Precise Modeling of Proton Exchange Membrane Fuel Cell Using the Modified Bald Eagle Optimization Algorithm

Abstract

The proton exchange membrane fuel cell (PEMFC) is a green energy converter that is based on the chemical reaction process. The behavior of this system can change with time due to aging and operating conditions. Knowing the current state of this system requires an accurate model, and an exact PEMFC model requires precise parameters. These parameters should be identified and used to properly fit the polarization curve in order to effectively replicate the PEMFC behavior. This work suggests a precise unknown PEMFC parameter extraction based on a new metaheuristic optimization algorithm called the modified bald eagle search algorithm (mBES). The mBES is an optimization algorithm based on the principles of bald eagle behavior that combines local search and global search to achieve a balance between the exploration and exploitation of search spaces. It is a powerful and efficient technique for optimization problems where accurate and near-optimal solutions are desired. To approve the accuracy of the proposed identification approach, the proposed algorithm is compared to the following metaheuristic algorithms: bald eagle search algorithm (BES), artificial ecosystem-based optimization (AEO), leader Harris Hawk’s optimization (LHHO), rain optimization algorithm (ROA), sine cosine algorithm (SCA), and salp swarm algorithm (SSA). This evaluation process is applied to two commercialized PEMFC stacks: BCS 500 W PEMFC and Avista SR-12 PEM. The extracted parameters’ accuracy is measured as the sum of square errors (SSE) between the results produced by the optimizer and the experimental data in the objective function. As a result, the proposed PEMFC optimizing model outperforms the comparison models in terms of system correctness and convergence. The proposed extraction strategy, mBES, obtained the best results, with a fitness value of 0.011364 for the 500 W BCS and 0.035099 for the Avista SR-12 500 W PEMFC.

1. Introduction

Energy is essential for human development and progress as it powers advancements in our economy, technology, and transportation. Energy sources significantly impact global relations and a country’s growth. Access to reliable and affordable energy is essential for economic and social progress, enabling countries to develop their infrastructure, create jobs, and improve quality of life. Renewable energy sources are growing significantly due to their ability to provide sustainable, clean, and affordable energy [1]. Renewable energy sources are essential for human development, helping promote economic and social progress, creating jobs, and improving the quality of life worldwide [2]. Integrating renewable energy sources is essential for promoting economic, social, and environmental advancement in countries worldwide. Renewable energy sources provide sustainable, clean, and affordable energy, which can help create jobs, improve quality of life, and contribute to overall development. In addition, investing in renewable energy sources provides security against the volatile prices associated with fossil fuels, allowing countries to achieve economic, social, and environmental progress sustainably and cost-effectively. Currently, renewables contribute to 26% of global electricity generation, and 16% of the end-user energy is generated from renewables [3].

Fuel cells (FC) are a promising renewable energy source that can provide a secure, sustainable, and cost-effective form of energy. Fuel cells are an efficient and clean energy source, and their use can help countries economically, socially, and environmentally progress [4]. Because of its benefits and wide range of applications, PEMFCs that employ polymers as electrolytes are becoming more prominent [5,6]. The pros of fuel cells include the fact that they are efficient and clean sources of energy, they help promote economic, social, and environmental progress, they are secure and cost-effective, and they can provide sustainable power. The cons include the high costs associated with fuel cells and the need for ongoing maintenance and technical expertise. Fuel cells have many applications, including portable electronics, stationary power generation, transportation, and auxiliary power units (APUs). Fuel cells also provide clean and efficient energy in residential and commercial buildings. Fuel cells come in several types, including PEM [7], phosphoric acid (PA) [8], molten carbonate (MC) [9,10], and solid oxide (SO) [11]. Each type has different characteristics, allowing for various applications and performance. The advantages of PEMFCs compared to other fuel cell types include their low operating temperature, fast start-up times, relatively low cost of components, and high efficiency. PEMFCs are also environmentally friendly, emitting very low levels of pollutants. Creating accurate PEMFC models is essential to analyze and optimize their performance. By better understanding the principles behind PEMFCs and their functionality, it is possible to develop more efficient, reliable, and environmentally friendly versions of the technology. Optimization algorithms are critical in determining the parameters of PEMFCs. By optimizing the parameters of PEMFCs, such as design, materials, operating temperature, and production rate, it is possible to improve the efficiency and cost-effectiveness of the technology [12]. This technology is getting more and more attention due to the considerable advantages. Various research on the development of this type of FCs has been published. The perovskite proton conductivity has been analyzed in [13]. The authors recommend the Y-doped BaZrO₃ perovskite material for electrochemical hydrogen FCs to help increase their efficiency. In addition, the progress in barium zirconate proton conductors has been reviewed in [14,15]. The authors explained the current investigations on this topic that help enhance the hydrogen devices.

