TELKOMNIKA ISSN: 1693-6930
Battery State of Charge Estimation with Extended Kalman Filter using Third ... Low Wen Yao 407
L S
S
I b
R
16 Based on the results from curve fitting method, the series resistance and RC parallel
networks parameters can be identified as tabulated in Table 2. The validation of battery model is made by comparing experimental and simulation results of random test as shown in Figure 6. It
can be seen that a significant diverge exist when SoC is below 20 . However, since electric vehicle is usually operated within 30 to 100 SoC [17], the accuracy of model is still
considered acceptable. The comparative analysis shows that the root-mean-square RMS of modeling errors is 32.265 mV. Based on the good match between experiment and simulation
results, the developed model is validated.
Table 2. Parameters for Third order Thevenin Equivalent Circuit Model
Parameters Value 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. Experimental and simulation results of random test
4. Extended Kalman-Filter for State-of-Charge Estimation
State-space model for battery as expressed in Eqs. 8 and 9 are utilized to estimate the SoC. The typical state-space representation for a nonlinear system is expressed as Eq.
17, where k is the time index, x
k
is the nonlinear state, u
k
is the control input, y
k
is the system output,
w
k
is a discrete time process white noise with covariance matrix Q, and v
k
is a discrete time measurement white noise with covariance matrix
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
1
17
In this application, nonlinear state is defined as Eq. 6, control input is defined as battery current, and system output is defined as battery terminal voltage. By applying Jacobian
matrix of partial derivatives of function f and g with respect to x
k-1
and u
k-1
, state-space equations are transformed 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 where,
ISSN: 1693-6930
TELKOMNIKA Vol. 13, No. 2, June 2015 : 401 – 412
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 denoted in Eq. 8 and Eq. 9, the matrix A
k
, B
k
, C
k
, D
k
are expressed in Eqs. 20- 23 respectively.
3 3
2 2
1 1
1 1
1 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 initialization of EKF algorithm is given by Eq. 24, where
P
+
is the prediction error covariance matrix.
T
xˆ x
xˆ x
E P
, x
E xˆ
, k
24
The computation of EKF algorithm consists of five steps. The variable which computed before system measurement
priori is denoted by superscript “–” whereas variable which computed after system measurement
posteriori is denoted by superscript “+”. i State estimation time update:
1 1
1 1
k k
k k
k
u B
xˆ A
xˆ
25 where
k
x ˆ
is priori state estimate at step k given the process prior to step k, whereas
1
k
x ˆ
is the posteriori state estimate at step k–1.
ii Error covariance time update:
Q A
P A
P
k k
k k
T
1 1
1
26 where
k
P
is the priori error covariance at step k whereas is
1
k
P
the posteriori error covariance at step
k–1. iii Calculation of Kalman gain:
1
R C
P C
C P
K
T k
k k
T k
k k
27
TELKOMNIKA ISSN: 1693-6930
Battery State of Charge Estimation with Extended Kalman Filter using Third ... Low Wen Yao 409
iv State estimate measurement update:
k k
k k
k k
k k
u D
xˆ C
y K
xˆ xˆ
28 In this stage, posteriori state is estimated.
y
k
is the measurement output. In this case, y
k
is the real-time terminal voltage of battery. v Error covariance measurement update:
k k
k k
P C
K I
P
29 In this stage, posteriori error covariance is estimated. The computing step is then
repeated again from i to v.
5. Result and Validation
