Hybrid parameter estimation and sensitivity analysis of PEM fuel cells using Rüppell’s fox optimizer and Sobol metrics

Abstract

This study proposes an advanced approach for optimizing and estimating the unknown parameters of proton exchange membrane fuel cells (PEMFCs) using the recently developed Rüppell’s Fox Optimizer (RFO). The RFO is applied to minimize the sum of squared voltage errors between experimental and modeled data obtained under various operating conditions, subject to realistic system constraints. The algorithm’s performance is validated using three widely recognized PEMFC benchmark cases: the Ballard Mark V, Horizon H-12 Stack, and Temasek 1 kW unit. These cases encompass different operating conditions, including variations in cell temperature and the partial pressures of hydrogen and oxygen, ensuring a comprehensive evaluation of the method’s robustness and adaptability. Comparative analyses are conducted against recently published optimization techniques to assess the reliability, convergence behavior, and estimation accuracy of the RFO. Statistical performance indicators further substantiate the algorithm’s capability in achieving precise parameter identification. Results demonstrate that the RFO offers a promising and efficient alternative for nonlinear, multi-parameter PEMFC modeling, providing accurate representations of experimental data and stable convergence across diverse test scenarios.

Introduction

Motivation

The global transition from fossil fuels to renewable energy sources is driven by the depletion of conventional fuel reserves, rising energy costs, and the imperative need to address environmental concerns such as climate change and carbon emissions¹. Among renewable sources, solar and wind energies are indispensable; however, their intermittent nature necessitates reliable energy storage solutions. Hydrogen has emerged as a promising candidate offering high energy density, clean combustion, and adaptability across various applications². Serving in fuel cells (FCs), hydrogen offers a zero-emission energy technique, efficient energy conversion method, producing only water as a byproduct³. FCs are gaining prominence for their efficiency and operational versatility, with several advantages such as quick startup, recognition to load changes, and inherent safety due to low operating temperatures and pressures⁴. Major types based on the electrolyte used include alkaline FCs, proton exchange membrane (PEMFCs), phosphoric acid FCs, solid oxide (SOFCs), molten carbonate FCs, and direct methanol FCs)⁵. Among these, PEMFCs are the widely employed promising solution, especially in transportation, due to their compact size, high power density and scalability⁶. As global demand for clean and reliable energy continues to rise, PEMFCs play a crucial role in fulfilling sustainability goals while enhancing energy efficiency and performance⁷.

Literature survey

The complex and nonlinear characteristics of PEMFCs present notable challenges for accurate modeling and control^8,9,10. One of the major technical obstacles lies in their unregulated output voltage, which drops nonlinearly with increasing load due to a combination of three kinds of losses¹¹. Activation losses which cause an initial sharp voltage degradation, Ohmic losses which are accompanied by a steady decrease, and concentration losses which result in a prompt fall at higher current densities¹². Precise modeling of PEMFCs is essential, especially for simulation and control strategies to anticipate and optimize PEMFC’s performance under various operating circumstances^13,14. Numerous modeling approaches have been evolved, generally classified as analytical or electrochemical, and further splitted into white-box, black-box, and grey-box models¹⁵. In this regard, white-box models rely exclusively on physical principles but are often mathematically complicated and hard to solve¹⁶. On the other side, black-box models are driven from data-centric approaches proposing strong predictive capabilities within trained datasets yet lack generalization outside those parameters. Contrarily, grey-box models effectively emulate PEMFCs performance by combining physical laws with empirical data, especially for predicting polarization and voltage-current characteristics¹⁷.

PEMFCs models rely heavily on the accurate estimation of critical parameters such as temperature, membrane thickness, gas pressures, humidity levels, and other internal states¹⁸. Conventional deterministic methods often fail to provide reliable solutions due to the interdependency and nonlinear features of these parameters¹⁹. As a result, researchers have increasingly turned to metaheuristic optimization algorithms (MOAs), such as genetic algorithms, particle swarm optimization, and other nature-inspired techniques, which are well-suited for exploring high-dimensional, non-differentiable, and stochastic solution spaces²⁰. These algorithms provide efficient and adaptive tools for identifying unknown PEMFCs’ parameters, enhancing model accuracy and supporting better control, efficiency, and reliability in real-world applications. Additionally, MOAs are utilized in several power system optimization frameworks such as developed and demonstrated in²¹.

In this context, artificial rabbits optimizer is utilized in²² to define the seven unknown parameters of the PEMFCs using sum of square error (SSE) metric. Same as reported using Circulatory system-based optimization²³, Pied kingfisher optimizer²⁴, differential evolution (DE) ameliorated meta-heuristics algorithm²⁵, and horned lizard defense tactics optimizer²⁶. Moreover, a new variant of the artificial hummingbird algorithm is employed in²⁷ to accelerate the convergence rate and alleviate fitness function evaluations during the iteration process. Additionally, the convergence speed and accuracy of the traditional artificial bee colony algorithm are boosted as proposed in²⁸ by the cross operation of genetic algorithm and DE. Improved walrus optimization algorithm integrated with levy flight mechanism is proposed in²⁹ validated by experimental setup. On the other side, various operating conditions of PEMFCs are investigated in³⁰ solved using exponential distribution optimizer while war strategy optimizer is utilized in³¹. Similarly, DE ameliorated algorithm and feed forward neural network pelican optimizer are simulated for the precise modelling of PEMFCs in the steady state as proposed in^32,33 respectively. In this regard, a novel methodology is presented in³⁴ incorporated by event-triggered and dimension learning mechanisms to reinforce the dynamic searching capabilities. Alongside the previous attempts, several optimization approaches are developed in^35,36,37 to tackle the parameters identification dilemma enhancing the computational efficiency of the executed algorithms. Subsequently, Table 1 reveals a brief comparison among several attempts tackled in the literature for PEMFCs parameters estimation. Clearly, the widespread optimized objective function (OF) is the minimization of the SSE between measured and estimated voltages while other formulas have reasonable potential. Among these, roots mean square error (RMSE), relative error (RE), mean absolute error (MAE), and mean absolute percentage error (MAPE). Furthermore, a modern OF called Huber loss function is exploited in³⁸ and considered as a novel outstanding formulation combining mean squared error (MSE) and MAE functions.

Research gap, paper organization and contribution

Building upon the preceding survey, numerous optimization algorithms have been utilized to estimate the optimal values of the seven uncertain parameters in the Mann model of PEMFC stacks. While existing methods have shown varied degrees of efficacy, there is still room for further progress in parameter optimization to improve model accuracy and resilience. Each optimization methodology has unique advantages and drawbacks, and no single solution is universally effective for all engineering optimization issues. Furthermore, there is no definite criterion for determining if a specific algorithm is best suited for a specific optimization problem with characteristics such as nonlinearity, convexity, decision variable separability, or modality. Until such mappings are clearly defined, ongoing efforts to create and test new optimization approaches are both required and justifiable. Despite the several existing methodologies that have successfully estimated the unknown parameters of PEMFC models, there is still significant room for improvement in precisely describing the complicated behavior of PEMFC stacks. As the area advances, a growing number of optimization algorithms are being created to solve these issues. In this context, the authors want to propose an efficient and cost-effective algorithm with greater applicability, while noting that, according to the No Free Lunch (NFL) theorem, no single optimization method outperforms others across all problem domains³⁹.

