Applied The mal Enginee ing 265 (2025) 125597
A ailable online 17 Janua y 2025
1359-4311/© 2025 The Au ho (s). Published by Else ie L d. This is an open access a icle unde he CC BY license (h p://c ea i ecommons.o g/licenses/by/4.0/).
Resea ch Pape
Op imal dispa ch o Li-Ion ba e y ene gy s o age, e iewing and
conside ing cycling and calenda ageing models
☆
And iy Vasylye
a,*
, Albe o Vannoni
a,b
, Alessand o So ce
a
a
TPG, DIME, Uni e si y o Geno a, I aly
b
Ma e ials and Gene a ion Technologies Depa men , RSE, Milan, I aly
ABSTRACT
The g owing sha e o enewable ene gy sou ces in he ene gy mix and he libe alisa ion o elec ici y ma ke s has d as ically a ec ed he ope a ion o elec ici y
gene a o s and g ids. The ansi ion om a ossil- uel-based ene gy sys em o a enewable one has been signi ican ly impac ing he ene gy ma ke , and ene gy s o age
sys ems ha e a pi o al ole o play. In he coming yea s, a la ge amoun o s o age capaci y is en isaged o be in eg a ed in o elec ici y g ids o sha e demand peaks,
mi iga e p ice ola ili y and ace he g owing demand o se ices o sys em ope a o s. In such a si ua ion, o p ope ly manage hese asse s, and hus gua an ee he
economic iabili y o ope a ing, i is essen ial o op imise hei dispa ch and de ine he bes possible scheduling conside ing any hidden cos s such as deg ada ion.
This pape ocuses on Li-ion Ba e y Ene gy S o age (BES), as he as es -deploying BES. A comp ehensi e li e a u e s udy is ca ied ou o p o ide a de ailed e iew
o ageing p ocess modelling. Bo h he cycling and he calenda ageing p ocesses a e in es iga ed conside ing he impac s o Dep h o Discha ge (DoD), S a e o Cha ge
(SoC), empe a u e (T), C
a e
, numbe o cycles (N
cycles
), and ime ( ). Mo eo e , he dependence o he e iciency o he cha ging and discha ging phases on he
cu en a e is ema kable. The dispa ch op imisa ion is gua an eed by a p oposed Mixed-In ege Linea P og amming op imisa ion algo i hm which conside s he
impac o deg ada ion cos bu is independen o he model selec ed om he li e a u e. The li e a u e e iew e eals ha p e ious s udies, dealing wi h ba e y
dispa ch op imisa ion, only include he cycling deg ada ion explici ly in he objec i e unc ion. In his wo k, he calenda deg ada ion, impac ing he ba e y capaci y
e en when he ba e y is no in use, is also conside ed. This equi es de elopmen o a s a egy o p ope ly weigh he calenda con ibu ion o ade, pa icula ly when
a bi age oppo uni ies a e limi ed because o small daily p ice luc ua ions. The p oposed s a egy in ol es he use o a ac o R o adjus he cos associa ed wi h
cycling. The op imal alue o his ac o is highly dependen on he economic condi ions o he ma ke , necessi a ing a sensi i i y analysis o e alua e i s impac .
1. In oduc ion
The impe a i e shi om ca bon-based o Renewable Ene gy Sou -
ces (RES) o mi iga e global wa ming aces challenges. Despi e he
inc easing use o RES, global g eenhouse gas emissions a e ising, as
epo ed by he IEA [1]. Thus, mo e ambi ious a ge s o ca bon
neu ali y ha e been se , such as he EU’s ‘Fi o 55
′
package [2]. Key
a ge s include eaching 40 % o he o al ene gy mix demand om RES
by 2030. Howe e , he s ochas ic and non–p og ammable na u e o
mos enewable ene gy sou ces poses signi ican challenges in mee ing
ins an aneous elec ici y gene a ion, demand, and g id secu i y
equi emen s.
In his scena io, he elec ici y g id is equi ed o be mo e lexible,
and la ge s o age sys ems a e conside ed essen ial o mi iga e he a i-
abili y o RES, empo a ily shi ing he load, a leas on a daily basis, and
p o iding se ices o he T ansmission and Dis ibu ion Sys em Ope a-
o s. Depending on he s o age echnology, he po en iali y o se ices
a ies, la ge pumped-hyd o powe plan s ha e been adi ionally
employed o black s a se ices, conges ion elie ing, eplacemen
ese e, equency es o a ion ese e, and e en equency con ainmen
ese e [3,4]. Mo e ecen ly, wi h he mo hballing o la ge o a ional
ine ia o spinning gene a o s, he demand o ul a as equency
egula ion se ices and syn he ic ine ia has isen and Ba e y Ene gy
S o age (BES) sys ems show a signi ican po en ial o play his ole
[5–7].
The BES po en ial is app ecia ed no only o i s abili y in as e-
quency egula ion. The independence o si e equi emen s and he
ele an ound- ip e iciency a e ema kable ea u es. Acco ding o he
Ne Ze o Emission by 2050 scena io o IEA, by 2030 680 GW o g id-
scale s o age sys ems mus be ins alled globally, while 16 GW had
al eady been ins alled in 2021 [8], and mos o his capaci y is om Li-
ion BES ha is app ecia ed o i s no able e iciency and educed cos .
This echnology makes up 92 % o ins alled g id-scale BES in he US [9].
Besides li hium-ion ba e ies, low ba e ies ha e eme ged ecen ly as a
b eak h ough echnology o s a iona y s o age as hey do no show
pe o mance deg ada ion o 25–30 yea s and a e capable o being sized,
allowing o a decoupling o in es men s in ene gy and powe capaci y,
☆
This a icle is pa o a special issue en i led: ‘Sus ainable Powe Sys ems’published in Applied The mal Enginee ing.
* Co esponding au ho .
E-mail add esses: [email p o ec ed] (A. Vasylye ), [email p o ec ed] (A. Vannoni).
Con en s lis s a ailable a ScienceDi ec
Applied The mal Enginee ing
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h ps://doi.o g/10.1016/j.appl he maleng.2025.125597
Recei ed 6 Ma ch 2024; Recei ed in e ised o m 19 Decembe 2024; Accep ed 15 Janua y 2025
Applied The mal Enginee ing 265 (2025) 125597
2
acco ding o ene gy s o age needs wi h limi ed in es men [8]. Finally,
BES sys ems a e also playing an inescapable ole in he mobili y sec o
ene gy ansi ion, including ma i ime, pa icula ly o essels wi h
hyb id p opulsion sys ems. The managemen o such ene gy sys ems
p esen s many challenges, such as planning he use o he BES sys em o
a oid pa ial loads on he engine [17]. Raggio M. e al. ha e discussed
he use o Ni-Zn ba e y ene gy s o age echnologies o educe peak
demand by combining ba e ies wi h mic o gas u bines [39]. Howe e ,
o he mos common BES ypes, ba e y li e is a se e e issue and, i BES
is no op imally managed, he ope a ional p o i s may no be enough o
pay back he in es men cos be o e he end o li e. Indeed, li espan is
limi ed by he chemical deg ada ion o he elec ochemical cells o he
ba e y pack. Many wo ks in he li e a u e in es iga e ageing o
li hium-ion BES iden i ying wo main ageing p ocesses: calenda and
cycling ageing; each o he p ocesses is in luenced by di e en idle and
cycling pa ame e s [10–12].
Calenda ageing is he p og essi e de e io a ion o a ba e y’s pe -
o mance o e ime, ega dless o i s ope a ional use. The main in lu-
encing ac o s a e empe a u e and a e age S a e o Cha ge (SoC).
