Table of Contents
Preface v
List of figures xiv
List of tables xxiv
Contributors xxvii
Part I Modelling
1 Telegrapher's equations for field-to-transmission line interaction 3
1.1 Transmission line approximation 3
1.2 Single-wire line above a perfectly conducting ground 5
1.2.1 Taylor, Satterwhite and Harrison model 6
1.2.2 Agrawal, Price and Gurbaxani model 10
1.2.3 Rachidi model 12
1.3 Contribution of the different electromagnetic field components 13
1.4 Inclusion of losses 13
1.5 Case of multi-conductor line 15
1.6 Time-domain representation of the coupling equations 17
1.7 Solutions with particular reference to time-domain numerical solutions 18
1.8 Application of theory to the case of lightning-induced voltages on distribution overhead lines 22
1.8.1 The LIOV code 22
1.8.2 The LIOV-EMTP-RV code 23
1.8.3 LEMP response of electrical distribution systems 29
1.9 Summary and concluding remarks 39
Acknowledgements 40
Bibliography 40
2 An affine arithmetic-based methodology for uncertain power flow and optimal power flow analyses 45
2.1 Introduction 45
2.2 Overview of existing approaches 46
2.2.1 Sampling methods 46
2.2.2 Analytical methods 47
2.2.3 Approximate methods 48
2.2.4 Non-probabilistic methods 49
2.2.5 AA-based methods 50
2.3 Mathematical background 51
2.3.1 PF analysis 51
2.3.2 OPF analysis 52
2.4 Self-validated computing 54
2.4.1 Interval arithmetic 54
2.4.2 Affine arithmetic 56
2.5 AA-based PF and OPF analyses 58
2.5.1 Theoretical framework 59
2.5.2 Applications 64
2.6 Numerical results 65
2.6.1 PF analysis 66
2.6.2 Reactive power dispatch 68
2.7 Computational requirements 69
2.8 Conclusions 71
Bibliography 73
3 DFT-based synchrophasor estimation processes for Phasor Measurement Units applications: algorithms definition and performance analysis 77
3.1 Literature review 78
3.2 Definitions 79
3.2.1 Signal model 79
3.2.2 Phasor 81
3.2.3 Synchrophasor 81
3.2.4 Frequency and rate of change of frequency 83
3.2.5 Phasor measurement unit 84
3.3 The discrete Fourier transform 86
3.3.1 From the Fourier transform to the DFT 86
3.3.2 DFT interpretation and relevant properties 87
3.3.3 DFT effects 89
3.3.4 DFT parameters 95
3.3.5 DFT calculation in real time 98
3.4 D FT-based SE algorithms 102
3.4.1 The Interpolated-DFT technique 103
3.4.2 The iterative-Interpolated DFT technique 110
3.5 Performance analysis of SE algorithm 112
3.5.1 The IEEE Std.C37.118 112
3.5.2 Performance assessment of the i-IpDFT SE algorithm 121
3.6 Conclusions 126
Bibliography 126
4 Modelling power systems with stochastic processes 131
4.1 Literature review 131
4.2 Outlines on SDEs 133
4.3 Design of SDE-based models 136
4.3.1 Method based on the solution of the stationary Fokker-Planck equation 136
4.3.2 Examples 139
4.3.3 Method based on translation processes 143
4.3.4 Application to wind speed modelling 144
4.4 Modelling power systems as SDAEs 145
4.4.1 Modelling stochastic perturbations in power systems 147
4.4.2 Examples 149
4.5 Time-domain integration of SDAEs 152
4.5.1 IEEE 145-bus 50-machine system 155
4.6 Conclusions 159
Bibliography 159
Part II Control
5 Optimization methods for preventive/corrective control in transmission systems 163
5.1 Formulation of a time-continuous dynamic optimization problem for corrective control 163
5.2 Formulation of a time-discrete static optimization problem for corrective control 166
5.3 Application to power system DAEs 169
5.3.1 Control variables 173
5.3.2 Control effort minimization 173
5.3.3 Kinetic energy cost function 174
5.3.4 Voltage penalty functions 175
5.3.5 Distance relays penalty function 176
5.4 Application of the proposed methodology for the corrective control of a realistically sized power system (test results) 178
5.5 Application to preventive control problems 183
Bibliography 186
6 Static and recursive PMU-based state estimation processes for transmission and distribution power grids 189
6.1 State estimation measurement and process model 190
6.1.1 Measurement model 191
6.1.2 Network observability 201
6.1.3 Process model 202
6.2 Static state estimation: the weighted least squares 203
6.2.1 Linear weighted least squares state estimator 204
6.2.2 Non-linear weighted least squares 205
6.3 Recursive state estimation: the Kalman filter 206
6.3.1 Discrete Kalman filter 206
6.3.2 Extended Kalman filter 210
6.3.3 Kalman Filter sensitivity with, respect to the measurement and process noise covariance matrices 211
6.3.4 Assessment of the process noise covariance matrix 212
6.4 Assessment of the measurement noise covariance matrix 212
6.5 Data conditioning and bad data processing in PMU-based state estimators 219
6.6 Kalman filter vs. weighted least squares 222
6.7 Numerical validation and performance assessment of the state estimation 223
