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A physics-based ac ional-o de equi alen ci cui model o ime and
equency-domain applica ions in li hium-ion ba e ies
Pablo Rod íguez-I u iagaa,∗, Da id Anseánb, Sal ado Rod íguez-Bolí a a, Manuela González b,
Juan Ca los Vie ab, Juan An onio López-Villanue aa
aDepa men o Elec onics and Compu e Technology, Facul y o Sciences, Uni e si y o G anada, G anada, 18071, Andalusia, Spain
bDepa men o Elec ical Enginee ing, Poly echnic School o Enginee ing, Uni e si y o O iedo, Gijon, 33204, As u ias, Spain
ARTICLE INFO
Keywo ds:
Li hium-ion ba e y
Equi alen ci cui model
F ac ional-o de model
Physics-based model
EIS
ABSTRACT
Equi alen ci cui models (ECMs) emain he mos popula choice o online applica ions in li hium-ion
ba e ies because o hei simple pa ame e iza ion and lowe compu a ional equi emen s in compa ison o
elec ochemical models. Ne e heless, s anda d ECMs lack physical insigh and ail o accu a ely ep oduce cell
beha io unde a wide ange o ope a ing condi ions. Fo his eason, he de elopmen o physics-in o med
ECMs becomes essen ial so as o p o ide a be e desc ip ion o he physical p ocesses while main aining a
educed compu a ional complexi y. In his a icle, we p opose a no el physics-based ECM de i ed di ec ly om
an elec ochemical model, so ha he e is a clea co ela ion be ween ci cui s a es and in e nal ba e y s a es,
as well as ci cui and physical pa ame e s. The p oposed model yields an RMS e o below 1.46 mV o cell
ol age, 0.28% o he su ace concen a ion in he ac i e ma e ial pa icles, 0.6% o he elec ode-a e aged
elec oly e concen a ion and 0.32 mV o he cha ge- ans e o e po en ials. Ano he key ea u e o his model
is he ela ionship be ween ci cui pa ame e s and hose iden i ied in equency-domain es s, which allows us
o cha ac e ize and alida e he model expe imen ally. We unde s and ha he p esen ed model cons i u es an
al e na i e o s anda d ECMs as well as elec ochemical models as i combines ad an ageous cha ac e is ics
om bo h o hem.
1. In oduc ion
As conce ns o e ene gy supply and en i onmen al issues g ow
la ge wo ldwide, elec ochemical ene gy s o age has become a subjec
o in ense esea ch. In pa icula , echa geable li hium-ion ba e ies
ha e ma e ialized as he leading s o age solu ion o a numbe o
applica ions [1–3], due o hei high ene gy densi y, high speci ic
ene gy, and low sel -discha ge [4]. The e o e, he de elopmen o
comp ehensi e ba e y models is c i ical o hei online moni o ing
by a Ba e y Managemen Sys em (BMS) [5], hus ensu ing hei sa e
ope a ion as well as a p olonged use ul li espan by accu a ely es i-
ma ing in e nal ba e y s a es [6]. Th ee b oad g oups a e commonly
conside ed in he ield o ba e y modeling: physics-based models,
equi alen ci cui models (ECMs) and da a-d i en models [4,7]. In
da a-d i en app oaches he ba e y is subs i u ed by a black-box model
ha is able o ep oduce i s beha io . Ne e heless, he alidi y o he
esul ing model elies hea ily on he aining da ase . This may lead o
p ac ical d awbacks, such as a conside ably ime-consuming aining
p ocess o o e i ing issues [7]. Consequen ly, elec ochemical and
∗Co esponding au ho .
E-mail add ess: [email p o ec ed] (P. Rod íguez-I u iaga).
equi alen ci cui models con inue o be o g ea esea ch in e es , and
we will ocus on hem in his s udy.
Physics-based app oaches model he physical p ocesses and elec-
ochemical eac ions ha occu in he cell, wi h he Doyle–Fulle –
Newman (DFN) [8,9] being conside ed he s anda d ba e y model.
Due o i s complex na u e, se e al app oxima ions ha e been p oposed
so as o main ain he desc ip ion o he physical p ocesses wi h a
es ained compu a ional load [10], wi h he Single Pa icle Model
(SPM) [11,12] being one o he mos widely used. The SPM makes
he assump ions ha each elec ode is composed o sphe ical ac i e
ma e ial pa icles wi h he same physical p ope ies and ha elec oly e
dynamics a e negligible [13], so he cu en dis ibu ion is uni o m
ac oss he elec ode. Consequen ly, only one ep esen a i e pa icle
om each elec ode needs o be aken in o accoun . This app oxi-
ma ion holds ue o low cu en a es [14]; howe e , a mode a e
and la ge cu en s he e ec s o concen a ion and po en ial g adi-
en s in he elec oly e should be accoun ed o . The e o e, se e al
app oaches ha e been p oposed o include elec oly e dynamics wi h
he goal o ex ending he ange o applicabili y o he SPM [14–18],
h ps://doi.o g/10.1016/j.es .2023.107150
Recei ed 22 Decembe 2022; Recei ed in e ised o m 16 Feb ua y 2023; Accep ed 14 Ma ch 2023
Jou nal o Ene gy S o age 64 (2023) 107150
2
P. Rod íguez-I u iaga e al.
usually by he names ‘enhanced SPM’ (eSPM) o ‘SPM wi h elec oly e
(SPMe)’, ia polynomial app oxima ions o he elec oly e concen a-
ion p o ile [15,16,18] o he nume ical esolu ion o he co esponding
PDE [14,17,18], wi h he o me app oach being less sui able o pulsed
cu en s [19]. The e also exis some al e na i es o he di ec nume ical
esolu ion o he PDEs, in which he ans e unc ions o he sys em
a e ob ained om elec ochemical models and hen ans o med in o a
disc e e s a e-space ep esen a ion wi h he Disc e e-Time Realiza ion
Algo i hm (DTRA) [20] o in o a SIMULINK [21] model ia simpli ied
ac ional-o de ans e unc ions [22–24]. Ne e heless, hese models
s ill equi e a p ecise se o physical pa ame e s, whose de e mina ion
om non-in asi e expe imen al measu emen s con inues o be an open
esea ch opic [25].
On he o he hand, ECMs app oxima e he elec ical beha io o a
ba e y cell by ha o a speci ic ci cui . This app oach is cha ac e ized
by educed memo y equi emen s and a low compu a ional load, he e-
o e making hem app op ia e o online BMSs [26–29]. Mo eo e , hei
disc e e s a e-space ep esen a ion is easily ob ained, hus allowing o
he implemen a ion o s a e obse e s and Kalman il e s [30–32] o
mi iga e he in luence o measu emen and p ocess noise. The challenge
is wo old when de eloping ECMs: i s , de e mining which ci cui
opology ep oduces ba e y beha io mo e accu a ely, and secondly,
ob aining he co esponding pa ame e alues o said opology om
expe imen al es s [33–35]. Mos ECMs include a ol age sou ce, which
co esponds o he open-ci cui ol age (OCV) as a nonlinea unc ion
o he cell s a e o cha ge (SOC), in se ies wi h he in e nal ohmic
esis ance and one o se e al pa allel RC elemen s ha model dynamic
beha io [30]. A a ia ion, use ul o ci cui simula o s, subs i u es
he independen OCV sou ce by a ol age sou ce dependen on he
s a e o cha ge o a capaci o which ep esen s ba e y SOC [36]. How-
e e , hese models in oduce a s ong sepa a ion be ween quasi-s a ic
elec ode he modynamics included in he OCV-SOC ela ionship [37]
and he dynamic beha io ep esen ed by he RC ne wo k, in which
di e en di usion p ocesses a e mixed and a di ec equi alence wi h
he in e nal s a es o he ba e y is no o en possible. Addi ionally,
physics-based models show ha elec ode po en ial is dependen on he
li hium concen a ion a he su ace o ac i e ma e ial pa icles a he
han hei a e age concen a ion, which is wha SOC s ands o . The
di e ence be ween bo h magni udes may be conside ably la ge a high
cu en a es, as p edic ed by he SPM [38]. Li e al. [39] ecen ly
p oposed a modi ica ion o he model by Chen and Rincon-Mo a [36]
by which he non-linea ol age sou ce depends on an in e media e
a iable named 𝑆𝑂𝐶𝑠𝑢𝑟𝑓 , which is ob ained as he esul o he se ies
connec ion o he SOC capaci o and a RC-elemen ha ep esen s
he non-uni o m concen a ion p o ile due o li hium di usion in he
ac i e ma e ial pa icles. Ne e heless, he ela ionship be ween ci cui
pa ame e s and hei physical coun e pa s is gene ally los , which
makes hese models inaccu a e ou side he ange in which hey ha e
been de e mined expe imen ally.
