Parameter Estimation of Fuel Cells Using a Hybrid Optimization Algorithm

Abstract

Because of the current increase in energy requirement, reduction in fossil fuels, and global warming, as well as pollution, a suitable and promising alternative to the non-renewable energy sources is proton exchange membrane fuel cells. Hence, the efficiency of the renewable energy source can be increased by extracting the precise values for each of the parameters of the renewable mathematical model. Various optimization algorithms have been proposed and developed in order to estimate the parameters of proton exchange membrane fuel cells. In this manuscript, a novel hybrid algorithm, i.e., Hybrid Particle Swarm Optimization Puffer Fish (HPSOPF), based on the Particle Swarm Optimization and Puffer Fish algorithms, was proposed to estimate the proton exchange membrane fuel cell parameters. The two models were taken for the parameter estimation of proton exchange membrane fuel cells, i.e., Ballard Mark V and Avista SR-12 model. Firstly, justification of the proposed algorithm was achieved by benchmarking it on 10 functions and then a comparison of the parameter estimation results obtained using the Hybrid Particle Swarm Optimization Puffer Fish algorithm was done with other meta-heuristic algorithms, i.e., Particle Swarm Optimization, Puffer Fish algorithm, Grey Wolf Optimization, Grey Wolf Optimization Cuckoo Search, and Particle Swarm Optimization Grey Wolf Optimization. The sum of the square error was used as an evaluation metric for the performance evaluation and efficiency of the proposed algorithm. The results obtained show that the value of the sum of square error was smallest in the case of the proposed HPSOPF, while for the Ballard Mark V model it was 6.621 × 10⁻⁹ and for the Avista SR-12 model it was 5.65 × 10⁻⁸. To check the superiority and robustness of the proposed algorithm computation time, voltage–current (V–I) curve, power–current (P–I) curve, convergence curve, different operating temperature conditions, and different pressure results were obtained. From these results, it is concluded that the Hybrid Particle Swarm Optimization Puffer Fish algorithm had a better performance in comparison with the other compared algorithms. Furthermore, a non-parametric test, i.e., the Friedman Ranking Test, was performed and the results demonstrate that the efficiency and robustness of the proposed hybrid algorithm was superior.

1. Introduction

Rapid urbanization is contributing to the depletion of fossil resources while the need for energy grows daily [1]. It has caused the world to change to renewable energy sources such as solar energy, geothermal energy, tidal energy, wind energy, and wave energy for generating electricity. Fuel cells are one of the most suitable and promising alternative energy sources, with numerous characteristics, such as a high efficiency, reliability, stability, durability, consistent power generation, and cleanliness. Similarly, the majority of conventional renewable energy sources, such as solar and wind, have experienced rapid expansion, and in recent years they have caught the attention of researchers and academics [2]. Fuel cells have several applications, including in transportation, power generation, heating companies, backup power supply, and so on. The main components of a fuel cell are an anode and a cathode with an electrolyte, H₂ (pure hydrogen), which feeds the anode, while O₂ (or oxygen) feeds the cathode. The fuel cell operates on the basis of a controlled chemical process [3]. In the presence of a catalyst, the chemical reaction between hydrogen and oxygen produces chemical energy, which is directly transformed into electrical energy.

Several types of electrolytes can be used in fuel cells, and different types of fuel cells can be constructed based on this, such as the Proton Exchange Membrane Fuel Cell (PEMFC). When it comes to energy generation, PEMFC has various advantages such as high efficiency, low operating temperature, less deterioration, higher stability, low noise, high power density, and being an ecofriendly energy resource. To clearly understand the phenomena of the dynamic processes that occur within the fuel cell, it is critical to perform PEMFC modelling and the construction of precise models, which saves time and effort [4]. The polarization curve, which describes the output voltage and current relationship, is the most essential PEMFC characteristic, and hence forms the basis of PEMFC modelling. Several attempts have been made to design and create a suitable model with greater accuracy and precision for PEMFC parameter extraction.

The statistics provided in any PEMFC datasheet are typically insufficient for establishing the optimal parameter configuration, and the results produced using the model and the data published in the manufacturer’s datasheet differ greatly. As a result, parameter estimation can be thought of as an optimization issue, and the best set of solutions can be found by combining several approaches. As PEMFC parameters are non-linear, conventional optimization methods have a lower accuracy and precision. However, meta-heuristic algorithms provide several advantages such as random starting estimate, convergence to global optimal solution, and tackling complicated issues. Many researchers are interested in identifying a PEMFC parameter estimation model with a high accuracy and efficiency. The mathematical model serves as the foundation for fuel cell design and integration. It also provides information on the physical events occurring in fuel cells. The electrochemical models of fuel cells contain significant experimental and empirical formulations [5].

Various optimization algorithms have been proposed and applied in order to estimate the PEMFC parameters, which can further be categorized into two categories: based on a single algorithm only and the combination of two or more algorithms, also known as hybrid algorithms. An adaptive RNA genetic algorithm (ARNA-GA) [6] based on a biological RNA mechanism was proposed for PEMFC parameter estimation enhancing the global search, maintaining population diversity and the avoidance of premature convergence. The adaptive method in this algorithm helps with the dynamic selection of the crossover operation, thus improving the search ability. An adaptive differential evolution algorithm (ADE) [7] was proposed to accurately estimate the parameters of PEMFC by adjusting the variation operators in a dynamic manner and the control parameters, thereby increasing the search efficiency. Another optimization algorithm, i.e., Grasshopper Optimization algorithm (GOA) [8], was applied in order to find the optimum values of the unknown parameters for PEMFC. GOA is inspired by the behavior of grasshopper groups in the surrounding environment. The Dragonfly algorithm (DA) [9], inspired by the static and dynamic swarming behavior of dragonflies, was applied in order to estimate the parameters of PEMFC. Because of the static swarming behavior of dragonflies, this algorithm has the advantage of high exploration. Another meta-heuristic algorithm, i.e., Chaotic Binary Shark Smell Optimization (CBSSO) [10] algorithm, based on the ability of sharks to smell while in search of their prey, was proposed in order to estimate the PEMFC parameters, and it has a better performance and takes less time to converge.

