1 Introduction

In the last decades, the necessities of using clean energy sources are rapidly increasing due to the ecological devastating impacts of the fossil fuels [1,2,3,4]. As a result, renewable energy sources (RESs) are targeted to somehow replace the conventional ones due to their advantageous characteristics. Virtually, no emissions, static nature of most RESs types, higher efficiency, availability upon wide range of output power (from mW to MW’s) and convenient for all applications (portable, transportation and stationery). Amongst the several alternatives of RESs; fuel cells (FCs), solar and wind have attracted decision-makers and industry-stakeholders to utilize them as prime energy supplies [1,2,3, 5].

Particularly, FCs have been considered as a new booming energy conversion source as they have penetrated several applications whether portable, stationery or transportation [1, 2, 6]. Depending on the electrolyte type, FCs are categorized and utilized in the market for different applications [1, 2]. Examples of FC types are; (i) Proton exchange membrane FCs (PEMFCs) [7, 8], (ii) Solid oxide FCs (SOFCs) [9, 10], (iii) Molten carbonate FCs (MCFCs) [11], (iv) Phosphoric acid FCs (PAFCs) [8], (v) Alkaline FCs (AFCs) [12, 13] and furthermore. Specifically, the basic characteristics of the latter-mentioned types are illustrated in Table 1.

Table 1 Basic features of five kinds of FCs

PEMFCs have distinguished by their flexibility, high power density, short startup time, fast response for load changes, low operating temperatures and pressures and no safety issues. Thus, they have been involved in the widest range of applications, mainly in transportation applications [1, 2, 14,15,16,17,18]. Nevertheless, the expensive cost of PEMFCs hinders their competitive penetration in the market [1, 2].

To appraise the features of PEMFCs systems, several modelling methods have been proposed, such as theoretical [19], empirical [20], and semi-empirical [21] models. Herein, a semi-empirical model, developed by Amphlett is adopted to simulate the polarization characteristics, which are represented by the output current versus voltage (I–V) curve, under various operating situations [21, 22].

In addition, precise PEMFC modelling is crucial for assessing the performance [23], optimum controlling [23], accurate simulating [24], and maximum power tracking [25, 26] of the PEMFCs units. Thus, in the recent years, numerous researchers have attempted to define the unknown parameters of PEMFCs’ model by the aid of several parameter estimation techniques. Examples of these techniques are electrochemical impedance spectroscopy-based approaches [27,28,29], black box-based methods [30, 31], adaptive filter-based techniques [32,33,34], current switching methods [35] and many more.

Nevertheless, these techniques are not broadly utilized because of their plain drawbacks, such as inflexibility and impracticability [7]. Moreover, the PEMFC’s model is high nonlinear, multi-variant and its variables are strongly coupled and severely affected by the operating conditions. Hence, the dynamics and accurate modelling have become more complicated and time-consuming when employed by such conventional techniques [36,37,38,39,40,41,42]. Therefore, there is an imperative demand to derive robust and feasible techniques to tackle such issue.

Since the rapid development of computer-based and artificial intelligence (AI)-based methods, meta-heuristic algorithms (MHAs) have attained distinct results when applied on several highly nonlinear optimization problems [43]. Principally, the problem of defining the unknown parameters of the PEMFCs can be easily dealt with as an optimization task. So, utilizing MHAs, as a reliable and precise tool to obtain the optimal solutions offers low computational effort and high accuracy.

Consequently, numbers of MHAs are employed for extracting the unspecified parameters of PEMFCs. Samples of such approaches are genetic algorithm [44, 45], differential evolution [46, 47], artificial bee colony [48], backtracking search algorithm [49], Biogeography-based optimizer [50], Artificial bee swarm optimizer [51], seeker optimizer [52], Artificial immune system [53], quantum-based optimizer [54], and etc.

Thereupon, an extensive survey article is urgently demanded to summarize these MHAs. A previous survey on several MHAs for PEMFC’s parameter estimation was performed in [5], where only fifteen algorithms are illustrated without methodical classification and comparison among various approaches. Furthermore, detailed discussions and simulation consequences of the algorithms have not been addressed. In addition, another recent survey was undertaken in [7], where only twenty-eight algorithms are discussed. However, it lacks details about the various PEMFC’s models in terms of their classification as well as their applications. Moreover, Amphlett model hasn’t been fully described in terms of mathematical representations.

Thus, a comprehensive literature survey on numerous MHAs implemented for extracting the unknown parameters of PEMFC’s model is undertaken. In-accordance, this paper can represent a unified reference for future in-depth research projects in the same field, while the major contributions can be epitomized as follows:

  1. (i)

    An overall discussion of various PEMFC models, besides their categories are presented,

  2. (ii)

    A summarized table gathering twenty-seven up-to-date models with their category-fulfill features,

  3. (iii)

    A total of thirty up-to-date MHAs with several simulation results are discussed, which are classified into four categories, swarm-based, nature-based, physics-based, and evolutionary-based, respectively, and

  4. (iv)

    A concluded Table offers the main characteristics of the MHAs, besides the technical specifications and operating conditions of each mentioned paper.

The rest of this paper is structured as follows: an overview of various PEMFC models found in the literature is revealed in Sect. 2. The mathematical model of PEMFC is described in Sect. 3. Section 4 announces some broadly utilized assessment criteria. Numerous MHAs for parameter identification of PEMFC are thoroughly illustrated in Sect. 5. The subsequent discussion is provided in Sect. 6. Finally, Sect. 7 represents the conclusion.

2 PEMFCs’ Generating Unit Models

Accurate modelling plays a significant role when the investigation of the FC’s performance is needed. FC models can be categorized in terms of scale, approach, state, spatial dimensions, and covered phenomena, as depicted in Fig. 1 [55, 56].

