A Modified Farmland Fertility Optimizer for Parameters Estimation of Fuel Cell Models

Abstract

This paper proposes a modified version of a well-known optimization technique called
Farmland Fertility Optimization algorithm (FFA). The modified FFA (MFFA) is
developed in order to improve the performance of conventional FFA. It is mainly based
on two stages. Firstly, the Levy flights are used to enhance the local searching
capability in the exploitation phase and the global searching capability in the
exploration phase. Secondly sine-cosine functions are used to create different
solutions which fluctuate outwards or towards the best possible solution. The
developed algorithm has been validated using ten benchmark functions and three
mechanical engineering benchmark optimization problems. After that, the newly
developed algorithm MFFA is used for extracting the effective unknown parameters of
Proton Exchange Membrane Fuel Cells (PEMFCs) models. The optimal extraction of
these parameters is essential to determine an accurate semi-empirical mathematical
model for PEMFC. The sum of squared errors (SSE) between the experimental data
and the corresponding calculated ones is adopted as the objective function. Four
different commercial PEMFC stacks are used to validate the effectiveness of the
developed algorithm. The results obtained by MFFA are compared with those obtained
by the conventional FFA and other well-known optimization techniques. Moreover, a
comprehensive statistical analysis is performed to determine the accuracy and
efficiency of the developed algorithm. The results prove the reliability and superiority of
the developed algorithm compared with the conventional FFA and other state-of-theart
optimizers.

Full text

ORIGINAL ARTICLE
A modified farmland fertility optimizer for parameters estimation
of fuel cell models
Ahmed S. Menesy
1
•Hamdy M. Sultan
1,2
•Ahmed Korashy
3,4
•Salah Kamel
3
•Francisco Jurado
4
Received: 17 January 2020 / Accepted: 10 February 2021
The Author(s), under exclusive licence to Springer-Verlag London Ltd. part of Springer Nature 2021
Abstract
This paper proposes a modified version of a well-known optimization technique called Farmland Fertility Optimization
algorithm (FFA). The modified FFA (MFFA) is developed in order to improve the performance of conventional FFA. It is
mainly based on two stages. Firstly, the Levy flights are used to enhance the local searching capability in the exploitation
phase and the global searching capability in the exploration phase. Secondly sine–cosine functions are used to create
different solutions which fluctuate outwards or towards the best possible solution. The developed algorithm has been
validated using ten benchmark functions and three mechanical engineering benchmark optimization problems. After that,
the newly developed algorithm MFFA is used for extracting the effective unknown parameters of Proton Exchange
Membrane Fuel Cells (PEMFCs) models. The optimal extraction of these parameters is essential to determine an accurate
semi-empirical mathematical model for PEMFC. The sum of squared errors between the experimental data and the
corresponding calculated ones is adopted as the objective function. Four different commercial PEMFC stacks are used to
validate the effectiveness of the developed algorithm. The results obtained by MFFA are compared with those obtained by
the conventional FFA and other well-known optimization techniques. Moreover, a comprehensive statistical analysis is
performed to determine the accuracy and efficiency of the developed algorithm. The results prove the reliability and
superiority of the developed algorithm compared with the conventional FFA and other state-of-the-art optimizers.
Keywords Modified Farmland Fertility Algorithm Metaheuristic optimization Proton exchange membrane fuel cells 
Parameter extraction
1 Introduction
Due to the increase of electric power demand and shortage
of fossil fuel, many researchers and decision-makers have
focused their attention on new technologies of renewable
energy sources. Nowadays, fuel cells (FCs) became one of
the most important renewable sources of energy thanks to
their sufficient operating efficiency, superior reliability,
applicability and easiness of use in various industrial areas.
Fuel cell (FC) can be defined as an electrochemical
&Francisco Jurado
fjurado@ujaen.es
Ahmed S. Menesy
ahmedsafy272@minia.edu.eg
Hamdy M. Sultan
hamdy.soltan@mu.edu.eg
Ahmed Korashy
ahmed.korashy2010@yahoo.com
Salah Kamel
skamel@aswu.edu.eg
1
Electrical Engineering Department, Faculty of Engineering,
Minia University, Minia 61111, Egypt
2
Electrical Power Systems Department, Moscow Power
Engineering Institute ‘‘MPEI’’, Moscow, Russia 111250
3
Electrical Engineering Department, Faculty of Engineering,
Aswan University, Aswan 81542, Egypt
4
Department of Electrical Engineering, University of Jae
´n,
EPS, 23700 Linares, Jae
´n, Spain
123
Neural Computing and Applications
https://doi.org/10.1007/s00521-021-05821-1(0123456789().,-volV)(0123456789().,-volV)

