In 1997 the authors of this book, J. Machowski, J.W. Bialek and J.R. Bumby, published a book entitled 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. Some 10 years after publication of that book we started work on a second edition. However, we quickly realized that the developments in the power systems industry over the intervening years required a large amount of new material. Consequently the book has been expanded by about a third and the word Control in the new title, Power System Dynamics: Stability and Control, reflects the fact that a 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. The new title also reflects a slight shift in focus from solely describing power system dynamics to the means of dealing with them. For example, we believe that the new WAMS technology is likely to revolutionize power system control. One of the main obstacles to a wider embrace of WAMS by power system operators is an acknowledged lack of algorithms which could be utilized to control a system in real time. This book tries to fill this gap by developing a number of algorithms for WAMSbased realtime control.
POWER SYSTEM DYNAMICS Stability and Control Second Edition Jan Machowski Warsaw University of Technology, Poland Janusz W Bialek The University of Edinburgh, UK James R Bumby Durham University, UK POWER SYSTEM DYNAMICS POWER SYSTEM DYNAMICS Stability and Control Second Edition Jan Machowski Warsaw University of Technology, Poland Janusz W Bialek The University of Edinburgh, UK James R Bumby Durham University, UK This edition first published 2008 C 2008 John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission 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 the Copyright, 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 as permitted 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 not be available in electronic books Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners The publisher is not associated with any product or vendor mentioned in this book This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought Library of Congress Cataloging-in-Publication Data Machowski, 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) Electric power system stability Electric power systems–Control I Bialek, Janusz W II Bumby, J R (James Richard) III Title TK1010.M33 2008 621.319 1–dc22 2008032220 A catalogue record for this book is available from the British Library ISBN 978-0-470-72558-0 Typeset in 9/11pt Times New Roman by Aptara Inc., New Delhi, India Printed in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire Contents About the Authors Preface xiii xv Acknowledgements xix List of Symbols xxi PART I INTRODUCTION TO POWER SYSTEMS 1.1 1.2 1.3 Introduction Stability and Control of a Dynamic System Classification of Power System Dynamics Two Pairs of Important Quantities: Reactive Power/Voltage and Real Power/Frequency 1.4 Stability of a Power System 1.5 Security of a Power System 1.6 Brief Historical Overview 3 9 12 Power System Components 2.1 Introduction 2.1.1 Reliability of Supply 2.1.2 Supplying Electrical Energy of Good Quality 2.1.3 Economic Generation and Transmission 2.1.4 Environmental Issues 2.2 Structure of the Electrical Power System 2.2.1 Generation 2.2.2 Transmission 2.2.3 Distribution 2.2.4 Demand 2.3 Generating Units 2.3.1 Synchronous Generators 2.3.2 Exciters and Automatic Voltage Regulators 2.3.3 Turbines and their Governing Systems 2.4 Substations 2.5 Transmission and Distribution