## 4 Experimental Results  
The experimental setup for the simulation of the interconnected tank system is detailed in Tables 6-9. Table 6 outlines the system parameters, including the tank area (A), interconnection cross-sectional area (\(A_{12}\)), drain pipe area (\(A_{2}\)), gravity (g), inflow rate (\(K_{1}\)), and valve gain/opening (a). These parameters are critical for establishing the equilibrium point of the interconnected tank system. Table 7 specifies the values used in the Ant Colony Optimization (ACO) algorithm, such as the number of ants (N), vaporization factor (\(\rho\)), scaling parameter (\(\zeta\)), number of iterations (T), step size (h), and the search space for the PID controller gains and fractional orders. Table 8 presents the parameters for the Grey Wolf Optimizer (GWO), detailing the population size (N), number of iterations (T), and search space for the controller parameters. Finally, Table 9 lists the parameters for the Flower Pollination Algorithm (FPA), including population size (N), number of iterations (T), coefficient (c), probability of change (p), scale parameter (\(\gamma\)), and search space for the controller gains and fractional orders. These tables collectively provide a comprehensive overview of the experimental conditions and algorithm-specific settings used in the optimization of the FOPID controller. For the FPA algorithm (Table 12), the ISE index again demonstrated the best performance, with an overshoot of 9.7737, an establishment time of 1.0704, and a rise time of 0.1514. The ITAE and IAE indices showed higher overshoot values and longer establishment times, while the ITSE index had a lower establishment time but a higher overshoot. Thus, the ISE index emerged as the most effective cost function for the FPA algorithm, ensuring a lower overshoot and reasonable establishment time. Overall, the results indicate that the ISE index generally performs better for both ACO and FPA algorithms, while the ITAE index is preferable for the GWO algorithm due to its balanced performance across different metrics. These findings highlight the importance of selecting appropriate performance indices as cost functions to optimize the tuning of fractional PID controllers using meta-heuristic algorithms. Using the IAE performance index as a cost function proved to be the least effective across all tuning algorithms. Figure 3 depicts the comparative results for the best-performing algorithms. The bar graph illustrates the overshoot, settling time, and rise time for the best results obtained by the ACO, GWO, and FPA algorithms. For the ACO algorithm, the overshoot was 10.6165%, the settling time was 9.07704 s, and the rise time was 1.27984 s. The GWO algorithm demonstrated an overshoot of 10.0831%, a significantly higher settling time of 12.5917 s, and a rise time of 1.22887 s. In comparison, the FPA algorithm achieved an overshoot of 10.7044%, a settling time of 9.77365 s, and a rise time of 1.51424 s. These results highlight the distinct performance characteristics of each algorithm, with the ACO algorithm showing a balance between overshoot and settling time, the GWO algorithm excelling in minimizing overshoot at the expense of a longer settling time, and the FPA algorithm providing a moderate performance across all metrics. This comparative analysis underscores the importance of selecting the appropriate algorithm based on the specific performance criteria required for optimal FOPID tuning. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Performance Index** & **Overshoot** & **Establishment Time** & **Rise Time** \\ \hline ITAE & 36.2235 & 1.5567 & 0.1534 \\ ISE & 9.7737 & 1.0704 & 0.1514 \\ ITSE & 10.8043 & 2.0958 & 0.1653 \\ IAE & 26.3363 & 2.7784 & 0.213 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: FPA algorithm results. Figure 3: Parameters of the system response of each algorithm with its best index.