Ci a ion: Ma ínez-López, J.;
Po al-Po as, K.; Fe nández-Gamiz,
U.; Sánchez-Díez, E.; Ola e, J.;
Jonsson, I. Vol age and O e po en ial
P edic ion o Vanadium Redox Flow
Ba e ies wi h A i icial Neu al
Ne wo ks. Ba e ies 2024,10, 23.
h ps://doi.o g/10.3390/
ba e ies10010023
Academic Edi o : Seiji Kumagai
Recei ed: 4 Decembe 2023
Re ised: 27 Decembe 2023
Accep ed: 4 Janua y 2024
Published: 9 Janua y 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/).
ba e ies
A icle
Vol age and O e po en ial P edic ion o Vanadium Redox Flow
Ba e ies wi h A i icial Neu al Ne wo ks
Joseba Ma ínez-López 1, Koldo Po al-Po as 1, Unai Fe nández-Gamiz 1,* , Edua do Sánchez-Díez 2,
Ja ie Ola e 2and Isak Jonsson 3
1Nuclea Enginee ing and Fluid Mechanics Depa men , Uni e si y o he Basque Coun y UPV/EHU,
Nie es Cano 12, 01006 Vi o ia-Gas eiz, Spain; [email p o ec ed] (J.M.-L.);
[email p o ec ed] (K.P.-P.)
2Cen e o Coope a i e Resea ch on Al e na i e Ene gies (CIC Ene giGUNE),
Basque Resea ch and Technology Alliance (BRTA), Ala a Technology Pa k, Albe Eins ein 48,
01510 Vi o ia-Gas eiz, Spain; [email p o ec ed] (E.S.-D.); jola e@cicene gigune.com (J.O.)
3Depa men o Mechanics and Ma i ime Sciences, Di ision o Fluid Dynamics, Chalme s
Uni e si y o Technology, SE-41296 Go henbu g, Sweden; [email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac : This a icle explo es he no el applica ion o a ained a i icial neu al ne wo k (ANN) in
he p edic ion o anadium edox low ba e y beha iou and compa es i s pe o mance wi h ha
o a wo-dimensional nume ical model. The aim is o e alua e he capabili y o wo ANNs, one o
p edic ing he cell po en ial and one o he o e po en ial unde a ious ope a ing condi ions. The
wo-dimensional model, p e iously alida ed wi h expe imen al da a, was used o gene a e da a o
ain and es he ANNs. The esul s show ha he i s ANN p ecisely p edic s he cell ol age unde
di e en s a es o cha ge and cu en densi y condi ions in bo h he cha ge and discha ge modes.
The second ANN, which is esponsible o he o e po en ial calcula ion, can accu a ely p edic he
o e po en ial ac oss he cell domains, wi h he lowes con idence nea high-g adien a eas such as he
elec ode memb ane and domain bounda ies. Fu he mo e, he compu a ional ime is subs an ially
educed, making ANNs a sui able op ion o he as unde s anding and op imisa ion o VRFBs.
Keywo ds: ANN; anadium edox low ba e y; nume ical model; cell po en ial; wo-dimensional;
o e po en ial; s a es o cha ge
1. In oduc ion
Deploying e ec i e and scalable ene gy s o age sys ems is becoming inc easingly im-
po an as we mo e owa d ca bon-neu al emissions and a sus ainable ene gy u u e [
1
,
2
].
The in eg a ion o in e mi en enewable ene gy sou ces in o ou ene gy in as uc u e
is impe a i e, conside ing he commi men o achie e ca bon neu ali y. The de elop-
men o la ge-scale ene gy s o age sys ems is i al o his in eg a ion and maximising
he use o enewable esou ces while main aining he s abili y o he elec ic g id. Redox
low ba e ies (RFBs) ha e become one o he on - unne s among he se e al a ailable
choices [
3
–
6
]. They a e dis inguished by he sepa a ion o he ene gy s o age capaci y om
he powe ou pu , because RFBs s o e ene gy in chemical solu ions con ained in ex e nal
anks, allowing he capaci y o be scaled independen ly o he powe densi y [
7
,
8
]. This
decoupling o ene gy and powe ende s RFBs excep ionally lexible o a wide ange o
applica ions.
