Fractional Order Fuzzy PID Control of Automotive PEM Fuel Cell Air Feed System Using Neural Network Optimization Algorithm

Abstract

The air feeding system is one of the most important systems in the proton exchange membrane fuel cell (PEMFC) stack, which has a great impact on the stack performance. The main control objective is to design an optimal controller for the air feeding system to regulate oxygen excess at the required level to prevent oxygen starvation and obtain the maximum net power output from the PEMFC stack at different disturbance conditions. This paper proposes a fractional order fuzzy PID controller as an efficient controller for the PEMFC air feed system. The proposed controller was then employed to achieve maximum power point tracking for the PEMFC stack. The proposed controller was optimized using the neural network algorithm (NNA), which is a new metaheuristic optimization algorithm inspired by the structure and operations of the artificial neural networks (ANNs). This paper is the first application of the fractional order fuzzy PID controller to the PEMFC air feed system. The NNA algorithm was also applied for the first time for the optimization of the controllers tested in this paper. Simulation results showed the effectiveness of the proposed controller by improving the transient response providing a better set point tracking and disturbance rejection with better time domain performance indices. Sensitivity analyses were carried-out to test the robustness of the proposed controller under different uncertainty conditions. Simulation results showed that the proposed controller had good robustness against parameter uncertainty in the system.

1. Introduction

In recent years, fuel cells gained a lot of interest as one of the most promising renewable energy sources because of its high efficiency, flexibility and sustainability. Fuel cells produce electricity via electrochemical reactions between hydrogen and oxygen. The byproducts of the electrochemical reactions are only water and heat so fuel cells are considered as clean energy sources. The most common type of fuel cells is the proton exchange membrane (PEM) fuel cell. Proton exchange membrane fuel cells (PEMFCs) are used in vehicular applications because of its high electrical efficiency [1].

A PEMFC stack works as an autonomous energy source in automotive systems where the compressor motor of the air feeding system is powered by the PEMFC stack acting as an auxiliary load. The air feeding system is one of the most important systems in the PEMFC stack that has a great impact on the stack performance because it consumes up to 30% of the fuel cell power [2,3]. The role of the air feeding system is to regulate the oxygen excess ratio also known as stoichiometry at it is a predefined value using compressed air to prevent both oxygen starvation and oxygen saturation and to obtain the maximum net power output from the PEMFC stack. Oxygen starvation occurs at a sudden increase in the fuel cell stack current causing damage of the fuel cell membrane and the catalyst layer leading to fuel cell damage. On the contrary, oxygen saturation, which means a high level of oxygen availability, increases the power consumption of the air compressor resulting in a reduction of the net power output of the fuel cell. The PEMFC air feeding system is a complex nonlinear multi-input multi-output (MIMO) system that may include parameter uncertainty, so an efficient controller is required for the precise regulation of the oxygen excess ratio at different disturbance levels.

For the PEMFC air feeding system, several control techniques have been investigated in the literature such as feedforward control [4,5,6,7], LQR/LQG control [8], feedforward plus PI feedback control [6], sliding mode control (SMC) [9], adaptive sliding mode observer based control [10], adaptive control [11], model predictive control (MPC) [12], time delay control (TDC) with static feedforward [13] and disturbance-observer-based control [2]. Recently, soft computing techniques gained a lot of interest for the control of the PEMFC air feeding system. A B-spline neuro controller (B-SNN) was proposed in [14], PID feedback control with a single-input single-output (SISO) fuzzy feedforward controller [15] and hybrid fuzzy-PID controller [16].

A fuzzy logic controller (FLC) is widely accepted as an efficient controller, which is capable of controlling system without knowledge of its underlying dynamics and without using extensive mathematical analysis. Applications of FLCs in the literature witness that FLC is very efficient for nonlinear and uncertain systems [17]. However, the design of FLC is difficult because it involves several parameters without a distinct method for tuning. The design parameters for FLC are input/output scaling factors, membership function parameters and the rule base. Several heuristic methods have been proposed for the design and tuning of FLCs usually involving trial and error methods. The use of metaheuristic optimization techniques is an efficient method for tuning FLC, which proved efficient for different applications in literature [18,19].

Fractional order controllers are a generalization of standard controllers by using fractional order calculus where the order of the differentiators or integrators is a fractional number rather than an integer number used in standard controllers. The use of fractional operators increases the degree of freedom of the controller allowing it to generate outputs, which cannot be generated using integer order operators. A fractional order PID (FOPID) controller was proposed by Podlubny [20] where it demonstrated better control performance compared to the standard integer order PID controller. As a result, Fractional order PID controller gained a lot of interest in different control applications [21,22,23,24,25].

The application of fractional order operators has been extended to be used with fuzzy logic controllers where it was firstly proposed by Das et al. in [26]. Results demonstrated the superiority of the fractional order fuzzy PID compared to the standard Fuzzy PID controller. As a result, the fractional order fuzzy PID (FOFPID) controller gained a lot of interest and it is considered as an active and promising research area for different control applications [27,28,29,30].

This paper proposes a fractional order fuzzy PID (FOFPID) controller as an efficient controller for the PEMFC air feeding system. The proposed controller is optimized using the neural network algorithm (NNA). NNA is a new metaheuristic optimization algorithm developed by Sadollah et al. [31]. Sadollah et al. concluded that the artificial neural networks (ANNs) could be modeled and used as a metaheuristic optimization algorithm for handling optimization problems. NNA was developed based on the structure and the operators of the artificial neural networks (ANNs) for solving optimization problems [31]. NNA is one of the parameter free metaheuristic optimization algorithms where it does not require the user to fine-tune any algorithm parameters.

