Ci a ion: Silaa, M.Y.; Ba ambones, O.;
Benche i , A. A No el Adap i e PID
Con olle Design o a PEM Fuel
Cell Using S ochas ic G adien
Descen wi h Momen um Enhanced
by Whale Op imize . Elec onics 2022,
11, 2610. h ps://doi.o g/10.3390/
elec onics11162610
Academic Edi o s: Nicu Bizon and
Mihai Op oescu
Recei ed: 28 July 2022
Accep ed: 18 Augus 2022
Published: 20 Augus 2022
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elec onics
A icle
A No el Adap i e PID Con olle Design o a PEM Fuel Cell
Using S ochas ic G adien Descen wi h Momen um Enhanced
by Whale Op imize
Mohammed Yous i Silaa 1,* , Osca Ba ambones 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,
1006 Vi o ia, Spain
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
*Co espondence: [email p o ec ed] (M.Y.S.); osca [email p o ec ed] (O.B.)
Abs ac :
This pape p esen s an adap i e PID using s ochas ic g adien descen wi h momen um
(SGDM) o a p o on exchange memb ane uel cell (PEMFC) powe sys em. PEMFC is a nonlinea
sys em ha encoun e s ex e nal dis u bances such as inle gas p essu es and empe a u e a ia ions,
o which an adap i e con ol law should be designed. The SGDM algo i hm is employed o minimize
he cos unc ion and adap he PID pa ame e s acco ding o he pe u ba ion changes. The whale
op imiza ion algo i hm (WOA) was chosen o enhance he adap i e a es in he o line mode. The
p oposed con olle is compa ed wi h PID s ochas ic g adien descen (PIDSGD) and PID Ziegle
Nichols uning (PID-ZN). The con ol s a egies’ obus nesses a e es ed unde a a ie y o empe -
a u es and loads. Unlike he PIDSGD and PID-ZN con olle s, he PIDSGDM con olle can a ain
he equi ed con ol pe o mance, such as as con e gence and high obus ness. Simula ion esul s
using Ma lab/Simulink ha e been s udied and illus a e he e ec i eness o he
p oposed con olle .
Keywo ds:
s ochas ic g adien descen wi h momen um; s ochas ic g adien descen ; PID con olle ;
whale op imiza ion algo i hm; p o on exchange memb ane uel cell
1. In oduc ion
1.1. Mo i a ions
The clima e has changed h oughou his o y; mos o hese sligh changes ha e been
caused by a small a ia ion in he ea h o bi cha ac e ized by an ab up inc ease in he
empe a u e [
1
]. Va ious e ec s ha e al eady begun o appea in a ious pa s o he wo ld,
causing se ious p oblems in he na u al en i onmen and peoples’ li es. In o de o emedy
his ma e and educe i s isks, all o e he wo ld, coal plan s a e being eplaced by sola
panels, sola he mal ene gy, wind u bines, and hyd ogen powe sou ces [
2
]. This la e
op ion may be he key o s o ing enewable ene gy and could, he e o e, change he ace o
he ene gy ansi ion. Hyd ogen accoun s o 75% o all ma e and is hough o be one o
he h ee elemen s ha we e c ea ed in he big bang; i accoun s o o e 90% o all a oms
in he uni e se and on ea h, and is easy o p oduce om many compounds, o ins ance,
wa e [
3
]. The cos o p oducing hyd ogen om enewable ene gy is an icipa ed o slump
by 50% by he middle o his cen u y, and his could clea he way o e en mo e g een
hyd ogen [
4
]. Fu he mo e, people may one day be able o p oduce hei own hyd ogen
a home [
5
]. The possibili y has encou aged scien is s and esea che s o look owa ds
o hyd ogen cells. P o on exchange memb ane uel cells (PEMFCs) a e among he mos
e icien elec ical gene a o s due o hei p ope ies such as high ene gy densi y, high
pe o mance, and high obus ness, esul ing in mul iple applica ions such as in mobiles,
ca s, ai c a , and space shu les [
6
–
10
]. The de elopmen o PEMFC sys ems o mul iple
applica ions has g own in ecen yea s, and esea che s ha e sugges ed a di e si y o con ol
Elec onics 2022,11, 2610. h ps://doi.o g/10.3390/elec onics11162610 h ps://www.mdpi.com/jou nal/elec onics
Elec onics 2022,11, 2610 2 o 21
echniques om di e en s a egies in o de o imp o e he ne ou pu powe . Howe e , o
ex ac he maximum ne powe , DC/DC con e e s a e equi ed o imp o e he e iciency
o he PEMFC s ack and o minimize he dec ease in powe gene a ion e iciency due o he
luc ua ion o empe a u e and gas p essu es [11].
