Proton Exchange Membrane Fuel Cell Stack Durability Prediction Using Arrhenius-Based Accelerated Degradation Model

Featured Application

Proton Exchange Membrane Fuel Cell (PEMFC) Stack.

Abstract

To expand the applications of proton exchange membrane fuel cell (PEMFC) stacks, it is essential to address the issue of their short lifetime. Various studies are being conducted to improve this limitation, and efficient methods for verifying durability in a short period of time are required. This study presents a novel dynamic load cycling protocol designed to emulate the real-world driving conditions of commercial vehicles. This protocol was employed as an accelerated degradation test for PEMFC stacks under two elevated temperatures (65 and 80 °C), each conducted for 1000 h. A bi-exponential model, incorporating Arrhenius principles, was fitted to the degradation data. During this process, a mixed effects modeling approach was employed to distinguish between fixed and random effects within the model parameters. The activation energy was consistent across all cells and was thus designated as a fixed effect. Activation energy, which predominantly affects the long-term durability of PEMFC stacks, was estimated as 0.808 eV. By applying this estimated value to the Arrhenius equation, we calculated the acceleration factors for the degradation of fuel cell performance. Specifically, the rate of voltage degradation was found to be approximately 1.516 times faster at 65 °C and 4.923 times faster at 80 °C, compared to the standard operating temperature of 60 °C. Additionally, Monte Carlo simulations were conducted to predict the failure-time distribution under normal use conditions, estimating a median lifetime of 3884 h, which corresponds to 155,360 km of driving. This methodology offers a reliable and time-efficient framework for assessing PEMFC durability, with significant implications for reducing testing costs and accelerating the development of hydrogen fuel cell technology.

1. Introduction

Fuel cell electric vehicles (FCEVs) are expected to play a crucial role in realizing a low-carbon transportation future within the burgeoning hydrogen economy. Proton exchange membrane fuel cells (PEMFCs), with their low emissions, high efficiency, and low-temperature operation, are gaining traction in the automotive industry. However, PEMFC lifetime remains a critical challenge that must be overcome to achieve their widespread commercial viability. Over prolonged operation periods, performance deteriorates due to intricate operational conditions and component ageing, eventually reaching the minimum acceptable threshold [1].

The U.S. Department of Energy (DOE) has set ambitious durability targets for transportation fuel-cell systems. To be competitive with internal combustion engines and other alternatives, these systems must achieve over 8000 h of operation with less than a 10% performance loss [2]. Furthermore, for heavy-duty vehicles, the target is even more demanding, requiring over 25,000 h of durability by 2030 [3]. Achieving these durability targets under realistic automotive operating conditions necessitates lengthy and expensive testing procedures. Consequently, accelerated stress tests (ASTs) have emerged as a vital tool for rapidly assessing and predicting fuel-cell durability [4].

ASTs accelerate the degradation processes in fuel cells by intensifying operational stressors such as temperature [5], humidity [6], and load cycling. The DOE has established standardized AST protocols for various PEMFC components, including electrocatalysts, catalyst supports, membrane chemical/mechanical, and stack drive-cycles [7]. These tests are divided into two main categories: accelerated life tests (ALTs) and accelerated degradation tests (ADTs). ALTs operate fuel cells until failure, providing insights into degradation mechanisms. By contrast, ADTs leverage degradation data to predict the time to failure.

ALTs have proven effective in identifying the root causes of performance degradation. For instance, Ramaswamy et al. [5] demonstrated that increasing the temperature from room temperature to 80 °C accelerates membrane degradation owing to increased oxygen reduction reaction (ORR) activity and peroxide generation.

Although ALTs are feasible for individual PEMFC components, their application to complete stacks is often impractical owing to the extended testing durations required. For example, Toyota conducted modified accelerated durability tests on 13-cell prototype fuel-cell stacks for commercial FCEVs, simulating 30,000 start-stop and 73,000 load cycles under realistic driving conditions. Despite meeting their target durability of 15 years and 200,000 km [8], with a cell-voltage drop of under 6% (at 2.2 A/cm²), assessing the ultimate lifetime requires more time. Birkner et al. [9] developed a Dynamic Accelerated Stress Test (AST) to evaluate the durability of fuel cell Membrane Electrode Assemblies (MEAs) for high-performance heavy-duty vehicles (HDVs). This test revealed that after 1000 h of accelerated aging, the maximum power density of the fuel cell decreased by about 10% from over 800 mW/cm², and the Electrochemical Surface Area (ECSA) decreased by about 25%. Schuettoff et al. [10] developed accelerated durability test protocols under realistic operating conditions. They performed a 5500 h evaluation on a reference stack and conducted four additional tests, each for 1200 h, focusing on various stress factors. These tests achieved an acceleration factor of 3–7 times, depending on the specific test. They particularly noted that relative humidity and temperature significantly influenced degradation. However, while these experiments accelerate the degradation of the fuel cell by increasing stress conditions, allowing for relative comparisons with existing stacks, accurately predicting the lifetime of fuel cells in real-world driving conditions remains a significant challenge.

