Computational Methods in Power System Analysis (eBook)

Preface 6
Contents 7
1 Introduction 10
Part IComputational Methods 12
2 Fundamental Mathematics 13
2.1 Complex Numbers 13
2.2 Vectors 14
2.3 Matrices 15
2.4 Graphs 17
References 18
3 Solving Linear Systems of Equations 19
3.1 Direct Solvers 20
3.1.1 LU Decomposition 20
3.1.2 Solution Accuracy 21
3.1.3 Algorithmic Complexity 21
3.1.4 Fill-in and Matrix Ordering 21
3.1.5 Incomplete LU decomposition 22
3.2 Iterative Solvers 22
3.2.1 Krylov Subspace Methods 23
3.2.2 Optimality and Short Recurrences 24
3.2.3 Algorithmic Complexity 24
3.2.4 Preconditioning 24
3.2.5 Starting and Stopping 26
References 27
4 Solving Nonlinear Systems of Equations 28
4.1 Newton--Raphson Methods 29
4.1.1 Inexact Newton 30
4.1.2 Approximate Jacobian Newton 31
4.1.3 Jacobian-Free Newton 31
4.2 Newton--Raphson with Global Convergence 32
4.2.1 Line Search 32
4.2.2 Trust Regions 34
References 35
5 Convergence Theory 36
5.1 Convergence of Inexact Iterative Methods 36
5.2 Convergence of Inexact Newton Methods 40
5.2.1 Linear Convergence 44
5.3 Numerical Experiments 45
5.4 Applications 49
5.4.1 Forcing Terms 49
5.4.2 Linear Solver 50
References 51
Part IIPower System Analysis 52
6 Power System Analysis 53
6.1 Electrical Power 55
6.1.1 Voltage and Current 55
6.1.2 Complex Power 56
6.1.3 Impedance and Admittance 57
6.1.4 Kirchhoff's Circuit Laws 58
6.2 Power System Model 58
6.2.1 Generators, Loads, and Transmission Lines 59
6.2.2 Shunts and Transformers 60
6.2.3 Admittance Matrix 61
6.3 Power Flow 62
6.4 Contingency Analysis 63
References 63
7 Traditional Power Flow Solvers 64
7.1 Newton Power Flow 64
7.1.1 Power Mismatch Function 65
7.1.2 Jacobian Matrix 66
7.1.3 Handling Different Bus Types 67
7.2 Fast Decoupled Load Flow 68
7.2.1 Classical Derivation 69
7.2.2 Shunts and Transformers 71
7.2.3 BB, XB, BX, and XX 72
7.3 Convergence and Computational Properties 76
7.4 Interpretation as Elementary Newton--Krylov Methods 76
References 77
8 Newton--Krylov Power Flow Solver 78
8.1 Linear Solver 78
8.2 Preconditioning 79
8.2.1 Target Matrices 80
8.2.2 Factorisation 80
8.2.3 Reactive Power Limits and Tap Changing 81
8.3 Forcing Terms 82
8.4 Speed and Scaling 83
8.5 Robustness 84
References 85
9 Contingency Analysis 87
9.1 Simulating Branch Outages 87
9.2 Other Simulations with Uncertainty 90
References 90
10 Numerical Experiments 91
10.1 Factorisation 91
10.1.1 LU Factorisation 92
10.1.2 ILU Factorisation 95
10.2 Forcing Terms 96
10.3 Power Flow 99
10.3.1 Scaling 102
10.4 Contingency Analysis 104
References 106
11 Power Flow Test Cases 107
11.1 Construction 107
References 109
Index 110