Red-Billed Blue Magpie Optimizer for Electrical Characterization of Fuel Cells with Prioritizing Estimated Parameters

Abstract

The red-billed blue magpie optimizer (RBMO) is employed in this research study to address parameter extraction in polymer exchange membrane fuel cells (PEMFCs), along with three recently implemented optimizers. The sum of squared deviations (SSD) between the simulated and measured stack voltages defines the fitness function of the optimization problem under investigation subject to a set of working constraints. Three distinct PEMFCs stacks models—the Ballard Mark, Temasek 1 kW, and Horizon H-12 units—are used to illustrate the applied RBMO’s feasibility in solving this challenge in comparison to other recent algorithms. The highest percentages of biased voltage per reading for the Ballard Mark V, Temasek 1 kW, and Horizon H-12 are, respectively, +0.65%, +0.20%, and −0.14%, which are negligible errors. The primary characteristics of PEMFC stacks under changing reactant pressures and cell temperatures are used to evaluate the precision of the cropped optimized parameters. In the final phase of this endeavor, the sensitivity of the cropped parameters to the PEMFCs model’s performance is investigated using two machine learning techniques, namely, artificial neural network and Gaussian process regression models. The simulation results demonstrate that the RBMO approach extracts the PEMFCs’ appropriate parameters with high precision.

1. Introduction

Among clean source alternatives, fuel cells (FCs) are thought to be a relatively new and quickly developing application of renewable energy-based conversion technology [1,2]. Owing to their interesting features, such as their small size, high power density, low operating pressures and temperatures, and the absence of dynamic elements, FCs have promising applications in smart grids, distributed generations, and transportation [3,4]. This electrochemical device, called FC for short, is a type of equipment that utilizes hydrogen gas as an input fuel to transform chemical energy into electric energy and heat, applying oxygen/air as an oxidant [5,6].

FCs are classified into various varieties, each with its own set of properties and applications. Here are a few common types: (i) proton exchange membrane FCs (PEMFCs) [7,8,9]; (ii) solid oxide FCs, which operate at high temperatures (600–900 °C) and are suited for stationary power generation and combined heat and power systems [10,11]; (iii) molten carbonate FCs, which run at high temperatures (600–700 °C), which are employed for large-scale stationary applications and can use natural gas as fuel [12]; (iv) phosphoric acid FCs, which operate at moderate temperatures (150–200 °C) and are frequently employed in stationary power applications, such as in commercial buildings [13]; (v) alkaline FCs, which function at relatively low temperatures (60–90 °C) and were formerly utilized in space applications and some ground vehicles [14]; (vi) direct methanol FCs, which are commonly used in portable applications and small electronics [15]; (vii) and reversible FCs, which can operate both as an FC and as an electrolyzer and are used for energy-storage applications [16].

One of the most common types of FCs is PEMFCs. Because the temperature and supply pressure can affect the output voltage, which can range from 0.9 to 1.23 V/cell, a series of PEMFCs is linked in order to increase the output voltage to the desired level [9,17,18,19]. A stack is a collection of PEMFCs connected in series/parallel for specific applications. Furthermore, because of polarization losses, their output voltage shows a non-linear relationship with the drawn load current. Stated otherwise, the activation voltage drop causes the PEMFCs’ output voltage to drop quickly at first; then, the ohmic voltage drop causes it to decrease linearly, and eventually, concentration losses cause it to decline exceedingly [20,21,22].

The model of PEMFCs has a set of unknown parameters in its mathematical form that are not specified in the fabricators’ datasheets. To accurately simulate the real behavior of PEMFCs, certain parameters must be optimally estimated. As a result, numerous attempts have been made to fully define the model’s ungiven parameters for the PEMFCs units. These efforts can essentially be divided into two categories: traditional and soft computing-based optimization frameworks [23,24]. Recently, machine learning has been used to achieve the same goal, as publicized in [25,26,27,28,29].

There are many optimization frameworks used to define the unknown parameters of PEMFCs stacks [4,30,31]. Among these recent optimizers are the artificial rabbits optimizer (ARO) [3], bonobo optimizer [32], chaotic Harris hawks optimizer (CHHO) [33], converged moth search algorithm [34], circle search algorithm (CSA) [35], transient search optimizer (TSO) [36], grey wolf optimizer [37,38,39], grasshopper optimizer (GHO) [40], whale optimization algorithm (WOA) [41], artificial bee colony differential evolution (DE) shuffled complex optimizer (ABDEO) [42], manta rays foraging optimizer (MRFO) [43], sine–cosine crow search algorithm [17], hybrid artificial bee colony DE optimizer [44], improved Archimedes optimizer [45,46], and Kepler optimizer [47]. In addition, recently, a number of cutting-edge, well-known optimization strategies have been made available to address the issue of extracting PEMFC’s parameters, including the atom search optimizer [48], equilibrium optimizer [49], gradient-based optimizer [50], shark smell optimizer [51], teamwork optimizer [52], modified farmland fertility optimizer (MFFO) [53], honey badger optimizer [54], two novel approaches reported in [55], human memory optimizer [56], and a reliable exponential distribution optimizer [57].

The “no free lunch” (NFL) theory [58] states that there is not a single algorithm that can handle every engineering optimization problem because every optimization technique has benefits and drawbacks for different tasks. There is no definitive solution yet, and choosing between the problems of optimization methods X and Y can be challenging depending on a number of factors, such as the degree of non-linearity, non-convexity, multi-modality, separability of the control variables, high dimensionality, etc. Until such a response is obtained in these endeavors, the attempts will persist. As shown above, there has been much success in determining these parameters; nonetheless, there is always room for improvement in order to more precisely address the ideal PEMFC stack model values.

The red-billed blue magpie optimizer (RBMO), developed in 2024 by Fu et al., can be used to solve problems involving continuous engineering optimization [59]. The mutually beneficial and effective predation habits of red-billed blue magpies were a model for the RBMO. The red-billed blue magpie’s hunting, locating, pursuing, attacking, and food-storage activities are all modeled mathematically in the RBMO’s procedures. It should be mentioned that the RBMO is an algorithm that uses swarm intelligence and is motivated by the red-billed blue magpie’s hunting strategy [59].

In this study, the RBMO is used to extract the parameters of PEMFCs in three test cases under various scenarios, along with implementing three other recent optimizers, namely, the dandelion optimization algorithm (DOA) [60], sinh-cosh optimizer (SCHO) [61], and growth optimizer (GO) [62]. The necessary verifications, including comparisons, are made using some specific measures. In addition to that, the principal performance of PEMFCs units under varied conditions is investigated and discussed.

