Ci a ion: Silaa, M.Y.; Ba ambones, O.;
De beli, M.; Napole, C.; Benche i , A.
F ac ional O de PID Design o a
P o on Exchange Memb ane Fuel
Cell Sys em Using an Ex ended G ey
Wol Op imize . P ocesses 2022,10,
450. h ps://doi.o g/10.3390/
p 10030450
Academic Edi o s: Giosue Giacoppo
and B uno Au i y
Recei ed: 31 Janua y 2022
Accep ed: 18 Feb ua y 2022
Published: 23 Feb ua y 2022
Publishe ’s No e: MDPI s ays neu al
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Copy igh : © 2022 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
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A ibu ion (CC BY) license (h ps://
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p ocesses
A icle
F ac ional O de PID Design o a P o on Exchange Memb ane
Fuel Cell Sys em Using an Ex ended G ey Wol Op imize
Mohammed Yous i Silaa 1,* , Osca Ba ambones 1,* , Mohamed De beli 1, C is ian Napole 1,*
and Aissa Benche i 2
1Enginee ing School o Vi o ia, Uni e si y o he Basque Coun y UPV/EHU, Nie es Cano 12,
01006 Vi o ia, Spain; [email p o ec ed]
2Telecommunica ions Signals and Sys ems Labo a o y (TSS), Ama Telidji Uni e si y o Laghoua , BP 37G,
Laghoua 03000, Alge ia; a.benche i @lagh-uni .dz
*Co espondence: [email p o ec ed] (M.Y.S.); osca [email p o ec ed] (O.B.);
[email p o ec ed] (C.N.)
Abs ac :
This pape p esen s a compa ison o op imize s o uning a ac ional-o de p opo ional-
in eg al-de i a i e (FOPID) and p opo ional-in eg al-de i a i e (PID) con olle s, which we e
applied o a DC/DC boos con e e . G ey wol op imize (GWO) and ex ended g ey wol op-
imize (EGWO) ha e been chosen o achie e sui able pa ame e s. This s a egy aims o imp o e and
op imize a p o on exchange memb ane uel cell (PEMFC) ou pu powe quali y h ough i s link wi h
he boos con e e . The model and con olle s ha e been implemen ed in a MATLAB/SIMULINK en-
i onmen . This s udy has been conduc ed o compa e he e ec i eness o he p oposed con olle s in
he ansien , accu acy in acking he e e ence cu en , s eady-s a e, dynamic esponses, o e shoo s,
and esponse ime. Resul s showed ha he combina ion EGWO-FOPID had signi ican ad an ages
o e he es o he op imized con olle s.
Keywo ds:
ex ended g ey wol op imize ; g ey wol op imize ; ac ional o de p opo ional in eg al
de i a i e; p o on exchange memb ane uel cell; DC/DC boos con e e
1. In oduc ion
Nowadays, ene gy esea ch mainly co e s wo issues. The i s one is linked o he
isk o deple ion o ossil and issile esou ces; he o he is en i onmen al. The sou ces
used oday a e wi h limi ed ese es, o bo h ossil uels (hyd oca bons, coal, e c.) and
issile uels (u anium). The use o hese ene gy sou ces also gene a es undesi able side
e ec s: emission o g eenhouse gases in he case o hyd oca bons and p oduc ion o was e
ha is di icul o ea in he case o nuclea powe [
1
,
2
]. Faced wi h he dec ease in
con en ional ene gy esou ces, i has become essen ial o ind ene gy al e na i es wi h
he same p ope ies as hyd oca bons in e ms o s o age and anspo [
3
]. In his con ex ,
hyd ogen u ns ou o be an ea nes candida e, e en i i is only an ene gy ca ie and no a
p ima y esou ce. Hyd ogen, which does no exis na u ally, can indeed be syn hesized
h ough enewable ene gies. In addi ion, i s unc ion as an ene gy ca ie , i s s o abili y can
be exploi ed o mee he equi emen s se by ene gy consume s [
4
]. The uel cells eme ge
as he missing link by ans o ming chemical ene gy in o elec ical ene gy wi h high
e iciency [
5
]. Hence, hey use he chemical ene gy o hyd ogen and oxygen o gene a e
elec ici y wi hou pollu ion, and he o he p oduc s a e jus plain wa e , and hea [
6
].
