Improved grass fibrous root algorithm for exergy optimization of a high-temperature PEMFC

Lu, Xiaohui; Ren, Jianglin; Guo, Lin; Wang, Peifang; Yousefi, Nasser Article Improved grass fibrous root algorithm for exergy optimization of a high-temperature PEMFC Energy Reports Provided in Cooperation with: Elsevier Suggested Citation: Lu, Xiaohui; Ren, Jianglin; Guo, Lin; Wang, Peifang; Yousefi, Nasser (2020) : Improved grass fibrous root algorithm for exergy optimization of a high-temperature PEMFC, Energy Reports, ISSN 2352-4847, Elsevier, Amsterdam, Vol. 6, pp. 1328-1337, https://doi.org/10.1016/j.egyr.2020.05.011 This Version is available at: https://hdl.handle.net/10419/244124 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ Energy Reports 6 (2020) 1328–1337 Contents lists available at ScienceDirect Energy Reports journal homepage: www.elsevier.com/locate/egyr Research paper Improved grass fibrous root algorithm for exergy optimization of a high-temperature PEMFC Xiaohui Lu a,∗, Jianglin Ren a, Lin Guo b, Peifang Wang a, Nasser Yousefi c aSchool of Earth Science and Engineering, Ministry of Education Key Laboratory of Integrated Regulation and Resource Development on Shallow Lakes, Hohai University, Nanjing, 210098, China bHenan Institute of Geological Survey, Zhengzhou, 450001, China cIslamic Azad University, Karaj Branch, Iran article info Article history: Received 4 April 2020 Received in revised form 10 May 2020 Accepted 14 May 2020 Available online xxxx Keywords: Proton exchange membrane fuel cell Grass fibrous root algorithm Improved Exergy analysis Irreversibility abstract This paper presents an exergy assessment for a proposed power generation system that is used the organic Rankine cycle to recycle the waste heat from a high-temperature proton exchange membrane fuel cell (HT-PEMFC). To do so, mathematical model for the studied PEMFC along with the water management system have been introduced. Parametric analysis has been directed to study the impact of different economic and thermodynamic parameters, like the fuel cell irreversibility, exergy efficiency, and its work. For optimal designing the PEMFC, its parameters have been optimized by considering three objective functions, i.e. irreversibility, exergy efficiency, and its work. The optimization process has been performed based on a new model of fibrous root optimization algorithm improved. Simulation results of the presented algorithm have been compared with empirical results, genetic algorithm, and the basic of fibrous root optimization algorithm. The optimized values of irreversibility, exergy efficiency, work for the proposed algorithm are achieved 0.012, -0.439, and -0.4993, respectively which has the best values compared with the other analyzed algorithms. ©2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). 1. Introduction Energy and different ways for access to it, is one of the most important concerns of 21st century. The competition for providing the required energy is not only a concern among the advanced countries, but also it is important for the developing and the third world countries as global communities (Ghadimi,2012). The rising energy prices of fossil fuels in the last decade, the environmental difficulties and difficulties of using nuclear energy and the limited and non-renewable nature of conventional fuels are some reasons that have expanded research into finding new sources of energy (Fei et al.,2019;Liu et al.,2020;Shamel and Ghadimi,2016). These resources must have some characteristics, such as easy accessibility, renewability, no environmental pollution, high energy, low cost, easy storage, and economical portability (Eskandari Nasab et al.,2014;Fan et al.,2020). A small group of energy resources currently falls into this category. One of the energies that has been extensively studied for many years is fuel cell energy. A fuel cell is a battery-like device that converts electrochemical energy directly into electricity (Chang et al.,2019;Aghajani and Ghadimi,2018). The overall structure ∗Corresponding author. E-mail address: [email protected] (X. Lu). and the basis of the fuel cell work is consistent with the reactions occurring in a galvanic or voltaic cell, albeit with a slight difference (Fei et al.,2019). The fuel cell simply includes two porous electrodes and a solid or liquid electrolyte that creates these closed orbital components for conducting ions (the same structure of our galvanic cell). But the source of energy in the fuel cell is the hydrogen element. Hydrogen as energy carrier in fuel cell comes from various