Semi-empirical PEM fuel cells model using whale optimization algorithm

Energy Conversion and Management 201 (2019) 112197

Contents lists available at ScienceDirect

Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Semi-empirical PEM fuel cells model using whale optimization algorithm a,⁎

b

Attia A. El-Fergany , Hany M. Hasanien , Ahmed M. Agwa

T

c,d

a

Electrical Power and Machines Department, Faculty of Engineering, Zagazig University, Zagazig 44519, Egypt Electrical Power and Machines Department, Faculty of Engineering, Ain Shams University, Cairo 11517, Egypt Electrical Engineering Department, Faculty of Engineering, Al-Azhar University, Cairo 11651, Egypt d Electrical Engineering Department, Faculty of Engineering, Northern Border University, Arar 73222, Saudi Arabia b c

A R T I C LE I N FO

A B S T R A C T

Keywords: Modelling of PEM fuel cell Polarization curves Parameters extraction Optimization methods Experimental results

The accuracy of the fuel cell model (FC) plays an important role, since it affects the simulation results. Obviously, this can appear in different systems of distributed generations and hybrid microgrids. This article presents a new application of the whale optimization approach (WOA) to obtain unidentified parameters of the proton exchange membrane (PEM) FC model. The ultimate objective of the current work is to develop a precise PEMFC model, which provides true results of modeling and simulation of these FCs. In this sense, the I-V characteristics of the PEMFC are non-linear, including seven unknown parameters due to the lack of data from the manufacturer. This problem can be expressed mathematically as a non-linear optimization problem, where the sum of the error squared between the measured FC voltage and its output voltage model defines the fitness value. WOA is applied directly to minimize the objective function under the predefined constraints. The estimated PEMFC model is certified by the numerical results shown, which are performed under various conditions of temperature and regulating pressures. The value of the proposed WOA-based PEMFC model is appraised by comparing the demonstrated results with the empirical results of several typical PEMFCs such as Ballard Mark V, the AVISTA SR12 PEM generator, the 250 W and the Horizon H-12 stacks. The results based on WOA are compared with other results based on optimization competing methods. The application of WOA can lead to the creation of a precise PEMFC model.

1. Introduction Clean energy sources have honored much interest throughout the world due to numerous key reasons, such as the depression of conventional fuels, the increase in their price and the trend of environmental concerns [1]. A good source of clean energy is the fuel cell (FC), which can convert the chemical energy of the fuel into electrical energy through chemical reactions [2,3]. Though, there are many types of FCs in industry based on their technology, proton exchange membrane (PEM) FCs have prominent characteristics such as low operating pressure and temperature, high level of efficiency, no wasted materials and so on [4]. The typical temperature level varies from 50 to 100 °C and its efficiency lies in the range [30, 60%] based on operating conditions [4,5]. The PEMFCs have been applied to different engineering applications, for instance, it is used in autonomous system to feed switched reluctance motor [6] and in power system applications as a distributed generation [7], micro-combined heat and power application [8] or involved in a microgrid [9]. The modeling of PEMFCs received a great attention over the last

⁎

decades [10]. The main target is to obtain an accurate PEMFC’s model that helps in efficiently simulating it using software programs and very close to experimental models. By this way, efforts and time can be saved [11]. Extracting the PEMFCs model to characterize their performances can be made by analytical, semi-empirical or purely empirical, and/or theoretical which can be realized by utilizing conventional analytical methods or meta-heuristic optimization algorithms. Among the empirical approaches: electrochemical impedance using broad-band current excitation [12], fractional order models [13], semi-empirical equations [14], and the generalized reduced gradient method [15]. Moreover, other analytical method supported by experimental validation has been presented in this regard to achieve a precise FC model [16,17]. However, these analytical methods rely on complex procedures and analyses [18]. In addition, conventional optimization methods have some demerits such as: (i) its high dependency on the initial conditions of the problem under study, (ii) the solution accuracy is based on the differential equations solver error, and (iii) it can get stuck in local minima instead of global minima, especially in solving

Corresponding author. E-mail address: [email protected] (A.A. El-Fergany).

https://doi.org/10.1016/j.enconman.2019.112197 Received 24 July 2019; Received in revised form 12 October 2019; Accepted 14 October 2019 0196-8904/ © 2019 Elsevier Ltd. All rights reserved.

Energy Conversion and Management 201 (2019) 112197

A.A. El-Fergany, et al.

2. Model of PEMFC

high non-linear optimization problems (NLOPs). For that reasons mentioned above, many meta-heuristic optimization algorithms have been applied to solve this optimization problem such as atom search optimizer [4], the authors [5] employed three algorithms, viz. shuffled frog-leaping (SFL), firefly, and imperialist competitive optimizers, harmony search algorithm [19], JAYA algorithm and Nelder-Mead simplex method [20], salp swarm algorithm [21], biogeography-based optimizer [22], genetic algorithm (GA) [23], neural network optimizer (NNO) [24], differential evolution (DE) algorithm [25], cuckoo search algorithm with explosion operator [26], artificial bee colony algorithm [27], grey wolf optimizer (GWO) [28], shark smell optimizer [29], backtracking search algorithm [30], teaching learning based optimization algorithm [11,31], seeker algorithm [32], multi-verse optimizer (MVO) [33], grasshopper optimizer (GHO) [34], hybrid vortex search algorithm and DE (VSDE) [35], and flower pollination algorithm (FPA) [36]. In general, these metaheuristic approaches have their own advantages and disadvantages, where someone can solve an engineering problem and does not solve another. The tremendous implementations of these meta-heuristic algorithms has motivated the authors to employ the whale optimization approach (WOA) to deal with this optimization problem and achieving a precise model for the PEMFCs. The WOA is considered a novel type of meta-heuristic optimization algorithms. It has inspired the abduction of whale family behavior and social activities. Proposed optimization algorithm presents obviously scenarios and technical procedures of the whales hunting process to the preys. The WOA was introduced by S. Mirjalili and others researchers [37]. Generally, this algorithm has several advantages that keeping it superior to others optimization algorithms like: (i) it has a higher speed of convergence, and (ii) It has a minimum number of parameters that must be adjusted to obtain the optimal solution. The WOA has been successfully utilized in solving many engineering optimization problems compromising several design problems [37]. Moreover, it has been efficiently applied to solve several power system optimization problems. In [38], an optimal control scheme is offered to improve the performance of grid-connected solar photovoltaic systems. Furthermore, the WOA is used to solve the automatic generation control problem of interconnected power systems in [39]. In addition, parameters of photovoltaic models are obtained by the WOA [40–42]. These salient applications confirm its high capability to deal with NLOPs. In this article, a novel attempt of the WOA is addressed with the intention of obtaining best values of uncertain parameters of the PEMFC model. The study aims at establishing an accurate model of the PEMFCs, which provides with true results of modeling and simulation of these FCs. In this regard, the I-V characteristics of the PEMFC is nonlinear including several unknown parameters due to the shortage of manufacturer's data. This problem can be expressed mathematically as a NLOP, where the fitness value is the sum of the squared errors (SSE) between the measured FC voltage and its voltage model. The WOA is applied directly to minimize the fitness function (FF) under the problem constraints. The estimated PEMFC model is validated by numerical results, which are demonstrated under various temperature and partial pressure conditions. The feasibility of the solution of the proposed model is valued by comparing the numerically realized results with the empirical results of four commercial PEMFCs. WOA-based results are compared with other optimization methods-based results. The application of the WOA can lead to creating an accurate PEMFC model. This article is arranged like this: Section 2 indicates the PEMFC mathematical simplified representation. In Section 3, the optimization problem is defined comprising the FF and set of predefined constraints. Section 4 depicts the WOA and associated procedures. In Section 5, the simulation results and its discussion are explored along with necessary comparisons. Lastly, Section 6 illustrates the conclusion.