A guably, anadium-based edox low ba e ies (VRFBs) a e he mos p omising
echnology o comme cial implemen a ion [
9
,
10
]. In en ed by M. Skyllas-Kazacos e al. [
11
]
in he 1980s, VRFBs employ only one single elemen , anadium, in di e en oxida ion
s a es o bo h elec oly es, hus educing he c oss-con amina ion isk [
12
]. Mo eo e , VRFB
sys ems exhibi no ewo hy cha ac e is ics in e ms o cyclabili y, ene gy e iciency, esponse
Ba e ies 2024,10, 23. h ps://doi.o g/10.3390/ba e ies10010023 h ps://www.mdpi.com/jou nal/ba e ies
Ba e ies 2024,10, 23 2 o 15
ime, sa e ope a ion, ema kable ene gy s o age capabili y, and inco po a ing eadily ecyclable
componen s o minimize o e all cos s o e he sys em li e ime [
11
,
13
–
17
]. This se s hem
apa om o he RFB echnologies, such as zinc-based chemis ies, which p esen hyd ogen
e olu ion in cha ging, dend i e o ma ion, and une en me al deposi ion, which hen lead
o cell ailu e [
3
,
18
,
19
]. O ganic-based RFBs ha e gained a en ion o hei use o abundan ,
non-expensi e o ganic compounds, bu hey s ill ace challenges a ising om cycling
s abili y and eac an decomposi ion [
6
,
20
,
21
]. A mo e in-dep h in es iga ion is impe a i e
o consolida e his echnology o u u e scalabili y and comme cial implemen a ion.
Figu e 1depic s a isual ep esen a ion o a single cell om a VRFB. The elec oly es
a e s o ed in wo sepa a e anks. The posi i e elec oly e con ains VO
+2
and VO
2+
ions,
while he nega i e elec oly e has V
2+
and V
3+
ions. Bo h elec oly es a e eci cula ed
by pumps in o he cell. The elec odes, inside he cell, p o ide an ac i e a ea o he
elec ochemical edox eac ions o occu . To p e en c oss-con amina ion and allow o
p o ons o pass and p ese e he cha ge conse a ion, an ion-selec i e memb ane is added
be ween bo h elec odes [22].
Ba e ies 2024, 10, x FOR PEER REVIEW 2 o 15
s a es o bo h elec oly es, hus educing he c oss-con amina ion isk [12]. Mo eo e ,
VRFB sys ems exhibi no ewo hy cha ac e is ics in e ms o cyclabili y, ene gy efficiency,
esponse ime, sa e ope a ion, ema kable ene gy s o age capabili y, and inco po a ing
eadily ecyclable componen s o minimize o e all cos s o e he sys em li e ime [11,13–
17]. This se s hem apa om o he RFB echnologies, such as zinc-based chemis ies,
which p esen hyd ogen e olu ion in cha ging, dend i e o ma ion, and une en me al
deposi ion, which hen lead o cell ailu e [3,18,19]. O ganic-based RFBs ha e gained a -
en ion o hei use o abundan , non-expensi e o ganic compounds, bu hey s ill ace
challenges a ising om cycling s abili y and eac an decomposi ion [6,20,21]. A mo e in-
dep h in es iga ion is impe a i e o consolida e his echnology o u u e scalabili y and
comme cial implemen a ion.
Figu e 1 depic s a isual ep esen a ion o a single cell om a VRFB. The elec oly es
a e s o ed in wo sepa a e anks. The posi i e elec oly e con ains VO
+2
and VO
2+
ions,
while he nega i e elec oly e has V
2+
and V
3+
ions. Bo h elec oly es a e eci cula ed by
pumps in o he cell. The elec odes, inside he cell, p o ide an ac i e a ea o he elec o-
chemical edox eac ions o occu . To p e en c oss-con amina ion and allow o p o ons
o pass and p ese e he cha ge conse a ion, an ion-selec i e memb ane is added be-
ween bo h elec odes [22].
Figu e 1. Schema ic o he VRFB cell employed, wi h he nega i ely cha ged luid symbolised in ed
and he posi i ely cha ged luid deno ed in g een. Addi ional de ails can be ound in he ex .