Fo hese easons, se e al wo ks combine an equi alen ci cui
model o elec oly e dynamics and ohmic losses wi h he ans e
unc ions o solid di usion in physics-in o med educed o de mod-
els [40–43]. These wo ks make use o Padé’s me hod in o de o
ob ain a ional app oxima ions o hese anscenden al ans e unc-
ions. Howe e , i has o be poin ed ou ha Padé’s app oximan s
a e calcula ed in he neighbo hood o a ce ain poin , ypically he
equency 𝑠= 0, so he accu acy o he app oxima ion is no gua an eed
wi hin he whole equency ange. Fu he mo e, i has been shown
ha di usion p ocesses a e mo e accu a ely modeled by a con inuous
dis ibu ion o ime cons an s [44], he e o e sugges ing ha ac ional-
o de ans e unc ions a e mo e app op ia e in his con ex . This is
also co obo a ed by Elec ochemical Impedance Spec oscopy (EIS)
es s [6,44–47]. The Nyquis plo o EIS da a shows se e al dep essed
semici cles and cons an -slope ails ha may be modeled elec ically
by Cons an -Phase Elemen s (CPEs) [48] and ZARC elemen s [49],
which a e he pa allel connec ion o a esis o and a CPE. The in-
oduc ion o ac ional-o de ci cui elemen s en ails he compu a ion
o ac ional-o de de i a i es, so se e al app oaches ha e been p o-
posed in he li e a u e o app oxima ing hei beha io in he ime
domain [49–53]. Mo eo e , ac ional-o de ECMs ha e been employed
success ully in conjunc ion wi h Kalman il e s in ecen yea s [54–56]
o he concu en es ima ion o ba e y s a e o cha ge and elec ical
pa ame e s.
Fo all he easons abo e, he de elopmen o a physics-based ECM,
which is able o p o ide in o ma ion abou in e nal ba e y s a es and
whose ci cui pa ame e s a e di ec ly co ela ed wi h physical pa am-
e e s, is an open esea ch opic. Fu he mo e, we also conside he
ela ionship be ween he ime-domain and equency-domain beha io s
o li hium-ion ba e ies o be a subjec o in e es . The e o e, in his
a icle we de i e a no el educed ac ional-o de ECM om an elec-
ochemical model, whose s a es and pa ame e s a e ela ed o hose
o he ba e y cell. Addi ionally, we ha e analyzed he co espondence
be ween said ci cui pa ame e s and hose iden i ied om EIS measu e-
men s, hus cla i ying he connec ion be ween equency-domain da a
and physical pa ame e s. Fo his pu pose, we ha e ob ained simpli ied
ans e unc ions om he SPMe and de e mined hei elec ical equi -
alen wi h ac ional-o de elemen s. Subsequen ly, he p oposed model
has been alida ed i s agains he SPMe o gi en a pa ame e se , and
nex wi h expe imen al da a by iden i ying equi alen ci cui pa ame-
e s om EIS measu emen s. The majo ad an age o his model o e
p e ious ECMs is he ex ended insigh in o he in e nal ba e y s a es
and physical p ocesses, whe eas i g oups physical pa ame e s in esis-
o s and ime cons an s and p esen s lowe compu a ional equi emen s
wi h espec o he SPMe, hus allowing o a simple pa ame e iza ion
p ocess as well as he implemen a ion o online es ima ion algo i hms.
The main con ibu ions o his a icle a e condensed as ollows:
1. De i ing educed ans e unc ions di ec ly om he SPMe and
es ablishing hei elec ical equi alen ia ZARC elemen s, as
well as he c i e ia o he alidi y o he employed app oxima-
ions.
2. Cons uc ing an equi alen ci cui model whose pa ame e al-
ues a e di ec ly co ela ed wi h hei physical coun e pa s, and
whose s a es con ain in o ma ion abou he in e nal s a es o he
ba e y cell.
3. P o iding a me hod o de e mine said equi alen ci cui pa am-
e e s om EIS da a, hus b idging he gap be ween he ime and
equency-domain beha io s o li hium-ion ba e ies.
In consequence, his pape is s uc u ed as ollows: he analy ical
de i a ion o he p oposed equi alen ci cui model is p esen ed in
Sec ion 2. The heo e ical alida ion agains an elec ochemical model
is ca ied ou in Sec ion 3, whe eas he expe imen al pa ame e iza ion
p ocess and esul s a e de ailed and discussed in Sec ion 4. Some inal
ema ks a e p o ided in he las sec ion.
2. Equi alen ci cui de elopmen
In his sec ion, he p ocedu e o ob ain simpli ied ans e unc ions
om he SPMe and hei elec ical equi alence is desc ibed wi h he
goal o de eloping an equi alen ci cui model. The main ad an age
o employing he SPMe ins ead o he DFN as in [20] lies in he ac
ha he o me emo es he coupling be ween he pa ial di e en ial
equa ions. The e o e, each di usion p ocess may be analyzed inde-
penden ly, hus p oducing conside ably simple ans e unc ions and
allowing o a di ec elec ical equi alence.
Jou nal o Ene gy S o age 64 (2023) 107150
3
P. Rod íguez-I u iaga e al.
2.1. SPMe model desc ip ion
Ma quis e al. [14] p oposed an SPMe de i ed as an asymp o ic
educ ion o he s anda d DFN model, which exp esses ba e y e minal
ol age as a unc ion o elec ode-a e aged quan i ies. In said model,
he ou pu ol age is spli in o he sum o i s componen s as in Eq. (1):
𝑉=𝑈𝑒𝑞 +𝜂𝑐+𝜂𝑟+𝛥𝛷𝑒+𝛥𝛷𝑠(1)
whe e he e m 𝑈𝑒𝑞 s ands o he equilib ium po en ial, 𝜂𝑐and 𝜂𝑟 ep e-
sen he ol age d op due o concen a ion g adien s in he elec oly e
and cha ge- ans e eac ions, espec i ely, and 𝛥𝛷𝑒and 𝛥𝛷𝑠a e he
ohmic losses in he elec oly e and solid, espec i ely. The de ailed
exp ession o each e m is shown in Eq. (2):
𝑈𝑒𝑞 =𝑂𝐶𝑃𝑝(𝜒𝑝|𝑟=𝑅𝑝) − 𝑂𝐶𝑃𝑛(𝜒𝑛|𝑟=𝑅𝑛)(2a)
𝜂𝑐=2𝑅𝑇
𝐹
(1 − 𝑡+)
𝑐𝑒,𝑡𝑦𝑝 (𝑐𝑒,𝑝 −𝑐𝑒,𝑛)(2b)
𝜂𝑟=2𝑅𝑇
𝐹[sinh−1 (𝐼𝑅𝑝
6𝑗0,𝑝𝜖𝑝𝐿𝑝𝐴)+ sinh−1 (𝐼𝑅𝑛
6𝑗0,𝑛𝜖𝑛𝐿𝑛𝐴)] (2c)
𝛥𝛷𝑒=𝐼
𝜅𝐴 (𝐿𝑛
3𝜖𝑏
𝑒,𝑛
+𝐿𝑠
𝜖𝑏
𝑒,𝑠
+𝐿𝑝
3𝜖𝑏
𝑒,𝑝 )(2d)
𝛥𝛷𝑠=𝐼
3𝐴(𝐿𝑛
𝜎𝑛
+𝐿𝑝
𝜎𝑝)(2e)
whe e 𝑂𝐶𝑃𝑝and 𝑂𝐶𝑃𝑛a e he open-ci cui po en ials o he posi-
i e and nega i e elec odes, espec i ely. 𝜒𝑝|𝑟=𝑅𝑝and 𝜒𝑛|𝑟=𝑅𝑛a e he
no malized li hium concen a ions in he posi i e and nega i e ac i e
ma e ial pa icles, espec i ely, e alua ed a hei su ace, wi h 𝑅𝑝and
𝑅𝑛being hei espec i e pa icle adius. Rega ding he elec oly e, 𝑡+is
he ca ion ans e ence numbe , 𝜅 ep esen s he elec ode conduc i i y
and 𝑐𝑒,𝑝 and 𝑐𝑒,𝑛 a e he elec ode-a e aged elec oly e concen a ions
in he posi i e and nega i e elec ode, espec i ely. 𝑗0,𝑝,𝑗0,𝑛,𝜎𝑝,𝜎𝑛𝜖𝑝
and 𝜖𝑛a e he exchange cu en densi ies, conduc i i ies and ac i e
ma e ial olume ac ions o bo h elec odes. 𝐿𝑛,𝐿𝑠,𝐿𝑝,𝜖𝑒,𝑝,𝜖𝑒,𝑠 and
𝜖𝑒,𝑛 ep esen he hickness and po osi ies o he posi i e elec ode,
sepa a o and nega i e elec ode, espec i ely. Las ly, 𝐴s ands o he
elec ode a ea, 𝑏is he B uggeman coe icien and 𝐼is he cu en
applied o he cell. No e ha we ha e changed he cu en sign c i e ion
o posi i e while cha ging in o de o ob ain impedance exp essions
wi h a posi i e eal pa .
