1. Introduction
The proton exchange membrane fuel cell (PEMFC) is an important renewable energy source that has attracted the attention of the world over the last decades. The main advantage is to convert hydrogen fuel into electrical energy, continuously and directly, for disposing of environmental pollution caused by the traditional energy sources [
1,
2]. Polarization curves that depict the relationship between current and voltage are so important and need accurate modeling. Several mechanistic and empirical, or semi-empirical, models have been designed to model the PEMFCs; among them, the semi-empirical model is more suitable for engineering purposes due to its publicity and reputation to solve problems more easily [
1,
3,
4,
5,
6,
7]. However, unfortunately, in this model, some non-mechanistic terms have to be accurately identified to design a model simulating the real PEMFCs performance better; this problem is known as the parameter estimation of PEMFCs.
This problem belongs to the optimization problems and could be tackled by a specific optimization algorithm, but not all these algorithms, especially traditional algorithms, could achieve the required accuracy due to the complex nonlinearity, multi-variability, and strong couple of the PEMFC system. Having a strong optimization algorithm to deal with this problem is inevitable. Therefore, in literature, researchers have tried new modern algorithms, namely metaheuristic algorithms, having a strong performance for solving several nonlinear optimization problems with high dimensions in a reasonable time [
8,
9,
10,
11]. Some of those that have worked to estimate these parameters will be reviewed in the rest of this section, to articulate their contributions besides their advantages and disadvantages.
In [
12], two metaheuristic algorithms, namely Harris Hawks’ optimization (HHO) and atom search optimization (ASO), have been investigated regarding their performance for finding the unknown parameters of the PEMFCs using the sum of square errors as the objective function. Those algorithms were assessed using three different commercial PEMFC stacks, 500 W SR-12 PEM, BCS 500-W PEM, and 250 W stack, under various operation conditions. In addition, the outcomes obtained by those algorithms have been extensively compared with those of several existing optimization techniques to determine their accuracy and speed; those experiments affirm that HHO is the best. However, this algorithm still suffers from falling into local optimal and low convergence speed. Those are observed from the employed maximum of function evaluations which might surpass 60,000 on account of the used population size.
For finding more accurate parameter values, in [
13], a new objective function has been designed and employed with the chaos embedded particle swarm optimization algorithm (CEPSO) for estimating the unknown parameters of three commercial PEMFCs stack. The findings have shown the superiority of this algorithm compared to some existing algorithms. This algorithm used a population size and a maximum iteration of 100 and 100, respectively, equal to 10,000 function evaluations. This large number notifies that this algorithm has low convergence speed as its main disadvantage. Additionally, Singla et al. [
14] have adopted a newly-published metaheuristic algorithm, known as black widow optimization (BWO), for finding the parameter estimation of the PEMFCs. BWO’s outcomes were compared with those obtained by five metaheuristic algorithms: particle swarm optimization (PSO), multi-verse optimizer (MVO), whale optimization algorithm (WOA), sine cosine algorithm (SCA), and grey wolf optimization (GWO). The experimental outcomes affirmed that BWO is better than all.
Zhu [
15] has employed another metaheuristic algorithm, known as the Adaptive Sparrow Search Algorithm (ASSA), to tackle this problem by minimizing the error between the measured and estimated current as of the objective function. The experimental findings for three case studies, Ballard Mark V, Horizon H-12, and NedStack PS6, elaborated the superiority of ASSA compared to three other algorithms. In [
16], the slime mold optimizer (SMA), which was recently proposed for tackling optimization problems and could fulfill superior outcomes, was employed for tackling this problem. The outcomes obtained using SMA outperform those of the compared algorithms. Diab et al. [
17] suggested a new parameter estimation model for PEMFCs based on the coyote optimization algorithm using the sum of square error as an objective function. In order to demonstrate its efficiency, it was evaluated using two PEMFCs stacks and compared four optimization algorithms to show its superiority.
Table 1 describes the contributions and disadvantages of some recently published metaheuristic algorithms for estimating the parameter of PEMFCs.
