Power systems electromagnetic transients simulation

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Accurate knowledge of electromagnetic power system transients is crucial to the operation of an economic, efficient and environmentally-friendly power system network, without compromising on the reliability and quality of the electrical power supply. Simulation has become a universal tool for the analysis of power system electromagnetic transients and yet is rarely covered in-depth in undergraduate programmes. It is likely to become core material in future courses.

The primary objective of this book is to describe the application of efficient computational techniques to the solution of electromagnetic transient problems in systems of any size and topology, involving linear and nonlinear components. The text provides an in-depth knowledge of the different techniques that can be employed to simulate the electromagnetic transients associated with the various components within a power system network, setting up mathematical models and comparing different models for accuracy, computational requirements, etc.

Written primarily for advanced electrical engineering students, the text includes basic examples to clarify difficult concepts. Considering the present lack of training in this area, many practising power engineers, in all aspects of the power industry, will find the book of immense value in their professional work.

computational requirements, etc.

Author Notes

Neville R. Watson received BE(Hons.) and PhD degrees in 1984 and 1988, respectively, from the University of Canterbury, New Zealand, where he is now a senior lecturer. He is co-author of three other books, has contributed several chapters to a number of edited books and has been published in nearly 120 other publications
Jos Arrillage received PhD and DSc degrees in 1966 and 1980, respectively, from UMIST, Manchester, UK, where he led the Power Systems Group between 1970 and 1974. Since 1975, he has been a Professor of Electrical Engineering at the University of Canterbury, New Zealand. He is the author of five other books, several book chapters and about 300 other publications. He is a Fellow of the IEE, the IEEE and the Academy of Sciences and Royal Society of New Zealand. He was the recipient of the 1997 Uno Lamm medal for his contributions to HVDC transmission

