## 6 Results
### Results of Case-1  
The proposed methodology for the IEEE 33-bus system is to multiples strategies presented in the literature, such as hybrid intelligent water drops and GA (GA-IWD) [31], mixed genetic algorithm with particle swarm optimization (GA-PSO) [32], loss sensitivity factor simulated annealing (LSFSA) [12], constructive heuristic vortex search algorithm (CHVSA) [33], teaching-learning based optimization (TLBO) [34], krill herd algorithm (KHA) [12], quasi-oppositional teaching-learning-based optimization (QOTLBO) [34], heuristic approach (AHA) [35], harmony search algorithm with particle swarm optimization (PSO) embedded artificial bee colony (HSA-PABC) [36], MINLP model [37], and mutated salp swarm algorithm (MSSA) [38]. For the IEEE 69-bus system, the proposed methodology is also compared to the same literature reported for Case-1 with the exception of the HSA-PABC approach since it did not report any information for this feeder. In addition, the hybrid teaching-learning-based optimization-grey wolf optimizer (HOGWO) reported by [39] has been included.  
**Remark 1**.: _The comparative shown for the distributed generators' optimal locations and sizes have been taken from the original reports. This research's main interest is to present an effective and robust optimization methodology via MI-SOCP to place and size DGs guaranteeing the global optimum finding when this is compared with the most classical and well-known metaheuristic approaches._  
Table 2 presents the results for the optimal placement and sizing of DERs of the proposed convex model and those reported in the literature.  
From Table 2, we can conclude that:  
* The proposed MI-SOCP model reaches the optimal solution of the relaxed convex model, which is indeed the optimal solution of the exact nonlinear model with final power losses of around 72.7853 kW and 69.4077 kW for the IEEE 33-bus and IEEE 69-bus systems, respectively. These results are the global optimum of the minimum of the studied problem;
* In the literature, multiple methodologies have identified the same combination of buses for the optimal location of DERs. Nevertheless, the final generated active power by DERs is not equal to the output provided by the proposed MI-SOCP convex model. This entails that the difference lies in implementing optimal power flow since the SOCP model can ensure the global optimum of the problem;* The solution of the proposed MI-SOCP model can ensure that it is the optimal global solution for the IEEE 33-bus and IEEE 69-bus systems. This is because it has evaluated all the possible nodal combinations by using an exhaustive search combined with a classical optimal power flow (OPF) model solved via interior point methods.  
Figure 4 depicts the power loss reductions reached for each methodology presented in Table 2. Note that the proposed convex model achieves the greatest reduction in power losses for the IEEE 33-bus system (see Figure 4a). While the IEEE 69-bus system, the MI-SOCP, and MSSA methodologies present the same power loss reduction (see Table 2), even their DERs sizing are different. This indicates that there are multiple global optima for this system, and there may be other sizes of DERs that reach the same power losses. The processing time for the IEEE 33- and 69-bus systems is 6.2 s and 49.7 s, respectively.