In this paper, the proposed fractional order fuzzy PID (FOFPID) controller is optimized using the NNA, where the NNA is used to optimize the input and output scaling factors, the membership function parameters of the controller inputs as well as the order of the fractional order differentiator and integrator. The main contributions of this paper are:

• A new application for the FOFPID and FOPID controllers is proposed to apply in the PEMFC air feed control to improve performance and robustness.
• This paper employs a direct discretization approach using an Al-Alawi operator for the first time to implement fractional order fuzzy PID controllers rather than indirect discretization approach based on Oustaloup’s recursive approximation.
• This paper is the first application of the NNA algorithm in controller design applications.
• The proposed NNA optimized FOFPID controller is tested for a constant set value for the oxygen excess ratio as well as the maximum power point operation by tracking a time varying set value for the oxygen excess ratio.
• Sensitivity analyses are performed to test the robustness of the proposed controller under various uncertainty conditions.

2. PEMFC Model

A PEM fuel cell stack consists mainly of three subsystems which are: (i) A hydrogen supply subsystem that feeds the anode side with hydrogen, (ii) an air feed subsystem that feeds the cathode by oxygen from the air and (iii) a humidification and thermal management subsystem that regulates the humidity and the temperature of the fuel cells, respectively. The main components of a PEMFC stack system are shown in Figure 1.

The air feeding subsystem has a great impact on the PEMFC stack performance because it consumes up to 30% of the fuel cell power [2,3]. The air feed subsystem consists of an electromechanical air compressor, which maintains the required oxygen pressure and mass flow rate in the cathode of PEMFC [2].

2.1. Air Feed System Model for PEMFC

There are several models for the PEMFC air feed system. Pukrushpan et al. introduced an accurate 9th order model for the air feed system in [4,5,32]. A reduced order model of Pukrushpan’s model was introduced by Suh in [33] where the 9th order model of Pukrushpan et al. was reduced into a 4th order model while preserving the dynamic behavior. Some assumptions have been assumed for PEMFC model reduction: The hydrogen subsystem dynamics are neglected by assuming perfect hydrogen supply control. Humidity and temperature variations are neglected by assuming perfect humidity and temperature control. The DC motor dynamics are neglected due to its small time constant compared to the mechanical system [33,34]. This model has been widely accepted by researchers for the design of the air feed system controller [16,35,36].

According to Suh’s model, the PEMFC air feed system equations are expressed as follows:

^x.=f(x)+_guu+_gωω

with a state vectorx=^{[_x1, _x2, _x3, _x4]T}, where_x1=_PO2is the partial pressure of oxygen in the cathode,_x2=_PN2is the partial pressure of nitrogen in the cathode,_x3=_ωcpis the angular velocity of the compressor,_x4=_Psmis the pressure of the supply manifold,u=_vcmis the compressor-motor voltage as the control input andω=_Istis the PEMFC stack current representing the measurable disturbance to the system. The components off(x) are [16,35,37]:

_f1(x)=_c1(_x4−χ)− _{c3 x1}α(_x1,_x2)_{c4 x1}+ _{c5 x2}+_c6

_f2(x)=_c8(_x4−χ)− _{c3 x2}α(_x1,_x2)_{c4 x1}+ _{c5 x2}+_c6

_f3(x)=−_{c9 x3}−_c10x3(^(_x4c11)_c12−1)_ψ3(_x3,_x4)

_f4(x)=_c14(1+_c15(^(_x4c11)_c12−1)).(_ψ3(_x3,_x4)−_c16(_x4−χ))

whereχ=_x1+_x2+_c2is the cathode pressure (_Pca) andα(_x1,_x2)is the total flow rate at the cathode outlet, which is given by:

α(_x1,_x2)={_c17χ^(_c11χ)_c18.^{(1−( _c11χ )_c12)0.5}for _c11χ>_{c19 c20}χfor _c11χ≤_c19

The input vectors_guand_gωare given by:

_gu=[0 0 _c13 0] ^T

_gω=[−_c7 0 0 0] ^T

The constants_ci,i=1, 2,…, 24 depend on the physical parameters of the PEMFC stack. The definition of these constants is given in Table 1 [16,35]. The values of the model parameters are shown in the Appendix A in Table A1 [16].

The measurement outputs vector is:

y=[_y1y2y3y4]=[_ψ1(_x1,_x2)_x4ψ3(_x3,_x4)_x3]

where_y1=_ψ1(_x1,_x2)is the stack voltage (_Vst) given by:

_Vst=n_vFC

where_vFCis the voltage of a single fuel cell andnis the number of fuel cells in the stack. The voltage of a single fuel cell is defined by:

_vFC=E−_vact−_vohm−_vconc

withEas the open circuit voltage and_vact,_vohmand_vconcare the activation, ohmic and concentration overvoltages, respectively. For more details about_ψ1(_x1,_x2) , the reader can refer to [4,5,7]._y3=_ψ3(_x3,_x4)is the airflow rate inside the compressor (_Wcp ) also known as the compressor flow map. It is approximated as follows [16,35]:

_ψ3 ^aproximated= _y3 ^max _x3x3 ^max(1−exp(−_rc(_sc+_x3 ²_qc−_x4)_sc+_x3 ²_qc−_x4 ^min) )

where,_rc=15,_sc=¹⁰⁵Pa and_qc=462.25 ^rad2/(^s2Pa).

2.2. Control Objective

The performance variables vector for the PEMFC stack system is defined by:

z=[_z1z2]=[_Pnetλo2]

where_z1=_Pnetis the net power output of the PEMFC stack and_z2=_λo2is the oxygen-excess ratio.

