Ci a ion: Elsha a, R.; Hançe lio˘gulla i,
A.; Rahebi, J.; Lopez-Guede, J.M. PV
Cells and Modules Pa ame e
Es ima ion Using Coa i Op imiza ion
Algo i hm. Ene gies 2024,17, 1716.
h ps://doi.o g/10.3390/en17071716
Academic Edi o : Abdul-Ghani Olabi
Recei ed: 15 No embe 2023
Re ised: 3 Ma ch 2024
Accep ed: 1 Ap il 2024
Published: 3 Ap il 2024
Copy igh : © 2024 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
ene gies
A icle
PV Cells and Modules Pa ame e Es ima ion Using Coa i
Op imiza ion Algo i hm
Ra a Elsha a 1, Aybaba Hançe lio˘gulla i 2, Ja ad Rahebi 3and Jose Manuel Lopez-Guede 4,*
1Depa men o Ma e ial Science and Enginee ing, Uni e si y o Kas amonua, Kas amonu 37150, Tu key;
[email p o ec ed]
2Depa men o Physics, Uni e si y o Kas amonu, Kas amonu 37150, Tu key; [email p o ec ed]
3Depa men o So wa e Enginee ing, Is anbul Topkapi Uni e si y, Is anbul 34087, Tu key;
[email p o ec ed]
4Depa men o Sys ems and Au oma ic Con ol, Facul y o Enginee ing o Vi o ia-Gas eiz, Uni e si y o he
Basque Coun y (UPV/EHU), C/Nie es Cano 12, 01006 Vi o ia-Gas eiz, Spain
*Co espondence: [email p o ec ed]
Abs ac : In ecen imes, he e ha e been no able ad ancemen s in sola ene gy and o he enewable
sou ces, unde sco ing hei i al con ibu ion o en i onmen al conse a ion. Sola cells play a
c ucial ole in con e ing sunligh in o elec ici y, p o iding a sus ainable ene gy al e na i e. Despi e
hei signi icance, e ec i ely op imizing pho o ol aic sys em pa ame e s emains a challenge. To
ackle his issue, his s udy in oduces a new op imiza ion app oach based on he coa i op imiza ion
algo i hm (COA), which in eg a es opposi ion-based lea ning and chaos heo y. Unlike exis ing
me hods, he COA aims o maximize powe ou pu by in eg a ing sola sys em pa ame e s e icien ly.
This s a egy ep esen s a signi ican imp o emen o e adi ional algo i hms, as e idenced by
expe imen al indings demons a ing imp o ed pa ame e se ing accu acy and a subs an ial inc ease
in he F iedman a ing. As global ene gy demand con inues o ise due o indus ial expansion and
popula ion g ow h, he impo ance o sus ainable ene gy sou ces becomes inc easingly e iden . Sola
ene gy, cha ac e ized by i s enewable na u e, p esen s a p omising solu ion o comba en i onmen al
pollu ion and lessen dependence on ossil uels. This esea ch emphasizes he c i ical ole o COA-
based op imiza ion in ad ancing sola ene gy u iliza ion and unde sco es he necessi y o ongoing
de elopmen in his ield.
Keywo ds: coa i op imiza ion algo i hm (COA); chaos heo y; opposi ion-based lea ning; sola
sys ems; op imiza ion o PV pa ame e s
1. In oduc ion
In ecen yea s, he inc ease in ene gy demand has caused en i onmen al conce ns o
inc ease. Al hough ossil uels a e ela i ely cheap, hey a e a con inuous sou ce o global
wa ming and ai pollu ion. Oil and gas ex ac ion p o ide ene gy esou ces, bu hey ha e
become he main ac o o en i onmen al pollu ion in he pas decades. Fu he mo e,
es ima es show ha ene gy sou ces such as oil and gas will un ou in he coming yea s, so
inding new and enewable ene gy sou ces is i al [1,2].
In ecen yea s, clean and enewable ene gies ha e de eloped and p og essed, so
now, hey a e a sui able al e na i e o ossil uels (oil and gas). Renewable ene gy, such as
he powe ob ained om he sun, wind, ides, wa e , ho sp ings, e c., has been a ac ing
he a en ion o scien is s. Among enewable ene gies, sola ene gy is mo e accessible
han o he ene gies and used in mos a eas o he ea h. Sola ene gy is a enewable
ene gy sou ce. Unlike elec ici y gene a ion by ossil uels such as gas and oil, sola
ene gy does no cause any en i onmen al pollu ion [
3
]. Es ima es show ha he sha e
o sola ene gy in elec ici y p oduc ion will each abou USD 194 billion by 2027, which
shows he impo ance o his ene gy [
4
]. Pho o ol aic sys ems help ans o m sola ene gy
Ene gies 2024,17, 1716. h ps://doi.o g/10.3390/en17071716 h ps://www.mdpi.com/jou nal/ene gies
Ene gies 2024,17, 1716 2 o 26
in o aluable elec ici y. Pho o ol aic (PV) sys ems con e sola ene gy in o elec ici y
using semiconduc o echnology [
5
]. P og ess in enewable ene gy has in luenced many
coun ies and go e nmen s o use his clean ene gy o p oduce elec ici y. Sola ene gy is
an economical and abundan clean ene gy sou ce. Due o he easy access o sunligh , sola
sys ems a e used in many coun ies in A ica, he Middle Eas , and Ame ica [6].
Sola ene gy con e sion in o elec ici y in sola cells needs de ices based on elec onic
semiconduc o s wi h a hin laye . A sola sys em based on semiconduc o c ys als is usually
made o wo ypes: mul i-c ys alline [
7
] and monoc ys alline [
8
] sola modules. Single-
c ys al semiconduc o s a e mo e e icien o gene a ing powe and ha e be e elec ical
p ope ies. Howe e , despi e ha ing a la ge p oduc ion capaci y, monoc ys alline PV
modules a e unp o i able because o he high cos o he c ys alline wa e -based echnology.
An al e na e p ocess o single-c ys al modules o sola powe p oduc ion is hin ilm
echnology. This me hod ypically uses amo phous silicon [
8
] o o he semiconduc o
ma e ials such as gallium a senide [
9
], coppe indium gallium selenide [
10
], o cadmium
ellu ide [11].
Sola ene gy is a clean, enewable sou ce because i di ec ly con e s sola ene gy in o
elec ici y using semiconduc o echnology in pho o ol aic de ices. The amoun o ene gy
p oduc ion om PV sys ems depends on wea he condi ions, sola adia ion, ambien
empe a u e, ype o modules, e c. [11].
An accu a e unde s anding o PV’s elec ical s uc u e and modules may help e-
sea che s inc ease powe p oduc ion e iciency. PV sys ems comp ise a powe con e e ,
a sola gene a o , and o he pa s connec ed o he powe g id by con ol ci cui s, powe
con e e s, o in e e s. The p oduc ion e iciency o PV de ices mus be inc eased h ough
e ec i e design and implemen a ion. PV powe gene a ion is non-linea by na u e and
is impac ed by en i onmen al a iables, including empe a u e, ligh in ensi y, and load
cha ac e is ics. The e o e, PV de ices need con ol ci cui s o high powe -gene a ion
pe o mance. Con ol ci cui s in PV de ices op imize cu en and ou pu ol age o inc ease
powe p oduc ion and sys em e iciency [10].
The wo main models o modeling sola PV cells include he double-diode model
(DDM) [
12
] and he single-diode model (SDM) [
13
]. DDM can e ec i ely ep esen a PV cell
since i is mo e accu a e han SDM. Planning and op imizing sola PV cells is complica ed
because he e a e abou en unknown a iables in his p oblem [
9
]. In es iga ions show
ha cu en cu es in e ms o ol age o I-V in PV cells ha e a non-linea and complex
na u e. A con olle is an essen ial componen in any PV ci cui . In pho o ol aic sys ems,
con olle s a e usually used o manage he cha ging o ba e ies o he powe supplies o
he g id [
14
]. Maximum powe poin acking (MPPT) is one o he capabili ies ha some
con olle s use o inc ease sola cells’ powe e iciency. MPPT algo i hms, especially in
la ge sys ems, a e essen ial in inc easing powe p oduc ion and managemen . Fo a sola
module, he MPPT acks he cu en and ol age o each he maximum powe and hen
abso bs i [
14
]. The e a e h ee ways o ind MPPT when sol ing non-linea equa ions
in a con olle . Nume ical app oaches [
10
], e olu iona y algo i hms [
15
], and analy ical
me hods [
16
] a e a ew o hese echniques. In o de o sol e non-linea calcula ions
wi h nume ous unknowns, he i s me hod, which mos ly uses linea calcula ions, mus
be ini ia ed. P edic ing he ini ial alue o sol ing he equa ion, which de e mines he
con e gence a e o he solu ion, is he mos di icul aspec o such app oaches. The second
app oach elies on local sea ch echniques. Al hough nume ical app oaches ha e se e al
limi a ions, hey can gene a e p ope ies o PV cells ha a e mo e p ecise han analy ical
me hods. Fi s , he con e gence o his echnique hea ily depends on he ini ial explana ion.
Second, hey a e p one o e o s, and hi d, ob aining he app op ia e model pa ame e s
akes ime and e o . In ecen yea s, op imiza ion and me a-heu is ic me hods ha e played
an essen ial ole in sol ing non-linea calcula ions and p o iding solu ions e y close o
he op imal solu ion. Using hese echniques, sea ching o he ideal solu ion akes li le
ime. Va ious me a-heu is ic echniques ha e ecen ly been employed o pinpoin sola
cell model pa ame e s in PV sys ems; o example, GA [
13
], PSO, MCDM [
12
], WOA [
17
],
Ene gies 2024,17, 1716 3 o 26
GWO [
18
], and IHHO-VMD [
16
] algo i hms a e used o his pu pose. Table 1p esen s
he applica ion o me a-heu is ic, bio-inspi ed, and hyb id op imiza ion algo i hms o
pa ame e ex ac ion o PV cell models.
Table 1. U ilizing me a-heu is ic op imiza ion algo i hms o pa ame e ex ac ion o PV cell models.
Re Aim Algo i hm Limi a ions Ad an ages Resul s
[10,13]
The s udy seeks o
c ea e a me hod o
designing
ze o-ene gy
esiden ial s uc u es
by employing
building
pe o mance
simula ion
echnology and
mul i-objec i e
op imiza ion, aiming
o a ain he mos
e icien
ene gy-sa ing
solu ions adap able
o a ious clima e
zones in China.
