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Comp ehensi e machine lea ning app oaches o modelling he s a e o
cha ge o li hium-ion ba e ies
Mi chell Raea, Michela O a ianib,c, Dominika Capko ac,d, Tomáš Kazdad,
Luigi Jacopo San a Ma iac, Ke in M. Ryanc, S e ano Passe inie, Mehakp ee Singha,∗
aMa hema ics Applica ions Conso ium o Science and Indus y (MACSI), Depa men o Ma hema ics and S a is ics, Uni e si y o Lime ick, Lime ick V94
T9PX, I eland
bDepa men o Applied Sciences, Technological Uni e si y o he Shannon, Moylish Campus, Lime ick V94EC5T, I eland
cDepa men o Chemical Sciences and Be nal Ins i u e, Uni e si y o Lime ick, Lime ick V94 T9PX, I eland
dDepa men o Elec ical and Elec onic Technology, Facul y o Elec ical Enginee ing and Communica ion, B no Uni e si y o Technology, B no, Czech Republic
eHelmhol z Ins i u e Ulm (HIU) ü Elek ochemische Ene giespeiche ung, Helmhol zs aße 11, 89081 Ulm, Ge many
H I G H L I G H T S
•MLP, LSTM and NARX a e used o model
SOC in LiBs wi h coa se da ase s.
•MLP and LSTM a e mo e adap able wi h
a smalle aining da ase .
•Highe co ela ion be ween SOC and EIS
da a is ound o EIS measu ed a 0%
SOC.
•S eng hs and weaknesses o he MLP,
LSTM and NARX a e discussed.
•E ec s o aining size, hidden laye s
and lea ning a es o ANNs a e dis-
cussed.
G R A P H I C A L A B S T R A C T
A R T I C L E I N F O
Da ase link:h ps://zenodo.o g/ eco ds/1336
1914
Keywo ds:
Machine lea ning
Deep a i icial neu al ne wo k
Li-ion ba e ies
S a e o cha ge
Ma hema ical modelling
Coa se da ase
A B S T R A C T
The ad ancemen o li hium-ion ba e ies (LIBs) is i al o achie ing ne -ze o emissions because i enables
enewable ene gy in eg a ion, suppo s elec ic ehicle (EV) adop ion, and p omo es cos -e ec i e and
sus ainable solu ions. The g owing demand o EVs and po able elec onics has ampli ied he need o eliable
ba e y managemen sys ems o ensu e sa e y and pe o mance. Machine lea ning (ML) me hods o modelling
he s a e o cha ge (SOC) in ba e ies a e gaining ac ion owing o hei adap abili y o di e se da ase s and
lowe compu a ional demands. Howe e , he challenge lies in selec ing he mos sui able ML a chi ec u e o a
speci ic applica ion. This s udy e alua es h ee ML app oaches o SOC modelling in LIBs: mul ilaye pe cep on
(MLP), long sho - e m memo y (LSTM), and nonlinea au o eg essi e wi h exogenous inpu (NARX) neu al
ne wo ks. The models we e es ed using an expe imen al da ase wi h mul iple inpu a iables, including
elec ochemical impedance spec oscopy da a, ol age, and capaci y om comme cial LIB cells. The esul s
show ha MLP and LSTM pe o m e ec i ely wi h smalle aining da ase s (14 samples), whe eas he NARX
model equi es mo e ex ensi e da a (34 ou o 67 samples) o accu acy. Addi ionally, he NARX model
showed g ea e sensi i i y o lea ning a e adjus men s and hidden laye con igu a ions, whe eas MLP and
LSTM main ained obus pe o mance ac oss a ying pa ame e s.
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (M. Rae), [email p o ec ed] (M. O a iani), [email p o ec ed] (D. Capko a), [email p o ec ed] (T. Kazda),
[email p o ec ed] (L.J.S. Ma ia), [email p o ec ed] (K.M. Ryan), [email p o ec ed] (S. Passe ini), [email p o ec ed] (M. Singh).
h ps://doi.o g/10.1016/j.jpowsou .2025.236929
Recei ed 15 Janua y 2025; Recei ed in e ised o m 14 Ma ch 2025; Accep ed 29 Ma ch 2025
Jou nal o Powe Sou ces 646 (2025) 236929
A ailable online 8 May 2025
0378-7753/© 2025 The Au ho s. Published by Else ie B.V. This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ ).
M. Rae e al.
1. In oduc ion
[1] The adop ion o elec ic ehicles (EVs), including e-scoo e s, e-
bikes, e- ucks, and o he o ms o e-mobili y, has expe ienced apid
g ow h wo ldwide o e he pas decade. By 2022, he numbe o
elec ic ca s alone on he oad exceeded 26 million, ma king a 60%
inc ease compa ed o 2021 and su passing mo e han i e imes he
s ock eco ded in 2018 (T ends in elec ic ligh -du y ehicles -- Global
EV Ou look 2023 -- Analysis - IEA). This g ow h is in line wi h he
a ge s ou lined in he Pa is Ac ion Agenda on Elec o-Mobili y and
Clima e Change (The Pa is Ag eemen | UNFCCC). Upon compa ing
he echnological eadiness o ba e ies a ailable in he ma ke , i is
e iden ha LIBs con inue o domina e [2,3]. Compa a i e analysis o
di e se ba e y chemis ies is oo ed in c i ical ac o s, including ene gy
densi y, powe densi y, cycle li e, calenda li e, and cos pe kWh. An
assessmen based on hese pa ame e s unde sco es he p eeminence o
LIBs as a leading echnology in he o eseeable u u e, boas ing supe io
ene gy densi y, e iciency, and an ex ended li espan. In addi ion, LIBs
display he po en ial o no ewo hy sho - e m enhancemen [2,4,5].
Ne e heless, he e ec i e pe o mance o he ba e y can be signi -
ican ly comp omised by sudden empe a u e changes, and inciden s
such as i e and ba e y des uc ion a e common consequences o
o e cha ging [1,2,4,6]. These aspec s aise se ious sa e y conce ns, pa -
icula ly when conside ing he use o LIBs in EVs. Mo eo e , mee ing
he inc eased demand o LIBs and he enla ged complexi y o he LIBs
a chi ec u e equi es he de elopmen o highly accu a e es ima ion o
and eliable ba e y managemen sys ems (BMSs). The BMS plays a i al
ole in es ima ing he s a e o cha ge (SOC) and s a e o heal h (SOH) o
he ba e y, espec i ely; he abili y o he ba e y o deli e i s speci ic
ou pu , and he emaining use ul li e [2]. Mo eo e , i can accu a ely
assess ba e y ageing and deg ada ion [7]. The SOC o a ba e y in use
is es ima ed using highly co ela ed ex e nal a iables, such as ol age,
cu en , cha ge, and impedance [7,8]. Ad anced ma hema ical models
can es ima e he ba e y SOC based on he a o emen ioned ex e nally
measu ed a iables. The bene i o aking ex e nal measu emen s is
ha he ba e y pe o mance is no a ec ed by he measu emen s, and
a iables can be obse ed wi hou disabling he ba e y [9].
