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# Degradation root cause analysis of PEM fuel cells using distribution of relaxation times

Jian Zuo, Nadia Yousfi Steiner, Zhongliang Li, Daniel Hissel

► To cite this version:

Jian Zuo, Nadia Yousfi Steiner, Zhongliang Li, Daniel Hissel. Degradation root cause analysis of PEM fuel cells using distribution of relaxation times. Applied Energy, 2025, 378, pp.124762. 10.1016/j.apenergy.2024.124762. hal-04965323

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Submitted on 25 Feb 2025

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# Highlights

## Degradation root cause analysis of PEM fuel cells using distribution of relaxation times

Jian Zuo, Nadia Yousfi Steiner, Zhongliang Li, Daniel Hissel

* PEM fuel cell degradation analysis based on impedance spectra.

* The polarization processes are quantified by distribution of relaxation times.

* Degradation root cause identified by contributions of overpotential losses.

* Two long-term durability tests are used to validate the degradation analysis strategy.
# Degradation root cause analysis of PEM fuel cells using distribution of relaxation times

Jian Zuo<sup>a</sup>, Nadia Yousfi Steiner<sup>a</sup>, Zhongliang Li<sup>a,*</sup>, Daniel Hissel<sup>a,b</sup>

<sup>a</sup>Université de Franche-Comté, FEMTO-ST, UTBM, CNRS, Belfort, France

<sup>b</sup>Institut Universitaire de France (IUF), France

## Abstract

Proton exchange membrane (PEM) fuel cells use hydrogen as fuel and produce only water and heat as byproducts. This makes them a promising technology for the worldwide energy transition. However, durability and efficiency remain two main obstacles to their widespread commercialization. Degradation of fuel cells is thus a key focus in recent studies. The majority of existing works use voltage or power as the global health indicator for evaluating the degradation of a running fuel cell. The use of global health indicators is limited in distinguishing the contributions of individual physicochemical degradation processes to the overall performance degradation. This work proposes to investigate the degradation root cause of PEM fuel cells using the distribution of relaxation times (DRT) analysis. DRT is a model-free approach that can separate the overlapped polarization processes by transforming the frequency-domain impedance data into distribution of time constants. Datasets from two long-term durability tests are used to calculate the DRT and identify the degradation root causes. Four distinct peaks are identified from the electrochemical impedance spectroscopy spectra, which correspond to the mass transport, charge transfer of the oxygen reduction reaction, proton transfer, and interface contact. The results show that the degradation of the fuel cell stacks is mainly caused by the degradation of the cathode catalyst layer and the gas diffusion layer. Specifically, these two processes account for 91.2% and 90.4% of the overall degradation in FC1 and FC2, respectively. Moreover, it is found that the mass transport resistance contributes a larger percentage under the dynamic current load (38.2%) compared to the constant load (23.1%). Thus, the oxygen diffusion process needs to be enhanced by proper operating strategy or improving the design of the cathode catalyst layer and gas diffusion layer for dynamic load applications. The proposed method provides a new perspective for the degradation analysis and prognostics of fuel cells, which can be further used for the development of efficient control and management strategies to ensure enhanced durability of fuel cells.

**Keywords:** PEM fuel cell, degradation root cause, distribution of relaxation times, electrochemical impedance spectroscopy, polarization resistance, overpotential losses

## 1. Introduction

Hydrogen is considered one of the most promising future energy carriers due to its high energy density and environmental benignity [1]. Energy devices that use hydrogen as fuel, such as fuel cells, are considered a key technology for the transition to a sustainable energy system [2]. Among different types of fuel cells, proton exchange membrane (PEM) fuel cells are being deployed in various applications due to their high power density, low operating temperature, and easy scale-up property [3]. However, low efficiency, durability, and reliability remain the main obstacles to the widespread commercialization of PEM fuel cells [4]. It is thus of high importance to measure and quantify the internal degradation of fuel cells to enable the development of efficient control and management strategies to enhance the long-term durability and reliability of fuel cells [5, 6].

The degradation of key components, such as the proton-exchange membrane, catalyst layer (CL), gas diffusion layer (GDL), and bipolar plate (BP), leads to the overall deterioration of fuel cell stacks [7, 8, 9]. In fuel cells, the

<footnote>*Corresponding author.
Email address: zhongliang.li@univ-fcomte.fr (Zhongliang Li)</footnote>
physicochemical processes that affect their performance are charge transport at the electrodes (mainly the cathode), proton transport through the electrolyte, and mass transport of the reactants [10, 11], while the global fuel cell performance can be often evaluated simply using polarization curves that describe the static relationship between the output voltage and the current [12]. Tremendous works have been done to understand/estimate fuel cell performance and degradation based on the polarization curves. The authors of Bressel et al. [13] proposed a polarization equation-based degradation model for fuel cells. By adopting an extended Kalman filter specific to the proposed degradation model, the global degradation of fuel cells and their trends could be accessed. In [14], the authors developed attention-based prognostic models for predicting voltage degradation under automotive profiles. A physically-based polarization curve model for automotive fuel cells was proposed in [15]. Then, the model was used to analyze the influence of fuel cell operation parameters on system efficiency and durability in the same study. Load allocation strategies were proposed by [16, 17] for managing the degradation of a multi-stack fuel cell system. The identified health indicator was obtained through the analysis of a fuel cell polarization curve model. A critical issue of using the polarization model is that the output voltage is a global performance indicator, which is a result of the complex interactions of multiple physicochemical processes that are not directly measurable. Thus, it fails to distinguish the contributions of individual physicochemical processes to the overall performance degradation.