P olonged s o age o ope a ions a ele a ed empe a u es accele a e
calenda ageing a ou ing chemical eac ions wi hin li hium-ion ba e-
ies ha comp omise pe o mance and cause capaci y loss. Simila ly,
main aining a li hium-ion ba e y a a high SoC o long pe iods con-
ibu es o he calenda ageing p ocess, inc easing he speed o he
a o emen ioned chemical eac ions wi hin he ba e y. Long- e m s o -
age ecommenda ions ad oca e keeping li hium-ion ba e ies a a low
a e age SoC [10,12].
Cycle ageing is caused by he epe i i e cha ge and discha ge cycles
inhe en in he ope a ional use o a ba e y. Each cycle causes a g adual
loss o capaci y and pe o mance. The Dep h o Discha ge (DoD), i.e., he
ex en he a io o ba e y capaci y ac ually discha ged in each cycle, has
a signi ican impac on cycle ageing. Deepe discha ge cycles gene ally
cause accele a ed capaci y ade, whe eas shallowe discha ge cycles,
whe e he ba e y is no ully discha ged, con ibu e o ex ending he li e
and he cumula i e ene gy exchanged a he end o li e. Cha ging pa-
ame e s such as ol age and cu en also a ec cycling ageing. Cha ging
Li-ion ba e ies a highe ol ages, o cu en a es, con ibu es o
deg ada ion o e ime. The o al numbe o cha ge/discha ge cycles, o
cycling his o y, has a signi ican e ec on wea [13–15]. Mo eo e ,
some au ho s p opose non-linea models conside ing ha he i s cycles
ha e a highe deg ada ion impac [10]; howe e , e en many linea
o mula ions a e p oposed, acco ding o which he deg ada ion a e o
each cycle is cons an [11].
An e ec i e model desc ibing he eal deg ada ion p ocess un il he
ba e y end-o -li e is essen ial o de elop a managemen s a egy able o
maximise he economic iabili y o ope a ing he ba e y also consid-
e ing he hidden cos s o deg ada ion [13]. Fo bo h cycling and cal-
enda deg ada ion p ocesses, se e al models ha e been p oposed in he
li e a u e, p esen ing di e en co ela ions ha exp ess he ela ionship
be ween di e en pa ame e s and ba e y deg ada ion. Sec ion 2 epo s
a de ailed li e a u e e iew o exis ing models, discussing how he
impac o di e en pa ame e s has been quan i ied and desc ibed
h ough equa ions.
In mos coun ies, g id-scale ba e ies mus ope a e in a libe alised
elec ici y ma ke design in which he mos common business model
consis s o shi ing he load i a bi age oppo uni ies subsis . The
d i ing ac o o a bi age is elec ici y p ice luc ua ion, which can be
cha ac e ised by a pe iod and a magni ude, consequen ly, he economic
iabili y o s o age s ongly depends on he ma ke scena io [16]. As
men ioned be o e, he p o ision o se ices is also mo e and mo e
p omising as a sou ce o e enue, howe e , i is cha ac e ised by
conside able unce ain y and dependency on he local egula o y
amewo k, and in ol es negligible amoun s o ene gy i compa ed o
a bi age.
The e o e, he ocus is on li hium-ion BES, selec ed as he mos
common ba e y echnology, while o a bi age oppo uni ies he
luc ua ions in he elec ici y day-ahead ma ke , which is common o
almos e e y ma ke design, a e conside ed as he base load o he
business models. Some pape s p esen op imisa ion models o he day-
ahead ma ke , conside ing he in luence o ba e y cycling as he main
deg ada ion e ec and ans o ming i in o a cos pa ame e o be
included in he op imisa ion [18–20]. Howe e , accoun ing also o
calenda ageing is no in es iga ed equally, o he bes o he au ho s’
knowledge, he only pape in eg a ing bo h cycle and calenda deg a-
da ion cos s in o i s op imisa ion model is by Bahloul M. e al. [21],
p esen ing an app oach ansla ing deg ada ion in o ma ion in o cos
in o ma ion o a hyb id PV-BES sys em. Howe e , cycling and calenda
cos s ha e been linea ised o educe he complexi y o he in o ma ion in
he op imise . In addi ion, he cos unc ions a e simpli ied as hey do
no ully accoun o a ious ba e y ope a ing pa ame e s such as DoD,
SoC
a g
, , T and C- a e. Amini A. e al. in es iga ed he impac o bo h
cyclic and calenda ageing du ing a ba e y li e cycle [22]. They de ined
he e ec o ope a ing pa ame e s on ba e y deg ada ion, o es ima e
he ba e y’s S a e o Heal h (SoH). Howe e , he cos -minimising
objec i e unc ion hey p oposed does no include he impac o dy-
namic ba e y deg ada ion cos s, as hey conside ed a ixed cos o
Nomencla u e
BES Ba e y Ene gy S o age
CAPEX Capi al Expendi u es
C a e Cu en Ra e
C a e e Re e ence Cu en Ra e
C ade Capaci y Fade
C adecyclig Capaci y ade due o Cycling
C adeidling Capaci y ade due o Idling
Cos cycle Cycle Cos unc ion
Cos idele Idle Cos unc ion
Cos cycle i ual Vi ual Cycle Cos
Cos deg Deg ada ion Cos unc ion
DoD Dep h o Discha ge
EoL End o Li e
iIn e es a e
LFP Li hium I on Phospha e
MILP Mixed In ege Linea P og amming
Ncycle Maximum Numbe o Cycles
ncycle Numbe o Cycles
η
cha ge Cha ge e iciency
η
discha ge Discha ge e iciency
OP Ope a ional P o i s
OPne Ne Ope a ional P o i s
Obji,j, Objec i e Func ion
R ac o Vi ual cycling cos ac o
SoC S a e o Cha ge
SoC e Re e ence S a e o Cha ge
ΔSoCiDisc e ised S a e o Cha ge
SoH S a e o Heal h
TTempe a u e
T e Re e ence Tempe a u e
ime
xOp imised bina y a iable
ZP Zonal P ice
A. Vasylye e al.
Applied The mal Enginee ing 265 (2025) 125597
3
main enance ope a ions p opo ional o he BES nominal powe .
Ins ead, his wo k de ines a comp ehensi e app oach o e alua e he
p o i abili y o a BES sys em in a ious scena ios, simul aneously
conside ing cycling and calenda deg ada ion. Disc e ising he objec i e
unc ion, and op imising i h ough an MILP algo i hm, also allows
conside a ion o nonlinea i ies o deg ada ion p ocesses. The pape o-
cuses on he poin o iew o a ba e y owne /manage ope a ing in a
libe alised elec ici y ma ke , assessing he economic iabili y o ene gy
a bi age in he wholesale day-ahead ma ke as a co e business. The
p oposed me hod is designed o be independen o speci ic ba e y
cha ac e is ics and e sa ile o any ope a ional cu es, i.e., deg ada ion
cu es o e iciency cu es.