List of figures	p. xiii
List of tables	p. xxi
Preface	p. xxiii
Acronyms and constants	p. xxv
1 Definitions, objectives and background	p. 1
1.1 Introduction	p. 1
1.2 Classification of electromagnetic transients	p. 3
1.3 Transient simulators	p. 4
1.4 Digital simulation	p. 5
1.4.1 State variable analysis	p. 5
1.4.2 Method of difference equations	p. 5
1.5 Historical perspective	p. 6
1.6 Range of applications	p. 9
1.7 References	p. 9
2 Analysis of continuous and discrete systems	p. 11
2.1 Introduction	p. 11
2.2 Continuous systems	p. 11
2.2.1 State variable formulations	p. 13
2.2.1.1 Successive differentiation	p. 13
2.2.1.2 Controller canonical form	p. 14
2.2.1.3 Observer canonical form	p. 16
2.2.1.4 Diagonal canonical form	p. 18
2.2.1.5 Uniqueness of formulation	p. 19
2.2.1.6 Example	p. 20
2.2.2 Time domain solution of state equations	p. 20
2.2.3 Digital simulation of continuous systems	p. 22
2.2.3.1 Example	p. 27
2.3 Discrete systems	p. 30
2.4 Relationship of continuous and discrete domains	p. 32
2.5 Summary	p. 34
2.6 References	p. 34
3 State variable analysis	p. 35
3.1 Introduction	p. 35
3.2 Choice of state variables	p. 35
3.3 Formation of the state equations	p. 37
3.3.1 The transform method	p. 37
3.3.2 The graph method	p. 40
3.4 Solution procedure	p. 43
3.5 Transient converter simulation (TCS)	p. 44
3.5.1 Per unit system	p. 45
3.5.2 Network equations	p. 46
3.5.3 Structure of TCS	p. 49
3.5.4 Valve switchings	p. 51
3.5.5 Effect of automatic time step adjustments	p. 53
3.5.6 TCS converter control	p. 55
3.6 Example	p. 59
3.7 Summary	p. 64
3.8 References	p. 65
4 Numerical integrator substitution	p. 67
4.1 Introduction	p. 67
4.2 Discretisation of R, L, C elements	p. 68
4.2.1 Resistance	p. 68
4.2.2 Inductance	p. 68
4.2.3 Capacitance	p. 70
4.2.4 Components reduction	p. 71
4.3 Dual Norton model of the transmission line	p. 73
4.4 Network solution	p. 76
4.4.1 Network solution with switches	p. 79
4.4.2 Example: voltage step applied to RL load	p. 80
4.5 Non-linear or time varying parameters	p. 88
4.5.1 Current source representation	p. 89
4.5.2 Compensation method	p. 89
4.5.3 Piecewise linear method	p. 91
4.6 Subsystems	p. 92
4.7 Sparsity and optimal ordering	p. 95
4.8 Numerical errors and instabilities	p. 97
4.9 Summary	p. 97
4.10 References	p. 98
5 The root-matching method	p. 99
5.1 Introduction	p. 99
5.2 Exponential form of the difference equation	p. 99
5.3 z-domain representation of difference equations	p. 102
5.4 Implementation in EMTP algorithm	p. 105
5.5 Family of exponential forms of the difference equation	p. 112
5.5.1 Step response	p. 114
5.5.2 Steady-state response	p. 116
5.5.3 Frequency response	p. 117
5.6 Example	p. 118
5.7 Summary	p. 120
5.8 References	p. 121
6 Transmission lines and cables	p. 123
6.1 Introduction	p. 123
6.2 Bergeron's model	p. 124
6.2.1 Multiconductor transmission lines	p. 126
6.3 Frequency-dependent transmission lines	p. 130
6.3.1 Frequency to time domain transformation	p. 132
6.3.2 Phase domain model	p. 136
6.4 Overhead transmission line parameters	p. 137
6.4.1 Bundled subconductors	p. 140
6.4.2 Earth wires	p. 142
6.5 Underground cable parameters	p. 142
6.6 Example	p. 146
6.7 Summary	p. 156
6.8 References	p. 156
7 Transformers and rotating plant	p. 159
7.1 Introduction	p. 159
7.2 Basic transformer model	p. 160
7.2.1 Numerical implementation	p. 161
7.2.2 Parameters derivation	p. 162
7.2.3 Modelling of non-linearities	p. 164
7.3 Advanced transformer models	p. 165
7.3.1 Single-phase UMEC model	p. 166
7.3.1.1 UMEC Norton equivalent	p. 169
7.3.2 UMEC implementation in PSCAD/EMTDC	p. 171
7.3.3 Three-limb three-phase UMEC	p. 172
7.3.4 Fast transient models	p. 176
7.4 The synchronous machine	p. 176
7.4.1 Electromagnetic model	p. 177
7.4.2 Electromechanical model	p. 183
7.4.2.1 Per unit system	p. 184
7.4.2.2 Multimass representation	p. 184
7.4.3 Interfacing machine to network	p. 185
7.4.4 Types of rotating machine available	p. 189
7.5 Summary	p. 190
7.6 References	p. 191
8 Control and protection	p. 193
8.1 Introduction	p. 193
8.2 Transient analysis of control systems (TACS)	p. 194
8.3 Control modelling in PSCAD/EMTDC	p. 195
8.3.1 Example	p. 198
8.4 Modelling of protective systems	p. 205
8.4.1 Transducers	p. 205
8.4.2 Electromechanical relays	p. 208
8.4.3 Electronic relays	p. 209
8.4.4 Microprocessor-based relays	p. 209
8.4.5 Circuit breakers	p. 210
8.4.6 Surge arresters	p. 211
8.5 Summary	p. 213
8.6 References	p. 214
9 Power electronic systems	p. 217
9.1 Introduction	p. 217
9.2 Valve representation in EMTDC	p. 217
9.3 Placement and location of switching instants	p. 219
9.4 Spikes and numerical oscillations (chatter)	p. 220