Electrochemical impedance spectroscopy (EIS) is an *in situ*, non-invasive characterization technique that can provide detailed information about the internal processes of fuel cells [18, 19]. To measure the EIS spectrum, a small amplitude alternative current (AC) perturbation is applied to a constant direct current (DC) with a changing frequency. By scanning the frequency range, the frequency-specific impedance of the fuel cell can be obtained [20]. The EIS can be achieved using a dedicated or the output power converter by directly injecting a controlled perturbation signal [21].

The interpretation of EIS spectra is not trivial since it is linked to the complex internal physicochemical processes of a fuel cell. Equivalent circuit model (ECM) is a commonly used method for interpreting EIS spectra [18]. In the ECM, the involved processes are correlated to the electric circuit components. Pan et al. [22] built an analytical ECM for fuel cells using charge transfer resistance in parallel with series resistance-inductance component for representing Faraday impedance. The model was further used to estimate the state of health of fuel cells by identifying the degradation parameters in ECM model. Kim et al. [23] studied the degradation of fuel cells by fitting the EIS spectra to an ECM. The identified degradation parameters in ECM were modeled as an empirical linear function of voltage. This model is further used to deduce the internal degradation states from the measured voltage. To improve the physical interpretation of the ECM, Giner-Sanz et al. [24] studied the mechanistic modeling of EIS spectra by comparing 12 different ECMs. The final ECM was selected based on the goodness of fit and the physical meaning of the parameters. Further, the ECM-based approach was used to quantify the effects of inlet air humidity on the performance of PEM fuel cells [25]. However, prior knowledge of the investigated fuel cells is required to configure proper ECMs as well as suitable initial values for the circuit elements' parameters. These make the use of ECM in quantifying the health state and degradation evolution of fuel cells challenging.

Recently, the distribution of relaxation times (DRT) analysis has obtained growing attention for its ability to separate the polarization processes of fuel cells without prior knowledge. DRT is an analysis method that turns the impedance data into a distribution of components with different time constants corresponding to the processes of the involved system [26]. DRT can be used for process identification in complex electrochemical systems such as fuel cells, where resistive-capacitive features are dominant [27]. It provides a way to separate these polarization processes based on their typical time constants. Multiple tools have been developed to calculate DRT from EIS data. Wan et al. [28] developed a graphical user interface (GUI)-based toolbox for DRT calculation based on Tikhonov regularization. A series of studies have been done regarding the calculation of DRT by the same group [29, 30]. Li et al. [31] built a robust and fast GUI-based toolbox considering both DRT calculation and equivalent circuit model fitting, with the flexibility of adapting to different electrochemical systems. Instead of relying on regularization or parameter adjustment, data-driven techniques such as the Loewner framework have also been proposed to calculate DRT [32].

Based on those tools, various DRT-based studies have been conducted on fuel cells, focusing on the internal processes characterization and diagnostics of fuel cells. Yuan et al. [33] quantified four polarization processes related to mass transport, charge transfer of the oxygen reduction reaction (ORR), proton transfer, and interface contact using DRT. Zhu et al. [34] compared the DRT peaks computed from the polarization curve and Tikhonov regularization-based deconvolution method using the EIS spectra. Characteristic frequency resolution is found to be an important factor for the accuracy of DRT results. Dierickx et al. [35] leveraged the model-free advantage of DRT, and demon-
<table><caption>Table 1: Characterization tests of the two fuel cell stacks.</caption><thead><tr><th></th><th>Test time (h)</th><th>Polarization test</th><th>EIS</th></tr></thead><tbody><tr><td>FC1</td><td>0, 48, 185, 348, 515, 658, 823, 991</td><td>Current ramp: from 0 to</td><td>0.7, 0.45, and 0.2 A.cm<sup>-2</sup>,</td></tr><tr><td>FC2</td><td>0, 35, 182, 343, 515, 666, 830, 1016</td><td>1 A.cm<sup>-2</sup> of 1000 s</td><td>from 50 mHz to 10 kHz</td></tr></tbody></table>

strated the combination of ECM estimation and DRT identification. Heinzmann et al. [36] separated and quantified the individual polarization processes of a PEM fuel cell. The identified polarization resistances are used as the initial values for identifying a transmission line model, which allows the revelation of electrodes' transport and reaction kinetics. The authors further extended their results to estimate the polarization curve, namely voltage-current characteristics of fuel cells [37]. The developed performance model was validated with experimental data, showing a good agreement over a wide range of operating conditions. Recently, works on using DRT as *in situ* diagnostics tools for fuel cells have been developed. Artigas et al. [11] developed a DRT-based strategy for managing the optimal water content of a PEM fuel cell. This is achieved by motoring the variation of the decomposed DRT peaks under different operating conditions, such that the optimal conditions can be identified. Zhao et al. [38] used DRT to diagnose the influence of working conditions on the consistency of the studied fuel cell. This approach enables to quantify the correlation between local voltage and internal states of the fuel cell which is vital for controlling the fuel cell and improving its performance.

