Fuel Cell Parameters Estimation via Marine Predators and Political Optimizers

Proton Exchange Membrane Fuel Cells (PEMFC) is considered a propitious solution for an environmentally friendly energy source. A precise model of PEMFC for accurate identification of its polarization curve and an in-depth understanding of all its operating characteristics attracted the interest of many researchers. In this paper, recent meta-heuristic optimization methods have been successfully applied to evaluate the unknown parameters of PEMFC models, particularly Marine Predators Algorithm (MPA) and Political Optimizer (PO) techniques. The proposed optimization algorithms have been tested on three different commercial PEMFC stacks, namely BCS 500-W, SR-12PEM 500 W, and 250 W stack under various operating conditions. The sum of square errors (SSE) between the results obtained by the application of the estimated parameters and the experimentally measured results of the fuel cell stacks was considered as the objective function of the optimization problem. In order to validate the effectiveness of the proposed methods, the results are compared with those obtained in the literature. Moreover, the I/V curves obtained by the application of MPA and PO showed a clear matching with datasheet curves for all the studied cases. Statistical analysis has been performed to evaluate the robustness of the MPA and PO techniques. Finally, the PEMFC model based on the MPA technique surpasses all compared algorithms in terms of the solution accuracy and the convergence speed. The obtained results confirmed the superiority and reliability of the applied approach of the MPA algorithm. The results prove that the MPA algorithm has a superior performance based on its reliability.