_z1=_y1ω−_c21u(u−_{c22 x3})

_z2= _c23c24ω(_x4−χ)

Oxygen starvation occurs when the value of_z2falls below 1, i.e., (_z2<1). Hence, the oxygen excess ratio_z2must be regulated at a certain point that prevents oxygen starvation at different disturbance conditions. For hydrogen/air fuel cells,_z2 ^ref=2 has been proposed as an optimal value [15,16,36]. Although, keeping the oxygen excess ratio at_z2 ^ref=2can avoid oxygen starvation, it cannot guarantee the maximum net power output from the fuel cell stack. The_z1/_z2 performance curve for different stack currents from 100 A to 300 A is shown in Figure 2.

Figure 2 show that the optimal operating point (_z1 ^⋆, _z2 ^⋆) depend on the stack current_Ist, meaning that for different values of the stack current_Ist, there exists an optimal operating point (_z1 ^⋆,_z2 ^⋆) between_z2=2and_z2=2.5where the maximum net power output is achieved. The optimal values_z1 ^⋆and_z2 ^⋆are functions of the stack current_Istand are given by:

_z2 ^⋆ =_φ1(ω)

_z1 ^⋆=_φ2(ω)

where_φ1(ω)and_φ2(ω)are approximated from the_z1/_z2 performance curve given in Figure 2._φ1(ω)is obtained using shape preserving interpolation while_φ2(ω)is a quadratic function with parameters obtained using the least squares method.

Hence, to obtain the maximum power output from the stack,_Z2refmust be determined based on the stack current_Istas follows:

_z2 ^ref=_z2 ^⋆

_z1 ^ref=_z1 ^⋆

The control objective is to design an optimal controller for the oxygen excess ratio_z2to regulate it at the required level to prevent oxygen starvation and obtain the maximum net power output_z1from the PEMFC stack at different disturbance conditions.

3. Air Feeding System Controller Design

The PEMFC air feeding system is a highly nonlinear MIMO system so an efficient controller is required for achieving the control objectives. This paper proposes a fractional order fuzzy PID controller as an efficient candidate for solving the PEMFC air feeding control problem. The proposed control scheme is shown in Figure 3. Fuzzy control simplifies the controller design procedures especially for complex nonlinear systems because FLCs apply the control actions in human-like thinking rather than a complex mathematical design [38]. The hybridization of fractional order operators for integration and differentiation with a fuzzy PID controller increases the degrees of freedom of the fuzzy controller allowing it to produce outputs, which cannot be produced with an integer order fuzzy controller.

3.1. Fractional-Order Operators and Its Discretization

Among the several definitions, the most common definitions for fractional order operators (differentiator/integrator) are the Grünwald-Letnikov (G-L) definition and Riemann-Liouville (R-L) definition [39].

The Grünwald-Letnikov (G-L) definition is given by:

_{a Dtr}f(t)=limh→01^hr∑i=0[(t−a)/h]⁽⁻¹⁾ⁱ(ri)f(t−ih)

where the time domain operator^Dris equivalent to the frequency domain operator^Sr,r∈[−1,1]. A positive value ofrimplies a fractional order differentiator while a negative value ofrimplies a fractional order integrator.

To obtain digital implementation of a fractional order controller (FOC), two discretization methods can be used: Direct discretization and indirect discretization [39]. Indirect discretization methods are two-step methods, where, the first step is to perform a frequency-domain approximation in a continuous time domain such as the Oustaloup’s band-limited rational approximation, the second step is to discretize the obtained fit s-transfer function. Several frequency-domain approximations can be used but the stable minimum-phase discretization cannot be guaranteed [39]. Direct discretization methods are used to obtain the discrete approximation transfer function directly.

Generally, direct discretization of the fractional-order differentiator/integrator^S±r, (r ∈ R), can be carried out using the generating functionS=ω(^z−1) . The generating function used and its expansion determine the form and the coefficients of the approximation [39]. Direct discretization methods include the direct power series expansion (PSE) of the Euler operator, continuous fractional expansion (CFE) of the Tustin operator and the numerical integration-based method [39].

In this paper, the direct discretization approach was used to obtain a discrete approximation of the fractional order operator^S±r,(0<r<1),in the infinite impulse response (IIR) form of discretization using the Al-Alaoui operator, which is a mixed scheme of the Euler and Tustin Operators. The Al-Alaoui operator as a generating function is given by:

ω(^z−1)=87T1−^z−11+^z−1/7

whereTis the sampling interval.

The discretized fractional-order operator is given by:

^D±r(z)=^{(ω(z−1))±r}=^{(87T1−z−11+z−1/7)±r}

Equation (22) is a rational discrete-time transfer function of infinite orders. CFE is an efficient way to approximate Equation (22) with a finite order rational one [39]. The resulting discrete transfer function approximating a fractional-order operator can be expressed as:

^D±r(z)≈^(87T)±rCFE_{{^{(1−z−11+z−17)±r}}p,q}=^(87T)±r_Pp(^z−1)_Qq(^z−1)=^(87T)±r_Pp(^z−1)_Qq(^z−1)=^(87T)±r_p0+_p1 ^z−1+_p2 ^z−2+…+_pm ^z−p_q0+_q1 ^z−1+_q2 ^z−2+…+_qn ^z−q

whereCFE{u}denotes the continued fraction expansion ofu;pandqare the order of approximation. Normally, it could be setp=q=n. The discretization of^Srresult is an infinite impulse response (IIR) form. An approximate rational function can be obtained by truncation.

The continued fractions expansion (CFE) of any well-behaved functionG(^z−1)is given by:

G(^z−1)≃_a0(^z−1)+_b1(^z−1)_a1(^z−1)+_b2(^z−1)_a2(^z−1)+_b3(^z−1)_a3(^z−1)+…

where the coefficients_aiand_biare either constants or rational functions of the variable^z−1.

The advantage of using the direct discretization method with the Al-Alaoui operator as a generating function is that it always gives discrete transfer functions with stable minimum phase characteristics, which is not always guaranteed when using the indirect discretization approach. The other advantage is that there is only one tuning knob [40,41].