Hybrid adaptive differential evolution (HADE) [11] based on biological genetic strategy, adaptive scaling factors, and dynamic crossover was proposed for PEMFC parameter estimation. The Bee Colony search method has been applied for improving the performance of local search. The modelling of PEMFC and parameter estimation was done using an optimization algorithm, known as the grouping-based global harmony search (GGHS) algorithm [12], based on the improvisation of a new harmony and using the best harmonies efficiently for finding an optimum solution. Another hybrid optimization algorithm i.e., cuckoo search algorithm with an explosion operator (CS-EO) [13], was proposed for finding an optimal solution to the PEMFC parameter estimation problem. This algorithm enhances the search ability with the help of the adaptive method and the local minima trap is avoided through the use of an explosion operator, which results in better convergence and accuracy. A hybrid algorithm based on Teaching−Learning-Based Optimization (TLBO) and Differential Evolution (DE) [14] was proposed for estimating the parameters of PEMFC. Nature-inspired TLBO is based on the teaching−learning technique occurring inside a classroom, and an individual population is used for finding an optimum solution, whereas the basis of DE is the evolution method of an individual’s population.

Therefore, the last few years have seen the development of multiple techniques for finding the optimal parameter set of PEMFC, beginning with conventional algorithms and moving to meta-heuristic algorithms. PEMFC parameter estimation was done by developing Black Widow Optimization (BWO) along with the comparison of its results with other different meta-heuristic algorithms under various operating temperatures. Various other meta-heuristic algorithms that have been developed are the Slime Mould (SM)-based optimization algorithm, Modified Monarch Butterfly Optimization (MMBO) algorithm, Backtracking Search Algorithm combined with Burger’s Chaotic map (BSABCM), Particle Swarm Optimization (PSO) [15], Genetic Algorithms (GA) [16], Chaotic Mayflies Optimization (CMO) algorithm [17], Grey Wolf Optimizer (GWO) [18], Whale Optimization Approach (WOA) [19], Salp Swarm Optimization algorithm (SSOA) [20], Effective Informed Adaptive PSO (EIA-PSO) [21], Cuckoo Search-Ant Colony Optimization (CS-ACO) [22], bi-inspired algorithm [23], Tribe PSO Algorithm (T-PSO) [24], Bi-Subgroup Optimization (BSO) [25], Sine Cosine Algorithm (SCA) [26], and Modified Owl Search Algorithm (DOSA) [27]. Another meta-heuristic algorithm, for example Puffer Fish (PF) [28], based on the behavior of male puffer fish in order to charm the female puffer fish by building special circular structures on the seabed, was developed for finding the optimal solution set to the problems of optimization. Various hybrid algorithms, i.e., Particle Swarm Optimization Grey Wolf Optimizer (PSOGWO) [29] and Grey Wolf Optimizer Cuckoo Search (GWOCS) [30], were also developed for the same reason and performed significantly better when identifying the optimal solution.

The major goal of this paper is to propose a new algorithm in order to estimate the parameters of PEMFC. The major contributions are listed below:

• A novel hybrid algorithm, i.e., HPSOPF based on Particle Swarm Optimization and Puffer Fish, is proposed for estimating the PEMFC parameters.
• For justification of the proposed hybrid algorithm, the mean and standard deviation (SD) are calculated using 10 benchmark functions.
• The Sum of Square Error (SSE) objective function is used for the performance evaluation and efficiency of the proposed algorithm for the parameter estimation of PEMFC.
• Ballard Mark V and Avista SR-12 models are the two datasheets used in order to estimate the PEMFC parameters.
• To check the performance and accuracy of the proposed algorithm, the computation time for both of the fuel cell models is calculated.
• To check the consistency and robustness of the proposed algorithm, the convergence curve, I–V, P–I curve, different operating temperature, and different pressure results are obtained.
• Non-parametric statistical test, i.e., Friedman Ranking Test, is done for finding the significance of the parameter estimation of PEMFC.

2. Materials and Methods

2.1. PEMFC Mathematical Modelling

The anode and cathode are two electrodes in PEMFC segregated by membranes with a polymer electrolyte, as illustrated in Figure 1. Hydrogen is injected by the anode, whereas oxygen is injected by the cathode, and a thin membrane forms the electrolyte which conducts ions and prevents the passage of electrons. The output voltage is generated when the flow of ions takes place through the electrolyte, whereas the outside circuit allows for the passage of electrons.

Electrochemical reactions occurring in PEMFC are shown in the following equations, where the anode and cathode side chemical reactions are represented in (1) and (2), respectively, and (3) shows the total chemical reaction or electrical energy generation [31].

$$H_2\to {2H}^{+}+2_e^{-}$$

(1)

$$\frac{1}{2}{O}_2+{2H}^{+}+2_e^{-}\to H_{2}O$$

(2)

$$H_2+\frac{1}{2}{O}_2\to H_{2}O$$

(3)

$$V_c=E_{\mathrm{N}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{s}\mathrm{t}}-V_{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}-V_{\mathrm{o}\mathrm{h}\mathrm{m}\mathrm{i}\mathrm{c}}-V_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}$$

(4)

There are three types of voltages constituted in the fuel cell terminal voltage ($V_c$), as illustrated in (4). $E_{\mathrm{N}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{s}\mathrm{t}}$ represents the reversible open circuit voltage, $V_{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}$ represents the activation voltage drop, $V_{\mathrm{o}\mathrm{h}\mathrm{m}\mathrm{i}\mathrm{c}}$ represents the ohmic voltage drop, and $V_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}$ represents the concentration voltage drop [32].

$$E_{\mathrm{N}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{s}\mathrm{t}}=E_r+\frac{R{T}_c}{ZF}\left[\ln \left(P_{H_2}\right)+\ln \left(\sqrt{P_{O_2}}\right)\right]$$

(5)

The reversible thermodynamic potential of the oxygen and hydrogen reaction is defined by the Nernst equation, as shown in (5), in which $E_r$ denotes the reference voltage, R denotes the universal gas constant, $T_c$ denotes temperature of cell (in Kelvin), Z denotes the number of electrons transferred, F denotes the faraday constant, and $P_{H_2}$ and $P_{O_2}$ denote hydrogen and oxygen pressure, respectively. After expansion, (5) is re-written in the following form [33],

$$E_{\mathrm{N}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{s}\mathrm{t}}=1.229-8.5\times {10}^{-4}\left(T_c-298.15\right)+4.385\times {10}^{-5}{T}_c\times \left(\ln \left(P_{H_2}\right)+\ln \left(\sqrt{P_{O_2}}\right)\right)$$

(6)

$$P_{H_2}=0.5\left({PH}_{anode}\times P_{H_{2}O}\right)\left[{\left(exp\left(\frac{1.635\left(\frac{i_c}{A}\right)}{{T_c}^{1.334}}\right)\times \frac{\left({PH}_{anode}\times P_{H_{2}O}\right)}{P_{anode}}\right)}^{-1}-1\right]$$

(7)

$$P_{O_2}=\left({PH}_{cathode}\times P_{H_{2}O}\right)\left[{\left(exp\left(\frac{4.192\left(\frac{i_c}{A}\right)}{{T_c}^{1.334}}\right)\times \frac{\left({PH}_{anode}\times P_{H_{2}O}\right)}{P_{cathode}}\right)}^{-1}-1\right]$$