NSGA-II
Challenges en ail he
necessi y o
enhancing he
building ene gy
calcula ion model o
encompass egional
nuances and he
simpli ica ion o
some pa ame e s o
esea ch p ac icali y.
The s eng hs o he
esea ch include he
c ea ion o a pa ame ic
design pla o m o
de e mine ene gy
consump ion h esholds
and pho o ol aic
eplacemen a es
speci ic o di e se
clima e zones, p o iding
aluable guidance o
policymake s and
s anda d-se ing bodies.
Fu he mo e, he s udy
unde sco es he
signi icance o holis ic
conside a ions beyond
me e ene gy usage in
esiden ial a chi ec u al
planning.
The indings indica e
ha nea -ze o ene gy
consump ion is iable in
selec clima e zones in
China, wi h de ined
pho o ol aic
eplacemen a es.
Addi ionally, he
esea ch o e s c ucial
guidance on ine- uning
design pa ame e s o
ha monize ene gy
e iciency, economic
iabili y, and esiden ial
com o , highligh ing
he need o a oid o e ly
zealous pu sui o
ze o-ene gy a ge s o
main ain a balanced
app oach.
[12]
The s udy aims o
compa e decision
ee and pa icle
swa m op imiza ion
algo i hms o
iden i ying op imal
sola powe plan
loca ions, p o iding
insigh s o
enewable ene gy
planning in I an.
PSO, MCDM
The esea ch
ecognizes
cons ain s such as
he una ailabili y o
speci ic da a such as
ansmission line
in o ma ion,
po en ially
impac ing he
p ecision o he
indings.
The a icle p esen s he
applica ion o he
decision ee me hod in
iden i ying p ime si es
o sola powe plan s,
o e ing a esh
app oach o
en i onmen al s udies.
Fu he mo e, i assesses
he e icacy o he
decision ee agains he
PSO me hod,
emphasizing he
decision ee’s
ad an age in his
pa icula domain.
The decision ee
me hod ou pe o ms
PSO in p edic ing
high-po en ial sola
ene gy a eas,
emphasizing i s
e ec i eness in
iden i ying op imal si es
o sola p ojec s.
Employing spa ial da a
mining echniques is
ad ised o imp o e si e
sui abili y assessmen
o powe plan s, wi h
eas e n and
sou heas e n I an
highligh ed as especially
a o able egions.
[17]
The s udy
in es iga es he
impac o PVs and
EVs on economic
emission dispa ch,
p oposes a modi ied
WOA o
op imiza ion, and
e i ies i s
pe o mance wi h
simula ions.
WOA
Cons ain s o he
s udy encompass he
in ica e na u e o
he op imiza ion
issue, non-linea and
non-con ex
cons ain s, and he
necessi y o
me iculous
conside a ion o
di e se ac o s like
al e poin loading
e ec s, es ic ed
a eas, and
ansmission losses.
The s eng hs o he
s udy a e ound in i s
capaci y o e icien ly
handle he in ica e and
con adic o y aims o
economic emission
dispa ching h ough he
in eg a ion o PVs and
EVs, along wi h
p oposing a cus omized
WOA algo i hm ha
su passes o he
op imiza ion me hods in
deli e ing supe io
quali y esul s.
The indings o he
s udy indica e ha
inco po a ing PVs and
EVs in o economic
dispa ching leads o
lowe emissions and
ene gy gene a ion cos s,
while he p oposed
modi ied WOA
algo i hm consis en ly
yields supe io quali y
esul s when compa ed
o o he op imiza ion
algo i hms employed o
economic load dispa ch.
Ene gies 2024,17, 1716 4 o 26
Table 1. Con .
Re Aim Algo i hm Limi a ions Ad an ages Resul s
[18]
The esea ch aims o
imp o e he
pe o mance o
MPPT con olle s by
op imizing hem
wi h he GWO
algo i hm and
assessing hei
e iciency compa ed
o adi ional
me hods in a ious
condi ions, ocusing
on esponse ime,
e iciency, and powe
gene a ion.
GWO
Challenges in he
s udy encompass he
need o ine- uning
he me aheu is ic
algo i hm when
applied wi hin a
PLC manda ing
his o ical i adia ion
da a ac oss di e se
wea he condi ions.
The s eng hs o he
s udy a e e iden in he
enhanced e icacy o he
MPPT con olle ,
op imized using he
GWO algo i hm,
esul ing in supe io
e iciency and powe
gene a ion when
con as ed wi h
con en ional
app oaches.
Fu he mo e, he
esea ch o e s aluable
obse a ions ega ding
he esponse ime o
a ious algo i hms
amids changing
condi ions.
The esea ch
demons a es ha
op imizing he MPPT
con olle wi h GWO
enhances e iciency and
powe gene a ion while
minimizing
o e shoo ing, wi h
GWO exhibi ing as e
esponse imes han
adi ional algo i hms.
[16]
The esea ch aims o
mi iga e he
a iabili y in
pho o ol aic ou pu
by sugges ing a
hyb id ene gy
s o age se up
s a egy.
IHHO-VMD
Challenges in ol e
he possibili y o
cons aining
pho o ol aic ou pu
and diminishing
powe gene a ion,
alongside he
es ic ed ene gy
s o age capaci y o
he HESS, indica ing
he need o
explo ing imp o ed
decomposi ion
echniques and
in eg a ing elec ic
hyd ogen in o HESS,
wa an ing u he
in es iga ion.
Bene i s encompass a
6.15% dec ease in he
hyb id ene gy s o age
sys em cos ela i e o
he o iginal algo i hm as
well as mi iga ed powe
luc ua ions, leading o
enhanced sys em
economy and s abili y.
The IHHO-VMD
algo i hm e ec i ely
educes ene gy s o age
sys em cos s, enhances
powe alloca ion, and
s abilizes pho o ol aic
g id-connec ed powe .
While MA helps
mi iga e powe
luc ua ions, challenges
emain wi h
pho o ol aic ou pu and
HESS capaci y,
wa an ing u he
explo a ion o imp o ed
decomposi ion me hods
and HESS expansion.
Finding app op ia e model pa ame e s is a challenging op imiza ion p oblem o
e alua ing PV pa ame e s. In gene al, classical op imiza ion algo i hms canno ind he
op imal solu ion. New me a-heu is ic algo i hms ha e demons a ed be e abili y in global
and local sea ch and sea ch he p oblem space mo e op imally. Op imiza ion me hods
such as gene ics and pa icles use basic mechanisms o sea ching, while op imizing he
pa ame e s o a PV is a complex p oblem wi h se e al a iables. In his esea ch, we
unde ook pa ame e op imiza ion o he SDM, DDM, and PV sola cell models o enhance
sola cell e iciency. We hus p opose a ailo ed COA [
19
], speci ically c a ed o imp o e
he e iciency o PV de ices. Mo eo e , we in oduce an enhanced e sion o he COA
ha os e s mu ual lea ning. Addi ionally, we conduc ed pa ame e es ima ion o PV
models, compa ing hem wi h ecen me a-heu is ic me hods. Finally, we he ein p esen
an upg aded i e a ion o he coa i op imiza ion algo i hm, which in eg a es chaos heo y,
o p o ide a comp ehensi e app oach o op imizing sola cell pa ame e s and imp o ing
e iciency. This s udy in oduces a no el op imiza ion s a egy aimed a enhancing he
e iciency o sola cells and op imizing hei pa ame e s. The main con ibu ions o his
manusc ip a e as ollows:
■Pa ame e op imiza ion o he SDM, DDM, and PV sola cell models;
Ene gies 2024,17, 1716 5 o 26
■Imp o ing he e iciency o sola cells;
■
In oducing a coa i op imiza ion algo i hm [
15
] designed o enhance he e iciency o
PV de ices;
■
In oducing an upg aded e sion o he coa i op imiza ion algo i hm ha os e s
mu ual lea ning;
■
Es ima ing pa ame e s o PV models and compa ing hem wi h ecen me a-
heu is ic me hods;
■
In oducing an enhanced e sion o he coa i op imiza ion algo i hm in eg a ed wi h
chaos heo y.
This esea ch wo k was compiled and is he ein p esen ed in i e sec ions. Sec ion 2
in oduces sola cells and hei componen s and ela ed wo ks o op imizing PV de ice
pa ame e s e iew. Sec ion 3shows he p oposed me hod o imp o ing he COA o he
PV pa ame e s’ op imal es ima ion. Sec ion 4explains es s and implemen a ion, and
he esul s a e analyzed and e alua ed. Finally, Sec ion 5includes he conclusions and
u u e wo k.
2. Rela ed Wo ks
A s a egy o op imizing he pho o ol aic model’s pa ame e s was p esen ed in e-
sea ch [
20
] ha applied no he n goshawk op imiza ion (NGO). In o de o de e mine
he iple-diode model’s pa ame e s (PV module), his esea ch applied an op imiza ion
algo i hm known as NGO. Th ee comme cial PV modules we e applied in he cu en
esea ch. The simula ion esul s demons a e ha NGO ou pe o med he o he op imiza-
ion algo i hms in e ms o speed and accu acy. Fu he mo e, wi h his echnique, he cos
unc ion o he Canadian Sola CS6K-M module may be educed o 0.000195.
A s a egy o op imizing he sola pho o ol aic models’ pa ame e s by applying
di e en ial e olu ion and queue sea ch op imiza ion was discussed in ano he pape [21].
The PV model has a mul i-model and non-linea speci ica ion, making i challenging o
de e mine i s ideal alues. The algo i hms employed o add ess his p oblem a e p one o
s ick in local op ima because o he non-linea na u e o he p oblem. Due o hei la ge
impac on he PV sys em’s cu en and powe gene a ion pe o mance, he pa ame e s’
app op ia e es ima ion is c ucial. In o de o ex ac he ideal PV pa ame e alues, his
s udy p o ided an enhanced queue sea ch op imiza ion (QSO) based on he di e en ial
e olu ion (DE) me hod. Thei me hod ou pe o med o he me hods like gene ic algo i hms
and pa icles o ob aining he bes pa ame e s, e.g., single-diode, double-diode, and PV
module models.