The e ms ha depend di ec ly on solid o elec oly e concen a ions
(i.e., 𝑈𝑒𝑞 and 𝜂𝑐) a e hose ha will exhibi a ime-domain ansien o
a equency esponse, he e o e hey will be analyzed in Sec ions 2.2
and 2.3. The e m 𝜂𝑟depends on said concen a ions indi ec ly, so
i will be s udied subsequen ly in Sec ion 2.4. The equi alen ci cui
model along wi h i s disc e e s a e-space ep esen a ion is p esen ed
in Sec ion 2.5 and i s co espondence wi h equency-domain da a is
analyzed in Sec ion 2.6.
2.2. Solid di usion ans e unc ion
To ob ain he equency esponse o he 𝑈𝑒𝑞 e m in Eq. (2), he
e alua ion o he li hium concen a ion a he su ace o he solid
pa icles o he elec odes is necessa y. I he pa icles a e assumed o
be sphe ical, he di usion p ocess aking place wi hin hem is desc ibed
by Eq. (3) [14]:
𝜕𝑐𝑠(𝑟, 𝑡)
𝜕𝑡 =𝐷𝑠
𝑟2
𝜕
𝜕𝑟 (𝑟2𝜕𝑐𝑠(𝑟, 𝑡)
𝜕𝑟 )(3)
wi h he ollowing bounda y condi ions [14]:
𝜕𝑐𝑠(𝑟, 𝑡)
𝜕𝑟 ||||𝑟=0
= 0,𝜕𝑐𝑠(𝑟, 𝑡)
𝜕𝑟 ||||𝑟=𝑅𝑠
=𝐼𝑅𝑠
3𝐴𝐹 𝐿𝑒𝐷𝑠𝜖𝑠
(4)
whe e 𝑟is he dis ance along he pa icle adius, 𝑐𝑠is he li hium
concen a ion in he solid, 𝐷𝑠is he li hium di usion coe icien in he
solid, 𝑅𝑠is he pa icle adius, 𝐴is he elec ode a ea, 𝐹is Fa aday’s
cons an and 𝐿𝑒is he elec ode leng h. The a ia ion in concen a ion
wi h espec o he ini ial alue is aken as he equa ion a iable, so as
o ha e ze o ini ial ime condi ions:
𝑐𝑠=𝑐𝑠−𝑐𝑠,0(5)
Upon sol ing, applying he bounda y condi ions in Eq. (4) and
e alua ing a 𝑟=𝑅𝑠, he ans e unc ion om he applied cu en
o he a ia ion in su ace concen a ion is ob ained in Eq. (6):
𝑐𝑠,𝑠(𝑠)
𝐼(𝑠)=𝜏𝑠
3𝜖𝑠𝐴𝐹 𝐿𝑒
1
√𝜏𝑠𝑠co h (√𝜏𝑠𝑠)− 1
(6)
whe e 𝜏𝑠is de ined as 𝑅2
𝑠
𝐷𝑠. I no malized concen a ion is conside ed
ins ead, he ans e unc ion may be ew i en as in Eq. (7):
𝐺𝑠(𝑠) = 𝜒𝑠,𝑠(𝑠)
𝐼(𝑠)=𝐾𝑠
√𝜏𝑠𝑠co h (√𝜏𝑠𝑠)− 1
(7)
whe e 𝜒𝑠,𝑠 =𝑐𝑠,𝑠(𝑠)∕𝑐𝑠,𝑚𝑎𝑥 and 𝐾𝑠=𝜏𝑠
3𝜖𝑠𝐴𝐹 𝐿𝑒𝑐𝑠,𝑚𝑎𝑥 .
Taking in o accoun ha o small alues o 𝑥,√𝑥co h (√𝑥)≈
1 + 𝑥
3−𝑥2
45 , one can s udy he beha io o 𝐺𝑠(𝑠)in he limi s 𝑠→0
and 𝑠→∞as in Eq. (8):
𝐺𝑠(𝑠) ≈ ⎧
⎪
⎨
⎪
⎩
3𝐾𝑠
𝜏𝑠𝑠+𝐾𝑠
5, 𝑠 →0
0, 𝑠 →∞
(8)
In o de o app oxima e he equency esponse o 𝐺𝑠(𝑠)be ween
hese wo limi s, in his a icle we p opose a ans e unc ion composed
o he addi ion o an in eg a o and a ZARC elemen as exp essed in
Eq. (9):
𝐾𝑠
√𝜏𝑠𝑠co h (√𝜏𝑠𝑠)− 1
≈3𝐾𝑠
𝜏𝑠𝑠+𝐾𝑠∕5
1 + (𝛽𝑠𝜏𝑠𝑠)𝛼𝑠(9)
The e o e, 𝛼𝑠and 𝛽𝑆need o be de e mined so as o ob ain he
closes app oxima ion o 𝐺𝑠(𝑠). Fo his pu pose, he equency esponse
o bo h ans e unc ions is plo ed in a Nyquis diag am and he
weigh ed impedance e o be ween he exac and he app oxima e
ans e unc ion is minimized wi hin a ange o equencies. In his
case, we ha e conside ed he in e al 𝜔=[1
𝜏𝑠
,103
𝜏𝑠]due o he ac
ha a equencies lowe han 2𝜋
𝜏𝑠 he in eg a o beha io is dominan ,
whe eas a equencies highe han 2𝜋⋅103
𝜏𝑠 he ampli ude o he e-
quency esponse app oaches 0. The esul ing alues a e 𝛼𝑠= 0.82 and
𝛽𝑠= 0.0207, and he Nyquis plo o he equency esponse o bo h
ans e unc ions is shown in Fig. 1.
No e ha Eq. (9) is exp essed in e ms o he no malized equency
𝜏𝑠𝑠, so his app oxima ion emains alid ega dless o he speci ic
physical pa ame e alues. Fu he mo e, he ac ha he iden i ied
o de exponen is no equal o 1 p o es ha a pa allel-RC elemen
would no be he mos app op ia e al e na i e o app oxima ing he
o iginal ans e unc ion.
Pa icula izing o he nega i e and posi i e pa icle, he e m 𝑈𝑒𝑞
om Eq. (2a) may be exp essed as in Eq. (10), aking in o accoun
ha he bounda y condi ion o he posi i e pa icle is nega i e i he
cu en applied o he cell 𝐼(𝑠)is conside ed posi i e while cha ging:
𝑈𝑒𝑞(𝑠) = 𝑂𝐶𝑃𝑝⎛⎜⎜⎝⎡⎢⎢⎣
−3𝐾𝑝
𝜏𝑝𝑠−𝐾𝑝∕5
1 + (0.0207𝜏𝑝𝑠)0.82 ⎤⎥⎥⎦
𝐼(𝑠)⎞⎟⎟⎠
−
𝑂𝐶𝑃𝑛([3𝐾𝑛
𝜏𝑛𝑠+𝐾𝑛∕5
1 + (0.0207𝜏𝑛𝑠)0.82 ]𝐼(𝑠))(10)
whe e 𝜏𝑝=𝑅2
𝑝
𝐷𝑝,𝜏𝑛=𝑅2
𝑛
𝐷𝑛,𝐾𝑝=𝜏𝑝
3𝜖𝑝𝐴𝐹 𝐿𝑝𝑐𝑝,𝑚𝑎𝑥 and 𝐾𝑛=𝜏𝑛
3𝜖𝑛𝐴𝐹 𝐿𝑛𝑐𝑛,𝑚𝑎𝑥 .
Jou nal o Ene gy S o age 64 (2023) 107150
4
P. Rod íguez-I u iaga e al.
Fig. 1. Nyquis plo o he equency esponse o he exac and app oxima e no malized
ans e unc ions in Eq. (9) o solid di usion. No e ha he eal and imagina y axes
ha e di e en scales.