(8)

$P_{H_2}$ and $P_{O_2}$ are shown in (7) and (8), respectively, where $P_{anode}$ represents the anode pressure from the input side, $P_{cathode}$ represents the cathode pressure from the input side, and the relative humidity of the vapor from the cathode and anode side of PEMFC is represented by ${PH}_{cathode}$ and ${PH}_{anode}$, respectively. $i_c$ represents the current generated by PEMFC, the surface area of the membrane is represented by A, and $P_{H_{2}O}$ is the pressure at water saturation, as represented in (9) [34].

$$P_{H_{2}O}=2.95\times {10}^{-2}\left(T_c-273.15\right)-9.18\times {10}^{-5}{\left(T_c-273.15\right)}^2+1.44\times {10}^{-2}{\left(T_c-273.15\right)}^2-2.18$$

(9)

$$V_{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}=\frac{R{T}_c}{z\alpha F}\ln \left(\frac{i_c}{i_O}\right)$$

(10)

The activation process leads to voltage loss, which is shown in (10), where $\alpha$ represents the transfer coefficient and $i_O$ represents the exchange current density, which can be expanded as shown below [34].

$$V_{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}=-\left[{\epsilon }_1+{\epsilon }_2{T}_c+{\epsilon }_3{T}_c\ln \left(C_{O_2}\right)+{\epsilon }_4{T}_c\ln \left(i_c\right)\right]$$

(11)

${\epsilon }_1,{\epsilon }_2,{\epsilon }_3{,\epsilon }_4$ represent semi-empirical coefficients on the PEMFC cathode side. The concentration of oxygen ($C_{O_2}$) is calculated as shown in (12).

$$C_{O_2}=\frac{P_{O_2}}{5.08\times {10}^6{exp}^{\left(\frac{-498}{T_c}\right)}}$$

(12)

The resistive ohmic voltage drop ($V_{\mathrm{o}\mathrm{h}\mathrm{m}\mathrm{i}\mathrm{c}}$) is illustrated in (13), in which $R_{MR}$ represents the membrane surface resistance and $R_{CR}$ represents the contact resistance.

$$V_{\mathrm{o}\mathrm{h}\mathrm{m}\mathrm{i}\mathrm{c}}=i_c\left(R_{MR}+R_{CR}\right)$$

(13)

$$R_{MR}=\frac{{\rho }_{M}l}{A}$$

(14)

The membrane resistance ($R_{MR}$) is calculated using (14), ${\rho }_M$ represents the membrane resistivity, and $l$ represents the membrane thickness. Furthermore, ${\rho }_M$ can be expressed as below [34].

$${\rho }_M=\frac{181.6\left(1+0.03\left(\frac{i_c}{A}\right)+0.0062\left(\frac{T_c}{303}\right){\left(\frac{i_c}{A}\right)}^{2.5}\right)}{\left(\lambda -0.634-3\left(\frac{i_c}{A}\right)\right)exp\left(418\left(\frac{T_c-303}{T_c}\right)\right)}$$

(15)

The adjustable empirical variable is denoted by $\lambda$. The value of the voltage concentration loss ($V_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}$) is calculated as shown in (16).

$$V_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}=\frac{R{T}_c}{z\alpha F}\ln \left(\frac{i_d}{i_d-i_c}\right)$$

(16)

$i_d$ is the current density when the reactant concentration is zero. It can be re-written as shown below.

$$V_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}=-b\ln \left(i_c-\frac{\frac{i_c}{A}}{i_{max}}\right)$$

(17)

$b$ represents the constant and $i_{max}$ represents the maximum current density. The evaluation of the value for $b$ can be done as shown in (18) [35].

$$b=\frac{R{T}_c}{z\alpha F}$$

(18)

In order to generate sufficient energy, multiple stacks need to be combined and connected in series or parallel, as the value of the single stack PEMFC current and output voltage is small. The output voltage ($V_{stack}$) of the stack of fuel cell in the case of series connection is shown in (19).

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

(19)

$n$ represents the number of cells that are connected in series. It can be re-written as shown below (20) [35].

$$V_{stack}=n\left(E_{\mathrm{N}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{s}\mathrm{t}}-V_{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}-V_{\mathrm{o}\mathrm{h}\mathrm{m}\mathrm{i}\mathrm{c}}-V_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}\right)$$

(20)

As per the equations mentioned above, seven parameters for PEMFC, i.e., ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3{,\epsilon }_4,\lambda ,R_{CR},b$, need to be estimated with accuracy and precision for the controlled operation and adequate modelling of the fuel cell. The proposed HPSOPF algorithm as well as other well-established meta-heuristic algorithms, i.e., PSO, GWO, GWOCS, PF, and PSOGWO, were used to estimate the parameters in order to find a suitable and promising alternative for non-renewable energy sources, and found it to be the proton exchange membrane fuel cell (PEMFC).

2.2. Problem Formulation

This paper proposes a new hybrid algorithm, i.e., HPSOPF, in order to estimate the parameters of the PEMFC. Against each current density input, prediction of the output voltage was done using optimization algorithms. SSE (sum of square error) is used as a metric in order to evaluate the predicted output voltage and experimental values of the output voltage, and its objective function is shown in (21).

$$\mathrm{SSE}=MIN(F={\sum }_{i=1}^N{(V_{actual}-V_i)}^2)$$

(21)

The constraints are as follows:

$${\epsilon }_{nmin}\le {\epsilon }_n\le {\epsilon }_{nmax}n=1:4$$

$${\lambda }_{min}\le \lambda \le {\lambda }_{max}$$

$$R_{CRmin}\le R_{CR}\le R_{CRmax}$$

$$b_{min}\le b\le b_{max}$$

n represents the number of data points, $V_{actual}$ is the experimental output voltage and $V_i$ is the predicted output voltage using various optimization algorithms. The main goal of this paper is to minimize the SSE value for obtaining a better performance as well as more accuracy and precision in order to estimate the parameters of PEMFC.