A p e ious s udy in oduced a uzzy sola PV and wind u bine sys em employing
pa icle swa m op imiza ion (PSO) [
22
] o boos e iciency. Howe e , accu a ely c ea ing a
powe o ecas ing model allows a esea che o egula e he andomized beha io o sola
and wind ene gy sou ces. Sola PV and wind o ecas ing algo i hms based on uzzy logic
may be e handle his unp edic able and andom aspec . Fu he mo e, he pe o mance o
he o ecas ing model was imp o ed by using hyb id uzzy–PSO in elligen o ecas ing,
which also enhanced he sys em’s es ic ions. Thei es s e ealed ha he p oposed
uzzy model was mo e e ec i e in boos ing he powe o sola and wind sys ems han he
uzzy model used in conjunc ion wi h a gene ic algo i hm (GA). In ano he s udy [
23
], he
op imiza ion o elec ici y p oduc ion by sola ene gy was p esen ed using imp o ed MPPT
echniques. In his esea ch, hey op imized he pa ame e s o he pho o ol aic module
wi h op imiza ion me hods.
In a esea ch pape [
24
], a hyb id app oach based on he ba algo i hms (BA) and
g asshoppe op imiza ion algo i hms (GOA) was p esen ed o maximize powe gene a ion
h ough sola pho o ol aics. This esea ch used a combined me a-heu is ic algo i hm o
ex ac he maximum powe om PV using he XSG con olle .
The p oposed algo i hm pe o med well o powe ex ac ion, acco ding o expe imen s,
and is mo e capable and e ec i e a boos ing powe han he BA and GOA algo i hms.
Ene gies 2024,17, 1716 6 o 26
One pape [
25
] alida ed he i e ly algo i hm’s e icacy in op imizing sola cell and
pho o ol aic module pa ame e s, especially compa ed o expe imen al da a and he exis ing
li e a u e, highligh ing i s e ec i eness in minimizing e o me ics and accu a ely ep oduc-
ing cu en - ol age cha ac e is ics unde a ying i adiance and empe a u e condi ions.
The esea ch om [
26
] endea o ed o c ea e a dependable app oach employing he
Lambe W unc ion o p ecisely es ima e single-diode PV pa ame e s, ackling issues
p esen in cu en me hodologies, such as inaccu acies in RMSE compu a ion and exces-
si e dependence on op imiza ion me hods. The sugges ed analy ical solu ion enhanced
pa ame e es ima ion accu acy, no ably obse able in single-diode PV equi alen ci cui s,
by ec i ying RMSE calcula ion inaccu acies iden i ied in he exis ing li e a u e.
The s udy endea o ed o de elop a dependable me hod o accu a ely es ima ing
PV pa ame e s, u ilizing an inno a i e hyb id s a egy ha in eg a es di e si ica ion and
in ensi ica ion mechanisms om di e en me aheu is ics (MHs). I ackled issues such
as compu a ional complexi y and pa ame e sensi i i y, highligh ing i s abili y o adap
o a ious op imiza ion challenges, explo e mul iple sea ch spaces simul aneously, and
enhance accu acy and eliabili y as e idenced by compa isons wi h al e na i e MHs and
benchma k unc ions [27].
The e e ence [
19
] p esen ed he a i hme ic op imiza ion algo i hm (IAOA) as a solu-
ion o imp o e he es ima ion o PV model pa ame e s, ackling issues such as pa ame e
sensi i i y and compu a ional complexi y. Despi e i s po en ial d awbacks, IAOA show-
cased no able p ecision and dependabili y in es ima ing sola cell pa ame e s, e icien ly
op imizing PV models ac oss a ious scena ios and su passing o he algo i hms in e ms
o accu acy and pe o mance.
The s udy o [
28
] p esen ed and assessed he moun ain gazelle op imize (MGO)
algo i hm’s e ec i eness in pinpoin ing PV model pa ame e s, speci ically a ge ing he
SDM and DDM o pho o ol aic sys ems. MGO showcased bene i s like apid p ocessing,
consis en con e gence, and p ecise esul s, ou pe o ming o he algo i hms wi h he lowes
RMSE ac oss 30 sepa a e i e a ions.
The use o he new op imiza ion echnique o p edic he ideal pa ame e s in sola
modules was obse ed in one s udy [
29
]. This s udy conside ed he Ha is hawks op i-
mize (HHO) o acqui e he PV sys ems’ model pa ame e s. The modi ied HHO o e s a
wo ldwide sea ch capaci y, high e iciency, and high con e gence speed compa ed o he
con en ional me hod. In addi ion, he esea ch e ealed ha he HHO me hod has a lowe
e o alue o ol age powe (P-V) and cu en - ol age (I-V) ea u es.
Ano he esea ch [
30
] p esen ed he e alua ion and imp o emen o he pho o ol aic
g id-connec ed sys em using he VPFOTADF con olle wi h he imp o ed e sion o
Wall’s algo i hm. This pape aimed o design sola cells o educe ha monic dis o ion and
imp o e he sola sys em’s pe o mance connec ed o he pho o ol aic g id using g oup
in elligence. The sola PV sys em has componen s such as a boos e con e e , pho o ol aic
a ay, mul i-le el in e e , and con olle . This s udy op imized he ampli ie con e e
using he imp o ed WOA algo i hm. E alua ion and es s showed ha hei me hod
inc eases he p oduc ion capaci y mo e han he WOA algo i hm.
In ano he s udy, an imp o ed a i hme ic op imiza ion algo i hm (AOA) was p o-
posed o ex ac ing pa ame e s o a single-diode pho o ol aic sola cell model [
31
]. The
expe imen al indings demons a ed ha IAOA ou comes a e mo e e ec i e and accu a e
han hose ob ained using AOA.
A me hod o de e mine he sola PV model’s op imal pa ame e s using he chimp
op imiza ion algo i hm (ChOA) was men ioned in one esea ch pape [
21
]. To p oduce
p ecise and us wo hy PV models, including single-diode, dual-diode, iple-diode, and
PV module models, his esea ch sugges ed a no el echnique called ChOA ha is inspi ed
by na u e. The undamen al di icul y in p edic ing he PV models’ pa ame e s using
op imiza ion me hods is con e gence o he local op imum. The e o e, his s udy in eg a ed
he bes dis inc i e aspec s o PSO and a local sea ch echnique. The es s e ealed ha
hei sugges ed algo i hm ou pe o ms he EHHOA, BMO, FPSO, CBBO, and GOTLA
Ene gies 2024,17, 1716 7 o 26
algo i hms in e ms o op imizing he h ee comme cial modules’ pa ame e s ha a e o en
used: KC200GT, SW255, and SM55 mul i-c ys al.
3. Me hodology
In he p esen s udy, he COA algo i hm was selec ed due o i s pionee ing me hodol-
ogy, which amalgama es elemen s o biological inspi a ion, popula ion-based op imiza ion,
modeling o na u al beha io s, mu ual lea ning, opposi ion-based lea ning, and in eg a ion
o chaos heo y. This comp ehensi e app oach empowe s he algo i hm o e ec i ely
na iga e in ica e sea ch spaces and disco e high-quali y solu ions, ende ing i a com-
pelling op ion o op imizing PV sys em pa ame e s. The bene i s o u ilizing he COA o
op imizing PV sys ems a e ou lined in Table 2.
Table 2. Ad an ages o he COA o PV Sys em Op imiza ion.
Re Ad an age Desc ip ion Bene i o PV Sys em Op imiza ion
[32] Biological inspi a ion Mimics coa is’ in elligen hun ing and
e asion beha io s.
No el op imiza ion pe spec i e, po en ially
leading o adap i e and
esou ce ul solu ions.
[15]Popula ion-based
app oach
Explo es mul iple solu ions
simul aneously.
E icien ly inds global op imums in complex
PV sys em p oblems.
[33]In eg a ion o na u al
beha io s
Models coa is’ hun ing and e asion
beha io s o op imiza ion.
Achie es as e con e gence and mo e
obus solu ions.
[34]Opposi ion-based
Lea ning
Di e si ies explo a ion by gene a ing
opposi e solu ions.
P e en s p ema u e con e gence and
encou ages explo a ion o di e se egions.
[35] Chaos heo y in eg a ion In oduces andomness o escape
local op ima.
Enhances explo a ion capabili ies and
a oids s agna ion.
The P-V and I-V cu e p ope ies o PV modules and sola cells a e c ea ed using
wo common ma hema ical models desc ibed in his sec ion. Fi s , he SDM model and
o mula ion a e explained, hen he DDM model is explained, and inally, he ci cui ’s
op imum pa ame e s a e disco e ed using he coa i op imiza ion algo i hm.
3.1. SMD Ci cui
Figu e 1shows he elec ical ci cui equi alen o SDM. In his ci cui , I
ph
ep esen s
he cu en p oduced by he pho ogene a ed cu en . The SDM amewo k is ela i ely
simple. Fi s , he gene a ed diode cu en I
d
, ou pu cu en I
L
, shun cu en I
sh
, and ho
Iph a e de ined.
Ene gies 2024, 17, x FOR PEER REVIEW 7 o 27
algo i hms in e ms o op imizing he h ee comme cial modules’ pa ame e s ha a e o -
en used: KC200GT, SW255, and SM55 mul i-c ys al.
3. Me hodology
In he p esen s udy, he COA algo i hm was selec ed due o i s pionee ing me hod-
ology, which amalgama es elemen s o biological inspi a ion, popula ion-based op imiza-
ion, modeling o na u al beha io s, mu ual lea ning, opposi ion-based lea ning, and in-
eg a ion o chaos heo y. This comp ehensi e app oach empowe s he algo i hm o e -
ec i ely na iga e in ica e sea ch spaces and disco e high-quali y solu ions, ende ing i
a compelling op ion o op imizing PV sys em pa ame e s. The bene i s o u ilizing he
COA o op imizing PV sys ems a e ou lined in Table 2.
Table 2. Ad an ages o he COA o PV Sys em Op imiza ion.
Re
Ad an age
Desc ip ion
Bene i o PV Sys em Op imiza ion
[32]
Biological inspi a-
ion
Mimics coa is’ in elligen hun ing
and e asion beha io s.
No el op imiza ion pe spec i e, po en ially leading o
adap i e and esou ce ul solu ions.
[15]
Popula ion-based
app oach
Explo es mul iple solu ions simul-
aneously.
E icien ly inds global op imums in complex PV sys-
em p oblems.