Recen applica ions o LIBs p esen new ci cums ances ha may
equi e unique o e ised models. The e o e, a ple ho a o SOC models
a e now a ailable o he public and indus y. Mos ypes o modelling
echniques all unde he ollowing umb ella e ms: di ec me hods,
adap i e me hods, bookkeeping me hods, and hyb id me hods. Exis ing
SOC es ima ion echniques, such as Coulomb coun ing, open-ci cui
ol age, and Kalman il e me hods, p esen signi ican d awbacks in
eal-li e applica ions. The limi a ions o hese example me hods include
he ollowing: Coulomb coun ing is suscep ible o cumula i e e o s
due o inaccu acies in cu en measu emen and ini ial SOC es ima ion;
open-ci cui ol age equi es he ba e y o be a es o accu a e
measu emen , which is imp ac ical o in-use BMS applica ions; and
Kalman il e me hods, while accu a e, a e compu a ionally in ensi e
and hea ily elian on p ecise ba e y models, making hem sensi i e
o pa ame e a ia ions and unce ain ies [10]. Ad ancemen s con inue
o be made in Kalman il e models [11] o educe compu a ional cos
and noise educ ion o imp o e modelling capabili ies. The me hods
used in esea ch on exis ing echniques can also be applied o imp o e
ML app oaches [12,13]. A i icial neu al ne wo ks (ANN) and ML
app oaches e ec i ely add ess hese challenges by modelling nonlinea
ba e y beha iou , educing suscep ibili y o noise, and adap ing o
di e se ope a ing condi ions. In addi ion, ANNs acili a e eal- ime
SOC es ima ion wi hou equi ing ba e y es . Se e al ad anced ANN
models de eloped in [12–14] add ess he challenges o eal-li e ap-
plica ions, such as noise educ ion, imp o ed ea u e ex ac ion, and
educed compu a ional cos . While ANNs o en exhibi limi a ions in
modelling wi h coa se da ase s, his pape p esen s imp o emen s on
ANN applica ions o coa se da a o SOC p edic ion using he mos basic
ANNs o highligh he impac o hei unique a chi ec u al di e ences.
ANN and ML me hods appeal o he scien i ic communi y because
o hei abili y o adap and de ec unde lying pa e ns using simple
compu a ions. ML models equi e no physical unde s anding o he
sys em and wo k e ec i ely wi h a much lowe compu a ional cos han
al e na i e models such as he Kalman il e me hod. These wo ac o s
ha e a ac ed he a en ion o many scien i ic esea che s [15,16]. In
ecen s udies [17–19], i has been demons a ed ha he e is po en ial
o implemen physics-in o med ANNs o modelling a ious in e nal
componen s o he ba e y, such as long- e m changes [17,18] o SOC,
as in [19]. Fo accu a e p edic ions, he la e used a la ge da ase ,
as opposed o he coa se da ase used in his s udy. Fu he uses o
machine lea ning in he ield o ba e ies and g eene ene gy include
he de elopmen and imp o emen o ene gy s o age ma e ials ia ML
me hods [20,21]. The ML me hods discussed in his s udy use an ANN
app oach wi hin he ML amily.
The aim o an ANN model is o mimic he na u al s uc u e and
p ocess, ha is, he p ocessing o in o ma ion by he b ain’s ne wo k o
neu al pa hways and synapses. ANNs a e designed o ain hemsel es
by implemen ing a lea ning algo i hm ha essen ially uses he e o
be ween he p edic ed ou pu (ini ially he esul o andom neu on
ac i a ion) and he ue ou pu o a aining da ase [22]. This ali-
da ion be ween he p edic ed and ue alues guides he hidden laye
o o m a bias, o weigh , on each synapse ha educes he ou pu
e o . I he s uc u e and ini ial condi ions o he ANN a e ela i ely
app op ia e, ex ensi e epe i ion will cause he ANN ou pu o con e ge
o a minimum e o , he eby p o iding an accu a e model o he
desi ed ou pu . This is es ed agains he ese ed es da a o judge
he model’s p edic i e abili y ai ly. ANNs ace limi a ions such as
being apped in local minima ha a e a om he global minimum,
o e i ing he model o he aining da a, and hus ailing o make
p edic ions wi h new da a [23].
A ple ho a o a ailable neu al ne wo k (NN) a chi ec u es can
ende he selec ion o an app op ia e design challenging. I is also
common o s uggle o ix he co ec laye sizes and lea ning algo-
i hms [16]. This is ue o indi iduals ha a e educa ed in he ield,
le alone indi iduals who a e no amilia wi h he needs and unc ions
o NN modelling [24]. While ini ial judgemen can na ow down he
selec ion o NN a chi ec u es, he alida ion o he emaining selec ion
and choice is o en accomplished by ial and e o [24]. The aim o
his s udy was o in es iga e h ee di e en neu al ne wo ks: mul ilaye
pe cep on (MLP), long sho - e m memo y (LSTM), and nonlinea
au o- eg essi e wi h exogenous inpu (NARX). These h ee models we e
selec ed o in es iga ion based on hei key di e ences: The MLP
was chosen as an indica o o he exis ing modi ied MLP designs and
se es as a s epping s one o he in es iga ion o mo e ad anced MLP
based ANN models. LSTM possesses he abili y o de ec bo h long-
and sho - e m ends; his cha ac e is ic is used o p o ide a basis o
in o ming he eade o he po en ial o u he in es iga ion o sub le
pa e ns be ween he inpu pa ame e s and SOC. NARX was selec ed o
in es iga e he empo al dependency o he da ase used in he ANN o
SOC p edic ion. The NARX eedback loop, as discussed la e , uniquely
p ese es he empo al ela ionship be ween he inpu and ou pu
alues o making he nex p edic ion. While nume ous al e na i e
ANN models exis in he li e a u e, he au ho s de e mined he basic
MLP, NARX, and LSTM models o be a sui able indica ion and s epping
s one o u u e esea ch in o mo e ad anced echniques, speci ically
o applica ions wi h coa se da ase s. The adap abili y and lexibili y o
he models we e assessed, and he pe o mances when p edic ing he
SOC we e analysed and compa ed. Simila o a p e ious s udy [25], he
lexibili y o di e en ML me hods is expanded by add essing di e en
aspec s. The sensi i i y and lexibili y o he lea ning a e, laye sizes,
Jou nal o Powe Sou ces 646 (2025) 236929
2
M. Rae e al.
aining pe cen ages, and o e all pe o mance we e compa ed o each
NN design. The EIS and ol age measu emen s o a Samsung INR18650-
35E LIB we e used o aining and alida ing he models. The ou comes
o his pape will in o m he eade o he beha iou o hese h ee
NN a chi ec u es and will guide owa ds an enligh ened decision abou
which model o use when i ing he da a. Fu he mo e, his pape
a emp s o ou line he possibili y o educing expe imen al labou ,
ma e ials, expenses, and was e.