The optimization of the design variables of PEMFCs can be achieved using both traditional and metaheuristic optimization techniques. Traditional techniques aim at finding a single optimal solution. In contrast, metaheuristic techniques are based on probabilistic methods and can quickly find a near-optimal solution. Conventional optimization techniques used for parameter estimation of PEMFC models have some drawbacks, including slow convergence time, lack of robustness to local minima, and difficulty handling complex nonlinear problems. In this context, many research papers study the PEMFC unknown parameters’ identification methods using metaheuristic optimization algorithms (MAs).

Table 1. Summary of PEMFC identification strategies.

Ref.	Author	Year	Used MOA	Used PEMFC Type
[16]	Sun et al.	2015	hybrid adaptive differential evolution (HADE)	R1, R2
[17]	Ali et al.	2017	grey wolf optimizer (GWO)	BCS 500 W-PEM SR-500 W 250 W-stack
[18]	Zhang and Wang	2018	co-evolution RNA with the genetic algorithm (coRNA-GA)	NA
[19]	Fathi et Hegazy	2018	multi-verse optimizer (MVO)	R1, R2
[20]	El-Fergany	2018	salp swarm algorithm (SSA)	NedStack PS6 BCS stack 500 W
[21]	Yang et al.	2020	improved barnacles mating optimizer (IBMO)	Horizon 500 W NedSstack PS6
[22]	Cao et al.	2020	improved whale optimization algorithm (IWOA)	NA
[23]	Salma et al.	2020	manta rays foraging optimizer (MRFO)	Ballard Mark V NedStack PS6 Horizon H-12
[24]	Qin et al.	2020	improved fluid search optimization (IFSO)	Horizon H-12 NedStack PS6 Ballard Mark V
[25]	Menesy et al.	2020	modified artificial ecosystem optimization (MAEO)	BCS 500 W SR-12 500 W 250 W-stack Temasek 1 kW
[26]	Diab et al.	2020	political optimizers (PO)	BCS 500 W SR-12PEM 500 W 250 W-stack
[27]	Hussein et al.	2021	modified artificial electric field algorithm (mAEFA)	NedStack PS6 SR-12 500 W
[28]	Syah et al.	2021	balanced water strider algorithm (bWSA)	NA
[29]	Fathy et al.	2021	LSHADE-EpSin optimization algorithm	250 W-stack NedStack PS6 BCS 500 W SR-12 500 W
[30]	Zhu et Yousefi	2021	adaptive sparrow search algorithm (ASSA)	Ballard Mark V Horizon H-12 NedStack PS6
[31]	Mossa et al.	2021	Harris Hawk’s and atom search optimization algorithms (HHO and SOA)	BCS 500-W SR-12 500 W 250 W-stack
[32]	Abaza et al.	2021	coyote optimization algorithm (COA)	250 W-stack Ned Stack PS6
[33]	Gouda et al.	2021	jellyfish search algorithm (JSA)	BCS 500 W 250 W-stack NedStack PS6
[34]	Hegazy et al.	2022	gradient-based optimizer (GBO)	250 W-stack BCS 500 W SR-12 500 W
[35]	Zhang et al.	2022	modified African vulture optimization algorithm (mAVOA)	SR-12 500 W BCS 500 W Temasek 1 kW
[36]	Chen et al.	2022	bi-subgroup algorithm (BSOA)	SR-12 500 W BCS 500 W Ballard Mark V
[37]	Han and Ghadimi	2022	improved honey badger algorithm (IHBA)	NA
[12]	Hegazy et al.	2022	bald eagle search algorithm (BES)	BCS 500 W NedStack PS6
[38]	Alsaidan et al.	2022	enhanced bald eagle search algorithm (eBES)	BCS 500 W 250 W-stack Horizon H12
[39]	Andrew et al.	2023	artificial rabbits optimization algorithm (ARO)	NedStack PS6 BCS stack 500 W Ballard Mark V
[40]	Hou et al.	2023	improved remora optimizer (IRO)	NedSstack PS6 Horizon 500 W
[41]	Sultan et al.	2023	standard and quasi oppositional bonobo optimizers	250 W-stack BCS 500 W SR-12 500 W Temasek 1 kW

summarizes some of these papers:

