Optimal PID plus fuzzy controller design for a PEM fuel cell air feed system using the self-adaptive...

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Optimal PID plus fuzzy controller design for a PEMfuel cell air feed system using the self-adaptivedifferential evolution algorithm

Hamed Beirami a,*, Ali Zargar Shabestari b, Mohammad Mahdi Zerafat c

a Department of Instrumentation and Automation Engineering, Shiraz University, Shiraz, Iranb Control Engineering Department, Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iranc Faculty of Advanced Technologies, Nano Chemical Engineering Department, Shiraz University, Shiraz, Iran

a r t i c l e i n f o

Article history:

Received 19 October 2014

Received in revised form

14 May 2015

Accepted 16 May 2015

Available online xxx

Keywords:

PEM fuel cell (PEMFC)

Air supply system

Maximum power tracking

Fuzzy logic control

Self-adaptive differential evolution

(SaDE) algorithm

* Corresponding author. Tel.: þ98 914124147E-mail address: [email protected]

http://dx.doi.org/10.1016/j.ijhydene.2015.05.10360-3199/Copyright © 2015, Hydrogen Ener

Please cite this article in press as: Beiramusing the self-adaptive differential evol10.1016/j.ijhydene.2015.05.114

a b s t r a c t

Various control strategies for the air feed to a PEM fuel cell are investigated. Feedback and

feedforward control strategies are used simultaneously in order to achieve maximum net

power and prevent oxygen starvation. The control objective is to adjust the oxygen excess

ratio using compressed air based on external disturbance variations. Maximum power

tracking is obtained using the optimal oxygen excess ratio which is also compared to the

constant oxygen excess ratio. In the feedforward strategy, a fuzzy logic controller and in

the feedback a filtered PID controller is applied. The controller parameters are optimized

simultaneously using a self-adaptive differential evolution algorithm. Simulation results

show that the proposed technique improves the fuel cell system performance and prevents

oxygen starvation by fixing the oxygen ratio at a proper setpoint.

Copyright © 2015, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights

reserved.

Introduction

The ever-increasing energy demand in industrial societies and

the predictable expiration of fossil fuel reserves has led to a

constant search toward new energy resources. Besides, due to

environmental pollution caused by fossil fuel combustion,

new energy sources such as solar, wind, geothermal or

hydrogen are considered as inevitable alternatives by energy

supervisors. In spite of boundless possibilities for the appli-

cation of these new energy sources and environmental

friendly features the main disadvantage lies in their limited

availability in specific situations.

9.om (H. Beirami).14gy Publications, LLC. Publ

i H, et al., Optimal PID pution algorithm, Interna

Fuel cells are proposed as one of these alternatives for

energy conversion systems during recent decades. Fuel cells

are electrochemical systems capable of chemical energy

conversion to the electrical form without combustion. High

efficiency, low process temperatures, low pollution, high

safety factors, flexibility of power generation and short

startup times are among the most important advantages of

these systems [1]. Hydrogen and oxygen are consumed to

produce electricity in a fuel cell and byproducts are discharged

as water and heat. However, short lifetime and high expenses

has impeded their vast applicability in common systems up to

now. As a result, in order to improve the performance,

enhance the lifetime and reduce the production costs, the

ished by Elsevier Ltd. All rights reserved.

lus fuzzy controller design for a PEM fuel cell air feed systemtional Journal of Hydrogen Energy (2015), http://dx.doi.org/

mailto:[email protected]

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http://dx.doi.org/10.1016/j.ijhydene.2015.05.114



i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y x x x ( 2 0 1 5 ) 1e1 32

optimization of system design, components and materials as

well as effective control strategies are essential.

Several control strategies are proposed for PEM fuel cells

suchasPI [2], LQG [1,3,4], fuzzy slidingmode [5], fuzzy [6,7], time

delay [8,9], and neural networks [10,11] which are not validated

in real plants. Other studies validated in plant-wide control

strategies can also bementioned as robust [12], gain-scheduled

control [13], sliding mode [14,15] and model predictive control

[16e18] which are applied for the manipulation of oxygen

excess ratio in PEM fuel cells with various degrees of success.

This study is focused on the design and validation of a PIDF

(PID with low band filter) and fuzzy controller optimized by

the self-adaptive differential evolution (SaDE) algorithm for

controlling the oxygen excess ratio in a PEM fuel cell. The

main control objectives are preventing oxygen starvation and

obtaining the maximum net power output. Oxygen starvation

is considered as a critical condition for fuel cell resulting in

rapid degradation of the membrane and catalyst layer [19],

which is controlled by keeping the oxygen excess ratio higher

than a minimum value.

