TELKOM
NIKA
, Vol.13, No
.2, June 20
15
, pp. 401 ~ 4
1
2
ISSN: 1693-6
930,
accredited
A
by DIKTI, De
cree No: 58/DIK
T
I/Kep/2013
DOI
:
10.12928/TELKOMNIKA.v13i2.1467
401
Re
cei
v
ed
Jan
uary 25, 201
5
;
Revi
sed Ma
rch 1
1
, 2015;
Acce
pted Ma
rch 2
7
, 2015
Battery State of Charge Estimation with Extended
Kalman Filter Using Third Orde
r Thevenin Model
Lo
w
We
n
Ya
o
1
, Wirun A/l Pra
y
un
2
, J.
A. Azi
z
*
3
, Tole Sutikno
4
1,2,
3
Department
of Electrical P
o
w
e
r En
gin
eer
i
ng, F
a
cult
y
of Electrical E
ngi
neer
ing,
Univers
i
ti T
e
knolo
g
i Mal
a
ysia,
8130
0 Skud
ai,
Johor, Mala
ys
i
a
4De
partment o
f
Electrical Eng
i
ne
erin
g, F
a
cul
t
y
of Industria
l T
e
chnolog
y,
Univers
i
tas Ah
mad Da
hla
n
, Jantura
n
, Umbu
lharj
o
55
16
4, Yog
y
ak
arta, Ind
ones
ia
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: juna
idi
@
fke.utm.m
y
A
b
st
r
a
ct
Lithi
u
m-io
n b
a
ttery has b
e
co
me t
he
ma
inst
rea
m
e
nergy
s
t
orage
el
ement
of the e
l
ectric
vehic
l
e.
One of the chal
len
ges in
electr
ic vehic
l
e d
e
vel
o
p
m
e
n
t
is the state-of-charg
e
estimatio
n
of b
a
ttery. Accurate
estimatio
n
of s
t
ate-of-charg
e
i
s
vital t
o
i
ndic
a
te the
r
e
mai
n
i
n
g ca
pacity
of the
batte
ry a
n
d
it w
ill
eve
n
tual
l
y
max
i
mi
z
e
th
e b
a
ttery perfor
m
a
n
ce a
nd e
n
sur
e
s the safe
op
eratio
n of the b
a
ttery. T
h
is pa
per stud
ied
on
th
e
app
licati
on of
extend
ed K
a
l
m
an-
filter a
nd th
i
r
d order T
h
eve
n
in e
q
u
i
va
l
ent
circuit mod
e
l i
n
state-of-char
ge
estimatio
n
of l
i
t
hiu
m
ferro
ph
osph
ate b
a
ttery. Rand
o
m
te
st and
puls
e
d
i
schar
ge test
are co
nd
ucted
to
obtai
n the
acc
u
rate b
a
ttery mo
de
l. T
he si
mu
lati
on a
nd
e
x
peri
m
e
n
tal r
e
sults are c
o
mp
ared to v
a
li
dat
e the
prop
osed state
-
of-charg
e
esti
mati
on
meth
od
.
Ke
y
w
ords
: Lit
h
ium
Ion Battery, Battery Managem
ent Syst
em, State of C
harg
e
, Extend
ed Kal
m
an F
ilt
er,
Energy Stora
g
e
System
1. Introduc
tion
Over the ye
ars, d
epletio
n of non-re
n
e
wa
bl
e en
ergy resource
s and the in
creme
n
t of
fossil fu
el p
r
i
c
e h
a
ve en
co
urag
ed th
e growin
g interests in rene
wa
b
l
e ene
rgy
sou
r
ce
s
espe
ciall
y
in tran
sp
ortat
i
on. Ele
c
tric
Vehicle
(EV)
is a
n
exam
pl
e of the
appli
c
ation
of ren
e
wa
ble e
n
e
r
gy
sou
r
ces in tra
n
sp
ortation. It is environm
e
n
tal fri
endly beca
u
se it neither con
s
ume
s
the petrol n
o
r
prod
uces the
gree
n hou
se
gaseou
s.
Lithium ion (Li-ion
)
battery has beco
m
e the main
strea
m
ene
rg
y storage el
ement in
electri
c
vehi
cle (EV). For instan
ce, lithium man
g
a
nate (LiM
n
2
O
2
) battery ha
s be
en u
s
ed
in
Nissa
n
Le
af
EV, Chevrol
e
t Volt and
Re
nault Flu
e
n
c
e
whe
r
e
a
s lithi
um ferro
pho
sph
a
te (LiFe
P
O
4
)
battery has b
een used in
BYD E6 [1].