The transfer function of the fractional order PID controller (P^{Iλ Dμ}) is given by:

_GFOPID(s)=_Kp+_Ki^sλ+_Kd ^sμ

where_Kp,_Ki,_Kdare proportional, integral and derivative gains respectively. μandλ are positive numbers that represent the order of differentiation and integration [30]. The control signal in the time domain representation given by:

_uFOPID(k)=_Kpe(k)+_Ki ^D−λe(k)+_Kd ^Dμe(k)

3.2. Fractional Order Fuzzy PID Controller

A fuzzy logic PID controller consists basically of a fuzzy PI and a fuzzy PD controller connected in parallel [18]. Hybridization of fractional order operators with a fuzzy controller is achieved by replacing the integer order differentiator and integrator at the input and the output of the FLC by a fractional order operator [26]. The use of fractional order operators adds extra degrees of freedom for tuning.

The structure of two inputs fractional order fuzzy PID controller with its tunable parameters is shown in Figure 4 where_GEand_GDEare the input scaling factors whileαandβare output scaling factors.^Dμis a fractional order differentiator with non-integer orderμwhile^D−λis a fractional order integrator with a non-integer orderλ . Integer order fuzzy PID controller can be obtained easily from a fractional order fuzzy PID controller by setting the order of the differentiator and integrator in Figure 4 to an integer value, i.e.,μ=1, λ=1. However, the use of fractional order operators increases the degrees of freedom (DOF) of the fuzzy controller allowing it to generate output values that cannot be generated using an integer order fuzzy controller.

The input scaling factors_GE,_GDEperform a scaling or normalization of the inputs from the real values into a normalized universe of discourse [−1,1], while the output scaling factorsα,β perform an inverse scaling or denormalization of the fuzzy controller output into applicable values suitable for the system. The performance of the fuzzy logic PID controller depends strongly on the values of these scaling factors [19]. Scaling factors has a global effect on the performance of the fuzzy controller. Therefore, these scaling factors must be properly tuned to achieve the desired system performance. Optimization algorithms represent an efficient tool for tuning the scaling factors of fuzzy controllers [19].

The inputs of the fractional order fuzzy PID controller are the errore(k)and the fractional derivative of error^Dμ(z)e(k), which are scaled using the input scaling factors_GEand_GDErespectively intoE(k)and^DμE(k). The output is the control signal_uFOFPIDwhich is scaled using the output scaling factorsαandβwhere:

e(k)=_z2ref−_z2(k)

fde(k)=^Dμ(z)e(k)=^{(ω(z−1))μ}e(k)

E(k)=_GE.e(k)

FDE(k)=^Dμ(z)E(k)=_GDE.^Dμ(z)e(k)

_uFOFPID(k)=α._uFIS(k)+β.^D−λ(z)_uFIS(k)=α._uFIS(k)+β.^{(ω(z−1))−λ} _uFIS(k)

where

_uFIS(k)=_fFuzzy(E(k),^Dμ(z)E(k))

whereD=ω(^z−1)is the generating function for the Al-Alaoui operator and^{(ω(z−1))μ}and^{(ω(z−1))−λ}are discrete transfer functions approximating the fractional-order differentiator and integrator respectively obtained using the Al-Alaoui operator._fFuzzyis a nonlinear function represent the fuzzy reasoning.

In this paper, seven membership functions (MFs) namely NB, NM, NS, Z, PS, PM and PB are used for the inputsE, ^DμEand the output_uFIS. Gaussian MFs are selected for the input variables. The Gaussian MF is defined by:

g(x;C,σ)= exp(−^(x−C)22σ ²)

where Cis the mean of the membership function and σis the standard deviation.

In this work, for computational efficiency, a zero-order Takagi-Sugeno-Kang (TSK) fuzzy inference is used, where the output of each rule is simply a constant or a singleton. The type and the parameters of the membership functions used affect the performance of the fuzzy controller. An optimization algorithm has been used for tuning the parameters of the membership functions [19,42]. The inputs and output membership functions for the fractional order fuzzy PID controller with its design parameters are shown in Figure 5.

Theithfuzzy rule used for the fractional order fuzzy PID controller has the following form:

Rule _Ri: IF E is _Ai AND ^DμE is _Bi THEN _uFIS is _yi.

where_Aiand_Biare Gaussian fuzzy sets, while_yi is a singleton. The complete rule base of the fractional order fuzzy PID controller with 49 rules is given in Table 2. This rule base has been selected according to [43].

4. Optimization Tool 4.1. Neural Network Algorithm (NNA)

Artificial Neural Networks (ANNs) map the input data to the target data through an iterative update of the weights_wij of the ANNs to reduce the mean square error between the predicted output and the target output. The neural network algorithm (NNA) is based on the concepts and the structure of the ANNs to generate new solutions where the best searching agent in the population is considered as the target and the procedures of the algorithm tries to make all the searching agents follow that target solution [31].

NNA is a population-based algorithm where it starts with an initial population of randomly generated solutions within the search space. Each individual or searching agent in the population is called a “pattern solution”, each pattern solution is a vector of1×Drepresenting the input data of the NNA.Pattern Solutio_ni =[_xi1, _{xi2, xi3,}……,_xiD].

To start the NNA optimization algorithm, a pattern solution matrix X with size_Npop×Dis randomly generated between the lower and upper bounds of the search space. The population of pattern solutionXis given by:

X=[_X1⋮_Xi⋮_XNpop]=[_x11x12⋯_x1D⋮_xi1⋮_xi2⋮⋯⋮_xiD⋮_xNpop1⋮_xNpop2⋮⋯⋮_xNpopD]

where

_xij=L_Bj+rand(U_Bj−L_Bj) , i=1,2,…,_Npop, j=1,2,…,D

whereLBandUBare1×Dvectors representing the lower and upper bounds of the search space.