After the introductionofdifferential evolution (DE)algorithm

in 1995 [20], many researchers have paid attention to the issue

and as a result many improved algorithms have been under

development since then. The SaDE algorithmused in this study

is developed by Qin et al. [21] in 2009 and applied in antenna

design [22,23], filter design [24] and transforming geocentric

Cartesian coordinates [25], thereafter. This algorithm is used in

control applications and fuzzy controller optimization for the

first time in this study.A feed forward controlleruses fuzzy logic

which is a representation of human thinking process. Fuzzy

logic was introduced by Zadeh [26,27], Mamdani and Assilian

[28] identified frameworks for fuzzy control strategy and devel-

oped the first industrially applied fuzzy controller and Beirami

and Zerafat [29] applied fuzzy control strategy for adaptation of

PID controller parameters in a heat exchanger system.

This study is organized as follows: Firstly, the mathemat-

ical model of the PEM fuel cell (PEMFC) air feed system and

control objective is presented (Section 2). In Section 3, the

Fig. 1 e Schematic of the PEM fuel cell and ancillary

components.

Please cite this article in press as: Beirami H, et al., Optimal PID pusing the self-adaptive differential evolution algorithm, Interna10.1016/j.ijhydene.2015.05.114

control scheme of fuzzy logic and PIDF are designed. The

self-adaptive differential evolution (SaDE) algorithm and

optimization of controller parameters are discussed in Section

4. Simulation results of the fuel cell model are given in Section

5. Finally, main conclusions are presented in Section 6.

The PEMFC model

The 75 kW PEM fuel cell model used in this study is developed

by Pukrushpan et al. [30,31]. The fuel cell stack consists of 381

cells, each with a 280 cm2 membrane. This model has been

widely accepted by researchers as a representative model to

indicate real fuel cells for control objectives. The stack is

combined with ancillary equipment for power generation.

Fig. 1 shows the components of the fuel cell system as well as

the chemical reactions involved. Increased pressures improve

the reaction rate and enhance the efficiency and the fuel cell

power density as well [32]. A cooling system is used to

decrease the temperature of compressed air. Upon cooling, air

humidity is increased to the desired value in order to prevent

electrolyte dehydration. In the anode section, hydrogen is

suppliedwhile flow rate and pressure aremanipulated using a

control valve. Hydrogen molecules are converted to 2Hþ and

two electrons on the anode catalytic layer:

H2/2Hþ þ 2e� (1)

Produced protons move toward the cathode catalytic layer

through the membrane. Since the membrane is not conduc-

tive to electrons, released electrons flow in the external cir-

cuit. In the cathode, oxygen passes through the gas layer and

reacts with protons and electrons in the catalyst layer

resulting in water production:

12O2 þ 2Hþ þ 2e�/H2O (2)

The general reaction taking place in the fuel cell is a

combination of Eqs. (1) and (2) accompanied by heat

generation:

H2 þ 12O2/H2Oþ heatþ electrical energy (3)

The PEMFC air feed system model

This section deals with the investigation of dynamic behavior

of variables concerning air flow control in order to prevent

oxygen starvation. Oxygen, nitrogen and vapor partial pres-

sures in the cathode are calculated using mass conservation

combined with ideal gas law:

dPO2

dt¼ RTfc

VcaMO2

�WO2 ;in �WO2 ;out �WO2 ;reacted

�(4)

dPN2

dt¼ RTfc

VcaMN2

�WN2 ;in �WN2 ;out

�(5)

dPv

dt¼ RTfc

VcaMv

�Wv;in �Wv;out þWv;gen þWv;membr �Wl;out

�(6)




i n t e r n a t i o n a l j o u r n a l o f h yd r o g e n e n e r g y x x x ( 2 0 1 5 ) 1e1 3 3

where, R is the universal gas constant, Vca is the cathode

volume; Tfc is the fuel cell temperature and MN2and MO2

are

oxygen and nitrogen molecular weights, respectively. The

oxygen and nitrogen outlet mass flow rates are denoted by

WO2 ;out and WN2 ;out. The inlet mass flow rates are also shown

by WO2 ;in and WN2 ;in. The rate of oxygen reaction is also given

by WO2 ;reacted. Wv;in is the mass flow rate of vapor entering and

Wv;out is the mass flow rate of vapor leaving the cathode.