There are several other parameter estimation PEMFC techniques based on metaheuristic algorithms: improved chaotic grey wolf optimization algorithm [
26], modified farmland fertility optimizer [
18], hunger games search algorithm [
27], improved version of the Archimedes optimization algorithm [
28], moth–flame optimization [
19], Levenberg–Marquardt backpropagation algorithm [
29], whale optimization algorithm [
30], marine predator algorithm optimizer [
31], pathfinder algorithm [
32], hybrid water cycle moth–flame optimization algorithm [
33], improved fluid search optimization algorithm [
34], Seeker optimization algorithm [
35], improved grass fibrous root optimization algorithm [
36], developed coyote optimization algorithm [
37], improved TLBO with elite strategy [
38], developed owl search algorithm [
39], modified artificial electric field algorithm [
40], Supply–Demand-Based Optimization Algorithm [
41], convolutional neural network optimized by balanced deer hunting optimization algorithm [
42], and chaos game optimization technique [
43].
The algorithms mentioned above have still suffered from two common problems, low convergence speed and falling into local minima, which prevent them from reaching accurate outcomes in fewer function evaluations. Therefore, in this paper, a new metaheuristic algorithm, namely the artificial gorilla troops optimizer (GTO), proposed recently for tackling the CEC optimization problems, has been adapted for tackling this problem due to its significant success achieved for the CEC problems. In addition, an effective modification has been performed on the GTO to improve its exploration and exploitation capability in a new strong variant called the modified GTO (MGTO). Four well-known commercial PEMFC stacks were employed to investigate the performance of the GTO and MGTO, and the obtained outcomes were compared with eight well-known metaheuristic algorithms to check its superiority for finding the unknown parameters which minimize the error between the measured and estimated current. Finally, those conducted experiments show that the MGTO is better than all the others for accuracy, convergence speed, and stability. The main contributions within this paper are:
To adapt the GTO for tackling the parameter estimation of PEMFC, in addition to making a strong modification to produce a new variant, abbreviated as MGTO, having better exploration and exploitation capabilities.
Comparing the performance of the GTO and MGTO with eight metaheuristic algorithms has shown that the MGTO is superior in terms of convergence speed, stability, and final accuracy.
The remainder of this paper is organized as follows:
Section 2 explains the mathematical model of the PEMFC;
Section 3 presents the standard GTO;
Section 4 discusses the steps of the proposed parameter estimation algorithm, MGTO; comparison and discussions are shown in
Section 5; and the last section involves the conclusion and future work.
2. The Mathematical Model of PEMFC
A PEMFC is compounded of two electrodes, an anode and a cathode, as well as an electrolyte between them, as depicted in
Figure 1 [
44]. The chemical reactions start with the hydrogen (
${H}_{2}$) converted at the anode to ions (
${H}^{+}$) and electrons (
${e}^{-}$) based on the catalyst layer action, as described in (1). Afterward, both
${H}^{+}$ and
${e}^{-}$ move to the cathode through the electrolyte and the external circuit, respectively. Thereafter, to generate water and heat, the protons and electrons react together with the oxygen in the catalyst layer of the cathode, as also depicted in
Figure 1 and described in (2); meanwhile, liberated electrons move through the external circuit and generate electricity. Ultimately, the overall reactions are described in (3) [
45,
46].
The output voltage of several fuel cells connected in series, while ignoring the irreversibility losses and entropy, is computed using the following equation according to several references [
36,
48,
49,
50]:
where
${N}_{cells}$ stands for the number of fuel cells connected in series,
$E$ indicates the open-circuit voltage per cell, and estimated by the following equation:
${v}_{act}$, which indicates the activation overpotential per cell, is computed by (9), and
${v}_{R}$ is computed using (11) to determine the ohmic voltage drop in the cells, and
${v}_{c}$ is used to compute the concentration over-potential in cells is calculated according to (13).
where
${T}_{fc}$ is a phrase about the operating FC temperature in Kelvin
$P{o}_{2}\text{}$,
${P}_{{H}_{2}}$, and
${P}_{H2O}$ are the partial pressure of the oxygen (
${O}_{2})$, the hydrogen (
${H}_{2})$, and
${H}_{2}O$, respectively.
