Handbook of Power System Engineering

Yoshihide Hase

I C E l t T E N N I A U

I C E N T E N N I A L

John Wiley & Sons, Ltd

Contents

PREFACE xix

ACKNOWLEDGEMENTS xxi

ABOUT THE AUTHOR xxiii

INTRODUCTION xxv

1 OVERHEAD TRANSMISSION LINES AND THEIR CIRCUIT CONSTANTS 1

1.1 Overhead Transmission Lines with LR Constants 1 1.1.1 Three-phase single circuit line without overhead grounding wire 1 1.1.2 Three-phase single circuit line with OGW, OPGW 8 1.1.3 Three-phase double circuit line with LR constants 9

1.2 Stray Capacitance of Overhead Transmission Lines 10 1.2.1 Stray capacitance of three-phase single circuit line 10 1.2.2 Three-phase single circuit line with OGW 16 1.2.3 Three-phase double circuit line 16

1.3 Supplement: Additional Explanation for Equation 1.27 17

CofFee break 1: Electricity, its substance and methodology 19

2 SYMMETRICAL COORDINATE METHOD (SYMMETRICAL COMPONENTS) 21

2.1 Fundamental Concept of Symmetrical Components 21 2.2 Definition of Symmetrical Components 23

2.2.1 Definition 23 2.2.2 Implication of symmetrical components 25

2.3 Conversion of Three-phase Circuit into Symmetrical Coordinated Circuit 26 2.4 Transmission Lines by Symmetrical Components 28

2.4.1 Single circuit line with LR constants 28 2.4.2 Double circuit line with LR constants 30 2.4.3 Single circuit line with stray capacitance C 33 2.4.4 Double circuit line with C constants 36

2.5 Typical Transmission Line Constants 38 2.5.1 Typical line constants 38 2.5.2 L, C constant values derived from typical travelling-wave

velocity and surge impedance 40

viii CONTENTS

2.6 Generator by Symmetrical Components (Easy Description) 41 2.6.1 Simplified symmetrical equations 41 2.6.2 Reactance of generator 43

2.7 Description of Three-phase Load Circuit by Symmetrical Components 44

3 FAULT ANALYSIS BY SYMMETRICAL COMPONENTS 4 5

3.1 Fundamental Concept of Symmetrical Coordinate Method 45 3.2 Line-to-ground Fault (Phase a to Ground Fault: 1 <frG) 46

3.2.1 Condition before the fault 47 3.2.2 Condition of phase a to ground fault 48 3.2.3 Voltages and currents at Virtual terminal point f in the

0 -1 -2 domain 48 3.2.4 Voltages and currents at an arbitrary point

under fault conditions 49 3.2.5 Fault under no-load conditions 50

3.3 Fault Analysis at Various Fault Modes 51 3.4 Conductor Opening 51

3.4.1 Single phase (phase a) conductor opening 51 3.4.2 Two-phases (phase b, c) conductor opening 57

Coffee break 2: Dawn of the world of electricity, from Coulomb to Ampere and Ohm 58

4 FAULT ANALYSIS OF PARALLEL CIRCUIT LINES

(INCLUDING SIMULTANEOUS DOUBLE CIRCUIT FAULT) 61

4.1 Two-phase Circuit and its Symmetrical Coordinate Method 61 4.1.1 Definition and meaning 61 4.1.2 Transformation process of double circuit line 63

4.2 Double Circuit Line by Two-phase Symmetrical Transformation 65 4.2.1 Transformation of typical two-phase circuits 65 4.2.2 Transformation of double circuit line 67

4.3 Fault Analysis of Double Circuit Line (General Process) 69 4.4 Single Circuit Fault on the Double Circuit Line 70

4.4.1 Line-to-ground fault (1 0G) on one side circuit 70 4.4.2 Various one-side circuit faults 73

4.5 Double Circuit Fault at Single Point f 73 4.5.1 Circuit 1 phase a line-to-ground fault and circuit 2 phases b

and c line-to-line faults at point f 73 4.5.2 Circuit 1 phase a line-to-ground fault and circuit 2 phase b

line-to-ground fault at point f (method 1) 74 4.5.3 Circuit 1 phase a line-to-ground fault and circuit 2 phase b

line-to-ground fault at point f (method 2) 75 4.5.4 Various double circuit faults at single point f 77

