Introduction

Due to a number of important factors, including the decline of conventional fuels, the trend of environmental concerns, and the rise in their price, clean energy sources have attracted significant interest around the world1. The fuel cell (FC), which can transform the chemical energy form into electrical form via chemical processes, is a reliable source of clean energy2. Even though there are myriads of FC types in the industry, Proton exchange membrane (PEM) FCs (PEMFCs) offer notable qualities such as low operating pressure and temperature, no wasted materials, and a high-efficiency level. Depending on the operating conditions, the regular temperature level ranges from 50 to 100 °C, and its efficiency is between 30 and 60%3. Over the past few decades, modeling of PEMFCs has attracted a lot of interest4. The major goal is to create a PEMFC’s model that is accurate, useful for software simulations, and closely resembles experimental models. Time and effort can be saved in this manner.

Each of these models has its own unique mathematical formulations, which include some unidentified parameters that are not displayed in the datasheets of manufacturer that are required for building an effective and reliable model5. Accordingly, myriads of handling methodologies was manifested to properly recognize the unidentified PEMFCs parameters such as adaptive filter-based6, electrochemical impedance spectroscopy-based techniques7,8, current switching methods9, and black box-based approaches10. However, these conventional optimizers are not commonly utilized to attain the adequate parameters of PEMFCs as they are inflexible and hard to implement. An electrochemical model that Mann et al.11 has developed in a semi-empirical formulation considering the steady state operation to emulate the PEMFC’s electrical characteristics. Mann's model has gained widespread acceptance over the last two decades for its ability to predict the PEMFC’s polarization properties with varied operating situations. However, Mann's unidentified parameters are tightly coupled and dramatically vary according to the load conditions which makes the model a nonlinearity issue. As a result, building the model using the aforementioned methods has become time-consuming and more complex12.

Recently, metaheuristic optimization methods have been used by many researchers to derive the PEMFC’s model required parameters owing to the major improvement of artificial intelligent-based methodologies. Metaheuristic techniques are the most effective and reliable method to use when estimating PEMFCs parameters as treated as one of optimization problems4,12,13. Regarding the optimization techniques used in this context, shark smell technique14, coyote optimization algorithm15,16, grey wolf algorithm (GWA)17, whale optimization algorithm (WOA)18, grasshopper optimization algorithm (GOA)19 bald eagle search optimizer20 and bonobo algorithm21. Besides, manta ray forage optimization (MRFO)22, pathfinder algorithm23, chaotic Harris hawk algorithm24, jellyfish searching algorithm25 and black widow algorithm26 were used to address the same problem. In addition to that, artificially ecosystem optimization27, tree-seed algorithm (TSA) and neural network algorithm28 have been applied for the same issue. Also, the researchers utilized the similar context of tree-growth algorithm29, flower pollination method30, political optimization algorithm, marine predator technique31 and slime mould optimization algorithm32 for parameter identification of PEMFCs. In33, a combination between teaching learning based optimizer and DE approach has been developed while a modified salp swarm optimizer has been presented to identify the optimal PEMFCs stack parameters in34.