ADTs offer a more time-efficient approach by accelerating degradation through increased stress and employing models to predict the system’s lifetime. Two primary approaches are used in degradation modeling: black-box and white-box [11]. The white-box approach analyzes individual components and their interactions, whereas the black-box approach treats the system as a single unit, focusing on output trends. Chen et al. [12] used a model based on thermodynamic and electrode kinetic principles to analyze 7000 h of durability evaluation data collected under a constant current of 8 mA/cm². The Particle Filter (PF) algorithm was applied to predict the RUL (Remaining Useful Life), and it was found that during the lifespan of the FC, catalyst degradation contributed to 84.3% of the voltage degradation. However, owing to the inherent complexity of PEMFC stacks, the black-box approach is often favored, and various models have been employed. Bea et al. [13] applied a bi-exponential model to predict the PEMFC stack degradation. Subsequently, Yuan et al. [14] enhanced it by integrating a hierarchical Bayesian framework to enhance the reliability of the PEMFC stack system. Ma et al. [15] employed a deep-learning approach, called the grid long short-term memory recurrent neural network, to investigate the performance degradation and predict the lifetime of multiple stacks, including 1.2 kW Ballard Nexa and Proton Motor 200 (PM 200) 25-kW fuel cells.

Accurate lifetime prediction requires substantial degradation data, which can be time-consuming to acquire under normal operating conditions. Therefore, accelerated testing, coupled with appropriate data interpretation for normal conditions, is essential. The Arrhenius model provides a framework for quantifying the influence of temperature on degradation rates [16,17]. Bea et al. [18] conducted ADTs on direct methanol fuel cells at 60, 70, and 80 °C to predict their lifespans and employed the Weibull–Arrhenius model to predict lifetime under normal conditions. Specifically, they applied a step-by-step approach to fit the model to the ADT data.

This study introduces a novel single-step approach for analyzing ADT data obtained from PEMFC stacks operated at two different temperatures. An Arrhenius-based bi-exponential mixed-effects model was developed, integrating the bi-exponential degradation, Arrhenius, and a non-linear mixed-effects (NLME) model. This approach allows for the simultaneous determination of both fixed and random effects within the model parameters. The remainder of the paper is organized as follows: Section 2 presents the durability test protocol, bi-exponential model for characterizing cell-voltage degradation, application of the Arrhenius model as an acceleration factor, and determination of fixed- and random-effects parameters using the NLME model. Section 3 investigates the degradation of the PEMFC stack under different temperatures and analyzes the acceleration factor through the parameter estimation of the ADT data. Based on the estimated parameter values, we conducted Monte Carlo simulations to predict the failure-time distribution. Finally, Section 4 concludes the paper and presents directions for future research.

2. Methodology

This section details the methodology employed for modeling the degradation data obtained through the accelerated testing of PEMFC stacks and evaluating their reliability under normal operating conditions. The section is structured as follows: Section 2.1 describes the durability test protocol, Section 2.2 introduces the bi-exponential model for characterizing cell-voltage degradation, Section 2.3 discusses the integration of the Arrhenius model to account for temperature-induced acceleration of the PEMFC stack, and Section 2.4 explains the application of a NLME model to distinguish between fixed and random effects within the model parameters. The proposed method operates under the assumption that individual cells within a PEMFC stack degrade independently.

2.1. Durability Test Protocol

A durability test protocol was developed based on the Korean-World Harmonized Vehicle Cycle (K-WHVC) to analyze the degradation characteristics of a PEMFC stack subjected to load cycles representative of commercial vehicle operation. The K-WHVC, a modification of the World Harmonized Vehicle Cycle (WHVC), provides a more accurate representation of real-world driving patterns for heavy-duty vehicles in Korea [19] (Figure 1).

