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Power system analysis is fundamental in the planning, design, and operating stages,and its importance cannot be overstated. This book covers the commonly requiredshortcircuit, load flow, and harmonic analyses. Practical and theoretical aspectshave been harmoniously combined. Although there is the inevitable computer simulation, a feel for the procedures and methodology is also provided, through examplesand problems. Power System Analysis: ShortCircuit Load Flow and Harmonicsshould be a valuable addition to the power system literature for practicing engineers,those in continuing education, and college students.
Power System Analysis Short-Circuit Load Flow and Harmonics J C Das Amec, Inc Atlanta, Georgia Marcel Dekker, Inc TM Copyright 2002 by Marcel Dekker, Inc All Rights Reserved New York • Basel 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 Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 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 Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 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 Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 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 Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 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 Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 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 Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 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 Unsymmetrical Fault Calculations 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 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 Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 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 Current Interruption in AC Networks 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 Application and Ratings of Circuit Breakers and Fuses According to ANSI Standards 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 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 Short-Circuit of Synchronous and Induction Machines 6.1 6.2 Reactances of a Synchronous Machine Saturation of Reactances Copyright 2002 by Marcel Dekker, Inc All Rights Reserved Contents 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 Conceptual and Analytical Differences Prefault Voltage Far-From-Generator Faults Near-to-Generator Faults Influence of Motors Comparison with ANSI Calculation Procedures Examples of Calculations and Comparison with ANSI Methods Calculations of Short-Circuit Currents in DC Systems 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 10 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 Procedure Examples of Calculations Thirty-Cycle Short-Circuit Currents Dynamic Simulation Short-Circuit Calculations According to IEC Standards 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Time Constants of Synchronous Machines Synchronous Machine Behavior on Terminal Short-Circuit Circuit Equations of Unit Machines Park’s Transformation Park’s Voltage Equation Circuit Model of Synchronous Machines Calculation Procedure and Examples Short-Circuit of an Induction Motor Short-Circuit Calculations According to ANSI Standards 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 ix DC Short-Circuit Current Sources Calculation Procedures Short-Circuit of a Lead Acid Battery DC Motor and Generators Short-Circuit Current of a Rectifier Short-Circuit of a Charged Capacitor Total Short-Circuit Current DC Circuit Breakers Load Flow Over Power Transmission Lines 10.1 Power in AC Circuits Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 676 Chapter 20 design of ST filters The shifted resonance frequencies should have at least 30 cycles difference between the adjacent and odd or even harmonics Even then, some amplification of the transformer switching inrush current will occur In this example, with three ST filters, the shifted frequencies are 271, 367, and 549 Hz The last two frequencies are close to the sixth and ninth harmonics A wider band can be attempted by slightly lowering the tuning frequency of the seventh and 11th ST filters The shifted frequencies can also be calculated from Eq (20.14) Effect of Tolerances on Filter