POWER SYSTEM TRANSIENTS Theory and Applications AKIH1RO AMETANI NAOTO NAGAOKA YOSHIHIRO BABA TERUO OHNO CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business
Contents Introduction List of Symbols List of Acronyms International Standards xv xix xxi xxiii 1. Theory of Distributed-Parameter Circuits and the Impedance/Admittance Formulas 1 1.1 Introduction 1 1.2 Impedance and Admittance Formula 2 1.2.1 Conductor Internal Impedance Z, 3 1.2.1.1 Derivation of an Approximate Formula 3 1.2.1.2 Accurate Formula by Schelkunoff 6 1.2.2 Outer-Media Impedance Z0 8 1.2.2.1 Outer-Media Impedance 8 1.2.2.2 Overhead Conductor 9 1.2.2.3 Pollaczek's General Formula for Overhead, Underground, and Overhead/Underground Conductor Systems 14 1.2.3 Problems 16 1.3 Basic Theory of Distributed-Parameter Circuit 17 1.3.1 Partial Differential Equations of Voltages and Currents 17 1.3.2 General Solutions of Voltages and Currents 18 1.3.2.1 Sinusoidal Excitation 18 1.3.2.2 Lossless Line 21 and Currents on a Semi-Infinite Line 23 1.3.3 Voltages 1.3.3.1 Solutions of Voltages and Currents 23 1.3.3.2 Waveforms of Voltages and Currents 24 1.3.3.3 Phase Velocity 25 1.3.3.4 Traveling Wave 27 1.3.3.5 Wave Length 28 1.3.4 Propagation Constants and Characteristic Impedance 28 1.3.4.1 Propagation Constants 28 1.3.4.2 Characteristic Impedance 31 1.3.5 Voltages and Currents on a Finite Line 32 1.3.5.1 Short-Circuited Line 32 1.3.5.2 Open-Circuited Line 35 1.3.6 Problems 38 v
vi Contents 1.4 Multiconductor System 38 1.4.1 Steady-State Solutions 38 1.4.2 Modal Theory 41 1.4.2.1 Eigenvalue Theory 41 1.4.2.2 Modal Theory 44 1.4.2.3 Current Mode 45 1.4.2.4 Parameters in Modal Domain 46 1.4.3 Two-Port Circuit Theory and Boundary Conditions 48 1.4.3.1 Four-Terminal Parameter 48 1.4.3.2 Impedance/Admittance Parameters 50 1.4.4 Modal Distribution of Multiphase Voltages and Currents 52 1.4.4.1 Transformation Matrix 52 1.4.4.2 Modal Distribution 53 1.4.5 Problems 55 1.5 Frequency-Dependent Effect 56 1.5.1 Frequency Dependence of Impedance 56 1.5.2 Frequency-Dependent Parameters 58 1.5.2.1 Frequency-Dependent Effect 58 1.5.2.2 Propagation Constant 59 1.5.2.3 Characteristic Impedance 61 1.5.2.4 Transformation Matrix 63 1.5.2.5 Line Parameters in the Extreme Case 68 1.5.3 Time Response 70 1.5.3.1 Time-Dependent Responses 70 1.5.3.2 Propagation Constant: Step Response 71 1.5.3.3 Characteristic Impedance 72 1.5.3.4 Transformation Matrix 74 1.5.4 Problems 77 1.6 Traveling Wave 77 1.6.1 Reflection and Refraction Coefficients 77 1.6.2 Thevenin's Theorem 79 1.6.2.1 Equivalent Circuit of a Semi-Infinite Line 79 1.6.2.2 Voltage and Current Sources at the End 79 Sending 1.6.2.3 Boundary Condition at the Receiving End 79 1.6.2.4 Thevenin's Theorem 82 1.6.3 Multiple Reflection 84 1.6.4 Multiconductors 88 1.6.4.1 Reflection and Refraction Coefficients 88 1.6.4.2 Lossless Two Conductors 88 1.6.4.3 Consideration of Modal Propagation 1.6.4.4 Consideration of Losses in a Velocities 91 Two-Conductor System 96 1.6.4.5 Three-Conductor System 99