Using the speed data from K-WHVC driving environments (urban, rural, and motorway), dynamic load cycles matched to the road driving conditions of a 25-ton hydrogen-powered commercial vehicle equipped with a 200 kW fuel cell were extracted through MATLAB/Simulink simulations (2024a). Referencing these cycles, Figure 2 illustrates the dynamic load cycles employed in the durability tests. Each cycle spans 30 min, corresponding to 20 km of driving, and incorporates various load stages, including idle-, medium-, and high-load conditions.

A zero-velocity condition in a vehicle translates to an idle mode for the power propulsion system. Consequently, the modified electric load demand schedule included an idle mode with a minimum current density of 0.02 A/cm² [21]. This minimum current density was selected based on the findings of Takahashi et al. [8], which indicate that the degradation slope is optimized at voltages below 0.9 V. Under these conditions, the maximum current density was determined to be 1.19 A/cm².

This durability test protocol was implemented on two PEMFC stacks, each comprising 10 cells. One stack was tested at 65 °C, whereas the other was subjected to a higher temperature of 80 °C. This temperature range aligns with typical PEMFC stack operating temperatures, which generally fall between 60 and 80 °C [22], with an upper limit of 80 °C [23]. Additional operating conditions maintained throughout the 1000 h evaluation period included 50% humidification for both the anode and cathode, and stoichiometric ratios of 1.5 for the anode and 2.0 for the cathode. To minimize the variability of operating conditions other than temperature and maintain consistency, humidification and stoichiometric ratios were set based on reference settings [24].

2.2. Bi-Exponential Model

This section introduces the bi-exponential model used to characterize the degradation paths of multiple cells within a PEMFC stack. This model combines two exponential functions to capture the degradation process, allowing for the representation of distinct degradation phases. For instance, one exponential function may model the initial rapid degradation, while the other captures the subsequent long-term degradation phase. The model describes the fuel-cell voltage over time t using the following equation:

$$y\left(t\right)={\varphi }_1{e}^{-{\gamma }_{1}t}+{\varphi }_2{e}^{-{\gamma }_{2}t},t\ge 0$$

(1)

where ${\varphi }_1$, ${\varphi }_2$, ${\gamma }_1$, and ${\gamma }_2$ are the model parameters to be estimated from the degradation data; ${\varphi }_1(>$0) and ${\varphi }_2(>$0) represent the initial proportions of two heterogeneous compounds; and ${\gamma }_1(>$0) and ${\gamma }_2(>$0) denote the respective rate constants for the two compounds. To be physically meaningful, both the initial proportions and rate constants must be positive [25].

2.3. Arrhenius-Based Bi-Exponential Model

The primary operational factors contributing to PEMFC stack degradation include temperature, humidity, and voltage [5,6,10]. In this study, temperatures were selected as the primary acceleration factor for ADT. The following Arrhenius equation provides a basis for understanding the relationship between reaction rates and temperature [26]:

$$\tau =Aexp\left(-{\displaystyle \frac{E_a}{k_{B}T}}\right)$$

(2)

where $\tau$ is the rate constant, $E_a$ is the activation energy of the reaction (typically expressed in electron-volts (eV)), $k_B$ is Boltzmann’s constant ($8.617\times {10}^{-5}$ electron-volts per Kelvin (eV/K)), T is the absolute Kelvin temperature, and A is a constant that depends on the product characteristics and testing method employed. Using Equation (2), the acceleration factor ($AF$) between a normal operating temperature ($T_0$) and an accelerated temperature (T) can be expressed using the Arrhenius equation as follows:

$$AF=Aexp\left(-{\displaystyle \frac{E_a}{k_{B}T}}\right)/Aexp\left(-{\displaystyle \frac{E_a}{k_B{T}_0}}\right)=exp\left({\displaystyle \frac{E_a}{k_B}}\left({\displaystyle \frac{1}{T_0}}-{\displaystyle \frac{1}{T}}\right)\right)$$

(3)

Given a normal operating temperature of 60 °C for the PEMFC stack, $T_0$ is 333.15 K (60 + 273.15). By incorporating the acceleration factor from Equation (3) into the t of Equation (1), the accelerated degradation model for the fuel cell can be expressed as follows:

$$y(t,T)={\varphi }_1{e}^{-{\gamma }_1\left(t·exp\left({\displaystyle \frac{E_{a1}}{k_B}}\left({\displaystyle \frac{1}{333.15}}-{\displaystyle \frac{1}{T}}\right)\right)\right)}+{\varphi }_2{e}^{-{\gamma }_2\left(t·exp\left({\displaystyle \frac{E_{a2}}{k_B}}\left({\displaystyle \frac{1}{333.15}}-{\displaystyle \frac{1}{T}}\right)\right)\right)},t\ge 0$$

(4)

2.4. NLME Model

NLME models are statistical models used to analyze data that exhibit both fixed and random effects in non-linear relationships [27]. These models find applications in various fields, including pharmacology, growth analysis, and ecology. They capture overall trends using fixed effects while accounting for individual variations with random effects. NLME models are particularly relevant to this study because durability tests are inherently subject to cumulative uncertainties arising from manufacturing processes, measurement techniques, and the operating environment, all of which can influence voltage decay over time.