[33]
In eg a ion o na -
u al beha io s
Models coa is’ hun ing and e asion
beha io s o op imiza ion.
Achie es as e con e gence and mo e obus solu-
ions.
[34]
Opposi ion-based
Lea ning
Di e si ies explo a ion by gene a -
ing opposi e solu ions.
P e en s p ema u e con e gence and encou ages ex-
plo a ion o di e se egions.
[35]
Chaos heo y in e-
g a ion
In oduces andomness o escape
local op ima.
Enhances explo a ion capabili ies and a oids s agna-
ion.
The P-V and I-V cu e p ope ies o PV modules and sola cells a e c ea ed using wo
common ma hema ical models desc ibed in his sec ion. Fi s , he SDM model and o mu-
la ion a e explained, hen he DDM model is explained, and inally, he ci cui ’s op imum
pa ame e s a e disco e ed using he coa i op imiza ion algo i hm.
3.1. SMD Ci cui
Figu e 1 shows he elec ical ci cui equi alen o SDM. In his ci cui , Iph ep esen s
he cu en p oduced by he pho ogene a ed cu en . The SDM amewo k is ela i ely
simple. Fi s , he gene a ed diode cu en Id, ou pu cu en IL, shun cu en Ish, and ho
Iph a e de ined.
Figu e 1. SDM ci cui s uc u e in PV sys ems [36].
Then, Ki chho ’s cu en law is used o ob ain IL using Equa ion (1). The equa ions
o he diode Shockley and Ohm laws a e, espec i ely, used o ob ain Id using Equa ion
(2) and 𝐼𝑠ℎ applying Equa ion (3).
Figu e 1. SDM ci cui s uc u e in PV sys ems [36].
Then, Ki chho ’s cu en law is used o ob ain I
L
using Equa ion (1). The equa ions o
he diode Shockley and Ohm laws a e, espec i ely, used o ob ain I
d
using Equa ion (2)
and Ish applying Equa ion (3).
IL=Iph −Id−Ish (1)
Ene gies 2024,17, 1716 8 o 26
Id=Isd ·expq·(VL+RS·IL)
n·k·T−1(2)
Ish =VL+RS·IL
Rsh
(3)
Equa ion (1) is expanded using Equa ions (2)–(4):
IL=Iph −Id·expq·(VL+RS·IL)
n·k·T−1−VL+RS·IL
Rsh
(4)
R
S
is a se ies esis o , R
sh
is a pa allel esis o , V
L
s ands o ou pu ol age k,q
s ands o ini ial cha ge (1.60217646
×
10
−19
), and nis he ideali y ac o o he diode
and e e s o Bol zmann’s cons an and equals (1.3806503
×
10
−23
J/K q). He e, SDM
iden i ies i e di e en unknown pa ame e s (
Iph
,
Id
,
RS
,
Rsh
, and n), and T ep esen s he
absolu e empe a u e.
3.2. DDM Ci cui
The co esponding elec ical ci cui o DDM is shown in Figu e 2. DDM can mo e
p ecisely depic he ol age–cu en ela ionship gi en he inhe en limi a ions o SDM. A
diag am o he DDM ci cui is shown in Figu e 2. Because DDM has an addi ional diode
(in pa allel) compa ed o SDM, Figu e 2demons a es how DDM di e s om SDM. I
L
is
calcula ed using Equa ion (5):
IL=Iph −Id1·expq·(VL+RS·IL)
n1·k·T−1−Id2·expq·(VL+RS·IL)
n2·k·T−1−VL+RS·IL
Rsh
(5)
He e,
Id2
s ands o sa u a ion cu en , n
1
and n
2
ep esen he ideal sa u a ion coe -
icien o wo diodes, and
Id1
is he diode emission cu en . DDM needs o ex ac se en
di e en pa ame e s (Iph,Id1,Id2,RS,Rsh,n1, and n2).
Ene gies 2024, 17, x FOR PEER REVIEW 8 o 27
𝐼𝐿=𝐼𝑝ℎ−𝐼𝑑−𝐼𝑠ℎ
(1)
𝐼𝑑=𝐼𝑠𝑑⋅[exp(𝑞⋅(𝑉𝐿+𝑅𝑆⋅𝐼𝐿)
𝑛⋅𝑘⋅𝑇 )−1]
(2)
𝐼𝑠ℎ=𝑉𝐿+𝑅𝑆⋅𝐼𝐿
𝑅𝑠ℎ
(3)
Equa ion (1) is expanded using Equa ions (2)–(4):
𝐼𝐿=𝐼𝑝ℎ−𝐼𝑑⋅[exp(𝑞⋅(𝑉𝐿+𝑅𝑆⋅𝐼𝐿)
𝑛⋅𝑘⋅𝑇 )−1]−𝑉𝐿+𝑅𝑆⋅𝐼𝐿
𝑅𝑠ℎ
(4)
RS is a se ies esis o , Rsh is a pa allel esis o , VL s ands o ou pu ol age k, q s ands
o ini ial cha ge (1.60217646 × 10−19), and n is he ideali y ac o o he diode and e e s o
Bol zmann’s cons an and equals (1.3806503 × 10−23 J/K q). He e, SDM iden i ies i e di e -
en unknown pa ame e s (𝐼𝑝ℎ, 𝐼𝑑, 𝑅𝑆, 𝑅𝑠ℎ, and n), and T ep esen s he absolu e empe -
a u e.
3.2. DDM Ci cui
The co esponding elec ical ci cui o DDM is shown in Figu e 2. DDM can mo e
p ecisely depic he ol age–cu en ela ionship gi en he inhe en limi a ions o SDM. A
diag am o he DDM ci cui is shown in Figu e 2. Because DDM has an addi ional diode
(in pa allel) compa ed o SDM, Figu e 2 demons a es how DDM di e s om SDM. IL is
calcula ed using Equa ion (5):
𝐼𝐿=𝐼𝑝ℎ−𝐼𝑑1⋅[exp(𝑞⋅(𝑉𝐿+𝑅𝑆⋅𝐼𝐿)
𝑛1⋅𝑘⋅𝑇 )−1]−𝐼𝑑2⋅[exp(𝑞⋅(𝑉𝐿+𝑅𝑆⋅𝐼𝐿)
𝑛2⋅𝑘⋅𝑇 )−1]−𝑉𝐿+𝑅𝑆⋅𝐼𝐿
𝑅𝑠ℎ
(5)
He e, 𝐼𝑑2 s ands o sa u a ion cu en , n1 and n2 ep esen he ideal sa u a ion coe -
icien o wo diodes, and 𝐼𝑑1 is he diode emission cu en . DDM needs o ex ac se en
di e en pa ame e s (𝐼𝑝ℎ, 𝐼𝑑1, 𝐼𝑑2, 𝑅𝑆, 𝑅𝑠ℎ, 𝑛1, and 𝑛2).
Figu e 2. DDM ci cui s uc u e in PV sys ems [36].
3.3. PV Module Modeling
Figu e 3 shows he PV module’s ci cui diag am wi h nume ous PV cells connec ed
in pa allel o se ies. IL is calcula ed using Equa ion (6):
𝐼𝐿=𝑁𝑃⋅𝐼𝑝ℎ−𝑁𝑃⋅𝐼𝑑⋅[exp(𝑞⋅(𝑉𝐿/𝑁𝑆+𝑅𝑆⋅𝐼𝐿/𝑁𝑃)
𝑛⋅𝑘⋅𝑇 )−1]−𝑁𝑃⋅𝑉𝐿/𝑁𝑆+𝑅𝑆⋅𝐼𝐿
𝑅𝑠ℎ
(6)
The e ms NS and NP ep esen he quan i y o pa allel and se ial connec ions be ween
PV cells. The SDM o PV modules is used in his esea ch. The PV module should de e -
mine i e unknowns (Iph, Id, RS, Rsh, and n).
Figu e 2. DDM ci cui s uc u e in PV sys ems [36].
3.3. PV Module Modeling
Figu e 3shows he PV module’s ci cui diag am wi h nume ous PV cells connec ed in
pa allel o se ies. ILis calcula ed using Equa ion (6):
IL=NP·Iph −NP·Id·expq·(VL/NS+RS·IL/NP)
n·k·T−1−NP·VL/NS+RS·IL
Rsh
(6)
The e ms N
S
and N
P
ep esen he quan i y o pa allel and se ial connec ions be ween
PV cells. The SDM o PV modules is used in his esea ch. The PV module should de e mine
i e unknowns (Iph,Id,RS,Rsh, and n).
Ene gies 2024,17, 1716 9 o 26
Ene gies 2024, 17, x FOR PEER REVIEW 9 o 27
Figu e 3. Equi alen ci cui o a PV module [36].
3.4. Objec i e Func ion
The disc epancy be ween he de e mined calcula ed alue and he ac ual measu ed
alue is objec i ely assessed using an objec i e unc ion. The calcula ed da a poin s o he
SDM, DDM, and PV modules and e o unc ions o he expe imen s a e ep esen ed by
Equa ions (7)–(9).
{𝑓𝑆𝐷(𝑉𝐿,𝐼𝐿,𝑿)=𝐼𝐿−𝐼𝑝ℎ+𝐼𝑑∗[exp(𝑞(𝑉𝐿+𝑅𝑔−𝐼𝐿)
𝑛−𝑘𝑇 )−1]+𝑉𝐿+𝑅𝑆⋅𝐼𝐿
𝑅Lh
𝑿={𝐼ph,𝐼𝑑,𝑅𝑆,𝑅sh ,𝑛}
(7)
{
𝑓𝐷𝐷(𝑉𝐿,𝐼𝐿,𝑿)=𝐼𝐿−𝐼𝑝ℎ+𝐼𝑑1⋅[exp(𝑞(𝑉𝐿+𝑅𝑆⋅𝐼𝐿)
𝑛1⋅𝑘𝑇 )−1]
+𝐼𝑑2⋅[exp(𝑞−(𝑉𝐿+𝑅𝑠−𝐼𝐿)
𝑛2⋅𝑘−𝑇 )−1]+𝑉𝐿+𝑅𝑆⋅𝐼𝐿
𝑅𝑠ℎ
𝑿={𝐼𝑝ℎ,𝐼𝑑𝑑,𝐼𝑑2,𝑅𝑆,𝑅𝑠ℎ,𝑛1,𝑛2}
(8)
{
𝑓𝑀𝐷(𝑉𝐿,𝐼𝐿,𝑿)=𝐼𝐿− 𝑁𝑃⋅𝐼𝑀ℎ+𝑁𝑃⋅𝐼𝑑⋅[exp(𝑞(𝑉𝐿/𝑁𝑠+𝑅𝑆⋅𝐼𝐿/𝑁𝑃)
𝑛⋅𝑘⋅𝑇 )−1]
+𝑁𝑃⋅𝑉𝐿/𝑁𝑆+𝑅𝑦⋅𝐼𝐿
𝑅𝑠ℎ
𝑿= {𝐼𝑝ℎ,𝐼𝑑,𝑅𝑆,𝑅𝑠ℎ,𝑛}
(9)
Equa ion (10) employs RMSE as an objec i e unc ion o objec i ely assess he o al
dispa i y be ween expe imen al and calcula ed da a.