Wv;gen, Wv;membr and Wl;out, represent the rate of vapor gener-

ated in the fuel cell, mass flow rate of water transferred

across the membrane and liquid water leaving the cathode,

respectively.

Increasing the oxygen excess ratio by increasing the inlet

oxygen mass inflow rate to the cathode is the most efficient

way to prevent oxygen starvation. Oxygen starvation is a

complicated phenomenon and an important reason of fuel

cell degradation [33e35]. It must be mentioned that oxygen

starvation risk is very high at high disturbance fluctuation

rates [36]. As a result, the air inflow to the cathode must be

increased in order to prevent sudden oxygen starvation during

fast disturbance fluctuations. Oxygen excess ratio is defined

as the ratio of oxygen supplied and oxygen made to react in

the fuel cell stack [14,6,30,37]:

lO2¼ WO2 ;in

WO2 ;reacted(7)

The inlet mass flow rates are also defined as:

WO2 ;in ¼ xO2Wca;in (8)

WN2 ;in ¼ �1� xO2

�Wca;in (9)

Wca;in ¼ 11þ uatm

kca;inðPsm � PcaÞ (10)

where, kca;in is the inlet orifice constant and Pca the cathode

pressure, given by:

Pca ¼ PO2þ PN2

þ Psat (11)

Oxygen mass fraction can also be calculated as:

xO2¼ ylO2

MO2

yO2MO2

þ �1� yO2

�MN2

(12)

where, oxygenmole fraction is denoted by yO2. Humidification

rate is also given by:

uatm ¼ Mv

Ma

Pv;ca;in

Pa;ca;in(13)

where,

Pv;ca;in ¼ fatmPsat

�Tca;in

�(14)

Pa;ca;in ¼ Patm � Pv;ca;in (15)

Ma ¼ yO2MO2

þ �1� yO2

�MN2

(16)

where, Ma and Mv are air and vapor molecular weights and

Pv;ca;in and Pa;ca;in are also vapor and dry air pressures, respec-

tively. Atmospheric pressures and air relative humidity at the

cathode inlet are also denoted by Patm and fatm. Tca;in is the inlet

flow temperature to the cathode, and PsatðTca;inÞ is also the


vapor saturation pressure, with the following temperature

dependency [31].

log10ðPsatÞ ¼��1:6� 10�10

�T4 þ �

3:85� 10�7�T3

� �3:39� 10�4

�T2 þ 0:143T� 20:29 (17)

The outlet mass flow rates are also given by:

WO2 ;out ¼MO2

PO2

MO2PO2

þMN2PN2

þMvPsatWca;out (18)

WN2 ;out ¼MN2

PN2

MO2PO2

þMN2PN2

þMvPsatWca;out (19)

Since, the pressure difference between the cathode and the

ambient is large in pressurized stacks. The total flow rate at

the cathode exit Wca;out is calculated using the nozzle flow

equation [38].

Wca;out ¼ CDATPcaffiffiffiffiffiffiffiffiffiRTfc

p �Patm

Pca

�1g

8<: 2gg� 1

"1�

�Patm

Pca

�g�1g

#9=;

12

for

�Patm

Pca

�>�

2gþ 1

� gg�1

(20)

and,

Wca;out ¼ CDATPcaffiffiffiffiffiffiffiffiffiRTfc

p g12

�2

gþ 1

� gþ12ðg�1Þ

for

�Patm

Pca

��

�2

gþ 1

� gg�1

(21)

where, g is the ratio of heat capacities for air, AT is the nozzle

opening area and CD is the discharge coefficient of the nozzle.

Finally, the total flow rate is determined using the simplified

orifice equation [31]:

Wca;out ¼ kca;out

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPca � Patm

p(22)

Oxygen consumption rate is proportional to the stack

current and is calculated based on fundamental electro-

chemical principles as follows:

WO2 ;reacted ¼ MO2

nIst4Fr

(23)

where, n is the number of stack cells, Fr Faraday constant and

Ist the stack current. Compressor rotational speed (ucp) is the

only dynamic state of the compressor model, which is

modeled using [31]:

ducp

dt¼ 1

Jcp

�tcm � tcp

�(24)

Where, the compressor motor inertia is denoted by Jcp and tcm

is the compressor motor torque and tcp is the external load

torque. tcm and tcp given as follows:

tcm ¼ hcm

kt

Rcm

�Vcm � kvucp

�(25)

tcp ¼ CpTatm

hcpucp

"�Psm

Patm

�g�1g

� 1

#Wcp (26)

where, Rcm, kv and kt are motor constants and hcm and hcp the

compressor and motor mechanical efficiencies, respectively.