A general NLME model for the j-th response on the i-th individual test item can be defined as

$$\begin{array}{cc}\hfill y_{ij}=f({\beta }_{ij},{\mathbf{v}}_{ij})+{ϵ}_{ij},& i=1,\dots ,M,j=1,\dots ,n_i\hfill \\ \hfill \mathrm{where}& {\beta }_{ij}=A_{ij}\theta +B_{ij}{b}_i\hfill \end{array}$$

(5)

where $y_{ij}$ is the j-th response for the i-th individual. Additionally, $f(·)$ is a non-linear function of ${\mathbf{v}}_{ij}$ and, in this study, corresponds to Equation (4). ${\beta }_{ij}$ is modeled using a linear mixed-effects model, where $\theta$ and $b_i$ are vectors of fixed and random effects associated with individual cell i, respectively. Specifically, it includes the parameters ${\varphi }_1,{\varphi }_2,{\gamma }_1$, and ${\gamma }_2$ from Equation (4). ${\mathbf{v}}_{ij}$ is the covariate vector with the temperature T and measurement time t, whereas ${ϵ}_{ij}$ is a normally distributed random error term.

In this study, the parameter estimation was performed using the $nlme$ function in R [28]. For simplicity and to avoid overfitting, the random effects $b_i$ were assumed to be independent, implying zero covariance between them. This assumption also contributes to computational efficiency.

3. Results and Discussion

3.1. PEMFC Stack Degradation

The durability experiments of the self-designed dynamic load cycle were conducted for more than 1000 h. Two 10-cell stacks were used, each evaluated under different temperature conditions. One experiment was conducted under 65 °C conditions, and another under 80 °C conditions. Other operating conditions, such as flow rate, humidification, and pressure, were kept identical. Figure 3 compares the I-V curves measured under identical conditions before and after the durability test. The beginning of test (BOT) performance of both stacks was similar. However, when comparing the end of test (EOT) performance, a greater performance degradation was observed under high-temperature conditions (80 °C). Furthermore, it was confirmed that the higher the current density, the greater the performance degradation.

Figure 4 illustrates the variability in cell voltages observed at different positions within the fuel-cell stacks, highlighting the influence of the load cycle protocol and operating conditions. Figure 4a shows the maximum power output (1.19 A/cm²), whereas Figure 4b shows the minimum power output (0.02 A/cm²). At the BOT and a current density of 1.19 A/cm², the average cell voltage for the 10-cell stack was 0.697 V at 65 °C and 0.674 V at 80 °C. By the EOT, these values decreased by 6.46% to 0.652 V at 65 °C, and by 11.15% to 0.599 V at 80 °C (Figure 4a). At a lower current density of 0.02 A/cm², the average BOT voltages were 0.926 V at 65 °C and 0.920 V at 80 °C. The corresponding EOT voltages showed reductions of 3.15% to 0.897 V at 65 °C and by 3.13% to 0.891 V at 80 °C (Figure 4b).

While minimal temperature-dependent voltage reduction was observed at the minimum power output (0.02 A/cm²), the impact of elevated temperature on voltage degradation was more pronounced at the maximum power output (1.19 A/cm²). This suggests that analyzing degradation at the maximum power output provides a more sensitive and relevant assessment of PEMFC stack reliability.

At maximum power output (1.19 A/cm²), the standard deviation of cell voltages at BOT was 6.659 mV at 65 °C and 2.848 mV at 80 °C. Similar standard deviations were observed at EOT (5.746 mV at 65 °C and 7.542 mV at 80 °C). These findings indicate negligible position-dependent voltage deviation within the 10-cell PEMFC stack. The consistent standard deviations throughout the stack’s lifetime, well within acceptable limits, suggest that degradation occurs uniformly across all cells, independent of their position within the stack [8].

