## 4 HFAJAYA for the Multi-Constraint OPF Problems
### OPF Solutions, Including Stochastic Solar and Wind Power
#### 4.2.4 Case 7  
Case 7 optimized the generating program for thermal and renewable energy producers in order to reduce the overall generation cost as determined by (50). The cost coefficients, in this case, were the same as in Case 1, and the probability density function (PDF) parameters are shown in Table 9. Moreover, Table 10 summarizes the optimal settings for all decision variables, reactive generator power (\(Q\)), total generating cost, and other important computed metrics. In the Table, \(V_{i}\) indicates the voltage at the \(i\)th bus. \(P_{loss}\) and \(V.D\) were computed using (30) and (31). It is worth noting that \(P_{ws,I}\) denotes the scheduled power from wind generator \(W_{GI}\), and so on. The lowest generation cost obtained using the generation schedules indicated in the table was 782.2795 USD/h.  
\[\begin{split} J_{7}=\sum\limits_{i=1}^{NG}F_{i}(P_{GI})=\sum \limits_{i=1}^{N_{T}}\left(\alpha_{i}+b_{i}P_{GI}+c_{i}P_{GI}^{2}\right)\\ +\sum\limits_{j=1}^{N_{W}}\left[C_{w,j}\left(P_{ws,j}\right)+C_{ Rw,j}\left(P_{ws,j}-P_{waw,j}\right)+C_{Pw,j}\left(P_{waw,j}-P_{ws,j}\right) \right]\\ +\sum\limits_{k=1}^{N_{S}}\left[C_{s,j}\left(P_{ss,k}\right)+C_{ Rs,k}\left(P_{ss,k}-P_{sav,k}\right)+C_{Ps,k}\left(P_{sav,k}-P_{ss,k}\right)\right] \end{split} \tag{48}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Variables** & **Optimal Values** \\ \hline \(P_{GI}\) (MW) & 134.90791 \\ \hline \(P_{G2}\) (MW) & 28.7687 \\ \hline \(P_{wsI}\) (MW) & 43.8862 \\ \hline \(P_{G3}\) (MW) & 10 \\ \hline \(P_{ws2}\) (MW) & 37.049 \\ \hline \(P_{ss}\) (MW) & 34.5606 \\ \hline \(V_{G1}\) (p.u.) & 1.0717 \\ \hline \(V_{G2}\) (p.u.) & 1.0568 \\ \hline \(V_{G5}\) (p.u.) & 1.0348 \\ \hline \(V_{G8}\) (p.u.) & 1.0547 \\ \hline \(V_{G11}\) (p.u.) & 1.0982 \\ \hline \(V_{G13}\) (p.u.) & 1.049 \\ \hline \(Q_{G1}\) (MVAR) & \(-2.33741\) \\ \hline \(Q_{G2}\) (MVAR) & 11.7952 \\ \hline \(Q_{ws1}\) (MVAR) & 22.4203 \\ \hline \(Q_{G3}\)(MVAR) & 40 \\ \hline \(Q_{ws2}\) (MVAR) & 30 \\ \hline \(Q_{ss}\) (MVAR) & 15.1297 \\ \hline Fuelvcost (USD/h) & 441.4626 \\ \hline Wind gen cost (USD/h) & 247.0741 \\ \hline Solar gen cost (USD/h) & 93.7428 \\ \hline \hline \end{tabular}
\end{table}
Table 10: The optimal values by HFAJAYA for Case 7.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{4}{c}{**Wind Power Generating Plants**} & \multicolumn{4}{c}{**Solar PV Plant**} \\ \hline
**Wind Farm** & \begin{tabular}{c} **No. of** \\ **Turbines** \\ \end{tabular} & \begin{tabular}{c} **Rated Power,** \\ **\(P_{uv}\) (MW)** \\ \end{tabular} & \begin{tabular}{c} **Weibull PDF** \\ **Parameters** \\ \end{tabular} & \begin{tabular}{c} **Weibull Mean,** \\ **Mwbl** \\ \end{tabular} & \begin{tabular}{c} **Rated Power,** \\ **\(P_{sr}\)** \\ **(MW)** \\ \end{tabular} & \
\begin{tabular}{c} **Legend normal** \\ **PDF** \\ **Parameters** \\ \end{tabular} &
\begin{tabular}{c} **Legend normal** \\ **PDF** \\ **Mlen** \\ \end{tabular} \\ \hline
1 (bus 5) & 25 & 75 & \(c=9,k=2\) & \(v=7.976\) m/s & 50 (bus 13) & \(\mu=6\), \(\sigma=0.6\) & \(G=483\) W/m\({}^{2}\) \\ \hline
2 (bus 11) & 20 & 60 & \(c=10,k=2\) & \(v=8.862\) m/s & 50 (bus 13) & \(\mu=6\), \(\sigma=0.6\) & \(G=483\) W/m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 9: PDF parameters of wind power and solar PV plants.