Fractional Order PID Design for a Proton Exchange Membrane Fuel Cell System Using an Extended Grey Wolf Optimizer

  Citation: Silaa, M.Y.; Barambones, O.; Derbeli, M.; Napole, C.; Bencherif, A. Fractional Order PID Design for a Proton Exchange Membrane Fuel Cell System Using an Extended Grey Wolf Optimizer. Processes 2022,10, 450. https://doi.org/10.3390/ pr10030450 Academic Editors: Giosue Giacoppo and Bruno Auvity Received: 31 January 2022 Accepted: 18 February 2022 Published: 23 February 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). processes Article Fractional Order PID Design for a Proton Exchange Membrane Fuel Cell System Using an Extended Grey Wolf Optimizer Mohammed Yousri Silaa 1,* , Oscar Barambones 1,* , Mohamed Derbeli 1, Cristian Napole 1,* and Aissa Bencherif 2 1Engineering School of Vitoria, University of the Basque Country UPV/EHU, Nieves Cano 12, 01006 Vitoria, Spain; [email protected] 2Telecommunications Signals and Systems Laboratory (TSS), Amar Telidji University of Laghouat, BP 37G, Laghouat 03000, Algeria; [email protected] *Correspondence: [email protected] (M.Y.S.); oscar[email protected] (O.B.); [email protected] (C.N.) Abstract: This paper presents a comparison of optimizers for tuning a fractional-order proportionalintegral-derivative (FOPID) and proportional-integral-derivative (PID) controllers, which were applied to a DC/DC boost converter. Grey wolf optimizer (GWO) and extended grey wolf optimizer (EGWO) have been chosen to achieve suitable parameters. This strategy aims to improve and optimize a proton exchange membrane fuel cell (PEMFC) output power quality through its link with the boost converter. The model and controllers have been implemented in a MATLAB/SIMULINK environment. This study has been conducted to compare the effectiveness of the proposed controllers in the transient, accuracy in tracking the reference current, steady-state, dynamic responses, overshoots, and response time. Results showed that the combination EGWO-FOPID had significant advantages over the rest of the optimized controllers. Keywords: extended grey wolf optimizer; grey wolf optimizer; fractional order proportional integral derivative; proton exchange membrane fuel cell; DC/DC boost converter 1. Introduction Nowadays, energy research mainly covers two issues. The first one is linked to the risk of depletion of fossil and fissile resources; the other is environmental. The sources used today are with limited reserves, for both fossil fuels (hydrocarbons, coal, etc.) and fissile fuels (uranium). The use of these energy sources also generates undesirable side effects: emission of greenhouse gases in the case of hydrocarbons and production of waste that is difficult to treat in the case of nuclear power [ 1 , 2 ]. Faced with the decrease in conventional energy resources, it has become essential to find energy alternatives with the same properties as hydrocarbons in terms of storage and transport [ 3 ]. In this context, hydrogen turns out to be an earnest candidate, even if it is only an energy carrier and not a primary resource. Hydrogen, which does not exist naturally, can indeed be synthesized through renewable energies. In addition, its function as an energy carrier, its storability can be exploited to meet the requirements set by energy consumers [ 4 ]. The fuel cells emerge as the missing link by transforming chemical energy into electrical energy with high efficiency [ 5 ]. Hence, they use the chemical energy of hydrogen and oxygen to generate electricity without pollution, and the other products are just plain water, and heat [ 6 ]. Scientists have already developed different types of fuel cells, characterized by the nature of the gases and electrolytes used, thereby determining their operating characteristics. Of all the existing families of fuel cells, the proton exchange membrane fuel cell (PEMFC) achieved the most attention from the researchers, which is considered the best appropriate for the automotive sector [ 7 ] and numerous fields [ 8 – 10 ]. The strong points of this fuel cells type are the relatively fast dynamic compared to other power generators and low Processes 2022,10, 450. https://doi.org/10.3390/pr10030450 https://www.mdpi.com/journal/processes Processes 2022,10, 450 2 of 17 operating temperature, from 40 ◦ C to 100 ◦ C, which facilitates its integration in a vehicle without specific thermal insulation [ 11 , 12 ]. As is common, hydrogen cells are nonlinear systems which are affected by mutable factors such as gases pressures and fluctuations of temperatures which eventually, reflect the output power. Consequently, to ensure an efficient power conversion from the PEMFC to the external circuit, an adaptation