Components The tolerances on capacitors and reactors will result in detuning Consider that components of the following tolerances are selected: Capacitors: + 5% Reactors: Ỉ2% Let the capacitance of the fifth and seventh filters increase by 5% and the inductance by 2% This is quite a conservative assumption for checking the detuning effect and resulting current distribution The series-tuned frequencies of the fifth and seventh filters will shift to a lower value The results of harmonic current flow are shown in Table 20-6 The harmonic distortion increases above the acceptable limits This points to the necessity of iterating the design with required tolerances and fine tuning the selected tuning frequencies Closer tolerances on the components is an option, but that may not be practical and economically justifiable Outage of One of the Parallel Filters Outage of one of the parallel ST filters should be considered It will have the following effects: The current loading of remaining filters in service may increase substantially and the capacitors and reactors may be overloaded Table 20-6 Harmonic Simulation with Tolerances on Filtersa (see text) h# Fifth ST filter Seventh ST filter 11th ST filter 2000-hp motor Supply system PCC 11 13 17 19 23 25 29 31 81.4 8.11 1.41 1.44 1.02 0.61 0.20 0.16 0.12 0.08 22.9 57.1 3.28 3.10 2.06 1.21 0.39 0.32 0.23 0.16 11.4 16.4 60.55 16.78 7.11 3.84 1.12 0.88 0.61 0.41 5.36 2.84 0.74 0.80 0.61 0.37 0.12 0.10 0.07 0.05 72.5 38.3 10.00 10.9 8.20 4.98 1.66 1.35 0.98 0.69 a ` Harmonic current flow in amperes Copyright 2002 by Marcel Dekker, Inc All Rights Reserved Harmonic Mitigation and Filters 677 The resonant frequencies will shift and may result in harmonic current amplification The harmonic distortion will increase Table 20-7 shows the effect of outage of one of the three filters at a time The harmonic distortion at the PCC increases in every case, though the filter components are not overloaded According to IEEE Standard 519 [1] it is permissible to operate the system on a short-term basis with higher distortion limits at the PCC, provided that the faulty unit is placed back in service quickly after rectification Sometimes, the outage of a filter may result in overloading of the remaining filters in service It then becomes necessary that parallel filters are also removed from service The filter protection and switching scheme are designed so that, with the outage of a unit, the complete system is shut down Operation with Varying Loads When load-dependent switching is required for reactive power compensation, multiple capacitor banks are switched in an ascending order, i.e., 5th, 7th, and 11th Generally, this will occur during start-up conditions; however, if sustained operation at reduced loads is required, it is necessary to control the harmonic distortion at each of the operating loads and switching steps The harmonic loads may or may not decrease in proportion to the overall plant load This adds another step in designing an appropriate passive filtering scheme to meet the TDD requirements Division of Reactive kvar Between Parallel Filter Banks When multiple parallel filters are required and the total kvar requirements are also known, it remains to find out the most useful distribution of kvar amongst the parallel filters In the above example, the 5th, 7th, and 11th filters are based on equal kvar This is too simplistic an approach, rarely implemented As filters should be sized to handle the harmonic loading, one approach would be to divide the Table 20-7 Example 20.1: Effect of Outage of One of the Parallel Filtersa Fifth ST out Seventh ST out 11th ST out Harmonic Seventh 11th ST Utility Fifth ST 11th ST Utility Fifth ST Seventh Utility order h # ST filter filter (PCC) filter filter (PCC) filter ST filter (PCC) 11 13 17 19 23 25 29 31 35 53.28 72.2 3.5 3.37 2.24 1.31 0.42 0.34 0.24 0.19 0.14 30.57 11.78 61.53 17.47 7.49 4.05 1.19 0.93 0.64 0.44 0.40 194 27.5 10.2 11.3 8.64 5.25 1.76 1.43 1.04 0.73 0.46 a ` Harmonic current in amperes Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 85.48 22.9 1.48 1.59 1.14 0.68 0.22 0.18 0.13 0.09 0.07 6.79 44.57 63.27 1.85 7.97 4.31 1.27 0.98 0.68 0.47 0.41 43.2 104.00 10.5 1.20 9.20 5.59 1.87 1.52 1.11 0.78 0.48 89.00 50.59 6.91 