## 5 PEM Fuel Cell and Supercapacitor Model  
The referenced HFCEV had a 68 kW PEM fuel cell (PFC) and 7 kWh maximum stored energy supercapacitor as the two main energy sources. PEM fuel was considered the main energy source for the calculation of the energy consumption and supercapacitor for the energy recovery of the HFCEV. The characteristics of the fuel cell are presented in the Appendix A. The stack of PFCs had 255 cells. Each cell considered had the same efficiency. Equation (18) represents the Nernst equation for the voltage of one cell without energy losses.  
\[V_{Nernst}=\ V_{0}\ +\frac{RT_{FC}}{2F}\ln\left(\frac{P_{H_{2}}P_{O_{2}}\ ^{0.5}}{P_{H_{2}O}}\right) \tag{18}\]  
Here, \(V_{0}\) denotes an open-circuit voltage with a value equal to 1.2 V; \(T_{FC}\) indicates the temperature of the PFC with a constant value of 50 \({}^{\circ}\)C. R is the universal gas constant; F is the Faraday constant; \(P_{H_{2}}\), \(P_{O_{2}}\), and \(P_{H_{2}O}\) are the pressures of hydrogen gas, oxygen, and water vapor, respectively. The operating pressure of the fuel-cell system was set at a constant 1.5 bar. Equations (19)-(21) illustrate the voltage losses of the fuel cell in the order of the activation losses, \(V_{act}\); concentration losses, \(V_{con}\); and ohmic losses, \(V_{\mathrm{ohm}}\), respectively. To simplify, the activation losses and concentration losses have been defined in the form of an empirical equation as a function of the current and temperature.  
\[V_{act}=\frac{RT_{FC}}{n\alpha F}\ln(\frac{I}{I_{0}}) \tag{19}\]  
\[V_{con}=1.1\times 10^{-4}-12\times 10^{-6}(T_{FC}-273)e^{(0.008)I} \tag{20}\]  
\[V_{\mathrm{ohm}}\ =\ \frac{t_{m}}{\sigma}I \tag{21}\]  
where \(I_{0}\) represents the referenced current, \(\alpha\) is the transfer coefficient, \(t_{m}\) denotes the membrane thickness with a value equal to 0.02 mm, and \(\sigma\) denotes the electrical conductivity. The total voltage of the PFC \(V_{FC}\) is presented in Equation (22) considering the voltage losses. Here, \(N\) represents the stack number, and the fuel-cell unit had 255 cells.  
\[V_{FC}=[V_{Nernst}-V_{act}-V_{con}-V_{ohm}]N \tag{22}\]  
The fuel consumption of the PEM fuel cell, \(HC\), was calculated using the linear function in Equation (23) based on the characteristics of the PFC.  
\[HC=k_{FC}V_{FC}I \tag{23}\]  
\begin{table}
\begin{tabular}{c c c} \hline Differential Ratio & \(\tau_{diff}\) & 1:4 \\ \hline Differential efficiency & \(\eta_{diff}\) & 0.9 \\ \hline Gear ratio of 1G & \(\tau_{1G}\) & 1:3.625 \\ \hline Gear efficiency of 1G & \(\tau_{1G}\) & 0.92 \\ \hline First gear ratio of 2G & \(\tau_{2G1}\) & 1:2.724 \\ \hline First gear efficiency of 2G & \(\eta_{2G1}\) & 0.92 \\ \hline Second gear ratio & \(\tau_{2G2}\) & 1:1.251 \\ \hline Second gear efficiency of 2G & \(\eta_{2G2}\) & 0.92 \\ \hline Minimum CVT speed ratio & \(\tau_{CVT,min}\) & 0.4 \\ \hline Maximum CVT speed ratio & \(\tau_{CVT,max}\) & 2.5 \\ \hline Maximum CVT efficiency & \(\eta_{CVT,max}\) & 0.96 \\ \hline Minimum CVT efficiency & \(\eta_{CVT,min}\) & 0.89 \\ \hline \end{tabular}
\end{table}
Table 2: The specifications of the powertrain layouts.  
Here, \(k_{FC}\) is the fuel consumption rate with respect to the power of the fuel cell, with a value equal to 0.01. By using Equation (24), it is possible to calculate the value of the charge voltage, \(V_{sp}\), of the supercapacitor with respect to the recovered current during braking.  
\[\frac{dV_{sp}}{dt}=\frac{1}{\mathcal{C}}\frac{dQ}{dt}=\frac{1}{\mathcal{C}_{sp}}I \tag{24}\]  
Here, \(C_{sp}\) represents the rated capacitance. The state of charge (SoC) rate of the supercapacitor was used for drawing a comparison of the energy recovery rates of the different architectures, according to Equation (25).  
\[SoC=\
icefrac{{\%}}{{\%}}\frac{{V_{sp}}^{2}}{V_{sp,maks}}100 \tag{25}\]