Dynamic Fractional-Order Model of Proton Exchange Membrane Fuel Cell System for Sustainability Improvement

Abstract

The proton exchange membrane fuel cell (PEMFC) stands at the forefront of advancing energy sustainability. Effective monitoring, control, diagnosis, and prognosis are crucial for optimizing the PEMFC system’s sustainability, necessitating a dynamic model that can capture the transient response of the PEMFC. This paper uses a dynamic fractional-order model to describe the behaviors of a stationary micro combined heat and power (mCHP) PEMFC stack. Based on the fractional-order equivalent circuit model, the applied model accurately represents the electrochemical impedance spectroscopy (EIS) and the dynamic voltage response under transient conditions. The applied model is validated through experiments on an mCHP PEMFC stack under various fault conditions. The EIS data is analyzed under different current densities and various fault conditions, including the stoichiometry of the anode and cathode, the stack temperature, and the relative humidity. The dynamic voltage response of the applied model shows good correspondence with experimental results in both phase and amplitude, thereby affirming the method’s precision and solidifying its role as a reliable tool for enhancing the sustainability and operational efficiency of PEMFC systems.

1. Introduction

As an increasing number of countries commit to reducing carbon dioxide (CO₂) emissions, the development of clean energy sources has become crucial [1]. Hydrogen fuel cells, known for their environmental friendliness due to their use of renewable sources and because water is their only discharge, have emerged as promising alternatives to engines powered by fossil fuels. While the direct availability of hydrogen in nature is scarce, advancements in leveraging renewable energy sources such as solar and wind power for hydrogen production offer a pathway to enhance its accessibility. This process, involving the conversion of surplus renewable energy into green hydrogen, not only addresses the intermittency of solar and wind energy but also contributes to a more sustainable and stable energy supply system [2,3,4,5,6,7]. The proton exchange membrane fuel cell (PEMFC) is particularly notable among various fuel cell types, distinguished by its superior efficiency and capacity to function effectively at lower temperatures [8]. Versatile in application, PEMFCs are employed in diverse fields, including mobile devices, distributed power stations, and automobiles, with notable implementations in fuel cell or fuel cell-based hybrid vehicles like Mirai and Citaro, where the fuel cell-fed DC/AC traction drive system is used [9,10,11,12]. Rapid advancements have been made in advanced AC electric motor control algorithms in recent years, leading to a significant maturation of motor control strategies in these vehicles [12,13,14,15,16,17]. These developments in AC electric motor technology have been instrumental in enhancing the performance and efficiency of fuel cell or fuel cell-based hybrid vehicles, contributing to their increasing viability in the market. Despite these technological strides, the high cost and limited lifecycle of PEMFCs continue to be the primary challenges impeding their widespread commercialization [18,19].

Within the PEMFC, a complex interplay of chemical, physical, and electrical activities take place, involving charge transfer, electrochemical reactions, and both mass and heat transfer. Consequently, developing an accurate model to describe PEMFC behavior is crucial yet challenging. Such models enable monitoring and control of PEMFC operations, enhancing reliability and durability through advanced diagnostic and prognostic techniques.

PEMFC modeling has been extensively studied using various approaches. One key method is computational fluid dynamic (CFD) modeling, involving 2D or 3D simulations of reactant flow and electrochemical reactions. For instance, a 3D PEMFC model was developed and the current distribution on the catalyst surface was investigated in [20]. In [21], the impact of contaminated fuel was explored using a 3D physical model, detailing the necessary equations for describing physical processes. In [22,23], the authors focused on the gas diffusion layer (GDL), examining GDL deformation and water formation. The CFD model provides detailed insights into the reactions and mass transfer within the PEMFC, enabling the discovery of numerous micromechanisms using this methodology. However, due to their computational complexity, CFD models are not suitable for real-time applications. Consequently, simpler models such as 1D or lumped parameter models have become the focus of research for their ease of integration into real-time control and monitoring systems. In [24], the physical models for both PEMFC stacks and auxiliary components were developed, including control equations and empirical formulas, which also extended to the control of the PEMFC system. The authors of [25] focused on a thermal model for a water-cooling PEMFC system, employing the equivalent thermal resistance method and neglecting the spatial distribution of temperature for simplicity and control feasibility, validated through experiments. Furthermore, the modeling approach in [26] considered the stack as four different control volumes with homogeneous parameters inside each. The model was calibrated and validated using experimental measurements, making it suitable for application in cathode humidity control.

