## 3 Materials and Methods
### Vortex Search Algorithm (VSA)  
The VSA is a solution based on a meta-heuristic approach that allows optimization of problems such as MINLP, whose operating principle is based on the search for vortices that adaptively change in size. It has two stages; the explorer stage is in charge of looking for a solution space, and the exploiting phase is in a load of evaluating the answers to find the optimal one within the solution space. This algorithm can model through hyper-ellipses enclosed one after another; the first solution space of the explorer stage defines a hyper-ellipse whose center is the first answer to the problem [33]. For the case under study, the dimension proposed is six; since it establishes a maximum of three possible PV systems, whose vectors must contain three different locations. These locations correspond to the nodes where the PV system localizes and three sizes according to their capacity. \(x^{\min}\) and \(x^{\max}\) represent the minimal and maximum value of the vector and are value between 2 and \(N\), with \(N\) being the maximum number of nodes in the network, where the minimum and maximum value of the PV system capacity is given between 0 and 2000 kW, as stated in [6]. Therefore, Equation (7) helps to find the center of the hyper-ellipse. \[\mu_{0}=\left(\frac{x_{1}^{\min}+x_{1}^{\max}}{2},\frac{x_{2}^{\min}+x_{2}^{ \max}}{2},\frac{x_{3}^{\min}+x_{3}^{\max}}{2},\frac{x_{4}^{\min}+x_{4}^{\max}} {2},\frac{x_{5}^{\min}+x_{5}^{\max}}{2},\frac{x_{6}^{\min}+x_{6}^{\max}}{2}\right) \tag{7}\]  
Furthermore, it has a set of neighboring solutions \(C_{i}^{t}(x)\) where \(i\) corresponds to the position of the potential solution and \(t\) represents the current iteration of the algorithm, which initializes at 0. Said solutions within the solution space, defined by dimension six, are randomly generated using a Gaussian distribution, represented by Equation (8). \[C_{i}^{t}(x)=p(x|\mu,\Sigma)=\frac{1}{\sqrt{(2\pi)^{d}|\Sigma|}}exp\left\{- \frac{1}{2}(x-\mu)^{T}\Sigma^{-1}(x-\mu)\right\} \tag{8}\]  
where \(d\) represents the dimension, \(x\) is the solution vector \(d\times 1\) of a random value, \(\mu\) is the vector \(d\times 1\) of the sample mean, and \(\Sigma\) is the covariance matrix. If the value of the diagonal elements of \(\Sigma\) are equal and those outside the diagonal are zero, then the resulting shape of the distribution will be a hyper-ellipse; therefore, the value of \(\Sigma\) can be calculated using equal variances with zero covariance, as presented in Equation (9). \[\Sigma=\sigma^{2}\cdot[I]_{d\times d} \tag{9}\]  
where \(\sigma\) represents the variance of the distribution, and \(I\) represents the identity matrix of dimension \(d\times d\). Equation (10) calculates the initial standard departure \(\sigma_{0}\) of the distribution. \[\sigma_{0}=\frac{\max\{x_{1}^{\max},x_{2}^{\max},x_{3}^{\max},x_{4}^{\max},x_{5 }^{\max},x_{6}^{\max}\}-\min\{x_{1}^{\min},x_{2}^{\min},x_{3}^{\min},x_{4}^{ \min},x_{5}^{\min},x_{6}^{\min}\}}{2} \tag{10}\]  
where \(\sigma_{0}\) is also considered the initial radius \(r_{0}\) of the outer hyper ellipse, that is, the first solution space [34]. As a first step, by multiplying the initial radius by Equation (12), this solution space is expanded to obtain complete coverage. The set of neighboring solutions obtained and contained in \(C_{i}^{t}(x)\), verified within the algorithm and taking into account the lower and upper limits set in \(\mu_{0}\), are defined by Equation (11). \[C_{i}^{t}(x)=\left\{ \begin{array}{cc}C_{i}^{t}(x),&x^{\min}\leq x\leq x^{ \max}\\ x^{\min}+\big{(}x^{\max}-x^{\min}\big{)}rand,&\text{otherwise}\end{array} \right. \tag{11}\]  
where \(rand\) is a function that generates a random number between 1 and 0 with a normal distribution; after it verifies the limits, at this point, the exploiter stage comes into operation. It selects the most optimal response within the solution space of \(C_{i}^{t}\) obtained in (11) so that it is the new center \(\mu\) of the hyper-ellipsoid [35].