element is required, and this can be done by inserting an electronic device between the power generator and the electrical load. This device is a static DC/DC converter equipped with an insulated gate bipolar transistor (IGBT) or metal oxide semiconductor field effect transistor (MOSFET) controlled by a command law [ 13 ]. This candidate connection is one of the most widely used power electronics circuits thanks to its high conversion efficiency and adjustable output voltage [ 14 ]. These DC/DC converters are electronic devices designed to regulate the output voltage against the input voltage and load current changes through the control of pulse width modulation PWM of the switch. This leads to the requirement of more advanced control methods to meet actual demand. Many control methods are developed in the literature to control DC-DC converters. For instance, authors of [ 15 ] applied a controller type proportional integral (PI) based on Ziegler–Nichols (ZN) tuning method to a DC/DC boost converter in order to stabilize the PEMFC output current. The proposed controller guarantees better performance in terms of rising time, settling time, steady-state error, and robustness even with large load variations. However, due to the obtained results, sharps undershoot of 3 A and overshoot of 8 A appears, which results from the obtained ZN method aggressive parameters. The authors of [ 16 ] implemented two different conventional control based on PI and PID in order to optimize the DC-DC buck converter performance. The control scheme was based on the ZN tuning method and genetic algorithm (GA). Simulation results showed that the PI and PID controllers using the GA gave satisfactory results in terms of rising time, steady-state error, settling time, low overshoot and low undershoot better than the provided by the conventional ZN tuning method. The authors of [ 17 ] implemented the GWO tuning for PID controller for DC/DC boost converter under a GA-PID and PSO-PID. Simulation results showed that the proposed GWO-PID has a low root mean squared error (RMSE) compared to the other algorithms. The authors of [ 18 ] controlled a DC/DC converter type buck based on PID combined with sliding mod (PID-SMC) in comparison to conventional sliding mode. The obtained simulation results showed that the proposed controller is better than the conventional SMC controller in terms of dynamic, static performance, and strong robustness under the periodically and irregularly load resistances. The authors of [ 19 ] applied a backstepping approach to a DC/DC boost converter in order to keep the PEMFC power system work at an optimum power point. The comparison against the PI showed that the backstepping approach gives fast and sufficient converging to the operating power point. The authors of [ 20 ] applied a total sliding-mode control (TSMC) for the voltage control of a DC/DC boost converter. Simulation results proved that the TSMC have low transient response time and high robustness in comparison with the conventional PI control and the SMC. The authors of [ 21 ] applied a quasi continuous high order sliding mode controller (QC-HOSM) to a DC/DC boost converter linked to PEMFC in order to reduce the chattering effects of the conventional sliding mode. Experimental results showed that the proposed control technique can achieve a chattering reduction up to 84%. The authors of [ 22 ] applied a robust integral fast terminal sliding mode combined with digital filter to a DC/DC boost converter in order to reduce the unwanted oscillation to improve the output power quality of the PEMFC. Experimental results showed that the proposed controller has significant advantages in term of rising time, robustness and a chattering reduction up to 91% could be achieved. With respect to state of the art, the main contribution of this paper is the design and implementation of a fractional order proportional-integralderivative optimized by an extended grey wolf optimizer (EGWO-FOPID) for enhancing the performance of the PEM fuel cell system. Comparison study with proportional-integralderivative optimized by grey wolf optimizer (GWO-PID), fractional order proportionalintegral-derivative optimized by grey wolf optimizer (GWO-FOPID), and proportional- Processes 2022,10, 450 3 of 17 integral-derivative optimized by an extended grey wolf optimizer (EGWO-PID), has been carried out in MATLAB/Simulink to validate the advantages of the proposed algorithm. Comparison results have demonstrated that the proposed controller can stabilize the PEMFC power system over the entire operating conditions and even in the presence of significant load variations. It has also been demonstrated that the proposed controller maintains the system’s robustness and provides better accuracy over the other controllers. The remainder of the paper is organized as follows. In Section 2, we discuss the fuel-cell type proton exchange membrane, as well as the mathematical equations related to its work that show the performance of the cell. Section 3is devoted to the control methodology design for the optimization of the PEMFC power system. Section 4focuses on the simulation results. 