# Hybrid GA-SOCP Approach for Placement and
## 5 Computational Implementation
### 21-Node Test Feeder  
Table 3 reports the numerical comparisons between the proposed hybrid optimization method and the literature approaches for the 21-node test feeder. Note that this problem's optimal solution corresponds to nodes 9, 12, and 16, with power injections of 0.8350 pu, 1.0258 pu, and 1.4632 pu, respectively.  
From Table 3, we can observe that:  
* All of the comparison methods, as well as the proposed approach, identifies the best optimal location in nodes 9, 12, and 16. However, due to the aleatory nature of the continuous metaheuristics (i.e., BHO, PSO, and CGA), both of them are stuck local optimums, and only the GA-PSO approach can deal with the global optimal solution.
* The mixed-integer quadratic programming (MIQP) model and the SBB solver (for the exact MINLP model) reach the global optimal solution with the lowest computational effort, which is attributable to the efficiency of the GAMS interface.
* The standard deviation of the proposed GA-SOCP programming is the lowest when compared with metaheuristics; this situation occurs because the proposed approach works with an exact optimization method based on second-order cone programming that has a unique solution for each combination of DGs provided by the GA in the master stage.  
Figure 3 presents the histogram after the 100 consecutive evaluations of the proposed methodology for optimal placement and sizing distributed generators in DC distribution networks in order to present the efficiency of the proposed GA-SOCP approach.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **Nodes** & \(\sigma\)**[p.u.]** & \(\mu\)**[p.u.]** & **min [p.u.]** & \(t_{ave}\)**[s]** \\ \hline
**GA-BHO** & [9,12,16] & \(2.2761\times 10^{-03}\) & 0.0368 & 0.0318 & 111.9440 \\
**GA-CGA** & [9,12,16] & \(1.3537\times 10^{-03}\) & 0.0329 & 0.0311 & 33.1974 \\
**GA-PSO** & [9,12,16] & \(1.8437\times 10^{-03}\) & 0.0319 & 0.0306 & 58.0934 \\
**MIQP** & [9,12,16] & \(---\) & 0.0306 & 0.0306 & 8.9688 \\
**SSB** & [9,12,16] & \(---\) & 0.0306 & 0.0306 & 5.3653 \\
**GA-SOCP** & [9,12,16] & \(1.1050\times 10^{-03}\) & 0.0309 & 0.0306 & 52.146 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of different approaches in the 21-node test feeder.  
Note that the histogram presented in Figure 3 shows that more than 90 solutions (93 exactly) reached by the proposed hybrid GA-SOCP method has the capability of finding the global optimal solution (i.e., 0.0306 pu) regarding power losses, which confirms the efficiency and robustness of the studied methodology for optimal placement and sizing DGs in DC distribution networks.  
**Remark 7**.: _We can ensure that the solution finds by the proposed GA-SOCP approach is the optimal one, since all the possibilities (i.e., 1140 options) for the location of DGs in the 21-node test feeder have been evaluated comprehensively via nested loops, which have confirmed that nodes 9, 12 and 16 are the best possible location for these DGs reaching a final power losses of \(0.0306\) pu. It is worth mentioning that this exhaustive evaluation takes about \(800\) s._  
Figure 4, which presents the percentage of improvement reached by the comparative methods in contrast to the proposed hybrid optimization approach, shows the total grid improvement about the power losses minimization.  
Figure 4 shows the positive impact that has the inclusion of distributed generation in DC distribution networks, since all of the comparative methods and the proposed hybrid approach present improvements that are higher than 88% with a total penetration of 60% in a distributed generation regarding total power consumption. In addition, these results confirm the global capabilities of the proposed hybrid GA-SOCP method for dealing with the problem of the optimal location and sizing DGs in DC feeders.  
Figure 4: Percentage of improvement regarding power losses minimization in the 21-node test feeder.  
Figure 3: Behavior of the proposed hybrid GA-PSO after 100 consecutive evaluations in the 21-node test feeder.