4.6 Simultaneous Double Circuit Faults at Different Points f, F on the Same Line 77 4.6.1 Circuit condition before fault 77 4.6.2 Circuit 1 phase a line-to-ground fault and circuit 2 phase b

line-to-ground fault at different points f, F 80 4.6.3 Various double circuit faults at different points 81

IX

5 PER UNIT METHOD A N D INTRODUCTION OF TRANSFORMER CIRCUIT 83

5.1 Fundamental Concept of the PU Method 83 5.1.1 PU method of Single phase circuit 84 5.1.2 Unitization of a Single phase three-winding transformer

and its equivalent circuit 85 5.2 PU Method for Three-phase Circuits 89

5.2.1 Base quantities by PU method for three-phase circuits 89 5.2.2 Unitization of three-phase circuit equations 90

5.3 Three-phase Three-winding Transformer, its Symmetrical Components Equations and the Equivalent Circuit 91 5.3.1 X — X — A-connected three-phase transformer 91 5.3.2 Three-phase transformers with various winding connections 97 5.3.3 Core structure and the zero-sequence excitation impedance 97 5.3.4 Various winding methods and the effect of delta windings 97 5.3.5 Harmonie frequency voltages/currents in the 0 -1 -2 domain 100

5.4 Base Quantity Modification of Unitized Impedance 101 5.4.1 Note on % IZ of three-winding transformer 102

5.5 Autotransformer 102 5.6 Numerical Example to Find the Unitized Symmetrical

Equivalent Circuit 104 5.7 Supplement: Transformation from Equation 5.18 to Equation 5.19 114

Coffee break 3: Faraday and Henry, the discoverers of the principle of electric energy application 11 6

6 The a-ß-0 COORDINATE METHOD (CLARKE COMPONENTS)

A N D ITS APPLICATION 119

6.1 Definition of cc-ß-0 Coordinate Method [a-ß-0 Components) 119 6.2 Interrelation Between cc-ß-0 Components and Symmetrical Components 120

6.2.1 The transformation of arbitrary waveform quantities 122 6.2.2 Interrelation between a-ß-0 and symmetrical components 123

6.3 Circuit Equation and Impedance by the a-ß-0 Coordinate Method 125 6.4 Three-phase Circuit in a-ß-0 Components 126

6.4.1 Single circuit transmission line 126 6.4.2 Double circuit transmission line 127 6.4.3 Generator 129 6.4.4 Transformer impedances and load impedances in the

a-ß-0 domain 130 6.5 Fault Analysis by a-ß-0 Components 131

6.5.1 The b-c phase line to ground fault 131 6.5.2 Other mode short-cireuit faults 133 6.5.3 Open-conductor mode faults 133

7 SYMMETRICAL A N D a-jS-0 COMPONENTS AS ANALYTICAL TOOLS

FOR TRANSIENT PH E N O M E N A 135

7.1 The Symbolic Method and its Application to Transient Phenomena 135 7.2 Transient Analysis by Symmetrical and a-ß-0 Components 137 7.3 Comparison of Transient Analysis by Symmetrical and a-/?-0 Components 138

Coffee break 4: Weber and other pioneers 141

CONTENTS

8 NEUTRAL G R O U N D I N G METHODS 143

8.1 Comparison of Neutral Grounding Methods 143 8.2 Overvoltages on the Unfaulted Phases Caused by

a Line-to-ground Fault 148 8.3 Possibility of Voltage Resonance 149 8.4 Supplement: Arc-suppression Coil (Petersen Coil)

