## 2 Simulation Model
### Fuel Cell Degradation Model
#### 2.3.2 Modeling Based on Empirical Parameters  
1. Low power  
Under this operating power, the fuel cell operates primarily in the activation overpotential regime. The degradation in this region is primarily caused by the degradation of the proton exchange membrane and the cathode catalyst. By analyzing the activation voltage formula, it is known that the main factors affecting the activation polarization voltage are the stack current density \(i_{stack}\), the exchange current density \(i_{0}\), and the internal current density \(i_{n}\). Among them, \(i_{0}\) has a functional relationship with ECSA as follows [26]:  
\[i_{0}=i_{0}^{Pt}ECSA \tag{8}\]  
\[ECSA=S_{0}\times e^{-k_{c}t} \tag{9}\]  
In the equation, A represents the ratio of the catalyst's active surface area to the single-cell area of the fuel cell. As shown in Figure 5, the typical active surface area of a fuel cell exhibits exponential characteristics [27]. Therefore, the activation voltage formula is as follows:  
\[V_{act}=\frac{RT}{\alpha\cdot n\cdot F}ln\left(\frac{i_{stack}+i_{n}}{i_{0} \times S_{0}\times e^{-k_{c}t}}\right)=V_{act,0}-A\times k_{c}\times t \tag{10}\]  
According to the reference model, the voltage decay in idle mode is determined by measuring the current density at 10 mA/cm\({}^{2}\). Therefore, at this current density, the polarization voltage coefficient is normalized to 1 [19]. The calculation of the polarization voltage coefficients at other current densities is as follows:  
\[\tau_{lp}(i)=0.0006i+0.9937 \tag{11}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Operating Condition** & \(\frac{d\delta}{dt}\) \\ \hline Idle Power (\%/h) & \(1.26\cdot 10^{-5}\) \\ Load Power (\%/cycle) & \(5.93\cdot 10^{-7}\) \\ High Power (\%/h) & \(1.47\cdot 10^{-5}\) \\ Start-Stop (\%/cycle) & \(1.96\cdot 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fuel cell baseline degradation rate. Figure 5: ECSA degradation rates. Medium power  
When operating in moderate power conditions, fuel cells experience more pronounced degradation factors. At this stage, the fuel cell stack exhibits higher ohmic losses. The ohmic resistance, which contributes to the ohmic losses, consists of three main components: ionic resistance \(R_{ion}\), electronic resistance \(R_{dce}\), and contact resistance \(R_{contact}\). The overall ohmic resistance can be determined via high-frequency resistance (HFR) experiments. Therefore, the formula for ohmic resistance is expressed as  
\[R_{ohm}=R_{ele}+R_{ion}+R_{contact} \tag{12}\]  
According to the results from reference [28], the ohmic resistance is a function that varies with time. Considering the degradation rate from the reference model, the coefficient for the high-power point can be set to 1. Therefore, the calculation of the ohmic resistance coefficient for fuel cells under moderate power conditions is as follows:  
\[\tau_{mp}(i)=0.0002i+0.8592 \tag{13}\]  
3. High power  
When fuel cells operate at high power levels, the ohmic losses and electrochemical losses in the fuel cell stack increase, leading to an elevated temperature. At this stage, the increased temperature in the fuel cell can potentially accelerate degradation phenomena. Degradation primarily occurs due to phenomena such as partial oxygen starvation, membrane flooding, and increased losses in kinetics, ohmic resistance, and mass transport. All these losses are directly proportional to the increase in current density. In the absence of temperature and humidity considerations, the degradation mechanisms in this scenario can be scaled with the current density:  
\[\tau_{hp}(i)=0.0002i+0.7674 \tag{14}\]  
4. Load Change  
The degradation caused by load variations is primarily a result of multiple factors interacting together. The main phenomena leading to membrane and catalyst degradation are associated with cathode and anode starvation, gas flow, water management, and thermal management issues.