# Hybrid GA-SOCP Approach for Placement and
## 3 Proposed Hybrid GA-SOCP Optimization Approach
### Slave Stage: Second-Order Cone Programming  
The SOCP is a branch of the convex optimization that allows transforming some nonlinear non-convex optimization problems into conic convex equivalents that guarantee the global optimum of the equivalent model [30]. In the case of the optimal power flow problem, the SOCP equivalent model has the same global optimum that the original nonlinear model, which is the main advantage of this technique in power system analysis [29]. Here, we present the SOCP version of the optimal power flow problem that addresses distributed generators' optimal sizing. For this purpose, let us suppose that all of the binary variables regarding the distributed generators' location, i.e., \(y_{i}\) has been fixed by the master stage, which will be presented in this study. In this sense, the SOCP deals with the Equality constraints (2) and (3), which are the only continuous nonlinear non-convex constraints.  
The convexification of the optimal power flow problem is made using the following auxiliary variable:  
\[z_{ij}=v_{i}v_{j}, \tag{10}\]  
note that, from (10), we know that \(z_{ii}=v_{i}^{2}\) and \(z_{jj}=v_{j}\), which implies that if we elevate both sides of (10) at the square, then we have:  
\[z_{ij}^{2}=z_{ii}z_{jj}. \tag{11}\]  
Now, in order to obtain an equivalent representation of (11), we use the hyperbolic relation of the product between two variables as follows:  
\[z_{ij}^{2} =z_{ii}z_{jj},\] \[=\frac{1}{4}\left(z_{ii}+z_{jj}\right)^{2}-\frac{1}{4}\left(z_{ii} -z_{jj}\right)^{2},\] \[(2z_{ij})^{2}+\left(z_{ii}-z_{jj}\right)^{2} =\left(z_{ii}+z_{jj}\right)^{2},\] \[\left\|\begin{matrix}2z_{ij}\\ z_{ii}-z_{jj}\end{matrix}\right\| =z_{ii}+z_{jj}. \tag{12}\]  
Note that Equation (12) is the conic equivalent of (11), since, to reach this result, only algebraic manipulations have been made; in addition, it is still non-convex due to the equality symbol; nevertheless, it can be relaxed with a lower-equal symbol as recommended in [13; 31; 32], which becomes (12) into a conic convex constraint, as presented below:  
\[\left\|\begin{matrix}2z_{ij}\\ z_{ii}-z_{jj}\end{matrix}\right\|\leq z_{ii}+z_{jj}. \tag{13}\]  
**Remark 2**.: _To obtain a SOCP equivalent of the optimal power flow problem, we can recur to substitute (11) into Expressions (1) to (3), which produce equivalent mixed-integer second-order cone programming (i.e., MISOCP) model to address the problem of the optimal placement and sizing of distributed generators in DC distribution networks._  
The complete MISOCP model is presented below:  
\[\min p_{\text{loss}}=\sum_{i\in\mathcal{N}}\sum_{j\in\mathcal{N}}G_{ij}z_{ij}, \tag{14}\]  
\[p_{i}^{s}+p_{i}^{gd}-p_{i}^{d}=\sum_{j\in\mathcal{N}}G_{ij}z_{ij},\;\forall i\in \mathcal{N} \tag{15}\]  
\[p_{ij}=g_{ij}\left(z_{ii}-z_{ij}\right),\;\forall\left(i,j\right)\in\mathcal{L} \tag{16}\]  
\[\left\|\begin{matrix}2z_{ij}\\ z_{ii}-z_{jj}\end{matrix}\right\|\leq z_{ii}+z_{jj},\forall\left(i,j\right)\in \mathcal{N} \tag{17}\]  
Expressions (4) to (9), (18)  
**Remark 3**.: _Observe that the slave optimization stage solves the proposed MI-SOCP model once all the binary variables have been assigned on the master stage with the main advantage that the SOCP structure of the power flow model guarantees the global optimum finding [30]._