Electrochemical Impedance Spectrum Equivalent Circuit Parameter Identification Using a Deep Learning Technique

Ci a ion: Zulue a, A.; Zulue a, E.;
Ola e, J.; Fe nandez-Gamiz, U.;
Lopez-Guede, J.M.; E xebe ia, S.
Elec ochemical Impedance Spec um
Equi alen Ci cui Pa ame e
Iden i ica ion Using a Deep Lea ning
Technique. Elec onics 2023,12, 5038.
h ps://doi.o g/10.3390/
elec onics12245038
Academic Edi o s: Yi Xie, Dan Dan
and Jiahao Liu
Recei ed: 26 Sep embe 2023
Re ised: 24 No embe 2023
Accep ed: 27 No embe 2023
Published: 18 Decembe 2023
Copy igh : © 2023 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/).
elec onics
A icle
Elec ochemical Impedance Spec um Equi alen Ci cui
Pa ame e Iden i ica ion Using a Deep Lea ning Technique
Asie Zulue a 1, Ekai z Zulue a 2, Ja ie Ola e 3, Unai Fe nandez-Gamiz 1, Jose Manuel Lopez-Guede 2,*
and Saioa E xebe ia 4
1Depa men o Ene gy Enginee ing, Uni e si y o he Basque Coun y (UPV/EHU),
01006 Vi o ia-Gas eiz, Spain; [email p o ec ed] (A.Z.); [email p o ec ed] (U.F.-G.)
2Depa men o Sys em Enginee ing and Au oma ic Con ol, Uni e si y o he Basque Coun y (UPV/EHU),
01006 Vi o ia-Gas eiz, Spain; [email p o ec ed]
3Cen 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]
4Depa men o Mechanical Enginee ing, Uni e si y o he Basque Coun y (UPV/EHU),
01006 Vi o ia-Gas eiz, Spain; [email p o ec ed]
*Co espondence: [email p o ec ed]; Tel.: +34-945014084
Abs ac : Physical models a e sui able o he de elopmen and op imiza ion o ma e ials and cell
designs, whe eas models based on expe imen al da a and elec ical equi alen ci cui s (EECs) a e
sui able o he de elopmen o ope a ion es ima o s, bo h o cells and ba e ies. This esea ch wo k
de elops an inno a i e unsupe ised a i icial neu al ne wo k (ANN) aining cos unc ion o
iden i ying equi alen ci cui pa ame e s using elec ochemical impedance spec oscopy (EIS) o
iden i y and moni o pa ame e a ia ions associa ed wi h di e en physicochemical p ocesses ha
can be ela ed o he s a es o ailu e modes in ba e ies. Many echniques and algo i hms a e used o
i a p ede ined EEC pa ame e , many equi ing high-human-expe ise suppo wo k. Howe e , once
he app op ia e EEC model is selec ed o model he di e en physicochemical p ocesses associa ed
wi h a gi en ba e y echnology, he challenge is o implemen algo i hms ha can au oma ically
calcula e pa ame e a ia ions in eal ime o allow he implemen a ion o es ima o s o capaci y,
heal h, sa e y, and o he deg ada ion modes. Based on p e ious s udies using da a augmen a ion
echniques, he new ANN deep lea ning me hod in oduced in his s udy yields be e esul s han
classical aining algo i hms. The da a used in his wo k a e based on an aging and cha ac e iza ion
da ase o 80 Ah and 12 V lead–acid ba e ies.
Keywo ds: neu al ne wo ks; deep lea ning; elec ical equi alen ci cui ; elec ochemical impedance
spec oscopy; model-based es ima o s; lead–acid ba e ies
1. In oduc ion
The lec u e no es and eco dings in ECE4710/5710: Modeling, Simula ion, and Iden i ica-
ion o Ba e y Dynamics p o ide a 360
◦
iew o he modeling echniques used in ba e ies [
1
].
Resea ch has been conduc ed o apply deep lea ning (DL) echniques o ba e y analyses,
such as in [
2
,
3
], al hough he e a e also esea ch s udies based on mo e con en ional neu-
al ne wo ks, such as in [
4
,
5
]. E en mo e con en ional algo i hms such as he ex ended
Kalman il e (EKF) a e sui able o when ba e y s a e p edic ions mus be calcula ed in
eal- ime applica ions [
6
]. A de ailed e iew o machine lea ning applica ions in ba e y
s a e analyses can be ound in [7].
Physical models a e sui able o he de elopmen and op imiza ion o ma e ials and
cell designs, whe eas models based on expe imen al da a and elec ical equi alen ci cui s
(EECs) a e sui able o he de elopmen o ope a ion es ima o s, bo h o cells and ba e ies.
A good example o physical models applied o ba e y design can be seen in [8].
Elec onics 2023,12, 5038. h ps://doi.o g/10.3390/elec onics12245038 h ps://www.mdpi.com/jou nal/elec onics
Elec onics 2023,12, 5038 2 o 13
The mos basic EECs consis o a ol age sou ce in se ies wi h a esis o , ep esen ing
he open-ci cui ol age and he in e nal esis ance o he cell, espec i ely. Addi ional
R-C-scale se ies a e o en added o simula e he cha ge and mass ans e phenomena