1.2. S a e o he A
In o de o ob ain g ea e powe con e sion om cells, many con ol echniques and
algo i hms ha e been adop ed in he li e a u e, such as neu al ne wo k con ol
(NNC) [12]
,
sliding mode con ol (SMC) [
13
], uzzy logic con ol (FLC) [
14
], adap i e con ol [
15
],
maximum powe poin acking (MPPT) echniques [
16
], e c. Hence, in Re . [
17
], a con-
en ional SMC was used in a compa ison o classical p opo ional–in eg al (PI) linea
con olle s. The p oposed echnique showed good esul s in e ms o obus ness agains
he ex eme load a ia ion. Howe e , despi e hese esul s, he SMC s ill expe iences he
cha e ing phenomenon. Hu Peng e al. [
18
] es ablished a dynamic model o he empe a-
u e mechanism o PEMFC based on con ol and designed a wo-dimensional inc emen al
uzzy con olle o he empe a u e o PEMFC wi h an in eg al link acco ding o he
es ablished empe a u e model and empi ical con ol ules. The esul s show ha he
model can simula e he dynamic cha ac e is ics o PEMFC, and when he empe a u e o
PEMFC is con olled wi hin he ideal wo king ange, he designed con olle can be used
o con ol he empe a u e o PEMFC in eal- ime; i also has s ong obus ness. In Re . [
19
],
a ac ional-o de p opo ional–in eg al-de i a i e (FOPID) con olle was applied o a
ou -swi ch buck-s ep-up DC/DC con e e in o de o s abilize he PEMFC ou pu powe .
Simula ion esul s showed ha he p oposed me hod achie ed be e pe o mance han
he in ege -o de con olle . Silaa e al. [
20
] used a high-o de sliding mode o keep he
PEMFC sys em wo king a an e icien powe poin and as a solu ion o he cha e ing
phenomenon. Expe imen al esul s showed ha he p oposed con ol p o ides a sa is ac-
o y esul in a e ms o educing he cha e ing e ec by up o 84%. Howe e , despi e
hese esul s, he p oposed con olle has a high-powe o e shoo agains he luc ua ing
load a ia ion. In Re . [
21
], an MPPT using pa icle swa m op imiza ion (PSO) combined
wi h a p opo ional in eg al de i a i e (PID) con olle was compa ed wi h he pe u bing
and obse ing (P&O) echnique and sliding mode con olle o a PEMFC. The simula ion
esul s showed ha he p oposed MPPT echnique achie es a low o e shoo , sho esponse
ime, and low oscilla ions a ound he MPP. Ahmed e al. [
22
] designed a hyb id sys em
consis ing o wind and pho o ol aic sys ems as he main sou ce o ene gy. The uel cell
is ins alled as a piece o seconda y equipmen o gua an ee a con inuous powe supply
and o add ess he e a ic na u e o wind/pho o ol aic supply. De beli e al. [
23
] used a
high-o de sliding mode compa ed o a con en ional SMC o keep he PEMFC ope a ing a
a e e ence cu en . Resul s showed ha he p oposed algo i hm was able o educe he
cha e ing impac by mo e han 82%, along wi h p o iding obus ness agains he load
a ia ion. In Re . [
24
], an adequa e powe poin was ob ained using a PID con olle . The
PID con olle was p og ammed o powe he uel cell by changing he boos con e e ’s
pulse-wid h modula ion (PWM). Resul s showed ha he goal was accomplished by he
p oposed app oach wi h be e dynamics and good acking e iciency. In Re . [
25
], a PSO