RMSE(𝑿)=√1
𝑁∑𝑘=1
𝑁 𝑓𝑘(𝑉𝐿,𝐼𝐿,𝑿)2
(10)
In his o mula, X ep esen s he solu ion consis ing o di e en unknown pa ame-
e s, and N shows he numbe o ac ual measu ed da a.
3.5. E o Me ics
The pape in es iga es PV pa ame e ex ac ion using he i e ly algo i hm, empha-
sizing i s accu acy in es ima ing pa ame e s o sola cells and PV modules. A h ee-s ep
me hodology assesses he algo i hm’s e icacy, compa ing i s pe o mance wi h o he
me hods and e alua ing e o me ics like RE and IAE. The esul s indica e he i e ly
algo i hm’s supe io pe o mance and accu acy, con i ming i s e ec i eness in PV pa am-
e e es ima ion compa ed o al e na i e algo i hms [37].
Figu e 3. Equi alen ci cui o a PV module [36].
3.4. Objec i e Func ion
The disc epancy be ween he de e mined calcula ed alue and he ac ual measu ed
alue is objec i ely assessed using an objec i e unc ion. The calcula ed da a poin s o he
SDM, DDM, and PV modules and e o unc ions o he expe imen s a e ep esen ed by
Equa ions (7)–(9).
SD(VL,IL,X)=IL−Iph +Id*expq(VL+Rg−IL)
n−kT −1+VL+RS·IL
RLh
X=nIph,Id,RS,Rsh ,no(7)
DD(VL,IL,X)=IL−Iph +Id1·hexpq(VL+RS·IL)
n1·kT −1i
+Id2·hexpq−(VL+Rs−IL)
n2·k−T−1i+VL+RS·IL
Rsh
X=nIph,Idd,Id2,RS,Rsh,n1,n2o
(8)
MD(VL,IL,X)=IL−NP·IMh +NP·Id·hexpq(VL/Ns+RS·IL/NP)
n·k·T−1i
+NP·VL/NS+Ry·IL
Rsh
X=nIph,Id,RS,Rsh,no
(9)
Equa ion (10) employs RMSE as an objec i e unc ion o objec i ely assess he o al
dispa i y be ween expe imen al and calcula ed da a.
RMSE(X) = 1
N∑N
k=1 k(VL,IL,X)2(10)
In his o mula, X ep esen s he solu ion consis ing o di e en unknown pa ame e s,
and Nshows he numbe o ac ual measu ed da a.
3.5. E o Me ics
The pape in es iga es PV pa ame e ex ac ion using he i e ly algo i hm, empha-
sizing i s accu acy in es ima ing pa ame e s o sola cells and PV modules. A h ee-s ep
me hodology assesses he algo i hm’s e icacy, compa ing i s pe o mance wi h o he me h-
ods and e alua ing e o me ics like RE and IAE. The esul s indica e he i e ly algo i hm’s
supe io pe o mance and accu acy, con i ming i s e ec i eness in PV pa ame e es ima ion
compa ed o al e na i e algo i hms [37].
In he e e ence [
20
], analysis o e o me ics such as RMSE e eals TSA’s supe io
accu acy in pa ame e ex ac ion compa ed o o he algo i hms. TSA exhibi s as e con-
Ene gies 2024,17, 1716 16 o 26
The undamen al necessi ies ou lined in his a icle a e p esen ed in Table 3.
Table 3. The ad an ages o sola PV sys ems.
Requi emen s Desc ip ion
Op imiza ion o sola PV pa ame e s Maximize he e iciency o sola PV sys ems by
accu a ely op imizing hei pa ame e s.
Reduc ion in en i onmen al impac Mi iga e en i onmen al pollu ion and educe eliance
on non- enewable ene gy sou ces like ossil uels.
Ad ancemen o
op imiza ion echniques
Rep esen s a signi ican ad ancemen in op imiza ion
echniques o sola PV sys ems.
Enhanced s abili y and accu acy
Imp o e he s abili y and accu acy o pa ame e
op imiza ion in PV modules and sola cells, he eby
inc easing he eliabili y and pe o mance o sola
ene gy sys ems.
3.8. The Challenges Tackled by This Pape
The challenges acked by his pape comp ise he ollowing:
•
Complexi y o Pa ame e Op imiza ion: The p ocess o op imizing pa ame e s wi hin
PV modules and sola cells en ails g appling wi h in ica e ma hema ical models and
inhe en unce ain ies s emming om luc ua ions in sola adia ion and empe a u e;
•
Requi emen o Imp o ed Op imiza ion Algo i hms: T adi ional op imiza ion algo-
i hms may encoun e di icul ies in adequa ely managing he in icacies o pa ame e
op imiza ion wi hin sola PV sys ems, unde sco ing he necessi y o no el app oaches
such as he COA algo i hm p oposed he ein;
•
A aining Global Op ima: The ask o iden i ying he global op imum solu ion o pa-
ame e op imiza ion in sola PV sys ems is a duous due o he exis ence o nume ous
local op ima and he expansi e sea ch space wi h high dimensions.
3.9. The Con ibu ions Made by This Pape o he Field
The con ibu ions made by his pape o he co esponding ield a e as ollows:
(1)
Inno a i e Op imiza ion S a egy: In oducing a esh op imiza ion s a egy u iliz-
ing he COA algo i hm and chao ic unc ions, which exhibi s supe io e ec i eness
compa ed o exis ing me a-heu is ic algo i hms in e ms o minimizing RMSE and
s anda d de ia ion;
(2)
Enhanced Pa ame e Op imiza ion: The p oposed app oach ensu es mo e p ecise
and consis en op imiza ion o pa ame e s wi hin SDM, DDM, and PV modules,
consequen ly imp o ing powe gene a ion e iciency and he o e all eliabili y o
sola PV sys ems;
(3) Po en ial o Fu u e Resea ch: The pape ou lines u u e esea ch p ospec s, including
he explo a ion o LSTM neu al ne wo ks o op imizing sola cell pa ame e s and
o ecas ing sola adia ion and panel empe a u e. This indica es p omising a enues
o ad ancing echniques in sola ene gy op imiza ion.
4. Resul s and Discussion
The sugges ed app oach is based on ou popula PV models ha we e applied and
p o en h ough SDM assaul in he DDM a ack in he RTC F ance cell, RTC F ance cell,
STP6-120/36 module, and Pho owa -PWP201 module. Cu en - ol age da a o DDM
and SDM we e measu ed on a 57 mm diame e comme cial silicon RTC. A 33
◦
C, F ench
sola cells we e measu ed below 1000 W/m
2
. A o al o 36 polyc ys alline silicon cells
comp ising a Pho owa -PWP201 module we e linked in se ies, and a 45
◦
C, he cu en -
ol age da a we e measu ed. Mo eo e , a o al o 36 polyc ys alline silicon cells, measu ed
a 55
◦
C, made up he STP6-120/36 modules. A ew ea lie in es iga ions we e s udied o
ga he he STP6-120/36 module’s cu en - ol age measu emen da a [
29
,
38
]. The p oposed
Ene gies 2024,17, 1716 17 o 26
algo i hm was implemen ed on he MATLAB 2021 pla o m using an In el
®
i7-HQ CPU
wi h 16 GB memo y. Th ough pa ame e op imiza ion o dis inc PV modules employing
he COA, he app oach no only boos s ou pu powe bu also educes e o s and s anda d
de ia ion, su passing adi ional algo i hms such as ITLBO, JSO, CPMPSO, WOA, SCA,
GNDO, and MJSO. Fu he mo e, i ou pe o med compe ing algo i hms in e ms o RMSE
and s anda d de ia ion, indica ing i s supe io pe o mance and sui abili y ac oss a ious
ci cui con igu a ions. Addi ionally, he me hod’s e icien execu ion ime u he bols e s
i s p ac ical applicabili y. In summa y, hese esul s unde sco e he e ec i eness and
e sa ili y o he p oposed me hod in ackling di e se op imiza ion challenges wi hin sola
ene gy sys ems.
4.1. The Range o Pa ame e s
Table 4demons a es he SM55 and ST40 pa ame e s, speci ically hei uppe and
lowe anges. In addi ion o he pa ame e s used o implemen he p oposed algo i hm, he
numbe o epe i ions is he conside ed a iable, and he popula ion size is 50. In he COA
algo i hm, has a andom alue be ween ze o and one, and I equals a andom numbe
ha equals 1 o 2. The alues o he chao ic unc ion in he p oposed me hod we e se a
z_0 = 0.125 and β= 2.59.
Table 4. Lowe and uppe ange in he modules [31].
Pa ame e s Low Range Uppe Range
Iph(A)0 2∗Isc
Isd(A)0100 ×10−6
Rs(Ω)0 2
Rsh(Ω)0 5000
n,n1,n21 4
4.2. Resul s Based on SDM
Table 5displays he op imal pa ame e s’ alues ex ac ed o he SDM ci cui in he
p oposed me hod, namely he imp o ed COA (ICOA), and compa es ITLBO, JSO, CPMPSO,
WOA, SCA, GNDO, MJSO, and COA. The expe imen s showed ha all me a-heu is ic
algo i hms excep WOA and SCA ha e minimized RMSE alues.
Table 5. Compa ison among op imal alues o pa ame e s in SDM.
