## 6 Results and Discussion
### Testing the Accuracy of the WSO-WOA Algorithm
#### 6.1.2 Cascading FOPI-FOPI Controller  
Figure 10 illustrates the convergence curves of the various algorithms used in this study. In the case of the FOPI-FOPI controller, the optimization process was more complex due to the increased number of decision-making variables, totaling 24. These variables are crucial in fine-tuning the controller's response to various system dynamics, making the optimization task more challenging. The algorithms were systematically applied to this problem, with each algorithm being executed ten times to ensure the reliability and consistency of the results. Each run consisted of 50 iterations. Table 5 provides a clear comparison of the performance of various optimization algorithms in tuning the FOPI-FOPI, with the WSO-WOA algorithm emerging as the top performer. The WSO-WOA achieved the best OF value of 34.15 p.u., a result that underscores its effectiveness in fine-tuning controller parameters. Following closely was the WSO algorithm, which secured a best fitness value of 37.054 p.u. Although this result is competitive, it still falls short of the WSO-WOA, reflecting the nuanced differences in how these algorithms navigate the search space. On the other end of the spectrum, the SCA algorithm recorded the weakest performance with a best fitness value of 41.39 p.u., which is a 21.2% increase over WSO-WOA's result. This disparity suggests that the WSO algorithm may struggle with more complex or multidimensional optimization tasks, whereas hybrid algorithms like WSO-WOA can excel better.  
Figure 10: Comparison of the convergence curves of the proposed WSO-WOA with other algorithms used in tuning the cascading FOPI-FOPI controller.  
Similarly, the GWO and WOA algorithms also lagged behind WSO-WOA, showing fitness value increases of approximately 19% and 17.3%, respectively. The WSO algorithm, with a comparatively smaller increase of 8.5%, was the closest competitor but still did not match the superior performance of WSO-WOA. Table 5 indicates that the WSO-WOA algorithm required approximately 5.7 h to achieve its best result, a duration that is 1.84 times longer than that of the WSO algorithm, 1.68 times longer than the WOA, 1.81 times longer than the GWO, and 2.04 times longer than the SCA. This extended convergence time highlights a key trade-off. While the WSO-WOA algorithm delivers superior accuracy and optimization performance, it does so at the cost of increased computational resources and time. However, the offline tuning process mitigates this drawback. In offline optimization scenarios, the emphasis is on achieving the highest possible accuracy and optimal solutions rather than minimizing computational time. As a result, the longer convergence time of the WSO-WOA can be considered acceptable, especially when the ultimate goal is to enhance the system's performance and stability. Moreover, the additional time invested in the tuning process is often justified by the substantial improvements in power quality and controller performance that the WSO-WOA algorithm provides. By efficiently navigating the solution space and avoiding local minima, the WSO-WOA ensures that the final solution is accurate and robust, making it a valuable tool for complex optimization problems where precision is paramount. Thus, while the increased computational demand may seem like a limitation, it is outweighed by the significant benefits in terms of enhanced system performance and reliability.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Parameter** & **SCA** & **GWO** & **WOA** & **WSO** & **WSO-WOA** \\ \hline \(K_{21}^{dc}\) & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 \\ \(K_{1}^{dc}\) & 0.15 & 0.15 & 0.15 & 0.15 & 0.15 \\ \(A_{1}^{dc}\) & 1.461 & 1.1454 & 1 & 1 & 1.022 \\ \(K_{P2}^{dc}\) & 1 & 1 & 1 & 1 & 1 \\ \(K_{2}^{dc}\) & 0 & 0.6591 & 0 & 0 & 0.0