Table 1 Compact evaluation of recent MOAs for estimating PEMFC parameters.

Motivated by this need, the present work explores the application of a recently developed heuristic algorithm -Rüppell’s fox optimizer (RFO)⁴⁷ - with the objective of producing results that are competitive with those reported in the literature. It draws its inspiration from Rüppell’s foxes’ daytime and nighttime hunting tactics, which mainly rely on their keen senses of smell, hearing, and vision. To simulate dynamic exploration and exploitation, these sensations and actions are abstracted mathematically. In order to search the issue space more efficiently, the RFO algorithm also simulates the foxes’ 260° vision and 150° hearing rotational capabilities. Furthermore, it simulates a strong sensory dominance by considering how sight, hearing, and smell vary with time and day/night, enabling the algorithm to modify its search approach.

In addition to the points mentioned above, it should be noted that, to the best of the authors’ knowledge and after an extensive literature review at the time of preparing this paper’s initial draft, this work represents the first attempt to employ the RFO algorithm for solving the PEMFC parameter identification problem, particularly in studying the dynamic model of PEMFCs. It can be concluded that each optimization approach possesses its own advantages and limitations, and no single algorithm can universally solve all engineering optimization problems. Moreover, there remains no definitive answer—or it is extremely difficult to establish—that a specific optimization algorithm A is inherently best suited for a given problem B with certain characteristics such as nonlinearity, convexity, variable separability, or modality. Until such a conclusion is reached, continued research efforts in this direction are warranted. Despite the existence of numerous strategies, as discussed earlier, that can estimate the uncertain parameters of PEMFC units with satisfactory accuracy, there remains room for improvement to achieve more precise modeling of PEMFC stacks.

It should be emphasized that the current research surpasses the straightforward application of a new optimizer such as RFO and incorporates multiple novel aspects such as.

• 1. First application of Rüppell’s Fox Optimizer (RFO) for PEMFC parameter estimation, offering adaptive exploration–exploitation balance and superior convergence performance.

• 2. Comprehensive validation on three distinct commercial PEMFC stacks, ensuring model robustness and generalizability.

• 3. Rigorous statistical benchmarking against 14 state-of-the-art metaheuristics, confirming competitive or superior accuracy with lower variance.

• 4. Integration of Sobol global sensitivity analysis to identify key parameters and enhance model interpretability.

• 5. Practical advancement toward simplified, efficient PEMFC models suitable for digital twin and control applications.

The remainder of this paper is organized as follows: Sect. "PEMFCs’ modelling and problem formulation" presents the problem formulation and the complete PEMFC stack model. Section "Procedures of the RFO" describes the RFO algorithm. Section "Numerical simulations, demonstrations and validations" discusses numerical results, experimental validation, and analytical evaluation. Section "Sobol indices for sensitivity analysis" provides Sobol-based sensitivity analysis. Finally, Sect. "Conclusions and future work" concludes the paper and outlines directions for future work.

PEMFCs’ modelling and problem formulation

This section describes the fundamental formulation of the PEMFC’s model, including the optimized OF and its corresponding constraints.

PEMFC stacks modelling

Modeling of PEMFCs involves developing mathematical and computational representations of the physical and electrochemical phenomena occurring within the cell^48,49,50. One widely used approach is the Mann model which is commonly applied in PEMFC performance simulations. Various voltage losses occur within the PEMFC, specifically activation losses $\:\left({V}_{act}\right)$ ohmic or resistive losses $\:\left({V}_{Ohm}\right)$ and concentration losses $\:\left({V}_{conc}\right)$⁵¹. Subsequently, the I-V polarization characteristics of a PEMFC may be divided into three distinct regions corresponding to these losses: activation, ohmic, and concentration.

Activation losses originate from the initially sluggish electrochemical reactions, causing a sharp voltage to drop during startup under light-load conditions. As the load increases, voltage declines more gradually and linearly due to internal resistance to proton and electron flow, known as ohmic losses. Under high-load conditions, the voltage decays steeply again due to excess water accumulation, which impairs reactant transport and leads to concentration losses at both electrodes^52,53. In this regard, Fig. 1 depicts the simple representation of PEMFC equivalent circuit with the typical I-V polarization curve. PEMFC voltage losses can significantly affect the performance and efficacy of the system, as they collectively lead to a reduction in the overall output voltage. To improve the efficiency of PEMFCs, it is crucial to thoroughly understand the nature of these losses and implement strategies to reduce them⁵⁴. To accomplish this, researchers and engineers employ various techniques such as developing advanced catalysts, optimizing flow field configurations, and enhancing the management of reactant gases⁵⁵.

Fig. 1

Electrical characteristics representation of PEMFC.

As shown in Fig. 1(a) and illustrated in (1), the output terminal voltage of the PEMFC $\:\left({V}_{FC}\right)$ can be computed^8,49,50.

$$\:{V}_{FC}={E}_{N}-{V}_{act}-{V}_{Ohm}-{V}_{conc}$$

(1)

where, $\:{E}_{N}$ defines the reversible open-circuit voltage that can be calculated from (2) when the operating temperature of the cell is below 100 °C^8,19,49.

$$\:{E}_{N}=1.229-8.5\times\:1{0}^{-4}\left(T-298.15\right)+4.385\times\:1{0}^{-5}T{ln}\left({P}_{H2}.\sqrt{{P}_{O2}}\right)$$

(2)

where, $\:{P}_{H2}$ and $\:{P}_{O2}$ symbolize the partial pressures of $\:{\text{H}}_{2}$ and $\:{\text{O}}_{2}$, respectively while $\:T$ denotes the cell temperature (K).

