soc estim ekf
TRANSCRIPT
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Energies2012, 5, 1-x manuscripts; doi:10.3390/en50x000x1
2
energies3ISSN 1996-10734
www.mdpi.com/journal/energies5
Article6
Estimation of State of Charge of Lithium-Ion Batteries used in7
HEV using Robust Extended Kalman Filtering8
Caiping Zhang1,
*, Jiuchun Jiang1, Weige Zhang
1, S.M. Sharkh
29
1School of Electrical Engineering, Beijing Jiaotong University, Beijing, 100044, China102
School of Engineering Sciences, University of Southampton, Highfield, Southampton, SO17 1BJ,11UK12
E-Mails: [email protected]; [email protected]; [email protected]
* Author to whom correspondence should be addressed; Tel.: +86-10-5168-3907; Fax: +86-10-5168-14
3907, Email: [email protected]
Received: / Accepted: / Published:16
17
Abstract: Robust extended Kalman filter (EKF) is proposed as a method for estimation of18
the state of charge (SOC) of lithium-ion batteries used in hybrid electric vehicles (HEV). An19
equivalent circuit model of the battery, including its electromotive force (EMF) hysteresis20
characteristic and polarization characteristics is used. The effect of the robust EKF gain21
coefficient on SOC estimation is analyzed, and an optimized gain coefficient is determined22
to restrain battery terminal voltage from fluctuating. Experimental and simulation results are23
presented. SOC estimates using the standard EKF are compared with the proposed robust24
EKF algorithm to demonstrate the accuracy and precision of the latter for SOC estimation.25
Keywords: lithium-ion batteries; SOC estimation; robust estimation; EKF; HEV26
27
1. Introduction28
Lithium-Ion batteries have become a promising alternative power source in electric vehicles (EV)29
and power assist units used in hybrid electric vehicles (HEV). Their advantages include high nominal30
cell voltage, high energy density, long life and not having a memory effect. As one of the power31
supplies on a vehicle, the performance of lithium-ion batteries will have direct impact on the driving32
performance of the vehicle. It is necessary for the battery to be effectively managed to improve its33
OPEN ACCESS
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performance and extend its lifetime. The main components of a battery management system include34
state of charge (SOC) estimation, cell balancing, thermal management, and safety control. The SOC,35
which describes the percentage of the battery available capacity to its rated capacity, is a key parameter36
in the battery management system. Accurate estimation of the SOC of the battery is also important for37
accurate simulation and optimization, and real time energy management of HEV and EV.38
Techniques for estimation of the SOC of a battery may be categorized as direct computational39
methods or intelligent computational methods. Direct computational methods calculate the SOC of the40
battery directly based on its relationship with measurable battery parameters, for example, using41
ampere hour counting, open-circuit voltage, or internal impedance [1-3]. These methods are42
extensively used in HEV and EV applications since they are easy to implement. However, they suffer43
from relatively poor accuracy due to accumulative errors, especially when ampere hour counting is44
used.45
Intelligent computational methods include those using artificial neural networks and extended46
Kalman filtering. The artificial neural networks approach has the advantage of adaptive learning, and47
can cope with the batterys nonlinear characteristics during charging and discharging. It has been48
investigated and used by many researchers [4, 5]. However, the algorithm requires a large amount of49
data for training, and the accuracy of these models is affected significantly by the training data and50
training method.51
In the extended Kalman filtering (EKF) approach the battery is regarded as a dynamic system, and52
the SOC is considered as an internal state of the dynamic system. Optimal state estimates can be53
obtained by adjusting the filter gain [6]. Using EKF to estimate the SOC of batteries has been the54
subject of extensive study in recent years due to its high accuracy and suitability for real time55implementation [7-9]. For an accurate estimate of the SOC, the EKF requires accurate system56
modeling as well as knowledge of the statistical properties of the system noise. However, in practice,57
the dynamic system model and the covariance matrices cannot be precisely determined especially in58
the environment of HEV including a variety of interference noises, e.g. Schottky noise, thermal noise59
and space electromagnetic noise, which may cause random modeling errors thus altering the statistical60