2.92 1.62 0.93 0.29 0.23 0.16 0.11 0.07 12.23 60.33 16.89 6.53 3.37 1.89 0.58 0.46 0.33 0.23 0.14 45.00 22.90 48.60 21.90 13.00 7.61 2.44 1.96 1.41 0.99 0.46 678 Chapter 20 Figure 20-6 Average losses in film-foil capacitor units, with variation of temperature required kvar based on the percentage of harmonic current that each filter will carry This will not be known in advance The other method is to proportionate the filters with respect to harmonic current generation, i.e., the lower order harmonics are higher in magnitude, so more kvar are allocated to a lower order filter Again some iteration will be required to optimize the sizes initially chosen, based on the actual fundamental and harmonic current loadings and the desired reactive power compensation Losses in the Capacitors The power capacitors have some active power loss component, though small Figure 20-6 shows the average losses versus ambient temperature for capacitors At an operating temperature of 408C, the loss is approximately 0.10 W/ kvar and increases to 0.28 W/kvar at À408C This loss should be considered in the filter design, by an equivalent series resistance inserted in the circuit 20.4 RELATIONS IN A ST FILTER The reactive power output of a capacitor at fundamental frequency is V =Xc In the presence of a filter reactor it is given by V2 XL À Xc V2 20:16ị ẳ Xc =n2 Xc n2 ẳ Â (reactive power without reactor) n À1 The reactive power output with a filter reactor tuned to, say 4.85 f is approximately 4% higher than without the reactor This is so because the voltage drop in the reactor is added to the capacitor voltage and its operating voltage is Sf ẳ Vc ẳ V ỵ VL ẳ V ỵ j!LV= j 1=j!Cịị ẳ n2 n2 Copyright 2002 by Marcel Dekker, Inc All Rights Reserved ð20:17Þ Harmonic Mitigation and Filters 679 The capacitors in a fifth harmonic filter tuned to 4.85 f operate at approximately 4% higher than the system voltage While selecting the voltage rating on the filter capacitors, the considerations are: Higher operating voltage due to presence of filter reactors Sustained upward operating voltage of the utility power supply system This may be due to location, e.g., close to generating stations, or may be due to voltage adjustment tap changing on transformers The higher voltages that will be imposed when one or two capacitor units in a parallel group go out of service The neutral unbalance detection schemes not take a bank out of service if one or two units in a parallel group go out of service, say due to fuse operation on the individual capacitor units If the problem cascades then a trip is initiated Generally, an operating voltage slightly higher than the nominal system voltage is selected, though the capacitors have a 10% overvoltage capability, Eq (18.16) This reduces the reactive capability at the voltage of application as the square of the voltages To meet the requirement of a certain reactive power output a larger number of units are then required The fundamental loading of the capacitors is given by " #2 " # Vc V2 n2 n2 ¼ ¼ Sf Xc Xc n2 À n À1 ð20:18Þ and the harmonic loading is 2 Ih Xc Ih V h2 ¼ h Sh h2 À ð20:19Þ When harmonic voltages and current flows are known form harmonic simulation, the harmonic loading can be found from hẳ1 X 20:20ị Vh Ih hẳ2 The fundamental frequency loading of the filter reactor is " # !" # 2 VL V c n2 Vc Sf n2 ¼ ¼ ¼ 2 XL Xc n n Xc n n À ð20:21Þ The harmonic loading for the reactor is the same as for the capacitor The increase in bus voltage on switching a capacitor at a transformer secondary bus is approximately given by %ÁV ¼ Kvarcapacitor Zt kVAt A flow chart of the design of ST filters is thus shown in Fig 20-7 Copyright 2002 by Marcel Dekker, Inc All Rights Reserved ð20:22Þ 680 Figure 20-7 Chapter 20 Flow chart for design of ST filters Copyright 2002 by Marcel Dekker, Inc All Rights Reserved Harmonic Mitigation and Filters 20.5 681 FILTERS FOR A FURNACE INSTALLATION Example 20.2 Figure 20-8 shows a furnace installation The total operating load is 150 MVA The PCC is the 230 kV side of 125/208 MVA transformer A reactive power compensation of 135 Mvar is required, which is provided by four ST filters formed with 54, 27, 27, and 27 Mvar capacitors, respectively, for second, third, fourth, and fifth harmonics These are