Equation (3) expresses the activation voltage loss ($\:{V}_{act}$) in a proton exchange membrane fuel cell (PEMFC). Activation losses occur due to the energy barrier that must be overcome for the electrochemical reactions (hydrogen oxidation and oxygen reduction) to proceed at the electrodes. This loss is most pronounced at low current densities, where reaction kinetics dominate the cell performance. As represented in (3), activation losses of voltage can be calculated by knowing some of the PEMFC unknown parameters^49,50. Equation (3) models how temperature, oxygen concentration, and current jointly influence the kinetic (activation) overpotential in a PEMFC, allowing accurate estimation of voltage losses associated with the electrode reaction mechanisms.

$$\:{V}_{act}=-[{\xi\:}_{1}+{\xi\:}_{2}\:.T+{\xi\:}_{3}\:.T.{ln}\left({C}_{O2}\right)+\:{\xi\:}_{4}\:.T.{ln}\left({I}_{FC}\right)]$$

(3)

where$\:,\:{I}_{FC}\:$stands for the PEMFC current and $\:{\xi\:}_{1}\:\text{t}\text{o}\:{\xi\:}_{4}$ refer to the coefficients. $\:{C}_{H2}$ and $\:{C}_{O2}$ denote the concentrations of hydrogen and oxygen (mol/cm³) respectively, that may be formulated as shown in (4).

$$\begin{aligned}\:{C}_{H2}=\frac{{P}_{H2}}{1.09\times\:{10}^{6}}.{e}^{\frac{-77}{T}}\\{C}_{O2}=\frac{{P}_{O2}}{5.08\times\:{10}^{6}}.{e}^{\frac{498}{T}}\end{aligned}$$

(4)

Additionally, the $\:{V}_{Ohm}$ can be evaluated from the PEMFC equivalent resistance $\:\left({R}_{Ohm}\right)$, as illustrated in (5).

$$\:{V}_{Ohm}={I}_{FC}({R}_{m}+{R}_{C})$$

(5)

where, $\:{R}_{m}$ and $\:{R}_{c}$ represent the membrane and contact resistances, respectively. Membrane resistance can be assessed using (6) as follows:

$$\:{R}_{m}=\frac{{\rho\:}_{m}\mathcal{l}}{{A}_{m}}$$

(6)

where, $\:{\rho\:}_{m}$ identifies the membrane resistivity (Ω.cm) which can be estimated from (7), $\:\mathcal{l}$ defines to the membrane thickness (cm), and $\:{A}_{m}$ indicates the cell active area (cm²).

$$\:{\rho\:}_{m}=\frac{181.6\left[1+0.03.J+0.062{\left(\frac{T}{303}\right)}^{2}{\left(J\right)}^{2.5}\right]}{\left[\lambda\:-0.634-3.J\right]{e}^{4.18.\left(\frac{T-303}{T}\right)}}$$

(7)

where, $\:J$ marks the current density (A/cm²), and $\:\lambda\:$ refers to the water content of the membrane.

The assessment formula for calculating the concentration losses $\:\left({V}_{conc}\right)$ is explained in (8).

$$\:{V}_{conc}=-\beta\:{ln}\left(1-\frac{J}{{J}_{max}}\right)$$

(8)

where, $\:\beta\:$ symbolizes a constant index, and $\:{J}_{max}$ indicates the maximum current density (A/cm²).

In standard practice, a PEMFC stack consists of $\:n$ cells connected in series, and the total stack voltage is expressed as shown in (9).

$$\:{V}_{stack}=n\:{V}_{FC}=n\:({E}_{N}-{V}_{act}-{V}_{Ohm}-{V}_{conc})$$

(9)

Equations (3)– (9) collectively describe the nonlinear behavior of PEMFC performance by integrating electrochemical kinetics, transport phenomena, and thermodynamic effects. This complete model serves as the foundation for parameter estimation and optimization using the proposed RFO framework.

Assuming uniform behavior across all cells and neglecting the interconnecting resistors, the previously described calculation is applied. A detailed examination of the mathematical PEMFC model specifically from Eq. (1) through (9) reveals that to construct a fully defined electrochemical model, seven unknown parameters must be estimated ($\:{\xi\:}_{1}$, $\:{\xi\:}_{2}$, $\:{\xi\:}_{3}$, $\:{\xi\:}_{4}$, $\:{R}_{C}$, $\:\lambda\:$, $\:\beta\:$)^{5,15,20,28,41,44,56,57}. The process of estimating these parameters in Mann’s model is inherently iterative, involving continuous cycles of model validation, parameter tuning, and refinement. Accurate and reliable parameter determination requires the integration of experimental measurements, numerical modeling, and optimization algorithms. Ultimately, the goal of parameter optimization whether in Mann’s model or any similar formulation is to determine the values that minimize the deviation between the model’s predicted outputs and actual experimental observations.

OF formulation

The evaluation criteria, along with the parameter identification method, are crucial for accurately estimating the unknown parameters and aligning the modeled I–V characteristics with the experimental data. Selecting an appropriate OF greatly facilitates the parameter estimation process and helps differentiate among various identification approaches based on the acceptability of results from both quantitative and qualitative perspectives. Among the numerous OFs reported in the literature, the most prevalent ones are outlined in (10). In the ongoing study, the SSE defined in (11) is employed to mathematically represent the selected OF^{11,12,24,29,38,58}.

$$\:OF=Minimize\left(SSE\right)$$

(10)

$$\:SSE=\sum\:_{i=1}^{Nn}{\left({V}_{m,i}-{V}_{e,i}\right)}^{2}$$

(11)

where, $\:{V}_{m,i}$ refers to the measured voltage of $\:{i}^{\text{t}\text{h}}$ reading, $\:{V}_{e,i}$ denotes the computed voltage of $\:{i}^{\text{t}\text{h}}$ reading, and $\:Nn$ indicates the number of readings captured from the experimental setup. The above-mentioned OF shall be constrained by a set of practical and operational minimum and maximum limits.

Procedures of the RFO

The RFO is a novel meta-heuristic algorithm to model the intelligent foraging behavior of Rüppell’s foxes⁴⁷. It is inspired by the day and night hunting strategies of Rüppell’s foxes, which rely heavily on their acute eyesight, hearing, and sense of smell. These senses and behaviors are mathematically abstracted to model dynamic exploration and exploitation. Also, the RFO algorithm models the rotational capabilities of the foxes’ eyes (260° vision) and ears (150° hearing) to search more effectively in the problem space. In addition, it models a significant sensory dominance considering sight, hearing, and smell changes over time and day/night, allowing the algorithm to adapt its search strategy. Moreover, a cognitive dynamic balance between exploration and exploitation is maintained based on simulated smell strength, growing as solutions approach optimal regions.

Step 1: initialization

Initially, the swarm of foxes are considered as randomly distributed initial solutions across the search space to promote diversity. Each fox (solution) is represented by a vector in a dimensional space.

$$\:{y}_{ij}\:=\:l{b}_{j}\:+\:{r}_{1}\:\times\:\:\left(u{b}_{j}\:-\:l{b}_{j}\right);\text{\:\:\:i}\in\:{N}_{Fox},j\in\:Dim$$

(12)

where, $\:{y}_{ij}$ represents the position of the i^th fox in dimension j; $\:l{b}_{j}$ and $\:u{b}_{j}$ refer to the lower and upper bounds for dimension j; r₁ is a uniform random variable; $\:{N}_{Fox}$ is the number of foxes in the population; $\:Dim$ is the number of decision variables (problem dimensions).

