applied
sciences
A icle
Compa ison o Lead-Acid and Li-Ion Ba e ies Li e ime
P edic ion Models in S and-Alone Pho o ol aic Sys ems
Rodol o Du o-López 1,* , Tomás Co és-A cos 2, Jesús Se gio A al-Se il 1and JoséL. Be nal-Agus ín1


Ci a ion: Du o-López, R.;
Co és-A cos, T.; A al-Se il, J.S.;
Be nal-Agus ín, J.L. Compa ison o
Lead-Acid and Li-Ion Ba e ies
Li e ime P edic ion Models in
S and-Alone Pho o ol aic Sys ems.
Appl. Sci. 2021,11, 1099. h ps://
doi.o g/10.3390/app11031099
Academic Edi o : Robe o Sale i
Recei ed: 17 No embe 2020
Accep ed: 21 Janua y 2021
Published: 25 Janua y 2021
Publishe ’s No e: MDPI s ays neu al
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Licensee MDPI, Basel, Swi ze land.
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A ibu ion (CC BY) license (h ps://
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1
Depa amen o de Ingenie ía Eléc ica, Uni e sidad de Za agoza, C/Ma ía de Luna, 3, 50018 Za agoza, Spain;
jsa al@uniza .es (J.S.A.-S.); jlbe nal@uniza .es (J.L.B.-A.)
2Escuela Uni e si a ia Poli écnica de La Almunia, EUPLA, Uni e sidad de Za agoza, C/Mayo , s/n,
La Almunia, 50100 Za agoza, Spain; co es@uniza .es
*Co espondence: du o@uniza .es
Abs ac :
Se e al models o es ima ing he li e imes o lead-acid and Li-ion (LiFePO
4
) ba e ies
a e analyzed and applied o a pho o ol aic (PV)-ba e y s andalone sys em. This kind o sys em
usually includes a ba e y bank sized o 2.5 au onomy days o mo e. The esul s ob ained by each
model in di e en loca ions wi h e y di e en a e age empe a u es a e compa ed. Two di e en
loca ions ha e been conside ed: he Py enees moun ains in Spain and Tindou in A gelia. Classical
ba e y aging models (equi alen ull cycles model and ain low cycle coun model) gene ally used
by esea che s and so wa e ools a e no adequa e as hey o e es ima e he ba e y li e in all cases.
Fo OPzS lead-acid ba e ies, an ad anced weigh ed Ah- h oughpu model is necessa y o co ec ly
es ima e i s li e ime, ob aining a ba e y li e o oughly 12 yea s o he Py enees and a ound 5 yea s
o he case Tindou . Fo Li-ion ba e ies, bo h he cycle and calenda aging mus be conside ed,
ob aining mo e han 20 yea s o ba e y li e es ima ion o he Py enees and 13 yea s o Tindou .
In he cases s udied, he li e ime o LiFePO
4
ba e ies is a ound wo imes he OPzS li e ime. As
nowadays he cos o LiFePO
4
ba e ies is a ound wo imes he OPzS ones, Li-ion ba e ies can be
compe i i e wi h OPzS ba e ies in PV-ba e y s andalone sys ems.
Keywo ds: lead-acid ba e ies; li hium ba e ies; pho o ol aic; s andalone sys ems; ba e y li e ime
1. In oduc ion
Renewable elec ici y gene a ion is widely used in u al a eas whe e he elec ical
g id is weak o nonexis en . S and-alone (o -g id) sys ems a e ypically powe ed by
a pho o ol aic (PV) gene a o wi h ba e y s o age. In his kind o sys em, he ba e y
echnology mos widely used is lead-acid. In some cases, a hyb id PV– ossil uel gene a o
(diesel o gasoline)–ba e y s o age sys em can be op imal— ha is, he sys em wi h lowe
cos s du ing he sys em’s li e ime. S andalone sys ems can be di ec cu en (DC) o al e n
cu en (AC) coupled [
1
]— ha is, he bus whe e componen s a e connec ed can be he
DC bus o he AC bus. DC coupled sys ems (Figu e 1) a e usual in low powe sys ems
( ypically lowe han 5 kW), while he AC-coupled sys em is commonly used in la ge
sys ems. The cha ge con olle is needed o a oid o e cha ge and o e -discha ge o he
ba e y, p e en ing p ema u e ailu e. Addi ionally, an in e e (DC/AC) is needed when
he e is AC load.