Despite the benefits of self-adaptive nature optimizers, the optimizers still in necessity for improving time load, statistical analysis, and convergence rate. Accordingly, this article characterizes a Gorilla Troops technique (GTT)35,36 that uses a gorilla approach to extract the FC parameters properly. The GTT depends on several distinct behaviors of the gorillas that are mathematically simulated. In this context, five regarding behaviors are considered including traveling to other gorillas, migration to a strange region, competing for adult females, escorting the silverback, and migration toward a specified spot. These five behaviors are divided into two stages. In the exploratory stage, dual implementation of the migration approach is adopted enables for preventing becoming trapped in the local minima. At first, migration to an uncharted location to boost GTT searching capacity. Secondly, another migration approach is adopted towards a known place that greatly improves the GTT's capacity to look for various optimization spaces. Thirdly, a strategy is modeled to improve the equilibrium between investigation and exploitation by moving to the other gorillas. Additionally, the exploitation stage employs the use of two strategies, which greatly improves search efficacy. This work develops a GTT to precisely estimate the PEMFC’s model parameters. The double execution of the mutation approach in the created GTT enables for boosting the exploitation phase and preventing becoming trapped in the local minima. The conventional GTT has been successfully implemented in solving different engineering optimization issues. In37, GTT has been utilized for optimal tunning of a cascaded controller with type of Proportional Integral (PI)-Fractional Order PID to stabilize the frequency response of a microgrid with two-area power systems. In38, GTT has been developed in power networks for handling the optimal power flow. In39, GTT has been carried out on electrical distribution networks for the allocations of different types of distributed energy sources considering their probabilistic nature simultaneously with the loading uncertainties. In40, GTT has been employed on electric power networks to enhance the whole network performance with the addition of Thyristor-Controlled Series Capacitor compensators. Based on these successful implementations of the GTT and its magnificent merits of simplicity, ease of implementation, and speed of convergence, this work develops a distinctive GTT to accurately predict the PEMFC’s model parameters. The GTT has been successfully applied using two practical PEMFCs modules of BCS 500W and Modular SR-12. They are also examined at various pressures and temperatures. Additionally, several new recently motivated optimization techniques are used for comparison validation, including supply demand optimizer (SDO), Flying Foxes Optimizer (FFO) and Red Fox Optimizer (RFO).

The main points of the article are summarized as follows: (i) an advanced Improved GTT (IGTT) with a Tangent Flight Strategy (TFS) is efficiently developed for optimal PEMFCs parameters estimation considering two common industrial modules of BCS 500W, and Modular SR-12, (ii) The proposed IGTT is precisely applied for optimal diagnostics of FC modules with varying PH2/PO2 and temperature levels, (iii) The proposed IGTT provides higher accuracy and robustness compared with recently employed techniques of SDO, FFO, and RFO, and (iv) The outcomes and statistical assessments manifest the proposed IGTT superiority compared with several previously reported results which demonstrate its promising features in defining the PEMFC’s model parameters.

The rest of this article can be arranged as follows: the model of PEMFCs is elaborated in "Model of PEMFCs", whilst "Proposed methodology" illustrates the GTT plus problem formulation. In "Simulation results and discussion", the simulation, and discussion of the GTT’s results and comparisons are revealed when applied to PEMFCs stack for parameters extraction, whereas the main conclusion of the article and future extension are given in "Conclusion".

Model of PEMFCs

PEMFCs stack operation

In FCs, hydrogen and oxygen gases could be employed as a sustainable fuel for generating electrical current depending on chemical processes. In these stacks, a positive anode and a negative cathode are separated by the electrolyte. The FC anode and cathode, respectively, would receive the hydrogen and oxygen directly. As indicated in Fig. 122, protons shall flow in a transversal manner along the short internally route after electrons are created because of a sufficient catalyst moving towards the cathode to supply the necessary electrical energy to the load in the outward channel. Heat is released during the chemical process that creates pure water.

Figure 1
figure 1

Fuel cell model.

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Anode side:

$${H}_{2}\to 2{H}^{+}+2{e}^{-}$$
(1)

Cathode side:

$${O}_{2}+4{e}^{-}\to 2{O}^{-}$$
(2)

Total chemical reaction:

$${H}_{2}+\frac{1}{2}{O}_{2}\to {H}_{2}O+Electric current+Temperature rising$$
(3)

Mathematical model of PEMFCs

To form the PEMFC’s model, the I-V characteristics (Polarization curves) could be mathematically displayed. The steady state performance of the PEMFCs is described using the electrochemical simplified model presented by Mann et al.11. This paradigm is widely employed in numerous literature analyses. The mathematical model for the stack’s output voltage (\({V}_{stack}\)) as illustrated in (4), which comprises of several series-connected cells (\({N}_{cells}\))11,19,41,42.