I. INTRODUCTION
The greenhouse gases and the depletion of fossil fuels have provoked the governments and industries to invest more in renewable energy sources (RES) such as PV, Wind, Tidal, Wave . . . etc. Utilizing such new RES into power grids takes new trends. It can be harnessed as a smart micro-grid or can be integrated as an isolated standalone AC-DC power grid [1]. However, due to its stochastic nature and during load peakhours, back-up supports are needed. Fuel cells are an elegant choice that can play an important role in such upcoming power grids.
The associate editor coordinating the review of this manuscript and approving it for publication was Lei Wang.
A reliable on-site emergency power supply is one of the most important systems used to assure the safety of nuclear reactors, which fulfills the safe shutdown, remove the heat after shutdown and plant confinement. In the case of Loss of Off-site Power Supply (LOPS) or plant blackout the reliable on-site emergency plays the main role in safe the plant and to protect the public and environment against radiation hazards as a consequence of LOPS. At the current situation, most of the nuclear reactors utilize independent diesel generator sets as on-site emergency power supply to act as a reliable electric source to guarantee the safety functions of the plant [2]. In the last decade the green energy resources became attractive to minimize the global warming and protect the environment from the effect of the burned of fossil fuel, that produced CO2 [3], [4]. Fuel cells system can be used to improve the backup power system performance in nuclear reactors. This will feed the reactor the emergency cooling system in cases of LOPS, Loss of Flow Accident and Loss of Coolant Accident (LOCA) and continue cooling the reactor core to prevent the core melting down and mitigate the consequences as a part of the defense in depth philosophy [2], [3].
Fuel cells can be categorized into numerous types such as Proton-Exchange Membrane Fuel Cells, from Perfluorosulfonic Acid (PFSA) to Hydrocarbon Ionomers, Direct Hydrocarbon Solid Oxide Fuel Cells, Solid Oxide Fuel Cells, Polybenzimidazole Fuel Cells, etc. [3]. On the other hand, there is a great work has been done in the last decade to use the hydrogen fuel cells in nuclear sites because nuclear energy can be used as the primary energy source for hydrogen production. Moreover, it is attractive because of the greenhouse gas emissions associated with nuclear energy production are much lower than those with conventional fossil fuel combustion. Nuclear energy is adaptable to large-scale hydrogen production [2]. Generally using fuel cells as a part of the electrical resources in the design, operation, and maintenance of nuclear power plants will enhance the reliability of on-site emergency power sources and save the plant against LOPS and plant blackout accidents.
Since 1990, fuel cell development has progressed rapidly. Car manufactures, and heating firms have discovered the technology and aim to benefit from its positive image. Fuel cell operation is based on the chemical reaction that occurs under controlled conditions. A Fuel cell consists of an electrode and a cathode with an electrolyte between them. The anode is fed by pure hydrogen (H2) or a flamed gas containing hydrogen, while oxygen (O2) or air is fed to the cathode. Depending on electrolyte, gases used as a fuel and the operating temperature, there are various classifications for the fuel cells. The polymer electrolyte fuel cell (PEFC) and proton exchange membrane fuel cell (PEMFC) are the most commonly used types. Its operating temperature is around 800C, and it can run on with normal air and reformed hydrogen as a fuel [5].
The mathematical model of the fuel cell is considered as the milestone on which the designing and testing of the fuel cell can be performed in an appropriate way. Moreover, a good mathematical model is essential to move forward the integration of the fuel cell besides supporting the designers with more information about the physical phenomena occurring inside it. The electrochemical model of the fuel cell has essential empirical and semi-empirical equations that depend on a combination set of unknown parameters. The inherited coupled parameters make the modeling of a fuel cell more difficult, which motivates the researcher to search for a suitable solution. Due to its sufficient way to obtain optimum solutions for complicated problems, Meta-heuristics can be adapted to provide robust parameter estimation for fuel cell modeling. From this fact, the no-free-lunch theorem has made a cogent remark that is employed by several optimization techniques to solve various engineering optimization problems [6], [7].
