1 Introduction ................................................. 1
2 The Solution of Linear Systems ............................... 3
2.1 Gaussian Elimination .................................... 4
2.2 LU Factorization ........................................ 9
2.2.1 LU Factorization with Partial Pivoting .......... 16
2.2.2 LU Factorization with Complete Pivoting ......... 20
2.3 Condition Numbers and Error Propagation ................ 22
2.4 Relaxation Methods ..................................... 23
2.5 Conjugate Gradient Methods ............................. 28
2.6 Generalized Minimal Residual Algorithm (GMRES) ......... 34
2.7 Problems ............................................... 40
3 Systems of Nonlinear Equations .............................. 45
3.1 Fixed Point Iteration .................................. 46
3.2 Newton-Raphson Iteration ............................... 53
3.2.1 Convergence Properties .......................... 56
3.2.2 The Newton-Raphson for Systems of Nonlinear
Equations ....................................... 57
3.2.3 Modifications to the Newton-Raphson Method ...... 60
3.3 Continuation Methods ................................... 62
3.4 Secant Method .......................................... 65
3.5 Numerical Differentiation .............................. 68
3.6 Power System Applications .............................. 72
3.6.1 Power Flow ...................................... 72
3.6.2 Regulating Transformers ......................... 80
3.6.3 Decoupled Power Flow ............................ 84
3.6.4 Fast Decoupled Power Flow ....................... 86
3.6.5 PV Curves and Continuation Power Flow ........... 89
3.6.6 Three-Phase Power Flow .......................... 96
3.7 Problems ............................................... 99
4 Sparse Matrix Solution Techniques .......................... 103
4.1 Storage Methods ....................................... 104
4.2 Sparse Matrix Representation .......................... 109
4.3 Ordering Schemes ...................................... 111
4.3.1 Scheme 0 ....................................... 119
4.3.2 Scheme I ....................................... 120
4.3.3 Scheme II ...................................... 126
4.3.4 Other Schemes .................................. 129
4.4 Power System Applications ............................. 130
4.5 Problems .............................................. 134
5 Numerical Integration ...................................... 139
5.1 One-Step Methods ...................................... 140
5.1.1 Taylor Series-Based Methods .................... 140
5.1.2 Forward-Euler Method ........................... 141
5.1.3 Runge-Kutta Methods ............................ 141
5.2 Multistep Methods ..................................... 142
5.2.1 Adams Methods .................................. 148
5.2.2 Gear's Methods ................................. 151
5.3 Accuracy and Error Analysis ........................... 152
5.4 Numerical Stability Analysis .......................... 156
5.5 Stiff Systems ......................................... 163
5.6 Step-Size Selection ................................... 167
5.7 Differential-Algebraic Equations ...................... 170
5.8 Power System Applications ............................. 173
5.8.1 Transient Stability Analysis ................... 173
5.8.2 Mid-Term Stability Analysis .................... 181
5.9 Problems .............................................. 185
6 Optimization ............................................... 191
6.1 Least Squares State Estimation ........................ 192
6.1.1 Weighted Least Squares Estimation .............. 195
6.1.2 Bad Data Detection ............................. 198
6.1.3 Nonlinear Least Squares State Estimation ....... 201
6.2 Linear Programming .................................... 202
6.2.1 Simplex Method ................................. 203
6.2.2 Interior Point Method .......................... 207
6.3 Nonlinear Programming ................................. 212
6.3.1 Quadratic Programming .......................... 213
6.3.2 Steepest Descent Algorithm ..................... 215
6.3.3 Sequential Quadratic Programming Algorithm ..... 220
6.4 Power System Applications ............................. 223
6.4.1 Optimal Power Flow ............................. 223
6.4.2 State Estimation ............................... 234
6.5 Problems .............................................. 239
7 Eigenvalue Problems ........................................ 243
7.1 The Power Method ...................................... 244
7.2 The QR Algorithm ...................................... 246
7.2.1 Shifted QR ..................................... 253
7.2.2 Deflation ...................................... 254
7.3 Arnoldi Methods ....................................... 254
7.4 Singular Value Decomposition .......................... 261
7.5 Modal Identification .................................. 264
7.5.1 Prony Method ................................... 266
7.5.2 The Matrix Pencil Method ....................... 268
7.5.3 The Levenberg-Marquardt Method ................. 269
7.5.4 The Hilbert Transform .......................... 272
7.5.5 Examples ....................................... 273
7.6 Power System Applications ............................. 278
7.6.1 Participation Factors .......................... 278
7.7 Problems .............................................. 280
Index ......................................................... 283
References .................................................... 289
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