Polarization curves or I-V characteristics can be exhibited mathematically to form the PEMFC model. The electrochemical simplified model offered by Mann et al. [43] is used to characterize the steadystate performance of the PEMFC in this article. This model is commonly used in many literature studies. The output voltage of the PEMFCs stack (VStack ) , which consists of a number of series connected cells (Ncells) , is mathematically modeled by (1) [34,43–45]. (1)

VStack = Ncells . (ENernst − vact − vΩ − vconc )

where ENernst is the Nernst voltage per cell, vact is the cell activation overpotential, νΩ is the cell ohmic voltage drop, and vconc is the concentration over-potential. Under a reference temperature of 25 °C, the voltage ENernst is determined by using (2) for the operating temperatures below 100 °C. The three voltage drop quantities are calculated by (3)-(5), respectively [43,44].

ENernst = 1.229 − 0.85 × 10−3 (Tfc − 298.15) + 4.3085 × 10−5Tfc ln (PH2 PO2 )

(2)

vact = −[ξ1 + ξ2 Tfc + ξ3 Tfc ln (CO2) + ξ 4 Tfc ln (Ifc )] where CO2 =

P O2 5.08 . 106

(3)

. exp(498/ Tfc )

vΩ = Ifc (Rm + R c ); Rm =

ρm l MA 2

2.5

( ) + 0.062 ( ) ( ) ⎤⎦ ( ) ⎤⎦. exp ⎡⎣4.18 ( ) ⎤⎦

181.6 ⎡1 + 0.03 ⎣ where ρm = ⎡λ − 0.634 − 3 ⎣

J ⎞ vconc = −β . ln ⎛1 − Jmax ⎠ ⎝ ⎜

I fc

Tfc

MA

303

I fc

MA

I fc

Tfc − 303

MA

Tfc

(4)

⎟

(5)

where Tfc is the working temperature of the FC (K) , PH2 and PO2 define the regulating pressures of hydrogen (H2) and oxygen (O2)(atm) ; respectively, Ifc is the operating current (A) , MA denotes the membrane area (cm2), CO2 is the concentration of O2 (mol/cm3) , ξ1 − ξ 4 are semiempirical coefficients, Rm and R c indicate the membrane and the leads’ ohmic resistances (Ω) ; respectively, l is the membrane thickness (cm) , ρm is the resistivity of the membrane (Ω.cm), λ is treated as an changeable parameter, β is treated as a constant within empirical range, and J and Jmax define the actual and maximum/thermal current densities (A/cm2), respectively. ξ1 – ξ4 represent parametric coefficients of the FC model which their values depend on the geometrical dimensioning and the material resources used in the manufacture of the PEMFCs stack [5,44]. On the other hand, it may be noting that the estimation of λ value is very difficult as λ is function of many factors such as cell relative humidity, and the stoichiometric ratio. Nevertheless, λ is considered here in this current study as adjustable parameter and It may be worth mentioning that the value of λ ranges from 13 to 23 which depends on the time service as well whereas, lower values points out high cell relative humidity and higher values indicates supersaturated conditions [5,34,43,44]. the aforementioned can be accepted, with the assumption of an appropriate membrane water content at all likely operating conditions is kept. On the other hand, β is a parameter for the Tafel equation (see (5)) which can be calculated using the formula depicted in (6). Practically, the FC performance often has a larger value than what the Eq. (6) estimates. Due to this, β is frequently found empirically/semi-empirically.

β=

R. Tfc 2αF

(6)

where R and F are ideal gas and Faraday’s constants; respectively, and 2

Energy Conversion and Management 201 (2019) 112197

A.A. El-Fergany, et al.

α defines charge transfer coefficient. A close look to (5) and (6), it is clear that the concentration voltage drop is changed linearly with temperature and actual current density. On other words, the concentration polarization voltage is expected to increase more at higher current densities and higher cell temperature. It is worth for noting here that seven parameters are essentially estimated to create an accurate model of the PEMFC for any further analysis in power system studies.

where X (t ) is the vector of whales location, t denotes the iteration counter, Xp (t ) is the vector of target prey, and A and C represent the coefficient vectors, which are determined by using (10)–(11) [37,38].

The PEMFC’s model comprises a high non-linear characteristic involving several uncertain parameters due to a shortage of manufacturer data. This represents a high level of difficulty to establish an accurate model. To obtain such accurate model, seven parameters are essentially estimated. The parameters estimation of the PEMFCs can be dealt as an optimization problem and the FF can be expressed as the sum squared error between the estimated model voltage and experimental FC voltage. In this current study, the aforementioned problem can be treated as a non-convex optimization problem and its FF is expressed based on the SSE between the practical terminal stack voltage and the corresponding estimated voltage by the PEMFC projected model. This FF is expressed in (7) as follows [21,34]: Nsamples

∑ m=1

⎞ [VFC, exp (m) − VFC , est (m)]2 ⎟ ⎠

(10)

C = 2. r

(11)

The vector a is linearly reduced from 2 to 0 with the course of iterations. This is carried out in the exploration and exploitation process). The vector r defines a randomized vector having values between [0, 1].