2.3. Elec oly e di usion ans e unc ion
Nex , he equency esponse o he 𝜂𝑐 e m in Eq. (2) mus be
de e mined in o de o calcula e he o e po en ial due o concen a-
ion g adien s in he elec oly e. Consequen ly, he elec ode-a e aged
li hium concen a ion in he elec oly e has o be e alua ed. Fo his
pu pose, he di usion p ocess desc ibed in Eq. (11) is analyzed [14]:
𝜖𝑒,𝑛
𝜕𝑐𝑒,𝑛(𝑥, 𝑡)
𝜕𝑡 =𝜖𝑏
𝑒,𝑛𝐷𝑒
𝜕2𝑐𝑒,𝑛(𝑥, 𝑡)
𝜕𝑥2− (1 − 𝑡+)𝐼
𝐴𝐹 𝐿𝑛
,0< 𝑥 < 𝐿𝑛(11a)
𝜖𝑒,𝑠
𝜕𝑐𝑒,𝑠(𝑥, 𝑡)
𝜕𝑡 =𝜖𝑏
𝑒,𝑠𝐷𝑒
𝜕2𝑐𝑒,𝑠(𝑥, 𝑡)
𝜕𝑥2, 𝐿𝑛< 𝑥 < 𝐿𝑛+𝐿𝑠(11b)
𝜖𝑒,𝑝
𝜕𝑐𝑒,𝑝(𝑥, 𝑡)
𝜕𝑡 =𝜖𝑏
𝑒,𝑝𝐷𝑒
𝜕2𝑐𝑒,𝑝(𝑥, 𝑡)
𝜕𝑥2− (1 − 𝑡+)𝐼
𝐴𝐹 𝐿𝑝
, 𝐿𝑛+𝐿𝑠< 𝑥 < 𝐿 (11c)
whe e 𝐷𝑒is he di usion coe icien o li hium in he elec oly e, x
is he linea dis ance along he cell leng h 𝐿=𝐿𝑛+𝐿𝑠+𝐿𝑝, and he
e e ence 𝑥= 0 is placed a he in e ace be ween he anode and he
cu en collec o . The bounda y condi ions a he endpoin s o he cell
a e he ollowing [14]:
𝜕𝑐𝑒,𝑛(𝑥, 𝑡)
𝜕𝑥 ||||𝑥=0
= 0,𝜕𝑐𝑒,𝑝(𝑥, 𝑡)
𝜕𝑥 ||||𝑥=𝐿
= 0,(12)
as well as con inui y in concen a ion and lux be ween he di e en
domains [14]:
𝑐𝑒,𝑛(𝑥, 𝑡)|||𝑥=𝐿𝑛
=𝑐𝑒,𝑠(𝑥, 𝑡)|||𝑥=𝐿𝑛
, 𝑐𝑒,𝑠(𝑥, 𝑡)|||𝑥=𝐿𝑛+𝐿𝑠
=𝑐𝑒,𝑝(𝑥, 𝑡)|||𝑥=𝐿𝑛+𝐿𝑠
𝜖𝑏
𝑛
𝜕𝑐𝑒,𝑛(𝑥,𝑡)
𝜕𝑥 ||||𝑥=𝐿𝑛
=𝜖𝑏
𝑠
𝜕𝑐𝑒,𝑠(𝑥,𝑡)
𝜕𝑥 ||||𝑥=𝐿𝑛
, 𝜖𝑏
𝑛
𝜕𝑐𝑒,𝑠(𝑥,𝑡)
𝜕𝑥 ||||𝑥=𝐿𝑛+𝐿𝑠
=𝜖𝑏
𝑝
𝜕𝑐𝑒,𝑝(𝑥,𝑡)
𝜕𝑥 ||||𝑥=𝐿𝑛+𝐿𝑠
(13)
P oceeding as in Sec ion 2.2, he a ia ion in concen a ion wi h
espec o he ypical li hium concen a ion in he elec oly e 𝑐𝑒,𝑡𝑦𝑝 is
aken as he equa ion a iable:
𝑐𝑒,𝑘(𝑥, 𝑡) = 𝑐𝑒,𝑘(𝑥, 𝑡) − 𝑐𝑒,𝑡𝑦𝑝 (14)
Addi ionally, he ollowing spa ial a iables a e conside ed o con-
enience om now on:
𝑥𝑛=𝑥, 𝑥𝑛∈ [0, 𝐿𝑛], 𝑥𝑠=𝑥−𝐿𝑛, 𝑥𝑠∈ [0, 𝐿𝑠], 𝑥𝑝=𝐿−𝑥, 𝑥𝑝∈ [0, 𝐿𝑝]
(15)
Taking he Laplace ans o m o Eq. (11) yields he ollowing se o
o dina y di e en ial equa ions:
𝜕2𝑐𝑒,𝑛(𝑥𝑛, 𝑠)
𝜕𝑥2
𝑛
−𝑠
𝜖𝑏−1
𝑒,𝑛 𝐷𝑒
𝑐𝑒,𝑛(𝑥𝑛, 𝑠) = (1 − 𝑡+)
𝐴𝐹 𝐿𝑛𝐷𝑒𝜖𝑏
𝑒,𝑛
𝐼(𝑠)(16a)
𝜕2𝑐𝑒,𝑠(𝑥𝑠, 𝑠)
𝜕𝑥2
𝑠
−𝑠
𝜖𝑏−1
𝑒,𝑠 𝐷𝑒
𝑐𝑒,𝑠(𝑥𝑠, 𝑠)=0 (16b)
𝜕2𝑐𝑒,𝑝(𝑥𝑝, 𝑠)
𝜕𝑥2
𝑝
−𝑠
𝜖𝑏−1
𝑒,𝑝 𝐷𝑒
𝑐𝑒,𝑝(𝑥𝑝, 𝑠) = (1 − 𝑡+)
𝐴𝐹 𝐿𝑝𝐷𝑒𝜖𝑏
𝑒,𝑝
𝐼(𝑠)(16c)
Sol ing Eq. (16) and applying he bounda y condi ions a he end-
poin s o he cell in Eq. (12) yields Eq. (17):
𝑐𝑒,𝑛(𝑥𝑛, 𝑠)=2𝐶𝑛cosh ⎛⎜⎜⎝
𝑥𝑛√𝑠
𝜖𝑏−1
𝑒,𝑛 𝐷𝑒⎞⎟⎟⎠
−(1 − 𝑡+)
𝐴𝐹 𝐿𝑛𝜖𝑒,𝑛
𝐼(𝑠)
𝑠(17a)
𝑐𝑒,𝑠(𝑥𝑠, 𝑠) = 𝐶𝑠,1exp ⎛⎜⎜⎝
𝑥𝑠√𝑠
𝜖𝑏−1
𝑒,𝑠 𝐷𝑒⎞⎟⎟⎠
+𝐶𝑠,2exp ⎛⎜⎜⎝
−𝑥𝑠√𝑠
𝜖𝑏−1
𝑒,𝑠 𝐷𝑒⎞⎟⎟⎠
(17b)
𝑐𝑒,𝑝(𝑥𝑝, 𝑠)=2𝐶𝑝cosh ⎛⎜⎜⎝
𝑥𝑝√𝑠
𝜖𝑏−1
𝑒,𝑝 𝐷𝑒⎞⎟⎟⎠
−(1 − 𝑡+)
𝐴𝐹 𝐿𝑝𝜖𝑒,𝑝
𝐼(𝑠)
𝑠(17c)
whe e 𝐶𝑛,𝐶𝑠,1,𝐶𝑠,2and 𝐶𝑝ha e o be de e mined om he bounda y
condi ions in Eq. (13). In his a icle, ins ead o sol ing o he coe i-
cien s di ec ly, we p opose a me hod o simpli y he sys em o equa ions
in Eq. (17) by de i ing he equi ed condi ions o educe he model. I
has o be no ed ha he conside a ion o spa ial a iables as in Eq. (15)
allows o he de ini ion o he ollowing imescales:
𝜏𝑒,𝑛 =𝐿2
𝑛
𝜖𝑏−1
𝑒,𝑛 𝐷𝑒
, 𝜏𝑒,𝑠 =𝐿2
𝑠
𝜖𝑏−1
𝑒,𝑠 𝐷𝑒
, 𝜏𝑒,𝑝 =𝐿2
𝑝
𝜖𝑏−1
𝑒,𝑝 𝐷𝑒
(18)
The a io be ween he li hium mig a ion imescales in he sepa a o
and he elec odes is de e mined by Eq. (19):
𝜏𝑒,𝑠
𝜏𝑒,𝑛,𝑝
=
𝐿2
𝑠𝜖𝑏−1
𝑒,𝑛,𝑝
𝐿2
𝑛,𝑝𝜖𝑏−1
𝑒,𝑠
(19)
I his a io is su icien ly close o 0, he elec oly e concen a ion in
he sepa a o may be conside ed o be in s eady s a e wi h espec o
ha in he elec odes. This esul s in a linea concen a ion p o ile along
he sepa a o leng h, by aking he limi 𝑠→0in Eq. (16b). The slope
𝑚o he concen a ion in he sepa a o is de e mined by he bounda y
condi ion ega ding lux con inui y in Eq. (13):
𝑚= 2𝐶𝑛
𝜖𝑏
𝑒,𝑛
𝜖𝑏
𝑒,𝑠 √𝑠
𝜖𝑏−1
𝑒,𝑛 𝐷𝑒
sinh (√𝜏𝑒,𝑛𝑠)(20)
A u he simpli ica ion is ca ied ou by calcula ing he a io be-
ween he concen a ion a he elec ode-sepa a o in e ace and he
o al concen a ion inc emen in he nega i e elec ode. The concen-
a ion a he elec ode-sepa a o in e ace is calcula ed assuming ha
i is equal o hal o he o al a ia ion in he sepa a o :
|||𝑐𝑒,𝑠|𝑥𝑠=0|||=𝑚𝐿𝑠
2=𝐶𝑛
𝜖𝑏
𝑒,𝑛
𝜖𝑏
𝑒,𝑠 √
√
√
√𝑠𝐿2
𝑠
𝜖𝑏−1
𝑒,𝑛 𝐷𝑒
sinh (√𝜏𝑒,𝑛𝑠)(21)
The concen a ion inc emen in he nega i e elec ode is calcula ed
di ec ly by subs i u ing in Eq. (17a):
|||𝛥 𝑐𝑒,𝑛|||= 2𝐶𝑛[cosh (√𝜏𝑒,𝑛𝑠)− 1](22)
The e o e, he co esponding a io is calcula ed as ollows:
||||
𝑐𝑒,𝑠|𝑥𝑠=0
𝛥 𝑐𝑒,𝑠 ||||=1
2
𝜖𝑏
𝑒,𝑛
𝜖𝑏
𝑒,𝑠
sinh (√𝜏𝑒,𝑛𝑠)
cosh (√𝜏𝑒,𝑛𝑠)− 1 √
√
√
√𝑠𝐿2
𝑠
𝐷𝑒𝜖𝑏−1
𝑒,𝑛
=1
2
𝜖𝑏
𝑒,𝑛
𝜖𝑏
𝑒,𝑠
co h (√𝜏𝑒,𝑛𝑠
2)√
√
√
√𝑠𝐿2
𝑠
𝐷𝑒𝜖𝑏−1
𝑒,𝑛
(23)