Accu
rate
st
ate-of-ch
a
rg
e (SoC) e
s
timati
on is cru
c
ial to
indicate the remaining capacity of the battery.
The accurate i
n
formation of SoC will
eventually
maximiz
e
battery performanc
e
and en
s
u
re the battery
s
a
fe operation.
SoC is th
e in
dicatio
n
of re
maining
batte
ry
cap
a
city which i
s
exp
r
e
s
sed in p
e
rcentage.
For in
stan
ce,
100% refe
r to fully charge
d whe
r
e
a
s 0
%
refer to full
y disch
arged.
Gene
rally, SoC
is d
e
fined
as the ratio of
the re
mainin
g charge
of the b
a
ttery a
nd the total
cha
r
ge
while
the
battery is fully charged at th
e same
spe
c
i
f
ic con
d
ition [1].
Several meth
ods h
a
ve be
en pro
p
o
s
ed
in previo
us
literature for S
o
C es
timation [2]–[8].
Disch
a
rg
e te
st metho
d
[2] is on
e of the
accu
rate a
p
p
roa
c
h
e
s to
cal
c
ulate So
C. In this
met
hod,
battery is discha
rge
d
u
n
d
e
r
spe
c
ific te
mperat
ure
a
nd current. T
he So
C is id
entified throu
g
h
discha
rge p
r
o
c
e
ss. However, this metho
d
is only
suitable for laborat
o
ry
study an
d
not suitable t
o
be used for real time SoC
estimation in
electri
c
vehi
cl
e.
Coul
omb cou
n
ting [3] is a
nother
popul
ar ap
pro
a
ch for SoC e
s
tim
a
tion. In this method,
SoC i
s
calcu
l
ated by a
ccumulating
ch
arge/di
scha
rg
e cu
rrent of
battery. Ap
plicatio
n of t
h
is
method en
abl
es re
al-time value of SoC to be ca
l
c
ulat
ed without th
e need of expen
sive devices.
Ho
wever, thi
s
metho
d
is
highly dep
en
dent on t
he
measured ba
ttery current
whi
c
h is di
stu
r
bed
measurement
noise. Th
e
measurement
drift would
eventually influen
ce the a
c
cura
cy of this
method. Mo
reover, initial
SoC of batte
ry is vita
l for this method
while the i
n
itial value of SoC
might not ready available in prac
tic
a
l
s
i
tuation [1],[2].
Neural network model and fuzzy logi
c [4],[5
] are also been
appli
ed for SoC estimation.
In these m
e
th
ods, a
n
inp
u
t to output rel
a
tionshi
p
is
est
ablished by u
s
ing
neu
ral n
e
twork o
r
fu
zzy
logic.
Neithe
r hypothe
sis n
o
r p
r
io
r kno
w
l
edge
of
batte
ry is
req
u
ire
d
to be
con
s
id
ered.
Ho
wev
e
r,
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 2, June 20
15 : 401 – 41
2
402
a gre
a
t traini
ng data a
r
e required to tra
i
n the neu
ral
netwo
rk
and f
u
zzy logic. It also n
eed
s a
lot
of comp
utatio
n and p
o
we
rf
ul pro
c
e
s
sing
chip
s such
a
s
DSP. Moreover, the e
s
timation error in
the training d
a
ta may influence the perf
o
rma
n
ce of these meth
od
s [1].
Kalman-filte
r
[6]–[8] is also
been used for So
C e
s
tim
a
tion. Kalma
n
-filter is a re
cursive
state e
s
timat
o
r
whi
c
h
esti
mates th
e
state by u
s
in
g
the info
rmati
on of th
e p
r
e
v
ious
estimat
ed
state and the
current me
a
s
ureme
n
t. Moreove
r
, an
o
p
timal state e
s
timation
can
be achi
eved
by
Kalman-filte
r
becau
se it ha
s con
s
ide
r
ed
the pr
ocess a
nd me
asure
m
ent noi
se
s i
n
the alg
o
rith
m.
Extended Kalman-filter
(EKF) is the no
nlinea
r vers
i
o
n of Kalman-filter and it is suitable to b
e
applie
d in th
e nonli
nea
r
system, su
ch
as b
a
ttery.
It is a ve
ry reli
able m
e
thod
becau
se it is
not
sen
s
itive to t
he n
o
ises an
d it do
es not
need
the
pre
c
ise valu
e of
initial SoC.
Besid
e
s from
EKF,
there
are
sev
e
ral
version
o
f
Kalman
-filter h
a
ve b
een
applie
d for S
o
C E
s
timatio
n
, su
ch
a
s
si
gma-
point Kalman
-filter [9], adaptive extende
d kalma
n
filter [10], and adaptive sig
m
a-p
o
int kal
m
an
filter [11]. Ho
wever,
comp
ared
to th
ese
algo
ri
thms
,
EKF has
a lower
c
o
mplexity. Thus
, a lower
co
st shall b
e
expecte
d for EKF SoC esti
mation syste
m
.