Fig. 1
figure 1

Various modelling categories of PEMFC

A summarized comparison among the various FC model techniques is tabulated in Table 2 [57].

Table 2 Summary of the comparative characteristics of various FC model techniques

It’s worth saying, some of the recent publications relative to the various FC model specifications are presented in Table 3.

Table 3 Various recent examples of PEMFC models

3 PEMFC Mathematical Representation

As previously illustrated, since the FC is a multi-physics device, there are various classifications of the PEMFC models. Each model is constructed to tackle a certain aspect of the PEMFC. In this article, the polarization features’ effects on the PEMFC performance are particularly addressed in terms of the thermodynamics potential, activation, ohmic and concentration losses. The difficulty of modelling the PEMFC polarization characteristics stems from their complexity, multi-variance, and strong coupling. However, the mathematical semi-empirical model deduced by Amphlett and Mann [21, 22] has gain the acceptance through its accurate forecasting the performance of PEMFC in form of I–V polarization curve under several steady state and dynamic operating conditions [5, 7]. Accordingly, numerous researchers have been biased to utilize this model for identifying the performance and the polarization characteristics of the PEMFC.

As depicted in Fig. 2, the I–V polarization curve of a single PEMFC can be divided into three regions: activation, ohmic and concentration losses, respectively. At the startup period with light load, a rapid decay of the PEMFC output voltage is noticed. This is due to the initial slow rate of the electrochemical reactions which is represented by the activation losses. Then, a linear decay of the output voltage due to the total resistance seen by the protons and electrons, which is represented by the ohmic losses. Again, the output voltage rapidly falls at higher load conditions due to the excessive water content reducing the concentration of the reactants in both electrodes. This voltage drop is represented by the concentration losses [1,2,3, 5, 7, 85].

Fig. 2
figure 2

A typical I–V curve of the PEMFC

Hence, the total output voltage of a single cell \({V}_{o/cell}\) is given by (1) [22]:

$${V}_{o/cell}={E}_{nernst}-{V}_{act}-{V}_{\Omega }-{V}_{con}$$

where \({E}_{nernst}\) is the cell open circuit voltage in (\(V\)) that is calculated by the Nernst equation. \({V}_{act}\) is the activation over-potential in (\(V\)). \({V}_{\Omega }\) is the ohmic voltage drop and \({V}_{con}\) is the concentration over-potential in (\(V\)).

When connecting \({N}_{C}\) multiple cells in series to form the stack, the stack output voltage \({V}_{o/stack}\) in (\(V\)) is described by (2).

$${V}_{o/stack}={N}_{C}\times {V}_{o/cell}$$

It’s worth mentioning, Eq. (2) assumes that all the connected cells exhibit the same polarization characteristics.

\({E}_{nernst}\) is calculated by the Nernst equation with the addition of the temperature variation effect, which is given by (3) [86].

$${E}_{nernst}=1.22-8.5\times {10}^{-4} (T-298.15)+4.3085\times {10}^{-5}\times T\left[ln\left({P}_{{H}_{2}}\sqrt{{P}_{{O}_{2}}}\right)\right]$$

where \(T\) is the cell operating temperature in (\(Kelvin\)) and \(T\le 100^\circ{\rm C}\). \({P}_{{H}_{2}}\) and \({P}_{{O}_{2}}\) are the partial regulating pressures of the hydrogen and oxygen in (\(atm\)), respectively.

Since \({P}_{{H}_{2}}\) and \({P}_{{O}_{2}}\) vary with the load current values, they are expressed as follows:

$${P}_{{H}_{2}}=0.5\times {R}_{{H}_{a}}\times {P}_{{H}_{2}O}\times \left[\left[\frac{1}{\frac{{R}_{{H}_{a}}\times {P}_{{H}_{2}O}}{{P}_{a}}\times exp\left(\frac{1.635{I}_{cell}}{A{T}^{1.334}}\right)}\right]-1\right]$$

While, calculating \({P}_{{O}_{2}}\) is dependable of whether the oxidant is pure oxygen or natural air. Thus, Eqs. (5) and (6) are applied for pure oxygen and natural air, respectively [86].

$${P}_{{O}_{2}}={R}_{{H}_{c}}\times {P}_{{H}_{2}O}\times \left[\left[\frac{1}{\frac{{R}_{{H}_{c}}\times {P}_{{H}_{2}O}}{{P}_{c}}\times exp\left(\frac{4.192{I}_{cell}}{A{T}^{1.334}}\right)}\right]-1\right]$$
$${P}_{{O}_{2}}={P}_{c}-{R}_{{H}_{c}}\times {P}_{{H}_{2}O}-\frac{0.79}{0.21}\times {P}_{{O}_{2}}\times exp\left[\frac{0.291{I}_{cell}}{A{T}^{0.832}}\right]$$

where \({R}_{{H}_{a}}\) and \({R}_{{H}_{c}}\) are the relative humidity of the vapor at anode and cathode, respectively. \({P}_{{H}_{2}O}\) is the saturation pressure of the water vapor in (\(\mathrm{atm}\)). \({P}_{a}\) and \({P}_{c}\) the anode and cathode inlet pressure in (atm), respectively. \({I}_{cell}\) and \(A\) are the cell current in (\(\mathrm{A}\)) and the membrane effective area in (\({\mathrm{cm}}^{2}\)), respectively.