Neutral Grounded Method 150

9 VISUAL VECTOR DIAGRAMS OF VOLTAGES A N D CURRENTS UNDER

FAULT CONDITIONS 151

9.1 Three-phase Fault: 34>S, 3</>G (Solidly Neutral Grounding System, High-resistive Neutral Grounding System) 151

9.2 Phase b-c Fault: 2<j)S (for Solidly Neutral Grounding System, High-resistive Neutral Grounding System) 152

9.3 Phase a to Ground Fault: ](j>G (Solidly Neutral Grounding System) 155 9.4 Double Line-to-ground (Phases b and c) Fault: 20G

(Solidly Neutral Grounding System) 157 9.5 Phase a Line-to-ground Fault: 1 <j)G (High-resistive Neutral

Grounding System) 160 9.6 Double Line-to-ground (Phases b and c) Fault: 2</>G (High-resistive

Neutral Grounding System) 162

Coffee break 5: Maxwell, the greatest scientist of the nineteenth Century 164

10 THEORY OF GENERATORS 169

10.1 Mathematical Description of a Synchronous Generator 169 10.1.1 The fundamental model 169 10.1.2 Fundamental three-phase circuit equations 171 10.1.3 Characteristics of inductances in the equations 173

10.2 Introduction of d -q-0 Method (d-q-0 Components) 177 10.2.1 Definition of d -q -0 method 177 10.2.2 Mutual relation of d -q -0 , a-b-c and 0 -1 -2 domains 179 10.2.3 Characteristics of d -q -0 domain quantities 179

10.3 Transformation of Generator Equations from a-b-c to d -q -0 Domain 1 81 10.3.1 Transformation of generator equations to d -q -0 domain 181 10.3.2 Unitization of generator d -q-0 domain equations 184 10.3.3 Introduction of d -q -0 domain equivalent circuits 188

10.4 Generator Operating Characteristics and it's Vector Diagrams on d-and q-axes piain 1 90

10.5 Transient Phenomena and the Generator's Transient Reactances 194 10.5.1 Initial condition just before sudden change 194 10.5.2 Assorted d-axis and q-axis reactances for transient phenomena 195

10.6 Symmetrical Equivalent Circuits of Generators 196 10.6.1 Positive-sequence circuit 197 10.6.2 Negative-sequence circuit 199 10.6.3 Zero-sequence circuit 202

10.7 Laplace-transformed Generator Equations and the Time Constants 202 10.7.1 Laplace-transformed equations 202

10.8 Relations Between the d -q -0 and a-ß-0 Domains 206

CONTENTS X I

10.9 Detailed Calculation of Generator Short-circuit Transient Current under Load Operation 206 10.9.1 Transient fault current by sudden three-phase terminal fault

under no-load condition 211 10.10 Supplement 1: The Equations of the Rational Function and

Their Transformation into Expanded Sub-sequential Fractional Equations 211 10.11 Supplement 2: Calculation of the Coefficients of Equation 10.120 212

10.11.1 Calculation of Equation (\)k}, k2, k3, k4lS, k4l - 8 212 10.11.2 Calculation of equation 10.120 Q) k5, k6, k7L ±S7 214

10.12 Supplement 3: The Formulae of the Laplace Transform 214

11 APPARENT POWER A N D ITS EXPRESSION IN THE 0 - 1 - 2

A N D d - q - 0 D O M A I N S 215

11.1 Apparent Power and its Symbolic Expression for Arbitrary Waveform Voltages and Currents 215 11.1.1 Definition of apparent power 215 11.1.2 Expansion of apparent power for arbitrary waveform

voltages and currents 217 11.2 Apparent Power of a Three-phase Circuit in the 0 -1 -2 Domain 217 11.3 Apparent Power in the d -q -0 Domain 220

Coffee break 6: Hertz, the discoverer and inventor of radio waves 222

12 GENERATING POWER A N D STEADY-STATE STABILITY 223

12.1 Generating Power and the P-ö and Q-d Curves 223 12.2 Power Transfer Limit between a Generator and Power System Network 226

1 2.2.1 Equivalency between one-machine to infinite-bus System and two-machine System 226

12.2.2 Apparent power of a generator 227 12.2.3 Power transfer limit of a generator (steady-state stability) 228 1 2.2.4 Visual description of generator's apparent power transfer limit 229