The chemical eac ions occu ing in he hal -cells and he ull cell a e as ollows:
Ca hode: VO2
+ + 2H+ + e
discha ge
⇌
cha ge
VO2+ + H2O (1)
Anode: V
2+
discha ge
⇌
cha ge
V
3+
+ e (2)
Full cell: VO2
+ + V
2
+ + 2H+
discha ge
⇌
cha ge
VO2+ + V
3+
+ H2O (3)
The expe imen al e alua ion o VRFB pe o mance is he con en ional app oach o
assess pe o mance, bu i has a conside able inancial bu den. As he use o VRFBs in
Figu e 1. Schema ic o he VRFB cell employed, wi h he nega i ely cha ged luid symbolised in ed
and he posi i ely cha ged luid deno ed in g een. Addi ional de ails can be ound in he ex .
The chemical eac ions occu ing in he hal -cells and he ull cell a e as ollows:
Ca hode : VO+
2+2H++e−discha ge
⇌
cha ge VO2++H2O (1)
Anode : V2+discha ge
⇌
cha ge V3++e−(2)
Full cell: VO+
2+V2++2H+discha ge
⇌
cha ge VO2++V3++H2O (3)
The expe imen al e alua ion o VRFB pe o mance is he con en ional app oach o
assess pe o mance, bu i has a conside able inancial bu den. As he use o VRFBs in a i-
ous applica ions has become inc easingly popula , he e is a g owing need o cos -e ec i e
me hods and s a egies o unde s and and op imise hei pe o mance and sys em in eg a-
ion. Nume ical models o e a apid and i e a i e design app oach, allowing esea che s
o easily modi y sys em pa ame e s, elec ode geome ies, and ope a ing condi ions. This
Ba e ies 2024,10, 23 3 o 15
app oach enables he explo a ion o a ious design con igu a ions and iden i ica ion o
op imal ope a ing condi ions mo e e icien ly han adi ional expe imen al me hods.
A wide ange o nume ical in es iga ions o VRFBs is a ailable in he public li e a u e,
anging om basic ze o-dimensional models [
23
–
25
] o he mos complex h ee-dimensional
models [26–31]. The use o nume ical models o simula e low ba e ies in ol es a ca e ul
balance be ween compu a ional cos and accu acy. Ze o- and one-dimensional models
a e pa icula ly help ul o p elimina y e alua ions, down-selec ion, and as explo a o y
s udies, bu ha e he limi a ion o o e simpli ying he complex beha iou o VRFBs,
neglec ing spa ial a ia ions and non-uni o mi ies wi hin he ba e y cell. The need o
compu a ional esou ces inc eases signi ican ly as he accu acy o nume ical simula ions is
imp o ed o include aspec s such as concen a ion p o iles, po en ial, and cu en densi y
dis ibu ions [
28
–
31
] o cap u e in ica e ime-dependen luid–memb ane in e ac ions.
Howe e , ecen ad ancemen s in A i icial Neu al Ne wo ks (ANNs) ha e enabled he
c ea ion o accu a e models ained on da a ob ained om Compu a ional Fluid Dynamics
(CFD) simula ions. These su oga e models o e he lexibili y and speed o low- ideli y
models, while inco po a ing many aspec s o ad anced compu a ional models. In he
ield o luid mechanics, se e al s udies ha e used ANNs o model di e en cases, such
as ac i e [
32
] and passi e low con ol de ices [
33
], ehicle ae odynamics [
34
], he mal
sys ems [35], and mul iphase-s a e sys ems [36], o men ion a ew.
The p ima y objec i e o his s udy was o model a VRFB using A i icial Neu al
Ne wo ks (ANNs). To accomplish his, nume ical simula ions we e pe o med o gene a e
aining and benchma k da a. By al e ing he cu en densi ies and s a es o cha ge in
hese simula ions, a comp ehensi e da ase was c ea ed. Speci ically, wo dis inc ANNs
we e designed and ained: one o p edic ing he ol age and he o he o p edic ing
he o e po en ial.
2. Ma e ials and Me hods
2.1. Go e ning Equa ions
The model p esen ed in his wo k consis s o h ee domains: posi i e elec ode, ion
exchange memb ane, and nega i e elec ode. The cu en wo k u ilizes a Na ion ype
ca ion exchange memb ane. This choice aligns wi h he con en ional app oach in he
li e a u e and is consis en wi h he expe imen al alida ion. Howe e , newly de eloped
anion exchange memb anes show p omising esul s in e ms o lowe anadium c osso e
and cos and enhanced H
+
pe meabili y [
37
]. The ollowing assump ions we e made in he
nume ical model:
•S a iona y condi ions;
•Incomp essible elec oly es;
•The luids we e assumed o be comple ely dilu ed;
•Side eac ions we e neglec ed;
•Bo h elec odes and he memb ane we e conside ed iso he mal;
•The p ope ies o he elec odes, elec oly e, and memb ane we e iso opic;
•Changes in he z-di ec ion o he cell we e igno ed (dep h in Figu e 1).