$R{H}_{c}$ indicates the relative moisture of vapor at the cathode (atm) and
$R{H}_{a}$ is the relative moisture of vapor at anode.
${P}_{c}$ is the inlet pressure of the cathode, while
${P}_{a}$ is the inlet pressure of the anode e in (atm). The concentration of the
${O}_{2}$ is symbolized using
${C}_{{O}_{2}}$ in
$\mathrm{mol}/{\mathrm{cm}}^{3}$.
${R}_{m}$ is the membrane’s resistance, while
$\text{}{R}_{c}$ is the resistance of connections.
${I}_{fc}$ is the FC operating current.
${\beta}_{2}$ stands for a parametric coefficient.
$l$ (CM) and
${\rho}_{m}$ are the thickness and the resistance of the membrane.
$J$ and
${J}_{max}$ are the density of the actual current and the maximum of
$J$ (
$\mathrm{A}/{\mathrm{cm}}^{2})$. It is obvious from the previously described equation that the values of seven unknown parameters (
${\xi}_{1}$,
${\xi}_{2}$,
${\xi}_{3}$,
${\xi}_{4}$,
$\lambda $,
${R}_{c}$, and
${\beta}_{2}$) have to be accurately extracted to build up an accurate PEMFC model. This problem is known as the parameter estimation of PEMFC and belongs to the optimization problem and hence could be solved using the metaheuristic algorithms. Therefore, in this paper, an effective metaheuristic algorithm, known as GTO, will be herein adapted, with some modification on its performance, for tackling this problem.
5. Findings and Discussions
Our proposed algorithm is validated using four commercial PEMFC stacks, 250 W stack, BCS-500W stack, and SR-12 stack, due to widespread use in the literature [
9,
34,
45,
52,
53,
54,
55,
56,
57,
58]. Those PEMFC stacks have I–V curves specified in the manufacturers’ datasheets and have to be estimated by finding the unknown parameters of its mathematical model. Each unknown parameter has a search boundary that contains the near-optimal solution in an unknown region within. Generally, the characteristics of the employed PEMFCs stack, and the lower and upper bound of the unknown parameters, are presented in
Table 2. Furthermore, the MGTO is compared with nine well-known optimizers to show its efficiency as a strong alternative to tackle the parameter estimation of PEMFC stacks; those algorithms are differential evolution (DE) [
59], grey wolf optimizer (GWO) [
59], hybrid DE with GWO (DEGWO) [
59], bonobo optimizer (BO) [
60], flower pollination algorithm (FPA) [
61], slime mold algorithm (SMA) [
16], seagull optimization algorithm [
62], horse herd optimization algorithm [
63], and classical GTO [
51]. Regarding the parameters of those algorithms, they are set as found in the cited paper, except classical GTO, such that its parameters assignment will be discussed within the next section.
All experiments conducted herein are implemented using the MATLAB platform, using a device with the capabilities:
Finally, there are two well-known metrics used to evaluate the accuracy of the obtained parameters for minimizing the error between the measured and estimated data; those metrics are mean absolute percentage error (MAPE) and mean absolute error (MAE), which are mathematically described using the following equation:
5.1. Parameter Settings
The classical GTO has three parameters,
p,
W, and
$\beta $, that have to be accurately picked to maximize its performance. Therefore, extensive experiments have been conducted using various values for each parameter. For example, the best value for both
p and
W has been picked after conducting extensive experiments with various values of 0.0, 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.99, and 1.0, which show that the best value for
p is of 0.01, as depicted in
Figure 3a, while the best for
W is competitive for 0.6, 0.8, and 0.99. Generally, values of 0.01 and 0.8 are used within the conducted experiments next for p and W, respectively. Likewise, for the parameter
$\beta $, several experiments have been conducted under virus values, and their outcomes have been depicted in
Figure 3c, which shows that a value of 2 for this parameter is better than all the other values.
