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Summary
Summary
Presenting the continuation and bifurcation-based approaches to assess power system voltage stability, this self-contained manual first provides basic definitions related to voltage stability based on IEEE/CIGRE voltage stability classification. Then the need for robust numerical techniques that are needed to address various aspects of voltage instability is articulated. It presents a tutorial introduction to the basic concepts in bifurcation theory and continuation methods. These methods lead to robust numerical techniques for voltage stability study. The book also provides details related to continuation power flow. In addition, it provides the approach to trace voltage stability boundary for changing system conditions and proposes a uniformed framework that provides computational approaches for both short-term and long-term voltage stability phenomena.
Table of Contents
Contents | p. V |
Preface | p. XI |
1 Introduction | p. 1 |
1.1 What is voltage stability? | p. 1 |
1.2 Voltage Collapse Incidents | p. 4 |
1.3 Two Bus Example | p. 5 |
1.3.1 Derivation for critical voltage and critical power | p. 7 |
1.3.2 Q-V curves | p. 11 |
1.3.3 Discussion on PV and QV Curves | p. 12 |
1.3.4 Maximum power and power flow Jacobian | p. 15 |
References | p. 16 |
2 Numerical Bifurcation Techniques | p. 19 |
2.1 Various Types of Bifurcation | p. 19 |
2.2 Bifurcation of Dynamical Systems | p. 22 |
2.2.1 Center manifold | p. 24 |
2.3 Detection of Bifurcation Points | p. 26 |
2.3.1 Static bifurcations | p. 26 |
2.3.2 Homotopy Method | p. 27 |
2.3.3 Continuation methods | p. 30 |
2.3.4 Curve Tracing | p. 32 |
2.3.5 Direct method in computing the Saddle node bifurcation point: a one step continuation | p. 38 |
2.4 Hopf Bifurcation | p. 40 |
2.4.1 Existence of Hopf bifurcation point | p. 40 |
2.4.1.1 Direct methods | p. 41 |
2.4.1.2 Indirect methods | p. 42 |
2.5 Complex Bifurcation | p. 42 |
References | p. 45 |
3 Continuation Power Flow | p. 49 |
3.1 Introduction | p. 49 |
3.2 Locally Parameterized Continuation | p. 49 |
3.3 Formulation of Power Flow Equations | p. 50 |
3.4 The Predictor-corrector Process | p. 51 |
3.4.1 Selecting the continuation parameter | p. 53 |
3.4.2 Identifying the critical point | p. 53 |
3.5 Examples | p. 54 |
3.6 Simultaneous Equilibria Tracing in Power Systems | p. 75 |
3.6.1 Total solution at an equilibrium | p. 76 |
3.6.2 Traditional approach | p. 76 |
3.7 Power Flow Methodology and Assumptions | p. 77 |
3.7.1 Nonlinearity in power flow | p. 78 |
3.7.2 Slack bus assumption | p. 79 |
3.7.3 PV bus assumption | p. 80 |
3.8 Total Power System Equilibria Solutions | p. 81 |
3.8.1 Formulation of power system DAE model | p. 82 |
3.8.1.1 Synchronous generators | p. 82 |
3.8.1.2 Excitation Control system | p. 83 |
3.8.1.3 Prime mover and speed governor | p. 84 |
3.8.1.4 Nonlinear load model | p. 85 |
3.8.1.5 LTC model | p. 86 |
3.8.1.6 Other models | p. 86 |
3.8.1.7 Network power equations | p. 89 |
3.8.1.8 Power system DAE model | p. 90 |
3.8.2 Bifurcation modeling of power system dynamics | p. 90 |
3.8.2.1 Saddle-node bifurcation | p. 91 |
3.8.2.2 Hopf bifurcation | p. 91 |
3.8.3 Manifold models in power systems | p. 92 |
3.8.3.1 Manifold | p. 92 |
3.8.3.2 Natural parameterization | p. 93 |
3.8.3.3 Local parameterization | p. 93 |
3.8.4 Equilibrium manifold Tracing of power systems | p. 95 |
3.8.5 Initialization for power system equilibrium tracing | p. 96 |
3.8.6 Continuation method with local parametrization | p. 98 |
3.8.7 Linerization of power system DAE | p. 99 |
3.8.8 Detection of Saddle Node Bifurcation with System Total Jacobian | p. 100 |
3.8.8.1 Detection of saddle-node bifurcation | p. 101 |
3.8.9 Limits implementation | p. 104 |
3.8.9.1 Governor limits | p. 104 |
3.8.9.2 AVR limits | p. 104 |
3.9 Numerical examples for EQTP | p. 108 |
References | p. 115 |
4 Sensitivity Analysis for Voltage Stability | p. 117 |
4.1 Introduction | p. 117 |
4.2 Given State Based Indices | p. 117 |
4.3 Large Deviation Based Indices | p. 120 |
4.4 Stability Studies via Sensitivity Analysis | p. 120 |
4.4.1 Identification of critical elements | p. 121 |
4.4.2 Eigenvalue sensitivity | p. 121 |
