## 2 Fuel Cell Hybrid Electric Vehicles Modeling
### Fuel Cell Model  
A proton exchange membrane fuel cell (PEMFC) was selected as the primary energy source due to its rapid startup capabilities, its high-power density, and the safety afforded by the solid electrolytes. The single fuel cell output voltage \(V_{fc\_cell}\) can be expressed as [29]:  
\[V_{fc\_cell}=E_{ner}-V_{act}-V_{ohm}-V_{con}, \tag{3}\]  
where \(E_{ner}\) is the theoretical thermodynamic electric potential, \(V_{act}\) is the activation loss, \(V_{ohm}\) is the ohmic loss, and \(V_{con}\) is the concentration loss. \(E_{ner}\) can be expressed as [30]:  
\[E_{Net}=E_{O}-0.85\times(T-298.15)\times 10^{-3}+\frac{R_{0}T}{2F}\big{(}\ln P _{H_{2}}+0.5\ln P_{O_{2}}\big{)}, \tag{4}\]  
where \(E_{O}\) is the reference potential per unit of activity, \(F\) is the Faraday constant, \(R_{0}\) is the molar gas constant, \(P_{H_{2}}\) and \(P_{O_{2}}\) are, respectively, the varying partial pressures of hydrogen and oxygen, which can be obtained using the look-up table method. The fuel cell stack output power \(P_{fc}\) can be calculated as:  
\[P_{fc}=NV_{fc\_cell}I_{fc}, \tag{5}\] \begin{table}
\begin{tabular}{c c c} \hline
**Component** & **Parameters** & **Values** \\ \hline \multirow{4}{*}{Vehicle} & Mass & 2250 kg \\  & Frontal area & 2.88 m\({}^{2}\) \\  & Rolling resistance coefficient & 0.009 \\  & Wheel radius & 0.358 m \\ \hline \multirow{2}{*}{Fuel cell stack} & Nominal power & 80 kW \\  & Number of cells & 300 \\ \hline \multirow{4}{*}{Battery} & Nominal energy & 11.2 kWh \\  & Battery capacity & 32 Ah \\  & Number of cells & 96 \\  & Nominal Voltage & 350 V \\ \hline \multirow{2}{*}{Motor} & Maximum speed & 11,000 rpm \\  & Maximum torque & 320 Nm \\ \hline Unidirectional DC/DC converter & Efficiency & 0.98 \\ \hline Bidirectional DC/DC converter & Efficiency & 0.95 \\ \hline \end{tabular}
\end{table} 
Table 1: Main parameters of fuel cell hybrid electric vehicle. where \(N\) is the number of single fuel cells, and \(I_{fc}\) is the single fuel cell current. The fuel cell system net output power depends on the air compressor power loss \(P_{ac}\) and on the power losses of other components \(P_{aux}\). The fuel cell system net output power \(P_{fcs}\) can be expressed as:  
\[P_{fcs}=P_{fc}-P_{aux}-P_{ac}. \tag{6}\]  
The fuel cell hydrogen consumption rate \(\dot{m}_{H_{2}}\) can be calculated as:  
\[\dot{m}_{H_{2}}=\int_{0}^{t}\frac{M_{H_{2}}N}{2F}I_{fc}dt, \tag{7}\]  
where \(M_{H_{2}}\) is the hydrogen molar mass. The characteristic curve of the hydrogen consumption rate under the variation of the net output power of the fuel cell system is shown in Figure 2.  
The fuel cell system efficiency \(\eta_{fcs}\) can be expressed as:  
\[\eta_{fcs}=\frac{P_{fcs}}{\dot{m}_{H_{2}}LHV}, \tag{8}\]  
where \(LHV\) is the hydrogen low heating value, which has a value of 120,000 kJ/kg. Figure 3 shows the relationship between the efficiency and the net output power when the fuel cell system is running. In order to improve the economy of the FCHEV, the fuel cell system should work more in the high-efficiency zone during the driving process. The part of the fuel cell system with an operating efficiency higher than 50% is defined as the high-efficiency operating area of the system, which is the operating area corresponding to a net output power of 8 to 36 kW in Figure 3.  
Since the degradation problem was considered, it was important to construct a fuel cell degradation model. As a direct reflection of the fuel cell performance, the fuel cell stack voltage can sensitively respond to internal chemical changes and physical damage. Thus, the fuel cell stack voltage was chosen to characterize the fuel cell performance degradation. The voltage degradation of a single fuel cell \(V_{cell}\) can be expressed as:  
\[V_{cell}=V_{fc\_cell}-V_{dec}, \tag{9}\]  
where \(V_{dec}\) is the voltage degradation loss. Based on the degradation rates of the fuel cell performance tested by [PERSON] et al.