The no free lunch (NFL) theorem states that two optimization algorithms are equivalent when averaged over all possible problems [42]. This means no algorithm can outperform all others for all problems, making algorithm selection critical for optimization success. The modified bald eagle search (mBES) is an effective and efficient technique [43]. mBES combines local and global search features with adaptive control gains to find accurate and near-optimal solutions. Therefore, this paper deploys this recent metaheuristic optimization algorithm to extract the parameters of a PEMFC optimally.

The novelty of the work is to provide a high-accuracy identification strategy to extract the parameters of a complex mode of PEMFC. This will be done by deploying for the first time the mBES. The performance of the proposed strategy will be compared with other published works to approve its contribution in better extracting the PEMFC parameters.

The rest of this paper can be organized as follows: Section 2 presents a detailed model of the PEMFC. Section 3 explains the mBES and the difference between it and the other BES versions. The data acquisition using the physical platform is presented in Section 4. The results and their discussion are provided in Section 4. This paper ends with a conclusion in Section 5.

2. Model of PEMFC

Fuel cell modeling is a crucial stage that enables the investigation and improvement of its performance [34]. The PEMFC model is composed of nonlinear differential equations that describe interior chemical reactions. These equations contain several empirical parameters that must be precisely stated to increase the model’s precision. As a result, determining these characteristics is critical for developing an accurate model. The dynamic model of PEMFC voltage can be expressed as follows [22]:

$$v_{FC}=U_{nernest}-v_{act}-v_{con}-v_{ohm}$$

(1)

where U_nernest represents the FC thermodynamic voltage, v_act, v_con, and v_ohm represent the activation voltage losses, the concentration voltage losses, and the ohmic voltage losses, respectively. The FC thermodynamic voltage (U_nernest) can be calculated as follows:

$$\begin{array}{l}{U}_{nernest}=1.229-\left(t-298.15\right)\times 0.85\times {10}^{-3}\\ +t\times 4.3085\times {10}^{-3}\left(\ln \left(P_{H_2}\right)+0.5\ln \left(P_{O_2}\right)\right)\end{array}$$

(2)

where t is the FC temperature in Celsius, $P_{H_2}$ and $P_{O_2}$ represent the partial pressures.

The activation voltage losses (v_act) can be expressed as follows:

$$v_{act}=-\left({\epsilon }_1-{\epsilon }_2\times t-{\epsilon }_3\times t\times \ln \left(C_{O_2}\right)+{\epsilon }_4\times t\times \ln \left(i_{FC}\right)\right)$$

(3)

where ε₁, ε₂, ε₃, and ε₄ are the semi-empirical parameters, $C_{O_2}$ is the oxygen concentration at the cathode’s side (mol.cm⁻³).

The concentration voltage losses caused by variations in hydrogen and oxygen concentrations or fuel crossover can be estimated as follows:

$$v_{con}=-b\times \ln \left(1-\frac{J}{J_{max}}\right)$$

(4)

where b is an empirical coefficient, J and J_max represent the current and maximum current densities (Acm⁻²).

The ohmic voltage losses (V_ohm) can be expressed as

$$v_{ohm}=i_{FC}\left(r_m+r_c\right)$$

(5)

where r_c and r_m are the resistances of the connectors and membrane, respectively. The resistance of the membrane. R_m can be expressed as

$$r_m={\rho }_m\left(\frac{1}{A_m}\right)$$

(6)

where ρ_m is the membrane-specific resistivity (ohm.cm), and A_m and l are the membrane surface (cm²) and thickness (cm), respectively.