A growing interest is seen in the use of DRT for interpreting the EIS spectra of fuel cells as well as related performance modeling. However, the majority of the existing works focus on the characterization of fuel cells, the degradation aspect has not been sufficiently explored yet. On this basis, this work aims to investigate the root cause of degradation in PEM fuel cell stacks using DRT analysis. First, the experimental work on two five-cell PEM fuel cell stacks is introduced. Then, the EIS validation and DRT calculation methods are presented. Next, the DRT analysis results on the measured EIS spectra are discussed. Based on the DRT results, the contribution of individual polarization processes to the overall performance degradation is quantified. Finally, the conclusions and future works are discussed. The main contribution of this work is highlighted as follows:

* A novel DRT-based approach is proposed to investigate the root cause of PEM fuel cell degradation.

* The increase in overpotential is obtained from DRT analysis and used to quantify the degradation contribution of individual polarization processes.

* The proposed approach is validated on both constant and dynamic load profiles. Moreover, the degradation contributions under those two load profiles are compared for guiding the component design of fuel cells.

## 2. Experimental

The datasets used in this work are collected from IEEE PHM 2014 data challenge [39]. Two long-term durability tests were performed respectively on two five-cell fuel cell stacks with a cell surface area of 100 cm². The first stack FC1 was operated under a constant current load of 70 A. The second stack FC2 was operated under a dynamic current load of 70 A with 7 A oscillations at a frequency of 5 kHz. The relative humidity of air was set to 50%. The inlet pressure of both sides was set to around 1280 mbar. The stack operating temperature is controlled by a water cooling system to around 55 °C. Electrochemical impedance spectroscopy and polarization tests are conducted periodically along the durability tests with durations of around one week. Note that the EIS is measured for the whole stack. Table 1 summarizes the specific information about the characterization tests.

The durability test durations for FC1 and FC2 are 991 and 1016 hours, respectively. This time difference is mainly caused by factors such as reactant supply and availability of experimental stuff. However, it can be seen that the time differences, namely 25 hours are relatively small compared to the total durability test time. Therefore, we assume that this time difference is neglectable for the following analysis.
### 3. Methodology

To analyze the degradation of fuel cells from the measured EIS spectra, the data validation and DRT computation methods are introduced in this section.

#### 3.1. Validation of EIS spectra

It is necessary to validate that the EIS spectra satisfy the linearity, stability, and causality, namely, the properties of linear time-invariant system to ensure the subsequent analysis is accurate and reliable [26].

**Linearity.** For a linear system, the combined response to multiple input signals is the sum of the response to each input signal. The linearity of EIS refers to the resulting impedance spectrum has to remain unchanged regardless of the input signal's amplitude. However, due to the inherent nonlinearity and complexity of fuel cell systems, a small AC perturbation is applied to the system to ensure the linearity of the EIS spectra.

**Stability.** Stability requires a system to be stable until a perturbation is applied. When exposed to perturbation, the system's response should converge to zero, allowing it to restore its initial state without altering its inherent characteristics during the measurement process.

**Causality.** The system's response should solely rely on the current and past input perturbations as well as its internal states. As a result, the system's response should be zero prior to the application of the input signal.

The Kramers-Kronig (KK) relation is introduced to validate the EIS spectra [40]. The KK relation establishes a connection between the real and imaginary parts of the impedance spectrum through semi-infinite integrals:

$$ 
\begin{aligned}
Z_{im}(\omega) &= -\frac{2\omega}{\pi} \int_{0}^{\infty} \frac{Z_{re}(\omega')-Z_{re}(\omega)}{\omega^{2}-\omega^{2}} d\omega' \\
Z_{re}(\omega) &= Z_{re}(\infty) + \frac{2}{\pi} \int_{0}^{\infty} \frac{\omega'Z_{im}(\omega')-\omega Z_{im}(\omega)}{\omega^{2}-\omega^{2}} d\omega'
\end{aligned}
\quad (1) $$

where $Z_{re}$ and $Z_{im}$ are the real and imaginary parts of the impedance spectrum, respectively. The infinity frequency range can not be measured in practice, thus, alternative methods are developed to ensure KK relations are met.

The validity of an impedance spectrum is analyzed by its reproducibility through a KK compliant equivalent circuit. Specifically, ohmic resistance $R_{ohm}$, and n RC electric circuit components (in parallel) are used in the Lin-KK test:

$$ Z_{\text{Lin-KK}} = R_{\text{ohm}} + \sum_{i=1}^{n} \frac{R_i}{1 + i\omega\tau_i} \quad (2) $$

where $\tau_i = R_i C_i$ is the time constant of the i-th RC circuit.

