Power System Analysis Short-Circuit Load Flow and Harmonics J C Das Amec, Inc Atlanta, Georgia Marcel Dekker, Inc New York • Basel TM Copyright © 2001 by Marcel Dekker, Inc All Rights Reserved ISBN: 0-8247-0737-0 This book is printed on acid-free paper Headquarters Marcel Dekker, Inc 270 Madison Avenue, New York, NY 10016 tel: 212-696-9000; fax: 212-685-4540 Eastern Hemisphere Distribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001 Basel, Switzerland tel: 41-61-261-8482; fax: 41-61-261-8896 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities For more information, write to Special Sales/Professional Marketing at the headquarters address above Copyright # 2002 by Marcel Dekker, Inc All Rights Reserved Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher Current printing (last digit): 10 PRINTED IN THE UNITED STATES OF AMERICA POWER ENGINEERING Series Editor H Lee Willis ABB Inc Raleigh, North Carolina Advisory Editor Muhammad H Rashid University of West Florida Pensacola, Florida Power Distribution Planning Reference Book, H Lee Willis Transmission Network Protection: Theory and Practice, Y G Paithankar Electrical Insulation in Power Systems, N H Malik, A A Al-Arainy, and M I Qureshi Electrical Power Equipment Maintenance and Testing, Paul Gill Protective Relaying: Principles and Applications, Second Edition, J Lewis Blackburn Understanding Electric Utilities and De-Regulation, Lorrin Philipson and H Lee Willis Electrical Power Cable Engineering, William A Thue Electric Systems, Dynamics, and Stability with Artificial Intelligence Applications, James A Momoh and Mohamed E El-Hawary Insulation Coordination for Power Systems, Andrew R Hileman 10 Distributed Power Generation: Planning and Evaluation, H Lee Willis and Walter G Scott 11 Electric Power System Applications of Optimization, James A Momoh 12 Aging Power Delivery Infrastructures, H Lee Willis, Gregory V Welch, and Randall R Schrieber 13 Restructured Electrical Power Systems: Operation, Trading, and Volatility, Mohammad Shahidehpour and Muwaffaq Alomoush 14 Electric Power Distribution Reliability, Richard E Brown 15 Computer-Aided Power System Analysis, Ramasamy Natarajan 16 Power System Analysis: Short-Circuit Load Flow and Harmonics, J C Das 17 Power Transformers: Principles and Applications, John J Winders, Jr 18 Spatial Electric Load Forecasting: Second Edition, Revised and Expanded, H Lee Willis 19 Dielectrics in Electric Fields, Gorur G Raju ADDITIONAL VOLUMES IN PREPARATION Protection Devices and Systems for High-Voltage Applications, Vladimir Gurevich Series Introduction Power engineering is the oldest and most traditional of the various areas within electrical engineering, yet no other facet of modern technology is currently undergoing a more dramatic revolution in both technology and industry structure But none of these changes alter the basic complexity of electric power system behavior, or reduce the challenge that power system engineers have always faced in designing an economical system that operates as intended and shuts down in a safe and noncatastrophic mode when something fails unexpectedly In fact, many of the ongoing changes in the power industry—deregulation, reduced budgets and staffing levels, and increasing public and regulatory demand for reliability among them—make these challenges all the more difficult to overcome Therefore, I am particularly delighted to see this latest addition to the Power Engineering series J C Das’s Power System Analysis: Short-Circuit Load Flow and Harmonics provides comprehensive coverage of both theory and practice in the fundamental areas of power system analysis, including power flow, short-circuit computations, harmonics, machine modeling, equipment ratings, reactive power control, and optimization It also includes an excellent review of the standard matrix mathematics and computation methods of power system analysis, in a readily-usable format Of particular note, this book discusses both ANSI/IEEE and IEC methods, guidelines, and procedures for applications and ratings Over the past few years, my work as Vice President of Technology and Strategy for ABB’s global consulting organization has given me an appreciation that the IEC and ANSI standards are not so much in conflict as they are slightly different but equally valid approaches to power engineering There is much to be learned from each, and from the study of the differences between them As the editor of the Power Engineering series, I am proud to include Power System Analysis among this important group of books Like all the volumes in the iii iv Series Introduction Power Engineering series, this book provides modern power technology in a context of proven, practical application It is useful as