## 6 Numerical Validation
### 21-Node Test Feeder  
For this test system, the initial power losses without distributed generation were about 0.2760 pu. When the MI-SDP method was applied to the 21-node test feeder, we found that the distributed generators must be located at nodes 9, 12, and 16 with power injections of 0.8350, 1.0258, and 1.4632 pu, respectively; these produce final power losses of about 0.0306 pu. This implies a total reduction of 88.91% in the power losses with respect to the best case. For comparative purposes, we used the general algebraic modeling system (GAMS) with different MINLP solvers to verify the efficiency of our MI-SDP approach. The numerical results are summarized in Table 3.  
From Table 3, we can observe that due to the strong non-convexities of the power flow problem being added to the binary nature of the problem of installations of DGs in DC current, all the solvers available in GAMS have the ability to find the global optimal solution of the studied problem. In addition, we can observe that in the case of the local solutions, i.e., KNITRO and LINDO solvers, they got stuck in a sub-optimal solution wherein node 17 was selected in the final report instead of node 16. This is important, as in the case of large-scale distribution networks, the local solutions can be far from optimal. On the other hand, to verify this in the 21-node test feeder, we exhaustively evaluated all the possible solution points, i.e., 1140 options in the MATLAB software using nested-loops, where we found that the solution reported by the proposed MI-SDP is indeed the global optimum.  
Regarding the comparison with metaheuristics in Figure 6, the best results reported for the 21-node test feeder in the literature are presented in comparison with our proposed MI-SDP model.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Solver** & **Nodes** & **Sizes [pu]** & \(\mathbf{p_{\text{loss}}}\) [pu]** \\ \hline BARON & \(\{9,12,16\}\) & \(\{0.8441,1.0254,1.4544\}\) & 0.0306 \\ BONMINH & \(\{9,12,16\}\) & \(\{0.8441,1.0254,1.4544\}\) & 0.0306 \\ DICOPT & \(\{9,12,16\}\) & \(\{0.8441,1.0254,1.4544\}\) & 0.0306 \\ KNITRO & \(\{9,12,17\}\) & \(\{0.9297,1.1491,1.2452\}\) & 0.0356 \\ LINDO & \(\{9,12,17\}\) & \(\{0.9297,1.1491,1.2452\}\) & 0.0356 \\ SBB & \(\{9,12,16\}\) & \(\{0.8441,1.0254,1.4544\}\) & 0.0306 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of the MI-SDP with the MINLP model solved in GAMS for the 21-node test feeder.  
Figure 6: Final power losses when heuristics and metaheuristics are compared to the MI-SDP approach.  
Note that results in Figure 6 show that some metaheuristics can find the global optimal solution of the problem. However, as mentioned in [18], at each simulation, the optimal solution has little changes, which implies that these present standard deviations different from zero. At the same time, the standard deviations of the HSDP, the HSQP, and the proposed approach are zero. This is because they work with convex solution spaces regarding the optimal power flow, which does not happen in the metaheuristic cases.  
To present in a graphical manner the proposed MI-SDP approach that has the ability to find the global optimal solution in the 21-node test feeder, we plotted the information obtained after the exhaustive evaluation of the solution space, i.e., 1140 possible locations for the constant power sources, in Figure 7.  
Note that information plotted in Figure 7 confirms that the global optimal solution in the case of the 21-node test feeder is about 0.0306 pu of power loss, which is evidenced as no one solution is lower than the dashed line (see upper and lower peaks in the shaded figure). It is important to mention that the exhaustive evaluation took about 1150 s, whereas the GAMS and the proposed MI-SDP approach did not take more than 10 s, which confirms the efficiency of the optimization models in contrast with heuristic approaches.