Cp and Vcm are the air heat capacity at constant pressure

and the inlet voltage to the compressor motor. Compressor




Fig. 2 e eThe lO2 performance curve for different

disturbance conditions.


flow (Wcp) is modeled by applying Jensen and Kristensen

nonlinear fitting method [3] as a function of pressure ratio

(Psm=Patm), upstream temperature (Tatm) and the compressor

rotational speed (ucp). Dynamical behavior of the supply

manifold air pressure is expressed as follows:

dPsm

dt¼ RTcp

MaVsm

�Wcp � kca;inðPsm � PcaÞ

�(27)

where,

Tcp ¼ Tatm þ Tatm

hcp

"�Psm

Patm

�g�1g

� 1

#(28)

where, Vsm and Tcp are the supply manifold volume and the

outlet gas temperature from the compressor, respectively.

Details of the compressor model used in this study can be

found elsewhere [31].

Control objective

Oxygen excess ratio (lO2 ) can be considered as an index of

system efficiency [30]. Oxygen starvation results in quick

degradation of the stack as a result of hot spot generation,

membrane surface burning and the reduction of power gen-

eration. The measure generally considered to prevent oxygen

starvation is simultaneous termination of reactant inflow and

outflow from the stack [31]. For lO2 <1, oxygen starvation

occurs. However, lO2[1means high oxygen availability at the

cathode which improves power generation in the stack. At the

same time, if oxygen availability is very high, net power (Pnet)

will be reduced due to the elevation of compressor power

consumption known as oxygen saturation. Since the air

compressor consumes the highest power among fuel cell

auxiliary components, others may be neglected in the calcu-

lation of net power:

Pnet ¼ Pst � Pcm (29)

where,

Pst ¼ VstIst (30)

and

Pcm ¼ tcmucp (31)

where, Pst and Pcm are the stack power generation and

compressor power consumption, respectively and Vst is also

the stack voltage.

The PEMFC system explained in Section 2 has two inputs;

One is the compressor motor voltage (Vcm) as the inlet to

the control system and the other is the stack current (Ist) as

the disturbance. Based on Eqs. (7)e(28), lO2 is dependent

on the compressor motor voltage and the stack current

and since Ist is the disturbance, lO2 must be manipulated

by Vcm.

The control objective is the maximum output power gen-

eration for various disturbance conditions. The relationship

between the net power output and lO2 for different distur-

bance conditions is called the performance curve. From Fig. 2,

the maximum power output for each current ends in lO2

(optimal lO2 shown by “*” in Fig. 2). Those lO2 values generating

the maximum power are in the 1.98e2.53 range for 100e300 A


stack currents. lO2 ¼ 2 is proposed in some studies [31] for

simplicity. In other studies [2,37,39], the reference lO2 is

assumed as a variable with the aim of maximum power gen-

eration for each load. In the first scenario, the control objective

is to keep lO2 at lO2 ¼ 2 which has been the subject of most

studies up to now. In the second scenario, a lookup table is

generated using lO2 which results in the maximization of Pnetfor the inlet Ist (Fig. 3). Obtained values are lO2 setpoints for

tracking. The controller task is also tracking the reference lO2

generated by the lookup table.

The control scheme

The fuel cell system has a complicated and nonlinear

behavior due to multiple inputs. So, in order to manipulate

the oxygen excess ratio as the control objective, a proper

control scheme is required to manipulate the compressor

motor voltage (Vcm). As a result, PIDF plus fuzzy controller is

implemented in this study with control parameters opti-

mized by SaDE algorithm. The proposed control scheme is

given in Fig. 4.

Feedforward fuzzy logic control

Fuzzy controllers are nonlinear having a special structure

with successful applications in engineering problems. These

controllers perform similar to expert humans while control-

ling a system.

In feedforward control, the measurement of stack current

as the inlet disturbance to the fuel cell and its implementation

as additional information results in the enhancement of PID

(feedback loop) control performance [40]. This measurement

can be considered as an early warning of lO2 deviation from

the setpoint. Based on this warning, the feedforward

controller will have sufficient time to vary the manipulated

variable (Vcm) before deviation.