Step 2: searching for prey in daylight

During the day, foxes primarily rely on sight, supplemented by hearing and eye/ear rotation to scan and hunt. Therefore, the RFO algorithm executes exploration and local exploitation in order to update the foxes’ position in the next iteration time using sighting-based navigation, hearing-based navigation, eye rotation scanning and ear rotation scanning.

Movement toward prey via sighting and hearing-based navigation

First, the sighting and hearing-based navigation is modelled as follows:

$$\:{y}_{i}\left(t+1\right)={y}_{i}\left(t\right)+\left(\left(4{Z}_{A}+1\right)\times\:{Z}_{B}\right)\times\:\left({Y}_{Best}+{y}_{rd1}-2{y}_{i}\left(t\right)\right)$$

(13)

where, $\:{y}_{i}\left(t\right)$and $\:{y}_{i}\left(t+1\right)$ are the current and updated positions of fox i; $\:{Y}_{Best}$ is the best solution found; $\:{Z}_{A}$ and $\:{Z}_{B}$ indicate two random coefficients following uniform distribution.

In this model, $\:{y}_{rd1}$ is a daytime-based randomly selected peer which is used for sight-based or hearing-based search during the daylight. It is used to simulate visual or auditory contact with a random fox during the day.

$$\:{y}_{rd1}=lbes{t}_{i}{\left(t\right)}^{Q}$$

(14)

where, $\:{lbest}_{i}{\left(t\right)}^{Q}$ indicates the position of the Q^th fox in the local population at iteration t while it is computed as follows:

$$\:lbes{t}_{i}{\left(t\right)}^{Q}=Ceil\left({N}_{Fox}\times\:rand\left(1,{N}_{Fox}\right)\right)$$

(15)

where, Ceil refers to the ceiling function. This encourages convergence and group tracking, mimicking a fox detecting a lead hunter.

Rotational search via eyesight and hearing

Second, foxes can rotate their eyes (260°) and ears (150°) independently to scan their surroundings. Thus, the eye and ear rotation scanning is modelled as follows:

$$\:{y}_{i}\left(t+1\right)=yRo{t}_{i}+Flag\times\:\text{{\rm\:B}}\times\:randn\left(1,Dim\right)$$

(16)

where, $\:\text{{\rm\:B}}$ is a small step size (1e-10); Flag symbolizes the direction of scan; $\:randn\left(1,Dim\right)$ indicates a normally distributed random vector; and $\:yRo{t}_{i}$_i is the rotation point which can be defined as follows:

$$\:yRo{t}_{i}=R.\left(\overrightarrow{p}-Ro{t}_{F}\right)+Ro{t}_{F}$$

(17)

where, $\:\overrightarrow{p}$ is the position vectors of the current and the global best of Rüppell’s foxes while $\:Ro{t}_{F}$ identifies the rotation feature matrix describing the center of rotation. Both symbols can be defined as follows:

$$\:\overrightarrow{p}=\left[{y}_{i}\left(t\right);y\_loca{l}_{i}\left(t\right)\right]$$

(18)

$$\:Ro{t}_{F}={\left[\begin{array}{c}{C}_{x}\\\:{C}_{y}\end{array}\:\begin{array}{c}{C}_{x}\\\:{C}_{y}\end{array}\:\begin{array}{c}{C}_{x}\\\:{C}_{y}\end{array}\:\begin{array}{c}..\\\:..\end{array}\:\begin{array}{c}..\\\:..\end{array}\:\begin{array}{c}{C}_{x}\\\:{C}_{y}\end{array}\right]}_{2\times\:Dim}$$

(19)

$$\:{C}_{x}=x\left(Ceil\left(\frac{Dim}{2}\right)\right);{C}_{y}=y\left(Ceil\left(\frac{Dim}{2}\right)\right);\text{}A={\left[{y}_{i}\left(t\right)\right]}_{1\times\:Dim};B={\left[{Y}_{Best}\right]}_{1\times\:Dim};$$

(20)

where, $\:y\_loca{l}_{i}\left(t\right)$ is the present local best of the fox i while $\:R$ defines the rotation angle matrix that can be defined as follows:

$$\:R=\left[\begin{array}{c}{cos}\left({\theta\:}_{Eye}\right)\\\:{sin}\left({\theta\:}_{Eye}\right)\end{array}\text{}\begin{array}{c}-{sin}\left({\theta\:}_{Eye}\right)\\\:{cos}\left({\theta\:}_{Eye}\right)\end{array}\right]$$

(21)

where, $\:{\theta\:}_{Eye}$ indicates the angle of rotation that Rüppell’s foxes activate the vision-sense as follows:

$$\:{\theta\:}_{Eye}={r}_{2}\times\:\left(2\times\:26{0}^{\circ\:}\times\:\frac{\pi\:}{36{0}^{\circ\:}}\right)$$

(22)

where, $\:{r}_{2}$ represents a randomly generated value via uniform distribution.

This eye rotation scanning enhances diversification via biologically inspired omnidirectional perception.

Sensory power model

Rüppell’s foxes dynamically rely on sight during the day and hearing at night. Their dominance shifts over time as they adapt to light, distance, and environmental complexity. To switch between these strategies, the RFO models a transition between exploration and exploitation across iterations using logistic sensory functions considering the sight power (s) and the hearing power (h) which are defined as follows:

$$\:s={\left(1+{e}^{0.01\times\:\left(\frac{TMX}{2}-t\right)}\right)}^{-1}$$

(23)

$$\:h={\left(1+{e}^{0.01\times\:\left(t-\frac{TMX}{2}\right)}\right)}^{-1}$$

(24)

where $\:t$ is the current iteration; $\:TMX$ is the total number of iterations. These S-shaped curves help balance exploration against exploitation over time.

Step 3: searching for prey at night

At night, hearing dominates; foxes still use sight and rotation to detect prey with slight modification in (13) by replacing daytime-based peer ($\:{y}_{rd1}$) by another randomly selected nighttime-based one ($\:{y}_{rd2}$) as follows:

$$\:{y}_{i}\left(t+1\right)={y}_{i}\left(t\right)+\left(\left(4{Z}_{A}+1\right)\times\:{Z}_{B}\right)\times\:\left({Y}_{Best}+{y}_{rd2}-2{y}_{i}\left(t\right)\right)$$

(25)

This nighttime-based selected peer ($\:{y}_{rd2}$) simulates how a fox hears another fox’s movement and orients its direction toward it during night.

$$\:{y}_{rd2}=lbes{t}_{i}{\left(t\right)}^{P}$$

(26)

where, $\:{lbest}_{i}{\left(t\right)}^{P}$ indicates the position of the $\:{P}^{th}$ fox in the local population at iteration t while it is computed as follows:

$$\:lbes{t}_{i}{\left(t\right)}^{P}=Floor\left(\left[{N}_{Fox}\times\:rand\left(1,{N}_{Fox}\right)\right]+1\right)$$

(27)

where, $\:Floor$ refers to the floor function.

On the other hand, the rotational search via eyesight and hearing remains similar to Eq. (16).