$${V}_{stack}={N}_{cells}.({E}_{Nernst}-{v}_{act}-{v}_{\Omega }-{v}_{conc})$$
(4)

whereas vact refers to the cell activation overpotential, \({E}_{Nernst}\) is the Nernst voltage per cell, \({v}_{conc}\) indicates the concentration over-potential, and \({v}_{\Omega }\) refers to the cell ohmic drop in voltage. The voltage \({E}_{Nernst}\) can be calculated using Eq. (2) under a reference temperature of 25 °C. Thus, these three voltages drop are provided as depicted in Eqs. (5)–(8)11,41.

$${E}_{Nernst}=-0.85\left({T}_{fc}-298.15\right)/1{0}^{3}+4.3085/1{0}^{5}\times {T}_{fc}\mathit{ln}\left({P}_{{H}_{2}}\sqrt{{P}_{{O}_{2}}}\right)+1.229$$
(5)
$${v}_{act}=-\left[{\xi }_{1}+{T}_{fc}\left({\xi }_{2}+{\xi }_{3}\mathit{ln}({C}_{{O}_{2}})+{\xi }_{4}\mathit{ln}({I}_{fc})\right)\right]$$
$${\mathrm{where }C}_{{O}_{2}}=\frac{{P}_{{O}_{2}}}{5.08\times 1{0}^{6}}.\mathit{exp}\left(\frac{498}{{T}_{fc}}\right)$$
(6)
$${v}_{\Omega }={I}_{fc}({R}_{m}+{R}_{c});{R}_{m}={\rho }_{m}l/{M}_{A}$$
$${\mathrm{where }\rho }_{m}=\frac{181.6\left[1+0.03{I}_{fc}/{M}_{A}+0.062{\left({T}_{fc}/303\right)}^{2}{\left({I}_{fc}/{M}_{A}\right)}^{2.5}\right]}{\left[\lambda -0.634-3{I}_{fc}/{M}_{A}\right].\mathit{exp}\left[4.18\times \left(({T}_{fc}-303)/{T}_{fc}\right)\right]}$$
(7)
$${v}_{conc}=-\beta .\mathit{ln}\left(1-J/{J}_{max}\right)$$
(8)

where \({P}_{O2}\) and \({P}_{H2}\) illustrate the regulating pressures of oxygen (O2) (atm) and hydrogen (H2), respectively, while \(T_{fc}\) represents the working temperature of the FC (K). Moreover, \({C}_{O2}\) manifests the concentration of O2 (mol/cm3), MA signifies the membrane area (cm), whereas \({I}_{fc}\) is its current (A) and ξ1 − ξ4 characterizes semiempirical coefficients3,41. Besides, l indicates the thickness of membrane (cm), whilst Rc and Rm reveal the leads and the membrane ohmic resistances (Ω); respectively. In addition to this, \({\rho }_{m}\) demonstrates the membrane resistivity (Ω.cm), β is handled as an empirical constant, and λ is treated as a changeable parameter, whilst Jmax and J describe the maximum and actual thermal current densities (A/cm2), respectively3,19.

$$\beta =\wp .{T}_{fc}/2\alpha F$$
(9)

where F, \(\wp\) and α represent Faraday’s, ideal gas constants, and charge transfer coefficient, respectively. It becomes clear that the temperature and current density can affect the concentrating voltage drop in linear relation based on a thorough understanding of Eqs. (5) and (6). To illustrate, at higher cell temperatures and larger current densities, the concentration polarization voltage is projected to be increased11,41. It can be deduced that the seven parameters are fundamentally approximated to construct an appropriate PEMFC’s model.