As \({P}_{{H}_{2}O}\) depends on the cell operating temperature, it’s given by:

$${log}_{10}({P}_{{H}_{2}O})=2.95\times {10}^{-2}\left(T-273.15\right)-9.18\times {10}^{-5}{\left(T-273.15\right)}^{2}+1.44\times {10}^{-7}{\left(T-273.15\right)}^{3}-2.18$$

To express the reactions slowness when starting to load the cell, the activation losses \({V}_{act}\) is given by:

$${V}_{act}=-[{\varphi }_{1}+{\varphi }_{2}T+{\varphi }_{3}Tln({Co}_{2})+{\varphi }_{4}Tln({I}_{cell})]$$

where \({\varphi }_{j} (j=1\dots .4)\) are defined as the semi-empirical coefficients in (\(V,V{K}^{-1},V{K}^{-1},V{K}^{-1}\)) and \({Co}_{2}\) represents the concentration of the oxygen at the catalytic layer of the cathode in (mol/cm3) which is expressed by:

$${Co}_{2}=\frac{{P}_{{O}_{2}}}{5.08\times {10}^{6}}\times exp\left(\frac{498}{T}\right)$$

Also, the concentration of the hydrogen at the catalytic layer of the anode \({C}_{{H}_{2}}\) in (mol/cm3) is described by [87]:

$${C}_{{H}_{2}}=\frac{{P}_{{H}_{2}}}{10.9\times {10}^{5}}\times exp\left(\frac{-77}{T}\right)$$

Moreover, the ohmic voltage drop \({V}_{\Omega }\) which exhibits a linear relation in the polarization curve is given by:

$${V}_{\Omega }={I}_{cell}({R}_{M}+{R}_{C})$$

where \({R}_{C}\) in (\(\Omega\)) represents the resistance shown by the electrons through the connections to the external circuit. \({R}_{M}\) in (\(\Omega\)) indicates the resistance shown by the protons, through the membrane active area \(A\) in (\({\mathrm{cm}}^{2}\)) and it’s described by:

$${R}_{M}={\rho }_{M}\left(\frac{l}{A}\right)$$

where the membrane thickness is symbolized by \(l\) in (\(\mathrm{cm}\)). The specific resistivity of the membrane is symbolized by \({\rho }_{M}\) in (\(\mathrm{\Omega cm}\)) and it’s given by:

$${\rho }_{M}=\frac{181.6\left[1+0.03\left(\frac{{I}_{cell}}{A}\right)+0.062{\left(\frac{T}{303}\right)}^{2}{\left(\frac{{I}_{cell}}{A}\right)}^{2.5}\right]}{\left[\gamma -0.634-3\left(\frac{{I}_{cell}}{A}\right)\right]exp\left(4.18\left(\frac{T-303}{T}\right)\right)}$$

where \(\gamma\) is a unitless parametric coefficient that indicates the water content in the membrane.

Lastly, the phenomenon, that affects the I–V curve when heavy loading the FC, is the concentration over-potential \({V}_{con}\) or mass transport losses and it’s determined by:

$${V}_{con}=-\beta ln\left(\frac{{J}_{Max}-J}{{J}_{Max}}\right)$$

where \(\beta\) is a parametric coefficient in (V), \({J}_{Max}\) and \(J\) are the maximum cell current density and the actual operating current density in (\(\mathrm{A }{\mathrm{cm}}^{-2}\)), respectively.

4 Model Identification and Assessment Criteria

It’s self-explanatory from the aforesaid formulas that for obtaining a fully defined electrochemical-based model, at least seven parameters (\({\varphi }_{1}\), \({\varphi }_{2}\), \({\varphi }_{3}\), \({\varphi }_{4}\), \({R}_{C}\), \(\gamma\), and \(\beta\)) are assigned. Nonetheless, the indispensability, difficulty, and complexity of the model identification process stem from the significant dependence of the model parameters on the operating conditions. As expected, the quality of the polarization curves and the heavy nonlinear characteristics of such model are significantly affected. In addition, the unknown parameters are strongly coupled and aren’t illustrated in the manufacturer’s specifications sheet [57]. Thus, to simply, accurately and with low time-effort define the unknown parameters, it is studied as an optimization problem and solved by numerous optimization techniques. Amongst the various optimization methods, AI-based approaches are widely utilized to obtain the anonymous parameters of the PEMFC model [43]. Summarily, the procedures of modelling of the PEMFCs stacks, depending on the extracted information from the datasheet and the experiment-based data, are depicted in Fig. 3.

Fig. 3
figure 3

The sequence of PEMFC modelling

Not only the identification method that affects the accuracy of the estimated parameters but also the assessment criteria play a vital role to precisely determine the unknown parameters and fit the calculated I–V curve to the experimental one. Proper picking of the objective function (OF) simplifies the parameter determination process and differentiates among the various model identification methods quantitatively and qualitatively in terms of the acceptable range of results. Thus, a summary of the most utilized OFs in the parameter estimation of PEMFC, is collected in Table 4.

Table 4 Summery of the popular-used OFs

5 MHAs for PEMFC’s model parameters identification

Amongst the various AI-based optimization techniques, MHAs have proven their accuracy and higher computation efficiency when compared with the other conventional optimization techniques [7]. Therefore, MHAs have been adopted to get the optimum solutions for several engineering problems such as power systems problems, as presented in [95,96,97,98,99,100,101,102,103,104]. Besides, depending on the no free-lunch theory [105], many researchers have utilized various algorithms for accurate and effective investigation of the polarization characteristics of the PEMFC, as reported in [5, 7, 16, 57]. It’s worth saying that this section highlights the most recent works regarding presenting a new MHA to identify the undefined parameters of the PEMFC based on the semi-empirical model given in [21, 22]. One can track the procedures depicted in Fig. 4 to implement the MHAs for obtaining the unknown parameters of the PEMFC model. MHAs can be classified into four categories: swarm-based, nature-based, physics-based, and evolutionary-based, respectively. The reader can get a brief knowledge about the technical specifications of the most widely employed PEMFCs types in the commercial market when perusing Table 5. Also, the lower and upper limits of the PEMFC unknown parameters, popularly used in the state-of-art, are encapsulated in Table 6 [57].

