## 5 Results and Discussions
### System 1: The IEEE 33-Bus
#### 5.1.3 Case 3: Simultaneous Minimization of APL, VD, and VSI\({}^{-1}\)  
In this case, the minimization of APL (OF\({}_{1}\)), VD (OF\({}_{2}\)), and VSI\({}^{-1}\) (OF\({}_{3}\)) is simultaneously considered. Thus, the weighting factors in Equation (18) are taken as \(\omega_{1}=1,\omega_{2}=0.65,\omega_{3}=0.35\), as in [32]. The QODELFA is implemented for three different values of power factor: Unity, 0.95 lag, and \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Technique** & **Optimal** & **Optimal Sizes** & **APL** & **VD** & **VSI\({}^{-1}\)** & **VSI** & **LR\%** & **SD** \\ \hline \multicolumn{10}{c}{**Case 2.1: DGs with unity power factor (PF = 1)**} \\ \hline \multirow{3}{*}{SIMBO-Q [28]} & 14 & 0.9029/0.0 & \multirow{3}{*}{92.50} & \multirow{3}{*}{0.0022} & \multirow{3}{*}{-} & \multirow{3}{*}{-} & \multirow{3}{*}{56.16} & \multirow{3}{*}{-} \\  & 26 & 1.4491/0.0 & & & & & & \\  & 31 & 0.9137/0.0 & & & & & & \\ \hline \multirow{3}{*}{QOSIMBO-Q [28]} & 12 & 1.3232/0.0 & \multirow{3}{*}{88.90} & \multirow{3}{*}{0.0022} & \multirow{3}{*}{-} & \multirow{3}{*}{-} & \multirow{3}{*}{57.87} & \multirow{3}{*}{-} \\  & 24 & 1.0223/0.0 & & & & & & \\  & 30 & 1.3735/0.0 & & & & & & \\ \hline \multirow{3}{*}{**QODELFA**} & **13** & **1.0204/0.0** & \multirow{3}{*}{**78.308**} & \multirow{3}{*}{**0.0055**} & \multirow{3}{*}{**1.0900**} & \multirow{3}{*}{**0.9175**} & \multirow{3}{*}{**62.88**} & \multirow{3}{*}{**3.45E-10**} \\  & **24** & **1.1504/0.0** & & & & & & \\  & **30** & **1.2702/0.0** & & & & & & \\ \hline \multicolumn{10}{c}{**Case 2.2: DGs with lagging power factor (PF = 0.95)**} \\ \hline \multirow{3}{*}{SIMBO-Q [28]} & 13 & 0.8813/0.2897 & \multirow{3}{*}{32.20} & \multirow{3}{*}{0.0003} & \multirow{3}{*}{-} & \multirow{3}{*}{-} & \multirow{3}{*}{84.74} & \multirow{3}{*}{-} \\  & 24 & 1.3048/0.4289 & & & & & & \\  & 30 & 1.5000/0.4930 & & & & & & \\ \hline \multirow{3}{*}{QOSIMBO-Q [28]} & 13 & 0.8303/0.2729 & \multirow{3}{*}{31.10} & \multirow{3}{*}{0.0003} & \multirow{3}{*}{-} & \multirow{3}{*}{85.26} & \multirow{3}{*}{-} \\  & 24 & 1.1239/0.3694 & & & & & & \\  & 30 & 1.2398/0.4075 & & & & & & \\ \hline \multirow{3}{*}{**QODELFA**} & **13** & **0.9001/0.2958** & \multirow{3}{*}{**29.231**} & \multirow{3}{*}{**0.0007**} & \multirow{3}{*}{**1.0331**} & \multirow{3}{*}{**0.9679**} & \multirow{3}{*}{**86.48**} & \multirow{3}{*}{**4.13E-08**} \\  & **24** & **1.1438/0.3759** & & & & & & \\  & **30** & **1.3214/0.4343** & & & & & & \\ \hline \multicolumn{10}{c}{**Case 2.3: DGs with lagging power factor (PF = 0.866)**} \\ \hline \multirow{3}{*}{**QODELFA**} & **13** & **0.7582/0.4378** & \multirow{3}{*}{**15.502**} & \multirow{3}{*}{**0.0003**} & \multirow{3}{*}{**1.0243**} & \multirow{3}{*}{**0.9763**} & \multirow{3}{*}{**92.65**} & \multirow{3}{*}{**3.71E-08**} \\  & **24** & **1.0273/0.5930** & & & & & & \\ \cline{1-1}  & **30** & **1.2139/0.7009** & & & & & & \\ \hline \hline \end{tabular} The sign “-” means unreported.  
\end{table} 
Table 4: Results for the IEEE 33-bus RDS with simultaneous minimization of OF\({}_{1}\) and OF\({}_{2}\). Figure 8: Convergence characteristics for Case 2 of the IEEE 33-bus system using QODELFA. 0.866 lag. The results obtained from the proposed QODELFA are presented in Table 5 and compared to those from three other methods. It can be observed that, with unity power factor DGs (Case 3.1), the APL is minimized to 77.408 kW by using QODELFA, which is much less than that of 98.20 kW from SIMBO-Q [28], 97.10 kW from QOSIMBO-Q [28], and approximately the same as that from SFSA [32]. However, the VD and VSI\({}^{-1}\) (in p.u.) obtained by QODELFA are not better than those obtained from both algorithms in [28]. Similar aspects can be remarked for Case 3.2 with DGs operating at PF = 0.95 lag when comparing the results obtained by QODELFA to those from methods in [28], where the ALP is reduced to 29.386 kW by the proposed algorithm, which is less than that of SIMBO-Q (32.40 kW) and QOSIMBO-Q (31.70 kW).