Network 2.5.1 Overhead Lines and Underground Cables 2.5.2 Transformers 2.5.3 Shunt and Series Elements 2.5.4 FACTS Devices 15 15 15 16 16 16 16 18 18 19 19 19 20 21 25 35 35 35 36 41 43 Contents vi 2.6 Protection 2.6.1 Protection of Transmission Lines 2.6.2 Protection of Transformers 2.6.3 Protection of Busbars 2.6.4 Protection of Generating Units 2.7 Wide Area Measurement Systems 2.7.1 WAMS and WAMPAC Based on GPS Signal 2.7.2 Phasors 2.7.3 Phasor Measurement Unit 2.7.4 Structures of WAMS and WAMPAC The Power System in the Steady State 3.1 Transmission Lines 3.1.1 Line Equations and the π -Equivalent Circuit 3.1.2 Performance of the Transmission Line 3.1.3 Underground Cables 3.2 Transformers 3.2.1 Equivalent Circuit 3.2.2 Off-Nominal Transformation Ratio 3.3 Synchronous Generators 3.3.1 Round-Rotor Machines 3.3.2 Salient-Pole Machines 3.3.3 Synchronous Generator as a Power Source 3.3.4 Reactive Power Capability Curve of a Round-Rotor Generator 3.3.5 Voltage–Reactive Power Capability Characteristic V(Q) 3.3.6 Including the Equivalent Network Impedance 3.4 Power System Loads 3.4.1 Lighting and Heating 3.4.2 Induction Motors 3.4.3 Static Characteristics of the Load 3.4.4 Load Models 3.5 Network Equations 3.6 Power Flows in Transmission Networks 3.6.1 Control of Power Flows 3.6.2 Calculation of Power Flows PART II 54 54 56 57 57 58 58 59 61 62 65 65 66 67 72 72 72 74 76 76 83 89 91 95 100 104 105 106 110 111 113 118 118 122 INTRODUCTION TO POWER SYSTEM DYNAMICS Electromagnetic Phenomena 4.1 Fundamentals 4.2 Three-Phase Short Circuit on a Synchronous Generator 4.2.1 Three-Phase Short Circuit with the Generator on No Load and Winding Resistance Neglected 4.2.2 Including the Effect of Winding Resistance 4.2.3 Armature Flux Paths and the Equivalent Reactances 4.2.4 Generator Electromotive Forces and Equivalent Circuits 4.2.5 Short-Circuit Currents with the Generator Initially on No Load 4.2.6 Short-Circuit Currents in the Loaded Generator 4.2.7 Subtransient Torque 127 127 129 129 133 134 140 146 149 150 Contents vii 4.3 Phase-to-Phase Short Circuit 4.3.1 Short-Circuit Current and Flux with Winding Resistance Neglected 4.3.2 Influence of the Subtransient Saliency 4.3.3 Positive- and Negative-Sequence Reactances 4.3.4 Influence of Winding Resistance 4.3.5 Subtransient Torque 4.4 Synchronization 4.4.1 Currents and Torques 4.5 Short-Circuit in a Network and its Clearing 152 153 156 159 160 162 163 164 166 Electromechanical Dynamics – Small Disturbances 5.1 Swing Equation 5.2 Damping Power 5.2.1 Damping Power at Large Speed Deviations 5.3 Equilibrium Points 5.4 Steady-State Stability of Unregulated System 5.4.1 Pull-Out Power 5.4.2 Transient Power–Angle Characteristics 5.4.3 Rotor Swings and Equal Area Criterion 5.4.4 Effect of Damper Windings 5.4.5 Effect of Rotor Flux Linkage Variation 5.4.6 Analysis of Rotor Swings Around the Equilibrium Point 5.4.7 Mechanical Analogues of the Generator–Infinite Busbar System 5.5 Steady-State Stability of the Regulated System 5.5.1 Steady-State Power–Angle Characteristic of Regulated Generator 5.5.2 Transient Power–Angle Characteristic of the Regulated Generator 5.5.3 Effect of Rotor Flux Linkage Variation 5.5.4 Effect of AVR Action on the Damper Windings 5.5.5 Compensating the Negative 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Angle of attack, 273 Angular momentum, 171 Annual energy demand, 15 Aperiodic mode, 504 Area control error (ACE), 343 Armature Leakage, 85, 136 flux