Appl. Sci. 2021,11, 1099. h ps://doi.o g/10.3390/app11031099 h ps://www.mdpi.com/jou nal/applsci
Appl. Sci. 2021,11, 1099 2 o 16
Figu e 1. Di ec cu en (DC) coupled s andalone pho o ol aic (PV) sys em.
In s andalone sys ems, di e en ypes o ba e ies can be used [
2
]. Lead-acid ba e ies
( al e- egula ed lead-acid ype, VRLA) a e he dominan echnology o pho o ol aic
o -g id applica ions [
3
] due o hei a o dable cos s o la ge ins alled capaci ies. Howe e ,
lead-acid ba e ies a e he o e all weakness o he PV sys em and end o be eplaced by
new echnologies such as Li-ion ba e ies [
4
], which can be compe i i e in some cases [
5
]
due o hei highe cycle li e, despi e hei highe cos .
The o al ba e y cos (including i s eplacemen du ing he sys em li e ime) is he high-
es in he sys em’s ne p esen cos (NPC); hus, in he op imiza ion p ocess o s andalone
sys ems, he accu a e es ima ion o ba e y li e is one o he mos c i ical issues. Signi ican
e o s in he ba e y li e ime p edic ion would lead o g ea e o s in he es ima ion o he
NPC.
Lead-acid ba e y aging ac o s a e cha ge and discha ge a es, cha ge (Ah) h oughpu ,
he ime be ween ull cha ge, ime a a low s a e o cha ge (SOC), and pa ial cycling. Se e al
esea che s ha e analyzed he lead-acid ba e y aging ac o s [
6
,
7
]. Classical models widely
used by esea che s and so wa e ools o es ima e he ba e y li e a e he “equi alen ull
cycles model” and he “ ain low cycle coun ing model” [8].
The equi alen ull cycles model coun s he ull cha ge (Ah h oughpu ) cycled by he
ba e y since he s a o i s li e ime, wi hou conside ing SOC, empe a u e, cu en , o
any o he a iable; when his alue eaches he cha ge, he ba e y can cycle, conside ing
he cycle li e shown in he manu ac u e da ashee ob ained unde s anda d es s, and
he end o he ba e y li e is eached. The ain low cycle coun model includes he e ec
o he dep h o discha ge (DOD). Ne e heless, eal ope a ing condi ions (cu en a e,
empe a u e, DOD, SOC, e c.) a e di e en om he labo a o y condi ions o he cycles
shown by he manu ac u e da ashee , so a signi ican e o in he li e ime p edic ion can
be ob ained.
Mos o he p e iously published s udies o he simula ion and/o op imiza ion o
sys ems wi h ba e y sys ems do no use ad anced models o es ima e ba e y li e ime. In
many cases, he ba e y deg ada ion is no conside ed o i s li e ime is es ima ed in ixed
alues based on he expe ience o he esea che [
9
–
20
]. In o he cases, ba e y li e ime
is es ima ed by using he equi alen ull cycles model [
21
–
25
]. In he bes cases, i may
be es ima ed by using he ain low cycle coun ing me hod [
26
–
29
]. Howe e , a ba e y’s
eal li e ime can di e om he es ima ed li e ime by many yea s using he men ioned
me hods, depending on he ope a ing condi ions. As p e iously men ioned, a high e o in
he es ima ion o he ba e y li e would imply a g ea e o in he es ima ion o he o al
Appl. Sci. 2021,11, 1099 3 o 16
cos o he ba e ies in he NPC o he sys em; he e o e, he eal le elized cos o Ene gy
(LCE) may be e y di e en om he expec a ion.
A much mo e accu a e lead-acid aging model (and also mo e complex and wi h highe
compu a ional di icul y) is he one desc ibed by Schi e e al. [
30
], called “weigh ed Ah
h oughpu model” and used by iHOGA so wa e [
31
]. The model is based on applying
weigh ing ac o s o he ba e y’s cha ge h oughpu o es ima e he los capaci y (consid-
e ing he di e en s ess ac o s o cycling and co osion). Schi e e al.’s model was used
o es ima e he ba e y li e in PV sys ems [
32
,
33
]. This model ob ained esul s e y simila
o he eal ones [
33
], while he equi alen ull cycles model and he cycle coun ing model
ob ained li e ime es ima ions ha can be, in some cases (PV-ba e y sys ems), wo o h ee
imes highe han he eal ba e y li e ime.