Like ANNs, in NNA each pattern solution_Xiwill have its corresponding weight_Wiwhere_Wi=^{[_wi1, _wi2,…,_wiNpop]T}. The weights arrayWis given by:

W= [_W1,…,_Wi,…,_WNpop]=[_w11⋯_wi1⋯_wNpop1w12⋯_wi2⋯_wNpop2⋮_w1Npop⋯⋮_wiNpop⋮⋯⋮_{wNpop Npop}]

whereWis a square matrix(_Npop×_Npop)of uniformly distributed random numbers between 0 and 1. The weight of the pattern solution is involved in the generation of a new candidate solution.

In NNA, the initial weights are random numbers and its value is updated as the iteration number increases according to the calculated error of the network. The weight values are constrained such that the summation of the weights for any pattern solution should not exceed one, defined mathematically as follows:

_wij∈U(0,1), i,j=1,2,3,…,_Npop

∑j=1_Npopwij=1, i=1,2,3,…, _Npop

These constraints for weight values are used to control the bias of movement and the generation of new pattern solutions. Without this constraint, the algorithm will be stuck in a local optimum solution [31].

The fitness_Ciof each pattern solution is computed by the evaluation of the objective function_fobjusing the corresponding pattern solution_Xi.

_Ci=_fobj(_Xi)= _fobj(_xi1, _xi2,…,_xiD), i=1,2,…, _Npop

where_fobjis the objective function.

After the fitness calculation for all pattern solutions, the pattern solution with the best fitness is considered as the target solution with a target position^XTarget, target fitness^FTargetand target weight^WTarget. The NNA models an ANN with_Npopinputs each input of D dimension(s) and only one target output^XTarget [31].

Inspired by the weight summation technique used in ANNs, the new pattern solution is generated as follows:

_X→jNew(k+1)=∑i=1_Npopwij(k)·_X→i(k), j=1,2,3,…,_Npop

_X→i(k+1)=_X→i(k)+_X→iNew(k+1), i=1,2,3,…,_Npop

where k is an iteration index.

After the new pattern solutions are generated from the previous population, the weight matrix is updated as well using the following equation:

_W→iUpdated(k+1)=_W→i(k)+2·rand·(^W→Target(k)−_W→i(k)), i=1,2,3,…,_Npop

where the constraints (37) and (38) must be satisfied during the optimization process.

For better exploration of the search space, a bias operator is used in the NNA algorithm. The bias operator is used to modify a certain percentage of the pattern solutions generated in the new population_X→i(k+1)as well as the updated weight matrix_WiUpdated(k+1) . The bias operator prevents the algorithm from premature convergence by modifying a certain number of individuals in the population to explore other places in the search space, which has not been visited by the population. For more details about the bias strategy, the reader can refer to reference [31].

A modification factor_βNNAis used to determine the percentage of the pattern solutions to be modified using the bias operator. The initial value of_βNNAis set to 1 meaning that all individuals in the population are biased. The value of_βNNAwill be adaptively reduced at each iteration using any possible reduction technique such as follows:

_βNNA(k+1)= 1−(kMax_iteration), k=1,2,3,…, Max_iteration

_βNNA(k+1)= _βNNA(k)·_αNNA, k=1,2,3,…, Max_iteration

where_αNNAis a positive number smaller than 1 originally selected as 0.99.

The reduction of the modification factor_βNNAis made to enhance the exploitation of the algorithm as the iterations increase by allowing the algorithm to search for the optimum solution near to the target solution especially at the final iterations.

Unlike ANNs, in NNA the transfer function operator is used to generate better-quality solutions. The transfer function operator (TF) is defined by the following equation:

_X→i*(k+1)=TF(_X→i(k+1))=_X→i(k+1)+2· rand·(^X→Target(k)−_X→i(k+1)),i=1,2,3,…,_Npop

Using the transfer function operator, theithupdated pattern solution_X→i(k+1)is transferred from its current position to a new updated position_X→i*(k+1)towards the target pattern solution^X→Target(k).

In NNA, at early iterations the bias operator has more chances to generate a new pattern solution meaning that more possible opportunities for discovering unvisited pattern solutions as well as using new weight values. As the iteration number increases, the chance of applying the bias operator decreases while the transfer function (TF) operator will have more chance enhancing the exploitation of the NNA especially at the final iterations.

NNA is considered as a dynamic optimization model because the generation of a new updated solution does not depend only on the previous value of that solution but also depends on all the population described mathematically as follows:

_X→i(k+1)=f(_X→i(k),X(k)), i=1,2,3,…,_Npop

where_X→i(k+1)and_X→i(k)are the next and current locations of theithpattern solution respectively.

4.2. Formulation of FOFPID Controller Design as an Optimization Problem

In this paper, the neural network algorithm (NNA) was used to optimize the fractional order fuzzy PID controller. NNA was used to obtain the optimal or suboptimal value of the four scaling factors{_GE, _GDE, α, β}, membership functions parameters for the two inputsEand^DμE{_C1, _C2,…, _C6, _σ1, _σ2,…, _σ8}as well as the order of the fractional order operators{μ, λ}. Each candidate pattern solution must contain these parameters of the FOFPID controllers as follows:

_X→i={_{GE i}, _{GDE i},_αi ,_βi,_{C1 i}, _{C2 i},…, _{C6 i}, _{σ1 i}, _{σ2 i},…, _{σ8 i},_μi, _λi}

Gaussian membership functions are used for the inputs of the FOFPID controller. The Gaussian membership function is characterized by two parameters, which are the center_Ciand the standard deviation_σi. In this paper, a technique for encoding the membership functions using the minimum number of parameters is used, where, the peer positive and negative membership functions have the same value of the mean_Ci, but with the opposite sign, and have the same standard deviation_σi as shown in Figure 5. This approach of encoding reduces the total number of the membership functions’ parameters to be optimized to half, reducing the dimension of the optimization problem leading to a reduction of the computational cost. The total problem dimension is 20. The encoding of the controller parameters into a pattern solution is given in Figure 6.