sources such as petroleum products (about 77%) coal (about 18%), water electrolysis (4%) and (1%) other sources. The major contributors to hydrogen carriers are hydrocarbons that are commonly found in fossil fuels, but the difference in their use in fuel cells is that there are no direct hydrocarbons burning in the machine and electricity generated by hydrogen interactions of Fossil fuel and air oxygen happens without combustion (Cold combustion process). Several kinds of fuel cells have been made that their difference is in their type of electrolyte. In addition, various combinations of fuel and oxidizer as well as anodic and cathode catalysts are also possible. Fuels can be diesel, gasoline, or methanol, and even natural gas and methane. Air, chlorine or dioxide-chlorine can be used as oxidants in the cell. A popular model of fuel cells is proton-exchange membrane fuel cell (PEMFC). The PEMFC is a good alternative to internal combustion engines in the transmission system. The PEMFC has higher efficiency than the other types of fuel cells with fast https://doi.org/10.1016/j.egyr.2020.05.011 2352-4847/©2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). X. Lu, J. Ren, L. Guo et al. / Energy Reports 6 (2020) 1328–1337 1329 Nomenclature A Membrane activity a ohmic loss ˙ Ex Exergy ratio (kw) F Faradays constant (Cmol−1) h Enthalpy (J/mol) I Irreversibility (kw) i Current density (A/cm2) i0Exchange current density (A/cm2) P Pressure (atm) ˙ QHeat generation R Universal gas constant (1/(mol ×K−1)) rfMolar flow rate (Kmol/s) T Temperature (◦C) tmembrane Membrane thickness (cm) V Cell potential (v) VrevReversible voltage VoOhmic voltage VaActivation voltage VcConcentration voltage W Power (w) x Mole fraction Greek letters: αTransfer coefficients βovervoltage constants of the concentration ηEfficiency λmembrane Membrane water content µChemical potential (J/mol) ςStoichiometric ratio σmembrane Membrane conductivity (1/(U−1× cm−1)) Subscripts: A Anode Act Activation C Cathode Ch chemical FC Fuel cell H2Hydrogen H2OWater in Inlet O2Oxygen out Outlet ph Physical ra Random value rev Reversible s Specific entropy, Jmol−1K−1 reaction and no contamination. These characteristics make the PEMFC as a popular fuel cell model in different applications. Generally, the mathematical model of the PEMFC is a critical case that should be considered for optimal designing and analysis of the fuel cell (Yu and Ghadimi,2019;Liu et al.,2017). There are various kinds of works have been done in this category in the literature. Cao et al. (Gollou and Ghadimi,2017) introduced an optimal model for the PEMFC stack. The main purpose was to present a developed model of seagull optimization algorithm to achieve the optimal parameters for PEMFC identification. The algorithm developed different mechanisms for increasing the algorithm convergence speed. The results were applied and compared with the empirical data from NedStack PS6 and BCS 500-W. The comparison results indicated that the presented technique gave well results from some other similar algorithms from the literature. Yu et al. (Aghajani and Ghadimi,2018) presented a parameter identification for the PEMFC using a developed Elman neural network and a combined version of the World Cup Optimization algorithm and the Fluid Search Optimization algorithm. The method was proposed for modifying the method efficiency for identifying the model parameters. Afterwards, four different operational conditions were adopted for the model analysis. Final results indicated the method excellence toward the others. Wang et al. (Mirzapour et al.,2019) presented a stacked long– short term memory (S-LSTM) model to fit the PEMFC stack degradation. In addition, the method presents the remaining useful life (RUL) estimation. The hyper parameters of the S-LSTM model were optimized based on a differential evolution algorithm. Simulation results showed that the presented model efficiency in comparison with other compared methods. Guo et al. (Hosseini Firouz and Ghadimi,2016) applied energy and exergy analysis for a hybrid HT-PMEFC system along with a two-stage thermoelectric generator (TTEG) by considering the Thomson effect. The paper achieved the relationships between the electric current and the inter-stage temperature of the TTEG and the working current density of HT-PEMFC. Simulation results showed that the proposed HT-PEMFC/TTEG system gives the maximum exergetic efficiency, electric efficiency, and power density compared with the basic HT-PEMFC system. Nalbant et al. (Hamian et al.,2018) proposed another arrangement for