## 4 Simulation Results
### Results on Fuel Cell Stack Design Optimization  
A proton exchange membrane fuel cell (PEMFC) [21; 22] stack design optimization problem [23; 24; 25] is considered here. This problem has been investigated in the fuel cell literature as a problem of practical importance for which the global minimum is believed to be mathematically intractable [24]. This is a constrained optimization problem where the task is to minimize the cost of building a PEMFC stack that meets specific requirements. The objective (cost) function is a function of three variables \(N_{\text{p}},N_{\text{s}},A_{\text{cell}}\):  
\[\text{cost}=K_{\text{n}}\times N_{\text{p}}\times N_{\text{s}}+K_{\text{diff} }\times\left|V_{\text{load,rated}}-V_{\text{load,mpp}}\right|+K_{\text{a}} \times A_{\text{cell}}+\mathcal{P},\]  
where \(N_{\text{s}}\) is the number of cells connected in series in each group; \(N_{\text{p}}\) is the number of groups connected in parallel; \(A_{\text{cell}}\) is the cell area; \(V_{\text{load,r}}\) is the rated (given) terminal voltage of the stack; \(V_{\text{load,mpp}}\) represents the output voltage at the maximum power point of the stack; \(P_{\text{load,r}}\) is the rated (given) output power of the stack; \(P_{\text{load,max}}\) is the maximum output power of the stack; \(K_{\text{n}},K_{\text{diff}},K_{\text{a}}\) are pre-determined constants [24] used to adjust the relative importance of the different components of the cost function; and \(\mathcal{P}\) represents a penalty term given by  
\[\mathcal{P}= \begin{cases}0&\text{if }P_{\text{load,max}}\geq P_{\text{load,r}}; \\ c(P_{\text{load,r}}-P_{\text{load,max}})&\text{otherwise}.\end{cases} \]  
\(P_{\text{load,max}}\) and \(V_{\text{load,mpp}}\) are obtained numerically from the following equation by iterating over the load current \(i_{\text{load,d}}\) (recall that power is voltage times current), using a step size of \(i_{\text{load}}=1\) mA:  
\[V_{\text{st}}=N_{\text{s}}\left\{E_{\text{Nernst}}-A\ln\left(\frac{i_{\text{ load,d}}/N_{\text{p}}+i_{\text{n,d}}}{i_{\text{0,d}}}\right)+\ B\ln\left(1-\frac{i_{\text{load,d}}/N_{\text{p}}+i_{ \text{n,d}}}{i_{\text{limit,d}}}\right)-(i_{\text{load,d}}/N_{\text{p}}+i_{ \text{n,d}})r_{\text{a}}\right\},\]  
where \(V_{\text{st}}\) is the stack voltage, \(E_{\text{Nernst}}\) is the Nernst e.m.f., \(A\) and \(B\) are constants known from electrochemistry, \(r_{\text{a}}\) is the area-specific resistance, and the \(i\)'s represent different types of current densities (the subscript d is used to indicate density) in the cell [21; 26]. The lower and upper bounds of \(N_{\text{s}}\), \(N_{\text{p}}\) and \(A_{\text{cell}}\) are provided in Table 6 and the numerical values of the parameters in Table 7. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Variable** & **Lower Bound** & **Upper Bound** \\ \hline \(N_{\text{s}}\) & \(1\) & 50 \\ \(N_{\text{p}}\) & \(1\) & 50 \\ \(A_{\text{cell}}\) (cm\({}^{2}\)) & 10 & 400 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Bounds of the design variables [23]. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline \(V_{\text{load,r}}\) & 12 V \\ \(P_{\text{load,r}}\) & 200 W \\ \(K_{\text{n}}\) & 0.5 \\ \(K_{\text{diff}}\) & 10 \\ \(K_{\text{a}}\) & 0.001 \\ \(c\) & 200 \\ \(r_{\text{a}}\) & 98.0\(\times 10^{-6}\) K\(\Omega\) cm\({}^{2}\) \\ \(i_{\text{limit,d}}\) & 129 mA/cm\({}^{2}\) \\ \(i_{\text{0,d}}\) & 0.21 mA/cm\({}^{2}\) \\ \(i_{\text{n,d}}\) & 1.26 mA/cm\({}^{2}\) \\ \(A\) & 0.05 V \\ \(B\) & 0.08 V \\ \(E_{\text{Nernst}}\) & 1.04 V \\ \hline \hline \end{tabular}
\end{table} 
Table 7: PEMFC parameters and coefficients. Tables 8 and 9 present results of the two algorithms on the fuel cell problem; 30 independent runs are executed for each of 13 PopSize-Gens combinations for either algorithm. For this problem, the success of a run is defined as the production of at least one solution with a fitness of 13.62 or lower [24]. For 12 of the 13 cases in Table 8, the mean of the best-of-run costs is better for Slyaya than for Jaya.