based on uzzy logic (FL) was planned o keep he PEMFC unning a an op imal powe
poin . Simula ion esul s showed a powe o e shoo and unde shoo o mo e han 63%. Fan
Liping e al. [
26
] es ablished a ma hema ical model o a PEMFC and designed an adap i e
FL con olle o cons an powe uel cells. Expe imen s show ha he designed con olle
can a ain a cons an powe ou pu o om he PEMFC. Li e al. [
27
] p oposed a uzzy
sliding mode con olle o con ol he ai supply low o he PEMFC s ack. The p oposed
con olle has good obus ness. The uzzy sliding mode con olle embeds a uzzy logic
in e ence mechanism in he con en ional SMC, esul ing in smoo h con ol. The esul s
show ha he uzzy syno ial con olle elimina es he cha e ing phenomenon o he adi-
ional syno ial con olle . The compa ison p o es ha he uzzy syno ial con olle can
signi ican ly imp o e he con ol pe o mance o a PEMFC. In Re . [
28
], a PID op imized
Elec onics 2022,11, 2610 3 o 21
by he g ey wol op imize (PID-GWO), FOPID op imized by he g ey wol op imize
(FOPID-GWO), and a PID op imized by an ex ended g ey wol op imize (PID-EGWO)
we e used o con ol a DC/DC boos con e e linked o a PEMFC. Simula ion esul s
show be e dynamics, good acking e iciency, and as e con e gence o he op imal
solu ion. In Re . [
29
], a model p edic i e con ol (MPC) me hod was designed o a DC/DC
boos con e e o keep a PEMFC wo king a an e icien powe s age. Expe imen al esul s
showed ha he MPC echnique is supe io o he PI echnique in e ms o acking accu acy,
o e shoo , and unde shoo .
1.3. Con ibu ions
The main con ibu ion o his pape is o design an adap i e PID con olle using
SGDM o con ol he DC/DC boos con e e o achie e sa e ope a ion o he PEMFC
sys em and o op imize he ou pu powe . The SGDM is used in o de o se he PID
adap i e gains acco ding o he change o dis u bances. Hence, he WOA algo i hm is used
o ind he op imal adap i e a es, and is he e o e injec ed in o he SGDM echnique.
This pape is di ided in o h ee sec ions. Sec ion 2desc ibes he ma hema ical model
o he PEM uel cell. Sec ion 3is de o ed o he con ol me hodology. Sec ion 4p esen s he
simula ion esul s and he conclusion.
2. PEM Fuel Cell Modeling
As shown in Figu e 1, a PEMFC is made up o wo pla es, wo elec odes, and wo hin
laye s o pla inum-based ca alys s sepa a ed by a memb ane. When he uel (hyd ogen) is
injec ed, i eac s elec ochemically o c ea e elec ici y [
30
]. The hyd ogen and he oxygen
a e ed h ough channels in he pla es. Hyd ogen lows on one side o he memb ane
and he oxygen on he o he . The ca alys spli s he hyd ogen molecule in o p o ons and
elec ons; he p o ons can pass h ough he memb ane, while he elec ons canno and mus
pass h ough an ex e nal ci cui , c ea ing use ul elec ici y. On he oxygen side o he
memb ane, he p o ons and he elec ons eac wi h he oxygen in he p esence o a second
ca alys laye , gene a ing wa e , hea , and elec ical ene gy [
31
]. The eac ions a he le el
o he PEMFC a e gi en in Equa ions (1)–(3) [32]:
Anode: 2H2=⇒4H++ 4e−(1)
Ca hode: 4H++O2+ 4e−=⇒2H2O(2)
Cell: 2H2+O2=⇒2H2O+ E.E (3)
Figu e 1. A c oss-sec ion o a PEMFC.