Jou nal o Ene gy S o age 64 (2023) 107150
5
P. Rod íguez-I u iaga e al.
One can e i y ha his a io educes o 𝐿𝑠𝜖𝑏
𝑒,𝑛
𝐿𝑛𝜖𝑏
𝑒,𝑠
o he equency
ange o in e es a ound 𝜔=1
𝜏𝑒,𝑛 . Taking his in o accoun , i 𝐿𝑠𝜖𝑏
𝑒,𝑛,𝑝
𝐿𝑛,𝑝𝜖𝑏
𝑒,𝑠
≪
1, he a ia ions in concen a ion in he sepa a o may be neglec ed
in compa ison o hose in he elec odes and, as a esul , he bound-
a y condi ion o he concen a ion in he nega i e elec ode may be
app oxima ed by 𝑐𝑒,𝑛(𝑥𝑛, 𝑠)|||𝑥𝑛=𝐿𝑛
= 0.
Upon applying said bounda y condi ion, he a ia ion in he elec-
oly e concen a ion is ob ained as a unc ion o 𝑥𝑛and 𝑠:
𝑐𝑒,𝑛(𝑥𝑛, 𝑠) = (1 − 𝑡+)
𝐴𝐹 𝐿𝑛𝜖𝑒,𝑛
𝐼(𝑠)
𝑠⎡⎢⎢⎢⎢⎣
cosh (𝑥𝑛√𝑠
𝜖𝑏−1
𝑒,𝑛 𝐷𝑒)
cosh (√𝜏𝑒,𝑛𝑠)− 1⎤⎥⎥⎥⎥⎦
(24)
Nex , he elec ode-a e aged alue o he a ia ion in elec oly e
concen a ion is calcula ed:
𝑐𝑒,𝑛(𝑠) = 1
𝐿𝑛∫𝐿𝑛
0
𝑐𝑒,𝑛(𝑥𝑛, 𝑠)𝑑𝑥𝑛
=(1 − 𝑡+)
𝐴𝐹 𝐿𝑛𝜖𝑒,𝑛
𝐼(𝑠)
𝑠⎡⎢⎢⎢⎣
1
√𝜏𝑒,𝑛𝑠co h (√𝜏𝑒,𝑛𝑠)− 1⎤⎥⎥⎥⎦
(25)
The e o e, he ans e unc ion om he applied cu en o he
elec ode-a e aged a ia ion in elec oly e concen a ion may be ex-
p essed as:
𝐺𝑒(𝑠) = 𝑐𝑒,𝑛(𝑠)
𝐼(𝑠)=𝐾𝑒,𝑛 ⎡⎢⎢⎢⎣
1 − √𝜏𝑒,𝑛𝑠co h (√𝜏𝑒,𝑛𝑠)
𝜏𝑒,𝑛𝑠√𝜏𝑒,𝑛𝑠co h (√𝜏𝑒,𝑛𝑠)⎤⎥⎥⎥⎦
(26)
whe e 𝐾𝑒,𝑛 =(1−𝑡+)𝜏𝑒,𝑛
𝐴𝐹 𝐿𝑛𝜖𝑒,𝑛 .
P oceeding as in Sec ion 2.2, one can s udy he beha io o 𝐺𝑒(𝑠)in
he limi s 𝑠→0and 𝑠→∞as in Eq. (27):
𝐺𝑒(𝑠) ≈ ⎧
⎪
⎨
⎪
⎩
−𝐾𝑒,𝑛
3, 𝑠 →0
0, 𝑠 →∞
(27)
In o de o app oxima e he equency esponse o 𝐺𝑒(𝑠)be ween
hese wo limi s, we p opose a ZARC elemen as exp essed in Eq. (28):
𝐾𝑒,𝑛 ⎡⎢⎢⎢⎣
1 − √𝜏𝑒,𝑛𝑠co h (√𝜏𝑒,𝑛𝑠)
𝜏𝑒,𝑛𝑠√𝜏𝑒,𝑛𝑠co h (√𝜏𝑒,𝑛𝑠)⎤⎥⎥⎥⎦
≈ − 𝐾𝑒,𝑛∕3
1 + (𝛽𝑒𝜏𝑒,𝑛𝑠)𝛼𝑒(28)
As in Sec ion 2.2,𝛼𝑒and 𝛽𝑒mus be de e mined so as o ob ain
he closes app oxima ion o 𝐺𝑒(𝑠). Fo his pu pose, he equency
esponse o bo h ans e unc ions is plo ed in a Nyquis diag am and
he impedance e o be ween he exac and he app oxima e ans e
unc ion is minimized wi hin a ange o equencies. In his case, we
ha e conside ed he in e al 𝜔=[1
103𝜏𝑒,𝑛
,103
𝜏𝑒,𝑛 ]and he esul ing alues
a e 𝛼𝑒= 0.9936 and 𝛽𝑒= 0.3983. The Nyquis plo o he equency
esponse o bo h ans e unc ions wi h a posi i e eal pa is shown
in Fig. 2.
A simila esul is ob ained o he posi i e elec ode wi h a plus
sign. As in Sec ion 2.2, Eq. (28) is exp essed in e ms o he no malized
equency 𝜏𝑒,𝑛𝑠, so his app oxima ion emains alid ega dless o he
speci ic physical pa ame e alues. In he in e es o simplici y, a alue
o 𝛼𝑒= 1 will be used u he in his a icle. Las ly, he o e po en-
ial due o concen a ion g adien s in he elec oly e 𝜂𝑐is calcula ed
acco ding o Eq. (2b):
𝜂𝑐(𝑠) = 2𝑅𝑇
𝐹
(1 − 𝑡+)
𝑐𝑒,𝑡𝑦𝑝 (𝑐𝑒,𝑝 −𝑐𝑒,𝑛)
=2𝑅𝑇
𝐹
(1 − 𝑡+)
𝑐𝑒,𝑡𝑦𝑝 (𝐾𝑒,𝑝∕3
1 + 𝛽𝜏𝑒,𝑝𝑠+𝐾𝑒,𝑛∕3
1 + 𝛽𝜏𝑒,𝑛𝑠)𝐼(𝑠)
(29)
Fig. 2. Nyquis plo o he equency esponse o he exac and app oxima e no malized
ans e unc ions in Eq. (28) o di usion in he elec oly e.
I 𝐿𝑛≈𝐿𝑝and 𝜖𝑒,𝑛 ≈𝜖𝑒,𝑝, he e ms co esponding o bo h elec odes
in Eq. (29) may be combined in o one single RC ne wo k.
2.4. Cha ge ans e o e po en ial
F om he exp essions o he solid and elec oly e concen a ions
de e mined in Sec ions 2.2 and 2.3, an accu a e app oxima ion o he
eac ion o e po en ials may be ob ained acco ding o Eq. (2c). Fo his
pu pose, he exchange cu en densi ies 𝑗0,𝑛 and 𝑗0,𝑝 a e calcula ed as
ollows [14]:
𝑗0,𝑛 =1
𝐿𝑛∫𝐿𝑛
0𝑚𝑛𝑐𝑛,𝑚𝑎𝑥√𝜒𝑠,𝑛(1 − 𝜒𝑠,𝑛)√𝑐𝑒,𝑛𝑑𝑥𝑛
𝑗0,𝑝 =1
𝐿𝑝∫𝐿𝑝
0𝑚𝑝𝑐𝑝,𝑚𝑎𝑥√𝜒𝑠,𝑝(1 − 𝜒𝑠,𝑝)√𝑐𝑒,𝑝𝑑𝑥𝑝
(30)
whe e 𝑚𝑛and 𝑚𝑝a e he eac ion a es o he nega i e and posi i e
elec ode, espec i ely. No e ha 𝜒𝑠,𝑛 and 𝜒𝑠,𝑝 ha e been de e mined
al eady in Sec ion 2.2 and do no depend on he 𝑥dimension in he
SPMe. The e o e, he nega i e elec ode exchange cu en densi y may
be ew i en as ollows:
𝑗0,𝑛 =𝑚𝑛𝑐𝑛,𝑚𝑎𝑥√𝜒𝑠,𝑛(1 − 𝜒𝑠,𝑛)√𝑐𝑒,𝑡𝑦𝑝
1
𝐿𝑛∫𝐿𝑛
0√1 + 𝑐𝑒,𝑛
𝑐𝑒,𝑡𝑦𝑝
𝑑𝑥𝑛(31)
Assuming ha he a ia ion in he elec oly e concen a ion is su -
icien ly smalle han he ypical concen a ion 𝑐𝑒,𝑡𝑦𝑝, he ollowing
app oxima ion can be made:
𝑗0,𝑛 ≈𝑚𝑛𝑐𝑛,𝑚𝑎𝑥√𝜒𝑠,𝑛(1 − 𝜒𝑠,𝑛)√𝑐𝑒,𝑡𝑦𝑝
1
𝐿𝑛∫𝐿𝑛
0(1 + 𝑐𝑒,𝑛
2𝑐𝑒,𝑡𝑦𝑝 )𝑑𝑥𝑛(32)
Consequen ly, he nega i e elec ode exchange cu en densi y may
be ew i en as a unc ion o he elec ode-a e aged a ia ion in he
elec oly e concen a ion 𝑐𝑒,𝑛, which was de e mined in he p e ious
Sec ion:
𝑗0,𝑛 ≈𝑚𝑛𝑐𝑛,𝑚𝑎𝑥√𝜒𝑠,𝑛(1 − 𝜒𝑠,𝑛)√𝑐𝑒,𝑡𝑦𝑝 (1 + 𝑐𝑒,𝑛
2𝑐𝑒,𝑡𝑦𝑝 )(33)
F om his exp ession o he exchange cu en densi y, he cha ge-
ans e o e po en ial may be calcula ed acco ding o Eq. (2c):
𝜂𝑟,𝑛 =2𝑅𝑇
𝐹sinh−1 (𝐼𝑅𝑛
6𝑗0,𝑛𝜖𝑛𝐿𝑛𝐴)(34)
This may also be exp essed ia he cha ge- ans e esis ance, in-
cluding i s explici dependence on he applied cu en as in Eq. (35)

Jou nal o Ene gy S o age 64 (2023) 107150
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P. Rod íguez-I u iaga e al.