Li-ion ba e ies ([
34
–
36
]) ha e a highe cycle li e, ene gy densi y, and ene gy e iciency,
and lowe main enance compa ed o lead-acid ba e ies. The LiFePO
4
(LFP) ype is he
mos used in o -g id sys ems. Li-ion ba e ies’ mos signi ican aging ex e nal ac o s a e
empe a u e, cha ge and discha ge a es, and DOD [
37
]. In simula ion and op imiza ion
o s andalone sys ems, Li-ion cell le el aging models [
38
] a e usually used due o hei
simplici y. Elec ochemical models a e usually e y complex, e en he mos simpli ied
ones [
37
,
39
,
40
], implying high calcula ion imes [
41
]. “Calenda ” aging occu s when a
ba e y is no being used while “cycle” aging occu s when he ba e y is unde cha ge o
discha ge cu en [
41
]. Cycle aging is a ec ed by he o al cha ge (Ah) h oughpu om
he s a o ba e y li e ime, he cu en , he ambien empe a u e, and he SOC. Calenda
aging main ac o s a e empe a u e and SOC [
38
]. A good example o Li-ion aging model
ex ma ix is shown in he wo k o Oya bide e al. [42].
The ain low cycle coun model is also used o Li-ion ba e ies. A henius kine ic-
based aging models [
38
] a e he mos used cycle aging models o Li-ion ba e ies. Fo
example, Wang e al.’s [
43
] model was ob ained by pe o ming many accele a ed cycling
es s o comme cial LiFePO
4
cells, ob aining a capaci y ade model ha akes in o accoun
he Ah- h oughpu and empe a u e o di e en cha ge/discha ge a es. Li-ion ba e y
calenda aging has been modeled by di e en esea che s [
38
]. Fo example, Pe i e al. [
44
]
used an exp ession based on A henius law, conside ing empe a u e, ime, and SOC.
In his wo k, we compa e he ba e y li e ime es ima ion o a PV-ba e y sys em used
o supply elec ici y o a household loca ed in wo di e en loca ions wi h e y di e en
a e age empe a u es, conside ing di e en models o he deg ada ion o lead-acid o
Li-ion ba e ies. In Sec ion 2, he models o he PV sys em componen s and he di e en
ba e y li e ime models a e shown. In Sec ion 3, we show he compa ison o he di e en
models applied in he PV-ba e y sys ems. Finally, he main indings and conclusions a e
discussed in Sec ion 4.
2. Ma e ials and Me hods
In his sec ion, he componen s’ models a e desc ibed, emphasizing he di e en
ba e y li e ime models used. The sys em’s simula ion was pe o med o e he cou se o a
whole yea in hou ly in e als ( = 0
. . .
. 8760 h), and he esul s we e ex apola ed o he
emaining yea s o he sys em’s li e ime.
2.1. Pho o ol aic Gene a o
I he e was no maximum powe poin acking in he con olle ( ypical in DC-coupled
sys ems), he PV ou pu cu en du ing ime ,
IPV ( )
(A), was calcula ed as ollows ([
33
]),
whe e he e ec o he module empe a u e is negligible:
IPV ( )=ISC·G( )(1)
whe e
ISC
is he sho cu cu en (A) o he PV module and G( ) (kW/m
2
) is he i adiance
o e he module su ace a ime .
Appl. Sci. 2021,11, 1099 4 o 16
The ou pu powe o he PV gene a o (W) o
NPV_p
s ings in pa allel was ob ained
by using Equa ion (2):
PPV ( )=NPVp·IPV( )·VDC( )· PV_loss (2)
whe e V
DC
( ) (V) is he ba e y bank ol age in DC-coupled sys ems and
PV_loss
is he loss
ac o (PV module misma ch o powe ole ance, losses due o di in he PV modules, and
losses in he wi es).
2.2. Cha ge Con olle
I p e en s o e cha ge and o e -discha ge o he ba e y. In lead-acid ba e ies, o e -
cha ge is a oided by cha ging ba e ies in h ee s ages (bulk, abso p ion, o boos and loa
s ages, Figu e 2). Abso p ion includes ba e y equaliza ion in some cycles, pe iodically.