The formulation of FOFPID controller design as an optimization problem is described as follows: Minimize

J=_fobj(_GE, _GDE, α , β ,_C1, _C2,…, _C6, _σ1, _σ2,…, _σ8,μ, λ )

Such that,

_{GE min}≤_GE≤_{GE max}

_{GDE min}≤_GDE≤_{GDE max}

_αmin≤α≤_αmax

_βmin≤β≤_βmax

0<_Ci≤1

_{σi min}≤_σi≤_{σi max}

0≤μ, λ≤1

With the constrains:

0<_C1 <_C2<_C3≤1

0<_C4 <_C5<_C6≤1

where,

J=ITAE= ∫t·|e| dt

is the integral of the time weighted squared error,eis the error signal andtis the time.

The detailed procedures for using NNA for the optimization of the FOFPID controller are described in Figure 7.

The optimized membership functions for both inputs of the FOFPID controller are shown in Figure 8. The optimal values forμ, λare:μ=0.8644andλ=1. Using the Al-Alawi operator, the truncated 5th order discrete transfer functions approximating^D0.8644and^D−1with a sampling intervalT= 0.001 s are:

^Dμ=^D0.8644=^{(87T1−z−11+z−1/7)0.8644}=2.537×¹⁰⁶−6.69×^{106 z−1}+6.163×^{106 z−2}−2.264×^{106 z−3}+2.483×^{105 z−4}+6291 ^z−55770−9515 ^z−1+4245 ^z−2−160.6 ^z−3−121.6 ^z−4+^z−5

^D−λ=^D−1=^{(87T1−z−11+z−1/7)−1}=0.7478−1.175 ^z−1+0.4782 ^z−2−0.001453 ^z−3−0.01458 ^z−4−0.000125 ^z−5854.6−2320 ^z−1+2221 ^z−2−865.4 ^z−3+108.6 ^z−4+^z−5

5. Simulation Results and Discussion

To verify the performance, the efficiency and the robustness of the proposed controller (NNA optimized fractional order fuzzy PID controller), detailed simulations using a MATLAB/SIMULINK environment were carried-out and analyzed. The performance verification was divided into three tasks. The first task was to test the controller for constant set point mode with_z2ref=2. The second task was to test the controller for maximum power point operation mode with_z2ref=_z2⋆, where_z2⋆∈[2, 2.5] is a function of the stack current. The third task was to test the robustness of the proposed controller against parameter uncertainty in the PEMFC stack system using the sensitivity analyses. For the validation of the simulation results, this work uses the same numerical values of the model parameters as well as the same profile of the disturbance used in a recent paper (reference [16]). Moreover, the simulation results are compared and verified to that of reference [16].

5.1. The First Task (Tracking Constant_z2ref)

In this task, the controller is tested by applying different values of the disturbance_Ist, which cover the whole range of the operation of the PEMFC stack while keeping the oxygen excess ratio at a constant set point value_z2ref=2. The profile of the disturbance, i.e., the PEMFC stack current_Ist is shown in Figure 9.

The PEMFC performance using four different controllers, which are the NNA optimized PID controller (NNA PID), NNA optimized fractional order PID controller (NNA FOPID), NNA optimized fuzzy PID controller (NNA FPID) and NNA optimized fractional order fuzzy PID controller (NNA FOFPID) is shown in Figure 10. Simulation results showed that a sudden increase in the stack current_Istrepresenting the disturbance to the system resulted in a sudden reduction of the oxygen excess ratio_z2. The proposed controller (NNA FOFPID) recovered from the disturbance effect faster than other controllers achieving the least settling time, rise time and fluctuations around the set point. At time = 20 s, a sudden reduction in the stack current caused a sudden increase in the oxygen excess ratio_z2. The proposed controller (NNA FOFPID) recovered from the disturbance effect faster than other controllers achieving the least settling time and fluctuations around the set point.

Simulation results showed that, the proposed NNA optimized fractional order fuzzy PID controller (NNA FOFPID) significantly improved the transient response of the PEMFC air feeding system by reducing the settling time and fluctuations around the set point compared to other controllers. Simulation results showed that the NNA FOPID controller could outperform the NNA PID controller. However, it could not outperform the NNA FPID controller. Simulation results showed that the proposed NNA FOFPID could outperform all the other three types of controllers achieving a better performance.

The variation of the stack voltage_Vstand the net power output_z1 of the PEMFC stack using the four controllers is shown in Figure 11 and Figure 12, respectively. It could be noticed that a sudden increase in the PEMFC stack current resulted in a sudden reduction in the oxygen excess ratio_z2reducing the stack voltage_Vst. Although, the reduction of the PEMFC stack current, at time = 20 s, resulted in an increase in the oxygen excess ratio_z2increasing the stack voltage_Vst, the net power output_z1of the stack was reduced because of the increased power consumption of the compressor motor. The compressor motor voltage_vcm using the proposed NNA optimized fractional order fuzzy PID (NNA FOFPID) controller is shown in Figure 13.

A performance comparison based on the time domain performance indices is given in Table 3. Results showed that, NNA optimized controllers could outperform the controllers presented in reference [16]. The proposed NNA optimized fractional order fuzzy PID controller (NNA FOFPID) was superior and achieved the best time domain performance indices.

5.2. The Second Task (MPPT)

In this task, the proposed NNA FOFPID controller was tested for the maximum power point operation for the PEMFC stack by tracking a time-varying set-value_z2 ^ref=_z2 ^⋆, where the set-value_z2 ^⋆is a function of the stack current_Istto obtain the maximum net power output_Pnet from the PEMFC stack as described in Section 2. The same profile of the disturbance used in task 1 was used in the maximum power point tracking mode (MPPT) task.