connected at the main 34.5-kV bus The capacitor banks are formed as follows: Second harmonic ST filter: double wye, grounded, three-series groups, each group containing eight units of 400-kvar capacitors of rated voltage 6.42 kV (Fig 20-9) Third, fourth, and fifth ST filters: single wye ungrounded, three-series groups, each group containing eight units of 400-kvar capacitors of rated voltage 6.42 kV The tuning frequencies for second, third, fourth, and fifth ST filters are 1.95, 2.95, 3.95, and 4.95 times the fundamental frequency, respectively The system is impacted with a harmonic spectrum during the melting cycle of the furnaces The results of harmonic flow calculations and TDD at the PCC are Figure 20-8 Single-line diagram of a furnace installation, showing ST filters Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 682 Chapter 20 Figure 20-9 Formation of a 57.6-Mvar double-wye, 34.5-kV capacitor bank for second harmonic filter (Example 20.2) shown in Table 20-8 This table shows harmonic current loading of the filters and harmonic currents fed into the 230-kV system We observe that each ST filter operates effectively providing a low impedance path for the harmonic it is intended to shunt away TDD at the 230-kV PCC is 1.32 Fig 20-9 shows formation of 2nd harmonic filter Dynamic Stresses Filters for furnace installations should receive special considerations with respect to transient surges Consider that a second harmonic ST filter is not installed in a Table 20-8 Harmonic Example 20.2: Harmonic Filters for Arc Furnace (Fig 20-8)a Second harmonic ST filter Third harmonic ST filter Fourth harmonic ST filter Fifth harmonic ST filter 192.20 6.83 0.91 0.54 5.08 1.07 134.2 1.63 0.75 6.18 0.77 10.00 59.5 2.34 13.39 0.69 6.69 3.87 97.78 28.89 a Harmonic Harmonic currents voltages at PCC at PCC (230-kV bus) (230-kV bus) 0.868 2.82 0.50 0.34 3.50 0.437 0.267 0.022 0.0337 8.37 ` Harmonic current flow in amperes; harmonic voltages in kV Load demand = 350.8 A, three-phase short-circuit current at PCC = 20 kA, ratio ISC/IL = 57, permissible total TDD = 6.0%, permissible TDD for h < 11 = 5%; calculated TDD = 1.32%, which is also total TDD Copyright 2002 by Marcel Dekker, Inc All Rights Reserved Harmonic Mitigation and Filters 683 furnace installation, and the lowest harmonic order filter is the third The shifted resonant frequency may coincide with one of the transformer inrush current harmonics or be close to it, and may increase the transformer inrush currents Though these last for a short duration, these will stress the filter reactor and capacitors Switching inrush currents were discussed in Sec 5.8 and it was noted that the presence of filter reactors lowers the frequency as well as the magnitude of the inrush currents However, their duration may increase, as the reactors are of high Q, giving less damping Synchronous switching and resistance switching are the options For normal switching it will be necessary to calculate the effect of switching and of transformer inrush currents, and resulting harmonic voltages, and apply these to the specifications of filter reactors and capacitors Attenuation of Harmonics In the above example we declared a 230-kV bus as the PCC If a 34.5-kV bus is declared as the PCC and the TDD is calculated, it will be higher than the TDD at the 230-kV bus The calculated TDD at the 34.5-kV bus is 1.35 This shows attenuation of harmonics in propagation through system elements, in this case, the transformer impedance The impact is the maximum at the point of injection and attenuation occurs as the harmonics are propagated into the system, unless there is amplification due to resonance Partial resonances are common when capacitors are used Noninteger Harmonics Furnace loads generate noninteger harmonics Generally, for a furnace installation the seventh harmonic filter is a high-pass filter (see Section 20.8) 20.6 FILTERS FOR AN INDUSTRIAL DISTRIBUTION SYSTEM Example 20.3 Example 19.2 for application of power capacitors in an industrial plant showed that depending on the operating condition, resonant frequencies swing over a wide spectrum and the harmonic distortion at the PCC is high The equivalent negative sequence current loading of generators is exceeded Even without harmonics, a