The abovementioned 1D or lumped parameter models can establish the relationship between the operating parameter and the voltage output. However, under dynamic conditions, they are often inadequate due to the omission of many transient phenomena. During transient operation, as load and current change rapidly, a mismatch between the load and the voltage output occurs, attributable to electrons and reactants stored in the PEMFC. Impedance, an important characteristic reflecting the internal condition of the PEMFC, can be obtained and analyzed at different frequencies, a process known as the electrochemical impedance spectroscopy (EIS). Consequently, the EIS has become a crucial diagnostic tool widely used in electrochemical apparatuses, including fuel cells and lithium-ion batteries [27,28]. In [29], the EIS was applied to a PEMFC subjected to various thermal and pressure stresses, with the findings closely aligning with theoretical expectations. Additionally, in [30] the authors observed an almost proportional relationship between resistance and voltage during long-term operation, suggesting its utility in prognosis and degradation prediction.

The equivalent circuit model (ECM) plays a vital role in the analysis and interpretation of EIS data. This model simplifies the PEMFC into an electrical circuit, allowing it to be represented solely by electrical elements while other PEMFC details are disregarded. Various types of ECMs have been developed for different applications, including PEMFCs, commonly incorporating electrical elements such as resistance, capacitance, and inductance. However, standard electrical elements cannot fully capture all characteristics of the PEMFC due to specific physical phenomena influencing its electrical properties. Therefore, additional nonlinear elements, including the constant phase element (CPE) and the Warburg element, are utilized. Such components are commonly identified as fractional-order or non-integer elements due to their integration of differential operations of fractional orders within the circuitry.

Models based on fractional orders have been utilized in the study of electrochemical apparatus [31,32,33,34]. In [32], a fractional-order model was employed for lithium-ion batteries, enabling the state of charge estimation. In [33], a PEMFC model using a fractional-order ECM was developed, accounting for both static and dynamic regimes. The EIS and dynamic responses of such a model were experimentally validated, demonstrating accuracy. In [34], the arrangement of the microporous layer in the PEMFC was explored by using the EIS and the fractional-order ECM to analyze internal features of the PEMFC under various conditions. Concurrently, the CPE was utilized in dynamic modeling of solid oxide fuel cells. Comparative analysis with integer-order models revealed that fractional-order models can provide more accurate results.

Current models, however, lack a specific focus on the diagnosis and dynamic voltage response of stationary micro combined heat and power (mCHP) applications. Additionally, the analysis and comparison of impedance phase and amplitude in this context have not yet been undertaken. With growing interest in hydrogen utilization and stationary applications, developing models that accurately depict the dynamic behaviors of PEMFCs becomes imperative. To this end, this paper investigates the ECM for the PEMFC, with its main contributions outlined as follows.

• A fractional-order ECM for the PEMFC is applied, enabling accurate calculation of the EIS and the dynamic voltage response.
• A mCHP PEMFC system is used to demonstrate the utility of the applied ECM. The applied ECM effectively simulates and analyzes EIS under a range of current densities and various fault conditions, enhancing diagnostic capabilities. The considered fault conditions include variations in the stoichiometry of cathode (SC), the stoichiometry of anode (SA), the cooling circuit temperature (CCT), and the relative humidity level (RHL), with each scenario undergoing detailed EIS analysis.
• The predictions of the phase and amplitude of the voltage response made by the applied ECM are compared with impedance data obtained from experiments, thereby confirming its reliability and applicability in real-world scenarios.

The remainder of this paper is structured as follows: Section 2 details the applied ECM. Section 3 elaborates on the experiments and operating conditions conducted with the mCHP PEMFC system. In Section 4, the EIS results of the applied ECM are validated based on experimental results, and the dynamic voltage response of the applied ECM is compared with the measured impedance data. Finally, Section 5 summarizes the main conclusions.