The model was based on wo ks published by Shah e al. [
38
] and Kneh e al. [
39
]. The
conse a ion o mass o he cha ged species can be exp essed by Equa ion (4).
∂
∂ (εci)+∇·
→
Ni=−Si(4)
Va iable
ε
e e s o he elec ode po osi y, c
i
is he concen a ion o species i, and S
i
is
he sou ce e m o he species (lis ed in Table 1).
Ba e ies 2024,10, 23 4 o 15
Table 1. Sou ce e ms o species in he posi i e and nega i e elec odes.
Sou ce Te m Posi i e Elec ode Nega i e Elec ode
SII (V(II) mass conse a ion equa ion) - →
ı/F
SIII (V(III) mass conse a ion equa ion) - −→
ı/F
SIV (V(IV) mass conse a ion equa ion) →
ı/F-
SV(V(V) mass conse a ion equa ion) −→
ı/F-
SH+(p o on concen a ion equa ion) - −2∇·→
ı/F
→
Ni
is he cha ged species lux desc ibed by he Ne ns –Planck equa ion, as shown in
Equa ion (5). →
Ni=−De
i∇ci−ziuiciF∇φl+→
uci(5)
The i s e m accoun s o di usion, whe e
De
i
is e ec i e di usi i y. In he second
e m (mig a ion e m), zi ep esen s he cha ge o species i,uiis he ionic mobili y, Fis he
Fa aday cons an , and
φl
is he liquid po en ial. In he hi d e m (con ec ion),
→
u
ep esen s
elec oly e eloci y.
The e ec i e di usi i y,
De
i
, was ob ained om he B uggemann co ela ion as
shown in Equa ion (6).
De
i=ε3/2Di(6)
The ionic mobili y u
i
was calcula ed using he Ne ns –Eins ein equa ion, as shown in
Equa ion (7), whe e Ris he uni e sal gas cons an and Tis he empe a u e.
ui=De
i
RT (7)
The elec oly e eloci y, ep esen ed by
→
u
in Equa ion (5) in he con ec ion e m, is
calcula ed by means o Da cy’s Law, as shown in Equa ion (8), whe e pis he p essu e
and
µ
is he dynamic iscosi y o he elec oly e, as indica ed in Table 2, among he o he
elec oly e p ope ies.
→
u=−K
µ∇p(8)
Table 2. Elec oly e p ope ies.
Te m Symbol Value
V(II) di usion coe icien DV22.4 ×10−10 m2s−1[40]
V(III) di usion coe icien DV32.4 ×10−10 m2s−1[40]
V(IV) di usion coe icien DV43.9 ×10−10 m2s−1[40]
V(V) di usion coe icien DV53.9 ×10−10 m2s−1[40]
HSO4−di usion coe icien DHSO−
41.33 ×10−9m2s−1[41]
SO42−di usion coe icien DSO2−
41.065 ×10−9m2s−1[41]
H+di usion coe icien DH+9.312 ×10−9m2s−1[41]
Dynamic iscosi y µ4.9238 ×10−3Pa s [42]
K ep esen s he po ous elec ode pe meabili y calcula ed using he Kozeny–Ca man
equa ion, as shown in Equa ion (9), whe e d
is he ib e diame e and k
ck
is he Kozeny–
Ca man cons an .
K=d2
ε3
16kck(1−ε)2(9)
Ba e ies 2024,10, 23 5 o 15
To ul il he condi ion o elec oneu ali y, Equa ion (10) was sol ed o all cha ged
species, excep o SO42−:
∑
i
zici=0 (10)
To in eg a e he species balance wi h elec ochemical eac ions and he cu en low
wi hin he elec ode du ing he cha ge/discha ge p ocesses, he cha ge conse a ion equa-
ion was sol ed, as shown in Equa ion (11).