paths,134 short circuit current, 133 time constant, 133 winding, 21 reactance, 81, 85, 135 reaction, 21, 79 Asynchronous operation, 210, 239, 245, 251 torque, 172 Attenuation constant, 66 Automatic generator control (AGC), 336, 346 Auto-reclosing, 54, 212 Automatic voltage regulator (AVR) modelling, 462, 527 load compensation, 23, 462 limiters, 24, 200 influence on fault current, 150 influence on steady state stability, 196 influence on transient stability, 216 influence on voltage stability, 317 modelling, 462, 526 Autotransformer, 37 Auxiliary services, 20 Backward swing, 400 Bang-bang control, 391, 397 Bandwidth of secondary control, 345 Betz limit, 266 Bias factors, 343, 366 Bifurcation point, 319 Blackouts, 323, 340 Blocking relay, 248 Boiler control, 475 boiler-follows-turbine, 475 turbine-follows-boiler, 475 integrated, 475 Braking resistor, 45, 49, 391 Buchholz protection, 57 Busbars, 35 protection, 57 Capacity factor, 269 Centre of power swing, 249 Classification of dynamics, Characteristic equation, 192, 492, 517, 602 Characteristic impedance, 67 Charging current line, 66 cable, 72 Classical model, 180, 456 Coherency, 567 based equivalents, 573 criterion, 571 electromechanical, 566 Power System Dynamics: Stability and Control, Second Edition Jan Machowski, Janusz W Bialek and James R Bumby C 2008 John Wiley & Sons, Ltd Index 624 Coherency (Continued ) exact, 567 recognition, 569 Combined cycle gas turbine (CCGT), 27 single shaft mode, 28 Combustion chamber, 27 Constant flux linkage, 127, 145, 153 Contingencies, Controller field-oriented, 289 vector, 289 Critical disturbance, fault clearing time, 210, 215, 230, 235 operating point, 176, 311 power demand, 310, 319 power state, 311 voltage, 311 Damped frequency, 611 Damper windings, 21,129, 136, 172, 186, 205 Damping coefficient, 171, 175, 193 power, 172, 240 ratio, 193, 256, 262, 611 torque, 205 Danish concept, 270 Decentralized control, 341 DC excitation system, 20, 463 DC offset, 129 Declaration area, 189, 209, 229 Defence plan, 347 Diagonalization, 496 Dimo’s method, 564 Direct method, 22, 230, 418 Disc averaging, 482 Discharge lighting, 105 Distance relay, 56 Distributed generation, 18 Distribution, 17, 19 Distribution transformer, 36 Disturbance critical, large, 207 small, 169 Dominant mode, 506 Droop, 32, 46, 336, 341 d,q axis, 76, 434 d,q to a,b transformation, 117, 436 0,d,q transformation, 436, 438 Economic dispatch, 336 Eigenvalues, 195, 491 Eigenvectors, 491 left and right hand, 492, 497 Electrical centre, 249 Electro-hydraulic governor, 29 Embedded generation, 18, 278 Emergency state, 10 Elimination of nodes, 230, 559 Energy function, 225 Environmental issues, 16 Exact coherency, 567 Excitation systems, 21, 384 load compensation, 23, 462 modelling, 462 Excitation transformer, 22, 40, 490 Extremis state, Equal-area criteria, 184, 209, 228, 388, 400, 419 Equilibrium point, 3, 176, 305, 339, 589 stable point, 178, 194, 306, 359 unstable point, 33, 178, 194, 306 Equivalents, 557 FACTS, 43, 119, 370, 391, 423,488 Fast-valving, 35, 244, 389 momentary, 389 sustained, 389 Fault clearing, 166, 168 critical clearing time, 210, 215 shunts, 208 unbalanced, 211 Feedback control, Fixed speed generator 269 Flux decrement145, 195, 215 Forward swing, 387, 400, 419, 423 Francis turbine, 476 Frequency collapse, 358, 363 Frequency control, 335 primary, 339, 346 secondary, 340, 346 tertiary, 345, 346 Frequency stiffness, 340 Fundamental system of solutions, 601 Gate