Fig. 4
figure 4

The main steps for implementing MHAs for PEMFC’s parameter estimation

Table 5 Summary of technical specifications of various commercial PEMFCs
Table 6 The minimum and maximum boundaries of the PEMFC’s unknown parameters

5.1 Swarm-Based MHAs

5.1.1 Grasshopper Optimizer (GO)

The GO imitates the behaviors of grasshopper swarms when searching for food. Substantially, to mathematically derive the GO equations, the intermediate forces among the agents are classified as attraction and repulsion forces [106]. Furthermore, GO merits can be summarized as follows: (i) avoiding falling into local minima as GO is strongly capable of balancing between exploration and exploitation, (ii) fast convergence and (iii) simple controlling variables [106, 112]. Thus, it has been utilized in the parameter identification of PEMFC, while promising results are incapsulated in Table 7 in the second row [112].

Table 7 PEMFC parameter estimation using swarm-based MHAs

5.1.2 Grey Wolf Optimizer (GWO)

The GWO mimics the intelligent attitude of grey wolves when attacking their preys. As they live in groups, their tasks are assigned according to the swarm hierarchy where the wolves are divided into four kinds: alpha, Beta, delta, and omega. The hunting process passes through three stages, pursuing and chasing, encircling, and attacking which mathematically represent the exploration, OF evaluation and exploitation phases, respectively [107, 113]. Due to GWO advantages such as simple tuning process and lower computational time and burden, it has been employed in the parameter estimation of the PEMFC model, while the results are gathered in Table 7 in the third row [107].

5.1.3 Salp Swarm Algorithm (SSA)

The SSA mimics the salps collective attitude when searching for food in oceans, where the swarm members are combined to form chains. Mathematically, the salp chains are modeled by classifying the group individuals into leader and followers. Thus, the leader position during the searching process can be formulated as in (15) [108, 114].

$${X}_{i}^{j}=\left\{\begin{array}{c}{S}_{j}+{b}_{1}\cdot\left[\left({HL}_{j}-{LL}_{j}\right)\cdot{c}_{2}+{LL}_{j}\right], {b}_{3}\ge 0\\ {S}_{j}-{b}_{1}\cdot \left[\left({HL}_{j}-{LL}_{j}\right)\cdot{c}_{2}+{LL}_{j}\right], {b}_{3}<0\end{array}\right.$$

where \({X}_{i}^{j}\) denotes the leader position, the food source position in the jth dimension is symbolized by \({S}_{j}\). \({HL}_{j}\) and \({LL}_{j}\) represent the higher and lower limits of the jth dimension, while \({b}_{1}\), \({b}_{2}\) and \({b}_{3}\) are three random numbers, respectively. SSA has the capability of effectively enhancing the initial haphazard solutions and rapidly converging into the optimum ones. As a result, it has been adopted to estimate the undefined parameters of the PEMFC, as elucidated in Table 7 in the fourth row [108].

5.1.4 Shark Smell Optimizer (SSO)

The SSO imitates the hunting mechanism of the sharks when sensing the prey smell. Employing their strong sense of smelling the prey blood, the hunting process depends on three assumptions: (i) the shark velocity is much greater than the prey velocity. So, the prey is assumed stationary. (ii) The blood is uninterruptedly flowed out from the prey to the sea and the propagation of the prey smell particles isn’t affected by the flow of the seawater. (iii) Only one prey (seeking environment) is existed in the search domain [115,116,117]. Accordingly, SSO exhibits the merits of high precision, low computational effort, and high convergence trend. Consequently, its results have outperformed the other conventional algorithms when utilized in the parameter estimation of the PEMFC model, as indicated in Table 7 in the fifth row [117].

5.1.5 Cuckoo Search Optimizer (CSO)

The CSO mimics the brood parasite attitude of cuckoos where the cuckoos put their eggs surreptitiously in the host birds’ nests. Fraudfully, the cuckoos try to empty the nests of the host birds out of their own eggs, keeping only the cuckoos’ eggs to enhance the hatching amount. The cuckoos implement a haphazard strategy to pick the host nest. Furthermore, the basic construction of CSO depends on three concepts, relying on which each cuckoo produces next solutions referring to (16) [118,119,120].

$${XP}_{i}^{t+1}={XP}_{i}^{t}+\lambda \oplus Levy (\varepsilon )$$

where \({XP}_{i}\) represents the ith egg position, \(\lambda >0\) indicates the step size, the element-wise multiplications are symbolized by \(\oplus\) and \(\varepsilon\) denotes the exponent of Lévy flight.

For the sake of effectively and robustly employ this technique in PEMFC parameter identification, a novel modified approach, called CS-EO, is developed in [119]. In CS-EO, CSO is merged with the explosion operator (EO), derived from fireworks algorithm (FWA). Consequently, the newly proposed hybrid algorithm can effectively improve the search capability and evade from being trapped into local minimum solutions. Eventually, the sixth row in Table 7 elucidates the obtained results from adopting CS-EO in identifying the PEMFC model unknown parameters [119].

