The technology of the fuel cell (FC) is an essential energy exporter due to its extraordinary production and reduced carbon effects. Also, in contrast to wind and photovoltaic power origins, the production of power from the FC is autonomous of the climatologic conditions. Thus, it can be used for perpetual power generation. Different models of the FC are revealed; their system is carried out based on the characteristics of the electrolyte applied. Among the numerous types of FCs are the chemical-based FC approach (CFC) (McLean et al., 2002), PEMFC (Eisman, 1989), solid-based oxide FC (SOFC) (Kawada et al., 1990), etc. One of the most well-known types of FCs is the PEMFC. Their active start is recognized because of their economic temperature and yield ranging between 30 and 60%. The PEMFCs are utilized in different disciplines (Messaoud et al., 2021). The escalating cost of human energy has a hazardous impression on the atmosphere. Electricity requirements contribute primarily to ecological degeneration while consuming nonrenewable supplies (Nain et al., 2021). The portion of renewable electricity from the universal energy production is 26% approximately. Various varieties of power references for providing hydrogen exist. Now, most of the hydrogen composition is generated by solar power (37%) and comes behind conventional fossil fuel (26%), as shown in Figure 1 (Fathy et al., 2020b). These H2 results guide to reasonable prices and more critical appropriate solutions via hydrogen in the real world (Kayfeci et al., 2019).

FC-based energy production methods can meet the anticipations of very low emanations and comparatively high conductivity (Mo et al., 2006). FC is characterized by more economical contamination and more extraordinary performance than traditional power origins; however, they have an excellent dynamic reaction, sound balance, and moderate noise (Ramos-Paja et al., 2010). Between several systems of FCs, because of their approximately low running temperature, quick reaction, small volume, high popular mass, no loss, and in case of explicit hydrogen zero emission is generated, PEMFC can be an excellent option for energy-producing origins in the future, particularly in the automation applications, shared power production, and transferable photoelectric applicability (Askarzadeh and Rezazadeh, 2011).

Despite substantial advancements in the previous few years, the financial performance of PEMFCs is still a point of contention among support and opposition (Alizadeh and Torabi, 2021). The improvement of PEMFC performance is critical for the marketing of the technology and achieving significant market adoption (Kahraman and Orhan, 2017). Generally, the performance of the FC is regarded as the essential aspect of end-user acceptability (J. Wang et al., 2018). Many published articles show that numerous structural and operational factors (parameters) highly influence the performance of the PEMFC. The most efficient methods that have successfully proved their ability to extract the parameters are the optimization methods (Eid et al., 2021; Hassan et al., 2021; Wang et al., 2021). This field still needs further investigation to find a more efficient approach to tackle this problem.

As presented in the relevant studies, Priya et al. (2015) presented a unique presentation for the efficient estimation of FC parameters. The parameters’ values of FC were determined based on the genetic algorithm, and the obtained results proved its ability to find better results than several other methods in this domain. İnci and Caliskan (2020) proposed a new enhanced energy extraction-based optimization method to tackle the FC parameters. The presented technique is based on using an improved cuckoo optimizer. The proposed technique achieved better convergence acceleration than traditional techniques. Kandidayeni et al. (2019) used several optimization techniques to solve PEMFC. The proposed method reduced squared errors among the included and measured voltage for two possible test cases. The proposed SFLA method got better results in terms of precision and repeatability than the other comparative methods.

Fathy et al. (2020a) introduced a hybrid of differential evolution and vortex search algorithms for determining the optimum parameters of the FC, called VSADE. The achieved results established the superiority of the introduced VSADE method. This study aimed to provide a new, simpler, and accurate model of the proton electrolyte membrane FC (Seleem et al., 2021). The suggested approach drastically lowers the number of unknown factors in such models, resulting in a more straightforward model. It discloses just four design factors within the model in this regard. This model’s great effectiveness is tested both in steady-state and dynamic operating situations. It is possible to build a highly exact PEMFC model using the suggested approach. Menesy et al. (2020) suggested an enhanced artificial ecosystem optimizer to determine the problem of the FC parameters. According to the results, it is proved that the presented optimizer has high performance in obtaining the optimal parameters compared with the other comparative methods.