occu ing a and a ound he elec oly e–elec ode in e ace.
One me hod o selec ing he equi alen ci cui is o simula e di e en ope a ing condi-
ions. Then, he pa ame e iza ion o he elec ical componen s ha cons i u e he ci cui is
usually pe o med using expe imen al ol age and cu en measu emen s [
9
,
10
]. Ano he
op ion is using impedance spec oscopy [
10
]. Spec oscopy is easie o pe o m and mo e
accessible. Howe e , due o he a iable na u e o he R-C ne wo k componen s conce ning
he ope a ing condi ions ( he s a e o cha ge (SOC), empe a u e, low a e, and cu en ), he
pa ame e iza ion p ocess becomes a cumbe some ask. The wide he ope a ing window
and he highe he desi ed accu acy, he g ea e he amoun o expe imen al da a equi ed
and he mo e complex he algo i hms needed o pe o m he dynamic pa ame e iza ion.
The associa ion o de e mina ion o an EEC o a spec um allows o ha e a model o
he beha io o he ba e y whe e di e en componen s o he ci cui a e iden i ied a each
equency ange, as de ailed by [11].
Elec ochemical impedance spec oscopy (EIS) can be used o eal- ime p edic ions by
in e p e ing pa ame e s om spec a [
12
]. Ba e y EIS da a ep esen a e y powe ul ool
o iden i ying ba e y EEC models, which help o e alua e di e en ba e y s a es and in
ope ando condi ions. Howe e , ini ially, he elec ochemical model is di icul o implemen
and is e y echnology-speci ic, as desc ibed in he s udy by Pille e al. [
13
]. The e o e, i
is necessa y o de elop speci ic p o ocols o ex ac ing EIS measu emen s o each ype o
elec ochemical ene gy s o age echnology, as demons a ed by Meddings e al. [
14
]. The
iden i ica ion p ocess o an EEC o be ex ac ed om EIS measu emen s usually equi es
human expe ise complemen a y o suppo -speci ic so wa e such as Z iew, which o e s
impedance- o gain-phase g aphing ools and guided ci cui selec ion ad ice, as seen in
he s udy by Csomós e al. [
15
]. Good e iews o he emaining ba e y li e p edic ion
echniques a e p esen ed in [16,17].
Addi ionally, he con inuous moni o ing o EIS da a allows o adjus ing and imp o -
ing he accu acy o EEC models based on con inuous models by allowing he iden i ica ion
o he ini ial condi ions, as iden i ied by Ola e e al. in [12].
A i icial neu al ne wo ks (ANNs) ha e been widely used in he de elopmen o
ba e y models o diagnosis and p ognosis, as epo ed by Lomba do e al. [
8
]. In hei
wo k, he au ho s applied NNs o pa ame e iden i ica ion because o he NNs’ abili y o i
non-linea ela ionships be ween inpu s (EIS da a) and ou pu s (equi alen ci cui pa ame-
e s), as demons a ed in he s udy by Chun e al. [
18
]. A ela ed s udy can be ound in [
19
].
Resea che s ha e p oposed a new loss unc ion o ain a neu al ne wo k o EEC pa ame e
p edic ions. Jimenez-Be mejo e al. [
20
] used NNs o ob ain a mo e accu a e es ima ion o
he ba e y SoC in elec ic ehicles, and Yang e al. [
21
] used backp opaga ion (BP) NNs o
es ima e he ba e y SoH in elec ic ehicles, using a la ge amoun o expe imen al da a o
eed he NNs.
ANNs can iden i y complex non-linea ela ionships be ween inpu s (expe imen al
EIS da a, in ou case) and ou pu s (EIS pa ame e s, in ou case). Howe e , ANNs equi e
ex ensi e amoun s o da a o iden i y complex ela ionships because he a chi ec u e and
numbe o weigh s a e e y la ge in hese cases. Ob aining expe imen al da a o all ba e y
ope a ing condi ions is e y ime-consuming; he e o e, we p opose ha an NN be ained
wi h an ini ial se o eal EIS da a augmen ed wi h syn he ic da a co esponding o no mal
alues o he equi alen ci cui model. We emphasize he possibili y o educing he
amoun o expe imen al da a by mo e e icien ly gene a ing syn he ic da a. This da a
augmen a ion p ocess mus be ca e ully ca ied ou , since he EEC pa ame e s a e e y
dependen on each o he , meaning i is no possible o achie e comple e s ochas ic da a
augmen a ion. As such, we p opose ha a easible da a augmen a ion p ocess is conduc ed
ia an expe imen al da a compa ison.
Elec onics 2023,12, 5038 3 o 13
A wo-s ep pa ame e iden i ica ion algo i hm o moni o EEC pa ame e a ia ions
was epo ed in a publica ion by Ola e e al. [
22
]. The au ho s’ goal wi h he p oposed
me hod was o au oma ically moni o he a ia ions in he EEC pa ame e s in ope a ion
wi h impedance senso s o in e he a ia ions in he ba e y heal h s a e o ailu e modes
