Preface ...................................................... xiii
Acknowledgments .............................................. xvii
Author ........................................................ xix
Abbreviations and Acronyms .................................... xxi
1 Introduction ................................................. 1
1.1 General ................................................. 1
1.2 Power System Stability .................................. 2
1.3 Power System Security ................................... 3
1.4 Monitoring and Enhancing System Security ................ 6
1.5 Emergency Control and System Protection ................. 7
1.6 Application of Energy Functions ......................... 8
1.7 Scope of This Book ..................................... 14
2 Review of Direct Methods for Transient Stability
Evaluations for Systems with Simplified Models .............. 17
2.1 Introduction ........................................... 17
2.2 System Model ........................................... 18
2.2.1 Synchronous Generators .......................... 18
2.2.2 Network Equations ............................... 21
2.2.3 Load Model ...................................... 22
2.2.4 Expressions for Electrical Power ................ 23
2.3 Mathematical Preliminaries ............................. 25
2.3.1 Equilibrium Points .............................. 26
2.3.2 Stability of Equilibrium Point .................. 27
2.3.3 Lyapunov Stability .............................. 27
2.3.4 Theorem on Lyapunov Stability ................... 27
2.4 Two-Machine System and Equal Area Criterion ............ 30
2.4.1 Equal Area Criterion ............................ 31
2.4.2 Energy Function Analysis of an SMIB System ...... 32
2.5 Lyapunov Functions for Direct Stability Evaluation ..... 34
2.5.1 Construction of Lyapunov Function ............... 38
2.6 Energy Functions for Multimachine Power Systems ........ 39
2.6.1 Characterization of Transient Stability ......... 39
2.6.2 Center of Inertia Formulations .................. 40
2.6.3 Energy Function Using COI Formulation ........... 43
2.7 Estimation of Stability Domain ......................... 44
2.7.1 Incorporating Transfer Conductances in Energy
Function ........................................ 44
2.7.2 Determination of Critical Energy ................ 46
2.7.2.1 Single-Machine System .................. 46
2.7.2.2 Multimachine System .................... 48
2.7.3 Potential Energy Boundary Surface ............... 48
2.7.4 Controlling UEP Method .......................... 50
2.7.5 BCU Method ...................................... 51
2.8 Extended Equal Area Criterion .......................... 53
2.8.1 Formulation ..................................... 53
2.8.2 Approximation of Faulted Trajectory ............. 54
2.8.3 Identification of Critical Cluster .............. 55
3 Structure Preserving Energy Functions for Systems with
Nonlinear Load Models and Generator Flux Decay .............. 57
3.1 Introduction ........................................... 57
3.2 Structure Preserving Model ............................. 57
3.3 Inclusion of Voltage-Dependent Power Loads ............. 61
3.4 SPEF with Voltage-Dependent Load Models ................ 62
3.4.1 Dynamic Equations of Generator .................. 62
3.4.2 Load Model ...................................... 63
3.4.3 Power Flow Equations ............................ 64
3.4.4 Structure Preserving Energy Functions ........... 64
3.4.5 Computation of Stability Region ................. 68
3.5 Case Studies on IEEE Test Systems ...................... 69
3.5.1 Seventeen-Generator System ...................... 70
3.5.2 Fifty-Generator System .......................... 74
3.6 Solution of System Equations during a Transient ........ 76
3.7 Noniterative Solution of Networks with Nonlinear
Loads .................................................. 77
3.7.1 System Equations ................................ 78
3.7.2 Dynamic Equations of Generators ................. 78
3.7.3 Power Flow Equations during a Transient ......... 79
3.7.4 Special Cases ................................... 81
3.7.5 Solutions of the Quartic Equation ............... 82
3.7.6 Network Transformation for Decoupling of Load
Buses ........................................... 83
3.7.7 Transformation of the Load Characteristics ...... 84
3.8 Inclusion of Transmission Losses in Energy Function .... 85
3.8.1 Transformation of a Lossy Network ............... 85
3.8.2 Structure Preserving Energy Function
Incorporating Transmission Line Resistances ..... 87
3.9 SPEF for Systems with Generator Flux Decay ............. 90
3.9.1 System Model .................................... 90
3.9.1.1 Generator Model ........................ 91
3.9.1.2 Load Model ............................. 92
3.9.1.3 Power Flow Equations ................... 92
3.9.1.1 Structure Preserving Energy Function ... 93
3.9.3 Example ......................................... 96
3.10 Network Analogy for System Stability Analysis .......... 97
4 Structure Preserving Energy Functions for Systems with
