# Hybrid GA-SOCP Approach for Placement and
## 2 MINLP Model  
The problem of the optimal placement and sizing of DGs in DC networks corresponds to an MINLP model since: (i) the problem of sizing of these power sources is modeled with binary variables, and (ii) the problem of the location of these DGs is modeled with continuous variables (power generation and voltage profiles). The optimization objective in this MINLP formulation corresponds to the minimization of the grid power losses produced by the current circulation in all the networks' branches. The mathematical structure of this objective function is presented in Equation (1). \[\min p_{\text{loss}}=\sum_{i\in\mathcal{N}}\sum_{j\in\mathcal{N}}G_{ij}v_{i}v_ {j}, \tag{1}\]  
where \(p_{\text{loss}}\) is the value of the objective function regarding power losses in all of the branches of the network, \(v_{i}\) and \(v_{j}\) are the voltage values at nodes \(i\) and \(j\), respectively, and \(G_{ij}\) is the conductance value taken from the admittance nodal matrix that defines the physical interconnection among all the nodes of the DC network. Observe that \(\mathcal{N}\) is the set that contains all of the nodes of the network. Regarding model constraints, these are related with operational conditions of the DC distribution networks. These constraints are presented below:  
\[p_{i}^{s}+p_{i}^{gd}-p_{i}^{d}=\sum_{j\in\mathcal{N}}G_{ij}v_{i}v_{j},\ \forall i\in \mathcal{N} \tag{2}\]  
\[p_{ij}=g_{ij}v_{i}\left(v_{i}-v_{j}\right),\ \forall\left(i,j\right)\in \mathcal{L} \tag{3}\]  
\[\sum_{i\in\mathcal{N}}p_{i}^{gd}\simeq\alpha\sum_{i\in\mathcal{N}}p_{i}^{d}, \tag{4}\]  
\[\sum_{i\in\mathcal{N}}y_{i}\leq N_{dg}^{\text{area}}, \tag{5}\]  
\[y_{i}p^{gd,\min}\leq p_{i}^{gd}\leq y_{i}p^{gd,\max}\ \forall i\in \mathcal{N} \tag{6}\]  
\[p_{ij}^{\min}\leq p_{ij}\leq p_{ij}^{\max}\ \forall\left(i,j\right)\in \mathcal{L} \tag{7}\]  
\[v^{\min}\leq v_{i}\leq v^{\max}\ \forall i\in\mathcal{N} \tag{8}\]  
\[y_{i}\in\left\{0,1\right\}\ \forall i\in\mathcal{N} \tag{9}\]  
where \(p_{i}^{g}\)\(p_{i}^{gd}\), \(p_{i}^{d}\), and \(p_{ij}\) represent the power generation in the voltage controlled source (slack nodes), power generation in distributed sources, power consumption in demand nodes, and power flow in distribution lines, respectively. \(\alpha\) is a parameter that defines the percentage of power penetration of renewable generation as function of the total demand; \(y_{i}\) is the binary variable associate with the location (\(y_{i}=1\)) or no (\(y_{i}=0\)) of a distributed generator at node \(i\); \(N_{dg}^{\text{area}}\) is the maximum number of distributed generators available for location in the DC network, \(p^{gd,\min}\) and \(p^{gd,\max}\) are the minimum and maximum power generation capabilities allowed for each distributed generator connected at node \(i\). \(p_{ij}^{\min}\) and \(p_{ij}^{\max}\) are the lower and upper bounds for the power flow in line that connects nodes \(i\) and \(j\), respectively; and, \(v^{\min}\) and \(v^{\max}\) are the minimum and maximum voltage bounds in all nodes of the DC distribution network, respectively. The complete interpretation of the MINLP model defined from (1) to (9) is as follows: Equation (1) defines the objective function of the problem regarding the minimization of the total grid power losses in the whole DC network; Equation (2) is the set of power balance equations applied at each node of the network; Equation (3) defines the amount of power that flows at each line of the DC grid. Expression (4) determines the maximum penetration of the distributed generation in the whole DC distribution network; Inequality constraint (5) bounds the number of generators available for location in the DC grid; inequalities (6), (7), and (8) present the upper and lower bounds for distributed generators, power flow in branches, and voltage variables, respectively. Finally, Expression (9) shows the binary nature of the decision variable regarding the location of distributed generators.