Modelling Techniques of Proton Exchange Membrane Fuel Cells (PEMFC): Electrical Engineer’s View ^†

Abstract

Proton exchange membrane fuel cells (PEMFCs) play a key role in hydrogen-based energy systems; however, accurate and practical modelling remains challenging due to system nonlinearities, parameter variability, and degradation effects. This paper presents a low-complexity parameter estimation methodology for a simplified PEMFC equivalent circuit model using current-switching techniques. The approach enables direct extraction of key parameters, including internal resistance and capacitance, from transient voltage responses without requiring complex optimization or large datasets. Experimental validation was conducted using 100 W and 1 kW PEMFC systems under current loading and interruption conditions. The results demonstrate good agreement between measured and simulated voltage responses, with a maximum error below 10% and typical error levels in the range of ~1.4–3%. Compared to conventional mechanistic and data-driven models, the proposed method significantly reduces computational complexity and measurement requirements while maintaining high predictive accuracy. Moreover, the combination of the simplified equivalent circuit model with current-switching-based parameter estimation offers an effective and practical tool for electrical engineers, enabling real-time monitoring, control-oriented modelling, and seamless integration with power electronic systems. The proposed approach is particularly suitable for applications in DC microgrids and digital twin-based monitoring of hydrogen energy systems.

1. Introduction

Hydrogen is the most abundant element in the universe, accounting for approximately 75% of normal matter by mass [1]. However, hydrogen rarely exists in nature in its pure molecular form (H₂) and therefore must be produced from other compounds such as water, hydrocarbons, or biomass. In the context of the global transition toward low-carbon energy systems, hydrogen is increasingly viewed as a promising energy carrier rather than a primary energy source. In hydrogen-based applications, fuel cells (FCs) convert the chemical energy of hydrogen directly into electrical energy. Compared to conventional power sources such as internal combustion engines and batteries, FCs offer several advantages, including high energy efficiency, zero carbon emissions at the point of use, and scalability across diverse applications. In particular, the production of green hydrogen through water electrolysis using surplus renewable energy from wind and solar systems has emerged as an important pathway for energy storage and deep decarbonization. This approach not only provides a clean hydrogen supply for PEMFC systems but also helps mitigate the intermittency and variability of renewable energy generation by converting excess electricity into a storable energy carrier [2]. Compared to conventional power sources such as internal combustion engines and batteries, FCs offer several advantages, including high energy efficiency, zero carbon emissions, and scalability across diverse applications [3,4,5]. FCs are employed across various sectors, such as transportation, stationary power generation, portable power systems, and aerospace technologies [6,7,8,9,10]. Due to these benefits, global interest in hydrogen has grown exponentially, leading many countries, including Australia, Japan, Korea, China, and Singapore, to incorporate hydrogen into their national energy strategies [11,12]. With the increasing adoption of FCs, regulatory standards have been established to ensure safe and efficient implementation.

Table 1. FC standards.

Standard Name	Summery	Organization
ISO 14687 [22]	Specify the hydrogen quality requirements for FC vehicles.	International Organization for Standardization
ISO 16111 [13]	Covers transportable gas storage devices for hydrogen.
ISO 23828 [15]	Methods for measuring energy consumption in FC vehicles.
ISO 17268 [16]	Define requirements for hydrogen refuelling connectors.
IEC 62282-3-100 [23]	Safety requirements for FC power systems.	International Electrotechnical Commission
IEC 62282-4-101 [24]	Performance evaluation techniques for FCs.
IEC 62282-7-1 [25]	Testing methods for PEMFCs.
NFPA 2 [26]	Hydrogen technologies code for generation, installation, storage, piping, use, and handling.	National Fire Protection Association
JIS C 8800 [27]	Safety requirements for stationary FC power systems.	Japanese Industrial Standards
JIS C 62282-5-100 [28]	Safety requirements for portable FC power systems.
KS R ISO 23273 [29]	FC road vehicles safety specifications.	Korean Agency for Technology and Standards.

summarizes the existing FC standards [13,14,15,16,17,18,19,20,21]. Among these, certain standards focus specifically on evaluating FC performance across different applications. As performance assessments—particularly during the design and early development stages—are often conducted through simulation-based analyses, the accuracy of these evaluations strongly depends on the fidelity of the employed FC models. Consequently, there is an increasing demand for advanced FC models capable of accurately capturing the dynamic response, transient characteristics, and nonlinear behaviour of FC systems under varying operating conditions. However, depending on the application, the interest performance characteristics of FCs vary. As a result, various FC equivalent circuit models have been developed, prioritizing the dynamics most relevant to the intended application.

In order to have a better understanding of the FC models, the basic operation of the FC needs to be analysed. As illustrated in Figure 1, a typical FC comprises three primary components: an anode, a cathode, and an electrolyte. Hydrogen is supplied to the anode, where it undergoes electrochemical oxidation, splitting into protons (H⁺) and electrons (e⁻), as represented by the following Reaction (1) [30]:

$${2\mathrm{H}}_2\to 4{\mathrm{H}}^{+}+4{\mathrm{e}}^{-}$$

(1)

The generated protons then migrate through the electrolyte (commonly referred to as the proton-exchange membrane) to the cathode. The membrane is selectively permeable, allowing only protons to pass while blocking electrons. Consequently, electrons travel through an external circuit, generating an electric current before reaching the cathode. At the cathode, protons, electrons, and oxygen (typically sourced from ambient air) combine to form water and heat as byproducts, as described by Reaction (2):

$${\mathrm{O}}_2+4{\mathrm{H}}^{+}+4{\mathrm{e}}^{-}\to 2{\mathrm{H}}_2\mathrm{O}+\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t}$$

(2)

This process, known as the oxygen reduction reaction (ORR), releases a considerable amount of thermal energy, which reduces the overall energy efficiency of the FC. Commercially available FCs can be classified based on their electrolyte materials, as detailed in

Table 2. FC types and applications.

FC Type	Electrolyte	Operating Temperature (°C)	Efficiency (%)	Applications
Proton Exchange Membrane Fuel Cell (PEMFC)	Polymer membrane, conducts H+	60–100	40–60	Vehicles, portable power, and small-scale stationary power.
Alkaline Fuel Cell (AFC)	Aqueous potassium hydroxide (KOH) solution, conducts OH−	60–90	50–60	Spacecraft, submarines, and specialized applications.
Phosphoric Acid Fuel Cell (PAFC)	Phosphoric acid (H3PO4), conducts H+	150–200	35–45	Medium-scale stationary power generation
Molten Carbonate Fuel Cell (MCFC)	Molten carbonate salts, conduct CO32−	600–700	50–60	Large-scale stationary power generation.
Solid Oxide Fuel Cell (SOFC)	Ceramic material, conducts O2−	800–1000	50–65	Large-scale stationary power generation with combined heat and power (CHP).
Direct Methanol Fuel Cell (DMFC)	Polymer membrane, conducts H+	60–130	20–40	Portable power.

[31,32,33]. Among these, PEMFCs are the most widely adopted due to their favourable characteristics, including high power density, low operating temperature, compact and lightweight design, rapid start-up capability, and high energy conversion efficiency.