Fi s , a comp ehensi e li e a u e e iew is conduc ed in Sec ion 2 o
p esen he na u e o deg ada ion phenomena and he models p oposed
in pas s udies discussing he impac o key ope a ing pa ame e s on
ba e y deg ada ion. The keywo ds ‘numbe o cycles’, ‘capaci y ade’,
‘cycling deg ada ion’, and ‘calenda deg ada ion’ om 2010 onwa ds
we e used o he li e a u e sea ch. This e iew was essen ial o wisely
model he BES, Sec ion 3, and ans o m a deg ada ion model in o
economic pa ame e s, such as cycle cos and calenda deg ada ion cos ,
used as inpu s o he op imisa ion algo i hm. The p oposed MILP al-
go i hm is desc ibed in Sec ion 4, he objec i e unc ion is de ined using
a disc e ised app oach, enabling he BES sys em o ope a e acco ding o
p ede ined ope a ing modes (i.e., ne powe ou pu le els), he eby
allowing holding non-linea i ies o complex objec i e unc ions and
cons ain s [23], o example hose ollowing he p oposed model o
e iciency. Mo eo e , his app oach well sui s he need o enhance he
e sa ili y and adap abili y o he p oposed op imisa ion amewo k o
any deg ada ion o e iciency model and u u e pu poses o in eg a ed
op imisa ion o ene gy a bi age and g id se ice p o ision. The op i-
misa ion me hodology is desc ibed in de ail, along wi h he manage-
men o he deg ada ion and e iciency cu e models inside he
op imisa ion app oach. The model conside s bo h cycling and calenda
ageing dependencies on he s a e o cha ge and dep h o discha ge, and
he dependency o e iciency on he cu en a e. Addi ionally, he
p og essi e ade in capaci y is accoun ed o o de e mine he ac ual
amoun o dispa chable ene gy. Sec ion 5 concludes he me hodology
p esen a ion by illus a ing he scena ios in es iga ed and he indica o s
assessed. Finally, Sec ion 6 applies he p esen ed op imise o a ious
ma ke scena ios, examining he impac o a e age p ice and p ice
a iabili y on economic Key Pe o mance Indica o s (KPIs), such as ca-
paci y ade and annual speci ic Ope a ional P o i s (OP). This sensi i i y
analysis es ablishes a ela ionship be ween ope a ing condi ions and
ma ke scena ios. Unde p o i able condi ions, ba e y managemen
becomes mo e s aigh o wa d, ensu ing ha he ba e y ope a es only
when ope a ional p o i s exceed he hidden cos s o deg ada ion. This
gua an ees ha , by he end o he ba e y’s li e, he in es men cos is
eco e ed, and ne p o i s a e gene a ed. Con e sely, when a bi age
oppo uni ies a e diminished due o a la e p ice p o ile, ba e y
managemen becomes mo e challenging. I he ba e y emains idle
unless po en ial ope a ional p o i s ou weigh deg ada ion cos s, calen-
da ageing will ul ima ely lead o highe economic losses by he end o
i s li espan.
2. BES ageing p ocesses e iew
BES pe o mance shows a decay o he nominal pa ame e s o e
ime, he ade o capaci y (i.e., he maximum amoun o ene gy ha can
be s o ed) due o he ageing p ocesses has a majo impac on he BES
ope a ing condi ions, while powe ade (i.e., he dec ease o maximum
powe achie able du ing he cha ging and discha ging phases) is
conside ed negligible [24]. The a ed capaci y con inuously dec eases
because o wo majo con ibu ions, i s he impac o each cycle, i.e.,
cycling ageing, and second in idle condi ions a deg ada ion occu s
de ined as calenda ageing. The S a e o Heal h (SoH), Eq. (1), is de ined
as he a io o a ed capaci y on he nominal alue, i.e., he
complemen a y o ade. Typically o mos li hium-ion BES and appli-
ca ions such as ene gy a bi age on a daily basis, o in eg a ion in sma
g ids, he End-o -Li e (EoL) is ixed a 20 % o ade, SoH =80 %. Unde
his h eshold alue, pe o mance is oo poo o keep BES ope a ing.
SoH =C a ed
Cnom
•100% =100% −C ade[%](1)
C ade =C adecycling +C adeidling [%](2)
Du ing bo h cycling and calenda deg ada ion p ocesses, di e en
ope a ing and idle pa ame e s s ongly impac he capaci y deg ada ion
a e, which educes he li espan o a ba e y, he di e en dependencies
a e shown in Table 1. Eq. (3) and Eq. (4) epo all he pa ame e s
impac ing he calenda and cycling capaci y ade, while he ela ionship
(e.g., linea o exponen ial) be ween he ade and hese pa ame e s is
epo ed in Table 1 acco ding o di e en e e ences. The model hen
selec ed o he analysis in Sec ion 4 is highligh ed in bold, likewise o
Tables 2 and 3 in Sec ions 2.1 and 2.2.
C adecycling = (DoD,C a e,T,SoCa g,ncycle)[%](3)
C adeidling =g(SoC,T, )[%](4)
EoL =∫Ncyclemax
0
’(DoD,C a e,T,SoCa g)⋅dncycle[%](5)
Ncyclemax =EoL
ʹ(DoD,C a e,T,SoCa g)(6)
EoL =∫ max
0
g’(SoC,T)⋅d [%](7)
max =EoL
gʹ(SoC,T)[h](8)
DoD =SoCs a −SoCend (9)
Equa ion (5) exp esses he cumula i e deg ada ion o he ba e y
be o e eaching end-o -li e, i is linea ly dependen on cycles al eady
pe o med, n
cycle
,N
cycle max
can be explica ed as in Eq. (6), whe e he
unc ion ’is de i a i e o wi h espec o n
cyclce
. EoL is he End-o -Li e
c i e ion, e.g., 20 % o capaci y ade. In Sec ion 3.1 he unc ion is
p esen ed speci ically o he adop ed model in he de elopmen o he
op imisa ion algo i hm. Analogously Eq. (7) de ines
max
, i.e., he cal-
enda li espan o he ba e y a e which he EoL is eached e en wi hou
pe o ming any cycle. I unc ion gis linea ly dependen on he ime ,
max
can be explica ed as in Eq. (8), whe e he unc ion g’is de i a i e o
gwi h espec o . In Sec ion 3.1 he calcula ion is p esen ed speci ically
o he model adop ed in he de eloped op imisa ion algo i hm. Fo ’
he uni o measu emen is exp essed in pe cen age poin s, pp, pe cycle,
while o g’pp pe hou . Eq. (9) de ines he Dep h o Discha ge (DoD) as
he di e ence be ween he ini ial and he inal S a e o Cha ge (SoC) o
a cycle.
2.1. Cycling capaci y ageing model
As a ba e y unde goes cha ging and discha ging cycles, i s elec-
odes slowly deg ade and become less e ec i e a holding and eleasing
ene gy, causing cycling ageing. Many au ho s ha e analysed he ac o s
in luencing cycling ageing [10–15,24,25,27–29]. All o hem ag ee ha
he DoD is o p ima y impo ance, some [10,27,28] desc ibe he impac
o empe a u e, ne e heless, his dependence is neglec ed by his pape
since a empe a u e con ol sys em is assumed o be in eg a ed wi h he
BES and i s impac as an auxilia y elec ical consump ion is conside ed
on he cha ging and discha ging e iciency model (Subsec ion 3.2). Since
he models ha e di e en dependencies on he same ope a ing pa am-
e e s, e.g., polynomial o exponen ial, i was decided o implemen a
A. Vasylye e al.
Applied The mal Enginee ing 265 (2025) 125597
4
uni o m nomencla u e o all i ing coe icien s o he a ious deg a-
da ion unc ions o be e compa ison and unde s anding o di e en
deg ada ion models. The subsc ip s o he i ing coe icien s e e o he
ope a ional pa ame e , o example, a
DoD
e e s o he i ing coe icien
ega ding he dependence on he dep h o discha ge. Only Eq. (12) has
been p esen ed wi h a speci ic discussion o he deg ada ion pa ame e s,
as i con ains coe icien s ha a e no easily gene alisable like hose in
o he models. This is because he p oposed model conside s combining
he con ibu ions o he wo deg ada ion mechanisms in o a single
deg ada ion unc ion.
To he bes o he au ho s’knowledge, he i s wo k in he open
li e a u e ha de ines a deg ada ion cu e, desc ibing he ela ionship
be ween he DoD and he admissible numbe o cycles N
cycle max
, was
p o ided in 2011 by Zhou e al. [25] o V2G (Vehicle o G id) appli-
ca ions. In his wo k, he impac o some pa ame e s ( empe a u e and
DoD) was in es iga ed o di e en ba e y ene gy s o age ypes
(li hium-ion, lead-acid, and nickel me al hyd ide). Conce ning he
empe a u e, Zhou sugges s using a hype bolic co ela ion, esul ing
om he A henius equa ion [30], meanwhile, he DoD dependence is a
loga i hmic, hus he numbe o cycles exponen ially inc eases as he
DoD dec eases.