The a
c
cura
cy of EKF So
C estim
a
tion
is highly de
p
ende
nt on the accu
racy o
f
battery
model. Thu
s
, Thevenin e
quivalent ci
rcuit model
provides go
od
predi
ction o
n
the runtime I-V
cha
r
a
c
t
e
ri
st
ic
of
bat
t
e
ry
.
P
r
ev
iou
s
st
u
d
ies
sh
o
w
th
at the accuracy of the p
r
edi
cted b
a
ttery
respon
se
is
enha
nced by
applying
hig
her
order of
Thevenin
eq
uivalent ci
rcu
i
t model [1
2]. It is
also
prove
n
that the third
orde
r T
heve
n
in eq
ui
valen
t
circuit mod
e
l is reliabl
e
to captu
r
e t
he
nonlin
ear dyn
a
mic cha
r
a
c
teristi
cs of Li
-i
on battery [12
],[13].
EKF SoC e
s
timation, whi
c
h i
s
ba
sed
on t
he first orde
r [14] a
nd the seco
nd order
Thevenin e
q
u
ivalent circuit model [15] are pre
s
e
n
ted in previ
ous literature
s
. Ho
wever,
at
pre
s
ent, the
r
e is
no
stud
y applying t
he EKF So
C estimatio
n
on the thi
r
d
ord
e
r T
hev
enin
equivalent
circuit mod
e
l. Con
s
id
erin
g the fact t
hat the third o
r
d
e
r
Theveni
n e
quivalent ci
rcuit
model h
a
s
b
e
tter a
ccu
ra
cy, in this pa
per, an EKF
SoC e
s
tima
tion for lithiu
m
-ion
battery
is
carrie
d out b
a
se
d on the
third ord
e
r
Thevenin
e
q
u
ivalent circu
i
t model. First, a third order
Thevenin
eq
u
i
valent ci
rcuit
model
is dev
elope
d b
a
sed
on
the
experimental
data
of battery te
st
s.
Then, the EKF algorithm
is applie
d on the stat
e
-
spa
c
e e
quati
ons of third
orde
r Theve
n
in
equivalent
circuit mod
e
l to
estimate the
SoC. The m
e
thod is th
en
validated by
comp
arin
g real
SoC value to the estimated
SoC value.
2. Batter
y
M
odeling
The third ord
e
r Th
evenin
equivalent circuit
m
odel i
s
illustrate
d in
Figure 1. Th
e
battery
model i
s
fo
rm
ed by a
n
op
e
n
ci
rcuit voltage (OCV
) sou
r
ce,
a seri
es
resi
stan
ce a
n
d
thre
e resi
st
or-
cap
a
cito
r (RC) pa
rallel
net
works in th
e
seri
es. T
he v
a
lue of
OCV i
s
no
nline
a
r
a
nd de
pen
den
t on
the SoC. Th
e
seri
es
re
si
stance (
R
S
) re
pre
s
ent
s the
internal
re
si
stance of batte
ry whe
r
e
a
s t
he
RC
parallel n
e
tworks (
R
1
,
R
2
,
R
3
,
C
1
,
C
2
,
C
3
) sim
u
late
the tran
sient
respon
se
of battery voltag
e.
Based o
n
Fig
u
re 1,
the followin
g
equati
ons
can b
e
o
b
tained:
S
L
RC
RC
RC
t
R
I
V
V
V
OCV
V
3
2
1
(1)
1
1
1
1
1
R
C
V
C
I
dt
dV
RC
L
RC
(2)
2
2
2
2
2
R
C
V
C
I
dt
dV
RC
L
RC
(3)
3
3
3
3
3
R
C
V
C
I
dt
dV
RC
L
RC
(4)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Battery State of Charge Est
i
m
a
tion with Extende
d Kal
m
an Filter usi
ng Third ... (Low Wen Yao)
403
Figure 1. Third orde
r Thev
enin eq
uivale
nt circuit mod
e
l
In ord
e
r to a
p
p
ly the b
a
ttery model i
n
E
K
F SoC
esti
mation al
go
rithm, the b
a
ttery mod
e
l
is tran
sfo
r
me
d as
state-sp
ace
equatio
n
s
. In th
is a
s
p
e
ct, the SoC
and the volta
ge drop a
c
ro
ss
RC
parallel n
e
tworks a
r
e
cho
s
e
n
a
s
th
e state va
ria
b
le. SoC i
s
expre
s
sed a
s
Eq. (5),
wh
ere
SoC
0
is the i
n
itial SoC,
C
N
is the
u
s
abl
e
ca
pacity
(in t
he unit
of Ah), and
I
L
is the battery
c
u
rrent
whi
c
h ha
s th
e neg
ative value du
rin
g
charg
e
an
d
p
o
sitive value
whe
n
disch
a
rge. The ove
r
all
state eq
uatio
n for thi
r
d o
r
d
e
r Th
evenin
equivalent circuit
mo
del
ca
n be fo
rmul
ated a
s
d
enote
d
in
Eq. (6) and E
q
. (7).