om he elec ochemical in e p e a ion o hese alues, as in he s udy by Chun e al. [18].
In his wo k, we p opose a new deep lea ning (DL) echnique o iden i ying an
equi alen ci cui ’s pa ame e s om a se ies o EIS da a using a ained ANN. Once he
ANN is ained, algo i hms o e y low compu a ional complexi y can be implemen ed,
allowing he design o sys ems o economically moni o ing he e olu ion o he EEC
pa ame e s associa ed wi h he ba e y s a es.
The s uc u e o his pape is as ollows. Sec ion 2poses he a ge p oblem, p o iding
he ounda ions and add essing he solu ion. The me hod p oposed o di ec ly lea n he
model om he EIS da a is p o ided in Sec ion 3. Sec ion 4 epo s he ob ained esul s,
while Sec ion 5discusses he esul s. Finally, Sec ion 6p esen s ou conclusions.
2. EEC Pa ame e Iden i ica ion P oblem
He e, we p opose a new DL NN-based algo i hm ha p edic s an equi alen ci cui ’s
pa ame e s ia EIS, as de ined in Sec ion 2.1. To achie e his objec i e, we designed a new
aining loss unc ion ha o ces he ANN o p opose an equi alen ci cui ’s pa ame e
se (see Tables 1and 2) ha explains he EIS da a as well as possible (see Sec ion 2.2). The
a iables de ined in he ma hema ical model a e p esen ed in Table 3.
Table 1. Pa ame e s ex ac ed om EIS spec a by adjus ing he Z iew so wa e (ini ial aw da a o
he ba e ies in one o six ounds o essays).
SoC R1 CPE1T CPE1P R2 CPE2T CPE2P R3
100% 0.0027176 7.17 0.85729 0.0092174 87.18 0.65421 -
80% 0.0027953 9.21 0.77865 0.0039696 184.13 0.61221 0.21606
60% 0.0031349 11.21 0.75909 0.0021683 218.8 0.56847 0.088716
40% 0.0033452 18.01 0.62091 0.0020905 229.5 0.5006 0.066692
20% 0.0039584 14.92 0.65745 0.0020599 199.4 0.38122 0.12304
0% 0.0046775 10.12 0.70804 0.0025044 152.2 0.29418 -
Table 2. EEC pa ame e p oposal.
Pa ame e Uni Range 1
R1Ω[0.00271760, 0.00467750]
L1H[10−6, 10−3]
R2Ω[0.00205990, 0.00921740]
T1secp1.Ω−1[7.17, 18.01]
p1- [0.62091000, 0.85729000]
R3Ω[0.06669200, ∞)
T2secp1.Ω−1[87.18, 229.50]
p2- [0.29418000, 0.65421000]
1
In his able, we lis he p oposed EEC pa ame e s. The ange o each pa ame e is a alue span wi hin which
he pa ame e can easonably be ound. The maximum and he minimum alues o hese anges a e ixed by he
expe imen al da a.
To compa e he esul s o his new algo i hm, we p opose using he same equi alen
ci cui and es s p esen ed in he wo k by Ola e e al. [22].
2.1. Elec ical Equi alen Ci cui P oposal
We p opose he EEC in Figu e 1a because he eal da a he ba e y essays p oduced,
as desc ibed in Sec ion 3, i well wi h wo pai s o cons an phase elemen s in pa allel
wi h a esis o . The ou pu s o he eal expe imen s (in black) wi h he heo e ical model
p edic ions ob ained wi h Z iew a e shown in Figu e 1b (in g een). The model ob ained
Elec onics 2023,12, 5038 4 o 13
using Z iew equi es expe human supe ision. The physical meanings o he EEC
pa ame e s a e desc ibed in [
23
]. He e,
R1
is he inne esis ance,
R2
is he cha ge ans e
esis ance o he nega i e elec ode,
CPE1
is associa ed wi h a dis ibu ion o elaxa ion
imes, and
CPE2
is an impedance ela ed o highe ime cons an s han
CPE1
. Finally,
R3
is
he esis ance associa ed wi h he cha ge ans e esis ance o he posi i e elec ode.
Table 3. EEC a iables.
Va iable 2Uni De ini ion
j - Imagina y uni numbe
w ad/s F equency
EIS(jw)ΩElec ochemical impedance Spec um o he ba e y
CPE1(jw)ΩFi s cons an -phase elemen
CPE2(jw)ΩSecond cons an -phase elemen
ZCPE1,R2Ω
Equi alen pa allel impedance o i s cons an elemen and
he second esis ance
ZCPE2,R3ΩEqui alen pa allel impedance o second cons an elemen
and hi d esis ance
2In his able, we lis he p oposed EIS equi alen ci cui ’s main a iables.
Elec onics 2023, 12, x FOR PEER REVIEW 5 o 13
Figu e 1. (a) P oposed EEC ci cui model and (b) adjus men o EIS spec a o 12 V block a 100% o
he s a e o cha ge (SoC) using Z iew ( e sion 3.5i) manual so wa e.
Figu e 1a shows he selec ed elec ical equi alen ci cui . The induc i e (see 𝐿) pa s
do no con ain aluable in o ma ion abou he ba e y deg ada ion p ocess because hey
a e e y dependen on he elec ical se up, so hey a e no conside ed.
Concep ually, i is good p ac ice o de i e all exp essions o impedance in he eal
and imagina y pa s because he cu en au oma ic diffe en ia ion algo i hms do no
suppo complex a iable ope a ions:
𝑟𝑒𝑎𝑙(𝐶𝑃𝐸)=󰇡