The DOA is used to solve problems involving a continuous optimization. The DOA mimics the wind-powered long-distance flight of dandelion seeds, which can be divided into three stages: rising, descending, and landing. Seeds rise spirally, descend slowly, and land randomly, with Brownian motion and Levy random walk describing their trajectory [60]. The DOA has been recently applied successfully to solve electric power system problems [63,64,65,66,67]. The GO is a metaheuristic algorithm that solves both continuous and discrete global optimization problems. This technique mimics natural growth processes, relying on biologically inspired mechanisms to effectively explore and exploit the solution space [62]. The GO stands out as a powerful tool in the field of optimization, combining natural inspiration with advanced algorithmic techniques to solve complex problems effectively. GO has recently been used to address a few engineering issues, according to published reports [62,68,69]. The SCHO is a metaheuristic algorithm inspired by the mathematical properties of the hyperbolic sine and cosine functions [61,70]. It is designed to solve optimization problems by effectively balancing exploration and exploitation of the search space. The SCHO is a flexible and powerful tool that can be used to solve a wide range of optimization issues. It makes use of the mathematical characteristics of hyperbolic functions to effectively traverse challenging environments and is applied to solve a few engineering and medical problems [70,71,72,73,74].

Now, let us highlight the key contributions of this article: (i) assessing RBMO’s performance to optimally give the values of unknown parameters in PEMFCs’ model by implementing three recent optimizers, namely, DOA, GO, and SCHO; (ii) carefully examining three real-world study cases such as Ballard Mark V, Temasek 1 kW, and Horizon H-12 under various operating scenarios; and (iii) conducting numerous comparisons and uncertainty assessments for the obtained results.

2. Mathematical Formulation of PEMFCs’ Modeling

This section discusses the basic formulation of the PEMFCs’ model. The modeling approach for PEMFCs involves developing computer and mathematical models that represent the physical and electrochemical processes that take place inside the FCs. The mathematical model developed to simulate the performance of PEMFCs is referred to as Mann’s model [75]. This model is widely used in the PEMFCs’ research field. The PEMFCs stacks contain a variety of voltage drops, including concentration voltage (${\mathrm{V}}_{\mathrm{c}}$), ohmic or resistive voltage (${\mathrm{V}}_{\mathsf{\Omega }}$), and activation voltage (${\mathrm{V}}_{\mathrm{a}}$).

Three different regions are identified on the I-V polarization curve of a single PEMFC: concentration, ohmic, and activation losses. Activation losses, a reflection of the slow electrochemical reactions at first, are the cause of the fast output voltage decay of the PEMFC when it is first starting up under light load. The entire resistance that the protons and electrons encounter, which is represented by ohmic losses, causes the output voltage to subsequently decrease linearly. Because of the increasing water content at higher load conditions, the output voltage drops quickly and lowers the reactant concentration in both electrodes. These voltage drops, which contribute to the overall voltage loss in the FC, may have a significant impact on the system’s performance and efficacy. A thorough understanding of these voltage losses, as well as specific attempts to reduce them, are required to improve the efficacy of PEMFCs stacks. Scientists and engineers use a range of strategies to achieve this goal, including catalyst development, improved flow field designs, and improved reactant gas management. As a result, the PEMFC’s terminal voltage is given as follows (1):

$$V_{fc}=E_N-\left(V_{\mathsf{\Omega }}+V_a+V_c\right)$$

(1)

For working temperatures below 100 °C, ${\mathrm{E}}_{\mathrm{N}}$, which represents the reversible open-circuit voltage, may be computed using the formula revealed in (2).

$$E_N=1.2290-8.5\times 10^{-4}\left(T_c-298.15\right)+4.385\times 10^{-5}{T}_c\mathit{ln}\left(P_{H2}.\sqrt{P_{O2}}\right)$$

(2)

Equation (3) gives an estimate of the activation voltage loss (${\mathrm{V}}_{\mathrm{a}}$).

$$V_a=-[{\xi }_1+\left({T_c\xi }_2+{\xi }_3\mathit{ln}\left(C_{02}\right)+{\xi }_4\mathit{ln}\left(I_{fc}\right)\right)]$$

(3)

The concentration of oxygen (mol/cm³), which is defined as

$$C_{O_2}={\displaystyle \frac{P_{O_2}}{5.08\times {10}^6}}{e}^{{\displaystyle \frac{498}{T}}}$$

(4)

The equivalent resistance of the FC is used to calculate $V_{\mathsf{\Omega }}$, which has the following definition:

$$V_{\mathsf{\Omega }}=I_{fc}(R_m+R_c)$$

(5)

Equations (6) and (7) can be used for ${\mathrm{R}}_{\mathrm{m}}$ calculations.

$$R_m={\displaystyle \frac{{\rho }_m\mathcal{l}}{A_m}}$$

(6)

where

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

(7)

The formula shown in (8) can be used to predict ${\mathrm{V}}_{\mathrm{c}}$.

$$V_c=-\beta \mathit{ln}\left(1-{\displaystyle \frac{J}{J_{max}}}\right)$$

(8)

Typically, the PEMFC stack is made up of a number of series n-cells, and the voltage across the stack is expressed as follows, assuming all cells behaved with the same performance.

$$V_{stack}=n.V_{fc}$$

(9)

The aforementioned calculation is applied under the presumption that each cell behaves in the same way and that the resistors connecting the cells are ignored. In a closer look at the above mathematical PEMFCs’ model beginning from (1) to (9), it is clear from the above-mentioned formulas that for obtaining a fully defined electrochemical-based model, seven unknown parameters (${\xi }_1$, ${\xi }_2$, ${\xi }_3$, ${\xi }_4$, $\lambda$, $R_C$, and $\beta$) should be estimated. Model validation, parameter optimization, and refinement cycles are all parts of the iterative process of parameter estimation for Mann’s model. To obtain precise and trustworthy parameters values that accurately and reliably describe the behavior of the PEMFCs units under investigation, it is necessary to combine experimental data, computational modeling, and optimization techniques. Certainly parameter optimization approaches, whether applied in Mann’s model or any other mathematical model, pursue the recognized parameters values that minimize the discrepancy between model predictions and experimental results of voltage dataset points.