a reference book as well as for selfstudy and advanced classroom use The series includes books covering the entire field of power engineering, in all its specialties and subgenres, all aimed at providing practicing power engineers with the knowledge and techniques they need to meet the electric industry’s challenges in the 21st century H Lee Willis Preface Power system analysis is fundamental in the planning, design, and operating stages, and its importance cannot be overstated This book covers the commonly required short-circuit, load flow, and harmonic analyses Practical and theoretical aspects have been harmoniously combined Although there is the inevitable computer simulation, a feel for the procedures and methodology is also provided, through examples and problems Power System Analysis: Short-Circuit Load Flow and Harmonics should be a valuable addition to the power system literature for practicing engineers, those in continuing education, and college students Short-circuit analyses are included in chapters on rating structures of breakers, current interruption in ac circuits, calculations according to the IEC and ANSI/ IEEE methods, and calculations of short-circuit currents in dc systems The load flow analyses cover reactive power flow and control, optimization techniques, and introduction to FACT controllers, three-phase load flow, and optimal power flow The effect of harmonics on power systems is a dynamic and evolving field (harmonic effects can be experienced at a distance from their source) The book derives and compiles ample data of practical interest, with the emphasis on harmonic power flow and harmonic filter design Generation, effects, limits, and mitigation of harmonics are discussed, including active and passive filters and new harmonic mitigating topologies The models of major electrical equipment—i.e., transformers, generators, motors, transmission lines, and power cables—are described in detail Matrix techniques and symmetrical component transformation form the basis of the analyses There are many examples and problems The references and bibliographies point to further reading and analyses Most of the analyses are in the steady state, but references to transient behavior are included where appropriate v vi Preface A basic knowledge of per unit system, electrical circuits and machinery, and matrices required, although an overview of matrix techniques is provided in Appendix A The style of writing is appropriate for the upper-undergraduate level, and some sections are at graduate-course level Power Systems Analysis is a result of my long experience as a practicing power system engineer in a variety of industries, power plants, and nuclear facilities Its unique feature is applications of power system analyses to real-world problems I thank ANSI/IEEE for permission to quote from the relevant ANSI/IEEE standards The IEEE disclaims any responsibility or liability resulting from the placement and use in the described manner I am also grateful to the International Electrotechnical Commission (IEC) for permission to use material from the international standards IEC 60660-1 (1997) and IEC 60909 (1988) All extracts are copyright IEC Geneva, Switzerland All rights reserved Further information on the IEC, its international standards, and its role is available at www.iec.ch IEC takes no responsibility for and will not assume liability from the reader’s misinterpretation of the referenced material due to its placement and context in this publication The material is reproduced or rewritten with their permission Finally, I thank the staff of Marcel Dekker, Inc., and special thanks to Ann Pulido for her help in the production of this book J C Das Contents Series Introduction Preface Short-Circuit Currents and Symmetrical Components 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Nature of Short-Circuit Currents Symmetrical Components Eigenvalues and Eigenvectors Symmetrical Component Transformation Clarke Component Transformation Characteristics of Symmetrical Components Sequence Impedance of Network Components Computer Models of Sequence Networks iii v 15 16 20 35 Unsymmetrical Fault Calculations 39 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 40 42 43 45 46 53 61 64 Line-to-Ground Fault Line-to-Line Fault Double Line-to-Ground Fault Three-Phase Fault Phase Shift in Three-Phase Transformers Unsymmetrical Fault Calculations System Grounding and Sequence Components Open Conductor Faults vii viii Contents Matrix Methods for Network Solutions 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 73 73 78 81 82 86 89 103 113 Current Interruption in AC Networks 116 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 Rheostatic Breaker Current-Zero Breaker Transient