Contents vii 1.6.4.6 Cascaded System Composed of the Different Numbers of Conductors 102 1.6.5 Problems 103 1.7 Nonuniform Conductors 104 1.7.1 Characteristic of Nonuniform Conductors 105 1.7.1.1 Nonuniform Conductor 105 1.7.1.2 Difference from Uniform Conductors 108 1.7.2 Impedance and Admittance Formulas 109 1.7.2.1 Finite-Length Horizontal Conductor 109 1.7.2.2 Vertical Conductor 112 1.7.3 Line Parameters 114 1.7.3.1 Finite Horizontal Conductor 114 1.7.3.2 Vertical Conductor 117 1.7.3.3 Nonparallel Conductor 119 1.7.4 Problems 119 1.8 Introduction of EMTP 122 1.8.1 Introduction 122 1.8.1.1 History of a Transient Analysis 122 1.8.1.2 Background of EMTP 123 1.8.1.3 EMTP Development 124 1.8.2 Basic Theory of EMTP 124 1.8.2.1 Representation of a Circuit Element by a Current Source and a Resistance 126 1.8.2.2 Composition of Nodal Conductance 128 1.8.3 Other Circuit Elements 129 1.8.4 Solutions of the Problems 131 References 136 2. Transients on Overhead Lines 141 2.1 Introduction 141 2.2 Switching Surge on Overhead Line 142 2.2.1 Basic Mechanism of Switching Surge 142 2.2.2 Basic Parameters Influencing Switching Surge 143 2.2.2.1 Source Circuit 143 2.2.2.2 Switch 146 2.2.2.3 Transformer 147 2.2.2.4 Transmission Line 147 2.2.3 Switching Surges in Practice 148 2.2.3.1 Classification of Switching Surges 148 2.2.3.2 Basic Characteristic of Closing Surge: Field Test Results 149 2.2.3.3 Closing Surge on a Single-Phase Line 151 2.2.3.4 Closing Surges on a Multiphase 2.2.3.5 Effect of Various Parameters on Line 153 Closing Surge 162
viii Contents 2.3 Fault Surge 166 2.3.1 Fault Initiation Surge 166 2.3.2 Characteristic of a Fault Initiation Surge 169 2.3.2.1 Effect of Line Transposition 169 2.3.2.2 Overvoltage Distribution 169 2.3.3 Fault-Clearing Surge 172 2.4 Lightning Surge 175 2.4.1 Mechanism of Lightning Surge 2.4.2 Modeling 2.4.2.1 Lightning Generation 177 of Circuit Elements 179 Current 179 2.4.2.2 Tower and Gantry 180 2.4.2.3 Tower Footing Impedance 182 2.4.2.4 Arc Horn 184 2.4.2.5 Transmission Line 185 2.4.2.6 Substation 185 2.4.3 Lightning Surge Overvoltage 185 2.4.3.1 Model Circuit 185 2.4.3.2 Lightning Surge Overvoltage 187 2.4.3.3 Effect of Various Parameters 188 2.5 Theoretical Analysis of Transients: Hand Calculations 194 2.5.1 Switching Surge on an Overhead Line 195 2.5.1.1 Traveling Wave Theory 195 2.5.1.2 Lumped-Parameter Equivalent with Laplace Transform 202 2.5.2 Fault Surge 206 2.5.3 Lightning Surge 208 2.5.3.1 Tower Top Voltage 208 2.5.3.2 Two-Phase Model 208 2.5.3.3 No Back Flashover 210 2.5.3.4 Case of a Back Flashover 212 2.5.3.5 Consideration of Substation 212 2.6 Frequency-Domain Method of Transient Simulations 215 2.6.1 Introduction 215 2.6.2 Numerical Fourier/Laplace Transform 215 2.6.2.1 Finite Fourier Transform 215 2.6.2.2 Shift of Integral Path: Laplace Transform 217 2.6.2.3 Numerical Laplace Transform: Discrete Transform 218 Laplace 2.6.2.4 Odd-Number Sampling: Accuracy Improvement 218 2.6.2.5 Application of FFT: Fast Laplace Transform (FLT) 221 2.6.3 Transient Simulation 228 2.6.3.1 Definition of Variables 228