3. Problem Formulation and Constraints

The assessment criteria, in addition to the identification procedure, play an important role in precisely determining the best values of the unknown parameters and fitting the calculated I-V curve to the experimental one. A well-chosen fitness function (FF) streamlines the process of determining parameters and distinguishes between different model identification techniques in terms of the permissible ranges of results both quantitatively and qualitatively. The most common and widely used FF available in the literature are given in (10) for a fair comparisons with others. The sum of squared deviations (SSD), which is estimated using (11), is adapted to mathematically describe the FF used in this study [1,3,17,23,26,34,36,37,40,41,42,43,47,54].

$$FF=Minimize\left(\sum _{j=1}^{Nm}{\left(V_{m,j}-V_{e,j}\right)}^2\right)$$

(10)

The above-mentioned FF is subjected to a set of operating and practical min/max constraints; they are mentioned in (11).

$$\mathrm{s}.\mathrm{t}. \left\{\begin{array}{c}{{\xi }_i}^{min} \le {\xi }_i\le {{\xi }_i}^{max} , \forall i\in \left\{1,\mathrm{2,3},4\right\}\\ {\lambda }^{min} \le \lambda \le {\lambda }^{max}\\ {R_C}^{min} \le R_C\le {R_C}^{max}\\ {\beta }^{min} \le \beta \le {\beta }^{max}\end{array}\right.$$

(11)

As the reader can see, seven unknown parameters must be precisely defined in order to verify the simulation’s accuracy and provide a good representation of the behaviors of PEMFCs stack under study. It could be important to note that the RBMO is self-constrained all inequality constraints, confirming that there is no chance of breaks outside of the working min/max range, which means no need to penalize the FF.

4. Procedures of the RBMO

Swarm intelligence is used by the red-billed blue magpie (RBMO) algorithm to simulate its activities of hunting, pursuing, attacking, and storing food. Fruits, insects, and small vertebrates are the main food sources for this adaptable predator. When roaming in small groups, they locate food sources more easily and collaborate to become more productive. In addition, they store food for later use and obtain nutrition from the earth and trees. The phases of exploration and exploitation are included in the RBMO mathematical model. Like other meta-heuristic competing optimizers, the RBMO initializes a random candidate solution in the manner described below, as depicted in (12).

$$X_{i,j}=L_{B,j}+{rand}_1\left(\dots \right)\left(H_{B,j}-L_{B,j}\right), \forall j\in D \mathrm{and} \mathrm{j} \in Pop$$

(12)

Red-billed blue magpies forage for food in small groups (2–5 agents), as described in (13), or clusters (more than 10 agents), as revealed in (14), employing walking, leaping, and tree-searching techniques. Because of their adaptability, food is always available despite shifting environmental conditions and resource availability.

$$X_i\left(t+1\right)=X_i\left(t\right)+r_2\left({\displaystyle \frac{1}{p}}\sum _{m=1}^p{X}_m\left(t\right)-X_{rs}\left(t\right)\right)$$

(13)

$$X_i\left(t+1\right)=X_i\left(t\right)+r_3\left({\displaystyle \frac{1}{C}}\sum _{m=1}^C{X}_m\left(t\right)-X_{rs}\left(t\right)\right)$$

(14)

The red-billed blue magpie is an expert predator that uses a variety of hunting techniques, such as flying, leaping, and rapid pecking. Its main objectives are larger prey in clusters and small animals or plants in small groups, which is described mathematically in (15) and (16), respectively. Thus, it can secure food in a variety of situations thanks to its adaptable nature, as specified in (15) and (16).

$$X_i\left(t+1\right)=X_F\left(t\right)+w\left(t\right).r_3\left({\displaystyle \frac{1}{C}}\sum _{m=1}^C{X}_m\left(t\right)-X_{rs}\left(t\right)\right)$$

(15)

$$X_i\left(t+1\right)=X_F\left(t\right)+w\left(t\right).r_2\left({\displaystyle \frac{1}{p}}\sum _{m=1}^p{X}_m\left(t\right)-X_{rs}\left(t\right)\right)$$

(16)

The most crucial parameter in the RBMO, $\mathrm{w}\left(\mathrm{t}\right)$, is obtained and updated repeatedly, as explained in (17), and it is in charge of striking a balance between exploration and exploitation (this factor is modified from the original paper [59], which is $\left(1-{\displaystyle \frac{\mathrm{t}}{{\mathrm{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\right).\mathrm{e}\mathrm{x}\mathrm{p}\left[{\displaystyle \frac{2\mathrm{t}}{{\mathrm{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\right]$).

$$w\left(t\right)=exp\left[{-\left({\displaystyle \frac{bt}{T_{max}}}\right)}^b\right]$$

(17)

As illustrated in (18), red-billed blue magpies conceal food and save data for globally ideal values.

$$X_i\left(t+1\right)=\left\{\begin{array}{c}{X}_i\left(t\right) \forall { FF}_i^{old}>{ FF}_i^{new}\\ X_i\left(t+1\right) Otherwise\end{array}\right.$$

(18)

The time complexity in the worst case of BIG O(…) of the RBMO comprises the following: (i) the initialization complexity and (ii) the main loop complexity. The reader may refer to Figure 1 for further illustrations. The initialization complexity of the RBMO is $\mathrm{O}\left(\mathrm{P}\mathrm{o}\mathrm{p}.\mathrm{D}\right)$. The main loop complexity of the RBMO is $\mathrm{O}\left(2.\mathrm{P}\mathrm{o}\mathrm{p}.\mathrm{D}.{\mathrm{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)$ for exploration and exploitation stages. Thus, the overall time complexity, on the other hand, is equal to $\mathrm{O}\left(\mathrm{P}\mathrm{o}\mathrm{p}.\mathrm{D}+2.\mathrm{P}\mathrm{o}\mathrm{p}.\mathrm{D}.{\mathrm{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)\approx \mathrm{O}\left(\mathrm{P}\mathrm{o}\mathrm{p}.\mathrm{D}.{\mathrm{T}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)$.

Further details about the RBMO and its distinguishing characteristics compared to other meta-heuristics, along with its pseudo-code, are available in [59]. Figure 1 depicts the general procedures of the RBMO, showing the algorithm’s framework.