equipment that directly converts the chemical energy of
hydrogen fuel gas into electric power through a chemical
reaction. FC device consists of anode and cathode elec-
trodes separated by an electrolyte and a catalyst layer.
Hydrogen gas can flow through the anode and oxygen gas
through the cathode, while the polymer electrolyte between
them plays an important role in facilitating the exchange of
positive and negative electric charges. Accordingly, the
negative electrons will complete the cell cycle in the
external load producing electric energy [1,2].
FCs have been widely used in many industrial, resi-
dential and commercial applications due to the above-
mentioned advantages compared with other renewable
energy sources [2,3]. There are many types of fuel cells
depending on the electrolyte type, while the proton
exchange membrane fuel cell (PEMFC) is considered the
most widely used type and has been commercialized for the
portable and vehicular applications [4,5]. PEMFC operates
at low temperatures and pressures as well allows fast start-
up, the start-up time for it is roughly one second. In the
normal conditions of pressure and temperature, a simple
FC usually generates a very small terminal voltage
(0.5–0.9 V), and this low potential is not sufficient for
practical industry applications. Therefore, a number of FCs
are arranged in series to form a stack to get the desired
output voltage, to be used in high generation systems [6].
An effective model of FC should be demonstrated
before prior to the installation phase of actual FC system to
have a simple tool, which makes the testing, performance
evaluation, and simulation of the FC much easier [7]. Over
the past decades, modelling and simulation of polarization
characteristics of PEMFCs have attracted the attention of
many researchers to understand the phenomena that hap-
pens within the FC stack. A number of design parameters
are not reported in the datasheet of the manufacturer and
their values have to be determined. Therefore, to develop
an accurate model of FC, a set of unknown model
parameters must be identified precisely by applying state-
of-the-art optimization techniques.
Conventional analytical optimization methods have
been utilized to tackle the PEMFC model problem [5,8,9].
Additionally, to validate the steady-state operation of
PEMFCs based on experimental measurements, various
empirical models have been reported [1,10,11]. Further-
more, semi-empirical models have been reported, in which
dual dimensional modeling [12], spectroscopy and char-
acterization [13], and fractional order aspect [14] have
been utilized to analyze the PEMFC model. Developing a
precise and reliable PEMFC model based on optimally
determined parameters is considered a very challenging
issue. The polarization curve of PEMFCs is a highly non-
linear curve, which makes deterministic optimization
methods deteriorate their performances while tackling such
highly nonlinear, complex, multivariable systems. To
overcome such drawbacks, meta-heuristic algorithms have
been recommended to deal with such optimization
problems.
In the literature, several meta-heuristic optimization
techniques have been utilized to extract the precise values
of the unknown parameters of PEMFC models. For
example, a Hybrid Genetic Algorithm (HGA) was pro-
posed in [15] to improve the accuracy of parameter esti-
mation. The Particle Swarm Optimization (PSO) method
presented in [16] identified the FC parameters from three
measured data set; good results were obtained in compar-
ison with other conventional optimization techniques. A
grouping-based global harmony search GGHS optimization
method was presented in [3] for identifying the values of
the PEMFC model parameters: The Ballard PEMFC was
considered for validating the results. The differential
evaluation (DE) algorithm and the hybrid adaptive DE
algorithm were developed for extracting the effective
parameters of PEMFC in [17,18]. The authors of [19]
applied circular genetic operator-based RNA Genetic
Algorithm (cRNA-GA) to determine the best parameters
for PEMFC. Many metaheuristic optimization techniques
have been presented to estimate the optimal parameters’
values of different PEMFC stacks, i.e. the Seeker Opti-
mization Algorithm (SOA) [20], Modified Artificial
Ecosystem Optimization (MAEO) [21], eagle strategy
based on JAYA algorithm and Nelder-Mead simplex
method (JAYA-NM) [22], Multi-verse Optimizer (MVO)
[23], Chaotic Harris Hawks Optimization (CHHO) [24],
hybrid Teaching Learning Based Optimization—Differen-
tial Evolution algorithm (TLBODE) [25], Shark Smell
Optimizer (SSO) [26], Bonobo Optimizer (BO) [27],
Cuckoo search algorithm with explosion operator (CS-EO)
[28], Equilibrium Optimizer (EO) [29], selective hybrid
stochastic strategy [30], Bird Mating Optimizer (BMO)
[31], Grasshopper Optimizer (GHO) [32] and Tree Growth
Algorithm (TGA) [33].
Taking into consideration the aforementioned, it was
observed that MHA inspired by natural strategies are fea-
sibly declared. Thanks to the ‘‘no free lunch’’ (NFL) theory
[34], which is based on the fact that ‘‘no single optimiza-
tion technique can tackle all optimization problem and
garntuee the same performance’’, it can be proposed that
there is a chance for enhancing existing optimization
algorithms or developing novel ones. The above-mentioned
clarifies why the Farmland Fertility algorithm (FFA) has
been developed and presented as a competitor to other
optimization algorithms for identifying the unknown
parameters of PEMFCs. This paper proposes and applies a
modified version of FFA (MFFA) to efficiently extract
seven unknown parameters of PEMFC. The optimal
parameters obtained by MFFA will help to demonstrate an
Neural Computing and Applications
123

accurate PEMFC model which simulates the electrical
characteristics and operation of actual PEMFCs under
various operating scenarios. Four different models of PEM
fuel cells are studied to demonstrate effectiveness of
MFFA. Moreover, a comprehensive statistical analysis is
presented to assess the accuracy of the developed
algorithm.
The rest of the paper is divided as follows. The math-
ematical model of PEMFC stacks and the objective func-
tion of the identification problem are presented in Sect. 2.
FFA and MFFA are illustrated in Sect. 3. The study cases,
results and discussion under various operating conditions
are presented in Sect. 4. The main conclusions drawn from
this study are provided in Sect. 5.
2 Problem statement
2.1 Basic concept of PEMFC
The construction of a typical PEMFC includes anode and
cathode electrodes separated by a polymer electrolyte
membrane. This membrane allows flowing of protons and
arrests electrons. The typical construction of PEMFC is
shown in Fig. 1.
The hydrogen gas is supplied to anode side and oxygen/
air gas is supplied to cathode side. The overall chemical
reactions that happens at electrodes sides of PEMFC can be
illustrated as follows [35]:
Side of anode :H2!2Hþþ2eð1Þ
Side of cathode: 2Hþþ1
2O2þ2e!H2Oð2Þ
Total chemical reaction :H2þ1
2O2
!H2OþEnergy þHeat ð3Þ
The energy part mentioned in Eq. (3) defines the gen-
erated electrical energy resulting from the transfer of
hydrogen electrons from anode side to cathode side via the
external circuit (load).
2.2 Mathematical model of PEMFC stacks
The actual generated voltage and current of each single
PEMFC are too small. Thus, a large number of FCs are
stacked together and connected in series and/or parallel to
obtain high value of voltage or current. Amphlett et al. [36]
proposed a mathematical electrochemical model for
PEMFC. When a group of identical Fuel cells N
cells
are
connected in series, the produced terminal voltage of the
stack can be calculated as follows [24],
Vstack ¼Ncells Vcell ð4Þ
where N
cells
is the number of FCs connected in series, and
V
cell
denotes the produced terminal voltage of each single
FC.
The produced terminal voltage of each single FC can be
calculated as [5,9,36]:
Vcell ¼ENernst Vactivation Vohmic Vconcent ð5Þ
where E
Nernst
represents the open-circuit (OC) voltage,
V
activation
denotes the activation over potential per cell,
V
ohmic
is the resistive voltage drop per cell and V
concent
is
the concentration overvoltage per cell. E
Nernst
is called
reversible potential which can be calculated as [37],
ENernst ¼1:229 0:85 103Tfc 298:15ðÞþ4:3085
105Tfc ln PH2
ðÞþln ffiffiffiffiffiffiffiffi
PO2
p

ð6Þ
where T
fc
is the cell temperature in Kelvin; P
H2
and P
O2
are
the partial pressures of hydrogen gas and oxygen gas at the
input channels of the FC stack (atm), respectively. In the
FC operation, hydrogen and pure oxygen are supplied to
the PEMFC stack, the reactants’ partial pressures at the
inlet channels can be mathematically expressed as follows
[38],
PH2¼0:5RH
aPH2O
ðÞexp
1:635 Ifc
=
A

T1:334
fc
0
@1
ARHaPH2O
ðÞ
Pa
0
@1
A
1
1
2
6
43
7
5
ð7Þ
Water and
Heat
e
-
H
+
Oxygen/Air
Inlet
Hydrogen
Inlet
e
-
e
-
Anode
Cathode
Catalytic layers
Membrane
+ +
+ +
+ +
-
-
-
-
-
-
--
e
-
Load
Hydrogen
Outlet
Fig. 1 Typical construction of PEMFC
Neural Computing and Applications
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PO2¼RHcPH2O
ðÞexp
4:192 Ifc
=
A