Step 4: locating prey with the smell feature

Foxes use their smell to detect nearby or buried prey as this sense intensifies as they close in. This characteristic adaptively shifts from exploration to exploitation based on proximity. Therefore, the new positions can be upgraded as follows:

$$\:{y}_{i}\left(t+1\right)=\left\{\begin{array}{c}{y}_{rd1}+\mathcal{B}\times\:randn\left(1,Dim\right)\\\:{y}_{i}\left(t\right)+\rho\:\times\:\left(\left(\left({Y}_{Best}-{y}_{i}\left(t\right)\right)\times\:{r}_{3}\right)+\left({y}_{rd1}-{y}_{i}\left(t\right)\right)\right)\times\:{r}_{4}\end{array}\right.\:\begin{array}{c}if\:{r}_{5}<smell\\\:if\:{r}_{5}\ge\:smell\end{array}$$

(28)

where, $\:{r}_{3}$, $\:{r}_{4}$, and $\:{r}_{5}$ are random numbers regarding uniform distribution; $\:\mathcal{B}$ is a small local perturbation factor while $\:smell$ is a threshold value based on distance to prey; and $\:\rho\:$ stands for a cognitive coefficient as a smell-guided exploitation. Both are defined as follows:

$$\:smell=0.1\times\:a{cos}\left(2\times\:{\left(1+{e}^{0.01\times\:\left(\frac{TMX}{2}-t\right)}\right)}^{-1}\right)$$

(29)

$$\:\rho\:=2\times\:\left(\frac{\left|\left(2\times\:{r}_{6}\right)-{r}_{7}+{r}_{8}\right|}{{r}_{9}\times\:\left({e}_{o}+{r}_{10}\times\:\left({e}_{1}-{e}_{o}\right)\right)}\right)$$

(30)

where $\:{r}_{6}$ to $\:{r}_{10}$ are random numbers regarding uniform distribution; $\:{e}_{o}$ and $\:{e}_{1}$ indicate positive factors to be specified ($\:{e}_{0}$ = 1, and $\:{e}_{1}$ = 3).

Step 5: movement toward the best Fox

Foxes often imitate or follow the best-performing member of the group to maximize food success. This step encourages exploitation by aggregating around the best-known solutions as follows:

$$\:{y}_{i}\left(t+1\right)=\left\{\begin{array}{c}{y}_{i}\left(t\right)+2\times\:\left({y}_{rd1}-{y}_{i}\left(t\right)\right)\times\:{r}_{12}+2\times\:\left(\left({Y}_{Best}-{y}_{i}\left(t\right)\right)\times\:{r}_{11}\right)\\\:{y}_{i}\left(t\right)+3\times\:\left(lbes{t}_{i}{\left(t\right)}^{F}-lbes{t}_{i}\left(t\right)\right)\times\:{r}_{14}+2\times\:\left(\left({Y}_{Best}-{y}_{i}\left(t\right)\right)\times\:{r}_{15}\right)\end{array}\right.\:\begin{array}{c}if\:{r}_{13}\ge\:0.1\\\:if\:{r}_{13}<0.1\end{array}$$

(31)

$$\:lbes{t}_{i}{\left(t\right)}^{F}=Ceil\left({N}_{Fox}\times\:rand\left(1,{N}_{Fox}\right)\right)$$

(32)

where $\:{r}_{11}$ to $\:{r}_{15}$ are random numbers regarding uniform distribution.

Step 6: animal behavior in the worst case

In the wild, when a Rüppell’s fox fails to catch prey using its primary sensory strategies (sight, hearing, or smell), it doesn’t remain stationary or repeat failed tactics. Instead, it may abandon its current trail, move toward a better-performing peer, or search locally near that successful peer’s location. This mimics observational learning and opportunistic behavior, where the fox leverages the success of others to inform its next move in an uncertain environment:

$$\:{y}_{i}\left(t+1\right)={y}_{best-near}+\delta\:\times\:randn\left(1,Dim\right)$$

(33)

where, $\:{y}_{best-near}$ regards the best neighboring fox (prey location); and $\:\delta\:$ denotes a small perturbation for neighborhood search. These mimic local following and clustering around the elite fox to enhance convergence.

Step 7: termination

Foxes stop hunting when prey is caught, or conditions change. In RFO, the algorithm stops the search when the maximum iterations (TMX) are reached. Figure 2 displays the main steps of the RFO while Figs. 3 and 4 display its mechanism regarding daylight and night hunting strategies.

Fig. 2

Main steps of the RFO.

Time complexity of the RFO is estimated as follows: (1) Fitness evaluations: typically, every agent evaluates the fitness once per iteration ◊ $\:{N}_{Fox}\times\:\text{D}\text{i}\text{m}$. (2) Position updates: updating all agent positions typically costs $\:O(TMX\cdot\:{N}_{Fox}\cdot\:Dim)$. (3) Selection/sorting/elite update: $\:O(TMX.{N}_{Fox}.log({N}_{Fox}\left)\right)$; if only a best-so-far is tracked, this is $\:\text{O}\left({N}_{Fox}\right)$. (4) Other bookkeeping: additional $\:O(TMX.{N}_{Fox}.Dim)$or occasional expensive local search calls (if present) which add to complexity. Thus, the overall worst-O(…) is:

$$\:O\:\left(\dots\:\right)=O\left(TMX\cdot\:{N}_{Fox}\right)\text{\hspace{0.25em}}+\text{\hspace{0.05em}}O\left(TMX\cdot\:{N}_{Fox}\cdot\:Dim\right)\text{\hspace{0.05em}}+\text{\hspace{0.05em}}O\left(TMX.{N}_{Fox}.\:log\left({N}_{Fox}\right)\right)\cong\:O(TMX\cdot\:{N}_{Fox}\cdot\:Dim)\text{\hspace{0.25em}}$$

(34)

Hence the runtime is dominated by $\:O(TMX\cdot\:{N}_{Fox}\cdot\:Dim)$ as indicated in Eq. (34).

Fig. 3

RFO daylight hunting strategy.

Fig. 4

RFO night hunting strategy.

Numerical simulations, demonstrations and validations

This section focuses on assessing the performance of the RFO by estimating the uncertain parameters of three PEM generating units: the Ballard Mark V, Horizon H-12 Stack, and Temasek 1 kW unit. Table 2 reveals the technical specifications of various commercial PEMFC stacks used as test cases in this study^{11,13,15,29,31,43,59}.

Table 2 Technical specifications of various commercial PEMFCs units.

Table 3 summarizes the lower and upper bounds of the unknown parameters for the PEMFC stacks, which are commonly used in advanced technologies and state-of-the-art studies to enable fair comparisons with other approaches^{5,8,10,14,15,22,27,41,44,53,60}.

Table 3 Limits of the pemfc’s ungiven 7 parameters.

