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      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

may be of short-term (1 day to 1 week), medium-term (1 month), or long-term (1 or more years), we are interested only in the short-term model. As aforementioned, the problem may be stochastic since power demands dt and inflows ajt are all random variables. However, a determinist model is suitable to understand the problem and its practical implementation. The situation becomes even more problematic when introducing other renewable energies, such as wind generation and photovoltaic solar generation. Chapter 9 presents the hydrothermal dispatch problem.

      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

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