󰇢




(6)
𝑖𝑚𝑎𝑔(𝐶𝑃𝐸)=󰇡




󰇢




(7)
𝑟𝑒𝑎𝑙(𝐶𝑃𝐸)=󰇡




󰇢




(8)
𝑖𝑚𝑎𝑔(𝐶𝑃𝐸)=󰇡




󰇢




(9)
𝑟𝑒𝑎𝑙𝑍

,

=

∙(

)(

(

))

(

)
(

(

))



(

) (10)
𝑖𝑚𝑎𝑔𝑍

,

=

∙(

)(

(

))(

)
(

(

))



(

) (11)
𝑟𝑒𝑎𝑙𝑍

,

=

∙(

)(

(

))

(

)
(

(

))



(

) (12)
𝑖𝑚𝑎𝑔𝑍

,

=

∙(

)(

(

))(

)
(

(

))



(

) (13)
Figu e 1. (a) P oposed EEC ci cui model and (b) adjus men o EIS spec a o 12 V block a 100% o
he s a e o cha ge (SoC) using Z iew ( e sion 3.5i) manual so wa e.
Table 1shows he e e ence pa ame e s ex ac ed om he EIS spec a o a lead–acid
ba e y analyzed by adjus ing he Z iew manual so wa e. These e e ence pa ame e s a e
conside ed o gene a e he syn he ic da a.
Elec onics 2023,12, 5038 5 o 13
The ma hema ical model is de ined ia Equa ions (1)–(3):
EIS(jw)=R1+jL1w+ZCPE1,R2+ZCPE2,R3(1)
ZCPE1,R2=R2CPE1(jw)
CPE1(jw)+R2(2)
ZCPE2,R3=R3CPE2(jw)
CPE2(jw)+R3(3)
The cons an phase elemen s’ impedances a e de ined in Equa ions (4) and (5):
CPE1(jw)=1
T1(jw)p1(4)
CPE2(jw)=1
T2(jw)p2(5)
Figu e 1a shows he selec ed elec ical equi alen ci cui . The induc i e (see
L1
) pa s
do no con ain aluable in o ma ion abou he ba e y deg ada ion p ocess because hey
a e e y dependen on he elec ical se up, so hey a e no conside ed.
Concep ually, i is good p ac ice o de i e all exp essions o impedance in he eal and
imagina y pa s because he cu en au oma ic di e en ia ion algo i hms do no suppo
complex a iable ope a ions:
eal(CPE1) = cosπ
2p1
T1wp1(6)
imag(CPE1) = sinπ
2p1
T1wp1(7)
eal(CPE2) = cosπ
2p2
T2wp2(8)
imag(CPE2) = sinπ
2p2
T2wp2(9)
ealZCPE1,R2=R2· eal(CPE1)(R2+ eal(CPE1))+imag2(CPE1)
(R2+ eal(CPE1))2+imag2(CPE1)(10)
imagZCPE1,R2=R2·imag(CPE1)((R2+ eal(CPE1))− eal(CPE1))
(R2+ eal(CPE1))2+imag2(CPE1)(11)
ealZCPE2,R3=R3· eal(CPE2)(R3+ eal(CPE2))+imag2(CPE2)
(R3+ eal(CPE2))2+imag2(CPE2)(12)
imagZCPE2,R3=R3·imag(CPE2)((R3+ eal(CPE2))− eal(CPE2))
(R3+ eal(CPE2))2+imag2(CPE2)(13)
The whole impedance can be de i ed ia Equa ions (14) and (15).
eal(EIS)=R1+ ealZCPE1,R2+ eal(ZCPE2,R3)(14)
imag(EIS)=L1w+imagZCPE1,R2+imag(ZCPE2,R3)(15)

