## 6 Results
### Results of Case-2  
In Case-2, the proposed convex model is compared to improved analytical (IA) [40], particle swarm optimization (PSO) [41], and MINLP. Table 3 shows the results for IEEE 33-buses and IEEE 69-buses systems for the optimal placement and sizing of DERs.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Loss [kW]** & **Location (Node)** & **Size [MW]** \\ \hline \hline \multicolumn{4}{c}{IEEE 33-bus system} \\ \hline GA-IWD [31] & 110.5100 & {11,16,32} & {1.2214,0.6833,1.2135} \\ GA-PSO [32] & 103.3600 & {11,16,32} & {0.9250,0.8630,1.2000} \\ LSPSA [12] & 82.0525 & {6,18,30} & {1.1124,0.4874,0.8679} \\ CHVSA [33] & 78.4534 & {6,14,31} & {1.1846,0.6468,0.6881} \\ TLBO [34] & 75.5400 & {10,24,31} & {0.8246,1.0311,0.8862} \\ KHA [12] & 75.4116 & {13,25,30} & {0.8107,0.8368,0.8410} \\ QOTLDO [34] & 74.1008 & {12,24,29} & {0.8808,1.0592,1.0714} \\ AHA [35] & 72.8340 & {13,24,30} & {0.7920,1.0680,0.10270} \\ HSA-PABC [36] & 72.8129 & {14,24,30} & {0.7550,1.0730,1.0680} \\ MINLP [37] & 72.7862 & {13,24,30} & {0.8000,1.0900,1.0500} \\ MSSA [38] & 72.7854 & {13,24,30} & {0.8010,1.0910,1.0530} \\ \hline MI-SOCP & 72.7853 & {13,24,30} & {0.8018,1.0913,1.0536} \\ \hline \hline \multicolumn{4}{c}{IEEE 69-bus system} \\ \hline GA-PSO [32] & 84.5909 & {21,61,63} & {0.9105,1.1926,0.8849} \\ GA-IWD [31] & 80.9100 & {20,61,64} & {0.9115,1.3926,0.8059} \\ TLBO [34] & 72.4157 & {15,61,63} & {0.5919,0.8188,0.9003} \\ LSPSA [12] & 72.1120 & {18,60,65} & {0.4204,1.3311,0.4298} \\ HOGWO [39] & 71.7281 & {18,61,62} & {0.5330,1.0000,0.7730} \\ QOTLBO [34] & 71.6345 & {18,61,63} & {0.5334,1.1986,0.5672} \\ AHA [35] & 69.6669 & {12,21,61} & {0.4710,0.3120,1.6890} \\ KHA [12] & 69.5730 & {12,22,61} & {0.4962,0.3113,1.7354} \\ MINLP [37] & 69.4090 & {11,17,61} & {0.5300,0.3800,1.7200} \\ CHVSA [33] & 69.4088 & {11,17,61} & {0.5284,0.3794,1.7186} \\ MSSA [38] & 69.4077 & {11,18,61} & {0.5260,0.3800,1.7180} \\ \hline MI-SOCP & 69.4077 & {11,18,61} & {0.5268,0.3801,1.7190} \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparative results for the optimal placement and sizing of distributed energy resources (DERs) for the IEEE 33-bus and IEEE 69-bus systems for Case-1.  
Note in Table 3 that the MI-SOCP convex model proposed has a superior performance for placement and sizing of multiple DERs for both test systems with real and reactive power capability when compared with the other three methodologies. In addition, it can also be observed that the MI-SOCP and MINLP present the same placement for the DERs in the IEEE 33-bus system, but the proposed convex model achieves a better solution than the MINLP model. This is because the MINLP model can not guarantee the global optimum. While for the IEEE 69-bus system, none of the other methodologies could find the combination of buses for the location of the DERs. The power loss reduction in Table 3 confirms that the MI-SOCP convex model gives a maximum reduction of power losses. However, the total DERs power for the proposed convex model in the IEEE 33-bus system is slightly higher than the MINLP model. While for the IEEE 33-bus system, the total DERs power is slightly lower in the MI-SOCP convex model. Finally, the IEEE 33- and 69-bus systems' processing time is 11.8 s and 75.1 s, respectively.