2. PEMFC Models

2.1. ECM

PEMFCs can be approximated using ECMs to describe their electrical features. Different models are suitable for various devices. The Randles model [35,36], which is one of the simplest ECM models, considers only a single capacitor, as illustrated in Figure 1a. This model features the capacitor C_dl, originating from the electron storage in the electrode’s double layer, and the resistance R_c, which is linked to the oxygen reduction reaction. Their parallel combination depicts the charge transfer process. R_m represents the resistance to proton movement through the membrane. The primary advantage of the Randles model lies in its simplicity, with only three elements. However, ignoring the small but crucial resistance and capacitor at the PEMFC anode introduces significant errors, making it inappropriate for the considered PEMFC system.

The 2-RC model offers enhanced precision for PEMFCs through its inclusion of both an anode and a cathode, in contrast to the Randles model [37,38], as depicted in Figure 1b. Such a model introduces the anode as a combination of a capacitor and a resistance in parallel, effectively capturing the anode’s capacity for electron storage and its reaction resistance. This model has been applied to some problems with accurate results [37]. However, it overlooks the potential for the additional mass diffusion impedance, which represents the resistance faced by gases traversing the gas diffusion layer (GDL), particularly in the cathode layer.

In this study, a more accurate ECM, named the 3-CPE model [39], is applied, as illustrated in Figure 1c. The impedances of the anode, cathode, and mass diffusion, alongside the membrane resistance, are used to represent the PEMFC impedance. The main advantages and disadvantages of the aforementioned ECMs are summarized in

Table 1. Main advantages and disadvantages of the ECMs.

Model	Advantage	Disadvantage
Randles model	Three unknown parameters. Easy to solve.	Lacks sufficient accuracy.
2-RC model	Five unknown parameters. Relatively easy to solve.	Excludes the gas transfer process.
3-CPE model	Incorporates three processes. Offers enhanced accuracy.	Requires solving 10 parameters.

.

The latter three components are each depicted by a parallel configuration of a CPE, whose impedance can be expressed as (1), and a resistance. The resistance in the mass diffusion impedance, denoted as R_diff, arises from the diffusion resistance of air and hydrogen through the GDL, while the CPE accounts for the gas storage within the porous materials:

$${{\rm Z}}_{CPE}\left(s\right)=\frac{1}{C{s}^{\alpha }}$$

(1)

where Z_CPE is the CPE’s impedance, s denotes the Laplace operator; C represents the effective capacitance of the CPE; and α denotes the CPE’s order, with its value required to be in the range from 0 to 1.

A CPE can be modeled as a network of numerous RC circuits and, when its order is set to 1, resembles a capacitor in behavior. The emergence of a CPE is attributed to the uneven surfaces of electrodes or the inhomogeneity within the dielectric material. A detailed account of the CPE’s impedance derivation is provided in [40]. The following equation can be used to calculate the circuit’s total impedance:

$${{\rm Z}}_{total}\left(s\right)=\frac{R_a}{1+R_a{C}_a{s}^{{\alpha }_a}}+R_m+\frac{R_c}{1+R_c{C}_c{s}^{{\alpha }_c}}+\frac{R_d}{1+R_d{C}_d{s}^{{\alpha }_d}}$$

(2)

where Z_total represents the circuit’s total impedance and the subscripts a, c, and d correspond to the impedances of the anode, cathode, and mass diffusion, respectively.

2.2. Parameters Identification

Upon the model’s establishment, the next step involves identifying the ECM’s parameters. The objective revolves around enhancing the model to mirror the experimental EIS data more accurately. To this end, the genetic algorithm (GA) is used for identifying approximate global optimal parameters. This step is then followed by the application of the Levenberg–Marquardt (LM) method to achieve precise parameter refinement [39].

In line with standard practices in model identification, the criterion to determine the “closeness” of fit in this study is to obtain the minimum squared error (MSE), expressed by the following equation:

$$min\left(J\right)=min\left\{{\displaystyle \sum _i\left[{\left(Re{\left(Z\right)}_i-\widehat{R}e{\left(Z\right)}_i\right)}^2+{\left(Im{\left(Z\right)}_i-\widehat{I}m{\left(Z\right)}_i\right)}^2\right]}\right\}$$

(3)

where $Re{\left(Z\right)}_i$ and $Im{\left(Z\right)}_i$ represent, respectively, the experiment data’s real and imaginary parts; $\widehat{R}e{\left(Z\right)}_i$ and $\widehat{I}m{\left(Z\right)}_i$ denote, respectively, the calculated real and imaginary parts from (2); and i represents the frequency’s sequence number.