However, despite these advantages, PEMFCs are subject to performance degradation over time, which remains a critical challenge for their widespread adoption. Degradation in PEMFCs arises from multiple electrochemical, mechanical, and thermal mechanisms, including catalyst layer degradation, membrane thinning and chemical decomposition, carbon support corrosion, and water management issues such as flooding and dehydration [34,35]. These effects lead to increased internal resistance, reduced electrochemical active surface area, and deterioration in mass transport characteristics, ultimately resulting in a gradual decline in output voltage and efficiency [36]. Furthermore, operating conditions such as load cycling, start–stop events, temperature fluctuations, and fuel impurities can accelerate degradation processes and introduce additional nonlinearities in the fuel cell behaviour [37]. From a modelling and control perspective, degradation significantly affects the accuracy and reliability of PEMFC models, as model parameters such as internal resistance, charge transfer resistance, and capacitance are time-varying and dependent on the health state of the FC [36]. Therefore, maintaining long-term modelling accuracy requires periodic recalibration and robust parameter estimation techniques that can capture these variations with minimal disruption to normal operation.

To accurately represent PEMFC behaviour, various modelling approaches have been developed, including mechanistic, data-driven, and empirical models, each offering distinct trade-offs between physical accuracy and computational complexity. Mechanistic models, particularly higher-dimensional (2D and 3D) Multiphysics frameworks, provide detailed insight into electrochemical, thermal, and fluid dynamic processes; however, they are computationally intensive and require extensive parameterization, making them unsuitable for real-time control and system-level applications. Data-driven models can effectively capture nonlinear behaviour with lower computational cost, but their performance depends heavily on large, high-quality datasets and they often lack interpretability and robustness outside trained operating conditions.

Consequently, there is a clear need for modelling approaches that balance accuracy, computational efficiency, and practical implementability, particularly for control-oriented and real-time applications. In this context, empirical equivalent circuit models offer a promising alternative, as they can represent the dominant electrical dynamics of PEMFCs using simplified structures. However, their accuracy is strongly dependent on reliable parameter estimation methods. This highlights the need for a parameter estimation approach that can determine model parameters accurately, repeatably, and with minimal impact on normal operation.

This paper presents a novel methodology for estimating the parameters of a simplified PEMFC equivalent circuit model using current-switching techniques. The proposed approach enables accurate and practical parameter identification suitable for real-time and control-oriented applications. The remainder of the paper is structured as follows: Section 2 reviews existing PEMFC modelling techniques. Section 3 presents the current-switching-based parameter estimation methodology. Section 4 describes the experimental validation and data analysis. Section 5 discusses the obtained results, and Section 6 concludes the paper and outlines future research directions.

2. PEMFC Modelling Frameworks

Figure 2 illustrates the electrical characteristic curve of a PEMFC, which exhibits a nonlinear current-voltage relationship [38]. This nonlinearity is primarily governed by three distinct voltage loss mechanisms: (1) activation losses due to electrochemical reaction kinetics, (2) ohmic losses arising from ionic and electrical resistances, and (3) concentration losses caused by mass transport limitations. Accounting for these losses, the terminal voltage (V_fc) of a PEMFC can be expressed as follows (3):

$$V_{fc}=E-V_{act,cell}-V_{ohm,cell}-V_{conc,cell}$$

(3)

where E represents the open-circuit voltage of the FC, while V_act,cell, V_ohm,cell, and V_conc,cell denote the activation, ohmic, and concentration overpotentials, respectively.

A wide range of PEMFC models has been developed to represent the loss mechanisms defined in (3) and to capture system behaviour under varying operating conditions. Over the past three decades, research efforts in PEMFC modelling have grown significantly, yielding numerous models in the literature. Broadly, these models can be categorized into three groups: (1) mechanistic models, (2) data-driven models, and (3) empirical models, as shown in Figure 3.

2.1. Mechanistic Models

Mechanistic models describe the internal physicochemical processes occurring within a PEMFC based on fundamental physical and electrochemical principles [39,40,41]. These models are formulated based on conservation laws of mass, momentum, energy, and charge [42], and typically incorporate electrochemical kinetics, fluid dynamics, and heat transfer to simulate coupled transport and reaction processes [43,44].

By explicitly capturing electrochemical reactions and energy transformations, mechanistic models enable detailed evaluation of performance metrics such as cell voltage, power density, and efficiency under varying operating conditions. This supports optimization of stack design, operating strategies, and material selection [45]. In addition, mechanistic simulations can assess electrode and electrolyte properties—including conductivity, catalytic activity, and stability—and help identify potential degradation and failure mechanisms [46,47]. Consequently, mechanistic models form an important theoretical basis for PEMFC design, optimization, and control, as well as for fault-tolerant and thermal management strategies.

Mechanistic modelling approaches often focus on one or more physical domains:

• Electrochemical Models capture: Reaction kinetics and charge transfer processes at the electrodes using the Nernst equation for equilibrium potential, the Tafel equation for activation overpotential, and the Nernst–Planck formulation for ionic transport [42,48].
• Thermodynamic Models: Analyse energy balance and heat generation mechanisms, providing insight into efficiency variations under different temperatures, pressures, and reactant conditions. Recent studies also highlight that effective thermal management strategies, such as liquid-cooled PEMFC systems, play a critical role in maintaining temperature uniformity, improving stack durability, and stabilizing electrochemical parameters during operation. They are particularly useful for thermal management design and optimization of operating parameters [44,49].
• Fluid Dynamics Models: Simulate the flow and distribution of reactant gases (hydrogen and oxygen) and water within the flow channels and porous media. These models employ Navier–Stokes and continuity equations to investigate gas transport, pressure drop, and water management, all of which are crucial for maintaining stable operation and preventing flooding or membrane dehydration [45].

Comprehensive mechanistic frameworks often couple these domains to capture Multiphysics interactions (heat–mass–charge transport with electrochemical kinetics). Based on spatial resolution and computational complexity, mechanistic models are commonly categorized as zero-dimensional (0D), one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D), each offering a trade-off between fidelity, interpretability, and computational cost [50]. A comparative summary of these mechanistic modelling approaches, including their key features, advantages, and limitations, is provided in

Table 3. Summary of the PEMFC mechanistic modelling techniques.

Model Type	Key Features/Governing Equations	Typical Applications	Advantages	Limitations
0D	Global mass and energy balance equations; empirical voltage–current relations	System-level simulations; control design; hybrid energy system studies	Simple, fast computation; suitable for real-time and control-oriented models	No spatial resolution; cannot capture local gradients or transient inhomogeneities
1D	Nernst, Ohm’s law, Tafel, Stefan–Maxwell, and Nernst–Planck equations	MEA design; steady-state performance evaluation	Good accuracy for through-plane transport; moderate computational cost	Assumes uniform in-plane conditions; limited spatial detail
2D	Coupled mass, momentum, charge, and energy equations; may include two-phase flow	Flow-field design; water and thermal management analysis	Captures in-plane and through-plane variations; realistic performance prediction	Higher computational cost; requires detailed parameterization
3D	Fully coupled Multiphysics equations (CFD/FEM/FVM solvers) for fluid flow, heat, and electrochemical reactions	Fundamental research; detailed design and degradation analysis	Most accurate; captures complex Multiphysics effects and nonuniformities	Extremely high computational demand; difficult to apply in control or real-time environments

.