Proposed methodology

Problem formulation and description

Due to a lack of manufacturer data, the PEMFC’s modelling has significant non-linear characteristics and various undetermined parameters. This signifies that creating an accurate model will be incredibly challenging. Seven parameters should be calculated. The optimization objective (FCF), in this purpose, is stated as the minimization of the sum squared error between the experimental FC voltage and estimated model voltage. Thus, the PEMFCs parameters estimation problem is approached as an objective target. The issue in the present work may be thought of as a non-convex optimization issue. The FCF is written in Eq. (10) as follows19,43.

$$FCF=Min(SSE)=Min\left({\sum }_{m=1}^{{N}_{samples}}{\left[{V}_{FC,exp}\left(m\right)-{V}_{FC,est}\left(m\right)\right]}^{2}\right)$$
(10)

where m expresses the iteration counter, \({N}_{samples}\) designates the number of measured voltage data, \({V}_{FC,est}\) manifests the FC estimated calculated voltage, and \({V}_{FC,exp}\) represents the measured output voltage of the model. The optimization objective is restricted with inequality constraints for unknown seven parameters which are the minimum and maximum limits of these parameters. The GTT is applied to optimize these seven unknown parameters which are namely, λ, ξ1 − ξ4, Rc, and β that obtain the best value of the SSE.

Gorilla troops technique

The GTT depends on several distinct behaviors of the gorillas that are mathematically simulated. Five behaviors are taken into account in this situation to optimize gorilla behavior: three for the exploration stage and two for the exploitation stage. These activities include migration to a strange region, traveling to other gorillas, migration toward a specified spot, competing for adult females and escorting the silverback. Two stages represent these strategic options that can be divided into the exploitation stage and exploration stage as will be manifested in the following subsections.

Exploration stage

Three distinct behaviors, in this stage, are elaborated: the first one is to manifest GTT exploration (which is movement to an unidentified end point), whereas the second tactic represents the traveling behavior to other gorillas. Furthermore, the third tactic aims at encouraging GTT competences in determining a myriad of calculation spaces that represents the migration toward a specified spot. Equation (11) can represent these three behaviors mathematically, where the movement to unidentified end point tactic, in this equation, is selected if a random number (rn) is smaller than a factor (Fr). Besides, the traveling to other gorillas or migration toward a specified is carefully selected if a random number equals/ (is more than) 50%.

$$GtX(Itn+1)=\left\{\begin{array}{l}LB+r{n}_{1}\times (UB-LB),Fr>rn,\\ Z\times X(Itn)\times Q+{X}_{r}(Itn)\times (r{n}_{2}-D\times \left(1-Itn/MxItn\right)),0.5\le rn,\\ X(Itn)+(X(Itn)-Go{X}_{r}(t)))\times r{n}_{3}-((X(Itn)-Go{X}_{r}(Itn)\times {Q}^{2}),0.5>rn\end{array}\right.$$
(11)
$$D=\mathit{cos}(2\times r{n}_{4})+1,$$
(12)
$$Q=D\times \left(1-Itn/MxItn\right)$$
(13)
$$Z=[-(D\times \left(1-Itn/MxItn\right)),D\times \left(1-Itn/MxItn\right)].$$
(14)

where rn, rn1, rn2, rn3, and rn4 illustrate random values among [0, 1], whilst X(Itn) and GtX(Itn + 1) define the full and forthcoming vectors of the gorilla's position. The arbitrary assignable variables Xr and GtXr could be used to ascertain a gorilla's current group and potential position. The factor (Fr) is, in the range [0:1] and characterizes the possibility of deciding on a migrating method to an unsettled location. The LB and UB are the variables ' minimum and maximum bounds. the variables D and Q could be determined mathematically by Eqs. (11)–(14). The maximum and present iteration number could be characterized by (Itn) and (MxItn), respectively. Besides, the symbol (Z) is \([-(D\times \left(1-Itn/MxItn\right)),D\times \left(1-Itn/MxItn\right)]\), while the symbol (s) is random values among [−1:1].