Integration of the GA and LM methods is employed for parameter optimization in this study. The GA is adept at conducting extensive global searches without initial values, beginning with a wide array of potential solutions and refining them via evolutionary processes. This strategy approximates a global optimum effectively. The LM method, renowned for its precision in nonlinear optimization, complements this by refining the globally optimized solution, although it risks converging to local minima if initial values are suboptimal [41]. The combination of these two methods aims to harness GA’s global search capabilities alongside LM’s precise local optimizations, providing both comprehensive exploration and meticulous refinement [42,43,44,45]. This dual approach is particularly beneficial for complex models, as demonstrated in our experiments where frequencies range from 0.1 Hz to 10,000 Hz across 51 data points.

In the practical application of this abovementioned method, as illustrated in Figure 2, the GA initially explores a broad spectrum of solutions, sifting through and eliminating the less effective ones. Solutions not meeting the fitness threshold undergo further enhancement through GA’s evolutionary mechanisms, such as reproduction, crossover, and mutation. This iterative process continues until an optimal solution is identified. This solution is then utilized as the initial input for the LM method, which iteratively refines it for greater accuracy. The LM method adjusts parameters to minimize the SE between iterations, proceeding until the error reduction meets a predefined precision criterion. This integrated approach is designed to achieve a globally optimal and high-precision solution for the fitting process, particularly in the complex scenarios encountered in PEMFC system studies.

3. Experiments and Operation Conditions

3.1. Experimental Test Bench

To develop diagnostic strategies for the new mCHP PEMFC application, aimed at preventing operational faults and enhancing the durability of the PEMFC system, experiments were conducted using the experimental test bench detailed in [46]. The adopted mCHP PEMFC system is designed by Riesaer Brennstoffzellentechnik GmbH and Inhouse Engineering GmbH, Germany [46]. The mCHP PEMFC system incorporates a stack of 12 PEMFC cells. Air is directed to the cathode inside the stack, and an anode receives a fuel mixture with a composition of 75% H₂ and 25% CO₂, emulating the reforming process of natural gas. Details on the stack’s characteristics and nominal operational parameters can be found in

Table 2. Parameters of investigated mCHP stack.

Parameter	Value
Cell number	12
Electrode active surface	196 cm2
Gas distributor plates	Graphite
Fuel	75% H2, 25% CO2
Coolant deionized water flow	3 L/min
Anode stoichiometry rate (H2 and CO2 mix)	1.3
Cathode stoichiometry rate (air)	2
Absolute pressure for H2 inlets	111 kPa
Absolute pressure for air inlets	106 kPa
Max. anode–cathode pressure gap	20 kPa
Cooling circuit’s temperature	70 °C
Anode relative humidity rate	50%
Cathode relative humidity rate	50%
Nominal Current	80 A

[46].

3.2. Fault Operating Conditions

To simulate various fault conditions of the mCHP system in the experiments, different operating parameters were adjusted to introduce specific faults. The failures were induced by altering four operational parameters: the SC, the SA, the CCT, and the RHL at both anode and cathode (controlled by adjusting the humidifier temperatures). As a result, eight distinct fault conditions were created by setting each of these four parameters either above or below their normal levels. The parameters are all modified around the usual operating parameter ranges provided by the PEMFC stack manufacturer, to ensure that poor PEMFC electrical performances are introduced while avoiding any complete collapse of the PEMFC (shutdown of the cell voltages that hinder the FC operation) and/or too severe irreversible FC material degradation. Therefore, several severe failure conditions mentioned in other research [47] are not considered in this work. Also, the faults corresponding to small deviations from the nominal conditions are more difficult to detect/diagnose than severe failures, thus the method is more sensitive to small faults, and it can anticipate more pronounced faults.

Table 3. The operation parameters applied to fault operating conditions.