part of the generator negative sequence capability may be utilized due to unbalance loads and voltages and system asymmetries The harmonic loading on the generators and harmonic distortion is reduced by turning capacitor banks at buses and into parallel fifth and seventh ST filters The following details are applicable for the final filter designs: Bus 2, fifth harmonic ST filter: five units of 300 kvar, 7.2 kV, per phase, total capacitor Mvar = 4.5, connected in ungrounded wye configuration; n = 4.85, C = 76.75 mF, L = 3.897 mH Buses and 3, seventh harmonic ST filter: three units of 300 kvar, 7.2 kV, per phase, giving 2.7 Mvar total, connected in ungrounded wye configuration; n = 6.75, C = 46.05 mF, L = 3.353 mH Bus 3, fifth harmonic ST filter: four units of 300 kvar, 7.2 kV, per phase, total capacitor Mvar = 3.6, n = 4.85, C = 61.40 mF, L = 4.872 mH, Q = 100 X=R ratio of reactors = 100 at fundamental frequency The harmonic current flow is studied under the same three conditions as in Example 19.2 and TDD at the Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 684 Chapter 20 Table 20-9 Harmonic order# Example 19.3: Harmonic Simulation with Filtersa Current in generator Current in generator 1b 37 24.4 2.03 3.76 2.83 2.03 1.94 1.75 35.6 104 42.3 3.99 1.76 1.60 1.81 1.73 31.7 27.3 2.3 2.87 4.0 8.20 2.89 1.30 3 48.7 26.3 27.2 15.2 12.1 4.96 9.53 6.19 6.78 4.13 1.24 1.17 0.75 0.54 0.54 0.48 7.05 25.81 5.45 0.21 0.56 0.45 0.47 0.44 12.8 5.44 3.70 4.15 0.90 2.13 0.95 1.80 Calculated TDD at PCC! 3.28 6.66 6.12 Permissible TDD at PCC! 7.5 7.5 11 13 17 19 23 25 a Current into the utility’s system (PCC) 51.7 Generator 24.9 out of 25.7 service 17.6 10.0 7.13 7.71 7.44 ` Harmonic current flow in amperes 1,2,3 refer to study cases, see Example 19.2 b PCC is calculated The normal load demand current Ir = 250 A in cases and 3, and 410 A in case 2, when No.2 generator is out of service The three-phase short-circuit current is 30.2 kA sym The results of calculation are shown in Table 20-9 The following observations are of interest When the 75-mile 115-kV line and its harmonic load is modeled, TDD increases over normal operating condition (See Example 19.3 for a description of the operating conditions for this system.) This shows the impact of harmonic loads that may be located at considerable distance from the consumer These should invariably be considered in an harmonic analysis study In operating condition 2, seventh harmonic ST filters on buses and are out of service TDD is slightly above the limits This is acceptable for short-term operation The smaller generator G2 rated at 47.97 MVA has a higher harmonic loading as compared to the larger generator G1 of 82 MVA The impedance modulus shows some interaction between ST filters and capacitors at motors Generally, it is desirable to observe one strategy of reactive power compensation in a distribution system, due to the problem of secondary resonance, discussed in Sec 20.7 The resonant frequency varies by a maximum of 1.2% in the three cases This can be compared to the much wider swings shown in Table 19-8, without filters The resulting harmonic current flows through the system and filters change with switching operation, yet the TDD at the PCC remains within acceptable limits (Table 20-9) Also, the negative sequence loadings of the generators are at safe levels 20.7 SECONDARY RESONANCE In the case where there are secondary circuits which have resonant frequencies close to the switched capacitor bank, the initial surge can trigger oscillations in the Copyright 2002 by Marcel Dekker, Inc All Rights Reserved Harmonic Mitigation and Filters 685 secondary circuits that are much larger than the switched circuit The ratio of these frequencies is given by s fc Lm Cm 20:23ị ẳ fm Ls Cs where fc is the coupled frequency, fm is the main circuit switching frequency, Ls and Cs are the inductance and capacitance in the secondary circuit, and Lm and Cm are the inductance and capacitance in the main circuit Figure 20-10(a) shows the circuit diagram and Fig 20-10(b) shows amplification of transient voltage in multiple capacitor circuits The amplification effect is greater when the natural frequencies of the Figure 20-10 (a) Circuit of a secondary resonance; (b) overvoltages due to secondary resonance (From Ref 4.) Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 686 Chapter 20 two circuits are almost identical Damping ratios of the primary and coupled circuits will effect the degree of interaction between the two circuits [4] 20.8 FILTER REACTORS Filter reactors for filter applications are subjected to high harmonic frequencies A harmonic current flow spectrum, based on the worst case operation, is normally required by a manufacturer for an appropriate design There are two basic forms of construction: (1) air-core reactors and (2) iron-core reactors The reactors can be designed considering higher harmonic losses and air-core types are generally preferred over iron-core reactors The earlier construction of air-insulated reactors, consisting of large conductors restrained in polyester or poured-in concrete has given way to small parallel conductors, epoxy insulated and encapsulated Aircore reactors can be designed to better tolerances as compared to iron-core reactors The change in magnetic material properties can give rise to wider fluctuations in the reactance value in iron-core reactors, though these are designed with lower flux densities, and are smaller in dimensions The reactors must withstand system` through fault symmetrical amperes for sec and also the mechanical stresses brought about by asymmetrical short-circuit currents Dynamic stresses due to switching and transformer inrush currents may have to be considered 20.8.1 Q Factor Apart from its impact on the filter performance, the Q factor determines the fundamental frequency losses and this could be an overriding consideration, especially when the reactors at medium-voltage level are required to be located indoors in metal or fiber-glass enclosures and space is at a premium Consider a second harmonic filter for the furnace installation in Fig 20-9 The capacitor is 0.00128 mF, the inductor is 0.01371 mH, i.e., inductive reactance = 5.1687 ohms A X=R of 50 gives a reactor resistance of 0.1032 ohms The fundamental frequency current is 1280 A This gives a loss of approximately 507 kW/hr (at fundamental), which is very substantial Equation (20.8) defines the filter Q based on the inductive or capacitive reactance at the tuned frequency (these are equal) The fundamental frequency losses and heat dissipation are of major consideration This does not mean that the effect on filter performance can be ignored The higher the value of Q, the more pronounced is the valley at the tuned frequency However, for industrial systems The value of R can be limited to the resistance built in the reactor itself Example 20.4 The effect of change in Q of the filter is examined in this example We have an X=R of 100 at fundamental frequency for the filter reactors in example 20.3 Thus, the resistance values of the reactors are: Fifth harmonic filter at bus = 0.01469 , Fifth harmonic filter at bus = 0.01836 , Seventh harmonic filter at bus and bus = 0.012642 Copyright 2002 by Marcel Dekker, Inc All Rights Reserved Harmonic Mitigation and Filters Table 20-10 Harmonic order 11 13 17 19 23 25 a 687 Effect of Change of Q: Harmonic Simulation, Condition of Example 20.3a Current in generator Current in generator Current into utility’s system (PCC) X/R =100 X/R =7–10 X/R =100 X/R =7–10 X/R =100 X/R =7–10 37 24.4 2.03 3.76 2.83 2.03 1.94 1.75 37.5 25.0 2.03 3.76 2.83 2.03 1.94 1.75 51.7 24.9 25.7 17.6 10.0 7.13 7.71 7.44 54.0 25.7 25.7 17.6 10.0 7.13 7.71 7.44 6.78 4.13 1.24 1.17 0.75 0.54 0.54 0.48 6.90 4.24 1.24 1.17 0.75 0.54 0.54 0.48 ` Harmonic current flow in amperes Figure 20-2 shows that the sharpness of tuning is dependent upon the resistance Harmonic load flow of Example 20-3 is repeated with tuning reactors of X= R ¼ 10 and the results are shown in Table 20-10 There is hardly an appreciable difference in the harmonic current flow In industrial systems the performance of single tuned filters will be, generally, indistinguishable for Q [Eq (20.8)] ¼ 20 to Q ¼ 100: The X/R of tuning reactors at 60-Hz is given by 3.07K0.377 where K is the threephase kVA ¼ Â2 X Â 10À3 (I is the rated current in amperes and X the reactance in ohms) X/R of a 1500 lVA reactor will be 50 while that of a 10MVA reactor it will bo 100 High X/R reactors can be purchased