Exploitation stage

Two tactics in this stage are proposed when the factor \(D\times \left(1-Itn/MxItn\right)\) is compared with the variable (Y). These two behaviors are the escorting the silverback and the competing for adult females. The first one is determined when the value of Y equals/ (is less than) the value of \(D\times \left(1-Itn/MxItn\right)\), the tactic of the silverback could be selected that can directs the others to food sources. This tactic can be represented mathematically as signified in (15) as follows:

$$GtX(Itn+1)=Q\times R(Itn)\times (X(Itn)-{X}_{sb})+X(Itn)$$
(15)
$$R(Itn)={\left({\left|(1/NG){\sum }_{i=1}^{NG}Gt{X}_{i}(Itn)\right|}^{{2}^{Q}}\right)}^{(\frac{1}{{2}^{Q}})}$$
(16)

where NG is the gorillas’ population; Xsb indicates the silverback (best solution); X(Itn) is the gorilla location vector; GtXi (Itn) signifies the gorilla position o in each iteration Itn.

If the value of Y is more than the term \(D\times \left(1-Itn/MxItn\right)\), the tactic of competing for adult females is selected38. This tactic can be represented mathematically as signified in (17) as follows:

$$GX(Itn)={X}_{sb}-({X}_{sb}\times L-X(Itn)\times L)\times A,$$
(17)
$$L=2\times r{n}_{5}-1$$
(18)
$$A=\beta \times E, E=\left\{\begin{array}{c}N{G}_{1}rn\ge 0.5\\ N{G}_{2}rn<0.5\end{array}\right.$$
(19)

where L is the force of impact; rn5 is random number from [0:1]; β is pre-optimization value which is specified and set to 3; The factor (A) vector is the violence level in a fight; and E is employed as imitator for the violence efficacy.

The GtX(Itn) solution will replace X(Itn) if the fitness value of GtX(Itr) is less than X(Itn).

Improved GTT incorporating tangent flight strategy

In this part, an improved version of the GTT (IGTT) incorporating Tangent Flight Strategy (TFS). The Cauchy is computed as follows, and its tangent function is the same for the TFS44:

$$f=\mathit{tan}\left(pp\times \frac{\pi }{2}\right),$$
(20)
$$pp=randn(1,Dim)$$
(21)

where pp is a uniformly distributed arbitrary number with a value in the interval [0, 1], and Dim is the number of dimensions in the function. This operation is capable of efficiently searching the search space. This function is periodic, and it does not break the balance between both exploration and exploitation. The TFS is added to Eq. (15) by the suggested IGTT approach. The gorilla and silverback's separation will narrow as a result of this modification, drastically reducing the ultimate step size and improving the objective value. This model may be explained mathematically as follows:

$$GtX(Itn+1)=(\frac{\mathrm{tan}(\pi \times \frac{2pp-1}{2})}{100})\times Q\times R(Itn)\times (X(Itn)-{X}_{sb})+X(Itn)$$
(22)

where pp is evaluated using Eq. (19). The key steps for the proposed IGTT are illustrated as depicted in Fig. 236. As shown, the five behaviors in optimizing the gorillas are highlighted in green.

Figure 2
figure 2

Steps of the IGTT for identifying the FC unknown parameters.

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Simulation results and discussion

Firstly, a comparative assessment of the developed IGTT and the original GTT are demonstrated on the ten common, and well-known benchmark mathematical models following to the Congress on Evolutionary Computation (CEC) 2017 unconstrained benchmark functions45. Their mathematical objective model, dimensions, ranges of the control variables and their optimal objective value are announced in Table 1. The first function (F1) represents a unimodal function while the second one (F2) is a multimodal function. The functions (F3–F6) represent mixed functions, and the functions (F7–F10) are composite functions.