2.3. Proposed Algorithm (HPSOPF)

This paper proposes a new hybrid algorithm, i.e., HPSOPF, based on the Particle Swarm Optimization (PSO) and Puffer Fish optimization algorithm (PF). PSO and PF are two well-known meta-heuristic algorithms that have distinct search techniques. The basis of PSO is bird flocking behavior and fish schooling, whereas Puffer Fish is a bio-inspired meta-heuristic optimization algorithm that is based on the behavior of male puffer fish in order to charm the female puffer fish [36]. Torquigeneral bomaculosus is the scientific name of a puffer fish with a length of 12 cm and 9.1 cm in males and females, respectively. The unique and special characteristic of this species is building large circular structures on the seabed. The circular structure created by the puffer fish includes a specific pattern in the center, sand peaks, and inner and outer circle. Satisfactory results were obtained for estimating the parameters of fuel cells using meta-heuristic algorithms, but there were some disadvantages such as convergence instability and the ability to fall easily into the trap of the local optimum. The genetic algorithm is time consuming as well as having premature convergence. There is partial optimism, a low rate of convergence, and less population diversity in Particle Swarm Optimization (PSO). This paper develops a new hybrid algorithm, i.e., HPSOPF, for obtaining better and highly efficient results in order to estimate the parameters of fuel cells. The pseudo code of HPSOPF is illustrated in Figure 2.

As prob denotes probability. Rand (0, 1) < prob increases the range of search, and every time there are no changes in order to avoid the algorithm getting stuck at the local minimum.

3. Results

3.1. Benchmark Test Function

Ten benchmark test functions were selected, as shown in

Table 1. Definition of the benchmark functions.

Function	Range
$F_1\left(k\right)={\sum }_{j=1}^m{k}_j^2$	[−100, 100]
$F_2\left(k\right)={\sum }_{j=1}^m\left|k\right|+{\prod }_{j=1}^m\left|x\right|$	[−10, 10]
$F_3\left(k\right)={\sum }_{j=1}^m{\left({\sum }_{j=1}^m{k}_j^2\right)}^2$	[−100, 100]
$F_4\left(k\right)={max}_j\left|k_j\right|,1\le j\le m$	[−100, 100]
$F_5\left(k\right)={\sum }_{j=1}^{m-1}\left[{100(k_{j+1}-k_j^2)}^2+{(k_j-1)}^2\right]$	[−30, 30]
$F_6\left(k\right)={\sum }_{j=1}^m{\left(\left|k_j+0.5\right|\right)}^2$	[−100, 100]
$F_7\left(k\right)={\sum }_{j=1}^m{ik}_j^4+random[0,1]$	[−1.28, 1.28]
$F_8\left(k\right)={\sum }_{j=1}^m-k_j\mathit{sin}\left(\sqrt{\left|k_j\right|}\right)$	[−500, 500]
$F_9\left(k\right)={\sum }_{j=1}^m\left[k_j^2-10\mathit{cos}\left(2\pi k_j\right)+10\right]$	[−5.12, 5.12]
$F_{10}\left(k\right)=-20\mathit{exp}\left(-0.2\sqrt{\left(1/m\right){\sum }_{j=1}^m{k}_j}\right)-exp\left(\left(1/m\right){\sum }_{j=1}^m\mathit{cos}\left(2\pi k_j\right)+20+e\right)$	[−32, 32]

, for evaluation of the proposed hybrid algorithm, i.e., HPSOPF, for the PEMFC parameter estimation. The ten benchmark test functions contained seven uni-modal functions and three multi-modal functions where every function’s dimension was set to 30.

Table 2. Results of the benchmark test functions.

Algorithms		F1	F2	F3	F4	F5	F6	F7	F8	F9	F10
PSO	mean	1.43 × 104	6.54 × 107	8.78 × 104	86.84	6.15 × 107	1.82 × 104	23.01	−4.32 × 103	4.33 × 102	16.95
	SD	5.29 × 103	2.11 × 108	1.26 × 104	4.84	3.44 × 107	4.64 × 103	6.78	3.66 × 102	22.61	2.62
GWO	mean	4.09 × 103	37.50	1.40 × 104	7.65	9.36 × 104	6.55 × 103	1.50 × 10−1	-5.05 × 103	1.76 × 102	7.61
	SD	1.15 × 104	8.35	2.84× 104	15.96	2.81 × 105	1.72 × 104	1.77 × 10−1	2.88 × 103	1.74 × 102	8.91
PF	mean	4.85 × 10−9	9.43 × 10−7	7.39 × 103	2.22 × 10−1	36.25	3.87	8.90 × 10−3	−7.89 × 104	8.28	1.20 × 10−5
	SD	6.96 × 10−9	5.92 × 10−7	7.13 × 103	2.45 × 10−1	8.35 × 10−1	2.84 × 10−1	3.80 × 10−3	9.22 × 102	13.44	8.12 × 10−6
GWOCS	mean	1.10 × 10−27	8.48 × 10−17	5.95 × 10−5	1.72 × 10−6	37.02	2.51	2.20 × 10−3	-8.48 × 103	1.59	1.20 × 10−13
	SD	1.13 × 10−27	3.47 × 10−17	1.56 × 10−4	1.87 × 10−6	5.66 × 10−1	6.16 × 10−1	1.16 × 10−3	2.39 × 103	4.9	1.84 × 10−14
PSOGWO	mean	1.45 × 10−28	4.13 × 10−17	1.35 × 10−5	6.44 × 10−7	27.52	9.78 × 10−1	1.55 × 10−3	-1.48 × 104	9.89 × 10−1	8.32 × 10−14
	SD	1.15 × 10−28	2.45 × 10−17	2.39 × 10−5	6.89 × 10−7	4.19 × 10−1	4.10 × 10−1	5.90 × 10−4	1.66 × 103	2.25	1.05 × 10−14
HPSOPF	mean	4.03 × 10−72	7.88 × 10−38	1.33 × 10−131	1.93 × 10−16	0	8.60 × 10−1	8.62 × 10−5	-1.73 × 105	8.21 × 10−8	4.74 × 10−15
	SD	9.96 × 10−72	2.45 × 10−37	3.70 × 10−131	4.69 × 10−16	0	4.62 × 10−1	4.20 × 10−5	3.20 × 102	2.88 × 10−15	4.79 × 10−14

represents the values of the mean and standard deviation (SD) calculated using 10 benchmark test functions, and it was clear that the proposed hybrid algorithm, i.e., HPSOPF, had a remarkably better performance than the various other meta-heuristic algorithms, i.e., PSO, PF, and GWO, and hybrid algorithms, i.e., GWOCS and PSOGWO. The coding of each program was done in MATLAB 2020a in Windows 10 with 8 GB RAM and a 2.50 GHz intel CPU processor. The feature evaluation limit for the benchmark test functions comparison was 1000 and the code of every algorithm was run 20 times with dim size 30.

3.2. PEMFC Parameter Estimation

The parameter search range for estimating the PEMFC parameters is represented in

Table 3. Range of the upper and lower bound.

