## 3 Results and Discussion
### Model Parameters Estimation  
While previous studies have employed a step-by-step procedure for fitting the bi-exponential model (1) to accelerated degradation paths of the cells [18], this study introduces a single-step approach using the integrated model (4) that combines the bi-exponential model with the Arrhenius acceleration factor. This single-step approach mitigates the potential for error accumulation inherent in multistep procedures, thereby enhancing the accuracy of the final prediction. In this study, relative voltage values were expressed as percentages. To account for cell-to-cell variability in degradation rates, most coefficients in the model (4) were designated as both fixed and random effects. However, the activation energy was treated only as a fixed effect because it was consistent across the stacks. Incorporating fixed and random effects into Equation (4) yielded the following modified model:  
\[ \begin{split} y_{ij}(t,T)&=(\phi_{1}+b_{1i})e^{-( \gamma_{1}+b_{2i})\left(t\cdot\exp\left(\frac{E_{4i}}{R_{2}}\left(\frac{1}{33315 }-\frac{1}{7}\right)\right)\right)}\\ &+(\phi_{2}+b_{3i})e^{-(\gamma_{2}+b_{4i})\left(t\cdot\exp\left( \frac{E_{4i}}{R_{2}}\left(\frac{1}{33315}-\frac{1}{7}\right)\right)\right)}+ \epsilon_{ij},\quad t\geq 0\end{split}  \tag{7}\]  
Table 2 presents the estimated fixed-effect coefficients obtained by fitting the ADT data to Equation (7). It includes the parameter estimates (estimate), their 95% confidence intervals (Lower and Upper), the standard errors (Std. Error), and the degrees of freedom (DF). The high t-values indicate that the parameters contribute significantly to the model. Furthermore, all \(p\)-values are below the commonly used significance level of 0.05, confirming their statistical significance. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Degradation Model**} & **Linear** & **Exponential** & **Power** & **Bi-Exponential** \\  & \(y=\beta_{1}t+\beta_{0}\) & \(y=\beta_{2}e^{\beta_{1}t}+\beta_{0}\) & \(y=\beta_{0}t^{\beta_{1}}\) & \(y=\beta_{1}e^{-\beta_{2}t}+\beta_{3}e^{-\beta_{4}t}\) \\ \hline
65 \({}^{\circ}\)C & 0.905 & 0.366 & 0.156 & **0.153** \\
80 \({}^{\circ}\)C & 1.010 & 0.869 & 1.044 & **0.458** \\ Total & 0.957 & 0.617 & 0.600 & **0.305** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: RMSE (Root Mean Square Error) values for degradation path models applied to each cell’s ADT data under 65 \({}^{\circ}\)C and 80 \({}^{\circ}\)C.  
Figure 5: Relative voltage degradation paths of PEMFC at 1.19 A/cm\({}^{2}\) and temperatures of 65 \({}^{\circ}\)C and 80 \({}^{\circ}\)C over 1000 h.  
The random effect coefficients \(b_{1i}\) follow a normal distribution with a mean of 0, and their covariance matrix is defined by diagonal elements of 0.137, \(3.451\times 10^{-12}\), \(4.100\times 10^{-9}\), and \(1.950\times 10^{-13}\), respectively. The final model is expressed as:  
\[y_{ij}(t,T) =(95.314+b_{1i})e^{-(1.474\times 10^{-5}+b_{2i})\left(t\cdot \exp\left(\frac{0.888}{R_{2}}\left(\frac{1}{3\pi\lambda 15}-\frac{1}{4}\right) \right)\right)} \tag{8}\] \[+(4.802+b_{3i})e^{-(1.529\times 10^{-2}+b_{4i})\left(t\cdot\exp \left(\frac{0.588}{R_{2}}\left(\frac{1}{3\pi\lambda 15}-\frac{1}{4}\right) \right)\right)}+\epsilon_{ij},\quad t\geq 0\]  
The degradation process is divided into two components. Comparing the parameters \(\phi_{1}\) and \(\phi_{2}\), it is observed that they represent approximately 95% and 5% of the total degradation, respectively. Since the value of \(\gamma_{2}\) is greater than \(\gamma_{1}\) and \(E_{a2}\) is lesser than \(E_{a1}\), it can be confirmed that most of the early degradation is reflected in \(\phi_{2}\), which accounts for the rapid performance drop at the beginning of the degradation process. The estimated activation energy \(E_{a1}\) affects long-term degradation and is more sensitive to temperature changes compared to \(E_{a2}\). This value corresponds to the reported activation energy for the ORR on platinum (Pt) and its alloys, which is approximately 0.8 eV [29; 30].