As mentioned before, metaheuristic optimization algorithms proved their ability to deal with various problems such as bioinformatics (Issa, 2021a; Issa and Abd Elaziz, 2020; Issa and Hassanien, 2017; Issa et al., 2018a; Issa et al., 2018b; Issa and Helmi, 2021; Issa et al., 2022), control engineering (Issa, 2021b; Issa et al., 2019), passive suspension system (Issa and Samn, 2022), and digital watermarking (Abualigah and Diabat, 2021; Issa, 2018). To estimate the model parameters of the PEMFC, an efficient method compared to the existing method is needed. This research work proposed a new parameter extraction technique to deal with the FC modeling optimization problem. The proposed method is based on the honey badger algorithm (HBA), a technique recently proposed by Hashim et al.(2021) inspired by the creative foraging habits of the honey badger in real life. The mathematical modeling of the HBA is produced using efficient search operators to deal with highly complicated problems which motivate to use it for the parameter estimation of PEMFCs. The balancing between diversification and intensification of the search space of the HBA is the main merit and motivation to use it in this work. The primary fitness function that is used in the proposed method is to minimize the integral squared errors. The high effectiveness of the presented technique is verified using dynamic and steady-state operating conditions. The results illustrated that the presented method using the HBA achieved promising results in comparison with several relevant study parameter extraction methods used in the literature.

The main contributions and novelties of this study are concluded as following:

1. The optimal values of the PEMFC model parameters were adjusted based on the HBA.

The remaining sections of this article are organized as follows: Section 2 proposes the background of the used optimization methods. Section 3 shows the procedure of the proposed FC parameter extraction using the honey badger algorithm. In Section 4, experiments and results are given. Finally, Section 5 presents the conclusions and future potential works.

In this part, the primary mathematical representation of the PEMFC design is explained. It includes a cathode, negative charges, charged anode, and electrolyte, as described in Figure 1. In the PEMFC system, the hydrogen data are divided into two main parts utilizing a catalyst: protons and electrons. Moreover, the cathode pulls the protons, and the electrons produce the output charge by moving along the exterior circuit. The mathematical notations of the chemical stability produced in the FC are presented as follows (Alizadeh and Torabi, 2021): H 2 → 2 H + + 2 e − , (1) O 2 + 4 e − → 2 O − 2 , (2) H 2 + 1 2 O 2 → H 2 O + e l e c t r i c a l   e n e r g y + H e a t . (3)

In the PEMFC system, three drops normally happen during the voltage process, called activation (V_act), ohmic (V_ohm), and concentration (V_con). Thus, Eq. 4 is used to calculate the FC terminal voltage. V F C = E N e r n e s t − V a c t − V o h m − V c o n , (4) where E_Nernest is the double-faced open circuit charge voltage, which is calculated as follows (Yuan et al., 2020): E N e r n e s t = 1.229 − 8.5 × 10 − 4 ( T − 298.15 ) + 4.385 × 10 − 5 T ⁡ ln ( P H 2 + 0.5 ⁡ ln ⁡ P O 2 ) , (5) where RP_O2 presents the pressure of the O2, RP_H2 presents the pressure of H₂, and T represents the cell temperature value used in this research. The activation voltage loss value (V_act) is calculated as follows: V a c t = − [ ξ 1 + ξ 2 T + ξ 3 T ⁡ ln ( C 02 ) +   ξ 4 T ⁡ ln ( I F C ) ] , (6) where I_FC is the present value of the FC and ξ₁, ξ₂, ξ₃, and ξ₄ denote the coefficient values. C_O2 presents the condensation value of oxygen (mol/cm³) calculated as follows: C O 2 = P O 2 5.08 ∗ 10 6 ∗ e ( − 498 T ) . (7)

The V_ohm is calculated using Eq. 8, which is resulted from the equivalent resistance of the FC value. V o h m = I F C ( R M + R C ) , (8) where R_C presents the connection resistance and R_M presents the membrane resistances calculated as follows: R M = ρ M . l A , (9) ρ M = 181.6 [ 1 + 0.03 ( I F C A ) + 0.0062 ( T 303 ) ( I F C A ) 2.5 ] [ λ m − 0.634 − 3 ( I F C A ) ] e x p [ 4.18 ( T − 303 T ) ] , (10) where ρ M presents the measure of the resisting power of the membrane (Ω.cm), l presents the density value of the membrane (cm), A presents the effective range of the cell (cm²), and λ m presents the membrane water fulfilled. The V_con value is calculated using Eq. 11. V c o n = − b ⁡ ln { 1 − ( I F C A ) I m a x } , (11) where b presents a constant value and I m a x is the present maximum destiny value. So, the stack includes a series value of n FCs, and the stack voltage value is calculated as follows. V s t a c k = n . V F C = n . ( E N e r n e s t − V a c t − V o h m − V c o n ) . (12)

Figure 2 presents the FC polarization detour.

