Power Systems Handbook Harmonic Generation Effects Propagation and Control Volume 3 Power Systems Handbook Series Author J.C Das Power System Studies, Inc., Snellville, Georgia, USA Volume 1: Short-Circuits in AC and DC Systems: ANSI, IEEE, and IEC Standards Volume 2: Load Flow Optimization and Optimal Power Flow Volume 3: Harmonic Generation Effects Propagation and Control Volume 4: Power System Protective Relaying Harmonic Generation Effects Propagation and Control Volume J.C Das CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2018 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 Printed on acid-free paper International Standard Book Number-13: 978-1-4987-4546-8 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the 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(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 Preface xi Preface to Volume 3: Harmonic Generation Effects Propagation and Control xiii Author xv Harmonics Generation 1.1 Sequence Components of Harmonics .3 1.2 Increases in Nonlinear Loads 1.3 Harmonic Factor 1.3.1 Equations for Common Harmonic Indices 1.4 Three-Phase Windings in Electrical Machines .6 1.4.1 Cogging and Crawling of Induction Motors 1.4.2 Harmonic Induction Torques 1.4.3 Harmonic Synchronous Torques 1.5 Tooth Ripples in Electrical Machines 11 1.6 Synchronous Generators Waveforms 12 1.7 Transformers: Harmonics 13 1.8 Harmonics due to Saturation of Current Transformers 15 1.9 Switching of Shunt Capacitor Banks 18 1.10 Subharmonic Frequencies 18 1.11 Static Power Converters 19 1.11.1 Single-Phase Bridge Circuit 19 1.11.1.1 Phase Control 22 1.11.1.2 Power Factor, Distortion Factor, and Total Power Factor 23 1.11.1.3 Harmonics on Output Side 25 1.11.2 Three-Phase Bridge Circuit 26 1.11.2.1 Cancellation of Harmonics due to Phase Multiplication 31 1.11.2.2 Effect of Source Impedance 31 1.11.2.3 Effect of Output Reactor .34 1.11.2.4 Effect of Load with Back EMF .34 1.11.2.5 Inverter Operation 35 1.11.3 Diode Bridge Converter 35 1.12 Switch Mode Power Supplies 35 1.13 Arc Furnaces 37 1.14 Cycloconverters 39 1.15 Thyristor-Controlled Reactor 41 1.16 Thyristor-Switched Capacitors 42 1.17 Pulse-Width Modulation 43 1.18 Adjustable Speed Drives 44 1.19 Pulse Burst Modulation 46 1.20 Chopper Circuits and Electrical Traction 46 1.21 Slip Frequency Recovery Schemes 48 1.22 Lighting Ballasts 49 1.23 Home Appliances 50 1.24 Voltage Source Converters 51 v vi Contents 1.24.1 Three-Level Converter 52 1.25 Wind Power Generation 54 Problems 56 References 56 Interharmonics and Flicker 59 2.1 Interharmonics 59 2.2 Generation of Interharmonics 59 2.2.1 Imperfect System Conditions 60 2.3 Interharmonics from ASDs 62 2.3.1 Interharmonics from HVDC Systems 63 2.3.1.1 DC Side 63 2.3.1.2 AC Side 64 2.3.2 Cycloconverters .64 2.4 Arc Furnaces .65 2.4.1 Induction Furnaces 68 2.5 Effects of Interharmonics 68 2.6 Reduction of Interharmonics 69 2.7 Flicker 70 2.7.1 Perceptible Limits 70 2.7.2 Planning and Compatibility Levels 71 2.7.3 Flicker Caused by Arcing Loads 72 2.8 Flicker Testing 74 2.9 Control of Flicker 75 2.9.1 STATCOM for Control of Flicker 76 2.10 Tracing of Flicker and Interharmonics 77 2.10.1 IEEE 519, 2014 Revision 78 2.11 Subsynchronous Resonance 79 2.11.1 Series Compensation of Transmission Lines 80 2.11.1.1 Subsynchronous Interharmonics (Subharmonics) 80 2.11.2 SSR Drive Systems 81 Problems 81 References 81 Estimation of Harmonics 85 3.1 Waveform without Ripple Content 86 3.1.1 Harmonic Estimation Using IEEE 519 Equations 88 3.2 Waveform with Ripple Content 88 3.2.1 Graphical Procedure for Estimating Harmonics with Ripple Content 89 3.2.2 Analytical Calculations 93 3.2.3 Effect of DC Reactor 94 3.3 Phase Angle of Harmonics 94 3.4 Measurements of Harmonics 102 3.4.1 Monitoring Duration 103 3.4.2 IEC Standard 6100 4-7 103 3.4.3 Measurement of Interharmonics 104 3.5 Measuring Equipment 105 vii Contents 3.5.1 Specifications of Measuring Instruments 105 3.5.1.1 Accuracy 105 3.5.1.2 Attenuation 106 3.5.1.3 Bandwidth 106 3.5.2 Presentation of Measurement Results 106 3.6 Transducers for Harmonic Measurements 108 3.6.1 Current Transformers 108 3.6.2 Rogowski Coils 109 3.6.3 Voltage Measurements 109 3.7 Characterizing Measured Data 109 3.8 Summation of Harmonic Vectors with Random Angles 112 Problems 113 References 113 Harmonic Resonance, Secondary Resonance, and Composite Resonance 115 4.1 Resonance in Series and Parallel Circuits 115 4.1.1 Series RLC Circuit 115 4.1.2 Parallel RLC Circuit 119 4.2 Practical LC Tank Circuit 120 4.3 Harmonic Resonance 123 4.3.1 Harmonic Resonance in Industrial Power Systems 125 4.3.2 Resonance at Even Harmonics 125 4.3.3 Transmission Systems 125 4.3.4 Elusiveness of Resonance in Power Systems 126 