## 3 The Proposed Optimization Hybrid Algorithm
### OPF Solutions IEEE 30-Bus Network [56]
#### 4.1.5 Case 5: Minimization of Fuel Cost and Voltage Deviation  
Among the critical indices of network security and continuation of supply to the customers is the magnitude of voltages of network buses. It is worth noting that adopting only one cost objective function in the OPF problem reaches a solution in which the voltage profile is unsatisfying. To this end, the present problem utilizes two objective functions: the fuel cost is minimized, the voltage profile is enhanced, and the voltage deviation on load buses does not violate one p.u. Equation (51) formulates the objective function of Case 5:  
\[\begin{split}& J_{5}=\sum\limits_{i=1}^{NG}\alpha_{i}+b_{i}P_{Gi}+c_{ i}P_{Gi}^{2}+\phi_{v}\sum\limits_{i=1}^{NL}\left|VL_{i}-1.0\right|+\lambda_{P}(\,P_{ Gi}-P_{G1}^{\text{lim}}\,)^{2}\\ &+\lambda_{V}\sum\limits_{i=1}^{NL}\left(\,VL_{i}-VL_{i}^{\text {lim}}\right)^{2}+\,\lambda_{Q}\sum\limits_{i=1}^{NG}\left(\,Q_{Gi}-Q_{Gi}^{ \text{lim}}\right)^{2}+\,\lambda_{S}\sum\limits_{i=1}^{NL}\left(\,S_{i}-S_{i} ^{\text{lim}}\right)^{2}\end{split} \tag{51}\]  
where, \(\phi_{v}\) = 100 [56].  
Table 3 provides the results of optimal settings of control variables when the TLTFWO is used for simulations. In addition, Table 8 compares the results of various algorithms. As is observed, TLTFWO has significantly reduced the value of the multi-objective function. The convergence curves of this function obtained by the TFWO, TLBO and TLTFWO algorithms in Case 5 are shown in Figure 9.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Optimizer** & \
\begin{tabular}{c} **Fuel Cost** \\ **(USD/h)** \\ \end{tabular} & **Emission (t/h)** &
\begin{tabular}{c} **Power** \\ **Losses (MW)** \\ \end{tabular} & **V.D. (p.u.)** & \(J_{4}\) \\ \hline EMSA [78] & 859.9514 & 0.2278 & 4.6071 & 0.7758 & 1044.2354 \\ MOALO [13] & 826.4556 & 0.2642 & 5.7727 & 1.2560 & 1057.3636 \\ MJaya [79] & 827.9124 & - & 5.7960 & - & 1059.7524 \\ MSA [56] & 859.1915 & 0.2289 & 4.5404 & 0.92852 & 1040.8075 \\ SpDEA [80] & 837.8510 & - & 5.6093 & 0.8106 & 1062.223 \\ QOMJaya [79] & 826.9651 & - & 5.7596 & - & 1402.9251 \\ TFWO & 860.1514 & 0.2292 & 4.5335 & 0.9145 & 1041.4914 \\ TLBO & 860.2684 & 0.2295 & 4.6002 & 0.9096 & 1044.2764 \\ TLTFWO & 859.0075 & 0.2289 & 4.5295 & 0.9279 & 1040.1875 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Optimal results of the present study in Case 4.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Optimizer** & \
\begin{tabular}{c} **Fuel Cost** \\ **(USD/h)** \\ \end{tabular} & **Emission (t/h)** &
\begin{tabular}{c} **Power** \\ **Losses (MW)** \\ \end{tabular} & **V.D. (p.u.)** & \(J_{5}\) \\ \hline PSO [81] & 804.477 & 0.368 & 10.129 & 0.126 & 817.0770 \\ PSO-SSO [81] & 803.9899 & 0.367 & 9.961 & 0.0940 & 813.3899 \\ BB-MOPSO [82] & 804.9639 & - & - & 0.1021 & 815.1739 \\ EMSA [78] & 803.4286 & 0.3643 & 9.7894 & 0.1073 & 814.1586 \\ TFWO [17] & 803.416 & 0.365 & 9.795 & 0.101 & 813.5160 \\ SpDEA [80] & 803.0290 & - & 9.0949 & 0.2799 & 831.0190 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Optimal results of the present study in Case 5.