Figure 5 depicts the cell-voltage degradation paths for the PEMFC operated at 1.19 A/cm² under 65 and 80 °C conditions. The conditions are classified by temperature using the same color, and the relative voltage degradation path of each cell in the 10-cell stack is represented with individual lines. Voltage degradation was measured relative to each cell’s initial voltage, with data sampled at 50-h intervals over the 1000 h durability evaluation period. As expected, the voltage drop was more significant at the higher operating temperature of 80 °C. The degradation profiles exhibit a characteristic two-stage pattern: an initial phase of rapid voltage decline followed by a slower, more gradual decrease. This pattern aligns with previous observations in PEMFC degradation studies.

To represent the degradation paths of fuel cells, four popular degradation models are employed: linear, exponential, power, and bi-exponential models.

Table 1. RMSE (Root Mean Square Error) values for degradation path models applied to each cell’s ADT data under 65 °C and 80 °C.

Degradation Model	Linear	Exponential	Power	Bi-Exponential
	$\mathit{y}={\mathit{\beta }}_{\mathbf{1}}\mathit{t}+{\mathit{\beta }}_{\mathbf{0}}$	$\mathit{y}={\mathit{\beta }}_{\mathbf{2}}{\mathit{e}}^{{\mathit{\beta }}_{\mathbf{1}}\mathit{t}}+{\mathit{\beta }}_{\mathbf{0}}$	$\mathit{y}={\mathit{\beta }}_{\mathbf{0}}{\mathit{t}}^{{\mathit{\beta }}_{\mathbf{1}}}$	$\mathit{y}={\mathit{\beta }}_{\mathbf{1}}{\mathit{e}}^{-{\mathit{\beta }}_{\mathbf{2}}\mathit{t}}+{\mathit{\beta }}_{\mathbf{3}}{\mathit{e}}^{-{\mathit{\beta }}_{\mathbf{4}}\mathit{t}}$
65 °C	0.905	0.366	0.156	0.153
80 °C	1.010	0.869	1.044	0.458
Total	0.957	0.617	0.600	0.305

presents the RMSE (Root Mean Square Error) values calculated after fitting the degradation models to each cell’s measured data. The RMSE formula is as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}$$

(6)

where $y_i$ represents the observed values, ${\widehat{y}}_i$ denotes the predicted values from each degradation model, and n is the total number of observations. Therefore, a smaller RMSE indicates a better model, making the bi-exponential model the most suitable for representing fuel cell degradation paths.

3.2. 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:

$$\begin{array}{cc}\hfill y_{ij}(t,T)& =({\varphi }_1+b_{1i})e^{-({\gamma }_1+b_{2i})\left(t·exp\left({\displaystyle \frac{E_{a1}}{k_B}}\left({\displaystyle \frac{1}{333.15}}-{\displaystyle \frac{1}{T}}\right)\right)\right)}\hfill \\ \hfill & +({\varphi }_2+b_{3i})e^{-({\gamma }_2+b_{4i})\left(t·exp\left({\displaystyle \frac{E_{a2}}{k_B}}\left({\displaystyle \frac{1}{333.15}}-{\displaystyle \frac{1}{T}}\right)\right)\right)}+{ϵ}_{ij},t\ge 0\hfill \end{array}$$

(7)

Table 2. Fixed-effect coefficients estimated from the accelerated degradation test.

		95% Confidence Intervals
Parameter	Estimate	Lower	Upper	Std. Error	DF	t-Value	p-Value
${\varphi }_1$	95.314	95.123	95.505	$9.794\times {10}^{-2}$	395	973.238	$0.000\times {10}^0$
${\gamma }_1$	$1.474\times {10}^{-5}$	$1.224\times {10}^{-5}$	$1.724\times {10}^{-5}$	$1.282\times {10}^{-6}$	395	11.499	$1.361\times {10}^{-26}$
${\varphi }_2$	4.802	4.609	4.995	$9.891\times {10}^{-2}$	395	48.552	$1.331\times {10}^{-168}$
${\gamma }_2$	$1.529\times {10}^{-2}$	$1.237\times {10}^{-2}$	$1.822\times {10}^{-2}$	$1.497\times {10}^{-3}$	395	10.213	$6.899\times {10}^{-22}$
$E_{a1}$	0.808	0.706	0.909	$5.188\times {10}^{-2}$	395	15.570	$5.813\times {10}^{-43}$
$E_{a2}$	0.528	0.356	0.701	$8.816\times {10}^{-2}$	395	5.994	$4.616\times {10}^{-9}$

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.