Parameters	NL	FSCH	FSCL	FSAH	FSAL	DTH	DTL	DRHH	DRHL
SC	2	2.6	1.6	2	2	2	2	2	2
SA	1.3	1.3	1.3	1.5	1.2	1.3	1.3	1.3	1.3
CCT	70 °C	70 °C	70 °C	70 °C	70 °C	72 °C	65 °C	70 °C	70 °C
RHL	50%	50%	50%	50%	50%	50%	50%	54%	46%

details these operating parameters under different conditions, where NL denotes the normal level, FSCH and FSCL indicate the faults in the cathode flow with the SC being higher and lower than its NL, respectively, FSAH and FSAL denote the faults in the anode flow with the SA being higher and lower than its NL, respectively, DTH and DTL represent the faults in the stack temperature fault with the CCT being higher and lower than its NL, respectively, DRHH and DRHL indicate the faults in the relative humidity fault with the RHL being higher and lower than its NL, respectively.

4. Experimental Results and Model Validation

4.1. EIS under Various Current Densities

In the course of PEMFC functioning, variations in current density are frequently observed at various levels. Since the behavior of PEMFC significantly differs under varying current density levels, it is important to explore the relationship between the current density level and the EIS of the stack. In the experiment, the current density was set at 102 mA/cm², 204 mA/cm², 306 mA/cm², 408 mA/cm², 510 mA/cm², and 612 mA/cm², respectively. The EIS results corresponding to these settings are presented in Figure 3. The parameter identification method shown in Figure 2 was applied to the EIS data. The results of the applied ECM, along with the identified parameters, are also included in Figure 3, and the detailed parameters are listed in

Table 4. The identified parameters of the applied ECM under different current densities.

Current Density (mA/cm2)	Ra	Ca	αa	Rm	Rc	Cc	αc	Rd	Cd	αd
102	0.0070	0.6830	0.6695	0.0103	0.0365	0.4455	0.9146	0.0079	9.8801	1.0000
204	0.0058	0.6385	0.7146	0.0075	0.0192	0.4015	1.0000	0.0099	9.6613	1.0000
306	0.0040	0.3845	0.7852	0.0062	0.0174	0.5166	0.9199	0.0133	8.8126	1.0000
408	0.0041	0.5353	0.7580	0.0055	0.0125	0.4589	1.0000	0.0134	6.0430	1.0000
510	0.0035	0.4694	0.7905	0.0052	0.0112	0.4770	1.0000	0.0144	5.0231	1.0000
612	0.0033	0.5245	0.7841	0.0049	0.0087	0.5367	1.0000	0.0184	4.9082	0.8871

.

As evident in Figure 3, the impedance obtained from the applied ECM closely matches the experimental data. At high frequencies, specifically in the region where the real part is less than 10 mOhm and the imaginary part approaches zero, a diminutive arc manifests, indicative of the anode’s impedance. This arc’s minimal extent corroborates findings from various studies suggesting greater efficiency of hydrogen oxidation at the anode’s catalyst compared to the cathode. Within the mid-frequency range, a distinct arc delineates cathode charge transfer impedance, while at the lower frequency end, a third arc highlights the mass diffusion impedance.

Comparative analysis of the EIS at various current densities, presented in Figure 3, indicates that the cathode arc diminishes with increasing current density, while the mass diffusion arc becomes more pronounced. This observation aligns with practical expectations. With the escalation of current density, there is an elevated consumption of oxygen, leading to a decrease in charge transfer resistance. However, this also results in increased water production within the stack, subsequently hindering gas diffusion towards the catalyst. Observations of the anode arc reveal minimal alterations, indicating that the anode impedance is relatively unaffected by current density variations. This phenomenon is further supported by data in Table 4. As the current density increases, the values of R_a, R_m, R_c, and C_d decrease, while the values of C_c and R_d increase. As previously mentioned, higher current densities lead to more water production, thereby increasing the humidity throughout the stack. This rise in humidity decreases the value of R_m, as water content aids proton movement across the membrane. Similarly, the values of R_a and R_c decrease, as the catalytic reactions are facilitated by higher humidity levels. However, increased humidity also raises the resistance for reactant gases reaching the reaction sites, due to water blocking the passages in the porous materials. Particularly, water management in the GDL is a key issue for large-scale commercial applications, because a large amount of water will be produced when the power density increases and a large amount of accumulation of liquid water in the GDL will lead to flooding and impede gas diffusion, resulting in rapid degradation of cell performance. Regarding the capacitance, the decrease in the value of C_d, which represents the gas storage ability, can be attributed to water occupying more space in the gas diffusion layer. Additionally, the rise in the value of C_c could be attributed to the double layer’s dielectric coefficient of the catalyst, influenced by the increased presence of water. As for the other parameters, namely C_a, α_a, α_c, and α_d, they show no significant changes, suggesting these metrics display limited sensitivity to shifts in the stack’s current density.