∇·→
ıl=−∇·→
ıs=→
ıR(11)
Equa ion (11) indica es ha he elec ochemical eac ion a e (i
R
) di ec ly co esponds o
he cha ges lea ing he elec oly e
→
ıl
, which in u n equa e o he cha ges en e ing he elec ode
→
ıs. Bo h he liquid and solid cu en densi ies a e exp essed by Equa ions (12) and (13).
→
il=F∑
i
zi
→
Ni(12)
→
is=−σe
s∇φs(13)
The e m
σe
s
, which co esponds o he e ec i e conduc i i y o he po ous elec ode,
was calcula ed using Equa ion (14), whe e
σs
is he elec ode bulk conduc i i y, lis ed in
Table 3, wi h o he pa ame e s ela ed o he elec odes.
σe
s=(1−ε)3
2σs(14)
Table 3. Elec ode p ope ies.
Te m Symbol Value
Elec onic conduc i i y σs1×103S m−1[42]
Po osi y ε0.929 [43]
Speci ic su ace a ea a1.62 ×104m2[43]
Kozeny–Ca man cons an kck 4.28 [42]
Elec ode ib e diame e d 1.76 ×10−5m [43]
An in eg a ion wi h he Bu le –Volme law, which cha ac e izes he elec ochemical
eac ions occu ing a he su ace o he po ous ca bon elec ode, is u ilised o con e ge he
conse a ion equa ions. Following his, he elec ochemical eac ion a e (i
R
) was calcula ed
o bo h elec odes, posi i e (“+”) and nega i e (“
−
”), as shown in Equa ion (15) and
Equa ion (16), espec i ely.
iR+=ai0,+exp(1−α+)Fη+
RT −exp−α+Fη+
RT (15)
iR−=ai0,−exp(1−α−)Fη+
RT −exp−α−Fη+
RT (16)
The speci ic su ace a ea o he elec ode was ep esen ed by a,
α
is he cha ge ans e
coe icien , and
η
deno es he o e po en ial.
i0,+
and
i0,−
, he exchange cu en densi ies,
a e exp essed as shown in Equa ions (17) and (18), whe e k
+
and k
−
a e he eac ion a e
cons an s o he posi i e and nega i e side, espec i ely.
i0,+=Fk+(cIV)(1−α+)(cV)α+(17)
i0,−=Fk−(cII)(1−α−)(cIII)α−(18)
Table 4lis s he kine ic pa ame e s used in Equa ions (17) and (18).
Ba e ies 2024,10, 23 6 o 15
Table 4. Kine ic pa ame e s.
Te m Symbol Value
Ca hodic ans e coe icien α+0.5 [42]
Anodic ans e coe icien α−0.5 [42]
S anda d a e cons an o posi i e eac ion k+6.8 ×10−7m s−1[40]
S anda d a e cons an o nega i e eac ion k−1.7 ×10−7m s−1[44]
S anda d equilib ium po en ial o posi i e side E′
+1.004 V [38]
S anda d equilib ium po en ial o nega i e side
E′
−−0.255 V [38]
The o e po en ial was de e mined o he posi i e and nega i e eac ions using
Equa ions (19) and (20).
η+=φs−φl−E+(19)
η−=φs−φl−E−(20)
The s anda d equilib ium po en ials E
+
and E
−
we e ob ained using he Ne ns equa-
ion, as shown in Equa ions (21) and (22).
E+=E′
++RT
FlncIII
cII (21)
E−=E′
−+RT
Fln cV·(cH+)2
cIV !(22)
The e ec i e conduc i i y o he memb ane
σe
m
can be modelled as shown in
Equa ion (23), whe e De
H+deno es he p o on e ec i e di usion coe icien .
σe
m=F2
RT z2
iDe
H+cH+(23)
2.2. Bounda y Condi ions
Figu e 1xand ycoo dina es we e aken as e e ences o he bounda y condi ion
desc ip ion. A x=x
0
, he anode ex e nal bounda y is se as an elec ical g ound, ha is,
he solid po en ial is equal o ze o:
φs=0x=x0(24)
The species luxes a he op and bo om o he memb ane and he ex e nal bounda ies
o he elec odes a e ze o, aside om he inle s and ou le s:
−→
n·
→
Ni=0
x=x0and x=x3
x=x1and x=x2(excep p o ons)
y=0 and y=he
(25)
A y= 0, a bounda y was se o he lux en e ing he cell h ough he elec odes.
ci=cin
i
→
n·→
u=Q
εweLe)x0<x<x1and x2<x<x3
y=0(26)
whe e Qis he olume ic low a e, w
cell
is he cell wid h, and L
e
is he elec ode hickness.