T1:334
fc
0
@1
ARHcPH2O
ðÞ
Pc
0
@1
A
1
1
2
6
43
7
5
ð8Þ
where P
a
is the input pressure around the anode, P
c
is the
input pressure at the cathode electrode; RH
a
and RH
c
are
the relative vapor humidity around the positive and nega-
tive electrodes, respectively; I
fc
is the current generated
from each cell (A), Ais the area of the membrane surface
(cm
2
), P
H2O
is the saturation pressure of water, which can
be defined as follows [38],
PH2O¼2:95 102Tfc 273:15ðÞ9:18
105Tfc 273:15ðÞ
2þ1:44
107Tfc 273:15ðÞ
32:18 ð9Þ
The voltage drop due to the activation mechanism V
ac-
tivation
can be expressed as follows [33],
Vactivation ¼n1þn2Tþn3Tln CO2
ðÞþn4Tln Ifc
ðÞ½
ð10Þ
where n
1
,n
2
,n
3
,n
4
denote semi-empirical coefficients and
need to be identified; CO
2
is the oxygen gas concentration
at cathode side in mol/cm
3
and can be mathematically
formulated as follows [24]:
CO2¼PO2
5:08 106exp498=Tfc
ðÞ ð11Þ
The ohmic voltage drop V
ohmic
occurs in the FC and can
be defined as follows [21,39],
Vohm ¼Ifc RMþRC
ðÞ ð12Þ
where R
M
represents the membrane surface resistance in
ohm (X), R
C
is the resistance of the connection that the
protons face during transferring through the membrane.
The resistance of the membrane can be calculated as fol-
lows [21,40],
RM¼qMl
Að13Þ
where q
M
is the specific resistance of membrane material
(Xcm), lis the thickness of the membrane (cm). q
M
can be
calculated as follows [21,40],
qM¼
181:61þ0:03 Ifc
A
þ0:062 Tfc
303

2Ifc
A

2:5
hi
k0:634 3Ifc
A

exp 4:18 Tfc303
Tfc
hi
ð14Þ
where kis an adjustable empirical parameter, which must
be identified. The voltage loss V
concent
due to concentration
process also occurs in the FC and can be expressed as
follows [21,40],
Vconcent ¼bln 1 J
Jmax
 ð15Þ
where bis an empirical parameter to be identified; Jand
J
max
denote the actual current density and the maximum
current density (A/cm
2
) of the fuel cell, respectively.
The previous Eqs. (1)–(15) include several unknown
parameters that must be extracted accurately. Generally,
the values of these parameters are not mentioned in the
datasheet of the manufacturer. Therefore, in order to ensure
a satisfactory modelling of PEMFC, accurate determination
of these parameters is very necessary. In particular, the
model of PEMFC has seven unknown parameters such as
f
1
,f
2
,f
3
,f
4
,k,R
C
, and b. The values of these seven
unknown parameters are perturbed within specified ranges
and then identified by means of the using the developed
MFFA meta-heuristic technique.
2.3 Objective function
The principle operation of the PEMFC illustrated in the
previous section depends on seven unknown variable
parameters. The main goal of this study is to extract the
optimal values of PEMFC parameters, which allows the
proposed model to match well with the measured data of
PEMFC. Therefore, the proposed objective function is a
measure of the quality of the extracted parameters. In this
work, the sum of the squared errors (SSE) between the
measured generated voltage of PEMFC and the simulated
one is defined as the objective function (OF) [25,41],
OF ¼MinSSEðxÞ¼Min X
N
i¼1
Vmeas iðÞVcal iðÞ½
2
!
ð16Þ
where Xdenotes the parameters vector need to be extrac-
ted, Nrepresents the number of measured data points, i
represents an iteration counter, V
meas
is the measured
generated voltage of PEMFC, and V
cal
denotes the calcu-
lated voltage of PEMFC, according to the following group
of practical inequality constraints,
ni;min nini;max;i¼1:4
RCmin RCRCmax
kmin kkmax
bmin bbmax
ð17Þ
where f
i, min
and f
i,max
are the lower and upper limits of
empirical coefficients; k
min,
and k
max
are the lower and
upper limits of water content; R
Cmin
and R
Cmax
define the
lower and upper limits of cell connections resistance; as
well as, b
min
and b
max
denote the lower and upper bounds
of the parametric coefficient.
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3 Optimization techniques
3.1 FFA
FFA is a new metaheuristic optimization technique, which
mimics the natural process of farmland fertility (FF). FFA
splits the farmland into various sections based on soil
quality. Two types of memory, internal and external, are
used for optimizing each section with high efficiency.
Furthermore, farmers have enough information on the
number of sections of their farmland and the corresponding
quality of the soil of each farmland section.
Farmers decide the type and the amount of materials
which should lead to change the quality of the soil and
enhance it in each farmland section. Moreover, a group of
memory units are located beside each farmland sec-
tion. This memory unit records the best quality for soil, and
it is called local memory. The local memory saves only the
best solutions of previous visits to each section individu-
ally. A series of memories called the global memory are
utilized for saving the optimal value of the soil quality in
all sections of the farmland. The global memory saves the
best solutions that find so far in all visits for all sections. In
the FFA, a combination between obtained solutions in
other farmland sections, current solutions in the global
memory and parts of the farmland that have the worst
quality, is made in order to obtain the better results [42]. In
other words, some of the best cases of each section are
stored in the local memory and best cases of all sections are
stored in global memory. So, global memory saves the best
solutions that ever found in all space searches and local
memory saves the best solutions that ever found in each of
the sections. Where the solutions are updated each iteration
until the conversion criteria are met.
3.1.1 First stage: values initialization
In this stage, the initial population are generated randomly
between upper and lower limits. All candidate solutions
should produce in this stage. At this stage, population size
is initialized in terms of the number of farmland sections
and the corresponding available solutions. The number of
initial population is calculated as:
N¼knð18Þ
where Nis the population size. kcalculates the number of
parts for the optimization problem. As a result, the whole
search space is divided into (k) sections that each section
has a specific number of solutions. nis the number of
solutions available for every farmland section. Random
population of search space is generated as follows:
Xij ¼Ljþrandð0;1ÞðUjLjÞð19Þ
where L
j
and U
j
, respectively, are the lower and upper
bounds of the design variable (X), that j=[1….D] repre-
sents dimension xand iis equal to[1….N], rand is random
numbers in the range of (0,1). Trial designs can be evalu-
ated by computing the objective function of the identifi-
cation problem (16).
3.1.2 Second stage: determination of soil quality for each
section of the farmland
For each section of the farmland, the quality of the soil can
be obtained using the average of the present solutions. The
available solutions of each section are selected in order to
determine their average quality, separately for each sec-
tion. That is:
Sections¼xðaÞð20Þ
a¼nðs1Þ:ns;s¼1;2;...k
fg ð21Þ
where sis the number of sections, xrefers to all candidate
solutions. For each farmland section, the quality of solu-
tions can be represented mathematically as follows:
Fit Sections¼MeanðallFitðxij Þin SectionsÞð22Þ
where Fit_Section
s
defines the average quality of the
solutions included in each part.
3.1.3 Third stage: updating the memories
In this phase, both global and local memories are updated
after calculating the average solution and the correspond-
ing solution quality far all parts of the farmland. The best
solutions of all sections are saved in the global memory,
while some of these solutions are kept in the local memory
corresponding to each farmland section. The two memories
are updated as well as the number of solutions of the best
local and global memories as follows:
Mlocal ¼roundðtnÞ0.1\t\1ð23Þ
MGlobal ¼roundðtnÞ0.1\t\1ð24Þ
where M
local
and M
Global
are the number of solutions in
local and global memories, respectively, based on the cost
function value. The t parameter is a constant number which
is selected between 0.1 and 1.
3.1.4 Fourth stage: change soil quality in each farmland
section
The worst quality part of the farmland and every present
solution in the worst farmland section are merged with one
of the solutions saved in the global memory as follows:
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123