6.7.1 Linear state estimation case studies 223
6.7.2 Non-linear SE case studies 232
6.8 Kalman filter process model validation 234
6.9 Numerical validation of Theorem 6.1 235
Bibliography 236
7 Real-time applications for electric power generation and voltage control 241
7.1 Introduction 241
7.2 Outlines of real-time system concepts 242
7.2.1 Real-time operating systems 244
7.2.2 Real-time communications 250
7.3 Voltage control 254
7.3.1 Excitation control systems 256
7.3.2 Secondary voltage control 259
7.3.3 Voltage control with distributed generation 265
7.4 Conclusions 270
Bibliography 270
8 Optimal control processes in active distribution networks 275
8.1 Typical architecture of ADN grid controllers 276
8.1.1 Control architecture 276
8.1.2 Controller's actions 278
8.2 Classic computation of sensitivity coefficients in power networks 280
8.3 Efficient computation of sensitivity coefficients of bus voltages and line currents in unbalanced radial electrical distribution networks 282
8.3.1 Voltage sensitivity coefficients 282
8.3.2 Current sensitivity coefficients 285
8.3.3 Sensitivity coefficients with respect to transformer's ULTC 286
8.4 Application examples 287
8.4.1 Distribution network case studies 287
8.4.2 Numerical validation 289
8.4.3 Voltage control and lines congestion management examples 295
8.5 Conclusions 308
Bibliography 308
Part III Stability Analysis
9 Time domain simulation for transient stability analysis 313
9.1 Introduction 313
9.2 Time-domain simulations and transient stability 315
9.3 Transient stability and high-performance computing 321
9.4 A new class of algorithms: from step-by-step solutions to parallel-in-time computations 324
9.5 Performances in parallel-in-time computations 332
9.6 Conclusions 335
Bibliography 335
10 Voltage security in modern power systems 339
1.0.1 Introduction 339
10.2 The power flow problem in rectangular coordinated 344
10.2.1 The power flow with SVC constraints 345
10.3 The OPF with SVC constraints 349
10.3.1 The maximum loadability with SVC constraints 350
10.3.2 Minimisation of the squared deviation of the bus voltage magnitude from a reference value 351
10.3.3 Constrained maximisation of the loadability with SVC 355
10.4 Solution of the optimisation problem 356
10.4.1 Primal-dual interior point method 356
10.4.2 Reduction of the linear system 359
10.5 Numerical, results 360
10.5.1 The New England 39 buses network case 361
10.5.2 The Italian case 363
10.6 Conclusions 367
Bibliography 367
11 Small-signal stability and time-domain analysis of delayed power systems 371
11.1 Introduction 371
11.1.1 Time-domain methods 372
11.1.2 Frequency-domain methods 372
11.2 A general model for power systems with time delays 373
11.2.1 Steady-state DDAE 374
11.3 Numerical techniques for DDAEs 376
11.3.1 Fade approximants 376
11.3.2 Numerical integration of DDAEs 378
11.3.3 Methods to approximate the characteristic roots of DDAEs 380
11.3.4 Discretization of the TIO 382
11.3.5 LMS approximation 384
11.4 Impact of delays on power system control 385
11.5 Case studies 388
11.5.1 IEEE 14-bus system 388
11.5.2 All-island 1479-bus Irish system 394
Bibliography 400
12 Shooting-based stability analysis of power system oscillations 405
12.1 Introduction 406
12.2 Mathematical background 408
12.2.1 The time-domain shooting method 408
12.2.2 The state transition matrix for hybrid dynamical systems 410
12.2.3 Bordering the Jacobian 413
12.2.4 The probe-insertion technique 414
12.3 Revisited PSM 417
12.3.1 Outlines of standard PSMs 417
12.3.2 From polar to rectangular coordinates 420
12.3.3 On the unit multipliers of the PSM periodic orbits 422
12.3.4 Bordering based on the COI 423
12.4 Case studies 424
12.4.1 IEEE 14-bus test system 424
12.4.2 WSCC 9-bus test system 426
12.4.3 A switching two-area PSM 427
12.5 Conclusions 431 Bibliography 431
Part IV Appendices
Appendix A Outlines of stochastic calculus 437
A.1 Stationary Markov processes 437
A.2 Regression theorem 438
A.3 Change of variables in stochastic calculus: the ltô formula 438
A.4 Memory less transformations: translation Processes 439
A.5 Fokker-Planck equation 440
Bibliography 440
Appendix B Data of lines, loads and distributed energy resources 441
B.1 IEEE 34-bus distribution test feeder data 441
B.2 IEEE 13-bus distribution test feeder data 442
B.3 IEEE 39-bus transmission test system data 445
Bibliography 446
Appendix C Proofs and tools for DDAEs 447
C.1 Determination of A0, A1 and A2 447
C.2 Chebyshev's differentiation matrix 448
C.3 Kroneckers product 448
Bibliography 449
Appendix D Numerical aspects of the probe-insertion technique 451
D.1 Parameters of the probe-insertion technique 451
D.2 Integration of (12.44) 451
Bibliography 452
Index 453
