Das Fundamental to the planning, design, and operating stages of any electrical engineering endeavor, power system analysis continues to be shaped by dramatic advances and improvements that reflect today’s changing energy needs Highlighting the latest directions in the field, Power System Analysis: Short-Circuit Load Flow and Harmonics, Second Edition includes investigations into arc flash hazard analysis and its migration in electrical systems, as well as wind power generation and its integration into utility systems Power System Analysis Designed to illustrate the practical application of power system analysis to real-world problems, this book provides detailed descriptions and models of major electrical equipment, such as transformers, generators, motors, transmission lines, and power cables With 22 chapters and appendices that feature new figures and mathematical equations, coverage includes: • Short-circuit analyses, symmetrical components, unsymmetrical faults, and matrix methods • Rating structures of breakers • Current interruption in AC circuits and short-circuiting of rotating machines • Calculations according to the new IEC and ANSI/IEEE standards and methodologies • Load flow, transmission lines and cables, and reactive power flow and control • Techniques of optimization, FACT controllers, three-phase load flow, and optimal power flow • A step-by-step guide to harmonic generation and related analyses, effects, limits, and mitigation, as well as new converter topologies and practical harmonic passive filter designs—with examples • More than 2000 equations and figures, as well as solved examples, cases studies, problems, and references Maintaining the structure, organization, and simplified language of the first edition, longtime power system engineer, J.C Das, seamlessly melds coverage of theory and practical applications to explore the most commonly required short-circuit, load-flow, and harmonic analyses This book requires only a beginning knowledge of the per-unit system, electrical circuits and machinery, and matrices, and it offers significant updates and additional information, enhancing technical content and presentation of subject matter As an instructional tool for computer simulation, it uses numerous examples and problems to present new insights while making readers comfortable with procedure and methodology Short-Circuit Load Flow and Harmonics Se c ond Ed i t ion Se cond Ed i t i on J C Das K11101 K11101_COVER_final.indd 6/9/11 12:09 PM 3RZHU6\VWHP $QDO\VLV 32:(5(1*,1((5,1* 6HULHV(GLWRU +/HH:LOOLV 4XDQWD7HFKQRORJ\ 5DOHLJK1RUWK&DUROLQD $GYLVRU\(GLWRU 0XKDPPDG+5DVKLG 8QLYHUVLW\RI:HVW)ORULGD 3HQVDFROD)ORULGD 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 Transformers: Principles and Applications, John J Winders, Jr 17 Spatial Electric Load Forecasting: Second Edition, Revised and Expanded, H Lee Willis 18 Dielectrics in Electric Fields, Gorur G Raju 19 Protection Devices and Systems for High-Voltage Applications, Vladimir Gurevich 20 Electrical Power Cable Engineering, Second Edition William Thue 21 Vehicular Electric Power Systems: Land, Sea, Air, and Space Vehicles, Ali Emadi, Mehrdad Ehsani, and John Miller 22 Power Distribution Planning Reference Book, Second Edition, H Lee Willis 23 Power System State Estimation: Theory and Implementation, Ali Abur 24 Transformer Engineering: Design and Practice, S.V Kulkarni and S A Khaparde 25 Power System Capacitors, Ramasamy Natarajan 26 Understanding Electric Utilities and De-regulation: Second Edition, Lorrin Philipson and H Lee Willis 27 Control and Automation of Electric Power Distribution Systems, James Northcote-Green and Robert G Wilson 28 Protective Relaying for Power Generation Systems Donald Reimert 29 Protective Relaying: Principles and Applications, Third Edition J Lewis Blackburn and Thomas J Domin 30 Electric Power Distribution Reliability, Second Edition Richard E Brown 31 Electrical Power Equipment Maintenance and Testing, Second Edition, Paul Gill 32 Electricity Pricing: Engineering Principles and Methodologies Lawrence J Vogt 33 Power System Analysis: Short-Circuit Load Flow