Elec onics 2023,12, 5038 6 o 13
2.2. Supe ised Ve sus Unsupe ised S a egies
We p opose a new unsupe ised aining algo i hm. The usual app oach is o ain
he a i icial neu al ne wo k wi h EIS empi ical da a as inpu s and EEC pa ame e s as
ou pu s, as seen in Figu e 2. This app oach has he disad an ages o he high cos s and ime
pe iods equi ed o ob ain a su icien da ase o aining. In his esea ch, we p oduced
essays o 5 mon hs. Each mon h, he ba e ies we e cha ged and discha ged a di e en
SoCs. The cos s o hese essays we e o he esea ch pe sonnel, he ba e y se s, he
labo a o y spaces, and he measu ing equipmen and e ige a ion sys ems’ amo iza ion.
The p ocess p oduced only 36 essays, including six di e en SoC le els (100% o 20%) and
six di e en imes ( om he s a o he i h mon h). The e o e, he essay measu emen s
we e expensi e and ime-consuming. Each impedance measu emen p oduced an EIS essay.
Due o hese ime and budge limi a ions, he amoun o da a was limi ed o 36 essays. As
seen in Equa ion (17),
PANN(EISq(jw
,
θ)
is an a i icial neu al ne wo k ha p edic s he
EEC pa ame e s:
→
P=[R1L1R2T1p1R3T2p2]
(16)
→
PANN =→
PANN(EISq(jw,→
θ)(17)
MSELoss→
θ=∑q=Npa e ns
q=1∑µ=N equency
µ=1
→
Pq−PANN(EISq(jwµ,→
θ))
2(18)
EISmodeljw,→
P=R1+jL1w+ZCPE1,R2+ZCPE2,R3(19)
MSELoss→
θ=∑q=Npa e ns
q=1∑µ=N equency
µ=1 ealEISqjwµ−
ealEISmodeljwµ,→
P
2
+
∑q=Npa e ns
q=1∑µ=N equency
µ=1
imagEISqjwµ−imagEISmodeljwµ,→
P
2(20)
Elec onics 2023, 12, x FOR PEER REVIEW 6 o 13
The whole impedance can be de i ed ia Equa ions (14) and (15).
𝑟𝑒𝑎𝑙(𝐸𝐼𝑆)=R+𝑟𝑒𝑎𝑙𝑍

,

+𝑟𝑒𝑎𝑙(𝑍

,

) (14)
𝑖𝑚𝑎𝑔(𝐸𝐼𝑆)=L
𝑤+𝑖𝑚𝑎𝑔𝑍

,

+𝑖𝑚𝑎𝑔(𝑍

,

) (15)
2.2. Supe ised Ve sus Unsupe ised S a egies
We p opose a new unsupe ised aining algo i hm. The usual app oach is o ain
he a i icial neu al ne wo k wi h EIS empi ical da a as inpu s and EEC pa ame e s as
ou pu s, as seen in Figu e 2. This app oach has he disad an ages o he high cos s and
ime pe iods equi ed o ob ain a sufficien da ase o aining. In his esea ch, we
p oduced essays o 5 mon hs. Each mon h, he ba e ies we e cha ged and discha ged a
diffe en SoCs. The cos s o hese essays we e o he esea ch pe sonnel, he ba e y se s,
he labo a o y spaces, and he measu ing equipmen and e ige a ion sys ems’
amo iza ion. The p ocess p oduced only 36 essays, including six diffe en SoC le els
(100% o 20%) and six diffe en imes ( om he s a o he i h mon h). The e o e, he
essay measu emen s we e expensi e and ime-consuming. Each impedance measu emen
p oduced an EIS essay. Due o hese ime and budge limi a ions, he amoun o da a was
limi ed o 36 essays. As seen in Equa ion (17), 𝑃(𝐸𝐼𝑆(𝑗𝑤,𝜃) is an a i icial neu al
ne wo k ha p edic s he EEC pa ame e s:
𝑃
󰇍

