Mathematical Programming for Power Systems Operation: From Theory to

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Alejandro Garcés, PhD, is a Professor at Universidad Tecnológica de Pereira in Colombia. Previously, he was a research fellow at the Norwegian University of Science and Technology in Trondheim-Norway, and an External Consultant for the Latin-American Organization of Energy and the Inter-American Development Bank. He is also Senior member of the IEEE, and Associate Editor of different IEEE and IET journals. In 2021 he was awarded with the Georg Forster Research fellow at the Alexander von Humboldt Foundation in Germany to continue his research in collaboration with TU-Dortmund.

Acknowledgment ix Introduction xi 1 Power systems operation 1 1.1 Mathematical programming for power systems operation 1 1.2 Continuous models 3 1.2.1 Economic and environmental dispatch 3 1.2.2 Hydrothermal dispatch 3 1.2.3 Effect of the grid constraints 5 1.2.4 Optimal power flow 5 1.2.5 Hosting capacity 7 1.2.6 Demand-side management 7 1.2.7 Energy storage management 9 1.2.8 State estimation and grid identification 9 1.3 Binary problems in power systems operation 11 1.3.1 Unit commitment 12 1.3.2 Optimal placement of distributed generation and capacitors 12 1.3.3 Primary feeder reconfiguration and topology identification 13 1.3.4 Phase balancing 13 1.4 Real-time implementation 14 1.5 Using Python 15 Part I Mathematical programming 17 2 A brief introduction to mathematical optimization 19 2.1 About sets and functions 19 2.2 Norms 22 2.3 Global and local optimum 24 2.4 Maximum and minimum values of continuous functions 25 2.5 The gradient method 26 2.6 Lagrange multipliers 32 2.7 The Newton's method 33 2.8 Further readings 35 2.9 Exercises 35 3 Convex optimization 39 3.1 Convex sets 39 3.2 Convex functions 45 3.3 Convex optimization problems 47 3.4 Global optimum and uniqueness of the solution 50 3.5 Duality 52 3.6 Further readings 56 3.7 Exercises 58 4 Convex Programming in Python 61 4.1 Python for convex optimization 61 4.2 Linear programming 62 4.3 Quadratic forms 67 4.4 Semidefinite matrices 69 4.5 Solving quadratic programming problems 71 4.6 Complex variables 74 4.7 What is inside the box? 75 4.8 Mixed-integer programming problems 76 4.9 Transforming MINLP into MILP 79 4.10 Further readings 80 4.11 Exercises 81 5 Conic optimization 85 5.1 Convex cones 85 5.2 Second-order cone optimization 85 5.2.1 Duality in SOC problems 90 5.3 Semidefinite programming 92 5.3.1 Trace, determinant, and the Shur complement 92 5.3.2 Cone of semidefinite matrices 95 5.3.3 Duality in SDP 97 5.4 Semidefinite approximations 98 5.5 Polynomial optimization 102 5.6 Further readings 105 5.7 Exercises 106 6 Robust optimization 109 6.1 Stochastic vs robust optimization 109 6.1.1 Stochastic approach 110 6.1.2 Robust approach 110 6.2 Polyhedral uncertainty 111 6.3 Linear problems with norm uncertainty 113 6.4 Defining the uncertainty set 115 6.5 Further readings 121 6.6 Exercises 121 Part II Power systems operation 125 7 Economic dispatch of thermal units 127 7.1 Economic dispatch 127 7.2 Environmental dispatch 133 7.3 Effect of the grid 136 7.4 Loss equation 140 7.5 Further readings 143 7.6 Exercises 143 8 Unit commitment 145 8.1 Problem definition 145 8.2 Basic unit commitment model 146 8.3 Additional constraints 150 8.4 Effect of the grid 151 8.5 Further readings 153 8.6 Exercises 153 9 Hydrothermal scheduling 155 9.1 Short-term hydrothermal coordination 155 9.2 Basic hydrothermal coordination 156 9.3 Non-linear models 159 9.4 Hydraulic chains 162 9.5 Pumped hydroelectric storage 165 9.6 Further readings 168 9.7 Exercises 169 10 Optimal power flow 171 10.1 OPF in power distribution grids 171 10.1.1 A brief review of power flow analysis 173 10.2 Complex linearization 177 10.2.1 Sequential linearization 181 10.2.2 Exponential models of the load 182 10.3 Second-order cone approximation 184 10.4 Semidefinite approximation 188 10.5 Further readings 190 10.6 Exercises 190 11 Active distribution networks 195 11.1 Modern distribution networks 195 11.2 Primary feeder reconfiguration 196 11.3 Optimal placement of capacitors 200 11.4 Optimal placement of distributed generation 203 11.5 Hosting capacity of solar energy 205 11.6 Harmonics and reactive power compensation 208 11.7 Further readings 212 11.8 Exercises 212 12 State estimation and grid identification 215 12.1 Measurement units 215 12.2 State estimation 216 12.3 Topology identification 221 12.4 Ybus estimation 224 12.5 Load model estimation 228 12.6 Further readings 231 12.7 Exercises 232 13 Demand-side management 235 13.1 Shifting loads 235 13.2 Phase balancing 240 13.3 Energy storage management 246 13.4 Further readings 249 13.5 Exercises 249 A The nodal admittance matrix 253 B Complex linearization 257 C Some Python examples 263 C.1 Basic Python 263 C.2 NumPy 266 C.3 MatplotLib 268 C.4 Pandas 268 Bibliography 271 Index 281