# Optimal Siting and Sizing of FACTS in Distribution Networks Using the Black Widow Algorithm
## 5 Numerical Results  
For the computational implementation of the proposed master-slave strategy, MATLAB software, Mathworks, Natick, Massachusetts, USA (version 2021_b_) was employed on a PC with an AMD Ryzen 7 3700 2.3 GHz processor and 16.0 GB RAM, running a 64-bit version of Microsoft Windows 10 Single Language. The BWO and the successive approximations power flow method were implemented with our own scripts. The parametrization of the BWO includes 100 repetitions, 10 individuals in the population, and 1000 iterations. The following simulation scenarios were considered to validate the effectiveness of our approach to locating and sizing FACTS in electrical distribution networks. 1. C1: The validation of the proposed optimization approach with respect to the literature reports in the case of the IEEE 33- and 69-bus grids regarding the installation of SVCs, which the authors of [25] have denoted as D-STATCOMs. For comparison purposes, this research uses the vortex search algorithm (VSA), the hybrid genetic algorithm with the particle swarm optimizer (GA/PSO), and the solution of the exact MINLP model in GAMS. In order to make fair comparisons, all the listed combinatorial optimizers were set with the same population sizes, number of iterations, and stopping criteria. 2. C2: The evaluation of the proposed optimization approach, considering all three FACTS analyzed (i.e., VSC, UPFC, and TCSC devices) in the IEEE 33-, 69, and 85-bus grids. Regarding the optimization algorithms used for comparison, it is essential to mention the following. 1. The VSA is a combinatorial method from the family of physics-inspired optimizers that models the behavior of fluids being shaken in pipes [41]. This behavior is modeled through Gaussian distributions that allow exploring and exploiting the solution space in optimization problems, starting with the highest possible hyper-ellipse that covers the entire multi-dimensional space, where the hyper-center represents the best current \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Par.** & **Value** & **Unit** & **Par.** & **Value** & **Unit** \\ \hline \(C_{\text{KWh}}\) & 0.1390 & USD/kWh & \(T\) & 365 & Days \\ \(\Delta_{h}\) & 0.50 & h & \(k_{1}\) & 6/2190 & 1/Days \\ \(k_{2}\) & 10 & Years & — & — & — \\ \hline \hline \multicolumn{6}{c}{**FACTS parameters**} \\ \hline
**Device** & \(\mathbf{\omega_{1}}\)**(USD/Mvar\({}^{3}\))** & \multicolumn{2}{c}{\(\mathbf{\omega_{2}}\)**(USD/Mvar\({}^{2}\))**} & \(\mathbf{\omega_{1}}\)**(USD/Mvar)** \\ \hline SVC & 0.30 & \(-\)305.10 & 127,380 \\ TCSC & 1.50 & \(-\)713.00 & 153,750 \\ UPFC & 0.30 & \(-\)269.10 & 188,220 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parametrization of the objective function considering different FACTS devices. Figure 3: Active and reactive power demand in a Colombian electrical distribution grid. solution [42]. The main idea is that all solutions are uniformly distributed in this hyper-ellipse with a Gaussian distribution. Then, during the exploration process, the radius of the hyper-ellipse is continuously reduced via Gamma functions, which allows starting to exploit the solution space in the most promising region of solutions [43]. 2. The GA/PSO approach is a combination of the classical genetic algorithm (GA) with the particle swarm optimizer (PSO) in a master-slave connection, where the GA is used to define the set of nodes where the FACTS must be installed (using discrete codification). In contrast, the PSO approach is entrusted with exploring the solution space to define the best FACTS sizes [25]. The main characteristic of this master-slave optimizer is the use of two nature-inspired algorithms to address complex combinatorial optimization problems, where the GA is selected due to its excellent performance for discrete variables [44], and the PSO is chosen based on its effectiveness in dealing with continuous variables [45]. 3.