5.1.6 Whale Optimizer (WO)

The WO is inspired by the smart attitude of humpback whales when hunting a swarm of small fishes at the ocean surface, which is called bubble net feeding. These whales start the hunting process by diving deeply and surrounding the prey by bubbles. Thereafter, the whales arise to the surface for trapping the small fishes. Consequently, the hunting process mainly includes three phases, encircling prey, making bubbles with various shapes, and looking for the prey. Mathematically, the whales update their positions based on (17)–(18). Upon ending the encircling phase, the whales will try to attack the victim through creating bubbles which mathematically refers to the local exploitation [110, 121].

$$Y\left(t+1\right)={Y}_{v}\left(t\right)-A\cdot B$$
$$B=\left|D\cdot {Y}_{v}\left(t\right)-Y(t)\right|$$

where \(Y(t)\) is the position vector of the whales, \(t\) refers to the iteration counter, \({Y}_{v}\left(t\right)\) is the position vector of the target victim and \(A\) and \(D\) represents the coefficient vectors.

Due to the robustness, accurateness, and effectiveness of WO, it has been applied for the sake of estimating the unknown parameters of several commercial types of PEMFC, as indicated in Table 7 in the seventh row [110].

5.1.7 Bonobo Optimizer (BO)

The BO is inspired by the social attitudes and reproductive process of Bonobos. Essentially, the Bonobos mating process depends on the fission–fusion concept which describes the techniques used by Bonobos groups for looking for food and other. Moreover, the Bonobos lifestyle is classified into two stages. Firstly, the positive stage (PS) at which all living circumstances are available like food, proper mating, and protection. Secondly, the negative stage (NS) which refers to the lack of these circumstances [122]. Recently, the authors in [122] have utilized BO for the sake of proper identifying the unknow parameters of three commercial PEMFCs, due to its ability to converge to the global solution smoothly and rapidly, while the results are indicated in Table 7 in the eighth row.

5.1.8 Harris Hawk’s Optimizer (HHO)

The HHO mimics the hunting strategy of Harris hawks when chasing their preys (rabbits). Mathematically, like any population-based approaches, HHO is divided into two phases, exploration, and exploitation. In the exploration phase, Harris hawks employ two tactics to discover the prey where the first tactic supposes that the hawks’ position is near to the group members and the prey. On the other hand, the second one assumes that the hawks allocate on stochastic trees [123, 124]. Furthermore, an improved HHO using chaotic equations, called chaotic HHO (CHHO), is presented in [86] where the convergence trend is enhanced. As foreseen, CHHO has been employed for identifying the unknown parameters of several commercial types of PEMFCs, while the results are revealed in Table 7 in the ninth row.

5.1.9 Coyote Optimizer (CO)

Basically, the CO imitates the coyotes’ attitude not only while chasing preys but also the community framework of coyotes and the experiences interchange by the coyotes. Mathematically, the coyote’s population consists of \({N}_{p}\) packs, each pack has \({N}_{c}\) coyotes which is a static number and fixed for all packs. Moreover, each coyote represents a feasible solution for the optimization task and its social circumstance is the cost of the fitness function [125, 126]. CO exhibits advantageous features such as low computational effort, simple tuning variables, and fast convergence trend. Consequently, it has been applied for defining the unspecified parameters of the commercial 250 W PEMFCs stack, while the results are capsulated in Table 8 [127].

Table 8 PEMFC parameter estimation using CO

5.1.10 Manta Rays Foraging Optimizer (MRFO)

The MRFO is inspired from the menta ray’s mechanism when searching for food (plankton). Hence, the foraging mechanism is subdivided into three phases such as (i) chain foraging, (ii) cyclone foraging, and (iii) somersault foraging. In the chaining phase, the positions of menta rays are updated to the best position of plankton where the plankton exists with high concentration using the mathematical equations given in (19)–(20) [78, 128].

$$X_{i,d} \left( {it + 1} \right) = \left\{ {\begin{array}{*{20}l} {F_{i,d} \left( {it} \right) + r\cdot \left( {F_{b,d} \left( {it} \right) - F_{i,d} \left( {it} \right)} \right) + w\left( {F_{b,d} \left( {it} \right) - F_{i,d} \left( {it} \right)} \right) } \hfill & {i = 1} \hfill \\ {F_{i,d} \left( {it} \right) + r\cdot\left( {F_{i - 1,d} \left( {it} \right) - F_{i,d} \left( {it} \right)} \right) + w\left( {F_{b,d} \left( {it} \right) - F_{i,d} \left( {it} \right)} \right)} \hfill & {i = 2, \ldots N_{pop} } \hfill \\ \end{array} } \right.$$

where \({F}_{i,d}\left(it\right)\) denotes the location of the ith member at itth iteration in dth dimension. \(r\) is a randomly generated vector from 0 to 1, \(w\) represents a weight coefficient and \({F}_{b,d}\left(it\right)\) refers to the plankton best location (plankton with high concentration). \({N}_{pop}\) denotes the population size.

It’s worth saying that MRFO requires lesser effort to fine-adjust its controlling parameters. therefore, the authors in [78] have applied it for extracting the unknown parameters of three commercial types of PEMFCs under various operating conditions, where the results are indicated in Table 7 in the tenth row.

5.1.11 Monarch Butterfly Optimizer (MBO)

The MBO mimics the migration mechanism of monarch butterflies which can be achieved by following the subsequent rules. Starting by dividing the whole population into individuals located only in Land 1 or Land 2. Ending by assuming that the movement to the next generation of the MB individuals, having the best fitness, is automatically performed and there is no operator can change them. Hence, this will prevent any deterioration for the MB population and maintain the effectiveness of the population while increasing the generations [129, 130].

However, a major drawback of MBO is that it sometimes gets trapped into local optimum which leads to premature convergence. As a result, a modified MBO (MMBO) has been utilized in [130] to solve this issue. In MMBO, two mechanisms have been integrated with the basic MBO where the first is the mutation operator and the second is the anti-cosine operator, While the procedures of MMBO are revealed in Fig. 5. Accordingly, MMBO has been utilized for tackling the parameter estimation problem of the PEMFCs, while the results are depicted in Table 7 in the eleventh row [130].