2.1.1. Zero-Dimensional (0D) Models

Zero-dimensional or lumped-parameter models describe the overall behaviour of the PEMFC using averaged parameters and global conservation equations without spatial discretization [51,52]. These models represent the cell as a single control volume, typically expressed through algebraic or low-order differential equations. Despite their simplicity, 0D models are capable of predicting key global performance characteristics such as polarization curves, overall efficiency, and thermal response [53]. Due to their computational efficiency, they are widely used in system-level simulations, control design, and hybrid energy management studies, where detailed spatial information is not essential.

2.1.2. One-Dimensional (1D) Models

One-dimensional models provide a more detailed description by resolving variations along the through-plane direction—typically from the anode gas diffusion layer (GDL) to the cathode GDL—while assuming uniform conditions in the other two dimensions. The pioneering 1D model by Bernardi and Verbrugge described gas transport and electrochemical reactions across the membrane–electrode assembly (MEA) using coupled equations for species diffusion, charge conservation, and heat balance [54]. Later, Amphlett et al. [43] incorporated the Stefan–Maxwell equations to describe mass transfer, the Nernst–Planck equation to characterize ionic conduction, and the Tafel equation to model activation losses. Further enhancements by Mann et al. [44] extended the framework to generalized steady-state operation under various operating pressures and temperatures.

While 1D models offer good predictive accuracy for through-plane transport processes and are suitable for design and optimization of MEA materials, they neglect in-plane gradients, making them less effective for analysing local nonuniformities in gas distribution, temperature, or humidity.

2.1.3. Two-Dimensional (2D) Models

Two-dimensional models expand upon 1D formulations by accounting for variations both through the membrane and along the gas flow direction [55]. This enables the simultaneous evaluation of reactant distribution, temperature gradients, and liquid water accumulation within the flow channels and porous media. Early two-dimensional steady-state PEMFC models incorporated coupled gas diffusion, electrochemical reaction kinetics, and heat transfer across the flow channels and electrode layers [56]. Subsequent developments introduced more detailed mass and energy transport formulations, improving the accuracy of thermal and species distribution predictions [57]. Further advancements extended these frameworks to two-phase two-dimensional models, enabling the investigation of anisotropic transport properties, liquid water effects, and temperature distribution under varying current densities [58]. Although computationally more demanding than one-dimensional approaches, two-dimensional models provide a more realistic representation of fuel cell behaviour, particularly for performance prediction and flow-field design optimization.

2.1.4. Three-Dimensional (3D) Models

Three-dimensional models provide the most comprehensive and physically accurate representation of PEMFC operation by resolving transport and reaction processes in all spatial directions [59]. They couple electrochemical kinetics, mass and heat transfer, and fluid flow using computational fluid dynamics (CFD) and finite element (FEM/FVM) techniques [60]. Recent 3D Multiphysics models (e.g., Berning et al. [61]) have been employed to analyse water generation, thermal management, and mechanical compression effects in full-scale PEMFCs. These models can accurately capture local phenomena such as gas starvation, membrane dehydration, and liquid water flooding, which are difficult to analyse using lower-dimensional approaches.

However, their high computational cost and the need for extensive physical parameters make them impractical for real-time control or embedded applications. They are primarily used for fundamental research, material development, and system design optimization, where detailed spatial information is essential.

2.2. Data-Driven Models

Data-driven models predict the behaviour of PEMFCs by directly learning relationships between system inputs and outputs using empirical data, rather than solving detailed physicochemical equations [62,63,64,65]. Unlike mechanistic models that explicitly represent electrochemical and transport processes, data-driven frameworks infer system dynamics implicitly through statistical or machine-learning-based pattern recognition.

These models employ diverse algorithms, including artificial neural networks (ANNs), support vector regression (SVR), decision trees (DTs), random forests (RFs), Gaussian process regression (GPR), and deep learning (DL) architectures such as convolutional (CNN) and recurrent neural networks (RNNs), including long short-term memory (LSTM) and gated recurrent unit (GRU) variants [66]. These approaches have been successfully applied for performance prediction, fault detection, state-of-health (SOH) estimation, prognostics, and control-oriented modelling of PEMFCs [67].

Recent studies further demonstrate that data-driven and machine learning-based models are increasingly applied in PEMFC systems for performance prediction, degradation estimation, and real-time monitoring. Deep learning architectures, particularly RNN-based models such as LSTM and GRU, have shown strong capability in capturing transient voltage behaviour and degradation trends under dynamic operating conditions. For example, GRU-based frameworks have achieved high prediction accuracy with reduced computational complexity, making them suitable for online degradation prediction [66]. In addition, data-driven models based on operational variables have been successfully used to predict PEMFC power output and support real-time monitoring and energy management applications.

For instance, Zhang et al. developed an LSTM-based framework to predict dynamic voltage response under variable load conditions, accurately capturing nonlinear transients and hysteresis effects [68]. Similarly, Liu et al. demonstrated the use of Bayesian-optimized deep neural networks (DNNs) for transient voltage prediction, achieving reduced mean-absolute-error and enhanced robustness to measurement uncertainty [69].

Beyond conventional supervised learning, unsupervised and reinforcement learning (RL) paradigms have been introduced for anomaly detection and adaptive control. Clustering algorithms such as k-means, self-organizing maps (SOMs), and autoencoders identify degradation patterns in sensor data, while RL-based controllers autonomously optimize operational parameters to enhance durability and efficiency [68,70].

Although these methods exhibit high computational efficiency during inference, their predictive accuracy depends strongly on the quality, representativeness, and volume of training data. Overfitting, data imbalance, and sensor noise can compromise model generalization. Moreover, most data-driven architectures behave as “black boxes,” limiting interpretability and trustworthiness when extrapolated beyond trained domains [71].

To address these shortcomings, recent studies focus on hybrid and physics-informed modelling frameworks that integrate mechanistic knowledge into data-driven architectures. Physics-informed neural networks (PINNs) and Gray-box hybrid models embed physical constraints (e.g., mass- or charge-balance equations) into the learning process, preserving interpretability while maintaining computational speed [72]. For example, coupling a reduced-order electrochemical model with an LSTM surrogate allows accurate transient prediction with a fraction of the computational cost [73].

Additionally, digital-twin platforms—which synchronize real-time operational data with continuously updated ML models—are emerging for online monitoring, predictive maintenance, and control optimization of PEMFC systems [74,75]. These frameworks dynamically recalibrate model parameters from sensor feedback, offering a pathway toward self-adaptive and resilient FC operation.

Despite these advantages, data-driven approaches rely heavily on the availability of large, high-quality datasets and often operate as black-box models, limiting interpretability and robustness when extrapolated beyond trained conditions. A comparative summary of commonly used data-driven modelling techniques is provided in

Table 4. Summary of the PEMFC data-driven modelling techniques.

