## 3 Results and Discussion
### Failure-Time Analysis  
To derive the failure-time distribution under normal use conditions, we used the Monte Carlo simulation, based on the estimated parameters of the degradation model, identified as the most suitable for the ADT experiment data. The simulation was based on a multivariate normal distribution, with the mean derived from the fixed-effect parameter estimates, and the variance-covariance matrix was obtained from the random-effect estimates. Using these estimates, 10,000 degradation paths were simulated through Monte Carlo simulation. In other words, Generate \(N\) (large number: e.g., \(N=10,000\)) simulated degradation paths using the parameter estimators (\(\hat{\phi}_{1(1)}\), \(\hat{\gamma}_{1(1)}\), \(\hat{\phi}_{2(1)}\), \(\hat{\gamma}_{2(1)}\), \(\hat{E}_{a1(1)}\), \(\hat{E}_{a2(1)}\)),..., (\(\hat{\phi}_{1(N)}\), \(\hat{\gamma}_{1(N)}\), \(\hat{\phi}_{2(N)}\), \(\hat{\gamma}_{2(N)}\), \(\hat{E}_{a1(N)}\), \(\hat{E}_{a2(N)}\)) of (\(\hat{\phi}_{1}\), \(\hat{\gamma}_{1}\), \(\hat{\phi}_{2}\), \(\hat{\gamma}_{2}\), \(\hat{E}_{a1}\), \(\hat{E}_{a2}\)) from a multivariate normal distribution with mean (fixed effects) and variance-covariance matrix (random effects) [31]. The pseudo-failure is determined when the degradation path reaches the given failure threshold (90%) (Figure 8).  
Figure 6: Model fit (8) (blue lines) to the relative voltage degradation data (red dots) for each cell at 1.19 A/cm\({}^{2}\) over 1000 h under accelerated conditions (65 and 80 \({}^{\circ}\)C).  
Figure 7: Histograms of residuals resulting from model (8) for CELL8 under each condition (65 and 80 \({}^{\circ}\)C).  
Using the Monte Carlo simulation, we estimated 10,000 failure-times based on the given failure criterion. As the number of simulation iterations increases, the accuracy of the estimation improves. The resulting distribution of the estimated failure-times is shown in Figure 9. Table 3 summarizes the estimated failure-times from the 1 st to the 50 th percentile, illustrating the failure-time distribution. As shown, the estimated failure-time at the 50 th percentile is 3895 h, which represents the central lifetime. Since the evaluation protocol simulates a driving speed of 40 km/h, this corresponds to an estimated driving distance of 155,800 km.  
Accelerated tests on fuel cells have traditionally focused on observing the rapid degradation of performance under accelerated stress conditions [8; 9; 10]. However, this study advances the field by not only observing performance degradation but also developing methods to quantify the acceleration factor. Specifically, the intrinsic activation energy values of PEMFCs were derived using a non-linear mixed-effects model. Activation energy is a key parameter that influences the rate of fuel cell performance degradation under high-temperature conditions. By applying the Arrhenius equation, we can derive temperature-specific acceleration factors. As a result, fuel cell performance was evaluated up to the failure threshold within a short time, which enabled the simulation of failure-times under  
Figure 8: Simulated degradation paths from model (8) under normal conditions (60 \({}^{\circ}\)C).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Percentile** & **1** & **5** & **10** & **20** & **30** & **40** & **50** \\ \hline Lifetime (h) & 2812 & 3116 & 3273 & 3481 & 3636 & 3761 & 3895 \\ Distance (km) & 112,480 & 124,640 & 130,920 & 139,240 & 145,440 & 150,440 & 155,800 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Percentile estimation of the fuel cell at normal use conditions.  
Figure 9: Simulated failure-time distribution under normal conditions (60 \({}^{\circ}\)C).  
normal operating conditions through model fitting. This highlights the potential for a more efficient and accurate assessment of fuel cell durability.