The PEMFC performance using the proposed controller and the other controllers for the maximum power point tracking mode (MPPT) is shown in Figure 14. Simulation results showed that the proposed NNA optimized fractional order fuzzy PID controller (NNA FOFPID) could outperform the other controllers achieving a better set point tracking with the least settling time and minimal fluctuations around the time-varying set value for both positive and negative set point changes achieving a better transient response. Results showed that the proposed NNA optimized fractional order PID controller (NNA FOPID) was better in both set point tracking and the disturbance rejection.

The variation of the stack voltage_Vst in the MPPT operation mode using the four controllers is shown in Figure 15. By comparing Figure 11 and Figure 15, it could be noticed that the stack voltage_Vstin the case of the MPPT operation mode was larger than that in case of the constant set point operation mode. The net power output_z1 of the PEMFC stack is shown in Figure 16. Simulation results showed that using a time-varying set-value_z2 ^ref=_z2 ^⋆, the net power output of the PEMFC stack was maximized. The compressor motor voltage_vcm in the MPPT operation mode using the NNA optimized fractional order fuzzy PID controller (NNA FOFPID) is shown in Figure 17.

A performance comparison based on the time domain performance indices for tracking a time-varying set-value_z2 ^ref=_z2 ^⋆ is given in Table 4. The proposed NNA optimized fractional order fuzzy PID (NNA FOFPID) controller was superior and achieved the best time domain performance indices.

5.3. The Third Task (Sensitivity Analysis)

Sensitivity analyses were carried-out for testing the robustness of the proposed NNA optimized FOFPID controller against system parameters changes. The system parameters were varied independently by ±25% of their nominal values without changing the optimized parameter of the proposed NNA FOFPID controller. The time domain performance indices (ISE, IAE, ITSE and ITAE) for the nominal PEMFC air feeding system as well as the perturbed systems are shown in Table 5. The performance of the system with the different considered parameter uncertainty is shown in Figure 18.

The results of Table 5 and Figure 18 showed that applying ±25% uncertainties in_Jcp,_Rcm,_Vsm,_kt,_kvand_Tatmcaused the time domain performance indices, overshoots, undershoots and settling time to deviate from their nominal values. However, these deviations were slight within an acceptable range and the system was dynamically stable. Sensitivity analyses showed that the PEMFC air feeding system with an NNA optimized FOFPID controller had satisfactory robustness against the considered parameter uncertainty range. It can be concluded that the NNA optimized FOFPID controller parameters obtained with the nominal system parameters can be used without retuning or resetting even the system parameters change in a considerable range.

6. Conclusions In this paper, a fractional order fuzzy PID controller was proposed as an efficient controller for the PEMFC air feeding system. The proposed controller was optimized using the neural network algorithm (NNA). NNA was used to obtain the optimal value of the controller scaling factors and the order of the fractional differentiator and integrator as well as the optimal parameters of the input membership functions. Detailed simulation using a MATLAB/SIMULINK environment was carried-out to test the performance of the proposed NNA optimized FOFPID controller for different modes of operation of the PEMFC stack. Simulation results proved the efficiency and the superiority of the proposed NNA optimized FOFPID controller over other types of controllers. The proposed controller achieved a better set point tracking and disturbance rejection with minimal fluctuations around the set value with better transient response and minimum time domain performance indices. Sensitivity analyses were carried-out to test the robustness of the proposed controller against parameter uncertainty in the PEMFC air feeding system. Future research will concentrate on modifying the original NNA algorithm for improving its convergence with applications to PEMFC control using new control schemes.

Figure 1. The main components of a PEMFC stack system.

Figure 2. Thez1/z2performance curves for the PEMFC stack at different levels of disturbance. (a) Maximum power point (z1⋆, z2⋆) for different levels of disturbance (Ist). (b)z1⋆, z2⋆as a function of the disturbance (Ist).

Figure 3. The proposed NNA optimized fractional order Fuzzy PID control scheme for the PEMFC stack.

Figure 4. Fractional order fuzzy PID controller with tunable parameters.

Figure 5. Input and output MFs for the fractional order fuzzy PID controller with its design parameters.

Figure 6. The encoding of FOFPID controller parameters into a pattern solution.

Figure 7. The procedures of FOFPID controller optimization using the NNA algorithm.

Figure 8. The optimized input MFs for the FOFPID controller.

Figure 9. The disturbance ω (Ist).

Figure 10. The response ofz2using four NNA optimized controllers for task 1 (z2ref=2).

Figure 11. The stack voltage variation using the four NNA optimized controllers for task 1 (z2ref=2).

Figure 12. The net power output (z1) using the four NNA optimized controllers for task 1 (z2ref=2).

Figure 13. The compressor motor voltage (vcm) using the proposed controller for task 1 (z2ref=2).

Figure 14. The response ofz2using four NNA optimized controllers for task 2 (MPPT).

Figure 15. The stack voltage variation using the four NNA optimized controllers for task 2 (MPPT).

Figure 16. The net power output comparison for constantz2ref=2and maximum power point operationz2ref=z2⋆.

Figure 17. The compressor motor voltage (vcm) using the proposed controller for task 2 (MPPT).

Figure 18. Sensitivity analyses for the PEMFC air feeding system with the proposed NNA optimized FOFPID controller for 25% uncertainty in different PEMFC stack parameters. (a) Uncertainty inJcp. (b) Uncertainty inRcm. (c) Uncertainty inVsm. (d) Uncertainty inkt. (e) Uncertainty inkv. (f) Uncertainty inTatm.