Model Type	Learning Principle/Algorithms	Application Areas	Advantages	Limitations
Statistical Regression [76,77]	Linear/nonlinear regression, polynomial fitting	Basic performance modelling	Simple, interpretable coefficients	Limited nonlinear accuracy
Support Vector Regression (SVR) [78]	Kernel-based supervised learning	Voltage prediction, fault detection	Good generalization on small datasets	Sensitive to kernel choice
Tree-Based Models (RF, GBM) [79,80]	Ensemble decision trees	Fault diagnosis, SOH estimation	Interpretable, robust to noise	May require large datasets
Artificial Neural Networks (ANNs) [55,81]	Feedforward MLPs	Static I–V prediction	Flexible nonlinear mapping	Data-intensive, limited extrapolation
Deep Learning (CNN, RNN, LSTM, GRU) [82,83,84]	Hierarchical and temporal learning	Transient voltage, degradation forecasting	Captures dynamic behaviour, high accuracy	High training cost, “black box” nature
Gaussian Process Regression (GPR) [39,85]	Probabilistic nonparametric learning	Uncertainty quantification, hybrid surrogates	Provides confidence bounds	Poor scalability for large datasets
Hybrid/Physics-Informed Models [73,85,86]	PINNs, gray-box coupling	Digital twins, control optimization	Combines interpretability and adaptability	Complex training, requires domain knowledge
Reinforcement Learning (RL) [74,87]	Q-learning, actor-critic algorithms	Online efficiency optimization	Learns optimal control autonomously	Data-hungry, stability issues

. This highlights the need for hybrid or simplified modelling approaches that can maintain both accuracy and practical applicability.

2.3. Empirical Models

Empirical models are derived from experimental data fitted to simplified mathematical representations of PEMFC behaviour. Unlike mechanistic or data-driven models, empirical approaches utilize limited experimental data to develop computationally efficient, reduced-order models. Their accuracy is inherently constrained by the quality and operating range of the experimental data used for calibration [71]. Consequently, empirical models may exhibit reduced reliability when applied outside their original validation conditions.

Among the various PEMFC modelling approaches, empirical models have found extensive application in electrical engineering and power electronics domains due to their ability to represent fundamental FC characteristics using basic electrical components. In practical power electronic systems, PEMFCs are commonly interfaced with DC–DC converters, inverters, and hybrid energy storage systems, exposing them to operating conditions such as high-frequency current ripple, pulsed current excitation, and rapid load transients. As a result, the suitability of an empirical PEMFC model is largely determined by its capability to capture the dynamic voltage response of the FC under electrically induced disturbances, in addition to steady-state behaviour.

Empirical PEMFC models typically employ resistive, capacitive, and semiconductor-based elements to approximate activation losses, ohmic losses, concentration polarization, and transient voltage dynamics. Based on their structure and level of abstraction, four primary categories of empirical models have been reported in the literature for electrical and power electronics applications: (1) electronic circuit-based models, (2) frequency response equivalent circuit models, (3) original electric equivalent circuit models, and (4) simplified equivalent circuit models. A comparative summary of these model types is provided in

Table 5. Description of PEMFC empirical equivalent circuit model.

Model	Configuration	Description
Electronic circuit-based model		Uses standard circuit elements to emulate PEMFC electrical behaviour under varying operating conditions.
Frequency response equivalent circuit-based model		Represents PEMFC dynamics using impedance-based (EIS) frequency-domain characteristics.
Original electric equivalent circuit-based model		Maps electrochemical processes to equivalent electrical components with physical interpretation.
Simplified equivalent circuit-based model		Reduced-order RC-based model capturing dominant PEMFC dynamics with low complexity.

, and their applications, advantages and limitations are provided in

Table 6. Summary of the PEMFC empirical models.

Model	Advantages	Limitations	Typical Applications
Electronic circuit-based model	Good dynamic representation	Good dynamic representation	Good dynamic representation
Frequency response equivalent circuit-based model	Captures frequency dynamics	Captures frequency dynamics	Captures frequency dynamics
Original electric equivalent circuit-based model	High accuracy	High accuracy	High accuracy
Simplified equivalent circuit-based model	Low complexity, fast computation	Low complexity, fast computation	Low complexity, fast computation

. While all four model categories aim to replicate the electrical behaviour of PEMFCs, they differ significantly in modelling complexity, physical relevance, and applicability to power electronics studies.

2.3.1. Electronic Circuit-Based Models

Electronic circuit-based models emulate PEMFC behaviour using conventional circuit elements such as resistors, capacitors, diodes, and controlled sources [88]. These models are particularly well suited for system-level simulations and converter-interfaced studies, as they can directly represent voltage fluctuations arising from current ripple and pulsed loading imposed by switching power converters. Their moderate accuracy and compatibility with simulation platforms such as MATLAB/Simulink R2022b, PSPICE, and PSIM make them attractive for control and power management investigations.

2.3.2. Frequency Response Equivalent Circuit Models (FRECMs)

FRECMs are derived from electrochemical impedance spectroscopy (EIS) data and characterize the dynamic behaviour of PEMFCs over a range of frequencies [89]. By fitting impedance spectra with networks of resistors, capacitors, inductors, and constant phase elements, FRECMs provide valuable insight into frequency-dependent processes such as charge transfer dynamics, double-layer capacitance, and mass transport effects. These models are particularly effective for analysing converter–FC interactions and current ripple effects within specific frequency bands, although their parameter identification is computationally intensive and often valid only for limited operating conditions.

2.3.3. Original Electric Equivalent Circuit Models

Original electric equivalent circuit models establish direct analogies between electrochemical phenomena and electrical components, enabling the representation of activation kinetics, membrane proton transport, and diffusion processes through physically motivated circuit elements [90]. While these models offer enhanced physical fidelity and are useful for examining the impact of transient current excitation on internal FC processes, their high complexity and extensive parameter requirements limit their suitability for large-scale power electronics simulations and iterative converter design studies.

2.3.4. Simplified Equivalent Circuit Model

Simplified equivalent circuit models provide reduced-order representations that capture the dominant steady-state and transient characteristics of PEMFCs using a minimal set of circuit elements [91]. Typically consisting of an ideal voltage source in series with a resistance and a parallel RC network, these models are computationally efficient and well suited for control design, real-time simulation, and system-level analysis in power electronics applications. Although simplified models do not accurately represent high-frequency electrochemical phenomena, they offer an effective compromise between modelling fidelity and computational efficiency when evaluating converter-induced current ripple, pulsed loading, and energy management strategies.

As schematically represented in Figure 4, this model topology incorporates: an ideal voltage source (E) represents the open circuit voltage of the PEMFC, a series resistance (R_r) represents the internal resistance of the FC and a parallel RC branch (R_a and C_a) models the charge transfer resistance and capacitive effects of the FC. Under steady-state operation, the capacitor C_a becomes fully charged, causing all current to flow through Ra. However, during transient load changes, C_a introduces a delay in the terminal voltage response of the FC.

While this model effectively emulates the electrical behaviour of PEMFCs, its accuracy is highly dependent on the precision of the equivalent circuit parameters. Therefore, accurate parameter identification is critical for reliable model performance.

Recent research has increasingly focused on improving the applicability of empirical equivalent circuit models for real-time and control-oriented applications. In particular, advanced parameter estimation techniques based on optimization algorithms and real-time identification methods have been proposed to enhance model accuracy under dynamic operating conditions. Recent work in PEMFC parameter estimation using optimization methods demonstrates that improved parameter identification can significantly enhance model robustness and predictive accuracy across different operating scenarios [92].

Furthermore, the study by Mei et al. (2024) [93] highlights the importance of accurate voltage model parameter extraction for improving real-time performance and reliability of PEMFC systems.