5.2. Test Case 1: 500 W Stack
This section investigates the performance of our proposed algorithm, MGTO, using the first test case based on using a well-known commercial PEMFC stack called 500 W stack. At the onset, all algorithms have been prepared using the same population size and maximum of function evaluation of 25 and 5000 to achieve a fair comparison, and then all are executed 25 independent runs. The outcomes of those runs have been analyzed in terms of the best, average, worst, and standard deviation (SD), which are presented in
Table 3. In addition, this table presents the best-obtained parameters by each algorithm. As a result of observation, the MGTO could be the best for all those terms and the GTO comes in the second rank as the second-best one after the MGTO, while the SOA is the worst. Moreover, this table shows that the MGTO is better for the two additional metrics: MAE and MAPE.
To measure the convergence speed, five-number summary, and CPU time of the MGTO,
Figure 4 is presented to expose all those outcomes for each algorithm. From this figure, it is observed that the MGTO is the best in terms of faster reaching the near-optimal solution and the five-number summary depicted in the boxplot. Broadly speaking,
Figure 4a, which depicts the outcomes of various algorithms on this test case using the boxplot, shows that the proposed algorithm is the best, and the GTO is the second-best one, while the SOA is the worst one. Regarding the convergence speed shown in
Figure 4c, the MGTO could come true with the best convergence speed, and the GTO is the second-best one, while the FPA is the worst one. For CPU time depicted in
Figure 4b, the MGTO could come as the eighth one after the GWO, DEGWO, DE, FPA, SMA, SOA, and BO, but its superiority for the other metrics, apart from converging the CPU time with the best eight algorithms, makes it the best for tackling this problem.
Furthermore,
Figure 5a,b depicts the I–V and I–P polarization curves obtained by depicting the estimated and measured data; the best-so-far parameters estimated by the MGTO receive the estimated data points. From this figure, the MGTO could significantly find accurate parameters that minimize the error between measured and estimated data.
In
Table 4, the
p-value under the Wilcoxon rank-sum test, which determines if the outcomes obtained by the MGTO on test case 1 are significantly different from those of each rival algorithms, are shown. This table shows that all
p-values are less than 0.05, making the acceptance moves toward the alternative hypothesis, which says that there is a significant difference between the outcomes obtained by the MGTO and each of the others.
5.3. Test Case 2: 250 W Stack
Another well-known commercial PEMFC stack, called 250 W stack, is used to investigate the performance of the MGTO compared with some of the rival algorithms. The outcomes obtained by running the MGTO and the other rival algorithms 30 independent times on this stack are given in
Table 5. This table shows that the MGTO is the best for all employed metrics, except MAE and MAPE, which are better for GTO.
In addition, the boxplot, the convergence speed, and CPU time of each algorithm are presented in
Figure 6, which shows the superiority of MGTO in terms of the five-number summary depicted using the boxplot, and the convergence speed, while their performance is competitive for CPU time. Broadly explaining,
Figure 6a shows that the proposed algorithm is the best and GTO is the second-best, while SOA is the worst. Additionally,
Figure 7 is presented to show I–V and I–P curves between measured and estimated data points. From this figure, it is obvious that estimated data points are highly consistent with those obtained practically.
Finally, the Wilcoxon rank-sum test is used to see if the outcomes of the MGTO are different from the rival algorithms or not. The outcome results from applying the Wilcoxon rank-sum test are presented in
Table 6, which shows that the alternative hypothesis is ace with all the rival algorithms because the
p-value of each algorithm is less than 5%.