4.3.5 Shifted Resonance Frequencies of ST Filters 127 4.3.6 Switched Capacitors 127 4.3.7 Nearby Harmonic Loads 128 4.3.8 Subsynchronous Resonance Series Compensated Lines 128 4.3.9 Ferroresonance 128 4.4 Secondary Resonance 128 4.5 Multiple Resonances in a Distribution Feeder 130 4.6 Part-Winding Resonance in Transformer Windings 130 4.7 Composite Resonance 134 Problems 135 References 136 Harmonic Distortion Limits 137 5.1 Standards for Harmonic Limits 137 5.1.2 IEEE Standard 519 137 5.2 IEEE 519 Current and Voltage Limits 137 5.3 PCC 140 5.4 Applying IEEE 519 Limits 140 5.5 Time Varying Characteristics of Harmonics 142 5.6 IEC Harmonic Current Emission Limits 144 5.7 Voltage Quality 146 5.7.1 IEEE 519 146 5.7.2 IEC Voltage Distortion Limits 147 5.7.3 Limits on Interharmonics 148 viii Contents 5.8 I EEE 519 (2014) 149 5.8.1 Measurement Window Width 149 5.8.2 Very Short Time Harmonic Measurements 149 5.8.3 Short-Time Harmonic Measurements 149 5.8.4 Statistical Evaluation 150 5.9 Commutation Notches 150 Problems 154 References 154 Effects of Harmonics 155 6.1 Rotating Machines 156 6.1.1 Pulsating Fields and Torsional Vibrations 156 6.1.1.1 Synchronous Machines 157 6.1.2 Subharmonic Frequencies and Subsynchronous Resonance 158 6.1.3 Increase of Losses 159 6.1.4 Effect of Negative Sequence Currents 159 6.1.5 Insulation Stresses 160 6.1.6 Bearing Currents and Shaft Voltages 162 6.1.7 Effect of Cable Type and Length 162 6.2 Transformers 162 6.2.1 Calculations from Transformer Test Data 164 6.2.2 Liquid-Filled Transformers 166 6.2.3 Underwriter’s Laboratories K-Factor of Transformers 168 6.3 Cables 169 6.4 Capacitors 171 6.5 Electromagnetic Interference 174 6.6 Overloading of Neutral 174 6.7 Protective Relays and Meters 175 6.8 Circuit Breakers and Fuses 176 6.9 Telephone Interference Factor 177 Problems 179 References 179 Harmonic Penetrations (Propagation) 181 7.1 Harmonic Analysis Methods 181 7.1.1 Frequency-Domain Analysis 182 7.1.2 Frequency Scan 183 7.1.3 Voltage Scan 184 7.1.4 Phase Angle of Harmonics 184 7.1.5 Newton–Raphson Method 184 7.1.6 Three-Phase Harmonic Load Flow 187 7.1.7 Time Domain Analysis 188 7.1.8 Switching Function 189 7.2 Harmonic Modeling of System Components 190 7.2.1 Transmission Lines 190 7.2.1.1 Transmission Line Equations with Harmonics 191 7.2.2 Underground Cables 192 7.2.3 Filter Reactors 195 Contents ix 7.2.4 Transformers 195 7.2.5 Induction Motors 197 7.2.6 Generators 198 7.3 Load Models 199 7.4 System Impedance 200 7.5 Three-Phase Models 201 7.5.1 Noncharacteristic Harmonics 202 7.5.2 Converters 203 7.6 Modeling of Networks 205 7.6.1 Industrial Systems 205 7.6.2 Distribution Systems 205 7.6.3 Transmission Systems 206 7.6.4 Sensitivity Methods 206 7.7 Power Factor and Reactive Power 208 7.8 Shunt Capacitor Bank Arrangements 211 7.9 Unbalance Detection 216 7.10 Study Cases 217 Problems 235 References 236 Harmonic Mitigation and Filters 237 8.1 Mitigation of Harmonics 237 8.2 Single-Tuned Filters 238 8.2.1 Tuning Frequency 240 8.3 Practical Filter Design 241 8.3.1 Shifted Resonant Frequencies 247 8.3.2 Effect of Tolerances on Filter Components 247 8.3.3 Outage of One of the Parallel Filters 248 8.3.4 Operation with Varying Loads 249 8.3.5 Division of Reactive kvar between Parallel Filter Banks 249 8.3.6 Losses in the Capacitors 250 8.3.7 Harmonic Filter Detuning and Unbalance 250 8.4 Relations in an ST Filter 251 8.4.1 Number of Series and Parallel Groups 255 8.5 Filters for an Arc Furnace 256 8.6 Filters for an Industrial Distribution System 261 8.7 Band Pass Filters 262 8.8 Filter Reactors 263 8.8.1 Q Factor 264 8.9 Double-Tuned Filter 265 8.10 Damped Filter 266 8.10.1 Second-Order High-Pass Filter 268 8.10.2 Type C Filter 269 8.11 Design of a Second-Order High-Pass Filter 270 8.12 Zero Sequence Traps 271 8.13 Limitations of Passive Filters 273 8.14 Active Filters 274 8.14.1 Shunt Connection 275 8.14.2 Series Connection 275 367 Appendix B: Solution to the Problems L = 0.00222 H/mile C = 0.01366 × 10−6 F/mile Z = 403 Ω γ = zy = [(0.14852 + j 0.837)( j 5.15 × 10−6 ]1/2 = (0.184 + j 2.084) × 10−3/miles λ= 2π = 3014 miles β ν = f λ = 1.808 × 105 miles/s fosc = = 3016 Hz(300 miles-line) LC Also Z s = Z c sin h( γl ) = 0.14852 + j 0.837 sin h(0.0552 + j 0.6252) j 5.15 × 10−6 = (405 − j 36)sin h(0.0552 + j 0.6252) sin h(0.0552 + j 0.6252) = sin h(0.0552)cos ( 0.6252 ) + j cos h(0.0552)sin ( 0.6252 ) = ( 0.0552 )( 0.8108) + j (1.0)( 0.5853) = 0.045 + j 0.5853 Then, Z s = (405 − j 36)(0.045 + j 0.5833) = 39.224 + j 234.6 Y p = tanh( γl/2)/Z c tan h( γl/2) = sin h( γl/2) = cos h( γl/2) sin h( γl/2) = sin h(0.0275 + j 0.326) = (0.0275)(0.947) + j (1)(0.32) = 0.026 + j 0.32 cos h( γl/2) = cosh(0.275)(0.947) + j (0.0275)(0.32) = 0.947 + j 