Parameters	Upper Bound	Lower Bound
${\epsilon }_1$	−0.08532	−1.1997
${\epsilon }_2\times {10}^{-3}$	6.00	0.8
${\epsilon }_3\times {10}^{-5}$	9080	3.60
${\epsilon }_4\times {10}^{-4}$	−0.954	−2.60
$\lambda$	24.00	10.00
$R_{CR}\times {10}^{-4}$	8.00	1.00
b	0.5	0.0136

for both of the models, i.e., Ballard Mark V and Avista SR-12.

Table 4. Manufacturer data sheets.

Model	Ballard Mark V Model	Avista SR-12 Model
$n$	35	48
$l\left[\mathsf{\mu }\mathrm{m}\right]$	178	25
$A\left[{\mathrm{c}\mathrm{m}}^2\right]$	50.6	62.5
$J_{max}\left[\mathrm{A}/{\mathrm{c}\mathrm{m}}^2\right]$	1.5	0.672
$P_{H_2}$[bar]	1	1.476
$P_{O_2}$[bar]	1	0.209
Power [W]	1000	500
$T$ [K]	343.15	323.15

represents the manufacturer data sheet for both of the models. Coding for each program was done in MATLAB 2020a and each program was run 20 times. The PEMFC parameter estimation was done with the proposed HPSOPF and a comparison of its performance and efficiency was further done with various other algorithms, i.e., PSO, PF, GWO, PSOGWO, and GWOCS. Parameter estimation for PEMFC using both the models at STC (standard temperature condition) is represented in

Table 5. Parameter estimation of PEMFC using the Ballard Mark V model.

Parameter/Algorithms	${\mathit{\epsilon }}_1$	${\mathit{\epsilon }}_2$	${\mathit{\epsilon }}_3$	${\mathit{\epsilon }}_4$	$\mathit{\lambda }$	${\mathit{R}}_{\mathit{C}\mathit{R}}$	b	SSE	Comp Time (s)
PSO	−1.0701	0.0026	5.5241 × 10−5	−1.8076 × 10−4	15.6439	0.00043	0.2249	2.4324 × 10−4	4.97
GWO	−1.0390	0.0024	6.9452 × 10−5	−1.9387 × 10−4	16.8429	0.00038	0.2925	3.1629 × 10−5	4.85
PF	−1.0651	0.0024	6.7059 × 10−5	−1.4820 × 10−4	18.9925	0.00044	0.2451	2.7562 × 10−5	2.71
GWOCS	−1.0539	0.0027	5.1428 × 10−5	−1.7231 × 10−4	14.6350	0.00034	0.1957	6.5922 × 10−8	2.64
PSOGWO	−1.0727	0.0022	6.3245 × 10−5	−1.8235 × 10−4	15.1396	0.00047	0.2682	4.8720 × 10−8	2.42
HPSOPF	−1.0059	0.0016	3.4698 × 10−5	−1.6129 × 10−4	14.6827	0.00031	0.2156	6.6216 × 10−9	2

and

Table 6. Parameter estimation of the Avista SR-12 model.

Parameter/Algorithms	${\mathit{\epsilon }}_1$	${\mathit{\epsilon }}_2$	${\mathit{\epsilon }}_3$	${\mathit{\epsilon }}_4$	$\mathit{\lambda }$	${\mathit{R}}_{\mathit{C}\mathit{R}}$	b	SSE	Comp Time (s)
PSO	−0.9645	0.0012	8.4215 × 10−5	−1.2348 × 10−4	13.0845	0.00031	0.2869	3.12 × 10−3	4.89
GWO	−0.9128	0.0022	8.2682 × 10−5	−1.4452 × 10−4	15.3653	0.00034	0.3566	2.99 × 10−4	4.74
PF	−0.8796	0.0014	9.7458 × 10−5	−2.0712 × 10−4	11.0000	0.00021	0.1800	1.88 × 10−4	2.76
GWOCS	−0.9212	0.0012	9.5168 × 10−5	−2.5841 × 10−4	17.4133	0.00042	0.2195	6.33 × 10−7	2.68
PSOGWO	−1.0617	0.0023	9.6387 × 10−5	−1.2701 × 10−4	15.8417	0.00046	0.2511	5.29 × 10−7	2.40
HPSOPF	−1.0432	0.0014	7.8963 × 10−5	−1.0965 × 10−4	14.8896	0.00044	0.2416	5.65 × 10−8	2.12

.

3.3. Solution Accuracy Analysis

The values of the objective function, computational time (s), and SSE were calculated after the PEMFC parameter estimation, as represented in Table 5 and Table 6. Bar graphs corresponding to the SSE value and computational time (s) are shown in Figure 3 and Figure 4, respectively, for the Ballard Mark V model. Bar graphs corresponding to the SSE value and computational time (s) are shown in Figure 5 and Figure 6, respectively, of the Avista SR-12 model. It is very clear from the bar graphs that the SSE value and computational time (s) of both the models were the lowest in the case of the new proposed hybrid algorithm. The proposed algorithm is an efficient and better algorithm than other well-known meta-heuristic algorithms such as PSO, PF, GWO, GWOCS, and PSOGWO, for estimating the PEMFC parameters.

3.4. Convergence Analysis

The proposed algorithm convergence graph and its further comparison with other well-known meta-heuristic algorithms, i.e., PSO, PF, GWO, GWOCS, and PSOGWO, are shown in Figure 7 and Figure 8. From this, it is clear that the proposed hybrid algorithm has a higher convergence pace than other meta-heuristic algorithms, thus proving it has more accuracy and precision. The values of power, absolute error, and voltage are calculated and represented in

Table 7. Calculated values for the voltage, power, and absolute error for the Ballard Mark V model.

Current Measured (A)	Voltage Measured (V)	Voltage Calculated (V)	Voltage Absolute Error	Power Measured (W)	Power Calculated (W)	Power Absolute Error
5.4	0.92	0.9211	1.19 × 10−3	4.97	4.97394	7.92 × 10−4
10.8	0.88	0.8721	9.06 × 10−3	9.50	9.41868	8.63 × 10−3
16.2	0.85	0.8451	5.80 × 10−3	13.77	13.69062	5.80 × 10−3
21.6	0.82	0.8176	2.94 × 10−3	17.71	17.66016	2.82 × 10−3
27.0	0.79	0.7858	5.34 × 10−3	21.96	21.2166	3.50 × 10−2
32.4	0.77	0.7682	2.34 × 10−3	24.95	24.88968	2.42 × 10−3
37.8	0.74	0.7361	5.30 × 10−3	27.97	27.82458	5.23 × 10−3
43.2	0.72	0.7112	1.24 × 10−2	31.10	30.72384	1.22 × 10−2
48.6	0.69	0.6887	1.89 × 10−3	33.53	33.47082	1.77 × 10−3
54.0	0.66	0.6562	5.79 × 10−3	35.64	35.4348	5.79 × 10−3
59.4	0.62	0.6175	4.05 × 10−3	36.83	36.6795	4.10 × 10−3
64.8	0.60	0.5918	1.39 × 10−2	38.88	38.34864	1.39 × 10−2
70.2	0.55	0.5507	1.27 × 10−3	38.61	38.65914	1.27 × 10−3
Sum of AE			7.12 × 10−2			9.98 × 10−2

for the Ballard Mark V model and in

Table 8. Calculated values for the voltage, power, and absolute error for the Avista SR-12 model.