Algo i hms Iph(A)Isd(A)Rs(Ω)Rsh(Ω)nRMSE
ITLBO [21] 0.76078 3.11 ×10−70.03654 52.8897 1.47726
0.000773006
JSO [22] 0.76079 3.11 ×10−70.03654 52.8882 1.47727
0.000773006
CPMPSO [39] 0.76078 3.11 ×10−70.03654 52.8897 1.47726
0.000773006
WOA [38] 0.76162 3.86 ×10−70.03530 45.9308 1.49953
0.001085820
SCA [29] 0.7582 4.09 ×10−70.03595 68.8388 1.50500
0.002483415
GNDO [30] 0.76078 3.11 ×10−70.03654 52.8897 1.47726
0.000773006
MJSO [31] 0.76078 3.11 ×10−70.03654 52.8897 1.47726
0.000773006
COA [15] 0.76078 3.11 ×10−70.03654 52.8897 1.47726
0.000773006
ICOA
(p oposed me hod) 0.76078 3.11 ×10−70.03654 52.8897 1.47726
0.000773006
Figu e 9displays he I-V and P-V cha ac e is ic cu es based on he ICOA’s ex ac ed
op imal pa ame e s o SDM. I is clea om Figu e 9 ha he e is a good ag eemen
be ween he measu ed and simula ed ICOA da a.
Ene gies 2024,17, 1716 18 o 26
Ene gies 2024, 17, x FOR PEER REVIEW 18 o 27
Table 4. Lowe and uppe ange in he modules [31].
Pa ame e s
Low Range
Uppe Range
𝐼𝑝ℎ(A)
0
2∗Isc
𝐼𝑠𝑑(A)
0
100 × 10−6
𝑅𝑠(Ω)
0
2
𝑅𝑠ℎ(Ω)
0
5000
n, 𝑛1,𝑛2
1
4
4.2. Resul s Based on SDM
Table 5 displays he op imal pa ame e s’ alues ex ac ed o he SDM ci cui in he
p oposed me hod, namely he imp o ed COA (ICOA), and compa es ITLBO, JSO,
CPMPSO, WOA, SCA, GNDO, MJSO, and COA. The expe imen s showed ha all me a-
heu is ic algo i hms excep WOA and SCA ha e minimized RMSE alues.
Table 5. Compa ison among op imal alues o pa ame e s in SDM.
Algo i hms
𝑰𝒑𝒉(𝑨)
𝑰𝒔𝒅(𝑨)
𝑹𝒔(𝛀)
𝑹𝒔𝒉(𝛀)
n
RMSE
ITLBO [21]
0.76078
3.11 × 10−7
0.03654
52.8897
1.47726
0.000773006
JSO [22]
0.76079
3.11 × 10−7
0.03654
52.8882
1.47727
0.000773006
CPMPSO [39]
0.76078
3.11 × 10−7
0.03654
52.8897
1.47726
0.000773006
WOA [38]
0.76162
3.86 × 10−7
0.03530
45.9308
1.49953
0.001085820
SCA [29]
0.7582
4.09 × 10−7
0.03595
68.8388
1.50500
0.002483415
GNDO [30]
0.76078
3.11 × 10−7
0.03654
52.8897
1.47726
0.000773006
MJSO [31]
0.76078
3.11 × 10−7
0.03654
52.8897
1.47726
0.000773006
COA [15]
0.76078
3.11 × 10−7
0.03654
52.8897
1.47726
0.000773006
ICOA (p oposed
me hod)
0.76078
3.11 × 10−7
0.03654
52.8897
1.47726
0.000773006
Figu e 9 displays he I-V and P-V cha ac e is ic cu es based on he ICOA’s ex ac ed
op imal pa ame e s o SDM. I is clea om Figu e 9 ha he e is a good ag eemen be-
ween he measu ed and simula ed ICOA da a.
(a)
(b)
Ene gies 2024, 17, x FOR PEER REVIEW 19 o 27
(c)
Figu e 9. Compa ison o he measu ed and es ima ed da a ob ained by ICOA o SDM: (a) I-V cha -
ac e is ic, (b) P-V cha ac e is ic, and (c) IAE cu es.
4.3. Resul s Based on DDM
Table 6 compa es he alue o op imal pa ame e s ex ac ed o he DDM ci cui us-
ing he p oposed me hod, ITLBO, JSO, CPMPSO, WOA, SCA, GNDO, MJSO, and COA.
Table 6. Compa ison among op imal alues o pa ame e s o DDM.
Algo i hms
𝑰𝒑𝒉(𝑨)
𝑰𝒔𝒅𝟏(𝑨)
Rs(Ω)
Rsh(Ω)
n1
𝑰𝒔𝒅𝟐(𝑨)
n2
RMSE
ITLBO [21]
0.7608
2.47 × 10−7
0.0368
53.9599
1.4579
4.78 × 10−7
1.9949
0.000742264
JSO [22]
0.7608
5.38 × 10−7
0.0371
54.4640
1.7980
1.61 × 10−7
1.4262
0.000754167
CPMPSO [39]
0.7608
7.03 × 10−8
0.0378
56.2715
1.3642
1.00 × 10−6
1.7963
0.000741937
WOA [38]
0.7608
2.67 × 10−7
0.0368
51.8538
1.4662
4.10 × 10−8
1.6133
0.000776464
SCA [29]
0.7684
0.00 × 10+0
0.0324
38.3064
1.1740
3.84 × 10−7
1.4970
0.007351184
GNDO [30]
0.7608
1.00 × 10−6
0.0373
55.6033
1.9051
1.40 × 10−7
1.4130
0.000742327
MJSO [31]
0.7608
7.03 × 10−8
0.0378
56.2715
1.3642
1.00 × 10−6
1.7963
0.000741937
COA [15]
0.7607
7.01 × 10−8
0.0377
56.2716
1.3641
1.00 × 10−6
1.7964
0.000741936
ICOA (p oposed
me hod)
0.7608
7.02 × 10−8
0.0377
56.2714
1.3642
1.00 × 10−6
1.7962
0.000741936
The compa ison and es s esul s show ha in he DDM ci cui , he p oposed me hod
p o ided mo e op imal alues o he DDM ci cui han he JSO, CPMPSO, WOA, SCA,
GNDO, MJSO, and COA me hods. The RMSE e o index in he p oposed me hod shows
a lowe alue. The COA algo i hm anks second a e he p oposed me hod, and he
CPMPSO and MJSO algo i hms ank hi d.
Figu e 10 displays he I-V and P-V cha ac e is ic cu es based on he ICOA’s ex-
ac ed op imal pa ame e s o DDM. I is clea om Figu e 10 ha he e is a good ag ee-
men be ween he measu ed and simula ed ICOA da a.
Figu e 9. Compa ison o he measu ed and es ima ed da a ob ained by ICOA o SDM: (a) I-V
cha ac e is ic, (b) P-V cha ac e is ic, and (c) IAE cu es.
4.3. Resul s Based on DDM
Table 6compa es he alue o op imal pa ame e s ex ac ed o he DDM ci cui using
he p oposed me hod, ITLBO, JSO, CPMPSO, WOA, SCA, GNDO, MJSO, and COA.
Table 6. Compa ison among op imal alues o pa ame e s o DDM.
Algo i hms Iph(A)Isd1(A)Rs(Ω)Rsh(Ω)n1Isd2(A)n2RMSE
ITLBO [21] 0.7608
2.47
×
10
−70.0368 53.9599 1.4579
4.78
×
10
−71.9949
0.000742264
JSO [22] 0.7608
5.38
×
10
−70.0371 54.4640 1.7980
1.61
×
10
−71.4262
0.000754167
CPMPSO [39] 0.7608
7.03
×
10
−80.0378 56.2715 1.3642
1.00
×
10
−61.7963
0.000741937
WOA [38] 0.7608
2.67
×
10
−70.0368 51.8538 1.4662
4.10
×
10
−81.6133
0.000776464
SCA [29] 0.7684 0.00 ×10+0 0.0324 38.3064 1.1740
3.84
×
10
−71.4970
0.007351184
GNDO [30] 0.7608
1.00
×
10
−60.0373 55.6033 1.9051
1.40
×
10
−71.4130
0.000742327
MJSO [31] 0.7608
7.03
×
10
−80.0378 56.2715 1.3642
1.00
×
10
−61.7963
0.000741937
COA [15] 0.7607
7.01
×
10
−80.0377 56.2716 1.3641
1.00
×
10
−61.7964
0.000741936
ICOA
(p oposed me hod) 0.7608
7.02
×
10
−80.0377 56.2714 1.3642
1.00
×
10
−61.7962
0.000741936
The compa ison and es s esul s show ha in he DDM ci cui , he p oposed me hod
p o ided mo e op imal alues o he DDM ci cui han he JSO, CPMPSO, WOA, SCA,
GNDO, MJSO, and COA me hods. The RMSE e o index in he p oposed me hod shows
Ene gies 2024,17, 1716 19 o 26
a lowe alue. The COA algo i hm anks second a e he p oposed me hod, and he
CPMPSO and MJSO algo i hms ank hi d.
Figu e 10 displays he I-V and P-V cha ac e is ic cu es based on he ICOA’s ex ac ed
op imal pa ame e s o DDM. I is clea om Figu e 10 ha he e is a good ag eemen
be ween he measu ed and simula ed ICOA da a.
Ene gies 2024, 17, x FOR PEER REVIEW 20 o 27
Figu e 10. Compa ison o he measu ed and es ima ed da a ob ained by ICOA o DDM model: (a)
I-V cha ac e is ic, (b) P-V cha ac e is ic, and (c) IAE cu es.
4.4. Resul s Based on STP6-120/36
Table 7 compa es he alue o op imal pa ame e s o he STP6-120/36 ci cui in he
p oposed me hod wi h simila me a-heu is ic me hods. The conduc ed es s show ha in
he STP6-120/36 ci cui , he p oposed me hod’s RMSE alue was lowe han he me a-
heu is ic algo i hms JSO, CPMPSO, WOA, SCA, GNDO, MJSO, and COA. In hese expe -
imen s, he COA me hod’s e o is he second lowes in e ms o minimum. On he o he
hand, he expe imen s show ha he CPMPSO, ITLBO, MJSO, and GNDO me hods ank
hi d o RMSE e o .
Table 7. Compa ison among op imal pa ame e alues in STP6-120/36.