4.2. EIS under Considered Fault Conditions

Eight distinct fault conditions described in Section 3.2 were tested by changing four parameters. The EIS data recorded under these various fault conditions can be successfully simulated using the applied ECM [39].

The EIS results under SC fault conditions are illustrated in Figure 4. It is evident that the results of the applied ECM closely match the experimental results, showing clear regularity. The overall EIS is smallest when the SC is higher than the NL, moderate under NLs, and largest when the SC is lower than the NL. Both the cathode and diffusion arcs vary in amplitude; specifically, as the SC declines, the enlargement of the cathode arc occurs at a more gradual pace relative to that of the mass diffusion arc. This trend is also quantitatively supported by the identified parameters of the applied ECM listed in

Table 5. The identified parameters of the applied ECM under different fault conditions.

Parameters	Ra	Ca	αa	Rm	Rc	Cc	αc	Rd	Cd	αd
NC	0.0049	0.8076	0.7242	0.0058	0.0119	0.4638	1.0000	0.0128	6.4416	1.0000
FSCH	0.0052	1.0031	0.6921	0.0059	0.0098	0.5210	1.0000	0.0095	11.4115	0.8468
FSCL	0.0045	1.1765	0.6897	0.0055	0.0136	0.5266	1.0000	0.0273	4.6193	1.0000
FSAH	0.0038	0.5252	0.7688	0.0058	0.0122	0.4325	1.0000	0.0125	5.9386	1.0000
FSAL	0.0017	0.3565	0.8647	0.0056	0.0082	0.5526	1.0000	0.0394	13.8461	0.5000
DTH	0.0048	0.5956	0.7450	0.0063	0.0104	0.4134	1.0000	0.0169	9.0078	0.8005
DTL	0.0029	1.2408	0.7053	0.0050	0.0132	0.6097	0.9771	0.0184	5.0479	1.0000
DRHH	0.0023	0.3316	0.8299	0.0053	0.0149	0.6334	0.9120	0.0175	5.1134	0.9946
DRHL	0.0030	1.0328	0.7172	0.0051	0.0141	0.7087	0.9365	0.0196	4.8441	1.0000

. With the decrease in the SC, because of the limited oxygen concentration at the catalyst, the resistance associated with cathode charge transfer escalates, which markedly affects the mass diffusion impedance in such instances. The resistance of mass diffusion increases with a decrease in SC, while the capacitance diminishes. This change is attributed to the reduced availability of oxygen in the cathode, making mass diffusion the primary limiting factor, thus elevating the diffusion resistance. Additionally, the decrease in capacitance is linked to insufficient air storage in the cathode.

Figure 5 shows the EIS results subjected to FSAH and FSAL scenarios. In these scenarios, the SA is higher and lower than the NL, respectively. Contrary to the SC fault scenarios, variations in anode and cathode impedances are minimally observed under SA fault conditions. However, there is a significant difference in the mass diffusion impedance. Notably, under the condition of low SA, the mass diffusion arc becomes substantially larger. Table 5 illustrates that a decrease in SA contributes to a significant rise in the mass diffusion resistance and capacitance. This is due to a combination of limited hydrogen availability at the anode and oxygen excess at the cathode. The insufficient hydrogen supply leads to an increase in mass diffusion resistance, while the unused oxygen at the cathode boosts its air storage capacity, thereby augmenting the capacitance.