Current Measured (A)	Voltage Measured (V)	Voltage Calculated (V)	Voltage Absolute Error	Power Measured (W)	Power Calculated (W)	Power Absolute Error
1.004	43.17	43.04	3.02 × 10−3	43.36	43.21216	3.42 × 10−3
3.166	41.14	41.17	7.29 × 10−4	130.25	130.34422	7.23 × 10−4
5.019	40.09	40.04	1.25 × 10−3	201.21	200.96076	1.24 × 10−3
7.027	39.04	39.03	2.56 × 10−4	274.33	274.26381	2.41 × 10−4
8.958	37.99	37.98	2.63 × 10−4	340.31	340.22484	2.50 × 10−4
10.97	37.08	37.1	5.39 × 10−4	406.77	406.987	5.33 × 10−4
13.05	36.03	36.05	5.55 × 10−4	470.19	470.4525	5.58 × 10−4
15.06	35.19	35.20	2.84 × 10−4	529.96	530.112	2.87 × 10−4
17.07	34.07	34.09	5.87 × 10−4	581.57	581.9163	5.95 × 10−4
19.07	33.02	33.04	6.05 × 10−4	629.69	630.0728	6.08 × 10−4
21.08	32.04	32.06	6.24 × 10−4	675.40	675.8248	6.29 × 10−4
23.01	31.20	31.24	1.28 × 10−3	717.91	718.8324	1.28 × 10−3
24.94	29.80	29.84	1.34 × 10−3	743.21	744.2096	1.34 × 10−3
26.87	28.96	28.99	1.03 × 10−3	778.16	778.9613	1.03 × 10−3
28.96	28.12	28.17	1.77 × 10−3	814.36	815.8032	1.77 × 10−3
30.81	26.3	26.32	7.60 × 10−4	810.30	810.9192	7.64 × 10−4
32.97	24.06	24.12	2.49 × 10−3	793.26	795.2364	2.49 × 10−3
34.90	21.40	21.44	1.87 × 10−3	746.86	748.256	1.87 × 10−3
Sum of AE			1.93 × 10−2			1.96 × 10−2

for the Avista SR-12 model. Furthermore, the P−I and V−I characteristic curves of PEMFC are illustrated in Figure 9a–d, respectively, which justifies the accuracy of the proposed HPSOPF.

Table 9. Parameter estimation of PEMFC at different operating temperatures for the Ballard Mark V model.

Temperature (Kelvin)	Parameter/Algorithms	PSO	GWO	PF	GWOCS	PSOGWO	HPSOPF
293.15	${\epsilon }_1$	−1.1326	−1.0216	−1.0633	−1.1343	−1.0241	−1.0137
	${\epsilon }_2$	0.0012	0.0010	0.0023	0.0026	0.00034	0.0014
	${\epsilon }_3$	4.2570 × 10−5	5.4195 × 10−5	6.4659 × 10−5	5.7921 × 10−5	5.1986 × 10−5	6.8914 × 10−5
	${\epsilon }_4$	−1.7265 × 10−4	−1.9506 × 10−4	−1.5468 × 10−4	−1.6315 × 10−4	−1.5499 × 10−4	−1.8425 × 10−4
	$\lambda$	17.5144	16.5712	15	12.6258	14.1545	14.2584
	$R_{CR}$	0.00041	0.00032	0.00045	0.00040	0.00026	0.00024
	b	0.2546	0.2488	0.3421	0.1965	0.3785	0.3697
	SSE	2.5925 × 10−4	2.8137 × 10−5	2.7437 × 10−5	7.0130 × 10−8	6.5215 × 10−8	5.8921 × 10−9
393.15	${\epsilon }_1$	−1.0686	−1.0844	−1.0158	−1.0352	−1.1436	−1.0496
	${\epsilon }_2$	0.0033	0.0013	0.0027	0.0034	0.0031	0.0020
	${\epsilon }_3$	5.6819 × 10−5	5.1894 × 10−5	6.7859 × 10−5	6.8406 × 10−5	5.7629 × 10−5	6.8941 × 10−5
	${\epsilon }_4$	−1.4625 × 10−4	−1.8152 × 10−4	−1.9255 × 10−4	−1.9135 × 10−4	−1.7365 × 10−4	−1.5625 × 10−4
	$\lambda$	14	12.9558	13.5158	14.4154	14.6215	15.5815
	$R_{CR}$	0.00028	0.00046	0.00047	0.00035	0.00040	0.00034
	b	0.3427	0.1327	0.3251	0.2055	0.3963	0.2453
	SSE	3.4633 × 10−4	2.1255 × 10−5	2.0181 × 10−5	6.9857 × 10−8	6.4140 × 10−8	6.2395 × 10−9
443.15	${\epsilon }_1$	−1.1425	−1.0542	−1.0365	−1.0154	−1.0453	−1.0280
	${\epsilon }_2$	0.0022	0.0030	0.0015	0.0010	0.0028	0.0036
	${\epsilon }_3$	6.8921 × 10−5	6.7470 × 10−5	5.6289 × 10−5	6.6953 × 10−5	6.4636 × 10−5	6.4270 × 10−5
	${\epsilon }_4$	−1.5146 × 10−4	−1.7266 × 10−4	−1.6420 × 10−4	−1.8250 × 10−4	−1.8942 × 10−4	−1.9562 × 10−4
	$\lambda$	13.6547	14.6889	13.8487	15.4126	17.4995	14.5146
	$R_{CR}$	0.00039	0.00041	0.00029	0.00030	0.00025	0.00036
	b	0.2066	0.3542	0.3285	0.2040	0.3546	0.1565
	SSE	2.7841 × 10−4	3.0326 × 10−5	2.9713 × 10−5	6.8570 × 10−8	5.5521 × 10−8	5.2563 × 10−9

and

Table 10. Parameter estimation of PEMFC at different operating temperatures for the Avista SR-12 model.