t
N
L
dt
C
t
I
SoC
SoC
0
0
)
(
3600
100
(5)
T
RC
RC
RC
V
V
V
SoC
x
3
2
1
(6)
L
N
I
C
C
C
C
x
R
C
R
C
R
C
x
3
2
1
3
3
2
2
1
1
1
1
1
3600
100
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
(7)
Then, the
sta
t
e-sp
ace eq
u
a
tion at time
step
k
ca
n
be e
x
p
r
ess
e
d
as
Eq
. (
8
)
an
d Eq
. (
9
)
by includi
ng the time interv
al
∆
t:
1
3
2
1
1
3
3
2
2
1
1
3600
100
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
k
L
N
k
k
I
C
t
C
t
C
t
C
t
x
R
C
t
R
C
t
R
C
t
x
(8)
k
k
L
S
k
t
I
R
x
SoC
OCV
V
1
1
1
(9)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 2, June 20
15 : 401 – 41
2
404
3. Param
e
ter
Extrac
tion o
f
Ba
tte
ry
Model
The pa
rame
terizatio
n
of third ord
e
r Thevenin
equivalent ci
rcuit
model
is fairly
s
t
raightforward. In this
aspec
t, eac
h
paramete
rs of
battery mod
e
l ca
n be i
d
entified from
the
experim
ental
data of battery test. In this pap
er, pa
ramete
rizatio
n
pro
c
e
s
ses
are a
rra
nge
d
as
follow:
(i) Battery
tes
t
s
.
(ii) Usable
cap
a
city.
(iii) OCV-S
o
C
rel
a
tionship.
(iv
)
Serie
s
re
sista
n
ce (
R
S
) a
nd
RC p
a
rall
el n
e
tworks pa
ra
meters (
R
1
, R
2
, R
3
, C
1
, C
2
, C
3
).
3.1. Batter
y
Tests
The
paramet
erization
of b
a
ttery sta
r
ts
with
b
a
ttery t
e
sts.
The
ex
perim
ental
set up
for
battery test i
s
sho
w
n in
F
i
gure
2. In th
is pa
per, 3.2
V
, 18Ah lithium ferro p
h
o
s
ph
ate battery is
applie
d. An el
ectro
n
ic loa
d
, IT8514
C,
with the
rating
o
f
120V, 24
0A, 1200
W i
s
used to di
scha
rge
the battery. A
data a
c
q
u
isit
ion devi
c
e, DAQ NI92
19,
from Nation
al Instrum
ent
is use
d
to colle
ct
and sto
r
e the
measure
m
en
t data into co
mputer. NI
9
2
19 is
capa
ble
of processin
g
more th
an
100
sampl
e
s pe
r
se
con
d
with t
he a
c
curacy
of up to
5
de
cimal
pla
c
e
s
. LabVIEW is
use
d
to
sto
r
e
the
battery data
acq
u
ire
d
fro
m
NI921
9
DAQ
.
In this p
ape
r, the sa
mplin
g rate
is
set t
o
6
sampl
e
s
per
minute. Hi
gh
er
sam
p
ling
rate is
not p
r
eferabl
e in
t
h
is expe
rime
nt be
cau
s
e it
requires a la
rger
memory space.
Figure 2. Experime
n
tal set
up for battery
test
In this pape
r, two battery tests a
r
e pe
rforme
d for ba
ttery modelin
g purp
o
se. The first
test is p
u
lse
discha
rge te
st. The test is made in
ord
e
r to ide
n
tify the tran
sient
respon
se
an
d
dynamic b
e
h
a
vior of batte
ry. Pulse discharg
e
test
co
nsi
s
ts of a se
quen
ce of co
nstant di
scha
rge
curre
n
t and
rest pe
riod
a
s
sho
w
n i
n
Fig
u
re
3(a
)
. The
pulse di
scharged te
st is
st
arted
with a f
u
lly
cha
r
ge
d batt
e
ry. The batt
e
ry is
then
di
scharged
wit
h
a sp
ecifi
c
c
onsta
nt cu
rre
nt to redu
ce
10%
of the
nomin
al capa
city. Afterwa
r
ds,
a rest
pe
riod
is appli
ed f
o
r th
e b
a
ttery to a
c
hieve
its
equilibrium
state before t
he next discharge. The discharge-re
st
cycle is repeated until battery
voltage dro
p
s to 2 V. The
curre
n
t of 6A (0.333
C), 9A
(0.5C) and 1
8
A (1C) are applie
d in pul
se
discha
rge te
st in order to find out the dynamic
b
ehavi
o
r of battery in different C-rate.