at a cost premium Thus, selection of X/R of the reactor depands upon: Initial capital investment Active energy losses Effectiveness of the filtering The optimization of filter admittance and Q for the impedance angle of the network and are required for the transmission systems The optimum value of Q is given by [4]: ỵ cos m 20:24ị Qẳ 2m sin m where m is the network impedance angle Consider a frequency variation of Ỉ1%, a temperature coefficient of 0.02% per degree Celsius, and a temperature variation of Ỉ308C on the inductors and capacitors, then from Eq (20.13) ¼ 0:006 For an impedance angle m ¼ 808, the optimum Q from Eq (20.24) is 99.31 The higher the tolerances on components and frequency deviation, the lower the value of Q 20.9 DOUBLE-TUNED FILTER A double-tuned filter is derived from two ST filters, and is shown in Fig 20-11 Its R–X plot and Z–! plots are identical to that of two ST filters in parallel, as shown in Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 688 Figure 20-11 Chapter 20 Equivalent circuits of two ST parallel filters and a single double-tuned filter Fig 20-3 The advantage with respect to two ST filters is that the power loss at fundamental frequency is less and one inductor instead of two are subjected to full impulse voltage In Fig 20-12 the BIL (basic insulation level) on reactor L2 is reduced while reactor L1 sees the full impulse voltage This is an advantage in Figure 20-12 Z–! plots of two parallel ST filters and equivalent double-tuned filter (Example 20.5) Copyright 2002 by Marcel Dekker, Inc All Rights Reserved Harmonic Mitigation and Filters 689 high-voltage applications The following equations [5] transform two ST filters of different frequencies into a single double-tuned filter: ð20:25Þ C1 ẳ Ca ỵ Cb La Ca Lb Cb ị2 20:26ị Ca ỵ Cb ị2 La ỵ Lb ị " # " # " # a2 ð1 À x2 Þ À x2 að1 À aÞð1 À x2 Þ R2 ẳ Ra Rb ỵ R1 ỵ aị2 ỵ x2 ị ỵ aị2 ỵ x2 ị ỵ aị2 ỵ x2 ị L2 ẳ 20:27ị Ca Cb Ca ỵ Cb ịLa ỵ Lb ị ð20:28Þ ðLa Ca À Lb Cb Þ2 " # " # " # a2 x4 ð1 À x2 Þ ð1 À x2 Þ ð1 À x2 Þð1 À ax2 Þ R3 ẳ Ra ỵ Rb ỵ R1 ỵ x2 ị1 ỵ ax2 ị ỵ ax2 ị2 ỵ x2 ị ỵ ax2 ị2 ỵ x2 ị C2 ẳ 20:29ị L1 ẳ La Lb La ỵ Lb where C aẳ a Cb 20:30ị s Lb Cb xẳ La Ca 20:31ị Generally, R1 is omitted and R2 and R3 are modified so that the impedance near resonance are practically the same Note that inductor L1 will have some resistance, which is considered in the above equations Example 20.5 Consider the fifth and seventh filter section for a high-voltage application, at 50 Hz, numerical values as shown in Fig 20-11 It is converted into a double-tuned filter and the response is compared with the original ST parallel filters Use of the above equations gives the following numerical values for the filter: C1 ¼ 0:34 mF C2 ¼ 7:931 mF L1 ¼ 0:329 H L2 ¼ 0:039 H R1 ¼ 2:07 R2 ¼ 1:527 R3 ¼ 1:232 The response of the two filters (without external connections) is superimposed in Fig 20-12 20.10 DAMPED FILTERS Figure 20-13 shows four types of damped filters The first-order filter is not used as it has excessive loss at fundamental frequency and requires a large capacitor The second-order high pass is generally used in composite filters for higher frequencies Copyright 2002 by Marcel Dekker, Inc All Rights Reserved 690 Chapter 20 Figure 20-13 Circuits of damped filters: (a) first-order filter; (b) second-order filter; (c) third-order filter; (d) type-C filter If it were to be used for the full spectrum of harmonics, the capacitor size would become large and fundamental frequency losses in the resistor would be of consideration This will be illustrated with an example The filter is more commonly described as a second-order high-pass filter The third-order filter has a substantial reduction in fundamental frequency losses, due to the presence of C2 which increases the filter impedance; C2 is very small compared to C1 The filtering performance of type-C filters lies between that of second- and third-order filters C2 and L2 are series tuned at fundamental frequency and the fundamental frequency loss is reduced Also see Ref [5] Copyright 2002 by Marcel Dekker, Inc All Rights Reserved ... 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. .. 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... 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