In summary, while mechanistic and data-driven models provide valuable insights into PEMFC behaviour, their practical implementation in parameter estimation and real-time control applications remains limited. Mechanistic models, although physically accurate, involve complex Multiphysics formulations and require extensive computational resources, making them unsuitable for real-time applications and online parameter identification. Data-driven approaches, while computationally efficient during inference, rely heavily on large datasets and often lack interpretability and robustness under varying operating conditions and degradation.

In contrast, empirical equivalent circuit models simplify PEMFC behaviour into electrically interpretable components, enabling direct correlation between measurable voltage–current responses and model parameters. This reduced-order representation significantly lowers computational complexity while preserving the dominant dynamic characteristics of the system. As a result, empirical models—particularly simplified equivalent circuit models—are highly suitable for real-time applications and parameter estimation. Their parameters can be extracted using transient voltage responses obtained from controlled current perturbations, without requiring complex optimization procedures or large datasets.

Therefore, empirical modelling provides a practical and effective framework for parameter estimation, which motivates the use of current-switching-based techniques in this work. The detailed experimental procedure and parameter calculation methodology are presented in Section 3.

3. Current Switching Techniques for PEMFC Parameter Estimation

Accurate parameter identification is a crucial step in developing reliable simplified equivalent circuit models of PEMFCs. The model parameters—such as internal resistance, charge transfer resistance, and double-layer capacitance—directly influence the fidelity of the simulated dynamic and steady-state behaviour. Traditional parameter estimation approaches often rely on steady-state polarization curves or frequency response analysis; however, these methods can be time-consuming, require specialized equipment, and may not effectively capture transient characteristics under realistic operating conditions.

To address these limitations, current switching techniques have emerged as an efficient and practical approach for PEMFC parameter estimation. In this method, the FC is subjected to controlled current perturbations or step changes, and the corresponding transient voltage response is analysed. The dynamic voltage behaviour obtained from these tests provides valuable insight into the electrochemical and electrical dynamics of the PEMFC, enabling accurate extraction of equivalent circuit parameters.

Two primary tests can be classified under the current switching technique:

• Current Loading Technique
• Current Interrupting Technique

Figure 4 illustrates the block diagram of the experimental setup used to perform these tests. In each test, the voltage response of the FC is recorded following a current transition, and the parameters of the simplified equivalent circuit model are subsequently determined based on the observed transient behaviour.

3.1. Current Loading Technique

In the current loading test, the PEMFC initially operates under open-circuit conditions. Once a steady-state voltage is achieved, a constant current load is suddenly applied to the system. This abrupt change in load current induces a transient response in the terminal voltage, reflecting the internal electrochemical and capacitive dynamics of the FC. Figure 5 depicts the typical terminal voltage response of the PEMFC during the current loading test. The different stages of this voltage profile, including the instantaneous voltage drop, transient recovery, and steady-state stabilization, can be mathematically characterized using the simplified PEMFC equivalent circuit.

The PEMFC steady-state voltage under open-circuit conditions can be expressed as follows:

$$E_0=V_{\left(t<t_0\right)}$$

(4)

Following the sudden application of a constant load, the transient terminal voltage response of the PEMFC can be formulated as:

$$V_{\left(t\right)}={\displaystyle \frac{R_L}{R_L+R_a+R_r}}\left(1-e^{-{\displaystyle \frac{t-t_0}{\tau }}}\right)E_0+{\displaystyle \frac{R_L}{R_L+R_r}} e^{-{\displaystyle \frac{t-t_0}{\tau }}}{E}_0$$

(5)

Here, τ denotes the time constant of the equivalent circuit, which is determined as:

$$\tau ={\displaystyle \frac{C_a(R_a{R}_L+R_a{R}_r)}{R_L+R_a+R_r}}$$

(6)

The load resistance (R_L) can be calculated using the steady-state voltage and current as (7):

$$R_L={\displaystyle \frac{V_{SS}}{I}}$$

(7)

As illustrated in Figure 5, when the load is abruptly connected to the PEMFC at t = t₀, the initial voltage drop can be expressed as:

$$V_{(t={t_0}^{+})}={\displaystyle \frac{R_L}{R_L+R_r}} E_0$$

(8)

During steady-state operation, C_a becomes fully charged, and the current flows entirely through the R_a. The corresponding terminal voltage is defined as:

$$V_{SS}={\displaystyle \frac{R_L}{R_L+R_r+R_a}} E_0$$

(9)

From the PEMFC terminal voltage profile obtained during the current loading test, the quantities V_ss, V_(t=t0), V_(t<t0), and τ can be extracted. These values are then used to determine the parameters of the simplified PEMFC equivalent circuit as follows.

The open-circuit voltage (E₀) is obtained directly from:

$$E_0=V_{\left(t<t_0\right)}$$

(10)

The internal resistance (R_r) is determined by combining (8) and (10), as shown in (11):

$$R_r={\displaystyle \frac{V_{(t<t_0)}}{V_{(t={t_0}^{+})}}} R_L-R_L$$

(11)

The charge transfer resistance (R_a) is subsequently calculated using (9) and (11):

$$R_a={\displaystyle \frac{R_L{E}_0}{V_{SS}}} R_L-R_r$$

(12)

Finally, the double-layer capacitance (C_a) is obtained from (6), (11) and (12):

$$C_a={\displaystyle \frac{\tau (R_L+R_a+R_r)}{R_a{(R}_L+R_r)}}$$

(13)

3.2. Current Interruption Technique

In the current interruption test, the PEMFC is initially operated under a constant current load. Once steady-state conditions are achieved, the load is abruptly disconnected while recording the corresponding transient voltage response. Figure 6 illustrates the typical voltage profile observed during this test. Each stage of the voltage response can be mathematically described using the simplified PEMFC equivalent circuit model.

During steady-state operation prior to interruption, C_a in the equivalent circuit is fully charged, resulting in zero current flow through the capacitive branch. The terminal voltage at this stage can be expressed as:

$$V_{(t<t_0)}=E_0-I\left(R_a{+R}_r\right)=I{R}_L$$

(14)

where I represent the steady-state current prior to load disconnection. The load resistance (R_L) can thus be determined as:

$$R_L={\displaystyle \frac{V_{(t<t_0)}}{I}}$$

(15)

After the load is disconnected at t = t₀, the transient voltage response of the PEMFC is characterized by:

$$V_{\left(t\right)}=E_0-I{R}_a{e}^{-{\displaystyle \frac{t-t_0}{\tau }}}$$

(16)

Here τ denotes the time constant of the equivalent circuit following the current interruption, defined as:

$$\tau =R_a{C}_a$$

(17)

Immediately after the load disconnection, the terminal voltage can be expressed using (16) as:

$$V_{(t={t_0}^{+})}=E_0-I{R}_a$$

(18)

The open-circuit voltage (E₀) of the PEMFC is then obtained from the final steady-state voltage:

$$E_0=V_{SS}$$

(19)

These equations enable the determination of the key parameters of the simplified equivalent circuit. The internal resistance R_r is calculated by combining (14) and (18):

$$R_r={\displaystyle \frac{V_{(t={t_0}^{+})}-V_{(t<t_0)}}{I}}$$

(20)

The charge transfer resistance R_a is determined from (18) and (19):

$$R_a={\displaystyle \frac{V_{SS}-V_{(t={t_0}^{+})}}{I}}$$

(21)

Finally, using the time constant τ derived from the transient voltage response, the double-layer capacitance (C_a) can be calculated as:

$$C_a={\displaystyle \frac{\tau }{R_a}}$$

(22)

4. Experimental Procedure

The proposed parameter estimation techniques were experimentally validated using a laboratory test setup, as illustrated in Figure 7. The setup consists of Horizon FC Technologies (Nagpur, India)’ H-100 (100 W) and H-1000 (1 kW) PEM FC stacks, a regulated hydrogen supply system, a controllable DC electronic load, and a digital oscilloscope for data acquisition.

