## 4 HFAJAYA for the Multi-Constraint OPF Problems
### OPF without Stochastic Renewable Energy
#### 4.1.1 Case 1  
The first target function, which was solitary, was to minimize the power generation's overall fuel cost, which may be expressed using a quadratic curve the below [18]:  
\[J_{1}=\sum_{i=1}^{NG}F_{i}(P_{Gi})=\sum_{i=1}^{NG}\Bigl{(}a_{i}+b_{i}P_{Gi}+c_{i} P_{Gi}^{2}\Bigr{)} \tag{25}\]  
where \(P_{Gi}\) and \(F_{i}\) are the output and fuel cost of the \(i\)th generator. Moreover, \(a_{i}\), \(b_{i}\), and \(c_{i}\) are the cost coefficients of the \(i\)th unit, respectively. Finally, \(NG\) is the whole quantity of generators. In [59], the cost coefficient values were accessible. The results of this case, as given in Table 3, indicates that the proposed method can achieve better solution in compared to the state-of-the-art algorithms.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Variables Optimal**} & \multicolumn{3}{c}{**Limits**} & \multicolumn{6}{c}{**The Cases**} \\ \cline{2-9}  & **Min** & **Max** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline \(P_{G1}\) (MW) & 50 & 250 & 177.0297 & 139.9985 & 198.7328 & 102.6069 & 176.3325 & 126.7899 \\ \hline \(P_{G2}\) (MW) & 20 & 80 & 48.7224 & 54.9972 & 44.8834 & 55.5590 & 48.8232 & 52.0824 \\ \hline \(P_{G5}\) (MW) & 15 & 50 & 21.3825 & 24.1391 & 18.4734 & 38.1133 & 21.6338 & 30.2948 \\ \hline \(P_{G8}\) (MW) & 10 & 35 & 21.2881 & 4.9826 & 10.0000 & 35.0000 & 22.2391 & 34.9995 \\ \hline \(P_{G11}\) (MW) & 10 & 30 & 11.9890 & 18.6929 & 10.0001 & 30.0000 & 12.2210 & 25.0682 \\ \hline \(P_{G13}\) (MW) & 12 & 40 & 12.0017 & 17.3272 & 12.0000 & 26.6501 & 12.0000 & 20.0045 \\ \hline \(V_{G1}\) (p.u.) & 0.95 & 1.1 & 1.0836 & 1.0735 & 1.0816 & 1.0698 & 1.0423 & 1.0737 \\ \hline \(V_{G2}\) (p.u.) & 0.95 & 1.1 & 1.0605 & 1.0565 & 1.0579 & 1.0576 & 1.0228 & 1.0574 \\ \hline \(V_{G5}\) (p.u.) & 0.95 & 1.1 & 1.0337 & 1.0298 & 1.0302 & 1.0359 & 1.0145 & 1.0323 \\ \hline \(V_{G8}\) (p.u.) & 0.95 & 1.1 & 1.0382 & 1.0405 & 1.0371 & 1.0438 & 1.0055 & 1.0406 \\ \hline \(V_{G11}\) (p.u.) & 0.95 & 1.1 & 1.0991 & 1.0915 & 1.0995 & 1.0834 & 1.0733 & 1.0405 \\ \hline \(V_{G13}\) (p.u.) & 0.95 & 1.1 & 1.0519 & 1.0650 & 1.0633 & 1.0573 & 0.9873 & 1.0242 \\ \hline \(T_{6-9}\) (p.u.) & 0.9 & 1.1 & 1.0713 & 1.0616 & 1.0417 & 1.0653 & 1.0999 & 1.1000 \\ \hline \(T_{6-10}\) (p.u.) & 0.9 & 1.1 & 0.9183 & 0.9112 & 0.9692 & 0.9205 & 0.9000 & 0.9503 \\ \hline \(T_{4-12}\) (p.u.) & 0.9 & 1.1 & 0.9774 & 1.0028 & 0.9951 & 0.9901 & 0.9381 & 1.0334 \\ \hline \(T_{28-27}\)(p.u.) & 0.9 & 1.1 & 0.9743 & 0.9743 & 0.9776 & 0.9750 & 0.9710 & 1.0048 \\ \hline \(Q_{C10}\) (MVAR) & 0.0 & 5.0 & 2.6745 & 1.1882 & 4.7749 & 4.9569 & 4.9638 & 3.1583 \\ \hline \(Q_{C12}\) (MVAR) & 0.0 & 5.0 & 1.2842 & 1.2617 & 1.9178 & 0.1636 & 0.0080 & 0.0291 \\ \hline \(Q_{C15}\) (MVAR) & 0.0 & 5.0 & 4.2681 & 4.7107 & 3.7824 & 4.4512 & 5.0000 & 3.9138 \\ \hline \(Q_{C17}\) (MVAR) & 0.0 & 5.0 & 5.0000 & 4.2344 & 4.5810 & 5.0000 & 0 & 4.9986 \\ \hline \(Q_{C20}\) (MVAR) & 0.0 & 5.0 & 4.2732 & 4.2153 & 4.3853 & 4.2168 & 5.0000 & 4.9972 \\ \hline \(Q_{C21}\) (MVAR) & 0.0 & 5.0 & 4.9997 & 4.9164 & 4.9552 & 5.0000 & 5.0000 & 4.9994 \\ \hline \(Q_{C23}\) (MVAR) & 0.0 & 5.0 & 3.4064 & 3.0390 & 2.8829 & 3.2778 & 5.0000 & 4.3156 \\ \hline \(Q_{C24}\) (MVAR) & 0.0 & 5.0 & 4.9966 & 4.8430 & 4.9368 & 5.0000 & 5.0000 & 4.9969 \\ \hline \(Q_{C29}\) (MVAR) & 0.0 & 5.0 & 2.6155 & 2.5157 & 2.6963 & 2.5602 & 2.6384 & 2.6202 \\ \hline \(\text{Cost}\) (USD/h) & - & - & 800.4800 & 646.5020 & 832.1798 & 859.0165 & 803.7036 & 825.0311 \\ \hline Emission (t/h) & - & - & 0.3659 & 0.2835 & 0.4378 & 0.2289 & 0.3638 & 0.2597 \\ \hline Power losses (MW) & - & - & 9.0134 & 6.7375 & 10.6897 & 4.5293 & 9.8496 & 5.8393 \\ \hline V.D. (p.u.) & - & - & 0.9047 & 0.9041 & 0.8578 & 0.9327 & 0.0948 & 0.2956 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The optimal values for the six cases by HFAJAYA.