and Harmonics, Second Edition, J C Das 3RZHU6\VWHP $QDO\VLV 6KRUW&LUFXLW/RDG)ORZDQG+DUPRQLFV 6HFRQG(GL W L RQ -&'DV $PHF,QFRUSRUDWHG 7XFNHU*HRUJLD 86$ CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2012 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 20110517 International Standard Book Number-13: 978-1-4398-2080-3 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Series Introduction xxi Preface to the Second Edition xxiii Preface to the First Edition xxv Author xxvii Short-Circuit Currents and Symmetrical Components 1.1 Nature of Short-Circuit Currents 1.2 Symmetrical Components 1.3 Eigenvalues and Eigenvectors 1.4 Symmetrical Component Transformation 1.4.1 Similarity Transformation 1.4.2 Decoupling a Three-Phase Symmetrical System 10 1.4.3 Decoupling a Three-Phase Unsymmetrical System 14 1.4.4 Power Invariance in Symmetrical Component Transformation 14 1.5 Clarke Component Transformation 15 1.6 Characteristics of Symmetrical Components 16 1.7 Sequence Impedance of Network Components 19 1.7.1 Construction of Sequence Networks 20 1.7.2 Transformers 21 1.7.2.1 Delta–Wye or Wye–Delta Transformer 22 1.7.2.2 Wye–Wye Transformer 23 1.7.2.3 Delta–Delta Transformer 24 1.7.2.4 Zigzag Transformer 24 1.7.2.5 Three-Winding Transformers 25 1.7.3 Static Load 29 1.7.4 Synchronous Machines 29 1.8 Computer Models of Sequence Networks 34 1.9 Structure and Nature of Electrical Power Systems 35 1.9.1 Power System Component Models 37 1.9.2 Smart Grids 37 1.9.3 Linear and Nonlinear Systems 37 1.9.3.1 Property of Decomposition 38 1.9.4 Static and Dynamic Systems 39 1.10 Power System Studies 39 Problems 40 Bibliography 42 Unsymmetrical Fault Calculations 43 2.1 Line-to-Ground Fault 43 2.2 Line-to-Line Fault 45 2.3 Double Line-to-Ground Fault 47 2.4 Three-Phase Fault 48 2.5 Phase Shift in Three-Phase Transformers 49 2.5.1 Transformer Connections 49 vii viii Contents 2.5.2 Phase Shifts in Winding Connections 50 2.5.3 Phase Shift for Negative Sequence Components 51 2.6 Unsymmetrical Fault Calculations 56 2.7 System Grounding 63 2.7.1 Solidly Grounded Systems 64 2.7.2 Resistance Grounding 67 2.7.2.1 High-Resistance Grounded Systems 67 2.7.2.2 Coefficient of Grounding 73 2.8 Open Conductor Faults 74 2.8.1 Two-Conductor Open Fault 74 2.8.2 One-Conductor Open Fault 75 Problems 78 References 81 Bibliography 81 Matrix Methods for Network Solutions 83 3.1 Network Models 83 3.2 Bus Admittance Matrix 84 3.3 Bus Impedance Matrix 88 3.3.1 Bus Impedance Matrix from Open-Circuit Testing 89 3.4 Loop Admittance and Impedance Matrices 90 3.4.1 Selection of Loop Equations 92 3.5 Graph Theory 92 3.6 Bus Admittance and Impedance Matrices by Graph Approach 95 3.6.1 Primitive Network 95 3.6.2 Incidence Matrix from Graph Concepts 97 3.6.3 Node Elimination in Y-Matrix 102 3.7 Algorithms for Construction of Bus Impedance Matrix 103 3.7.1 Adding a Tree Branch to an Existing Node 103 3.7.2 Adding a Link 105 3.7.3 Removal of an Uncoupled Branch 106 3.7.4 Changing Impedance of an Uncoupled Branch 107 3.7.5 Removal of a Coupled Branch 107 3.8 Short-Circuit Calculations with Bus Impedance Matrix 114 3.8.1 Line-to-Ground Fault 115 3.8.2 Line-to-Line Fault 115 3.8.3 Double Line-to-Ground Fault 115 3.9 Solution of Large Network Equations 125 Problems 125 Bibliography 127 Current Interruption in AC Networks 129 4.1 Rheostatic Breaker 129 4.2 AC Arc Interruption 131 4.2.1 Arc Interruption Theories 132 4.2.1.1 Cassie’s Theory 132 4.2.1.2 Mayr’s Theory 132 4.2.1.3 Cassie–Mayr Theory 132 4.3 Current-Zero Breaker 133 Contents ix 4.4 Transient Recovery Voltage 135 4.4.1 First Pole to Clear Factor 136 4.5 Terminal Fault 139 4.5.1 Four-Parameter Method 139 4.5.2 Two-Parameter Representation 140 4.6 Short-Line Fault 141 4.7 Interruption of Low Inductive Currents 142 4.7.1 Virtual Current Chopping 145 4.8 Interruption of Capacitive Currents 145 4.9 TRV in Capacitive and Inductive Circuits 147 4.10 Prestrikes in Breakers 149 4.11 Overvoltages on Energizing High-Voltage Lines 149 4.11.1 Overvoltage Control 151 4.11.2 Synchronous Operation 152 4.11.3 Synchronous Capacitor Switching 152 4.11.4 Shunt Reactors 153 4.12 