The technical specifications of the H-100 and H-1000 PEMFC stacks are summarized in

Table 7. FC specifications.

Parameter	Value
	H-100 FC	H-1000 FC
Rated Power	100 W	1000 W
Type of the FC	PEM	PEM
Performance	12 V at 8.3 A	28.8 V at 35 A
Max. Stack temperature	65 °C	650 °C
Efficiency of the stack	40% at 12 V	40% at 28.2 V
External power supply	13 V (±1 V), 5 A	13 V (±1 V), 8 A
H2 pressure	0.45-0.55 bar	0.45–0.55 bar
Hydrogen purity	≥99.995% dry H2	≥99.995% dry H2
Flow rate at max output	1.3 L/min	13 L/min
Cooling	Integrated cooling fan	Integrated cooling fan

. To ensure experimental reproducibility and consistency in parameter extraction, the PEMFC stacks were operated under controlled conditions throughout all tests. As summarized in Table 7, both the 100 W and 1 kW systems were supplied with hydrogen at a pressure of 0.45–0.55 bar and a purity of at least 99.995% dry H₂, while stack cooling was provided by integrated cooling fans. The maximum stack operating temperature was maintained at 65 °C in accordance with the manufacturer specifications, and the external auxiliary power supply was regulated at 13 V (±1 V). All experiments were conducted at an ambient laboratory temperature of approximately 22–25 °C with a typical relative humidity of approximately 50–65%.

During all experiments, the electronic load was operated in constant-current mode so that the imposed current steps were well defined and repeatable. Maintaining these operating conditions was important because variations in temperature, gas pressure, fuel quality, and load stability can influence the transient voltage response of the PEMFC and therefore affect the accuracy of the extracted equivalent circuit parameters.

No active humidity control was externally imposed beyond the manufacturer-configured stack operating arrangement; therefore, the reported results correspond to the nominal operating conditions of the commercial PEMFC systems used in this study. The test parameters and conditions corresponding to each experiment are presented in

Table 8. Test conditions.

Test #	Current Loading Technique
	Initial Current (A)	Final Current (A)
1	0	1
2	0	2
Test #	Current Interrupt Technique
	Initial Current (A)	Final Current (A)
3	1	0
4	2	0

.

As shown in Figure 8, For each test scenario, the terminal voltage profiles of the PEMFC were recorded during both current loading and current interruption operations. The measured transient voltage responses were then analysed to calculate the equivalent circuit parameters based on the equations derived in Section 3. These experimentally obtained results were compared with theoretical model responses to verify the accuracy and validity of the proposed parameter estimation methodology.

4.1. Current Loading Test

In the current loading test, the equivalent circuit parameters were calculated using Equations (7), (8) and (11).

Table 9. Experimental data and equivalent circuit parameters extracted from the current loading test.

PEMFC Details		Test #	${\mathit{V}}_{(\mathit{t}<{\mathit{t}}_0)}$ (V)	${\mathit{V}}_{(\mathit{t}={{\mathit{t}}_0}^{+})}$ (V)	${\mathit{V}}_{\mathit{S}\mathit{S}}$ (V)	$\mathit{\tau }$ (S)	${\mathit{R}}_{\mathit{a}}$ (Ω)	${\mathit{R}}_{\mathit{r}}$ (Ω)	${\mathit{C}}_{\mathit{a}}$ (F)
Model	Power
H-100	100 W	1	19.4	17.4	15.97	0.11	1.59	1.84	0.075
		2	19.45	15.4	14.68	0.11	0.455	1.93	0.254
H-1000	1 kW	1	46.6	45.88	38.5	0.07	7.5	0.6	0.011
		2	46.7	45.25	37.8	0.07	3.85	0.6	0.022

presents the equivalent circuit parameters obtained from each current loading test for the 100 W and 1 kW PEMFCs.

The measured and simulated voltage responses corresponding to the current loading test are shown in Figure 9 for the 100 W and 1 kW FCs, respectively. The simulated curves closely follow the experimental results, confirming that the simplified equivalent circuit accurately represents the PEMFC’s dynamic response under step current loading conditions.

4.2. Current Interruption Test

In the current interruption test, the equivalent circuit parameters were determined using Equations (18), (19) and (22).

Table 10. Experimental data and equivalent circuit parameters extracted from the current interruption test.

PEMFC Details		Test #	${\mathit{V}}_{(\mathit{t}<{\mathit{t}}_0)}$ (V)	${\mathit{V}}_{(\mathit{t}={{\mathit{t}}_0}^{+})}$ (V)	${\mathit{V}}_{\mathit{S}\mathit{S}}$ (V)	$\mathit{\tau }$ (S)	${\mathit{R}}_{\mathit{a}}$ (Ω)	${\mathit{R}}_{\mathit{r}}$ (Ω)	${\mathit{C}}_{\mathit{a}}$ (F)
Name	Power
H-100	100 W	3	16.34	17.4	19.4	0.17	2	1.06	0.085
		4	15.8	17.4	19.4	0.21	1	0.8	0.21
H-1000	1 kW	3	38.3	38.8	46.6	0.13	7.8	0.5	0.017
		4	37.85	38.64	46.6	0.12	3.98	0.39	0.03

presents the equivalent circuit parameters obtained from each current interruption test for the 100 W and 1 kW PEMFCs.

The corresponding measured and simulated transient voltage profiles for the 100 W and 1 kW PEMFCs are illustrated in Figure 10, respectively. The close alignment between the experimental and simulated voltage responses demonstrates that the proposed equivalent circuit effectively captures the transient behaviour of the PEMFC following current interruption.

Table 11. Comparison of computational complexity of PEMFC modelling approaches.

Model Type	Typical Number of Parameters	Mathematical Complexity	Typical Computational Requirement
3D Mechanistic Models [59,60,61]	30–50+	Coupled PDEs (mass, charge, energy transport)	High computational cost; CFD/FEM simulation required
1D/2D Mechanistic Models [52,54,55,56,57,94]	15–30	Nonlinear differential equations	Moderate–high computational effort
Data-Driven/ML Models [66,67,69,95]	10–100+ (network weights)	Statistical/ML inference	Fast inference but requires large training datasets
Original Equivalent Circuit Models [89,90]	6–10	Nonlinear algebraic equations	Low computational cost
Proposed Simplified Equivalent Circuit Model	3 parameters (Rr, Ra, Ca)	Simple algebraic expressions	Very low computational cost; suitable for real-time applications

summarizes the maximum error observed between the PEMFC responses and the simplified equivalent circuit predictions under each test condition. As shown, the maximum error remains below 10%, confirming the reliability and accuracy of the proposed method for determining the equivalent circuit parameters of the simplified equivalent circuit model.

