Computational Techniques for Voltage Stability

Assessment and Control

Venkataramana Ajjarapu Iowa State University

Arnes, Iowa, U.S.A.

4y Spri ringer

Contents

Contents V

Preface XI

1 Introduction 1 1.1 What is voltage stability? 1 1.2 Voltage Collapse Incidents 4 1.3 Two Bus Example 5

1.3.1 Derivation for critical voltage and critical power 7 1.3.2 Q-Vcurves 11 1.3.3 Discussion on PV and QV Curves 12 1.3.4 Maximum power and power flow Jacobian 15

References 16

2 Numerical Bifurcation Techniques 19 2.1 Various Types of Bifurcation 19 2.2 Bifurcation of Dynamical Systems 22

2.2.1 Center manifold 24 2.3 Detection of Bifurcation Points 26

2.3.1 Static bifurcations 26 2.3.2 Homotopy Method 27 2.3.3 Continuation methods 30 2.3.4 Curve Tracing 32 2.3.5 Direct method in Computing the Saddle node bifurcation

point: a one step continuation 38 2.4 Hopf Bifurcation 40

2.4.1 Existence of Hopf bifurcation point 40 2.4.1.1 Direct methods 41 2.4.1.2 Indirect methods 42

2.5 Complex Bifurcation 42 References 45

3 Continuation Power Flow 49

3.1 Introduction 49 3.2 Locally Parameterized Continuation 49 3.3 Formulation of Power Flow Equations 50 3.4 The Predictor-corrector Process 51

3.4.1 Selecting the continuation parameter 53 3.4.2 Identifying the critical point 53

3.5 Examples 54 3.6 Simultaneous Equilibria Tracing in Power Systems 75

3.6.1 Total Solution at an equilibrium 76 3.6.2 Traditional approach 76

3.7 Power Flow Methodology and Assumptions 77 3.7.1 Nonlinearity in power flow 78 3.7.2 Slack bus assumption 79 3.7.3 PV bus assumption 80

3.8 Total Power System Equilibria Solutions 81 3.8.1 Formulation of power system DAE model 82

3.8.1.1 Synchronous generators 82 3.8.1.2 Excitation Control system 83 3.8.1.3 Prime mover and speed governor 84 3.8..1.4 Nonlinear load model 85 3.8.1.5 LTC model 86 3.8.1.6 Other modeis 86 3.8.1.7 Network power equations 89 3.8.1.8 Power system DAE model 90

3.8.2 Bifurcation modeling of power system dynamics 90 3.8.2.1 Saddle-node bifurcation 91 3.8.2.2 Hopf bifurcation 91

3.8.3 Manifold modeis in power Systems 92 3.8.3.1 Manifold 92 3.8.3.2 Natural parameterization 93 3.8.3.3 Local parameterization 93

3.8.4 Equilibrium manifold Tracing of power Systems 95 3.8.5 Initialization for power system equilibrium tracing 96 3.8.6 Continuation method with local parametrization 98 3.8.7 Linerization of power system DAE 99 3.8.8 Detection of Saddle Node Bifurcation with System Total

Jacobian 100 3.8.8.1 Detection of saddle-node bifurcation 101

3.8.9 Limits implementation 104 3.8.9.1 Governor limits 104 3.8.9.2 AVR limits 104

3.9 Numerical examples for EQTP 108

VII

References 115

Sensitivity Analysis for Voltage Stability 117 4.1 Introduction 117 4.2 Given State Based Indices 117 4.3 Large Deviation Based Indices 120 4.4 Stability Studies via Sensitivity Analysis 120

4.4.1 Identification of critical elements 121 4.4.2 Eigenvalue sensitivity 121 4.4.3 Modal analysis 122 4.4.4 Sensitivity analysis via CPF 123 4.4.5 Tangent vector, right eigenvector, and right singular vector

ofJ 124 4.4.6 Voltage stability index from the tangent vector 125 4.4.7 Sensitivity analysis from the tangent vector 127 4.4.8 Bus sensitivities 127 4.4.9 Branch sensitivities 129 4.4.10 Generator sensitivities 131 4.4.11 Qualitative vs. quantitative sensitivities 133