Algo i hms
𝑰𝒑𝒉(𝑨)
𝑰𝒔𝒅(𝑨)
𝑹𝒔(𝛀)
𝑹𝒔𝒉(𝛀)
N
RMSE
ITLBO [21]
7.47528
1.93 × 10−6
0.16891
570.1972
44.80042
0.014251063
JSO [22]
7.47525
1.93 × 10−6
0.16890
571.5660
44.80254
0.014251066
CPMPSO [39]
7.47528
1.93 × 10−6
0.16891
570.1975
44.80042
0.014251063
WOA [38]
7.50318
3.27 × 10−6
0.15781
307.7831
46.40846
0.017581962
SCA [29]
7.56027
1.70 × 10−6
0.17318
323.9495
44.38346
0.052443825
GNDO [30]
7.47528
1.93 × 10−6
0.16891
570.1972
44.80042
0.014251063
MJSO [31]
7.47528
1.93 × 10−6
0.16891
570.1975
44.80042
0.014251063
COA [15]
7.47528
1.92 × 10−6
0.16891
570.1975
44.80041
0.014251063
ICOA (p oposed
me hod)
7.47528
1.93 × 10−6
0.16891
570.1974
44.80041
0.014251063
Figu e 10. Compa ison o he measu ed and es ima ed da a ob ained by ICOA o DDM model:
(a) I-V cha ac e is ic, (b) P-V cha ac e is ic, and (c) IAE cu es.
4.4. Resul s Based on STP6-120/36
Table 7compa es he alue o op imal pa ame e s o he STP6-120/36 ci cui in he p o-
posed me hod wi h simila me a-heu is ic me hods. The conduc ed es s show ha in he
STP6-120/36 ci cui , he p oposed me hod’s RMSE alue was lowe han he me a-heu is ic
algo i hms JSO, CPMPSO, WOA, SCA, GNDO, MJSO, and COA. In hese expe imen s, he
COA me hod’s e o is he second lowes in e ms o minimum. On he o he hand, he
expe imen s show ha he CPMPSO, ITLBO, MJSO, and GNDO me hods ank hi d o
RMSE e o .
Acco ding o he expe imen a ion, he wo s algo i hm o his si ua ion is he SCA
algo i hm, which has he highes e o among he compa ed algo i hms.
Figu e 11 displays he I-V and P-V cha ac e is ic cu es based on he ICOA’s ex ac ed
op imal pa ame e s o STP6-120/36. I is clea om Figu e 11 ha he e is a good ag eemen
be ween he measu ed and simula ed ICOA da a.
Ene gies 2024,17, 1716 20 o 26
Table 7. Compa ison among op imal pa ame e alues in STP6-120/36.
Algo i hms Iph(A)Isd(A)Rs(Ω)Rsh(Ω)NRMSE
ITLBO [21] 7.47528 1.93 ×10−60.16891 570.1972
44.80042 0.014251063
JSO [22] 7.47525 1.93 ×10−60.16890 571.5660
44.80254 0.014251066
CPMPSO [39] 7.47528 1.93 ×10−60.16891 570.1975
44.80042 0.014251063
WOA [38] 7.50318 3.27 ×10−60.15781 307.7831
46.40846 0.017581962
SCA [29] 7.56027 1.70 ×10−60.17318 323.9495
44.38346 0.052443825
GNDO [30] 7.47528 1.93 ×10−60.16891 570.1972
44.80042 0.014251063
MJSO [31] 7.47528 1.93 ×10−60.16891 570.1975
44.80042 0.014251063
COA [15] 7.47528 1.92 ×10−60.16891 570.1975
44.80041 0.014251063
ICOA
(p oposed me hod) 7.47528 1.93 ×10−60.16891 570.1974
44.80041 0.014251063
Ene gies 2024, 17, x FOR PEER REVIEW 21 o 27
Acco ding o he expe imen a ion, he wo s algo i hm o his si ua ion is he SCA
algo i hm, which has he highes e o among he compa ed algo i hms.
Figu e 11 displays he I-V and P-V cha ac e is ic cu es based on he ICOA’s ex-
ac ed op imal pa ame e s o STP6-120/36. I is clea om Figu e 11 ha he e is a good
ag eemen be ween he measu ed and simula ed ICOA da a.
Figu e 11. Compa ison o he measu ed and es ima ed da a ob ained by ICOA o STP6-120/36
model, (a) I-V cha ac e is ic, (b) P-V cha ac e is ic, and (c) IAE cu es.
4.5. Ranking
This sec ion discusses he anking o he p oposed algo i hm and compe ing algo-
i hms such as WOA, GWO, HHO, AVOA, and COA in wo e o indica o s, namely
RMSE and s anda d de ia ion, o SMD, DDM, and PV modules, espec i ely, which a e
shown in Figu es 12 and 13.
Figu e 11. Compa ison o he measu ed and es ima ed da a ob ained by ICOA o STP6-120/36
model, (a) I-V cha ac e is ic, (b) P-V cha ac e is ic, and (c) IAE cu es.
4.5. Ranking
This sec ion discusses he anking o he p oposed algo i hm and compe ing algo-
i hms such as WOA, GWO, HHO, AVOA, and COA in wo e o indica o s, namely RMSE
and s anda d de ia ion, o SMD, DDM, and PV modules, espec i ely, which a e shown
in Figu es 12 and 13.
Ene gies 2024,17, 1716 21 o 26
Ene gies 2024, 17, x FOR PEER REVIEW 22 o 27
Figu e 12. Ranking o algo i hms in e ms o he minimum RMSE e o .
Figu e 13. Algo i hm anking based on s anda d de ia ion (SD) index.
In he expe imen s, F iedman’s es was applied o ank di e en algo i hms. In
F iedman’s es , any algo i hm ha shows a lowe alue has a be e anking o ind he
op imal solu ion. A lowe numbe means he algo i hm has ob ained a be e anking in
inding he op imal solu ion.
Acco ding o he expe imen s, i he ci cui is SDM- ype, he anking o WOA, GWO,
HHO, AVOA, JSO, COA algo i hm, and he p oposed me hod is 1.79, 1.87, 1.98, 1.73, 1.56,
1.48, and 1.36, espec i ely. In his case, he p oposed me hod ob ained he bes anking,
and HHO showed he wo s pe o mance.
Expe imen s pe aining o he DDM ci cui show ha he anks o algo i hms, includ-
ing WOA, GWO, HHO, AVOA, JSO, COA, and he p oposed me hod o calcula ing he
minimum RMSE a e 2.54, 2.73, 2.36, 1.73, 1.52, 1.39, and 1.24, espec i ely, and he p o-
posed me hod anks bes in minimizing RMSE. The PV ci cui ’s a ing alues o WOA,
GWO, HHO, AVOA, JSO, COA, and he p oposed me hod a e 1.63, 1.88, 1.79, 1.82, 1.64,
1.25, and 1.13, espec i ely, so he p oposed me hod again showed he op pe o mance.
Figu e 12. Ranking o algo i hms in e ms o he minimum RMSE e o .
Ene gies 2024, 17, x FOR PEER REVIEW 22 o 27
Figu e 12. Ranking o algo i hms in e ms o he minimum RMSE e o .
Figu e 13. Algo i hm anking based on s anda d de ia ion (SD) index.
In he expe imen s, F iedman’s es was applied o ank di e en algo i hms. In
F iedman’s es , any algo i hm ha shows a lowe alue has a be e anking o ind he
op imal solu ion. A lowe numbe means he algo i hm has ob ained a be e anking in
inding he op imal solu ion.
Acco ding o he expe imen s, i he ci cui is SDM- ype, he anking o WOA, GWO,
HHO, AVOA, JSO, COA algo i hm, and he p oposed me hod is 1.79, 1.87, 1.98, 1.73, 1.56,
1.48, and 1.36, espec i ely. In his case, he p oposed me hod ob ained he bes anking,
and HHO showed he wo s pe o mance.
Expe imen s pe aining o he DDM ci cui show ha he anks o algo i hms, includ-
ing WOA, GWO, HHO, AVOA, JSO, COA, and he p oposed me hod o calcula ing he
minimum RMSE a e 2.54, 2.73, 2.36, 1.73, 1.52, 1.39, and 1.24, espec i ely, and he p o-
posed me hod anks bes in minimizing RMSE. The PV ci cui ’s a ing alues o WOA,
GWO, HHO, AVOA, JSO, COA, and he p oposed me hod a e 1.63, 1.88, 1.79, 1.82, 1.64,
1.25, and 1.13, espec i ely, so he p oposed me hod again showed he op pe o mance.
Figu e 13. Algo i hm anking based on s anda d de ia ion (SD) index.
In he expe imen s, F iedman’s es was applied o ank di e en algo i hms. In
F iedman’s es , any algo i hm ha shows a lowe alue has a be e anking o ind he
op imal solu ion. A lowe numbe means he algo i hm has ob ained a be e anking in
inding he op imal solu ion.
Acco ding o he expe imen s, i he ci cui is SDM- ype, he anking o WOA, GWO,
HHO, AVOA, JSO, COA algo i hm, and he p oposed me hod is 1.79, 1.87, 1.98, 1.73, 1.56,
1.48, and 1.36, espec i ely. In his case, he p oposed me hod ob ained he bes anking,
and HHO showed he wo s pe o mance.
Expe imen s pe aining o he DDM ci cui show ha he anks o algo i hms, includ-
ing WOA, GWO, HHO, AVOA, JSO, COA, and he p oposed me hod o calcula ing he
minimum RMSE a e 2.54, 2.73, 2.36, 1.73, 1.52, 1.39, and 1.24, espec i ely, and he p oposed
me hod anks bes in minimizing RMSE. The PV ci cui ’s a ing alues o WOA, GWO,
HHO, AVOA, JSO, COA, and he p oposed me hod a e 1.63, 1.88, 1.79, 1.82, 1.64, 1.25, and
1.13, espec i ely, so he p oposed me hod again showed he op pe o mance.
Ene gies 2024,17, 1716 22 o 26
Based on he simula ions conduc ed in MATLAB, he p oposed me hod anks i s in
he op imiza ion o SDM, DDM, and PV pa ame e s. The p oposed me hod also inc eases
he op imal calcula ion ank in hese h ee ci cui s by 8.1%, 10.79%, and 9.6%, espec i ely,
compa ed o he COA algo i hm. Figu e 13 displays he a e age ank o me a-heu is ic
algo i hms and he p oposed algo i hm in he s anda d de ia ion index.