Table 1 Detailed definition of the ten common, well-known mathematical models under consideration.
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Table 2 shows the performance study of the developed IGTT and the original GTT for ten common, well-known mathematical models and how it compares to two well-established, and well-known optimizers such as GWA46 and particle swarm optimization (PSO)47. Also, the best regarding convergence characteristics are displayed in Fig. 3. This Table clearly shows that the IGTT performs and operates more effectively than the original GTT, PSO and GWA in the tested mathematical functions, demonstrating the robustness of IGTT in finding the best answer to these mathematical functions. From this Table, the suggested IGTT outperforms the standard GTT in 92.5% of the statistical indices of the investigated benchmark functions for the best, mean, worst, and standard deviations. Similarly, compared to the PSO, the developed IGTT outperforms it in 92.5% of the statistical indices of the investigated benchmark functions. Compared to the GWA, the IGTT outperforms it in 87.5% of the statistical indices of the investigated benchmark functions.

Table 2 Performance study of the IGTT, GTT, PSO and GWA for ten common, well-known mathematical models.
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Figure 3
figure 3

Best convergence curves of the IGTT, GTT, PSO and GWA for benchmarks.

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After that, two instances of common commercial PEMFCs stacks are discussed: the BCS 500-W and the Modular SR-12 PEM units in this study to manifest the performances of the developed IGTT to obtain parameter extraction of FC. In addition to this, different new recently inspired optimizers are implemented for comparative validation which are the original GTT, SDO, FFO and RFO. The compared techniques are performed in the MATLAB environment (MATLAB 2017b) using PC with Intel(R) Core(TM) i7-3632QM CPU @ 2.20 GHz and 8 GB RAM. For fair comparisons, similar circumstances are taken into considerations with 50 solutions as a population size and 100 iterations as a maximum number. It is commonly known that meta-heuristics have a high level of randomness. As a result, the exhibited minimal SSE results are obtained after 100 separate executions, as well as efficiency measurements to verify the correctness of the metrics set by: STD, maximum, and mean values. Table 3 displays the practical boundaries for the unidentified parameters for BCS 500 W and Modular SR-12.

Table 3 Datasheets for BCS 500W and modular SR-12 with the practical boundaries for the unidentified parameters.
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Test case 2: BCS 500 module

For the first model, based on the American Company BCS Technologies as the main manufacturer, the BCS 500 W PEMFC stack is considered where the maximum current represents 30 A, and its rated power represents 500 W48. The developed IGTT, the original GTT, SDO, FFO and RFO are performed to obtain the best parameter extraction for this model. Table 4 records their best obtained values and regarding SSE objective are illustrated besides, it tabulates the achieved parameters and the accompanying obtained SSE objectives using different published outcomes of recently inspired optimizers. The compared techniques are shuffled multi-simplexes search (SMS) algorithm2, WOA18, ant lion optimizer (ALO)27, GOA27, multi-verse optimizer (MVO)27, SSO43, sine STA (STSA)49, MRFO49, equilibrium optimization (EO)49, Improved Heap-based optimizer (IHBO)49, HHO50, atom search optimizer (ASO)50, moth-fame optimizer (MFO)51, SSO52, modified HHO (MHHO)53, fractional-order MHHO (FMHHO)53, vortex search algorithm (VSA)54 and VSA and differential evolution (VSDE)54.

Table 4 Extracted parameters using the proposed IGTT, recent, and reported optimizers for the BCS 500W stack.
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From this table, the developed IGTT derives the best performance with the smallest SSE objective value of 0.0016977 compared to GTT, SDO, FFO and RFO. The original GTT achieves SSE value of 0.0116982 where FFO, SDO and RFO obtain SSE values of 0.0119, 0.0163 and 0.0212, respectively. Moreover, compared to published results, the developed IGTT declares very high outperformance over WOA18, GOA27, MVO27, SSO43, STSA49, MRFO49, EO49, HHO50, ASO50, SSO52, MHHO53, HHO53 and VSA54. Also, the GTT shows a significant supremacy compared to ALO27, MFO51 and VSDE54 which obtain SSE objectives of 0.0119, 0.0119 and 0.01214, respectively. Additionally, the developed IGTT shows a small comparable preponderance compared to SMS2, IHBO49 and FMHHO53 which obtain SSE objectives of 0.0169778, 0.0117 and 0.01177, respectively.