The adaptive differential evolution algorithm (ADE) is one of the competitive methods which have been used for solving the parameter estimation for PEMFC [8]. The main contribution of the proposed ADE method is to decrease premature convergence and increase search efficiency. Hybrid adaptive differential evolution is introduced in [9]. It is a combination set between biological genetic strategy and bee colony foraging method. The first method enhances the parametric scaling for dynamic cross-over probability, while the former method improves the weak local search. Hence, the ADE enhances the performance of the optimization techniques. A grouping-based global harmony search algorithm (GGHS) has been adopted for obtaining a precise estimation for PEMFC parameters, as reported in [10]. The algorithm performance was compared with different methods such as particle swarm optimization (PSO) and seeker optimization algorithm (SOA). From the comparison, it had been concluded that the GGHS platform exhibits better performance than other algorithms [10]. The genetic algorithm has been applied for parameter estimation of PEMFC [11]- [13]. In [11], a new formulation based on a genetic algorithm (GA) is used to deal with fuel cell parameter evaluation. The main advantage of this method is lower complexity, less time consumption, enhancing accuracy and ease of implementation. A hybrid combination set between teaching learning-based optimization method (TLBO) and Differential Evaluation Algorithm (DE) is introduced in [14] to obtain a proper estimation for the parameter model of PEMFC. The (TLBO-DE) performance is compared with different optimization algorithms. The TLBO-DE proves its accuracy and robustness, besides its ability to obtain an optimum solution with lesser computation time. The Grey Wolfe Optimization (GWO) algorithm is used for identifying the PEMFC model parameters [15]. An experimental test is performed to prove a superior performance for GWO to other optimization methods such as Antlion Optimizer (ALO), and Dragonfly Algorithm (DA). New meta-heuristic approaches have been employed to adapt the PEMFC model parameter such as Grasshopper Optimization Algorithm (GOH), Slap Swarm Optimizer (SSO) and Shark Smell Optimizer (SSO) [16]- [18]. The advantages of these methods are better  convergence to an optimum solution, tuning its controlling parameters with the low effort of computation, and faster process execution. In [19], JAYA algorithm was deployed to estimate the PEMFC parameters effectively. Compared to other optimization techniques, JAYA has better convergence time, accuracy, and stability. In [20], the Cuckoo Search (CS) algorithm is used to obtain the parameters of the PEMFC. The author has proposed an explosion operator to fine-tune the step size of the CS. The Cuckoo Search Algorithm with Explosion Operator (CS-EO) proves its ability to avoid precipitate convergence and enhances the overall performance of the CS. According to [21], the hybrid optimizer based on the vortex search algorithm (VSA) and differential evolution (DE) has been developed to evaluate the optimum uncertain parameters of the PEMFC. Both VSA and DE had been incorporated to increase the attitude of VSA preventing its local-optima by promoting the operating of exploitation followed in VSA based on DE. From [22], it has been applied prepared and utilized the Modified Artificial Ecosystem Optimization (MAEO) to estimate the parameters of PEMFC. In this technique, the MAEO was very effective to increase the attitude of AEO for introducing a very fast process of convergence to avoid the local optima. In [23], the authors proposed and utilized the Hybrid Grey Wolf  Optimizer (HGWO) to solve the problems of the parameters extraction of PEMFC. The HGWO has been used to combine the crossover and mutation operators during the optimization evaluation to enhance the ability of the search potential avoiding the trapping in local-optima. Moreover, Harris Hawks Optimization (HHO) algorithm has been applied for extracting the parameters of the PEMFC model in Refs. [31]- [34]. The obtained results in the reported references [31]- [34] prove the ability of the HHO to estimate the parameters of PEMFC. A comparison between the reported results from references [31]- [34] and those of the MPA and PO algorithms will be presented in the results section to evaluate the effectiveness of each algorithm. All these algorithms are presented and applied for estimating the parameters of the PEMFC Model for improving the estimation accuracy. However, most of these works could not strengthen the estimation accuracy. Therefore, it is necessary to present and validate recent methods that have the ability to accurately estimate the parameters of the PEMFC Model with good convergence characteristics.
Recently one of the outstanding optimization techniques that have been discovered not a long time ago is the MPA [24]. It has a superior performance to act with global optimization problems rather than the other optimization techniques. It possesses a better exploration of the optimum solution without getting stacked to the local search. Moreover, a contemporary optimization method has been proposed to solve complicated problems; this method is the PO method [25]. It is a simple structure method. Its advantages underlying, fewer controlling parameters, self-adaption to its parameters, can be adapted easily with other methods to obtain better convergence for optimal solutions. However, this method is not deeply applied to solve electrical engineering problems.
In this paper, MPA and PO have been used for estimating the parameters of a number of commercial proton exchange membrane fuel cells. To validate the effectiveness of the MPA and PO methods, comparisons and different operating scenarios have been studied. The contribution of this work includes the implementation of two unprecedented optimization algorithms, namely MPA and PO, for defining the equivocal parameters of the fuel cell model and comprehensive comparison with other competitive techniques that have been provided in the literature.