Du ing he second and hi d s ages, he cu en is limi ed using he pulse wi h modula ion
(PWM) echnique. The limi s be ween s ages a e ol age se poin s, while ad anced con-
olle s compa e he minimum ba e y s a e o cha ge (SOC) o he p e ious discha ge wi h
SOC se poin s o de e mine he cha ge s ages o be applied. In his wo k, SOC con olle s
pe o med he boos ing s age i he SOC du ing he las discha ge was lowe han a speci ic
alue (usually 70%), and equaliza ion was pe o med i ba e y SOC du ing discha ge was
lowe han ano he speci ic alue (usually 40%).
Figu e 2. Cha ge con olle cha ging s ages.
Li hium ba e ies equi e a speci ic cha ge con olle , as he e is a e y li le ol age
di e ence o a high SOC di e ence (s a e o unc ion is es ima ed in Li-ion ba e ies),
and hey canno accep o e cha ge; he e o e, he e is no equaliza ion s age and no loa
s age, jus a cons an ol age/cons an cu en (CV/CC) cha ge algo i hm. Some con olle
models a e p og ammable and can be used o lead-acid o o li hium ba e ies.
The cha ge con olle p e en s o e -discha ge by disconnec ing he ba e y when a
speci ic se poin ol age (o SOC) is eached and econnec ing a e pa ial echa ging when
ano he highe speci ic se poin ol age (o SOC) is eached.
2.3. In e e
Al hough cha ge con olle s include ba e y o e -discha ge p o ec ion, s andalone
in e e s also include his ea u e (in some cases, hey a e di ec ly connec ed o he ba e y).
The in e e e iciency depends on i s ou pu powe , as shown in Figu e 3. Many esea che s
and so wa e ools use a cons an alue o he in e e e iciency, leading o signi ican
e o s when he AC load p o ile has g ea peaks and alleys.
Appl. Sci. 2021,11, 1099 5 o 16
Figu e 3. Typical in e e e iciency [45].
2.4. Ba e y
Du ing each ime s ep, he s a e o cha ge SOC ( ) (pe uni ) was calcula ed om he
p e ious ime s ep SOC, adding o sub ac ing he cha ge o he ba e y cu en :
SOC( )=SOC( −∆ )+Z
−∆
Ib(τ)
CN
dτ(3)
whe e
∆
is he leng h o he ime s ep (h),
Ib( )
(Equa ion (4)) is he cu en ha e ec i ely
a ec s he ba e y cha ge, C
N
is he nominal capaci y o he ba e y (Ah), and
τ
is he ime
bew een −∆ and .
Ib( )=Iba ( )·ηba _ch;Iba ( )>0(cha ge)
Ib( )=Iba ( )
ηba _d;Iba ( )<0(discha ge)(4)
whe e
Iba ( )
(posi i e cha ging, nega i e discha ging) is he ba e y cu en ,
ηba _ch
is he
cha ging e iciency, and
ηba _d
is he discha ging e iciency. Usually bo h e iciencies a e
conside ed o be he same, equal o he squa e oo o he ound ip e iciency.
Ba e y deg ada ion models a e shown in he nex subsec ions. In all he models, i
he ba e y li e ime es ima ion in yea s was highe han he loa ing li e (shown in he
manu ac u e da ashee ), he loa ing li e was used as he ba e y li e ime. In he upcoming
subsec ions, he ba e y ol age will be modelled.
2.4.1. Equi alen Full Cycles Model
This me hod es ima es he end o he ba e y li e ime when a speci ied numbe o ull
cha ge–discha ge cycles ( ega dless o he ope a ing condi ions) a e eached (Z
IEC
), de ined
by he IEC s anda d [46].
Du ing he simula ion, o e e y ime s ep, he equi alen numbe o ull cycles since
he beginning (ZN) was calcula ed as ollows:
ZN( +∆ )=ZN( )+|Ib( )|·∆
CN
(5)
whe e |
Ib( )
| is he absolu e alue o he discha ge cu en . When Z
N
( ) = Z
IEC
, he end
o he li e ime o he ba e y has been eached.
2.4.2. Rain low Cycles Coun ing Model
This model is mo e complex and p ecise as i conside s he dep h o discha ge (DOD)
o he cha ge/discha ge cycle, and is based on Downing’s algo i hm [
47
]. This me hod is
based on coun ing, du ing one yea o he sys em’s simula ion, he cha ge/discha ge cycles

Appl. Sci. 2021,11, 1099 6 o 16
Z
i
co esponding o each ange o he DOD (spli in o min e als). The e a e a numbe o
cycles o ailu e (CFi) ob ained om he manu ac u e da ashee (Figu e 4).