Scien is s ha e al eady de eloped di e en ypes o uel cells, cha ac e ized by he na u e
o he gases and elec oly es used, he eby de e mining hei ope a ing cha ac e is ics. O
all he exis ing amilies o uel cells, he p o on exchange memb ane uel cell (PEMFC)
achie ed he mos a en ion om he esea che s, which is conside ed he bes app op ia e
o he au omo i e sec o [
7
] and nume ous ields [
8
–
10
]. The s ong poin s o his uel
cells ype a e he ela i ely as dynamic compa ed o o he powe gene a o s and low
P ocesses 2022,10, 450. h ps://doi.o g/10.3390/p 10030450 h ps://www.mdpi.com/jou nal/p ocesses
P ocesses 2022,10, 450 2 o 17
ope a ing empe a u e, om 40
◦
C o 100
◦
C, which acili a es i s in eg a ion in a ehicle
wi hou speci ic he mal insula ion [
11
,
12
]. As is common, hyd ogen cells a e nonlinea
sys ems which a e a ec ed by mu able ac o s such as gases p essu es and luc ua ions
o empe a u es which e en ually, e lec he ou pu powe . Consequen ly, o ensu e an
e icien powe con e sion om he PEMFC o he ex e nal ci cui , an adap a ion elemen
is equi ed, and his can be done by inse ing an elec onic de ice be ween he powe
gene a o and he elec ical load. This de ice is a s a ic DC/DC con e e equipped wi h an
insula ed ga e bipola ansis o (IGBT) o me al oxide semiconduc o ield e ec ansis o
(MOSFET) con olled by a command law [
13
]. This candida e connec ion is one o he
mos widely used powe elec onics ci cui s hanks o i s high con e sion e iciency and
adjus able ou pu ol age [
14
]. These DC/DC con e e s a e elec onic de ices designed o
egula e he ou pu ol age agains he inpu ol age and load cu en changes h ough
he con ol o pulse wid h modula ion PWM o he swi ch. This leads o he equi emen
o mo e ad anced con ol me hods o mee ac ual demand. Many con ol me hods a e
de eloped in he li e a u e o con ol DC-DC con e e s. Fo ins ance, au ho s o [
15
]
applied a con olle ype p opo ional in eg al (PI) based on Ziegle –Nichols (ZN) uning
me hod o a DC/DC boos con e e in o de o s abilize he PEMFC ou pu cu en . The
p oposed con olle gua an ees be e pe o mance in e ms o ising ime, se ling ime,
s eady-s a e e o , and obus ness e en wi h la ge load a ia ions. Howe e , due o he
ob ained esul s, sha ps unde shoo o 3
A
and o e shoo o 8
A
appea s, which esul s
om he ob ained ZN me hod agg essi e pa ame e s. The au ho s o [
16
] implemen ed
wo di e en con en ional con ol based on PI and PID in o de o op imize he DC-DC
buck con e e pe o mance. The con ol scheme was based on he ZN uning me hod
and gene ic algo i hm (GA). Simula ion esul s showed ha he PI and PID con olle s
using he GA ga e sa is ac o y esul s in e ms o ising ime, s eady-s a e e o , se ling
ime, low o e shoo and low unde shoo be e han he p o ided by he con en ional
ZN uning me hod. The au ho s o [
17
] implemen ed he GWO uning o PID con olle
o DC/DC boos con e e unde a GA-PID and PSO-PID. Simula ion esul s showed
ha he p oposed GWO-PID has a low oo mean squa ed e o (RMSE) compa ed o he
o he algo i hms. The au ho s o [
18
] con olled a DC/DC con e e ype buck based
on PID combined wi h sliding mod (PID-SMC) in compa ison o con en ional sliding
mode. The ob ained simula ion esul s showed ha he p oposed con olle is be e han
he con en ional SMC con olle in e ms o dynamic, s a ic pe o mance, and s ong