Temperature (Kelvin)	Parameter/Algorithms	PSO	GWO	PF	GWOCS	PSOGWO	HPSOPF
273.15	${\epsilon }_1$	−0.9125	−0.8813	−0.9236	−1.0155	−1.0792	−1.0285
	${\epsilon }_2$	0.0010	0.0013	0.0016	0.0027	0.0018	0.0021
	${\epsilon }_3$	9.3255 × 10−5	9.4699 × 10−5	9.1652 × 10−5	7.9745 × 10−5	8.1145 × 10−5	7.7626 × 10−5
	${\epsilon }_4$	−2.6185 × 10−4	−1.2152× 10−4	−1.9128 × 10−4	−2.7016 × 10−4	−2.4216 × 10−4	−2.4236 × 10−4
	$\lambda$	17	18.2674	12.8591	14.2577	15	16.9778
	$R_{CR}$	0.00031	0.00046	0.00048	0.00034	0.00030	0.00040
	b	0.1359	0.2413	0.3239	0.3616	0.2463	0.3798
	SSE	2.4595 × 10−3	3.5955 × 10−4	2.5790 × 10−4	6.1246 × 10−7	5.6329 × 10−7	5.8517 × 10−8
373.15	${\epsilon }_1$	−1.0158	−1.0265	−0.9615	−0.9281	−0.8299	−1.0523
	${\epsilon }_2$	0.002	0.0032	0.0028	0.0014	0.0026	0.0015
	${\epsilon }_3$	9.84 × 10−5	7.13 × 10−5	8.42 × 10−5	9.28 × 10−5	9.43 × 10−5	9.79 × 10−5
	${\epsilon }_4$	−1.43 × 10−4	−2.82 × 10−4	−2.40 × 10−4	−2.64 × 10−4	−1.12 × 10−4	−1.72 × 10−4
	$\lambda$	15	16.8412	11	17.1963	16.987	15
	$R_{CR}$	0.00041	0.00022	0.00040	0.00035	0.00026	0.00043
	b	0.3241	0.3658	0.1492	0.2810	0.2615	0.3958
	SSE	3.1358 × 10−3	2.5189 × 10−4	2.0126 × 10−4	5.6585 × 10−7	4.9952 × 10−7	6.4166 × 10−8
423.15	${\epsilon }_1$	−0.9357	−1.0875	−1.0413	−0.8137	−0.9013	−0.8594
	${\epsilon }_2$	0.0011	0.0032	0.0010	0.0016	0.0021	0.0023
	${\epsilon }_3$	7.9897 × 10−5	9.4056 × 10−5	9.5378 × 10−5	8.7902 × 10−5	9.1288 × 10−5	9.6525 × 10−5
	${\epsilon }_4$	−2.8125 × 10−4	−2.3413 × 10−4	−1.2846 × 10−4	−1.3153 × 10−4	−2.9711 × 10−4	−2.1987 × 10−4
	$\lambda$	16.6125	17.5142	13.5845	13.1278	11.9254	14.8014
	$R_{CR}$	0.00030	0.00026	0.00027	0.00034	0.00040	0.00046
	b	0.1325	0.2472	0.1240	0.0348	0.3188	0.1576
	SSE	2.7827 × 10−3	3.9142 × 10−4	2.4783 × 10−4	5.7463 × 10−7	4.8362 × 10−7	5.7366 × 10−8

represent the parameter estimation of the fuel cell at different operating temperatures for both of the fuel cell models.

Table 11. Parameter estimation of PEMFC at different pressures for the Ballard Mark V model.

Pressure (atm)	Parameter/Algorithms	PSO	GWO	PF	GWOCS	PSOGWO	HPSOPF
1	${\epsilon }_1$	−1.13658	−1.02874	−1.03658	−1.14963	−0.9745	−1.02458
	${\epsilon }_2$	0.0011	0.0016	0.0032	0.0022	0.0025	0.0034
	${\epsilon }_3$	4.6148 × 10−5	5.0545 × 10−5	5.6188 × 10−5	6.8424 × 10−5	6.1637 × 10−5	5.4198 × 10−5
	${\epsilon }_4$	−1.5846 × 10−4	−1.8246 × 10−4	−1.9216 × 10−4	−1.6226 × 10−4	−1.5125 × 10−4	−1.7182 × 10−4
	$\lambda$	12	14.2541	15.4692	16.5045	14.1278	13.0585
	$R_{CR}$	0.00024	0.00035	0.00034	0.00026	0.00041	0.00027
	b	0.2347	0.2463	0.3622	0.2379	0.1643	0.2125
	SSE	3.2641E-04	3.6246 × 10−5	2.9236 × 10−5	6.6745 × 10−8	6.3326 × 10−8	5.6146 × 10−9
2	${\epsilon }_1$	−1.168102	−1.15487	−1.1442	−1.01578	−1.05146	−1.136214
	${\epsilon }_2$	0.0031	0.0024	0.0025	0.0034	0.0012	0.0014
	${\epsilon }_3$	6.2972 × 10−5	6.2841 × 10−5	5.7941 × 10−5	5.4348 × 10−5	6.1688 × 10−5	4.8153 × 10−5
	${\epsilon }_4$	−1.4298 × 10−4	−1.5969 × 10−4	−1.7124 × 10−4	−1.5874 × 10−4	−1.7137 × 10−4	1.6385 × 10−4
	$\lambda$	13.0955	15.9810	15.2151	12.4322	15.0453	16.7446
	$R_{CR}$	0.00027	0.00042	0.00030	0.00046	0.00033	0.00024
	b	0.1459	0.2695	0.2156	0.3879	0.3413	0.2712
	SSE	2.7463 × 10−4	3.1965 × 10−5	2.1259 × 10−5	6.2893 × 10−8	5.6385 × 10−8	6.4692 × 10−9
3	${\epsilon }_1$	−1.02545	−1.054281	−1.04748	−1.0695158	−1.13545	−1.08745
	${\epsilon }_2$	0.0031	0.0034	0.0025	0.0030	0.0017	0.0011
	${\epsilon }_3$	4.5628 × 10−5	5.2482 × 10−5	6.2782 × 10−5	6.4955 × 10−5	5.1898 × 10−5	6.4819 × 10−5
	${\epsilon }_4$	−1.6843 × 10−4	−1.8463 × 10−4	−1.7365 × 10−4	−1.6513 × 10−4	−1.7365 × 10−4	−1.5179 × 10−4
	$\lambda$	16.5421	16.2484	14.6987	15.5584	15.1364	15.4589
	$R_{CR}$	0.00045	0.00034	0.00028	0.00025	0.00043	0.00040
	b	0.2812	0.3425	0.3125	0.2821	0.2165	0.4137
	SSE	3.1562E-04	2.9770 × 10−5	2.8236 × 10−5	5.8146 × 10−8	5.4628 × 10−8	6.0326 × 10−9

and

Table 12. Parameter estimation of PEMFC at different pressures for the Avista SR-12 model.