HT-PEMFC to give a satisfying technology for cogeneration usages. The research presented a mathematical model along with energy and exergy analysis based on the principles of electrochemistry and thermodynamics. Finally, some key operating parameters were examined. When energy has been converted in a PEMFC, a determinative value of heat has been generated. During this process, the most extractable work is called exergy, i.e. exergy determines the reversibility of a process during the enhancing of the entropy. However, exergy does not include system features, it characterizes the system and its environment. The reference state of the system is adopted to measure enthalpy and entropy, and the input energy can be changed by varying different parameters like input mass flow rate, pressure, and temperature. Furthermore, internal energy difference between the inputs and the outputs, and also the mechanical work of the system are the principal cases in exergy calculation (Leng et al.,2018). In other words, exergy is the maximum produced work of a fuel cell (Akbary et al., 2019;Ebrahimian et al.,2018). This research performs an energetic analysis and the effect of its parameters on thermodynamic irreversibility will be implemented (Saeedi et al.,2019;Gao et al., 2019). In this research, a methodology has been introduced an exergy synthesis for optimizing the fuel cell function. A PEMFC unit consists of three distinct parts. Anode (negative electrode), electrolyte and cathode (positive electrode). The way the fuel cell reacts is that the hydrogen (H2) loses its electron at the anode with the intervention of a catalyst and becomes a proton ion (H+) and free electron (e−). The proton moves to the cathode via electrolyte. The electrolyte is a solid or liquid compound designed to allow only the proton (and not the electron) to pass through it. Platinum is usually used as a catalyst in the vicinity of anode or hydrogen ion. The electron is also directed to the cathode through an external circuit (here a wire connected to a light source such as a lamp). The ions and electrons then react with the cathode 1330 X. Lu, J. Ren, L. Guo et al. / Energy Reports 6 (2020) 1328–1337 to produce oxygen. The movement of electrons over the external circuit generates a current that is usable in electrical devices and the water in the cathode can also be reused. Nickel is usually used as a catalyst at the cathode pole (Kwon et al.,2015). For handling the performance of the fuel cell, diffusion and reaction rates are adopted that must be compatible with together. There are several works that have been introduced about the optimizing the exergy for the fuel cells. These methods are usually categorized into two different classes: empirical-based approaches and mechanistic models (Ehyaei and Rosen,2019). The optimal identification of the fuel cell parameters are like momentum, heat transfer, mass, and diffusion electrochemical reactions (Haghighi and Sharifhassan,2016). In addition, analyzing the exergy of the system principally increases the fuel cell efficiency. due to the complexity of these kinds of problems, most of the performed optimization algorithms stuck in the local minimum value that gives a weak or even wrong solution. To resolve this shortcoming, in this paper, a new improved version of Grass Fibrous Root Optimization Algorithm (GRA) has been presented. The objective of this study is to optimize and synthesis of the exergy for a PEMFC based on optimal control of some characteristics of the PEMFC. To do so, the study uses three main objectives in the optimization: the exergy efficiency, the work efficiency, and the thermodynamic irreversibility. The contributions of the paper are summarized below: –A technique is adopted for Exergy analysis on a PEMFC. –A modified metaheuristic method is used for the model optimizing. –The metaheuristic is used for the PEMFC optimal identification. –The algorithm is validated by metaheuristics. –The method has compared with three different methods from the literature. 