obus ness unde he pe iodically and i egula ly load esis ances. The au ho s o [
19
]
applied a backs epping app oach o a DC/DC boos con e e in o de o keep he PEMFC
powe sys em wo k a an op imum powe poin . The compa ison agains he PI showed
ha he backs epping app oach gi es as and su icien con e ging o he ope a ing powe
poin . The au ho s o [
20
] applied a o al sliding-mode con ol (TSMC) o he ol age
con ol o a DC/DC boos con e e . Simula ion esul s p o ed ha he TSMC ha e low
ansien esponse ime and high obus ness in compa ison wi h he con en ional PI con ol
and he SMC. The au ho s o [
21
] applied a quasi con inuous high o de sliding mode
con olle (QC-HOSM) o a DC/DC boos con e e linked o PEMFC in o de o educe
he cha e ing e ec s o he con en ional sliding mode. Expe imen al esul s showed ha
he p oposed con ol echnique can achie e a cha e ing educ ion up o 84%. The au ho s
o [
22
] applied a obus in eg al as e minal sliding mode combined wi h digi al il e o a
DC/DC boos con e e in o de o educe he unwan ed oscilla ion o imp o e he ou pu
powe quali y o he PEMFC. Expe imen al esul s showed ha he p oposed con olle
has signi ican ad an ages in e m o ising ime, obus ness and a cha e ing educ ion
up o 91% could be achie ed. Wi h espec o s a e o he a , he main con ibu ion o
his pape is he design and implemen a ion o a ac ional o de p opo ional-in eg al-
de i a i e op imized by an ex ended g ey wol op imize (EGWO-FOPID) o enhancing
he pe o mance o he PEM uel cell sys em. Compa ison s udy wi h p opo ional-in eg al-
de i a i e op imized by g ey wol op imize (GWO-PID), ac ional o de p opo ional-
in eg al-de i a i e op imized by g ey wol op imize (GWO-FOPID), and p opo ional-
P ocesses 2022,10, 450 3 o 17
in eg al-de i a i e op imized by an ex ended g ey wol op imize (EGWO-PID), has been
ca ied ou in MATLAB/Simulink o alida e he ad an ages o he p oposed algo i hm.
Compa ison esul s ha e demons a ed ha he p oposed con olle can s abilize he
PEMFC powe sys em o e he en i e ope a ing condi ions and e en in he p esence o
signi ican load a ia ions. I has also been demons a ed ha he p oposed con olle
main ains he sys em’s obus ness and p o ides be e accu acy o e he o he con olle s.
The emainde o he pape is o ganized as ollows. In Sec ion 2, we discuss he
uel-cell ype p o on exchange memb ane, as well as he ma hema ical equa ions ela ed
o i s wo k ha show he pe o mance o he cell. Sec ion 3is de o ed o he con ol
me hodology design o he op imiza ion o he PEMFC powe sys em. Sec ion 4 ocuses
on he simula ion esul s.
2. PEM Fuel Cell Modeling
As shown in Figu e 1. A PEM uel cell is a gene a o o elec ical ene gy. I di ec ly
con e s he chemical ene gy o he uel (hyd ogen) in o elec ical ene gy using he ca a-
lys [
23
–
25
]. I is a sys em ha p oduces no pollu ion and i ually no noise since i does
no ha e any mo ing mechanical componen s, such as u bines and mo o s. In addi ion, an
elec ic cu en is p oduced as long as he cell is join ly supplied wi h uel (hyd ogen) and
oxidize (oxygen in he ai ) [
25
]. Tha is wha di e en ia es i om di e en powe gene a-
o s and o he cells. The chemical eac ion a he le el o he PEMFC can be ep esen ed in
he ollowing Equa ions (1)–(3) [26,27].
Anode: 2H2=⇒4H++ 4e−(1)
Ca hode: 4H++O2+ 4e−=⇒2H2O(2)
O e all eac ion: 2H2+O2=⇒2H2O+ Ene gy + Hea (3)
Figu e 1. C oss sec ion o a single PEMFC.
