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Power System Dynamics and Stability Jan Machowski Warsaw University of Technology, Poland Janusz W. Bialek James R. Bumby University of Durham John Wiley & Sons Chichester New York Weinheim Brisbane Singapore .Toronto Copyright 0 1997 by John Wiley & Sons Ltd, Baffins Lane, Chichester, West Sussex PO19 lUD, England National 01243 779777 International (+44) 1243 779777 e-mail (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on http://www.wiley.co.uk or http://www.wiley.com 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, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London, UK WIP 9HE. without the permission in writing of the publisher. Other Wiley Editorial Ojjces John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, USA VCH Verlagsgesellschaft mbH, Pappelallee 3, D-69469 Weinheim, Germany Jacaranda Wiley Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Re Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario M9W ILI, Canada Library of Congress Cataloguing-in-Publication Data Machowski, J. p. cm. Power system dynamics and stability/J. Machowski, J. Bialek, J.R. Bumby. Includes bibliographical references and index. ISBN 0 471 97174 X (PPC) I. Electric power system stability. 0 471 95643 0 (PR) 2. Electric power systems. I. Bialek, J. 11. Bumby. J.R. (James Richard) 111. Title. TK1010.M33 1997 621.319'1 - dc20 96-39033 CIP British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0 471 97174 X (PPC) 0 471 95643 0 (PR) Foreword Professor William Fairney, F. Eng, F.1.E.E Director of Plant Development & Construction, National Power plc Visiting Professor in Engineering at Durham University It gives me great pleasure to write this foreword as I have been associated with Durham University for over ten years and become well acquainted with the authors. This work is a state-of-the-art exposition on the dynamics and stability of power systems, the complexity of which is not exceeded by any other dynamical system. Massive computer power can now be brought to bear on solving power system equations but this was not available when these problems first started to be addressed in the 1920’s and 30’s. It is a tribute to the pioneers of those days, who first formulated the complex equations, that they were able to solve them with paper, pen and mechanical calculators, and gain an understanding of the fundamental issues relating to power system dynamics. To enable calculations to be made at all, the models had to be simple, and conservative stability margins had to be used. When I joined the electricity supply industry in 1965 and first became interested in power system dynamics, the stability of the CEGB network could only be solved on the huge mainframe computer based in Park Street, London. My interest was in the use of the newly invented thyristor for generator excitation, and the ability to explore a wide range of control settings was desirable. The frustration and high cost of sequential programming with punched card input data led to the search for alternative solutions. Necessity being the mother of invention, I was able to profit from the on-line satellite links to large mainframes in the USA which were just being introduced. Since that time processing power has mushroomed and computer size shrunk, to the point where today, a power system engineer can sit at his desk and optimise the performance of large power networks in real time. Much of this development work in the UK has been carried out at Durham University, stimulated by research contracts with power system utilities. The privatisation of the electricity supply industry has resulted in a large number of generators and distribution utilities where power requirements are co-ordinated commercially through the electricity pool. The National Grid Company has the task of co-ordinating the technical performance xii Fore word of the network and of ensuring that technical needs are reflected in the commercial arrangements for the transmission of electricity. This book will become a major reference work in future years for all involved in power system dynamics. Its comprehensive coverage of all aspects of the subject, plus its progressive approach from simplicity to complexity makes excellent reading on this fascinating subject. List of symbols a bar on top of a symbol denotes a phasor or a complex number (e.g. 7, 3); an arrow on top of a symbol denotes a spatial vector (e.g. F); lower case symbols normally denote instantaneous values (e.g. v, i); upper case symbols normally denote rms or peak values (e.g. V, I); bold face denotes a matrix or a vector (e.g. Y); subscripts, A, B, C refer to three-phase axes