properties of the system noise. Using regular EKF may result in accumulative errors in the states61
estimates under the above conditions, and may even cause filter divergence.62
In this paper we propose a robust EKF in which the state estimation is decoupled from the bias state63
estimation. The state estimation error is gradually reduced to a minimum under a certain criterion even64
if the system model contains indeterministic information.65
This paper is organized as follows. The next section discusses the modeling of the lithium-ion66
battery used in HEV as the foundation for SOC estimation. EMF hysteresis is included in the battery67
model to improve SOC estimation. Section 3 introduces the robust state estimation EKF method. SOC68
estimation based on the proposed battery model is performed. The effects of the bias vector on state69
estimation and the acquisition of the bias constant are subsequently addressed. And then the70
optimization of Kalman gain is proposed. The results of simulation and tests are discussed in section 4.71
72
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2. Modeling of the lithium-ion battery73
74
Modeling of the battery aims to find the relationship between currents and voltages measured at the75
terminals of the battery. Besides concentration and activation polarization effects lithium ion batteries76exhibit an equilibrium potential hysteresis phenomenon. The hysteresis characteristics of the lithium-77
ion battery have a notable effect on SOC estimation accuracy, and it is therefore important that the78
model should incorporate these characteristics.79
2.1. Hysteresis characteristics80
The cell equilibrium potential depends on its charge and discharge history, and in some batteries81
exhibits hysteresis. There are many publications on the hysteresis characteristics for nickel-hydrogen82
batteries; a detailed study is reported in [10, 11]. A preliminary study of lithium-ion battery hysteresis83
characteristics is reported in [12-15].84A series of experiments were conducted in order to study the open circuit voltage characteristics of85
the battery and the influence of the current on its hysteresis during charging and discharging. The86
lithium-ion battery studied in this paper is composed of 16 cells in series. Figure 1 (a) shows the open-87
circuit cell voltage versus SOC under the same charge/discharge current of 30 A. Figure 1 (b) shows88
the open circuit voltage versus SOC for different charge/discharge currents. The testing procedure in89
Figure 1 (a) was as follows: at room temperature, the battery was fully charged, left it in the open-90
circuit state for 2 hours, and then discharged by 10 % of the rated capacity at a current of 30A.The91
battery is then kept in the open-circuit state and the open circuit voltage was observed. After 10 hours,92
the measured battery voltage was regarded to be the equilibrium potential of the battery since its93
growth rate was negligible. The process was subsequently repeated to get the equilibrium voltage94
curve during discharge as shown in Figure 1 (a). Similarly the battery was charged, and the95
equilibrium open circuit voltage of the battery was measured every 10% SOC to obtain the open circuit96
voltage curve during charging. Using the same discharge test procedure described above, the97
equilibrium potential was also measured when the discharge current was 100 A as shown in Figure98
1(b).99
Figure 1. Open-circuit cell voltages as a function of SOC at room temperature100
101
From Figure 1 (a), it is evident that the equilibrium open circuit voltage of a cell for a given SOC102
during charge and discharge are different. The OCV measured during charge is higher than that103
0 0.2 0.4 0.6 0.8 13.3
3.4
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
SOC(a)
OCV(V)
OCV after discharge
OCV after charge
0 0.2 0.4 0.6 0.8 13.3
3.4
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
SOC(b)
OCV(V)
OCV after 30A discharge
OCV after 100A discharge
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measured during discharge. It is evident in Figure 1(b) that the equilibrium potentials at the discharge104
current of 30 A and 100 A are nearly the same and the biggest difference is only 3mV, which means105
the equilibrium potential of the battery is essentially independent of the battery current, and the106
difference of equilibrium potential between charge and discharge is an inherent characteristic of the107
battery itself. Thus, for a specified open-circuit voltage there are two different SOC values, which108
introduce uncertainty in SOC determination using the OCV method. Therefore, it is important to109
analyze the hysteresis characteristics of the battery to reduce SOC estimation error.110
