# A Sea Horse Optimization-Based Approach for PEM Fuel Cell Model Parameter Estimation
## 4 Simulation Study and Results
### Parametric Analysis of SHO
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{\lambda}\) & **Time (s)** & **SSE** & \(\mathbf{R^{2}}\) & **Explanation** \\ \hline  & & & & Exploitation dominant. The step length is very short, so the algorithm searches more around the current region. Exploration is limited; the solution is stable, but the probability of reaching the global minimum is low. \\ \hline
1.1 & 1.70 & 0.000031 & 0.999346 & Near balance. Exploration ability has increased, but exploitation remains strong. Solution quality is high and time is acceptable. \\ \hline
1.3 & 1.72 & 0.000019 & 0.999498 & Exploration-oriented balance. Step lengths are longer, enabling broader search and better solution quality. Slight increase in convergence time. \\ \hline
1.5 & 1.74 & 0.000010 & 0.999791 & Optimal balance. Perfect harmony between exploration and exploitation. Best results in both accuracy and convergence. Most recommended value in the literature. \\ \hline
1.7 & 1.83 & 0.000037 & 0.999213 & Exploration dominant. Longer jumps enable broad search but delay convergence and increase error. \\ \hline
1.9 & 1.85 & 0.000063 & 0.998640 & Over-exploration. Very long jumps cause unstable roaming in the solution space. Exploitation is reduced, solution quality decreases. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The effect of the \(\lambda\) parameter on the SHO algorithm’s performance. This indicates that medium-sized random jumps balance both the exploration capacity and the convergence stability of SHO. While solution quality decreases with lower \(\beta_{t}\) values (e.g., 0.2) due to limited movement, higher \(\beta_{t}\) values (1.2) result in deterioration in convergence stability and increased error values due to excessive jump growth. The findings in Table 4 demonstrate that, within the framework of the exploration-exploitation balance emphasized in the literature, careful tuning of the \(\beta_{t}\) parameter is critical to the performance of the SHO algorithm. Another important parameter is the \(r_{1}\) parameter in the algorithm. This parameter is a random number and determines which movement pattern an individual will adopt in a given situation. In this study, the effects of different distributions of the \(r_{1}\) parameter on the SSE, \(R^{2}\), and convergence time is investigated. The results of tests conducted with different distributions are summarized below:  
* \(r_{1}=\text{randn}()\): Spiral and Brownian motions are balanced. In this reference case, SSE = 0.00001 and \(R^{2}=0.9998\) are achieved. * \(r_{1}=\text{rand}()\): Because all individuals demonstrate spiral motion, the algorithm converges quickly, but its exploration capability is somewhat reduced. Nevertheless, high accuracy and a quick solution are achieved. * \(r_{1}=0\): In this case, where all individuals explored only with Brownian motion, solution quality decreases significantly (SSE = 0.00409, \(R^{2}=0.912\)). This result demonstrates that excessive exploration bias can limit solution accuracy. * \(r_{1}\sim\text{Beta}(2,5)\): In this distribution with a higher exploration bias, a slight increase in the SSE value is observed, while the convergence time is extended. This demonstrates \begin{table}
\begin{tabular}{l l l l l} \hline \hline \(\beta_{t}\) & **Time (s)** & **SSE** & \(R^{2}\) & **Explanation** \\ \hline
0.2 & 1.66 & 0.000055 & 0.998830 & Low \(\beta_{t}\), narrower movement, shorter time but moderate solution quality \\ \hline
0.5 & 1.71 & 0.000033 & 0.999302 & Improvement starts with increasing \(\beta_{t}\) \\ \hline
0.8 & 1.82 & 0.000012 & 0.999733 & Best performance: strong exploration, good convergence \\ \hline
1.0 & 1.79 & 0.000040 & 0.999147 & Slight degradation begins, not too high \\ \hline
1.2 & 1.80 & 0.000051 & 0.998914 & Excessive jumps, performance degradation observed \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The effect of the \(\beta_{t}\) parameter on the SHO performance algorithm.