3. Modified Bald Eagle Optimization

The authors in [38] proposed an enhanced BES algorithm by adding a new phase based on the Levy function. The enhanced bald eagle search (EBES) is an optimization algorithm modeled after the hunting strategy of the bald eagle. The hunting process is divided into three stages: selecting a space, searching the space, and swooping to catch the fish. EBES can effectively find near-optimal solutions to complex nonlinear optimization problems by repeating these steps in succession. The proposed mBES in this work is based on changing the updating in each phase. The main differences can be illustrated as shown in Figure 1.

Each stage has been associated with a guiding factor that allows it to converge on the best solution. The transaction operator directs the space selection stage, enabling it to explore the search space better and escape from local solutions. Its effect decreases with increasing iterations. Similarly, the C1 gain guides the search stage in the first iterations to explore the search space better, and its effect will be increased with the increased number of iterations. The enhancement operator (EO) enhances the eagle’s movement during this phase by increasing the accuracy of this stage as a function of the iterations.

• Select stage: the new generated positions are generated as follows:

$$P_{t+1}=P_{best}+TF\times rand\times \left(P_{mean}-P_t\right)$$

(7)

where TF is an adaptive factor that enhances the exploitation and exploration phases. It can be expressed as

$$TF=k\left(1+\frac{t_{max}-t}{t_{max}+t}\right)$$

(8)

where k represents a control parameter [1.5, 2], t_max and $t$ are the numbers and the current iteration, respectively.

b.

Search stage: the newly generated positions in this phase are provided as follows:

$$P_{t+1}=P_t+y^i\times \left(P^i-P^{i+1}\right)+x^i\times \left(P^i-P_{mean}\right)$$

(9)

$$y^i=\frac{r^i\times \sin \left({\theta }^i\right)}{max\left(\left|r^i\times \sin \left({\theta }^i\right)\right|\right)};{\theta }^i=C_1\times \pi \times rand;r^i={\theta }^i\times R\times rand$$

(10)

$$x^i=\frac{r^i\times \cos \left({\theta }^i\right)}{max\left(\left|r^i\times \cos \left({\theta }^i\right)\right|\right)}$$

(11)

where R represents a constant [0.5, 2], and $C_1$ is a control parameter that can be obtained as follows:

$$C_1=k_2\left(1+\frac{t_{max}-t}{t_{max}+t}\right)$$

(12)

where k₂ represents a control parameter [5, 10].

c.

Search stage: the positions in this phase are updated as follows:

$$P_{t+1}=rand\times P_{best}+x{1}^i\times \left(P^i-c_1\times P_{mean}\right)+y{1}^i\times \left(P^i-EO\times P_{best}\right)$$

(13)

$$x{1}^i=\frac{r^i\times \cosh \left({\theta }^i\right)}{max\left(\left|r^i\times \cos \mathrm{h}\left({\theta }^i\right)\right|\right)};{\theta }^i=C_2\times \pi \times rand;r^i={\theta }^i$$

(14)

$$y{1}^i=\frac{r^i\times \sinh \left({\theta }^i\right)}{max\left(\left|r^i\times \sin \mathrm{h}\left({\theta }^i\right)\right|\right)}$$

(15)

where C₂ represents a control parameter [5, 10], and $EO$ is the enhancement operator that can be expressed as follows:

$$y{1}^i=2+\sin \left(2.5+\raisebox{1ex}{$t$}\!\left/ \!\raisebox{-1ex}{$t_{max}$}\right.\right)$$

(16)

4. Results and Discussions

Two types of PEMFC are considered: Avista SR-12 500 W and 500 W BCS. The specifications of PEMFCs are shown in

Table 2. The specifications of SR-12 PEM 500.