h¼arandð1:1Þð25Þ
Xnew ¼hðXij XMGlobalÞþXij ð26Þ
where his a decimal number, ais a number in the range of
(0,1), X
MGlobal
is a randomly selected solution from the
global memory, X
ij
is a solution belonging to the worst
farmland section, which can be chosen to make modifica-
tions and X
new
is the new trial design generated by per-
turbing optimization variables. X
new
is a new solution that
obtained by applied changes. After making the changes in
the worst part of farmland, other sections should be com-
bined with available solutions in the entire search space.
Available solutions in other sections are determined by
Eqs. (27) and (28). Available solutions in other sections
can be calculated as:
h¼brandð0:1Þð27Þ
Xnew ¼hðXij XujÞþXij ð28Þ
where X
uj
represents a randomly selected solution over the
available solutions in the whole search space and bis an
arbitrary number in the range of (0,1).
3.1.5 Fifth stage: combination of soil
Each soil within the sections are combined based on the
best available cases in their local memory (BestLocal). Not
all available solutions are combined with local memory in
all sections, some of the available solutions in all places are
combined with the best solution ever found (BestGlobal)to
improve quality of existing solutions in each section. The
combination of a candidate solution with BestLocal or
BestGlobal is done as follows:
Xnew ¼Xijþw1ðXij BestGlobalÞQrand
Xijþrandð0:1ÞðXij BestLocalÞQ\rand
no
ð29Þ
where X
new
is a new trial design generated with the applied
perturbations, while X
ij
is the selected solution to be per-
turbed based on the information available from all sections.
Qis a parameter between (0,1). This parameter determines
amount of combination of solutions with best global
(BestGlobal). x1 is a parameter of the FFA. This parameter
is significantly reduced as the FFA optimization process
progresses. The w
1
parameter is updated as follows [24]:
w1¼w1Rv:0\Rv\1ð30Þ
where the R
v
parameter is a constant number is selected
between 0 to 1. In this paper, the R
v
value is selected to be
0.1.
3.1.6 Sixth stage: final conditions
At this stage, all available solutions in the search space is
accomplished. The FFA algorithm terminates when the
limit number of optimization iterations is completed.
1. MFFA
MFFA was developed in order to improve FFA’s
performance. For that purpose, the following modifi-
cations in the fourth and fifth stages of conventional
FFA were implemented.
•In Fourth Stage, The Levy flight motion strategy
replaced Eqs. (25) and (27).
In this stage, Eqs. (25) and (27) are modified
using a random walk strategy called Levy flights.
This random is a random process consists of taking
a series of consecutive random steps. However, the
randomization has a significant role in exploitation
and exploration. The modified equations can be
expressed as follows:
h¼aLevyðNvarÞð31Þ
h¼bLevyðNvarÞð32Þ
where Levy flights is used to enhance the algo-
rithm’s local searching capability in the exploita-
tion phase and global searching capability in the
exploration phase. The Levy flight can be calcu-
lated as follows [43]:
LevyðNvarÞ¼0:01 rr1d
rr2
jj
1
b
ð33Þ
where rr
1
and rr
2
are two randomly distributed
numbers in the range [0,1], bis a constant and dis
mathematically expressed as follows:
d¼
Cð1þbÞsin pb
2

C1þb
2

b2b1
2
ðÞ
0
@1
Að34Þ
where
CðxÞ¼ðx1Þ!ð35Þ
•In Fifth Stage, the sine–cosine method replaces
Eq. (29).
In this stage, sine–cosine function is added to
(29). Sine–cosine function is used to create differ-
ent solutions that fluctuate outwards or towards the
best possible solution. The modified equation can
be expressed as follows:
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123

H¼Xnew¼Xij þr1sinðr2ÞðXijBestGlobal ðbÞÞ:Q[rand
Xnew¼Xij þr1cosðr2ÞðXijBestLocalðbÞÞ:Q[rand
no
ð36Þ
r1¼2It ð2=MaxItÞð37Þ
r2¼ð2pi) rand (0,1) ð38Þ
where It is the present iteration, MaxIt denotes the
limit number of iterations, and rand is an arbitrary
number in the range (0,1) [44]. The overall search
process of the proposed MFFA is presented
graphically in Fig. 2.
Start
Calculate initial values (number
of sections and algorithm
parameters) with Eq. (18)
Generate initial
population with Eq. (19)
Calculate soil quality in
each part of the farmland
with Eqs. (21) & (22)
Update Local Memory and
Global Memory with Eqs.
(23) & (24)
Changes on the considered
sections by using all available
solutions in the section with Eqs.
(27) & (28)
Changes on the considered
sections by using best available
solutions in external memory with
Eqs. (25) & (26)
Evaluate all new trial designs and
replace previous solutions of
lower quality
Changes of solution based on the
best existing solutions in the local
memory with Eqs. (29) & (30)
Changes of solution based on the
best existing solutions in the global
memory with Eqs. (29) & (30)
Evaluate all new trial designs
and replace previous solutions of
lower quality
End
Q>= rand
Convergence
criterion satisfied?
YesNo
Yes
No
(a)
Fig. 2 Flowchart for athe FFA,
bthe proposed MFFA
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4 Validation of MFFA
4.1 Validation using test functions
In this section, the accuracy and reliability of the proposed
MFFA has been validated through the application on ten
different standard benchmark functions. The simulation
procedure has been carried out using MATLAB program.
The MFFA algorithm was implemented in MATLAB. The
optimization runs were performed on a laptop with a
2.4 GHz Intel Core i3 -M370 CPU and 4.00 GB RAM. The
obtained results by the application of MFFA have been
compared with those obtained from the conventional FFA.
Table 1shows the used benchmark functions for the
Start
Calculate initial values (number
of sections and algorithm
parameters) with Eq. (18)
Generate initial
population with Eq. (19)
Calculate soil quality in
each part of the farmland
with Eqs. (20) & (22)
Update Local Memory and
Global Memory with Eqs.
(23) & (24)
Changes on the considered
sections by using all available
solutions in the section with Eqs.
(28), (32), (33), (34) & (35)
Changes on the considered
sections by using best available
solutions in external memory with
Eqs. (26), (31), (33), (34) & (35)
Evaluate all new trial designs and
replace previous solutions of
lower quality
Changes of solution based on the
best existing solutions in the local
memory with Eqs. (30), (36),
(37), &(38)
Changes of solution based on the
best existing solutions in the
global memory with Eqs. (30),
(36), (37), &(38)
Evaluate all new trial designs
and replace previous solutions of
lower quality
End
Q>= rand
Convergence
criterion satisfied?
YesNo
Yes
No
(b)
Fig. 2 continued
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123