Multiple independent runs are conducted, and the most optimally cropped results are presented in the following subsection. Numerical simulations are carried out using MATLAB R2023b on a Windows 10 Enterprise laptop equipped with an Intel^® Core™ i7-7700HQ CPU @ 2.80 GHz and 16 GB of RAM. The RFO is configured with 50 individuals and a maximum of 100 iterations for each test case—values determined through trial and error. It is worth noting that simulation time is not considered critical in the proposed approach, as the design parameters are determined offline. The cropped best values of the 7 ungiven parameters as extracted by the RFO are specified in Table 4. It may be noted that all the reported values of PEMFC’s parameters are with the lower and upper limits mentioned in Table 3 which indicates that no violations of these self-constrained limits For clarity, the RFO’s convergence trends (Magnifier close to the end of the iteration between 90 and 100 iterations is shown with data tip of best SSE’s value) are shown in the plots depicted in Fig. 5(a)-(c) for Ballard Mark V, Horizon H-12 Stack, and Temasek 1 kW, respectively. In which, the best, average and worst convergences are plotted for easy reference. The reader may observe that the RFO accelerates steadily and stably to the final best SSE values in a smooth fashion, signaling the quality of the RFO’s procedures.

Table 4 RFO’s best estimating parameters of the three PEMFCs under study.

The performance of PEMFC stacks can be evaluated based on the first law of thermodynamics, with voltage efficiency $\:\left({{\upeta\:}}_{stack}\right)$ offering a more precise approximation. Assuming that the hydrogen flow rate is governed by the load demand, the fuel utilization factor ($\:{{\upmu\:}}_{\text{F}}$) can be considered constant, typically around 95% (i.e. 0,95). The maximum voltage ($\:{\text{v}}_{\text{m}\text{a}\text{x}}$) that a PEMFC can generate based on the higher heating value of hydrogen is generally 1.48 V per cell. Notably, the theoretical maximum voltage corresponding to the lower heating value is 1.23 V per cell. In (35), the value of $\:{\text{v}}_{\text{m}\text{a}\text{x}}$ is taken as 1.48 VPC (volts per cell).

$$\:{{\upeta\:}}_{stack}={{\upmu\:}}_{F}\times\:\frac{{V}_{stack}}{{\text{v}}_{\text{m}\text{a}\text{x}}\times\:n}$$

(35)

Subsequently, the efficiency of the examined PEM generating units is analyzed to illustrate how it varies under different conditions, including changes in partial pressures, operating temperatures, and loading scenarios.

Fig. 5

Trends of pattern convergence of RFO’s behavior.

The following subsections provide a detailed discussion of the numerical results and compare the performance of the RFO with other challenging published results by other optimizers per each test case. More specifically, the internal voltage drops inside the stack, I-V polarization curves and biased errors are reported and analyzed per each test case.

Test case # 1: Ballard mark V

It is important to note that although The Mark V is no longer manufactured commercially, its historical availability and free data make it a common research reference. A well-known stack design developed by Ballard Power Systems, the Ballard Mark V PEMFC stack is widely utilized in research and prototype for stationery and transportation applications. The Mark V still has an impact on fuel cell research even though it is a legacy design. The normal operating temperature range for the Ballard Mark V PEMFC stack is 60 to 80 °C. Although low-temperature PEMFCs often function in the 60–100 °C range, the Ballard Mark V is a particular model that is made for this smaller range for maximum endurance and performance. The rated voltage ranges from 25 to 30 V with operating temperature of 343.0 K. Further ratings and construction data for this stack are available in Table 2 (1st row). The parameters of the evaluated model and the fitness of the solution are shown in Table 4 (1st row of the Table). With a result of 0. 853,607,516 $\:{V}^{2}$, it is clear that the RFO approach performed better than the closest best/lowest value of the SSE. The Ballard Mark-V’s Fig. 6(a)–(c) compares the calculated model to the real I-V and I-P plot readings, and internal voltage drops within the stack, respectively. There are thirteen observations in this specific experiment. It is clear that there is an accurate fit between the calculated and measured voltages. By utilizing the best obtained parameters of PEMFC stack, the estimated I-V and I-P polarization curves versus the measured datasets could be visualized to represent the principal characteristics as indicated in Fig. 6(a)-(b), respectively. It is worth stating that comprehensive discussions and analyses are made for this test case which may be extended to the next other test cases in this section of the paper’s text. The reader may observe that the I-V curve starts with a steep drop in voltage at low current, indicating activation losses, mainly due to sluggish electrode kinetics. The central portion is more linear, representing ohmic losses, which are due to ionic resistance in the membrane and electronic resistance in the electrodes. Then, the tail-end flattens out or drops faster, showing mass transport losses caused by concentration gradients and limitations in reactant delivery (primarily hydrogen and oxygen). To summarize, these curves show the typical voltage and power response of the stack as current increases. In addition to that, power increases with current to a peak and then decreases (See Fig. 6(b)). One more point worth mentioning is that the peak in this curve corresponds to the maximum power point (MPP). After MPP, the PEMFC stack suffers from excessive voltage drops, meaning efficiency falls despite increased current. The internal voltage declines may be detailed as shown in Fig. 6(c). This curve may represent voltage degradation with increasing current density across internal cells.

Fig. 6

Principal curves at given conditions for Ballard Mark V.

Figure 7(a) shows how voltage performance changes with stack temperature. The reader may observe that as temperature increases, voltage at a given current generally improves due to enhanced electrode kinetics and membrane conductivity. On the other hand, risk of drying and thermal stress is raised if not controlled. Higher hydrogen/oxygen pressures boost performance of the PEMFC as depicted in Fig. 7(b). In which, elevated pressure improves voltage across all current densities improves voltage across all current densities. The latter is due to the increased reactant availability, reducing mass transport losses. Figure 7(c)-(d) shows how increasing partial pressures and temperature influence efficiency. It can be observed that moderate temperature rise improves efficiency due to reduced losses and higher-pressure results in higher efficiency, particularly at moderate loads at a level of 45%.

The voltages are once more computed at 45, 60, and 75 °C with a constant pressure of ($\:{P}_{H2}$/$\:{P}_{O2}$= 1.5/1 atm) in order to examine the effects of temperature and pressure on PEMFC performance. The results of the temperature effect on PEMFC performance are graphically displayed in Fig. 7(a)–(c) in terms of I–V and efficiency, respectively as generated by the RFO. Temperature causes the PEMFC’s output voltage to increase. But while keeping the temperature at 343 K, Fig. 7(b)–(d) shows how changing partial pressures impacts stack performance (I–V and efficiency as well). As the pressure increases, so does the voltage.

Fig. 7

Key performance of Ballard Mark V at varied conditions.

