## 5 Results and Discussions
### System 2: The IEEE 69-Bus
#### 5.2.3 Case 3: Simultaneous Minimization of APL, VD, and VSI\({}^{-1}\)  
The simultaneous minimization of APL (OF\({}_{1}\)), VD (OF\({}_{2}\)) and VSI\({}^{-1}\) (OF\({}_{3}\)) is considered in this case. Accordingly, 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 applied for three different values of power factor: Unity, 0.95 lag, and 0.82 lag. The results obtained from the proposed QODELFA are illustrated in Table 8 and compared to those from methods in [28; 32]. Figure 11: Convergence characteristics for Case 1 of the IEEE 69-bus system using QODELFA. It may be observed that, by applying QODELFA with unity power factor DGs (Case 3.1), the APL is reduced to 72.295 kW, which is lower than that of 80.0 kW from SIMBO-Q [28], 79.70 kW from QOSIMBO-Q [28], and 72.445 kW from SFSA [32]. However, the VD and VSI\({}^{-1}\) (in p.u.) obtained by QODELFA in this case are not better than those obtained from the algorithms in [28; 32]. \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** \\ \cline{3-10}  & **Locations** & **(MW/MVAr)** & **(kW)** & **(p.u.)** & **(p.u.)** & **(p.u.)** & **(p.u.)** & **LR\%** & **SD** \\ \hline \hline \multicolumn{10}{c}{**Case 2.1: DGs with unity power factor (PF = 1)**} \\ \hline SIMBO-Q [28] & 16 & 0.7693/0.0 & & & & & & \\ \cline{3-10}  & 59 & 0.7233/0.0 & 78.100 & 0.00100 & - & - & 65.29 & - \\  & 61 & 1.4597/0.0 & & & & & & \\ \hline \multirow{3}{*}{QOSIMBO-Q [28]} & 18 & 0.6987/0.0 & & & & & & \\ \cline{3-10}  & 59 & 0.7037/0.0 & 77.400 & 0.00100 & - & - & 65.6 & - \\ \cline{3-10}  & 61 & 1.5000/0.0 & & & & & & \\ \hline \multirow{3}{*}{**QODELFA**} & **11** & **0.6616/0.0** & & & & & & & \\ \cline{3-10}  & **20** & **0.4554/0.0** & **72.154** & **0.00150** & **1.0535** & **0.9492** & **67.93** & **4.17E-05** \\ \cline{3-10}  & **61** & **1.9201/0.0** & & & & & & \\ \hline \multicolumn{10}{c}{**Case 2.2: DGs with lagging power factor (PF = 0.95)**} \\ \hline SIMBO-Q [28] & 14 & 0.7976/0.2622 & & & & & & \\ \cline{3-10}  & 61 & 0.6549/0.2153 & 25.900 & 0.00020 & - & - & 88.48 & - \\ \cline{3-10}  & 62 & 1.3615/0.4475 & & & & & & \\ \hline \multirow{3}{*}{QOSIMBO-Q [28]} & 14 & 0.8167/0.2685 & & & & & & \\ \cline{3-10}  & 61 & 1.5000/0.4930 & 24.600 & 0.00020 & - & - & 89.06 & - \\ \cline{3-10}  & 64 & 0.4615/0.1517 & & & & & & \\ \hline \multirow{3}{*}{**QODELFA**} & **11** & **0.5859/0.1926** & & & & & & & \\ \cline{3-10}  & **18** & **0.4359/0.1433** & **20.806** & **0.00014** & **1.0235** & **0.9770** & **90.75** & **3.21E-04** \\ \cline{3-10}  & **61** & **1.9080/0.6272** & & & & & & & \\ \hline \multicolumn{10}{c}{**Case 2.3: DGs with lagging power factor (PF = 0.82)**} \\ \hline \multirow{3}{*}{**QODELFA**} & **11** & **0.5077/0.3544** & & & & & & & \\ \cline{3-10}  & **18** & **0.3875/0.2593** & **4.302** & **0.00010** & **1.0234** & **0.9771** & **98.08** & **2.96E-04** \\ \cline{3-10}  & 61 & **1.6959/1.1837** & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results for the IEEE 69-bus RDS with simultaneous minimization of OF\({}_{1}\) and OF\({}_{2}\). Figure 12: Convergence characteristics for Case 2 of the IEEE 69-bus system using QODELFA. For Case 3.2 with DGs operating at PF = 0.95 lag, and by comparing the results obtained by QODELFA to those from methods in [28; 32], it can be remarked that the APL is reduced to 20.774 kW by the proposed algorithm, which is less than that of SIMBO-Q (30.90 kW) and QOSIMBO-Q (25.70 kW), and the same as that of SFSA. Additionally, the value of VD obtained by QODELFA in this case (0.00015 p.u.) is better than the other three algorithms in [28; 32].