## 5 Conclusions  
This study suggested a novel modified ITFWO algorithm for optimizing various complex OPF problems such as piecewise quadratic and quadratic objective functions, total cost while considering emissions, and losses and valve point effects in the IEEE 30-bus network with PV and WT units while satisfying security, physical and feasibility limits. Firstly, the various complex OPF problems have been illustrated as real-world optimization problems with different limits in a typical network. OPF with the various complex cost functions has been efficiently solved through the proposed ITFWO method whose computational efficiency, robustness and applicability have been also evidenced. ITFWO has efficiently fulfilled the objective to discover near-global optimal or optimum solutions of the non-linear test functions of the typical power network more effectively than previous optimal solutions and confirms the optimization power of the ITFWO method in comparison with the other optimization techniques based on the result quality for the various complex OPF problems. An equation of this nature cannot be solved using conventional methods, such as the equal consumed energy increase ratio law, when the constraint is complex and the cost function is not convex. In terms of solving such problems, the proposed ITFWO provides a feasible and effective reference scheme. It is found that the proposed ITFWO provides the lowest minimum of total cost among all the heuristic optimization techniques and confirms its capability in yielding a suitable balance between exploitation and exploration with better performance in terms of efficiency and robustness.  
It has been found that the proposed ITFWO algorithm performs better than the other algorithms. This algorithm beats the original TFWO and a lot of other optimization algorithms in recent papers. In light of the ITFWO's success in solving various OPF problems, it should also be applicable to other optimization problems. As part of our future studies, we will use the proposed algorithm to solve problems related to micro-grid power dispatch, global optimization of overcurrent relays, and dust control systems. Furthermore, ITFWO can solve complex hydrothermal scheduling, dynamic OPF, and optimal reactive power dispatch (ORPD). The author is particularly interested in the field of intelligent control of industrial dust in environmental protection, which is one of the areas of future research he plans to pursue.  
This research is supported by the Deanship of Scientific Research at Majmaah University under Project Number No. R-2023-336.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Type** & **Method** & **Min** & **Mean** & **Max** & **Std.** & **Time (s)** \\ \hline \multirow{4}{*}{1} & TFWO & 800.7494 & 800.9724 & 801.3421 & 0.47 & 24 \\  & ITFWO & 800.4787 & 800.5890 & 800.6625 & 0.15 & 27 \\  & TFWO & 646.9958 & 647.3823 & 647.5492 & 0.61 & 26 \\  & ITFWO & 646.4799 & 646.5597 & 646.6483 & 0.19 & 22 \\  & ITFWO & 832.6795 & 832.9914 & 833.4756 & 0.52 & 27 \\  & ITFWO & 832.1611 & 832.2704 & 832.37921 & 0.12 & 25 \\  & TFWO & 1041.6999 & 1041.9880 & 1042.3729 & 0.35 & 25 \\  & ITFWO & 1040.1889 & 1040.2974 & 1040.3582 & 0.18 & 28 \\  & TFWO & 813.9110 & 814.3185 & 814.7001 & 0.44 & 28 \\  & ITFWO & 813.2267 & 813.4023 & 813.5199 & 0.21 & 23 \\  & ITFWO & 965.9551 & 966.4769 & 966.9814 & 0.52 & 24 \\  & ITFWO & 964.2606 & 964.4160 & 964.5976 & 0.23 & 24 \\  & TFWO & 782.3182 & 782.5711 & 782.8745 & 0.43 & 29 \\  & ITFWO & 781.9943 & 782.1624 & 782.3004 & 0.25 & 33 \\  & ITFWO & 810.8986 & 811.1852 & 811.5639 & 0.39 & 31 \\  & ITFWO & 810.1812 & 810.3045 & 810.4496 & 0.16 & 27 \\ \hline \hline \end{tabular}
\end{table}
Table 10: The optimal solutions to show the optimization power of the TFWO and ITFWO algorithms.