The HBA was used for tuning the best parameters’ values of PEMFCs where each agent has a total of the seven parameters (λ, Rc, ξ₁, ξ₂, ξ₃, ξ₄, and b), and the best agent is the agent that produces the best fitness. The fitness function used to evaluate the search is the sum square error (SSE) function, which represents the integral square of the subtraction between experimental and estimated voltages. The representation of the SSE function as proposed in Eq. (20), where Vexp and Vest represent the experimental and estimated voltages, respectively. S S E = ( V e x p − V e s t ) 2 . (20)

Figure 4 shows the flowchart of parameter estimation of PEMFCs based on the HBA. The initial random solutions are initialized according to satisfying conditions and input into the HBA’s block. The output of the HBA is the best solution found that achieves the smallest SSE.

The efficiency of the created HBA used for estimating PEMFC model parameters is assessed in this part using three datasets: PEMFC 250-W stack, NedStack PS6, and BCS 500-W, and their electrical specification are listed in Table 1. In addition, Table 2 presents the parameters’ boundaries (lower bound and higher bound) (Zhang and Liu, 2010).

The HBA is compared to other MH techniques such as grey wolf optimization (GWO) (Ali et al., 2017; Mirjalili et al., 2014), Hunger Games Search (HGS) (Yang et al., 2021), sine cosine algorithm (SCA) (Mirjalili, 2016), and Harris hawk optimization (HHO) (Heidari et al., 2019) to demonstrate its capability. The values of each algorithm’s parameters are assigned depending on the algorithm’s original implementation. The conventional settings for the number of populations and iterations are 50 and 500, respectively.

Tables 3–7 and Figures 5–9 using the three datasets show the comparison between the HBA and other approaches. Table 3 shows the estimated parameters derived by each algorithm and their SSE values in general. The performance of the HBA in terms of SSE is superior to other MH approaches among the datasets studied, as can be seen from these results. HBA’s SSE value with BCS 500-W, 250-W, and NedStack PS6 is, for example, 0.0118, 0.3378, and 1.38E+00, respectively.

Furthermore, the values of SSE over the iterations for SCA, GWO, HGS, HHO, and HBA are presented in Figure 5 among the three datasets for SCA, GWO, HGS, and HBA to justify the produced HBA’s convergence rate. These charts show that the HBA has a higher convergence rate than other approaches, especially in the NedStack PS6 dataset. Figure 6 also displays the voltage and measured I/V polarization percentage errors, where the percentage error was estimated as the difference between estimated and measured voltage relative to the measured voltage. The percentage error of the HBA is nearly -0.9E-3 to 4.8E-3 for BCS 500 W and from -0.012 to.015 for 250 W module, according to these curves. Finally, the percentage errors for the NedStack PS6 range from -0.012 to +0.012.

The obtained results in Figure 8A,B depict the effect of the three temperature values on the I/V and I/P polarization curves, respectively. The pressures RP_O2 and RP_H2 were set to justify the effect of temperature on the performance of the PEMFC stack as shown in Figure 9A,B.

Various statistical parameters such as mean, standard deviation, best, and worst of the SSE are calculated to further examine the effectiveness of HBA as a PEMFC model, as shown in Table 6. From these metrics, it is clear that the HBA is better at finding optimal settings than other approaches, as evidenced by the examined datasets. We also utilized the Wilcoxon nonparametric test to see if there was a significant difference between the HBA and other approaches. Table 7 shows the Wilcoxon test p-value at a significance level of 0.05. These results show that there is a considerable difference in overall datasets between the HBA and other approaches.

In this study, an alternative approach for estimating the model parameters of a PEMFC under various operating conditions is described. This method is based on the HBA. This algorithm has proven its efficacy in a variety of applications, which prompted us to use it. The main motivation for using the HBA for estimating the PEMFC’s parameters is the advantage of balance between exploration and exploitation of the search space, which avoids trapping in local minima. A set of experimental series has been conducted utilizing three datasets entitled 250-W stack, BCS 500-W, and NedStack PS6 to justify the usage of the HBA to determine the PEMFC’s parameters (i.e., λ, Rc, ξ₁, ξ₂, ξ₃, ξ₄, and b). HBA’s results have also been compared to those of other metaheuristic techniques such as HGS and SCA. In terms of performance measures, the results showed that the HBA outperformed other MH approaches. The findings revealed that the presented approach produced promising results and outperformed the other pproaches. The main limitation of using the HBA for estimating the parameters of PEMFCs is it was tested on three modules only. More modules are needed to be used in the experimental tests for efficient verification of the performance of the HBA. Apart from the findings generated by the HBA, it may be employed in a variety of applications, such as PV parameter estimation, mechanical engineering, and other challenges such as cloud computing and picture segmentation.