2. System model For modeling and synthetizing performance of the exergy for the analyzed HT-PEMFC, some considerations have been presented. In the system, the gases are considered ideal and the analyzed fuel cell is considered to be worked under steadystate conditions. Because the value of the produced water is too low, it has been neglected in the evaluations. Here, the kinetic energies and the potential have been neglected. During the exergy calculation, the humidity in the input H2and O2is ignored. Table 1 indicates the studied PEMFC thermodynamic characteristics (Akkar and Mahdi,2017). In the following, by considering the Faraday’s constant, we can determine the value of the generated H2by electrical current and the value of O2generated by the achieved H2and humidity quantity. The exergy is a proper economical criterion to analyze the system that depends to the irreversibility, enthalpy, work, and entropy of a system. Here, an optimal designing for exergy based on three functions with some predefined constraints has been introduced. Fig. 1 shows the exergy process in the PEMFC. As can be seen from Fig. 1, the electrochemical reaction between H2and O2at the catalyst layer is obtained as follows (Reddy and Jayanti,2012): H2+O→H2O+Heat +Electricity (1) The exergy of the fuel cell during the operation, mass, and heat is as follows: ∑˙ Exin =∑˙ Exout +˙ I(2) Table 1 Thermodynamic characteristics of the considered HT-PEMFC (Akkar and Mahdi, 2017). Parameter Value Unit The temperature for humidification of the feed gases 28 ◦C O2utilization ratios 50 % H2utilization ratios 80 % The thickness for the membrane 0.018 and 0.022 cm The current density range [0.02, 1.2] A ×cm2 The temperature of the dead state 25 ◦C Dead state pressure 1 atm Heat loss ratio (rHL) 20 % The pressure of the Exergy for the analysis 1, 2 and 3 atm Operating temperatures of the cells 119.85, 139.85, 159.85 and 179.85 ◦C ∑˙ Exin mass =∑˙ Exwork +∑˙ Exout mass −∑˙ ExHeat +˙ IFC (3) And the exergy value for the input and the output of the system are: ∑˙ Exin mass =˙ Exin O2+˙ Exin H2=(˙ rf×ex)in H2+(˙ rf×ex)in O2(4) ∑˙ Exmass,out =˙ Exout O2+˙ Exout H2+˙ Exout H2O=(˙ rf×ex)in H2 +(˙ rf×ex)in O2+(˙ rf×ex)in H2O(5) There are several parameters that are adopted for determining the exergy. For instance, potential exergy, kinetic exergy, physical exergy, and chemical exergy. The physical and chemical exergies can be considered by the following (Ishihara et al.,2004): Ex =∑ j xjECH j+RT0∑ j xjln xj    ExCH +(h−h0)−T0(s−s0)    ExpH (6) The rate of molar flow for O2, H2, and the produced water are achieved as follows (Reddy and Jayanti,2012): ˙ nout H2O=1 2×i×F−1=˙ nreacted O2=˙ 2nreacted O2(7) ˙ nin H2=˙ nreacted H2+˙ nout H2(8) ˙ nin O2=˙ nreacted O2+˙ nout O2(9) The following equation describes the wasted exergy over the work (Reddy and Jayanti,2012): ˙ WFC =∑˙ Exwork (10) And the Nernst equation adopted for obtaining the reversible cell voltage is as follows (Lee et al.,2004): Vr=1.2+43.1×10−5×TFC ×[ln (PH2)+ln PO] −85 ×10−5(TFC −298.15)(11) Furthermore, the partial pressure of the H2and the O2are obtained as follows (Amphlett et al.,1995): PH2= 1−Psat PA 1+(xA 2)×(1+ςA (ςA−1))×PA(12) X. Lu, J. Ren, L. Guo et al. / Energy Reports 6 (2020) 1328–1337 1331 Fig. 1. The PEMFC exergy balancing. PO2= 1−Psat PC 1+(xC 2)×(1+ςC (ςC−1))×PC(13) The irreversibility reduces the voltage value of the real cell voltage than the equilibrium state. in this study, ohmic loss, activation loss, and concentration loss are considered and modeled that are consequently given below (Rowe and Li,2001): VO=tmembrane ×i×[(0.005 ×λmembrane −0.003)exp (4.2−T−1 FC )]−1 (14) Va=(αA+αC αA×αC)×RTFC F×rf ×ln (i i0)(15) VC=i×(β1×i imax )β2 (16) where, λmembrane ={0.043 +17.81 ×a−39.85 ×a2−39.85 ×a30<a≤1 14 +1.4×(a−1)1<a≤3 (17) and i0stands for the exchange current density as follows (Alberro et al.,2015): i0(T)=10.8×10−22 ×exp (0.086 ×TFC )(18) The ohmic loss model is as follows (Berning et al.,2002): a=xH2O×P Psat (19) The constants for the overvoltage concentration (β1and β2) are obtained as follows (Haghighi and Sharifhassan,2016): β1 = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ if PO2 0.12 +Psat <2atm (71.6×10−5×TFC −0.62)(PO2 0.12 +Psat )+(1.68 −14.5×10−4×TFC ) else (86.6×10−6×TFC −0.07)(PO2 0.12 +Psat )+(0.54 −16 ×10−5×TFC ) (20) In addition, the fuel cell consuming work, i.e. ˙ WFC based on Akkar and Mahdi (2017) is considered by the following: ˙ WFC =V(i)×i=i×[Vrev−VO−Va−VC](21) Therefore, the exergy for the heat loss is obtained by the following (Alberro et al.,2015): ∑˙ ExHeat =rHL ×˙ QFC (22) where, ˙ QFC stands for the heat generation in the fuel cell and is obtained by the following: ˙ QFC =˙ Qirrev+˙ Qrev(23) where (Alberro et al.,2015): ˙ Qirrev=(−VCell −∆GT F×rf)×i(24) ˙ Qrev= −i×TFC ×∆ST F×rf (25) where, ∆GTrepresents the difference of the Gibbs free energy and ∆STstands for the entropy change and is obtained as follows (Tian et al.,2020): ∆ST=12414.8−9967.35 ×ln TFC (26) With regardless of the amount of the generated water by the electrolyte, the enthalpy of water generation can be achieved as follow (Akkar and