A single PEM uel cell ol age
VFC
is he sum o ou e ms: he no-load ol age
ENe
,
he ac i a ion o e ol age
Vac
(o ac i a ion d op), he ohmic o e ol age
Vohm
(o ohmic
d op) and he o e ol age concen a ion
Vcon
(o d op in concen a ion), which a e de ined
by he ollowing exp ession [28]:
VFC =ENe −Vac −Vohm −Vconc (4)
2.1. Ne ns Po en ial
The chemical ene gy eleased can be calcula ed by he change in Gibbs ee ene gy
(
4g
), which is he di e ence be ween he ene gy o he p oduc s and he ene gy o
he eac an s. In he case o he PEMFC he a ia ion o his ee ene gy is gi en in
Equa ion (5) [29,30]:
P ocesses 2022,10, 450 4 o 17
4g = (g )p oduc s −(g ) eac an s = (g )2H2O−(g )2H2−(g )O2(5)
The a ia ion o Gibbs ee ene gy depends on empe a u e and p essu e as gi en in
Equa ion (6) [31]:
4g =4g0
−RTln"PH2P1
2O2
PH2O#(6)
whe e
4g
is he a ia ion o Gibbs ee ene gy a s anda d condi ions p essu e 1 (ba ),
which depends on he empe a u e
T
exp essed in Kel in.
PH2
,
PO2
and
PH2O
a e he
p essu es o hyd ogen, oxygen and wa e apo , espec i ely.
R
is he uni e sal gas
cons an (8.31451 J
·
kg
−1·
K
−1
). Fo e e y hyd ogen mole, wo elec ons pass by he ex e nal
elec ical ci cui , and he elec ical wo k is equal o he change in Gibbs ee ene gy i he
sys em has no lossless, he elec ical wo k pe o med is gi en in Equa ion (7) [32]:
4g =nFE (7)
whe e
F
is Fa aday’s cons an (96,485 Coulombs/mole), which ep esen s he elec ic cha ge
o an elec on mole.
n
co esponds o he numbe o moles o elec ons in he eac ion.
E
is
he open ci cui ol age o he PEMFC. The PEMFC open ci cui ol age can he e o e be
exp essed as Equa ion (8) [31]:
ENe =−4g
2F=−4g0
2F+RT
2Fln"PH2P1
2O2
PH2O#(8)
In p ac ice, he ope a ion o PEMFC is accompanied by losses, pa o he chemical
ene gy is con e ed in o hea . The e m
−4g
2F
a ies depending on he ope a ing poin . I
is equal o 1.229 ol s a he s anda d s a e (25
◦
C) and 1 ba . We can exp ess he ension
E
in he o m [33]:
ENe =1.299 −0.85 ·10−3·(T−298.15) + 4.3085 ·10−5Tln(PH2) + 1
2·ln(PO2)(9)
2.2. The Ac i a ion Pola iza ion
The ac i a ion losses occu due o he kine ics o he eac ions aking place a he
elec ode. They can be calcula ed using Equa ion (10) [34].
Vac =ζ1+ζ2·T+ζ3·T·ln(CO2) + ζ4·T·ln(I)(10)
whe e he pa ame e s
ζ1
,
ζ2
,
ζ3
and
ζ4
a e pa ame ic coe icien s de e mined by he
cons uc o ,
I
is he cu en o he PEMFC, and
CO2
is he oxygen concen a ion in he
ca alys s (mol·cm−3) and i could be calcula ed using Equa ion (11) [34,35].