validation of the proposed MFFA. The values of MFFA
parameters are adjusted as follows: w=1, Q= 0.6 and
b= 0.8, a= 0.9, the maximum number of iterations is set
equal to 100; the number of sections is 2 each of which
includes 50 designs. The values of w,Q,b, and aare taken
as reported in the conventional FFA algorithm [42],
therefore a sensitivity analysis have not been performed in
the present study. Other control parameters for both FFA
Table 1 The used benchmark functions for the MFFA validation
Function no Function name Formulation Range
F01 Sphere fðxÞ¼P
d
i¼1
x2
i
[2100 100]
F02 Bohachevsky fðxÞ¼x2
1þ2x2
20:3 cosð3px1þ3px2Þþ0:3[2100 100]
F03 Matyas fðxÞ¼0:26ðx2
1þx2
1Þ0:48x1x2[210 10]
F04 Camel fðxÞ¼4x2
12:1x4
1þ1
3x6
1þx1x24x2
2þ4x4
2[255]
F05 Rastrigin fðxÞ¼10dþP
d
i¼1
x2
i10 cosð2pxiÞ[25.12 5.12]
F06 Zakharov fðxÞ¼P
d
i¼1
x2
iþð
P
d
i¼1
0:5ixiÞ2þð
P
d
i¼1
0:5ixiÞ4[25 10]
F07 Rosenbrock fðxÞ¼P
d1
i¼1
100ðxiþ1x2
iÞ2þðxi1Þ
hi [210 10]
F08 Ackley
fðxÞ¼20 exp 0:2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
dX
d
i¼1
x2
i
v
u
u
t
0
@1
Aexp 1
dX
d
i¼1
cosð2pxiÞ
!
þ20 þexpð1Þ
[232 32]
F09 Rotated hyper ellipsoid fðxÞ¼P
d
i¼1P
i
i¼1
x2
i
[265 65]
F10 Dixon-price fðxÞ¼ðxi1Þ2þP
d
i¼2
ð2x2
ixi1Þ2[210 10]
Table 2 Statistical analysis of
the MFFA and FFA
optimization results for the ten
different benchmark functions
Function Algorithm Min Max Mean SD Median
F01 MFFA 2.04857e-05 5.84043 0.36094 96.14431 0.01323
FFA 121.55465 470.44426 297.71958 6883.55051 290.88327
F02 MFFA 3.91989e-05 7.32211 0.58612 151.43442 0.01952
FFA 189.65473 494.13335 309.74001 7060.27000 304.29073
F03 MFFA 2.09775e-07 0.02625 0.00181 0.50249 7.12299e-05
FFA 1.57946 4.85930 3.24225 72.38020 3.27665
F04 MFFA 3.35859e-08 0.00474 0.00051 0.09723 4.69233e-05
FFA 0.38089 1.19601 0.75926 18.48263 0.72000
F05 MFFA 7.86557e-08 0.01865 0.00113 0.28779 8.29815
FFA 0.53350 1.34280 0.85278 20.97061 0.82383
F06 MFFA 1.71118e-06 0.59071 0.05506 11.69888 0.00909
FFA 0.95141 3.34174 2.00285 47.80885 1.92781
F07 MFFA 1.51261e-07 0.02678 0.00222 0.59250 0.00011
FFA 1.49826 5.47930 3.44832 86.53964 3.49250
F08 MFFA 1.59482e-06 0.34888 0.02712 6.14196 0.00301
FFA 14.43323 50.43123 31.791794 831.30395 30.88348
F09 MFFA 4.58577e-06 1.75140 0.08239 26.47897 0.00271
FFA 67.16994 195.86020 125.65062 2865.27019 128.22838
F10 MFFA 1.98513e-07 0.07565 0.00421 1.19399 0.00024
FFA 1.85292 4.90931 3.2451 70.07138 3.11935
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Fig. 3 The comparison results
of best cost for the MFFA and
FFA on the used benchmark
functions: aF01, bF02, cF03,
dF04, eF05, fF06, gF07,
hF08, iF09, jF10
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and MFFA are adjusted on trial and error basis over several
independent runs and the performance of the algorithms is
observed until fine results are generated. The adjusting
parameters for both FFA and MFFA are kept constant for
all studied cases. The value of the optimization programs
have been implemented 50 individual runs and the obtained
results are listed. Table 2provides the statistical analysis of
the MFFA and FFA optimization results for the ten dif-
ferent benchmark functions. The statistical results of the
validation are based on the best, worst, mean, and standard
deviation (SD) for the cost function. From Table 2,itis
clearly noticed that the MFFA gives the minimum values
of the SD compared with FFA, which proves the high
accuracy and reliability of MFFA in solving these
functions.
Figure 3shows the convergence curves for the used ten
standard test functions using MFFA and FFA. It can be
observed that the MFFA gives better and fast solution in all
used test functions compared with FFA. According to the
above validation, it can be confirmed that the proposed
MFFA has high stability and robustness in solving different
complex optimization problems.
5 Validation using mechanical engineering
design problems
For more validation of the proposed modification, the
MFFA has been utilized for solving the design optimiza-
tion problem for different mechanical systems, namely
Welded Beam Design (WBD), Compression Spring Design
(CSD), and Pressure Vessel Design (PVD). In principle, the
WBD is an optimization problem of the manufacturing cost
[45]. The design problem is described graphicaly with the
help of Fig. 4. The optimization problem includes four
parameters that have to be optimaly obtained. These
parameters are: the length (l), the height (t), the thickness
(b) and weld thickness (h). Mathematical explanation of the
optimization problem is reported in [45]. The results of the
optimization problem for WBD obtained using MFFA and
conventional FFA optimization techniques are listed in
Table 3. The optimization programs have been imple-
mented 50 individual runs and the obtained results are
reported. The best, worst, mean, and standard deviation
(SD) for the objective function are listed in the table. In
addition, the obtained values for the design parameters for
the best case are reported. The convergence curves for
MFFA and FFA are shown in Fig. 5. It is clearly noticed
that the proposed MFFA gives the best results compared
with the conventional FFA.
The compression spring design problem is described in
Fig. 6. The objective function in the CSD optimization
problem is to minimize the weight of the spring [46–48].
The diameter of the wire (d), coil diameter (D) and the
number of coils in the spring (N) are the three parameters
that have to be optimally determined. The mathematical
representation of the optimization problem, constraints,
and objective function are reported in [47]. The results of
statistical measure based on the values of the objective
function obtained over the 50 runs and the values of the
t
b
h
l
L
Buckling
Load
Fig. 4 Welded beam design problem
Table 3 Optimization results of WBD problem based on different optimization algorithms
Algorithm hltb Best Worst Mean SD Median
MFFA 0.205392 3.261251 9.036199 0.2057542 1.6958249 2.060094 1.801081 9.160092 1.779142
FFA 0.203712 3.322194 9.056177 0.2064882 1.7066521 2.155681 1.850743 9.5583599 1.824364
Fig. 5 Convergence curves of MFFA and FFA for WBD problem
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design parameters for the best case are listed in Table 4.
The convergence curves for both MFFA and FFA algo-
rithms for the best case are shown in Fig. 7.
The pressure vessel design problem is described in
Fig. 8, where the parameters of the optimization are four,