Recovery Voltage The Terminal Fault The Short-Line Fault Interruption of Low Inductive Currents Interruption of Capacitive Currents Prestrikes in Breakers Overvoltages on Energizing High-Voltage Lines Out-of-Phase Closing Resistance Switching Failure Modes of Circuit Breakers 117 118 120 125 127 127 130 133 134 136 137 139 Application and Ratings of Circuit Breakers and Fuses According to ANSI Standards 145 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 Network Models Bus Admittance Matrix Bus Impedance Matrix Loop Admittance and Impedance Matrices Graph Theory Bus Admittance and Impedance Matrices by Graph Approach Algorithms for Construction of Bus Impedance Matrix Short-Circuit Calculations with Bus Impedance Matrix Solution of Large Network Equations 72 Total and Symmetrical Current Rating Basis Asymmetrical Ratings Voltage Range Factor K Capabilities for Ground Faults Closing–Latching–Carrying Interrupting Capabilities Short-Time Current Carrying Capability Service Capability Duty Requirements and Reclosing Capability Capacitance Current Switching Line Closing Switching Surge Factor Out-of-Phase Switching Current Rating Transient Recovery Voltage Low-Voltage Circuit Breakers Fuses 145 147 148 148 149 153 153 155 160 162 163 168 173 Short-Circuit of Synchronous and Induction Machines 179 6.1 6.2 180 182 Reactances of a Synchronous Machine Saturation of Reactances Harmonic Mitigation and Filters 697 Figure 20-17 Delta–wye or zig-zag transformer used as neutral trap in a three-phase fourwire system serving nonlinear loads Some of the limitations of the passive filters are apparent from the examples [7] These can be summarized as follows: Passive filters are not adaptable to the changing system conditions and once installed are rigidly in place Neither the tuned frequency nor the size of the filter can be changed so easily The passive elements in the filters are close tolerance components A change in the system or operating condition can result in detuning and increased distortion This can go undetected, unless there is on-line monitoring equipment in place The design is largely affected by the system impedance To be effective, the filter impedance must be less than the system impedance, and the design can become a problem for stiff systems In such cases a very large filter will be required This may give rise to overcompensation of reactive power, and overvoltages on switching and undervoltages when out of service Often, passive filters will require a number of parallel shunt branches Outage of a parallel unit totally alters the resonant frequencies and harmonic current flows This may increase distortion levels beyond permissible limits Power losses in the resistance elements of passive filters can be very substantial for large filters The parallel resonance between filter and the system (for single- or doubletuned filters) may cause amplification of currents of a characteristic or noncharacteristic harmonic A designer has a limited choice in selecting 698 Chapter 20 20.14 the tuned frequency to avoid all possible resonances with the background harmonics System changes will alter this frequency to some extent, however carefully the initial design might have been selected Damped filters not give rise to a system parallel resonant frequency; however, these are not so effective as a group of ST filters The impedance of a high-pass filter at its notch frequency is higher than the corresponding ST filter The size of the filter becomes large to handle the fundamental and the harmonic frequencies The aging, deterioration, and temperature effects detune the filter in a random manner (though the effect of maximum variations can be considered in the design stage) If the converters feed back dc current into the system (with even harmonics), it can cause saturation of the filter reactor with resulting increase in distortion Definite-purpose breakers are required To control switching surges, special synchronous closing devices or resistor closing is required (see Chap 4) The grounded neutrals of wye-connected banks provide a low-impedance path for the third harmonics Third-harmonic amplification can occur in some cases Special protective and monitoring devices (not discussed) are required ACTIVE FILTERS By injecting harmonic distortion into the system, which is equal to the distortion caused by the nonlinear load, but of opposite polarity, the waveform can be corrected to a sinusoid The voltage distortion is caused by the harmonic currents flowing in the system impedance If a nonlinear current with opposite polarity is fed into the system, the voltage will revert to a sinusoid Active filters can be classified according to the way these are connected in the circuit [8,9]: in