Fig. 5
figure 5

MMBO flowchart

5.1.12 Black Widow Optimizer (BWO)

The BWO imitates the distinguished mating attitude of black widow spiders. Fundamentally, this reproductive behavior is composed of an exclusive phase called cannibalism. Thanks to this phase, the individuals with unwanted fitness value are excluded from the circle, resulting in early convergence. While the main phases of BWO are revealed in Fig. 6 [131, 132].

Fig. 6
figure 6

BWO flowchart

Consequently, BWO has been applied for the parameter estimation of two commercial PEMFCs, while the results are depicted in Table 7 in the twelfth row [132].

5.1.13 Sparrow Search Optimization Algorithm (SSOA)

Fundamentally, the SSOA simulates the collective wisdom, countering the predators and foraging attitudes of sparrows. Specially, the sparrows are divided into two types of producers and scroungers where the producers seek for obtain their food source. On the other hand, the scroungers are fed by the producers [133, 134]. Despite the advantages of SSOA, such as accuracy, stability and robustness, the convergence speed may be slowed down due to haphazardness walk strategy. Thus, an improved version of SSOA, called improved SSOA (ISSOA), has been introduced in [134] to refine this problem. Basically, ISSOA combines the basic SSA with an adaptive learning factor. As a result, ISSOA has been utilized for extracting the unknown parameter of the PEMFCS, while the results are indicated in Table 7 in the thirteenth row [134].

5.1.14 Jellyfish Search Optimizer (JSO)

The JSO is motivated by the search mechanism and movement of jellyfish in the ocean. Specifically, JSO is composed of three concepts: (a) jellyfish track only one governing rule either the ocean current or inside-swarm movement (passive or active) and a time control technique. (b) Jellyfish are interested towards the locations which contains large amount of food. (c) The food quantity is assigned, and its corresponding cost fitness value is determined accordingly [135, 136]. Fewer control parameter and lesser computational efforts and random trials are such the JSO merits. Hence, the authors in [136] has applied JSO in the field of estimating the PEMFCs unknown parameters, while the outcomes are encapsulated in Table 7 in the fourteenth row.

5.1.15 Pathfinder Optimizer (PFO)

The PFO emulates the haphazard motion and performance of animal’s groups following their leader according to their adjoining place, searching for food location or prey. Primarily, PFO is based on the animal competitors’ movement which are gathered in groups. On the other side, the way to the target may be picked by the collaboration between the leader and some competitors who have sufficient information. Moreover, there are four parameters that has been tuned to refine the competitors’ attitude in the exploration phase, which are (a) the oscillation frequency of the competitors, (b) the competitors’ vibration, (c) the communication parameter, and (d) the attraction parameter [137, 138].

PFO has been utilized for extracting the unknown parameters of two commercial PEMFCs. Besides, it has been involved for investigating the steady state and dynamic operation of the PEMFCs, while the results are tabulated in Table 7 in the fifteenth row [138].

5.2 Nature-Based MHAs

5.2.1 Flower Pollination Optimizer (FPO)

Principally, FPO is inspired by the pollination behavior of flowers. Hence, the pollination process can be classified into two techniques, self-pollination (abiotic) and cross pollination (biotic). Herein, the self-pollination, in which pollens of the same flowers inseminate to emanate new flowers via wind as the pollinating medium. While, in the cross pollination, pollens of various flowers inseminate to emanate new flowers via bats, honeybees and birds as the pollinating medium [139].

FPO is characterized by enhancing the convergence tendency, simple variables tuning effort and effectively balancing the global exploration and the local exploitation. Consequently, the authors in [109], have utilized FPO for the parameters estimation purpose of the PEMFC, while the results are indicated in Table 9 in the second row.

Table 9 PEMFC parameter estimation using nature-based MHAs

5.2.2 Neural Network Optimizer (NNO)

The NNO emulates the artificial neural network (ANN) attitude, in which obtaining the incoming data and the desired data is established first and thereafter forecasting the relation between them. Referring to the ANN framework, it can be divided into two frames feed-forward and frequent NNs. Moreover, it depends on its framework whichever open-loop or closed-loop (feedback). Specifically, in NNO, the targeted solution at a certain iteration is regarded as the output data and the principal aim is to reduce the deviation between this targeted solution and the other forecasted solutions [140, 141].

Additionally, low tuning efforts of the NNO controlling variables and low computation burdens are such the NNO benefits. Hence, the authors in [141] have implemented the NNO to properly define the unknown parameters of the PEMFC, while the results are indicated in Table 9 in the third row.

5.2.3 Firefly Optimizer (FFO)

The FFO is inspired by the social attitude of fireflies when courting mates. Basically, FFO procedures relies on three fundamental assumptions, (i) all the fireflies are considered as hermaphrodite where all the agents in the population seek for the same target. (ii) the attraction possibility of the fireflies is proportional to the intensity of their flashlights. (iii) the brightness strength of a firefly is determined by the objective function [111, 142].

Moreover, FFO procedures are revealed in Fig. 7. In addition, as tabulated in Table 9 in the fourth row, FFO has proven reasonable results in commercial test cases regarding the PEMFC parameter identification [111].

Fig. 7
figure 7

FFO flowchart

5.2.4 Artificial Ecosystem Optimizer (AEO)

The AEO is inspired by the energy flow in an ecological system on the earth. Specifically, AEO imitates three distinguished attitudes of the living creatures, production, consumption, and decomposition, as revealed in Fig. 8. Mathematically, the production target is to enhance the exploration and exploitation characteristics. Meanwhile, the consumption’s endeavor is to improve the exploration capability. Finally, the decomposition’s objective is to promote the exploitation ability [143, 144].