Mahdi,2017): ∆Hg,T=∆Hf i−∆Hvap T(27) 1332 X. Lu, J. Ren, L. Guo et al. / Energy Reports 6 (2020) 1328–1337 where, ∆Hvap T=68260.58+3.70×10−4×T3 FC −0.48×T2 FC −152.43×TFC (28) And the overall produced heat by a PEMFC is as follows (Akkar and Mahdi,2017). ˙ QFC = −VCell ×i−∆Hg,T×i F×rf (29) Therefore, the thermodynamic irreversibility of the system will be as follows (Akkar and Mahdi,2017): ˙ IFC =(∑˙ Exmass,in +∑˙ ExHeat )−(∑˙ Exmass,out +∑˙ Exwork) (30) And the exergy performance of the fuel is formulated as follows (Akkar and Mahdi,2017): ηexergy = ˙ WFC ˙ Exmass,in (31) 3. Improved Grass Fibrous Root Optimization Algorithm Generally speaking, in solving optimization problems, the problem we face has infinite solutions which make us to find the best solution among them. Indeed, different search mechanisms have been performed to these problems and are of the same kind and as such, the algorithms are so useful that explore a large part of the answers to achieve the final solution. The best and most efficient algorithms must have a number of features. For example, high exploration and exploitation that classical optimization algorithms often lack these capabilities in a balanced way. For instance, they do not have the capability of Global search to extract. The mechanism of such algorithms is Local search. Random search algorithms also have a good global search but ultimately cannot achieve the required convergence. In fact, the way in which these algorithms operate intelligently and ultimately converge is the same as meta-evolutionary and evolutionary algorithms. There are different types of metaheuristics which have been proposed in the field of optimization, such as: pigeoninspired optimization algorithm (Springer et al.,1991), quantum invasive weed optimization (QIWO) (Cui et al.,2019), Emperor Penguin Optimizer (EPO) (Razmjooy and Ramezani,2014), world cup optimization (WCO) (Dhiman and Kumar,2018;Bandaghiri et al.,2016;Razmjooy et al.,2017;Shahrezaee,2017), Collective Animal Behavior (CAB) (Tian et al.,2020), states of matter search (SMS) (Cuevas et al.,2020), and Grass Fibrous Root Optimization Algorithm (GRA) (Cuevas et al.,2018). The common point of these algorithms is to refer to a natural and real phenomenon or social reactions. For example, the QIWO algorithm is derived from invasive growth of the weeds, and the WCO is inspired by the social and humankind challenges to achieve the champion cup, which makes these phenomena a mathematical model and a problemsolving method. In 2017, Akkar and Mahdi (Cuevas et al.,2018) introduced a new metaheuristic algorithm based on the fibrous root system of the grass. The GRA is significantly inspired by the grass plants regeneration, progress, and their fibrous root system. The grass plants are basically replicated based on two procedures. The first one is based on the subterranean stems that is often performed underground by sending out roots and shooting the nodes that is called rhizomes. The second procedure is based on the stems which grow below the surface. The mentioned procedures are continuously developing the secondary roots to replace the vanished primary roots. The hair roots have been produced by the secondary roots. The explained roots are often adopted for local and global searching of the mineral and the water resources. These mechanisms are utilized to model the optimal searching of the GRA. This algorithm gives two different procedures for global search based on the survived grasses and the best achieved grass developed and reproduced. Besides, the local has also two different searching methods including secondary hair roots and regenerated secondary roots. The following subsection, explains briefly about the basic and the improved model of GRA. 3.1. Mathematical model of GRA The GRA algorithm, just like other metaheuristics, starts by an initial population swarm (grass swarm) that is generated randomly and uniformly in the solution space that is initiated by seeding process (pop). After starting the optimization process, a new population (PopNew) has been generated that is limited in the rangePopL New≤PopNew≤PopH New, where PopL Newand PopH New represent the lower and the upper ranges of the population variables. The new swarm contains some parameters, such as the best value (Gbest ): Gbest =min (f(swarm)) ∈Rd(32) where, ddescribes the problem dimension, and fstands for the mean square error (MSE) function. Another element for the PopNewis the number of grasses (Gr) that is obtained by the Gbest by stolons that are often deviated by the basic grass (GrN) including