2. Machine lea ning (ML)
This sec ion p o ides a b ie o e iew o ML concep s o p epa e
he eade o elabo a ions and jus i ica ions made in he Resul s
and Discussions sec ion. The sec ion in oduces basic concep s using
an example o a eed o wa d NN, including he ac i a ion unc ion,
op imisa ion algo i hms, and weigh s. We hen in oduce he h ee ML
app oaches applied in his s udy, ha is, MLP, LSTM, and NARX NNs,
and he de ails o each ne wo k-speci ic laye p ocess.
ML app oaches use ma hema ical algo i hms o ain a pa icula
model such ha i can p edic he ou pu o a ce ain p ocess gi en
speci ied inpu a iables [15,22,26–29]. The NN ield o ML is inspi ed
by he na u al p ocesses o o ganic b ains [22]. Tha is, an NN com-
p ises se e al laye s ha con ain a collec ion o synapses o nodes
linked oge he ia neu al pa hs o connec ions. Hence, NNs in ML a e
o en called ANNs o dis inguish hem om biological ones. Ex e nal in-
o ma ion is p ocessed as i lows om he ini ial inpu laye ⟶ hidden
laye s ⟶ ou pu o decision laye . An ANN uses echnology o cons uc
an in o ma ion p ocessing sys em ha mimics his s uc u e and uses
ma hema ical unc ions o in e p e and assign da a h oughou he
e ol ing ANN. An ANN can be s uc u ed wi h a ious a chi ec u es
and unc ions ha uniquely p ocess he inpu da a o p o ide an ou pu .
Thei di e ences in design a e use ul o he speci ic cha ac e is ic
beha iou s o he da ase . Selec ing he app op ia e a chi ec u e will
help he model pe o m wi hin easonable e o bounds. Howe e ,
simila o he human b ain, any ANN equi es aining o unc ion
app op ia ely. T aining an ANN in ol es educing o inc easing he
weigh o bias o each synapse wi hin he ne wo k such ha he ou pu
app oaches a minimum e o . The weigh ep esen s he impo ance o
co ela ion s eng h be ween he node and ou pu , and i is adjus ed
a e e e y i e a ion. Unlike he weigh s, he bias is mo e igid and
po ays a i mly implemen ed ela ion be ween nodes and ou pu s; i
is no adjus ed as eely as he weigh s. No all NNs include bias. Fo
mos NNs, he ini ial dis ibu ion o node weigh s wi hin he hidden
laye is andom, and hey a e adjus ed when he e o is p opaga ed
back h ough he ne wo k. The ou pu accu acy is a ma e o assigning
he co ec weigh s o each indi idual synapse, hus iden i ying he ue
unde lying pa e ns in he da ase . Many ma hema ical algo i hms ha e
been de eloped o de e mine he op imal weigh dis ibu ion; hese a e
called aining algo i hms [30,31]. The mos sui able aining algo i hm
depends on he beha iou o he da ase and he NN a chi ec u e and
is o en de e mined a e explo ing many possibili ies [30].
2.1. Basic a chi ec u e: Feed o wa d neu al ne wo k (FNN)
The FNN is he mos basic NN s uc u e, as shown in Fig. 1(a).
I consis s o an inpu laye , a hidden laye , and an ou pu laye . The
hidden laye does no need o be ‘‘ ully connec ed’’ o be classi ied as an
FNN; a ully connec ed laye will ha e all synapses connec ed o e e y
synapse o he subsequen laye . Any NN inpu laye mus connec o
he hidden laye ia ac i a ion unc ions and assign weigh s o each
node. The inal hidden laye o any NN mus be ully connec ed o
he ou pu laye , in his case a single node ha p esen s he p edic ed
SOC. Fu he mo e, he hidden laye can consis o any numbe o
nodes; howe e , ewe nodes will desi ably educe he numbe o
compu a ions. An e o e m is ound by compa ing he p edic ed and
expe imen al ou pu s, and hen he e o is ed h ough he ne wo k o
adjus he weigh s using he aining algo i hm.
2.2. Pa e n ecogni ion and da a ex ac ion
ANNs a e amous o hei abili y o p ocess da a and e ol e in o
models ha can accu a ely p edic an ou come beyond he comp e-
hension o al e na i e physics-in o med models. The hidden laye o
an ANN is labelled as such o i s unobse able na u e. The speci ic
pa e ns, ends, and ela ionships ha he ANN de ec s du ing he
aining p ocess a e hidden wi hin he hidden laye s o he ANN and
canno be ex ac ed. Much esea ch is unde way o unco e knowledge
in ANNs [32–34]. The ecogni ion o new pa e ns by an ANN is mos
ob ious du ing he ine- uning o he ANN pa ame e s. As one begins o
op imise he model, he e o cu e (di e ence be ween he p edic ed
and known da a poin s) will smoo h. The smoo hing o he e o
cu e is indica i e o an ANN lea ning new and sub le ends wi hin
a da ase . A key pa ame e a ec ing he de ec ion o sub le pa e ns
is he lea ning a e. As he lea ning a e dec eases, an ANN is able o
na iga e in o small local e o minima, whe eas a la ge lea ning a e
would cause he ANN o oscilla e a ound he minimum o skip o e i
en i ely. Despi e di icul ies in ex ac ing he inne wo kings o ANNs,
he abili y o o m hese sub le pa e ns ha exceed he comp ehension
o physical and ma hema ical heo ies is ex emely aluable and ma ks
a co ne s one o he success o ANN modelling pe o mances.
2.3. Ac i a ion unc ions
An ac i a ion unc ion maps he alues om he espec i e nodes o
a speci ic domain. I s pu pose is o manipula e he da a such ha i i s
a desi ed dis ibu ion o imp o e he unc ionali y o he model- aining
algo i hm. Popula ac i a ion unc ions applied in NNs include he sig-
moid (𝜎), an-sigmoid ( ansig), ec i ied linea (ReLU), and hype bolic
an ( anh) unc ions [35,36]. The sigmoid unc ion con e s he inpu
da a o 𝑥∈ [0,1] along he sigmoid cu e gi en by
𝜎(𝑥) = 1
1 + 𝑒𝑥.
Al e na i ely, he ansig unc ion maps 𝑥∈ [−1,1] using a combina-
ion o he an and sigmoid unc ions:
ansig(𝑥) = 2
1 + 𝑒−2𝑥− 1.
Bo h a e good o aking any inpu alue. Howe e , he ReLU unc ion
is s ic e because i maps all nega i e inpu alues o 0 and all o he
alues emain he same.
Finally, he anh unc ion is he amilia unc ion
anh(𝑥) = 𝑒𝑥−𝑒−𝑥
𝑒𝑥+𝑒−𝑥,
which also maps any inpu 𝑥∈ [−1,1] along he al e na i e anh(𝑥)
cu e.