4.5 Margin Sensitivity 133 4.5.1 Transfer margin estimation 136 4.5.2 Multi-parameter margin sensitivity 138 4.5.3 Sensitivity formulas 139

4.6 Test System Studies 142 4.6.1 Two bus example: 142 4.6.2 The New England System 147

4.6.2.1 Exciter parameters 147 4.6.2.2 Network parameters 149 4.6.2.3 Load (scenario) parameters 151 4.6.2.4 Multiple-parameter variations 153

References 154

Voltage Stability Margin Boundary Tracing 157 5.1 Introduction 157 5.2 Natural Parameterization for Margin Boundary Tracing 158

5.2.1 Load parameter space 158 5.2.1 Control parameter space 159

5.3 Formulation of Margin Boundary Tracing 160 5.3.1 Margin boundary manifold of power System 160 5.3.2 Characterization of margin boundary 160

5.3.2.1 Characterization of saddle node bifurcation related margin boundary tracing 160

VIII

5.3.3 Margin boundary tracing 161 5.3.3.1 Augmentation for bifurcation characterization 161 5.3.3.2 Augmentation for local parameterization 162

5.3.4 Basic Steps Involved in the Margin Boundary Tracing 164 5.3.5 Practical implementation 164

5.3.5.1 Implementation of reduced method 165 5.4 Examples 167

5.4.1 Series compensation between bus 6 and bus 31 176 5.4.2 Shunt Compensation 176 5.4.3 Multiple contingencies 177 5.4.4 Boundary tracing with respect to generation control

Parameters 178 5.4.4.1 Load margin vs adjustment of Ka of AVR System 178 5.4.4.2 Load margin versus adjustment of Vref of AVR systeml79

5.4.5 Control combination 180 5.4.6 Advantages of margin boundary tracing 181

5.5 Formulationof Vortage Stability Limited ATC 181 5.6 Scenario Parameters 184 5.7 Scenario According to Simultaneous Multi-area Transactions. 185

5.7.1 Determination of KLi 187

5.7.2 Determination of KGi 190

5.8 Numerical Example 191 5.8.1 Description of the Simulation system 191 5.8.2 Emergency transmission load relief 201

5.8.2.1 Single transaction case 201 5.8.2.2 Simultaneous transaction case 202

5.8.3 Reactive power Support 202 5.8.3.1 Single transaction case 202 5.8.3.2 Simultaneous transaction case 203

5.8.4 Control combination 204 5.9 Conclusion 205 References 206

6 Time Domain Simulation 207 6.1 Introduction 207 6.2 Explicit and Implicit Methods 208

6.2.1 Explicit method 208 6.2.2 Implicit method 209 6.2.3 Stiffness and Numerical Stability 209

6.3 Decoupled Time Domain Simulation 212

IX

6.4 Numerical Examples 217 6.4.1 Two bus System 217 6.4.2 New England 39-bus System 223

6.5 Quasi-Steady-State Simulation (QSS) 226 6.5.1 Problem Formulation 227 6.5.2 Steps involved in QSS Method 228 6.5.3 Implementation of the Continuation Method in QSS 230 6.5.4 Consideration of Load Change with respect to Time 230 6.5.5 Numerical Results 232

6.5.5.1 2-bus system 232 6.5.5.2 CQSS Simulation for New England 39-bus System.... 233

References 236

Appendix 239 A. Data of 2-bus test System 239

AI. One line diagram 239 A2. The IEEE format: Base case power flow data of the

2-bus system 239 A3. The dynamic data of the 2-bus system 240

B. Data ofNew England test system 241 Bl. One line diagram 241

B2. The IEEE format: Base case power flow data of the New England system 242

B3. The Dynamic Data of the New England System 245

Index 249