2. PEM Fuel Cell Modeling As shown in Figure 1. A PEM fuel cell is a generator of electrical energy. It directly converts the chemical energy of the fuel (hydrogen) into electrical energy using the catalyst [ 23 – 25 ]. It is a system that produces no pollution and virtually no noise since it does not have any moving mechanical components, such as turbines and motors. In addition, an electric current is produced as long as the cell is jointly supplied with fuel (hydrogen) and oxidizer (oxygen in the air) [ 25 ]. That is what differentiates it from different power generators and other cells. The chemical reaction at the level of the PEMFC can be represented in the following Equations (1)–(3) [26,27]. Anode: 2H2=⇒4H++ 4e−(1) Cathode: 4H++O2+ 4e−=⇒2H2O(2) Overall reaction: 2H2+O2=⇒2H2O+ Energy + Heat (3) Figure 1. Cross section of a single PEMFC. A single PEM fuel cell voltage VFC is the sum of four terms: the no-load voltage ENer , the activation overvoltage Vact (or activation drop), the ohmic overvoltage Vohm (or ohmic drop) and the overvoltage concentration Vcon (or drop in concentration), which are defined by the following expression [28]: VFC =ENer −Vact −Vohm −Vconc (4) 2.1. Nernst Potential The chemical energy released can be calculated by the change in Gibbs free energy ( 4gf ), which is the difference between the energy of the products and the energy of the reactants. In the case of the PEMFC the variation of this free energy is given in Equation (5) [29,30]: Processes 2022,10, 450 4 of 17 4gf= (gf)products −(gf)reactants = (gf)2H2O−(gf)2H2−(gf)O2(5) The variation of Gibbs free energy depends on temperature and pressure as given in Equation (6) [31]: 4gf=4g0 f−RTln"PH2P1 2O2 PH2O#(6) where 4gf is the variation of Gibbs free energy at standard conditions pressure 1 (bar), which depends on the temperature T expressed in Kelvin. PH2 , PO2 and PH2O are the pressures of hydrogen, oxygen and water vapor, respectively. R is the universal gas constant (8.31451 J · kg −1· K −1 ). For every hydrogen mole, two electrons pass by the external electrical circuit, and the electrical work is equal to the change in Gibbs free energy if the system has no lossless, the electrical work performed is given in Equation (7) [32]: 4gf=nFE (7) where F is Faraday’s constant (96,485 Coulombs/mole), which represents the electric charge of an electron mole. n corresponds to the number of moles of electrons in the reaction. E is the open circuit voltage of the PEMFC. The PEMFC open circuit voltage can therefore be expressed as Equation (8) [31]: ENer =−4gf 2F=−4g0 f 2F+RT 2Fln"PH2P1 2O2 PH2O#(8) In practice, the operation of PEMFC is accompanied by losses, part of the chemical energy is converted into heat. The term −4gf 2F varies depending on the operating point. It is equal to 1.229 volts at the standard state (25 ◦ C) and 1 bar. We can express the tension E in the form [33]: ENer =1.299 −0.85 ·10−3·(T−298.15) + 4.3085 ·10−5Tln(PH2) + 1 2·ln(PO2)(9) 2.2. The Activation Polarization The activation losses occur due to the kinetics of the reactions taking place at the electrode. They can be calculated using Equation (10) [34]. Vact =ζ1+ζ2·T+ζ3·T·ln(CO2) + ζ4·T·ln(I)(10) where the parameters ζ1 , ζ2 , ζ3 and ζ4 are parametric coefficients determined by the constructor, I is the current of the PEMFC, and CO2 is the oxygen concentration in the catalysts (mol·cm−3) and it could be calculated using Equation (11) [34,35]. CO2=PO2 5.08 ·106·e(−498 T)(11) 2.3. The Ohmic Losses The ohmic losses occur due to the electrical resistance of the different elements of the PEMFC. They have two origins: the internal resistance of the electrolyte membrane Rmem and the resistance that occurred due to the contact between the bipolar plates and the carbon electrodes Rcon. These losses can be calculated using Equation (12) [34]: Vohm =I·(Rmem +Rcon)(12) Processes 2022,10, 450 5 of 17 where Rmem =σmem ·l A(13) The parameter σmem is the specific resistance of the membrane ( Ω· cm), A is the single cell active surface in cm 2 , l is the membrane thickness in (cm). The following expression for the specific resistance is used [34,36]: σmem =181.6[1+0.03(I A) + 0.062(T 303)2·(I A)2.5] [δ−0.634 −3(I A)] ·exp [4.18(T−303)/T](14) The parameter δ is an amenable parameter with a maximum value of 23. This parameter depends on the membrane fabrication process and is a function of the relative humidity and the stoichiometric rate of the inlet hydrogen gas pressures the anode. Under ideal humidity conditions (100%), this parameter may have a value ranging from 14 to 20. 