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.
1.2 Continuous models
1.2.1 Economic and environmental dispatch
The economic and environmental dispatch of thermal units is one of the most classic problems in power systems operation. It consists of minimizing the operating costs or the total CO2 emissions, subject to physical constraints such as the power balance and the maximum generation capacity. For the economic dispatch, each generation unit has a cost function fi, which is usually quadratic and depends on the power generated by each unit. Thus, the objective is to minimize the total cost (or emissions), subject to power balance, as presented below:
where pi is the power generated by each thermal unit, and d is the total demand. Environmental dispatch introduces quadratic or exponential functions in the objective function, but the problem’s structure is the same. Moreover, power flow constraints can be introduced into the model, although, in that case, it is more precise to name the problem as an optimal power flow (OPF). Chapter 7 presents the economic and environmental dispatches, while Chapter 10 presents the OPF problem.
Another problem closely related to the economic dispatch of thermal units is the unit commitment. This problem considers not only the operating costs but also the start-up and shut-down costs of thermal units. Therefore, the problem becomes binary and dynamic. This problem is studied in Chapter 8.
1.2.2 Hydrothermal dispatch
The economic dispatch problem may include hydroelectric power plants; said plants, generate two fundamental changes into the model. On the one hand, the model becomes dynamic since current operational decisions affect the future operation of the system. On the other hand, the problem becomes stochastic, because the inflows are usually random variables, especially in long-term models. The later aspect is usually solved by an accurate forecasting of the loads and the inflows; hence, it is possible to formulate a deterministic problem, called hydrothermal dispatch or hydrothermal coordination. The basic model has the following structure, namely:
Where i represents thermal units and j enumerates hydroelectric units; t represents the time, thus, pit is the power generated by the thermal unit i at time t. The values of ajt, vjt, pjt, qjt and sjt are respectively, the inflow, volume, power, outflow, and spillage of the hydroelectric unit j at time t. Figure 1.2, which is self-explanatory, shows these variables.
Figure 1.2 Schematic representation of the variables associated to a hydroelectric generation unit.
In this model, g represents the relation between generated power, outflow, and water volume stored in the dam. Although the planning horizon
1.2.3 Effect of the grid constraints
Both the economic dispatch of thermal units and the hydrothermal dispatch are relatively simple problems. However, the transmission grid introduces additional constraints. Let us consider, for example, a power system with three operation areas, as shown in Figure 1.3.
Figure 1.3 Economic dispatch by areas considering