Du ing his s udy, he ansig ac i a ion unc ion was iden i ied as
he common unc ion wi h a high accu acy ou pu be ween all models.
Using consis en ac i a ion unc ions whe e possible helped s eamline
he compa ison o a chi ec u al impac s be ween he models.
2.4. Adam op imise
Lea ning algo i hms in NNs a e an essen ial componen o ML. The
pu pose o he lea ning algo i hm is o guide he ne wo k owa ds
minimal e o by compa ing he ne wo k ou pu and he ue ou -
pu and using s a is ical e o measu emen s o adjus he ne wo k
acco dingly [37]. The common s ochas ic g adien descen (SGD) is
he basis om which he Adam algo i hm was cons uc ed [38]. I
is a g adien -based op imisa ion me hod wi h simple compu a ions
ha can be labelled as an adap i e momen es ima ion algo i hm –
‘‘Adam’’ [38]. The Adam op imise upda es he weigh s o he NN wi h
Eq. (1) [39]:
𝑤𝑡=𝑤𝑡−1 −𝛼𝑚𝑡
√𝑣𝑡+𝜖
,(1)
Jou nal o Powe Sou ces 646 (2025) 236929
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M. Rae e al.
Fig. 1. Schema ic ep esen a ion o (a), he basic eed o wa d NN a chi ec u e showing he nodes and weigh s o each laye connec ed by s aigh line neu al pa hs; and (b), he
a chi ec u e o a gene al mul ilaye pe cep on NN.
whe e 𝑤 is he weigh ec o o he hidden laye and 𝛼 is he ini ial
lea ning a e, which is cons an in his applica ion. 𝑚 and 𝑣 a e mo ing
a e ages and 𝜖 is a small posi i e cons an , ypically 10−8, o a oid
any di ision by ze o (exploding g adien ). The mo ing a e ages 𝑚𝑡
and 𝑣𝑡 a e he exponen ial a e ages o he g adien along 𝑤𝑡 and he
exponen ial a e ages o he squa es o g adien s along 𝑤𝑡 (o he wise
known as agg ega e - he exponen ial a e age o he p e ious g adien
𝑔𝑡−1 and p esen g adien 𝑔𝑡). 𝑚𝑡 and 𝑣𝑡 a e hen gi en as
𝑚𝑡=𝛼𝑚𝑡−1 + (1 − 𝛼)𝑔𝑡,
𝑚𝑡=𝑚𝑡
(1 − 𝛼𝑡),(2)
𝑣𝑡=𝜇𝑚𝑡−1 + (1 − 𝜇)𝑔𝑡2,
𝑣𝑡=𝑣𝑡
(1 − 𝜇𝑡).(3)
whe e 𝛼 and 𝜇 a e he hype pa ame e s o he Adam op imise . The
lea ning a e 𝛼 and g adien decay a e 𝜇 o he selec ed NNs in his
s udy a e displayed in Table 1.
The aining o he models in his s udy used he Adam op imise
and he emaining aining pa ame e s we e iden i ied h ough ial and
e o as desc ibed in Sec ion 2.6.
2.5. App oach 1: Long sho e m memo y (LSTM)
A long sho - e m memo y (LSTM) NN ollows he simple ecu -
en neu al ne wo k (RNN) a chi ec u e wi h an addi ional memo y
componen . Tha is, he LSTM NN includes he inpu , LSTM, hidden,
and ou pu laye s. The pu pose o he LSTM laye is o hold da a in
s o age o housands o ime s eps o allow long- e m ends o be
acknowledged. This helps he model a oid anishing and exploding
g adien s and conside s he sou ces o long- e m ends.
2.5.1. LSTM laye p ocess
The LSTM p ocess is essen ially a eedback loop wi h a ew addi-
ional componen s ha enable long- e m memo y s o age. Each i e -
a ion c ea es a ‘‘cell s a e’’ which is used o in o m he LSTM laye
in he ollowing i e a ion. The cell s a e is a unc ion o ou ga es
ha in e ac wi h he inpu da a: o ge ga e ( ), e ain ga e (i), cell
candida e ga e (g), and ou pu ga e (o). Fig. 2 p esen s he low o
in o ma ion h ough he LSTM laye . The inpu is copied ou imes and
plugged in o each ga e, whe e i is combined wi h he p e ious hidden
s a e. Using 𝐱𝑡 as he cu en inpu ec o , 𝐖 and 𝐑 as he inpu and
ecu en weigh ec o s, espec i ely, and he bias 𝑏 o each node, he
ollowing symbolic exp essions desc ibe he da a p ocessing h ough
he LSTM laye :
i ga e ⟶𝑖𝑡=𝜎(𝑊𝑖𝐱𝑡+ R𝑖𝐡𝑡−1 +𝑏𝑖),(4)
ga e ⟶𝑓𝑡=𝜎(𝑊𝑓𝐱𝑡+ R𝑓𝐡𝑡−1 +𝑏𝑓),(5)
g ga e ⟶𝑔𝑡= anh (𝑊𝑔𝐱𝑡+ R𝑔𝐡𝑡−1 +𝑏𝑔),(6)
o ga e ⟶𝑜𝑡=𝜎(𝑊𝑜𝐱𝑡+ R𝑜𝐡𝑡−1 +𝑏𝑜).(7)
Feeding he p e ious hidden s a e ℎ𝑡−1 and cu en inpu s 𝐱𝑡 h ough
hese ga es as in Fig. 2, he cu en hidden s a e ℎ𝑡 is calcula ed by
elemen -wise mul iplica ion and ma ix addi ion:
𝐡𝑡=𝑜𝑡⋅ anh (𝐜𝑡).(8)
Simila ly, he cu en cell s a e (𝐜𝑡) is de e mined by elemen -wise
mul iplica ion:
𝐜𝑡=𝑓𝑡⋅𝐜𝑡−1 +𝑖𝑡⋅𝑔𝑡.(9)
The o ge ga e yields a ec o 𝑓𝑡 wi h alues be ween 1 ( e ain)
and 0 ( o ge ), which co espond o each node o he inpu ec o 𝐱𝑡.
This is done using 𝐡𝑡−1 om he p e ious i e a ion, hence he ‘‘sho -
e m memo y’’ om he LSTM model. Howe e , he ou pu ga e simply
passes he conca ena ion o 𝑥𝑡 and 𝐡𝑡−1 o yield he 𝑜𝑡 ec o which is
ed in o he cu en 𝐡𝑡 and passed on o he nex i e a ion; hence he e
is also ‘‘long- e m memo y’’ in he LSTM model.
2.6. App oach 2: Mul ilaye pe cep on (MLP)
The MLP has se e al hidden laye s wi h ully connec ed synapses
(nodes). In his s udy, he MLP has one inpu laye , wo ully connec ed
hidden laye s, and one ou pu laye (see Fig. 1(b)). The ne wo k u ilises
he ‘‘Adam’’ aining op imise in MATLAB (see Sec ion 2.4) o i s
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Fig. 2. In o ma ion low h ough LSTM laye . 𝑐𝑡 is cell s a e a i e a ion , ℎ𝑡 is he hidden s a e ou pu a i e a ion , -1 deno es in o ma ion om he p e ious i e a ion, 𝑥𝑡 is
he cu en inpu a iables o i e a ion . The ga es a e deno ed as ollows: o ge ga e ( ), e ain ga e (i), cell candida e ga e (g), and ou pu ga e (o).