=󰇟𝑅
𝐿𝑅𝑇𝑝𝑅𝑇𝑝󰇠 (16)
𝑃
󰇍

 =𝑃
󰇍

(𝐸𝐼𝑆(
𝑗
𝑤,𝜃
) (17)
𝑀𝑆𝐸

󰇡
𝜃
󰇍
󰇍

󰇢=∑∑ 󰇻𝑃
󰇍


−𝑃

(𝐸𝐼𝑆

(𝑗𝑤

,
𝜃
󰇍
󰇍

))󰇻







(18)
𝐸𝐼𝑆
𝑗
𝑤,𝑃
󰇍

=R+
𝑗
L𝑤+𝑍

,

+𝑍

,

(19)
𝑀𝑆𝐸𝜃
=∑∑ 󰇻𝑟𝑒𝑎𝑙󰇡𝐸𝐼𝑆
𝑗
𝑤󰇢−






𝑟𝑒𝑎𝑙󰇡𝐸𝐼𝑆
𝑗
𝑤,𝑃
󰇍

󰇢󰇻+
∑∑ 󰇻𝑖𝑚𝑎𝑔󰇡𝐸𝐼𝑆
𝑗
𝑤󰇢−𝑖𝑚𝑎𝑔󰇡𝐸𝐼𝑆
𝑗
𝑤,𝑃
󰇍

󰇢󰇻





 (20)
Figu e 2. Supe ised aining model.
Figu e 2. Supe ised aining model.
We p opose a di e en mean loss unc ion (an unsupe ised aining me hod) o
educe he p edic ion e o , as shown in Figu e 3. Table 4de ines he main a iables o he
EEC neu al ne wo k p edic o , including he inpu s and ou pu s.
Elec onics 2023,12, 5038 7 o 13
Elec onics 2023, 12, x FOR PEER REVIEW 7 o 13
We p opose a diffe en mean loss unc ion (an unsupe ised aining me hod) o
educe he p edic ion e o , as shown in Figu e 3. Table 4 de ines he main a iables o he
EEC neu al ne wo k p edic o , including he inpu s and ou pu s.
Figu e 3. Unsupe ised aining model.
Table 4. The pa ame e s o he p oposed neu al ne wo k o EEC pa ame e p edic ions. These
pa ame e s a e de ined in he elec ical scheme in Figu e 1a.
Pa ame e Uni Range De ini ion
𝑟𝑒𝑎𝑙󰇡𝐸𝐼𝑆
𝑗
𝑤󰇢 - [0, 1]
I is a 121- e m inpu ec o . I con ains he eal pa o
𝐸𝐼𝑆
𝑗
𝑤, and hese inpu s a e no malized.
𝑖𝑚𝑎𝑔󰇡𝐸𝐼𝑆
𝑗
𝑤󰇢 - [0, 1]
I is a 121- e m inpu ec o . I con ains he imagina y
pa o 𝐸𝐼𝑆
𝑗
𝑤, and hese inpu s a e no malized.
𝑅 Ω [0, 1] The i s esis ance. The alue is no malized.
𝐿 H [0, 1] The se ial induc ance. The alue is no malized.
𝑅 Ω [0, 1]
The second esis ance. I is in pa allel wi h he i s
cons an -phase elemen .
𝑇 sec
p1.
Ω
−1
[0, 1] The in e se gain o he i s cons an -phase elemen . I
is a no malized alue.
𝑝 - [0, 1]
The powe index pa ame e o he i s cons an -phase
elemen . I is a no malized alue.
𝑅 Ω [0, 1]
The hi d esis ance. I is in pa allel wi h he second
cons an -phase elemen . I is a no malized alue.
𝑇 sec
p1.
Ω
−1
[0, 1]
The in e se gain o he second cons an -phase elemen .
I is a no malized alue.
𝑝 - [0, 1]
The powe index pa ame e o he second cons an -
phase elemen . I is a no malized alue.
𝑤 ad/sec [0.0628, 63323] wi h an
exponen ial s ep o Δ𝑤
The equency sample. The e a e 121 diffe en
equency samples.
Δ𝑤 ad/sec Δ𝑤=𝑒 


