# Hybrid GA-SOCP Approach for Placement and
## 5 Computational Implementation
### 69-Node Test Feeder  
Table 4 presents the numerical results of the proposed and comparative methods. These results allow observing that:  
* All of the comparative methods are stuck in locally optimal solutions, including the MIQP approach and the MINLP model solved in GAMS through the SBB solver since these finds final power losses about 0.1660 pu and 0.1574 pu, respectively. However, the best optimal solution is reached by the GA-SOCP method with 0.1573 pu.
* Some of the metaheuristic approaches (i.e., GA-CGA and GA-PSO) find the best positions for installing the GDs being these nodes 21, 61, and 64; nevertheless, due to the random procedures in the slave stages, these find local solutions.
* Even if standard deviations of the GA-PSO and GA-CGA are lower than the proposed GA-SOCP, these values imply that these methods are always trapped in the local solutions, and they do not have the capacity of scope from these solutions to explore other promissory regions of the solution space.  
Note that the best optimal solution reported by the GA-SOCP proposed approach connects DGs in nodes 21, 61, and 64 with generation capabilities of about 1.4140 pu, 10.2630 pu, and 3.8803, respectively. Once again, we can confirm that this solution is the global optimum, since, after the exhaustive evaluation of the solution space (i.e., 50116 possible locations) via nested loops, the same set of nodes and power generations have been identified as the best global solution. It is important to highlight that this exhaustive evaluation takes about 7000 s to revise the whole solution space, which implies that 66.91 s reported by the proposed GA-SOCP approach demonstrates its efficiency and robustness global finding capabilities.  
Figure 5 presents the histogram with the optimization behavior of the proposed GA-SOCP method for locating and sizing DGs in DC distribution networks.  
From Figure 5, it can be noted that more than 70% of the solutions that were found by the proposed GA-SOCP method find the global optimal solution, and 24% of the additional solutions are better than the mean solutions reported in Table 4 (see \(\mu\) column). These results confirm the proposed GA-SOCP optimization approach's efficiency and robustness for solving the optimal placement and sizing of DGs in DC distribution networks with global optimum finding capabilities.  
Figure 6 presents the percentage of power losses improvement reached by the comparative and the proposed optimization approaches.  
Note that the percentage of reduction of power losses reported in plot 6 confirms the global optimization capabilities of the proposed GA-SOCP approach, since it is the only one methodology that reports the highest percentage of reduction with 89.78%, only followed by the SBB methodology with 89.77% of reduction.  
\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** & [23,61 67] & 2.5207\(\times 10^{-03}\) & 0.1633 & 0.1593 & 713.7193 \\
**GA-CGA** & [21,61,64] & 3.4801\(\times 10^{-04}\) & 0.1648 & 0.1603 & 218.0169 \\
**GA-PSO** & [21,61,64] & 5.4023\(\times 10^{-04}\) & 0.1689 & 0.1588 & 378.4731 \\
**MIQP** & [22,61,64] & \(---\) & 0.1660 & 0.1660 & 24.1868 \\
**SBB** & [22,61,65] & \(---\) & 0.1574 & 0.1574 & 12.4270 \\
**GA-SOCP** & [21,61,64] & \(3.1295\times 10^{-03}\) & 0.1591 & 0.1573 & 66.9100 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of different approaches in the 69-node test feeder.