Test case # 2: horizon H-12 stack

The H-12 is a 12 W (approx. 7.8 V @ 1.5 A) air-cooled and air-breathing hydrogen FC stack, designed and built by Horizon Fuel Cell Technologies for compactness and suitable for educational and small-scale power experiments with start-up time ≤ 30 s. The operating temperature is 5–30 °C ambient, max stack 55 °C and efficiency ~ 40% at full load. Other specifications of this stack can be found early in this paper in Table 2. Meanwhile, Fig. 8(a)-(b) shows a comparison of computed and measured I-V and I-P characteristics. In this case, there are fifteen experimental dataset points. These two subfigures verify that the theoretical representation is accurate and that the RFO produces findings that correspond to the actual operating performance of the PEMFC stack. Figure 8(c) visually depicts the internal VDs in this PEMFC stack. The voltage losses are expected to increase as the current rises.

Fig. 8

Principal curves at given conditions for Horizon H-12 stack.

A number of simulations with pressure constants of $\:{P}_{O2}$=1 atm and $\:{P}_{H2}$=0.4935 atm are conducted at various temperatures: 30, 40, and 55 °C. The output voltage values and efficacy following temperature changes are shown in Fig. 9(a)-(c). The output voltage, power, and efficiency all increase as the temperature does. Additionally, while keeping the temperature at 29 °C (302.15 K) and the oxygen pressure at 1 atm constant, the simulations are run at different hydrogen ($\:{P}_{H2}$=0.4 and 0.55 atm). The voltage and power variations as a function of pressure change are shown in Fig. 9(b)-(d) as generated by the RFO. Although the effect is less pronounced than when the temperature changes, the voltage and power increase as the pressure increases.

Fig. 9

Key performance of Horizon H-12 stack at varied conditions.

Test case # 3: Temasek 1 kW

This 1 kW PEMFC stack frequently appears in academic and engineering publications for performance analysis and study. Complete data about the specifications of this stack is mentioned in Table 2. The computed model is contrasted with the actual I-V and I-P plot readings, and internal voltage drops within the stack in Fig. 10(a), (b), and (c). There are twenty dataset points in this specific experiment. It is clear that there is an accurate fit between the calculated and measured voltages. The voltages are once more computed at 45, 65, and 85 °C with a constant pressure of ($\:{P}_{H2}$/$\:{P}_{O2}$= 0.5/0.5 atm) in order to examine the effects of temperature and pressure on PEMFC performance. The results of the temperature effect on PEMFC performance are graphically displayed in Fig. 11(a)–(c) in terms of I-V characteristics and efficiency trends as generated by the RFO. Higher temperature drives the PEMFC’s output voltage to increase. On the other hand, Fig. 11(b)-(d) shows how stack performance is impacted by changing partial pressures while keeping the temperature at 323 K constant. As the pressure increases, so does the voltage.

Fig. 10

Principal curves at given conditions for Temasek 1 kW stack.

Fig. 11

Key performance of Temasek 1 kW stack at varied conditions.

Comparisons and validations

At this stage, validations and verifications of RFO’s results are anticipated to signify its performance. Statistical measures over the 50 independent runs of the RFO indicating the best, average, worst values of SSE along with standard deviation (StD) are revealed in Table 5. In addition to that, the average elapsed CPU time is mentioned in the last column for easy reference by other researchers.

Table 5 Statistical measures over the 50 independent runs.

The reader can see from a closer look at Table 5 that the RFO’s performance is resilient because the mean value of the SEE is rather near to the best values. Once more, very small or negligible StD values support the same conclusion. In addition to that, further comparisons with recent published results by other competing optimization frameworks are arranged in Table 6. In this Table, numerous final results of other challenging optimizers which recently reported in literature. The tabulated results indicate and signify the quality of the results produced by the RFO.

Table 6 Qualified analysis among RFO’s results and others.

To provide a concise and effective statistical comparison for non-parametric evaluation without overloading the reader with lengthy details, t-tests are conducted between the RFO results and those of competing optimizers based on the generated voltage values for the Ballard Mark V fuel cell model (representative cases and the same can be extended to the other two test cases straightforward). The results, summarized in Table 7, present the t-statistic, degrees of freedom (DF), p-value, and the 95% confidence interval (CI) for each pairwise comparison. As shown, all p-values are considerably greater than 0.05, indicating no statistically significant difference between the voltage outputs produced by RFO and the other optimization algorithms. Moreover, The Kruskal-Wallis test which an extension of the Wilcoxon rank sum test to more than two groups and a nonparametric variant of the traditional one-way ANOVA is used. When there are two or more groups in the data, the Kruskal-Wallis test is valid. In order to determine if the samples originate from the same population (or, conversely, from distinct populations with the same distribution), it compares the medians of the data groups. Last two columns summarize the outcomes of Mann-Whitney test. In which, fail to reject H0: no significant difference between RFO and any of the other competing optimizers.

Table 7 t-test results and Kruskal-Wallis ANOVA for Ballard mark V test case.

The statistical outcomes in Table 7 indicate that the voltage outputs obtained using the RFO algorithm are statistically indistinguishable from those produced by the other optimization methods. The extremely high p-values (all > 0.89) confirm that there is no significant difference between RFO and its counterparts at the 95% confidence level. This suggests that RFO achieves comparable voltage prediction accuracy to the other tested algorithms while maintaining stable and consistent performance. In practical terms, these results demonstrate that RFO can reliably reproduce voltage responses similar to established optimizers, reinforcing its validity and robustness for the Ballard Mark V fuel cell case. These findings confirm the robustness and reliability of the RFO method in reproducing consistent voltage responses for the Ballard Mark V fuel cell model. Furthermore, when combined with its superior convergence stability and solution consistency observed across test runs, RFO demonstrates strong potential as an efficient and dependable optimization framework for fuel cell performance modeling.

Hardware-in-the-loop (HIL) implementation perspective

Although this study focuses primarily on simulation-based validation of the RFO for PEMFC parameter identification, future work will expand this framework to include real-time verification⁶¹ using a hardware-in-the-loop (HIL) environment. The projected HIL configuration will connect an actual PEMFC stack to real-time control hardware and data collecting devices, resulting in a closed-loop interaction between the physical system and its digital twin model. This setup will allow for the online estimate of important parameters such as membrane resistance, activation losses, and concentration overpotentials while the load, temperature, and gas flow conditions alter dynamically. Implementing the RFO on an HIL platform also allows for evaluation of its resilience, convergence speed, and real-time adaptability in the presence of sensor noise and operational disturbances. Such integration is expected to enhance the practical relevance of the proposed optimization approach and strengthen its potential for deployment in advanced PEMFC monitoring, diagnostics, and control applications.

Sobol indices for sensitivity analysis

The Sobol method is a variance-based global sensitivity analysis technique^62,63. It quantifies how much each input parameter (alone or in combination with others) contributes to the output variance of a mathematical or computational model. This method works with linear or nonlinear, black-box models. Sobol indices can capture both main effects and interaction effects via first-order and higher-order effects⁶².