II. PROBLEM FORMULATION A. BASIC PHYSICAL OPERATION
Fuel cells are considered a direct method of converting chemical energy into electrical energy. The construction of a typical proton exchange membrane (PEM) fuel cell is described in Fig. 1. As seen from the figure, the PEM fuel cell model comprises two electrodes (anode and cathode), between which a catalyst and membrane layers are stacked. In addition, at the anode and cathode sides, two channels are used for supplying hydrogen and air, which will be diffused through the electrodes.
A simple way of understanding the base operation of the fuel cell is to say that the hydrogen gas is being 'burnt' or combusted in the simple chemical reaction described as follows [26]: However, in this case, instead of heat energy being released, electrical energy is generated. The reaction takes place at the anode and cathode and can be declared as follows: At the anode of the fuel cell, the hydrogen gas ionizes, releasing electrons and creating H + ions (or protons).
This reaction releases electrical energy presented by the negative electrons e − . At the cathode side, oxygen reacts with the electrons which are taken from the injected air at the electrode and the H + ions produced from the electrolyte, to finally form water, and it can be expressed as:

B. MATHEMATICAL MODEL OF PEMFC
An electrochemical-based model for PEMFC has been adopted by Amphlett et al. [27], which considered a number of fuel cells N cells connected in series to form a fuel cell stack system. This model is a helpful tool for engineers interested in evaluating the performance of PEMFC and optimizing the system parameters. The output voltage across the terminals of the fuel cell stack can be presented as follows [9]: where V act presents the activation voltage drop caused by the kinetics of the chemical reactions around the surface of the electrodes, which causes a sharp drop in the I/V polarization curve of the fuel cell stack at lower currents [12]. V ohmic presents the ohmic voltage drop, which results from the resistance of transferring the protons and electrons in the electrolyte. For intermediate currents, the ohmic voltage drops smoothly and linearly as a result of the ohmic losses. V con is the concentration voltage drop, which appears due to the sophisticated processes of transport, and which lets the output voltage of the fuel cell fall sharply another time at higher currents [28]. The E Nernst represents the fuel cell reversible voltage in an open circuit electrodynamic balance and is calculated using (5) [9]- [11]: where T is the operating temperature of the fuel cell in Kelvin; P H 2 and P O2 denote for the partial pressures of hydrogen and oxygen, respectively. The partial pressure of the gases depends on the nature of the components of the chemical reaction. If the components of the chemical reaction are air and hydrogen, then according to [15], [29], the partial pressure of each reactant can be estimated by the following formulas: where: The saturation pressure of the water vapor P sat H 2O is estimated by the following expression: 18 (8) when the reactants are oxygen and hydrogen, then the formula (8) can be as follows:  The partial pressure of hydrogen PH2 in both conditions can be calculated from the following expression: where RH c and RH a are the relative humidity of vapor at the cathode and anode, respectively. While P a and P c are the channel pressure (atm) at the anode and cathode, respectively; P N 2 is the partial pressure of nitrogen at the flow channel of gas at the cathode (atm) and A is the active surface area of the membrane.
The activation voltage drop V act can be determined as follows [11]: where ξ 1 , ξ 2 , ξ 3 , ξ 4 present semi-empirical coefficients; I fc is the output current from the fuel cell stack; C O2 is the concentration of oxygen at the surface of the cathode (mol.cm −3 )  and is calculated as [11], [15], [29]: The ohmic loss V ohmic is calculated depending on the fundamentals of Ohm's law and directly depends on the current density, and it can be written as the following: where R M and R c present the resistance of the membrane and the equivalent resistance that the protons face when transported through the membrane and it is considered as a constant value. Accordingly, the resistance of the membrane surface can be given from the following expression: where l denotes the effective thickness of the membrane surface (cm), A is the area of the membrane surface (cm 2 ), and ρ M denotes the resistivity of the membrane against the flow of electrons ( .cm) and is calculated empirically for Nafion membrane from the following expression [12], [27], [28]: where λ denotes to an adjustable parameter, which indicates the water content of the membrane material. The value of λ can be adjusted between 13 and 24 [9], [10]. The last part of these losses is the concentration voltage drop V con , which appears due to the changes in the concentration of hydrogen and oxygen or fuel crossover and is calculated according to the following expression: where β denotes the adjusting parametric coefficient; J and J max denote the current density and the maximum current density (A cm −2 ), respectively.

C. OBJECTIVE FUNCTION
From the mathematical expressions described by equations (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16), it is obviously noticed that the operation and performance characteristics of the PEMFC stack system are originally depended on a number of parameters. A part of these parameters are not available in the datasheet of the manufacturer and have to be carefully estimated to guarantee an accurate representation of the PEMFC, which gives results that match with experimental data. After the closer study of the above-mentioned equations, it is found that a set of parameters (ξ 1 , ξ 2 , ξ 3 , ξ 4 , β, R c and λ) are not recognized in the datasheet and have to be extracted. The degree of matching between the model of the PEMFC and the experimental data is obtained by calculating the differences between the output voltage of the proposed model and that measured experimentally under different operating currents. In this paper, the sum of squared errors (SSE) between the measured values of voltage and the values of the output voltage of the PEMFC model is considered as the objective function (OF).
The objective function to minimize the SSE is commonly used in many works of literature [8], [9], [14] is expressed by the following: The objective function of (17) is ruled by the following constraints: where X is a vector of the seven unknown parameters that have to be determined, V meas is the output voltage obtained experimentally from actual PEMFC, V est is the output voltage obtained from the proposed model and N is the length of the experimental data series used for validation. MPA and PO methods are proposed for determining the optimal values of these parameters of the PEMFC model to give a high agreement with the results of the actual fuel cell stacks.

III. MPA
The proposed Marine Predators Algorithm (MPA) has been introduced by Faramarzi et al. [24]. The MPA has been implemented to imitate the strategy of foraging optimally for the Marine Predators (MP) to detect their Prey as the following: In the case of a low concentration of prey, the MP follow the Lévy behavior. In the case of abundant Prey, the MP follow the Brownian movements' behavior [24], [30]. According to the environmental effects, the velocity-ratio v from the Prey to MPA can be changed based on the behaviors of Lévy and Brownian as follows: • In the low velocity-ratio v < 0.1, the most suitable behavior for MP is Lévy. On the other hand, the Prey is proceeding in Brownian/Lévy behavior; • In the low velocity-ratio v = 1, in the case of the Prey can move in Lévy behavior, therefore the MP have to follow the Brownian behavior; • In the high velocity-ratio v > 10, the most suitable behavior for MP is to be without moving. On the other hands, the Prey is proceeding in Brownian/Lévy behavior.
The scheme of the MPA methodology can be proposed in the following steps: Firstly, the Prey in a group can be initiated through a search-space based on formula (19): where X min and X max represent the lower & upper boundaries and rand is a random number that take a range from 0 to 1. Then, the fitness of MP can be determined. According to the survival of the fittest theory , the fittest solution is considered as a top MP to structure a matrix that can be defined as Elite.
The Elite matrix can be represented in the following form: The main iteration-loop of the MPA can be provided into three-phases depending on the velocity-ratio that are presented as follows:

A. EXPLORATION-PHASE
This phase occurs in high velocity-ratio, and it can be written as follows: where R B is a vector of random numbers and it depends on the nominal distribution introducing the Brownian movement. ''⊗'' provides the entry wise multiplications. P = 0.5 and it is constant. R is a random numbers vector that takes a range [0, 1]. This phase takes place in the one-third of iterations when the step-size or the velocity of motion is high for high exploration ability. ''Iter'' is the present iteration.

B. INTERMEDIATE-PHASE
This phase occurs in unit velocity-ratio. The exploration is gradually changed to exploitation and it is proposed as follows: -In the first-half of the population: -In the second-half of the population: where − → R L is proposed based on the Lévy-flight behavior. In the intermediate-phase, the first half of Prey can proceed with Lévy steps. On the other hand, the second half utilizes Brownian steps. The CF is considered as an adaptive parameter to control the step-size of MP motions. The CF can be determined by the following Eq. (28):

C. EXPLOITATION-PHASE
This phase occurs in low velocity-ratio and it is prepared as follows: One of the critical points that can be taken into consideration is the manner of the MP can be affected by environmental problems like the eddy formation or Fish Aggregating Devices (FADs). According to [24], the MP consume 80% of their time in the vicinity of FADs looking for the Prey. The rest of MP time is consumed in another environment to find their Prey. In search space, the FADs effects are considered as trapping. Thereby, the FADs are deemed as local optima. The FADs process is formulated as follows: where FADs = 0.2 is the impact of FADs on the optimization process. − → U is the binary with the arrays having 0 and 1. r is − → X max and − → X min are including the lower and upper boundaries of the dimensions. r 1 and r 2 are random indexes of Prey matrix. MPA has a well memory that can be useful to remember the old positions of the Prey. Therefore, based on the fitness values, each present and old solutions can be compared, and the best one can be saved at each iteration.
The MPA implementation procedure can be summarized in the flowchart shown in Fig. 2.

IV. PO
The proposed PO technique is considered as a physics-based algorithm combined with swarm-based characteristics which are concerned mainly with finding global optima. The PO has been implemented by Q. Askari et al. [25], and it has been inspired by the different phases of the Politics process. Politics process can be divided into two main attitudes: every individual optimizes its good-will to win the election, and every party attempt to find the maximum number of seats in parliament to form a government. However, the PO is created as a consequence of five-phases involving party formation and constituency allocation, election campaign, party switching, inter-party election, and parliamentary affairs [25].
The scheme of the PO methodology can be proposed in the following process using the variables that defined in Table 1 [25]: Firstly, the phase of party formation and constituency allocation executes only once for the purpose of initialization and the others of the four-phases perform in a loop.

A. PARTY FORMATION AND CONSTITUENCY ALLOCATION
The β is sectioned into n political parties. Every party β i is composed of n members. Every j th member p j i is deemed as a potential solution. This potential solution can represent as an election candidate. It is supposed n constituencies, and j th member of every party contests election from the j th constituency C j .
The fittest member of a party is expressed as the party leader. The election of the party leader is determined using (33) as follows: Therefore, all the party leaders' β * can be introduced by (34) as the following expression:  After selection, the winners from all the constituencies become the parliamentarians C * as shown in (35) as the following formula:

B. ELECTION CAMPAIGN ''EXPLORATION AND EXPLOITATION''
This phase-step assists nominees to make their performance better in the election. In this regard, there are three parts in this phase as following: -A new position updating strategy called recent past-based position updating strategy (RPPUS) that can produce a suitable learn from the previous election. This strategy can be written by (36)   -The comprehensive analysis with the constituency winner is proposed mathematically based on the updating of the candidate position.