of a generator; subscripts “d” and “q” refer to the direct- and quadrature-axis components. Symbols: The most important symbols are defined below: B, - magnetising susceptance of a transformer; Bsh - susceptance of a shunt element; D - damping coefficient; Ek - kinetic energy of the rotor relative to the synchronous speed; E, - potential energy of the rotor with respect to the equilibrium point ef - field voltage vf referred to the fictitious q-axis armature coil; e4 - steady-state emf induced in the fictitious q-axis armature coil proportional to the field winding e& - transient emf induced in the fictitious d-axis armature coil proportional to the flux linkages ed - transient emf induced in the fictitious q-axis armature coil proportional to the field winding ej - subtransient emf induced in the fictitious d-axis armature coil proportional to the total q-axis ef - subtransient emf induced in the fictitious q-axis armature coil proportional to the total d-axis E - steady-state internal emf; Ef - excitation emf proportional to the excitation voltage V,; Elm - amplitude of the excitation emf; Ed - d-axis component of the steady-state internal emf proportional to the rotor self-linkages due self-flux linkages; of the q-axis coil representing the solid steel rotor body (round-rotor generators only); flux linkages; rotor flux linkages (q-axis damper winding and q-axis solid steel rotor body); rotor flux linkages (d-axis damper winding and field winding); - - to currents induced in the q-axis solid steel rotor body (round-rotor generators only); xvlll Llst of symbols E, - q-axis component of the steady-state internal emf proportional to the field winding self-flux linkages (i.e. proportional to the field current itself): E - transient internal emf proportional to the flux linkages of the field winding and solid steel rotor body (includes armature reaction); EL - d-axis component of the transient internal emf proportional to flux linkages in the q-axis solid steel rotor body (round-rotor generators only); Eb - q-axis component of the transient internal emf proportional to the field winding flux linkages; E - subtransient internal emf proportional to the total rotor flux linkages (includes armature reaction); E: - d-axis component of the subtransient internal emf proportional to the total flux linkages in the q-axis damper winding and q-axis solid steel rotor body; E: - q-axis component of the subtransient internal emf proportional to the total flux linkages in the d-axis damper winding and the field winding; E, - resultant air-gap emf; E, - amplitude of the resultant air-gap emf; EG - vector of the generator emf's f - mains frequency; f. - rated frequency; Fj - magnetomotive force (mmf) due to the field winding; F, - armature reaction mmf; F,,, - ac armature reaction mmf (rotating); cudc Fad. F,, - d- and q-axis components of the armature reaction mmf; F, - resultant mmf; GF~ - core loss conductance of a transformer; G,j1 - conductance of a shunt element; H - inertia constant; iA, is, ic - instantaneous currents in phases A, B, and C; iAdc, i8dc. icdc - dc component of the current in phases A, B. and C; i~,~, is,,, ic., - ac component of the current in phases A, B, and C; id, i, -currents flowing in the fictitious d- and q-axis armature coils; iD, iQ - instantaneous d- and q-axis damper winding current; i/ - instantaneous field current of a generator; iAsC - vector of instanteneous phase currents; i/DQ - vector of instanteneous currents in the field winding and the d- and q-axis damper windings; iudy - vector of armature currents in the rotor reference frame; I - armature current; Id, I, - d- and q-axis component of the armature current; Ts, fR - currents at the sending and receiving end of a transmission line; iR, iE - vector of complex current injections to the retained and eliminated nodes; iG, I, - vector of generator and load currents; AIL - vector of load corrective currents; J - moment of inertia; kp", k~v - voltage sensitivities of the load (the slopes of the real and reactive power demand kp/, kQ/ - frequency sensitivities of the load (the slopes of the real and reactive power demand -1 I, - - dc armature reaction mmf (stationary); - - characteristics as a function of voltage); characteristics as a function of frequency); Llst of symbols xix KE, - steady-state synchronising power coefficient (the slope of the steady-state power angle curve KE; - transient synchronising power coefficient (the slope of the transient power angle curve KE, - transient synchronising power coefficient (the slope of the transient power angle curve Ki - reciprocal of droop for the i-th generating unit; KL - frequency sensitivity coefficient of the