## 3 The Proposed Optimization Hybrid Algorithm
### OPF Problem Solution in the Presence of WT and PV Units  
#### 4.2.1 Case 7  
Minimizing the Generation Cost When Incorporating WT and PV Generation  
In this case, the TLTFWO helps find the minimum fuel, wind, and PV costs defined by Equation (53) for a system with WT and PV units.  
\[\begin{split}& J_{7}=\sum\limits_{i=1}^{NG}\alpha_{i}+b_{i}P_{Gi}+c_{i} P_{Gi}^{2}+\left(\sum\limits_{i=1}^{NW}F\cos t(WT_{i})\right)+\left(\sum \limits_{i=1}^{NV}F\cos t(PV_{i})\right)\\ &\lambda_{P}(\,P_{G1}-P_{G1}^{\text{lim}}\,)^{2}+\,\lambda_{V} \sum\limits_{i=1}^{NL}\,(\,VL_{i}-VL_{i}^{\text{lim}}\,)^{2}+\,\lambda_{Q} \sum\limits_{i=1}^{NG}\,(\,Q_{Gi}-Q_{Gi}^{\text{lim}}\,)^{2}+\,\lambda_{S} \sum\limits_{i=1}^{NTL}\,(\,S_{i}-S_{i}^{\text{lim}}\,)^{2}\\ \end{split} \tag{53}\]  
\(NW\) and \(NV\) are the number of WT and PV units in this equation. Further, \(F\cos t(WT_{i})\) and \(F\cos t(PV_{i})\) express the output power generation cost of the \(i\)th WT and PV units, respectively.  
Cost coefficients, in this case, are similar to Case 1, and PDF parameters are given in Table 10. Table 11 provides the optimal solutions of TLTFWO obtained for more than thirty runs. As observed, incorporating the optimal parameters helps decrease the objective function significantly compared to TFWO and TLBO. Moreover, Figure 11 compares convergence behavior in Case 7 between TFWO, TLBO and TLTFWO algorithms.  
Figure 10: Convergence trends for Case 6.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Wind Power Generating Plants**} & \multicolumn{3}{c}{**Solar PV Plant**} \\ \hline
**Wind Farm** & \begin{tabular}{c} **No. of** \\ **Turbines** \\ \end{tabular} & \begin{tabular}{c} **Rated Power,** \\ **\(P_{cur}\) **(MW)** \\ \end{tabular} & \begin{tabular}{c} **Weibull PDF** \\ **Parameters** \\ \end{tabular} & \begin{tabular}{c} **Weibull Mean,** \\ **\(\mathbf{Mwbl}\)** \\ \end{tabular} & \begin{tabular}{c} **Rated Power,** \\ **\(\mathbf{P_{sr}}\) **(MW)** \\ \end{tabular} & \
\begin{tabular}{c} **Lognormal Polar,** \\ **Parameters** \\ \end{tabular} &
\begin{tabular}{c} **Lognormal Mean,** \\ **\(\mathbf{Mlgn}\)** \\ \end{tabular} \\ \hline
1 (bus 5) & 25 & 75 & \(c=9,k=2\) & \(v=7.976\) m/s & 50 (bus 13) & \(\sigma=0.6,\,\mu=6\) & \(G=483\) W/m2 \\
2 (bus 11) & 20 & 60 & \(c=10,k=2\) & \(v=8.862\) m/s & 50 (bus 13) & \(\sigma=0.6,\,\mu=6\) & \(G=483\) W/m2 \\ \hline \hline \end{tabular}
\end{table}
Table 10: PDF parameters of WT and PV units [19].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & \begin{tabular}{c} **TFWO** \\ \end{tabular} & \
\begin{tabular}{c} **TLBO** \\ \end{tabular} &
\begin{tabular}{c} **TLFWO** \\ \end{tabular} \\ \hline \(P_{G1}\) (MW) & 134.90791 & 134.90791 & 134.90793 \\ \(P_{G2}\) (MW) & 29.0275 & 28.2868 & 27.0466 \\ \(P_{us1}\) (MW) & 44.0282 & 43.6213 & 42.9326 \\ \(P_{G3}\) (MW) & 10 & 10 & 10 \\ \(P_{us2}\) (MW) & 37.1649 & 36.8238 & 36.2236 \\ \(P_{ss}\) (MW) & 34.0406 & 35.5344 & 38.0833 \\ \(V_{G1}\) (p.u.) & 1.0718 & 1.0723 & 1.072 \\ \(V_{G2}\) (p.u.) & 1.0568 & 1.0573 & 1.057 \\ \(V_{G5}\) (p.u.) & 1.0349 & 1.0352 & 1.0348 \\ \(V_{G8}\) (p.u.) & 1.0702 & 1.0398 & 1.0395 \\ \(V_{G11}\) (p.u.) & 1.0981 & 1.0996 & 1.0999 \\ \(V_{G13}\) (p.u.) & 1.0489 & 1.0548 & 1.0559 \\ \(Q_{G1}\) (MVAR) & \(-2.31923\) & \(-1.91971\) & \(-1.91987\) \\ \(Q_{G2}\) (MVAR) & 11.8198 & 13.2443 & 13.2115 \\ \(Q_{us1}\) (MVAR) & 22.4185 & 23.1879 & 23.2748 \\ \(Q_{G3}\)(MVAR) & 40 & 35.0704 & 34.6188 \\ \(Q_{us2}\) (MVAR) & 30 & 30 & 30 \\ \(Q_{ss}\) (MVAR) & 15.0849 & 17.4102 & 17.8624 \\ Fuelvlvcost (USD/h) & 442.3257 & 439.8602 & 435.7669 \\ Wind gen cost (USD/h) & 247.9662 & 245.3840 & 240.9739 \\ Solar gen cost (USD/h) & 92.0150 & 97.2115 & 105.2384 \\ Total Cost (USD/h) & 782.3068 & 782.4558 & 781.9791 \\ Emission (t/h) & 1.76196 & 1.76213 & 1.76245 \\ Power losses (MW) & 5.7692 & 5.7741 & 5.7941 \\ _V.D._ (p.u.) & 0.45405 & 0.46348 & 0.46546 \\ \hline \hline \end{tabular}
\end{table}
Table 11: Optimal variables in Case 7.