a step size less than PopH Newas follows: Gr =   (0.5×pop)×(Avg(MSE) Avg(MSE)+min (MSE))   (33) where, Avgand the mindescribe the average value and the minimum value, respectively. It can be observed that the maximum generated new grass branches are equal to (0.5×pop) and obtained when the minimum of MSE is too small. The final mechanism of the PopH Newis the new grass equal to (pop–Gr −1) that is deviated randomly by the survived best initial grasses (Sde). All the new branch grasses deviated by the Gbest are considered as follows: GrN=ones (Gr,1)×Gbest +2×max (PopH New)×(σ(Gr,1)−0.5)×Gbest (34) And the survived best initial grasses are formulated as follows: Sde =GrN+2×max (PopH New)×(σ(pop–Gr −1,1)−0.5)×PopH New (35) where, σstands for a random value between 0 and 1, ones (.)determines the one’s column vector, and GrNdescribes the (pop–Gr −1) highest MSE initial population. The new population (PopNew) can be also modeled as follows: PopNew= [Gbest ;GrN;Sde](36) The new regenerated population (PopNew) have been evaluated to obtain the minimum valued MSE grass and limited it in the range PopL Newand PopH New. Here, if new Gbest gives better value than the previous one, it will be substituted by the best new grass (solution), else, the absolute rate of decrease in MSE has been evaluated. If the rate gives less value than or equal to a predefined tolerance value (ε), a global stack (stackg) has been increased and after the stackgreached to its maximum predefined value, then the next local search starts as follows: M=min i=1,...,pop(MSE) (37) X. Lu, J. Ren, L. Guo et al. / Energy Reports 6 (2020) 1328–1337 1333 bestmin =min j=1,...,iter(M) (38) ⏐⏐⏐⏐ mini=1,...,pop (MSE)−Gbest mini=1,...,pop(MSE)⏐⏐⏐⏐≤ε(39) As before mentioned, there is also local search mechanism in the optimization that contains two parts, hair roots loop and secondary roots loop. The secondary roots are defined by random numbers which will have a number of hair roots equal to d. The updating model of hair root location equal to the secondary roots number (S) is given below. mGbest (1,i)=Avg(Gbest )+Gbest (1,i)+C2×(σ−0.5)(40) C=[C1,C2,...,C10](41) C2=C×(1+(∥σ×10∥)) (42) i=1,2,...,d,k=1,2,...,S(43) where, mGbest describes the locally modified Gbest ,S∈ [0,d] stands for the number of secondary generated roots, Cdescribes the analyzed step size vector equation, and C2will be a random element of C. If the calculated mGbest gives a value less than the Gbest value, it will be replaced by it, otherwise, MSE absolute rate of decreasing has been evaluated. if the rate is less than ε, then the local stack counter (stackl) will be increased by one, if stackl gives the maximum predefined value, the hair root loop will be stopped and a new secondary root loop has been started. At the final of each iteration, the stopping criteria (εSC ) will be checked. 3.2. Improved Grass Fibrous Root Optimization Algorithm (IGRA) However, GRA gives good results in optimization, it has a big drawback in providing a proper convergence, i.e. it has premature convergence for some problems. To resolve this problem, two mechanism have been adopted that are explained in the following: The first mechanism is Opposition-Based Learning (OBL). The OBL is a technique based on the oriental philosophy conception (Tizhoosh,2005). By considering the OBL mechanism, opposite positions are obtained based on candidate solutions that helps to enhance the exploration in the algorithm to escape from the local optima for selecting the best solution for the nextgeneration candidate. Let us consider an integer, xin the range xϵ[xl,xu].xis the opposite value of the integer xand can be formulated as follows: x=xl+xu−x(44) With extending the above equation into ndimensions, xi=xi l+xi u−xi(45) where, x,x∈Rn. After optimization, the best grass in the range xand xobtained from the objective function has been extracted and saved as the main solution candidate and the other has been removed, i.e. if f(x)>f(x), xwill be stored and xwill be removed and if (x)<f(x), xwill be stored and xwill be removed. Here, 60% of the population has been generated by the OBL mechanism. There is also a filtering for removing In this step, the out-of-range solutions to obtain the optimal values. The second mechanism is Lévy flight (LF) that is regularly adopted for developing metaheuristics (Choi and Lee,1998). Random walk technique is the heart of the LF mechanism which is used for modifying the search capability. The mathematical model of LF mechanism is given below: LF (w)≈w−(τ+1)(46) Fig. 2. The flowchart diagram of the proposed