5. Numerical Simulations, Demonstrations, and Validations

This section evaluates the performance of the RBMO in conjunction with implementing three recent optimizers, named the DOA [60], the GO [62], and the SCHO [61], with other recently published results using three test cases under various scenarios. In these test scenarios, Ballard Mark V, Temasek 1 kW, and Horizon H-12 are the PEMFCs units that were used to attain the above-mentioned goal. Numerous autonomous attempts were conducted, and the most optimally cropped outcomes are documented in the upcoming subsection. The numerical simulations were performed using MATLAB Software version R2023b on a laptop running with Windows 10 Enterprise and outfitted with an Intel(R) Core(TM) i7-7700HQ CPU @ 2.80GHz, 2.80 GHz, and 16 GB of RAM. The implemented optimizers’ adjusted settings for every test case are 50 populations and a maximum of 100 iterations for the DOA, the SCHO, and the GO for fair comparisons. However, the adapted control settings of the RBMO are $b=2.2$,$T_{max}=100$, and $Pop=50$. It might be important to note that these values were arrived at through a process of trial-and-error procedures. Furthermore, it should be noted that the simulation time is not regarded as a crucial component in the suggested strategy because the design parameters are decided offline. Only the RBMO’s findings (data tip) are displayed in plots in order to provide a clear vision and because the other optimizers’ results are extremely comparable to them except for convergence patterns. The following sections of the study go into greater depth on the numerical results and comparisons among the optimizers that were used and implemented.

5.1. Test Cases and Their Associated Results

The technical specifications of several commercial PEMFCs stacks are revealed in

Table 1. Technical specifications of various commercial PEMFCs units.

PEMFCs’ Unit	Specs
	$\mathit{n}, \mathbf{c}\mathbf{e}\mathbf{l}\mathbf{l}\mathbf{s}$	${\mathit{A}}_{\mathit{m}}, {\mathbf{c}\mathbf{m}}^2$	$\mathcal{l}, \mathsf{\mu }\mathbf{m}$	${\mathit{J}}_{\mathit{m}\mathit{a}\mathit{x}}, \mathbf{m}\mathbf{A}/{\mathbf{c}\mathbf{m}}^2$	$\mathit{T}, \mathbf{K}\mathbf{e}\mathbf{l}\mathbf{v}\mathbf{i}\mathbf{n}$	${\mathit{P}}_{{\mathit{H}}_2}, \mathbf{a}\mathbf{t}\mathbf{m}$	${\mathit{P}}_{{\mathit{O}}_2}, \mathbf{a}\mathbf{t}\mathbf{m}$
Ballard Mark V	35	50.6	178	1500	343	1.0	1.0
Temasek 1 kW	20	150	51	1500	323	0.5	0.5
Horizon H-12	13	8.1	25	246.9	302.15	0.4935	1.0

for the test cases under study, and their principle data are obtained from [17,23,37,40,41,42,47,54].

Table 2. Limits of the PEMFC’s 7 unknown parameters.

Limits	${\mathsf{\xi }}_1$ ($\mathbf{V})$	${\mathsf{\xi }}_2 \times {10}^{-3} (\mathbf{V}/\mathbf{K})$	${\mathsf{\xi }}_3 \times {10}^{-5}$ (V/K)	${\mathsf{\xi }}_4 \times {10}^{-5}$ (V/K)	${\mathit{R}}_{\mathit{C}} \left(\mathbf{m}\mathsf{\Omega }\right)$	$\mathsf{\lambda }$	$\mathit{\beta }$ (V)
$L_B$	−1.1997	0.8000	3.6000	−26.0000	0.1000	13.0000	0.0136
$H_B$	−0.8532	6.0000	9.8000	−9.5400	0.8000	23.0000	0.5000

recapitulates the lower and higher bounds of the PEMFCs’ stack unknown parameters, which are frequently used in cutting-edge technology and in the state of the art [1,3,17,23,26,34,36,37,40,41,42,43,47,54] to ensure fair comparisons with other competitors.

The patterns of the RBMO, the DOA, the GO, and the SCHO’s convergence signatures for the test cases under study, which show the FF’s final cropped best values as the data tip by the RBMO only, are shown in Figure 2a–c. The reader may observe that the RBMO approach was able to obtain extremely competitive results with fine convergence and steady progress at iteration 50. When the parallel pool is enabled with four working processors, the CPU average processing times over 100 implementations of the RBMO approach for finalizing this optimization task are 2.1 s, 3.1 s, and 2.5 s for Ballard Mark V, Temasek 1 kW, and Horizon H-12, respectively.

5.1.1. Test Case 1: Ballard Mark V

The Ballard Mark-V 5 kW (its specs and datasheet are given in Table 1, the third row from the top of the table) PEMFC’s stack modeling is adapted to the RBMO-based technique plus DOA, GO, and SCHO;

Table 3. RBMO’s performance in comparison to other recent optimizers for Ballard Mark V.

	Parameter	${\mathit{\epsilon }}_1 \left(\mathbf{V}\right)$	${\mathit{\epsilon }}_2.{10}^{-3} (\mathbf{V}/\mathbf{K})$	${\mathit{\epsilon }}_3.{10}^{-5} (\mathbf{V}/\mathbf{K})$	${\mathit{\epsilon }}_4.{10}^{-5} (\mathbf{V}/\mathbf{K})$	$\mathit{\lambda }$	${\mathit{R}}_{\mathit{c}} \left(\mathbf{m}\mathsf{\Omega }\right)$	$\mathit{\beta } \left(\mathbf{V}\right)$	$\mathit{S}\mathit{S}\mathit{D} \left({\mathbf{V}}^2\right)$
Optimizer
CSA [35]		1.18130	3.569096	3.9929	−16.2830	23.00	0.1000	0.0136	0.853601
MRFO [41]		1.08980	3.8249	7.7306	−16.2830	23.00	0.1000	0.0136	0.853661
NNO [76]		0.97997	3.6940	9.0870	−16.2800	23.00	0.1000	0.0136	0.863697
WOA [41]		1.197800	4.4183	9.7214	−16.2730	23.00	0.1002	0.0136	0.853766
GhO [40]		0.85300	3.4170	9.8000	−15.9500	22.84	0.1000	0.0136	0.853661
ETSO [36]		−0.85340	2.5591	3.6100	−16.2870	23.00	0.1000	0.0136	0.853600
ARO [3]		−1.158859	3.5208	4.0526	−16.7251	23.99 *	0.1000	0.015884	0.813912
IAHA [4]		−1.0130	4.0000	8.9800	−16.3000	23.0000	0.1000	0.0136	0.853608
HBO [54]		−1.1997	4.33453	9.20688	−16.283	23.0000	0.1000	0.0136	0.853608
TSO [54]		−0.8552	2.72227	4.86143	−16.2831	23.0000	0.1000	0.0136	0.853608
ESMO [54]		−0.8532	2.54055	3.60422	−16.2824	23.0000	0.1000	0.0136	0.853608
GO		−1.19263	3.52557	3.61898	−16.00296	22.99967	0.12442	0.01368	0.865333
SCHO		−1.19392	3.98211	7.07842	−14.80570	22.42048	0.12154	0.01463	0.957596
DOA		−0.85745	2.93264	6.31831	−16.28304	23.00000	0.10000	0.01360	0.853608
RBMO		−1.000618	3.83643	9.79532	−16.28296	22.99999	0.10000	0.01360	0.853608