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, $3.451\times {10}^{-12}$, $4.100\times {10}^{-9}$, $\mathrm{and}1.950\times {10}^{-13}$, respectively. The final model is expressed as:

$$\begin{array}{cc}\hfill y_{ij}(t,T)& =(95.314+b_{1i})e^{-(1.474\times {10}^{-5}+b_{2i})\left(t·exp\left({\displaystyle \frac{0.808}{k_B}}\left({\displaystyle \frac{1}{333.15}}-{\displaystyle \frac{1}{T}}\right)\right)\right)}\hfill \\ \hfill & +(4.802+b_{3i})e^{-(1.529\times {10}^{-2}+b_{4i})\left(t·exp\left({\displaystyle \frac{0.528}{k_B}}\left({\displaystyle \frac{1}{333.15}}-{\displaystyle \frac{1}{T}}\right)\right)\right)}+{ϵ}_{ij},t\ge 0\hfill \end{array}$$

(8)

The degradation process is divided into two components. Comparing the parameters ${\varphi }_1$ and ${\varphi }_2$, 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 ${\varphi }_2$, 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]. This finding is consistent with the observation by Chen et al. [12] that catalyst degradation contributes significantly (84.3%) to the overall voltage degradation in PEMFC.

Substituting the estimated $E_{a1}$ value into Equation (3) enables the calculation of the acceleration factors. The degradation rate is accelerated by 1.516 times at 65 °C and 4.923 times at 80 °C, relative to the normal operating temperature of 60 °C.

Figure 6 illustrates the excellent agreement between the fitted model (8) (blue lines) and experimental relative voltage data (red dots) for each cell at 1.19 A/cm² over 1000 h under accelerated conditions of 65 °C and 80 °C. The histogram of residuals shown in Figure 7 further supports the model’s accuracy, using CELL8 as a representative example. The symmetric distribution of residuals around zero, observed for all cells at both temperatures, confirms the model’s ability to capture the degradation trends in the experimental data. The generally larger deviations observed at 80 °C compared to those at 65 °C can be attributed to the more pronounced voltage drop at the higher temperature. The accelerated degradation at 80 °C leads to a wider range of voltage values, resulting in larger residuals. By contrast, the less severe degradation at 65 °C results in smaller deviations.

3.3. 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., $N=\mathrm{10,000}$) simulated degradation paths using the parameter estimators (${\widehat{\varphi }}_{1\left(1\right)},{\widehat{\gamma }}_{1\left(1\right)},{\widehat{\varphi }}_{2\left(1\right)},{\widehat{\gamma }}_{2\left(1\right)},{\widehat{E}}_{a1\left(1\right)},{\widehat{E}}_{a2\left(1\right)}$), …, (${\widehat{\varphi }}_{1\left(N\right)},{\widehat{\gamma }}_{1\left(N\right)},{\widehat{\varphi }}_{2\left(N\right)},{\widehat{\gamma }}_{2\left(N\right)},{\widehat{E}}_{a1\left(N\right)},{\widehat{E}}_{a2\left(N\right)}$) of (${\widehat{\varphi }}_1,{\widehat{\gamma }}_1,{\widehat{\varphi }}_2,{\widehat{\gamma }}_2,{\widehat{E}}_{a1},{\widehat{E}}_{a2}$) 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).

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. Percentile estimation of the fuel cell at normal use conditions.

Percentile	1	5	10	20	30	40	50
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

summarizes the estimated failure-times from the 1st to the 50th percentile, illustrating the failure-time distribution. As shown, the estimated failure-time at the 50th 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 normal operating conditions through model fitting. This highlights the potential for a more efficient and accurate assessment of fuel cell durability.

4. Conclusions

This study successfully modeled the degradation of a PEMFC stack under accelerated conditions using a novel approach, combining an NLME model with an Arrhenius-based bi-exponential model. By incorporating the Arrhenius equation, the study quantified the acceleration factor at different temperatures, providing a valuable methodology for predicting the long-term durability of PEMFC stacks under normal operating conditions. This approach significantly reduces the time required for validating PEMFC durability and offers a practical tool to minimize the time and costs associated with long-term durability testing.

However, this study primarily focused on the overall voltage degradation of the PEMFC stack and did not explicitly investigate the acceleration factors for various underlying degradation mechanisms. Future research will delve deeper into the physicochemical and electrochemical processes responsible for voltage decay to determine the acceleration factors for individual components within the PEMFC stack. This will enable a more comprehensive understanding of degradation phenomena and further improve the accuracy of lifetime predictions for PEMFC stacks.