0.0088 tan h ( γl/2 ) = 0.031 + j 0.338 Yp = 0.031 + j 0.338 = 2.341 × 10−6 + j 8.348 × 10−4 ≈ j 8.348 × 10−4 405 − j 36 368 Appendix B: Solution to the Problems Then, the line impedance at fundamental frequency is Z s( h )Y p( h ) + Y p( h ) Z s( h )Y p( h ) + 2 = (405 − j 36)( j 8.348 × 10−4 ) + ( j 8.348 × 10−4 )[(405 − j 36)( j 8.348 × 10−4 ) + 2] = 1.03 + j 0.338 −2.822 × 10−4 + 1.695 × 10−3 ) = 98.46 − j 600.36 This shows the procedure: a computer program will run these calculations at close ­increment of frequency to capture the series and shunt resonance frequencies; plots as demonstrated in Figure 7.5 B.8 Solutions to the Problems: Chapter B.8.1 Problem 8.1 THD i = ∑I h=2 I1 I rms = ∑I h = 59.1% = 242.78 A h =1 h I rms = I 1 + THD 2i = 1.162 As the filter is 5 Mvar Xc = kV 13.82 = = 38.09 Ω Mvar As the tuning frequency of the filter is n = 10.65: XL = Xc = 0.336 Ω n2 369 Appendix B: Solution to the Problems The output of the filter is n2 × = 5.044 Mvar n2 − B.8.2 Problem 8.2 As per IEEE capacitor bank ratings, Irms is no more than 1.35 times the fundamental c­ urrent This limit is met as calculated in Problem 8.1 Capacitor rms voltage is within 1.1 pu limit The capacitor voltage is calculated in Table B.7 using the following expression: Vch = I h X c / h We have already calculated Xc in Problem 8.1 Vc,rms = ∑V ch = 13.944 kV h =1 This is 1.0104 times the rated voltage and, therefore, is acceptable The peak capacitor voltage should be within 1.2 pu limit as per IEEE ∑V ch = 15.583 h =1 This is 1.129 times the rated and, therefore, is acceptable TABLE B.7 Problems 8.1 and 8.2 Harmonic Order 11 13 17 19 23 25 29 31 35 37 Harmonic Current (A) Harmonic Voltage (kV) I h (pu) 20 45 100 50 10 12 2 1 0.264 0.424 0.600 0.254 0.038 0.027 0.034 0.010 0.004 0.004 0.002 0.002 0.095 0.214 0.476 0.238 0.047 0.038 0.057 0.019 0.009 0.009 0.005 0.005 370 Voltage Appendix B: Solution to the Problems 0.04 0.02 0.06 0.08 Time (s) (50 Hz basis) FIGURE B.9 Output waveform of a three-level inverter, Problem 8.3 Finally, the kvar limit is 1.365 pu This is calculated as follows: QC ,pu =  I h2 ,pu    = 1.03 pu  h  h =1 ∑ (See Table B.7.) Thus, none of the limits are exceeded B.8.3 Problem 8.3 The operation time in each level is 90° (Figure B.9) The switching angles in a cycle are α1 = 45 α = 135 α = 225 α = 315 371 Appendix B: Solution to the Problems Thus, the harmonics are h3 = sin(3 × 90 / 2) = sin(90 / 2) h5 = sin(5 × 90 / 2) = sin(90 / 2) h7 = Considering harmonics only up to 7th the THD = 43.7% Index Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively A Accuracy classification, of current transformers, 15 of measured data, 109 potential transformer, 109, 109f requirements, of measuring instruments, 105 Active and passive filters, hybrid connections of, 276, 276f Active current shaping, 283 Active filters, 237, 274 combination of, 277–278 hybrid connection, 276–277 series connection, 275–276 shunt connection, 275 Adjoint network, 206, 208f Adjustable speed drives (ASDs), 4, 19, 44–46, 46t, 62, 128, 161f, 263, 283, 284f cycloconverters, 64–65 interharmonics from HVDC systems, 63 AC side, 64 DC side, 63 Admittance matrix, 182–183 Air-gap flux distribution, 11 Amplitude, 44, 332 Apparent power, 23, 210 Arc furnace, 37–39, 65–68, 72 filters for, 256–260 induction furnaces, 68 Armature reaction, 6, 7f ASDs, see Adjustable speed drives (ASDs) Asymmetry, 31, 201 Attenuation, 106, 106t B Back EMF, effect of load with, 34–35, 34f Backward Fourier transform, 332 Band pass filters, 262–263, 262f, 267, 269 Bandwidth, 106, 117, 119, 120 Basic insulation level (BIL), 263, 266 Bearing currents and shaft voltages, 162 Bipolar junction transistors (BJTs), 174, 277 BPA (Bonneville Power Administration), 182 Bridge circuit single-phase, 19, 20f output side, harmonics on, 25–26 phase control, 22–23, 24f power factor, distortion factor, and total power factor, 23–25 three-phase, 26 back EMF, effect of load with, 34–35 inverter operation, 35 output reactor, effect of, 34 phase multiplication, cancellation of harmonics due to, 31 source impedance, effect of, 31–34 C Cables, 169–170 high-pressure fluid-filled cables, 194 low-pressure oil-filled cables, 194 pipe-type cables, 194 type and length, 162 underground, 192–195 XD cables, 194 XLPE cables, 194 Capacitive voltage transformers, 109 Capacitor, 171–174 -clamped multilevel inverters, 286 estimating resonance with, 220 filter capacitors, 262 frequency scan, 220 fundamental loading of, 255 fuseless capacitor banks, 216 harmonic analysis with, 302–306 harmonic analysis without, 298–302 harmonic study with, 220 individual capacitor-can protection, 211, 213 losses in, 250 power capacitors, 18 reactive power of, 269 series capacitors, 