	Avista SR-12 500 W	500 W BCS
No of cells	48	32
Area	62.5 cm2	64 cm2
$l$	$25\mathsf{\mu }\mathrm{m}$	$178\mathsf{\mu }\mathrm{m}$
$P_{H_2}^{*}$	1.476 bar	1 bar
$P_{o_2}^{*}$	0.2095 bar	1 bar
Temperature	323 K	333 K
RHa (%)	100
RHa (%)	100

. The optimal parameters for both PEMFC have been defined by using five recent optimization methods, namely, bald eagle search algorithm (BES), equilibrium optimizer (EO), coot algorithm (COOT), ant lion optimizer (ALO), and heap-based optimizer (HBO). Avista SR-12 PEM fuel cell stack with a power rating of 500 W. The current deduced from the cell varied between 0–34 A, and the variation in terms of voltage was between 23–43 VDC. The fuel for the cell was passed through an in–house developed chamber to ensure that the hydrogen gas was humidified before entering the cell. This step was critical in ensuring the membrane was well humidified to allow an increase in protonic conductivity but a reduction in resistance on the electrolyte. Attached to the cell are a boost converter, a battery, and a load cell. The hydrogen gas entering the cell was varied in terms of pressure and flow rate to determine the effect of operating conditions on the overall cell performance. The cell comprised 48 cells with the active area denoted as 62.5 cm². The highest cell current was 42 A, with the stack temperature varied between 65–80 °C.

To guarantee equal comparison, the number of populations is fixed at 25, whereas the maximum number of iterations (n_max) is chosen as 250. During the optimization process, the seven unknown parameters of the PEMFC are selected as decision variables, whereas the objective function that needs to be minimum is the sum square error (SSE) between the estimated cell voltage and the measured cell voltage. The maximum and minimum limits of the unknown parameters are presented in

Table 3. Maximum and minimum limits of the parameters.

Parameter	${\mathit{\xi }}_1$	${\mathit{\xi }}_2$	${\mathit{\xi }}_3$	${\mathit{\xi }}_4$	$\mathit{\lambda }$	$\mathit{B}$	$\mathit{R}$
Max.	−1.19969	0.001	3.6 × 10−5	−2.6 × 10−4	10	0.0136	1 × 10−4
Min.	0.8532	0.005	9.8 × 10−5	−9.54 × 10−5	24	0.5	8 × 10−4

.

Table 4. Optimal values of the PEMFC model parameters using different optimization methods.

	AEO	BES	LHHO	ROA	SCA	SSA	mBES
500 W BCS PEM fuel cell
${\xi }_1$	−1.1628849	−0.8532	−1.03402	−0.88767	−1.19422	−1.19945	−0.8628
${\xi }_2$	0.0033088	0.002163	0.002901	0.002488	0.003269	0.003652	0.002193
${\xi }_3$	5.06 × 10−5	3.60 × 10−5	4.95 × 10−5	5.11 × 10−5	4.22 × 10−5	6.66 × 10−5	3.61 × 10−5
${\xi }_4$	−0.0001936	−0.00019	−0.00019	−0.00019	−0.00019	−0.00019	−0.00019
$\lambda$	22.032114	22.00802	17.94862	22.06159	18.12214	22.99359	22.00799
$B$	0.0001	0.0001	0.000103	0.000213	0.000117	0.000293	0.0001
$R$	0.0174447	0.017434	0.014742	0.016856	0.014458	0.017099	0.017434
Avista SR-12 500 W PEM fuel cell
${\xi }_1$	−1.0934124	−0.854333	−0.8532	−0.86028	−0.8532	−0.86422	−0.8532
${\xi }_2$	0.0033993	0.0023694	0.002892	0.002318	0.002234	0.002425	0.002278
${\xi }_3$	6.26 × 10−5	4.38 × 10−5	7.78 × 10−5	3.93 × 10−5	3.50 × 10−5	4.54 × 10−5	3.81 × 10−5
${\xi }_4$	−0.0001022	−0.000102	−0.0001	−0.0001	−0.00011	−0.0001	−0.0001
$\lambda$	23	23	19.72667	23	14.07028	22.80896	23
$B$	0.1470375	0.1470379	0.146961	0.146045	0.145591	0.147495	0.147078
$R$	0.0005703	0.0005703	0.000518	0.00064	0.000145	0.000537	0.000582

shows the optimal values of the PEMFC parameters after 30 runs. The obtained results are evaluated statistically, as demonstrated in

Table 5. Statistical assessments for considered optimization algorithms.