Elec onics 2022,11, 2610 4 o 21
2.1. PEMFC S a ic Model
Acco ding o [
32
], he Ne ns
(ENe n)
equa ion desc ibes he cell’s elec ochemical
he modynamic po en ial and gi es he ela ionship be ween he open-ci cui ol age
o elec ochemical cells unde s anda d condi ions and non-s anda d condi ion (
ENop
).
The equa ion is as ollows [32]:
ENop =1.299 −0.85 ·10−3·(T−T ) + 4.3085 ·10−5Tln(PH2) + 1
2·ln(PO2)(4)
whe e
T
is he cell ope a ing empe a u e,
T
ep esen s he e e ence empe a u e in
Kel in, which is equal o 298.15 K a 25
°
C, and
PO2
and
PH2
ep esen he inle oxygen and
hyd ogen gas p essu es, espec i ely [32].
The ollowing exp ession can de ine he ou pu ol age o a single PEMFC [33].
VS c =ENop −EAc −EOhm −ECon (5)
whe e
EAc
,
EOhm
, and
ECon
ep esen he pola iza ion po en ials o he ol age losses ha
a e gene a ed by he e e sibili y o he sys em.
The ac i a ion pola iza ion
EAc
is due o he kine ics o he eac ions aking place a he
elec ode/memb ane eac ion in e ace. This loss can be calcula ed using Equa ion (6) [33]:
Eac =γ1+γ2·T+γ3·T·ln(CO2) + γ4·T·ln(I c)(6)
whe e he pa ame e s
γ1
,
γ2
,
γ3
, and
γ4
a e he pa ame ic coe icien s o each PEMFC
model; CO2is he oxygen concen a ion in he ca alys s (mol/cm3).
The
EOhm
pola iza ion is caused by he elec ical esis ance o he di e en elemen s o
he cell. This loss has wo o igins: he equi alen esis ance o he memb ane o p o on con-
duc ion
Rmem
and he con ac esis ance
Rcon
be ween he bipola pla es and he elec odes.
The EOhm ol age can be calcula ed using Equa ion (7) [33]:
EOhm =I c ·(Rmem +Rcon)(7)
whe e
Rmem =Γmem ·l
A(8)
whe e
l
is he memb ane hickness (
µ
m),
A
is he cell ac i e a ea (cm
2
), and
Γmem
is he
speci ic esis ance o he memb ane, which is ob ained by he ollowing [34]:
Γmem =181.6[1+0.03(I c
A) + 0.062(T
303)2(I c
A)2.5]
[ψ−0.634 −3(I c
A)] ·exp [4.18(T−303)/T]
(9)
whe e
ψ
is he wa e con en in he memb ane, assuming a minimum and maximum alue
o 0 and 24, espec i ely.
The concen a ion pola iza ion
ECon
is caused by he a ia ion in he concen a ion o
eagen s on he elec ode. This loss can be calcula ed using Equa ion (10) [34]:
Econ =δ·ln1−J
Jmax (10)
whe e
δ
,
J
, and
Jmax
a e he cons an pa ame e s, he cu en densi y, and he maximum
cu en densi y, espec i ely.
A single PEMFC ou pu ol age unde s anda d condi ions does no exceed 1.29 V.
In o de o p oduce he equi ed amoun o powe , i is necessa y o ha e cells in se ies,
which inally o ms a s ack. The ea e , he powe gene a ed by he PEMFC s ack is gi en
in Equa ion (11) [35]:
Ps ack =VS c ·I c ·NCell (11)
Elec onics 2022,11, 2610 5 o 21
whe e
I c
ep esen s he single cell cu en and
NCell
ep esen s he numbe o s ack laye s.
The PEMFC s a ic model is ep esen ed in Figu e 2.
Figu e 2. PEMFC s a ic model.
The PEMFC pa ame e s used in he simula ion a e gi en in Table 1:
Table 1. PEMFC pa ame e s.