Fig. 3. (a) Full equi alen ci cui model. (b) Simpli ied equi alen ci cui model.
𝑅𝑐𝑡,𝑛(𝐼) = 2𝑅𝑇
𝐹 𝐼0,𝑛 [𝐼0,𝑛
𝐼sinh−1 (𝐼
𝐼0,𝑛 )], 𝐼0,𝑛 =6𝑗0,𝑛𝜖𝑛𝐿𝑛𝐴
𝑅𝑛
(35)
A simila esul is ob ained o he posi i e elec ode. We con-
side his o be a majo ad an age o employing elec ode-a e aged
quan i ies as in [14], which allows o an accu a e es ima ion o he
eac ion o e po en ial om only he elec ode-a e aged elec oly e
concen a ion ins ead o ha ing o sol e o he concen a ion p o ile
along he 𝑥dimension. Taking in o accoun he cu en dependence o
he eac ion o e po en ials is especially c i ical in applica ions whe e a
wide ange o cu en a es is expec ed.
2.5. Equi alen ci cui model
Acco ding o he ou pu ol age exp ession in Eq. (1) and he esul -
ing equa ions o he indi idual e ms in Eq. (2), he equi alen ci cui
model in Fig. 3-(a) may be cons uc ed, whe e he esis o s 𝑅𝑜ℎ𝑚,𝑒 =
1
𝜅𝐴 (𝐿𝑛
3𝜖𝑏
𝑒,𝑛
+𝐿𝑠
𝜖𝑏
𝑒,𝑠
+𝐿𝑝
3𝜖𝑏
𝑒,𝑝 )and 𝑅𝑜ℎ𝑚,𝑠 =1
3𝐴(𝐿𝑛
𝜎𝑛
+𝐿𝑝
𝜎𝑝)co espond o he
e ms in Eqs. (2d) and (2e).
Fo he ime-domain implemen a ion o he ci cui , a se o se ially
connec ed pa allel RC b anches will be employed as an app oxima ion
Table 1
Pa ame e equa ions o he 7-RC app oxima ion as a unc ion o 𝛼[53].
Pa ame e Exp ession
𝑟1=𝑟70.14(1 − 𝛼)2
𝑟2=𝑟60.22(1 − 𝛼)−0.08(1 − 𝛼)3
𝑟3=𝑟5(0.12 + 0.057𝑒3.4𝛼)(1 − 𝛼)
𝑟41−2(𝑟1+𝑟2+𝑟3)
𝑡1=1/𝑡71.4⋅10−8𝑒19𝛼(1.6−𝛼)
𝑡2=1/𝑡6
0.078𝛼5.63
0.026+𝛼3.67
𝑡3=1/𝑡5
0.56𝛼2.27
0.4+𝛼1.3
𝑡41
o he ZARC elemen . In pa icula , we will use he con inuous app ox-
ima ion by 7 RC elemen s ha we p esen ed in [53]. This app oach
is he mos app op ia e o he ime-domain simula ion o he ZARC
elemen as we showed in [56], due o he ac ha i a oids he issues
caused by he speci ica ion o a memo y leng h in he G ünwald–
Le niko app oach. Taking he ac ional o de 𝛼, he esis o 𝑅𝑍𝐴𝑅𝐶
and he ime cons an 𝜏𝑍𝐴𝑅𝐶 as inpu s, he pa ame e alues a e
di ec ly calcula ed as 𝑅𝑖=𝑅𝑍𝐴𝑅𝐶 ⋅𝑟𝑖(𝛼)and 𝜏𝑖=𝜏𝑍𝐴𝑅𝐶 ⋅𝑡𝑖(𝛼) o
se en-elemen ne wo k acco ding o Table 1 [53]:
Jou nal o Ene gy S o age 64 (2023) 107150
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P. Rod íguez-I u iaga e al.
As a esul , he app oxima e disc e e s a e-space ep esen a ion o a
ZARC elemen is shown in Eq. (36):
𝑥𝑍𝐴𝑅𝐶 (𝑘) = [𝑖1(𝑘)𝑖2(𝑘) … 𝑖7(𝑘)]𝑇
𝑥𝑍𝐴𝑅𝐶 (𝑘) = 𝐴𝑍𝐴𝑅𝐶 𝑥𝑍𝐴𝑅𝐶 (𝑘− 1) + 𝐵𝑍𝐴𝑅𝐶 𝑢(𝑘− 1)
𝑦𝑍𝐴𝑅𝐶 (𝑘) = 𝐶𝑍𝐴𝑅𝐶 𝑥𝑍𝐴𝑅𝐶 (𝑘)
(36)
whe e 𝑥𝑍𝐴𝑅𝐶 is he s a e ec o , 𝑢is he inpu cu en and 𝑦𝑍𝐴𝑅𝐶 is he
o al ol age di e ence in he subci cui , wi h he ollowing s a e-space
ep esen a ion ma ices:
𝐴𝑍𝐴𝑅𝐶 =𝑑𝑖𝑎𝑔 [exp (−𝛥𝑡
𝜏1)exp (−𝛥𝑡
𝜏2)… exp (−𝛥𝑡
𝜏7)]
𝐵𝑍𝐴𝑅𝐶 =[1 − exp (−𝛥𝑡
𝜏1)1 − exp (−𝛥𝑡
𝜏2)… 1 − exp (−𝛥𝑡
𝜏7)]𝑇
𝐶𝑍𝐴𝑅𝐶 =[𝑅1𝑅2…𝑅7](37)
whe e 𝛥𝑡 is he sampling ime.