namely the shell thickness (T
s
), head thickness (T
h
), vessel
length (L), and radius (R). The mathematical expression of
the optimization problem is described in [49]. The statis-
tical measures for both MFFA and FFA are listed in
Table 5. In addition, the values of the optimized parameters
for the best case are also reported in the table. The con-
vergence characteristics of the proposed algorithms for the
PVD problem are shown in Fig. 9.
As the proposed modification proved its superiority over
the conventional FFA algorithm in reaching the minimum
value of the objective function and improving the shape of
the convergence curve, in the next section the proposed
MFFA will be applied for solving the optimization problem
of extracting the unknown parameters of proton exchange
membrane fuel cells.
D
N
N
Fig. 6 Compression spring
design problem
Table 4 Optimization results of CSD problem based on different optimization algorithms
Algorithm DDNBest Worst Mean SD Median
MFFA 0.05172789 0.3577067 11.22866 0.01266349 0.01271748 0.01270933 0.001607903 0.0127169
FFA 0.05429324 1.300000 9.675990 0.01266571 0.01425211 0.01279099 0.0236840 0.0127206
Fig. 7 Convergence curves of MFFA and FFA for WBD problem
R
Ts
L
Th
R
Fig. 8 Pressure vessel design problem
Table 5 Optimization results of PVD problem based on different optimization algorithms
Algorithm T
s
T
h
RLBest Worst Mean SD Median
MFFA 1.09378 0 65.225439 10.00034 2302.54733 3625.84614 2634.2574 54,837.3332 2302.63404
FFA 0 0 64.908593 200 2302.60038 2315.94947 2304.1655 214.171194 2303.56132
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6 Results and discussion
To validate the stability and viability of MFFA in
extracting unknown parameters mentioned in the mathe-
matical model of PEMFC stack, four commercial PEMFC
stacks were considered in this study [20–22]; namely,
250 W PEMFC stack, BCS-500 W FC, SR-12-500 W FC
and Temasek 1 kW FC stack. The lower and upper bounds
of these unknown parameters are given in Table 6[19,22].
The datasheet characteristics of the proposed PEMFC
stacks considered in this study are summarized in Table 7.
In order to account for the stochastic nature of the proposed
optimization technique, 50 independent optimization runs
starting from randomly generated initial populations were
performed. The optimal solution corresponds to the mini-
mum value of the cost function SSE (Eq. (16)) over the 50
independent. Moreover, to ensure the validity of the pro-
posed MFFA, the dynamic response of the studied PEMFC
stacks was verified. The voltage versus current (I–V) and
the power versus current (I–P) polarization curves obtained
by the optimized parameters are compared with their
counterparts given in the datasheet of each PEMFC stack.
In each case study, the results obtained by the application
of the proposed MFFA technique are compared with the
results provided in the literature.
6.1 Application of MFFA for extraction PEMFC
parameters
The MFFA was used for extracting the optimal values of
the unknown parameters of the PEMFC according to the
search boundaries listed in Table 6. Figure 10 compares the
cost function (SSE) convergence trends for the best opti-
mization runs of the developed MFFA and the conven-
tional FFA for all PEMFC stacks identification problems.
From Fig. 10a for 250 W PEMFC stack, it is clearly
noticed that in the case of MFFA, the optimal value of the
cost function is obtained after 20 iterations, while in the
case of FFA, the best value of the objective function is
reached after 91 iterations. Similarly, for all other PEMFCs
stacks, the MFFA reaches the minimum cost function after
few numbers of optimization iterations compared to FFA.
Also, it can be seen that MFFA minimized the SSE cost
function within only 37 iterations for the BCS 500 W
stack, 18 iterations for the SR-12 500 W stack, and 58
iterations for the Temasek 1 kW stack. As the PEMFC
optimization problem is offline, so the time elapsed until
reaching the optimal solution is not important. The main
contribution is that the proposed MFFA successes to reach
the minimum value of the cost function (SSE) for all types
of PEMFCs under consideration. It can be concluded that
MFFA smoothly advanced towards the optimum in all
identification problems.
In order to validate the reliability and stability of the
developed MFFA optimization technique in identifying the
unknown parameters of the PEMFC stacks, a statistical
analysis was carried out on the optimized values of SSE
Fig. 9 Convergence curves of MFFA and FFA for PVD problem
Table 6 Bounds of PEMFC
stacks unknown parameters Parameter n
1
n
2
n
3
n
4
kR
c
b
Lower bound -1.1997 0.001 3.6 910
–5
-2.6 910
–4
10 0.0001 0.0136
Upper bound -0.8532 0.005 9.8 910
–5
-9.54 910
–5
23 0.0008 0.5
Table 7 Design specifications
of PEMFC stacks under study PEMFC type 250 W stack BCS 500 W SR-12 PEM 500 W Temasek 1 kW
N(cells) 24 32 48 20
A(cm
2
) 27 64 62.5 150
l(lm) 127 178 25 51
J
max
(mA/cm
2
) 860 469 672 1500
P
H2
(atm) 1 1 1.47628 0.5
P
o2
(atm) 1 0.2095 0.2095 0.5
T(K) 343.15 333 323 323
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cost function obtained over the 50 independent runs. A
parametric and nonparametric statistical study was per-
formed for the four fuel cell stacks considered in this study.
The following metrics were analyzed in order to assess the
effectiveness of the proposed MFFA algorithm: the best
value of the objective function, the worst value of the
objective function, mean value of SSE, median, standard
deviation (SD), relative error of the objective function
Fig. 10 Comparison of best-run convergence curves of MMFA and FFA for the: a250 W PEMFC stack, bBCS-500 W PEMFC stack, cSR-12
500 W PEMFC stack, dTemasek 1 kW PEMFC stack
Table 8 Statistical analysis of
the FFA optimization results for
the different PEMFC stacks
250 W stack SR-12 500w stack BCS 500 W stack Temasek 1 kW stack
Min 0.64499 1.05813 0.01335 0.79256
Max 0.99607 1.12655 0.13828 0.88132
Mean 0.73087 1.08204 0.03881 0.80981
Median 0.71276 1.07571 0.02958 0.80592
SD 6.06642 1.90882 2.78582 1.47419
RE 6.65811 1.13005 95.32894 1.08812
MAE 0.08589 0.02391 0.02546 0.01725
RMSE 0.1048 0.03048 0.03753 0.02259
Eff 88.77453 97.81929 47.86181 97.90064
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(RE), mean absolute error (MAE), and root mean square
error (RMSE). The mathematical formula of these metrics
can be represented as [21]:
SD ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P50
i¼1SSEiSSE