3. Problem formulation and description

⎛ FF = Min (SSE ) = Min ⎜ ⎝

A = 2a. r − a

4.2. Bubble net attacking the prey The attacking strategy demonstrates the exploitation process or the local search of the proposed algorithm. In this issue, two methods are utilized to describe the humpback whales bubble behavior. These methods are formulated in two schemes as: 1. Encircling Shrinkage strategy: In this scheme, the humpback whales swim about the prey establishing the lessening circles. In other words, this procedure can be obtained mathematically by reducing the vector a linearly from 2 to 0 with the course of iterations and |A| < 1. 2. Spiral updating location: These whales possess the ability to swim up to the victim forming a spiral configuration. The location of the humpback whales is continuously updated by using (12)-(13).

(7)

where Nsamples indicates the number of voltage data measured, m defines the iteration counter, VFC, exp is the practical measured output voltage of the FC(V) , and VFC, est is the calculated voltage of the estimated FC model (V). This FF is subjected to the min/max limits of unknown seven parameters as inequality constraints. These seven unknown parameters are namely; ξ1 − ξ 4 , λ , R c , and β which are optimized by the WOA for their best values. The proposed WOA is directly applied to the SSE to find the best values of these seven unidentified uncertain parameters. In this regard, the optimized problem under study is solved using MATLAB environment [46].

X (t + 1) = D′. e bl . cos(2πl) + Xp (t )

(12)

XD′ = |Xp (t ) − X (t )|

(13)

where b is an exponent constant, which defines the spiral logarithmic shape, and l is a uniform random number between [−1, 1]. The humpback whales propose these two methods simultaneously through the attacking strategy. In this regard, a probability of 50% is selected to perform shrinking encircling process and the same probability is joined the spiral model. The mathematical model of this behavior is built by using (14) [34].

X (t + 1) =

∀ p < 0.5 ⎧ Xp (t ) − A. D ⎨ D′. e bl . cos(2πl) + Xp (t ) ∀ p ⩾ 0.5 ⎩

(14)

where p defines the probability percentage that lies in the range [0, 1]. 4. WOA 4.2.1. Looking for the victim In this course of action, the proposed whales look for the prey indicating the exploration process of the proposed algorithm. The condition |A| > 1 can illustrate this search practice. The location of the humpback whales is updated by (15)-(16).

In fact, the whales live generally in the oceans and they are considered as the largest mammals worldwide. They grow to reach 30 m and 180 t weight. One type of these whales is called the humpback whale, which has a special type of hunting process. The hunting behavior is named bubble net feeding. These whales hunt a group of small fishes at the ocean surface. The hunting procedure can be done by diving to a deep position and creating bubbles around the prey. Then the whales swim to the surface for hunting the small fishes. The travel from the deep position to the surface is performed in a group, which provides bubbles in a circular or spiral form. The WOA model points out the foraging process, which involves encircling the prey, creating bubbles maneuver with their shapes, and also searching for the prey. The mathematical WOA model can be written as follows [37]:

D = |C . Xr (t ) − X (t )|

(15)

X (t + 1) = Xr (t ) − A. D

(16)

where Xr (t ) represents a random location vector. A detailed flowchart of WOA is demonstrated in Fig. 1. As all metaheuristic algorithms, the WOA starts with a randomized initial population of the agents through the search region and finally ends with the best agent solution or the place of victim Xp which represents the optimal solution of the defined problem.

4.1. Encircling the prey

5. Simulation results, demonstration and comparisons

Actually, the humpback whales have the capability to identify the position of the prey and then enclose them in circles. In the WOA model, the location of the best agent solution so far is selected as the target prey. The encircle movement of the whales around the prey can be expressed in by using (8) and (9).

D = |C . Xp (t ) − X (t )|

(8)

X (t + 1) = Xp (t ) − A. D

(9)

To appraise the value of the proposed WOA in generating the unknown parameters of the PEMFC model for effective simulation and modeling, three test cases are performed on commercial typical ones such as Ballard Mark V, AVISTA SR-12 PEM generator, 250 W and Horizon H-12 12 W stacks with the given experimental I/V data curves. Datasheets of these four commercial PEMFCs stacks and the typical boundaries of the seven unidentified parameters are concluded from [28,29,34,36]. It is well-known that nature of the parameter estimation 3

Energy Conversion and Management 201 (2019) 112197

A.A. El-Fergany, et al.

the correct match between the measured voltage data and the corresponding estimated values by the WOA based model. The obtained best results by the WOA are compared with those attained by the GHO [34], which are arranged in Table 1. Additionally, the plots of various internal voltage losses in the stack versus the stack current are illustrated in Fig. 2(c). It may be useful to state that the GHO has the following controlling variables: Npop equals 60 and Max_Iter is equal to 100. The computation time for the GHO to implement such optimization process is 8.50 s. On the other hand, NNO has NPOP and Max_Iter equal 50, and 100; respectively and requires 20.0 s to implement such optimization process. The reader can see that the WOA generates very competitive results compared to the GHO with a lower SSE value. At this moment, the simulated performance at different operating temperatures (30oC, 50oC and 75oC) are depicted in Fig. 3(a) with keeping partial pressures constant (PH 2/ PO2 =1/1 atm). On the other hand, under regulating the partial pressures with constant temperature of 70oC, the depicted I/V polarization characteristics are illustrated in Fig. 3(b). 5.2. Test case 2: Performance assessment on AVISTA SR-12 500 W modular The AVISTA SR-12 500 W modular PEMFCs stack has 48 cells connected in series with a membrane thickness of 25 µm, MA = 62.5 cm2 , 672 mA/cm2 maximum current density, and maximum working stack current is 30 A. By implementing the WOA 100 independent trials, the best results of these seven unknown parameters are recorded in Table 2. The best value of SSE is equal to 0.0111 V 2 which is indicated in the convergence curve shown in Fig. 4(a). The average CPU processing time to execute one run over the 200 iterations is 38.3 s re-implemented on the above-mentioned Lenovo laptop with activating parallel processing feature. On the other hand, the computation burdens of GWO and GHO are 35.40 s and 12.50 s; respectively. The GWO has population size of 20 agents and a Max_Iter of 500. The changeable parameters of SFL are: population size = 50, Max_Iter = 100, memeplexes = 5, mems = 10, and evolutions = 10. The SFL converges in 45.0 s. The VSDE algorithm is executed for 500 iterations, and the population size = 50. In addition, the average elapsed time taken by the NNO to realize such best settings is 33.0 s. The characteristics of I/V polarization plots comprising the experimental data along the corresponding estimated ones are depicted in Fig. 4(b). It is well-meaning here to confirm that the computed results using the WOA-based model are coincide to that obtained by the experimental results. This marks the high accurateness of the proposed PEMFC model. In addition, plots of various internal voltage losses in the stack versus the stack current are revealed in Fig. 4(c). The simulated performance at different operating temperatures (30oC, 50oC and 75oC) are depicted in Fig. 5(a) with keeping partial pressure constant(PH 2/ PO2 = 1.47628/0.2095 atm). On the other hand, under regulating the partial pressures with constant temperature of 60oC, the depicted I/V polarization characteristics are demonstrated in Fig. 5(b). The reader may realize that these curves are extremely smooth under different operating conditions providing with the trust of the high efficiency of the proposed WOA-based model.