Analogously, he elec odes ha e a p essu e ou le a y=h
cell
, and he lux o he species
caused by di usion is neglec ed.
p=pou
−De
i∇ci·→
n=0)x0<x<x1and x2<x<x3
y=he(27)
Ba e ies 2024,10, 23 7 o 15
A cons an cu en densi y was applied o he ex e nal bounda y o he ca hode using
Equa ion (28).
−→
n·
→
is=ia g x=x3(28)
whe e i
a g
deno es he use -de ined cu en applied o he bounda y. The sign o his pa am-
e e de e mines whe he he cell is in cha ge o discha ge. This leads o he applica ion o an
elec ical insula ion o he uppe and lowe bounda ies o he memb anes and elec odes.
−→
n·
→
is=0
−→
n·
→
il=0
x0<x<x3
y=0 and y=he(29)
The geome ical dimensions and ope a ing condi ions o he cell a e lis ed in Table 5.
Table 5. Ope a ing condi ions and cell geome ical pa ame e s.
Te m Symbol Value
Tempe a u e T298 K
S a e o Cha ge SOC 50%
Volume ic low a e Q60 mL min−1
Ou le p essu e pou 0 Pa
Elec ode hickness Le0.003 m [42]
Elec ode wid h we0.025 m [42]
Elec ode leng h he0.02 m [42]
Memb ane hickness Lm125 µm [42]
2.3. Nume ical Model
The cell was modelled using he comme cial so wa e COMSOL Mul iphysics 5.5 wi h
i s inco po a ed physics packages, including Da cy’s law and e ia y cu en dis ibu ion.
By employing he ini e elemen me hod, he model ea u ed a s uc u ed mesh made o
4616 quad a ic elemen s. The compu a ional app oach adhe ed o a ela i e e o se a
1.0 ×10−6.
2.4. Neu al Ne wo k
In he p esen s udy, a mul ilaye pe cep on wi h backp opaga ion (MLP-BP) is
used, which is a mul ilaye model wi h hidden laye s. In his model, he ou pu
y
is
es ima ed using Equa ion (30). The ou pu o each hidden neu on is calcula ed wi h he
sigmoid unc ion de ined in Equa ion (31), which ecei es as inpu he pos synap ical
hi
o
each
i
neu on om he p e ious laye , calcula ed wi h he linea combina ion de ined in
Equa ion (32)
, whe e
x
ep esen s he inpu s o he laye s,
ω
he weigh s o he laye s, and
θ he biases.
y=
i=Nhidden
∑
i=1
ωi·gi→
x+θ(30)
gi→
x=1
1+e−hi(31)
hi→
x=
j=Nhidden
∑
j=1
ω′
i,j·xj+θ′
i(32)
The comme cial so wa e MATLAB 2022a [
45
], comme cial code wi h i s Deep Lea n-
ing oolbox [46], was used o design and ain he ANN.
The numbe o hidden laye s and neu ones in each ne wo k depended on he com-
plexi y o he magni ude conside ed. Hence, wo di e en ne wo ks we e ained o he
wo a ge ed ANN models: one o ol age p edic ion, as shown in Figu e 2, and he o he
o o e po en ial p edic ion, as shown in Figu e 3. The aining da a we e spli in o 70%
aining, 20% alida ion, and 10% es ing o bo h ne wo ks.
Ba e ies 2024,10, 23 8 o 15
Ba e ies 2024, 10, x FOR PEER REVIEW 8 o 15
wo a ge ed ANN models: one o ol age p edic ion, as shown in Figu e 2, and he o he
o o e po en ial p edic ion, as shown in Figu e 3. The aining da a we e spli in o 70%
aining, 20% alida ion, and 10% es ing o bo h ne wo ks.
The i s ne wo k (ANN1) p edic s he ol age o he cell unde cha ge and discha ge
egimes o diffe en cha ge and cu en densi ies. ANN1 is a ela i ely simple ANN wi h
a single inpu laye comp ising h ee pa ame e s: SoC, Cu en densi y, Cha ging o Dis-
cha ging, a single hidden laye wi h eigh nodes, and an ou pu laye wi h a single node.