Procedures of Sobol analysis

Total output variance decomposition: The Sobol method decomposes the total variance $\:V\left(Y\right)$ of a model output $\:Y=f({X}_{1},\:{X}_{2},...,{X}_{k})$ into portions attributed to: (i) each individual input (first-order effects), (ii) interactions among inputs (higher-order effects) as follows:

$$\:V\left(Y\right)=\sum\:_{i}{V}_{i}+\sum\:_{i<j}{V}_{ij}+\sum\:_{i<j<k}{V}_{ijk}+\dots\:+{V}_{12\dots\:k}$$

(36)

First-order Sobol index $\:{S}_{i}$: Measures the main effect of input $\:{X}_{i}$​ on output - how much variance ($\:{V}_{i}$) in the output is caused by $\:{X}_{i}$ by itself, ignoring interactions as explained in (37).

$$\:{S}_{i}=\frac{{V}_{i}}{V\left(Y\right)}$$

(37)

Total-order Sobol index $\:{ST}_{i}$: Measures the total effect of $\:{X}_{i}$, including both its individual contribution and all interactions with other variables as demonstrated in (38).

$$\:{ST}_{i}=1-\frac{{V}_{\sim{i}}}{V\left(Y\right)}$$

(38)

where $\:{V}_{\sim{i}}$ is the variance of the output when $\:{X}_{i}$ is fixed.

It is worth mentioning that the ordinary range of $\:{S}_{i}$ and $\:{ST}_{i}$ varies from 0 to 1. The general procedures of Sobol analysis are depicted in Fig. 12.

Fig. 12

Procedures of Sobol analysis.

Interpreting the $\:{\varvec{S}}_{\varvec{i}}$ and$\:{\:\varvec{S}\varvec{T}}_{\varvec{i}}$ values

The interpretations of possible scenarios of cropping the $\:{S}_{i}$ and$\:{\:ST}_{i}$Values are demonstrated in Table 8.

Table 8 Interpreting the $\:{S}_{i}$ and$\:{\:ST}_{i}$ values.

Sobol indices for the studied test cases

In this study, Jansen estimator to compute $\:{S}_{i}$ and $\:{ST}_{i}$ is used⁶⁴. ±10% parameter perturbation based on RFO-optimal values is applied with 1000 samples to assess sensitivity over the models. In addition to that, Latin Hypercube Sampling (LHS) to approximate Sobol points is used which is commonly used in engineering, uncertainty quantification, and sensitivity⁶⁵. By utilizing the reported optimal values of the 7 parameters given in Table 4, the cropped values of $\:{S}_{i}$ and $\:{ST}_{i}$ are stipulated in Table 9.

Table 9 Sobol indices based on LSH with 1000 samples per each case.

It may be emphasized that a perturbation range of ± 10% is adopted to capture a realistic level of parameter uncertainty while ensuring the model operates within its valid range. This interval reflects typical variability arising from manufacturing tolerances, environmental conditions, and parameter estimation errors. Broader ranges (e.g., ± 20–30%) are evaluated but led to unrealistic parameter combinations that did not represent physically plausible scenarios. The 1000-sample LHS scheme was chosen based on convergence testing and evidence from comparable sensitivity analyses. LHS provides effective coverage of the multidimensional input space with fewer samples than random sampling. Preliminary tests with 250, 500, and 1500 samples showed that Sobol first- and total-order indices stabilized beyond roughly 800–1000 samples, indicating diminishing returns in accuracy. Accordingly, the selected ± 10% perturbation and 1000-sample LHS design offer an optimal balance between computational efficiency and result reliability.

A closer look at Table 9, the following observations can be concluded for each test case:

Ballard mark V

ε₁ has high total-order influence ($\:{ST}_{i}$=0.8612), but relatively low first-order ($\:{S}_{i}$=0.1835) which indicate strong interactions. ε₂ shows Similar pattern; not strong on its own ($\:{S}_{i}$=0.1084), but significant interactions. ε₃ to β have very low influence; can potentially be fixed or excluded to simplify the model. Thus, ε₁ and ε₂ are the only parameters worth optimal tuning and the others barely contribute.

Horizon H-12

ε₁ dominates both directly and totally ($\:{S}_{i}$ ≈ $\:{ST}_{i}$ ≈ 0.62), suggesting low interaction. ε₂ has Moderate influence. However, ε₃ to β, All very low - especially $\:{ST}_{i}$ for several parameters (i.e. entirely inactive in the model).

Temasek 1 kW

ε₁ and ε₂ again dominate; large $\:{S}_{i}$, and $\:{ST}_{i}$ gaps imply strong interactions. ε₃ to β have a near-zero S-values (likely estimation noise), and very low ST-values (i.e. negligible impacts).

To sum up, ε₁ and ε₂ dominate in most cases which make them as the most influential parameters. Their total-order indices are consistently > 0.7, indicating strong direct and interaction effects. ε₃ to β show weak influence, Most have low $\:{S}_{i}$ and $\:{ST}_{i}$ values (often < 0.1), suggesting negligible effect on output. Some even show small value of $\:{S}_{i}$ close to ZERO. One more point, large gap between low $\:{S}_{i}$ and $\:{ST}_{i}$ indicates nonlinear interactions or strong cross-dependencies, especially for ε₁ and ε₂. The latter mentioned are supported by the reported in the literature as in^3,12,59,60.

Conclusions and future work

In this study, the performance of proton exchange membrane fuel cells (PEMFCs) is evaluated under steady-state operating conditions with varying temperature and pressure parameters. A comprehensive sensitivity analysis using Sobol indices is performed to quantify the influence of the model’s uncertain parameters. Three benchmark PEMFC test cases are analyzed to assess the effectiveness of the proposed Rüppell’s Fox Optimizer (RFO). Multiple validation procedures, comparative analyses, and statistical performance indices are employed to verify the accuracy and robustness of the obtained results. Key characteristics of the I–V and I–P polarization curves are examined, highlighting internal voltage losses and their contributing mechanisms under different operating conditions. Furthermore, the Sobol sensitivity results (first-order $\:{S}_{i}$ and total-order $\:S{T}_{i}$) reveal that parameters $\:{\epsilon\:}_{1}$and $\:{\epsilon\:}_{2}$consistently exhibit the highest sensitivity, confirming their dominant influence on PEMFC model behavior. Their total-order indices are consistently > 0.7, indicating strong direct and interaction effects. ε₃ to β show weak influence, Most have low $\:{S}_{i}$ and $\:{ST}_{i}$ values (often < 0.1), suggesting negligible effect on output. Some even show small value of $\:{S}_{i}$ close to ZERO. One more point, large gap between low $\:{S}_{i}$ and $\:{ST}_{i}$ indicates nonlinear interactions or strong cross-dependencies, especially for ε₁ and ε₂. Once again, the study validated the RFO for PEMFC parameter identification using numerical intensive simulations. Future work will integrate the RFO with real-time verification in a hardware-in-the-loop environment.