76, 80, 81 bank arrangements, 211–216 shunt capacitors, 18, 125 switched capacitors, 127, 128 thyristor-switched, 42, 43f, 76 Cascaded multilevel inverters, 286 373 374 Case rupture, 211, 213, 216 Characteristic harmonics, 19, 30, 54, 63 Chopper circuits and electrical traction, 46–48 CIGRE type-C load model, 199, 199f Circuit breakers and fuses, 176–177 Cogging, 9, 158 Common harmonic indices, equations for, 5–6 Commutation current, 88, 89, 92f Commutation notches, 150–154, 189 Composite resonance, 134–135 Conduction angle, 34, 41f Constant parameter model (CP), 194 Converter, 203–204 cycloconverters, 39–41, 40f, 59, 64–65 line-commutated converters, 25 multilevel converters, 285, 288 six-pulse current source converters, 85, 86, 91f as a source of harmonic generation and harmonic power flow, 26f static power converters, see Static power converters three-level converter, 52 thyristor-based converters, 25 12-pulse thyristor converter, 283, 285f voltage source converters, 51, 52f, 53f three-level converter, 52–54 Core-type transformers, 14, 196, 197f Cross-linked polyethylene (XLPE) cables, 193–194 C-type filter, 260 Current transformers, 17f, 108 harmonics due to saturation of, 15–18 Cycloconverters, 39–41, 40f, 59, 64–65, 282 D Damped filter, 266, 266f, 267 second-order high-pass filter, 268–269 type C filter, 269–270 Delta connection, 214 Delta–delta connection, 27, 31 Delta–wye transformers, 28, 31, 271, 273f, 343 Diode bridge converter, 35, 36t Diode-clamped multilevel inverter, 285 Discrete Fourier transform, 334, 336, 337, 338f, 339 leakage, 337 picket fence effect, 337–339 Displacement power factor, 23, 209, 211 Distortion analyzers, 105 Distortion factor, 23–25, 44 Distortion limits, harmonic, 137 commutation notches, 150–154 Index IEC harmonic current emission limits, 144–146 IEEE Standard 519, 137, 149 applying IEEE 519 limits, 140–142 current and voltage limits, 137–140 measurement window width, 149 short-time harmonic measurements, 149 statistical evaluation, 150 very short time harmonic measurements, 149 point of common coupling (PCC), 140 standards for harmonic limits, 137 time varying characteristics of harmonics, 142–144 voltage quality, 146 IEC voltage distortion limits, 147 IEEE 519, 146 limits on interharmonics, 148–149 Distribution feeder, multiple resonances in, 130 Distribution systems, 115, 126f, 205–206 Disturbing frequency, 63, 352 Double-tuned filter, 265–266, 265f, 266f Double-wye banks, 214, 250 Doubly fed induction generator (DFIG), 306, 308, 310f E Effects of harmonics, 155 cables, 169–170 capacitors, 171–174 circuit breakers and fuses, 176–177 electromagnetic interference, 174 neutral currents, 174–175 protective relays and meters, 175–176 rotating machines, 156 bearing currents and shaft voltages, 162 cable type and length, effect of, 162 increase of losses, 159 insulation stresses, 160–161 negative sequence currents, effect of, 159–160 pulsating fields and torsional vibrations, 156–158 subharmonic frequencies and subsynchronous resonance, 158 telephone interference factor (TIF), 177–179, 179f transformers, 162 liquid-filled transformers, 166–168 transformer test data, calculations from, 164–166 Underwriter’s Laboratories K-factor of transformers, 168–169 375 Index Electrical arc furnaces (EAFs), 59, 65, 65f, 73f harmonic filter configuration to avoid magnification of interharmonics in, 67f installation, 65f spectrum of harmonic and interharmonic emission from, 66f Electrical machines three-phase windings in, cogging and crawling of induction motors, harmonic induction torques, 8–9, 9f harmonic synchronous torques, 9–11, 9f tooth ripples in, 11–12 Electromagnetic interference (EMI), 33, 174, 283 EMTP, 189, 194, 195, 252 EPRI project, 181 Equipment, measuring, 105 presentation of measurement results, 106–108 specifications of measuring instruments, 105 accuracy, 105 attenuation, 106 bandwidth, 106 Estimation of harmonics, 85 measured data, characterizing, 109–111 measurements of harmonics, 102 IEC Standard 6100 4–7, 103–104 measurement of interharmonics, 104–105 monitoring duration, 103 measuring equipment, 105 presentation of measurement results, 106–108 specifications of measuring instruments, 105–106 phase angle, 94–102 ripple content, waveform with, 88 analytical calculations, 93 DC reactor, effect of, 94 graphical procedure, 89–93 ripple content, waveform without, 86 harmonic estimation using IEEE 519 equations, 88 summation of harmonic vectors with random angles, 112–113 transducers for harmonic measurements, 108 current transformers, 108 Rogowski coils, 109 voltage measurements, 109 Even symmetry, 326–327 Exact π model, 194 Expulsion-type fuses, 213 EXT (extraction of fundamental frequency component), 279 Extruded