CO2=PO2
5.08 ·106·e(−498
T)(11)
2.3. The Ohmic Losses
The ohmic losses occu due o he elec ical esis ance o he di e en elemen s o
he PEMFC. They ha e wo o igins: he in e nal esis ance o he elec oly e memb ane
Rmem
and he esis ance ha occu ed due o he con ac be ween he bipola pla es and he
ca bon elec odes Rcon. These losses can be calcula ed using Equa ion (12) [34]:
Vohm =I·(Rmem +Rcon)(12)
P ocesses 2022,10, 450 5 o 17
whe e
Rmem =σmem ·l
A(13)
The pa ame e
σmem
is he speci ic esis ance o he memb ane (
Ω·
cm),
A
is he single
cell ac i e su ace in cm
2
,
l
is he memb ane hickness in (cm). The ollowing exp ession
o he speci ic esis ance is used [34,36]:
σmem =181.6[1+0.03(I
A) + 0.062(T
303)2·(I
A)2.5]
[δ−0.634 −3(I
A)] ·exp [4.18(T−303)/T](14)
The pa ame e
δ
is an amenable pa ame e wi h a maximum alue o 23. This pa ame-
e depends on he memb ane ab ica ion p ocess and is a unc ion o he ela i e humidi y
and he s oichiome ic a e o he inle hyd ogen gas p essu es he anode. Unde ideal
humidi y condi ions (100%), his pa ame e may ha e a alue anging om 14 o 20.
2.4. The Concen a ion Pola iza ion
The concen a ion losses a e caused by he a ia ion in he concen a ion o eac-
an s. These losses can be calcula ed using Equa ion (15) [
34
,
36
]; whe e
ψ
,
J
and
Jmax
a e,
espec i ely, a cons an pa ame e , he cu en densi y and he maximum cu en densi y.
Vcon =ψ·ln1−J
Jmax (15)
2.5. PEM Fuel Cell S ack Ou pu Powe
The ou pu ol age unde he load is app oxima ely 0.6–0.7 V [
37
,
38
]. The e o e, i
is necessa y o ha e cells in se ies, which inally o m a “s ack” o achie e he su icien
ol age and he amoun o powe needed. The powe gene a ed by he PEMFC s ack
can be calcula ed using Equa ion (9); whe e
Nc
ep esen s he numbe o cells used in he
s ack [36].
Ps ack =VFC ·I·Nc(16)
The da a and cha ac e is ics o he PEMFC conside ed in he simula ion a e shown
in Table 1.
Table 1. The PEMFC model pa ame e s.
Pa ame e Value
A162 cm2
β23
l175 ×10−6cm
ψ0.1 V
Rc0.0003
Jmax 0.062 A·cm−1
Nc10
ζ10.9514 V
ζ2−0.00312 V/K
ζ3−7.4 ×10−5V/K
ζ41.87 ×10−4V/K
3. Con ol Design Me hodology
The ol age deli e ed by he PEMFC is con inuous and o low ampli ude. In o de o
aise i in o a highe alue, a s ep-up con e e is used. In gene al, he s ep-up con e e
is he easies way o inc ease he ol age o a DC powe supply, and p omises high e i-
ciency [
39
]. This sec ion de e mines he con e e s uc u e adop ed and p esen s some
exis ing con ol echniques ha allow he PEMFC o ope a e a an adequa e powe poin .
As shown in Figu e 2, he closed loop consis s o a PEMFC powe sys em, a DC/DC boos
con e e , a con ol echnique and inally a load.
P ocesses 2022,10, 450 6 o 17
Figu e 2. P inciple o indi ec adap a ion wi h con ol echnique.
3.1. Boos Con e e S a e Space Modeling
Supposed ha he boos con e e ope a es in he con inuous conduc ion mode
(CCM) [
40
] which includes wo sequences depending on whe he he con ollable swi ch is
closed o open as shown in Figu e 3. In o de o model he con e e , one applies he laws
o Ki chho o he elec ic ci cui s cha ac e izing he wo ope a ing sequences [41,42].