2.4. The Concentration Polarization The concentration losses are caused by the variation in the concentration of reactants. These losses can be calculated using Equation (15) [ 34 , 36 ]; where ψ , J and Jmax are, respectively, a constant parameter, the current density and the maximum current density. Vcon =ψ·ln1−J Jmax (15) 2.5. PEM Fuel Cell Stack Output Power The output voltage under the load is approximately 0.6–0.7 V [ 37 , 38 ]. Therefore, it is necessary to have cells in series, which finally form a “stack” to achieve the sufficient voltage and the amount of power needed. The power generated by the PEMFC stack can be calculated using Equation (9); where Nc represents the number of cells used in the stack [36]. Pstack =VFC ·I·Nc(16) The data and characteristics of the PEMFC considered in the simulation are shown in Table 1. Table 1. The PEMFC model parameters. Parameter Value A162 cm2 β23 l175 ×10−6cm ψ0.1 V Rc0.0003 Jmax 0.062 A·cm−1 Nc10 ζ10.9514 V ζ2−0.00312 V/K ζ3−7.4 ×10−5V/K ζ41.87 ×10−4V/K 3. Control Design Methodology The voltage delivered by the PEMFC is continuous and of low amplitude. In order to raise it into a higher value, a step-up converter is used. In general, the step-up converter is the easiest way to increase the voltage of a DC power supply, and promises high efficiency [ 39 ]. This section determines the converter structure adopted and presents some existing control techniques that allow the PEMFC to operate at an adequate power point. As shown in Figure 2, the closed loop consists of a PEMFC power system, a DC/DC boost converter, a control technique and finally a load. Processes 2022,10, 450 6 of 17 Figure 2. Principle of indirect adaptation with control technique. 3.1. Boost Converter State Space Modeling Supposed that the boost converter operates in the continuous conduction mode (CCM) [ 40 ] which includes two sequences depending on whether the controllable switch is closed or open as shown in Figure 3. In order to model the converter, one applies the laws of Kirchhoff to the electric circuits characterizing the two operating sequences [41,42]. First sequence is characterized by u= 1, the switch closed and the diode open. The equations which govern the converter are given by:      diL dt =1 L(VFC) dVo dt =1 RC (−Vo) (17) If we set x= [x1,x2]T= [iL,Vo]T, then the expression (17) can be written: "˙ x1 ˙ x2#=0 0 0−1 RC .x1 x2+1 L 0VFC (18) The second operating sequence is characterized by u= 0, the switch open and the diode closed. The system of equations which governs the converter in the “off” state is presented below:      diL dt =1 L(VFC −Vo) dVo dt =1 C(iL−io) (19) If we set x= [x1,x2]T= [iL,Vo]T, then the expression (19) can be written: "˙ x1 ˙ x2#=0−1 L 1 C−1 RC .x1 x2+1 L 0VFC (20) In state space description, if the state equations of two modes are described as following [ 41 ]: ˙ x=A1x+B1u(Switch“1”) ˙ x=A2x+B2u(Switch“0”)(21) Processes 2022,10, 450 7 of 17 Then the average state space model is given by: ˙ x=¯ Ax +¯ Bu (22) where, ¯ A=A1d+A2(1−d)and ¯ B=B1d+B2(1−d) Averaging the state space matrix of two different working modes using Equations (18), (20)–(22), we get the average model as a function of the duty cycle [41,43].              "˙ x1 ˙ x2#="0−(1−d) L (1−d) C−1 RC #.x1 x2+1 L 0VFC y=0 1 .x1 x2 (23) Figure 3. Basic electrical diagram of the boost converter linked to PEMFC. 3.2. Fractional Order PID Controller In 1999, Podlubny [ 44 ] proposed the PIλDµ controller, a generalization of the classical PID controller, comprising a fractional integration of order λ and a fractional derivation of order µ , thus widening the field of application of fractional calculus to the command theory, which has directed several researchers to a new line of research which is the adjustment of the fractional-order PIλDµ controller [ 44 ]. The following form gives the output equation of the fractional-order controller in the time domain: u=kpe(t) + kiD−λ te(t) + kdDµ te(t)(24) where kp is the proportional constant, ki is the integrating constant, kd is the differentiating constant, λ is the fractional order of the integrating action, and µ is the fractional order of the differentiating action. By comparison with the conventional PID controller [ 45 ], fractional-order controllers have in addition two other parameters noted λ and µ , which present the order of integration and derivation, respectively. Depending