## 2 MINLP Model
### Restrictions  
The optimal location problem and sizing of PV systems to minimize energy losses must accomplish different restrictions. Equation (2) describes the power balance that occurs in the distribution system. \[\mathbb{S}_{ih}^{s}+\mathbb{S}_{ih}^{pv}-\mathbb{S}_{ih}^{d}=\mathbb{V}_{ih} \sum_{j\in N}\Big{(}\mathbb{Y}_{ij}\mathbb{V}_{ih}\Big{)}^{*},\,\forall\{i, j\,\epsilon\,N\,\,\,\&\,\,\,h\,\epsilon\,H\} \tag{2}\] \begin{table}
\begin{tabular}{c l c} \hline \hline
**Acronym** & **Optimization Method** & **Ref.** & **Year** \\ \hline GA & Genetic Algorithm & [16] & 2012 \\ PSO & Particle Swarm Optimization & [16] & 2012 \\ LSFSA & Loss Sensitivity Factor Simulated Annealing & [17] & 2013 \\ TLBO & Teaching Learning Based Optimization & [18] & 2014 \\ PMC & Parallel Monte Carlo & [4] & 2014 \\ HSA & Harmony Search Algorithm & [19] & 2014 \\ QOTLBO & Quasi-Oppositional Teaching Learning Based Optimization & [20] & 2014 \\ HSA-PABC & Harmony Search Algorithm and Particle Artificial Bee Colony Algorithm & [21] & 2014 \\ MINLP & Mixed-Integer Nonlinear Programming Formulation & [10] & 2014 \\ REPSO & Rank Evolutionary Particle Swarm Optimization & [22] & 2015 \\ RBFNN-PSO & Radial Basis Function Neural Network and Particle Swarm Optimization & [23] & 2015 \\ AHA & Algorithmic Heuristic Approach & [24] & 2016 \\ GA-IWD & Genetic Algorithm and Intelligent Water Drops & [25] & 2016 \\ KHA & Krill-Herd Algorithm & [26] & 2016 \\ SOS & Symbiotic Organism Search & [27] & 2017 \\ PBL-PSO & Population-Based Incremental Learning and Particle Swarm Optimizer & [4] & 2018 \\ ABCA & Artificial Bee Colony Algorithm & [28] & 2018 \\ MOHTLBOGWO & Multi-Objective Hybrid Teaching–Learning Based Optimization-Grey Wolf Optimizer & [29] & 2019 \\ MSSA & Mutated Salp Swarm Algorithm & [30] & 2019 \\ CHVSA & Constructive Heuristic Vortex Search Algorithm & [31] & 2019 \\ GAMS & General Algebraic Modeling System & [14] & 2020 \\ CBGA-VSA & Chu and Beasley Genetic Algorithm and Vortex Search Algorithm & [5] & 2020 \\ DSCA-SOCP & Discrete Sine Cosine Algorithm and Second-Order Cone Programming & [6] & 2021 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Methods recently used to reduce energy losses in distribution systems. where \(\mathbb{S}_{ih}^{s}\) is the complex power of the slack node connected to bus \(i\) in period \(h\), \(\mathbb{S}_{ih}^{pv}\) is the complex power of the PV system connected to bus \(i\) in period \(h\), and \(\mathbb{S}_{ih}^{d}\) is the complex power node demanded \(i\) in the term time \(h\). The second restriction corresponds to the voltage of each one of the system's nodes and is described by Equation (3). \[V_{\min}\leq|\mathbb{V}_{ih}|\leq V_{\max},\ \forall\{i\ e\ N\ \ \&\ h\ e\ H\} \tag{3}\]  
where \(V_{\min}\) and \(V_{\max}\) represent the lower and upper voltage bounds for each system node in different periods. The third restriction refers to the generation capacity of a PV system and is described by Equation (4). \[z_{i}P_{\min}^{pv}\leq P_{i}^{pv}\leq z_{i}P_{\max}^{pv},\ \forall\{i\ e\ N\} \tag{4}\]  
where  
\[z_{i}\ \epsilon\{0,1\},\forall\{i\ e\ N\} \tag{5}\]  
\(z_{i}\) describes the binary variable of the problem associated with the location of the PV systems; if \(z_{i}=1\), this represents the installation of a PV system in node \(i\). Otherwise, \(z_{i}=0\), \(P_{\min}^{pv}\) and \(P_{\max}^{pv}\) are the minimum and maximum active power limits that a PV system can inject, and \(P_{i}^{pv}\) is the active power injected by a system PV in node \(i\). It considers that the PV systems will only have an active power injection; therefore, for this model \(Q_{i}^{pv}=0\)[14]. Finally, Equation (6) describes the maximum number of PV systems set up in the distribution system. \[\sum_{i\in N}z_{i}\leq N_{pv} \tag{6}\]  
where \(N_{pv}\) is the maximum number of PV systems to install.