system real power demand; KT - reciprocal of droop for the total system generation characteristic; 1 - length of a transmission line; LA,,, LBB, Lcc, Lf f, LDD, LQQ - self-inductances of the windings of the phase windings A, B, C, Ld, Lq - inductances of the fictitious d- and q-axis armature windings; LA, L$, L&‘, Ly - d- and q-axis transient and subtransient inductances; Lxy, where x, y E (A, B, C, D, Q, f] and x # y, are the mutual inductances between the windings Ls - minimum value of the self-inductance of a phase winding; ALs - amplitude of the variable part of the self-inductance of a phase winding; LR - submatrix of the rotor self and mutual inductances; Ls - submatrix of the stator self and mutual inductances; LSR. LRS - submatrices of the stator-to-rotor and rotor-to-stator mutual inductances; M - inertia coefficient; M,, MD, MQ - amplitude of the mutual inductance between a phase winding and, respectively, N - generally, number of turns of a winding; p - number of poles; P,,, - accelerating power; PD - damping power; P, - electromagnetic air-gap power; PB~~~ - critical (pull-out) air-gap power developed by a generator; PE~(S), PE#(S’), PE;(S’) - air-gap power curves assuming E, = constant, E’ = constant and E$ = PL - real power absorbed by a load (Chapter 7) or total system load (Chapter 8); P, - mechanical power supplied by a prime mover to a generator; P, - real power demand at rated voltage; PR - real power at the receiving end of a transmission line; Pr/, Pr//v Pr///, P,/V - contribution of the generating units remaining in operation towards covering the real power imbalance during the first, second, third and fourth stage of load frequency control; P,/, PS//, Ps//l, PSlv - contribution of the system towards covering the real power imbalance during the first, second, third and fourth stage of load frequency control; Ps - real power at the sending end of a transmission line (Chapter 3) or real power supplied by a source to a load (Chapter 7); PS~L - surge impedance (natural) load; P, - real power supplied to the infinite busbar; PS~,(S) - curve of real power supplied to the infinite busbar assuming E, = constant; PT - total power generated in a system; P,ie - net tie-line interchange power; PE, (8)); PEp‘)); pEl(8’)); the field winding, and the d- and the q-axis damper winding; denoted by the indices as described above; the field winding, and the d- and the q-axis damper winding; constant; xx List of symbols Pv, (6) - air-gap power curve assuming V, = constant; Pv,,, - critical value of PV, (6); QC - reactive power generated by a source (the sum of QL and the reactive power loss in the QL - reactive power absorbed by a load; Qn - reactive power demand at rated voltage; QR - reactive power at the receiving end of a transmission line; Qs - reactive power at the sending end of a transmission line (Chapter 3) or reactive power R - resistance of the armature winding of a generator; r - total resistance between (and including) the generator and the infinite busbar; RA, RB, Rc. RD, RQ, Rf - resistances of the phase windings A, B, C, the d- and q-axis damper RABC - diagonal matrix of phase winding resistances; R,DQ - diagonal matrix of resistances of the field winding and the d- and q-axis damper windings; s - Laplace operator; s - slip of induction motor; s,, - critical slip of induction motor; S, - rated apparent power of a generator; S~HC - short-circuit power; r - time; T:, T: - short-circuit d-axis transient and subtransient time constants; Tho, T:" - open-circuit d-axis transient and subtransient time constants; Ti, T: - short-circuit q-axis transient and subtransient time constants; Tio, Tto - open-circuit q-axis transient and subtransient time constants; T, - armature winding time constant; T - transformation matrix between network (a,b) and generator (d,q) co-ordinates; VA, VB, vc, vf - instantaneous voltages across phases, A, B. C and the field winding; vd, vq - voltages across the fictitious d- and q-axis armature coils; V~BC - vector of instantaneous voltages across phases, A, B, and C; V~DQ - vector of instantaneous voltages across the field winding and the d- and q-axis damper V - Lyapunov function; V,, - critical value of the voltage; vd, 8, - direct and quadrature-axis component of the generator terminal voltage; V, - voltage aplied to the field winding; V, - voltage at the generator terminals; V, - infinite busbar voltage; V,d, 8,, - direct- and quadrature-axis component of the infinite busbar voltage; Vs, VR - voltage at the sending and receiving end of a transmission line; Vsh - local voltage at the point of installation of a shunt element; Vi = ViL& - complex voltage at node i; VR, vE - vector of