12.3 Supplement: Derivation of Equation 12.17 231

13 THE GENERATOR AS ROTATING MACHINERY 233

13.1 Mechanical (Kinetic) Power and Generating (Electrical) Power 233 1 3.1.1 Mutual relation between mechanical input power and

electrical Output power 233 13.2 Kinetic Equation of the Generator 235

1 3.2.1 Dynamic characteristics of the generator (kinetic motion equation) 235

13.2.2 Dynamic equation of generator as an electrical expression 237 13.2.3 Speed governors, the rotating speed control equipment for

generators 237

Coffee break 7: Heaviside, the great benefactor of electrical engineering 241

14 TRANSIENT/DYNAMIC STABILITY, P-Q-V CHARACTERISTICS

A N D VOLTAGE STABILITY OF A POWER SYSTEM 245

14.1 Steady-state Stability, Transient Stability, Dynamic Stability 245 14.1.1 Steady-state stability 245

CONTENTS

14.1.2 Transient-state stability 245 14.1.3 Dynamic stability 246

14.2 Mechanical Acceleration Equation for the Two-generator System, and Disturbance Response 246

14.3 Transient Stability and Dynamic Stability (Case Study) 247 14.3.1 Transient stability 248 14.3.2 Dynamic stability 249

14.4 Four-terminal Circuit and the P-3 Curve under Fault Conditions 250 14.4.1 Circuit 1 251 14.4.2 Circuit 2 252 14.4.3 Trial calculation under assumption of xi = xi = x, x^ = x'2 — x' 253

14.5 P-Q-V Characteristics and Voltage Stability (Voltage Instability Phenomena) 254 14.5.1 Apparent power at sending terminal and receiving terminal 254 14.5.2 Voltage sensitivity by small disturbance AP, AQ 255 14.5.3 Circle diagram of apparent power 256 14.5.4 P-Q-V characteristics, and P-V and Q-V curves 257 14.5.5 P-Q-V characteristics and voltage instability phenomena 258

14.6 Supplement 1: Derivation of Equation 14.20 from Equation 14.19 262 14.7 Supplement 2: Derivation of Equation 14.30 from Equation 14.18 (2) 262

15 GENERATOR CHARACTERISTICS WITH AVR A N D STABLE

OPERATION LIMIT 263

15.1 Theory of AVR, and Transfer Function of Generator System with AVR 263 15.1.1 Inherent transfer function of generator 263 15.1.2 Transfer function of generator + load 265

15.2 Duties of AVR and Transfer Function of Generator + AVR 267 15.3 Response Characteristics of Total System and

Generator Operational Limit 270 15.3.1 Introduction of s functions for AVR + exciter + generator + load 270 15.3.2 Generator operational limit and its p-q coordinate expression 272

15.4 Transmission Line Charging by Generator with AVR 274 15.4.1 Line charging by generator without AVR 275 15.4.2 Line charging by generator with AVR 275

15.5 Supplement 1: Derivation of Equation 15.9 from Equations 15.7 and 15.8 275 15.6 Supplement 2: Derivation of Equation 15.10 from

Equations 15.8 and 15.9 276

Coffee break 8: The symbolic method by complex numbers and Arthur Kennelly, the prominent pioneer 277

16 OPERATING CHARACTERISTICS A N D THE CAPABILITY

LIMITS OF GENERATORS 279

16.1 General Equations of Generators in Terms of p-q Coordinates 279 16.2 Rating Items and the Capability Curve of the Generator 282

16.2.1 Rating items and capability curve 282 16.2.2 Generator's locus in the p-q coordinate plane under various

operating conditions 285

XIII

16.3 Leading Power-factor (Under-excitation Domain) Operation, and UEL Function by AVR 287 16.3.1 Generator as reactive power generator 287 16.3.2 Overheating of Stator core end by leading power-factor

Operation (low excitation) 289 16.3.3 UEL (under-excitation limit) protection by AVR 292 16.3.4 Operation in the over-excitation domain 293