Table 1.Comparison of the cell OCV during charging and discharging at SOC=0.5111
Test conditionsOCV after
Charge/(V)
OCV after
Discharge/(V)
Difference
/(mV)
Average
OCV
/(V)
At room temperature, left at
open-circuit state for 10 h
3.979 3.955 24 3.967
112
Table 1 shows the equilibrium potential measured during charging process and discharging process113
when the SOC is 0.5. We find that the OCV after charging is 24 mV higher than that measured after114
discharging. The difference of 24 mV compared to the whole working voltage range of the battery is115
small (the voltage range of single battery over the whole SOC working range is around 1200 mV).116
However, the gradient of the measured OCV of the battery is relatively small in the middle region of117
SOC and a small hysteresis of 24 mV can cause a significant error in the SOC estimate based on the118
OCV curve. The practical capacity of the battery used in the experiment is 94 Ah, and the change in119
capacity per 1 mV change of open-circuit voltage is 0.19Ah mV-1 in the vicinity of the 50% depth of120
discharge (DOD). A 24 mV uncertainty in the OCV means a 4.56 Ah capacity estimate uncertainty,121
with a significant corresponding SOC uncertainty of around 4.85%. It is therefore important that the122
hysteresis characteristics are included in the battery model used to estimate the SOC.123
2.2. Model formulation124
The proposed equivalent circuit model including OCV hysteresis is shown in Figure 2. It comprises125
three parts: (1) open-circuit battery voltage Voc, which is composed of an average equilibrium potential126
Veand a hysteresis voltageVh, (2) internal resistance Ricomprising the Ohmic resistance Roand the127
polarization resistances, Rpa and Rpc. Rpa represents effective resistance characterizing activation128
polarization and Rpc represents the effective resistance characterizing concentration polarization, (3)129
effective capacitances Cpa and Cpc, which are used to describe the activation polarization and130
concentration polarization, and used to characterize the transient response of the battery. In addition,131
ideal diodes are added so that different resistance parameters are used during charging and discharging.132
The resistances connected in series with the diodes have additional subscripts to indicate charging or133
discharging. But in the equations these subscripts are omitted for conciseness.134
By analyzing the hysteresis characteristics as the function of SOC, the electrical behavior of the135
circuit can be expressed as follows:136
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,max( )( ( ) )
pa
pa
pa pa pa
pc
pc
pc pc pc
h h h
t e h pa pc o
V IV
R C C
V IV
R C C
V sign I V sign I V
V V V V V IR
(1)137
whereIis the current through the battery, Vh,maxrepresents the maximum hysteresis voltage of the138
battery as a function of SOC, which is defined as s,is hysteresis coefficient, and Vtis the terminal139
voltage of the battery. In addition, the average open-circuit voltage of the battery is considered as the140
equilibrium potential Ve in this paper. The identification of the model parameters was described in141
detail in a previous paper by the authors [16].142
Figure 2. Proposed equivalent circuit model for the lithium-ion battery143
Ve
Vh(a)
RSD
Rod Rpad
CpcCpa
Voc
Vt
Roc
Rpac
Rpcd
Rpcc
IVpcVpa
144
145
3. SOC estimation using robust extended Kalman filtering146
3.1. Robust state estimation147
The main idea of robust state estimation is that the modeling errors are regarded as constant bias148
state vectors. The dynamic states of the system and bias states are estimated separately, and the149
dynamic states are subsequently corrected using the bias states estimated values. To get the solution to150
states and bias estimation of a dynamic system, Friedland proposed decoupling the bias estimation151
from the state estimation to reduce computational complexity [17]. Hsieh and Chen generalized152
Friedlands filter and proposed an optimal solution for a two-stage Kalman estimator by applying a153
two-stage U-V transformation [18, 19]. To reduce the number of arithmetic operations in speed and154
rotor flux estimation of induction machine, Hilairet proposed modified optimal two-stage Kalman155
estimator derived from Hsieh and Chen algorithm [20]. Considering the nonlinear characteristics of the156
lithium-ion battery model, we propose in this paper a robust state estimation including EKF, which is157
suited for nonlinear systems. If modeling errors are neglected, the nonlinear system of interest can be158defined by:159
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1 1 1
,
( , , )
( , , )
~ (0, )
~ (0, )
k k k k
k k k k k
k x k
k k
x f x u w
y h x u v
w Q
v R