Consequen ly, he s a e ec o and he s a e equa ion o he ECM
a e as ollows:
𝑥(𝑘) = [𝜒𝑝(𝑘)𝑥𝑍𝐴𝑅𝐶,𝑝(𝑘)𝑐𝑒,𝑝(𝑘)𝜒𝑛(𝑘)𝑥𝑍𝐴𝑅𝐶,𝑛(𝑘)𝑐𝑒,𝑛(𝑘)]𝑇
𝑥(𝑘) = 𝐴𝑥(𝑘− 1) + 𝐵𝑢(𝑘− 1), 𝑢(𝑘) = 𝐼(𝑘)
(38)
whe e
𝐴=𝑑𝑖𝑎𝑔 [1𝑑𝑖𝑎𝑔(𝐴𝑍𝐴𝑅𝐶,𝑝) exp (−𝛥𝑡
𝛽𝑒𝜏𝑒,𝑝 )1𝑑𝑖𝑎𝑔(𝐴𝑍𝐴𝑅𝐶,𝑛) exp (−𝛥𝑡
𝛽𝑒𝜏𝑒,𝑛 )]
𝐵=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
−𝐾𝑝𝛥𝑡
3𝜏𝑝
−𝐵𝑍𝐴𝑅𝐶,𝑝
𝐾𝑒,𝑝
3[1 − exp (−𝛥𝑡
𝛽𝑒𝜏𝑒,𝑝 )]
𝐾𝑛𝛥𝑡
3𝜏𝑛
𝐵𝑍𝐴𝑅𝐶,𝑛
−𝐾𝑒,𝑛
3[1 − exp (−𝛥𝑡
𝛽𝑒𝜏𝑒,𝑛 )]
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(39)
No e he minus sign in he e ms co esponding o he posi i e
pa icle and he elec oly e concen a ion in he nega i e elec ode
in ma ix 𝐵. Be o e e alua ing he ou pu ol age exp ession, i is
con enien o calcula e explici ly he su ace concen a ion o bo h
pa icles as well as he exchange cu en densi y in bo h elec odes:
𝜒𝑠,𝑝(𝑘) = 𝜒𝑝(𝑘) + 𝐶𝑍𝐴𝑅𝐶,𝑝𝑥𝑍𝐴𝑅𝐶,𝑝(𝑘)
𝜒𝑠,𝑛(𝑘) = 𝜒𝑛(𝑘) + 𝐶𝑍𝐴𝑅𝐶,𝑛𝑥𝑍𝐴𝑅𝐶,𝑛(𝑘)
𝑗0,𝑝(𝑘) = 𝑚𝑝𝑐𝑝,𝑚𝑎𝑥√𝜒𝑠,𝑝(𝑘)(1 − 𝜒𝑠,𝑝(𝑘))√𝑐𝑒,𝑡𝑦𝑝 (1 + 𝑐𝑒,𝑝(𝑘)
2𝑐𝑒,𝑡𝑦𝑝 )
𝑗0,𝑛(𝑘) = 𝑚𝑛𝑐𝑛,𝑚𝑎𝑥√𝜒𝑠,𝑛(𝑘)(1 − 𝜒𝑠,𝑛(𝑘))√𝑐𝑒,𝑡𝑦𝑝 (1 + 𝑐𝑒,𝑛(𝑘)
2𝑐𝑒,𝑡𝑦𝑝 )
(40)
Taking all he p e ious equa ions in o accoun , he ou pu ol age
exp ession is as ollows:
𝑣(𝑘) = 𝑂𝐶𝑃𝑝(𝜒𝑠,𝑝(𝑘)) − 𝑂𝐶𝑃𝑛(𝜒𝑠,𝑛(𝑘)) + 2𝑅𝑇
𝐹
(1 − 𝑡+)
𝑐𝑒,𝑡𝑦𝑝 (𝑐𝑒,𝑝(𝑘) − 𝑐𝑒,𝑛(𝑘))
+2𝑅𝑇
𝐹[sinh−1 (𝐼𝑅𝑝
6𝑗0,𝑝(𝑘)𝜖𝑝𝐿𝑝𝐴)+ sinh−1 (𝐼𝑅𝑛
6𝑗0,𝑛(𝑘)𝜖𝑛𝐿𝑛𝐴)]
+𝐼
𝜅𝐴 (𝐿𝑛
3𝜖𝑏
𝑒,𝑛
+𝐿𝑠
𝜖𝑏
𝑒,𝑠
+𝐿𝑝
3𝜖𝑏
𝑒,𝑝 )
+𝐼
3𝐴(𝐿𝑛
𝜎𝑛
+𝐿𝑝
𝜎𝑝)(41)
I bo h elec odes a e assumed o ha e simila physical and spa ial
p ope ies, he ECM can be u he simpli ied o ha in Fig. 3-(b), hus
educing in hal he numbe o equi ed s a es.
The de eloped ECM may be in e p e ed as he combina ion o he
wo discussed app oaches in Sec ion 1: on he one hand, he anscen-
den al ans e unc ions a e ob ained di ec ly om he elec ochemical
model and hen app oxima ed wi h ac ional-o de ans e unc ions
ins ead o Padé’s app oximan [40–43] in ou case. On he o he
hand, he s anda d RC equi alen ci cui model om [36] is spli in o
wo sec ions as in [39], so ha he le mos one accoun s o he
li hium di usion wi hin he ac i e ma e ial pa icles. Consequen ly, he
li hium concen a ion is e alua ed a hei su ace, hus es ablishing a
mo e accu a e co espondence wi h he manne in which he elec ode
po en ial is calcula ed in elec ochemical models. Howe e , a be e
desc ip ion o he di usion p ocess is achie ed wi h a ZARC elemen
ins ead o a RC ne wo k, as indica ed by he op imal o de exponen
𝛼= 0.82. The e o e, his model, s a ed in his way, may be use ul
o mos o he applica ions whe e ECMs a e employed p o iding
addi ional physical insigh , and may also be implemen ed in s anda d
ci cui simula o s by subs i u ing he ZARC elemen by i s mul iple-RC
app oxima ion [53], hus allowing o a simple simula ion o cons an -
ol age phases by subs i u ing he cu en sou ce in Fig. 3 by a ol age
sou ce.
2.6. F equency domain applica ion
F om he ob ained ans e unc ions, i is possible o de e mine
he medium and low equency esponse o he cell di ec ly gi en
i s physical pa ame e s. Howe e , he high- equency ansien e ec s
o he cha ge ans e p ocesses a e usually no conside ed in he
DFN o SPMe due o hem being o de s o magni ude as e han
di usion dynamics. Consequen ly, in o de o accu a ely ep oduce EIS
da a, a CPE is usually connec ed in pa allel wi h he cha ge- ans e
esis ance 𝑅𝑐𝑡 in he ci cui shown in Fig. 3-(b). The small-signal alue
o 𝑅𝑐𝑡 is calcula ed by aking in o accoun ha o small alues o 𝑥,
𝑥−1 sinh−1 (𝑥)≈ 1:
𝑅𝑐𝑡(𝐼≈ 0) = 2𝑅𝑇
𝐹 𝐼0
, 𝑍𝑐𝑡(𝑠) = 𝑅𝑐𝑡
1 + (𝜏𝑐𝑡𝑠)𝛼𝑐𝑡 (42)
Con e sely, i he equency-domain beha io o he ba e y cell is
cha ac e ized expe imen ally in an EIS es , he e will be a subs an ial
o e lap be ween he e ec s o bo h elec odes and only he e ec i e
pa ame e s o he simpli ied ECM in Fig. 3-(b) will be iden i iable.
Gi en ha an EIS es consis s o small-signal a ia ions a ound a
ce ain ope a ing poin , he ECM is linea ized a ound 𝑆𝑂𝐶 =𝑆𝑂𝐶𝐸𝐼𝑆 .
Using he ac ha cell SOC emains unchanged, one can w i e:
𝑂𝐶𝑉 (𝜒𝑠) = 𝑂𝐶𝑉 (𝑆𝑂𝐶𝐸𝐼𝑆 ) + 𝜕𝑂𝐶𝑉
𝜕𝑆𝑂𝐶 ||||𝑆𝑂𝐶𝐸𝐼𝑆
𝜒𝑠(43)
whe e 𝜕𝑂𝐶𝑉
𝜕𝑆𝑂𝐶 |||𝑆𝑂𝐶𝐸𝐼𝑆
is de e mined empi ically om he OCV-SOC ela-
ionship and he equency esponse o 𝜒𝑠may be app oxima ed by ha
o a ZARC elemen . The e o e, a possible ans e unc ion o i ing
EIS da a is composed o he addi ion o an ohmic esis o , a high-
equency ZARC elemen o cha ge ans e p ocesses, a mid- equency
RC elemen o he di usion in he elec oly e and a low- equency
ZARC elemen o he solid di usion, as shown in Eq. (44):
𝑍(𝑠) = 𝑅𝑜ℎ𝑚 +𝑅𝑐𝑡
1 + (𝜏𝑐𝑡𝑠)𝛼𝑐𝑡 +𝑅𝑒
1 + 𝜏𝑒𝑠+𝜕𝑂𝐶𝑉
𝜕𝑆𝑂𝐶 ||||𝑆𝑂𝐶𝐸𝐼𝑆
𝐾𝑠
1 + (𝜏𝑠𝑠)𝛼𝑠(44)
whe e and he pa ame e s o be iden i ied a e 𝑅𝑜ℎ𝑚,𝑅𝑐𝑡,𝜏𝑐𝑡,𝛼𝑐𝑡,𝑅𝑒,
𝜏𝑒,𝐾𝑠and 𝜏𝑠, al hough 𝜏𝑐𝑡 and 𝛼𝑐𝑡 will no be used in he ECM.
Addi ionally, he alue o he exchange cu en 𝐼0may be calcula ed
di ec ly om 𝑅𝑐𝑡, so as o ake in o accoun he cu en dependence o
he cha ge ans e esis ance o la ge-signal ope a ion.
We belie e ha he co espondence be ween he ime and
equency-domain beha io s o he cell is an ad an age o e elec-
ochemical models, as i also p o ides a consis en cha ac e iza ion
me hod since physical pa ame e s a e g ouped in esis o s and ime
cons an s.
Jou nal o Ene gy S o age 64 (2023) 107150
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P. Rod íguez-I u iaga e al.
Table 2
Pa ame e se om [14].