16.4 V - Q (Voltage and Reactive Power) Control by AVR 293 16.4.1 Reactive power distribution for multiple generators and

cross-current control 293 16.4.2 P-f control and V-Q control 295

16.5 Thermal Generators' Weak Points (Negative-sequence Current, Higher Harmonie Current, Shaft-torsional Distortion) 296 16.5.1 Features of large generators today 296 16.5.2 The thermal generator: smaller /2-withstanding capability 297 16.5.3 Rotor overheating caused by d.c. and higher harmonic

currents 299 16.5.4 Transient torsional twisting torque of TG coupled shaft 302

16.6 General Description of Modern Thermal/Nuclear TG Unit 305 16.6.1 Steam turbine (ST) unit for thermal generation 305 16.6.2 Combined cycle System with gas/steam turbines 306 16.6.3 ST unit for nuclear generation 309

16.7 Supplement: Derivation of Equation 16.14 310

17 R-X COORDINATES A N D THE THEORY OF DIRECTIONAL

DISTANCE RELAYS 313

17.1 Protective Relays, Their Mission and Classification 313 1 7.1.1 Duties of protective relays 314 17.1.2 Classification of major relays 314

17.2 Principle of Directional Distance Relays and R-X Coordinates Plane 315 17.2.1 Fundamental funetion of directional distance relays 315 17.2.2 R-X coordinates and their relation to P-Q coordinates and p-q

coordinates 316 17.2.3 Characteristics of DZ-Rys 317

17.3 Impedance Locus in R-X Coordinates in Case of a Fault (under No-Ioad Condition) 318 17.3.1 Operation of DZ(S)-Ry for phase b-c line-to-line fault (2</>S) 318 17.3.2 Response of DZ(G)-Ry to phase a line-to-ground fault (1 <pG) 321

17.4 Impedance Locus under Normal States and Step-out Condition 325 17.4.1 R-X locus under stable and unstable conditions 325 17.4.2 Step-out detection and trip-lock of DZ-Rys 328

17.5 Impedance Locus under Faults with Load Flow Conditions 329 17.6 Loss of Excitation Detection by DZ-Rys 330

17.6.1 Loss of excitation detection 330 1 7.7 Supplement 1: The Drawing Method for the Locus Z = A / ( l — keis)

of Equation 17.22 332 17.7.1 The locus for the case S: constant, k: 0 to 00 332 17.7.2 The locus for the case k: constant, S: 0 to 360° 332

17.8 Supplement 2: The Drawing Method for Z = 1/(1 / Ä + 1 /B) of Equation 17.24 334

xiv CONTENTS

Coffee break 9: Steinmetz, prominent benefactor of circuit theory and high-voltage technology 335

18 TRAVELLING-WAVE (SURGE) PHENOMENA 339

1 8.1 Theory of Travelling-wave Phenomena along Transmission Lines (Distributed-constants Circuit) 339 1 8.1.1 Waveform equation of a transmission line

(overhead line and cable) and the image of a travelling wave 339 18.1.2 The general Solution for voltage and current

by Laplace transforms 345 18.1.3 Four-terminal network equation between two arbitrary points 346 18.1.4 Examination of line constants 348

1 8.2 Approximation of Distributed-constants Circuit and Accuracy of Concentrated-constants Circuit 349

18.3 Behaviour of Travelling Wave at a Transition Point 351 18.3.1 Incident, transmitted and reflected waves at a transition point 351 1 8.3.2 Behaviour of voltage and current travelling waves at typical

transition points 352 18.4 Behaviour of Travelling Waves at a Lightning-strike Point 354 18.5 Travelling-wave Phenomena of Three-phase Transmission Line 356

18.5.1 Surge impedance of three-phase line 356 18.5.2 Surge analysis by symmetrical coordinates

(lightning strike on phase a conductor) 357 18.6 Line-to-ground and Line-to-line Travelling Waves 358 18.7 The Reflection Lattice and Transient Behaviour Modes 361