Mallon e al. [26], p oposed new coe icien s o he equa ion e-
po ed by Zhou o desc ibe he DoD in luence on he numbe o cycles. In
hei wo k, he empe a u e dependence was no men ioned. The p o-
posed equa ions we e used in hea y-du y elec ic powe ain
applica ions, assessing he impac o he ins alla ion o sola panels on
he bus’s oo and sides on he ba e y’s li espan. Eq. (10) desc ibes he
ageing ela ionship p e iously sugges ed by Zhou e al. and la e
adop ed and e ised by Mallon.
C adecycling =aDoD⋅DoDbDoD ⋅e(cDoD⋅(1−DoD) )⋅ncycle (10)
S oe e al. [10,14,24] analysed in de ail he in luence o mo e,
compa ed o Zhou, ope a ing and idling pa ame e s, de eloping models
no only o ba e y capaci y ade bu also o ba e y powe deg ada-
ion, o speci ic li hium-ion phospha e/g aphi e ba e ies (LiFePO
4
o
LFP). They adop ed an equi alen -elec ical ci cui pe o mance-
deg ada ion modelling app oach o de elop he li e ime models,
conside ing bo h e ec s o cycling and calenda p ocesses; in Sec ion
2.1.2 he calenda ageing model will be discussed in mo e de ail. The
i s applica ion in which hey had o de elop he deg ada ion model
ela es o an in eg a ed wind powe plan o compensa e o he s o-
chas ic na u e o wind ene gy [10]. The dependence on he numbe o
cycles is polynomial (deg ee 0.5), while o quan i y he dependence on
he DoD he au ho s used a powe -law unc ion and he empe a u e
impac on capaci y ade is desc ibed by an exponen ial unc ion. Finally,
he las pa ame e conside ed by S oe and colleagues is he a e age
S a e o Cha ge (SoC
a g
) du ing he cycle and i has a dec easing expo-
nen ial impac : a high a e age SoC gene ally has a posi i e e ec on he
deg ada ion a e inc easing he li espan.
This model wi h mino a ia ions is used in o he wo ks [14,24,31]
o pe o m echno-economic op imisa ion o a Vi ual Powe Plan (VPP)
dispa ch, equency egula ion in he Danish elec ici y ma ke o li e-
ime es ima ion o ully elec ic ehicles. In he nex equa ion, Eq. (11),
he ela ionship desc ibed by S oe e al. is explica ed.
C adecycling =a⋅(aDoD⋅DoDbDoD )⋅e(aSoC⋅SoCa g)⋅nan
cycle⋅(aT⋅ebT⋅T)(11)
Xu e al. [12] in hei i s wo k analyse bo h ageing p ocesses,
conside ing all he ope a ing and idling pa ame e s. This model in-
eg a es bo h heo e ical conside a ions and empi ical e idence in i s
analysis. The deg ada ion model ocuses on he wo p ima y s ess
unc ions: calenda ageing and cycling ageing. I is used o a echno-
economic analysis o P ima y F equency Regula ion (PFR) and
Table 1
Cycling and idling pa ame e s in di e en deg ada ion models.
Cycling Ageing Calenda Ageing
Re No. DoD n
Cycle
SoC
a g
T C
a e
SoC T ime
[25] exp linea −hype bolic − − − −
[26] exp linea − − − − − −
[10,14,24] powe oo -squa e exp exp −exp exp exp
[12,27] exp exp exp exp exp exp exp exp
[11,28] poly linea ¡poly ¡poly poly linea
[15] poly linea − − − poly −linea
[13] exp linea − − − − − −
Table 2
The i ing coe icien o each Cycling ageing model.
Re e ence
Coe icien [25] [26] [10,14,24] [31] [12,27,29] [11,28] [15] [13]
a− − 2.64e-2 2.4e-4 −¡− −
aDoD 2.88e-1 7.3e-1 1.23e-2 2.982e-2 8.95e4 ¡4.72e-5 4.83e-4 5.2e-5
bDoD 7.95e-1 6.79e-1 7.162e-1 4.904e-1 −4.86e-1 9.62e-5 2.38e-5 6.8e-1
cDoD 0−1.61e1 − − − 7.28e-4 ¡− − 1.6e1
aSoC − − − 1.943e-2 −1.04e1 ¡− −
ac a e − − − − 2.63e-1 ¡− −
aT− − 4e-3 −6.93e-2 3.62e-3 − −
bT− − 1.705e-2 2.717e-2 −¡1.05e-1 − −
cT− − − −1.93 − −
an− − 0.5e-1 0.5e-1 −¡− −
Table 3
The i ing coe icien o each calenda ageing model.
Re e ence
Coe icien [12,27,29] [10,14,24] [31] [11,28] [15]
aSoC 1.04e1 1.639e-1 1.9e-2 6.02e-6 4.34e-6
bSoC −7.38e-3 8.23e-1 1.35e-5 2.73e-5
cSoC − − 5.195e-1 1.85e-5 1.45e-5
aT6.93e-2 1.977e-11 3.258e-9 2.31e-3 −
bT−7.51e-2 5.087 ¡4.01e-2 −
cT− − 2.95e-1 ¡1.21 −
a 1 8e-1 8e-1 11
A. Vasylye e al.
Applied The mal Enginee ing 265 (2025) 125597
5
Dynamic F equency Regula ion (DFR) p o ision by s o age. The models’
i ing pa ame e s a e de ined o Li hium Manganese Oxide (LMO).
Unlike he o he p oposed models, he calenda and cycling ageing does
no impac linea ly on he aded capaci y ha is exp essed as in Eq. (12).
C ade =1−psei⋅e− sei⋅ d+ (1−psei)⋅e− d(12)
Whe e:
p
SEI
and
SEI
a e wo cons an s cha ac e ising di e en BES models
and co ela ed wi h he o ma ion o a combina ion o s ess unc ions,
While
d
is a combined deg ada ion unc ion consis ing o wo e ms:
cycling
is he con ibu ion due o he cycling ageing p ocess, and
calenda
is
he con ibu ion due o he calenda ageing p ocess.
In his subsec ion he cycling con ibu ion is analysed, while he
pa ame e s impac ing
calenda
will be discussed in subsec ion 2.2.
cycling
,
Eq. (10) is he p oduc o 4 unc ions desc ibing he impac o he DoD,
SoC, C
a e
, and empe a u e.
cycling
has a polynomial dependence on he
DoD ha plays he majo ole, Eq. (11). The o he pa ame e s ha e an
exponen ial impac on
cycling
when hey de ia e om he e e ence
alues Eq. (13–17). Fo S oe e al. [10], du ing cycling, a high s a e o
cha ge has a nega i e e ec on li e consump ion, while a low s a e o
cha ge has a posi i e e ec , while o Xu e al. [12] he e is a symme -
ical end wi h espec o he 50 % s a e o cha ge, inc easing he impac
o ex eme SoC. This a icle is he only s udy epo ing he dependence
on he C
a e
(i.e., he cu en a e a which he cha ge and discha ge
phases occu , de ined as a a io be ween he cha ging/discha ging
powe and he nominal capaci y)
cycling = DoD(DoD)⋅ SoC(SoCa g)⋅ C a e (C a e)⋅ T(T)⋅ncycle (13)
DoD(DoD) = (aDoD⋅DoDbDoD +cDoD)−1(14)
SoC(SoC) = eaSoC⋅(SoC−SoC e a g )2
(15)
C a e (C a e) = eaC a e ⋅(C a e−C a e e )(16)
T(T) = eaT⋅(T−T e )⋅T e
T(17)
In ano he wo k by he same au ho s [27], hey again pe o m a
echno-economic analysis o PFR and DFR, bu hey conside di e en
li hium-ion ba e y echnologies such as Li hium I on Phospha e (LFP),
Li hium Nickel Manganese Cobal Oxide (NMC), and Li hium Manga-
nese Oxide (LMO).