Ba e y du a ion, in yea s, can be calcula ed as ollows:
Li eb=1
∑m
i=1
Zi
CFi
(6)
Figu e 4. Lead-acid ba e y: cycles o ailu e s. dep h o discha ge (DOD) [48].
2.4.3. Lead-Acid Ba e ies: Schi e e al.’s Weigh ed Ah-Th oughpu Model
The weigh ed Ah- h oughpu model p esen ed by Schi e e al. [
30
] assumes ha
ope a ing condi ions a e ypically mo e se e e han hose used in s anda d es s o cycling
and loa li e ime.
This model uses weigh ing ac o s o he cha ge h oughpu o e he ba e y li e o
model he los capaci y due o he di e en aging mechanisms. These weigh s depend on
he DOD, he cu en a e, he acid s a i ica ion, and he ime since he las ull cha ging.
Ba e y ol age a each ime s ep was calcula ed depending on i he ba e y was
Iba ( )>
0 (cha ging), Equa ion (7), o
Iba ( )<
0 (discha ging), Equa ion (8), using he
Shephe d model [49]:
Vb( )=V0−gDOD( )+ρc( )Iba ( )
CN+ρc( )McIba ( )
CN SOC( )
Cc−SOC( )(7)
Vb( )=V0−gDOD( )+ρd( )Iba ( )
CN+ρd( )MdIba ( )
CN DOD( )
Cd( )−DOD( )(8)
whe e
V0
(V) is he open-ci cui equilib ium cell ol age a he ully cha ged s a e,
g
(V)
is an elec oly e p opo ionali y cons an ,
ρc( )
and
ρd( )
(
Ω
Ah) ep esen he agg ega ed
in e nal esis ance du ing cha ge o discha ge, and
Cc
and
Cd( )
ep esen he no malized
capaci y o he ba e y du ing cha ge o discha ge.
This model es ima es he capaci y loss by co osion,
∆Cc( )
and he capaci y loss
by cycling (deg ada ion),
∆Cd( )
. Each hou , he emaining ba e y capaci y,
C em( )
,
could be es ima ed as he no malised ini ial ba e y capaci y
Cd(0)
minus he capaci y
loss by co osion and deg ada ion. The end o he ba e y li e was conside ed o be when
Cd( )= 0.8 CN.
C em( )=Cd(0)−∆Cc( )−∆Cd( )(9)
Appl. Sci. 2021,11, 1099 7 o 16
Capaci y loss by deg ada ion
∆Cd( )
was calcula ed coun ing he weigh ed numbe
o cycles, wi h he impac o he SOC, he discha ge cu en , and he acid s a i ica ion.
A he same ime, he capaci y loss by co osion was es ima ed, which is p opo ional
o he e ec i e co osion laye hickness, which g ows du ing he li e ime o he ba e y
depending on he co osion ol age o he posi i e elec ode and empe a u e. This is a
complex model wi h many equa ions; u he de ails can be ound in [30].
2.4.4. LFP Li-Ion Models
In Li-ion ba e ies, du ing each ime s ep, he capaci y loss
Q( )
(pe cen age capaci y
ade) can be calcula ed as he sum o cycle capaci y ade and calenda capaci y ade:
Q( )=Qcal( )+Qcyc( )(10)
Cycle capaci y ade is a ec ed by he cha ge cycled, which depends on he numbe o
cycles Nand DOD, and i is also a ec ed by o he ac o s such as SOC, cu en and ambien
empe a u e T. Calenda aging depends on he empe a u e, SOC, and ime [38].
Di e en models can calcula e he cycling capaci y ade. Elec ochemical models
a e usually e y complex. One o he mos simpli ied elec ochemical models is As aneh
e al.’s Li-ion ba e y li e ime p edic ion model [
37
,
39
], which in eg a es he simpli ied
single pa icle model (SSPM) and educed-o de model (ROM) o p edic solid elec oly e
in e phase g ow h (SEI), gi ing good esul s o mode a e cycling cu en s; howe e , i
includes many a iables ha a e dependen on he speci ic chemis y o he ba e y and a e
di icul o es ima e.
I mus be se led ha each aging model is in insically ela ed o each cell ype and,
he e o e, he same aging model canno be alid o all he Li-ion models.