The
se
con
d
test is the
random
test i
n
which
the
battery is ra
ndomly
cha
r
ged a
n
d
discha
rge
d
o
v
er a
certai
n
perio
d of tim
e
as ill
ust
r
ate
d
in Fig
u
re
3(b). Thi
s
te
st is ma
de in
order
to evaluate the accu
ra
cy of t
he devel
oped thi
r
d o
r
der b
a
ttery model. In thi
s
test, battery is
loade
d with variou
s
curre
n
ts, whi
c
h in
clu
de 3A (0.1
67
C), 6A (0.3
33
C), 9A (0.5
C), 12A (0.667
C),
18A (1
C) an
d
36A (2C) of curre
n
t. More
over, som
e
of the cha
r
ging
con
d
ition
s
are also in
clu
d
e
d
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Battery State of Charge Est
i
m
a
tion with Extende
d Kal
m
an Filter usi
ng Third ... (Low Wen Yao)
405
Figure 3. Voltage an
d cu
rrent profile
s for (a
)
pulse discha
rge te
st, and (b
) ran
d
o
m
test
3.2. Usable
Capa
cit
y
Usable
ca
pa
city of battery varied
accordin
g to
ch
arge/di
scha
rg
e current. It is n
o
t
necessa
rily e
qual to
the
n
o
minal
ca
pa
city. In th
is a
s
pect, the
u
s
a
b
le
cap
a
city i
s
lo
we
r fo
r hi
gh
cha
r
ge/di
sch
a
rge
current.
Based
on th
e expe
rime
ntal
result from battery
tes
t, the
relationship
betwe
en u
s
a
b
le ca
pa
city and current i
s
illustrate
d
in Figure 4(a). T
he usable
ca
pacity of battery
is expre
s
sed
as Eq. (10
)
.
L
L
N
I
.
.
I
.
.
C
0017
0
exp
44
13
4932
0
exp
559
4
(10)
3.3. OCV-So
C Rela
tionsh
i
p
Open
ci
rcuit
voltage i
s
def
ined
as th
e t
e
rmin
al volta
ge of
battery
at ch
arg
e
e
q
u
ilibriu
m
con
d
ition. Th
e value of OCV is dire
ctly depend
ent
o
n
the value of SoC. In this paper, O
C
V is
identified fro
m
the p
u
lse
d
i
scharge
te
st whe
n
the battery
ha
s re
st.
Several re
st
times are
ap
pli
ed
in pulse disch
a
rge te
st in determini
ng O
C
V (i.e. 30 minutes for 0.3
3
C, 60 minut
es for 0.5
C
, an
d
45 min
u
tes for 1
C
). The
relation
shi
p
b
e
twee
n O
C
V
and S
o
C i
s
illustrate
d in
Figure 4
(
b).
As
sho
w
n
in th
e
figure, lithi
um
ferro
pho
sph
a
te batte
ry h
a
s
a flat
OCV value
withi
n
the
SoC ra
nge
of 40-90 %.
By using
cu
rve fitting, a fifth-order poly
nom
ial e
quati
on can b
e
formulated to
re
pre
s
ent
the OCV
-
So
C rel
a
tion
shi
p
as d
enote
d
in Eq. (11).
The pa
ram
e
ters i
n
Eq. (1
1) are tabul
ated in
Table 1.
6
5
2
4
3
3
4
2
5
1
a
SoC
a
SoC
a
SoC
a
SoC
a
SoC
a
OCV
(11
)
Table 1. Para
meters of Eq. (11)
Parameter Value
a
1
4.513
х
10
-
10
a
2
–1.295
х
10
-
7
a
3
1.505
х
10
-
5
a
4
–8.927
х
10
-
4
a
5
2.764
х
10
-
2
a
6
2.918
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 2, June 20
15 : 401 – 41
2
406
Figure 4. Ral
a
tionship bet
wee
n
(a
) usa
b
le ca
pa
city and cu
rrent, and (b
) ope
n ci
rcuit voltage
and
state-of
-charge
3.4. Series Resista
n
ce an
d RC Paralle
l Net
w
o
r
ks P
a
rameters
The tra
n
sie
n
t voltage re
sp
onse for di
scharg
e
an
d re
st are illu
stra
ted in Figu
re
5. The
seri
es re
si
sta
n
ce
and
RC parallel net
works pa
ra
m
e
ters can b
e
identified from the tra
n
sient
voltage
re
spo
n
se
du
ring
th
e rest
peri
o
d
[16]. The
volta
ge a
c
ro
ss RC pa
rallel
net
works for load
e
d
and re
st con
d
itions
a
r
e de
noted as
Eq. (12
)
,
wher
e
i
= 1,
2, 3,
t
0
is the
begi
nnin
g
time,
t
d
is
th
e
discha
rge e
n
d
ing time and
t
r
is the rest endin
g
time of the period.