Contents ix 2.6.3.2 Subroutine to Prepare F(co) 229 2.6.3.3 Subroutine FLT 230 2.6.3.4 Remarks of the Frequency-Domain Method 230 References 230 3. Transients on Cable Systems 233 3.1 Introduction 233 3.2 Impedance and Admittance of Cable Systems 234 3.2.1 Single-Phase Cable 234 3.2.1.1 Cable Structure 234 3.2.1.2 Impedance and Admittance 234 3.2.2 Sheath Bonding 235 Model of a Cross-Bonded Cable 238 3.2.3 Homogeneous 3.2.3.1 Homogeneous Impedance and Admittance 238 3.2.3.2 Reduction of Sheath 243 3.2.4 Theoretical Formula of Sequence Currents 246 3.2.4.1 Cross-Bonded Cable 246 3.2.4.2 Solidly Bonded Cable 251 3.3 Wave Propagation and Overvoltages 256 3.3.1 Single-Phase Cable 256 3.3.1.1 Propagation Constant 256 3.3.1.2 Example of Transient Analysis 258 3.3.2 Wave Propagation Characteristic 260 3.3.2.1 Impedance: R, L 263 3.3.2.2 Capacitance: C 264 3.3.2.3 Transformation Matrix 264 3.3.2.4 Attenuation Constant and Propagation Velocity 264 3.3.3 Transient Voltage 265 3.3.4 Limitation of the Sheath Voltage 269 3.3.5 Installation of SVLs 271 3.4 Studies on Recent and Planned EHV AC Cable Projects 272 3.4.1 Recent Cable Projects 273 3.4.2 Planned Cable Projects 275 3.5 Cable System Design and Equipment Selection 277 3.5.1 Study Flow 277 Phenomenon 278 3.5.2 Zero-Missing 3.5.2.1 Sequential Switching 280 3.5.3 Leading Current Interruption 281 3.5.4 Cable Discharge 284 3.6 EMTP Simulation Test Cases 285 References 287
X Contents 4. Transient and Dynamic Characteristics of New Energy Systems... 291 4.1 Wind Farm 291 4.1.1 Model Circuit of Wind Farm 291 4.1.2 Steady-State Analysis 294 4.1.2.1 Cable Model 294 4.1.2.2 Charging Current 298 4.1.2.3 Load-Flow Calculation 301 4.1.3 Transient Calculation 303 4.2 Power-Electronics Simulation by EMTP 306 4.2.1 Simple-Switching Circuit 306 4.2.2 Switching-Transistor Model 307 4.2.2.1 Simple-Switch Model 308 4.2.2.2 Switch with Delay Model 312 4.2.3 MOSFET 314 4.2.3.1 Simple Model 315 4.2.3.2 Modified Switching Device Model 316 4.2.3.3 Simulation Circuit and Results 321 4.2.4 Thermal Calculation 329 4.3 Voltage Regulation Equipment Using Battery in a DC Railway System 331 4.3.1 Introduction 331 4.3.2 Feeding Circuit 333 4.3.3 Measured and Calculated Results 336 4.3.3.1 Measured Results 336 4.3.3.2 Calculated Results of Conventional System 336 4.3.3.3 Calculated Results with Power Compensator 340 4.4 Concluding Remarks 343 References 344 5. Numerical Electromagnetic Analysis Methods and Their Applications to Transient Analyses 345 5.1 Fundamentals 345 5.1.1 Maxwell's Equations 345 5.1.2 Finite-Difference Time-Domain Method 346 5.1.3 Method of Moments 355 5.2 Applications 363 5.2.1 Grounding Electrodes 363 5.2.2 Transmission Towers 367 5.2.3 Distribution Lines: Lightning-Induced Surges 371 5.2.4 Transmission Lines: Propagation of Lightning Surges in the Presence of Corona 375 5.2.5 Power Cables: Propagation of Power Line Communication Signals 379