9.4.1 Interpolation and chatter removal	p. 222
9.5 HVDC converters	p. 230
9.6 Example of HVDC simulation	p. 233
9.7 FACTS devices	p. 233
9.7.1 The static VAr compensator	p. 233
9.7.2 The static compensator (STATCOM)	p. 241
9.8 State variable models	p. 243
9.8.1 EMTDC/TCS interface implementation	p. 244
9.8.2 Control system representation	p. 248
9.9 Summary	p. 248
9.10 References	p. 249
10 Frequency dependent network equivalents	p. 251
10.1 Introduction	p. 251
10.2 Position of FDNE	p. 252
10.3 Extent of system to be reduced	p. 252
10.4 Frequency range	p. 253
10.5 System frequency response	p. 253
10.5.1 Frequency domain identification	p. 253
10.5.1.1 Time domain analysis	p. 255
10.5.1.2 Frequency domain analysis	p. 257
10.5.2 Time domain identification	p. 262
10.6 Fitting of model parameters	p. 262
10.6.1 RLC networks	p. 262
10.6.2 Rational function	p. 263
10.6.2.1 Error and figure of merit	p. 265
10.7 Model implementation	p. 266
10.8 Examples	p. 267
10.9 Summary	p. 275
10.10 References	p. 275
11 Steady state applications	p. 277
11.1 Introduction	p. 277
11.2 Initialisation	p. 278
11.3 Harmonic assessment	p. 278
11.4 Phase-dependent impedance of non-linear device	p. 279
11.5 The time domain in an ancillary capacity	p. 281
11.5.1 Iterative solution for time invariant non-linear components	p. 282
11.5.2 Iterative solution for general non-linear components	p. 284
11.5.3 Acceleration techniques	p. 285
11.6 The time domain in the primary role	p. 286
11.6.1 Basic time domain algorithm	p. 286
11.6.2 Time step	p. 286
11.6.3 DC system representation	p. 287
11.6.4 AC system representation	p. 287
11.7 Voltage sags	p. 288
11.7.1 Examples	p. 290
11.8 Voltage fluctuations	p. 292
11.8.1 Modelling of flicker penetration	p. 294
11.9 Voltage notching	p. 296
11.9.1 Example	p. 297
11.10 Discussion	p. 297
11.11 References	p. 300
12 Mixed time-frame simulation	p. 303
12.1 Introduction	p. 303
12.2 Description of the hybrid algorithm	p. 304
12.2.1 Individual program modifications	p. 307
12.2.2 Data flow	p. 307
12.3 TS/EMTDC interface	p. 307
12.3.1 Equivalent impedances	p. 308
12.3.2 Equivalent sources	p. 310
12.3.3 Phase and sequence data conversions	p. 310
12.3.4 Interface variables derivation	p. 311
12.4 EMTDC to TS data transfer	p. 313
12.4.1 Data extraction from converter waveforms	p. 313
12.5 Interaction protocol	p. 313
12.6 Interface location	p. 316
12.7 Test system and results	p. 317
12.8 Discussion	p. 319
12.9 References	p. 319
13 Transient simulation in real time	p. 321
13.1 Introduction	p. 321
13.2 Simulation with dedicated architectures	p. 322
13.2.1 Hardware	p. 323
13.2.2 RTDS applications	p. 325
13.3 Real-time implementation on standard computers	p. 327
13.3.1 Example of real-time test	p. 329
13.4 Summary	p. 330
13.5 References	p. 331
A Structure of the PSCAD/EMTDC program	p. 333
A.1 References	p. 340
B System identification techniques	p. 341
B.1 s-domain identification (frequency domain)	p. 341
B.2 z-domain identification (frequency domain)	p. 343
B.3 z-domain identification (time domain)	p. 345
B.4 Prony analysis	p. 346
B.5 Recursive least-squares curve-fitting algorithm	p. 348
B.6 References	p. 350
C Numerical integration	p. 351
C.1 Review of classical methods	p. 351
C.2 Truncation error of integration formulae	p. 354
C.3 Stability of integration methods	p. 356
C.4 References	p. 357
D Test systems data	p. 359
D.1 CIGRE HVDC benchmark model	p. 359
D.2 Lower South Island (New Zealand) system	p. 359
D.3 Reference	p. 365
E Developing difference equations	p. 367
E.1 Root-matching technique applied to a first order lag function	p. 367
E.2 Root-matching technique applied to a first order differential pole function	p. 368
E.3 Difference equation by bilinear transformation for RL series branch	p. 369
E.4 Difference equation by numerical integrator substitution for RL series branch	p. 369
F MATLAB code examples	p. 373
F.1 Voltage step on RL branch	p. 373
F.2 Diode fed RL branch	p. 374
F.3 General version of example F.2	p. 376
F.4 Frequency response of difference equations	p. 384
G FORTRAN code for state variable analysis	p. 389
G.1 State variable analysis program	p. 389
H FORTRAN code for EMT simulation	p. 395
H.1 DC source, switch and RL load	p. 395
H.2 General EMT program for d.c. source, switch and RL load	p. 397
H.3 AC source diode and RL load	p. 400
H.4 Simple lossless transmission line	p. 402
H.5 Bergeron transmission line	p. 404
H.6 Frequency-dependent transmission line	p. 407
H.7 Utility subroutines for transmission line programs	p. 413
Index	p. 417

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