The impact of stack temperature is also investigated. The EIS results under various temperature fault conditions are illustrated in Figure 6. Notably, the cathode arc enlarges with the decline in temperature. Additionally, in scenarios of both elevated and reduced temperatures, the mass diffusion arc becomes more distinct at high frequency. Importantly, when the imaginary part is equal to zero, a notable shift in the value of R_m is observed at high frequency, which differs from other conditions. According to Table 5, the value of R_m decreases with a drop in temperature. This trend is presumably connected to the water content and additional membrane’s physical characteristics, as higher temperatures typically result in lower humidity within the membrane. Given that the temperature of the stack influences every internal process, notably water transfer from the cathode to the anode, the impedance variations in each component are composite in nature. Whether at higher or lower temperatures, the gas diffusion resistances surpass those seen in a normal scenario. This indicates that normal temperature conditions favor gas flow through the GDL.

Evaluating the influence of RHL on the mCHP PEMFC system involves adjusting the RHL in the incoming gas. In Figure 7, the EIS results across varied RH fault conditions are shown. The impedance is noticeably lower in normal conditions than in conditions of low or high humidity. Deviations from normal humidity levels cause an expansion of both cathode and mass diffusion arcs, with changes in the mass diffusion impedance proving to be more pronounced than those affecting the cathode impedance. In scenarios of reduced RHL, the impedance is notably higher compared to conditions of increased RHL. Referring to Table 5, under RHL fault conditions, it is observed that an escalation in the resistances associated with cathode and mass diffusion increase. This indicates that both the cathodic reactions and the mass diffusion in the GDL rely on maintaining a suitable humidity level.

4.3. Dynamic Voltage Response

Since the applied ECM accurately simulates the impedance of the PEMFC across various frequencies, the dynamic voltage response can be determined by converting the identified circuit’s impedance from the frequency domain to the time domain.

Based on (2), the total impedance can be written in four parts, as follows:

$${{\rm Z}}_{total}\left(s\right)=Z_a\left(s\right)+Z_{R_m}+Z_c\left(s\right)+Z_d\left(s\right)$$

(4)

According to the circuit depicted in Figure 1c, the total overpotential of the PEMFC can be expressed as the sum of the overpotentials of its four components. The overpotential of each component is derived from the inverse Laplace transform of their respective impedances. As Z_a(s), Z_c(s), and Z_d(s) share a similar structure, the subscript can be omitted, allowing for a generalized expression of the impedance for each part, as follows:

$${\rm Z}\left(s\right)=\frac{U\left(s\right)}{I\left(s\right)}=\frac{R}{1+RC{s}^{\alpha }}$$

(5)

From (5), the following equation can be obtained as follows:

$$s^{\alpha }U\left(s\right)+bU\left(s\right)=aI\left(s\right)$$

(6)

where a$=$ 1/C and b $=$ 1/RC.

Applying the inverse Laplace transform to (6), the following time domain differential equation can be derived as follows:

$${}_{0}D_t^{\alpha }u\left(t\right)+bu\left(t\right)=ai\left(t\right)$$

(7)

where the ${}_{0}D_t^{\alpha }$ is the α order differential operator in the region [0, t].

As mentioned above, the order α, identified from the EIS curve, ranges between 0 and 1. Consequently, Equation (7) is classified as a fractional-order differential equation. To effectively tackle this type of equation, defining the fractional differential operator is essential. Three definitions that are widely used, as noted in [48], include the Grunwald–Letnikov, Riemann–Liouville, and Caputo definitions. The Grunwald–Letnikov definition, renowned for its extensive application in numerous studies, is utilized in this study to calculate the fractional-order differential equation. Thus, the fractional-order differential term in (7) can be calculated in the discretization formula, as follows:

$${}_{0}D_t^{\alpha }u\left(k\right)=h^{-\alpha }{\displaystyle \sum _{j=0}^k{\omega }_j^{\alpha }u\left(k-j\right)}$$

(8)

where h is the time step size, i.e., the time interval between two points, k represents the ordinal number of the sampling point, ${\omega }_j^{\alpha }$ is the weight coefficient, which can be calculated as follows:

$${\omega }_0^{\alpha }=1,{\omega }_j^{\alpha }=\left(1-\frac{\alpha +1}{j}\right){\omega }_{j-1}^{\alpha },j=1,2,3,\dots$$

(9)