Say u dino e al. [11,28] p esen ed a polynomial model o desc ibe
he dependence on bo h cycling and calenda ageing mechanisms. The
model was used o op imise he size o he ene gy s o age sys em
conside ing he impac o deg ada ion o e he yea s. Fou di e en
li hium-ions we e compa ed conside ing di e en i ing coe icien s o
he models: Li hium I on Phospha e (LFP), Li hium Nickel Manganese
Cobal Oxide (NMC), Li hium Manganese Oxide (LMO), and Li hium-
Ti anium-Oxide (LTO). The only wo pa ame e s ha we e aken in o
conside a ion we e he DoD and he empe a u e, o bo h a quad a ic
unc ion was u ilised o desc ibe he impac on he capaci y deg ada ion.
Fallahi a e al. [15] u ilise Say u dino e al. models o cha ac e ise
he capaci y deg ada ion unc ion o cycling and calenda ade, while
de ining he i ing coe icien s o hei speci ic ba e ies. The deg a-
da ion model is applied o es ablish he op imal scheduling in mic og id
applica ions, aking in o accoun capaci y ade. Va ious li hium-ion
echnologies a e conside ed, including Li hium I on Phospha e (LFP),
Li hium Nickel Manganese Cobal Oxide (NMC), Li hium Manganese
Oxide (LMO), and Li hium-Ti anium-Oxide (LTO). The nex equa ion,
Eq. (18), explica es he ela ionship o Say u dino e al. and Fallahi a
e al. as desc ibed abo e. Fo his case s udy, he second-o de poly-
nomial used o desc ibe he dependence on Tis equal o 1, neglec ing he
empe a u e e ec .
C adecycling =(aDoD⋅DoD2+bDoD⋅DoD)⋅(aT⋅T2+bT⋅T+cT)⋅ncycle (18)
Lee e al. [13] conside ed only he dependence o DoD on capaci y
deg ada ion, exp essed by an exponen ial ela ionship. The p oposed
model was used o de ine he op imum way o manage he cycling cos
du ing scheduling p oblems. In he nex equa ion, Eq. (19), he ela-
ionship desc ibed ea lie o he Lee e al. models can be seen.
C adecycling =aDoD⋅DoDbDoD ⋅ecDoD⋅(1−DoD)⋅ncycle (19)
Table 2 shows he coe icien s o he models discussed abo e, hus
making i easie o he eade o compa e di e en ones. I is impo an
o conside whe he in he di e en wo ks he coe icien s a e i ed o
pe cen age DoD o dimensional alues.
In Fig. 1 he e iewed cycling ageing models a e compa ed calcu-
la ing he maximum numbe o cycles N
cycle max
ha a ba e y can
pe o m be o e eaching he EoL c i e ion, all models’ i ing coe icien s
o di e en ba e ies a e conside ed o Li-ion echnology. The main
models p esen ed in his pa ag aph a e ep esen ed wi h con inuous
lines wi h a di e en ma ke om he o he s [11–14,25], while hose
based on hese models bu sugges ing di e en i ing coe icien s ha e
he same ma ke bu a do ed line [15,26]. The loga i hmic scale
adop ed on he y-axis is signi ican o he conside able p edic ion gap
be ween some models.
Only o he S oe e al. models [10,14,24] does he p og essi e
numbe o cycles ha e a nonlinea impac on he deg ada ion; his
means ha he i s cycles ha e a highe deg ada ion a e han he las
cycles, his in o ma ion is no di ec ly ep esen able in he igu e. To
plo he ela i e cu e, N
cycle max
is calcula ed nume ically, h ough Eq.
(5), o di e en DoD assuming C
a e
equal o 1, empe a u e equal o
25 ◦C, and an in e media e alue o SoC
a g
equal o 60 %.
Xu e al., S oe e al., Say u dino e al., and Fallahi a e al.
[11,15,28] also epo in he same a icles an analysis o he calenda
deg ada ion model; hese e e ences allow o consis ency o models as
hey e e o he same li hium-ion ba e y ene gy s o age echnology and
a e e iewed in de ail in he subsequen sec ion.
2.2. Calenda capaci y ade model
Calenda ageing is a na u al deg ada ion beha iou o he chemical
componen inside he ba e y ene gy s o age, i is due o he occu ence
o colla e al eac ions gene a ed by he he modynamic ins abili y o
cons i uen ma e ials [10]. Fu he mo e, he he modynamic s abili y o
Fig. 1. Maximum numbe o cycles a ixed ope a ing condi ions, in unc ion o
main ope a ing pa ame e DoD.
A. Vasylye e al.
Applied The mal Enginee ing 265 (2025) 125597
6
he nega i e elec ode is pi o al since g aphi e is no elec ochemically
s able when used wi h mos elec oly e ypes. Calenda ageing is
gene ally less in es iga ed han cycling in he open li e a u e
[15,24,27,28], ne e heless a p ope model, also including his e ec , is
essen ial i he BES is o ced o long idling pe iods, e.g., because he
ma ke is no p o i able enough o pe o m a bi age.
S oe e al., Say u dino e al., and Fallahi a e al. poin ou ha an
idling SoC is he key pa ame e and ag ee on he de imen al e ec o a
high SoC. As summa ised in Table 1, acco ding o S oe e al. [10,14,24],
Say u dino e al. [11,28] and Fallahi a e al. [15] a second-o de
polynomial dependence is de ined while o Xu e al. [12,27] he
dependence o capaci y ade on SoC is a symme ical exponen ial end
wi h espec o 50 %, so desc ibing a non-mono onic end ha implies
an op imal SoC (50 %) ha minimises calenda ageing du ing idling
pe iods. E en when i conce ns he in luence o empe a u e on he
calenda capaci y ade S oe e al. [10,14,24] and Xu e al. [12,27]
main ain he same exponen ial dependence, while Say u dino e al.
[11,28] and Fallahi a e al. [15] main ain a second-o de dependence.
The p oposed models di e in he kind o dependency on ime, i
[15,27,28] p opose a linea ela ionship, he dependence indica ed by
S oe [10] epo s a powe law dependence wi h exponen a equal o 0.8
a i ming ha calenda deg ada ion p og essi ely slows down o e
ime.
Eq. (20) desc ibes S oe e al.’s model; hen Eq. (21) desc ibes Xu
e al.’s model o he linea ised ime deg ada ion unc ion [12]; and,
inally, Eq. (22) explica es Say u dino e al.’s and Fallahi a e al.’s
models [11,15].
C adeidling = (aT⋅ebT⋅T+cT)⋅(aSoC⋅ebSoC⋅SoC +cSoC)⋅ a (20)
calenda =a ⋅eaSoC⋅(SoC−SoC e )2
⋅eaT⋅(T−T e )⋅T e
T⋅ (21)
C adeidling =(aSoC⋅SoC2+bSoC⋅SoC +cSoC)⋅(aT⋅T2+bT⋅T+cT)⋅ (22)
Fig. 2 shows he ends associa ed wi h he e ised calenda ageing
models by calcula ing, ixing empe a u e a 25 ◦C, he maximum li e-
ime
max
ha can be achie ed be o e he ba e y eaches he EoL c i-
e ion. In Eq. (5) he x-axis shows he SoC a which he ba e y is
main ained o e he yea s, and i can be seen ha Xu e al. [12,27] ha e
p oposed a non-mono onic model, wi h a maximum o a e age SoC
equal o 50 %. Xu’s model is compa able o he o he models o a high
a e age SoC idling pe iod, while o low SoC i de e s signi ican ly,
being mo e cau iona y in e ms o li espan expec ancy. Analogously o
cycling ageing, o plo he Xu cu e
max
was assessed o di e en SoC
nume ically, h ough Eq. (7), assuming T equal o 25 ◦C.