Table 1
Adjus ed aining pa ame e s o each model om
he Adam op imise algo i hm. Whe e 𝛼 is he lea n-
ing a e (cons an ), 𝜇 is he g adien decay ac o
(cons an ), and N is he numbe o i e a ions du ing
aining.
Op imised Model
Pa ame e s MLP LSTM NARX
𝛼0.05 0.012 0.081
𝜇0.9 0.9 0.9
Shu le none none none
N700 1,000 10,000
lexibili y so ha he compa ison be ween MLP and LSTM ne wo ks
could be made mo e ob ious. The Adam op imise is o en used because
o i s abili y o implemen an adap i e lea ning a e. Howe e , o
simplici y o analysis, he lea ning a e was kep cons an . The comple e
aining pa ame e s a e displayed in Table 1.
The e ec i e implemen a ion o any NN equi es expe imen a ion
wi h key componen s, o hype pa ame e s, o he a chi ec u e. Fo he
MLP, he key componen s we e he hidden laye (HL) size, numbe o
nodes wi hin each HL, lea ning a e 𝛼, and numbe o i e a ions (N).
This expe imen al phase can be di icul because he pa ame e s may
be sensi i e. Gene ally, a model ha wo ks well wi h low sensi i i y
o hese pa ame e s is much easie o i . The selec ion o alues o
he pa ame e s was conduc ed unde ial and e o . Each pa ame e
was de e mined based on a balance be ween he model pe o mance
me ics and compu a ional cos . The models we e assessed as po en ial
BMSs; he e o e, he model should be highly e icien — sui able o
scaling up o eal-li e applica ions. The same p ocess was applied o
he de e mine he pa ame e s o he LSTM and NARX seen in Table 1
2.7. App oach 3: Nonlinea au o eg essi e wi h exogenous inpu (NARX)
The NARX model p edic s he ou pu da a using delayed inpu da a.
In his case, NARX uses a 1:2 ime-s ep delay. Tha is, he NN uses he
inpu s om sample 𝑠−1 o p edic he sample 𝑠 ou pu . The NARX NN in
his s udy was cons uc ed using he MATLAB Deep Lea ning Toolbox.
The pa ame e s o he NARX model we e adjus ed acco ding o Table
1. Excluding he ime-delay componen , he NARX model in his case
wo ks like he FNN model wi h one HL.
3. Elec ochemical impedance spec oscopy (EIS)
EIS is a powe ul and non-des uc i e echnique o in es iga ing he
ba e y li e ime, elec ochemical mechanisms, ma e ial anspo p op-
e ies, eac ion kine ics, beha iou o po ous elec odes, and SOC/SOH
es ima ion. The esponse o he sys em o a pe iodic AC signal a a spec-
i ied ange o equencies wi h a small ampli ude was in es iga ed. In
Table 2
Speci ica ions o he in es iga ed Samsung INR18650-35E cells.
Pa ame e Value
Nominal Capaci y 3.40 Ah
Nominal Vol age 3.60 V
Maximum Vol age 4.20 V
Minimum Vol age 2.65 V
Nominal Cha ging Cu en 1.70 A
Nominal Discha ging Cu en 8.00 A
he gal anos a ic mode, he cu en is measu ed a he applied po en ial
in he equency domain. The impedance (𝛺) can be de i ed om he
a io o he ol age- o-cu en ampli ude and he phase lag o he ou pu
and inpu . The impedance can be spli in o he eal pa (Re(𝛺)) and
imagina y pa (Im(𝛺)), which can be ep esen ed in Nyquis plo s (
Fig. 3(a)), om which g ea insigh can be gained [40–44]. The 𝑥-axis
ep esen s he eal pa o he impedance, while he 𝑦-axis illus a es
he nega i e alue o he imagina y pa o he impedance. The high-
equency egion indica es he Ohmic esis ance o he ba e y, he
mid- equency semici cle signi ies he double-laye capaci ance e ec ,
and he low- equency ail is a ibu ed o he di usion p ocesses wi hin
he ac i e ma e ial o he ba e y.
4. Expe imen al design
In his sec ion, he de ails o he expe imen al p ocedu e a e ou -
lined sys ema ically, including he speci ica ions o he equipmen and
i ems used h oughou . The sec ion also desc ibes he manipula ion o
da a in p epa a ion o i s use in NN models.
4.1. Expe imen al me hodology
The Li-ion ba e ies conside ed in his s udy we e comme cial cylin-
d ical cells INR18650-35E p oduced by Samsung. The anode ma e ial is
based on g aphi e and he ca hode ma e ial is based on li hium-nickel-
manganese-cobal -oxides (NMC). The speci ica ions o he Samsung
INR18650-35E cells a e lis ed in Table 2.
The measu emen me hodology was se up as ollows:
1. Two cycles using a cha ging and discha ging cu en o 0.1C;
2. Measu emen o EIS e e y 25% o SOC in a ange o 0–100%
SOC du ing cha ging and discha ging a 0.1C;
3. Cycling o he cell using a cha ging and discha ging cu en o
0.5C o 50 cycles using 80% o he ol age ange;
4. Repea e e y 50 cycles: wo cycles using a cu en o 0.1C
and measu emen o EIS e e y 25% SOC du ing cha ging and
discha ging a 0.1C.
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Fig. 3. (a): Typical Nyquis plo he Li-ion ba e y a 25 % SOC. (b): Deg ada ion o Samsung INR18650-35E Ba e y Cell O e Pe iod o 100 Cycles cha ged and discha ged a
0.5 C. (c): Changes in cha ging and discha ging cu es o Samsung INR18650-35E cell o a capaci y check be o e EIS measu emen a 0.1 C be o e cycling, a e 50 cycles, and
a e 100 cycles. (d): Compa ison o Samsung INR18650-35E cell EIS cu es measu ed a 0% SOC be o e cycling, a e 50 cycles, and a e 100 cycles.
A ZKETECH EBC-X ba e y es e was used o ba e y cycling. The
EIS measu emen s du ing 0.1C cycling we e pe o med a he Biologic
VMP3 measu emen s a ion in he equency ange om 100 kHz o
30 mHz wi h a ol age ampli ude o 10 mV. I is help ul o no e he e
ha he abo e p ocedu e yielded measu emen s made be o e cycling,
a e 50 cycles, and a e 100 cycles; hus, h ee cycles wo h o EIS,
ol age, and capaci ance da a we e used o he modelling o he LIB
SOC (as seen in all he igu es o he supplemen a y da a).