Detailed Generator and Load Models ......................... 105
4.1 Introduction .......................................... 105
4.2 System Model .......................................... 106
4.2.1 Generator Model ................................ 106
4.2.2 Excitation System Model ........................ 107
4.2.3 Load Model ..................................... 108
4.2.4 Power Flow Equations ........................... 108
4.3 Structure Preserving Energy Function with Detailed
Generator Models ...................................... 109
4.3.1 Structure Preserving Energy Function ........... 109
4.3.2 Simpler Expression for SPEF .................... 112
4.4 Numerical Examples .................................... 114
4.4.1 SMIB System .................................... 114
4.4.2 Ten-Generator, 39-Bus New England Test System .. 115
4.4.3 Variation of Total Energy and Its Components ... 121
4.5 Modeling of Dynamic Loads ............................. 122
4.5.1 Induction Motor Model .......................... 124
4.5.2 Voltage Instability in Induction Motors ........ 126
4.5.3 Simpler Models of Induction Motors ............. 128
4.5.4 Energy Function Analysis of Synchronous and
Voltage Stability .............................. 128
4.5.4.1 Computation of Equilibrium Points ..... 130
4.5.4.2 Computation of Energy at UEP .......... 132
4.5.5 Dynamic Load Models in Multimachine Power
Systems ........................................ 135
4.6 New Results on SPEF Based on Network Analogy .......... 136
4.6.1 Potential Energy Contributed by Considering
the Two-Axis Model of the Synchronous
Generator ...................................... 140
4.7 Unstable Modes and Parametric Resonance ............... 144
4.7.1 Normal Forms ................................... 145
4.7.2 Fast Fourier Transform of Potential Energy ..... 146
4.7.2.1 Results of the Case Study ............. 146
5 Structure Preserving Energy Functions for Systems with
HVDC and FACTS Controllers ................................. 149
5.1 Introduction .......................................... 149
5.2 HVDC Power Transmission Links ......................... 149
5.2.1 HVDC Systems and Energy Functions .............. 149
5.2.2 HVDC System Model .............................. 150
5.2.2.1 Converter Model ....................... 150
5.2.2.2 DC Network Equations .................. 152
5.2.2.3 Converter Control Model ............... 152
5.2.3 AC System Model ................................ 155
5.2.3.1 Generator Model ....................... 155
5.2.3.2 Load Model ............................ 156
5.2.3.3 AC Network Equations .................. 156
5.2.4 Structure Preserving Energy Function ........... 156
5.2.5 Example ........................................ 160
5.2.5.1 Auxiliary Controller .................. 161
5.2.5.2 Emergency Controller .................. 162
5.2.5.3 Case Study and Results ................ 162
5.3 Static Var Compensator ................................ 163
5.3.1 Description .................................... 163
5.3.2 Control Characteristics and Modeling of SVC
Controller ..................................... 164
5.3.3 Network Solution with SVC: Application of
Compensation Theorem ........................... 166
5.3.3.1 Calculation of Φsvc in Control Region .. 167
5.3.3.2 Network Solution ...................... 168
5.3.4 Potential Energy Function for SVC .............. 169
5.3.5 Example ........................................ 171
5.3.6 Case Study of New England Test System .......... 172
5.3.6.1 Network Calculation with Multiple
SVCs .................................. 173
5.3.6.2 Structure Preserving Energy Function .. 174
5.3.6.3 Results and Discussion ................ 175
5.4 Static Synchronous Compensator ........................ 175
5.4.1 General ........................................ 175
5.4.2 Modeling of a STATCOM .......................... 176
5.4.3 STATCOM Controller ............................. 178
5.4.4 Potential Energy Function for a STATCOM ........ 180
5.5 Series-Connected FACTS Controllers .................... 181
5.5.1 Thyristor-Controlled Series Capacitor .......... 182
5.5.1.1 Power Scheduling Control .............. 182
5.5.1.2 Power Swing Damping Control ........... 183
5.5.1.3 Transient Stability Control ........... 183
5.5.2 Static Synchronous Series Compensator .......... 184
5.6 Potential Energy in a Line with Series FACTS
Controllers ........................................... 185
5.6.1 Thyristor-Controlled Series Capacitor .......... 186
5.6.2 Static Synchronous Series Compensator .......... 187
5.6.3 Potential Energy in the Presence of CC and CA
Controllers .................................... 188
5.6.3.1 Potential Energy with CC Control ...... 188
5.6.3.2 Potential Energy with CA Control ...... 189
5.7 Unified Power Flow Controller ......................... 189
5.7.1 Description .................................... 189
5.7.2 Energy Function with Unified Power Flow
Controller ..................................... 191
6 Detection of Instability Based on Identification of