	AEO	BES	LHHO	ROA	SCA	SSA	mBES
500 W BCS PEMFC
Best	0.011364	0.011364	0.026542	0.012081	0.049496	0.012964	0.011364
worst	0.021357	0.027315	5.486003	0.338949	1.17559	0.520671	0.011375
Mean	0.011859	0.011961	1.23466	0.040992	0.415628	0.113427	0.011364
StD	0.001938	0.002862	1.69796	0.058075	0.275373	0.123744	1.99 × 10−6
Median	3.756 × 10−6	8.19 × 10−6	2.88307	0.003373	0.07583	0.015313	3.96 × 10−12
Variance	0.011377	0.011364	0.448936	0.024738	0.343643	0.066428	0.011364
Avista SR-12 500 W PEMFC
Best	0.035199	0.035199	0.035654	0.036048	0.2593	0.03539	0.035099
worst	0.03521	0.03584	6.375827	0.062539	3.2553	0.130003	0.035099
Mean	0.0352	0.035221	0.428004	0.0478	1.363295	0.049698	0.035099
StD	2.6759 × 10−6	0.000115	1.313477	0.006892	0.86374	0.020633	0.009039392
Median	8.2179 × 10−5	8.2173 × 10−5	3.035408738	0.000185808	0.864106877	0.000771983	8.17106 × 10−5
Variance	0.035199	0.035199	0.054157	0.047752	1.047143	0.041905	0.035099

.

Considering Table 5, for BCS PEMFC, the mean cost function values ranged from 1.23466 and 0.011364. The minimum mean cost function value of 0.011364 is obtained by mBES, followed by AEO (0.011859), whereas LHHO gets the largest mean cost function value of 1.23466. The standard deviation values ranged between 1.99 × 10⁻⁶ and 1.69796. The minimum standard deviation value of 1.99 × 10⁻⁶ is achieved by mBES, followed by AEO (0.001938), whereas LHHO obtains the largest standard deviation value of 1.69796.

In the case of Avista SR-12, the mean cost function values ranged from 0.035099 and 1.363295. The minimum mean cost function value of 0.035794 is obtained by mBES, followed by AEO (0.0352), whereas SCA gets the largest value of 1.363295. The standard deviation values ranged between 0.001557 and 0.045321. The standard deviation value using mBES is 0.009039392. It can be deduced that for both types of PEMFC under investigation, the mBES exhibited the best performance when compared to the original BES, AEO, LHHO, ROA, SCA, and SSA. The details of the objective function values for different algorithms throughout 30 runs are demonstrated in

Table 6. Absolute error in the cell voltage with different optimization algorithms: 500 W BCS PEMFC.

	Current	AEO	BES	LHHO	ROA	SCA	SSA	mBES
1	0.6	0.000449	0.01856	0.176832	0.29631	0.203719	0.153741	0.000684
2	2.1	0.005712	0.008385	0.000077	0.00394	0.207919	0.004179	0.006087
3	3.58	0.00085	0.003324	0.064753	0.100468	0.195486	0.060034	0.00045
4	5.08	0.001855	0.006595	0.091348	0.146669	0.187165	0.080579	0.001556
5	7.17	0.003301	0.009023	0.103009	0.169884	0.174051	0.086683	0.003193
6	9.55	0.013715	0.019009	0.105914	0.173786	0.148555	0.086146	0.013812
7	11.35	0.011475	0.015847	0.089904	0.153137	0.138088	0.069226	0.01169
8	12.54	0.009307	0.012895	0.075989	0.134267	0.131172	0.055449	0.009576
9	13.73	0.012773	0.015483	0.066092	0.118161	0.118012	0.04626	0.013075
10	15.73	0.09977	0.098667	0.071659	0.032446	0.21266	0.089031	0.099462
11	17.02	0.017474	0.017504	0.028032	0.057538	0.0826	0.01309	0.017752
12	19.11	0.014316	0.012639	0.004663	0.00704	0.062269	0.014248	0.014489
13	21.2	0.013877	0.010674	0.034407	0.04283	0.034634	0.036834	0.013886
14	23	0.007684	0.003482	0.063637	0.0911	0.011174	0.058405	0.00752
15	25.08	0.005033	0.000398	0.086036	0.137323	0.030954	0.070134	0.00468
16	27.17	0.001937	0.005267	0.093596	0.170614	0.092163	0.064829	0.002329
17	28.06	0.005083	0.006611	0.082232	0.170661	0.134076	0.04723	0.005334
18	29.26	0.000361	0.004968	0.011746	0.115701	0.24725	0.032702	0.000846
SSE		0.011388	0.011993	0.118099	0.338949	0.406351	0.084183	0.011364
RMSE		0.025153	0.025812	0.081	0.137224	0.15025	0.068387	0.025126
MAE		0.012498	0.014963	0.06944	0.117882	0.133997	0.059378	0.012579

.