5.4. Test Case 3: SR-12 PEMFC Stack
In this section, an additional commercial PEMFC stack, namely SR-12 500 W, utilized widely in the literature, is used to affirm the effectiveness of our proposed algorithm; the proposed and rival algorithms have been executed for 30 independent runs under the same settings and their analyzed outcomes are given in
Table 7. This table (
Table 7) shows that the proposed algorithm, MGTO, is the best in terms of the best, average, worst, SD, MAE, and MAPE, while the SOA is the worst for all those metrics. In addition,
Figure 8 has been presented to show the performance of the proposed algorithm graphically compared to the others in terms of CPU time, convergence speed, and five-number summary. It is concluded from this figure that the MGTO is the best for the five-number summary, shown in
Figure 8a, and the convergence speed, displayed in
Figure 8c, while its CPU time is competitive with the others, as depicted in
Figure 8b. Furthermore,
Table 8 presents the outcomes resulting from applying the Wilcoxon rank-sum test on the outcomes obtained by the proposed algorithm against those of each one of the rival algorithms. According to the outcomes presented in this table, the MGTO’s outcomes are significantly different from those obtained by the others because the
p-value under each rival algorithm is less than 0.05. This makes the alternative hypothesis in the Wilcoxon rank-sum test accepted.
Figure 9 is presented to show the consistency of the estimated I–V and I–P curves against the measured ones, which affirms that the estimated parameters by the MGTO could reach estimated characteristics that are highly consistent with the measured ones and hence it is a strong alternative to all the existing parameter estimation techniques.
5.5. Accumulative Grade Point Assessment
In [
64], a new assessment mechanism, known as an accumulative grade point assessment (CGPA), has been proposed to rank and evaluate the performance of the various algorithms based on six factors:
Absolute Error (E);
Computational Time (t);
Standard Deviation of Error (${\sigma}_{e}$);
Standard Deviation of Time (${\sigma}_{e}$);
Consistency of Rs (λRs);
Consistency of Rsh (λRsh).
Herein, this mechanism is employed to evaluate the performance of the proposed algorithms relative to the others. However, only the first four factors are employed because it is corresponding to the current problem, parameter estimation of the PEMFC, while the other two factors correspond to the parameter estimation of the photovoltaic model. This mechanism calculates the grade point assessment (GPA) for each factor from those four factors using the following formula:
where
$\beta $ indicates the obtained value by an algorithm for an arbitrary factor,
$w$ is the weight of this factor in proportion to the others,
${\beta}_{max}$ and
${\beta}_{min}$ are the maximum and minimum values obtained by the algorithms for this factor. For the first factor, the GPA weightage was set to 2 because the absolute error is considered the most important factor used to measure the efficiency of the algorithms for tackling this problem, while the other factors were set to 0.025. Finally, the total GPA (T-GPA) is a phrase about the average GPA values obtained on the four factors. In
Table 9,
Table 10 and
Table 11, the total GPA for those four factors on three investigated test cases are presented. After observing those tables, it is concluded that the MGTO could reach the best T-GPA compared to the others although the proposed could not achieve the best value for the computational time for those test cases. Finally, the accumulative GPA values (CGPA), which result from calculating the average of the T-GPA values obtained by each algorithm on three investigated test cases, are introduced in
Table 12, which affirms the superiority of the MGTO.
6. Conclusions and Future Work
This paper presents a new parameter estimation technique for the PEMFC based on the artificial gorilla troops optimizer (GTO), which has been recently proposed for tackling global optimization problems. However, unfortunately, the GTO still suffers from falling into local optima and low convergence speed, so it is modified in this paper by replacing the exploitation operator with a new one, aiding in disposing of those problems. This modified variant, abbreviated MGTO, and the standard GTO are herein assessed using three well-known PEMFC stacks, 250 W stack, BCS-500W stack, and SR-12 stack, compared with eight optimization algorithms, SOA, DE, DEGWO, GWO, FPA, SMA, BO, and HOA, under various performance metrics such as best, average, worst, SD, CPU time, convergence curve, MAE, and MAPE, in addition to a statistical test, namely the Wilcoxon rank-sum test. The outcomes of the MGTO are better than those of the compared algorithms for the employed performance metrics on all investigated PEMFCs, except the CPU time which is competitive among the algorithms. Our future work involves finding another way to make further improvements to the MGTO to reduce the consumed CPU time, while keeping or improving the current accuracy.