To contrast the robustness validation of the developed IGTT with GTT, SDO, FFO and RFO, Fig. 4 describes the best SSE values of a 30 run times sample. As shown, the relative optimum SSE values are related to the developed IGTT where the attained results by the developed IGTT always supersede the GTT, SDO, FFO and RFO as represented in that figure. Not only that, but Table 5 illustrates their comparative assessment for the BCS 500W Stack with several other published results through the best, mean, worst and STD over the separate runs. As shown, the proposed IGTT has the best effectiveness since it acquires the least good, mean, worst and STD values of 0.011697781, 0.014329, 0.02699 and 0.0053594, respectively.

Figure 4
figure 4

Sample of the best 30 runs of the proposed IGTT, GTT, SDO, FFO and RFO for BCS 500W stack.

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Table 5 Comparative assessment of the proposed IGTT with reported optimizers for the BCS 500W stack.
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Also, Fig. 5 depicts the best convergence curves related to the IGTT, GTT, SDO, FFO and RFO for BCS 500W Stack. As shown, the IGTT has the fastest response in finding the minimum SSE in approximately 30% of the iteration’s axis.

Figure 5
figure 5

Best convergence patterns of the IGTT, GTT, SDO, FFO and RFO for BCS 500W stack.

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Based on the IGTT parameters extraction for BCS 500W Stack, Fig. 6 shows the regarding I/V and P/V characteristics compared to the related experimental recordings (The regarding values are tabulated in the appendix, kindly refer to Table A.1). As shown, excellent fittings among the simulated and measured I/V and P/V characteristics are observed.

Figure 6
figure 6

I/V and P/V curves based on the IGTT’s parameters extraction for BCS 500W stack.

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In contrast, the described polarization characteristics in terms of I/V and I/P plots are shown in Fig. 7a–c. First, the I/V curves are plotted under the pressures of \({\mathrm{P}}_{{\mathrm{H}}_{2}}/{\mathrm{P}}_{{\mathrm{O}}_{2}}\) of 1.000/0.2095 bar, 1.5/1.0 bar, and 2.5/1.5 bar; respectively, at a constant cell temperature of 333 K which are shown in Fig. 7a. Then, the temperature’s variations are simulated at 303 K, 333 K and 373 K; respectively at constant partial pressures as specified in the datasheet (i.e. \({\mathrm{P}}_{{\mathrm{H}}_{2}}/{\mathrm{P}}_{{\mathrm{O}}_{2}}\)=1.0/0.2095) which are depicted in Fig. 7b. In addition to that, the I/P curves are plotted under varied temperatures at 60 °C, 70 °C, and 80 °C, respectively as depicted in Fig. 7c. These curves are exceptionally smooth under various operating situations, offering confidence in the IGTT-based model's high efficiency.

Figure 7
figure 7

Performance curves based on the IGTT’s parameters extraction for BCS 500W stack under varied conditions.

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Test case 2: modular SR-12

The parameter extraction algorithms are thoroughly validated using the Modular SR-12 PEMFCs to verify how well the IGTT based-parameter extraction approach performs. The IGTT, GTT, SDO, FFO and RFO are performed to obtain the best parameter extraction for this model where their best obtained values and regarding SSE objective are tabulated in Table 6. In addition to this, different published outcomes of recently inspired optimizers are added in this table such as WOA18, flower pollination algorithm (FPA)30, MRFO49, EO49, STSA49, MFO51, SSO52 and interior search algorithm (ISA)55.