Figu e 2 shows he a chi ec u e o ANN1.
Figu e 2. A chi ec u e o he ANN o ol age p edic ion.
The second ne wo k (ANN2) aims o p edic he spa ial o e po en ial o he cell o
a cons an S a e o Cha ge o 50% which is expec ed o be subs an ially less con inuous
compa ed o ANN1. Hence, added complexi y is equi ed in ANN2 which con ains h ee
hidden laye s (wi h 8, 16, and 8 nodes) and an ou pu laye wi h a single node o he
o e po en ial. Figu e 3 shows he a chi ec u e o he ANN2.
Figu e 3. A chi ec u e o he ANN o o e po en ial p edic ion.
Figu e 2. A chi ec u e o he ANN o ol age p edic ion.
Ba e ies 2024, 10, x FOR PEER REVIEW 8 o 15
wo a ge ed ANN models: one o ol age p edic ion, as shown in Figu e 2, and he o he
o o e po en ial p edic ion, as shown in Figu e 3. The aining da a we e spli in o 70%
aining, 20% alida ion, and 10% es ing o bo h ne wo ks.
The i s ne wo k (ANN1) p edic s he ol age o he cell unde cha ge and discha ge
egimes o diffe en cha ge and cu en densi ies. ANN1 is a ela i ely simple ANN wi h
a single inpu laye comp ising h ee pa ame e s: SoC, Cu en densi y, Cha ging o Dis-
cha ging, a single hidden laye wi h eigh nodes, and an ou pu laye wi h a single node.
Figu e 2 shows he a chi ec u e o ANN1.
Figu e 2. A chi ec u e o he ANN o ol age p edic ion.
The second ne wo k (ANN2) aims o p edic he spa ial o e po en ial o he cell o
a cons an S a e o Cha ge o 50% which is expec ed o be subs an ially less con inuous
compa ed o ANN1. Hence, added complexi y is equi ed in ANN2 which con ains h ee
hidden laye s (wi h 8, 16, and 8 nodes) and an ou pu laye wi h a single node o he
o e po en ial. Figu e 3 shows he a chi ec u e o he ANN2.
Figu e 3. A chi ec u e o he ANN o o e po en ial p edic ion.
Figu e 3. A chi ec u e o he ANN o o e po en ial p edic ion.
The i s ne wo k (ANN1) p edic s he ol age o he cell unde cha ge and discha ge
egimes o di e en cha ge and cu en densi ies. ANN1 is a ela i ely simple ANN
wi h a single inpu laye comp ising h ee pa ame e s: SoC, Cu en densi y, Cha ging o
Discha ging, a single hidden laye wi h eigh nodes, and an ou pu laye wi h a single node.
Figu e 2shows he a chi ec u e o ANN1.
The second ne wo k (ANN2) aims o p edic he spa ial o e po en ial o he cell o
a cons an S a e o Cha ge o 50% which is expec ed o be subs an ially less con inuous
compa ed o ANN1. Hence, added complexi y is equi ed in ANN2 which con ains h ee
hidden laye s (wi h 8, 16, and 8 nodes) and an ou pu laye wi h a single node o he
o e po en ial. Figu e 3shows he a chi ec u e o he ANN2.
The ANNs a e benchma ked using he s anda d Pea son p oduc -momen co ela ion
coe icien R.
Ba e ies 2024,10, 23 9 o 15
3. Resul s
3.1. Model Valida ion
The nume ical model was alida ed using expe imen al da a om You e al. [
28
],
which in ol ed placing a 5 cm
2
cell in a s a ic solu ion and measu ing he cha ge-discha ge
cu es a wo di e en cu en densi ies: 40 mA cm
−2
and 80 mA cm
−2
. Figu e 4illus a es
he excellen ag eemen be ween he nume ical esul s o he in-house simula ion and
expe imen al da a om You e al. [
28
]. The model demons a ed an a e age ela i e e o o
1.6% when calcula ing he ol age, which is compa able o he le el o ag eemen be ween
he nume ical simula ions and expe imen s desc ibed in You e al. [
28
]. The ela i e e o
has been calcula ed by means o he ollowing exp ession:
Rela i e e o (%)=Nume ical model alue −Expe imen al alue
Expe imen al alue ×100 (33)
Ba e ies 2024, 10, x FOR PEER REVIEW 9 o 15
The ANNs a e benchma ked using he s anda d Pea son p oduc -momen co ela-
ion coefficien R.