Fig. 1. Detailed flowchart of the proposed WOA.

is an off-line task; the computation time is not a matter. However, to show the superiority of the WOA over other competing methods and following anonymous respected reviewers’ comments, the authors has considered this point. The simulations are performed using MATLAB platform Version: 9.4.0.813654 (R2018a) under Microsoft Windows 10 Pro Version 10.0 (Build 16299) 64-bit. The MATLAB code is executed on an Intel(R) Core™ i7-4510U CPU @ 2.00 GHz 2.6 GHz Lenovo laptop equipped with 16 GB of RAM and running the windows operating system. The final accepted WOA controlling parameters are: number of whales equals 50 and the number of iterations is equal to 200. For all test cases, the WOA is implemented 100 independent runs and the statistical measures (best, worst, and standard deviation (StD)) are made per each case study to investigate its robustness. In the subsequent subsections, the associated trend of convergence (best) of SSE and the I/V polarization curves are shown. All convergence plots are shown in semi-logy graphs. The anticipated test cases are demonstrated under steady-state conditions as follows:

5.3. Test case 3: Performance assessment on 250 W stack 5.1. Test case 1: Performance assessment on Ballard Mark V This stack has 24 cells connected in series with a membrane thickness of 178 µm, MA = 27 cm2 , 23 A thermal current and 680 mA/cm2 maximum current density. The cropped best value of the SSE is equal to 0.3372 and the average CPU processing time to implement one run over the 200 iterations is 13.5 s to reach this value with activating the parallel processing feature. The best values of this stack are arranged in Table 3 along with those obtained by other competing optimization algorithms in the literature such as GWO [28], MVO [33], GHO [34], and VSDE [35]. The adopted settings for the MVO are: Pop. size = 50,

This PEMFCs stack has 35 cells connected in series with a membrane thickness of 178 µm, MA = 50.6 cm2 , thermal rated current of 70 A and 1.5 A/ cm2 maximum current density. Fig. 2(a) illustrates the trend of SSE convergence over the 200 iterations with steadily convergence. The average CPU processing time to performed one run over the 200 iterations is 5.80 s re-implemented on the above-mentioned Lenovo laptop with activating parallel processing feature. The I/V polarization curves of this Ballard Mark V stack are depicted in Fig. 2(b) indicating 4

Energy Conversion and Management 201 (2019) 112197

A.A. El-Fergany, et al.

Fig. 2. Demonstrations of Ballard Mark V: (a) trend of convergence. (b) I/V polarization plots. (c) internal voltage losses.

are very competitive in the view of lesser SSE and best statistical measures. The trend of SSE minimizations is shown in Fig. 6(a) and the I/V polarization characteristics showing the measured and estimated data are illustrated in Fig. 6(b). In which, matching between experimental and WOA-model results is indicated. Additionally, the plots of various internal voltage losses stack versus the stack current are demonstrated in Fig. 2(c). The simulated performance at different operating temperatures (30oC, 50oC and 75oC) are depicted in Fig. 7(a) with keeping partial pressure constant (PH 2/ PO2 =1/1 atm). On the other hand, under regulating the partial pressures with constant temperature of 65oC, the I/V polarization characteristics are illustrated in Fig. 7(b). From the above-mentioned demonstrations for the studied cases, the proposed WOA have succeeded in estimating the optimal values of the PEMFCs unknown seven parameters and within the predefined constraints with lesser best SSE. The superiority of the WOA-based PEMFC model to other optimization methods-based model reflects the high performance of the WOA and its excellent design by the designers. Moreover, it has a minimum number of parameters to be adjusted or

Table 1 Optimum values of unknown parameters together with the statistical measurements of ESS compared to others for case 1. Parameter

WOA

NNO [24]

GHO [34]

ξ1 (V )

−1.1978

−0.97997

−0.8532

ξ2 (V / K )

4.4183e-3

3.6946 e-3

3.4173e-3

ξ3 (V / K )

9.7214e-5

9.0871 e-5

9.8000e-5

ξ4 (V / K )

−16.2730e-5

−16.2820 e-5

−15.9555e-5

λ R c (mΩ) β SSE (best) SSE (Worst) SSE (StD)

23.000 0.1002 0.0136 0.8537 0.9420 0.0239

23.0000 0.1000 0.0136 0.8536 0.8707 0.0085

22.8458 0.1000 0.0136 0.8710 0.9097 0.0113

and Max_Iter = 2000 with convergence time of 135.50 s. On the other hand, the elapsed CPU computational times for GWO and GHO are 150.66 s and 9.30 s; respectively. It can be seen that the results of WOA

Fig. 3. Performance assessment under various operating conditions for Ballard Mark V: (a) different temperatures. (b) regulating partial pressures. 5

Energy Conversion and Management 201 (2019) 112197

A.A. El-Fergany, et al.