Pressure (atm)	Parameter/Algorithms	PSO	GWO	PF	GWOCS	PSOGWO	HPSOPF
1	${\epsilon }_1$	−0.9857	−0.9264	−0.8552	−0.9112	−0.8499	−1.02614
	${\epsilon }_2$	0.0027	0.0020	0.0024	0.0014	0.0016	0.0028
	${\epsilon }_3$	9.1365 × 10−5	9.8987 × 10−5	8.6987 × 10−5	9.4370 × 10−5	7.9187 × 10−5	8.5897 × 10−5
	${\epsilon }_4$	−2.1928 × 10−4	−1.2973 × 10−4	−1.9121 × 10−4	−2.7905 × 10−4	−2.9240 × 10−4	−1.0156 × 10−4
	$\lambda$	17.2367	15.6904	11.0012	16.5977	15	13.5196
	$R_{CR}$	0.00025	0.00035	0.00021	0.00042	0.00043	0.00031
	b	0.30547	0.2647	0.39712	0.20452	0.1684	0.2326
	SSE	2.9813 × 10−3	3.1246 × 10−4	2.8562 × 10−4	5.1025 × 10−7	4.9968 × 10−7	6.4516 × 10−8
2	${\epsilon }_1$	−0.9645	−0.9054	−0.953978	−0.81236	−1.0499	−1.09536
	${\epsilon }_2$	0.0020	0.0026	0.0013	0.0027	0.0019	0.0017
	${\epsilon }_3$	7.1968 × 10−5	9.3655 × 10−5	9.2417 × 10−5	8.9058 × 10−5	7.5962 × 10−5	8.6045 × 10−5
	${\epsilon }_4$	−2.5584 × 10−4	−2.9578 × 10−4	−1.3221 × 10−4	−1.8778 × 10−4	−2.8763 × 10−4	−1.4896 × 10−4
	$\lambda$	18.2212	17.2971	16	16.9248	11.2301	12.0596
	$R_{CR}$	0.00031	0.00026	0.00030	0.00043	0.00037	0.00041
	b	0.1842	0.2793	0.1204	0.3126	0.3427	0.2168
	SSE	2.8845 × 10−3	2.5953 × 10−4	2.1563 × 10−4	6.4328 × 10−7	5.9246 × 10−7	6.2235 × 10−8
3	${\epsilon }_1$	−0.8037	−0.9364	−0.9625	−0.8297	−1.0242	−0.9987
	${\epsilon }_2$	0.0011	0.0023	0.0016	0.0024	0.0020	0.0013
	${\epsilon }_3$	8.1369 × 10−5	8.8969 × 10−5	9.4963 × 10−5	7.2674 × 10−5	9.1757 × 10−5	8.1479 × 10−5
	${\epsilon }_4$	−1.7458 × 10−4	−1.4369 × 10−4	−2.6549 × 10−4	−2.8666 × 10−4	−1.2879 × 10−4	2.6934 × 10−4
	$\lambda$	11	10.6584	12.7251	13	16.8796	17.3274
	$R_{CR}$	0.00045	0.00040	0.00038	0.00027	0.0003	0.00046
	b	0.3613	0.2987	0.1254	0.1626	0.2842	0.3165
	SSE	3.0636 × 10−3	3.5413 × 10−4	2.7846 × 10−4	5.5648 × 10−7	5.1872 × 10−7	5.2982 × 10−8

show the parameter estimation of PEMFC at different pressures for both of the fuel cell models. From this, it is very clear that the proposed algorithm has better accuracy and performance than the compared algorithms. The comparison of both of the sheets was done at different temperature and pressure to check the reliability of the model.

3.5. Statistical Analysis

Table 13. Friedman ranking test.

Algorithm	Friedman Ranking
HPSOPF	1
PSOGWO	2
GWOCS	3
PF	4
GWO	5
PSO	6

shows the statistical analysis results from the PEMFC parameter estimation done using the Friedman Ranking Test [37,38,39,40] for both of the fuel cell models. First place is secured by HPSOPF, followed by PSOGWO, GWOCS, PF, GWO, and PSO. For both fuel cell models, HPSOPF is shown to perform significantly better than various other meta-heuristic algorithms, because it is more efficient, accurate, precise, and robust.

4. Conclusions

For attaining the best result for the parameter estimation of PEMFC, a novel hybrid approach called HPSOPF has been suggested in this study. The proposed algorithm is better than the compared algorithms. Two datasheets are considered in this work, i.e., Ballard Mark V and Avista SR-12. The following is a list of conclusions drawn from the results:

• Firstly, the proposed algorithm is applied on the 10 benchmark test functions to justify the algorithm. The mean and standard deviation values are calculated and it is seen that the proposed algorithm achieved better results than the other metaheuristic algorithms considered.
• Parameter estimation of PEMFC for both the models is done and the SSE and computation time are calculated. The SSE of the proposed algorithm for the Ballard Mark V model is 6.621 × 10⁻⁹ and for the Avista SR-12 model it is 5.65 × 10⁻⁸.
• The computation time for both models is also calculated. The computation time of the Ballard Mark V model is 2 s and for the Avista SR-12 model it is 2.12 s. It is clearly indicated that the proposed algorithm is better than the other metaheuristic algorithms considered.
• It is clearly demonstrated that hybrid algorithms have a faster pace of convergence when compared with other metaheuristic algorithms using convergence graphs, P−I and V−I curves, different operating temperatures, and different pressure.
• Furthermore, a non-parametric test is carried out, i.e., Friedman Ranking Test. From this test, the first rank is secured by HPSOPF and then continued by PSOGWO, GWOCS, PF, GWO, and PSO, respectively. This test is conducted to check the efficiency, robustness, and performance of the proposed algorithm.

HPSOPF highlights the advantages of hybridization; hence, it can form the reference for different new hybrid algorithms. The proposed algorithm also can also be observed in different domains such as multi optimization and image processing. Finally, hybrid algorithms can also provide improved accuracy over traditional algorithms. By leveraging the strengths of multiple algorithms, hybrid algorithms can provide more accurate results than a single algorithm alone. This can be particularly helpful for applications where accuracy is essential, such as in medical diagnosis and fraud detection. Overall, hybrid algorithms offer a variety of benefits over traditional algorithms. They can provide better results, faster processing times, and improved accuracy, making them a great choice for various applications [41,42,43,44,45,46,47]. The proposed algorithm can be further used in different applications such as solid oxide fuel cell, economic load dispatch, and the parameter estimation of solar PV models.