 The equency sampling s ep.
𝜇∈0,1,2,..,𝑁

2.3. Da a Augmen a ion P oposal
We ha e p oposed a da a augmen a ion algo i hm o gene a e syn he ic aining om
i e essays wi h diffe en s a es o cha ge (SoCs) om 20% o 100% (see Table 1 o he
whole se o essays used).
The gene a ion p ocess is de ined in Algo i hm 1. The objec i e is o gene a e h ee
diffe en da ase s: a aining da ase wi h 20,000 syn he ic equi alen ci cui s, a alida ion
da ase o 2500 syn he ic equi alen ci cui s, and a es da ase o 500 syn he ic equi alen
ci cui s. The diffe ence be ween he EIS alues o a selec ed expe imen al essay and he
syn he ic equi alen ci cui is de ined as he absolu e alue o he diffe ence be ween hese
wo EIS alues, which is di ided by he absolu e alue o he selec ed expe imen al essay’s
Figu e 3. Unsupe ised aining model.
Table 4. The pa ame e s o he p oposed neu al ne wo k o EEC pa ame e p edic ions. These
pa ame e s a e de ined in he elec ical scheme in Figu e 1a.
Pa ame e Uni Range De ini ion
ealEISjwµ - [0, 1] I is a 121- e m inpu ec o . I con ains he eal pa o EISjwµ, and
hese inpu s a e no malized.
imagEISjwµ - [0, 1] I is a 121- e m inpu ec o . I con ains he imagina y pa o
EISjwµ, and hese inpu s a e no malized.
R1Ω[0, 1] The i s esis ance. The alue is no malized.
L1H [0, 1] The se ial induc ance. The alue is no malized.
R2Ω[0, 1] The second esis ance. I is in pa allel wi h he i s
cons an -phase elemen .
T1
sec
p1.Ω−1[0, 1] The in e se gain o he i s cons an -phase elemen . I is a
no malized alue.
p1- [0, 1] The powe index pa ame e o he i s cons an -phase elemen . I is a
no malized alue.
R3Ω[0, 1] The hi d esis ance. I is in pa allel wi h he second cons an -phase
elemen . I is a no malized alue.
T2
sec
p1.Ω−1[0, 1] The in e se gain o he second cons an -phase elemen . I is a
no malized alue.
p2- [0, 1]
The powe index pa ame e o he second cons an -phase elemen . I is
a no malized alue.
wµ ad/sec [0.0628, 63323] wi h an
exponen ial s ep o ∆wµThe equency sample. The e a e 121 di e en equency samples.
∆wµ ad/sec ∆wµ=e
15µ
N equency −5The equency sampling s ep.
µ∈n0,1,2, . . . , N equencyo
2.3. Da a Augmen a ion P oposal
We ha e p oposed a da a augmen a ion algo i hm o gene a e syn he ic aining om
i e essays wi h di e en s a es o cha ge (SoCs) om 20% o 100% (see Table 1 o he
whole se o essays used).
The gene a ion p ocess is de ined in Algo i hm 1. The objec i e is o gene a e h ee
di e en da ase s: a aining da ase wi h 20,000 syn he ic equi alen ci cui s, a alida ion
da ase o 2500 syn he ic equi alen ci cui s, and a es da ase o 500 syn he ic equi alen
ci cui s. The di e ence be ween he EIS alues o a selec ed expe imen al essay and he
syn he ic equi alen ci cui is de ined as he absolu e alue o he di e ence be ween hese
wo EIS alues, which is di ided by he absolu e alue o he selec ed expe imen al essay’s
EIS alue (see line 9 in Algo i hm 1). The maximum accep ed di e ence le el is equal o
30%. The eco ds o each da ase we e andomly selec ed.
Elec onics 2023,12, 5038 8 o 13
Algo i hm 1: Da a augmen a ion p ocess o syn he ic essay gene a ion
Inpu s:
The se o essays wi h each pa ame e se . The e a e 5 di e en essays, wi h di e en SoCs.
•EISµ(jw),µ∈{20%, 40%, 60%, 80%, 100%}
•Equi alen Ci cui Pa ame e µ= [R1µ, R2µ, R3µ, T1µ, p1µ, T2µ, p2µ, L1µ]
The numbe o essays gene a ed.
•Ngene a edEIS
The maximum di e ence le el.
•Jmax
Ou pu s:
The se o syn he ic essays. The e a e Ngene a edEIS di e en essays.
•EISsyn he ic,q(jw), q ∈{1, Ngene a edEIS}
•Equi alen Ci cui Pa ame e q= [R1q, R2q, R3q, T1q, p1q, T2q, p2q, L1q]
1: p ocedu e
2: Ob ain he maximum and he minimum alues o each EEC pa ame e . These alues a e
calcula ed om Equi alen Ci cui Pa ame e µ.
◦R1max, R2max, R3max, T1max, p1max, T2max, p2max, L1max
◦R1min, R2min, R3min, T1min, p1min, T2min, p2min, L1min
3: o q om 1 o Ngene a edEIS
4: Nex Ci cui = False
5: while no Nex Ci cui
6: µselec ed = mod(µ,Ca dinal(µ))
7: Gene a e an equi alen pa ame e se (Ci cui Pa ame e q) based
on Equi alen Ci cui Pa ame e µselec ced,
The pa ame e s a e calcula ed wi h a pseudo andom uni o m alue be ween he maximum and
he minimum alues calcula ed in line 2.
8: Gene a e syn he ic EISsyn he ic,q(jw). See Equa ion (1).
9: Calcula e di e ence le el as ollows:
J=100µ=N equency
∑
µ=1
|EISsyn he ic,q(jwµ)−EISµselec ed(jwµ)|
|EISµselec ed(jwµ)|
10: i J<Jmax
11: Nex Ci cui =T ue
12: end i
13: S o e EISsyn he ic,q(jw), and S o e Equi alen Ci cui Pa ame e µselec ced
14: end while
15: end o
16: e u n EISsyn he ic, q(jw), Ci cui Pa ame e q, q ∈{1, Ngene a edEIS}
3. Ma e ials and Me hods
The da a used in his wo k we e based on he aging and measu emen s o a se o
80–100 Ah and 12 V lead–acid ba e ies. The es ing p o ocol included a CC cha ge s age
and a CV cha ge s age. Pe iodic impedance measu emen s we e pe o med a di e en