Additionally, to further validate the accuracy of the derived model, the characteristic voltage–current (V–I) curves of the actual PEMFCs (obtained from manufacturer datasheets) were compared with those generated from the equivalent circuit model. Figure 11 depicts these comparisons for the 100 W and 1 kW PEMFCs, respectively. As shown, the characteristic curves derived from the equivalent circuit are nearly identical to the actual PEMFC curves, thereby confirming the validity and accuracy of the model and the computed equivalent circuit parameters.

5. Evaluation and Implementation Potential

A variety of PEMFC equivalent circuit models have been proposed by researchers to capture the electrical and dynamic characteristics of FCs, depending on the intended application. Within the broader PEMFC modelling framework presented in Section 2, the proposed approach belongs to the category of empirical equivalent circuit models but introduces a practical parameter identification strategy specifically tailored for electrical engineering applications. In the context of electrical engineering, the use of a simplified equivalent circuit model provides an effective means to represent the behaviour of a PEMFC using only basic circuit elements—resistors and capacitors. This approach enables accurate replication of both steady-state and transient responses, without requiring detailed consideration of the complex electrochemical processes occurring within the FC.

Compared with mechanistic models, which typically require the numerical solution of coupled nonlinear partial differential equations describing mass transport, electrochemical kinetics, and thermal dynamics, the proposed simplified modelling framework significantly reduces computational complexity. Data-driven models also require large datasets and computationally intensive training procedures. In contrast, the proposed method determines model parameters directly from experimentally measured transient voltage responses using current-switching tests. As a result, the approach provides a balance between modelling accuracy, computational efficiency, and practical applicability for system-level studies.

To quantify the computational advantages of the proposed modelling framework, a comparison of modelling complexity and computational requirements between commonly used PEMFC modelling approaches is presented in Table 11. Mechanistic models typically require the numerical solution of coupled nonlinear partial differential equations representing mass transport, electrochemical kinetics, and thermal dynamics, often involving tens of physical parameters and high computational cost. Data-driven models also require large datasets and computationally intensive training procedures. In contrast, the proposed simplified equivalent circuit model represents the PEMFC using only three key parameters (Rr, Ra, and Ca) and can be evaluated using simple algebraic expressions derived from transient voltage responses. As a result, the proposed method significantly reduces computational complexity and enables simulation and parameter estimation within milliseconds, making it particularly suitable for real-time control, digital twin implementation, and power electronics applications.

Such models are particularly advantageous for system-level analyses, including integration studies of FCs in DC microgrids, hybrid power systems, and power electronic interfaces, where electrical engineers primarily focus on dynamic response, stability, and control aspects rather than electrochemical kinetics. However, the accuracy of any equivalent circuit model strongly depends on the precision of the identified parameters. Therefore, reliable and repeatable parameter extraction techniques are essential for ensuring that the simplified model faithfully represents real-world PEMFC performance.

In this study, current switching techniques—specifically the current loading and current interruption methods—were utilized to determine the equivalent circuit parameters of 100 W and 1 kW commercial PEMFC stacks. The results presented in the previous section demonstrate that parameters obtained from these methods enable the simplified equivalent circuits to accurately reproduce both the dynamic and steady-state characteristics of the FCs. As shown in

Table 12. Comparison of the performance of the current switching methods.

Test #	Current Switching	FC Power	Maximum Error (%)
	Technique
1	Current Loading	100 W	2.56
		1 kW	5.95
2		100 W	2.88
		1 kW	1.43
3	Current Interrupting	100 W	2.39
		1 kW	6.85
4		100 W	2.98
		1 kW	9.08

, the maximum error remains below 10% across all test cases, with several cases exhibiting significantly lower error levels (e.g., ~1.4–3%), confirming good agreement between experimental and modelled results. The observed modelling error can arise from several sources. First, measurement noise and sampling limitations in the voltage acquisition system may introduce small uncertainties when capturing the transient voltage response, particularly during rapid voltage transitions immediately following current switching events. Second, the simplified equivalent circuit model inherently represents complex electrochemical processes using lumped parameters, which can introduce approximation errors, particularly under highly nonlinear operating conditions. For example, concentration losses and spatial effects within the fuel cell are not explicitly modelled, which may contribute to discrepancies at higher current densities. In addition, variations in operating conditions such as temperature, hydrogen pressure, and membrane hydration can influence internal resistances and dynamic behaviour, potentially leading to parameter estimation deviations. These limitations represent the typical trade-off between modelling simplicity and physical accuracy in reduced-order models. Future improvements may include incorporating adaptive parameter updating strategies, additional nonlinear elements to better represent concentration effects, and enhanced filtering techniques to reduce measurement noise during transient detection. The excellent agreement between the simulated and measured voltage profiles, as well as the close match between the modelled and datasheet characteristic curves, further validates the reliability of these techniques.

The accuracy and robustness of the proposed parameter estimation approach can also be evaluated in the context of recent polarization loss decomposition techniques reported in the literature. For example, the method presented in [96] employs a systematic decomposition of activation, ohmic, and concentration losses using electrochemical measurements such as Tafel analysis and high-frequency resistance, achieving voltage prediction errors below 1% across the full operating range. In comparison, the proposed simplified equivalent circuit model adopts a reduced-order representation, where polarization losses are incorporated into lumped parameters, particularly through the concept of equivalent resistance. Despite this simplification, the proposed method achieves a comparable level of predictive accuracy while significantly reducing computational complexity and measurement requirements. This highlights that, although detailed electrochemical decomposition approaches provide higher physical interpretability, simplified models combined with appropriate parameter estimation techniques can achieve similar performance for system-level applications. Furthermore, polarization loss decomposition techniques provide a pathway for enhanced health state monitoring by enabling quantification of degradation-related parameters such as electrochemical surface area (ECSA) and equivalent resistance. In this context, for degradation-aware recalibration, such techniques can provide quantifiable ageing indicators, which can be integrated with the proposed modelling framework to support advanced applications such as digital twins and predictive maintenance in PEMFC systems.

Thermal effects also play an important role in the degradation and durability of PEMFC systems. Temperature variations influence electrochemical reaction kinetics, membrane hydration, catalyst activity, and internal resistances, which can directly affect the stability of equivalent circuit parameters over time. Recent studies have shown that temperature-dependent state-of-health (SOH) models can be used to systematically quantify the sensitivity of PEMFC degradation mechanisms to thermal operating conditions. For example, adaptive SOH temperature sensitivity analysis has been proposed to evaluate the influence of temperature on PEMFC ageing behavior and to develop durability improvement strategies through dynamic parameter compensation. In this context, the simplified equivalent circuit model and the low-complexity parameter estimation method proposed in this work can complement such SOH-based approaches. By periodically estimating the equivalent circuit parameters under varying operating conditions, it becomes possible to track temperature-induced parameter drift and integrate these observations into adaptive health monitoring frameworks. This combination can enable real-time degradation monitoring and durability management for PEMFC systems used in dynamic environments such as microgrids, transportation systems, and hybrid energy storage applications.

The key advantages of the current switching techniques include their simplicity, low cost, and high repeatability. The required instrumentation is minimal—typically a programmable DC load and a voltage measurement device—making the method suitable for both laboratory testing and field implementation. Moreover, the technique does not require complex electrochemical impedance measurements or invasive system modifications, which enhances its applicability to commercial PEMFC systems.

