Some of the problems previously described are binary. These problems appear in electrical systems, both in the planning and the operation stages. In the planning stage, the problems of transmission expansion planning and distribution planning are typical. Heuristic techniques usually solve these problems together with mixed-integer programming approximations [3]. Although mixed-integer problems are non-convex, they can be solved efficiently using methodologies such as the branch-and-bound algorithm. We present a brief introduction of this method in Chapter 4. Besides, several modules can be called from Python to solve these types of problems. We used Gurobi and Mosek for this task, although there are plenty of options available. Binary operation problems include unit commitment and phase balancing in power distribution networks.
1.3.1 Unit commitment
As aforementioned, the unit commitment problem consists in determining the order of starting and stopping of the thermal units, taking into account the costs of turning-ON and turning-OFF, as well as the starting ramps. The optimization model is similar to the economic dispatch, but in this case, there are binary variables sk related to the state ON or OFF of each thermal unit. The most simplified model is presented below:
where f represents operative costs, h are costs of turning-ON and turning-OFF, r are starting ramps, and pmin, pmax are the operation limits of each unit. Binary problems are usually difficult; however, it is possible to find either convex approximation or MILP equivalents as presented in Chapter 5.
1.3.2 Optimal placement of distributed generation and capacitors
We can use the same convex approximations of the optimal power flow problem in problems such as the optimal placement of distributed generation in power distribution networks. The problem consists of determining the placement and capacity of distributed generation, subject to physical constraints, such as voltage regulation and transmission lines’ capacity. Besides, the problem includes binary constraints related to the placement of the distributed generation. The objective function is usually total power loss, although other objectives, such as reliability and optimal costs, are also suitable.
Another binary problem closely related to the OPF is the optimal placement of capacitors. In this problem, fixed or variable capacitors are placed along the primary feeder to minimize power loss. The capacitors’ location and the number of fixed capacitors in each node are binary variables considered in the model. Chapter 11 examines the optimal placement of capacitors and the optimal placement of distributed generation.
1.3.3 Primary feeder reconfiguration and topology identification
The topology of a grid is not constant, especially in power distribution networks. There are several sectionalizing switches along with the primary feeders that permit transferring load among circuits. A feeder reconfiguration is an operating model that determines the optimal topology to minimize power loss. This model is non-linear, non-convex, and binary. In addition, it has a constraint related to the connectivity and radiality of the graph that is tricky to represent in equations. All these aspects are studied in Chapter 11.
Like the state estimation is the inverse of the optimal power problem, there is a problem that can be considered as the inverse of the primary feeder reconfiguration. It is the topology identification in power distribution grids, which takes measurements of current and voltages in different parts of the grid and determines the state of each sectionalizing switch. This problem is studied in Chapter 12.
1.3.4 Phase balancing
Phase balancing is a combinatorial problem that consists of phase swapping of the loads and generators to reduce power loss. Despite being a classic problem, it is still relevant since the unbalance is a common phenomenon in microgrids, especially when single-phase photovoltaic units are included. Because it is a combinatorial problem, phase balancing requires heuristic algorithms with high computational effort. However, it is possible to generate simplified instances of the problem, as presented in Chapter 13.
Each three-phase node has six possible configurations as depicted in Figure 1.6. The problem consists of defining the phase in which each load or generator is connected in order to reduce the grid’s total losses. Therefore, the problem is combinatorial since there are 6n possible configurations, where n is the number of three-phase nodes. The problem is nontrivial even in small networks; for example, a microgrid with 10 nodes will have 6 × 107 possible configurations.
Figure 1.6 Set of possible configurations in a three-phase node.
Phase-balancing problems appear in many applications such as aircraft electric systems [4], and in power distribution grids with high penetration of electric vehicles [5]. Due to its combinatorial nature, the problem necessitates the use of heuristics [6], and meta-heuristics [7] as well as expert systems [8]. Modern approaches include the uncertainty associated to the load and generator [9].
1.4 Real-time implementation
A receding horizon control can implement most of the optimization algorithms presented in this book. Figure 1.7 depicts the main architecture for this simple but powerful strategy, for real-time implementation of operation models. These optimization models may be the optimal power flow, economic dispatch, energy storage management, or a mixed model that includes multiple models. An unbiased forecast predicts variables such as wind speed, solar radiation, and power demand. Moreover, a state estimator gives accurate measurements of the system variables.
Figure 1.7 A possible architecture for implementing an optimization model for power systems operation.
(Equation 1.10) represents the optimization model, where x is the vector of decision variables for each time t, and α are the parameters predicted by the forecast module. Of course, this forecast may change since renewable resources and loads may be highly variable in modern power systems. Therefore, the optimization model must be continuously executed and the solution updated.