18.7.1 The reflection lattice 361 18.7.2 Oscillatory and non-oscillatory convergence 362

18.8 Supplement 1: General Solution Equation 18.10 for Differential Equation 18.9 362

18.9 Supplement 2: Derivation of Equation 18.19 from Equation 18.18 363

19 SWITCHING SURGE PHENOMENA BY CIRCUIT-BREAKERS

A N D LINE SWITCHES 365

19.1 Transient Caiculation of a Single Phase Circuit by Breaker Opening 365 1 9.1.1 Caiculation of fault current tripping (single phase circuit) 365 19.1.2 Caiculation of current tripping (double power source circuit) 369

19.2 Caiculation of Transient Recovery Voltages Across a Breaker's Three Poles by 3</>S Fault Tripping 374 19.2.1 Recovery voltage appearing at the first phase (pole) tripping 375 19.2.2 Transient recovery voltage across a breaker's three poles

by 3</>S fault tripping 377 19.3 Fundamental Concepts of High-voltage Circuit-breakers 384

19.3.1 Fundamental concept of breakers 384 19.3.2 Terminology of switching phenomena and breaker

tripping capability 385 1 9.4 Actual Current Tripping Phenomena by Circuit-breakers 387

19.4.1 Short-circuit current (lagging power-factor current) tripping 387 19.4.2 Leading power-factor small-current tripping 389 19.4.3 Short-distance line fault tripping (SLF) 394

CONTENTS x v

19.4.4 Current chopping phenomena by tripping small current with lagging power factor 394

19.4.5 Step-out tripping 396 19.4.6 Current-zero missing 396

19.5 Overvoltages Caused by Breaker Closing (Close-switching Surge) 397 19.5.1 Principlesof overvoltage caused by breaker closing 397

1 9.6 Resistive Tripping and Resistive Closing by Circuit-breakers 399 19.6.1 Resistive tripping and closing 399 19.6.2 Overvoltage phenomena caused by tripping of breaker

with resistive tripping mechanism 401 19.6.3 Overvoltage phenomena caused by closing of breaker

with resistive closing mechanism 403 19.7 Switching Surge Caused by Line Switches (Disconnecting Switches) 406

19.7.1 LS switching surge: the mechanism appearing 407 19.8 Supplement 1: Caiculation of the Coefficients k\ — kÄ of Equation 19.6 408 19.9 Supplement 2: Caiculation of the Coefficients k\ — k(, of Equation 19.17 408

Coffee break 10: Fortescue's symmetrical components 409

20 OVERVOLTAGE PHENOMENA 411

20.1 Classification of Overvoltage Phenomena 411 20.2 Fundamental (Power) Frequency Overvoltages

(Non-resonant Phenomena) 411 20.2.1 Ferranti effect 411 20.2.2 Self-excitation of a generator 413 20.2.3 Sudden load tripping or load failure 414 20.2.4 Overvoltages of unfaulted phases by one line-to-ground fault 415

20.3 Lower Frequency Harmonie Resonant Overvoltages 415 20.3.1 Broad-area resonant phenomena

(lower order frequency resonance) 415 20.3.2 Local area resonant phenomena 417 20.3.3 Interrupted ground fault of cable line in a neutral

ungrounded distribution System 419 20.4 Switching Surges 419

20.4.1 Overvoltages caused by breaker closing (breaker closing surge) 420 20.4.2 Overvoltages caused by breaker tripping (breaker tripping surge) 421 20.4.3 Switching surge by line Switches 421

20.5 Overvoltage Ph enomena by Lightning Strikes 421 20.5.1 Direct strike on phase conduetors (direct flashover) 422 20.5.2 Direct strike on O G W or tower strueture (inverse flashover) 422 20.5.3 Induced strokes (electrostatic induced strokes,

electromagnetic induced strokes) 423

21 INSULATION COORDINAT ION 4 2 5

21.1 Overvoltages as Insulation Stresses 425 21.1.1 Conduction and insulation 425 21.1.2 Classification of overvoltages 426

21.2 Fundamental Concept of Insulation Coordination 431 21.2.1 Concept of insulation coordination 431 21.2.2 Specific principles of insulation strength and breakdown 431

xvi CONTENTS

21.3 Countermeasures on Transmission Lines to Reduce Overvoltages and Flashover 432 21.3.1 Countermeasures 433

21.4 Overvoltage Protection at Substations 436 21.4.1 Surge protection by metal-oxide surge arresters 436 21.4.2 Separation effects of Station arresters 441 21.4.3 Station protection by OGWs, and grounding resistance reduction 443