50 1
sð39Þ
RE ¼P50
i¼1SSEiSSEmin
ðÞ
SSEmin
ð40Þ
MAE ¼P50
i¼1SSEiSSEmin
ðÞ
50 ð41Þ
RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P50
i¼1SSEiSSEmin
ðÞ
2
50
sð42Þ
Eff:¼1
50 X
50
i¼1
SSEmin
SSEi
100 ð43Þ
where SSE
i
represents the objective function value after
completing each individual run, SSE
min
denotes the mini-
mum value of objective function overall. SSE is the mean
value of objective function over the 50 runs of optimization
program. The values of the metrics under consideration for
the different PEMFC stacks are summarized in Tables 8
and 9for the FFA and MFFA, respectively. It can be
admitted that the small values of MAE and RMSE validate
that the computed data match well with that provided in the
measurements. The proposed MFFA clearly outperformed
standard FFA because it achieves lower errors in all
identification problems. The simulation results of each
PEMFC stack will be presented briefly in the next
subsections.
(1) Test case 1
In this case study, the 250 W PEMFC stack was
characterized according to the specifications given in
the second column of Table 8. The datasheet
specifications provided the (I–Vcurve) polarization
characteristics. The measured data at (1/1 bar and
343.15 K) were utilized to extract the optimal values
of the model parameters. Table 10 shows the 250 W
PEMFC parameters that yield the minimum values of
Table 9 Statistical analysis of
the MFFA optimization results
for the different PEMFC stacks
E 250 W stack SR-12 500w stack BCS 500 W stack Temasek 1 kW stack
Min 0.64202 1.05677 0.01157 0.7910
Max 1.08608 1.08185 0.37161 0.90154
Mean 0.72676 1.06172 0.04103 0.79794
Median 0.70322 1.05979 0.01588 0.79284
SD 9.96261 0.53064 7.02396 1.61756
RE 6.59912 0.23396 127.3005 0.43855
MAE 0.08474 0.00494 0.02946 0.00694
RMSE 0.13003 0.00721 0.07552 0.01745
Eff 89.69824 99.53666 65.23262 99.16654
Table 10 Comparison of the 250 W PEMFC stack parameters identified by MFFA, FFA and other metaheuristic algorithms used in literature
n
1
n
2
910
–3
n
3
910
–5
n
4
910
–4
kbR
c
910
–4
SSE
MFFA -1.00183 2.61199 3.99591 -1.55872 23 0.05455 1.00000 0.64202
FFA -0.85320 2.8153 9.80000 -0.954000 18.48953 0.01360 8.0000 0.64499
AEO [21]-1.11090 4.26165 4.98499 -2.13929 19.39091 0.02434 4.78500 0.64295
HHO [24]-1.16200 4.01800 9.74537 -1.33452 16.68106 0.04784 7.00848 1.02274
TGA [33]-1.1914 4.1129 6.0573 -1.7090 18.689 0.0544 4.8527 0.7496
RGA [30]-1.1568 3.4243 6.4161 -1.1544 12.8989 0.0343 1.4504 8.4854
MPSO [22]-0.9479 3.0835 7.799 -1.8800 20.7624 0.0296 2.8666 9.7539
ARNA-GA [16]-0.9470 3.0586 7.6059 -1.8800 23.00 0.0329 1.1026 2.9518
JAYA-NM [17]-1.19966 3.55 6.00 -1.200 13.2287 0.03334 1.00 5.2513
HGA [9]-0.944957 3.01801 7.401 -1.880 23.00 0.02914489 1.00 4.8469
SGA [9]-0.9473 3.0641 7.7134 -1.9390 19.7650 0.0240 2.7197 5.6530
HADE [12]-0.8532 2.81009 8.09203 -1.2870 14.0448 0.0335374 1.00 7.9908
TLBO-DE [22]-0.853200 2.65052 8.001574 -1.360144 15.651416 0.0364609 1.00 7.2776677
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SSE cost function over the 50 independent runs of
MMFA and standard FFA. Moreover, MFFA and
FFA results are compared with those quoted in
literature for other metaheuristic algorithms such as
HGA and SGA [9], HADE [12], ARNA-GA [16],
JAYA-NM [17], TLBO-DE [22]. Remarkably, the
proposed MFFA algorithm obtained the lowest value
of SSE cost function overall followed by standard
FFA.
Figure 11 compares the measured I–Vand I–
Ppolarization curves for the 250 W PEMFC stack
with those predicted by the numerical model includ-
ing the values of parameters extracted by MFFA. It
can be seen that the numerical model reproduced
very well the experimental data, thus confirming that
MFFA minimized the error cost function SSE.
(2) Test case 2
In this case study, the other three FC stacks are used to
examine the effectiveness and stability of the proposed
MFFA-based PEMFC mathematical model. Table 11
compares the values of extracted parameters identified by
MFFA for the BCS 500 W PEMFC stack with those
identified by other algorithms such as GWO [29], SSO [19]
and CS-EO [20]. The table shows the SSE value for the
best optimization run. Similarly, the results of MFFA are
compared with those of GWO [29], SSO [19] and CS-EO
[20] for the SR-12 500 W PEMFC stack in Table 12.
Finally, Table 8compares the identified parameters for the
Temasek-1 kW PEMFC stack by the proposed MFFA,
GWO [29] and SSO [19]. Again, the proposed MFFA
algorithm was able to identify FC parameters that lead to
have the lowest SSE values overall. This confirms the
efficiency of the present approach. Curves obtained for the
three PEMFC stacks of this case study by giving in input to
the FC model the extracted parameter values listed in
Tables 11,12 and 13 are in very good agreement with the
experimentally measured data as it appears from Fig. 12.
Slightly higher deviations occur in the case of the Temasek
1 kW stack.
6.2 Simulation under different operating
scenarios
In this section, the values of the unknown parameters of the
PEMFC stacks identified in the previous sections are used
for evaluating the performance of the proposed model
under variations in the cell temperature and the partial
pressures of the reactants in the inlet channels. For the sake
of brevity, the I–V and I–P characteristics for two FC
Fig. 11 Comparison of estimated and measured voltage and power
polarization curves of the 250 W FC stack: (a) I–Vplot, (b) I–Pplot
Table 11 Comparison of the BCS-500 W PEMFC stack parameters identified by MFFA, FFA and other metaheuristic algorithms used in
literature
n
1
n
2
910
–3
n
3
910
–5
n
4
910
–4
kbR
c
910
–4
SSE
MFFA -1.05361 3.68756 8.75376 -1.93209 21.21654 0.01625 1.13256 0.01157
FFA -0.85320 3.71723 8.05285 -1.83520 22.45941 0.01416 8.0000 0.01335
MAEO [21]-0.85596 2.73328 6.634280 -1.92816 20.702572 0.016023 1.004526 0.01157
TGA [33]-0.970482 2.952169 5.9528506 -1.838608 22.50299 0.018229 3.8311999 0.083525
GWO [29]-1.0180 2.3151 5.240 -1.2815 18.8547 0.0136 7.503 7.1889
SSO [19]-1.018 2.3151 5.240 -1.2815 18.8547 0.0136 7.5036 7.1889
CS-EO [20]-1.1365 2.9254 3.7688 -1.3949 18.5446 0.013600 8.00 5.5604
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stacks will be examined under temperature variation while