dielectric (XD) cables, 193, 194 F Fast Fourier transform (FFT) algorithm, 189, 339–342 Filter impedance, 239, 267, 268 Filters active filters, 274 combination of, 277–278 hybrid connection, 276–277 series connection, 275–276 shunt connection, 275 for arc furnace, 256–260 band pass filters, 262–263 damped filter, 266 second-order high-pass filter, 268–269 type C filter, 269–270 double-tuned filter, 265–266 filter reactors, 195, 263 Q factor, 264–265 frequency domain, corrections in, 280 harmonic mitigation at source, 282 active current shaping, 283 input reactors to pulse-width modulation ASDs, 283 parallel connected 12-pulse converters with interphase reactor, 283 phase multiplication, 283 for industrial distribution system, 261–262 instantaneous reactive power, 280–282 multilevel inverters, 283–290 passive filters, limitations of, 273–274 practical filter design, 241 division of reactive kvar between parallel filter banks, 249–250 effect of tolerances on filter components, 247–248 harmonic filter detuning and unbalance, 250–251 losses in the capacitors, 250 operation with varying loads, 249 outage of one of the parallel filters, 248–249 shifted resonant frequencies, 247 second-order high-pass filter, design of, 270–271 single-tuned (ST) filters, 238, 238f number of series and parallel groups, 255 relations in, 251 tuning frequency, 240–241 time domain, corrections in, 278–280 zero sequence traps, 271–273 Finite element (FE) method, 193 376 Flicker, 38, 70 caused by arcing loads, 72–74 control of, 75–46 perceptible limits, 70–71 planning and compatibility levels, 71–72 STATCOM for control of, 76, 77f subsynchronous resonance (SSR), 79 drive systems, 81 transmission lines, series compensation of, 80–81 testing, 74–75 tracing of flicker and interharmonics, 77–79 Flicker and interharmonics, tracing of, 77–79 Form factor, 20 Fourier analysis, 1, 6, 23, 27, 323 even symmetry, 326–327 half-wave symmetry, 328 harmonic spectrum, 328–330 odd symmetry, 326 orthogonal functions, 323 periodic functions, 323 Fourier expansion, 25 Fourier integral, 332 Fourier series, 2–3, 28, 349 and coefficients, 323–326 complex form of, 330–332 constructing, from graphs and tables, 328–330 of input current, 349 of square-wave function, 328 Fourier spectrum, 332 Fourier transform, 111, 280, 332–334 backward, 332 discrete, 334 leakage, 337 picket fence effect, 337–339 fast Fourier transform (FFT), 339–342 inverse, 332 Frequency-dependent cable model (FD), 194 Frequency domain, 182–183 corrections in, 280 Frequency scan, 183–184, 193f Fuseless capacitor banks, 216, 250 G Gap flux distribution due to tooth ripples, 12f Gate turn-off thyristors (GTOs), 25, 48, 81 Generator neutrals, 13 Generators, 198–199 synchronous generators, 159 synchronous generators waveforms, 12–13 turbine generator shaft, 80 Index Graphical procedure for estimating harmonics with ripple content, 89–93 Grounded capacitors, 214 Grounding grid, 214–215 H Half power frequencies, 117, 119, 120, 356 Half-wave symmetry, 327, 328 Hall effect probes, 108 Harmonic analysis methods, 181 at 50% generation, 319–321 frequency-domain analysis, 182–183 frequency scan, 183–184 generation, at the maximum, 315–319 Newton–Raphson method, 184–187 phase angle of harmonics, 184 switching function, 189–190 three-phase harmonic load flow, 187–188 time domain analysis, 188–189 voltage scan, 184 Harmonic analyzers, 105 Harmonic distortions, 126 Harmonic factor, common harmonic indices, equations for, 5–6 Harmonic filter detuning and unbalance, 250–251 Harmonic frequencies, 63, 64, 109, 177, 198, 199 resistance at, 199 Harmonic induction torques, 8–9, 9f Harmonic loading, 255 Harmonic losses, defined, 156 Harmonic loss factor, 163 Harmonic magnitude, 34 Harmonic mitigation at source, 282 active current shaping, 283 input reactors to pulse-width modulation ASDs, 283 parallel connected 12-pulse converters with interphase reactor, 283 phase multiplication, 283 Harmonic modeling of system components, 190 filter reactors, 195 generators, 198–199 induction motors, 197–198 transformers, 195–197 transmission lines, 190–192 underground cables, 192–195 Harmonic penetrations, 181 load models, 199–200 networks, modeling of, 205 distribution systems, 205–206 Index industrial systems, 205 sensitivity methods, 206–208 transmission systems, 206 power factor and reactive power, 208–211 shunt capacitor bank arrangements, 211–216 study cases, 217–235 system impedance, 200–201 three-phase models, 201 converters, 203–204 noncharacteristic harmonics, 202–203 unbalance detection, 216 Harmonic power flowchart, 186f Harmonic resonance, 123, 124f elusiveness of resonance in power systems, 126 at even harmonics, 125 in industrial power systems, 125 nearby harmonic