21.5 Insulation Coordination Details 446 21.5.1 Definition and some principal matters of Standards 446 21.5.2 Insulation configuration 447 21.5.3 Insulation withstanding level and BIL, BSL 449 21.5.4 Standard insulation levels and the principles 450 21.5.5 Comparison of insulation levels for Systems under

and over 245 kV 450 21.6 Transfer Surge Voltages Through the Transformer, and

Generator Protection 456 21.6.1 Electrostatic transfer surge voltage 456 21.6.2 Generator protection against transfer surge voltages through

transformer 464 21.6.3 Electromagnetic transfer voltage 465

21.7 Internal High-frequency Voltage Oscillation of Transformers Caused by Incident Surge 465 21.7.1 Equivalent circuit of transformer in EHF domain 465 21.7.2 Transient oscillatory voltages caused by incident surge 465 21.7.3 Reduction of internal oscillatory voltages 470

21.8 Oil-filled Transformers Versus Gas-filled Transformers 471 21.9 Supplement: Proof that Equation 21.21 is the Solution of Equation 21.20 473

Coffee break 11: Edith Clarke, the prominent woman electrician 474

22 W A V E F O R M DISTORTION A N D LOWER ORDER HARMONIC

RESONANCE 4 7 5

22.1 Causes and Influences of Waveform Distortion 475 22.1.1 Classification of waveform distortion 475 22.1.2 Causes of waveform distortion 477

22.2 Fault Current Waveform Distortion Caused on Cable Lines 478 22.2.1 Introduction of transient current equation 478 22.2.2 Evaluation of the transient fault current 481 22.2.3 Waveform distortion and protective relays 484

23 POWER CABLES 4 8 5

23.1 Power Cables and Their General Features 485 23.1.1 Classification 485 23.1.2 Unique features and requirements of power cables 488

23.2 Circuit Constants of Power Cables 491 23.2.1 Inductances of cables 491 23.2.2 Capacitance and surge impedance of cables 495

23.3 Metallic Sheath and Outer Covering 498 23.3.1 Role of metallic sheath and outer covering 498 23.3.2 Metallic sheath earthing methods 499

CONTENTS xvii

23.4 Cross-bonding Metallic-shielding Method 500 23.4.1 Cross-bonding method 500 23.4.2 Surge voltage analysis on the cable sheath circuit

and jointing boxes 501 23.5 Surge Voltages Arising on Phase Conductors and Sheath Circuits 504

23.5.1 Surge voltages at the cable connecting point m 504 23.5.2 Surge voltages at the cable terminal end point n 506

23.6 Surge Voltages on Overhead Line and Cable Combined Networks 507 23.6.1 Overvoltage behaviour on cable line caused by lightning

surge from overhead line 507 23.6.2 Switching surges arising on cable line 508

23.7 Surge Voltages at Cable End Terminal Connected to GIS 509

Coffee break 1 2: Park's equations, the birth of the d -q -0 method 512

24 APPROACHES FOR SPECIAL CIRCUITS 513

24.1 On-Ioad Tap-changing Transformer (LTC Transformer) 513 24.2 Phase-shifting Transformer 515

24.2.1 Introduction of fundamental equations 516 24.2.2 Application for loop circuit line 518

24.3 Woodbridge Transformer and Scott Transformer 519 24.3.1 Woodbridge winding transformer 519 24.3.2 Scott winding transformer 522

24.4 Neutral Grounding Transformer 522 24.5 Mis-connection of Three-phase Orders 524

24.5.1 Cases 524

Coffee break 13: Power System engineering and insulation coordination 529

APPENDIX A - MATHEMATICAL FORMULAE 531

APPENDIX B - MATRIX EQUATION FORMULAE 533

ANALYTICAL METHODS INDEX 539

COMPONENTS INDEX 541

SUBJECT INDEX 545