Table 3 shows he coe icien s o he models discussed abo e, hus
making i easie o he eade o compa e di e en ones. I is impo an
o conside in di e en wo ks i he coe icien s a e i ed o pe cen age
SoC o dimensional alues. Acco ding o he o he models, i is always
ad isable o keep he ba e y a a low SoC o educe he a e o deg a-
da ion o e ime. As o he cycling models, he same logic is used o
isualise he models men ioned: con inuous lines o o iginal models
[10–13,25], while do ed lines o he same models bu e ised wi h
di e en coe icien s [15,26]. I is impo an o unde line ha each
au ho de ines models conside ing di e en ime uni s, his can lead o
di icul y in compa ing di e en models. In his wo k, in o de o make a
be e compa ison, he calenda capaci y ade models a e epo ed wi h
an hou ly ime uni , o de ine he idle cos pe hou in acco dance wi h
ime disc e isa ion o he op imisa ion ca ied ou in Sec ion 4, as he
Day Ahead Ma ke (DAM) elec ici y ma ke is based on hou ly ime
esolu ion.
3. BES model
This sec ion desc ibes he BES model. The MILP algo i hm is inde-
penden o he BES model. Indeed, he p oposed app oach can be
eplica ed o di e en ba e ies jus by modi ying he assump ion p e-
sen ed in his sec ion.
3.1. Adop ed cycle and calenda ade model
Fo he analysis ca ied ou in his pape , a model is selec ed ha
consis en ly desc ibes he calenda and cycling ageing; among he
models e iewed in Sec ion 2, S oe [10,14,24], Xu [12,27,29], and
Say u dino [11,28] a e eligible op ions. Since he impac o he a e age
SoC on he cycling ag eeing is no ce ain, con o e sial, and in any case
o seconda y impo ance, he model p oposed by Say u dino [11] was
selec ed, because o i s independence om his pa ame e .
Consequen ly, Eq. (6), 8) indica e he heo e ical limi s o cycles and
idling ime independen ly o he conside ed ageing model, conside ing
an EoL c i e ion o 20 % o capaci y ade. Adop ing he coe icien s
p oposed by Say u dino e al. [11] and epo ed in Table 2, he cycling
and calenda limi s a e quan i ied espec i ely in 4109 cycles wi h DoD
=80 % and 2.3•10
4
h (app oxima ely 25 yea s) o idling a SoC =20 %.
Table 2 epo s he coe icien s o he Eq. (23), 24).
Ncyclemax(DoD) = EoL
aDoD⋅DoD2+bDoD⋅DoD (23)
max(SoC) = EoL
(aSoCa ⋅SoC2+bSoC⋅SoC +SoC)(24)
3.2. E iciency model
When conside ing a BES ope a ing on he elec ici y g id, i is
impo an o conside he global e iciency, om al e na e cu en o
al e na e cu en (AC-AC) which is lowe han he alue p o ided by
some manu ac u e s conce ning he ba e y i sel (DC-DC). Rancilio
e al. [32,33] iden i y he C
a e
and he SoC as he ac o s wi h he
highes impac on AC-AC e iciency. Howe e , he esul s epo ed show
ha he e ec o SoC is negligible i compa ed o he C
a e
con ibu ion.
The empe a u e a which he ba e y ope a es du ing i s li e ime has a
s ong e ec on i s ope a ing pa ame e s, li e ime, and on i s ade a e.
Yang T. e al. conduc ed an expe imen al campaign wi h 21,700Li-ion
ba e y packs, unde se e al ope a ing condi ions, e.g., C
a e
and
layou pack con igu a ion [34]. Thei wo k poin ed ou ha a high C
a e
can lead o excessi e ba e y hea ing, so o keep he empe a u e wi hin
op imal ope a ing limi s i is necessa y o conside in eg a ing Ba e y
The mal Managemen Sys ems (BTMS). Se e al solu ions o BTMS we e
Fig. 2. Li espan a ixed idling condi ions o he ba e y ene gy sys em, in
unc ion o he idling S a e o Cha ge (SoC).
A. Vasylye e al.
Applied The mal Enginee ing 265 (2025) 125597
7
e iewed in he li e a u e by Kuma Thaku A. e al. and Haosong H.
e al. [35,36]. Acco ding o he e iews, an ai -condi ioned sys em may
only be su icien o quasi-s a ic applica ions, C
a e
limi equal o 1, o
as cha ge and discha ge applica ions i is necessa y o conside highe -
pe o mance sys ems such as solid–liquid phase ansi ion. As he p o-
posed sys em has o ope a e in he DAM elec ici y ma ke , he e is no
eal ad an age in ins alling ba e ies wi h high C
a e
, as he ma ke ime
uni is one hou . The ai condi ioning in he ba e y pack is he e o e
conside ed su icien o main ain he empe a u e limi s. The e ec o
he auxilia ies consump ion was successi ely in eg a ed in o he e i-
ciency cu e be ween 0 ◦C and 1 ◦C. The dependence o e iciency on he
C
a e
is hen conside ed in e pola ing he epo ed expe imen al da a
[32,33]. Consis en ly wi h he e e ence, discha ge, and cha ge e i-
ciencies a e assumed o be equal. The associa ed end epo s a
maximum app oxima i ely o C
a e
=0.5, mo ing owa d highe cu -
en causes a sligh dec ease because o he pa asi ic cu en losses and
g ea e e o equi ed o main ain he cell design empe a u e.
Dec easing C
a e
below 0.2 o ces he ene gy con e sion auxilia y sys-
ems o ope a e in s ong o -design condi ions. Consequen ly, a signi -
ican d op in cha ging and discha ging o e all e iciency occu s a
educed C
a e
.Table 4 epo s he e iciency alues as a unc ion o he
disc e isa ion adop ed and C
a e
.
4. MILP op imise accoun ing o deg ada ion cos s
4.1. F om an ageing model o a cos quan i ica ion
The p esen sec ion ocuses on he op imisa ion algo i hm. The
op imise implemen a ion can be gene alised ega dless o he speci ic
model selec ed o desc ibe he ageing p ocess o he ac o s impac ing
he BES e iciency. In he p e ious sec ions 3.1 and 3.2, he deg ada ion
and e iciency models we e selec ed o conduc a sensi i i y analysis o
he MILP model unde a ying ma ke scena ios.
The pu pose o his sec ion is o illus a e how ageing, and e iciency
a e conside ed wi hin he op imisa ion s age. To conside he deg ada-
ion cos associa ed wi h he capaci y ade due o cycling and idling, ad
hoc cos unc ions a e de ined. These unc ions, Eq. (25–26), exp ess he
cos [
€
] associa ed wi h each uni , i.e., a cycle o cycling ageing and an
idling hou o calenda ageing.
Cos cycle(DoD) = CAPEX
Nmax(DoD)(25)
Cos calenda (SoC) = CAPEX
max(SoC)(26)
Fig. 3(a) exp esses he cycle cos acco ding o he model e iewed in
Sec ion 2.1 wi h he same assump ions made o plo ing Fig. 1. Anal-
ogously, in Fig. 3(b), he calenda cos is exp essed, o he models
e iewed in Sec ion 2.2, as a unc ion idling SoC wi h he same as-
sump ions made o plo Fig. 2. Addi ionally, o he p e iously assumed
alues, CAPEX has been ixed a 150 k
€
/MWh [37]. In he p e ious
igu es, all he models analysed a e p esen ed o a gene al compa ison.