(2)160
wherexkis the vector of dynamic states, ukis the control input, wkrepresents process noise which is161
assumed to be discrete-time Gaussian zero-mean white noise with covariance of Qx,k, vk represents162
measurement noise which is assumed to be discrete-time Gaussian white noise with zero mean and a163
covarianceRk,Fk-1andBk-1are system matrices.164
( , ,0) ( )
( , ,0) ( )
[ ( , ,0) ]
k k
k k k k k k k
x x
k k k k k k k k
k k k k k k k k k
k k k k
h hy h x u x x v
x v
h x u C x x M v
C x h x u C x M v
C x z v
(3)165
wherek
z and kv are defined as follows:166
( , ,0)
~ (0, )
k k k k k k
T
k k k k
z h x u C x
v N M R M
(4)167
To allow for modeling errors constant matrices C A and F are introduced such that the168
linearized systems equations with modeling error is are given by:169
1 1 1 1 1( A) ( F)k k k k k k x A x F u w (5)170
( C)k k k k k y C x z v (6)171
where:172 T1 nA [a a ] ,
T
1 nF [f f ] , T
1 nC [c c ] .173
where a ni R , f q
i R , c m
i R , 1, ,i n .174
Define a constant bias vector ( )b n n q mk
R such that T T T T T T T1 n 1 1b [a a f f c c ]k q m . The equations (5)175
and (6) can be rearranged as follows:176
1 1 1 1 1 1 1k k k k k k k k x A x F u B b w (7)177
k k k k k k k y C x D b z v (8)178
where 1 11
1 1
0 0
0 0
T T
k k
k T T
k k
n n
x uB
x u
,0 0 0
0 0 0
Tm k
k T
m k
m
xD
x
.179
The vector bkcontains the constant bias states, which includes Gaussian white noise in practice, and180
can be expressed by:181
1 1k k kb b (9)182where k is Gaussian zero-mean white noise, and183
( )Ti j b ijE Q , ( ) ( ) ( ) 0T T T
i j i j i jE w v E w E v .184
The state estimation of the system with modeling error is converted to state estimation of the system185
with constant bias through the transformation. The robust state estimation combined EKF based on186
separate bias estimator are derived in detail in appendix A as described in [22]. Combining the187equations in appendix A, the estimates of x can be given by:188
1 1 1 1 1 1
k k k k k k k x A x F u B b
(10)189
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, 1 , 1 1 1 , 1 1 , 1
T T
x k k x k k k b k k x kP A P A B Q B Q
(11)1901
, , ,( )
T T
x k x k k k x k k kK P C C P C R (12)191
, , ,( )x k x k k x kP I K C P
(13)192
, ,k x k k b k K K V K (14)193
( ( , ,0) )k k k k k k k k x x K y h x u D b (15)194
From equations (10) to (15), it is shown that the bias information 1 , 1 1
T
k b k k B Q B is added to estimate of195
x . The vectorbkappears as an input to the system and thus its associated noise Qbmust be included in196
the estimate of P. In equations (10) to (15), the matrices Bk and Dk contains the states 1kx
, which is197
used instead of the statesxkin this paper:198
1 1
1 1
0 0
0 0
T T
k k
k T T
k k
n n
x uB
x u
1
1
0 0 0
0 0 0
Tm k
k Tm k
m
xD
x
.199
From the equations, it is inferred that the dynamic states of the system and the bias states are200
decoupled. They can be estimated based on the proposed robust estimation algorithm, which reduces201
the dimensions of the dynamic state equations, and further decreases the computation compared to the202
virtual noise compensation algorithm described in [23].203
3.2. SOC estimation based on the proposed battery model204
SOC can be regarded as a state variable, which is added to the proposed battery model. Discretizing205
the equivalent model of the battery, and the system can be expressed by [24]:206
, , 1 1
, , 1 1
, , 1 ,max 1
1 1
,
exp( / ( )) (1 exp( / ( )))exp( / ( )) (1 exp( / ( )))
exp( ) (1 exp( )) ( ) ( )
/
pa k pa k pa pa k pa pa pa
pc k pc k pc pc k pc pc pc
h k h k h k
k k i k N
t
V V t R C I R t R C V V t R C I R t R C
V V t t sign I V s
s s I t C
V
, , ,( )
k e k k o pa k pc k h k V s I R V V V
(16)207
where t is the sampling interval;ks represents the SOC, i stands for columbic efficiency, which208
is a function of the battery current. Using curve fitting of the experiment data, the equilibrium potential209
Veas the function of SOC was determined as follows:2104 3 2
( ) 5.2 0.86 12 15 59e k k k k k V s s s s s (17)211
To estimate the SOC using the Robust EKF, the mathematical model of the battery in (17) needs to212
be rearranged as:213
1 1 1 1
( , )
k k k k k
k k k
x A x F u
y h x u
(18)214
Where215T
pa pc hx V V V s , u I , ty V .216
1
exp( / ( )) 0 0 0
0 exp( / ( )) 0 0
0 0 0
0 0 0 1
pa pa
pc pc
k
t R C
t R CA
t
exp(- ),217
T
1 (1 exp( / ( ))) (1 exp( / ( ))) 0 /k pa pa pa pc pc pc i N F R t R C R t R C t C .218
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The largest source of error is the SOC estimation error. There are two main possible sources of219
modeling error. One is the change of the internal resistance caused by the effect of the working220
environment such as temperature and aging of the battery. The other arises from inaccuracy of the221