the other two FCs will be tested under variation of oxygen
and hydrogen pressures. The I–V and I–P polarization
curves of the BCS 500 W PEMFC stack and 250 W
PEMFC stack at 303, 323, and 353 K are presented in
Fig. 13.
The effect of changing the pressures of the inlet gases
(P
H2
/P
O2
) and keeping the FC temperature at the values
indicated in the datasheet of each PEMFC type tested in
this study was analyzed. Figure 14a, b shows the I–Vand
I–Ppolarization curves of the SR-12 500 W PEMFC stacks
at pressures of (1/0.2075 bar), (1.5/1 bar) and (2.5/1.5 bar)
when the FC temperature was kept constant at 333 K.
Similarly, the I–V and I–P polarization curves of Temasek
1 kW PEMFC are shown in Fig. 14c, d, at the constant
temperature of 323 K and for reactants pressures of (1/
0.2075 bar), (1.5/1 bar) and (2.5/1.5 bar), respectively.
7 Conclusion
This study presented a modified farmland fertility opti-
mization algorithm (MFFA) developed in order to improve
the performance of conventional FFA technique. The
modified algorithm has been validated using mathematical
test benchmark functions and several mechanical engi-
neering design optimization problems. The proposed
MFFA provided a robust and reliable performance
regarding the original FFA algorithm. Therefore, the new
algorithm was used for identifying seven unknown
parameters of PEMFC models. The results obtained by
MFFA were comprehensively compared with those
obtained by different optimization techniques. Moreover,
parametric and nonparametric statistical analysis were
performed to show the effectiveness of MFFA in estimat-
ing the parameters of PEMFC. The obtained results show
Table 12 Comparison of the SR-12 500 W PEMFC stack parameters identified by MFFA, FFA and other metaheuristic algorithms used in
literature
n
1
n
2
910
–3
n
3
910
–5
n
4
910
–4
kbR
c
910
–4
SSE
MFFA -0.88835 3.23679 8.4475397 -0.95410 22.93903 0.17486 6.92348 1.05677
FFA -1.09383 3.09783 9.80000 -2.60000 21.89913 0.168339 8.0000 1.05813
MAEO [21]-0.86068 2.77134 6.19649 -0.954009 22.98870 0.175366 6.70732 1.05663
TGA [33]-1.112395 3.85466 4.36985 -0.96448 23 0.18307 2.188689 1.10408
GWO [29]-0.9664 2.2833 3.400 -0.954 15.7969 0.1804 6.6853 1.517
SSO [19]-0.9664 2.2833 3.400 -0.954 15.7969 0.1804 6.6853 1.517
CS-EO [20]-1.0353 3.3540 7.2428 -0.954 10 0.1471 7.1233 7.5753
Table 13 Comparison of the Temasek-1 kW PEMFC stack parameters identified by MFFA, FFA and other metaheuristic algorithms used in
literature
n
1
n
2
910
–3
n
3
910
–5
n
4
910
–4
kbR
c
910
–4
SSE
MFFA -0.90347 3.82665 8.47514 -2.29347 13.32511 0.07049 1.00051 0.7910
FFA -1.19970 8.0000 9.80000 -0.95400 13 0.50000 1.00000 0.79256
TGA [33]20.87218 2.52655 3.81895 22.42319 14.79207 0.066572 0.89434 0.79692
GWO [29]21.0299 2.4105 4.00 20.9540 10.0005 0.1274 1.0873 1.6481
SSO [19]21.0299 2.4105 1.00 20.9540 10.0005 0.1274 1.0873 1.6481
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that the MFFA is more efficient than conventional FFA and
other well-known optimization techniques, in extracting
PEFMC parameters in terms of the minimum values of
SSE and convergence characteristics. In future work, the
developed MFFA can be utilized to solve other optimiza-
tion problems. Moreover, it can be applied to simulate the
dynamic behavior of solid-oxide FC.
Fig. 12 Comparison of estimated and measured voltage and power polarization curves: aBCS 500 W stack I–V plot, bBCS 500 W stack I–
Pplot, cSR-12 500 W FC I–Vplot, dSR-12 500 W FC I–Pplot, eTemasek 1 kW stack I–V plot, fTemasek 1 kW stack I–Pplot
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Fig. 13 Characteristics of the FC stack under temperature variation: aI–Vcurves for BCS 500 W PEM, bI–Pcurves for BCS 500 W PEM, cI–
Vcurves for 250 W PEMFC, dI–Pcurves for 250 W PEMFC
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Fig. 14 Characteristics of the FC stack under pressure variation: aI–Vcurves for SR-12 500 W PEMFC, bI–Pcurves for SR-12 500 W PEMFC
PEM, cI–Vcurves for Temasek 1 kW PEMFC, dI–Pcurves for Temasek 1 kW PEMFC
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List of symbols FC: Fuel cell; PEMFC: Proton exchange membrane
fuel cell; SSE: Sum of squared errors; V
stack
: PEMFC stack voltage
(V); E
Nernst
: Nernst voltage of a single FC (V); N
cells
: Number of cells
in the stack; v
act
,v
ohm,
and v
con
: Activation, ohmic, and concentration
over potential, respectively (V); T: FC operating temperature (K); P
H2
and P
O2
: Hydrogen and oxygen partial pressures, respectively (atm);
f
1
,f
2
,f
3
, and f
4
: Parametric adjustable parameters for a particular FC;
I
fc
:PEMFC stack current (A); R
c
and R
M
:Resistance due to
concentration and transfer of proton, respectively (X); q
M
:
Specific resistance of the membrane (Xcm); l:Membrane thickness
(lm); A:Effective electrode area (cm
2
); k:Adjustable parameter
describes the water content in the membrane; b:Parametric
coefficient (V); Jand J
max
:Actual and maximum current density
of the FC stack, respectively (A cm
-
2); N: Population size; k:
Number of sections; n:Number of solutions available in each
section; maxiter:Maximum number of iterations; L
j
and U
j
:Lower
and upper limits of the design variable, respectively; Fit_Sections:
Average quality of the solutions in each section; M
local
and
M
Global
:Number of solutions in local and global memories,
respectively; t:Constant between 0.1 and 1; h:Decimal number; a:A
number in the range of (0,1); X
MGlobal
:A randomly selected solution
from the global memory; X
new
:A new solution that obtained by
applied changes; b:An arbitrary number in the range of (0,1);
Best
Local
:The best available solution in the local memory;
Best
Global
:The best solution ever found; Q:A parameter between
(0,1); w
1
:A parameter of the FFA reduced as the optimization
process progresses; R
v
:A constant number between 0 to 1; rr
1
and rr
2
:Two randomly distributed numbers between [0,1]; MaxIt:
Maximum number of iterations; rand:A random number
between (0,1)
Acknowledgment The authors thank the support of the National
Research and Development Agency of Chile (ANID), ANID/Fondap/
15110019.
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