C. PARTY SWITCHING (BALANCING EXPLORATION AND EXPLOITATION)
In this phase, an adaptive factor λ party switching rate is decreased in linear form from its maximum value to 0 during the iterations process. The probabilistic selection of p j i is turned to few randomly elected party β r that can be changed with the least fit p q r of that party. The calculation of q index of β r can be written as follows:

D. ELECTION (FITNESS DETERMINATION)
The election is imitated by estimating the fitness of all the candidates contesting in a constituency and proclaiming the winner. In this process, c * j indicates the winner of j th constituency (C j ). This phase can be modeled mathematically by (39) as the following expression:

E. PARLIAMENTARY AFFAIRS (EXPLOITATION AND CONVERGENCE)
The government is created, in accordance with an inter-party election. The party leaders and the constituency winners are made a decision by applying (33) and (39). Every winner c * j renews its position with reference to a randomly selected winner c * r and if it causes any modification in fitness of c * j thereafter the situation and fitness of c * j are renewed. The PO implementation procedure can be summarized in the flowchart shown in Fig. 3 [25].

V. RESULTS
The simulation tests have been carried out to validate the applied optimization algorithms (MPA and PO). Both algo- rithms have been applied for estimating the parameters of PEM fuel cells. The two algorithms have been tested for estimating the parameters of three different modules of fuel cells, namely BCS 500W, SRR_12 modular, and 250W stack. The datasheet parameters of these commercial PEMFC stacks are obtained from [9]- [11], [14], [15], [22] and are presented in Table 2. Moreover, the estimated model parameters are ξ 1 , ξ 2 , ξ 3 , ξ 4 , β, R C , and λ in PEMFC, as shown in Fig. 4.
The upper and lower limits of the unknown parameters for all case studies are given in [8], [10], [18], and [19] and presented in Table 2 (last three columns in the right).   The results of the two algorithms have been compared with each other and with those obtained using other techniques from the literature. Moreover, the optimized parameters using MPA and PO methods have been used to estimate the performance and characteristics of the PEMFC at different operating conditions. Furthermore, the characteristics have been compared with the measured data of each module.
For the simulation, a dedicated software program for fuel cell parameter extraction problem is developed in MATLAB for MPA and PO based upon their theories of operation which are described before. Simulations are performed using an Intel R core TM i5-4210U CPU, 1.7 GHz, 8 GB RAM Laptop.