The sensitivity of the parameter estimation methods to measurement noise differs between the current loading and current interruption techniques. The current interruption method relies on capturing an instantaneous voltage jump immediately after current disconnection to estimate resistive components, making it more sensitive to high-frequency measurement noise, sampling resolution, and switching transients. In contrast, the current loading method extracts parameters from the transient voltage evolution over a finite time window, allowing implicit averaging of noise and therefore offering improved robustness in practical implementations. However, the loading method may be more sensitive to slow measurement drift and load regulation accuracy. This trend is also reflected in the results, where the current loading method generally exhibits improved accuracy compared to the current interruption method, as illustrated in Figure 9 and Figure 10. Overall, the current loading technique is generally more robust to noise, whereas the current interruption method provides faster estimation but requires higher measurement fidelity and careful signal conditioning.

Real-Time Monitoring and Digital-Twin Application

As discussed, the simplified equivalent circuit parameters can be determined solely from the voltage response to a sudden change in load current. In practical FC applications, such transient conditions naturally occur during normal operation. For example, in vehicular systems, rapid load variations arise during acceleration or braking events, while in microgrid environments, they occur due to sudden demand fluctuations. These naturally occurring load transients are analogous to the controlled current switching conditions applied in this study.

Consequently, by capturing the voltage and current profiles during these regular operational events, it becomes possible to estimate the equivalent circuit parameters in real time without interrupting or perturbing the PEMFC’s normal function. This non-intrusive approach allows continuous model updating, enabling online monitoring and digital-twin-based applications aimed at maintaining an accurate, up-to-date representation of the PEMFC’s electrical behaviour.

The implementation of this real-time monitoring strategy requires only the integration of voltage and current sensing modules with data communication capabilities to stream measurements to a local controller or a cloud-based monitoring platform. Using the acquired voltage and current data, the simplified equivalent circuit parameters of the PEMFC can be computed based on the parameter extraction methodologies described earlier. The updated parameters can then be compared with previously identified values to track parameter drift, performance degradation, and the evolution of the state of health (SOH) under varying operating and environmental conditions, such as temperature, humidity, and pressure.

This approach offers a scalable and cost-effective pathway for in situ diagnostics, predictive maintenance, and adaptive control of PEMFC systems. By leveraging operational transients as natural excitation signals, the method eliminates the need for specialized test sequences and supports the development of self-updating digital twins, thereby enhancing the long-term reliability and efficiency of PEMFC-powered energy systems.

It should be noted that the simplified equivalent circuit model adopted in this study does not explicitly include elements to represent concentration losses. In PEMFCs, concentration losses become significant at high current densities due to mass transport limitations, leading to a nonlinear voltage drop in the high-current region of the polarization curve. In the proposed model, these effects are not separately parameterized but are implicitly captured within the lumped parameters, particularly the internal resistance and the equivalent RC network.

As a result, while the model demonstrates good accuracy under low to moderate current operating conditions, its ability to precisely reproduce the voltage behaviour at high current densities may be limited. This reflects the inherent trade-off of simplified models, where reduced complexity is achieved at the expense of detailed electrochemical representation. Nevertheless, for the intended applications of this work—such as control-oriented modelling, real-time parameter estimation, and system-level analysis—the dominant system dynamics are adequately captured.

Furthermore, the experimental validation in this study is limited to low- and medium-power PEMFC systems (100 W and 1 kW). The applicability of the proposed method to higher-power or multi-stack systems requires further investigation, as additional factors such as thermal gradients, stack non-uniformities, and balance-of-plant interactions may influence parameter estimation accuracy. Future work may focus on extending the model by incorporating additional nonlinear elements to explicitly account for concentration losses, particularly for operation near the limiting current region or under high-power transient conditions.

6. Conclusions and Future Directions

This paper presented a practical and computationally efficient methodology for parameter estimation of a simplified equivalent circuit model of proton exchange membrane fuel cells (PEMFCs) using current-switching techniques. The proposed approach enables accurate extraction of key model parameters, including internal resistance, charge transfer resistance, and double-layer capacitance, based on transient voltage responses under controlled current perturbations.

Experimental validation was conducted using 100 W and 1 kW PEMFC systems under both current loading and current interruption tests. The results demonstrated good agreement between measured and simulated responses, with a maximum error below 10% and several cases achieving significantly lower error levels, confirming the effectiveness of the proposed approach.

From an application perspective, the proposed method is particularly well suited for real-time and control-oriented implementations, including integration with power electronic converters, hybrid energy storage systems, and DC microgrids. Its low computational requirement and reliance solely on measurable electrical signals make it highly suitable for deployment in digital twin platforms, condition monitoring systems, and advanced energy management strategies in hydrogen-based power systems. This positions the proposed approach as a practical tool for supporting emerging hydrogen energy infrastructures, where reliable and scalable modelling is essential for system integration and operational optimisation.

Practical implementation of the proposed modelling approach in large-scale PEMFC systems may also present several engineering challenges. In real-world applications such as DC microgrids or hybrid energy systems, PEMFC stacks are typically interfaced with power electronic converters that introduce additional dynamic interactions, including switching ripple, control delays, and measurement latency. These factors may influence the accuracy and stability of real-time parameter estimation if not properly considered. In addition, scalability to higher-power multi-stack systems may introduce non-uniform operating conditions, thermal gradients, and balance-of-plant interactions that can affect parameter consistency across the stack.

The proposed current-switching-based parameter estimation technique is fundamentally based on analysing the transient voltage response of a fuel cell to controlled current perturbations. In principle, this methodology can be extended to other fuel cell technologies such as solid oxide fuel cells (SOFCs) and molten carbonate fuel cells (MCFCs). However, key physical differences may limit direct applicability. High-temperature fuel cells exhibit significantly slower electrochemical and thermal dynamics, meaning that transient responses occur over longer time scales and are strongly coupled with thermal behaviour. In addition, their impedance characteristics often include inductive and diffusion-related effects that are not adequately captured by simple RC-based equivalent circuit models. Therefore, while the general concept remains valid, extending the method to these systems would require modified equivalent circuit structures, longer observation windows, and the inclusion of thermal and diffusion-related dynamics.

Despite these advantages, several limitations remain. The current validation is limited to low- and medium-power PEMFC systems (100 W and 1 kW), and further investigation is required to assess scalability to high-power and multi-stack configurations, where factors such as thermal gradients, stack non-uniformities, and balance-of-plant interactions may influence parameter estimation accuracy. Furthermore, the simplified equivalent circuit model does not explicitly represent concentration losses, which may affect accuracy under high-current operating conditions. Additionally, the impact of long-term degradation, parameter drift, and highly dynamic operating conditions on estimation accuracy has not yet been fully explored.

Future Work

Future research directions emerging from this work include:

• Extending the proposed methodology to high-power and multi-stack PEMFC systems to evaluate scalability and industrial applicability;
• Investigating degradation-aware and adaptive parameter estimation techniques for long-term operation;
• Integrating artificial intelligence and machine learning methods to enhance real-time parameter tracking and predictive control;
• Validating the approach under realistic renewable energy profiles and dynamic load conditions;
• Developing digital twin frameworks for PEMFC systems using the proposed modelling and estimation approach;
• Conducting techno-economic and life-cycle assessments to evaluate the benefits compared to conventional battery energy storage systems.