## 5 Results and Discussions
### System 1: The IEEE 33-Bus
#### 5.1.3 Case 3: Simultaneous Minimization of APL, VD, and VSI\({}^{-1}\)
The values of VD and VSI\({}^{-1}\) obtained from QODELFA in this case are slightly higher than those obtained from both algorithms in [28]. When the power factor is set to 0.866 lag (Case 3.3), the APL, VD, and VSI\({}^{-1}\) are improved to 15.498 kW, 0.0003, and 1.0242 p.u., respectively. This verifies that better solutions can be achieved when choosing the best value of DGs' power factor. Besides that, for all cases (3.1, 3.2, and 3.3), the robustness by the small SDs and the fast convergence of the proposed algorithm are validated as given in Table 5 and Figure 9, respectively. The voltage profiles for Case 3 of the IEEE 33-bus system are shown in Figure 10, where the voltages along the feeder are well enhanced for all subcases, especially for Case 3.3. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Technique**} & **Optimal** & **Optimal Sizes** & **APL** & **VD** & **VSI\({}^{-1}\)** & **VSI** & **LR\%** & **SD** \\  & **Locations** & **(MW/MVAr)** & **(kW)** & **(p.u.)** & **(p.u.)** & **(p.u.)** & **(p.u.)** & **LR\%** & **SD** \\ \hline \hline \multicolumn{10}{c}{**Case 3.1: DGs with unity power factor (PF = 1)**} \\ \hline \multirow{3}{*}{SIMBO-Q [28]} & 12 & 1.3482/0.0 & & & & & \\  & 24 & 1.3805/0.0 & 98.20 & 0.00081 & 1.0370 & 0.9643 & 53.46 & - \\  & 30 & 1.5000/0.0 & & & & & & \\ \hline \multirow{3}{*}{QOSIMBO-Q [28]} & 12 & 1.3465/0.0 & & & & & & \\  & 24 & 1.3043/0.0 & 97.10 & 0.00088 & 1.0383 & 0.9631 & 53.98 & - \\  & 30 & 1.5000/0.0 & & & & & & \\ \hline \multirow{3}{*}{SFSA [32]} & 13 & 0.9647/0.0 & & & & & & & \\  & 24 & 1.1337/0.0 & 77.410 & 0.00623 & 1.0891 & 0.9182 & 63.31 & 2.73E-07 \\  & 30 & 1.3018/0.0 & & & & & & \\ \hline \multirow{3}{*}{**QODELFA**} & **13** & **0.9647/0.0** & & & & & & & \\  & **24** & **1.1334/0.0** & **77.408** & **0.00621** & **1.0891** & **0.9182** & **63.31** & **2.62E-10** \\  & **30** & **1.3017/0.0** & & & & & & & \\ \hline \multicolumn{10}{c}{**Case 3.2: DGs with lagging power factor (PF = 0.95)**} \\ \hline \multirow{3}{*}{SIMBO-Q [28]} & 13 & 0.9429/0.3099 & & & & & & & \\  & 24 & 1.3271/0.4362 & 32.40 & 0.0003 & 1.0234 & 0.9771 & 84.74 & - \\  & 30 & 1.4429/0.4742 & & & & & & \\ \hline \multirow{3}{*}{QOSIMBO-Q [28]} & 13 & 0.9890/0.2952 & & & & & & & \\  & 24 & 1.3928/0.4578 & 31.70 & 0.0003 & 1.0235 & 0.9770 & 85.26 & - \\  & 30 & 1.4193/0.4665 & & & & & & \\ \hline \multirow{3}{*}{SFSA [32]} & 13 & 0.9174/0.3015 & & & & & & & \\  & 24 & 1.1463/0.3768 & 29.383 & 0.0007 & 1.0312 & 0.9697 & 86.07 & 1.47E-07 \\  & 30 & 1.3157/0.4324 & & & & & & \\ \hline \multirow{3}{*}{**QODELFA**} & **13** & **0.9169/0.3013** & & & & & & & \\  & **24** & **1.1466/0.3768** & **29.386** & **0.0007** & **1.0311** & **0.9698** & **86.07** & **1.97E-09** \\  & **30** & **1.3167/0.4327** & & & & & & \\ \hline \multicolumn{10}{c}{**Case 3.3: DGs with lagging power factor (PF = 0.866)**} \\ \hline \multirow{3}{*}{**QODELFA**} & **13** & **0.7911/0.4567** & & & & & & & \\  & **24** & **1.0411/0.5991** & **15.498** & **0.0003** & **1.0242** & **0.9764** & **92.65** & **3.52E-08** \\ \cline{1-1}  & **30** & **1.2431/0.7178** & & & & & & \\ \hline \hline \end{tabular}  
* The sign “-” means unreported.  
\end{table} 
Table 5: Results for the IEEE 33-bus RDS with simultaneous minimization of OF\({}_{1}\), OF\({}_{2}\), and OF\({}_{3}\).