Interval Methods for Uncertain Power System Analysis

Alfredo Vaccaro, PhD, is a Full Professor of Electric Power Systems in the Department of Engineering at the University of Sannio. He is Editor-in-Chief of Technology and Economics of Smart Grids and Sustainable Energy, Springer Nature and Chair of the PES-IEEE Task Force on Enabling Paradigms for High-Performance Computing in Wide Area Monitoring Protective and Control Systems.

About the Author ix

Preface xi

Acknowledgments xiii

Acronyms xv

Introduction 1

1 Introduction to Reliable Computing 3

1.1 Elements of Reliable Computing 4

1.2 Interval Analysis 7

1.3 Interval-Based Operators 8

1.4 Interval Extensions of Elementary Functions 9

1.5 Solving Systems of Linear Interval Equations 11

1.6 Finding Zeros of Nonlinear Equations 15

1.7 Solution of Systems of Nonlinear Interval Equations 16

1.8 The Overestimation Problem 20

1.9 Affine Arithmetic 22

1.9.1 Conversion Between AA and IA 25

1.9.2 AA-Based Operators 25

1.9.3 Chebyshev Approximation of Univariate Nonaffine Functions 28

1.9.4 Multiplication of Affine Forms 31

1.9.5 Effects of Recursive Solution Schemes 35

1.10 Integrating AA and IA 35

2 Uncertain Power Flow Analysis 37

2.1 Sources of Uncertainties in Power Flow Analysis 39

2.2 Solving Uncertain Linearized Power Flow Equations 41

2.3 Solving Uncertain Power Flow Equations 46

2.3.1 Optimization-Based Method 48

2.3.2 Domain Contraction Method 52

3 Uncertain Optimal Power Flow Analysis 59

3.1 Range Analysis-Based Solution 61

3.1.1 Optimal Economic Dispatch 63

3.1.2 Reactive Power Dispatch 66

3.2 AA-Based Solution 70

4 Uncertain Markov Chain Analysis 75

4.1 Mathematical Preliminaries 77

4.2 Effects of Data Uncertainties 78

4.3 Matrix Notation 79

4.4 AA-Based Uncertain Analysis 80

4.5 Application Examples 83

4.5.1 Case Study 1: Grid Resilience Analysis 83

4.5.2 Case Study 2: Energy Storage Model 84

4.5.3 Summary 86

5 Small-Signal Stability Analysis of Uncertain Power Systems 87

5.1 Problem Formulation 89

5.2 The Interval Eigenvalue Problem 90

5.3 Applications 92

5.3.1 Case Study 1 92

5.3.2 Case Study 2 93

6 Uncertain Power Components Thermal Analysis 95

6.1 Thermal Rating Assessment of Overhead Lines 96

6.1.1 Sources of Data Uncertainties 98

6.1.2 AA-Based Thermal Rating Assessment 99

6.1.3 Application Examples 100

6.2 Thermal Rating Assessment of Power Cables 104

6.2.1 Thermal Modeling of Power Cables 105

6.2.2 Sources of Data Uncertainties 107

6.2.3 Tolerance Analysis of Cable Thermal Dynamics by IA 108

6.2.4 Application Examples 109

References 112

Index 119