4.4.3 Modal analysis | p. 122 |
4.4.4 Sensitivity analysis via CPF | p. 123 |
4.4.5 Tangent vector, right eigenvector, and right singular vector of J | p. 124 |
4.4.6 Voltage stability index from the tangent vector | p. 125 |
4.4.7 Sensitivity analysis from the tangent vector | p. 127 |
4.4.8 Bus sensitivities | p. 127 |
4.4.9 Branch sensitivities | p. 129 |
4.4.10 Generator sensitivities | p. 131 |
4.4.11 Qualitative vs. quantitative sensitivities | p. 133 |
4.5 Margin Sensitivity | p. 133 |
4.5.1 Transfer margin estimation | p. 136 |
4.5.2 Multi-parameter margin sensitivity | p. 138 |
4.5.3 Sensitivity formulas | p. 139 |
4.6 Test System Studies | p. 142 |
4.6.1 Two bus example | p. 142 |
4.6.2 The New England system | p. 147 |
4.6.2.1 Exciter parameters | p. 147 |
4.6.2.2 Network parameters | p. 149 |
4.6.2.3 Load (scenario) parameters | p. 151 |
4.6.2.4 Multiple-parameter variations | p. 153 |
References | p. 154 |
5 Voltage Stability Margin Boundary Tracing | p. 157 |
5.1 Introduction | p. 157 |
5.2 Natural Parameterization for Margin Boundary Tracing | p. 158 |
5.2.1 Load parameter space | p. 158 |
5.2.1 Control parameter space | p. 159 |
5.3 Formulation of Margin Boundary Tracing | p. 160 |
5.3.1 Margin boundary manifold of power system | p. 160 |
5.3.2 Characterization of margin boundary | p. 160 |
5.3.2.1 Characterization of saddle node bifurcation related margin boundary tracing | p. 160 |
5.3.3 Margin boundary tracing | p. 161 |
5.3.3.1 Augmentation for bifurcation characterization | p. 161 |
5.3.3.2 Augmentation for local parameterization | p. 162 |
5.3.4 Basic Steps Involved in the Margin Boundary Tracing | p. 164 |
5.3.5 Practical implementation | p. 164 |
5.3.5.1 Implementation of reduced method | p. 165 |
5.4 Examples | p. 167 |
5.4.1 Series compensation between bus 6 and bus 31 | p. 176 |
5.4.2 Shunt Compensation | p. 176 |
5.4.3 Multiple contingencies | p. 177 |
5.4.4 Boundary tracing with respect to generation control parameters | p. 178 |
5.4.4.1 Load margin vs adjustment of Ka of AVR system | p. 178 |
5.4.4.2 Load margin versus adjustment of V[subscript ref] of AVR system | p. 179 |
5.4.5 Control combination | p. 180 |
5.4.6 Advantages of margin boundary tracing | p. 181 |
5.5 Formulation of Voltage Stability Limited ATC | p. 181 |
5.6 Scenario Parameters | p. 184 |
5.7 Scenario According to Simultaneous Multi-area Transactions | p. 185 |
5.7.1 Determination of K [subscript Li] | p. 187 |
5.7.2 Determination of K [subscript Gi] | p. 190 |
5.8 Numerical Example | p. 191 |
5.8.1 Description of the simulation system | p. 191 |
5.8.2 Emergency transmission load relief | p. 201 |
5.8.2.1 Single transaction case | p. 201 |
5.8.2.2 Simultaneous transaction case | p. 202 |
5.8.3 Reactive power Support | p. 202 |
5.8.3.1 Single transaction case | p. 202 |
5.8.3.2 Simultaneous transaction case | p. 203 |
5.8.4 Control combination | p. 204 |
5.9 Conclusion | p. 205 |
References | p. 206 |
6 Time Domain Simulation | p. 207 |
6.1 Introduction | p. 207 |
6.2 Explicit and Implicit Methods | p. 208 |
6.2.1 Explicit method | p. 208 |
6.2.2 Implicit method | p. 209 |
6.2.3 Stiffness and Numerical Stability | p. 209 |
6.3 Decoupled Time Domain Simulation | p. 212 |
6.4 Numerical Examples | p. 217 |
6.4.1 Two bus system | p. 217 |
6.4.2 New England 39-bus system | p. 223 |
6.5 Quasi-Steady-State Simulation (QSS) | p. 226 |
6.5.1 Problem Formulation | p. 227 |
6.5.2 Steps involved in QSS Method | p. 228 |
6.5.3 Implementation of the Continuation Method in QSS | p. 230 |
6.5.4 Consideration of Load Change with respect to Time | p. 230 |
6.5.5 Numerical Results | p. 232 |
6.5.5.1 2-bus system | p. 232 |
6.5.5.2 CQSS Simulation for New England 39-bus system | p. 233 |
References | p. 236 |
Appendix | p. 239 |
A Data of 2-bus test system | p. 239 |
A1 One line diagram | p. 239 |
A2 The IEEE format: Base case power flow data of the 2-bus system | p. 239 |
A3 The dynamic data of the 2-bus system | p. 240 |
B Data of New England test system | p. 241 |
B1 One line diagram | p. 241 |
B2 The IEEE format: Base case power flow data of the New England system | p. 242 |
B3 The Dynamic Data of the New England System | p. 245 |
Index | p. 249 |