Table 2 Optimal values of unknown parameters together with the SSE statistical measures compared to others for case 2. Parameter

WOA

SFL [5]

NNO [24]

GWO [28]

GHO [34]

VSDE [35]

ξ1 (V )

−0.8902

−0.9657

−1.0596

−0.9664

−1.1997

−0.8576

ξ2 (V / K )

3.3088e-3

3.080

3.7435e-3

2.2833e-3

4.2695e-3

3.0100e-3

ξ3 (V / K )

9.75455e-5

7.2236

9.6902e-5

3.4000e-5

9.8000e-5

7.7800e-5

ξ4 (V / K )

−10.3300e-5

−19.3000

−19.3020e-5

9.5400e-5

−10.1371e-5

9.5400e-5

λ R c (mΩ) β SSE (best) SSE (Worst) SSE (StD)

22.8311 0.5677 0.1464 0.0111 0.0183 0.0018

20.8862 0.1000 0.0161 0.0117 0.0117 5.039e-8

20.8772 0. 1000 0.0161 0.0117 0.013670 0.0054

15.7969 0.6685 0.1804 1.5170 NR* NR*

23.0000 0.4638 0.1486 0.0478 0.0599 0.0036

23.0000 0.1339 0.1516 1.26600 NR* NR*

*NR means not reported.

fine-tuned plus its high and smooth convergence speed, as demonstrated in the simulation results.

Then, the proposed WOA is applied to the Horizon H-12 PEMFC to determine its optimal parameters by minimizing the SSE. In this regard, the best value of SSE reaches 0.1160 V 2 . The average CPU processing time to implement one run over the 200 iterations is 12.5 s to reach this value. The optimal design variables of this stack are listed in Table 5. To validate the WOA results, GA is employed as a well-established evolutionary based optimization algorithm which is well-known to other researchers (i.e. no extra materials are required). Detailed results of WOA along with GA are depicted in Tables 4 and 5. The approved GA controlling parameters are: population size = 50, Iterations = 200, crossover = 0.85 and mutation = 0.01 with convergence time 218.50 s for the processing of this optimization problem. The I/V polarization characteristics showing the measured and estimated data are illustrated in Fig. 9(a). It is worth noting here that the experimental results are very close to the results of the WOA model. Simulated performance at different operating temperatures (25oC, 40oC and 55oC) are illustrated in Fig. 9(b) with keeping the partial pressure is constant (PH 2/ PO2 =0.5/1 atm). On the other hand, under regulating the partial pressures with constant temperature of 29oC, the I/V polarization characteristics are illustrated in Fig. 9(c). It can be realized that these I/V characteristics are very close to each other under different pressure due to the narrow

5.4. Test case 4: Performance assessment on Horizon H-12 12 W stack The main goal of this test case study is to verify the application of WOA to efficiently determine the unknown parameters of a Horizon H12 PEMFC stack in our laboratory. Fig. 8 displays an experimental setup of the aforementioned PEMFC stack in our laboratory L107 at the Electrical Engineering Department, Faculty of Engineering, Northern Border University, Saudi Arabia. The H-12 PEMFC stack is implemented at a normal pressure of 0.5 bar and a room temperature of 29 °C. This FC is loaded gradually up to 2.00 A (a temporary overload is attempted). A thermometer is placed on the fan outlet for a measurement of the stack temperature. The hydrogen pressure is controlled by a pneumatic valve and measured by a pneumatic gage. The principal specifications of this fuel cell include Ncells = 13, MA = 8.1 cm2, Jmax = 246.9 mA/cm2, Tfc = 302.15 K, l= 25 μm, PH2 = 0.4935 atm, PO2 = 1 atm [47]. In this experimental test, 20 measurements of I/V characteristics are recorded under steady state conditions as enumerated in Table 4.

Fig. 4. Demonstrations of AVISTA SR-12 500 W modular: (a) trend of convergence. (b) I/V polarization plots. (c) internal voltage losses. 6

Energy Conversion and Management 201 (2019) 112197

A.A. El-Fergany, et al.

Fig. 5. Performance assessment under various operating conditions for AVISTA SR-12 500 W modular: (a) different temperatures. (b) regulating partial pressures. Table 3 Optimal values of unknown parameters together with the SSE statistical measures compared to others for case 3. Parameter

WOA

GWO [28]

MVO [33]

GHO [34]

VSDE [35]

ξ1 (V )

−0.9565

−1.0564

−0.9182

−0.8532

−1.1921

ξ2 (V / K )

3.2221e-3

3.7953e-3

3.1299e-3

3.1237 e-3

3.199e-3

ξ3 (V / K )

8.2328e-5

9.8000e-5

8.7031e-5

9.8000 e-5

3.7990e-5

ξ4 (V / K )

−17.5410e-5

11.7550e-5

−18.0253e-5

−17.0112 e-5

−18.7000e-5

λ R c (mΩ) β SSE (best) SSE (Worst) SSE (StD)

20.4470 0.1082 0.0152 0.3372 0.5029 0.0493

23.0000 0.1088 0. 0136 1.1505 1.2329 0.0921

15.1921 0.4223 0.01800 3.5846 11.420 200.9

18.3441 0.1000 0.0136 0.3395 0.4986 0.0467

22.8170 0.1202 0.02903 1.0526 1.1875 0.1768

Fig. 6. Demonstrations of 250 W stack: (a) trend of convergence. (b) I/V polarization plots. (c) internal voltage losses.

constraints with lower best SSE. The superiority of the WOA-based PEMFC model to other optimization methods-based model reflects the high performance of the WOA and its excellent design by the designers. Moreover, it has a minimum number of parameters to be adjusted or

range of this PEMFC, which varies from 0.4 to 0.55 bar. From the above-mentioned demonstrations for the studied cases, the proposed WOA have succeeded in estimating the optimal values of the PEMFCs unknown seven parameters and within the predefined 7

Energy Conversion and Management 201 (2019) 112197

A.A. El-Fergany, et al.