Out-of-Phase Closing 154 4.13 Resistance Switching 155 4.14 Failure Modes of Circuit Breakers 156 4.15 Operating Mechanisms-SF6 Breakers 159 4.16 Vacuum Interruption 160 4.17 Stresses in Circuit Breakers 161 Problems 162 References 163 Bibliography 164 Application and Ratings of Circuit Breakers and Fuses according to ANSI Standards 165 5.1 Total and Symmetrical Current Rating Basis 166 5.2 Asymmetrical Ratings 167 5.2.1 Contact Parting Time 167 5.3 Voltage Range Factor K 170 5.4 Circuit Breaker Timing Diagram 170 5.5 Maximum Peak Current 173 5.6 Permissible Tripping Delay 174 5.7 Service Capability Duty Requirements and Reclosing Capability 174 5.7.1 Transient Stability on Fast Reclosing 175 5.8 Capacitance Current Switching 178 5.8.1 Switching of Cables 183 5.8.2 Effects of Capacitor Switching 186 5.9 Line-Closing Switching Surge Factor 187 5.9.1 Switching of Transformers 188 5.10 Out-of-Phase Switching Current Rating 188 5.11 Transient Recovery Voltage 189 5.11.1 Short-Line Faults 193 5.11.2 Oscillatory TRV 195 5.11.3 Initial TRV 196 5.11.4 Adopting IEC TRV Profiles in IEEE Standards 196 5.11.5 Definite Purpose TRV Breakers 198 1019 Appendix F: Limitation of Harmonics Arc furnaces cause flicker because the current drawn during melting and refining periods is erratic and fluctuates widely and the power factor is low There are other loads that can generate flicker, for example, large spot welding machines often operate close to the flicker perception limits Industrial processes may comprise a number of motors having rapidly varying loads or starting at regular intervals, and even domestic appliances such as cookers and washing machines can cause flicker on weak systems However, the harshest load for flicker is an arc furnace During the melting cycle of a furnace, the reactive power demand is high Figure 17.26 gives typical performance curves of an arc furnace, and Figure 17.25 shows that an arc furnace current is random and no periodicity can be assigned (thus Fourier analysis cannot be used), yet some harmonic spectrums have been established and Table 17.6 shows typical harmonics during melting and refining stage Note that even harmonics are produced during melting stage The high reactive power demand and poor power factor cause cyclic voltage drops in the supply system Reactive power flow in an inductive element requires voltage differential between sending end and receiving ends and there is reactive power loss in the element itself When the reactive power demand is erratic, it causes corresponding swings in the voltage dips, much depending upon the stiffness of the system behind the application of the erratic load This voltage drop is proportional to the shortcircuit MVA of the supply system and the arc furnace load For a furnace installation, the short-circuit voltage depression (SCVD) is defined as SCVD ¼ 2MWfurnace MVASC (F:8) where MW is the installed load of the furnace in MWfurnace MVASC is the short-circuit level of the utility supply system This gives an idea whether potential problems with flicker can be expected An SCVD of 0.02–0.025 may be in the acceptable zone between 0.03 and 0.035 in the borderline zone, and above 0.035 objectionable [9] When there are multiple furnaces, these can be grouped into one equivalent MW Example 20.2 describes the use of tuned filters to compensate for the reactive power requirements of an arc furnace installation The worst flicker occurs during the first 5–10 of each heating cycle and decreases as the ratio of the solid–toliquid metal decreases The significance of DV=V and a number of voltage changes are illustrated with reference to Figure F.5 [4] This shows a 50 Hz waveform, having a 1.0 average voltage with a relative voltage change Dv=" v ¼ 40% and with 8.8 Hz rectangular modulation It can be written as & ' 40   signum[2p  8:8  t] v(t) ¼  sin (2p 50t) ỵ 100 (F:9) Each full period produces two distinct changes: one with increasing magnitude and one with decreasing magnitude Two changes per period with a frequency of 8.8 Hz give rise to 17.6 changes per second F.5.1 Control of Flicker The response of the passive