s a e-o -cha ge le els, a each
∆
h% = 20 in he discha ge p ocess. A each SOC le el, a 12 h
elaxa ion ime was es ablished be o e pe o ming he impedance measu emen , which
was conduc ed unde an exci a ion cu en o 50 mA and in he equency ange o 10 mHz
o 10 kHz.
As al eady ou lined in Sec ion 2, he EIS spec a ex ac ed om he es we e adjus ed
o a speci ic EEC o iden i y an EEC pa ame e . The p oposed neu al ne wo k’s a chi ec u e
has wo inpu s, whe e he i s inpu is he eal pa o he EIS spec a and he second inpu
is he imagina y pa o he EIS spec a, wi h each inpu ha ing 121 equency samples.
These wo inpu s a e conca ena ed in one ea u e ec o . A e his conca ena ion, he e
is a ully connec ed laye wi h 100 neu ons and a ReLU laye . A e his, he e a e h ee
blocks o ully connec ed laye s wi h a ReLU laye o 10 neu ons. Finally, he e is a ully
connec ed laye and a sigmoid laye o eigh neu ons. The ou pu laye has eigh neu ons
Elec onics 2023,12, 5038 9 o 13
because he a i icial neu on ne wo k has o p edic he pa ame e s o he equi alen ci cui
(see Equa ion (16)). The a chi ec u e has 25,600 lea nable pa ame e s. In Figu es 4and 5,
we ha e a ached a desc ip ion o he whole a chi ec u e.
Elec onics 2023, 12, x FOR PEER REVIEW 9 o 13
was conduc ed unde an exci a ion cu en o 50 mA and in he equency ange o 10
mHz o 10 kHz.
As al eady ou lined in Sec ion 2, he EIS spec a ex ac ed om he es we e adjus ed
o a speci ic EEC o iden i y an EEC pa ame e . The p oposed neu al ne wo k’s
a chi ec u e has wo inpu s, whe e he i s inpu is he eal pa o he EIS spec a and he
second inpu is he imagina y pa o he EIS spec a, wi h each inpu ha ing 121
equency samples. These wo inpu s a e conca ena ed in one ea u e ec o . A e his
conca ena ion, he e is a ully connec ed laye wi h 100 neu ons and a ReLU laye . A e
his, he e a e h ee blocks o ully connec ed laye s wi h a ReLU laye o 10 neu ons.
Finally, he e is a ully connec ed laye and a sigmoid laye o eigh neu ons. The ou pu
laye has eigh neu ons because he a i icial neu on ne wo k has o p edic he
pa ame e s o he equi alen ci cui (see Equa ion (16)). The a chi ec u e has 25,600
lea nable pa ame e s. In Figu es 4 and 5, we ha e a ached a desc ip ion o he whole
a chi ec u e.
Figu e 4. The p oposed a chi ec u e wi h a di ec ional acyclical g aph o he p oposed neu al
ne wo k.
Figu e 5. Summa y in o ma ion o each laye o he p oposed neu al ne wo k.
Figu e 4. The p oposed a chi ec u e wi h a di ec ional acyclical g aph o he p oposed neu al ne wo k.
Elec onics 2023, 12, x FOR PEER REVIEW 9 o 13
was conduc ed unde an exci a ion cu en o 50 mA and in he equency ange o 10
mHz o 10 kHz.
As al eady ou lined in Sec ion 2, he EIS spec a ex ac ed om he es we e adjus ed
o a speci ic EEC o iden i y an EEC pa ame e . The p oposed neu al ne wo k’s
a chi ec u e has wo inpu s, whe e he i s inpu is he eal pa o he EIS spec a and he
second inpu is he imagina y pa o he EIS spec a, wi h each inpu ha ing 121
equency samples. These wo inpu s a e conca ena ed in one ea u e ec o . A e his
conca ena ion, he e is a ully connec ed laye wi h 100 neu ons and a ReLU laye . A e
his, he e a e h ee blocks o ully connec ed laye s wi h a ReLU laye o 10 neu ons.
Finally, he e is a ully connec ed laye and a sigmoid laye o eigh neu ons. The ou pu
laye has eigh neu ons because he a i icial neu on ne wo k has o p edic he
pa ame e s o he equi alen ci cui (see Equa ion (16)). The a chi ec u e has 25,600
lea nable pa ame e s. In Figu es 4 and 5, we ha e a ached a desc ip ion o he whole
a chi ec u e.
Figu e 4. The p oposed a chi ec u e wi h a di ec ional acyclical g aph o he p oposed neu al
ne wo k.
Figu e 5. Summa y in o ma ion o each laye o he p oposed neu al ne wo k.
Figu e 5. Summa y in o ma ion o each laye o he p oposed neu al ne wo k.
The aining p ocess is an ADAM algo i hm wi h 60 epochs, wi h a mini-ba ch size o
100 syn he ic ci cui s. The ini ial lea ning a io is equal o 10
−3
. The g adien decay ac o
is se o 0.9 and he squa ed decay ac o is se o 0.999. The epsilon alue (small cons an )
is se o 10−8.
4. Resul s
We compa ed he no malized heo e ical EIS da a wi h he EIS da a gene a ed using
he pa ame e s p edic ed by he neu al ne wo k. We plo ed he heo e ical EIS da a (in ed)
calcula ed using Equa ions (1)–(5) wi h he EEC pa ame e p edic ed by he neu al ne wo k.
When he neu al ne wo k akes an expe imen al EIS da um as an inpu , i p edic s he EEC
pa ame e se (see Table 1). We ained he neu al ne wo k o p edic pa ame e s o ma ch
he expe imen al EIS da a ia Equa ions (1)–(5), no he EEC pa ame e s. In Figu e 6, he
axes a e he no malized eal pa s o he EIS da a (x-axis) and he imagina y pa s o he
EIS da a (y-axis). The igu es do no ha e uni s because he eal pa s we e no malized
om [0 o 1] and he imagina y pa s om [
−
1 o 0]. The scales we e no malized wi h he