Pa ame e Value
A162 cm2
l175 ·10−6cm
ψ23
δ0.1 V
Rcon 0.0003
Jmax 0.062 A·cm−1
NCell 10
γ10.9514 V
γ2−0.00312 V/K
γ3−7.4 ·10−5V/K
γ41.87 ·10−4V/K
2.2. PEMFC Dynamic Model
In a PEMFC, he wo elec odes a e sepa a ed by a solid memb ane ha allows p o ons
o pass and blocks he low o elec ons. The elec ons low om he anode h ough he
ex e nal cha ge and a e collec ed a he su ace o he ca hode, o which he hyd ogen
p o ons a e a ac ed a he same ime. Thus, wo cha ged laye s o opposi e pola i ies a e
o med ac oss he po ous bounda y be ween he ca hode and he memb ane. The laye s
a e known as “double elec ochemical laye s” and can s o e elec ical ene gy and beha e
like a supe capaci o [
36
]. The PEMFC equi alen ci cui showing his e ec is p esen ed
in Figu e 3.
Figu e 3. PEMFC equi alen ci cui .
Elec onics 2022,11, 2610 6 o 21
The elec odes o a PEMFC a e po ous. The capaci ance is e y la ge and can be
o he o de o se e al a ads.
Rac
and
Rconc
a e he ac i a ion equi alen esis ance
and he concen a ion equi alen esis ance, espec i ely [
37
]. By using Ki chho ’s law,
he dynamical equa ion o he model is ep esen ed by [38]:
dVd
d = ( I
C−Vd
τ)(12)
whe e
Vd
is he dynamical ol age ac oss he capaci o ,
C
is he equi alen capaci o , and
τ
is he PEMFC ime cons an , which is gi en by he ollowing equa ion [38]:
τ=C·(Rac +Rconc) = C·(EAc +ECon
I)(13)
The e o e, he PEMFC ol age is gi en by he equa ion below [38,39]:
VS c =ENe n −Vd−IRohm (14)
whe e
Rohm,ac ,conc
ep esen he ohmic, ac i a ion, and concen a ion esis ances, espec i ely.
Using Equa ions (12) and (13) and he Laplace ans o ma ions in Equa ion (14),
he PEMFC ol age is gi en as ollows [39]:
VS c =ENe n −EAc +ECon
sC ·(EAc +ECon) + 1+Rohm·I(15)
Acco ding o [
39
], PEMFC inle gas p essu es a e a iable in di e en condi ions.
In o de o calcula e he dynamic pa ial p essu es, each indi idual gas is conside ed
sepa a ely and he ideal gas equa ion is applied o each one [
40
]. The pa ial gas p essu es
a e gi en as ollows [41]:
PH2=
1
KH2
(1+τH2)·(qH2−2·I·K )(16)
PO2=
1
KO2
(1+τO2)·(qO2−2·I·K )(17)
whe e
τH2=Van
R·T·KH2
τO2=Van
R·T·KO2
(18)
By using he p e ious equa ions, he PEMFC dynamic model can be ep esen ed, as in
Figu e 4.
Figu e 4. PEMFC dynamic model.
The meanings o he a iables used in Equa ions (16)–(18) a e lis ed in Table 2.
Elec onics 2022,11, 2610 7 o 21
Table 2. PEMFC nomencla u e.
Va iable Meaning
KH2Hyd ogen al e mola cons an (kmol/a m·s)
KO2Oxygen al e mola cons an (kmol/a m·s)
τH2Hyd ogen ime cons an (s)
τO2Oxygen ime cons an (s)
qH2Mola low a e o hyd ogen (Kmol/s)
qO2Mola low a e o oxygen (Kmol/s)
K Modeling cons an (Kmol/s·A)
RUni e sal gas cons an (1·a m/Kmol·K)
Van Volume o he anode (cm3)
3. Con ol Me hodology
In his sec ion, an adap i e PID using SGDM and SGD is designed in o de o s abilize
he PEMFC and keep i ope a ing a a e e ence cu en
I e
. The closed loop sys em
consis s o a PEMFC s ack, a DC/DC boos con e e , a
P
&
O
MPPT echnique, a con olle ,
and inally, a load, as shown in Figu e 5.
Figu e 5. Closed loop sys em.
