapid growth of renewablegeneration, driven by global warming concerns, is changing the fundamental characteristics ofthe system. Currently wind power is the dominant renewable energy source and wind generatorsusually use induction, rather than synchronous, machines. As a signiﬁcant penetration of suchgeneration will change the system dynamics, the new material in Chapter 7 is devoted entirely towind generation. The third factor to be taken into account is the fallout from a number of highly publicized black-outs that happened in the early years of the new millennium. Of particular concern were the autumn2003 blackouts in the United States/Canada, Italy, Sweden/Denmark and the United Kingdom,the 2004 blackout in Athens and the European disturbance on 4 November 2006. These blackoutshave exposed a number of critical issues, especially those regarding power system behaviour atdepressed voltages. Consequently, the book has been extended to cover these phenomena togetherwith an illustration of some of the blackouts. It is important to emphasize that the new book is based on the same philosophy as the previousone. We try to answer some of the concerns about the education of power system engineers. Withthe widespread access to powerful computers running evermore sophisticated simulation packages,there is a tendency to treat simulation as a substitute for understanding. This tendency is especiallydangerous for students and young researchers who think that simulation is a panacea for everythingand always provides a true answer. What they do not realize is that, without a physical understandingof the underlying principles, they cannot be conﬁdent in understanding, or validating, the simulationresults. It is by no means bad practice to treat the initial results of any computer software with ahealthy pinch of scepticism.
xvi Preface Power system dynamics are not easy to understand. There are a number of good textbooks whichdeal with this topic and some of these are reviewed in Chapter 1. As the sy