A. PEMFC OF BCS 500W
To test and validate the proposed optimization algorithms, the proposed algorithms have been applied with the problem formulation PEM fuel cell of BCS 500W. The selected BCS 500W is studied because several studies have been introduced earlier for this purpose, but the majority have failed in achieving an accurate estimation for the parameters. The results of applying the MPA and PO algorithms to estimate the finest values of BCS 500W stack parameters are illustrated in Table 3. As shown from the table, the results gained by the MPA are better than those obtained by the PO technique and also are better than the other methods from literature. The convergence curves of the MPA and PO have been shown in Fig. 5, which reports that the MPA has the best convergence curve with respect to the speed of convergence and reaches the best minimum value of the objective function. From this figure, the MPA optimization algorithm reaches a minimum value of 0.011556305 while the PO reaches its minimum, which equal to 0.0115564179. It should be noted that the small difference between the results of the two algorithms. Table 3 shows the comparison between the estimated parameters of the PEMFC model using MPA and PO and other techniques. The results of MPA is the best one as compared with the latest published papers in the literature [22].
To validate more the effectiveness of MPA and PO optimization algorithms, the obtained results have been used to estimate the characteristics of the PEMFC by estimating the voltage and power curves versus the current. Furthermore, the estimated characteristics have been compared with the measured one for both models of MPA and PO as shown in Fig. 6. Figure 6.a) shows a very good match between the estimated and measured characteristics. It should be noted that the coincidence of the two curves of the estimated models of MPA and PO with the measured one because the values of SSE for MPA and PO are 0.0115564179 and 0.011556305, respectively. Additionally, figure 6.b) shows the squared error for both models based on MPA and PO techniques. Also, Table 4 listed the squared error between the estimated and measured performance of BCS 500w based on PO and MPA. Table 4 listed the obtained results of the estimated voltage using models based on MPA and PO algorithms and the squared error at each current value. Moreover, the characteristics of the PEMFC have been plotted at different operating condition of the pressure of PH2 /PO2 of 1.000/ 0.2075bar, 1.5/1.0bar, 2.0/1.25 and 2.5/1.5bar; and temperature of 303K, 313K, 323K and 333K as shown from Fig. 7 for only the MPA-based model.
As known, the performance of these algorithms is based on randomness and the sole best fitness value (SSE) in one of the runs cannot assure the acceptable performance of the optimization algorithms. As mentioned earlier, there is no guarantee that the algorithms can repeat the same results overruns. In order to appreciate the stability and accurateness as well as give a clear assessment of the proposed MPA and PO in obtaining the exact values of PEMFC unknown parameters, sensitivity and statistical analysis is studied for all tested PEMFC stacks. Fig. 8 shows the convergence curve of 30 independent runs. From the figure, over the 30 runs; the MPA reached the same optimized solution while the PO algorithm results are varying each run around its best solution. The results are very essential which prove that the MPA is stability and reliability of the MPA.
Statistical analysis is completed for assurance and valuation the robustness concert of the optimization algorithms. So, MPA and PO algorithms are performed 30 individual runs. The finest objective function is logged and stated. Furthermore, statistical indices of Mean, Standard deviation (SD), relative Error (RE), root mean square error (RMSE), the minimum and maximum over 30 runs are calculated. The results of MPA and PO algorithms are recorded in Table 5. The listed results of the table demonstrate that the MPA technique has the best performance during this test while the value of SD, RE and RMSE of the PO algorithm is relatively high as shown from the table because it fails to optimize the problem in number of runs. So, the MPA is an effective algorithm for solving the optimization problem of parameters' identification of various mathematical models of BCS 500W.
Furthermore, the Wilcoxon signed-rank test is performed to validate the MPA and PO algorithms. The reported results are listed in Table 5. The results show that P-value for both   specification of the SR-12PEM 500 W are listed in Table 2. The results of the estimated parameters have been listed in Table 6. Also, Table 4 consists of a comparison between the PO and MPA algorithms and with the obtained results by other researchers. From the table, the best solution has been reached by the MPA and equals 1.056628334025551 while the best optimal value with the PO is 1.056628334030994. Moreover, the table shows that the other researchers with other optimization techniques could not reach the same solution. Furthermore, a comparison between PO and MPA with respect to the convergence characteristics has been shown in Fig. 9. The figure shows that the convergence speed of the MPA is better than that of the PO.
The simulation tests have been done to test and evaluate the robustness of the MPA and PO optimization techniques with this case of study. The MPA and PO have been applied for 30 runs. The convergence curves of the 30 runs have been shown in Fig. 10 for both algorithms. The figure shows that the MPA has the ability to reach the same best solution over 30 runs while figure10.b) shows the PO algorithm reaches to different values of the best solution in the number of runs. This concludes that the MPA is the best choice for estimating the parameters of the PEMFC model.
The results of the characteristics of SR-12PEM 500 W based on the estimated parameters using MPA and PO and the experimental data have been shown in Fig. 11.a). The figure shows that the obtained characteristics from the proposed MPA optimization algorithm introduce a high matching degree with the experimental data. Furthermore, the squared error between the measured voltages and the estimated based on MPA and Po algorithms have been illustrated in figure 11.b) and table 7. Moreover, due to the Exploration and exploitation characteristics of MPA, which is related to global search; the MPA keeps improving a longer time than PO.
Statistical results of the applied MPA and PO methods for SR-12PEM 500 W based on 30 individual runs have been  Table 8 which are produced based on the Wilcoxon test demonstrate that the results of MPA and PO algorithms are statistically significant.
For more validating, the estimated model based on the MPA is used for plotting the characteristics at different operating conditions such as the variation of temperature and pressure, as shown in Fig. 12.

C. 250 W STACK
For more justification, the two algorithms of MPA and PO have been applied for extracting the model parameters of the 250 W stack. The data specifications of the 250 W stack have been listed in Table 2. The results of the estimated parameters have been concluded in Table 9. From the table; it is revealed that the MPA algorithm has the best result concerning reaching the minimum value of the objective function. The best optimal value of MPA is equal to 0.5940499653, while the best value with PO equals 0.644205811. Also, the table presents a comparison with other techniques from the literature. From the comparison, the optimal values of the objective function obtained by the MPA and PO are better than the other techniques.
In order to analyze the convergence characteristics of the two algorithms, the convergence curves of the PO and MPA algorithms via iterations have been shown in Fig. 13. The figure shows that the MPA has a better convergence speed of solving the optimization problem compared with the PO technique.
Another time, the robustness and probability of finding the optimal answer by the MPA and PO algorithms have been tested by finding the best solution of 30 independent runs. The results of these runs have been shown in Fig. 14. Figure 14.a) proves the robustness of the MPA optimization algorithm for finding the best parameters of the PEMFC model. While figure 14.b) shows the results of PO algorithm which confirms the results have been varied each run around the best one.
The validation of the results has been proved by plotting the voltage and power characteristics versus the current for both models of MPA and PO, as shown in Fig. 15. From the figure, the precise matching between the estimated characteristics and the experimental data of the fuel cell can be easily investigated. Table 10 shows the squared error between the estimated and measured performance of 250 W stack based on PO and MPA. Furthermore, Table 11 shows the statistical measurement of the planned MPA and PO methods for 250 W stack based on 30 individual runs. The statistical results prove that the both algorithms of MPA and PO have robust performance with consideration of superiority of the MPA algorithm. Furthermore, the characteristics of the module with the variation of the temperature and pressure have been shown through Fig. 16.