3.1. DC/DC Boos Con e e
A s ep-up DC/DC con e e is a simple elec onic ci cui ha combines a swi ching
elemen wi h a coil, a capaci o , and a diode, as shown in Figu e 6. This con e e wo ks in
o de o ob ain a DC ou pu ol age highe han he inpu ol age. The impo an poin
o unde s and abou he ci cui is ha he posi ions o he swi ching elemen ( ansis o ),
coil, and diode a e di e en [
42
]. When he swi ch is u ned
ON
and he cu en lows in,
he coil s o es ene gy; when he swi ch is u ned
OFF
, he s o ed ene gy is eleased and he
induced cu en lows in a di ec ion ha p e en s he cu en change. The longe he swi ch
is
ON
, he highe he ou pu ol age is, and he longe he swi ch is
OFF
, he lowe he
ou pu ol age is. The equi ed ou pu ol age can be ob ained by con olling he
ON
/
OFF
ime (du y cycle). The boos con e e s a e space is gi en in Equa ion (19) [43]:
Elec onics 2022,11, 2610 8 o 21
Figu e 6. S ep-up DC/DC con e e diag am.
"˙
x1
˙
x2#="0−(1−d)
L
(1−d)
C−1
RC #.x1
x2+1
L
0VS c
y=0 1 .x1
x2
(19)
whe e x= [x1,x2]T= [iL,Vou ]T.
The s ep-up DC/DC con e e pa ame e s used in he simula ion a e illus a ed in
Table 3.
Table 3. DC/DC boos con e e pa ame e s.
Pa ame e Value
Induc ance (L) 6.9 ·10−2H
Capaci o (C) 150 ·10−7F
Max Sw 10 kHz
Max in ol age 25 V
Max in cu en 18 A
Max ou ol age 80 V
Max ou cu en 2 A
3.2. Adap i e PID Using SGD
The adap a ion op imum me hod used o he PID con olle , known as SGD, consis s
o i e a i ely adjus ing all o he p oposed con olle coe icien s acco ding o hei calcu-
la ed g adien s in o de o minimize he e o unc ion. The disc ee PID gene al o m is
gi en as ollows [44]:
u(k) = u(k−1) + Kp[e(k)−e(k−1)] + Kie(k) + Kd[e(k)−2e(k−1) + e(k−2)] (20)
whe e
Kp
,
Ki
, and
Kd
ep esen he p opo ional, in eg al, and de i a i e gains, espec i ely,
and kis he ime ins ance.
The gene al ma hema ical o mula o he SGD is gi en as ollows [45]:
w(k+1) = w(k)−µ∂L(k)
∂w(k)(21)
Elec onics 2022,11, 2610 9 o 21
whe e
µ∈(
0,1
)
is he adap i e a e and
L
is he loss unc ion, which is de ined as
ollows [45]:
L(k) = 1
2e(k)2=1
2( (k)−y(k))2=1
2(I e (k)−IL(k))2(22)
The e o e, he upda ed gains o he PID con olle using SGD a e gi en by he ollow-
ing equa ion:
Kp,i,d(k+1) = Kp,i,d(k)−µp,i,d
∂L(k)
∂Kp,i,d(k)(23)
whe e subsc ip s
p
,
i
, and
d
indica e he alues o
Kp
,
Ki
, and
Kd
, espec i ely. Equa ion (23)
can be exp essed by he ollowing o mula:
∆Kp,i,d(k) = −µp,i,d
∂L(k)
∂Kp,i,d(k)=−µp,i,d
∂L(k)
∂IL(k)
∂IL(k)
∂u(k)
∂u(k)
∂Kp,i,d(k)(24)
Using Equa ions (20) and (22), he disc e e pa ial de i a i e e ms o Equa ion (24)
can be exp essed by he ollowing o mulas:
∂L(k)
∂IL(k)=−e(k) = −(I e (k)−IL(k))
∂IL(k)
∂u(k)=IL(k)−IL(k−1)
u(k)−u(k−1)
∂u(k)
∂Kp(k)=e(k)−e(k−1)
∂u(k)
∂Ki(k)=e(k)
∂u(k)
∂Kd(k)=e(k)−2e(k−1) + e(k−2)
(25)
The e o e, by subs i u ing o mulas o Equa ion (25) in o Equa ion (24), he adap i e
gains can be exp essed as ollows:
∆Kp(k) = µp·e(k)·IL(k)−IL(k−1)
u(k)−u(k−1)·[e(k)−e(k−1)] (26)
∆Ki(k) = µi·e(k)·IL(k)−IL(k−1)
u(k)−u(k−1)·e(k)(27)
∆Kd(k) = µd·e(k)·IL(k)−IL(k−1)
u(k)−u(k−1)·[e(k)−2e(k−1) + e(k−2)] (28)
3.3. Adap i e PID Using SGDM
While SGD is a e y popula op imiza ion me hod, i s lea ning p ocess can some imes
be slow. The momen um me hod is designed o speed up lea ning (Figu e 7), especially
when dealing wi h g adien s wi h high cu a u e, small bu consis en g adien s, o noisy
g adien s [46].