To prevent under- or over-fitting, Schönleber et al. [40] proposed a ratio criterion $\mu$ to determine the optimal number of RC elements as follows:

$$ \mu = 1 - \frac{\sum_{R_i>0} |R_i|}{\sum_{R_i<0} |R_i|} \quad (3) $$

The test gradually increases the number of $\mu$ (normally a maximum number is used) until $\mu < c$ is reached, where c is a predefined threshold ($c = 0.24$ is used in this work based on the calculation results). Finally, the residual between the fitted model and measured data can be computed using the following formulas to assess the quality of the data:

$$ 
\begin{aligned}
\Delta Z_{re} &= \frac{Z_{re}(\omega) - \hat{Z}_{re}(\omega)}{|Z(\omega)|} \\
\Delta Z_{im} &= \frac{Z_{im}(\omega) - \hat{Z}_{im}(\omega)}{|Z(\omega)|}
\end{aligned}
\quad (4) $$

where $Z_{re}(\omega)$ and $Z_{im}(\omega)$ are the real and imaginary parts of the measured EIS spectrum. $\hat{Z}_{re}(\omega)$ and $\hat{Z}_{im}(\omega)$ are the real and imaginary parts of the estimated spectrum according to the RC circuits.
## 3.2. Distribution of relaxation time

The relation between the impedance ($Z(\omega)$) and the DRT $h(\tau)$ is given by the following equation [26]:

$$Z(\omega) = R_0 + \int_0^\infty \frac{h(\tau)}{1 + i\omega\tau} d\tau \qquad (5)$$

where $R_0$ is the ohmic resistance, $f$ is the frequency, and $\omega = 2\pi f$ is the angular frequency. Since the impedance of resistor and capacitor in parallel (R//C) connected component is $\frac{R}{1+i\omega\tau}$ ($\tau = RC$, is the time constant), the Eq. (5) is equivalent to the impedance of an ohmic resistance plus the infinite sum of RC elements.

Usually, the frequency data is recorded on the logarithmic scale, thus it is beneficial to compute the DRT equation in logarithmic form. This can be done by introducing a new variable $u = \ln(\tau)$:

$$Z(\omega) = R_0 + \int_{-\infty}^{\infty} \frac{\gamma(\tau)}{1 + i\omega\tau} d\ln(\tau) = R_0 + R_p \int_{-\infty}^{\infty} \frac{g(\tau)}{1 + i\omega\tau} d\ln(\tau) \qquad (6)$$

where $\gamma(\tau) = \tau h(\tau)$, $g(\tau)$ is the normalized distribution ($\int_{-\infty}^{\infty} \frac{g(\tau)}{1+i\omega\tau} d\ln(\tau) = 1$). $R_p$ is the polarization resistance. $R_p$ can be approximated by the area under each DRT peak as follows:

$$R_p = \int_{\tau_l}^{\tau_u} \gamma(\ln(\tau)) d\ln(\tau) \qquad (7)$$

where $\tau_l$ and $\tau_u$ are the lower and upper limits of the relaxation time, respectively.

According to Eq. (6), the EIS spectrum is decomposed into a series of RC components, corresponding to different time constants. In the above formulation, the transfer function of a resistive-capacitive element is used, thus the devolution of resistive-inductive features is not directly applicable [26]. However, the pure inductive features can be incorporated into Eq. (6) by adding a new impedance term $i\omega L_0$:

$$Z(\omega) = R_0 + i\omega L_0 + R_p \int_{-\infty}^{\infty} \frac{g(\tau)}{1 + i\omega\tau} d\ln(\tau) \qquad (8)$$

where $L_0$ is the inductance. This inductive behavior is mostly associated with the connecting wires of the fuel cell.

**DRT calculation.** It is noted that the calculation of DRT from Eqs. (6, 8) is an ill-posed problem. Thus, numerical solutions based on various regularization methods are used, such as Tikhonov regularization (TR), maximum entropy, and Fourier transform [41]. Among these methods, TR is the most widely used method for DRT calculation due to its simplicity and high accuracy.

The implementation of DRT calculation in this study is based on an open-sourced toolbox developed by Wan et al. [28]. The computation of DRT is implemented based on the TR method. The radial basis function (RBF) kernel is used to approximate the DRT (see Eq. (9)) and the Gaussian function $\exp(-(\mu x)^2)$ is used in this work. In TR, the regularization parameter $\lambda$ and the full-width half maximum (FWHM) coefficient are two key parameters that need to be identified to ensure the accuracy of the DRT results. For $\lambda$, three values are tested ($\lambda = 1.5 \times 10^{-3}, 1.5 \times 10^{-4}, 1.5 \times 10^{-5}$) and the results are compared in terms of DRT peaks and the error between the estimated EIS and the measured EIS. According to the results shown in Figs. 1a), b), and c), $\lambda = 1.5 \times 10^{-4}$ is the best value that gives the lowest error. For the FWHM coefficient, we have compared the values of 0.1, 0.5, and 1.0. The results are shown in Figs. 1d), e), and f). It is observed that the FWHM coefficient of 0.5 is the best value that gives the most accurate DRT results.

$$\gamma(\ln \tau) = \sum_{m=1}^{M} x_m \phi_{\mu}(\ln \tau - \ln \tau_m) \qquad (9)$$

where $\phi_{\mu}(\ln \tau - \ln \tau_m)$ is a radius basis function with center time constant $\tau_m$. $\mu$ is the shape parameter, $x_m$ is the coefficient of RBF, and $M$ is the number of RBF which is the same as the measurement points in the EIS spectrum.
<figure><img src="image_3.png" /><figcaption>Figure 1: Parameters identification for DTR calculation. a) DRT under different regularization parameters $\lambda$, b) Real part error of estimated EIS to the measured EIS, c) Imagine part error of estimated EIS to the measured EIS; d) DRT under different FWHM coefficients, e) Real part error of estimated EIS to the measured EIS, f) Imagine part error of estimated EIS to the measured EIS.</figcaption></figure>

# 4. Results and discussion

## 4.1. Characteristics of the measured EIS spectra

First, the characteristics of the measured EIS spectra are analyzed. In PEM fuel cells, the hydrogen oxidation reaction is known to be faster than the oxygen reduction reaction at the cathode [42]. It is recognized that the EIS spectrum of a fuel cell mainly reflects the characteristics of the cathode side [23]. A schematic of a typical EIS spectrum and the related polarization processes are demonstrated in Fig. 2. The overall EIS spectrum of a fuel cell is dominated by resistive-capacitive features. Two semicircles are observed in the EIS spectrum, which is related to the charge transfer process during ORR (at high frequency) and the oxygen transport process in the gas diffusion layer and cathode catalyst layer (at lower frequency). The proton transfer at cathode CL is located at a higher frequency range since it is a fast process.