Fig. 3 e eCurve fitting for the optimal air excess ratio with disturbance variation.

Fig. 4 e eThe proposed control scheme.

Table 1 e The fuzzy rules.

If the stack current(Ist) is:

Then the compressor motor voltage(vcm) is:

Very Low (VL) Very Low (VL)

Low (L) Low (L)

Medium (M) Medium (M)

High (H) High (H)

Very High (VH) Very High (VH)


Fuzzy logic control designThe fuzzy logic controller used in this study is a Mamdani

fuzzy inference systemwith a single input and a single output.

The stack current (Ist) is selected as the input and the

compressor motor voltage (Vcm) as the output (Uff ). The input

range is 100e300 A and the output range varies in the

100e235 V range. For input and output variables, triangular

membership functions and for the sides, S-shape and Z-shape

functionalities are assumed. Twelve membership function

parameters are optimized using the SaDE algorithm. The

input and output fuzzy linguistic variables are denoted by

Very High (VH), High (H), Medium (M), Low (L) and Very Low

(VL). Five IF-THEN rules are used in the fuzzy logic system

given in Table 1. Deffuzification is also implemented by the

center of gravity (COG) technique [41].

Feedback PIDFcontrol

A feedforward controllermust be accompanied by a feedback

controller so that the steady-state offset value is ensured to

be zero [40]. Feedback control has proved its performance in

many applications with the exception of cases with very

slow feedback dynamics. It is worth mentioning that,

imprecise adjustment of controller parameters can result in

instability [40].

The main problem of the differentiating part in a PID

controller is the amplification of high frequency noise due to

measurement inefficacy resulting in high output values. To


overcome this problem, a low-band filter is used to limit

the high frequency gain and noise [42]. The transfer func-

tion of the PID controller with a low-band differentiating

filter is:

u ¼ KP þ KI1sþ KD

N

1þN 1s

(32)

KP, KI and KD are the proportional, integral and differential

gains, respectively and N is the filter coefficient.

Optimization tools

DE algorithm

DE is a population-based, stochastic algorithm using global

optimization which is implemented for solving several




Fig. 5 e Memory of success and failure: (a) memory of success, (b) memory of failure.


engineering problems due to simplicity and high efficiency.

More information on DE applications can be found elsewhere

[43,44]. In DE algorithm, the population consists of NP vectors

as Xi;G ði ¼ 1;…;NPÞ. Each Xi;G is a D-dimensional vector rep-

resenting a candidate solution. The Xi;G vectors are initially

randomized within the search space with a uniform distri-

bution. The population of the forthcoming generations is

produced by mutation, crossover and selection operators.

Based on these operators, various DE algorithms are gener-

ated. Different strategies are in mutation with the most

common as:

Vi ¼ XGr1þ F

�XG

r2� XG

r3

�; r1sr2sr3 (33)

Three vectors (r1, r2 and r3) are thus selected randomly

from the population and the r2 � r3 vector is multiplied by an

F factor and added to r1 to generate the trial vector. F is the

mutation control parameter which is a positive number

close to unity. The selection of the appropriate mutation

strategy is dependent on the problem. The mutation oper-

ator in DE, has a more pronounced role compared with other

operators contrary to other algorithms and is identified by

trial and error. In order to remove this shortcoming, a

strategy candidate pool consisting of a number of various

strategies is applied instead of the time consuming trial and

error strategy. Upon mutation, the crossover operator is

implemented to generate offsprings from the previous trial

population. In this algorithm, two crossover operators are


defined; One is exponential and the other binomial crossover

which is used here:

uji;G ¼

(vji;G; if

�randj½0;1Þ �CR

�or

�j¼ jrand

�xji;G; otherwise

; j¼ 1;2;…;D (34)

The crossover rate (CR) is randomly chosen from the [0, 1)

range showing the parameters’ share copied from the mutant

vector. jrand is a random integer from the [1, D] range. The se-

lection process in DE is different from other algorithms. In

other evolutionary algorithms, the remainingmembers for the

next generation are selected based on probability, while se-

lection in DE is performed deterministically between the

offspring and parent vectors based on fitness. This approach is

usually called a selection tournament given as follows:

Xi;Gþ1 ¼Ui;G; if f

�Ui;G

� � f�Xi;G

�Xi;G; otherwise

(35)

where, fðXi;GÞ and fðUi;GÞ are fitness functions of the offspring

and parent vectors. The stopping criterion for DE is usually

the number of generations or the number of objective func-

tion evaluations.