* Out of the range of $\lambda$ (13 to 23), as stated in Table 2; thus, it is unfeasible answer for SSD’s value.

provides the appraised model’s parameters and the solution’s fitness (the last row of the table). The outcomes of the competitors from the literature are placed alongside them. It was clear that the DOA method outperformed the closest best FF, with a value of 0.853608 ${\mathrm{V}}^2$. Figure 3a,b of the Ballard Mark-V shows the calculated model versus the actual readings of the V-I plots and internal voltage drops inside the stack, respectively. In this particular experiment, there are thirteen observations for voltage dataset points. It is evident that the computed voltage and the measured voltages fit each other accurately, as confirmed in Figure 3a.

With the purpose of investigating how temperature and pressure affect PEMFCs’ performance, the voltages are again calculated at 45, 60, and 75 degrees Celsius, with a constant pressure of ($P_{H2}$ = /$P_{O2}$= = 1.5/1 atm). Figure 3c shows the findings of the temperature's impact on the performance of the PEMFCs graphically. The PEMFC’s output voltage rises in conjunction with temperature. However, Figure 3d illustrates how altering partial pressures affects stack performance while maintaining a constant temperature of 343 Kelvin. The voltage rises in concert with the pressure.

5.1.2. Test Case 2: Temasek 1 kW

The Temasek 1 kW PEMFCs stack (its specs and datasheet are given in Table 1, the second row from the bottom of the table), modeling is adapted to the RBMO-based technique;

Table 4. RBMO’s performance in comparison to other recent optimizers for Temasek 1 kW.

	Parameter	${\mathit{\epsilon }}_1 \left(\mathbf{V}\right)$	${\mathit{\epsilon }}_2.{10}^{-3} (\mathbf{V}/\mathbf{K})$	${\mathit{\epsilon }}_3.{10}^{-5} (\mathbf{V}/\mathbf{K})$	${\mathit{\epsilon }}_4.{10}^{-5} (\mathbf{V}/\mathbf{K})$	$\mathit{\lambda }$	${\mathit{R}}_{\mathit{c}} \left(\mathbf{m}\mathsf{\Omega }\right)$	$\mathit{\beta } \left(\mathbf{V}\right)$	$\mathit{S}\mathit{S}\mathit{D} \left({\mathbf{V}}^2\right)$
Optimizer
SSO [51]		−1.0299	2.4105	4.00000	−9.5400	10.0005	0.10870	0.1274	1.6481
CHHO [33]		−1.0944	4.4282	8.76560	−21.4650	18.6392	0.18910	0.1016	0.80234
MFFO [53]		−0.9035	3.8267	8.47510	−22.9347	13.3251	0.10010	0.0705	0.791000
QOBO [77]		−1.1997	3.8220	3.60000	−22.9500	13.0000	0.10000	0.0680	0.783040
MAEA [78]		−0.8544	3.5766	7.88880	−22.9258	13.0017	0.10000	0.0683	0.79096
MAEO [78]		−1.11706	3.8290	4.56767	−2.26309	22.5232	0.10192	0.11019	0.79243
HHO [78]		−0.85320	3.49910	7.21118	−2.44049	22.9963	0.18265	0.07124	0.80553
GO		−0.91402	3.41643	9.73056	−9.54000	13.03231	0.10076	0.16117	0.597749
SCHO		−0.87344	2.80162	6.26615	−10.09827	18.00000	0.10000	0.18541	0.731239
DOA		−1.16873	3.61233	5.66745	−9.54000	22.29625	0.10000	0.19962	0.601922
RBMO		−0.91921	3.33183	9.04072	−9.54000	13.00000	0.10000	0.16105	0.597504

provides the estimated model parameters and the solution’s fitness (the last row of the table). The outcomes of the competitor techniques from the literature, plus the two other methods implemented, are positioned alongside them. It was clear that the RBMO-based framework outperformed the closest best FF with a value of 0.597504 ${\mathrm{V}}^2$, much less than the other competitors' results. Similar to the above-mentioned test case 1, Figure 4a,b show the calculated model versus the actual readings of the V-I plots and internal voltage drops inside the stack, respectively. In this particular experiment, there are twenty observations. It is evident that the computed voltage and the measured voltages fit each other accurately. With the purpose of investigating how temperature and pressure affect PEMFCs’ performance, the voltages are again calculated at 45, 60, and 75 degrees Celsius, with a constant pressure of (${\mathrm{P}}_{\mathrm{H}2}$ = /${\mathrm{P}}_{\mathrm{O}2}$ = 0.5/0.5 atm). Figure 4c shows the findings of the temperature impact on the performance of the PEMFCs graphically. The PEMFC’s output voltage rises in conjunction with temperature. However, Figure 4d illustrates how altering partial pressures affects stack performance while maintaining a constant temperature of 323 Kelvin. The voltage rises in concert with the pressure.

5.1.3. Test Case 3: Horizon H-12

In this scenario, 13 FCs connected in series are investigated. The H-12 PEMFC has a thickness of 25 m and a surface area of 8.1 cm². Its specifications and datasheet are given in Table 1, the last row of this table. This device has a nominal power of 12 W (7.8 V and 1.5 A), a hydrogen input pressure of 0.45–0.55 bar, and a maximum operating temperature of 55 °C Overall, 246.9 mA/cm² is the greatest current density. Figure 3c depicts the RBMO’s convergence pattern, which indicates the final cropped value of the SSD over 100 distinct runs.

Table 5. RBMO’s performance in comparison to other recent optimizers for H-12.