It is noticed that the inductive features are observed in the ultra-high and ultra-low frequency range. The ultra-high frequency inductive part is related to the measuring instrument and load cables. While the ultra-low frequency inductive part is complicated and related to multiple factors. In general, it is related to three factors: 1) multiple-step reactions with adsorbed intermediates on Pt surface, 2) platinum oxide growth, and 3) water transport in the membrane [43]. However, it is still unclear which factor is the dominant one.

There are four types of polarization processes involved in the cathode side that are closely related to fuel cell performance (Fig. 2): 1) process 1 refers to proton transport in the proton-exchange membrane as well as the electrode and membrane contact process; 2) process 2 refers to proton transport in the ionomer of cathode CL; 3) process 3 refers to charge transfer during ORR at the cathode CL; 4) process 4 refers to mass (oxygen) transport in the cathode GDL and CL. These physicochemical processes are overlapped in the EIS spectra, which makes it difficult to analyze. The sum of the resistances of these processes is defined as the total polarization resistance as shown in Fig. 2. Another important factor is the high-frequency resistance (ohmic resistance), which is related to the resistance of membrane, CL, BPs, and the contact resistance between the components. It is of high importance to separate these processes to analyze their influence on fuel cell performance. This will be addressed in the following sections.
<figure><img src="image_4.png" /><figcaption>Figure 2: EIS spectra and the key processes - demonstrate with *I* = 70 A at *t* = 0 h (only cathode is considered due to the sluggish of ORR). The key polarization processes involved in the cathode side are: process 1: the conduction of proton in PEM as well as the electrode and membrane contact, process 2: proton transport in the cathode CL, process 3: charge transfer during ORR at cathode CL, process 4: oxygen transport in the GDL and cathode CL.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 3: Lin-KK validation of EIS data.</figcaption></figure>

## 4.2. Validation

The linearity, time-invariant, and causality of the measured EIS data have to be validated before the subsequent analysis. The Lin-KK method introduced in Section 3 is used to validate the measured EIS data. Fig. 1 shows part of the validation results. It is observed that in both datasets (FC1 and FC2), the calculated error is below 1% in general, which is acceptable.

## 4.3. DRT analysis

After validating the EIS spectra, the distribution of relaxation times is computed for the measured EIS spectra. First, the decomposed peaks are assigned to the physicochemical processes of the fuel cell. Then, the root causes of fuel cell performance degradation are analyzed and quantified using the identified polarization resistances. The DRT results are presented for DC current $I = 0.7$ A.cm$^{-2}$ to demonstrate the decomposition of the EIS spectra.

### 4.3.1. Process assignment

The gas diffusion in cathode GDL and CL, charge transfer kinetics at the cathode, proton conduction in the ionomer of the cathode catalyst layer, and electrode and polymer membrane contact process are the four polarization processes within a PEM fuel cell that can be separated by DRT analysis. Different processes are associated with different time constants that can be identified from the DRT results.

Fig. 4 presents the decomposed DRT peaks over different DC currents of FC1 at $t = 0$ h. According to Fig. 4a), P2 is dominated at low current load, P1 gradually increases with the current load. While P3 and P4 are relatively stable with the change in current load. To quantify the polarization resistance of each process, the area under each DRT peak is calculated according to Eq. (7) and presented in Fig. 4b). It is observed that the polarization resistance of P2 decreased sharply with the increase of current load, which indicates that P2 might be associated with charge transfer kinetics during ORR at the cathode catalyst layer. On the other hand, the polarization resistance of P1 increased with the increase of current load, which indicates that P1 might be associated with oxygen diffusion (mass transport) in the cathode catalyst layer and gas diffusion layer. Note that the polarization resistance of P1 at 45 A is between those at 20 A and 70 A. This indicates that adding more impedance measurements at different DC points could be beneficial for identifying the variation trend of P1 with increasing currents, which needs to be considered in future experimental designs. The polarization resistances of P3 and P4 are relatively small compared with that of P1 and P2. $R_{P3}$ presents a slight decrease trend with the increase of current load, while $R_{P4}$ is relatively stable. Thus, P3 is related to proton conduction in the ionomer of the cathode catalyst layer, and P4 is related to the electrode and polymer membrane contact process. A similar decomposition conclusion is given by Yuan et al. [33]. The characteristic time constant of each peak can be identified from Fig. 4a). The oxygen diffusion has the largest time constant, that is, the lowest frequency among the investigated processes, which is consistent with the literature [33, 34].
<figure><img src="image_6.png" /><figcaption>Figure 4: Decomposed DRT peaks over different current loads (EIS of FC1, measured at t=0 h).</figcaption></figure>