SaDE algorithm

Estimation of DE parameters entails a trial and error approach

which imposes an extra computational load on the solution

procedure. The SaDE algorithm can adjust F and CR during the




Fig. 6 e eSaDE algorithm.


Please cite this article in press as: Beirami H, et al., Optimal PID plus fuzzy controller design for a PEM fuel cell air feed systemusing the self-adaptive differential evolution algorithm, International Journal of Hydrogen Energy (2015), http://dx.doi.org/10.1016/j.ijhydene.2015.05.114



Fig. 7 e eEncoding of candidate solutions.


algorithm performance and can select the mutation strategy.

In SaDE, both the generation of trial vectors and control

parameters become gradually self-adapted by learning from

previous generations. In this paper, a strategy candidate

pool is used that consists of four generation strategy trial

vectors as follows:

Table 2 e Parameters used in the simulation.

Symbol Parameter

n Number of cells in fuel cell stack

R Universal gas constant

Ra Air gas constant

Patm Atmospheric pressure

Tatm Ambient temperature

Tfc Fuel cell temperature

Fr Faraday constant

Ma Air molar mass

MO2 Oxygen molar mass

MN2 Nitrogen molar mass

Mv Vapor molar mass

g Ratio of specific heats of air

Jcp Compressor inertia

Kv Motor constant

Kt Motor constant

Rcm Motor constant

Cp Specific heat of air at constant pressure

hcp Compressor efficiency

hcm Motor mechanical efficiency

Vca Cathode volume

Vsm Supply manifold volume

Kca; in Cathode inlet orifice constant

Kca; out Cathode outlet orifice constant

AT Cathode outlet throttle area

CD Cathode outlet throttle discharge coefficie

ra Air density

YO2 Oxygen mole fraction

fatm Average ambient air relative humidity

N Derivative filter time constant

ui;j ¼xr1 ;j þ F

�xbest;j � xi;j

�þ F�xr1 ;j � xr2 ;j


�; if rand½0; 1

xi;j; otherwise


“DE/rand/bin”:

ui;j ¼xr1 ;j þ F

�xr2 ;j � xr3 ;j

�; if rand½0;1Þ<CR or j ¼ jrand

xi;j; otherwise(36)

“DE/rand-to-best/2/bin”:

SI units Value

381

Jmol�1K�1 8.31451

Jmol�1K�1 286.9

Pa 101.325

K 298.15

K 353.15

Cmol�1 96.487

Kgmol�1 28.97 � 10�3

Kgmol�1 32 � 10�3

Kgmol�1 28 � 10�3

Kgmol�1 18.02 � 10�3

1.4

Kgm2 5 � 10�5

Vrad�1s 0.0153

N-mA�1 0.0225

U 1.2

Jkg�1K�1 1004

80%

98%

m3 0.01

m3 0.02

Kgs�1Pa�1 0.3629 � 10�5

Kgs�1Pa�1 0.2177 � 10�5

m2 0.00175

nt 0.0124

Kgm�3 1.23

0.21

0.5

s 100

Þ<CR or j ¼ jrand (37)




Fig. 8 e eThe optimizedmembership functions of the fuzzy

feedforward controller (a) input membership functions, (b)

output membership functions.


“DE/rand/2/bin”:

ui;j ¼xr1 ;j þ F

�xr2 ;j � xr3 ;j


�; if rand½0;1Þ<CR or j ¼ jrand

xi;j; otherwise(38)

“DE/current-to-rand/1”:

Ui;G ¼ Xi;G þ K�Xr1 ;G � Xi;G

�þ F�Xr2 ;G � Xr3 ;G

�(39)

A probability value is assigned to the selection of each

strategy so that the summation of these probabilities equals

unity. Initially, these probabilities are equally assumed as 1=k

and will be modified by the evolution of the algorithm. The

probability of the K th strategy being selected is Pk

(k ¼ 1; 2;…;K) where K is the number of strategies. In the

generation G, after the calculation of the whole trial vectors,

the total number of generated vectors by the K th strategy

which could successfully enter the next generation are

denoted as nsk;G and the others as nfk;G. Registration of these

numbers for decision making in the upcoming generations

can identify the memory of success and failure (Fig. 5)

being responsible for the registration of nsk;G and nfk;G for a

number of previous generations equal to the learning period

(LP). After the initial LP generations, the probability of different

strategies being selected will be updated at each subsequent

generation based on the memory of success and failure:

Pk;G ¼ Sk;GPKk¼1Sk;G

(40)

where,

Sk;G ¼PG�1

g¼G�LPnsk;gPG�1g¼G�LPnsk;g þ

PG�1g¼G�LPnfk;g

þ ε ðk ¼ 1;2;…;K;G> LPÞ (41)


and Sk;G is the success rate of generated trial vectors with the K

th strategy in previous LP generations from generation G

successfully entered the next generation.