IGRA. w=A×|B|−(1 τ)(47) σ2={sin(π×τ/2) 2(1+τ)/2×Γ(1 +τ) τ×Γ((1 +τ)/2)}2τ−1 (48) where, wdescribes the step size, τstands for the LF index (her, τ=1.5) (Li et al.,2018), A∼N(0, σ2), B∼N(0, σ 2), and Γ(.) indicates the Gamma function. By considering the above assumptions, the updated formula for the new branch grasses, survived best initial grasses, and hair root location are consequently as follows: Gr+ N=ones (Gr,1)×Gbest +2×LF(δ)×max (PopH New) ×(σ(Gr,1)−0.5)×Gbest (49) S+ de =GrN+2×max (PopH New)×LF(δ) ×(σ(pop–Gr −1,1)−0.5)×PopH New(50) mGbest (1,i)=Avg(Gbest )+Gbest (1,i)+C2×LF(δ)×(σ−0.5) (51) Fig. 2 indicates the flowchart diagram of the presented IGRA. 3.3. The performance verification of the proposed algorithm To verify the efficiency of the presented IGRA, some benchmark functions have been adopted and the results compared with some other optimization algorithms from the literature. The iteration number for all the algorithms is considered 100. The compared algorithms are States of Matter (SMS) (Cuevas et al.,2020), Lion Optimization Algorithm (LOA) (Yazdani and Jolai,2016), Thermal Exchange Optimization (TEO) (Kaveh and Dadras,2017), Socio Evolution and Learning Optimization Algorithm (SELO) (Kumar et al.,2018), and the basic Grass Fibrous Root Optimization Algorithm (GRA) (Cuevas et al.,2018). The formulation for the adopted test functions has been explained 1334 X. Lu, J. Ren, L. Guo et al. / Energy Reports 6 (2020) 1328–1337 Fig. 3. The polarization diagram for the analyzed algorithms with considering P=1 and T =199.85 ◦C. below: F1(x)= D−1 ∑ i=1(100 (x2 i−xi+1)+(xi−1)2)(52) F2(x)= D ∑ i=1 x2 i(53) F3(x)=10D+ D ∑ i=1(x2 i−10cos (2πxi))(54) F4(x)= −20exp ⎛ ⎝−0.2   √1 D D ∑ i=1(x2 i)⎞ ⎠ −exp (1 D D ∑ i=1 (cos (2πxi)))+20 +e(55) In the above equations, F1is the Rosenbrock function that is restricted in the range [−2.045, 2.045], F2and F3describe the Sphere function and the Rastrigin function in the range in the interval [−512, 512], respectively, and F4determines the Ackley function in the interval [−10, 10]. Table 2 shows the median value (MV) and the standard deviation value (SD) for the algorithms. As can be observed, the value of both MV and SD for the proposed IGRA is lower than the other compared algorithms which shows its higher accuracy and precision toward the others. 4. Simulation results This section presents the results and the discussion of the parametric studies using the mathematical model in details. The performance assessment parameters are taken as fuel cell irreversibility, exergy efficiency, and its work. To analyze of the proposed system efficiency, it is simulated based on Matlab Simulink software. The method is first coded and simulated in the Matlab platform and then, its results are compared with some other methods including Genetic Algorithm (GA) (Haghighi and Sharifhassan,2016), the basic Grass Fibrous Root Optimization Algorithm (GRA) (Cuevas et al.,2018), and also the experimental results extracted from Ubong data (Ubong et al.,2009). The polarization diagram for the analyzed algorithms with considering T=199.85◦Cand P=1 is given in Fig. 3. Indeed, Fig. 3 shows the performance of PEMFC by different algorithms. The cell performance is slightly improved from IGRA to GA based method, but it is clear that the best state for the real output among the all compared methods is the suggested IGRA. Fig. 4. The impact of pressure and temperature on the exergy performance. Fig. 5. The simulation results for the thermodynamic irreversibility toward the current density. As it is clear, the proposed IGRA gives the best fittest results to the empirical data. This can be observed in Fig. 4. As can be seen, by assuming the constant membrane thickness of 0.017 cm, enhancing the current density makes the efficiency of the exergy to get decreased, while enhancing the temperature increases the efficiency of the exergy. As can be observed, increasing the temperature makes a sensible rising on the mass exergy and the ˙ WFC of the feed and consequently the ratio of ˙ WFC /˙ Exin mass. Here, the effect of the pressure on the exergy efficiency has been neglected. More analysis has been done by analyzing the thermodynamic irreversibility that is determined by the input and the output consumed exergy and work. The simulation results for the thermodynamic irreversibility (˙ IFC) toward the current density have been shown by different pressure and temperature values in Fig. 5. As can be seen, the more the current density value increased, the more the thermodynamic irreversibility value increases. Figure also shows