Based on (7) into (8), the following equation can be obtained as follows:

$$h^{-\alpha }u\left(k\right)+h^{-\alpha }{\displaystyle \sum _{j=1}^k{\omega }_j^{\alpha }u\left(k-j\right)}+bu\left(k\right)=ai\left(k\right)$$

(10)

According to (10), the overpotential at step k can be calculated as follows:

$$u\left(k\right)=\frac{ai\left(k\right)-h^{-\alpha }{\displaystyle \sum _{j=1}^k{\omega }_j^{\alpha }u\left(k-j\right)}}{h^{-\alpha }+b}$$

(11)

Since (11) is a general equation to represent the overpotentials of the anode, cathode, and mass diffusion components, the overall overpotential response can be accordingly expressed as follows:

$$u\left(t\right)=u_a\left(t\right)+u_c\left(t\right)+u_d\left(t\right)+R_{m}i\left(t\right)$$

(12)

During the EIS experiment, a sinusoidal current is applied to the PEMFC system, with both current and voltage being recorded. The voltage response can be calculated based on the frequency and amplitude of the current, using the aforementioned equations. For instance, considering the EIS under normal conditions, the comparison of the experimental current, the experimental voltage response, and voltage response of the applied ECM at a frequency of 0.1 Hz are illustrated in Figure 8.

As observed, the phase and amplitude of the voltage response described by the applied ECM closely resemble those of the experimental data. The phase difference, defined as the disparity between the phases of the voltage and current, should align with the impedance angle. Similarly, the ratio of the voltage response’s amplitude to the current’s amplitude should correspond to the impedance’s magnitude. In Figure 8, the phase difference is nearly zero for both the model and the experiment, and the impedance amplitude is approximately 32 mOhm. This concurrence suggests a good correlation with the EIS data.

When the frequency changes from 0.1 Hz to 10,000 Hz, as in the experiments, the calculated phase differences and impedance amplitudes can be compared with the experiment results, as shown in Figure 9 and Figure 10, respectively. It can be seen that both the phase difference and the amplitude of the voltage response from the applied ECM correspond well with the experimental results. However, there is a relatively larger error in the phase difference at frequencies above 2000 Hz. This discrepancy could be attributed to certain differences between the applied ECM and the actual PEMFC. However, the impedance amplitude of the applied ECM closely aligns with the experimental data across all frequencies, indicating that the model is capable of providing accurate voltage responses over a wide frequency range.

5. Conclusions

The PEMFC plays an important role in advancing energy sustainability, and developing a dynamic model that captures the transient response of the PEMFC is crucial. In this paper, a novel dynamic fractional-order model is developed based on experimental data from a stationary mCHP PEMFC system, demonstrating potential applications in diagnostic analysis. The applied ECM encompasses the impedances related to the anode, cathode, and mass diffusion, in addition to the membrane resistance. A novel parameter identification method is introduced, integrating the GA and the LM method to effectively determine the parameters of the applied ECM. EIS data under various current levels have been thoroughly analyzed. Parameters for eight different fault conditions, including SC, SA, CCT, and RHL, have been accurately fitted and examined. Additionally, the voltage response predicted by the model has been validated against experimental results, confirming the accuracy and suitability of the applied ECM. The main conclusions of this study are as follows:

• The applied ECM incorporates the membrane resistance along with the impedances related to the anode, cathode, and mass diffusion, and accurately represents the stationary PEMFC application. Furthermore, the novel parameter identification method, integrating the GA and the LM method, proves highly effective in fitting the parameters of the applied ECM.
• As the current density increases, the internal reactions influence the water content in both the anode and cathode, consequently affecting the impedance of various components. The mass diffusion resistance, in particular, is closely related to the current density level. Additionally, the characteristics of different parts undergo changes under fault conditions, making them useful indicators for diagnosing and signaling fault progression.
• The voltage response of the applied dynamic fractional-order model closely aligns with experimental results. Both the phase and amplitude of the impedance are accurately represented compared to experimental data across a wide frequency range. Hence, this dynamic model can be integrated with other static models to comprehensively simulate the behavior of the PEMFC.

The applied ECM holds potential for application to other types of fuel cells or in various fields where similar problems exist. Future work will focus on integrating the applied ECM with additional models to comprehensively represent the entire characteristics of the PEMFC system.