compensating devices is slow When it is essential to compensate load fluctuations within a few milliseconds, SVCs are required Referring to Figure 13.12, 1020 Appendix F: Limitation of Harmonics 1.5 v = 1.0 Δv = 0.4 1.2 1.0 Voltage normalized (V) 0.8 0.5 –0.5 –1.0 –1.5 0.05 0.1 0.15 0.2 Time (s) 0.25 0.3 0.35 0.4 FIGURE F.5 Modulation with rectangular voltage change DV=V ¼ 40%, 8.8 Hz 17.6 changes per second (From IEC, Electromagnetic compatibility (EMC)—Part 4: Testing and measurement techniques (Section 15: Flicker meter— Functional and design specifications), 2003.) large TCR flicker compensators of 200 MW have been installed for arc furnace installations Closed-loop control is necessary due to the randomness of load variations and complex circuitry is required to achieve response times of less than one cycle Significant harmonic distortion may be generated, and harmonic filters will be required TSCs have also been installed and these have inherently one cycle delay as the capacitors can only be switched when their terminal voltage matches the system voltage Thus, the response time is slower SVCs employing TSCs not generate harmonics, but the resonance with the system and transformer needs to be checked We discussed STATCOM in Chapter 13 It has been long recognized that reactive power can be generated without the use of bulk capacitors and reactors and STATCOM makes it possible As discussed in Chapter 13, it is capable of operating with leading or lagging power factors With the design of high bandwidth control capability, STATCOM can be used to force three-phase currents of arbitrary waveshape through the line reactance This means that it can be made to supply non-sinusoidal, unbalanced, randomly fluctuating currents demanded by the arc furnace With a suitable choice of dc capacitor, it can also supply the fluctuating real power requirements, which cannot be achieved with SVCs Harmonics are not of concern (Figure 17.36) The instantaneous reactive power on the source side is the reactive power circulating between the electrical system and the device, while reactive power on the output side is the instantaneous reactive power between the device and its load There is no relation between the instantaneous reactive powers on the load and source side, and the instantaneous imaginary power on the input is not equal to the instantaneous reactive power on the output (Chapter 15) The STATCOM for furnace compensation may use vector control based on the concepts of instantaneous active and reactive power, ia and ib (Chapter 20) 1021 Appendix F: Limitation of Harmonics STATCOM Flicker without compensator 90 R = 1– Flicker with compensator × 100 100 80 SVC 70 60 50 40 30 20 10 0 10 15 20 25 30 Flicker frequency (Hz) FIGURE F.6 Flicker factor R with STATCON and SVC Figure F.6 shows flicker reduction factor as a function of flicker frequency STATCOM versus SVC [10] Flicker mitigation with a fixed reactive power compensator and an active compensator—a hybrid solution for welding processes—is described in [11] References IEEE IEEE recommended practice and requirements for harmonic control in electrical systems, 1992 Standard 519 IEC Electromagnetic compatibility Part 3: Limits (Section 2: Limits for harmonic current emission (equipment input current #16 A per phase)), 1995 Standard 61000-3-2 IEC Electromagnetic compatibility Part 3: Limits (Section 3: Limitations of voltage fluctuations and flicker in low-voltage supply systems for equipment with rated current 16 A per phase and not subjected to conditional connection, Geneva), 2008 Standard 61000-3-3 IEC Electromagnetic compatibility (EMC) Part 4: Testing and measurement techniques (Section 15: Flicker meter—Functional and design specifications), 2003 IEEE IEEE interharmonics task force working document IH0101, 2000 IEEE Recommended practice for measurement and limits of voltage fluctuations and associated light flicker on AC power systems Standard 1453, 2004 IEC Electromagnetic compatibility (EMC) Part 3-11: Limits: Limitation of voltage fluctuations and flicker in low-voltage supply systems for equipment with rated current