Fig. 8
figure 8

The distinct behaviors of the living creatures

Moreover, a new modified AEO, called improved AEO (IAEO), is presented in [144], which overcomes the lack of accurate solution related to the basic AEO. Consequently, IAEO has been utilized in the parameter estimation of the PEMFCs. This is due to its ability to effectively fit the computed voltage datasets to the experimental ones, while the results are depicted in Table 7 in the fifth row [144].

5.2.5 Moth-Flame Optimizer (MFO)

Mainly, the MFO mimics the distinguished navigation techniques of moths in night where they can fly with the aid of the moonlight. Using transverse orientation technique, moths fly by keeping a certain angle with respect to the moon [145, 146]. Recently, a proposed algorithm, called without certainty MFO (WCMFO), has been presented in [146] which integrates the uncertainty measurement with the conventional algorithm. Hence, the suggested algorithm has gained more robustness to disturbance with respect to the conventional one. Thus, WCFMO has been employed for extracting the unknown parameters of the PEMFCs with various numbers of unknowns and two OFs, while the results are depicted in Table 10 [146].

Table 10 PEMFC parameter estimation using WCMFO

5.2.6 Marine Predator Optimizer (MPO)

The MPO is inspired from the foraging process implemented by the marine predators to catch their preys. Particularly, the predators follow the Levy mechanism in case of lack of prey. On the other hand, in case of plentiful prey, the predators follow Brownian movements’ mechanism. According to the ecological impacts, the relative velocity of the prey \(v\) with respect to the predators can be varied relying on the Levy and Brownian mechanisms. Moreover, MPO has exhibited an enhanced global and local search besides, the fast convergence trend [147, 148]. Thus, MPO has been applied for defining the unknown parameters of several commercial PEMFCs, while the results are encapsulated in Table 9 in the sixth row [148].

5.2.7 Slime-Mould Optimizer (SMO)

The SMO is inspired by the natural oscillation state of slime mould. Basically, SMO employs adaptive weights to imitate the concept of generating positive and negative feedback of the spread wave of slime mould. Basically, these feedbacks depend on the bio-oscillator to establish the optimal path for linking food with perfect explorative capability and exploitative tendency [149, 150]. Recently, SMO has been utilized for identifying the unknown parameters of the PEMFCs, while the results are depicted in Table 9 in the seventh row [150].

5.3 Physics-Based MHAs

5.3.1 Multi-Verse Optimizer (MVO)

The MVO is extracted from the multi-verse theory which states that various universes are generated from numerous big-explosions. Each universe is related to each explosion. MVO construction relies on three cosmological concepts, white, black and worm holes. Mathematically, the exploration phase is represented by white and black holes, while the worm holes indicate the exploitation phase [151, 152].

MVO has shown advantageous features such as, simple implementation, less tuning parameters, and less computational effort. Thus, the authors in [152] have encouraged to employ it for identifying the unknown parameters of the PEMFC model, where the results are depicted in Table 11 in the second row.

Table 11 PEMFC parameter estimation using physics-based MHAs

5.3.2 Atom Search Optimizer (ASO)

Principally, the atoms are the preliminary part that form all the substances. The ASO is inspired by the atoms motion which are moving sustainably based on the classical mechanics. While moving, there are two types of forces that affect the atoms interactions with each other. The first type is the interaction forces deduced from Lennard–Jones potential. While the second one is the constraint forces generated from Bond-Length potential. Referring to Newton’s second law, the atom acceleration \({a}_{i}\) is a function of the atom mass \({m}_{i}\), the interaction \({IF}_{i}\) and the constraint \({CF}_{i}\) forces, as given in (21) [153, 154].


Accordingly, ASO is distinguished by its simple construction and smooth convergent rate. Hence, as expected, it has proven such robustness and accurateness in determining the unknown parameters of the PEMFC model, as illustrated in [154] and revealed in Table 11 in the third row.

5.3.3 Vortex Search Optimizer (VSO)

The VSO simulates the vortex pattern generated by cephalic flow of stirred fluids. This pattern is considered as a set of overlapped circles coordinated in a two-dimensional searching area. Like other metaheuristic approaches, Elementally, VSO is divided into two phases, generation, and replacement. In the generation phase, the existing solution is utilized to produce a group of solutions, while the existing solution is updated in the replacement phase [47, 155]. Furthermore, to improve the computational efficiency and the capability to evade from the local minima, a novel approach called vortex search differential evolution (VSDE) is illustrated in [47]. Thus, VSDE has exhibited robustness and effectiveness in investigating the PEMFC electrical characteristics, while the consequences are arranged in Table 11 in the fourth row.

5.3.4 Fluid Search Optimizer (FSO)

Principally, the FSO simulates the Bernoulli’s law to evaluate the fluid speed relying on the fluid pressure. Where, improving the fluid speed, diminishes the fluid pressure and the potential energy. Herein, the fluid pressure is considered as the fitness function value where improving the fluid pressure, reduces its velocity [156, 157]. Despite all the afore-stated procedures, FSO has an obvious drawback which is the premature convergence. Hence, an improved FSO (IFSO) is proposed in [157] to tackle this problem. Particularly, IFSO is composed of two enhancement techniques, that have been integrated with the basic FSO, which are quasi-oppositional based learning and chaotic concept. IFSO has been applied for tackling the parameter estimation problem of the PEMFCs, while the results are indicated in Table 11 in the fifth row [157].