The model by Say u dino e al. is selec ed o he op imisa ion phase
and is highligh ed wi h a hicke ed line.
4.2. De ini ion o i ual cycle cos
Because o calenda ageing, he li e ime is no exclusi ely ela ed o
he BES ope a ion, in o de o maximise he ne ope a ing p o i s o e
he whole li e ime i is ad an ageous o in oduce a i ual cos
inc easing o dec easing a i icially he ac ual cycle cos du ing he
op imisa ion p ocess adop ing di e en ope a ional s a egies.
Cos cycle i ual(DoD,R ac o )=Cos cycle(DoD)⋅R ac o (27)
The Cycle Cos de ined in Eq. (27) is mul iplied by R
ac o
which can
inc ease o educe he cos associa ed wi h one cycle a ixed DoD
c ea ing a i ual cos . R
ac o
is de ined as non-nega i e eal and he
sensi i i y o his pa ame e is in es iga ed in his pape in he ange
be ween 0 and 2 wi h a disc e isa ion s ep o 0.1. R
ac o
equal o ze o
means pe o ming he analysis wi h he cycle cos equal o ze o, so he
ba e y will pe o m all he cycles ha pay o he ine iciencies o he
sys em, while R
ac o
equal o 2 means ha he i ual cycle cos is double
he eal cos . The objec i e o a a iable i ual cycle cos is o e alua e
whe he he o e all BES pe o mance can be enhanced by inc easing he
numbe o ope a ing cycles du ing he low p o i abili y ma ke , educing
he ope a ional losses, a oiding ha , wi h he BES idling, he EoL is
eached due o calenda ageing. On he o he hand, in ma ke pe iods
cha ac e ised by high p o i abili y, an a i icial inc ease in cycle cos s
can lead o maximising inal ne OP by selec ing jus he mo e p o i able
cycles. I mus be s essed ha he change in cycling cos is jus a ma e
o ine uning one o he main op imisa ion pa ame e s, while he ac ual
deg ada ion cos is compu ed wi hou conside ing he R ac o o co ec ly
e alua e he economic pe o mance.
4.3. Op imisa ion algo i hm desc ip ion
The MILP algo i hm is implemen ed h ough a disc e ised app oach
unc ion since i is demons a ed ha i is capable o handling he
complexi y o he p oblem wi hou losing e iciency in inding he
op imal solu ion [23]. Mo eo e , i can be adop ed in u u e wo ks o
in eg a ed op imisa ion o he day-ahead and ancilla y se ice ma ke s
[38]. Eq. (28) illus a es a gene al de ini ion o he MILP op imisa ion
p oblem. Looking a he e ms in he equa ion:
T
is he ec o coe icien
o he objec i e unc ion, and xis he op imised a iable; in his wo k i
is bina y. To desc ibe he p oblem conside ing echnical and economic
cons ain s, ma ix Ais used o he linea cons ain s de ini ion and lb
and ub o he lowe and uppe bounds.
The p oblem o an op imal a bi age s a egy is desc ibed by he
objec i e unc ion, i.e., he ne ope a ional p o i s de ined by Eq. (29),
ha accoun o he Cos
cycle
and he Cos
idle
, as de ined by Eq. (25–26)
and ep esen ed in Fig. 3. In Eq. (29) wo e ms can be dis inguished, he
i s conce ns he ope a ional p o i s (OP
) ha a e exclusi ely ela ed o
he cha ging and discha ging phases, and he second quan i ies he
deg ada ion cos (Cos
deg
), de e mined as he sum o he con ibu ions
o bo h cycling and idling ageing. Eq. (25) and (26) espec i ely o each
ime, . In his way, he algo i hm de ines he economic op imum ha
also pays back he cycle cos associa ed wi h deg ada ion.
Indeed, capaci y ade is a cos since, once he EoL c i e ion is
eached, he ba e y mus be eplaced. While OP ep esen s he ac ual
cash low o he ope a o , wha is wo h maximising is he OP
ne
o
a oid pe o ming cycles cha ac e ised by oo low OP o jus i y he li e-
ime consump ion o he BES. Such a s a egy gua an ees maximising
ea nings, ne o CAPEX, o e a long pe iod. On a yea ly basis, he
summa ion o Cos
deg
can be conside ed as a p o ision o new in es -
men a he end o BES li e o amo isa ion o al eady paid CAPEX. Eq.
(29) shows how he ope a ing phases o he ba e y a e di ided and how
he objec i e unc ion coe icien s (
T
) a e de ined. In he ma ix ep e-
sen a ion, i can be seen ha he elemen s on he diagonal ep esen he
idling phase (i=j), while he uppe ma ix ep esen s he cha ging phase
Table 4
E iciency alues o he disc e ised C
a e
, o MILP op imisa ion [32,33].
C
a e
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
η
0.09 45.79 74.40 86.75 93.83 94.49 93.13 91.96 90.94
A. Vasylye e al.
Applied The mal Enginee ing 265 (2025) 125597
8
(i < j), and inally he lowe ma ix ep esen s he discha ging phase (i
>j). In (30), he e a e: I he inal s a e o cha ge eached by he ba e y, j
he ini ial s a e o cha ge, and inally op imised hou .
Since he op imisa ion is ca ied ou conside ing he day ahead
ma ke (DAM), he ime disc e isa ion is se o one hou in acco dance
wi h he ma ke ime esolu ion. Op imisa ion o dispa ch is pe o med
subsequen ly day by day, conside ing a o ecas ing olling ho izon, H, o
36 h as he bes ade-o be ween compu a ional ime and global op i-
mum iden i ica ion [23].
min
x Txs. .{A⋅x≤b
lb ≤x≤ub (28)
OPNe =∑
H=36
=1(OP −Cos deg )
=∑
H=36
=1((Re enuedisch −Cos ch) − (Cos cycle i ual +Cos calenda ))
(29)
The p oblem o mula ion o he MILP is p esen ed in Eq. (29), While
he men ioned disc e ised cha ac e o he p oblem can be app ecia ed
h ough he ma ix app oach o Fig. 4 ha p o ides a isualisa ion o he
op imised bina y a iable x, which mul iplies he coe icien ma ix
de ined in Eq. (30). Fo each ime s ep , a squa ed ma ix is de ined,
columns indica e he ini ial SoC while ows de e mine he inal. A each
ime s ep, one and only one elemen o he ma ix mus be se o 1
selec ing a uni ocal ope a ional mode (i.e., a SoC a ia ion). In Fig. 4,
ows a e cha ac e ised by index iand columns by index j. The ma ix
dimension depends on he disc e isa ion esolu ion o SoC; by adop ing a
ine s ep he numbe o columns and ows o N
SoC
inc eases and he
compu a ional e o ises. As a me e example, in Fig. 4, SoC is dis-
c e ised wi h a s ep o 20 pe cen age poin s, as he selec ed mode is i =1
and j =1, which means ha he ba e y is main ained a 100 % SoC.
Analogously o x, he Obj a ay and i s e ms, shown in Eq. (30), can
be de ined wi h 3D ma ix shape. Fo each posi ion in he ma ix,
Cos
cycle
is de e mined by he associa ed ΔSoC,Cos
idle
will be de ined by
a diagonal ma ix de e mined by he SoC associa ed wi h he index, and
OP is compu ed om he ΔSoC o each ope a ional mode, he elec ici y
zonal p ice (ZP), which depends on he ime, and he cha ging o dis-
cha ging e iciency, depending on C
a e
and so on ΔSoC. Eq. (30) epo s
in de ail how each elemen o Obj is compu ed acco ding o he objec i e
unc ion. Obj is hen eshaped as a 1-D a ay be o e unning he MILP
algo i hm.