A small constant value, ε ¼ 0:01 is added to SaDE in order to

prevent null probability. It is obvious that a larger Sk;Gindicates a higher probability value for the selection of the Kth

strategy in the current generation. At the next step, control

parameters will become self-adapted. The selection of NP is

totally dependent on the nature of the problem being

performed through simple estimations. Among other param-

eters, CR is very sensitive to the problem type and can have a

significant role in obtaining the correct solution. The F value

has a great impact on the rate of convergence, which is

selected randomly with a probability proportional to a

normal distribution with an average of 0.5 and a variance

equal to 0.3, denoted by N (0.5, 0.3). The control parameter K in

“DE/current-to-rand/1” strategy is also selected randomly

from the [0, 1] range.

The crossover rate control parameter is selected with a

normal distribution probability and a standard variance equal

to 0.1. CRm is also given an initial value of 0.5 equal for all

strategies. The CR value with a successful trial vector in the

previous generation is registered in the crossover rate mem-

ory ðCRmemoryk Þ for every strategy and the previous generation

LP. The CRm value for every generation and the K th strategy is

calculated by averaging the registered values in the CRmediank

memory and CR is also obtained using a normal distribution

probability with a CRm average and a variance equal to 0.1. The

flowchart of the SaDE algorithm is given in Fig. 6. More in-

formation on SaDE can be found elsewhere [21].

Encoding

A number of parameters are required for encoding the

candidate answers for fuzzy and PIDF controllers. Simulta-

neous optimization of fuzzy and PIDF controllers is performed

to obtain improved results. In this study, optimization of

proportional (KP), Integral (KI) and differential (KD) coefficients

of the PIDF controller and membership functions of the fuzzy

controller are performed using SaDE algorithm.

Encoding the position of membership functions has been

performed by different techniques in the literature [45,46].

Fig. 4 shows the encoding approach used in this study. The

position of membership functions are represented by aji,

where j shows the number of εf and i a special point from εf .

The initial and end points of triangular membership functions

are shown by i¼ 1 and i¼ 3, respectively and the point εf ¼ 1 is

represented by i ¼ 2. For Z-Shape membership functions, the

point separated from unity (Fig. 7) is i ¼ 1 and the point

touching zero is indicated by i ¼ 2. For S-shape membership

functions, the point separated from zero is i ¼ 1 and the point

touching unity is i ¼ 2. In many studies, membership function

parameters for each εf are defined independently in the




Fig. 9 e eThe comparison of three control strategies with a constant oxygen excess ratio: (a) disturbance variation, (b) stack

voltage variation, (c) oxygen excess ratio variation using PIDF plus Fuzzy FF, (d) oxygen excess ratio variation using PI þ FF,

(e) oxygen excess ratio variation using FF.


candidate solution for the optimization problem leading to an

extra step for changing or removing impossible solutions

bringing about large computational loads. Special rules are

devised in order not to confront impossible solutions. We

assume that the sequence of membership function linguistic

variables does not change and each value from the output

range has at least one nonzero member in one of its


membership functions in order to simplify the solutions

encoding and algorithm acceleration by eliminating impos-

sible solutions. To this end, the sequence of points indicates

the position of membership functions (Fig. 7).

The optimization problem parameters for the fuzzy

controller are x1 through x12 where xi shows the relative dis-

tance between the membership function points in the [0, 1]




Fig. 10 e The magnified plot of oxygen excess ratio variations at: (a) 10 s, (b) 20 s.

Table 3 e Performance index comparison with theconstant oxygen excess ratio.

Performance index IAE ITAE ISE

PIDF þ Fuzzy FF 0.1069 1.234 0.02029

PI þ FF 0.1557 2.060 0.02873

FF 0.7413 8.371 0.06231


range. The membership function point values are calculated

as follows: a1 is the initial point in the range and the other

points are previous values plus Li ¼ xi=P

xi � L, where L is the

length of the controller output range. Finally, x13, x14 and x15are defined for proportional, Integral and differential param-

eters of the PIDF controller, respectively.