complex voltages at the retained and eliminated nodes; W - work; W - Park's modified transformation matrix; xd, xh, x&' - total d-axis synchronous, transient and subtransient reactance between (and including) the generator and the infinite busbar; network); supplied by a source to a load (Chapter 7); winding, and the field winding; windings; - - - - - - - List of symbols xxi XdpRE, xLF, x~posT - pre-fault, fault and post-fault value of xL; x,, xi, xC - total q-axis synchronous, transient and subtransient reactance between (and including) X, - armature reaction reactance (round-rotor generator); Xc - reactance of a series compensator; X1 - armature leakage reactance of a generator; XD - reactance corresponding to the flux path around the damper winding; Xf - reactance corresponding to the flux path around the field winding; Xd, X& X: - d-axis synchronous, transient and subtransient reactance; X,, Xi, X: - q-axis synchronous, transient and subtransient reactance; XsHC - short-circuit reactance of a system as seen from a node; YT - admittance of a transformer; Y - admittance matrix; YGG, YG~, Y'G - admittance submatrices where subscript G corresponds to fictitious generator nodes and subscript L corresponds to all the other nodes (including generator terminal nodes); Yi, =-Gij + jBij element of the admittance matrix; YRR. YEE, RE, YER - complex admittance submatrices where subscript E refers to eliminated and the generator and the infinite busbar; - - - subscript R to retained nodes; Z = d-; Z, - characteristic impedance of a transmission line; ZT = RT + jXT - series impedance of the transformer; Au - rotor speed deviation equal to (w - w3); y - instantaneous position of the generator d-axis relative to phase A; yo - position of the generator d-axis at the instant of fault; @, - armature reaction flux; @ad, Qos - d- and q-axis component of the armature reaction flux; = R, + jX, - internal impedance of the infinite busbar; - - ac armature reaction flux (rotating); - dc armature reaction flux (stationary); Of - excitation (field) flux; QA, QB, QC - total flux linkage of phases A, B, and C; QM, QSB, QCC - self-flux linkage of phases A, B, and C; Qaacr - rotor flux linkages produced by Qooc; Qadcr - rotor flux linkages produced by @,,dc; U,, - rotor flux linkages produced by the total armature reaction flux; WD, QQ - total flux linkage of damper windings in axes d and q; Qd, Qq - total d-and q-axis flux linkages; Uf - total flux linkage of the field winding; *fa - amplitude of the excitation flux linkage with armature winding; Q~A, Q~B, Q~c - excitation flux linkage with phases A, B, and C; Q~ec - vector of phase flux linkages; Q~DQ - vector of flux linkages of the field winding and the d- and q-axis damper windings; Qodq - vector of armature flux linkages in the rotor reference frame; re - electromagnetic torque; r, - fundamental-frequency subtransient electromagnetic troque; r2, - double-frequency subtransient electromagnetic torque; rd, r, - direct- and quadrature-axis component of the electromagnetic torque; xxii List of symbols rR, r, - subtransient electromagnetic torque due to stator and rotor resistances; E - rotor acceleration; 'pw - power factor angle at the generator terminals; S - power (or rotor) angle with respect to the infinite busbar; S, - power (or rotor) angle with respect to the voltage at the generator terminals; ?J.~ - stable equilibrium value of the rotor angle; 8' - transient power (or rotor) angle between E' and V,; Sf, - angle between the resultant and field mmf's; AR - frequency bias factor; w - angular velocity of the generator (in electrical radians); w, - synchronous angular velocity in electrical radians (equal to 2xf); wd - angular velocity of rotor swings in electrical radians; $2 - rotation matrix; p - static droop of the turbine-governor characteristic; p~ - droop of the total system generation characteristic; 19 - transformation ratio; y - propagation constant of a transmission line; B - phase constant of a transmission line (Section 3.1); 3 - reluctance; %d, RQ - reluctance along the direct- and quadrature-axis; Abbreviations: ac - alternating current; ACE - area control error; AGC - Automatic Generation Control; AVR - Automatic Voltage Regulator; d - direct axis of a generator; dc - direct current; emf - electro-motive force; FACTS - Flexible AC Transmission Systems HV - high voltage LFC - load frequency control mmf - magneto-motive force; PSS - power system stabiliser q - quadrature axis of a generator; rms - root-mean-square; rpm - revolutions per minute; rhs - right-hand-side; SMES - superconducting magnetic energy storage STATCOM - static compensator; SVC - Static VAR Compensators; UPFC - unified power flow controller. [...]