Contents xi 5.2.6 Air-Insulated Substations 385 5.2.7 Wind Turbine Generator Towers 387 References 389 6. Electromagnetic Disturbances in Power Systems and Customers 393 6.1 Introduction 393 6.2 Disturbances in Power Stations and Substations 394 6.2.1 Statistical Data of Disturbances 394 6.2.1.1 Overall Data 394 6.2.1.2 Disturbed Equipments 395 6.2.1.3 Surge Incoming Route 397 6.2.2 Characteristics of Disturbances 397 6.2.2.1 Characteristics of Lightning Surge Disturbances 397 6.2.2.2 Characteristics of Switching Surge Disturbances 398 6.2.2.3 Switching Surge in DC Circuits 402 6.2.3 Influence, Countermeasures, and Costs of Disturbances 403 6.2.3.1 Influence of Disturbances on Power System Operation 403 6.2.3.2 Countermeasures Carried Out 405 6.2.3.3 Cost of Countermeasures 406 6.2.4 Case Studies 407 6.2.4.1 Case No. 1 408 6.2.4.2 Case No. 2 410 6.2.4.3 Case No. 3 411 6.2.5 Concluding Remarks 412 6.3 Disturbances in Customers and Home Appliances 413 6.3.1 Statistical Data of Disturbances 413 6.3.2 Breakdown Voltage of Home Appliances 415 6.3.2.1 Testing Voltage 415 6.3.2.2 Breakdown Test 416 6.3.3 Surge Voltages and Currents into Customers due to Nearby Lightning 416 6.3.3.1 Introduction 416 6.3.3.2 Model Circuits for Experiments and EMTP Simulations 417 6.3.3.3 Experimental and Simulation Results 425 6.3.3.4 Concluding Remarks 429 6.3.4 Lightning Surge Incoming from a Communication Line 429 6.3.4.1 Introduction 429
xii Contents 6.3.4.2 Protective Device 430 6.3.4.3 Lightning Surge 430 6.3.4.4 Concluding 6.4 Analytical Method of Solving Induced Remarks 433 Voltages and Currents 435 6.4.1 Introduction 435 6.4.2 F-Parameter Formulation for Induced and Currents 439 Voltages 6.4.2.1 Formulation of F-Parameter 439 6.4.2.2 Approximation of F-Parameters 440 6.4.2.3 Cascaded Connection of Pipelines 440 6.4.3 Application Examples 441 6.4.3.1 Single Section Terminated by R, and R2 441 6.4.3.2 Two-Cascaded Sections of a Pipeline (Problem 6.1) 446 6.4.3.3 Three-Cascaded Sections of a Pipeline 453 6.4.4 Comparison with a Field-Test Result 454 6.4.4.1 Comparison with EMTP Simulations 454 6.4.4.2 Field-Test Result 454 6.4.5 Concluding Remarks 459 Solution of Problem 6.1 460 Appendix 6.A.1 Test Voltage for Low-Voltage Control Circuits in Power Stations and Substations (JEC-0103-2004) 461 6.A.2 Traveling Wave Solution 464 6.A.3 Boundary Conditions and Solutions of a Voltage and a Current 464 6.A.4 Approximate Formulas for Impedance and Admittance 465 6.A.5 Accurate Solutions for Two-Cascaded Sections 466 References 467 7. Problems and Application Limits of Numerical Simulations 471 7.1 Problems of Existing Impedance Formulas Used in Circuit Theory-Based Approaches 471 7.1.1 Earth-Return Impedance 471 7.1.1.1 Carson's Impedance 471 7.1.1.2 Basic Assumption of the Impedance 472 7.1.1.3 Nonparallel Conductor 472 7.1.1.4 Stratified Earth 473 7.1.1.5 Earth Resistivity and Permittivity 473 7.1.2 Internal Impedance 473 7.1.2.1 Schelkunoff's Impedance 473 7.1.2.2 Arbitrary Cross-Section Conductor 473
Contents xiii 7.1.2.3 Semiconducting Layer of Cable 474 7.1.2.4 Proximity Effect 474 7.1.3 Earth-Return Admittance 474 7.2 Existing Problems in Circuit Theory-Based Numerical Analysis 475 7.2.1 Reliability of a Simulation Tool 475 7.2.2 Assumption and Limit of a Simulation Tool 475 7.2.3 Input Data 476 for Power 7.3 Numerical Electromagnetic Analysis System Transients 476 References 477 Index 481