The empe a u e was held cons an a 22
◦C. h oughou he en i e y
o he expe imen al p ocedu e. The e o e, he esul s ob ained om he
expe imen a e alid o 22
◦C. The e was no alida ion o he esul s
a di e en empe a u es.
4.2. Expe imen al da a
Expe imen al da a om Samsung INR18650-35E cells wi h NMC
chemis y, as desc ibed abo e, we e used in his s udy. The deg ada ion
o he cell cha ged a 0.1 C and discha ged a 0.5 C du ing 100 cycles
is shown in Fig. 3(b). The ini ial discha ge capaci y was 2.66 Ah,
and a e 100 cycles, he capaci y dec eased o 2.41 Ah, ep esen ing a
capaci y e en ion o 90%. A he end o cycling, he ba e y cell did
no each i s end-o -li e.
Two cycles a 0.1 C o cha ge and discha ge we e pe o med be o e
cycling, a e 50, and 100 cycles. The i s cycle was a p e-condi ioning
cycle o ese he ‘cumula i e his o y’ o he cell, and he second cycle
was a measu emen o he ac ual cha ge and discha ge capaci y p io
o he EIS measu emen a di e en SOCs. The second cha ge and
discha ge cycles wi h ega d o ba e y deg ada ion a e illus a ed in
Fig. 3(c). The discha ge capaci y a 0.1 C be o e cycling was 3.38 Ah,
and in a capaci y check a e 100 cycles, he capaci y dec eased o
3.29 Ah.
A compa ison o he EIS cu es measu ed du ing ba e y ageing
a 0% SOC is depic ed in Fig. 3(d). The cu e shape is consis en
du ing deg ada ion; howe e , he main di e ence is he inc ease in he
ohmic esis ance wi h ba e y deg ada ion. No e also ha an inc ease
in ohmic esis ance is obse ed be o e cycling and a e 50 cycles, bu
demons a es s abili y be ween 50 and 100 cycles, as seen in Fig. 3(d).
4.3. Da a p ocessing
The size o he expe imen al da a used o his s udy is coa se com-
pa ed o many NN esea ch pape s ela ed o SOC es ima ion [45–50].
Typical esea ch pape s ha e published compa isons and pe o mances
o LIB models using da ase s wi h hund eds o housands o ba e y
cycles. The da ase in his s udy includes 69 samples sou ced om h ee
ba e y cha ge/discha ge cycles. A ‘‘sample’’ e e s o an obse a ion a
a speci ic ins an , o he wise in e p e ed as a ow in he o al da ase
whe e each column ep esen s a a iable ( hus, a 69 × 7 ma ix is
he da ase o his s udy). This s udy p o ides a compa ison o h ee
NNs designed o model he SOC o a comme cialised LIB. The esul s
o his s udy p o ide a good e e ence o each model’s abili y o e -
ec i ely lea n unde lying pa e ns wi h a small da ase . The measu ed
inpu a iables in his s udy include he cha ge and discha ge ol age
(𝑉+
and 𝑉−, espec i ely), and he eal and imagina y pa s o he
impedance (Re(𝛺) and Im(𝛺), espec i ely). EIS was no measu ed
in nea -con inuous disc e ised s eps, such as he ol age and cha ge.
Ra he , i was measu ed a speci ic SOC in e als h oughou he cycle:
0%, 25%, 50%, 75%, and 100% SOC. Each EIS measu emen yielded
an a ay o Re(𝛺) and Im(𝛺) alues co esponding o an a ay o
spec oscopy measu emen equencies. NNs equi e an equal numbe
o samples o he inpu and ou pu by design; he e o e, he EIS da a
we e o ganised such ha h ee samples o 0% SOC EIS da a we e
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Table 3
The a chi ec u e and ne wo k pa ame e s ha ga e op imal ou pu accu acy when using 50%
da a o aining.
LSTM
Hidden Laye LSTM Laye I e a ions Lea n a e (𝛼)
50 4 1000 0.012
MLP
Hidden Laye 1 Hidden Laye 2 I e a ions Lea n a e (𝛼)
3 2 700 0.05
NARX
Time S ep Delay Hidden Laye 1 I e a ions Lea n Ra e
1:2 10 10000 0.081
s acked o ming a o al o 69 eadings (∼ 23 measu emen s a each
0% SOC cycle), and conca ena ing his wi h 69 samples o he ol age
and cha ge eadings aken a e en in e als h oughou he ull h ee
cycles. The eason o using only EIS measu ed a 0% SOC is explained
in Sec ion 5.2.
Fu he mo e, no malisa ion by max-scaling is pe o med on each
ea u e o he da ase indi idually. The pu pose o his is o map all
a iables o he se 𝑆∈ [−1,1] in which he sigmoid unc ion can
be success ully applied. The max-scaling echnique had g ea e success
in inc easing he ou pu accu acy compa ed o o he no malisa ion
echniques such as linea scaling.
5. Resul s and discussion
In his sec ion, he h ee p oposed ML app oaches will be compa ed
wi h expe imen al da a o e alua e hei accu acy and e iciency. I is
wo h men ioning ha he accu acy o each echnique depends hea ily
on he pe cen age o da a conside ed o aining and alida ing he
models. The e o e, he accu acy and adap abili y o all echniques will
be es ed by conside ing di e en a ios o da ase s. In addi ion, he
sensi i i y analysis o each model in ol es a ying he alue o he
lea ning a e 𝛼 and he hidden laye sizes. Fu he mo e, he e a e
de ia ions om he pa ame e speci ica ions in Table 1.
5.1. Gene al pe o mance
The di e en a chi ec u es o MLP, LSTM, and NARX NNs p esen
a ious limi a ions and ad an ages. The h ee NN a chi ec u es wo k
e ec i ely and wi hin ypical e o bounds when ope a ing wi h a
aining size ≥70% (≈ 49 samples o mo e). Howe e , educing he
aining size o 50% a ec s he pe o mance o e e y echnique o
di e en ex en s. The quan i a i e SOC ou pu ob ained o each model
is shown in Fig. 4(a). One can obse e ha i is di icul o isualise he
accu acy o di e en app oaches. The e o e, he ela i e e o s (abso-
lu e di e ence be ween he expe imen al and model alues, all di ided
by he expe imen al alues) ob ained using di e en app oaches we e
es ima ed and a e shown in Fig. 4(b). The esul s e eal ha he MLP
shows mo e accu a e esul s han he LSTM and NARX echniques. In
addi ion, he aining pa ame e s and a chi ec u al designs (quan i a-
i ely) wi h op imal pa ame e s o each model a e displayed in Tables
1and 3; wi h he espec i e pe o mance s a is ics displayed in Table
4. Again, he quan i a i e analysis demons a es ha he MLP app oach
ou pe o ms bo h LSTM and NARX echniques. This is likely due o he
mo e simplis ic na u e o he MLP applied o a simple da ase and he
abili y o LSTM o de ec mo e sub le unde lying pa e ns. Howe e ,
he nonlinea au o eg essi e wi h exogenous inpu (NARX) NN shows
less accu acy han bo h MLP and LSTM in e ms o p edic ing he SOC
(see Fig. 4).