The cons ain s o consis ency be ween he inal SoC o ime and he
ini ial SoC a he ime +1 is exp essed by Eq. (31) while Eq. (32)
imposes selec ing one, and only one, ope a ional mode a e e y ime
s ep, inally, Eq. (33) imposes he SoC a he i s ime s ep a he i s day
and gua an ees ha he ini ial SoC, i=1, a he i s hou o he day
n
is
equal o he inal SoC a he 24 h hou , j=24, o he day
n–1
, a de ailed
desc ip ion o he olling ho izon op imisa ion logic is p o ided in [23].
xis imposed as a bina y a iable.
Fig. 3. Cycle cos associa ed wi h each speci ic occu ed cycle (a). Calenda cos associa ed wi h each speci ic idle hou (b). The CAPEX (Capi al Expendi u e) cos
was assumed o be
€
150 k/MWh, o calcula e cycle and calenda unc ion cos s [37].
Fig. 4. Ma ix isualisa ion o op imisa ion a iable x.
Obji,j, =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
ΔSoCi,j
100 ⋅Cnom⋅
η
disch(ΔSoCi,j)⋅ZP −Cos cycle i ual(DoDi,j,R ac o ) − Cos calenda (SoCmeani,j),ΔSoCii>j
−Cos calenda (SoCi,j),ΔSoCii=j
(ΔSoCi,j
100⋅
η
ch(ΔSoCi,j)⋅Cnom⋅ZP −Cos calenda (SoCmeani,j))ΔSoCii<j
(30)
A. Vasylye e al.
Applied The mal Enginee ing 265 (2025) 125597
9
∑
NSoC
j=1
xn,j, =∑
NSoC
i=1
xi,n, +1;∀ ∈T∧ ∀n∈NSoC (31)
∑
NSoC
i=1∑
NSoC
j=1
xi,j, =1;∀ ∈T(32)
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
∑NSoC
j=1x1,j,1=1,dayn=1
(∑NSoC
j=1xn,j,1)dayn
=(∑NSoC
i=1xi,n,24)dayn−1
∀n∈NSoC,dayn>1(33)
5. In es iga ed ma ke scena ios and KPIs
This sec ion de ines he inpu and bounda ies o he op imise
desc ibed in Sec ion 3, he ageing model p oposed by Say u dino [11]
in Subsec ion 4.1, and he e iciency o mula ion p esen ed in Subsec-
ion 4.2 o a 1 MWh/1 MW Li-ion BES in di e en eal ma ke scena ios.
Fi s , he I aly NORD elec ici y ma ke zone is assumed as a case s udy,
and di e en scena ios a e c ea ed conside ing day-ahead elec ici y
p ices o yea s 2022 and 2023, each epea ed se e al imes un il he
EoL. The dispa ch o BES is op imised on a daily basis, wi h a 36 h ime
ho izon and pe ec p ice knowledge. The BES- a ed capaci y is
con inuously upda ed consis en ly wi h he adop ed ageing model and
once he EoL (20 %) is eached, he p ocess s ops. Conce ning he
me hodology p esen ed in Sec ion 3.2, he SoC disc e isa ion is limi ed
be ween 20 % and 100 % wi h 10-pe cen age-poin s eps. This analysis
aims o assess how he cos s associa ed wi h deg ada ion can a ec he
economic pe o mance o he ba e y.
Subsequen ly, i is he e o e essen ial o ca y ou a sensi i i y
analysis conside ing di e en ma ke scena ios because i is impo an
o check i he e is any ela ionship be ween he mos impo an
pa ame e in he ma ke when i comes o a bi age, i.e., he p ice
luc ua ion magni ude, and he op imum R
ac o
. Fo his analysis,
di e en ma ke scena ios we e e alua ed, cha ac e ised by bo h yea s
and di e en ma ke zones: 4 yea s om 2019 o 2022 and 7 di e en
ma ke zones ( he no h o I aly, I a-NORD, Sicily, I a-SICI, Po ugal, PT,
Poland, PL, G eece, GR, No way NO1, and Sweden SE3).
Fo u he analysis, di e en KPIs we e conside ed: li espan, op i-
mised R
ac o
, cycling and calenda pe cen age ade, cha ged and
discha ge ene gy, yea ly OP and inally he P esen Value o Ope a ion
P o i s PV
OP
as de ined in Eq. (34) o di e en yea ly in e es a es i(0
%, 5 % and 10 %).
PVOP =∑
li espan−1
yea =0
OPyea
(1+i)yea (34)
6. Resul s
Fi s , he I alian elec ici y ma ke , NORD bidding zone, is consid-
e ed o ocus on he p ice scena ios de ined by he yea s 2022 and 2023,
ep esen a i e o p o i able and non-p o i able condi ions o ba e y
ope a ions espec i ely. A his s age, he R
ac o
is main ained equal o 1.
Fig. 5 shows how ba e y deg ada ion is di ided be ween cycle and
calenda ade be o e eaching he EoL c i e ium and associa ed Ope a-
ional P o i s OP.
Fig. 5 shows he cumula i e ope a ional p o i s Eq. (34), plo ed as
lines on he igh y-axis, di e en ly colou ed acco ding o he in e es
a e, and he deg ada ion, ep esen ed wi h blue and ed a eas o
cycling and calenda con ibu ion espec i ely, on he le y-axis he
cumula i e capaci y ade can be ead Eq. (2) while on he igh he
co esponding cumula i e deg ada ion cos s a e epo ed which a e
equal o CAPEX as ade eaches he EoL. Sub igu es (a) and (b) epo he
esul s o wo di e en scena ios in which he 2022 and 2023 hou ly
ma ke p ices a e epea ed un il he BES EoL. 2022 was cha ac e ised by
high p ices, which a e commonly associa ed wi h inc eased a iabili y
[16], i.e., he main d i ing ac o o ene gy a bi age [16]. In 2022 he
a e age daily a iabili y o elec ici y p ice, i.e., he di e ence be ween
he maximum and minimum daily p ices, sco ed on a e age 161.52
€
/
MWh, while in 2023 i was limi ed o 78.42
€
/MWh. Consequen ly, BES
ope a es much mo e equen ly in he 2022 scena io pe o ming 0.84
equi alen cycles pe day agains 0.49 i 2023 p ices a e adop ed.
This is immedia ely e lec ed in he ageing p ocess; in he i s case,
he li espan is es ima ed a 8.0 yea s wi h a ele an impac o cycling
(63.8 %) on he o e all capaci y ade. In he second scena io, BES las s
longe (11.4 yea s), and cycling ageing has a lowe impac on he o al
ade, bu he pe cen age is educed o 52.3 %. Howe e , wha is
impo an is o look a he cumula i e sum o OP a he end o li e: 234.9
k
€
and 126.2 k
€
o he wo scena ios espec i ely. Conside ing ha he
CAPEX o a 1 MWh BES is assumed o be 150 k
€
[37], i is possible o
conclude ha he op imisa ion o ope a ions gua an ees 10.63 k
€
/yea
o ne p o i s in he 2022 scena io. Con e sely, conside ing he p ices o
2023, i minimises he losses o 2.09 k
€
/yea . The op imisa ion app oach
p esen ed in he p e ious sec ion gua an ees ha BES pe o ms a cycle
only i he a bi age e enues (OP) o e come he associa ed deg ada ion
cos . Indeed, in Fig. 5, he cumula i e ope a ional p o i lines a e always
abo e he blue a ea. Howe e , he p esence o calenda ageing jeopa -
dises he iabili y o his app oach and uning o R
ac o
is needed.
Fig. 5. Cycling and calenda con ibu ion o capaci y ade (le y-axis) and cumula i e OP o 2022 (a) and 2023 (b) scena ios. Dashed ho izon al lines indica e he
CAPEX (150 k
€
[37]) and he cumula i e OP a he EoL, hus he gap be ween hese highligh s he ne p o i s o economic loss.
A. Vasylye e al.