<table><caption>Table 2: Decomposed processes and characteristic relaxation times of FC1 and FC2 under 0.7 A.cm⁻² (t = 0 h).</caption><thead><tr><th rowspan="2">Peak</th><th rowspan="2">Process</th><th colspan="2">Time constant (s)</th><th colspan="2">Polarization resistance (Ω)</th></tr><tr><th>FC1</th><th>FC2</th><th>FC1</th><th>FC2</th></tr></thead><tbody><tr><td>P1</td><td>Oxygen diffusion in cathode CL and GDL</td><td>4.69×10<sup>-2</sup></td><td>4.77×10<sup>-2</sup></td><td>3.98×10<sup>-3</sup></td><td>4.32×10<sup>-3</sup></td></tr><tr><td>P2</td><td>Charge transfer during ORR at cathode CL</td><td>4.24×10<sup>-3</sup></td><td>4.74×10<sup>-3</sup></td><td>6.12×10<sup>-3</sup></td><td>6.28×10<sup>-3</sup></td></tr><tr><td>P3</td><td>Proton transport inside cathode ionomer</td><td>5.26×10<sup>-4</sup></td><td>4.80×10<sup>-4</sup></td><td>6.98×10<sup>-4</sup></td><td>5.55×10<sup>-4</sup></td></tr><tr><td>P4</td><td>Electrode and membrane contact</td><td>1.02×10<sup>-4</sup></td><td>1.43×10<sup>-4</sup></td><td>2.26×10<sup>-4</sup></td><td>1.74×10<sup>-4</sup></td></tr></tbody></table>

### 4.3.2. Degradation analysis

The decomposed DRT peaks during the durability test for FC1 and FC2 are shown in Fig. 5. Four peaks are identified for each dataset as demonstrated in the previous section. Table 2 summarizes the physicochemical process of each peak, the characteristic relaxation time, and the polarization resistance ($R_p$) at $t = 0$ h. $R_p$ is computed by integrating the area under each DRT peak (Eq. (7)). According to [26], the time constants of gas diffusion and charge transfer during ORR are in the range of 0.02-0.15 s and 0.002-0.02 s, respectively. The time constant presented in Table 2 is in the same order of magnitude as the literature.

To quantify the influence of the decomposed processes on overall fuel cell performance, the polarization resistance variation trend is computed and shown in Fig. 6. In both datasets, the mass transport resistance $R_{mt}$ and the charge transfer resistance $R_{ct}$ show an obvious increasing trend during the durability test. While the polarization resistances of P3 and P4 did not show a clear monotonous varying trend. Besides, the high-frequency resistances $R_{hf}$, as shown in Figs. 6b), d) increases with the degradation of the fuel cell. Thus, $R_{mt}$, $R_{ct}$, and $R_{hf}$ are selected as the degradation indexes. Among them, the mass transport resistance is the most sensitive to the degradation of the fuel cell which presents the largest variation (up to 23% for FC1 and 37% for FC2). From the voltage measurements, the degradation rate of FC1 is estimated as 132 µV/h/stack, while the degradation rate of FC2 is calculated as 169 µV/h/stack. The degradation rate of FC2 is higher than that of FC1, which is consistent with the EIS analysis results. However, the variation of the degradation indexes is not equivalent to their contribution to fuel cell voltage degradation, which will be addressed in the following section.

As demonstrated in Section 4.1, the mass transport resistance is mainly related to GDL and CL. The increase in mass transport resistance indicates the degradation of the cathode GDL and CL. Similarly, The increase in charge transfer resistance indicates the degradation of the cathode CL. The high-frequency resistance ($R_{hf}$) is the sum of contributions from contact resistance and ohmic resistance of various components in the fuel cell, including resistance of membrane, catalyst layer, bipolar plates, and the contact resistance between them. Among these components, the resistance of the membrane is the main contributor to $R_{hf}$. Thereby, the degradation of $R_{hf}$ indicates the degradation of fuel cell membrane.

In the following, the contribution of each process to the overall fuel cell performance will be quantified. Then the root causes of fuel cell degradation will be analyzed.
<figure><img src="image_7.png" /><figcaption>Figure 5: Decomposed DRT peaks at *I* =0.7 A.cm⁻² during the durability test. a) FC1, b) FC2.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6: Polarization resistances variation trend based on the DRT results at DC current *I* = 0.7 A.cm⁻² (FC1 and FC2). a) Resistances variation of contact process *R*<sub>co</sub>, of proton transport in the cathode ionomer *R*<sub>pt</sub>, of charge transfer during ORR at cathode *CL* *R*<sub>ct</sub>, and of oxygen diffusion in cathode GDL and *CL* *R*<sub>mt</sub> for FC1; b) ohmic resistance variation trend for FC1; c) resistances variation of contact process *R*<sub>co</sub>, of proton transport in the cathode ionomer *R*<sub>pt</sub>, of charge transfer during ORR at cathode *CL* *R*<sub>ct</sub>, and of oxygen diffusion in cathode GDL and *CL* *R*<sub>mt</sub> for FC2; d) ohmic resistance variation trend for FC2.</figcaption></figure>
### 4.4. Contribution of internal processes to fuel cell degradation