5.2. Elec ochemical impedance spec oscopy (EIS) da a ele ance
NN a chi ec u es ypically equi e da a o ha e inpu s and ou pu s
o he same leng h. In his s udy, he o al da a leng h was 69. The
ol age and capaci ance obse a ions we e made a mo e equen ly
Table 4
Compa ison o pe o mance me ics be ween MLP,
LSTM, and NARX models using he mean absolu e
e o (MAE), adjus ed coe icien o de e mina ion
(Adj. 𝑅2
o Adj. R-squa ed), and s anda d de ia ion
o e o (SD) o each model.
S a is ic LSTM MLP NARX
SD 0.0328 0.0318 0.0595
Adj. 𝑅20.990 0.999 0.947
MAE 0.0328 0.0098 0.0357
han he impedance spec oscopy, bu he da a we e comp essed o
align wi h he ewe impedance alues. Impedance spec oscopy was
only obse ed unde i e condi ions: when he s a e o cha ge was
equal o 100%, 75%, 50%, 25%, and 0%; hence, EIS was eco ded
i e imes du ing cha ging and ou imes du ing discha ge. EIS was
measu ed ac oss a spec um o 23 di e en equencies (100 kHz o
30 mHz) and epea ed a he a o emen ioned SOC alues. Howe e ,
he EIS obse a ions showed a de ia ion be ween he di e en SOC
alues owing o he li hia ion/deli hia ion o he elec ode ma e ials.
The e o e, a co ela ion ma ix was used o de e mine which SOC
yielded he highes co ela ion be ween EIS and SOC.
The EIS analysis ou lined he signi icance o he SOC alue when EIS
is measu ed. Table 5 shows ha he EIS measu ed when SOC = 0% has
he g ea es co ela ion o he SOC p edic ion and yields he highes
accu acy when implemen ed in o he MLP model. This is a esul o
he speci ic in e nal p ope ies obse ed a 0% SOC. The e o e, only
he 0% SOC EIS da a seen in Fig. 3(d) a e used in he model; all o he
EIS da a can be ound in he supplemen a y ma e ial. The pu pose o
selec ing ewe EIS measu emen s is o educe he numbe o inpu
ea u es, simpli y he model, and imp o e pe o mance.
Fu he mo e, o he s udies ha e shown ha he speci ic equencies
in EIS a e ela ed o speci ic in e nal componen s o he ba e y. Gen-
e ally, he EIS Nyquis plo displays he h ee key in e nal componen s
o he ba e y. Fig. 3(a) shows he cha ac e is ic cu e ha ela es o
he ollowing componen s: The in e cep along he ho izon al axis by
he semici cle ela es o he Ohmic esis ance o he ba e y and co -
esponds o high- equencies; he semici cle, i sel , is ep esen a i e o
he double-laye capaci ance e ec and co esponds o mid- equencies;
and he di usion p ocess o he ba e y cell ma e ial is indica ed by
he s aigh ail o he igh o he semici cle and co esponds o he
low- equency EIS [51].
Howe e , in his case, o a ange he da a such ha all a iables had
he same leng h, he 23 EIS alues eco ded a 0% SOC be o e cycling,
a e 50 cycles, and a e 100 cycles we e o de ed sequen ially and
yielded 69 samples. The e o e, he ol age and capaci ance we e com-
p essed o i he 69 samples. This shows ha he EIS measu ed once pe
cycle a 0% SOC can p oduce highly accu a e SOC p edic ions. A ecen
s udy by Buchicchio e al. [52] measu ed EIS a mul iple in e mi en
pe iods wi hin he ba e y SOC cycle, making he expe imen al p ocess
e y ime-consuming. Howe e , he esul s s ipula ed abo e ejec he
necessi y o equi ing mo e han one EIS measu emen pe cycle. This
esul is simila o ha obse ed in Babaeiyazdi e al. [51] and u he
sugges s ha EIS measu ed a mid-high equencies is mo e s ongly
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Fig. 4. (a): Compa ison o SOC ou pu s o LSTM, MLP and NARX models agains he expe imen al esul s. (b): Compa ison o mean absolu e e o s (MAE) o LSTM, MLP, and
NARX models. This was he esul o 50% o he da ase being used o aining.
Table 5
Model accu acy when using he EIS measu ed a di e en
SOC alues.
MLP
EIS measu ed a : Adj R-squa ed RMSE
100% SOC 0.991 0.0253
75% SOC 0.991 0.0263
50% SOC 0.995 0.0195
25% SOC 0.996 0.0176
0% SOC 0.999 0.0108
co ela ed o he SOC han he EIS measu emen s a lowe equencies,
depending on empe a u e.
These esul s can be used o signi ican ly educe he expe imen-
al labou o ob aining ba e y da a o SOC p edic ion by educing
he numbe o eco ded obse ables. Reducing he numbe o inpu
a iables also educes he compu a ional cos o he ANN model.
5.3. Compu a ional cos s
The es ing o di e en ML app oaches is done in e ms o com-
pu a ional cos . Va ying he numbe o laye s, numbe o ac i a ion
unc ions (neu ones) in each laye , and ype o ac i a ion conside ed
has a signi ican impac on he compu a ional cos s o ML app oaches.
The compu a ional cos s o an ANN can be p esen ed as he o al
du a ion o ime o as he numbe o compu a ions pe o med in
he ne wo k. The la e is mo e complica ed bu p o ides a mo e
gene alised alue independen o compu e p ocessing powe and ini-
ialisa ion cos s. The numbe o compu a ions pe i e a ion (o epoch)
can be app oxima ed as he numbe o nodes (synapses) plus he
numbe o neu al pa hs (connec ions) ha connec hem. Mul iplying
his by he numbe o i e a ions in aining yields he app oxima ed
compu a ional cos in e ms o he numbe o compu a ions. This elim-
ina es he di e ences owing o he coding layou and o ma be ween
each model. Howe e , i assumes ha all compu a ions a e he same;
ha is, applying he ansig o anh unc ion o a ec o ep esen s
he same compu a ional cos as mul iplying he ec o by single- alued
weigh s. This limi a ion is o e come by assuming ha mos models
ha e a simila a io be ween complica ed and simple compu a ions.
The MLP, LSTM, and NARX NNs ha e compu a ion imes ha a e
less han 1 min. This was due o he size o he da ase used in his
analysis. The compa ison o du a ion in seconds and he numbe o
compu a ions be ween each model ha e con adic ing esul s in Table
6 because o he di e ence in coding s uc u e and unc ions used
in MATLAB. The ini ialisa ion o each model is di e en and has a
signi ican con ibu ion o he o al un ime because o he ela i ely
sho aining du a ion. The e o e, he app oxima e numbe o compu-
a ions should be ega ded as a mo e accu a e ep esen a ion o he
compu a ional cos di e ences be ween models.