Wang e al.’s Cycle Aging Model
Wang e al. [
43
] ob ained he cycle capaci y ade o comme cially a ailable 2.2 Ah
cells om A123 Sys ems, o di e en C- a es and empe a u es:
Qcyc( )=B·exp−31700 +370.3·C a e
RT (Ah)0.55 (11)
whe e Bis he p e-exponen ial ac o ha depends on C a e (h
−1
) (cu en h ough he ba -
e y in A di ided by i s nominal a ed capaci y in Ah), Ris he gas cons an
(8.314 J/mol-K)
,
and Tis he ambien empe a u e (K).
Ah =N·DOD·CN
is he o al Ampe e-hou (Ah)
h oughpu o he ba e y.
G oo e al.’s Cycle Aging Model
G oo e al. [
50
] model was ob ained a e es ing comme cial LFP cells o di e en
cu en s I(A) and empe a u es T(◦C):
Qcyc( )=a·eb·I·Tc·I2+d·I+e+ (12)
whe e a,b,c,d,e, and a e i pa ame e s.
Pe i e al.’s Calenda Aging Model
Calenda capaci y ade is modelled by Pe i e al. [
44
] using an exp ession based on
A henius law, dependen on empe a u e, ime, and SOC:
Qcal( )=Bcal(SOC)·exp−Ea_cal(SOC)
RT  zcal (SOC)(13)
Appl. Sci. 2021,11, 1099 8 o 16
whe e
Bcal(SOC)
(Ah/s
Zcal
) is he p e-exponen ial ac o depending on SOC,
Eacal (SOC)
is
he ac i a ion ene gy (Jmol
−1
), and
zcal(SOC)
is a dimensionless cons an (assumed o be a
alue o 0.5).
Wang e al.’s Calenda Aging Model Combined wi h Pe i e al.’s Calenda Model
Cycle aging mainly occu s when he ba e y is cha ging. Pe i e al. [
44
] pos ula es
ha i only happens when he cu en is abo e a gi en limi
Ilimi
(A), using in his case he
Wang model, while when cu en is lowe han he limi calenda aging occu s, using, in
said case, Equa ion (13). The limi cu en depends on he ba e y capaci y o ackle high
cha ging a es.
3. Resul s
An o -g id household AC load p o ile om a p e ious publica ion [
33
] has been
conside ed o compa e he ba e y li e ime es ima ion o he PV-ba e y sys em. The
a e age measu ed AC load was 3.61 kWh/day. The sys em is loca ed in he Py enees
moun ains, in A agon, Spain (la i ude 42.772
◦
, longi ude
−
0.334
◦
) wi h an a e age ou doo
annual empe a u e o 5.1
◦
C. Fo compa ison, he same sys em is conside ed o be in
a dese ic place (Tindu , A gelia, la i ude 27.669
◦
, longi ude
−
8.144
◦
), wi h an a e age
ou doo annual empe a u e o 23.1 ◦C.
Di e en ba e y li e ime es ima ion models will be used.
Fo lead-acid ba e ies:
•Equi alen ull cycles model;
•Rain low cycle coun model;
•Schi e e al.’s weigh ed Ah- h oughpu model.
Fo LiFePO4ba e ies:
•Equi alen ull cycles model;
•G oo e al.’s cycle aging model.
Wang e al.’s calenda aging model combined wi h Pe i e al.’s calenda model
conside ed an Ilimi o 5% o he ba e y bank’s nominal capaci y ( ha is, cu en a C- a e
o 20 h, C20).
The nominal ol ages a e 48 VDC and 230 VAC.
The PV modules conside ed ha e a peak powe o 100 W, 12 V nominal ol age, and
sho -ci cui cu en o 6.79 A. In he Py enees, he PV gene a o is composed o ou
se ial
×
se en pa allel PV modules ( o al 2800 W), while in Tindou (highe i adia ion)
i is composed o ou se ial
×
six pa allel PV modules ( o al 2400 W). A loss ac o o
PV_loss = 0.8 was conside ed.
Two ypes o ba e y banks we e conside ed o bo h loca ions. The ba e y bank size
was selec ed, conside ing ha abou 2.5 days o au onomy a e equi ed.