As demonstrated in the results of $I = 0.7$ A.cm$^{-2}$, the DRT results of other DC currents ($I = 0.2, 0.45$ A.cm$^{-2}$) can also be calculated for further analysis. Based on the DRT results, the polarization losses (overpotential) caused by each internal process can be estimated by [44]:

$$ \eta_{pj}(i) = \int_{i_l}^{i_u} R_{pj}(i)di = \sum_{i_l}^{i_u} R_{pj}(i)\Delta i \quad (10) $$

where $\eta_{pj}(i)$ is the polarization losses caused by process j. $R_{pj}(i)$ is the identified polarization resistance of process j at current i. $i_l$ and $i_u$ are the lower and upper current limits, respectively. The integral is approximated by the sum of the resistances at different current points. $\Delta i$ is the current increment. The lower the value of $\Delta i$, the more accurate the estimation of the polarization losses. Recall that the EIS is measured for three DC currents, $I = 0.2, 0.45, 0.7$ A.cm$^{-2}$. In Eq. (10), $\Delta i = (0.2), (0.2, 0.45 - 0.2)$, and $(0.2, 0.45 - 0.2, 0.7 - 0.45)$ for $I = 0.2, 0.45, 0.7$ A.cm$^{-2}$, respectively. Similarly, the losses caused by high-frequency resistance ($R_{hf}$) can be estimated by:

$$ \eta_{hf}(i) = \int_{i_l}^{i_u} R_{hf}(i)di = \sum_{i_l}^{i_u} R_{hf}(i)\Delta i \quad (11) $$

Then, the total voltage loss due to those polarization resistances and the high-frequency resistance are expressed as:

$$ \eta_{tot}(i) = \eta_{pj}(i) + \eta_{hf}(i) \quad (12) $$

In general, the polarization equation of a fuel cell can be expressed as:

$$ E_{fc}(i) = E_{ocv} - \eta_{tot}(i) \quad (13) $$

where $E_{ocv}$ is the open circuit voltage. Then the estimated voltage losses $\Delta E^{<>} (i)$ can be calculated by substituting Eqs. (10, 11, 12, 13) into Eq. (14):

$$ \Delta E^{<>} (i) = E_{fc}^{<0>} - E_{fc}^{<>} = (E_{OCV}^{<0>} - E_{OCV}^{<>} + (\eta_{tot}^{<>} (i) - \eta_{tot}^{<0>} (i)) \quad (14) $$

where $\Delta E^{<>} (i)$ is the voltage loss from initial test to time t at current density i. Finally, the contribution of a certain process to the overall voltage degradation can be accessed by comparing the contributions of the corresponding resistance to the overall voltage losses. Specifically, the degradation contribution of OCV, that is $C_{OCV}$, polarization process pj, that is $C_{pj}$, and HFR $C_{hf}$, that is $C_{hf}$, can be quantified by following Eqs. (15), (16), and (17), respectively.

$$ C_{OCV} = 100 \times \frac{E_{OCV}^{<0>} - E_{OCV}^{<>} \%}{\Delta E^{<>} (i)} \quad (15) $$

$$ C_{pj} = 100 \times \frac{\eta_{pj}^{<>} (i) - \eta_{pj}^{<0>} (i) \%}{\Delta E^{<>} (i)} \quad (16) $$

$$ C_{hf} = 100 \times \frac{\eta_{hf}^{<>} (i) - \eta_{hf}^{<0>} (i) \%}{\Delta E^{<>} (i)} \quad (17) $$

According to the measured polarization data, the OCVs in FC1 and FC2 are nearly constant. Thus, the degradation contribution of OCV is 0.0%. Eqs. (16, 17) are used to compute the polarization losses caused by each internal process and the high-frequency resistance for FC1 and FC2.
<figure><img src="image_9.png" /><figcaption>Figure 7: Contribution of internal processes to the degradation of FC1. a) Voltage losses caused by high-frequency resistance, b)voltage losses caused by mass transport process (P1), c) voltage losses caused by charge transfer process (P2), d) Contribution (percentage) of internal processes to fuel cell degradation.</figcaption></figure>

On FC1. The overpotential losses caused by the high-frequency resistance and the identified processes, that are mass transport process (P1) and charge transfer process (P2), are calculated using eqs. (10, 11). To quantify the contribution of each factor to the overall fuel cell degradation, the overpotential of *t* = 0 h is set as the benchmark. The overpotential losses at the end of the test (*t* = 991 h) are compared with the benchmark to quantify the individual degradation contribution. Fig. 7 summarizes the obtained results. Specifically, the high-frequency resistance contributes by 3.5 mV to the overall degradation of FC1 (8.8%). The polarization processes (P1 and P2) account for 91.2% in the overall degradation of FC1. Among them, the mass transport process (P1) contributes by 9.33 mV (23.1%) to the overall degradation, while the charge transfer process (P2) contributes by 27.5 mV (68.1%) to the overall degradation. The charge transfer process is the dominant factor that contributes to the degradation of FC1. This corresponds to the degradation of the cathode catalyst layer.

According to the recorded voltage data of FC1, the overall stack voltage decreases by 100 mV during the durability test. This value is larger than the sum of the degradation contributions of the internal processes (40.33 mV). This may be due to the lack of sufficient DC current points (the EIS measurement starts from 0.2 A.cm⁻², and only three DC currents are considered) which underestimates the degradation.

On FC2. We further quantify the individual contribution of each process to the overall degradation of FC2. As shown in Fig. 8, the high-frequency resistance contributes by 10.8 mV (9.6%) to the overall degradation of FC2. The polarization processes (P1 and P2) account for 90.4% of the overall degradation of FC2. Among them, the mass transport process (P1) contributes by 42.7 mV (38.2%) to the overall degradation, while the charge transfer process (P2) contributes by 58.3 mV (52.2%) to the overall degradation. Similar to FC1, the charge transfer process is the dominant factor that contributes to the degradation of FC2, which indicates the degradation of the cathode catalyst layer. Again, the overall voltage degradation of FC2 from the recorded voltage data is 187 mV, while the sum of the degradation contributions of the internal processes is 112 mV. This may be caused by the limited DC current points in the EIS measurement, as discussed in the previous section.

Fig. 9 summarizes the contribution of each internal process to the overall degradation of FC1 and FC2. No
<figure><img src="image_10.png" /><figcaption>Figure 8: Contribution of internal processes to the degradation of FC2. a) Voltage losses caused by high-frequency resistance, b)voltage losses caused by mass transport process (P1), c) voltage losses caused by charge transfer process (P2), d) Contribution (percentage) of internal processes to fuel cell degradation.</figcaption></figure>

contribution is assigned to P3 and P4 since no obvious degradation trend is observed in them, as discussed in the previous Section 4.3.2. The charge transfer process is the dominant factor that contributes to the degradation of the fuel cell. Followed by the mass transport process. One interesting observation is that the mass transport process contributes more to the overall degradation of FC2 than FC1. In PEM fuel cells, the mass transport process is influenced by the cathode gas diffusion layer and catalyst layer where the oxygen diffusion in the catalyst layer is the dominant factor [24]. This indicates that the oxygen diffusion process needs to be enhanced for dynamic load applications, either by building a proper operating strategy or by improving the material design of the cathode catalyst layer and gas diffusion layer. It is noticed that the degradation contribution of the membrane is relatively lower compared with the catalyst layer. This is linked to the durability test duration and the operating load profile. The studied tests are performed under normal durability test profiles and the test duration is around 1,000 hours, thus no obvious degradation is observed in the membrane.

To sum up, the degradation quantification of the internal processes can provide valuable information for internal degradation states, moreover, it can be used to guide the development and operation optimization of fuel cells under harsh conditions such as dynamic loads. However, the degradation in the anode catalyst layer is not directly reflected in the presented results since the main degradation originates from the cathode of fuel cells. The DRT-based approach is considered to be able to analyze the degradation of the anode side if significant degradation is observed and reflected in the impedance spectrum. Nonetheless, this needs to be verified with additional experimental investigations which will be considered in future work.

## 5. Conclusion

This work investigates the degradation root cause analysis of PEM fuel cell stacks using *in situ* EIS measurements. Model-free DRT analysis is used to interpret the EIS spectra and to identify the degradation root cause of the fuel cell. Two long-term durability tests are used to verify the method. The main conclusions are as follows:
<figure><img src="image_11.png" /><figcaption>Figure 9: Contribution of internal processes to overall fuel cell degradation.</figcaption></figure>

* The frequency domain EIS spectrum can be decomposed into four DRT peaks in the time domain, which are assigned to oxygen diffusion in the cathode gas diffusion layer and catalyst layer (P1), charge transfer during ORR at the cathode catalyst layer (P2), proton transport inside cathode ionomer (P3), and electrode and membrane contact process (P4).

* According to the computed polarization resistances of the identified processes, the mass transport resistance and the charge transfer resistance present a clear increasing trend which are identified as the root cause of fuel cell degradation.

* Based on the computed overpotential losses caused by each internal process, the charge transfer process is the dominant factor that contributes to the degradation of the fuel cell. Followed by the mass transport process. In FC1, the charge transfer and mass transport processes account for 68.1% and 23.1% of the overall degradation, respectively. In FC2, the charge transfer and mass transport processes account for 52.2% and 38.2% of the overall degradation, respectively. The high-frequency resistance contributes less to the overall degradation of the fuel cell. Moreover, the dynamic load profile accelerates the degradation of the mass transport process compared to the constant load.

Prognostics and health management is an efficient technique for handling the durability and reliability challenges in PEM fuel cells. It consists of observation, analysis, and action phases. The observation phase provides and preprocessing durability datasets for the following phases. The analysis phase conducts system health condition assessment, diagnosis, and predicting future health status. Finally, the action phase integrates the analyzed health status and future degradation information to prescribe proper operating actions for enhancing system durability and reliability. Future works will focus on developing a physics-based degradation model that links fuel cell output voltage and identified DRT indexes to provide real-time internal degradation information. This is of high importance for the development of energy management strategies, which fall within the action phase of fuel cell prognostics and health management, to enhance the durability and reliability of fuel cell systems.

## Acknowledgments

This work has been supported by France 2030/PEPR Hydrogène décarboné/Project DuraSyS-PAC managed by the French Research Agency (Reference ANR-22-PEHY-0002), ANR ERC Tremplin StG grant (ANR-23-ERCB-0004-01), and the EIPHI Graduate School (contract ANR-17-EURE-0002) and the Region Bourgogne Franche-Comté.
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