PEMFC Model Constants
C1=RTst Kca,inMO2 Vca(xO2,atm1+ωatm)	C14=RTatmγMa,atm Vsm
C2=Psat	C15=1ηcp
C3=RTstVca	C16=Kca,in
C4=MO2	C17=CD AT(RTst)0.5(2γγ−1)0.5
C5=MN2	C18=1γ
C6=Mv Psat	C19=(2γ+1)γγ−1
C7=RTstn4FVca	C20=CD ATRTstγ0.5 (2γ+1)γ+12γ−2
C8=RTst Kca,inMN2 Vca(1−xO2,atm1+ωatm)	C21=1Rcm
C9=ηcm kt kvJcp Rcm	C22=kv
C10=Cp TatmJcp ηcp	C23=Kca,in(xO2,atm1+ωatm)
C11=Patm	C24=nMO24F
C12=γ−1γ	xO2,atm=yO2,atm MO2Ma,atm
C13=ηcm ktJcp Rcm	ωatm=MvMa,atmϕatm PsatPatm−ϕatm Psat

a: Adopted with permission from Reference [16] Copyright (2017) Elsevier.

uFIS		E
		NB	NM	NS	Z	PS	PM	PB
DμE	NB	NB	NB	NB	NB	NM	NS	Z
	NM	NB	NB	NB	NM	NS	Z	PS
	NS	NB	NB	NM	NS	Z	PS	PM
	Z	NB	NM	NS	Z	PS	PM	PB
	PS	NM	NS	Z	PS	PM	PB	PB
	PM	NS	Z	PS	PM	PB	PB	PB
	PB	Z	PS	PM	PB	PB	PB	PB

Controller	ISE	IAE	ITSE	ITAE
PID [16]	0.0627	0.2903	NA	2.2741
FLC [16]	0.5045	1.1047	NA	8.0201
HFPID [16]	0.0249	0.1005	NA	0.6781
NNA PID	0.03711	0.1995	0.1032	1.356
NNA FOPID	0.02652	0.1261	0.07036	0.8443
NNA FPID	0.014	0.09013	0.06015	0.6539
NNA FOFPID (proposed)	0.009186	0.05291	0.04193	0.3639

Controller	ISE	IAE	ITSE	ITAE
HFPID [16]	NA	NA	NA	NA
NNA PID	0.04931	0.1938	0.062	1.034
NNA FOPID	0.03104	0.1238	0.04675	0.7024
NNA FPID	0.03671	0.1127	0.03346	0.4368
NNA FOFPID (proposed)	0.02459	0.0701	0.02513	0.2619

Parameter	% Change	ISE	IAE	ITSE	ITAE
Nominal	0	0.02459	0.0701	0.02513	0.2619
Jcp	+25%	0.0239	0.07184	0.03055	0.3069
	−25%	0.0263	0.07017	0.01969	0.2255
Rcm	+25%	0.0256	0.07463	0.03141	0.3158
	−25%	0.0239	0.06669	0.01997	0.225
Vsm	+25%	0.03594	0.08698	0.03287	0.3187
	−25%	0.01389	0.05485	0.01815	0.2263
kt	+25%	0.02459	0.0701	0.02513	0.2619
	−25%	0.02459	0.0701	0.02513	0.2619
kv	+25%	0.03929	0.1005	0.04467	0.519
	−25%	0.01774	0.05987	0.02576	0.2671
Tatm	+25%	0.0313	0.07813	0.02625	0.2707
	−25%	0.01966	0.06367	0.02441	0.2571

Author Contributions

M.S.A. has proposed, conceptualization, simulations, analysis and written the manuscript. H.-J.Z. and Y.-X.S. validated the main idea and results, supervision and checked the whole manuscript. All authors organized and refined the manuscript in present form.

Funding

The APC was funded by Wuhan University of Technology.

Acknowledgments

M.S. AbouOmar acknowledges the Chinese Scholarship Council (CSC) and the Egyptian Government for the financial support. The authors would like to acknowledge the reviewers for their time and valuable comments regarding a high-quality paper.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

Table

Table A1.System Parameters for Simulation b.

Table A1.System Parameters for Simulation b.

Fuel Cells (FC) Data
n	Number of cells in fuel cell stack	381	-
Vca	The volume of the cathode	0.01	m3
CD	Throttle discharge coefficient for the cathode outlet	0.0124	-
AT	Cathode outlet throttle area	0.00175	m2
Vsm	Supply manifold volume	0.02	m3
Tst	Fuel cell temperature	353.15	K
Kca,in	Cathode inlet orifice constant	0.3629 ×10-5	Kgs-1 Pa-1
Air & Steam Properties
γ	Ratio of specific heats of air	1.4	-
MN2	Nitrogen molar mass	28 ×10-3	Kgmol-1
MO2	Oxygen molar mass	32 ×10-3	Kgmol-1
Mv	Vapor molar mass	18.02 ×10-3	Kgmol-1
Ma,atm	Air molar mass	28.97 ×10-3	Kgmol-1
Tatm	Atmospheric temperature	298.15	K
Patm	Atmospheric pressure	1.01325 ×105	Pa
Cp	Specific heat of air at constant pressure	1004	JKg-1 K-1
ϕatm	Average relative humidity of the ambient air	0.5	-
yO2,atm	Oxygen mole fraction	0.21	-
Electrochemistry
F	Faraday constant	96.487	Cmol-1
R	Universal gas constant	8.31451	Jmol-1 K-1
Compressor (CP)
ηcp	Compressor efficiency	80%	-
Jcp	Compressor inertia	5 ×10-5	Kgm2
Compressor Motor (CM)
Rcm	Compressor motor resistance	0.82	Ω
kt	Motor constant	0.0225	NmA-1
kv	Motor constant	0.0153	Vrad-1s
ηcm	Motor mechanical efficiency	98%	-

b: Adopted with permission from reference [16] Copyright (2017) Elsevier.

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