Table 6 Extracted parameters using the GTT, recent, and reported optimizers for Modular SR-12.
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As demonstrated, when compared to SDO, FFO, and RFO, the IGTT besides GTT have the capability to achieve the best performance with the smallest SSE target. Additionally, the formed IGTT claims the best performance with the least SSE when compared to published findings. The IGTT with GTT, SDO, FFO, and RFO's best SSE values from a sample of 30 runs are shown in Fig. 8 for the robustness comparison. The relative optimum SSE values are related to the IGTT, where the IGTT's obtained results always take precedence over the GTT, SDO, FFO, and RFO given in that figure. Additionally, Table 7 compares their evaluation of the Modular SR-12 Stack with several other published results using the best, mean, worst, and STD across many runs. As evident, the IGTT has the best effectiveness since it acquires the least good, mean, worst and STD values of 1.421E-4, 19.593E-4, 281.22E-4 and 5.277E-3, respectively.

Figure 8
figure 8

Sample of the best 30 runs of the GTT SDO, FFO and RFO for modular SR-12.

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Table 7 Comparison of the IGTT with reported optimizers for Modular SR-12 stack.
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Additionally, Fig. 9 shows the IGTT, GTT, SDO, FFO, and RFO for Modular SR-12 stack's finest convergence properties. As demonstrated, the zooming part in this figure is dedicated for illustrating the capability of the IGTT and GTT in faster reaching the optimal solution after only 35% of the iteration journey while the SDO approaches to a very close value after 92% of the journey. On the other side, RFO and FFO fails to achieve a close value through 100% of the iteration journey.

Figure 9
figure 9

Best convergence curves of the IGTT, GTT, SDO, FFO and RFO for modular SR-12.

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Interestingly, Fig. 10 depicts the relevant I/V and P/V curves in comparison to the relevant experimental recordings based on the IGTT’s parameters extraction for Modular SR-12 Stack. The generated and measured I/V and P/V curves provide excellent fits, as seen. Confirmation of this, Fig. 11 displays the regarding absolute errors between the experimental and the simulated curves (The regarding values are tabulated in the appendix, see Table A.2). It can be noticed that the maximum error of the voltage data points is lesser than 0.016% while the maximum error of the power data points is lesser than 0.015%.

Figure 10
figure 10

I/V and P/V curves based on the IGTT parameters extraction for modular SR-12.

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Figure 11
figure 11

Absolute errors of the voltage and output power based on the IGTT parameters extraction for modular SR-12 stack.

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Conclusion

In this article, an advanced IGTT incorporating a Tangent Flight Strategy (TFS) within the exploitation stage has been employed to effectively extract the PEMFC model's unidentified parameters. A precise model of the PEMFCs is created via the IGTT that delivers accurate simulation and modelling results for two industrial FCs with type of BCS 500W and Modular SR-12 Stacks. Through the IGTT development, the output voltage model of FC's total squared error between the measured and its optimally estimated is minimized for both PEMFCs stacks. Following the CEC 2017, a comparison of the established IGTT and the original GTT is performed against ten unrestricted benchmark functions. In 92.5%, 87.5%, and 92.5% of the statistical indices, the suggested IGTT beats the traditional GTT, the GWA and the PSO. Computer simulations are used to show the robustness of the PEMFC’s model across a range of temperature and pressure variations. By contrasting the numerical modeling outcomes against the experimental findings of the commercial PEMFCs stacks under study, the suggested model's effectiveness is assured. For each case study, the simulated results based on the IGTT are compared to several new optimization techniques of SDO, FFO and RFO. The employed IGTT provides higher accuracy and robustness compared with recently techniques of SDO, FFO, and RFO. Additionally, the outcomes from IGTT are contrasted with those from other optimization techniques. The outcomes and statistical assessments manifest the IGTT superiority compared with several previously reported results which demonstrate its promising features in defining the PEMFC’s model parameters. This leads to the IGTT-based model having a significant advantage over other optimizing method-based models from the literature. Consequently, the IGTT application can offer a precise PEMFC’s model.

Based on the successful application of the IGTT for PEMFC modules parameter estimation in this paper, as a future research trend, the IGTT algorithm is recommended to be employed to solve further advanced engineering problems especially in power systems such as controllers design for power system stability including renewable sources, battery models identification, optimal operation of power systems with renewable sources penetrations.