D. RESULTS DISCUSSION
In the previous sub-sections, the results of the application of both MPA and PO algorithms have been presented. The two algorithms have been applied to estimate three various cases of the PEMFC. The results started with the convergence  curves of the two algorithms to solve the optimization problem. Then the minimum value of each algorithm has been presented and compared with those of other reported optimization algorithms. Considering the convergence characteristics, the MPA algorithm has the best one as confirmed from figures of the convergence curves. Moreover, MPA can reach in all studied cases to the finest value of the objective function as confirmed from tables 3, 4 and 5. While, the results of the PO are better than those of the other reported methods as shown from table 3, 4 and 5 but its results are the second ones after those of the MPA algorithms. So, the MPA and PO algorithms have the ability to solve the presented optimization problem to estimate the parameters of PEMFC with the superiority of the MPA algorithm.
Furthermore, the obtained optimized parameters have been used to estimate the V/I and P/I characteristics of the studied PEMFC cases. The obtained results based on the MPA and PO estimated parameters confirm the coincide of the estimated PEMFC characteristics with those of the measured and datasheet.
The merits of the MPA can be shown by focusing on the MPA flowchart of figure 2. The MPA algorithm has three phases that improve its characteristics of global solution (exploration) w.r.t the first and second phases and local solution (exploitation) considering the third phase which enhances its characteristics when MPA applied 30 individual runs with the same obtained results. Although its limitation can be well-thought-out in the future work once it applies for more complex optimization problems with large a variable. PO optimization problem has the merits of the global solution (exploration) characteristics; this phase gives it the ability to reach the finest solution. However; its ability to locate the local solution (exploitation) is not very accurate based on its fifth phase. So, the PO algorithm can reach acceptable results with respect to those of the reported references as listed in table 3, 4, and 5.
Statistical analysis for both algorithms of MPA and PO have been performed. The results prove the robustness of the two algorithms with the superiority of the MPA algorithm as proved from table 5, 6, and 7. Also, the same results can be investigated from figures 8, 10 and 14. It should be noted that the main reason is the good exploration and exploitation performance of the MPA.

VI. CONCLUSION AND FUTURE DIRECTIONS
A PEMFC is a nonlinear complicated dynamic system, which involves many interrelated parameters. This paper comprises the formulation of an optimization problem, which is devoted to optimal identification of the seven unknown parameters of the PEMFC. The MPA and PO optimization techniques have been utilized for solving the optimization problem, while the fitness function is presented by the sum of square errors (SSE) between the actual and estimated models. The proposed methods introduced the high performance with high matching degree respecting the measured data of different fuel cell stacks. The MPA and PO proved their effectiveness in reaching the optimal solution in a better way compared with the results in the literature. Moreover, the statistical tests have been performed to validate the robustness of the two algorithms. From the comparison, it is concluded that the MPA is an accurate method which can precisely extract the parameters of the PEMFC with different cases of study. Therefore, it is recommended that the MPA algorithm can be implemented for solving sophisticated highly integrated optimization problems. From a practical point of view, the estimated model can be used online with PEMFC for fault diagnosis and condition monitoring. Moreover, the estimated model also may be helpful in designing the real-time control PEMFC systems as well as system analysis. The future work, the analysis of the parameter's variations of the PEMFC model far from the standard operating conditions considering the presence of measuring noise should be studied. Furthermore, the application of other recent optimization algorithms such as Slime Mould Algorithm (SMA) and hybrid techniques to enhance the estimation process is one of the future directions. Additionally, more interest will be focused to enhance the performance of the PEMFC in the microgrid operation.