Figu e 7. SGDM s. SGD.
Elec onics 2022,11, 2610 16 o 21
de ail, he inc ease o capaci ance leads o a change in he sys em dynamics and slowe
sys em esponse. In he opposi e case, when he capaci ance dec eases, sys em esponse is
as e and he e is a esidual ipple inc ease.
Figu e 14.
(
a
) PEMFC cu en con ol obus ness analysis: e ec o L a ia ion; (
b
) PEMFC cu en
con ol obus ness analysis: e ec o C a ia ion.
Figu e 15 exhibi s he demeano o he boos con e e ou pu signals unde PIDS-
GDM, PIDSGD, and PID-ZN. Acco ding o his igu e, smoo h and g adual mo emen s o
he desi ed alue a e achie ed using he PIDSGDM, PIDSGD, and PID-ZN. Fu he mo e,
PIDSGDM, and PIDSGD ha e as con e gence, which is clea ly p esen ed in his igu e.
On he deep side, acco ding o he cu en and powe esul s, i is no able ha he h ee
con olle s ha e app oxima ely he same pe o mance in e ms o o e shoo and unde -
shoo . On he o he hand, he PIDSGDM and PIDSGD can success ully bea he ol age
o e shoo , unlike he PID-ZN con olle . Howe e , hese o e shoo s and unde shoo s can
heo e ically be conside ably educed by using an ex a capaci o on he inpu side o he
con e e . Ne e heless, adding ex a capaci ance may lead o a change in he sys em
dynamics, slowe sys em esponse, and s eady-s a e e o s.
Elec onics 2022,11, 2610 17 o 21
Figu e 15.
(
a
) S ep-up con e e ou pu cu en ; (
b
) s ep-up con e e ou pu ol age; (
c
) s ep-up
con e e ou pu powe .
The ob ained adap a ion gains
Kp
,
Ki
, and
Kd
o he PIDSGDM and PIDSGD a e
shown in Figu e 16.
Figu e 16. (a) Adap i e gain Kp; (b) adap i e gain Ki; (c) adap i e gain Kd.
Acco ding o hese esul s, he p oposed PIDSGDM dynamic pe o mance, especially
as - acking ime, a oids he gas s a a ion phenomena ha occu when gas ansmission
lags behind he speed o cu en change, and he e is no appa en pe o mance e ec on
he ex e nal cha ac e is ics such as he ou pu ol age and ou pu powe o he PEMFC.
This abili y is mo e signi ican in he case o s epped a ia ion, which is conside ed he
ha des case.
Du ing he simula ion, he con ol pe o mance was u he alida ed using a Simulink
p o ile [
15
,
54
]. The p o ile collec s pe o mance da a while simula ing and c ea es a
epo based on his da a, known as a simula ion p o ile, which p o ides he amoun
o Simulink ime spen pe o ming each unc ion o simula e he model. The p o ile
assis s in iden i ying he model componen s ha need he mos ime o simula e, and
Elec onics 2022,11, 2610 18 o 21
hus, whe e o concen a e model op imiza ion e o s. Acco ding o he Simulink p o ile ,
he compu a ional complexi y e ms o he con ol s a egies a e summa ized in Table 7.