0
exp
0
exp
1
0
0
L
r
d
i
i
d
d
RC
RC
L
d
i
i
i
L
RC
I
,
t
t
t
,
C
R
t
t
t
V
V
I
,
t
t
t
,
C
R
t
t
R
I
V
i
i
i
(12
)
Figure 5. Tra
n
sie
n
t voltage respon
se fo
r pulse disch
a
rge te
st
By applying
MATLAB cu
rve fitting tool, tran
sient volt
age
re
spon
se
for rest p
e
rio
d
ca
n be
rep
r
e
s
ente
d
by Eq. (13).
d
d
d
S
t
t
t
c
b
t
t
c
b
t
t
c
b
b
OCV
V
3
3
2
2
1
1
exp
exp
exp
(13)
Afterwa
r
ds, t
he p
a
ra
mete
rs fo
r thi
r
d
o
r
de
r Th
eveni
n
eq
uivalent circuit
m
odel
ca
n b
e
identified a
s
denote
d
in Eqs. (14
)
-(1
6),
whe
r
e
i
= 1, 2
,
and 3.
0
exp
-
1
t
t
c
I
b
R
d
i
L
i
i
(14
)
i
i
i
R
c
C
1
(15
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Battery State of Charge Est
i
m
a
tion with Extende
d Kal
m
an Filter usi
ng Third ... (Low Wen Yao)
407
L
S
S
I
b
R
(16
)
Based o
n
the results fro
m
curve fitting met
hod, the seri
es resi
stan
ce and
RC pa
rall
el
netwo
rks
pa
rameters ca
n be
id
entified as
ta
bulate
d
i
n
Ta
ble
2. Th
e validatio
n o
f
battery m
o
d
e
l i
s
made by com
parin
g expe
ri
mental an
d si
mulation resu
lts of ran
dom
test as
sho
w
n in Figu
re 6.
It
can b
e
seen
that a signifi
cant diverg
e e
x
ist when
So
C is b
e
low
2
0
%. Howeve
r, sin
c
e ele
c
tric
vehicle i
s
usually operate
d
within 30
% to 100
% SoC [17], the accu
ra
cy of model is
still
con
s
id
ere
d
a
c
ceptabl
e. T
he comp
arati
v
e analysi
s
shows that th
e ro
ot-m
e
an-squ
a
re
(RMS
) of
modelin
g e
r
rors is 32.26
5
mV. Base
d
on the
goo
d match betwe
en
expe
rime
nt
and simul
a
tion
results, the d
e
velope
d mo
del is validate
d
.
Table 2. Para
meters for Th
ird order T
h
e
v
enin Equival
ent Circuit M
odel
Parameters
Val
ue
R
1
0.006
Ω
R
2
0.003
Ω
R
3
0.002
Ω
C
1
2127.949 F
C
2
37348.281 F
C
3
286996.625 F
R
S
0.003
Ω
Figure 6. Experime
n
tal and
simu
lation
re
sults of rando
m test
4. Extende
d Kalm
an-
F
ilte
r
for Sta
t
e-
of
-Ch
a
rge Es
tim
a
tion
State-sp
ace
model fo
r
bat
tery a
s
exp
r
e
s
sed i
n
Eq
s.
(8) an
d (9) are utilized to
e
s
timate
the SoC. T
h
e typical
stat
e-spa
c
e
rep
r
ese
n
tation
fo
r a n
onlin
ear system
is
e
x
presse
d a
s
Eq.
(17
)
, where
k
is th
e time
in
dex,
x
k
i
s
th
e
nonlin
ear stat
e,
u
k
i
s
the
control
input, y
k
is t
h
e
sy
st
e
m
output,
w
k
is
a di
screte
time p
r
o
c
ess wh
ite noi
se
with
cova
rian
ce
matrix
Q
, an
d
v
k
is a di
screte
time measure
m
ent white n
o
ise
with cov
a
rian
ce m
a
tri
x
R
.
