## 3 The Proposed Optimization Hybrid Algorithm
### Discussions  
Table 13 lists the results related to the cost's minimum, maximum, standard deviation, and mean values. According to this table, the TLTFWO provides more suitable solutions than its counterparts, i.e., PSO [87] (population size = 60), GA [88] (population size = 80), TFWO, and TLBO. Furthermore, even the worst solution of the proposed TLTFWO is more desirable than the best solutions of the PSO, GA, TFWO, and TLBO algorithms. So, TLTFWO is preferred when dealing with OPF problems in reality. Additionally, there is a small difference between the worst, average, and best solutions of the TLTFWO, showing its stability and reliability. The time required to converge to the optimal solution is also acceptable regarding the TLTFWO algorithm.  
Figure 12: Convergence trends for Case 8.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **Min** & **Mean** & **Max** & **Std.** & **Time (s)** \\ \hline \multicolumn{5}{c}{Case \(1\)} \\ \hline PSO & 801.1419 & 801.8326 & 802.9401 & 1.03 & 28 \\ GA & 801.6345 & 802.5472 & 803.2009 & 1.48 & 36 \\ TFWO & 800.8426 & 801.2513 & 801.5878 & 0.56 & 30 \\ TLBO & 800.9923 & 801.2958 & 801.6004 & 0.45 & 33 \\ TLTFWO & 800.4780 & 800.6012 & 800.7639 & 0.14 & 30 \\ \hline \multicolumn{5}{c}{Case 2} \\ \hline PSO & 647.5328 & 647.9796 & 648.6117 & 0.74 & 31 \\ GA & 647.9935 & 648.7213 & 649.5020 & 1.21 & 35 \\ TFWO & 646.9425 & 647.3011 & 647.7208 & 0.42 & 30 \\ TLBO & 647.5263 & 647.8730 & 648.4102 & 0.38 & 33 \\ TLTFWO & 646.4715 & 646.5819 & 646.7009 & 0.17 & 30 \\ \hline \multicolumn{5}{c}{Case 3} \\ \hline PSO & 832.9628 & 833.4139 & 833.8996 & 1.01 & 32 \\ GA & 833.6085 & 834.8323 & 836.0047 & 1.95 & 38 \\ TFWO & 832.6598 & 832.9418 & 833.3994 & 0.61 & 29 \\ TLBO & 832.7624 & 832.9771 & 833.4825 & 0.72 & 34 \\ TLTFWO & 832.1584 & 832.2837 & 832.4035 & 0.19 & 30 \\ \hline \multicolumn{5}{c}{Case 4} \\ \hline PSO & 1045.3157 & 1046.7329 & 1047.3214 & 1.61 & 33 \\ GA & 1045.9559 & 1047.1046 & 1048.3610 & 1.14 & 40 \\ TFWO & 1041.4914 & 1042.5326 & 1043.4218 & 1.52 & 31 \\ TLBO & 1044.2764 & 1045.4579 & 1046.8018 & 1.49 & 35 \\ TLTFWO & 1040.1875 & 1040.3267 & 1040.4793 & 0.23 & 29 \\ \hline \multicolumn{5}{c}{Case 5} \\ \hline PSO & 815.3592 & 816.5410 & 817.7862 & 1.75 & 32 \\ GA & 816.6919 & 817.7764 & 819.2102 & 2.32 & 35 \\ TFWO & 814.2310 & 815.6249 & 816.5998 & 1.40 & 30 \\ TLBO & 815.3880 & 816.4528 & 817.6781 & 1.44 & 34 \\ TLTFWO & 813.1829 & 813.3613 & 813.4817 & 0.15 & 30 \\ \hline \multicolumn{5}{c}{Case 6} \\ \hline PSO & 970.9024 & 973.1466 & 974.1565 & 1.39 & 36 \\ GA & 973.4101 & 975.6303 & 976.8919 & 2.16 & 40 \\ TFWO & 967.1586 & 967.8543 & 968.4628 & 0.55 & 30 \\ TLBO & 971.4777 & 972.0647 & 972.9847 & 0.81 & 33 \\ TLTFWO & 964.2506 & 964.3928 & 964.5224 & 0.15 & 32 \\ \hline \hline \end{tabular}
\end{table}
Table 13: Results of various parameters obtained by TTWO, TFWO, and TLBO algorithms.  
Moreover, the first benefit of using renewable energy sources can be understood by comparing the fuel cost calculated in the two studied cases, 1 and 7. The optimized calculation cost in case 1 for the proposed algorithm equals 800.4780 USD/h. In contrast, the value of the fuel cost calculated in case 7 of the article for the same system with renewable energy is equal to 781.9791 USD/h, which has a significant reduction. On the other hand, with the optimal use of renewable energy sources in the energy system, pollution can be effectively reduced. For example, by comparing cases 7 and 8, it can be seen that by considering the amount of production pollution as an objective function, the amount of pollution has been reduced effectively. For the proposed algorithm, it has decreased from the value of 1.76245 t/h to a much lower value and almost half equal to 0.88144 t/h. If we had used fossil fuel sources instead of these renewable energy production units, we would never have been able to reduce the amount of production pollution to this extent.