Simulation results

The PIDF plus Fuzzy control scheme is applied to the 75 kW

PEMFC system. The numerical parameters used in the

Fig. 11 e ePIDF plus Fuzzy FF control scheme for the case of m

variation, (b) stack voltage variation.


simulation are given in Table 2. PIDF controller parameters

and the parameters regarding the membership functions of

the fuzzy controller are optimized using SaDE algorithm. The

optimized membership functions for the fuzzy controller are

given in Fig. 8. The main control objectives are: (a) Preventing

oxygen starvation by keeping lO2 constant at lO2 ¼ 2, and (b)

acquiring the maximum power tracking by the optimal lO2 to

achieve the best system performance.

Controller performance with constant lO2

The first simulation is performed by keeping lO2 constant

and stepwise changing of the disturbance covering the

whole fuel cell performance range (Fig. 9). Despite load

fluctuations, the control system adjusts lO2 at the setpoint.

Fig. 9 (c)e(d) shows the simulation results for controlling lO2

using feedforward [30,31], PI plus feedforward [30,31] and

PIDF plus fuzzy feedforward control strategies. Also, Fig. 10

indicates a magnified version of the oxygen excess ratio

graph using various control techniques. Based on the re-

sults, the proposed techniques have improved values of

aximum power tracking: (a) optimal oxygen excess ratio




Fig. 12 e eThe comparison of fuel cell net power for changing the oxygen excess ratio.


rising time, undershoot and overshoot compared with pre-

vious studies [30,31]. The comparison of the total perfor-

mance criteria is given in Table 3. As it is obvious, the PIDF

plus fuzzy feedforward has a better response compared to

other techniques.

Besides, the proposed technique has a better performance

toward the instantaneous disturbance rise. The settling time

for PI plus feedforward and pure feedforward are 230 and

430 ms, respectively reduced to 100 ms in the PIDF plus fuzzy

feedforward control scheme (Fig. 10(a)). A sudden decrease of

the disturbance results in 110 and 60 ms settling times for PI

plus feedforward and feedforward control schemes, respec-

tively reduced to 20 ms in the PIDF plus fuzzy feedforward

type (Fig. 10(b)). The results show that the PIDF plus fuzzy

feedforward controller is a promising alternative for oxygen

supply to the fuel cell especially in case of continual distur-

bance variations.

Variable lO2 tracking for maximum power efficiency

Based on Section 2.2, the variable reference lO2 based on

disturbance variations is derived in order to achieve the

maximum efficiency and reduction of fuel cell expenses. lO2

must be manipulated according to disturbance variations in

order to obtain the maximum net power. However, the risk of

oxygen starvation in case of sudden high power demands is

increased as a result of oxygen shortage in the stack. As a

result, the controller capacity for tracking the variable lO2

setpoint is very important. Using the same disturbance

variation characteristics in the first simulation, the proposed

controller is tested for the case of variable lO2 (Fig. 11). The

results show that the proposed control scheme has per-

formed successfully for setpoint tracking and noise rejection.

Finally, the net power efficiency is calculated for all tech-

niques (Fig. 12). The results show that the second scenario

(maximum power tracking) results in a higher net power by

1% compared with the first scenario (constant oxygen excess

ratio). Based on the results, the proposed control strategy has

a higher efficiency compared with other techniques in both

scenarios.


Conclusions

In this study, the hybrid PIDF plus fuzzy feedforward control

technique is developed for oxygen excess ratio to manipulate

the air inflow to the PEMFC. Controller parameters are opti-

mized simultaneously using SaDE algorithm. The control

objective is preventing oxygen starvation and obtaining the

maximum fuel cell efficiency. The results show that, the

proposed control scheme performs better compared with

other methods for fuel cell control under various disturbance

fluctuations. Also, this technique reduces the settling time,

overshoot and undershoot significantly in comparison with

other techniques. The calculation of the optimal oxygen

excess ratio as the system setpoint results in the enhance-

ment of net power by 1% compared with other control stra-

tegies or constant oxygen excess ratio resulting in an

improved fuel cell efficiency.

Acknowledgments

We gratefully acknowledge the financial support provided by

Fuel cell Steering Committee of the Iranian Organization of

Renewable Energies (SUNA).

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