R
,
~
v
,
Q
,
~
w
v
u
,
x
g
y
w
u
,
x
f
x
k
k
k
k
k
k
k
k
k
k
0
0
1
(17
)
In this application, nonlin
ear state i
s
defi
ned a
s
Eq. (6), cont
ro
l input is defined a
s
battery
curre
n
t, and
sy
ste
m
outp
u
t is d
e
fined
as
batt
e
ry terminal
voltage. By a
p
p
lying
Ja
cobi
an
matrix of
partial derivatives
of func
tion
f
a
nd
g
with res
p
ec
t t
o
x
k-1
and
u
k-1
, state-spa
c
e
equatio
ns a
r
e
transfo
rmed
as Eq. (18
)
.
k
k
k
k
k
k
k
k
k
k
k
k
v
u
D
x
C
y
w
u
B
x
A
x
1
(18
)
whe
r
e,
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 2, June 20
15 : 401 – 41
2
408
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
u
u
,
x
g
D
,
x
u
,
x
g
C
,
u
u
,
x
f
B
,
x
u
,
x
f
A
(19
)
As de
noted
i
n
Eq. (8)
and
Eq. (9
), the
matrix
A
k
,
B
k
,
C
k
,
D
k
are e
x
presse
d in
Eqs. (20)-
(23) res
p
ec
tively.
3
3
2
2
1
1
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
R
C
t
R
C
t
R
C
t
A
k
(20)
T
N
k
C
t
C
t
C
t
C
t
B
3
2
1
3600
100
(21)
1
1
1
SoC
OCV
C
k
(22
)
s
k
R
D
(23
)
The initiali
zati
on of EKF
al
gorithm
is given by Eq.
(24
)
, wh
ere
P
0
+
is the
predi
ction e
r
ro
r
c
o
varianc
e
matrix.
T
x
ˆ
x
x
ˆ
x
E
P
,
x
E
x
ˆ
,
k
0
0
0
0
0
0
0
0
(24
)
The comp
uta
t
ion of EKF algorithm
con
s
ists of five st
eps. Th
e vari
able which computed
before
sy
ste
m
mea
s
u
r
em
ent (
pr
i
o
r
i
) i
s
denote
d
by
sup
e
rscript
“–”
wh
erea
s
variable
whi
c
h
comp
uted after sy
stem me
asu
r
em
ent (
po
s
t
e
r
io
r
i
) is d
enoted by su
perscript “+”.
(i) State esti
m
a
tion ti
m
e
update:
1
1
1
1
k
k
k
k
k
u
B
x
ˆ
A
x
ˆ
(25
)
whe
r
e
k
x
ˆ
is p
r
i
o
ri stat
e e
s
timate at ste
p
k
given th
e p
r
ocess p
r
io
r to step
k
, wh
erea
s
1
k
x
ˆ
is the po
steri
o
ri state e
s
ti
mate at step
k–1
.
(ii) Error
co
va
rian
ce tim
e
update:
Q
A
P
A
P
k
k
k
k
T
1
1
1
(26
)
whe
r
e
k
P
is the
priori
error
covarian
ce
at step
k
wh
ere
a
s is
1
k
P
the po
sterio
ri e
rro
r
covari
an
ce at
step
k–
1
.
(iii) Calculatio
n of Kalm
an
gain:
1
R
C
P
C
C
P
K
T
k
k
k
T
k
k
k
(27)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
1693-6
930
Battery State of Charge Est
i
m
a
tion with Extende
d Kal
m
an Filter usi
ng Third ... (Low Wen Yao)
409
(iv) State esti
m
a
te
m
easurem
ent update
:
k
k
k
k
k
k
k
k
u
D
x
ˆ
C
y
K
x
ˆ
x
ˆ
(28)
In this sta
ge,
poste
rio
r
i stat
e is e
s
timate
d.
y
k
is the
m
easure
m
ent o
u
tput. In this
ca
se,
y
k
is the real
-tim
e terminal vol
t
age of battery.
(v) Error cova
rian
ce m
easu
r
em
ent updat
e:
k
k
k
k
P
C
K
I
P
(29
)
In this
stage
, poste
riori
e
rro
r
covari
an
ce i
s
e
s
timat
ed. The
co
m
puting
step i
s
then
repe
ated ag
ai
n from (i) to (v).