Pa ame e Uni s Desc ip ion Value
Nega i e elec ode
𝐿𝑛mThickness 1⋅10−4
𝑅𝑛mPa icle adius 1⋅10−5
𝐷𝑛m2∕s Solid di usi i y 3.9⋅10−14
𝜖𝑛– Ac i e ma e ial olume ac ion 0.6
𝑐𝑛,𝑚𝑎𝑥 mol∕m3Maximum li hium concen a ion 2.4983 ⋅104
𝜎𝑛S∕m Solid conduc i i y 100
𝜖𝑒,𝑛 – Po osi y 0.3
𝑚𝑛(A∕m2)(m3∕mol)1.5Reac ion a e 2⋅10−5
Posi i e elec ode
𝐿𝑝mThickness 1⋅10−4
𝑅𝑝mPa icle adius 1⋅10−5
𝐷𝑝m2∕s Solid di usi i y 1⋅10−13
𝜖𝑝– Ac i e ma e ial olume ac ion 0.5
𝑐𝑝,𝑚𝑎𝑥 mol∕m3Maximum li hium concen a ion 5.1218 ⋅104
𝜎𝑝S∕m Solid conduc i i y 10
𝜖𝑒,𝑝 – Po osi y 0.3
𝑚𝑝(A∕m2)(m3∕mol)1.5Reac ion a e 6⋅10−7
Sepa a o
𝐿𝑠mThickness 2.5⋅10−5
𝜖𝑒,𝑠 – Po osi y 1
O e all
𝑐𝑒,𝑡𝑦𝑝 mol∕m3Typical elec oly e concen a ion 1⋅103
𝐷𝑒m2∕s Typical elec oly e di usi i y 5.34 ⋅10−10
𝜅S∕m Typical elec oly e conduc i i y 1.1
𝑡+– T ans e ence numbe 0.4
𝑏– B uggeman coe icien 1.5
𝐴m2Elec ode a ea 2.8359 ⋅10−2
𝑄Ah Cell capaci y 0.68
3. Theo e ical esul s and discussion
In his sec ion, he pe o mance o he p oposed ECM is alida ed
agains he SPMe om [14] o di e en cu en p o iles. Fo he sim-
ula ion o he elec ochemical model, we ha e used PyBaMM (Py hon
Ba e y Ma hema ical Modeling) [57]. PyBaMM is a ba e y modeling
so wa e implemen ed in Py hon designed o simpli y he compa ison
o s anda d ba e y models by p o iding an in e ace o disc e iza ion
me hods and nume ical sol e s. In his case, we ha e employed a
ypical disc e iza ion consis ing o 20 poin s in each domain as well
as 20 poin s o bo h pa icles.
The se o physical pa ame e s is shown in Table 2.
Be o e ca ying ou he simula ions, he alidi y o he app oxima-
ions de i ed in Sec ion 2.3 is e i ied:
𝐿2
𝑠𝜖𝑏−1
𝑒,𝑛,𝑝
𝐿2
𝑛,𝑝𝜖𝑏−1
𝑒,𝑠
= 0.034 ≪1,
𝐿𝑠𝜖𝑏
𝑒,𝑛,𝑝
𝐿𝑛,𝑝𝜖𝑏
𝑒,𝑠
= 0.041 ≪1(45)
Nex , he pe o mance o he p oposed ECM has been alida ed
agains he ci ed SPMe model by compa ing hei simula ion esul s
o dis inc ope a ion scena ios. In his case, we ha e employed ou
cons an -cu en discha ges a 2C, 1C, C/2 and C/5 ollowed by a 30-
min es whe e C- a e is he measu emen o he cha ge and discha ge
cu en wi h espec o i s nominal capaci y. Addi ionally, we ha e also
conside ed a d i ing cycle (US06) in o de o es ou model unde
dynamic ope a ing condi ions.
The simula ion esul s o he cons an -cu en discha ges a e shown
in Fig. 4, plo ed wi h espec o no malized ime as a unc ion o he C-
a e. I is obse ed ha he p oposed model is able o ep oduce he cell
ol age accu a ely du ing he discha ge ega dless o he cu en a e.
We a ibu e his mainly o he p ecise calcula ion o he cha ge- ans e
eac ion o e po en ials as de ailed in Sec ion 2.4, which p esen a
highly nonlinea dependency on cu en . Fu he mo e, he elaxa ion
p o ile is also accu a ely modeled due o he in oduc ion o ac ional-
o de ci cui elemen s o accoun o he solid di usion p ocess. The
ol age esponse o he ba e y and he US06 cu en p o ile a e shown
in Fig. 5. I can also be obse ed ha he p oposed ECM is able
o yield a g ea ly accu a e e minal ol age in dynamic condi ions.
Fu he mo e, he compa ison be ween he ba e y in e nal s a es and
hei equi alen in he p oposed ECM is shown in Fig. 6. Gi en ha
he p oposed model has been analy ically de i ed om he SPMe, i is
able o p o ide in o ma ion abou he in e nal s a es o he ba e y. We
belie e his is a quali a i e ad an age o e s anda d equi alen ci cui
models [26], which a e cons uc ed o ep oduce ba e y ol age only.
Ne e heless, a ho ough compa ison be ween he accu acy o di e en
ECMs wi h espec o expe imen al da a cons i u es a sepa a e s udy
ha is beyond he scope o his a icle. The e could be many ac o s
in ol ed in he disc epancies be ween model esul s and expe imen al
da a, such as empe a u e, hys e esis and cu en - a e e ec s, as well as
he pa ame e iza ion p ocess and he ope a ing condi ions unde which
he models a e es ed.
The e o esul s be ween he SPMe and he p oposed ECM a e sum-
ma ized in Table 3. I has o be poin ed ou ha no only does he model
p o ide an accu a e app oxima ion o he e minal ol age, bu also
he in e nal s a es o he ba e y, namely he su ace concen a ion o
he solid pa icles and he elec ode-a e aged elec oly e concen a ion,
which esul s in a p ecise calcula ion o he cha ge- ans e eac ion
o e po en ials. The maximum ol age e o occu s immedia ely a e
he end o he discha ge, due o he ac ha he p oposed app oxima-
ion o he solid di usion ans e unc ion is less accu a e a highe
equencies, as obse ed in Fig. 1. This also explains he sligh ly highe
e o in he su ace concen a ion o he nega i e pa icle in compa ison
o he posi i e pa icle, gi en i s la ge solid di usion ime cons an .
The elec ode-a e aged elec oly e concen a ion is also modeled wi h
g ea accu acy: aking in o accoun ha i s ypical concen a ion is
𝑐𝑒,𝑡𝑦𝑝 = 1000 mol/m3, he ela i e RMS and maximum e o s a e below
0.6% and 3% espec i ely. No e ha he esul s o he elec oly e
concen a ion a e equal in bo h elec odes due o he ac ha hei
hickness and po osi y ha e he same alues in his pa ame e se ; his
is no a gene al esul none heless. In conclusion, he p oposed ECM
is able o accu a ely model he ou pu ol age as well as he in e nal
s a es wi h espec o an SPMe.
Las ly, a compa ison on he compu a ional equi emen s o bo h
models is ca ied ou . We ha e de e mined ha he p oposed ECM
is abou 3 o 4 imes quicke on a e age o he same imes ep and
simula ion p o ile; howe e , we belie e ha he main ad an age o ou
model wi h espec o he SPMe is he numbe o s a es equi ed. Fo a
ypical disc e iza ion, such as he one conside ed he e, 100 s a es mus
be s o ed and upda ed e e y imes ep o he SPMe, whe eas only 18
a e necessa y o he ull ECM and 9 o he simpli ied e sion i he
7-RC app oxima ion o he ZARC elemen is employed. Ne e heless,
o he con inuous app oxima ions by 5 [53] and 3 [49] RC ne wo ks
ha e also been epo ed in ecen li e a u e, so should he a ailable
memo y be igh ly cons ained, he numbe o necessa y s a es may
be po en ially educed o 14 o 10 o he ull ECM and 7 o 5 o
he simpli ied ECM, wi hou a majo loss o accu acy. This is a c ucial
ea u e gi en he limi a ions on memo y and compu a ion powe in
on-boa d BMSs.
Fu he mo e, al hough some s a e obse e s ha e been p oposed
o he SPM and SPMe [15,42], employing an ECM makes i simple
o implemen a Dual F ac ional-O de Ex ended Kalman Fil e o he
concu en es ima ion o s a e o cha ge and ci cui pa ame e s, as we
p esen ed in [56], wi h he ad an age ha he iden i ied pa ame e s
a e di ec ly ela ed o hei physical coun e pa s in his model.
4. Expe imen al applica ion
In his sec ion, he simpli ied e sion o he p oposed equi alen
ci cui model is compa ed o expe imen al da a by iden i ying he
equi alen ci cui pa ame e s om an EIS es .
Jou nal o Ene gy S o age 64 (2023) 107150
9
P. Rod íguez-I u iaga e al.
Fig. 4. Te minal ol age o he SPMe and he p oposed ECM o a cons an -cu en discha ge a (a) 2C, (b) 1C, (c) C/2 and (d) C/5 espec i ely, ollowed by a 30-min es .
Cu es ha e been plo ed wi h espec o no malized ime, i.e. ime di ided by he nominal discha ge du a ion acco ding o he C- a e.
Fig. 5. Te minal ol age and inpu cu en o he SPMe and he p oposed ECM o he US06 d i ing cycle.