# A Multi-Period Optimal Reactive Power Dispatch Approach Considering Multiple Operative Goals
## 1 Introduction  
The ORPD can be seen as an AC optimal power flow problem with a particular set of control variables, which include shunt compensations, transformer tap changers, and voltage generator set points. In this case, the active power outputs of all generators, with the exception of the slack bus, are kept constant [1]. The ORPD problem plays an important role in the economic and secure operation of power systems, since it is one of the ways to solve the problems of reactive power management and voltage control [2]. The main idea behind the ORPD problem is finding an optimal schedule for the reactive power control devices within a network in such a way that operational constraints are met while also optimizing a given objective function [3]. Numerous ORPD formulations and solution methods have been reported in the literature to address specific instances of the problem [4]. These include different objective functions, controls, and system constraints, as well as distinct mathematical characteristics and computational requirements [5]. The resulting optimization problems go by many names, depending on the particular objective function being addressed and the constraints under consideration. Given the nonlinear and nonconvex nature of the ORPD problem, numerous meta-heuristic techniques have also been proposed for its solution. In [6], the authors proposed a particle swarm optimization (PSO) approach to deal with the ORPD problem. PSO is inspired by the sociological behavior associated with schools of fish and flocks of birds. Some modifications have been proposed to this technique to better handle the ORPDproblem, such as parallel vector-evaluated PSO [7], coordinated aggregation PSO [8], fractional PSO [9], and turbulent crazy PSO [10]. Genetic algorithms (GAs) are metaheuristic algorithms that mimic the process of Darwinian natural selection. In this case, a set of individuals or candidate solutions must go through stages of selection, crossover, and mutation in order to improve a given fitness function. In [11], the authors proposed a GA to solve the ORPD problem, seeking loss minimization as well as the maximization of the voltage stability margin. For this last objective, the L-index of the system was used. A self-adaptive real coded GA was also implemented in [12] to approach the ORPD problem. Differential evolution (DE) is also a population-based technique that iteratively improves a candidate solution based on an evolutionary process. In [13; 14; 15], the authors used DE to solve the ORPD problem. The mean-variance mapping optimization algorithm has also been applied to tackle the ORPD problem [16]. This metaheuristic algorithm is based on a mapping function used to generate new candidate solutions based on the mean and variance of a set that contains the current best solutions of the algorithm. Other metaheuristic techniques applied to the ORPD problem include the seeker optimization algorithm [17], gravitational search algorithm [18], harmony search algorithm [19], teaching-learning-based optimization [20], krill herd algorithm [20], chaotic bat algorithm [21], evolutionary algorithm [22], heap-based optimizer [23], hierarchical distributed optimization [24], and gray wolf optimizer algorithm [25; 26]. Hybrid approaches combine characteristics of two or more metaheuristic techniques to take advantage of their strengths. In [27; 28; 29], the authors proposed hybrid versions of PSO and the firefly algorithm to solve the ORPD problem. A detailed description of the techniques applied to the ORPD problem is outside the scope of this paper; nonetheless, a detailed classification and comparison can be found in [30]. The common feature of the aforementioned methodologies is the fact that they approach the static version of the ORPD problem; that is to say, the optimization is carried out over a single time period. Real-life ORPD problems must consider a multi-period approach and are, therefore, often significantly more challenging than the classically considered problems [31].