turn-off thyristor, 44 Gas turbines, 26 Index Gauss-Seidel, 123 General solution, 602 Generating unit, 20 Generation characteristic, 336 Generator emfs, 78, 137, 143 establishing initial values, 143 subtransient, 140, 449 steady-state, 99, 447 transient, 142, 447 Generator equivalent circuits, 454 steady-state, 143 subtransient, 141, 143 transient, 142, 143 Generator reactance, 136, 138 Generator transformer21, 36 Generator tripping, 392 preventive, 393 restitutive, 394 Governor control valves, 26, 389, 471, 473 Governor systems, 32, 480 Heat recovery boiler, 27 Hybrid network equations, 116 Hydraulic turbines, 28 modelling, 476 transfer function, 479 Ideally stiff load, 300 Impulse turbine, 29 Incremental model, 371, 577 Induction generator, 265 doubly fed, 271 fixed speed, 269 variable speed, 271 Induction motor, 103, 282 easy starting, 107 equivalent circuit, 106 heavy-starting, 107 modelling, 485 slip, 106, 486 stalling voltage, 107 torque-slip curve, 106 Inertia coefficient, 172 Inertia constant, 171 Infinite busbar, 146, 163, 167, 173 Integration constants, 604 Intercept valves, 26, 389, 472 Interconnected systems, 340, 344, 364 Interface error, 353 625 Internal power factor angle, 82 Islanded systems, 335, 361 Jacobian, 116 Kaplan turbine, 29 Line trap, 55 Leakage flux, 72, 76, 459 Leakage reactance, 81, 460 Load angle, 68, 118 Load characteristics, 104, 110 effect of tap-changer, 111 frequency, 104, 112, 358 static, 110 voltage, 105, 107, 108, 110 Load modelling, 104, 111, 485 composite load, 104, 485 constant current, 111 constant power, 111 constant impedance, 105, 111, 324 exponential, 112 frequency dependent, 112 polynomial model, 111 stiffness, 321 voltage sensitivity, 105, 318 Load sensitivity frequency, 104, 339 voltage, 104 Load shedding, 348, 360 Load reference set point, 39 Local measurements, 400, 410, 421, 428 Logarithmic spiral, 613 Lyapunov’s first method, 513 Lyapunov’s second (direct) method, 222, 407 Magnetizing reactance, 106, 274 Main stop valves, 34 Matrix factorization, 547, 553 partial inversion, 543, 559, 600 Mechanical-hydraulic governor, 29, 470 Mechanical time constant, 171 Mho relay, 248, 250 Modal analysis, 513, 528, 570, 579 matrix, 501 reduction, 558 variables, 501, 580, 587 Mode shape, 254, 509 Index 626 Modulation controller, 47, 401, 404, 405 Multimachine systems, 220, 230, 307, 318, 352, 371, 491 Natural frequency, 193 damped/undamped, 193, 256, 262 Natural load, 67 Negative damping, 187, 204, 218, 384, 422 Network equations, 113, 442, 450, 546, 553 Network feasibility, 299, 317 Network loadability, 300 Network splitting, 251 Network reduction, 557–558 Newton method, 123, 539, 551 Nodal admittance matrix, 75, 113 Nodal impedance matrix, 114 Nodal network equations, 113 Nodes: PV, PQ, slack, 122 Non-intervention rule, 342 Normal state, 10 Nose curves, 301 Numerical integration, 536 Off-nominal transformation ratio, 72, 74 Oscillatory mode, 504 Out-of-step protection (relaying), 244 Out-of-step tripping, 245 Overexcitation, 103 Overhead line, 68 Park’s transformation, 438 Partial inversion, 543, 559, 600 Partially equidistant point, 590 Participation factor, 511 Particular solution, 606 Partitioned solution method, 541 Participation factor, 511, 588 Pelton Wheel, 28 Penstock, 28 Per unit systems, 439, 593 Pitch control, 237, 482 Phase angle regulator (TCPAR), 52, 370, 423 Phase constant, 66 Phase-shifting transformer , 40, 423, 488 Phasor Diagram, 86, 95, 141, 161, 180, 197, 288, 