loads, 128 shifted resonance frequencies of ST filters, 127 switched capacitors, 127 transmission systems, 125 Harmonic slip, 198 Harmonic spectrum, 62f, 63, 182, 290t, 314f, 328 from an actual adjustable speed drives, 62f, 63 constructing Fourier series from graphs and tables, 328–330 Harmonic synchronous torques, 9–11, 11f Harmonic voltage factor (HVF), 157, 157f H-bridge (HB) converters, 288 multilevel inverter based upon series connections of, 288f High-voltage DC (HVDC) applications, 240 Home appliances, 50–51 hth-order harmonic voltage, HVDC light, 52 HVDC systems, 138, 182 interharmonics from, 63 AC side, 64 DC side, 63 HVDC transmission, 24, 25, 35, 51 Hybrid connection of active and passive filters, 276–277, 276f I IEC 61000-2-2, 143, 147 IEC 61000-3-2, 139, 145t IEC 61000-3-6, 113, 141 IEC 61000-4-7, 149 IEC 61000-4-15, 148 IEC 61000-4-30, 149 377 IEC 61200-2-2, 147t IEC 61200-2-4, 147t IEC harmonic current emission limits, 144–146 IEC voltage distortion limits, 147 IEEE Standard 519, 105, 137, 146, 149 applying IEEE 519 limits, 140–142 current and voltage limits, 137–140 harmonic estimation using IEEE 519 equations, 88 IEEE 519–2014, 148 measurement window width, 149 short-time harmonic measurements, 149 statistical evaluation, 150 very short time harmonic measurements, 149 Incandescent lighting, 1, 50 Index of merit, Individual capacitor-can fusing, 211 Induction furnace, 68 system configuration for, 68f Induction generator effect, 80 Induction motors, 197–198 cogging and crawling of, equivalent circuit of, 197f Industrial distribution system, filters for, 261–262 Industrial power systems, 60, 205 harmonic resonance in, 125 Instantaneous active power, 280, 282 Instantaneous reactive power (IRP), 279, 280–282 Insulated gate bipolar transistors (IGBTs), 160, 174 Insulation stresses, 160–161 Integrated gate bipolar transistors (IGCTs), 25 Interarea mode oscillations, 18 Interharmonics, 54 from adjustable speed drives (ASDs), 62 cycloconverters, 64–65 from HVDC systems, 63–64 arc furnaces, 65 induction furnaces, 68 defined, 59 effects of, 68–69 generation of, 59 imperfect system conditions, 60–61 from HVDC systems, 63 AC side, 64 DC side, 63 limits on, 148–149 measurement of, 104–105 reduction of, 69 Interunit mode oscillations, 18 Inverse Fourier transform, 332 Inverter operation, 35, 162 378 L Leakage, 337 Lighting ballasts, 49, 50t Linear time-invariant loads, Line-commutated converters, 25, 52 Line-commutated inverters, 209 Line-to-line terminal voltage, 12 Liquid-filled transformers, 166–168 LLCL filter, 262 Load-generated harmonic, 115, 123, 124, 127, 155, 171 Load loss, stray loss component of, 164 Load models, 199–200 Local mode oscillations, 18 Low-pressure oil-filled cables, 194 M Magnetic circuits, 1, 81 Magnetic voltage transformers, 109 Measurements of harmonics, 102 IEC Standard 6100 4–7, 103–104 interharmonics, measurement of, 104–105 monitoring duration, 103 Microprocessor based online meters, 105 Mitigation of harmonics, 237 MMFs, 7–8, 12 MOS turn-off thyristors (MTOs), 25 Motor distortion index (MDI), 156 Multilevel converters, 285, 288 Multilevel inverters, 283–290, 288f diode-clamped, 285 operation of, 289t Multiple harmonic sources, 94, 183, 188 Multiple resonances in distribution feeder, 130 Multiple series groups, 214 N National Electric Code (NEC), 175 Nearby harmonic loads, 128 Negative sequence currents, effect of, 159–160 Networks, modeling of, 205 distribution systems, 205–206 industrial systems, 205 sensitivity methods, 206–208 transmission systems, 206 Neutral currents, 174–175, 175f, 271 Neutral point-clamped (NPC) inverter, 285 Newton–Raphson method, 184–187, 186f, 188 NGH-SSR scheme, 158 Nitrogen gas, 194 Index Nodal voltages, vector of, 183 Noncharacteristic harmonics, 19, 40, 54, 85, 139, 188, 202–203 Noninteger harmonics, 3, 59 Nonlinearity, Nonlinear loads, 2, 4, 175, 203 Norton equivalent, 35, 182, 204, 275 Notches, commutation, 33, 150 O Odd symmetry, 326–327 Online measurement of harmonics, 85 Orthogonal functions, 323, 324f Oscillating torques, 158 Output ac power, 19 Output reactor, effect of, 34 Output side, harmonics on, 25–26 Overlap angle, 23, 31, 33f P Parallel connected 12-pulse converters with interphase reactor, 283 Parallel filter banks, division of reactive kvar between, 249–250 Parallel filters, outage of one of, 248–249, 249t Parallel resonance frequency, 262 Parallel RLC circuit, 119–120, 120f Parasitic magnetic fields, Partial weighted harmonic distortion (PWHD) of current, Part-winding resonance, 18 in transformer windings, 130–132 Passive filters, 237 limitations of, 273–274 Peninsula grounding, 214–215 Periodic functions, 323, 324f, 337 Periodic waveform, sinusoidal components of, Phase-angle