3. Resul s
3.1. Model Valida ion
The nume ical model was alida ed using expe imen al da a om You e al. [28],
which in ol ed placing a 5 cm
2
cell in a s a ic solu ion and measu ing he cha ge-dis-
cha ge cu es a wo diffe en cu en densi ies: 40 mA cm
−2
and 80 mA cm
−2
. Figu e 4
illus a es he excellen ag eemen be ween he nume ical esul s o he in-house simula-
ion and expe imen al da a om You e al. [28]. The model demons a ed an a e age el-
a i e e o o 1.6% when calcula ing he ol age, which is compa able o he le el o ag ee-
men be ween he nume ical simula ions and expe imen s desc ibed in You e al. [28]. The
ela i e e o has been calcula ed by means o he ollowing exp ession:
Rela i e e o (%) = Nume ical model alue − Expe imen al alue
Expe imen al alue ×100 (33)
(a) (b)
Figu e 4. Compa ison o expe imen ally ob ained cha ge-discha ge cu es om [42] and simula ed
cu es in a 5 cm
2
cell o : (a) a cu en densi y o 40 mA cm
−2
; (b) a cu en densi y o 80 mA cm
−2
.
3.2. A i icial Neu al Ne wo k Valida ion
To e alua e he accu acy o he ANN p edic ions, he co ela ion coefficien (R- alue)
o he es se is conside ed, since hese cases a e unknown o he ne wo k, and he e o e
de e mine he gene alisabili y o he p oposed ne wo ks. The R- alues a e aining
ANN1 and ANN2 a e shown in Figu e 5.
The R- alues o ANN1 and ANN2 we e 0.99927 and 0.99516. Bo h models p o ide
ela i ely high co ela ion coefficien s, indica ing a high le el o con idence in ANN p e-
dic ion. I should be no ed ha he wo g aphs do no ha e he same e ical scale and
ha ANN1 has no ou lie s, whe eas he mo e complex ANN2 has a subs an ial se o ou -
lie s a he ex emes.
To u he analyse he pe o mance o he wo neu al ne wo ks, Figu e 6 shows a
compa ison be ween he p edic ions o ANN1 o bo h cha ging and discha ging, oge he
wi h he CFD esul s o he conside ed cases. The ANN1 p edic ion is illus a ed as a
su ace, and he CFD esul s a e illus a ed using black ma ke s. The possibili y o span-
ning a con inuous and smoo h su ace be ween CFD cases enables he p edic ion o a well-
o mula ed ANN o ha e ew ou lie s, which is he case o he ANN1 model.
Figu e 4. Compa ison o expe imen ally ob ained cha ge-discha ge cu es om [
42
] and simula ed
cu es in a 5 cm
2
cell o : (a) a cu en densi y o 40 mA cm
−2
; (b) a cu en densi y o 80 mA cm
−2
.
3.2. A i icial Neu al Ne wo k Valida ion
To e alua e he accu acy o he ANN p edic ions, he co ela ion coe icien (R- alue)
o he es se is conside ed, since hese cases a e unknown o he ne wo k, and he e o e
de e mine he gene alisabili y o he p oposed ne wo ks. The R- alues a e aining ANN1
and ANN2 a e shown in Figu e 5.
The R- alues o ANN1 and ANN2 we e 0.99927 and 0.99516. Bo h models p o ide
ela i ely high co ela ion coe icien s, indica ing a high le el o con idence in ANN p edic-
ion. I should be no ed ha he wo g aphs do no ha e he same e ical scale and ha
ANN1 has no ou lie s, whe eas he mo e complex ANN2 has a subs an ial se o ou lie s a
he ex emes.
To u he analyse he pe o mance o he wo neu al ne wo ks, Figu e 6shows a
compa ison be ween he p edic ions o ANN1 o bo h cha ging and discha ging, oge he
wi h he CFD esul s o he conside ed cases. The ANN1 p edic ion is illus a ed as
a su ace, and he CFD esul s a e illus a ed using black ma ke s. The possibili y o
spanning a con inuous and smoo h su ace be ween CFD cases enables he p edic ion o a
well- o mula ed ANN o ha e ew ou lie s, which is he case o he ANN1 model.