Fi s sequence is cha ac e ized by
u=
1, he swi ch closed and he diode open. The
equa ions which go e n he con e e a e gi en by:
diL
d =1
L(VFC)
dVo
d =1
RC (−Vo)
(17)
I we se x= [x1,x2]T= [iL,Vo]T, hen he exp ession (17) can be w i en:
"˙
x1
˙
x2#=0 0
0−1
RC .x1
x2+1
L
0VFC (18)
The second ope a ing sequence is cha ac e ized by
u=
0, he swi ch open and he
diode closed. The sys em o equa ions which go e ns he con e e in he “o ” s a e is
p esen ed below:
diL
d =1
L(VFC −Vo)
dVo
d =1
C(iL−io)
(19)
I we se x= [x1,x2]T= [iL,Vo]T, hen he exp ession (19) can be w i en:
"˙
x1
˙
x2#=0−1
L
1
C−1
RC .x1
x2+1
L
0VFC (20)
In s a e space desc ip ion, i he s a e equa ions o wo modes a e desc ibed as ollowing [
41
]:
˙
x=A1x+B1u(Swi ch“1”)
˙
x=A2x+B2u(Swi ch“0”)(21)
P ocesses 2022,10, 450 7 o 17
Then he a e age s a e space model is gi en by:
˙
x=¯
Ax +¯
Bu (22)
whe e, ¯
A=A1d+A2(1−d)and ¯
B=B1d+B2(1−d)
A e aging he s a e space ma ix o wo di e en wo king modes using
Equa ions (18), (20)–(22), we ge he a e age model as a unc ion o he du y cycle [41,43].
"˙
x1
˙
x2#="0−(1−d)
L
(1−d)
C−1
RC #.x1
x2+1
L
0VFC
y=0 1 .x1
x2
(23)
Figu e 3. Basic elec ical diag am o he boos con e e linked o PEMFC.
3.2. F ac ional O de PID Con olle
In 1999, Podlubny [
44
] p oposed he
PIλDµ
con olle , a gene aliza ion o he classical
PID con olle , comp ising a ac ional in eg a ion o o de
λ
and a ac ional de i a ion o
o de
µ
, hus widening he ield o applica ion o ac ional calculus o he command heo y,
which has di ec ed se e al esea che s o a new line o esea ch which is he adjus men o
he ac ional-o de
PIλDµ
con olle [
44
]. The ollowing o m gi es he ou pu equa ion o
he ac ional-o de con olle in he ime domain:
u=kpe( ) + kiD−λ
e( ) + kdDµ
e( )(24)
whe e
kp
is he p opo ional cons an ,
ki
is he in eg a ing cons an ,
kd
is he di e en ia ing
cons an ,
λ
is he ac ional o de o he in eg a ing ac ion, and
µ
is he ac ional o de o
he di e en ia ing ac ion.
By compa ison wi h he con en ional PID con olle [
45
], ac ional-o de con olle s
ha e in addi ion wo o he pa ame e s no ed
λ
and
µ
, which p esen he o de o in eg a ion
and de i a ion, espec i ely. Depending on he a ia ion o hese wo pa ame e s, we can
dis inguish di e en possibili ies o ac ional o de con olle [44].
As indica ed in Figu e 4, he ac ional o de
PIλDµ
con olle gene alizes he classical
PID
con olle and ex ends i om he poin o a plane. This expansion could p o ide
much mo e lexibili y in he design o PID con ol. Clea ly, by choosing
(λ
,
µ) = (
1,1
)
, a
classic
PID
co ec o can be eco e ed and using
(λ
,
µ) = (
1.0
)
and
(λ
,
µ) = (
0.1
)
, we ge
con olle s classic
PI
and
PD
, espec i ely. In o he wo ds, all hese ypes o classical,
n
con olle s a e special cases o he ac ional PIλDµcon olle gi en in Equa ion (24).
P ocesses 2022,10, 450 8 o 17
Figu e 4. Types o con olle s acco ding o λand µ.
3.3. Op imiza ion Using EGWO Me hod
The g ey wol op imize (GWO) is an in elligen swa m echnique de eloped in 2014
by Seyedali Mi jalili [
46
], which mimics he leade ship hie a chy o wol es ha a e well
known o hei g oup hun ing. In his algo i hm, he popula ion is di ided in o ou
g oups: alpha
(α)
, be a
(β)
, del a
(δ)
and omega
(ω)
. The i s h ee mos i al wol es
guide he las weak wol es
ω
o p omising a eas o he sea ch space. One o he exci ing
eali ies o he social li e o hese wol es is hei igo ous social hie a chical s uc u e in he
g oup, as shown in Figu e 5.
Figu e 5. G ey wol hie a chy.
