## 6 Numerical Validation
### 69-Node Test Feeder  
In the 69-node test feeder case, the base case without distributed generation produces total power losses of about 1.5385 pu. When the MI-SDP model is implemented, the optimal nodes to locate GDs are 21, 61, and 64 with power generations of 1.4140, 10.2627, and 3.8803 pu, respectively. These power injections produce final power losses of 0.1573 pu, which implies that these generators allow the reduction of total grid power losses by about 89.78%, which corresponds with the results reported in [17; 18] using the HSDP and HSQP approaches, respectively. It is worth mentioning that the solution determined by the proposed MI-SDP approach is indeed the optimum global, as it was verified by implementing an exhaustive approach that evaluates all the possibles node combinations for locating DGs, i.e., 50,116 options for this test feeder. Table 4 reports the numerical results obtained by different solvers in the GAMS package. Figure 7: Solutions reached by the exhaustive evaluation in the 21-node test feeder. From Table 4, we can observe that only two solvers in the GAMS determined the global optimum regarding final power losses (see BONMINH and SBB solvers). However, the BONMINH solver identified the same nodes of the proposed MI-SDP approach, with some variations in the total power injections, while the SBB changed node 21 for the node 22, with similar results regarding power generation when compared with the BONMINH and the MI-SDP approaches. These variations may be attributed to the higher connectivity between nodes 21 and 22, as observed by the small resistance value of 0.014 pu (hence, higher connection) between nodes as shown in Table 4. This means that there are multiples power injections that can achieve the same objective function value, as shown in [17]. **Remark 5**. : _Despite some GAMS's solvers reaching the global optimum of the problem, this cannot always be guaranteed. Furthermore, as the solution space grows, these solvers may fall to local optima, whereas the proposed technique will continue to reach the global optimum._  
Regarding the comparison with metaheuristics, Figure 8 reports the final power losses between the MI-SDP and the comparative approaches. Based on the results reported in Figure 8, we can confirm that when the solution space increases for the metaheuristic approaches, it is difficult to achieve the global optimum, as it gets stuck in a local optimum. This occurs due to not having a sense-solve optimal power flow problem with continuous metaheuristics, as in the literature, this problem can be solved precisely with convex formulations. **Remark 6**. : _Note that HSDP and HSQP methods also achieve the global optimum of the problem. However, when presenting these results, it cannot be ensured their solutions are the global optimum, whereas the proposed method \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Solver** & **Nodes** & **Sizes [pu]** & \(\mathbf{p_{\text{loss}}}\) [pu]** \\ \hline DICOPT & \(\{24,61,66\}\) & \(\{1.4969,11.6024,2.4577\}\) & 0.1577 \\ CONOPT & \(\{21,61,66\}\) & \(\{1.4969,11.6024,2.4577\}\) & 0.1577 \\ MINOS & \(\{24,61,64\}\) & \(\{1.4552,10.2855,3.8163\}\) & 0.1574 \\ SNOPT & \(\{26,61,68\}\) & \(\{1.3981,12.0000,2.1589\}\) & 0.1599 \\ SBB & \(\{22,61,64\}\) & \(\{1.4961,10.2446,3.8163\}\) & 0.1573 \\ BONMINH & \(\{21,61,64\}\) & \(\{1.4973,10.2434,3.8163\}\) & 0.1573 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the MI-SDP with the MINLP model solved in GAMS for the 69-node test feeder. Figure 8: Final power losses when heuristics and metaheuristics are compared to the MI-SDP approach in the 69-node test feeder. can ensure it.