## 5 Results and Discussions
### System 2: The IEEE 69-Bus
#### 5.2.1 Case 1: APL Minimization
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Technique**} & **Optimal** & **Optimal Sizes** & **APL** & **VD** & **VS1\({}^{-1}\)** & **VSI** & **LR\%** & **SD** \\  & **Locations** & **(MW/MVA)** & **(kW)** & **(p.u.)** & **(p.u.)** & **(p.u.)** & **(p.u.)** & **LR\%** & **SD** \\ \hline \multicolumn{8}{c}{**Case 1.1: DGs with unity power factor (PF = 1)**} \\ \hline \multirow{3}{*}{QOSIMBO-Q [28]} & 9 & 0.8336/0.0 & \multirow{3}{*}{71.000} & \multirow{3}{*}{0.0071} & \multirow{3}{*}{1.1131} & \multirow{3}{*}{0.8984} & \multirow{3}{*}{68.44} & \multirow{3}{*}{-} \\  & 18 & 0.4511/0.0 & & & & & & \\  & 61 & 1.5000/0.0 & & & & & & \\ \hline \multirow{3}{*}{MINLP [13]} & 11 & 0.5300/0.0 & \multirow{3}{*}{69.590} & \multirow{3}{*}{-} & \multirow{3}{*}{-} & \multirow{3}{*}{-} & \multirow{3}{*}{-} & \multirow{3}{*}{69.07} & \multirow{3}{*}{-} \\  & 17 & 0.3800/0.0 & & & & & & \\  & 61 & 1.7200/0.0 & & & & & & \\ \hline \multirow{3}{*}{KHA [31]} & 12 & 0.4962/0.0 & \multirow{3}{*}{69.563} & \multirow{3}{*}{-} & \multirow{3}{*}{1.0887} & \multirow{3}{*}{0.9185} & \multirow{3}{*}{69.08} & \multirow{3}{*}{-} \\  & 22 & 0.3113/0.0 & & & & & & \\  & 61 & 1.7354/0.0 & & & & & & \\ \hline \multirow{3}{*}{SFSA [32]} & 11 & 0.5273/0.0 & \multirow{3}{*}{69.428} & \multirow{3}{*}{0.00518} & \multirow{3}{*}{1.0886} & \multirow{3}{*}{0.9186} & \multirow{3}{*}{69.14} & \multirow{3}{*}{5.24E-08} \\  & 18 & 0.3805/0.0 & & & & & & \\  & 61 & 1.7198/0.0 & & & & & & \\ \hline \multirow{3}{*}{**QODELFA**} & **11** & **0.5267/0.0** & \multirow{3}{*}{**69.426**} & \multirow{3}{*}{**0.00519**} & \multirow{3}{*}{**1.0887**} & \multirow{3}{*}{**0.9185**} & \multirow{3}{*}{**69.15**} & \multirow{3}{*}{**3.16E-10**} \\  & **18** & **0.3806/0.0** & & & & & & \\  & **61** & **1.7189/0.0** & & & & & & \\ \hline \multicolumn{8}{c}{**Case 1.2: DGs with lagging power factor (PF = 0.95)**} \\ \hline \multirow{3}{*}{SIMBO-Q [28]} & 19 & 0.5656/0.1859 & \multirow{3}{*}{23.100} & \multirow{3}{*}{0.00075} & \multirow{3}{*}{1.0281} & \multirow{3}{*}{0.9727} & \multirow{3}{*}{89.73} & \multirow{3}{*}{-} \\  & 61 & 1.5000/0.4930 & & & & & & \\  & 64 & 0.4227/0.1404 & & & & & & \\ \hline \multirow{3}{*}{SFSA [32]} & 11 & 0.5435/0.1786 & \multirow{3}{*}{20.727} & \multirow{3}{*}{0.00033} & \multirow{3}{*}{1.0234} & \multirow{3}{*}{0.9772} & \multirow{3}{*}{90.79} & \multirow{3}{*}{6.51E-06} \\  & 17 & 0.4132/0.1358 & & & & & & \\  & 61 & 1.8728/0.6156 & & & & & & \\ \hline \multirow{3}{*}{**QODELFA**} & **11** & **0.5597**/**0.1839** & \multirow{3}{*}{**20.716**} & \multirow{3}{*}{**0.00027**} & \multirow{3}{*}{**1.0235**} & \multirow{3}{*}{**0.9770**} & \multirow{3}{*}{**90.79**} & \multirow{3}{*}{**4.33E-08**} \\  & **18** & **0.4172**/**0.1371** & & & & & & \\  & **61** & **1.8775**/**0.6171** & & & & & & \\ \hline \multicolumn{8}{c}{**Case 1.3: DGs with lagging power factor (PF = 0.82)**} \\ \hline \multirow{3}{*}{IA [39]} & 17 & 0.5100/0.3560 & \multirow{3}{*}{-} & \multirow{3}{*}{-} & \multirow{3}{*}{-} & \multirow{3}{*}{-} & \multirow{3}{*}{97.74} & \multirow{3}{*}{-} \\  & 50 & 0.6798/0.4745 & & & & & & \\  & 61 & 1.6999/1.1865 & & & & & & \\ \hline \multirow{3}{*}{**QODELFA**} & **11** & **0.4986/0.3480** & \multirow{3}{*}{**4.286**} & \multirow{3}{*}{**0.00012**} & \multirow{3}{*}{**1.0234**} & \multirow{3}{*}{**0.9771**} & \multirow{3}{*}{**98.09**} & \multirow{3}{*}{**1.84E-05**} \\  & **18** & **0.3762/0.2559** & & & & & & \\ \cline{1-1}  & **61** & **1.6869/1.1774** & & & & & & \\ \hline \hline \end{tabular} The sign “-” means unreported.  
\end{table} 
Table 6: Results for the IEEE 69-bus RDS with OF\({}_{1}\). The convergence characteristics of QODELFA given in Figure 11 validate its effectiveness, where the proposed algorithm can quickly converge to the optimal solution for all cases (1.1, 1.2, and 1.3).