5. Result and Validation
In this sectio
n, the state-spa
c
e
equati
ons fo
r third
orde
r Th
eve
n
in
equival
e
n
t
circuit
model
are e
m
ployed fo
r
SoC e
s
timati
on. The
vali
dation
of EKF SoC e
s
timation i
s
d
one
by
comp
ari
ng
experim
ental S
o
C
and
the
e
s
timated
SoC. In this a
s
pe
ct, the
experi
m
ental So
C
i
s
measured
by
usin
g di
scha
rge te
st meth
o
d
. Eq. (5)
is us
ed in
disc
harge tes
t
method
with the pre
-
kno
w
n valu
e of SoC.
5.1. Selectio
n of Initial Condition and
Noise Cov
a
riances
The initial st
ate (
x
0
), e
rro
r cov
a
rian
ce
(
P
0
), pro
c
e
s
s noise
covari
ance (
Q
) an
d sen
s
o
r
noise c
o
var
i
anc
e
(
R
) are
chosen
as den
oted in
Eqs.
(30)-(3
3). T
h
e
initial So
C fo
r EKF al
gorit
hm
is
s
e
t to 60%.
T
x
0
0
0
60
0
(30
)
0001
0
0
0
0
0
0001
0
0
0
0
0
0001
0
0
0
0
0
2500
0
.
.
.
P
(31)
0001
0
0
0
0
0
0001
0
0
0
0
0
0001
0
0
0
0
0
0001
0
.
.
.
.
Q
(32
)
004
0
.
R
(33
)
5.2. Validation of EKF Sta
t
e-of-Ch
a
rge
Estimation
Ran
dom te
st
is u
s
e
d
to
evaluate th
e pe
rform
a
n
c
e
of EKF SoC e
s
timati
on. The
experim
ental
SoC for random test
is
compared to estimated SoC
illustrated in
Figure 7(a).
As
illustrate
d in Figure 7(a
)
, althoug
h the initial SoC
for EKF algorithm has deviat
e
significantly to
the real SoC, EKF is
still able to estimat
e
the ac
curat
e
value
of SoC withi
n
a
short time. In this
asp
e
ct, the root-me
an-sq
uare
(RMS
)
SoC estim
a
tion error i
s
3.
5934 %. Moreover, the m
odel
output from E
K
F estimatio
n
is also well matche
d
wit
h
the mea
s
ured value of b
a
ttery voltage
as
s
h
ow
n
in
F
i
gu
r
e
8(
a
)
.
The EKF SoC estimatio
n
techni
que is f
u
rthe
r validated with pul
se
discharge te
sts. The
experim
ental
SoC an
d EKF estimate
d SoC for p
u
lse
discha
rge te
sts of 0.33
C,
0.5C an
d 1
C
are
sho
w
n i
n
Fi
gure
7(b), Fi
gure
7(c),
a
nd Figu
re
7(d) respe
c
tively. The RMS
error fo
r S
o
C
estimation i
s
tabulated i
n
Table 3. T
h
e
com
parative analysi
s
sho
w
s th
e go
od
match
betwe
en
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 16
93-6
930
TELKOM
NIKA
Vol. 13, No. 2, June 20
15 : 401 – 41
2
410
experim
ent a
nd estimat
e
d
SoC for pul
se di
schar
ge
test with RMS erro
r of less than 2
%
.
More
over, th
e model o
u
tp
uts for EKF e
s
timation a
r
e
also
well mat
c
he
d with the
measure
d
value
of battery vol
t
age fo
r p
u
lse di
scharge
tests of
0.3
3
C
, 0.5
C
a
nd
1C
as sho
w
n
in Fi
gure 8
(
b),
Figure 8(c), a
nd Figu
re 8(d
)
re
spe
c
tively.
Table 3. RMS
SoC Estimati
on error fo
r Pulse
Disch
a
rg
e Test
s
Curre
nt (C
)
RMS erro
r for S
o
C Es
ti
mati
on (%)
0.33 1.417
0.5 1.881
1 1.611
Based
on t
he go
od m
a
tch b
e
twe
e
n
expe
rimen
t
and EKF
estimate
d
SoC, the
perfo
rman
ce
of the develo
ped SoC e
s
ti
mation metho
d
is validated.
Figure 7. Co
mpari
s
o
n
bet
wee
n
experi
m
ental SoC a
nd EKF estim
a
ted SoC for
(a)
rand
om te
st,
(b) 0.3
3
C p
u
l
s
e di
scharge
test, (c) 0.5
C
pulse
disch
a
rge test, and (d) 1C p
u
lse d
i
scharge test
Figure 8. Co
mpari
s
o
n
bet
wee
n
experi
m
ental and
si
mulation voltage
s for (a
) random te
st, (b)
0.33C p
u
lse discha
rge te
st, (c) 0.5C p
u
l
s
e di
scharge
test, and (d
) 1C pul
se di
scharg
e
test
Evaluation Warning : The document was created with Spire.PDF for Python.