Wi h his no ed, he MLP has he lowes compu a ional cos among
he h ee models. Howe e , LSTM has a simila app oxima e numbe o
compu a ions and a signi ican di e ence in du a ion. The eason he
LSTM has a longe du a ion han NARX is pa ly due o he di e en
ini ialisa ion p ocesses; howe e , i may also be no ed ha he LSTM
will ha e a g ea e numbe o complica ed compu a ions compa ed o
he NARX. This is because he LSTM laye in he LSTM model duplica es
he inpu ec o and applies ou ac i a ion unc ions (Eqs. (4), (5), (6),
and (7)) h ough he ou ga es, as demons a ed in Sec ion 2.5.1 and
Fig. 2.
I is wo h no ing ha he alues eco ded in Table 6 a e ob ained
om he op imal model con igu a ion. When es ing he e ec o di e -
en lea ning a es (𝛼) in he ollowing sec ion, he compu a ional cos
was much g ea e o allow he con e gence o he e o cu e.
5.4. Pa e n ecogni ion and sensi i i y o lea ning a e
The lea ning a e (𝛼) is a alue ha signi ies he signi icance o
he e o p oduced a e each i e a ion and is hus di ec ly linked o
he co ec ion s ep sizes ha he model applies o he weigh s pe
i e a ion. Consis en and high pe o mance o a la ge ange o 𝛼 alues
is cha ac e is ic o lexible and adap able NN model s uc u es. MLP
and LSTM exhibi good pe o mance o a la ge ange o 𝛼 han he
NARX model. None o he h ee models can i a model wi h 𝛼⪆1. The
esul s om he di e en lea ning a es o each model a e abula ed
in Table 7. Beyond he obse a ions om Fig. 5, he e a e wo mo e
in e es ing poin s. Fi s , he LSTM e o cu e (Fig. 5b) is smoo hed
when 𝛼 dec eases, which ep esen s he abili y o he LSTM model o
de ec mo e sub le unde lying pa e ns/ ends in he da a when gi en
a smalle lea ning a e and g an ed a much la ge numbe o i e a ions
o aining (and compu a ional cos exceeding 10 min). Second, he
NARX model (Fig. 5c), while ini ially exhibi ing he same beha iou as
he LSTM, e en ually eaches a u ning poin whe e he e o begins
o inc ease o 𝛼⪅10−4. Howe e , he MLP model (Fig. 5a) does
no exhibi ei he o hese cha ac e is ics; a he , i emains ela i ely
unhinde ed by a dec ease in 𝛼. LSTM has he mos complica ed p o-
cessing o da a owing o he LSTM laye . The LSTM laye is p ima ily
ocused on unco e ing long- and sho - e m ela ionships and pa e ns
and assigning a deg ee o signi icance o each. This p ocess is likely o
be he cause o he success o LSTM in sub le pa e n ecogni ion.
To his ex en , he NARX NN demons a ed he highes sensi i i y
o he lea ning a e. The LSTM NN has he mos e ec i e ela ionship
wi h 𝛼, and he MLP NN demons a es he lowes sensi i i y o 𝛼, bu
does no indica e an abili y o 𝛼 o make signi ican imp o emen s in
he model accu acy.
Jou nal o Powe Sou ces 646 (2025) 236929
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M. Rae e al.
Table 6
Compu a ional cos o MLP, LSTM, and NARX models in [(numbe o compu a ions) ×(Numbe o i e a ions)]
and elapsed ime (seconds) o he comple e aining p ocess wi h N i e a ions. Values we e eco ded using
he 50/50 aining/ es ing a io and op imal aining pa ame e s (Table
1).
Model: MLP LSTM NARX
App oxima e numbe o compu a ions (70) × (700) (1100) × (1000) (26) × (10000)
To al T aining Du a ion (seconds) 5 11 3
Fig. 5. E o plo s o di e en lea ning a es, 𝛼, on he MLP (a), LSTM (b), and NARX (c) models.
Table 7
Pe o mance in RMSE and MAE o each
model using dec easing 𝛼 as seen in Figu e
5.
MLP
𝛼RMSE MAE
5 0.2692 0.2322
0.5 0.0247 0.0201
0.00005 0.0302 0.0246
LSTM
1.2 0.2692 0.2322
0.12 0.0247 0.0241
0.00012 0.0138 0.0109
NARX
8.1 0.4362 0.3620
0.81 0.0182 0.0110
0.081 0.0275 0.0200
0.00081 0.0973 0.0648
5.5. Analysis o model gene alisa ion o unseen da a based on da a size
The a io be ween he aining and es ing da ase po ions had a
signi ican impac on he accu acy o each model ou pu . The impac
o a ying he aining size o each model’s da ase is summa ised
in Fig. 6 by plo ing he changes in RMSE 6(a) and SD 6(b). This
shows ha he MLP model is he leas sensi i e o educing he aining
size compa ed o he LSTM and NARX models. MLP ecognising he
ela ionships be ween inpu and ou pu a iables o smalle da ase s
sugges s supe io gene alisa ion abili ies. Fu he mo e, he size o he
da ase is mos impac ul o he NARX model’s abili y o gene alise
o unseen da a, demons a ed by he g ea es dec ease in pe o mance
when he size o he aining da a was dec eased.
In Table 8(b), he pe o mance o he LSTM model is consis en and
highly accu a e when mo e han 25% o he da a a e used o aining.
This is signi ican because 25% o he o al da ase comp ises 18
samples, which does no make a ull cycle. The e o e, LSTM success ully
modelled he SOC o he ba e y wi hou da a o explain he beha iou
o a ull cycle. Howe e , he RMSE inc eased signi ican ly om 0.0267
o 0.1630 when he aining size dec eased om 25% o 20%. The
changes in he ela i e e o o he LSTM model ou pu a e highligh ed
in Fig. 7(b). Simila ly, he da a in Table 8(a) show ha he MLP NN
success ully p edic s he SOC when using 20% o mo e o he da a o
aining.
In o he wo ds, he 14 samples used in he aining p ocess allowed
he MLP NN o p edic he ollowing 53 samples wi h an RMSE o
0.0189 and an SD o 0.0312. The dec ease in accu acy o MLP is
highligh ed by he jump in he SD om 0.0312 o 0.1906 when he
aining size dec eases om 20% o 15%. This change is e iden in he
ela i e e o plo ed in Fig. 7(a). By con as , he NARX NN did no
exhibi lexibili y wi h espec o he aining/ es ing a io. Table 8(c)
shows he lowes accep able model accu acy when he aining da a
is 50% o he o al da ase . The e o e, he NARX NN demons a ed
he leas lexibili y wi h aining da a sizes. This de iciency is clea ly
p esen ed in Fig. 7(c).
This beha iou is a s ong indica ion o he abili y o he ANN model
o unde s and he key ela ionships be ween he inpu and ou pu
da ase s. The MLP and LSTM models ecognise he key pa e ns and
Jou nal o Powe Sou ces 646 (2025) 236929
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