451 measurement unit (PMU), 61 Physical reduction, 558 Point of common connection, 278 Pole-slip protection, 245, 249 Potier reactance, 460 Power angle, 92, 97, 101, 171 Power-angle characteristic classical, 183 steady-state, 103 transient, 179, 183, 209 Power capability curve, 91 Power conditioning system, 48 Power flow, 118 Power invariance, 595 Power line carrier, 55 Power swing, 185, 191, 215, 219 detection, 248 Power system stability, angle, 9, 207 voltage, 9, 299 frequency, 9, 335 Power system stabilizer, 23, 47, 383, 429, 470 Power system structure, 19 Primary control, 339 Protection differential current, 54 directional comparison, 55 distance, 56 local back-up, 43 pole-slip, 244 remote back-up, 54 Pull-out power, 177 Quadrature booster, 40, 52 q axis, 76 Rayleigh distribution, 268 Reactive power, 70, 91, 109, 305 Regulation power, 344, 362 Reheat stop valves, 26 Reliability of supply, 15 Reluctance power, 91 Reserve of secondary control, 345 Resynchronization, 237, 393 Rotation matrix, 441 Rotational voltage, 87, 486 sampling, 482 Rotor angle, 101, 118, 170 Rotor screening, 135 Rotor swing, 184, 207, 218 Run-up control, 34 Roots distinct, 604 repeated, 605 Index Saturation characteristics, 458 factor, 459 parameter sensitivity, 461 Secondary control, 340 Security, assessment, 10 dynamic, 11 small signal, 10 static, 10 transient, 11 voltage, 10 Self-excitation, 329 synchronous, 332 asynchronous, 333 Sensitivity analysis, 319, 510 Series capacitor, 41 thyristor controlled (TCSC), 50, 416 Series transformer, 40 Servomotor, 26, 473 Short circuit current, 128, 134, 146, 149 in network, 167 power, 108, 396 reactance, 396 Shunt capacitor, 42, 45, 395 Shunt compensation, 41, 425 Shunt reactor, 25 Shut down wind speed, 267 Simulation methods partitioned (alternating), 541 simultaneous (combined), 553 Slack bus (node), 122 Slip frequency, 271 Small signal stability, 491 Sparse matrix, 114, 547, 560 Speed deviation, 170 Speed measuring device, 26 Speeder gear, 29 Speed-droop characteristic, 34, 336, 338 coefficient, 33 Speed reference set-point, 30 Speed ratio, 266 Spinning reserve, 337 Stability enhancement, 383 Stable equilibrium point, 181, 210, 227, 589 Stall control, 273 Standard form, 609 627 State control, 370, 410 matrix, 500 space equation, 515 steady, 65, 135 subtransient, 125, 140 transient, 135, 141 variable, 3, 397, 500, 541 vector, 3, 500 Static compensator (STATCOM), 47, 395, 327, 488 Static VAR compensator (SVC), 45, 395, 488 Steady-state stability, 69, 177, 196, 491 condition, 190, 517 limit, 178 margin, 178 regulated system, 196 unregulated system, 177 Steam turbine, 25 cross-compound, 26 condenser, 26 modelling, 470 non-reheat, 25 overspeed control, 34 regulated characteristic run-up control, 34 reheat, 25 stages, 25 tandem-compound, 25, 472 transfer function, 472 unregulated characteristic, 31 valves, 26, 389, 475 Step-up transformer, 18 Stiffness, 349, 344 Subtransmission network, 17 Subsynchronous resonance, 334 Substation, 21, 35 Subtransient reactance, 136 d axis, 136, 139 q axis, 137, 139 Subtransient saliency, 137, 148, 156 Subtransient time constant, 139 d axis, 139, 446 q axis, 139, 446 Superconducting magnetic energy storage (SMES), 49, 395 Supplementary control, 383 Surge impedance, 66 Surge impedance load (SIL), 67 Survival wind speed, 267 Swing equation, 169, 452, 514 Index 628 Swing frequency, 189 Symmetrical components, 158 Synchronization, 163, 237 condition, 239 currents, 164, 238 