The hun ing s a egy and wol es’ social hie a chy a e modeled o design he GWO
op imiza ion algo i hm. This algo i hm includes he ollowing s eps [46,47]:
• Social hie a chy
• P ey sea ch (explo a ion)
• Follow, hun and app oach he p ey
• Pu sue, ci cle and ha ass he p ey un il hey s op mo ing
• A ack on he p ey
Figu e 6gi es he lowcha o he GWO op imiza ion me hod. The ma hema ical
Equa ions which go e n he GWO algo i hm can be summa ized as ollows:
−→
D=|−→
C−→
Xp( )−−→
X( )|(25)
−→
X( +1) = |−→
Xp( )−−→
A−→
D|(26)
whe e
indica es he cu en i e a ion,
−→
A
and
−→
C
a e ec o s coe icien s,
−→
Xp
he posi ion
ec o o he p ey, −→
Xis he posi ion ec o .
P ocesses 2022,10, 450 9 o 17
The ec o s, −→
Aand −→
Ca e calcula ed as ollows:
−→
A=2−→
a( )−→
2−−→
a( )(27)
−→
C=2−→
1(28)
whe e,
−→
a
linea incline ec o dec eased om 2 o 0, and
−→
1−→
2
a e andom ec o s in
[
0.1
]
.
−→
Dα=|−→
C1−→
Xα( )−−→
X( )|(29)
−→
Dβ=|−→
C2−→
Xβ( )−−→
X( )|(30)
−→
Dδ=|−→
C3−→
Xδ( )−−→
X( )|(31)
−→
X( +1) =
−→
X1+−→
X2+−→
X3
3(32)
whe e,
Xα( )
ep esen s he posi ion o he
α
,
Xβ( )
indica es he posi ion o he
β
,
Xδ( )
is
he posi ion o
δ
,
C1−3
a e andom ec o s and
X
indica es he posi ion o he cu en solu ion.
The ex ended GWO is he same as he o iginal, whe e he di e ence is adding h ee
pa ame e s (
αE
,
βE
and
δE
) called he emphasis coe icien s o he upda ed posi ion o
Equa ion (32). The e o e, he ex ended, upda ed posi ion can be exp essed as
Equa ion (33) [48,49]:
−→
X( +1) = αE
−→
X1+βE
−→
X2+δE
−→
X3
3(33)
whe e,
αE>βE>δE
In his pape , he EGWO and GWO algo i hms a e implemen ed o une he FOPID
and PID con olle s pa ame e s in he o line mode in o de o ensu e an op imal con ol
pe o mance unde he a ia ions o he ope a ing condi ions. The i s s ep is o ini ialize a
andom wol popula ion based on he uppe and lowe bounds o he a iables, uni o mly
dis ibu ed in he sea ch space
D
, and ix he s op c i e ion. Second, e alua e he objec i e
unc ion o each wol . Thi d, choose he i s h ee bes wol es and sa e hem unde
α
,
β
and
δ
. Fou h, upda e he posi ion o he es o he popula ion (wol es). Fi h, upda e
o pa ame e s
a
,
A
and
C
. I he s opping c i e ion is no sa is ied, go o he second s ep;
o he wise, he p og am ends, and he op imal solu ion is p oduced. In he simula ion, he
popula ion size is he numbe o sea ch agen s, which is 30 and he numbe o i e a ions
equal o 40 and 200 o EGWO and GWO, espec i ely. The emphasis coe icien s used in
he EGWO a e chosen as αE=1.2, βE=1.1, and δE=0.9.
The implemen a ion o he EGWO Algo i hm-based op imiza ion con ol scheme o
DC-DC con e e is o minimize ITAE while, p opo ional gain
kp
, in eg al gain
ki
and
de i a i e gain
kd
o he PID con olle , in addi ion
λ
and
µ
o he FOPID a e aken as
decision a iables. Acco ding o Ziegle Nichols echnique [
19
] and Podlubny [
44
], he
a ia ions anges o he FOPID and PID decision a iables used in he simula ion a e gi en
in Table 2.
P ocesses 2022,10, 450 16 o 17
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