# A Sea Horse Optimization-Based Approach for PEM Fuel Cell Model Parameter Estimation
## 4 Simulation Study and Results
### PEM Fuel Cell Model Parameter Estimation  
In this study, for PEMFC modeling, the Ballard Mark V model is employed. The model parameters and operating conditions are set as follows: \(N=1\), \(T=343\) K, \(P_{\mathrm{O}_{2}}=1\) bar, \(R_{\mathrm{H}_{2}}=1\) bar, \(J_{\mathrm{max}}=1.5\) A/cm\({}^{2}\), \(l=0.0178\) cm, \(A=50.6\) cm\({}^{2}\). Reference experimental measurements were obtained from the polarization curve in [28].  
For the Ballard Mark V model, experimental measurements reported in the literature are used. The population size is set to 30, and the maximum number of iterations is limited to 5000. Since the performance of an optimization method is limited by parameter limits, The search area of the Ballard Mark V PEMFC model is limited, using the limit values available in the literature for the parameters to be optimized [1; 2; 6; 12; 23; 35; 36; 37]. Boundary limits for unknown parameters in the PEMFC model are given in Table 7.  
The following results were obtained from the optimization process: RMSE = 0.001322, MAE = 0.001079, \(R^{2}\) = 0.999791, true/model ratio = 0.9992 (seahorse success rate), convergence time: 1.74 s, best SSE = 0.000010. Best Parameters (ObjectivePosition): \(\epsilon_{1}=-0.853200\), \(\epsilon_{2}=0.003250\), \(\epsilon_{3}=0.000036\), \(\epsilon_{4}=-0.000095\), \(\lambda=12.068511\), \(R_{\mathrm{c}}=0.000293\), \(\beta=0.013642\).  
To assess the impact of varying the parameter search limits on model accuracy, sensitivity analysis was performed using limits narrowed and expanded by \(\pm 10\%\) in addition to  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(r_{2}\) Distribution & **Best SSE** & **Time (s)** & \(R^{2}\) & **Explanation** \\ \hline Fixed (\(r_{2}=0.5\)) & 0.020477 & 1.61 & 0.562087 & Low accuracy, fast runtime, but weak exploration–exploitation balance. \\ \hline Beta(5,2) (exploitation–biased) & 0.009850 & 4.15 & 0.789031 & Better accuracy, but high exploitation increases the risk of local minima. \\ \hline Beta(2,5) (exploration–biased) & 0.000029 & 4.28 & 0.999375 & Very high accuracy with best balance, but the longest runtime. \\ \hline Uniform \(U(0,1)\) & 0.000010 & 1.74 & 0.999791 & Most balanced structure, high accuracy and short runtime, recommended default setting. \\ \hline \hline \end{tabular}
\end{table}
Table 6: Effect of \(r_{2}\) distribution on SHO algorithm performance.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model Parameter** & \(\epsilon_{1}\) & \(\epsilon_{2}\) & \(\epsilon_{3}\) & \(\epsilon_{4}\) & \(\lambda\) & \(R_{c}\) & \(\beta\) \\ \hline Lower Bound & \(-1.1997\) & 0.001 & 0.000036 & \(-0.00026\) & 10 & 0.0001 & 0.0136 \\ Upper Bound & \(-0.8532\) & 0.005 & 0.000098 & \(-0.0000954\) & 24 & 0.0008 & 0.5 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Boundary limits for unknown parameters in PEMFC model [1; 2; 6; 12; 23; 35; 36; 37].  
the literature limits. The results obtained with \(\pm\)10% narrowing or expanding of the limit values are shown in Table 8.  
According to the results obtained, when the limits in the literature were used, the lowest error value was obtained, RMSE was 0.001322, and the best SSE was 0.00001. In cases where the boundaries were narrowed and expanded, model accuracy remained acceptable, with RMSEs of 0.002709 and 0.002629, respectively. These findings indicate that the SHO algorithm has low sensitivity to small changes in parameter boundary values, thus achieving high solution stability and identification reliability.  
The comparison between the simulated model and the experimental data and while the SHO algorithm converges over 5000 iterations is depicted in Figure 4. A gradual decrease in the error value is observed, and finally, the best SSE of 0.000010 is reached.