on the variation of these two parameters, we can distinguish different possibilities of fractional order controller [44]. As indicated in Figure 4, the fractional order PIλDµ controller generalizes the classical PID controller and extends it from the point to a plane. This expansion could provide much more flexibility in the design of PID control. Clearly, by choosing (λ , µ) = ( 1,1 ) , a classic PID corrector can be recovered and using (λ , µ) = ( 1.0 ) and (λ , µ) = ( 0.1 ) , we get controllers classic PI and PD , respectively. In other words, all these types of classical, n controllers are special cases of the fractional PIλDµcontroller given in Equation (24). Processes 2022,10, 450 8 of 17 Figure 4. Types of controllers according to λand µ. 3.3. Optimization Using EGWO Method The grey wolf optimizer (GWO) is an intelligent swarm technique developed in 2014 by Seyedali Mirjalili [ 46 ], which mimics the leadership hierarchy of wolves that are well known for their group hunting. In this algorithm, the population is divided into four groups: alpha (α) , beta (β) , delta (δ) and omega (ω) . The first three most vital wolves guide the last weak wolves ω to promising areas of the search space. One of the exciting realities of the social life of these wolves is their rigorous social hierarchical structure in the group, as shown in Figure 5. Figure 5. Grey wolf hierarchy. The hunting strategy and wolves’ social hierarchy are modeled to design the GWO optimization algorithm. This algorithm includes the following steps [46,47]: • Social hierarchy • Prey search (exploration) • Follow, hunt and approach the prey • Pursue, circle and harass the prey until they stop moving • Attack on the prey Figure 6gives the flowchart of the GWO optimization method. The mathematical Equations which govern the GWO algorithm can be summarized as follows: −→ D=|−→ C−→ Xp(t)−−→ X(t)|(25) −→ X(t+1) = |−→ Xp(t)−−→ A−→ D|(26) where t indicates the current iteration, −→ A and −→ C are vectors coefficients, −→ Xp the position vector of the prey, −→ Xis the position vector. Processes 2022,10, 450 9 of 17 The vectors, −→ Aand −→ Care calculated as follows: −→ A=2−→ a(t)−→ r2−−→ a(t)(27) −→ C=2−→ r1(28) where, −→ a linear incline vector decreased from 2 to 0, and −→ r1−→ r2 are random vectors in [ 0.1 ] . −→ Dα=|−→ C1−→ Xα(t)−−→ X(t)|(29) −→ Dβ=|−→ C2−→ Xβ(t)−−→ X(t)|(30) −→ Dδ=|−→ C3−→ Xδ(t)−−→ X(t)|(31) −→ X(t+1) = −→ X1+−→ X2+−→ X3 3(32) where, Xα(t) represents the position of the α , Xβ(t) indicates the position of the β , Xδ(t) is the position of δ , C1−3 are random vectors and X indicates the position of the current solution. The extended GWO is the same as the original, where the difference is adding three parameters ( αE , βE and δE ) called the emphasis coefficients to the updated position of Equation (32). Therefore, the extended, updated position can be expressed as Equation (33) [48,49]: −→ X(t+1) = αE −→ X1+βE −→ X2+δE −→ X3 3(33) where, αE>βE>δE In this paper, the EGWO and GWO algorithms are implemented to tune the FOPID and PID controllers parameters in the offline mode in order to ensure an optimal control performance under the variations of the operating conditions. The first step is to initialize a random wolf population based on the upper and lower bounds of the variables, uniformly distributed in the search space D , and fix the stop criterion. Second, evaluate the objective function for each wolf. Third, choose the first three best wolves and save them under α , β and δ . Fourth, update the position of the rest of the population (wolves). Fifth, update of parameters a , A and C . If the stopping criterion is not satisfied, go to the second step; otherwise, the program ends, and the optimal solution is produced. In the simulation, the population size is the number of search agents, which is 30 and the number of iterations equal to 40 and 200 for EGWO and GWO, respectively. The emphasis coefficients used in the EGWO are chosen as αE=1.2, βE=1.1, and δE=0.9. The implementation of the EGWO Algorithm-based optimization control scheme for DC-DC converter is to minimize ITAE while, proportional gain kp , integral gain ki and derivative gain kd for the PID controller, in addition λ and µ for the FOPID are taken as decision variables. According to Ziegler Nichols technique [ 19 ] and Podlubny [ 44 ], the variations ranges of the FOPID and PID decision variables used in the simulation are given in Table 2. Processes 2022,10, 450 16 of 17 4. Ventura, M. Long term storability of hydrogen peroxide. In Proceedings of the 41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Tucson, AZ, USA, 10–13 July 2005; p. 4551. 5. 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