Читать книгу Mathematical Programming for Power Systems Operation - Alejandro Garcés Ruiz - Страница 12

Mathematical optimization is a fundamental tool for the electrical supply chain, from generation through transmission, distribution, and end-use. It may also be used, in different time frames, from a few milliseconds to several years. This book concentrates on optimization problems for power systems operation. These problems are usually continuous and have a time frame from several minutes to one day. Optimization problems with faster dynamics lie in the control and stability analysis, whereas problems with slower dynamics are planning problems.

Mathematical optimization problems associated with power system operation have existed since the beginning of operations research as an independent area, back in the middle of the 20th century. However, modern technologies such as renewable energies and electric vehicles; and current concepts, such as smart-grids, active distribution networks, and microgrids, have created a renewed interest in mathematical optimization applied to power systems. Smart-grids implicate a massive use of technologies such as power electronics, communications, and advanced metering. However, the smart aspect of these grids comes from mathematical techniques such as mathematical optimization, that manage these technologies, in order to improve the efficiency, reliability, security, and resilience of the system.

Figure 1.1 depicts schematically four common types of mathematical optimization models. These are linear programming (LP), mixed-integer linear programming (MIP), non-linear programming (NLP), and mixed-integer non-linear programming (MINLP). Another classification is to separate them into convex and non-convex problems. The former include LPs and some NLPs; the latter is the rest of the problems. Convex problems are well-behaved in the sense that they have theoretical guarantees, such as global optimum and practical algorithms with fast convergence rate. The first part of the book presents these theoretical aspects. However, not all power systems operation problems are convex; therefore, we need to develop convex approximations for those problems, most of them based on conic optimization as presented in Chapter 5.

Figure 1.1 Types of optimization models.

A power system is quite complex, and therefore, modeling and implementing mathematical optimization problems are equally complex. We need to gain experience in the complex art of modeling and solving mathematical optimization problems for power system applications. Our approach is to create toy-models for each problem. These are simplified models that allow us to understand the central issues and do numerical experiments. In the following sections, we briefly describe each of these toy-models, explained in detail in the second part of the book.