torque, 164, 238 Synchronous compensator, 42 Synchronous generator, 19 emfs, 85, 95, 99 equivalent circuits, 87, 100 leakage flux, 76 modelling, 433 on no-load, 77 parameters, 139 power supplied to system, 102, 103 protection, 57 round-rotor, 76 salient pole, 83 saliency, 76 steady state, 135 subtransient, 135 torque, 82, 88 transient state, 135 Synchronous reactance, 82 d axis, 85 q axis, 85 Synchronous speed, 28, 77, 163, 170, 239 Synchronizing power steady state, 178 transient, 184 Tap-changing transformers, 37 Tertiary control, 345 Thyristor, 43 conventional, 43 gate turn-off thyristor (GTO), 44 Tie-line, 17, 120, 342 control, 336, 343 oscillations, 364 Time constant, 128 armature, 133 d-axis, 133 mechanical, 171 q-axis, 138 Torque asynchronous, 172 in round-rotor generator, 82 in salient pole generator, 88 mechanical, 171 subtransient, 150 synchronous, 82, 88 Torque angle, 83 Torsional oscillation, 253 Torsional fatigue, 237, 253, 263 Tower shadow, 482 Trajectory, 3, 223, 567, 591 Transfer admittance, 231, 561 capability, 523 equivalent network, 207, 560 Transformation matrix, 437 Transformer emf, 442 Transformers, 36 booster, 40 π equivalent, 73 autotransformers, 41 core loss, 72 equivalent circuit, 73, 75 excitation current, 73 ideal, 73 leakage flux, 72 magnetization current, 72 modelling, 74 no-load test, 73 off-nominal taps, 74 phase shifting, 75 short-circuit test, 73 short-circuit voltage, 74 tap-changing, 37 three-winding transformers, 36 two-winding transformers, 36 unit transformer, 20 Transmission, 16 angle, 66 Transmission lines, 65 long, 66 lossless, 67 medium-length, 67 short-length, 67 π equivalent, 67 Transient droop, 479 Transient power angle, 178, 215 Transient reactance d axis, 136, 139, 444 q axis, 138, 139, 444 Transient saliency, 179, 455 Transient stability, 9, 207 margin, 210, 215 Transient time constant open-circuit, 139, 445 short-circuit, 139, 445 Index Trumpet characteristic, 349 Turbine governor, 19, 34, 470 modelling, 470 Turbogenerator, 20 Two-reaction theory, 84 Unbalanced faults, 211 Underexcitation, 103 Underdamped second – order system, 609 Underground cables, 35, 72 Unified power controller (UPFC), 52, 119, 423, 488 Uniform damping, 517 Unit transformer, 20 Unstable equilibrium point, 179, 195, 228, 589 Vandermonde’s determinant, 603 Vector control, 289 Voltage collapse, 299, 310, 312 Voltage sensitivities, 104 Voltage-sensitive loads, 315 Voltage source converter, 47 629 Voltage stability, 9, 299 criteria, 305 index, 320 proximity index, 320 Watt governor, 29 Water starting time, 474 Weibull distribution, 268 Wicket gates, 28 Wide area measurement system (WAMS), 58 monitoring, 59 measurement, protection, and control, (WAMPAC), 59, 378, 410 Wind power, 265 shear, 482 speed, 266, 482 turbines, 265, 481 Wronskian, 603 Zhukov’s method, 563 ... 7.9 .2 Variable-Speed Induction Generators 7.10 Influence of Wind Generators on Power System Stability 26 5 26 5 26 9 27 4 27 7 28 0 28 2 28 4 28 6 28 7 28 9 29 0 29 1 29 2 29 3 29 4 29 4 29 4 29 6 29 6 Voltage Stability. .. Operation and Resynchronization 6.5.1 Transition to Asynchronous Operation 6.5 .2 Asynchronous Operation 6.5.3 Possibility of Resynchronization 20 7 20 7 20 7 21 2 21 5 21 5 21 6 22 0 22 2 22 3 22 5 22 7 22 8 23 0 23 7... Quality 2. 1.3 Economic Generation and Transmission 2. 1.4 Environmental Issues 2. 2 Structure of the Electrical Power System 2. 2.1 Generation 2. 2 .2 Transmission 2. 2.3 Distribution 2. 2.4 Demand 2. 3