	Parameter	${\mathit{\epsilon }}_1 \left(\mathbf{V}\right)$	${\mathit{\epsilon }}_2.{10}^{-3} (\mathbf{V}/\mathbf{K})$	${\mathit{\epsilon }}_3.{10}^{-5} (\mathbf{V}/\mathbf{K})$	${\mathit{\epsilon }}_4.{10}^{-5} (\mathbf{V}/\mathbf{K})$	$\mathit{\lambda }$	${\mathit{R}}_{\mathit{c}} \left(\mathbf{m}\mathsf{\Omega }\right)$	$\mathit{\beta }\left(\mathbf{V}\right)$	$\mathit{S}\mathit{S}\mathit{D} \left({\mathbf{V}}^2\right)$
Optimizer
WOA [41]		−1.1870	2.66970	3.6000	−9.5400	13.8240	0.8000	0.1598	0.1160
MRFO [43]		−1.0630	2.36410	4.3272	−9.5400	19.8150	0.2853	0.1829	0.0966
ASSA [79]		−1.1300	2.44000	3.5700	−9.5400	18.7900	0.7140	18.1700	0.0970
PFA [23]		−1.1113	2.05730	3.6000	−9.5400	22.9999	0.1058	0.1868	0.0965
ETSO [36]		−1.032285	2.729677	7.7200	−9.5400	22.99898	0.112417	0.186888	0.09653
TSO [36]		−10.8532	1.571852	3.6100	−9.5400	13.02437	0.327874	0.175274	0.09685
HHO [36]		−0.861488	1.93210	6.0300	−9.5400	13.54238	0.1974	0.174067	0.09657
PSO [36]		−1.034754	2.54490	6.3200	−9.5400	23.0000	0.8000	0.182704	0.09658
ABDEO [42]		−0.85435	20.09613	6.76763	−9.54000	23.00000	0.10000	0.18685	0.096536
AEO [42]		−1.00278	27.44907	8.53668	−9.54000	18.13900	0.10000	0.18230	0.096536
SCE [42]		−0.85575	15.77536	3.60145	−9.54000	23.00000	0.10000	0.18685	0.096536
ELBO [42]		−0.97712	23.73787	6.46218	−9.54000	23.00000	0.10000	0.18685	0.096536
GO		−1.18668	3.38134	8.80335	−9.54000	22.57700	0.10186	0.03510	0.0616502
SCHO		−1.04862	2.50321	5.75161	−9.75943	14.80501	0.46664	0.03153	0.0672907
DOA		−0.86428	1.75798	4.76938	−9.54000	21.80941	0.13155	0.034797	0.601255
RBMO		−0.97738	1.98296	3.68744	−9.54000	22.99970	0.10000	0.03505	0.0616371

contains further information and comparisons between the RBMO, implementing algorithms, and other standing optimization techniques. In this situation, the RBMO-based method produced significantly superior results than the other analyzed algorithms. The RBMO method reduced the FF by roughly 37.5% less than the other analyzed techniques.

Meanwhile, Figure 5a depicts the comparison of the calculated and measured power. In this situation, there are fifteen experimental readings. These two figures ensure the accuracy of the theoretical representation and the usefulness of the RBMO approach in producing results that correspond to the actual operating performance of the PEMFC stack. The internal voltage drops (VDs) losses in the stack are depicted graphically in Figure 5b, comprising the total voltage drop (TVD). The losses are seen to grow as the current increases.

Several simulations are executed at different temperatures of 30 °C, 40 °C, and 55 °C, with pressure constants of ${\mathrm{P}}_{\mathrm{O}2}$ = 1 bar and ${\mathrm{P}}_{\mathrm{H}2}$ = 0.4935 bar. Figure 5c,d illustrate the values of the output voltage after altering the temperature. When the temperature rises, so do the output voltage and power. Furthermore, the simulations are performed at various hydrogen pressures (${\mathrm{P}}_{\mathrm{H}2}$ = 0.4 and 0.55 bar) while maintaining a constant temperature of 29 °C and an oxygen pressure of 1 bar. Figure 5c,d demonstrates the voltage and power changes as a function of pressure change. When the pressure increases, the voltage and power increase, although with a lower effect than when the temperature changes.

5.2. Validations by Some Measures

Table 6. Detailed dataset points of actual and estimated voltage points determined by the RBMO.

Ballard Mark V				Temasek 1 kW				Horizon H-12
${\mathit{I}}_{\mathit{m},\mathit{j}} \left(\mathbf{A}\right)$	${\mathit{V}}_{\mathit{m},\mathit{j}} \left(\mathbf{V}\right)$	${\mathit{V}}_{\mathit{e},\mathit{j}} \left(\mathbf{V}\right)$	${∆\mathit{V}}_{\mathit{B}\mathit{E}},\mathit{j}$	${\mathit{I}}_{\mathit{m},\mathit{j}} \left(\mathbf{A}\right)$	${\mathit{V}}_{\mathit{m},\mathit{j}} \left(\mathbf{V}\right)$	${\mathit{V}}_{\mathit{e},\mathit{j}} \left(\mathbf{V}\right)$	${∆\mathit{V}}_{\mathit{B}\mathit{E}},\mathit{j}$	${\mathit{I}}_{\mathit{m},\mathit{j}} \left(\mathbf{A}\right)$	${\mathit{V}}_{\mathit{m},\mathit{j}} \left(\mathbf{V}\right)$	${\mathit{V}}_{\mathit{e},\mathit{j}} \left(\mathbf{V}\right)$	${∆\mathit{V}}_{\mathit{B}\mathit{E}},\mathit{j}$
5.0600	33.2500	32.9392	0.3108	0.8322	18.2517	18.5784	−0.3267	0.1040	9.5800	9.7198	−0.1398
10.6260	30.8000	31.0698	−0.2698	3.7448	17.5222	17.5809	−0.0587	0.2000	9.4200	9.4477	−0.0277
16.1920	29.7500	29.8076	−0.0576	6.3800	17.1493	17.1880	−0.0387	0.3090	9.2500	9.2524	−0.0024
20.2400	28.7000	29.0379	−0.3379	9.5700	16.8575	16.8590	−0.0015	0.4030	9.2000	9.1235	0.0765
27.8300	28.0000	27.7503	0.2497	14.1470	16.4684	16.5030	−0.0345	0.5100	9.0900	8.9999	0.0901
34.4080	26.6000	26.7082	−0.1082	19.0014	16.1928	16.1966	−0.0038	0.6140	8.9500	8.8938	0.0562
37.4440	26.2500	26.2340	0.0160	24.2718	15.8848	15.9076	−0.0228	0.7030	8.8500	8.8097	0.0403
43.0100	25.2000	25.3594	−0.1594	28.9875	15.6903	15.6720	0.0183	0.8060	8.7400	8.7171	0.0229
48.0700	24.5000	24.5439	−0.0439	33.1484	15.4957	15.4757	0.0200	0.9080	8.6500	8.6281	0.0219
56.1660	23.8000	23.1583	0.6417	38.0028	15.3174	15.2563	0.0611	1.0760	8.4500	8.4823	−0.0323
61.2260	22.0500	22.2109	−0.1609	43.6893	15.1391	15.0080	0.1311	1.1270	8.4100	8.4373	−0.0273
67.2980	21.0000	20.9302	0.0698	48.8211	14.9445	14.7892	0.1554	1.2880	8.2000	8.2885	−0.0885
71.8520	19.6000	19.7503	−0.1503	53.6754	14.7824	14.5850	0.1974	1.3900	8.1200	8.1858	−0.0658
$Sum\left(\left|{∆V}_{BE}\right|\right)$, V			0.1982	58.5298	14.5879	14.3822	0.2057	1.4500	8.1100	8.1206	−0.0106
				63.3842	14.3771	14.1798	0.1973	1.5780	8.0500	7.9635	0.0865
				68.6547	14.1015	13.9596	0.1420	$Sum\left(\left|{∆V}_{BE}\right|\right)$, V			0.0526
				73.2316	13.8422	13.7671	0.0750
				78.0860	13.5341	13.5612	−0.0271
				81.6921	13.2261	13.4067	−0.1806
				$Sum\left(\left|{∆V}_{BE}\right|\right)$, V			0.1095