Critical Cutsets ........................................... 195
6.1 Introduction .......................................... 195
6.2 Basic Concepts ........................................ 196
6.3 Prediction of the Critical Cutset ..................... 198
6.3.1 Analysis ....................................... 198
6.3.2 Case Study ..................................... 203
6.3.3 Discussion ..................................... 203
6.4 Detection of Instability by Monitoring Critical
Cutset ................................................ 203
6.4.1 Criterion for Instability ...................... 204
6.4.2 Modification of the Instability Criterion ...... 206
6.5 Algorithm for Identification of Critical Cutset ....... 207
6.6 Prediction of Instability ............................. 209
6.7 Case Studies .......................................... 210
6.7.1 Ten-Generator New England Test System .......... 210
6.7.2 Seventeen-Generator IEEE Test System ........... 213
6.7.3 Discussion ..................................... 214
6.8 Study of a Practical System ........................... 215
6.8.1 Discussion ..................................... 217
6.9 Adaptive System Protection ............................ 219
6.9.1 Discussion ..................................... 225
7 Sensitivity Analysis for Dynamic Security and Preventive
Control Using Damping Controllers Based on FACTS ........... 227
7.1 Introduction .......................................... 227
7.2 Basic Concepts in Sensitivity Analysis ................ 228
7.3 Dynamic Security Assessment Based on Energy Margin .... 229
7.3.1 Transient Energy Margin ........................ 229
7.3.2 Computation of Energy Margin ................... 229
7.3.2.1 Evaluation of Path-Dependent
Integrals ............................. 231
7.3.2.2 Computation of Energy Margin Based
on Critical Cutsets ................... 231
7.4 Energy Margin Sensitivity ............................. 232
7.4.1 Application to Structure Preserving Model ...... 232
7.5 Trajectory Sensitivity ................................ 235
7.5.1 Sensitivity to Initial Condition Variations .... 235
7.5.2 Discussion ..................................... 236
7.6 Energy Function-Based Design of Damping Controllers ... 236
7.6.1 Series FACTS Controllers ....................... 237
7.6.1.1 Linearized System Equations ........... 238
7.6.1.2 Synthesis of the Control Signal ....... 242
7.6.1.3 Case Study of a 10-Machine System ..... 244
7.6.2 Shunt FACTS Controllers ........................ 245
7.6.2.1 Linear Network Model for Reactive
Current ............................... 247
7.7 Damping Controllers for UPFC .......................... 251
7.7.1 Discussion ..................................... 254
8 Application of FACTS Controllers for Emergency Control—1 ... 255
8.1 Introduction .......................................... 255
8.2 Basic Concepts ........................................ 256
8.3 Switched Series Compensation .......................... 257
8.3.1 Time-Optimal Control ........................... 257
8.4 Control Strategy for a Two-Machine System ............. 260
8.5 Comparative Study of TCSC and SSSC .................... 264
8.5.1 Control Strategy ............................... 265
8.6 Discrete Control of STATCOM ........................... 269
8.7 Discrete Control of UPFC .............................. 272
8.7.1 Application of Control Strategy to SMIB
System with UPFC ............................... 278
8.7.2 Discussion ..................................... 278
8.8 Improvement of Transient Stability by Static Phase-
Shifting Transformer .................................. 280
8.9 Emergency Control Measures ............................ 282
8.9.1 Controlled System Separation and Load
Shedding ....................................... 283
8.9.2 Generator Tripping ............................. 284
9 Application of FACTS Controllers for Emergency Control-II .. 285
9.1 Introduction .......................................... 285
9.2 Discrete Control Strategy ............................. 285
9.3 Case Study I: Application of TCSC ..................... 289
9.3.1 Fault at Bus 26: Without Line Tripping ......... 289
9.3.2 Fault at Bus 26: Cleared by Line Tripping ...... 292
9.4 Case Study II: Application of UPFC .................... 292
9.4.1 Single UPFC .................................... 293
9.4.2 Multiple UPFC .................................. 298
9.4.3 Practical Implementation ....................... 299
9.5 Discussion and Directions for Further Research ........ 302
References .................................................... 305
Appendix A: Synchronous Generator Model ....................... 315
Appendix B: Boundary of Stability Region: Theoretical
Results ........................................... 327
Appendix С: Network Solution for Transient Stability
Analysis .......................................... 331
Appendix D: Data for 10-Generator System ...................... 341
Index ......................................................... 345
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