series connection in parallel shunt connection hybrid connections of active and passive filters 20.14.1 Shunt Connection As we have seen, the voltage in a weak system is very much dependent on current, while a stiff system of zero impedance will have no voltage distortion Thus, provided that the system is not too stiff, a nonsinusoidal voltage can be corrected by injecting proper current A harmonic current source is represented as a Norton equivalent circuit, and it may be implemented with a voltage-fed PWM inverter to inject a harmonic current of the same magnitude as that of the load into the system, but of harmonics of opposite polarity A shunt connection is shown in Fig 20-18(a) The load current will be sinusoidal, so long as the load impedance is higher than the source impedance: Ifl ¼ Ih for jZL j > jZs j; Ih % ð20:52Þ Harmonic Mitigation and Filters Figure 20-18 699 (a) Shunt connection of an active filter, (b) series connection of an active filter In Chap 17 we studied two basic types of converters-current and voltage A converter with dc output reactor and constant dc current is a current harmonic source A converter with a diode front end and dc capacitor has a highly distorted current depending on the ac source impedance, but the voltage at rectifier input is less dependent on ac impedance This is a voltage harmonic source It presents a low impedance and shunt connection will not be effective A shunt connection is more suitable for current source controllers where the output reactor resists the change of current 20.14.2 Series Connection Figure 20-18(b) shows a series connection A voltage Vf is injected in series with the line and it compensates the voltage distortion produced by a nonlinear load A series active filter is more suitable for harmonic compensation of diode rectifiers where the dc voltage for the inverter is derived from a capacitor, which opposes the change of the voltage Thus, the compensation characteristics of the active filters are influenced by the system impedance and load This is very much akin to passive filters; however, active filters have better harmonic compensation characteristics against the impedance 700 Chapter 20 variation and frequency variation of harmonic currents The control systems of the active filters have a profound effect on the performance and a converter can have even a negative reactance The active filters by themselves have the limitations that initial costs are high and not constitute a cost-effective solution for nonlinear loads above approximately 500 kW, though further developments will lower the costs and extend applicability 20.14.3 Hybrid Connection Hybrid connections of active and passive filters are shown in Fig 20-19 Figure 20-19(a) is a combination of shunt active and shunt passive filters Figure 20-19(b) shows a combination of a series active filter and a shunt passive filter while Fig 2019(c) shows an active filter in series with a shunt passive filter The combination of shunt active and passive filters has already been applied to harmonic compensation of large steel mill drives [10] Addition of a large shunt capacitor will reduce the load resistance and Eq (20.52) is no longer valid The shunt passive filter will draw a large source current from a stiff system and may act as a sink to the upstream harmonics It is required that in a hybrid combination the filters share compensation properly in the frequency domain In a series connection, the active filter is connected in series with the passive filter, both being in parallel with the load, as shown in Fig 20-19(c) With suitable control of the active filter, it is possible to avoid resonance and improve filter performance The active filter can be either voltage or current controlled In currentmode control the inverter is a voltage source to compensate for current harmonics In voltage-mode control the converter is a voltage-source inverter controlled to compensate for the voltage harmonics The advantage is that the converter itself is far smaller, only about 5% of the load power The active filter in such schemes regulates the effective source impedance as experienced by the passive filter, and the currents are forced to flow in the passive filter rather than in the system This makes the passive filter characteristics independent of the actual source impedance and a consistent performance can be obtained 20.14.4 Combination of Active Filters A combination of series and shunt active filters is shown in Fig 20-20 This looks similar to the unified power controller discussed in Chap 13, but its operation is different [10] A series filter blocks harmonic currents flowing in and out of the distribution feeders It detects the supply current and is controlled to present a