... Demand The demand for electrical power is never constant and changes continuously throughout the day and night The changes in demand of individual consumers may be fast and frequent but as one moves up the power system structure (Figure 2.1) from individual consumers, through the distribution network, to the transmission level, the changes in demand become smaller and smoother as individual demands... unique features power system dynamics can be conveniently divided into groups characterised either by their cause, consequence, time frame, physical character or the place in the system that they occur Of prime concern is the way the power system will respond to both a changing power demand and to various types of disturbance; the two main causes of power system dynamics A changing power demand introduces... time as the first interconnected power systems were constructed As power systems developed the interest in their behaviour grew until power system dynamics became a scientific discipline in its own right Perhaps the greatest contribution in developing the theoretical foundations of power system dynamics has been from research workers in those countries whose power systems cover large geographic areas,... in mathematical modelling of power systems with the main monographs on this topic being written by Anderson & Fouad (1977), Arillaga & Arnold (1983 and 1990) and Kundur (1994) Another category of books use Lyapunov’s direct method to analyse the electromechanical stability of power systems The main texts here are those written by Pai (1981 and 1989), Fouad & Vittal (1992) and Pavella & Murthy (1994)... classification of power system dynamics 1.2 Brief historical overview 2 Power System Components Structure of the electrical power system Generating units 2.2.1 Synchronous generators 2.2.2 Exciters and automatic voltage regulators 2.2.3 Turbines and their governing systems 2.3 Substations 2.4 Transmission and distribution network 2.4.1 Overhead lines and underground cables 2.4.2 Transformers 2.4.3 Shunt and series... a power system function and the effect they have on both power system operation and control Structure of the electrical power system 2.1 7 Structure of the Electrical Power System The basic structure of a contemporary electrical power system is illustrated schematically in Figure 2.1 Conventionally the power system is divided into three parts, one concerned with generation, one with transmission and. .. of large-scale power systems Examining the chapters in more detail, Chapter 1 classifies power system dynamics and provides a brief historical overview Chapter 2 contains a brief description of the major power system components, including modem FACTS devices Chapter 3 introduces steady-state models and their use in analysing the performance of the power system Chapter 4 analyses the dynamics following... lightning and switching overvoltages, occur almost exclusively in the network and basically do not propagate beyond the transformer windings The electromagnetic phenomena involve mainly the generator armature and damper windings and partly the network The electromechanical phenomena, that is the rotor oscillations and accompanying network power swings, involve mainly the rotor field and damper windings and. .. when power systems were run by vertically integrated utilities and operated with large stability margins are all but over In the present climate of deregulation and privatisation, the utilities are often separated into generation, transmission and distribution so as to help promote economic efficiency and encourage competition Coupled with the difficulty of obtaining new rights of way for expanding... comprehensive text by Kundur contains an excellent overview of modelling and analysis of power systems and constitutes the basic monograph on power system dynamics The fast electromagnetic phenomena, like wave and switching transients, are described by Greenwood (1971) From the 1940s until the 1960s power system dynamics were generally studied using physical (analogue) models of the system However rapid . Machowski, J. p. cm. Power system dynamics and stability/ J. Machowski, J. Bialek, J.R. Bumby. Includes bibliographical references and index. ISBN 0 471 97174 X (PPC) I. Electric power. is the way the power system will respond to both a changing power demand and to various types of disturbance; the two main causes of power system dynamics. A changing power demand introduces. exposition on the dynamics and stability of power systems, the complexity of which is not exceeded by any other dynamical system. Massive computer power can now be brought to bear on solving power system