difference, 112–113 Phase angle of harmonics, 94–102, 184 Phase control, 22–23, 24f Phase multiplication, 283 cancellation of harmonics due to, 31 Picket fence effect, 111, 337–339 Point of common coupling (PCC), 68, 77, 85, 103, 137–138, 140, 211, 237, 249 Potential transformer accuracy, 109f Power factor, 23–25, 27 and reactive power, 208–211 Power loss, 269 Power transformers, 15 Practical LC tank circuit, 120–123 379 Index Protective relays and meters, 175–176 Pulsating fields and torsional vibrations, 156 synchronous machines, 157–158 Pulse burst modulation, 46, 174 Pulse-width modulation (PWM), 25, 43–44, 45f, 54, 161f, 162, 237 Quality factor, 117, 264–265 of tuning reactor, 239 Rotating machines, 156 bearing currents and shaft voltages, 162 cable type and length, effect of, 162 increase of losses, 159 insulation stresses, 160–161 negative sequence currents, effect of, 159–160 pulsating fields and torsional vibrations, 156 synchronous machines, 157–158 subharmonic frequencies and subsynchronous resonance, 158 Rotor resistance at harmonic frequencies, 198 R S Rated steady-state voltage, 263 Ratio correction factor (RCF), 108, 108f Reactive kvar, division of between parallel filter banks, 249–250 Reactive power, 210 of capacitor, 269 power factor and, 208–211 Rectifier–dc link-converter system, 61 Resistance at harmonic frequencies, 198, 199 Resistor, loss in, 269 Resonance, 115 composite, 134–135 distribution feeder, multiple resonances in, 130 harmonic, see Harmonic resonance parallel resonance frequency, 262 part-winding, 18 in transformer windings, 130–132 secondary, 128–130 in series and parallel circuits, 115 parallel RLC circuit, 119–120 series RLC circuit, 115–119 subsynchronous, see Subsynchronous resonance (SSR) Resonant circuits, 118 Ripple coefficient, 89 Ripple content, 94 waveform with, 88 analytical calculations, 93 effect of DC reactor, 94 graphical procedure for estimating harmonics, 89–93 waveform without, 86 harmonic estimation using IEEE 519 equations, 88 Ripple factor, 20–21, 31 Ripple ratio, 89 Rogowski coils, 109 Root mean square (RMS), Secondary resonance, 128–130 Second-order high-pass filter, 268–269, 272f design of, 270–271 Self-contained liquid-filled systems, see Lowpressure oil-filled cables Sensitivity methods, 206–208 Sequence components of harmonics, 3–4 Sequence of harmonics, 4, 4t Series active filters (SAFs), 76, 275 Series and parallel groups, number of, 255 Series connection, 275–276, 288f Series RLC circuit, 115–119 Shaft voltages, bearing currents and, 162 Shielded cable, 214 Shifted resonance frequencies, 247 of single-tuned (ST) filters, 127 Short-circuit voltage depression (SCVD), 73 Shunt capacitor banks arrangements, 211–216 switching of, 18 three-phase connections of, 215f Shunt connection, 275, 275f Simplest harmonic model, 188 Simplified capacitance model of two-winding transformer, 196f Single point grounding, 214 Single-tuned (ST) filter, 238, 238f, 240, 243, 251 flowchart for design of, 253f number of series and parallel groups, 255 shifted resonance frequencies of, 127 ST harmonic filters, 119 tuning frequency, 240–241 Sinusoidal components of periodic waveform, Sinusoidal flux wave, 13 Sinusoidal magnetizing current, 13 Sinusoidal voltage, 1, Six-pulse converter, 34, 35, 176 switching function of, 189f Skew symmetric function, 326 Q 380 Slip frequency recovery schemes, 48–49 Solar inverters, 293, 294f configuration of solar generating plant, 293–295 harmonic analysis with capacitors, 302–306 harmonic analysis without capacitors, 298–302 load flow study, 295–298 Source impedance, effect of, 31–34 Space vector of imaginary power, 281 Spectrum analyzers, 105 Speed–torque curve, 8, 10 Spiral-shaped impedances, 200, 201 Square-wave function, Fourier series of, 328 STATCOM, 54, 66, 312, 313t, 314f, 315t for control of flicker, 76, 77f Static power converters, 19 diode bridge converter, 35 single-phase bridge circuit, 19 distortion factor, 23–25 harmonics on output side, 25–26 phase control, 22–23, 24f power factor, 23–25 total power factor, 23–25 three-phase bridge circuit, 26 cancellation of harmonics due to phase multiplication, 31 effect of load with back EMF, 34–35 inverter operation, 35 output reactor, effect of, 34 source impedance, effect of, 31–34 Static var compensators (SVCs), 42, 75, 76, 77f, 115 Stator resistance, 198 Storage oscilloscope, 105 Stray loss component of load loss, 164 Stresses, insulation, 160–161 Subharmonic frequencies, 18–19, 158 Subsynchronous frequency components, 80 Subsynchronous resonance (SSR), 79, 80, 158 drive systems, 81 transmission lines, series compensation of, 80 subsynchronous interharmonics, 80–81 Switched capacitors, 127 Switching function, 60, 189–190 of