that at the same temperature, the thermodynamic irreversibility is proportional to the pressure. The study also analysis the effect of membrane thickness and the current densities changes on the exergy efficiency. Fig. 6 gives the simulation results for the effect of the membrane thickness variations on the exergy efficiency by operating pressure fixed on 3 atm. As it is clear from Fig. 6, increasing of the temperature makes the exergy efficiency increasing. Besides, due to the increasing of E˙ xmass,in which makes the work decreasing to input exergy ratio, increasing the current density, enhances the thermodynamic irreversibility. In this study, 3 single-objective optimizations are adopted for the system and the results are optimized by the proposed IGRA. Table 3 indicates the feasible range of parameters for the studied system (Akkar and Mahdi,2017). Based on Akkar and Mahdi (2017), the pressure of the system is considered in the range 1 X. Lu, J. Ren, L. Guo et al. / Energy Reports 6 (2020) 1328–1337 1335 Table 2 The numerical results of the analyzed algorithms on the adopted functions. Benchmark IGRA GRA (Cuevas et al., 2018) LOA (Yazdani and Jolai, 2016) SMS (Cuevas et al., 2020) SELO (Kumar et al., 2018) SMS (Cuevas et al., 2020) F1MV 0.00 0.00 1.42e−4 4.38e−4 1.17e−9 0.00 SD 0.00 0.00 4.29e−5 9.44e−4 5.24e−9 0.00 F2MV 4.28 5.16 52.45 42.15 12.30 11.85 SD 2.86 5.19 32.08 40.25 4.92 6.34 F3MV 0.00 2.12 30.11 45.27 10.52 3.82 SD 0.00 1.50 7.25 15.46 4.83 3.17 F4MV 0.00 2.89e−17 4.62e−2 7.41 3.98e−4 4.86e−16 SD 0.00 0.00 3.72e−2 3.83 3.15e−4 0.00 Table 3 The Optimization results of the Irreversibility, Work, and Exergy efficiency. The objective Function Current density (A/cm2) Pressure (atm) The fitness value (W/cm2) GA GRA IGRA Irreversibility 0.06 3 0.015 0.012 0.012 Work 1 2 −0.4490 −0.4899 −0.4993 Exergy efficiency 0.06 3 −0.437 −0.439 −0.439 Fig. 6. The simulation results for the temperature toward exergy efficiency during the membrane thickness variations. atm and 3 atm, temperature is limited in the range 125 ◦C and 185 ◦C, current density is considered in the interval 0.05 A/cm2 and 1.3 A/cm2, and membrane thickness is limited between 0.02 cm and 0.03 cm. In this research, the population size and the number of iterations for all the algorithms are considered 100, the mutation rate is considered 0.6, the crossover rate is assumed 0.4, and ε=0.02. It should be note that the parameters values for the optimization are achieved based on trials and error. Table 3 illustrates the final results of applying the analyzed algorithms on the fitness functions. These results have been obtained for the PEMFC with 0.018 cm membrane thickness at 180 ◦C. The current density and pressure of the objective functions are supposed similar to each other. The results show higher efficiency with −0.4993W/cm2for the proposed IGRA. Furthermore, the irreversibility value is 0.012 W/cm2which is achieved by the GRA and the suggested IGRA and shows the system efficiency. Also, exergy efficiency for the suggested IGRA is the best compared with the GA. For more clarification, the results of the fitness value for irreversibility, work, and exergy efficiency functions are given in Fig. 7–Fig. 9, respectively. Fig. 7. The convergence diagram of the fitness for the irreversibility term. The irreversibility convergence of the system is given in Fig. 7. It is obvious that the presented algorithm gives the minimum value with minimum iteration for the convergence (35 iterations). More information about this optimization is given in Table 3. Table declares that the value of the irreversibility at iteration 100 is 0.012 w/cm2that is a proper performance for the results. As it is clear from Fig. 8, single-objective optimization of the work term diagram, the best minimum value and the best convergence is achieved by the proposed IGRA that is converges in iteration 33 toward the GA that gives the worst result with larger convergence steps that in iteration 100. The prominent ability of the proposed algorithm in convergence than the basic GRA is obvious from the diagram. Finally, the optimal efficiency for the exergy of the system is analyzed based on its convergence. The details of this term are illustrated in Table 3 and Fig. 9. As can be seen, the GRA based algorithms has better results than the GA for exergy efficiency, however, the presented IGRA has faster convergence to the fixed value toward the others (iteration 62). 5. Conclusion Among different new generation systems, fuel cell-based systems are a good option for cogeneration for reasons such as high