Alejandro Garcés Ruiz - Mathematical Programming for Power Systems Operation

141 128

142 129

143 130

144 131

145 132

146 133

147 134

148 135

149 136

150 137

151 138

152 139

153 140

154 141

155 142

156 143

157 144

158 145

159 146

160 147

161 148

162 149

163 150

164 151

165 152

166 153

167 154

168 155

169 156

170 157

171 158

172 159

173 160

174 161

175 162

176 163

177 164

178 165

179 166

180 167

181 168

182 169

183 170

184 171

185 172

186 173

187 174

188 175

189 176

190 177

191 178

192 179

193 180

194 181

195 182

196 183

197 184

198 185

199 186

200 187

201 188

202 189

203 190

204 191

205 192

206 193

207 194

208 195

209 196

210 197

211 198

212 199

213 200

214 201

215 202

216 203

217 204

218 205

219 206

220 207

221 208

222 209

223 210

224 211

225 212

226 213

227 214

228 215

229 216

230 217

231 218

232 219

233 220

234 221

235 222

236 223

237 224

238 225

239 226

240 227

241 228

242 229

243 230

244 231

245 232

246 233

247 234

248 235

249 236

250 237

251 238

252 239

253 240

254 241

255 242

256 243

257 244

258 245

259 246

260 247

261 248

262 249

263 250

264 251

265 252

266 253

267 254

268 255

269 256

270 257

271 258

272 259

273 260

274 261

275 262

276 263

277 264

278 265

279 266

280 270

281 271

282 272

283 273

284 274

285 275

286 276

287 277

288 278

289 279

290 280

291 281

292 282

Throughout the writing of this book, I have received a great deal of support and assistance from many people. I would first like to thank my friends Lucas Paul Perez at Welltec, Adrian Correa at Universidad Javeriana in Bogotá-Colombia, Ricardo Andres Bolaños at XM (the transmission system operator in Colombia), Raymundo Torres at Sintef-Norway, and Juan Carlos Bedoya at the Pacific Northwest National Laboratory (USA), who, in 2020 (during the COVID-19 pandemic), agreed to discuss some practical aspects associated to power system operation problems. The discussions during these video conferences were invaluable to improve the content of the book. I am also very grateful to my students, who are the primary motivation for writing this book. Special thanks to my former Ph.D. students, Danilo Montoya and Walter Julian Gil. Finally, I want to thank the Department of Electric Power Engineering at the Universidad Tecnológica de Pereira in Colombia and the Von Humbolt Foundation in Germany for the financial support required to continue my research about the operation and control of power systems.

Alejandro Garcés

Electrification is the most outstanding engineering achievement in the 20th century, a well-deserved award if we consider the high complexity of generation, transmission, and distribution systems. An electric power system includes hundreds or even thousands of generation units, transformers, and transmission lines, located throughout an entire country and operated continuously 24 hours per day. Running such a complex system is a great challenge that requires using advanced mathematical techniques.

All industrial systems seek to increase their competitiveness by improving their efficiency. Electric power systems are not the exception. We can improve efficiency by introducing new technologies but also by implementing mathematical optimization models into daily operation. In every mathematical programming model, we require to perform four critical stages depicted in Figure . The first stage is an informed review of reality, identifying opportunities for improvement. This stage may include conversations with experts in order to establish the available data and the variables that are subject to be optimized. The second stage is the formulation of an optimization model as given below:

(0.1)

Where x is the vector of decision variables, f is the objective function and, Ω is the set of feasible solutions. Going from stage one (reality) to stage two (model) is more of an art than a science. One problem may have different models and different degrees of complexity. Practice and experience are required to master this stage, as some models are easier to solve than others. Subsequently, the third stage consists of the implementation of the mathematical model into a software. After that, the fourth stage is the analysis of results in the context of the real problem.

Figure 0.1Stages of solving an optimization problem.

This book will focus on stages two and three, associated with power system operations models. In particular, we are interested in models with a geometric characteristic called convexity , that present several advantages, namely:

We can guarantee the global optimum and unique solution under well-defined conditions. This aspect is interesting from both theoretical and practical points of view. In general, a global optimum advisable in real operation problems.

There are efficient algorithms for solving convex problems. In addition, we can guarantee convergence of these algorithms. This is a critical aspect for operation problems where the algorithm requires to be solved in real-time.

There are commercial and open-source packages for solving convex optimization models. In particular, we are going to use CvxPy, a free Python-embedded modeling language for convex problems.

Many power system operations problems are already convex; for example, the economic and environmental dispatches, the hydrothermal coordination, and the load estimation problem. Besides, it is possible to find efficient convex approximations to non-convex problems such as the optimal power flow.

In summary, convex problems have both theoretical and practical advantages for power systems operation. This book studies both aspects. The book is oriented to bachelor and graduated students of power systems engineering. Concepts related to power systems analysis such as per-unit representation, the nodal admittance matrix, and the power flow problem are taken for granted. A previous course of linear programming is desirable but not mandatory. We do not pretend to encompass all the theory behind convex optimization; instead, we try to present particular aspects of convex optimization which are useful in power systems operation. The book is divided into two parts: In the first part, the main concepts of convex optimization are presented, including a distinct chapter about conic optimization. After that, selected applications for power systems operation are presented. Most of the solvers for convex optimization allow mixed-integer convex problems. Therefore, we include models that can be solved in this framework too. The student is recommended to do numerical experiments in order to acquire practical intuition of the problems.