mathematical relationship between the equilibrium potential and SOC since the function is obtained by222
curve fitting the experiment data; the measurement error and fitting error may lead to estimation error223
of SOC. Referring to the system equations (5) and (6) including modeling error, and considering the224
SOC error components of the battery system, the bias matrix of the battery system can be defined as:225
0 0 0C 226
whereis a bias constant. The bias vector will be therefore given by:227T
[0 0 0 ]k
b 228
The procedure of SOC estimation using the proposed robust EKF algorithm is shown in Figure 3.229
Figure 3.Procedure of the Robust EKF algorithm in SOC estimation230
1 , 1 1 , 1
, , ,k b k k x k b P x P
Caluculation,k kB D
Prediction
, , , , ,k b k k x k
b P x P
Caluculation
, ,, , , ,k k x k b k k U S K K V
Caluculation,
t k
V
Caluculation
, , , , ,
k b k k x k b P x P
,k k k k
x x b b
231
3.3. Analysis of the effect of the bias vector on state estimation232
As described above, the values of the bias vector will have significant effect on robustness of SOC233
estimation using the Robust EKF algorithm. It is therefore necessary to investigate the influence of the234
bias constant variation on the estimation of the states of the battery system. The battery discharge235
current of 1C was used in the experiment. The results are shown in Figure 4.236
The bias constant as described in Section 3.2 reflects measurement error and coefficient fitting237
error of the battery system, which impact on the relationship between states (Vpa, Vpc, Vh and SOC)238
estimation of the battery dynamics and the parameteris illustrated in Figure 4. From Figure 4, it is239
seen that the polarization voltage Vpa and hysteresis voltage Vh estimate are hardly affected by the240
variation of the bias constant . But the change in the polarization voltage Vpcand SOC estimate for a241bias constant value of 0.5 are 0.157 V and 0.01, respectively compared to their value when =0. It is242
clear that the estimates of Vpc and SOC are influenced significantly by the bias constant. The243
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polarization voltage Vpcoverestimated when >0 compared and underestimated when 0, and overestimated when
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zero. The measured and estimated SOC of the battery for three rates of discharge of C/3, 1C and 1.5C,266
respectively, are plotted in Figure 5. From Figure 5(a), (b) and (c), it is clear that the estimated SOC267
are all less than the true value. The difference between the estimated and measured SOC increases with268
the battery current. Based on the analysis of the impact of the bias vector on SOC estimation as269
discussed in Section 3.3, the bias constant needs to be negative to reduce the steady error of SOC270
estimate. A bias constant ranging from -0.8 to 0 was selected based on the practical results in Figure271
5, and an optimal value of is ultimately chosen, based on Figure 6, to be -0.4.272
Figure 5. The experimental and estimated SOC of the battery with Robust EKF at =0 (SOC0=0.9)273
274
275
276
0 1000 2000 3000 4000 5000 6000 7000 80000
0.2
0.4
0.6
0.8
1
Time(s)
(a)
S
OC
true
Robust EKF (=0)
0 500 1000 1500 2000 2500 30000
0.2
0.4
0.6
0.8
1
Time(s)
(b)
SOC
true
Robust EKF( =0)
0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time (s) (c)
SOC
true
Robust EKF( =0)
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Figure 6. Modified result of Robust EKF estimation with different bias value at the discharge277
current of 1C (SOC0=0.9)278
279
3.5. Optimization of the filter gain coefficient280
The greatest advantage of EKF is to make the estimated state with initial error converge quickly to281
the true value of the state. While the SOC value describing the state of charge of the battery is a282
gradually changing state variable, the rate of change of the battery terminal voltage is relatively more283
rapid especially when the current changes direction between charging and discharging, leading to large284
fluctuations of the terminal voltage error of the battery model, which may further lead to oscillations of285
the SOC estimate thus enlarging the error. To resolve this problem, an optimal filter gain coefficient r286
is introduced to adjust the Kalman gain of the SOC in order to improve the stability of SOC estimation287
while ensuring the required precision. Based on this idea, the states estimation is changed to:288 ( ( , ,0) )k k k k k k k k k x x K y h x u D b
(19)289
where
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0
k
r
.290
Figure7. Effect of gain coefficient values on SOC estimation with Robust EKF291