Figure 2 shows the objective function values distribution for different algorithms throughout 30 runs 500 W BCS PEM fuel cell. The BES and the mBES from this figure have the lowest area, whereas the mBES from these figures have the lowest variations (most of the results are in the same bar). Figure 3 presents the same results for Avista SR-12 PEM fuel cell. The mBES also provided the best results.

The current density, voltage, and power characteristics of PEMFC using mBES curves are provided in Figure 4. There is an excellent agreement between the estimated data and measured data. This proves the superiority of the mBES in determining the unknown parameters of PEMFC. The absolute error in the cell voltage with different optimization algorithms is presented in Figure 5. For the first run and BCS PEMFC, as shown in Table 6 and Figure 2 and Figure 3, the SSE values ranged from 0.011364 and 0.406351. The minimum SSE value of 0.011364 is obtained using the mBES followed by AEO (0.011388), whereas SCA gets the worst SSE of 0.406351. Referring to Figure 5a, the maximum absolute error of 0.2963 is obtained by ROA. In the case of the first run and Avista SR-12, as demonstrated in Table 6 and

Table 7. Absolute error in the cell voltage with different optimization algorithms: SR-12 PEM fuel cell.

	Current Density	AEO	BES	LHHO	ROA	SCA	SSA	mBES
1	0.00615	0.04489	0.04487	0.14126	0.04761	0.11211	0.02351	0.00615
2	0.02665	0.00093	0.00094	0.04807	0.00633	0.11058	0.00841	0.02665
3	0.041	0.05729	0.0573	0.06861	0.04979	0.06272	0.03075	0.041
4	0.05371	0.09392	0.09393	0.08509	0.08679	0.02951	0.05911	0.05371
5	0.10086	0.04271	0.04271	0.08744	0.04642	0.1658	0.08516	0.10086
6	0.11398	0.02035	0.02035	0.06945	0.0228	0.1417	0.06168	0.11398
7	0.16031	0.00188	0.00187	0.05279	0.00445	0.11142	0.03018	0.16031
8	0.20787	0.00398	0.00398	0.04642	0.01216	0.10073	0.0138	0.20787
9	0.23411	0.03839	0.03839	0.08371	0.02711	0.13915	0.04732	0.23411
10	0.2829	0.0267	0.0267	0.05918	0.01	0.12311	0.01873	0.2829
11	0.30873	0.01714	0.01714	0.04159	0.00205	0.11351	0.0005	0.30873
12	0.32922	0.0474	0.04741	0.0651	0.02656	0.14523	0.02428	0.32922
13	0.36243	0.0034	0.00339	0.003	0.02612	0.10029	0.03588	0.36243
14	0.40344	0.02032	0.02033	0.01299	0.00265	0.13914	0.02065	0.40344
15	0.43623	0.00368	0.00367	0.02086	0.02435	0.13586	0.04768	0.43623
16	0.47108	0.04307	0.04306	0.0683	0.05758	0.13035	0.08473	0.47108
17	0.50511	0.0806	0.0806	0.10934	0.08345	0.14282	0.11166	0.50511
18	0.53832	0.02342	0.02341	0.04834	0.00632	0.2732	0.03167	0.53832
19	0.56498	0.01339	0.0134	0.02637	0.02965	0.36926	0.01091	0.56498
20	0.59122	0.06766	0.06766	0.08004	0.15094	0.57429	0.14489	0.59122
SSE		0.0352	0.0352	0.09477	0.05197	0.79429	0.06615	0.0351
RMSE		0.04195	0.04195	0.06884	0.05097	0.19928	0.05751	0.04195
MAE		0.03256	0.03256	0.0609	0.03616	0.16104	0.04458	0.03256

, the SSE values ranged from 0.0351 and 0.79429. The minimum SSE value of 0.0351 is obtained using the mBES followed by AEO (0.0352), whereas the worst SSE of 0.79429 is obtained by SCA. mBES obtains the best MAE of 0.0325, and SCA obtains the worst MAE of 0.161. Referring to Figure 5b, the maximum absolute error of 0.574 is obtained by SCA.