Fig. 7. Performance assessment under various operating conditions for test case 3: (a) different temperatures. (b) regulating partial pressures. Table 5 Optimal values of unknown parameters along with the SSE statistical measures for case 4. Parameter

WOA

GA

ξ1 (V )

−1.1870

−1.1042

ξ2 (V / K )

2.6697e−3

2.4540e−3

ξ3 (V / K )

3.6000e−5

3.9881e−5

ξ4 (V / K )

−9.5400e−5

−9.5400e−5

λ R c (mΩ) β SSE (best) SSE (Worst) SSE (StD)

13.8240 0.8000 0.1598 0.1160 0.1258 0.00224

14.6601 0.7981 0.1656 0.1184 0.1317 0.00362

is to examine the validity of the obtained uncertain seven parameters. Two extra real measurements are implemented at (29oC and 0.4 bar) and (39oC and 0.5 bar), which are listed in Table 6. In addition, these recorded measurements together with corresponding estimated data by the model are charted in Fig. 10(a)-(b). Two further statistical indicators are used to point out the accuracy of the developed PEMFC model: (i) mean absolute error (MAE), and (ii) mean absolute percentage error (MAPE). Both MAE and MAPE are statistically defined in (17) and (18); respectively.

Fig. 8. Experimental setup of horizon PEM fuel cell H-12 stack.

fine-tuned plus its high and smooth convergence speed, as demonstrated in the simulation results. Additional validations are performed to appraise the accuracy of the developed WOA-based PEMFC model with changes of the operating parameters with comparison with experimental measurements at various temperatures and hydrogen partial pressures. The principal target Table 4 H-12 stack measured and estimated data. I fc (A)

0.104 0.200 0.309 0.403 0.510 0.614 0.703 0.806 0.908 1.076 1.127 1.288 1.390 1.450 1.578 1.707 1.815 1.900 2.060 SSE

VFC , meas (V )

9.580 9.420 9.250 9.200 9.090 8.950 8.850 8.740 8.650 8.450 8.410 8.200 8.120 8.110 8.050 7.990 7.950 7.940 7.900

WOA

GA

VFC, est (V )

[VFC , meas − VFC , est ]2

VFC, est (V )

[VFC , meas − VFC , est ]2

9.731 9.450 9.246 9.110 8.981 8.870 8.784 8.691 8.605 8.471 8.432 8.313 8.239 8.196 8.107 8.018 7.944 7.886 7.777 0.1160

0.0227 0.0009 0.0000 0.0080 0.0119 0.0063 0.0043 0.0024 0.0020 0.0004 0.0005 0.0127 0.0142 0.0075 0.0032 0.0008 0.0000 0.0030 0.0152

9.749 9.467 9.262 9.126 8.996 8.884 8.798 8.704 8.617 8.481 8.442 8.321 8.246 8.203 8.112 8.022 7.947 7.888 7.777 0.1184

0.0286 0.0022 0.0001 0.0055 0.0088 0.0044 0.0027 0.0013 0.0011 0.0010 0.0010 0.0146 0.0159 0.0086 0.0038 0.0010 0.0000 0.0027 0.0151

8

Energy Conversion and Management 201 (2019) 112197

A.A. El-Fergany, et al.

Fig. 9. Demonstrations of Horizon H-12 stack: (a) I/V polarization plots. (b) internal voltage losses. (c) Performance assessment under different temperatures. (d) Performance assessment under regulating partial pressures.

varies from case to case, justifying the differences in CPU computation time for each case study. Frankly speaking, these algorithms stated in the literature might perform well or not as good as required by varying their changeable parameters which are realized by trials and errors methodology over many independent shots and much dependable on the designer experience. For the reader convenience and to avoid lengthy article, the detailed results and subsequent analysis shall will be addressed and discussed in a separate journal manuscript. As extension of this current work, transient model shall be addressed and to test the robustness of the developed methodology under various conditions such a series of fuel cell at various temperature and variable conditions at point by point V-I measurements.

Table 6 Validations under various operating temperature and hydrogen partial pressures. At 29 oC and 0.4 bar

At 39 oC and 0.5 bar

I fc (A)

VFC , meas (V )

VFC, est (V )

I fc (A)

VFC , meas (V )

VFC, est (V )

0.104 0.199 0.307 0.403 0.511 0.614 0.704 0.806 0.908 1.075 1.126 1.280 1.390 1.450 1.570

9.530 9.380 9.200 9.240 9.100 8.940 8.840 8.750 8.660 8.450 8.410 8.200 8.140 8.110 8.000

9.693 9.414 9.211 9.073 8.942 8.833 8.745 8.654 8.567 8.434 8.395 8.281 8.201 8.159 8.074 0.1379

0.097 0.115 0.165 0.204 0.249 0.273 0.326 0.396 0.500 0.621 0.711 0.797 1.006 1.141 1.370

9.870 9.840 9.770 9.700 9.610 9.590 9.500 9.400 9.260 9.050 8.930 8.830 8.540 8.420 8.270

9.949 9.877 9.719 9.622 9.529 9.484 9.396 9.294 9.165 9.034 8.946 8.868 8.693 8.589 8.420 0.1416

SSE(V 2) MAPE MAE (V)

MAE =

0.92% 0.0814

This article have proposed a novel attempt of the WOA aimed at an efficient extraction of the unknown parameters of the PEMFC model. The ultimate goal is to attain a precise model of the PEMFC, which provides with accurate results of modeling and simulation of these FCs. The several unknown parameters are ungiven in manufacturer's datasheets. The optimized problem was the total of squared error between the measured and its output voltage model of FC. The WOA was applied directly to minimize the FF under the problem constraints. The computer demonstrations have proven the strength of the proposed PEMFC model under different temperature and pressure conditions. The efficacy of the proposed model is evaluated by comparing its numerical model results with the experimental results of the four commercial PEMFCs stacks studied. The simulation results are coincide with the experimental results for all case studies. Moreover, The WOAbased results are compared with other optimization methods-based results. This results in a high superiority of the WOA-based model to other optimization methods-based models in the literature. Therefore, the application of the WOA can provide with an accurate PEMFC model.

0.94% 0.0853

Nsamples

1 Nsamples

MAPE =

SSE(V 2) MAPE MAE (V)

6. Conclusions

1 Nsamples

∑

|VFC, est (k ) − VFC, meas (k )|

k=1

Nsamples

∑ k=1

VFC, est (k ) − VFC, meas (k ) × 100 VFC , meas (k )

(17)

(18)

The calculated MAE and MAPE values are illustrated in Table 6. It is good to observe that the small values of both MAE and MAPE confirm the validity of the realized optimal parameters when they are used under other different operating conditions. It should be noted that the number of experimental data points 9

Energy Conversion and Management 201 (2019) 112197

A.A. El-Fergany, et al.