5.3.5 Equilibrium Optimizer (EO)

The EO emulates the control technique of the balance between mass and volume utilized to evaluate dynamic and equilibrium phases. Particularly, in EO, the search agent is represented by each particle with its concentration. Where, the search agents stochastically update their concentration with respected to the best results, called equilibrium candidates. Furthermore, the mass balance is mathematically described in (22) [158, 159].


where \(V\) refers to the control volume, \(V\frac{d{C}_{in}}{dt}\) represents the mass change rate and \({F}_{r}\) denotes the flow rate. \({C}_{e}\) is the equilibrium phase concentration, \({C}_{in}\) denotes the inside concentration and \({G}_{in}\) symbolizes the inside mass generation rate.

Due to the effectiveness, robustness, and fast convergence trend, EO has been employed for extracting the unknown parameters of various commercial PEMFCs, while the results are tabulated in Table 11 in the sixth row [159].

5.3.6 Gradient-Based Optimizer (GBO)

Mainly, the GBO is inspired by the gradient-based Newton’s concept. Especially, two worthy operators are utilized in GBO, called gradient search approach (GSA) and local escaping coefficient (LEC), respectively. Besides, a group of vectors to explore the search space. Moreover, the exploration features and the convergence trend can be enhanced by implementing the GSA. On the other hand, the LEC is utilized to make GBO evade the local optima problem [160, 161]. Since, GBO has a smooth transition between exploration and exploitation phases and fast convergence features, it has been employed for tackling the parameter estimation issue of the PEMFCs. Hence, the results are revealed in Table 11 in the seventh row [161].

5.4 Evolutionary-Based MHAs

5.4.1 Satin bowerbird Optimizer (SBO)

The SBO is inspired by the satin bowerbirds’ behavior when constructing their own bowers in such a manner to attract satin females during reproduction season. Consequently, various elements like sparkling materials, branches, flowers, and fruits are utilized while building bowers to entice satin females [162, 163]. Owing to the advantageous features of SBO such as algorithm stability and fast convergence trend, it has been employed in [164] to extract the accurate values of the PEMFC unknown parameters, as encapsulated in Table 12 in the second row. Moreover, SBO has shown a significant fitness between the experimental data and the computed ones.

Table 12 PEMFC parameter estimation using evolutionary-based MHAs

5.4.2 Shuffled Frog-Leaping Optimizer (SFLO)

The SFLO is a memetic evolution-inspired metaheuristic approach which merges the local search mechanism of the particle swarm optimizer (PSO) into the concept of integrating the data obtained from various local searches to a global solution. Mainly, in SFLO, the number of frogs (solutions) represent the agents (population size), where the agents are split into some subgroups called memeplexes. Every memeplex represents a set of frogs carrying out a local search [111, 165].

Finally, SFLO performance in terms of robustness, accuracy and reliability has been proven via parametric statistical tests while identifying the unspecified parameters of many PEMFC types. One can track the results obtained by SFLO in Table 12 in the third row [111].

6 Concluding Discussions

For the sake of helping the reader for simply browsing the various earlier-mentioned MHAs, a comprehensive summary is offered in Table 13. Furthermore, it gives a detailed comparison of thirty MHAs, besides four MHAs that don’t fit to any of the previously-mentioned MHAs’ categories. Essentially, the comparison is divided into two categories, MHA features and PEMFC characteristics. Especially, MHA features include year of application, controlling variables, number of independent runs and statistical performance tests. On the other hand, PEMFC characteristics are represented by model state, number of unknowns, OF, types of plotted curves, various operating conditions and lastly, number of measured I–V dataset points.

Table 13 Comprehensive summary of the MHAs used in PEMFC parameter estimation

As concluded from the mathematical model of PEMFC, the polarization characteristics depends mainly on the operating temperatures and the partial pressures of the fuel and oxidant. Hence, in order to make the reader fully grasp such dependence, two well-known commercial types of PEMFC have been studied under various operating conditions. Particularly, BCS 500 W and SR-12 500 W PEMFCs have been evaluated in terms of I–V and I–P curves, as revealed in Figs. 9a–d and 10a–d; respectively [148]. Generally, as revealed in Figs. 9a–d and 10a–d, it’s clear that the polarization characteristics and the output power of PEMFC are enhanced by increasing the operating temperature at constant pressure. Also, the polarization characteristics are promoted by rising the suppliants’ partial pressures at a constant temperature, as long the minimum and maximum limits haven’t been violated.

Fig. 9
figure 9

Principal characteristics of BCS 500 W PEMFC stack under various operating conditions: a I–V at varied temperature, b I–P at varied temperature, c I–V at varied pressure, d I–P at varied pressure

Fig. 10
figure 10

Principal characteristics of SR-12 500 W PEMFC stack under various operating conditions: a I–V at varied temperature, b I–P at varied temperature, c I–V at varied pressure, d I–P at varied pressure

7 Conclusions

In this paper, an inclusive survey of various models’ categories of PEMFC has been carried out. In which, 27 models related to such categories have been gathered and summarized. Besides, a detailed PEMFC mathematical model, widely utilized in identifying the PEMFC electrical characteristics, has been totally represented. In addition, a summary of various commercial types of PEMFCs, in terms of their datasheet-extracted parameters, is encapsulated. Moreover, as the paper core work, 34 MHAs, which have been employed for extracting the unknown parameters of PEMFC, have been thoroughly discussed. Particularly, the discussion has covered their based-category, their inspiration, their features, and lastly their results in PEMFC parameter estimation. Consequently, a comprehensive comparison, among these MHAs, has been applied for clearly help the reader to simply investigate their characteristics. Also, the impact of varying the operating conditions on PEMFC output voltage and power have been demonstrated for sake of elaboration.

Lastly, as a future-wise point of view, further development of newly designed PEMFC models, together with novel optimization techniques are crucial for more properly and accurately evaluate the PEMFC performance. In addition, developing new objective functions play a vital role in effectively and precisely assess the performance of the optimization methods.