•
The lead-acid ba e y bank, which consis s o 24
×
2 V OPzS [
30
] ( looded, ubula -
pla ed, deep cycle) comme cial ba e ies in se ial, C
N
= 270 Ah ( o al 12.96 kWh),
1258 equi alen ull cycles (CF s. DOD cu e shown in Figu e 4), loa li e o 20 yea s
a 20
◦
C (manu ac u e da ashee ) and ound ip e iciency o 85%. SOC o disconnec
load (SOC
min
) 20%. The pa ame e s o he Schi e model we e he ones used in [
30
].
•
Li-ion LFP ba e y consis s o a comme cial 48 V pack o C
N
= 213.3 Ah ( o al
10.24 kWh), 5600 cycles a 80% DOD, 4022 equi alen ull cycles, loa li e o 20 yea s a
20 ◦C
(manu ac u e da ashee does no show loa li e, bu ope a ing li e o 20 yea s
o s a iona y ba e y sys ems is usually conside ed [
51
]) and ound ip e iciency o
90%. SOCmin is 10%.
The nominal in e e powe is 600 W, and i s e iciency is shown in Figu e 3.
The annual a e age ambien empe a u e in he ba e y oom in he Py enees loca ion
was es ima ed o be highe han he ou doo empe a u e. We conside ed wo alues as
he a e age: 8 o 12
◦
C. In he dese ic place o Tindou , we conside ed ha in he ba e y
oom, he a e age empe a u e can be simila o he a e age ou doo empe a u e, 23.1
◦
C.
Appl. Sci. 2021,11, 1099 9 o 16
Conside ing A henius law, co ec ed loa li e a he di e en empe a u es conside ed
a e shown in Table 1.
Table 1. Floa li e a di e en empe a u es conside ing A henius law.
S anda d Py enees Tindu
A e age empe a u e (◦C) 20 8 12 23.1
Ba e y loa li e (yea s) 20 46.1 35.1 16.1
I adia ion o e he op imal slope (sou h o ien a ion) o bo h loca ions is shown in
Table 2.
Table 2. Mon hly a e age i adia ion da a om Pho o ol aic Geog aphical In o ma ion Sys em (PVGIS) [52], yea 2015.
Jan. Feb. Ma . Ap . May Jun. Jul. Aug. Sep. Oc . No . Dec.
Py enees, 65◦slope
i adia ion
(kWh/m2/day)
3.31 2.69 3.92 4.71 4.76 4.49 5.03 5.17 4.93 3.87 3.68 3.87
Tindu , 35◦slope i adia ion
(kWh/m2/day) 6.75 7.54 7.7 7.41 7.36 7.02 6.65 6.5 6.64 5.85 7.07 6.43
Table 3shows he esul s o he simula ion du ing a whole yea o he wo lo-
ca ions conside ed. I can be seen ha he ba e y bank cha ge/discha ge ene gy is
e y low ( oughly 700 kWh/y ), which implies a ound 60 equi alen ull cycles. The
cha ge/discha ge a es a e e y low, as is usual in PV-ba e y s and-alone sys ems: he
a e age cha ge a e is oughly 4% o he ba e y’s nominal capaci y, while he a e age
discha ge a e is a ound 1%.
Table 3. Resul s o he simula ion du ing a whole yea o he di e en sys ems.
Py enees Tindou
Ba e y Type Lead-acid Li-ion Lead-acid Li-ion
Load (kWh/y ) 1318 1318 1318 1318
PV gene a ion (kWh/y ) 2804 2804 3944 3944
Ba e y cha ge/discha ge ene gy
(kWh/y ) 740 717 767 669
Equi alen ull cycles pe yea 57.1 70 53.6 65.3
Hou s o ba e y cha ge pe yea 2261 2151 3102 2399
Hou s o ba e y discha ge pe yea 5757 5792 5050 5050
A e age cha ge a e (% o CN) 3.77 4.67 4.15 4.88
A e age discha ge a e (% o CN) 0.97 1.23 1.03 1.30
The hou ly simula ion o a whole yea is shown in Figu e 5(Py enees, lead-acid) and
6 (Tindu , Li-ion). Fo Tindou , he ba e y bank is almos all he ime a SOC highe han
80% (Figu e 6). Fo he Py enees, he ba e y bank is also mos o he ime a highe SOCs
han 80%; howe e , in win e , he e a e pe iods wi h lowe SOCs (Figu e 5), eaching 30%
du ing sho pe iods.
Appl. Sci. 2021,11, 1099 16 o 16
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