###### Abstract  
This paper proposes a new hybrid master-slave optimization approach to address the problem of the optimal placement and sizing of distribution static compensators (D-STATCOMs) in electrical distribution grids. The optimal location of the D-STATCOMs is identified by implementing the classical and well-known Chu and Beasley genetic algorithm, which employs an integer codification to select the nodes where these will be installed. To determine the optimal sizes of the D-STATCOMs, a second-order cone programming reformulation of the optimal power flow problem is employed with the aim of minimizing the total costs of the daily energy losses. The objective function considered in this study is the minimization of the annual operative costs associated with energy losses and installation investments in D-STATCOMs. This objective function is subject to classical power balance constraints and device capabilities, which generates a mixed-integer non-linear programming model that is solved with the proposed genetic-convex strategy. Numerical validations in the 33-node test feeder with radial configuration show the proposed genetic-convex model's effectiveness to minimize the annual operative costs of the grid when compared with the optimization solvers available in GAMS software. annual operational cost minimization; Chu and Beasley genetic algorithm (CBGA); daily active and reactive demand curves; distribution static compensators (D-STATCOMs); radial distribution networks; reactive power compensation +
Footnote †: journal: Applied Sci. 2021, 11, 40, 11, 12  
can operate continuously in a range of operations, permitting constant reactive power injections depending on the demand load behavior, and these can vary their outputs as a function of the grid operating conditions [12; 13]. This implies that these can be considered as dynamic reactive power compensators [14]. The general structure of a D-STATCOM is depicted in Figure 1, where it can be noted that the power electronic converter makes it possible to control the amount of reactive power injected into the grid [10]. In the literature, the problem of the optimal placement and sizing of D-STATCOMs has been studied with different optimization methodologies for medium-voltage applications [15]. Following are some of these approaches. In [9], analytical and heuristic optimization methods were used for the optimal placement and sizing of D-STATCOMs in distribution networks. In [9], fitness functions based on voltage stability and power loss indices were implemented to enhance distribution network performance. In [10], a multi-objective particle swarm optimizer was utilized to size and locate D-STATCOMs in a medium-voltage electrical network, using simultaneous system reconfiguration. Although in [10] a multi-objective optimizer was used (e.g., active power loss reduction, voltage stability index, and distribution line loadability factor), only the load condition was considered, which is not appropriate as it leads to selection of over-sized devices. In [16], a heuristic algorithm based on costs was developed to assess the performance in an isolated power system based on location and dimensioning of D-STATCOMs. Nevertheless, the analysis performed in [16] was greatly simplified as it modeled the power system as a unique nodal system. In [17], a genetic algorithm and optimal power flow implemented in DigSILENT software were presented to size and place D-STATCOMs in distribution networks. Although [17] used a real distribution grid to test the algorithm, a cost analysis was not shown. In [18], a fuzzy multi-objective algorithm using an ant colony optimization approach was presented to improve the distribution system's performance regarding locating and sizing photo-voltaic generations and D-STATCOMs. The fitness functions used in [16] were the reduction of power losses, improvement of the voltage profiles, and load balance increase.