shows the actual and estimated dataset voltage points using the RBMO. In this table, the average absolute voltage errors between actual and model calculations are 0.19815, 0.10947, and 0.05259 V for the Ballard Mark V, Temasek 1 kW, and Horizon H-12 test cases, respectively, as indicated. It is also worth noting that the number of readings for the Ballard Mark V, Temasek 1 kW, and Horizon H-12 test cases are 13, 20, and 15, respectively. It is likely to notice that the reported numbers only have four digits, which can lead to a little difference between the precise value of SSDs, as generated by the MATLAB calculations, and subsequent manual assessments. The biased voltage error $\left({∆\mathrm{V}}_{\mathrm{B}\mathrm{E}},\mathrm{j}\right)$ and mean absolute error $\left(\mathrm{M}\mathrm{A}\mathrm{E}\right)$ are computed as follows:

$${∆V}_{BE},j=V_{m, j}-V_{e, j}, \forall j\in N_m$$

(19)

$$MAE={\displaystyle \frac{\sum _{\mathrm{j}=1}^{N_m}\left|{∆V}_{BE},j\right|}{N_m}}$$

(20)

Figure 6a–c depicts the percentage of biased voltage against the drawn current at measured records for the three studied test cases. Upon closely examining Figure 6a–c, it becomes evident that the highest percentages of biased voltage per reading for Ballard Mark V, Temasek 1 kW, and Horizon H-12 are equal to +0.65%, +0.20%, and −0.14%, respectively.

In addition to the above-mentioned results, the mean absolute percentage errors (MAPE) for voltage dataset are estimated using (21).

$$MAPE={\displaystyle \frac{\sum _{k=1}^{N_m}\left|\frac{V_{m,k}-V_{e,k}}{V_{m,k}}\right|}{N_m}}\times 100$$

(21)

For the purpose of quantification, the Ballard Mark V, Temasek 1 kW, and Horizon H-12 test cases’ respective MAPE values are 0.7598%, 0.8259%, and 0.6003%. It appears that the audiences are aware of how tiny and unimportant the reported particular values of MAEs and MAPEs are. The latter makes it abundantly evident that the calculated and real voltage datasets match well, demonstrating the applicability of the applied RBMO in producing the optimal values for the seven unknown parameters.

The Pearson correlation coefficient $\mathsf{\rho }\left({\mathrm{V}}_{\mathrm{m}},{\mathrm{V}}_{\mathrm{e}}\right)$, between measured and simulated datasets, is calculated using the formula given in (22). The values of $\mathsf{\rho }\left({\mathrm{V}}_{\mathrm{m}},{\mathrm{V}}_{\mathrm{e}}\right)$ for the Ballard Mark V, Temasek 1 kW, and Horizon H-12 test cases are 0.9978, 0.9930, and 0.9915, respectively. The aforementioned discussion demonstrates the RBMO-based optimization framework’s dominance after adaptation. This has been performed for each of the three test scenarios that have been studied.

$$\mathsf{\rho }\left(V_{m }, V_{e }\right)={\displaystyle \frac{1}{N_m-1}}.\sum _{i=1}^{N_m}\left[\left({\displaystyle \frac{V_{m,i}-{\overline{V}}_m}{{\sigma }_m}}\right).\left({\displaystyle \frac{V_{e,i}-{\overline{V}}_e}{{\sigma }_e}}\right)\right]$$

(22)

Furthermore, a p-value with a 5% significance level is generated for the model’s derived voltage points versus the observed voltage datasets, which have values of $6.75\times {10}^{-12},$ 0.00, and $2.80\times {10}^{-14}$, for the studied PEMFCs units. These negligible numbers, which are very close to zero, indicate a significant joining and a low likelihood of detecting the null hypothesis.

5.3. Sensitivity Analysis of Extracted Parameters

Sensitivity analysis (SA) quantifies the variations in results due to parameter variations [80]. It aims to rank, screen, and map regions producing extreme output values [81]. Global SA (GSA) evaluates model sensitivity for input/parameter variations. The GSA can be used before model calibration to identify significant parameters and extreme values. The GSA methods are based on probabilistic or random sampling that can be applied to any mathematical model with multiple inputs and outputs. The main concept is to optimize the information extracted from an optimal number of model runs by defining the evaluated locations in the parameter space. These methods are tailored to models with moderate to large inputs and are ideal for identifying maximum or minimum outputs. In this effort, two models, namely, $\mathrm{G}\mathrm{P}\mathrm{R}$ and $\mathrm{a}\mathrm{n}\mathrm{n}$ models, are implemented to perform the GSA. First-order (${\mathrm{S}}_{\mathrm{i}}$) and total order (${\mathrm{S}\mathrm{T}}_{\mathrm{i}}$) indices are collected for both $\mathrm{G}\mathrm{P}\mathrm{R}$ and $\mathrm{A}\mathrm{N}\mathrm{N}$ models as well. They are called $\mathrm{g}\mathrm{p}\mathrm{r}{\mathrm{S}}_{\mathrm{i}}$, $\mathrm{g}\mathrm{p}\mathrm{r}{\mathrm{S}\mathrm{T}}_{\mathrm{i}}$, $\mathrm{a}\mathrm{n}\mathrm{n}{\mathrm{S}}_{\mathrm{i}}$, and $\mathrm{a}\mathrm{n}\mathrm{n}{\mathrm{S}\mathrm{T}}_{\mathrm{i}}$. In most circumstances, knowing ${\mathrm{S}}_{\mathrm{i}}$ and ${\mathrm{S}\mathrm{T}}_{\mathrm{i}}$ is sufficient to ascertain how sensitive the examined function is to each of the individual input parameters (${\mathsf{\xi }}_1$, ${\mathsf{\xi }}_2$, ${\mathsf{\xi }}_3$, ${\mathsf{\xi }}_4$, $\mathsf{\lambda }$, ${\mathrm{R}}_{\mathrm{C}}$, and $\mathsf{\beta }$). The sensitivity range is ±10% of the original estimated best parameters values by the RBMO, and the number of samples is equal to 100 when using the Sobol method and Jansen estimator.