Figu e 13 compa es he s anda d de ia ion (SD) o h ee modes, namely SDM, DDM,
and PV, o he SD alues o he p oposed me hod and me a-heu is ic algo i hms. The
s anda d de ia ion index is an essen ial and c i ical index o measu ing he op imiza ion
s abili y o op imizing he pa ame e s o DDM, PV, and SDM ci cui s. The ank o he
p oposed algo i hm in he s anda d de ia ion index is equal o 1.22, and i has he lowes
s anda d de ia ion among he compe ing algo i hms.
This means ha he p oposed algo i hm has mo e s abili y in e ms o op imizing he
pa ame e s o SDM, DDM, and PV ci cui s han he WOA, GWO, HHO, AVOA, JSO, and
COA algo i hms. The wo s algo i hm in e ms o s abili y in inding op imal solu ions
is he HHO algo i hm. The second me a-heu is ic algo i hm in e ms o s abili y is he
WOA algo i hm.
4.6. Time Complexi y
To ensu e equi able compa ison, we inco po a ed ou comes om al e na i e me hods
a he han solely elying on ex e nal e e ences. Ensu ing uni o mi y, we main ained
iden ical objec i e unc ions and pa ame e s ac oss all expe imen s as ou lined in he
o iginal pape s. Fu he mo e, each me hod ecei ed an equal numbe o a emp s o add ess
he p oblem, enabling a ai compa ison wi hin compa able compu a ional cons ain s.
By concen a ing on me hods add essing iden ical op imiza ion challenges, we assessed
he e ec i eness o ou p oposed model agains exis ing app oaches wi hin a con olled
amewo k. This me hod acili a ed an impa ial and s aigh o wa d e alua ion o he
s eng hs and limi a ions o each app oach.
The imp o ed COA algo i hm has mo e equa ions and complexi y han he COA algo-
i hm, so i was expec ed o be mo e ime-consuming han he COA algo i hm. Howe e ,
con a y o he expec a ion o he ICOA algo i hm, i needs ewe i e a ions o each an
e o le el because i has as e con e gence han he COA algo i hm. Figu e 14 compa es
he op imiza ion ime o he sola sys em pa ame e s in he p oposed me hod and o he
me a-heu is ic me hods.
Ene gies 2024, 17, x FOR PEER REVIEW 23 o 27
Based on he simula ions conduc ed in MATLAB, he p oposed me hod anks i s in
he op imiza ion o SDM, DDM, and PV pa ame e s. The p oposed me hod also inc eases
he op imal calcula ion ank in hese h ee ci cui s by 8.1%, 10.79%, and 9.6%, espec i ely,
compa ed o he COA algo i hm. Figu e 13 displays he a e age ank o me a-heu is ic
algo i hms and he p oposed algo i hm in he s anda d de ia ion index .
Figu e 13 compa es he s anda d de ia ion (SD) o h ee modes, namely SDM, DDM,
and PV, o he SD alues o he p oposed me hod and me a-heu is ic algo i hms. The
s anda d de ia ion index is an essen ial and c i ical index o measu ing he op imiza ion
s abili y o op imizing he pa ame e s o DDM, PV, and SDM ci cui s. The ank o he
p oposed algo i hm in he s anda d de ia ion index is equal o 1.22, and i has he lowes
s anda d de ia ion among he compe ing algo i hms.
This means ha he p oposed algo i hm has mo e s abili y in e ms o op imizing he
pa ame e s o SDM, DDM, and PV ci cui s han he WOA, GWO, HHO, AVOA, JSO, and
COA algo i hms. The wo s algo i hm in e ms o s abili y in inding op imal solu ions is
he HHO algo i hm. The second me a-heu is ic algo i hm in e ms o s abili y is he WOA
algo i hm.
4.6. Time Complexi y
To ensu e equi able compa ison, we inco po a ed ou comes om al e na i e me h-
ods a he han solely elying on ex e nal e e ences. Ensu ing uni o mi y, we main ained
iden ical objec i e unc ions and pa ame e s ac oss all expe imen s as ou lined in he o ig-
inal pape s. Fu he mo e, each me hod ecei ed an equal numbe o a emp s o add ess
he p oblem, enabling a ai compa ison wi hin compa able compu a ional cons ain s. By
concen a ing on me hods add essing iden ical op imiza ion challenges, we assessed he
e ec i eness o ou p oposed model agains exis ing app oaches wi hin a con olled
amewo k. This me hod acili a ed an impa ial and s aigh o wa d e alua ion o he
s eng hs and limi a ions o each app oach.
The imp o ed COA algo i hm has mo e equa ions and complexi y han he COA
algo i hm, so i was expec ed o be mo e ime-consuming han he COA algo i hm. How-
e e , con a y o he expec a ion o he ICOA algo i hm, i needs ewe i e a ions o each
an e o le el because i has as e con e gence han he COA algo i hm. Figu e 14 com-
pa es he op imiza ion ime o he sola sys em pa ame e s in he p oposed me hod and
o he me a-heu is ic me hods.
Figu e 14. Compa ison o he calcula ion ime o op imal PV pa ame e s.
Figu e 14. Compa ison o he calcula ion ime o op imal PV pa ame e s.
The expe imen s conduc ed in MATLAB show ha he p oposed algo i hm’s execu ion
ime is abou 6.34 s, and i needs less ime o each a ce ain e o le el han o he algo i hms.
Ene gies 2024,17, 1716 23 o 26
The pe o mance o he COA algo i hm is sligh ly wo se han he p oposed me hod in e ms
o ime index, bu i is in second place o execu ion ime. Among he compa ed algo i hms,
he wo s algo i hm in e ms o ime index is he HHO algo i hm. The eason behind he
signi ican execu ion ime o he HHO algo i hm is he la ge numbe o equa ions and he
high complexi y o his algo i hm.
5. Conclusions
Global powe demand is escala ing due o indus ial expansion and popula ion g ow h,
wi h elec ici y being he p ima y ene gy sou ce o indus ies. Howe e , con en ional
elec ici y p oduc ion me hods such as ossil uels pose en i onmen al challenges. T an-
si ioning o sola ene gy h ough PV modules o e s a enewable solu ion o mi iga e
en i onmen al pollu ion. Sola cells e icien ly con e sola ene gy in o elec ical ene gy,
ye hei ou pu luc ua es due o a ying adia ion in ensi y and angles, posing a chal-
lenge o maximizing powe gene a ion. To add ess his, he p oposed me hod u ilized
he COA o op imize pa ame e s o di e en PV modules, enhancing ou pu powe by
minimizing e o s and educing s anda d de ia ion compa ed o con en ional algo i hms.
Howe e , challenges pe sis , including unce ain y in inding op imal solu ions and e-
liance on p ecise inpu da a, which may hinde op imiza ion e icacy. Acknowledging
p ac ical obs acles like ha dwa e limi a ions and main enance complexi ies is c ucial o
eal-wo ld implemen a ion, unde sco ing he need o u he alida ion o scalabili y and
applicabili y in la ge-scale sola ene gy sys ems. Fu u e esea ch could ocus on mi iga ing
unce ain y, enhancing algo i hm obus ness, and in eg a ing p edic i e modeling ech-
niques like LSTM neu al ne wo ks o imp o e pa ame e op imiza ion and sola ene gy
o ecas ing, aiming o add ess he inhe en limi a ions and guide u u e ad ancemen s in
sola ene gy op imiza ion.
Au ho Con ibu ions: Concep ualiza ion, R.E. and A.H.; me hodology, A.H.; so wa e, R.E.; alida-
ion, A.H., J.R. and J.M.L.-G.; o mal analysis, R.E. and A.H.; in es iga ion, R.E.; esou ces, A.H.; da a
cu a ion, R.E.; w i ing—o iginal d a p epa a ion, R.E. and A.H.; w i ing— e iew and edi ing, J.R.
and J.M.L.-G.; isualiza ion, R.E.; supe ision, J.R. and J.M.L.-G.; p ojec adminis a ion, J.R.; unding
acquisi ion, J.M.L.-G. All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding: The au ho s we e suppo ed by he Vi o ia-Gas eiz Mobili y Lab Founda ion, an o ganiza-
ion o he go e nmen o he P o incial Council o A aba and he Ci y Council o Vi o ia-Gas eiz
h ough he ollowing p ojec g an (“U ilización de d ones en la mo ilidad de me cancías”).
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : The da a p esen ed in his s udy a e a ailable on eques om he
co esponding au ho . The da a a e no publicly a ailable due o p i acy.
Con lic s o In e es : The au ho s decla e no con lic s o in e es .
Abb e ia ions
De ini ions o abb e ia ions used h oughou he a icle.
COA Coa i op imiza ion algo i hm
PV Pho o ol aic
SDM Single-diode model
DDM Double-diode model
MPPT Maximum powe poin acking
IEA Ene gy agency
MPP Maximum powe poin
PWM Pulse wid h modula ion
GWO G ay wol op imiza ion
NGO No he n goshawk op imiza ion
EVs Elec ic ehicles
Ene gies 2024,17, 1716 24 o 26
HESS Hyb id ene gy s o age sys em
ICOA Imp o ed coa i op imiza ion algo i hm
SD S anda d de ia ion
MHs Me a-heu is ics
RE Rela i e e o
MJSO Modi ied a i icial jelly ish sea ch op imize
GNDO Gene alized no mal dis ibu ion op imiza ion
SCA Sine cosine algo i hm
JSO Jelly ish sea ch op imize
QSO Queue sea ch op imiza ion
DE Di e en ial e olu ion
PSO Pa icle swa m op imiza ion
GA Gene ic algo i hm
GOA G asshoppe op imiza ion algo i hms
BA Ba algo i hms
HHO Ha is hawks op imize
AOA A i hme ic op imiza ion algo i hm
ChOA Chimp op imiza ion algo i hm
WOA Whale op imiza ion algo i hm
IHHO-VMD Imp o ed Ha is hawk op imiza ion algo i hm– a ia ional mode decomposi ion
MCDM Mul iple-c i e ia decision making
NSGA-II Non-domina ed so ing gene ic algo i hm II
PLC P og ammable logic con olle
MGO Moun ain gazelle op imize
IAE Indi idual absolu e e o
RMSE Roo mean squa e e o
TSA T ee seed algo i hm
CPMPSO Classi ied pe u ba ion mu a ion-based pa icle swa m op imiza ion
ITLBO Imp o ed eaching-lea ning-based op imiza ion
AVOA A ican ul u es op imiza ion algo i hm
IAOA A i hme ic op imiza ion algo i hm
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