As ema ked, he complexi y e ms o he con ol echniques a e app oxima ely simila ,
anging be ween 0.88%–1.22%, which is a e y low complexi y e m compa ed o he
whole model.
Table 7. Compu a ional complexi y summa y.
Algo i hm Time Calls Time/Call Sel Time
PIDSGDM 0.058–1.22% 600127 0.0000000966 0.057–2.19%
PIDSGD 0.051–1.07% 570155 0.0000000897 0.048–1.84%
PID-ZN 0.020–0.88% 18005 0.0000011 0.0020–0.33%
5. Conclusions
In his pape , an adap i e PID using SGDM was p oposed ha is e ec i e and has a
low complexi y o implemen a ion. This is applied o a simple DC/DC boos con e e
in o de o achie e sa e ope a ion o a PEMFC sys em and o op imize he ou pu powe
quali y. The p oposed con olle was compa ed wi h he PIDSGD and PID-ZN echniques.
Based on he esul s, an o e shoo educ ion up o 98% and an unde shoo educ ion up o
94% we e achie ed. The simula ion compa ison demons a es ha he PIDSGDM can bea
dis u bances and o e s low esponse ime, high obus ness, and o e shoo and unde shoo ,
which leads o a minimiza ion o he powe losses. Consequen ly, as is demons a ed in
he esul s, he p oposed con olle p o ides sa e ope a ion and an op imal solu ion o a
PEMFC sys em a ec ed by ex e nal dis u bances.
Au ho Con ibu ions:
Concep ualiza ion, M.Y.S.; me hodology, M.Y.S.; so wa e, M.Y.S.; alida ion,
M.Y.S., O.B. and A.B.; o mal analysis, M.Y.S. and A.B.; in es iga ion, M.Y.S., O.B. and A.B.; esou ces,
O.B.; da a cu a ion, M.Y.S. and A.B.; w i ing—o iginal d a p epa a ion, M.Y.S.; w i ing— e iew
and edi ing, M.Y.S., O.B. and A.B.; isualiza ion, M.Y.S., O.B. and A.B.; supe ision, O.B.; p ojec
adminis a ion, O.B.; unding acquisi ion, O.B. All au ho s ha e ead and ag eed o he published
e sion o he manusc ip .
Funding:
The au ho s wish o exp ess hei g a i ude o he Basque Go e nmen h ough he
p ojec EKOHEGAZ (ELKARTEK KK-2021/00092), o he Dipu ación Fo al de Ála a (DFA) h ough
he p ojec CONAVANTER, and o he UPV/EHU h ough he p ojec GIU20/063 o suppo ing
his wo k.
Acknowledgmen s:
The au ho s wish o exp ess hei g a i ude o he Basque Go e nmen h ough
he p ojec EKOHEGAZ (ELKARTEK KK-2021/00092), o he Dipu ación Fo al de Ála a (DFA)
h ough he p ojec CONAVANTER, and o he UPV/EHU h ough he p ojec GIU20/063 o sup-
po ing his wo k.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Abb e ia ions
The ollowing abb e ia ions a e used in his manusc ip :
SGD S ochas ic g adien descen
SGDM S ochas ic g adien descen wi h momen um
WOA Whale op imiza ion algo i hm
ZN Ziegle Nichols
PEMFCs P o on exchange memb ane uel cells
SMC Sliding mode con ol
PI P opo ional–in eg al
Elec onics 2022,11, 2610 19 o 21
MPC Model p edic i e con ol
FOPID F ac ional-o de p opo ional-in eg al-de i a i e
MPPT Maximum powe poin echnique
PSO Pa icle swa m op imiza ion
PID P opo ional in eg al de i a i e
P&O Pe u b and obse e
PWM Pulse-wid h modula ion
FL Fuzzy logic
GWO G ey wol op imize
EGWO Ex ended g ey wol op imize
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