zero impedance to the fundamental frequency and high resistance to the harmonics The shunt filter absorbs the harmonics from the supply feeders and detects the bus voltage at the point of connection It is controlled to present infinite impedance to the fundamental frequency and low impedance to the harmonics The harmonic currents and voltages are extracted from the supply system in the time domain The electronics and power devices used in both types of converters for filters are quite similar, Fig 20-21, which shows three-phase voltage-source and current-source PWM converters The current-source active filter has a dc reactor with a constant dc current while the voltage-source active filter has a capacitor on the dc side with constant dc voltage An output filter is provided to attenuate the inverter switching Harmonic Mitigation and Filters Figure 20-19 701 (a), (b), and (c): Hybrid connections of active and passive filters effects In a current-source type, LC filters are necessary, Fig 20-21(b) Transient oscillations can appear because of resonance between filter capacitors and inductors The controls are implemented so that the inverter outputs a harmonic current equivalent but opposite to that of the load The source side current is therefore sinusoidal, but the voltage will be sinusoidal only if the source does not generate any harmonics Bipolar junction transistors are used with switching frequencies up to 50 kHz for modest ratings SCRs and GTOs are used for higher power outputs A further classification is based on the control system, i.e., time domain and frequency domain corrections 20.15 CORRECTIONS IN TIME DOMAIN Corrections in the time domain are based on holding instantaneous voltage or current within reasonable tolerance of a sine wave The error signal can be the 702 Chapter 20 Figure 20-20 Connections of a unified power quality conditioner difference between actual and reference waveforms Time-domain techniques can be classified into three main categories [9]: Triangular wave Hysteresis Deadbeat The error function can be instantaneous reactive power (IRP, described in Sec 20.17) or EXT (extraction of fundamental frequency component) For EXT the fundamental component of the distorted waveform is extracted through a 60-Hz filter and then the error function is eðtÞ ¼ f 60ðtÞ À f ðtÞ For IRP the error function is given by the difference between the instantaneous orthogonal transformation of actual and 60-Hz components of voltages and currents The triangular-wave method is easiest to implement, and can be used to generate two-state or three-state switching functions A two-state function can be connected positively or negatively, while a three-state function can be positive, negative, or zero (Fig 20-22) In the two-state system, the inverter is always on, Fig 20-22(a) The extracted error signal is compared to a high-frequency triangular carrier wave, and the inverter switches each time the waves cross The result is an injected signal that produces equal and opposite distortion In a three-state system (hysteresis method), preset upper and lower limits are compared to an error signal, Fig 20-22(b) So long as the error is within a tolerable band, there is no switching and the inverter is off The advantages of time-domain methods are fast response, though these are limited to one-node application, to which these are connected and take measurement from 20.16 CORRECTIONS IN THE FREQUENCY DOMAIN Fourier transformation is used to determine the harmonics to be injected The error signal is extracted using a 60-Hz filter and the Fourier transform of the error signal is Harmonic Mitigation and Filters Figure 20-21 703 (a) Voltage-source inverter active filter; (b) current-source active inverter filter taken The cancellation of M harmonics method allows for compensation up to the Mth harmonic, where M represents the highest harmonic to be compensated A switching function is constructed by solving a set of nonlinear equations to determine the precise switching times and magnitudes Quarter-wave symmetry is assumed to reduce the computations Because an error function is used, the system can easily accommodate system changes, but requires intense calculations and the time delays associated with it The computations increase with M and the increased computational requirements are the main disadvantage, though these can be applied in dispersed networks The predetermined frequency method injects specific frequencies into the system, which are decided in the design stage of the system, much like passive harmonic filtering This eliminates the need for real-time commutation of switching signals, but the harmonic levels present must be carefully evaluated beforehand and each filter designed for the