Figure 6 and Figure 7 demonstrate the objective function variation with BCS PEMFC and SR-12 PEEMFC, respectively. As shown in Figure 6a, for BCS PEMFC and 1st run, the SSE values converge to 0.01139, 0.01199, 0.1181, 0.33895, 0.40635, 0.08418, and 0.01136, respectively, for AEO, BES, LHHO, ROA, SCA, SSA, and mBES. mBES catches the optimal solution rapidly, whereas SCA and SSA need more time to reach its best solution. During run no 15, as demonstrated in Figure 6b, the SSE values converge to 0.01139, 0.01136, 1.88428, 0.03657, 0.3219, 0.03827, and 0.01136, respectively, for AEO, BES, LHHO, ROA, SCA, SSA, and mBES. mBES catches the optimal solution rapidly, whereas, for the second time, SCA and SSA need more time to reach its best solution.

As shown in Figure 7, the mBES with its fast convergence taken from the original BES and enhanced exploration and exploitation abilities obtained the best results.

ANOVA and Tukey tests are conducted to confirm the superiority of BES compared with other algorithms. An ANOVA test is a statistical method used to compare two or more groups. This test measures the mean differences between groups, assessing how likely each observed difference has arisen by random chance. ANOVA helps answer the question of whether the difference between the means of two or more samples is statistically significant. It works by comparing the variability between groups and the variability within groups to determine whether the differences between group means could have arisen by chance.

The results provided in

Table 8. ANOVA results.

	Source	df	SS	MS	F	Prob
SR	Columns	6	131.244	21.874	26.07	6.768 × 10−23
BCS		6	54.63	9.105	13.63	5.503 × 10−13
SR	Error	203	170.329	0.839
BCS		203	135.61	0.668
SR	Total	209	301.572
BCS		209	190.24

approve the difference in the results between the algorithms for both types of SR and BCS. The ANOVA ranking provided in Figure 8 confirms that the best performance is provided by the mBES, which provided the smallest mean fitness and lowest variations.

The results of ANOVA are approved by the Tukey test, as illustrated in Figure 9. The mean of the mBES group is significantly better than the BES group and relatively better than the AEO group for the SR type. Regarding the BCS type, the mBES mean group is lower than the other groups, confirming its ability to provide better performance.

5. Conclusions

The optimal parameter identification process of the PEMFC model has been investigated in this research work using different recent optimization algorithms. This paper suggested using the modified bald eagle search algorithm (mBES) to optimally extract the unknown parameters of two PEMFC types: BCS 500 W and SR 500 W. Six optimization algorithms to compare them with the proposed mBES, namely, bald eagle search algorithm (BES), artificial ecosystem-based optimization (AEO), leader Harris Hawk’s optimization (LHHO), rain optimization algorithm (ROA), sine cosine algorithm (SCA) and salp swarm slgorithm (SSA) have been considered. For BCS PEMFC, as an example, the mean cost function values ranged from 1.23466 and 0.011364. The minimum mean cost function value of 0.011364 is obtained by mBES, followed by AEO (0.011859), whereas LHHO gets the largest mean cost function value of 1.23466. The standard deviation values ranged between 1.99 × 10⁻⁶ and 1.69796. The minimum standard deviation value of 1.99 × 10⁻⁶ is achieved by mBES, followed by AEO (0.001938), whereas LHHO obtains the largest standard deviation value of 1.69796. This proves the superiority of mBES in extracting the parameters of BCS PEMFC. Based on the main contributions list, this paper deployed for the first time the mBES and provided a high-accuracy identification strategy to extract the parameters of a complex mode of PEMFC compared to other algorithms, such as the original BES, ROA, AEO, and other recent algorithms. Most of the used PEMFCs are of the more than 0.5 kW type, so investigating the proposed algorithm for PEMFCs that have bigger capacity is required. However, the data for these types are not available.