six-pulse converter, 189f Switching mode power supplies (SMPSs), 35–37, 37f Symmetrical component transformation, 183, 190 Symmetry, 201 even, 326–327, 327f half-wave, 328 odd, 326, 327f quarter-wave, 280 Synchronous generators, 2, 12–13, 159, 198 Index Synchronous machines, 157–158, 159t System harmonic impedance, 182 System impedance, 49, 77, 123, 200–201 T Tank circuit, 121 Tank resonant circuit, 121f Telephone interference factor (TIF), 177–179, 179f Third-order filter, 266f, 267 Three-level converter, 52 Three-phase connections of shunt capacitor banks, 215f Three-phase core-type transformers, 14 Three-phase harmonic load flow, 187–188 Three-phase models, 201 converters, 203–204 noncharacteristic harmonics, 202–203 Three-phase windings in electrical machines, cogging and crawling of induction motors, 8 harmonic induction torques, 8–9, 9f harmonic synchronous torques, 9–11, 9f Thyristor, 22, 25 Thyristor-based converters, 25 Thyristor controlled capacitor circuits, 42 Thyristor controlled reactor (TCR), 41–42, 41f, 256 Thyristor switched capacitors (TSCs), 42–43, 43f, 76 Thévenin/Norton equivalent, 204 Time domain analysis, 188–189 corrections in, 278–280 Time varying characteristics of harmonics, 142–144 Tolerances on capacitors and reactors, 247–248 Tooth ripples in electrical machines, 11–12 gap flux distribution due to, 12f and slotting, Torque ripples, 158 Torsional vibrations, pulsating fields and, 156 synchronous machines, 157–158 Total demand distortion (TDD), 6, 103, 138, 139, 143, 144f, 243, 245 Total harmonic distortion (THD), 5, 263, 315 Total harmonic loading, 269 Total harmonic voltage distortion (THD), 140, 146 Total interharmonic distortion (THID) factor, 69 Total power factor, 23–25, 211 Total subharmonic distortion factor (TSHD), 69 Transducers for harmonic measurements, 108 current transformers, 108 Rogowski coils, 109 voltage measurements, 109 381 Index Transfer impedance, 206 Transformers, 162, 195–197 calculations from transformer test data, 164–166 delta–wye transformers, 271, 273f harmonics in, 13–15 liquid-filled transformers, 166–168 magnetic voltage transformers, 109 part-winding resonance in, 130–133 and rotating machines, saturation of current transformers, 15–18 Underwriter’s Laboratories K-factor of transformers, 168–169 zigzag transformers, 271, 273f Transformer windings, part-winding resonance in, 130–132 Transient torques, 80 Transmission lines, 190 series compensation of, 80 subsynchronous interharmonics, 80–81 transmission line equations with harmonics, 191–192 Transmission systems, 115, 125, 206 Trapezoidal current waveform, 89f Triangular-wave method, 279 Triplen harmonics, 3–4, 23 Tuned angular frequency, 239 Tuning frequency, 240–241 effect of, 243 Tuning reactor, quality factor of, 239 Two-winding transformer, simplified capacitance model of, 196f Type C filter, 66, 67f, 260, 260t, 267, 269–270 U Unbalance detection, 216, 250 Underground cables, 192–195 Underwriter’s Laboratories K-factor of transformers, 168–169 Unsymmetrical waveforms, 4, 86f Voltage notch, 32, 32f depth of, 33 Voltage quality, 146, 237 IEC voltage distortion limits, 147 IEEE 519, 146 interharmonics, limits on, 148–149 Voltage scan, 184 Voltage source converters, 51, 53f principle of, 52f three-level converter, 52–54 Voltage source inverters (VSIs), 35, 44, 48, 277, 278f Voltage THD harmonic histogram of, 107f probability histogram of, 107f W Wave analyzers, 105 Waveform, 52 synchronous generators, 12–13 with ripple content, 88 analytical calculations, 93 effect of DC reactor, 94 graphical procedure for estimating harmonics, 89–93 without ripple content, 86 harmonic estimation using IEEE 519 equations, 88 of various disturbing loads, 141f Weighting factors, 140–141, 142t Wideband cable model, 194 Wind power generation, 54–55, 307 Wind power plant (WPP) harmonic analysis in, 306, 311–321 configuration, 308–310, 309f load flow study, 310–311 load flow results in, 311t WTG (Wind Turbine Generator), 308 Wye–wye connected transformers, 14, 15f, 31 Z V Voltage measurements, 103, 109 Zero sequence traps, 234, 271–273 Zigzag transformers, 271, 273f ... sinusoidal Tooth ripples and slotting may produce forward and reverse rotating harmonics Magnetic circuits Harmonic Generation Effects Propagation and Control can saturate and generate harmonics As an... characteristic and noncharacteristic harmonics Interharmonics and flicker Resonance, secondary resonance, and harmonic resonance Effects of harmonics Limitations of harmonics according to the IEEE and IEC... Short-Circuits in AC and DC Systems: ANSI, IEEE, and IEC Standards Volume 2: Load Flow Optimization and Optimal Power Flow Volume 3: Harmonic Generation Effects Propagation and Control Volume 4: