# Interpreting full-frequency impedance spectrum for PEM electrolyzers: Distribution of relaxation times-based modeling

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

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Jian Zuo, Nadia Yousfi Steiner, Zhongliang Li, Daniel Hissel. Interpreting full-frequency impedance spectrum for PEM electrolyzers: Distribution of relaxation times-based modeling. Applied Energy, 2025, 394, pp.126185. 10.1016/j.apenergy.2025.126185. hal-05087453

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Highlights

Interpreting full-frequency impedance spectrum for PEM electrolyzers: Distribution of relaxation times-based modeling

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

* Interpreting full frequency range impedance spectra using distribution of relaxation times.

* EIS characterization regarding temperature, cathode pressure, water flow rate, and current.

* Investigating the influence of varying operating conditions on PEMWEs.

* Investigating low-frequency inductive loops in a PEMWE stack.
# Interpreting full-frequency impedance spectrum for PEM electrolyzers: Distribution of relaxation times-based modeling

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 Marie et Louis Pasteur, FEMTO-ST, UTBM, CNRS, Belfort, F-90000, France

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

## Abstract

Among various hydrogen production technologies, proton exchange membrane water electrolyzers (PEMWEs) are promising thanks to their ability to operate at high and intermittent loads, high efficiency, and high hydrogen purity. The development and application of PEMWEs rely strongly on performance characterization and estimation techniques. Electrochemical impedance spectroscopy (EIS) is one of the most important non-invasive characterization tools for electrochemical devices such as PEMWEs. Nevertheless, modeling and interpreting the impedance spectrum remain an open challenge that hinders its application in PEMWEs. To bridge the gaps, a model-free distribution of relaxation times (DRT)-based approach is proposed to analyze EIS measured from in-operation PEMWEs. Moreover, the interpretation of the full frequency range including low-frequency inductive loops is investigated. To this end, experiments have been performed to measure the impedance spectra under different temperatures, cathode pressures, water flow rates, and current loads. Then, the DRT-based approach is applied to analyze the measured spectra. Conclusions have been drawn regarding the influence of various operating conditions on the performance of the PEMWE stack. Especially, the low-frequency inductive loops are systematically investigated for the first time to reveal their influencing factors and possible causes. The temperature is identified as the dominant influencing factor, followed by water flow rate and cathode pressure. This work provides useful insights into the PEMWE functionality through interpreting impedance spectra including low-frequency inductive loops and its application to PEMWEs.

**Keywords:** Proton exchange membrane water electrolyzer, electrochemical impedance spectroscopy, distribution of relaxation times, low-frequency inductive loop, polarization process, oxygen evolution reaction

## 1. Introduction

A promising and sustainable hydrogen production solution is electrolysis technology with renewable energy [1]. Among various types of electrolyzers, the proton exchange membrane electrolyzer (PEMWE) is one of the most promising technologies due to its ability to operate at high and intermittent loads, high efficiency, and high hydrogen purity [2]. A PEMWE stack consists of an anode bipolar plate, an anode gas diffusion layer, an anode catalyst layer, a proton-exchange membrane, a cathode catalyst layer, a cathode gas diffusion layer, and a cathode bipolar plate. The electrochemical reactions of a PEMWE are summarized as:

$$ 
\begin{array}{l}
\text{Anode: } 2H_2O(l) \rightarrow O_2(g) + 4H^+(aq) + 4e^- \\
\text{Cathode: } 4H^+(aq) + 4e^- \rightarrow 2H_2(g) \\
\text{Overall: } 2H_2O(l) \rightarrow 2H_2(g) + O_2(g).
\end{array} 
\qquad (1) $$

The reaction at the anode is called the oxygen evolution reaction (OER), and the reaction at the cathode is called the hydrogen evolution reaction (HER). OER is the dominant step due to its complex reaction mechanisms and sluggish

<footnote>*Corresponding author.</footnote>

<footnote>Email address: zhongliang.li@univ-fcomte.fr (Zhongliang Li)</footnote>
kinetics [3, 4]. Despite its potential as a promising hydrogen production technology, limited durability, low operational reliability, and high costs remain core barriers to developing and deploying PEMWEs [4, 5].

Non-destructive characterization and analysis technologies are indispensable for overcoming the durability and reliability challenges of PEMWEs [6]. The inherent multiphysics, multiscale characteristics, and elusive interactions complicate the challenge of decomposing and identifying the fundamental chemical and physical mechanisms of internal physicochemical processes. As a result, the non-destructive PEMWE analysis becomes a crucial task.

Polarization curves describe the relationship between the output voltage and current density, which is one of the most widely used non-destructive characterization tools in PEMWEs [7]. Temperature, pressure, and water flow rate are three key performance-related operating parameters in a PEMWE [8, 9]. Selamet et al. [10] studied the above three influencing factors on the performance of a PEMWE. By comparing the measured polarization curves, they found that the operating temperature is the most influential factor. The performance is improved with increasing temperature. The temperature also affects the two-phase water flow and the flow of the produced gas inside the PEMWE [8]. Increased temperature enhances bubble transport and reduces bubble accumulation, which can improve performance [11]. The operating temperature is also identified as an influential parameter of PEMWEs' performance in [12]. Compared with temperature, the influences of pressure and water flow rate are more complicated and less studied, particularly the water flow rate. Through a lumped polarization curve model, Correa et al. [13] identified that the increased pressure mainly increases the cathode polarization losses. However, the high-frequency resistance (HFR) shows a pressure dependence, with an optimal value observed at a medium cathode pressure of 5 bar [14]. The enhanced mass transport is reported with increasing cathode pressure [14]. The water flow rate can influence the mass transport in PEMWEs through bubble formation and removal [8, 15]. However, this is not well-revealed in the literature.

Although easy to acquire and interpret, the voltage obtained from polarization curves is a global performance indicator that fails to provide useful insights into the internal dynamic processes within a PEMWE. Electrochemical impedance spectroscopy (EIS) is a powerful non-invasive and *in-operando* characterization tool that carries abundant information about a PEMWE, including the polarization processes that are closely linked to the performance of a PEMWE [16]. To achieve EIS, a small amplitude of AC perturbation with a varying frequency is applied to the PEMWE, the impedance can be obtained using the recorded voltage and current signals [17]. Different from the polarization curve which directly reflects the PEMWE performance through voltage amplitude or a simple polarization curve model fitting procedure, interpreting impedance spectra can be challenging [18]. Equivalent circuit models (ECM) are one of the most commonly used methods for impedance modeling. ECM uses electrical elements such as resistors, capacitors, inductors, and Warburg elements to reproduce the measured spectrum [19, 20]. For example, a modified Randles circuit with a Warburg element was proposed in [21] to fit the EIS data of a PEMWE. ECM parameters such as HFR and mass transport resistance are obtained by fitting the ECM to the EIS data. However, ECM is a model-based method that requires prior knowledge of the investigated electrochemical system. Moreover, the ambiguity in model configurations, namely multiple equivalent circuits that may fit the same impedance spectrum, complicates the data interpretation [18, 22].

A model-free method called distribution of relaxation times (DRT) is receiving a growing research interest for its application in EIS interpretation. DRT transfers the frequency-domain impedance data into the time domain which enables the separation of various dynamic processes based on their time constants [23, 24]. Tikhonov regularization is one of the most used approaches in DRT inversion. Wan et al. [25] developed a graphical user interface (GUI)-based toolbox for DRT calculation using Tikhonov regularization. Li et al. [26] built a robust and fast GUI-based toolbox considering both DRT calculation and equivalent circuit model fitting. Probabilistic approaches such as the Gaussian process (GP) have also been deployed. “GP-DRT” and “fGP-DR” are two GP-based tools developed for DRT inversion [27, 28]. By assuming that the DRT is a GP, the approach allows for DRT calculation and uncertainty quantification. Until now, very few works have applied DRT to PEMWEs. Chan et al. [29] studied the separation of core electrochemical processes in a membrane electrode assembly by impedance spectra measured under varying operating conditions and catalyst loads. A Tikhonov regularization-based DRT inversion method is applied to a membrane electrode assembly in [30]. However, these studies only deal with resistive-capacitive parts of the impedance. The inductive regions have to be removed before performing DRT inversion.

Most existing approaches to the DRT inversion focus on solving a regression problem to ensure the accuracy of the distribution function [31]. However, our end use is to interpret a series of discrete generative processes. Further development to include a discrete view is required. Another pitfall is that most approaches only solve nonnegative
DRTs, namely they only deal with capacitive-resistive regions of the impedance spectrum. As a result, low-frequency inductive loops, which commonly exist but are largely unexplored in PEMWES, can not be effectively studied [32]. Moreover, it is reported that conducting impedance analysis by excluding inductive effects can lead to misleading information about the investigated system [33]. Indeed, the inductive loop phenomena are closely linked to the performance of electrochemical systems which need to be fully understood regarding their origin and influencing factors. Schiefer et al. [34] observed two low-frequency inductive processes depending on relative humidity and currents. Kulikovsky [35] proved that the proton transient conductivity is a cause for low-frequency inductive behavior using an analytical impedance model. An innovative non-electric EIS using air perturbation is proposed to investigate the origin of low-frequency inductive loops in PEM fuel cells [36]. Their results revealed that the variations of the oxygen concentration at the electrode are crucial cause of low-frequency loops that have not been detected previously. Generally, the inductive behaviors in PEM fuel cells have been intensively investigated. In comparison, limited work has been done for PEMWES. To date, only two references [32, 37] are found in PEMWES that involve this topic. Therefore, studying the impedance spectrum of PEMWES is necessary to investigate the low-frequency inductive loops and improve interpretation accuracy.

According to the literature review, the existing research gaps regarding performance-related studies in PEMWES include:

* Lack of sufficient investigation into applying non-invasive and in-operando characterization tools such as EIS to the characterization of PEMWES.

* Regarding the experimental scale of PEMWES, the majority of studies focus on a unit cell, the stack scale experimental works and studies are rare.

* Interpretation of impedance spectra in PEMWES remains a challenging task. Especially, the separation of dynamic polarization processes and the influencing factors of low-frequency inductive loops.

By addressing these research gaps, the contributions of this study are listed as follows:

* A DRT-based approach is applied to interpret the full frequency range of EIS spectra, including low-frequency inductive loops.

* Different polarization processes are separated and quantified using DRT-based timescale identification method.

* The influence of key operating conditions, namely, temperature, cathode pressure, and water flow rate is investigated using the DRT-based interpreting method.

* The causes of low-frequency inductive loops are studied by systematically varying the operational parameters and the impact of different currents, temperatures, cathode pressures, and water flow rates is quantified.

In the following, experimental works on two PEMWE stacks are described in Section 2. Next, the EIS interpretation methods, including EIS validation and DRT inversion, are demonstrated in Section 3. Then, the interpretation of the measured spectra under different temperatures, cathode pressures, and water flow rates is presented and discussed in Section 4. Finally, Section 5 summarizes the main discoveries of the proposed work.

## 2. Experimental

Experimental works are performed at two French national research laboratories, FCLAB and LAPLACE, to characterize and support the analysis of key operating conditions on the performance of PEMWE at the stack level. The experiments done in FCLAB are denoted test 1, and test 2 for LAPLACE.

**PEMWE stack.** Two commercial 1kW PEMWE stacks of the same size are used for the characterization tests. The PEMWE stack consists of 10 cells, with a nominal power load of 1 kW. Table 1 summarizes the key physical parameters of the PEMWE stack. Regarding the porous transport layers, the anode uses a traditional titanium mesh, while the cathode features a two-layer structure comprising a titanium fiber felt layer added to the titanium mesh.
<table><caption>Table 1: General physical parameters of the tested PEMWE stack.</caption><thead><tr><td>Parameters</td><td>Value</td><td>Unit</td></tr></thead><tbody><tr><td>Nominal load</td><td>1</td><td>kW</td></tr><tr><td>Number of cells</td><td>10</td><td></td></tr><tr><td>Cell area</td><td>25</td><td>cm<sup>2</sup></td></tr><tr><td>H<sub>2</sub> maximal production</td><td>225</td><td>Nl.h<sup>-1</sup></td></tr><tr><td>Operating temperature</td><td>30-70</td><td>°C</td></tr><tr><td>Operating pressure H<sub>2</sub></td><td>0-20</td><td>bar</td></tr><tr><td>Operating pressure O<sub>2</sub></td><td>1 (unpressurised)</td><td>atm</td></tr><tr><td>Stack dimensions</td><td>85×85×150</td><td>mm</td></tr><tr><td>Weight</td><td>2.5</td><td>kg</td></tr></tbody></table>

*Performed tests.* The homemade test benches from the two research laboratories are shown in Fig. 1. In test 1, the PEMWE stack is tested under different water flow rates, while the operating temperature and cathode pressure are kept constant. The stack operating temperature is maintained at 60 °C through the circulation pump as shown in Fig. 1a). The stack is operated at unpressurized mode, that is, both the pressures of the anode and cathode are kept at 1 atm. Four water flow rates, namely 0.3, 0.5, 0.7, and 1.0 l.min⁻¹ are tested. Specifically, the sequential test protocols performed for the new stack are listed as follows:

1) Break-in: operating the stack at 30 A for 8 hours.

2) Test 1.1, 1.2, 1.3, 1.4: set the water flow rate to 0.3, 0.5, 0.7, and 1.0 l.min⁻¹, respectively.

The EIS characterizations are performed for each test using a commercialized Zahner workstation. At each flow rate, the EIS tests are conducted at five DC currents, namely 5, 15, 25, 37.5, and 50 A, respectively. The frequency ranges from 50 mHz to 20 kHz and the AC amplitude perturbation is set to 10 % of the DC amplitude.

In test 2, the PEMWE stack is tested under different temperatures and pressures (only the cathode is pressurized, and the anode side is maintained at atmospheric pressure) to measure its performance. Specifically, the sequential test protocols performed for the new stack are listed as follows:

1) Break-in: operating the stack at 30 A for 8 hours.

2) Test 2.1: set the operating temperature to 60 °C, and the cathode pressure is 1 bar.

3) Test 2.2: set the operating temperature to 60 °C, and the cathode pressure is 5 bar.

4) Test 2.3, 2.4, and 2.5: set the cathode pressure to 3 bar, and control the operating temperature to 40 °C, 50 °C, and 60 °C, respectively.

For each test, the polarization curve and EIS characterizations are performed. The polarization curve is measured by ramping the current density from 0 to 50 A in 8,000 seconds, and the voltage is recorded at each current density. The EIS tests are conducted on four DC currents, namely 15, 25, 37.5, and 50 A. The frequency ranges from 50 mHz to 5 kHz. The AC amplitude perturbation is set to 10% of the DC amplitude. Note that for the stack operating temperature presented in the above tests, we mean to set the initial inlet water temperature to a specific value. The temperature within the stack may vary depending on the cell area and cell location.

# 3. Methodology

## 3.1. Validation of impedance spectrum

Before interpreting the measured spectra, it is essential to verify that they satisfy the criteria of linearity, stability, and causality to ensure the accuracy and reliability of the subsequent analysis. The validity of an impedance spectrum is analyzed by its reproducibility through a Kramers-Kronig (KK) compliant equivalent circuit. Specifically, ohmic resistance $R_{ohm}$, and m RC electric circuit components (in parallel) are used in the Lin-KK test:

$$Z_{\text{Lin-KK}} = R_{\text{ohm}} + \sum_{i=1}^{m} \frac{R_i}{1 + j\omega\tau_i} \quad (2)$$
<figure><img src="image_2.png" /><figcaption>Figure 1: Configuration of the homemade PEMWE stack test benches. a) Test bench 1, b) test bench 2.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: Impedance validation results, where $\Delta Z_{re}$ and $\Delta Z_{im}$ are the error of the real and imaginary parts of the impedance spectrum. Spectra measured at 25 A and water flow rate of a) 0.3 l.min$^{-1}$, b) 0.5 l.min$^{-1}$, c) 0.7 l.min$^{-1}$, and d) 1.0 l.min$^{-1}$; Spectra measured at 50 A and water flow rate of e) 0.3 l.min$^{-1}$, f) 0.5 l.min$^{-1}$, g) 0.7 l.min$^{-1}$, and h) 1.0 l.min$^{-1}$.</figcaption></figure>

where $\tau_i = R_i C_i$ is the time constant of the i-th RC circuit. $j^2 = -1$ is the imaginary unit.

To prevent under- or over-fitting, a ratio criterion $\mu$ is used to determine the optimal number of RC elements, which writes [38] :

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

The test gradually increases the number of $m$ (normally a maximum number is used) until $\mu < c$ is reached, where $c$ is a predefined threshold ($c = 0.24$).

The quality of the impedance data is assessed by the residual between the fitted model and the experimental data, which writes:

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

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. The measured spectrum is validated when the computed residuals are less than 1%, specifically when $\Delta Z_{re} \le 1\%$ and $\Delta Z_{im} \le 1\%$.

The measured impedance spectra are validated one by one using the above method. To demonstrate, Fig. 2 shows the validation error of six example spectra. It is seen that the measured spectra exhibit a high signal-to-noise ratio, with residuals less than 0.25 %, and thus are validated for the following analysis.

## 3.2. Equivalent circuit model

The characteristics of the impedance spectrum and corresponding equivalent circuit model need to be identified before calculating DRT. Fig. 3a) shows a typical impedance spectrum measured from the test PEMWE. The spectrum
includes three typical characteristic regions: the high-frequency inductive region, the middle-frequency resistive-capacitive region, and the low-frequency inductive region. Usually, the interpretation of the impedance spectrum refers to pure inductive and resistive-capacitive regions, excluding the low-frequency inductive region. To decompose the full frequency range of the measured impedance spectrum, an extended equivalent circuit model is constructed by adding negative resistances and constant phase elements to describe the low-frequency inductive loops. As shown in Fig. 3a), a pure inductor is chosen as the corresponding ECM for a high-frequency inductive loop. This region is associated with the connecting cables between the measuring equipment and PEMWE. The resistive-capacitive region is represented by a base element circuit composed of a resistor connected in parallel with a constant phase element (CPE), namely a ZARC circuit. It is noted that multiple base elements might be required to construct the ECM for representing the whole resistive-capacitive area.

When considering practical scenarios such as the dispersion phenomenon in porous electrodes, a CPE is used instead of a pure capacitor. The total impedance of the ZARC circuit $Z_{ZARC}$ is calculated as:

$$Z_{CPE}(\omega)^{-1} = q(j\omega)^n = q\omega^n(\cos(\frac{\pi}{2}n) + j\sin(\frac{\pi}{2}n)) \quad (5)$$

$$ \begin{aligned} Z_{ZARC}(\omega) &= \{R^{-1} + q\omega^n(\cos(\frac{\pi}{2}n) + j\sin(\frac{\pi}{2}n))^{-1} \\ &= \frac{R+qR^2\omega^n\cos(\frac{\pi}{2}n)}{1+2qR\omega^n\cos(\frac{\pi}{2}n)+q^2R^2\omega^{2n}} - j\frac{qR^2\omega^n\sin(\frac{\pi}{2}n)}{1+2qR\omega^n\cos(\frac{\pi}{2}n)+q^2R^2\omega^{2n}} \end{aligned} \quad (6) $$

where $\omega$ is the angular frequency, $R$ is the resistance, $q$ is the capacitance, and $n$ is the coefficient for characterizing the phase shift. $Z_{CPE}$ is the impedance of CPE.

The low-frequency inductive loops can be modeled by i) adding a parallel circuit of resistance and inductance, or ii) enabling negative resistance and capacitance in ZARC circuit [39]. The negative form of resistance and capacitance circuit is selected as indicated in Fig. 3a). This choice is supported by two observations [34]: i) the parallel circuit of inductor and resistor can hardly reproduce the measured impedance spectrum and the estimated inductance is physically meaningless, and ii) the negative resistance and capacitance combining the associated time constant corresponding to the physical meaningful resistance decay due to an increase in currents. To sum up, the ECM as shown in Fig. 3b) is selected and adopted in the DRT inversion algorithm as introduced in the next section.

### 3.3. Distribution of relaxation times

DRT transfers the impedance spectrum in the frequency domain into time representation. The total impedance is composed of an infinite series of resistive-capacitive relaxations, expressed as

$$Z(\omega) = \int_{-\infty}^{\infty} \frac{\gamma(\ln \tau)}{1 + j\omega\tau} d \ln \tau \quad (7)$$

where $\gamma(\ln \tau)$ is the distribution function in the time domain, namely DRT. The DRT deconvolution is to estimate $\gamma(\ln \tau)$ from measurements $Z(\omega)$.

As already discussed in the Introduction section, most existing approaches solve the DRT inversion problem from a regression point of view, namely to determine the magnitude of $\gamma_{true}$ as a function of $\ln(\tau)$. However, this contradicts the end use of DRT, that is to identify the individual physical processes.

Due to its high flexibility and accuracy, the dual inversion algorithm provided by Huang et al. [40] is adopted as the DRT calculation method. The deconvolution of DRT $\gamma_{true}$ from the measured impedance is approached using both regression and classification views. In regression, the objective is to determine the magnitude of $\gamma_{true}$ as a function of $\ln(\tau)$. In classification, the objective is to decide the discrete relaxations that compromise $\gamma_{true}$. The integration of these two views successfully solves Eq. (7) and satisfies the end users' needs, namely to identify and quantify the polarization processes of electrochemical systems such as PEMWEs. This approach overcomes the limits of the regression-centered DRT inversion which relies heavily on regularization parameter optimization. Moreover, this dual inversion algorithm allows for conveniently interpreting resistive-inductive processes which are usually not considered in existing methods.

A schematic of the dual inversion algorithm is illustrated in Fig. 4. Firstly, the continuous candidate DRTs are generated by varying the regularization parameters of the continuous DRT inversion algorithm, namely the hierarchical ridge regression model [41]. The lower the regularization parameter, the more peaks are identified in the final
<figure><img src="image_4.png" /><figcaption>Figure 3: a) A typical Nyquist plot of impedance spectrum measured from PEMWEs, with three typical characteristic areas: the high-frequency inductive loop, the resistive-capacitive, and the low-frequency inductive loop, b) the ECM proposed for fitting the spectrum.</figcaption></figure>

DRT. Next, the obtained continuous candidates are applied to identify and optimize discrete models. These discrete candidates are generated with different numbers of ZARC elements, namely *Q*. The ECM proposed in Fig. 3 is adopted in the calculation. Finally, the optimal discrete model is selected based on the computed evaluation metrics *B*<sub>Q</sub> (normalized Bayes factors). In general, those metrics are evaluated based on the simplicity and likelihood, namely the plausibility of the generated models. The model with the highest plausibility is selected as the most plausible representation of the impedance data. For more details on the dual inversion algorithm, please refer to [40].

## 4. Results and discussion

The interpretation of the measured EIS is investigated using the proposed DRT deconvolution method. The primary goal is to investigate the impacts of operating conditions on the physicochemical processes of PEMWEs through a DRT-based approach. Firstly, the DRT deconvolution results are demonstrated with the measured spectra. Next, the identified peaks of the measured impedance spectra are assigned to polarization processes in PEMWEs. Then, the influence of temperature, cathode pressure, and water flow rate on those processes is analyzed using the DRT analysis approach. After that, the influencing factors and possible causes of low-frequency inductive loops are investigated. Finally, the discussion on research limitations is provided to guide further development.

### 4.1. *Distribution of relaxation times-based EIS deconvolution*

*Demonstration of DRT deconvolution results.* To demonstrate the interpretation of the measured impedance spectra, the detailed DRT deconvolution results obtained via the dual inversion algorithm are shown in Fig. 5. The investigated spectrum is measured during test 1 as introduced in Section 2 at 15 A and water flow rate of 1.0 l.min⁻¹. The inversion of DRT for the spectrum follows the calculation steps as introduced in Fig. 4 5. According to the computed criteria for the candidate models as shown in Fig. 5a), *Q* = 5 is selected as the optimal number of peaks in the DRT. The final DRT and the decomposed peaks are presented in Fig. 5b). Furthermore, we have compared the estimated impedance
Step 1: Generate continuous candidate DRTs

<figure><img src="image_5.png" /></figure>

Step 2: Generate discrete candidate models

<figure><img src="image_6.png" /></figure>

Step 3: Model selection

<figure><img src="image_7.png" /><figcaption>Figure 4: Calculation procedure of dual inversion algorithm [40]. *Q* is the number of ZARC elements. *B*<sub>Q</sub> is normalized Bayes factors and three evaluation metrics are considered.</figcaption></figure>

<table><caption>Table 2: Summary of the identified peaks.</caption><thead><tr><td>Peaks</td><td>Physical processes/description</td></tr></thead><tbody><tr><td>P<sub>hf</sub></td><td>High-frequency inductive loop</td></tr><tr><td>P1</td><td>Proton transport in the ionomer of the cathode catalyst layer</td></tr><tr><td>P2</td><td>Charge transfer of HER</td></tr><tr><td>P3</td><td>Charge transfer of OER</td></tr><tr><td>P4</td><td>Mass transport process</td></tr><tr><td>P<sub>lf</sub></td><td>Low-frequency inductive loop</td></tr></tbody></table>

spectrum using the obtained ECM with the measured one as shown in Fig. 5c). It can be seen that the estimated spectrum agrees well with the experimental data. The interpretation of the physical meaning of the peaks as shown in Fig. 5b) will be further checked.

*Peak assignment.* After obtaining the DRTs, the next step is to assign the identified peaks to various polarization processes of PEMWEs. To illustrate, the DRTs of impedance spectra measured under 60 °C, 3 bar are collected to analyze the physical processes associated with the identified peaks.

Fig. 6a) shows the decomposed DRTs under different DC currents. According to Fig. 6a), a maximum of six peaks, namely $P_{hf}$, P1, P2, P3, P4, and $P_{lf}$, are identified. The peaks observed at the high frequencies $P_{hf}$ are linked to the connection wires between the spectrometer and PEMWE, namely parasitic effects [33]. For the spectra measured in test bench 1, the $P_{hf}$ is not observed and only pure inductance is observed at high frequencies. This indicates that $P_{hf}$ is not attributed to processes within PEMWEs and does not influence their performance. Therefore, the analysis of $P_{hf}$ will not be considered in the following sections. P1 is also observed at relatively high frequencies, next to $P_{hf}$ as observed in Fig. 6a). According to Fig. 6b), the resistance of P1 remains nearly the same for different DC currents. Thus, P1 is attributed to the proton transport in the ionomer of the cathode catalyst layer. P2 and P3 lie in the middle-frequency range. Their resistances decrease with increasing DC currents. This indicates that they are most likely linked to charge transfer processes. Since $R_{p3}$ is larger than $R_{p2}$, P3 is thus attributed to charge transfer during sluggish OER, and P2 is caused by charge transfer during HER. P4 presents the largest time constant among the capacitive-resistance region of the spectrum. Moreover, $R_{p4}$ increases slightly with increasing DC currents. Therefore, P4 is associated with the mass transport processes. $P_{lf}$ corresponds to the low-frequency inductive loop with the largest time constant. The detailed interpretation and possible causes of $P_{lf}$ will be further investigated. Table 2 summarizes the physical meaning of the identified peaks in DRT. In the following, the interpretation of impedance spectra measured under different operating conditions will be presented.
<figure><img src="image_8.png" /><figcaption>Figure 5: Demonstration of DRT deconvolution results using impedance spectra measured under the water flow rate of 1.0 l.min⁻¹. a) The computed criteria for model selection, b) the final DRT curve and the identified peaks obtained from the dual inversion algorithm. The polarization resistance associated with each peak is estimated using the area under each peak (τ = 1/(2πf), f is the frequency in Hz), and c) comparison of the measured impedance spectrum (indicated with legend “Measurement”) and the estimated spectrum using the equivalent circuit model (indicated with legend “ECM”).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 6: DRT inversion results and the calculated polarization resistances under each peak for impedance spectra measured under 60 °C, 3 bar. a) The decomposed DRTs, b) the polarization resistance of *P*<sub>lf</sub> and *P*<sub>1</sub>, c) the polarization resistance of *P*<sub>2</sub> and *P*<sub>2</sub>, and d) the polarization resistance of *P*<sub>4</sub> and *P*<sub>hf</sub>.</figcaption></figure>
<figure><img src="image_10.png" /></figure>

Figure 7: The measured impedance spectra, the estimated polarization and high-frequency resistances under different temperatures (the complete impedance spectra are shown in Fig. A.1). a) Spectra measured at 50 A under 40, 50, and 60 °C, b) resistance of the high-frequency inductive loop $P_{hf}$ over different currents, c) resistance of the proton transport inside ionomer P1 over different currents, d) resistance of the charge transfer of HER P2 over different currents, e) resistance of charge transfer of OER P3 over different currents, f) total charge transfer resistance (P2+P3) over different currents, g) resistance of the mass transport process P4 over different currents, h) resistance of the low-frequency inductive loop $P_{lf}$ over different currents, and i) high-frequency resistance over different currents. Figures b)-i) share the same legend.

## 4.2. Impact of temperature

The spectra measured at three different temperatures, shown in Fig. A.1, are analyzed to interpret the impact of temperature on PEMWE performance. The discussion focuses on analyzing the impact on key physicochemical processes, including proton transfer, charge transfer, and mass transport processes. According to the polarization resistance shown in Figs. 7c), the proton transfer resistance decreases with increased operating temperature, enhancing the proton transfer process. However, this enhancement diminishes as the DC increases. $R_{P1}$ is nearly constant at 37.5 and 50 A over the three operating temperatures. Fig. 7f) presents the sum of charge transfer resistance of Fig. 7d) and Fig. 7e), namely the total charge transfer resistance of the PEMWE. It is observed that the total charge transfer resistance decreases with increasing DC currents.

At relatively lower DC currents, namely 15 and 25 A, the mass transport resistance decreases with increasing temperatures, as shown in Fig. 7g). However, when the operating current further increases to 37.5 A, the mass transport resistance increases with increasing temperature. This is assumed to be associated with complex bubble removal behavior in PEMWES. At high DC currents, the increased hydrogen production may produce more bubbles, increasing the mass transport resistance. Similarly, the mass transport resistance increases at 50 A when the operating temperature increases from 50 to 60 °C.

For the resistance of low-frequency inductive loop, as shown in Fig. 7h), its value decreases with increasing temperature. Finally, the value of high-frequency resistance is presented in Fig. 7i). When increasing the operating temperature, $R_{HFR}$ shows a monotonic decreasing trend for all DC currents. Moreover, by comparing the variation of $R_{HFR}$ with the resistance of other peaks, it is seen that the decrease in $R_{HFR}$ is the dominant factor when increasing the operating temperature. The increase in electric conductivity of all cell components due to increasing temperature is the primary reason for the decrease in $R_{HFR}$.

## 4.3. Impact of cathode pressure

The results of different cathode pressures are summarized in Fig. 8. The complete set of the measured impedance spectra is shown in Fig. A.2. According to Figs. 8c), the polarization resistance of the proton transfer process shows
a relatively large variation under 15 and 25 A. In general, the proton transfer resistance decreases with increasing cathode pressure. This indicates an enhancement in the proton transport process.

The influence on charge transfer processes during HER, OER, and the total charge transfer resistance is shown in Figs. 8d), e), and f), respectively. According to Fig. 8f), the total charge transfer resistance decreases as the cathode pressure increases. However, the separated charge transfer resistance of the anode and cathode shows a different variation trend, as shown in Figs. 8d), e). For instance, On the cathode side, the charge transfer resistance of HER measured at 37.5 and 50 A remains similar as the cathode pressure increases. While on the anode side, the charge transfer resistance shows a decreasing trend, particularly for the spectrum at 37.5 and 50 A. The proposed DRT-based interpretation method enables the separation and quantification of various internal polarization processes of PEMWEs which is critical for material design and operation optimization of PEMWEs.

Regarding the influence on the mass transport process, a slightly different variation trend is observed under lower and higher currents according to Fig. 8g). For 15 A, the mass transport process is enhanced as the cathode pressure increases. For higher currents, namely 25, 37.5, and 50 A, the mass transport resistance shows a non-monotonic variation trend, indicating that the cathode pressure of 3 bar is optimal for the mass transport in the tested PEMWE. This difference in mass transport could be due to bubble removal phenomena. For low currents such as 15 A, a relatively small amount of bubbles is produced. Increasing the pressure can then aid in enhancing bubble removal and reducing mass transport resistance. However, a relatively large amount of bubbles is produced at higher currents, and increasing the pressure could distort the shape of the bubbles and hinder the bubble removal. Note that the bubbles are mainly produced at the anode side of the PEMWE stack. The cathode side might also have some bubbles present at the cathode side. The additional titanium fiber felt layer makes the mass transport process more complex. These effects at the cathode and anode sides lead to the non-monotonic behavior observed in Fig. 8g).

The influence on HFR is illustrated in Fig. 8i). At 15 A, the HFR decreases as the pressure increases. For higher currents, a non-monotonic variation trend is observed. Similar to the observation in Fig. 8g), the cathode pressure of 3 bar shows the lowest HFR among the investigated pressures. This could be linked to the contact effects of PEMWE core components such as the membrane and catalyst layer under different pressures. When operating with a pressurized cathode, the contact between the titanium mesh and the catalyst layer might be enhanced on the anode. However, the impacts on the cathode side might be more complex due to the additional titanium fiber felt structure. The produced hydrogen bubbles and water transfer may alter its conductivity. Nonetheless, further investigation is needed to fully understand the impact of the cathode pressure. Fig. 8h) shows that the resistance of the low-frequency inductive loop increases as pressure increases. This will be further investigated in the inductive loop analysis section.

### 4.4. Impact of water flow rate

The impacts of water flow rate are investigated by interpreting the impedance spectra measured under four water flow rates, namely 0.3, 0.5, 0.7, and 1.0 l.min⁻¹, as shown in Fig. A.3 and Fig. 9b). The obtained results are summarized in Fig. 9. Five peaks are identified according to the DRT deconvolution results which correspond to five distinct dynamic processes, namely P1 (proton transport inside ionomer), P2 (charge transfer of HER), P3 (charge transfer of OER), P4 (mass transport), and $P_{if}$ (low-frequency inductive loop). The variation of these processes regarding the water flow rates is represented by the polarization resistances, as shown in Figs. 9c)-h).

It is found that only the resistances of mass transport processes and low-frequency inductive loops show a clear variation trend versus increasing water flow rates, as shown in Figs. 9f) and g), respectively. The resistances of the remaining processes are nearly constant. The mass transport resistances, as shown in Fig. 9f) present a nonlinear variation trend versus different water flow rates. It is observed that the mass transport resistance at 0.7 l.min⁻¹ is the local minimum point for all spectra measured under different DC currents. This nonlinear behavior in mass transport resistance may be related to bubble removal in PEMWEs. Both excessively low and high water flow rates are unfavorable for bubble removal. Therefore, there is an optimal flow rate between these extremes that is most effective for bubble removal and results in the lowest mass transport resistance. This is the first time that such a relation between mass transport resistance and water flow rate has been reported thanks to the DRT-based analysis approach. According to these observations, we can help to reduce the mass transport resistance in PEMWE by setting the water flow rate to 0.7 l.min⁻¹. An interesting observation arises when comparing the mass transport behavior in PEMWEs with that in PEM fuel cells. In PEM fuel cells, under certain flow rates, the mass transport polarization increases with increasing current loads. However, this is not the case in PEMWE due to the mutual influence between bubble removal and water
<figure><img src="image_11.png" /></figure>

Figure 8: The measured impedance spectra and the estimated polarization and high-frequency resistances under different cathode pressures (the complete impedance spectra are shown in Fig. A.2). a) Spectra measured at 50 A under 1, 3, and 5 bar, b) resistance of the High-frequency inductive loop $P_{lf}$ over different currents, c) resistance of the proton transport inside ionomer P1 over different currents, d) resistance of the charge transfer of HER P2 over different currents, e) resistance of the charge transfer of OER P3 over different currents, f) total charge transfer resistance (P2+P3) over different currents, g) resistance of the mass transport process P4 over different currents, h) resistance of the low-frequency inductive loop $P_{lf}$ over different currents, and i) high-frequency resistance over different currents. Figures b)-i share the same legend.

flow rates. For example, at 1.0 l.min⁻¹, the mass transport resistance of 25 A gives the lowest values among all DC currents. The analysis of low-frequency inductive loops will be discussed in the subsequent section.

## 4.5. Inductive loop analysis

Some measured impedance spectra observe positive imaginary values at the extreme-low frequencies. This phenomenon, known as the low-frequency inductive loop, is identified as an independent peak, according to the previous DRT inversion results. Such low-frequency inductive loops also exist in PEM fuel cells and are well explained regarding their influencing factors and causes. However, the inductive loops in PEMWES are nearly unexplored and remain a mystery.

We have collected the low-frequency inductive loop deconvolution results under varying water flow rates, pressures, temperatures, and currents, as shown in Figs. 10a-c). In Fig. 10a), the low-frequency inductive loop starts to appear at 15 A when the water flow rate is 0.3 l.min⁻¹. It is seen that the $R_{lf}$ decreases as the water flow rate increases. In general, the related time constant, as shown in Fig. 10d), decreases as the water flow rate increases. Note that $R_{lf}$ (takes positive value) characterizes the effect of the low-frequency inductive loop, with its effect being directly proportional to the value of $R_{lf}$. As the cathode pressure increases, the resistance of the low-frequency inductive loop exhibits an upward trend (Fig. 10b)). Meanwhile, the corresponding time constants remain nearly unchanged as demonstrated in Fig. 10c). As shown in Fig. 10c), the $R_{lf}$ decreases with increasing operating temperature. The related time constants also remain nearly unchanged as shown in Fig. 10f). By comparing the resistance variation trend over different currents in Figs. 10a-c), it is found that $R_{lf}$ increases as the current increases.

The dominant influencing factors for the low-frequency inductive loop can be identified thanks to the quantitative interpretation results. Fig. 10g) illustrates the variation of $R_{lf}$, namely $\Delta R_{lf}$ under different DC currents. At current $i$, $\Delta R_{lf}^{<}$ is computed as $\Delta R_{lf}^{<} = R_{lf,max}^{<} - R_{lf,min}^{<}$, where $i = 15, 25, 37.5, 50$ A, $R_{lf,max}^{<}$ is the maximum $R_{lf}$ under current $i$, and $R_{lf,min}^{<}$ is the minimum $R_{lf}$ under current $i$. It is observed that temperature produces the largest $\Delta R_{lf}$ at all DC currents. Therefore, the temperature is the dominant factor for low-frequency inductive loops, followed by water flow rate, and cathode pressure. Note that similar observations on the influences of temperature and current density are
<figure><img src="image_12.png" /><figcaption>Figure 9: The measured impedance spectra and the estimated polarization and high-frequency resistances under different water flow rates (the complete impedance spectra are shown in Fig. A.3). a), b) Spectra measured at 37.5 and 50 A under the water flow rates of 0.3, 0.5, 0.7, and 1.0 l.min⁻¹, c) resistance of the proton transport inside ionomer P1 over different currents, d) resistance of the charge transfer of HER P2 over different currents, e) resistance of the charge transfer of OER P3 over different currents, f) resistance of the mass transport process P4 over different currents, g) resistance of the low-frequency inductive loop $P_{if}$ over different currents, and h) high-frequency resistance over different currents. Figures c)-h) share the same legend.</figcaption></figure>

also revealed in [32, 37] by Fraunhofer ISE. Moreover, advanced segment cell experiments are implemented in their investigations.

According to the obtained results, the production of reaction intermediates and water transport are the two most plausible causes of the low-frequency inductive loops since the temperature and water flow rate are the two most influential factors. Moreover, according to the conclusion in PEM fuel cells, the platinum oxide formation or relaxation of adsorbed intermediates for oxygen oxidation reaction, and water transport are also considered the main causes of low-frequency inductive loops [17, 34, 36]. The observed inductive behaviors are assumed to be closely linked to bubble dynamics. High current densities, low operating temperature, high cathode pressure, and low water flow rate conditions tend to cause accumulated bubbles within the PEMWE. The enhanced inductive loops under those conditions tend to support that the accumulated bubbles inside the PEMWE may enhance the low-frequency inductive loop. The link between the bubble dynamics and low-frequency inductive loops may be used to monitor the bubble dynamics. The study in [32] indicated that the low-frequency inductive loops are also linked to the membrane thickness. This indicates that the interfacial effects are also critical for the formation of low-frequency inductive loops. The influence of varying cathode pressure may also be linked to the interfacial effects and further influence the inductive loops. The modeling and quantification of inductive loops under varying operating conditions provide useful insights to identify their formation mechanisms. However, further experimental investigations are required to fully explain the inductive loops in PEMWEs.

## 4.6. Discussion on research limits

**Uncertainty quantification.** The proposed studies did not cover the uncertainty quantification for the impedance interpretation results. It could enhance the rigor of the proposed analysis, although the validation results provided in Fig. 2 already demonstrate very low error levels.

**Heterogeneity analysis.** It is crucial to check the impacts of operating conditions on the heterogeneity in PEMWEs such as temperature and current density [42]. The proposed EIS tests are performed on the entire stack, ignoring the internal heterogeneous factors. This is also recognized as a major limitation in the proposed studies.
<figure><img src="image_13.png" /><figcaption>Figure 10: Results of quantifying low-frequency inductive loops under different water flow rates, cathode pressures, temperatures, and currents. Water flow rate: a) Polarization resistance, d) time constant; Pressure: b) Polarization resistance, e) time constant; Temperature: c) Polarization resistance, f) time constant; g) Variation of $R_{if}$ over different temperatures, cathode pressures, water flow rates, and current densities.</figcaption></figure>

To demonstrate the heterogeneity in PEMWEs, the temperature differences between the anode outlet and inlet are illustrated in Fig. 11. It is seen that the temperature at the outlet is generally larger than that of the inlet. As indicated in Figs. 11a)-e), the temperature differences are enlarged with increasing operating current loads. Fig. 11e) lists the temperature differences for different operating conditions. It is observed that the differences increase with increasing cathode pressure and decreasing operating temperature. In particular, the largest temperature differences of 50 and 40 °C are above 10 °C.

In the performed tests, only the cathode side is pressurized. The pressure of the anode inlet and anode outlet is recorded and presented in Fig. 12. In general, the pressure at the outlet is smaller than that at the inlet. Nevertheless, the difference is very small. Fig. 12f) illustrates the pressure differences versus different DC currents in Fig. 12c). It can be seen that the variation of the pressure difference with current and other conditions is small. This means that the water is relatively well distributed within the tested stack.
<figure><img src="image_14.png" /><figcaption>Figure 11: Anode inlet and outlet temperatures during the characterization tests (obtained from the test bench 2). a) 60 °C and 1 bar, b) 60 °C and 3 bar, c) 60 °C and 5 bar, d) 50 °C and 3 bar, e) 40 °C and 3 bar, and f) the temperature differences between outlet and inlet at 50 A (the operating conditions are indicated in x-axis, for example, 60 °C1 stands for 60 °C and 1 bar. The temperature is computed by the average value during a EIS test).</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 12: Anode inlet ($O_2$) and outlet ($O_2$) pressure during the characterization tests (obtained from the test bench 2). a) 60 °C and 1 bar, b) 60 °C and 3 bar, c) 60 °C and 5 bar, d) 50 °C and 3 bar, e) 40 °C and 3 bar, and f) the pressure differences between anode inlet and outlet at 60 °C and 5 bar over different DC currents (the pressures are computed by the average values during EIS tests).</figcaption></figure>

**5. Conclusion**

This work investigates the interpretation of EIS in PEMWEs using a distribution of relaxation times-based approach. This approach is applied to study the influence of operating temperature, cathode pressure, and water flow rate on the internal polarization processes of a PEMWE stack. Experimental data acquired from two research laboratories supports the above investigations. The low-frequency inductive loops of impedance spectra in PEMWEs are quantitatively investigated for the first time. The main discoveries of this study are listed as follows:

* Five internal processes, namely proton transfer in the ionomer of the cathode catalyst layer, charge transfer of HER, charge transfer of OER, mass transport, and low-frequency inductive loops are separated from the measured impedance spectrum. The associated resistance of each process and the high-frequency resistance are obtained using the DRT inversion algorithm.

* The operating temperature mainly influences the high-frequency resistance. The high-frequency resistance is reduced due to improved electron conductivity at elevated temperatures. Increasing the operating temperature enhances the proton transport and charge transfer processes.

* Increasing the cathode pressure improves the proton transport and total charge transfer processes. However, the impacts of pressure on the charge transfer of HER and OER are different. Regarding the impacts on mass transport and high-frequency resistance, both show the lowest resistance at 3 bar for high currents.

* The water flow rate mainly influences the mass transport and low-frequency inductive loop. The water flow rate of 0.7 l.min⁻¹ gives the lowest mass transport resistance among the investigated flow rates.
* The low-frequency inductive loops are enhanced with decreasing water flow rate, increasing pressure, decreasing temperature, and increasing current load. The temperature is the dominant influencing factor, followed by water flow rate and pressure.

It should be emphasized that the impedance of this work is averaged over the total cell area, ignoring the heterogeneous factors within the stack such as gradients in temperature, bubble volume, and current density. The studies conducted in [42] provide valuable insights into overcoming this limit. Future work will investigate the origin and causes of inductive loops using segmented cell techniques.

## Acknowledgment

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 Chaire de Professeur Junior project (ANR-22-CPJ1-0074-01), the EIPHI Graduate School (contract ANR-17-EURE-0002) and the Region Bourgogne Franche-Comté.

## Appendix A. Nyquist plots of the measured impedance

The measured impedance spectra under different temperatures, cathode pressures, and water flow rates are shown in Figs. A.1, A.2, A.3.

<figure><img src="image_16.png" /><figcaption>Figure A.1: Nyquist plots measured under different temperatures. a) Measured at DC current 15 A, b) at DC current 25 A, c) at DC current 37.5 A, and d) at DC current 50 A.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Figure A.2: Nyquist plots measured under different cathode pressures. a) Measured at DC current 15 A, b) at DC current 25 A, c) at DC current 37.5 A, and d) at DC current 50 A.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Figure A.3: Nyquist plots measured under different water flow rates. a) Measured at DC current 5 A, b) at DC current 15 A, c) at DC current 25 A, and d) at DC current 37.5 A. The impedance is also measured at DC current 50 A as shown in Fig. 9b</figcaption></figure>

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