specific requirements See also Refs [11,12] 704 Chapter 20 Figure 20-22 20.17 (a) Two-step switching function; (b) three-step switching function INSTANTANEOUS REACTIVE POWER The signal method of control generates an error signal based on input voltage or current and a reference sinusoidal waveform A more elaborate function is the instantaneous power method which calculates the desired current so that the instan- Harmonic Mitigation and Filters 705 taneous active and reactive power in a three-phase system are kept constant, i.e., the active filter compensates for variation in instantaneous power [13] By linear transformation the phase voltages ea, eb, ec and load currents ia, ib, ic are transformed into an – (two-phase) co-ordinate system:    rffiffiffi  e    ¼ 2  e  3 0  1   À  ea  À pffiffi2ffi p2ffiffiffi  eb  3    ec À 2 ð20:53Þ and      rffiffiffi À À  ia   i      ¼ 2 pffiffi2ffi p2ffiffiffi  ib  ð20:54Þ  i    3     ic  0 À 2 The instantaneous real power p and the instantaneous imaginary power q are defined as       p   e e  i     ¼ ð20:55Þ  q   Àe e  i  Here, p and q are not conventional watts and vars The p and q are defined by the instantaneous voltage in one phase and the instantaneous current in the other phase p ¼ e i þ e [...]... 214 Short-Circuit Calculations According to ANSI Standards 219 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 219 220 222 222 223 231 233 233 235 236 261 262 Types of Calculations Impedance Multiplying Factors Rotating Machines Model Types and Severity of System Short-Circuits Calculation Methods Network Reduction Breaker Duty Calculations High X/R Ratios (DC Time Constant Greater than 45ms) Calculation... 10.7 10.8 10.9 10.10 11 12 14 331 334 336 345 346 347 349 352 356 Load Flow Methods: Part I 360 11.1 11.2 11.3 11.4 11.5 11.6 361 366 367 377 383 384 Modeling a Two-Winding Transformer Load Flow, Bus Types Gauss and Gauss–Seidel Y-Matrix Methods Convergence in Jacobi-Type Methods Gauss–Seidel Z-Matrix Method Conversion of Y to Z Matrix Load Flow Methods: Part II 391 12.1 12.2 12.3 391 393 12.4 12.5 12.6... Jacobian Matrix Simplifications of Newton–Raphson Method Decoupled Newton–Raphson Method Fast Decoupled Load Flow Model of a Phase-Shifting Transformer DC Models Load Models Impact Loads and Motor Starting Practical Load Flow Studies 395 397 405 408 408 411 413 415 422 424 Reactive Power Flow and Control 435 13.1 13.2 13.3 13.4 13.5 436 442 447 460 467 Voltage Instability Reactive Power Compensation Reactive... 16.6 16.7 16.8 16.9 16.10 525 527 528 528 536 539 545 545 547 551 Optimal Power Flow Decoupling Real and Reactive OPF Solution Methods of OPF Generation Scheduling Considering Transmission Losses Steepest Gradient Method OPF Using Newton’s Method Successive Quadratic Programming Successive Linear Programming Interior Point Methods and Variants Security and Environmental Constrained OPF Harmonics Generation... Capacitor Bank Arrangements Study Cases Harmonic Mitigation and Filters 664 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 20.10 664 665 668 678 681 683 684 686 687 689 Mitigation of Harmonics Band Pass Filters Practical Filter Design Relations in a ST Filter Filters for a Furnace Installation Filters for an Industrial Distribution System Secondary Resonance Filter Reactors Double-Tuned Filter Damped Filters... short-circuit current For simplification of empirical short-circuit calculations, the impedances of static components like transmission lines, cables, reactors, and transformers are assumed to be time invariant Practically, this is not true, i.e., the flux densities and saturation characteristics of core materials in a transformer may entirely change its leakage reactance Driven to saturation under high current... applied at the fault point In other words, the unbalance part of the network Short-Circuit Currents and Symmetrical Components 15 can be thought to be connected to the balanced system at the point of fault Practically, the power systems are not perfectly balanced and some asymmetry always exists However, the error introduced by ignoring this asymmetry is small (This may not be true for highly unbalanced ... 7.11 7.12 219 220 222 222 223 231 233 233 235 236 261 262 Types of Calculations Impedance Multiplying Factors Rotating Machines Model Types and Severity of System Short-Circuits Calculation Methods... overstated This book covers the commonly required short-circuit, load flow, and harmonic analyses Practical and theoretical aspects have been harmoniously combined Although there is the inevitable... can be experienced at a distance from their source) The book derives and compiles ample data of practical interest, with the emphasis on harmonic power flow and harmonic filter design Generation,