## 6 Results
### Comparison with a Hybrid Optimization Approach  
To demonstrate the effectiveness and robustness of the proposed MI-SOCP model to determine the optimal location and sizing of renewable DGs in AC distribution networks, here, we present a comparison with hybrid optimization approaches that combines a discrete metaheuristic with a SOCP formulation to locate and size DGs in electrical  
Figure 4: Power loss reduction for Case-1: (**a**) IEEE 33-bus system and (**b**) IEEE 69-bus system. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Loss [kW]**} & \multirow{2}{*}{**Location** **(Node)**} & \multirow{2}{*}{**Size [MW]**} & \multirow{2}{*}{**Size [MVAr]**} & \multirow{2}{*}{**DER Power** **[MVA]**} & \multirow{2}{*}{**Loss Red.** **[\%]**} \\  & & & & & & \\ \hline \multicolumn{7}{c}{IEEE 33-bus system} \\ \hline IA [40] & 23.05 & \{06,14,30\} & \{1.0590,0.7410,1.0590\} & \{0.6563,0.4592,0.6563\} & 2.859 & 89.09 \\ PSO [41] & 15.00 & \{13,24,30\} & \{0.7640,1.0680,1.0160\} & \{0.5350,0.6130,0.6910\} & 3.395 & 92.89 \\ MINLP [37] & 12.74 & \{13,24,30\} & \{0.7660,1.0440,1.1460\} & \{0.4110,0.5520,0.8590\} & 3.481 & 92.89 \\ \hline MI-SOCP & 11.7401 & \{13,24,30\} & \{0.7939,1.0700,1.0297\} & \{0.3734,0.5171,1.0115\} & 3.509 & 94.44 \\ \hline \multicolumn{7}{c}{IEEE 69-bus system} \\ \hline IA [40] & 5.0839 & \{17,50,61\} & \{0.6300,0.9000,0.9000\} & \{0.3900,0.5570,0.5570\} & 3.524 & 97.74 \\ PSO [41] & 5.0839 & \{18,50,61\} & \{0.7640,1.0680,1.0160\} & \{0.5350,0.6130,0.6910\} & 3.545 & 97.74 \\ MINLP [37] & 4.2695 & \{11,17,61\} & \{0.7660,1.0440,1.1460\} & \{0.4110,0.5520,0.8590\} & 3.123 & 98.1020 \\ \hline MI-SOCP & 4.2667 & \{11,18,61\} & \{0.4945,0.3783,1.6743\} & \{0.3533,0.2510,1.1954\} & 3.119 & 98.1033 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparative results for the optimal placement and sizing of DERs for the IEEE 33-bus and IEEE 69-bus systems for Case-2. distribution grids. This approach was initially presented in [21], and it works with the discrete version of the sine-cosine algorithm to determine the nodes where DGs will be located, while the SOCP is used to determine their optimal sizes (this approach is named DSCA-SOCP). In addition, we include comparisons with MINLP solvers available in the GAMS optimization tool. The renewable energy resources correspond to photovoltaic (PV) sources considering a typically daily generation curve and a daily load curve. These curves are presented in Figure 5. This comparison is considered the IEEE 69 node test feeder since these are more complex than the IEEE 33 node test feeder. Furthermore, the test feeder has about 50,116 possible combinations to locate three PV sources. In the case of MINLP solvers, we consider the following optimization tools available in GAMS: Solvers BONMIN, COUENNE, and DICOPT, respectively. The complete comparison of the proposed MI-SOCP model with the DSCA-SOCP and the GAMS solvers is reported in Table 4.  
The results in Table 4 demonstrate that: (i) With the location of three photovoltaic sources, the proposed approach, i.e., the proposed MI-SOCP and the DSCA-SOCP approaches, reduces the daily energy loss per day to about 638.1187 kWh/day, i.e., 23.93%; while the best GAMS approach using the COUENNE solver reach a reduction about 23.84%.The solutions found by the different MINLP solvers in GAMS demonstrate that these stuck in optimal local solutions when compared with the optimal solution found by the MI-SOCP and the DSCA-SOCP methodologies, and (ii) in all the solutions presented in Table 4 nodes higher than 60 show the high power injection regarding PV penetration, which allows observing that these nodes are more sensitive to power injections when compared with the remainder of buses.