Fig. 10. Polarization curves with the model estimation and the experimental data at (a) 29oC and 0.4 bar. (b) 39oC and 0.5 bar.

Declaration of Competing Interest

[14] Al-Baghdadi MARS. Modelling of proton exchange membrane fuel cell performance based on semi-empirical equations. Renew Energy 2005;30(10):1587–99. https:// doi.org/10.1016/j.renene.2004.11.015. [15] Geem ZW, Noh JS. Parameter estimation for a proton exchange membrane fuel cell model using GRG technique. Fuel Cell 2016;16(5):640–5. https://doi.org/10.1002/ fuce.201500190. [16] Aouali FZ, Becherif M, Ramadan HS, Emziane M, Khellaf A, Mohammedi K. Analytical modelling and experimental validation of proton exchange membrane electrolyser for hydrogen production. Int J Hydrogen Energy 2017;42(2):1366–74. https://doi.org/10.1016/j.ijhydene.2016.03.101. [17] Jiang Y, Yang Z, Jiao K, Du Q. Sensitivity analysis of uncertain parameters based on an improved proton exchange membrane fuel cell analytical model. Energy Convers Manage 2018;164:639–54. https://doi.org/10.1016/j.enconman.2018.03.002. [18] Asl SMS, Rowshanzamir S, Eikani MH. Modelling and simulation of the steady-state and dynamic behavior of a PEM fuel cell. Energy 2010;35(4):1633–46. https://doi. org/10.1016/j.energy.2009.12.010. [19] Askarzadeh A, Rezazadeh A. An innovative global harmony search algorithm for parameter identification of a PEM fuel cell model. IEEE Trans Power Electron 2012;59(9):3473–80. https://doi.org/10.1109/TIE.2011.2172173. [20] Xu S, Wang Y, Wang Z. Parameter estimation of proton exchange membrane fuel cells using eagle strategy based on JAYA algorithm and Nelder-Mead simplex method. Energy 2019;173:457–67. https://doi.org/10.1016/j.energy.2019.02.106. [21] El-Fergany AA. Extracting optimal parameters of PEM fuel cells using salp swarm optimizer. Renew Energy 2018;119:641–8. https://doi.org/10.1016/j.renene.2017. 12.051. [22] Niu Q, Zhang L, Li K. A biogeography-based optimization algorithm with mutation strategies for model parameter estimation of solar and fuel cells. Energy Convers Manage 2014;86:1173–85. https://doi.org/10.1016/j.enconman.2014.06.026. [23] Zhang L, Wanga N. An adaptive RNA genetic algorithm for modeling of proton exchange membrane fuel cells. Int J Hydrogen Energy 2013;38(1):219–28. https:// doi.org/10.1016/j.ijhydene.2012.10.026. [24] Fawzi M, El‐Fergany AA, Hasanien HM. Effective methodology based on neural network optimizer for extracting model parameters of PEM fuel cells. Int J Energy Res 2019. https://doi.org/10.1002/er.4809. [25] Sun Z, Wang N, Bi Y, Srinivasan D. Parameter identification of PEMFC model based on hybrid adaptive differential evolution algorithm. Energy 2015;90:1334–41. https://doi.org/10.1016/j.energy.2015.06.081. [26] Chen Y, Wang N. Cuckoo search algorithm with explosion operator for modeling proton exchange membrane fuel cells. Int J Hydrogen Energy 2019;44(5):3075–87. https://doi.org/10.1016/j.ijhydene.2018.11.140. [27] Zhang W, Wang N, Yang SP. Hybrid artificial bee colony algorithm for parameter estimation of proton exchange membrane fuel cell. Int J Hydrogen Energy 2013;38(14):5796–806. https://doi.org/10.1016/j.ijhydene.2013.01.058. [28] Ali M, El-Hameed MA, Farahat MA. Effective parameters’ identification for polymer electrolyte membrane fuel cell models using grey wolf optimizer. Renew Energy 2017;111:455–62. https://doi.org/10.1016/j.renene.2017.04.036. [29] Rao Y, Shao Z, Ahangarnejad AH, Gholamalizadeh E, Sobhani B. Shark Smell Optimizer applied to identify the optimal parameters of the proton exchange membrane fuel cell model. Energy Convers Manage 2019;182:1–8. https://doi.org/ 10.1016/j.enconman.2018.12.057. [30] Askarzadeh A, Coelho LS. A backtracking search algorithm combined with Burger's chaotic map for parameter estimation of PEMFC electrochemical model. Int J Hydrogen Energy 2014;39(21):11165–74. https://doi.org/10.1016/j.ijhydene. 2014.05.052. [31] Niu Q, Zhang H, Li K. An improved TLBO with elite strategy for parameters identification of PEM fuel cell and solar cell models. Int J Hydrogen Energy 2014;39(8):3837–54. https://doi.org/10.1016/j.ijhydene.2013.12.110. [32] Dai C, Chen W, Cheng Z, Li Q, Jiang Z, Jia J. Seeker optimization algorithm for global optimization: a case study on optimal modelling of proton exchange membrane fuel cell. Int J Electr Power Energy Syst 2011;33(3):369–76. https://doi.org/ 10.1016/j.ijepes.2010.08.032. [33] Fathy A, Rezk H. Multi-verse optimizer for identifying the optimal parameters of PEMFC model. Energy 2018;143:634–44. https://doi.org/10.1016/j.energy.2017. 11.014.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors frankly thank the anonymous reviewers and editors for sparing their valuable time and giving constructive comments and suggestions to criticize our article that definitely improve the paper technical quality and presentations and make it more attractive to the readers. References [1] Nejad HC, Farshad M, Gholamalizadeh E, Askarian B, Akbarimajd A. A novel intelligent-based method to control the output voltage of Proton Exchange Membrane Fuel Cell. Energy Convers Manage 2019;185:455–64. https://doi.org/10.1016/j. enconman.2019.01.086. [2] El-Hay EA, El-Hameed MA, El-Fergany AA. Optimized parameters of SOFC for steady state and transient simulations using interior search algorithm. Energy 2019;66:451–61. https://doi.org/10.1016/j.energy.2018.10.038. [3] El-Hay EA, El-Hameed MA, El-Fergany AA. Steady-state and dynamic models of solid oxide fuel cells based on satin bowerbird optimizer. Int J Hydrogen Energy 2018;43(31):14751–61. https://doi.org/10.1016/j.ijhydene.2018.06.032. [4] Agwa AM, El-Fergany AA, Sarhan GM. Steady-state modeling of fuel cells based on atom search optimizer. Energies 2019;12:1884. https://doi.org/10.3390/ en12101884. [5] Kandidayeni M, Macias A, Khalatbarisoltani A, Boulon L, Kelouwani S. Benchmark of proton exchange membrane fuel cell parameters extraction with metaheuristic optimization algorithms. Energy 2019;183:912–25. https://doi.org/10.1016/j. energy.2019.06.152. [6] El-Hay EA, El-Hameed MA, El-Fergany AA. Performance enhancement of autonomous system comprising proton exchange membrane fuel cells and switched reluctance motor. Energy 2018;63:699–711. https://doi.org/10.1016/j.energy.2018. 08.104. [7] Sun L, Jin Y, Pan L, Shen J, Lee KY. Efficiency analysis and control of a grid-connected PEM fuel cell in distributed generation. Energy Convers Manage 2019;195:587–96. https://doi.org/10.1016/j.enconman.2019.04.041. [8] Budak Y, Devrim Y. Investigation of micro-combined heat and power application of PEM fuel cell systems. Energy Convers Manage 2018;160:486–94. https://doi.org/ 10.1016/j.enconman.2018.01.077. [9] Bizon N, Mazare AG, Ionescu LM, Enescu FM. Optimization of the proton exchange membrane fuel cell hybrid power system for residential buildings. Energy Convers Manage 2018;163:22–37. https://doi.org/10.1016/j.enconman.2018.02.025. [10] Priya K, Sathishkumar K, Rajasekar N. A comprehensive review on parameter estimation techniques for proton exchange membrane fuel cell modelling. Renew Sustain Energy Rev 2018;93:121–44. https://doi.org/10.1016/j.rser.2018.05.017. [11] Guo C, Luc J, Tiana Z, Guob W, Darvishan A. Optimization of critical parameters of PEM fuel cell using TLBO-DE based on Elman neural network. Energy Convers Manage 2019;183:149–58. https://doi.org/10.1016/j.enconman.2018.12.088. [12] Ritzberger D, Striednig M, Simon C, Hametner C, Jakubek S. Online estimation of the electrochemical impedance of polymer electrolyte membrane fuel cells using broad-band current excitation. J Power Sources 2018;405:150–61. https://doi.org/ 10.1016/j.jpowsour.2018.08.082. [13] Taleb MA, Béthouxb OB, Godoy E. Identification of a PEMFC fractional order model. Int J Hydrogen Energy 2017;42(2):1499–509. https://doi.org/10.1016/j. ijhydene.2016.07.056.