With a closer look at

Table 7. Sensitivity indices based on GPR and ANN.

PEMFCs Unit	Index	${\mathit{\epsilon }}_1 \left(\mathbf{V}\right)$	${\mathit{\epsilon }}_2.{10}^{-3} (\mathbf{V}/\mathbf{K})$	${\mathit{\epsilon }}_3.{10}^{-5} (\mathbf{V}/\mathbf{K})$	${\mathit{\epsilon }}_4.{10}^{-5} (\mathbf{V}/\mathbf{K})$	$\mathit{\lambda }$	${\mathit{R}}_{\mathit{c}} \left(\mathbf{m}\mathsf{\Omega }\right)$	$\mathit{\beta } \left(\mathbf{V}\right)$
Ballard Mark 5	$gpr{S}_i$	−0.02011	0.14612	−0.04515	−0.02258	−0.02196	−0.02004	−0.01996
	$gpr{ST}_i$	0.73694	0.89428	0.12701	0.05336	0.01640	0.00002	0.00023
	Rank	2	1	3	4	5	7	6
	$ann{S}_i$	0.05374	0.19968	0.07201	0.09620	0.09291	0.08687	0.08374
	$ann{ST}_i$	0.80768	0.91263	0.13991	0.05326	0.01662	0.00002	0.00028
	Rank	2	1	3	4	5	7	6
Temasek 1 kW	$gpr{S}_i$	−0.01186	0.12336	−0.04709	−0.02796	−0.02626	−0.02619	−0.02688
	$gpr{ST}_i$	0.72454	0.87072	0.20621	0.01315	0.00056	0.00002	0.00143
	Rank	2	1	3	4	6	7	5
	$ann{S}_i$	0.04619	0.20495	0.07647	0.08417	0.08768	0.08673	0.07861
	$ann{ST}_i$	0.79104	0.88583	0.22618	0.01296	0.00056	0.00002	0.00164
	Rank	2	1	3	4	6	7	5
Horizon H−12	$gpr{S}_i$	0.25405	0.03101	−0.04507	−0.03886	−0.03850	−0.03864	−0.03890
	$gpr{ST}_i$	0.97969	0.67889	0.05194	0.00034	0.00001	0.00000	0.00107
	Rank	1	2	3	5	6	7	4
	$ann{S}_i$	0.29141	0.07841	0.10773	0.10332	0.10458	0.10452	0.09923
	$ann{ST}_i$	1.00166	0.72235	0.05722	0.00032	0.00001	0.00000	0.00121
	Rank	1	2	3	5	6	7	4

, the ranking of the seven parameters based on their significant impact on the performance of the model, it can be concluded that, in accordance with $\mathrm{g}\mathrm{p}\mathrm{r}$ and $\mathrm{a}\mathrm{n}\mathrm{n}$ $\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{s}$, the model is highly sensitive to variations in ${\mathsf{\epsilon }}_2,$ ${\mathsf{\epsilon }}_1$, and ${\mathsf{\epsilon }}_3$; moderately sensitive to ${\mathsf{\epsilon }}_4$; and less sensitive to $\mathsf{\lambda }$, $\mathsf{\beta }$, and ${\mathrm{R}}_{\mathrm{c}}$ for Ballard Mark V and Temasek 1 kW PEMFC generating units. On the other hand, for Horizon H-12, almost the same conclusion can be summed up for ${\mathsf{\epsilon }}_1,$ ${\mathsf{\epsilon }}_2$, and ${\mathsf{\epsilon }}_3$, with moderate sensitivity to $\mathsf{\beta }$ and less sensitivity to $\mathsf{\lambda }$, ${\mathsf{\epsilon }}_4$, and ${\mathrm{R}}_{\mathrm{c}}$. Bear in mind the H-12 unit has limited power and a limited operating range. One more note is that both $\mathrm{g}\mathrm{p}\mathrm{r}$ and $\mathrm{a}\mathrm{n}\mathrm{n}$ sensitivity-based models produce the same conclusion.

6. Conclusions and Suggestions

This study employs the RBMO to extract the unknown parameters, i.e., (${\xi }_1$, ${\xi }_2$, ${\xi }_3$, ${\xi }_4$, $\lambda$, $R_C$, and $\beta$) of the PEMFCs stacks under varying operating conditions. Furthermore, the fitness function is derived from the sum of square deviations between the computed and actual PEMFC stack voltages. The RBMO’s results on Ballard Mark V, Temasek 1 kW, and Horizon H-12 are three popular PEMFC units that were compared to three implemented algorithms, namely, DOA, GO, and SCHO, plus other recently published competitors. Some specific measures, including biased voltage errors, mean absolute voltage errors, mean absolute percentage errors, Pearson correlation coefficients, and p-values, are used to assess the outcomes of the RBMO’s procedure. It is evident that the highest percentages of biased voltage per reading for Ballard Mark V, Temasek 1 kW, and Horizon H-12 are, respectively, +0.65%, +0.20%, and −0.14%, which are negligible errors. Based on the three test cases under investigation, the comparison results show that the used RBMO is the best approach among the competing methods. In the last stage of this work, sensitivity analysis based on Gaussian process regression and artificial neural network models is used to prioritize the ranks of cropped defined parameters. This indicates that the PEMFCs’ model is highly sensitive to variations of ${\mathsf{\epsilon }}_2,$ ${\mathsf{\epsilon }}_1$ and ${\mathsf{\epsilon }}_3$; moderately sensitive to ${\mathsf{\epsilon }}_4$; and less sensitive to $\mathsf{\lambda }$, $\mathsf{\beta }$, and ${\mathrm{R}}_{\mathrm{c}}$. Based on the encouraging outcomes of its use, the RBMO can be applied to other domains like the parameter estimation of solar cells, tuning load frequency controllers of microgrids comprising 100% renewables, wind speed forecasts, and other renewable energy applications.