10

Energy Conversion and Management 201 (2019) 112197

A.A. El-Fergany, et al.

2018.5317. [41] Abd Elaziz M, Oliva D. Parameter estimation of solar cells diode models by an improved opposition-based whale optimization algorithm. Energy Convers Manage 2018;171:1843–59. https://doi.org/10.1016/j.enconman.2018.05.062. [42] Xiong G, Zhang J, Shi D, He Y. Parameter extraction of solar photovoltaic models using an improved whale optimization algorithm. Energy Convers Manage 2018;174:388–405. https://doi.org/10.1016/j.enconman.2018.08.053. [43] Mann RF, Amphlett JC, Hooper MAI, Jensen HM, Peppley BA, Roberge PR. Development and application of a generalised steady-state electrochemical model for a PEM fuel cell. J Power Sources 2000;86(1–2):173–80. https://doi.org/10. 1016/S0378-7753(99)00484-X. [44] Saleh IMM, Ali R, Zhang H. Simplified mathematical model of proton exchange membrane fuel cell based on horizon fuel cell stack. J Mod Power Syst Clean Energy 2016;4(4):668–79. https://doi.org/10.1007/s40565-016-0196-5. [45] Zhang G, Jiao K. Three-dimensional multi-phase simulation of PEMFC at high current density utilizing Eulerian-Eulerian model and two-fluid model. Energy Convers Manage 2018;176:409–21. https://doi.org/10.1016/j.enconman.2018.09. 031. [46] Release 2018a, ‘MATLAB’, The Math Works press, 2018, https://www.mathworks. com/, (Accessed on 5 April 2018). [47] https://www.horizoneducational.com/advancedproducts/h-12-pem-fuel-cell12w/, (Accessed 24 July 2019).

[34] El-Fergany AA. Electrical characterisation of proton exchange membrane fuel cells stack using grasshopper optimiser. IET Renew Power Gener 2018;12:9–17. https:// doi.org/10.1049/iet-rpg.2017.0232. [35] Fathy A, Abd Elaziz M, Alharbi AG. A novel approach based on hybrid vortex search algorithm and differential evolution for identifying the optimal parameters of PEM fuel cell. Renew Energy 2020;146:1833–45. https://doi.org/10.1016/j.renene. 2019.08.046. [36] Priya K, Rajasekar N. Application of flower pollination algorithm for enhanced proton exchange membrane fuel cell modelling. Int J Hydrogen Energy 2019;44:18438–49. https://doi.org/10.1016/j.ijhydene.2019.05.022. [37] Mirjalili S, Lewis A. The whale optimization algorithm. Adv Eng Softw 2016;95:51–67. https://doi.org/10.1016/j.advengsoft.2016.01.008. [38] Hasanien HM. Performance improvement of photovoltaic power systems using an optimal control strategy based on whale optimization algorithm. Electr Power Syst Res 2018;157:168–76. https://doi.org/10.1016/j.epsr.2017.12.019. [39] Hasanien HM. Whale optimisation algorithm for automatic generation control of interconnected modern power systems including renewable energy sources. IET Gener Transm Distrib 2018;12(3):607–14. https://doi.org/10.1049/iet-gtd.2017. 1005. [40] Elazab OS, Hasanien HM, Elgendy MA, Abdeen AM. Parameters estimation of single- and multiple-diode photovoltaic model using whale optimisation algorithm. IET Renew Power Gener 2018;12(15):1755–61. https://doi.org/10.1049/iet-rpg.

11