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0 100 200 300 400 500
Time (s)
SOC
TRUE
r=0.3
r=0.1
r=0.05
r=0.01
292
0 500 1000 1500 2000 2500 30000.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
SOC
true
=-0.05
=-0.2
=-0.4
=-0.8
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The value of the gain coefficientrdirectly affects the accuracy and stability of SOC estimation. If293
the value is too big, the fluctuation of SOC estimate might be enlarged, and the accuracy reduced. If it294
is too small, filtering convergence speed to the true value of the state may be decreased, and the295
accuracy of SOC estimation may also be affected.296
The convergence speed and estimation accuracy were all taken into consideration during the297
optimization of the gain coefficient. Based on comparison between experimental and simulation results298
with different rvalues of ranging from 0.01 to 0.5 were investigated.299
In Figure 7, the true SOC value during a hybrid pulse charge/discharge test is compared with the300
estimated SOC obtained from the Robust EKF algorithm for different gain coefficients assuming the301
same initial error. It is evident that the bigger the gain coefficient r, the faster the convergence speed to302
the true value of SOC, however, the estimated SOC is easily influenced by the terminal voltage of the303
battery. Taking r= 0.3 for example, during t= 60 s ~ 70 s, the changing from the discharging state to304
charging state causes the terminal voltage of the battery to rapidly increase, which results in the SOC305
being over estimated and vice versa. When r is small, SOC estimation result can reflect the true306
tendency well, but the speed of convergence to the true value is slow. The trend of the estimated SOC307
curve is the closest to the True plotted in Figure 7 when r= 0.01, however, the estimation error may308
be unacceptably large if the initial error is relatively large. A gain coefficient of 0.1 was ultimately309
selected to achieve a compromise between the speed of convergence and estimation accuracy.310
4. Results discussion311
The self-defined hybrid pulse power characterization (self-defined HPPC) shown in Figure 8 was312
applied for validating SOC estimation using the Robust EKF. Figure 9(a) shows a comparison between313
the SOC values estimated using both the Robust EKF and regular EKF. The estimation error is shown314
in Figure 9 (b). From Figure 9 we find that using regular EKF, the SOC estimation changes with the315
voltage fluctuations rapidly, and the maximum estimation error can be over 10 %, which does not316
satisfy the requirement of electric vehicle. Using the proposed Robust EKF algorithm, the estimation317
error is within 5% during the entire process of charging and discharging of the battery, thus meeting318
the SOC accuracy requirement.319
Figure 8.Profile of self-defined HPPC for one cycle320
321322
0 50 100 150 200 250 300 350 400 450 500-400
-300
-200
-100
0
100
200
300
400
500
Time (s)
Current(A)
Charge
Discharge
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Figure 9. Comparing results of SOC estimation with the proposed Robust EKF and EKF for self-323
defined HPPC cycles324
325
The DST driving cycles was also used to validate the SOC estimation performance using Robust326
EKF algorithm. The measured SOC and the estimated SOC for different SOC initial values are327
illustrated in Figure 10 (the true initial SOC value was 0.9). It is seen that the estimated SOC using the328
Robust EKF converges faster to the measured SOC values, and the estimation accuracy is significantly329
improved compared to SOC estimate using EKF only. When the initial SOC values are set to 0.8 and330
0.7, the SOC estimation error with Robust EKF falls to within 6 % of the true value after 300 s and 600331s correction, respectively. Furthermore, the SOC estimation error using the Robust EKF is enlarged332
when the SOC is in the range from 0.55 to 0.2. This is because the battery terminal voltage reduces and333
the output current supplied by the battery increase (for given power output) with the passage of time,334
which results in serious polarization of the battery, leading to large estimation error. The SOC335
estimation error using Robust EKF is controlled within 5 % during the entire DST driving cycles.336
Figure 10. Estimated and tested SOC for DST driving cycles (SOC0=0.9)337
338
0 1000 2000 3000 4000 5000 6000 7000 80000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time (s)
(a)
SOC
True
EKF
Robust EKF
( =-0.2, r=0.1)
0 2000 4000 6000 8000-15
-10
-5
0
5
10
15
Time (s)
(b)
Estimationerror(%)
EKF
Robust EKF
( =-0.2, r=0.1)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
SOC
True (SOCo=0.9)
Robust EKF(SOCo=0.9)
EKF(SOCo=0.9)Robust EKF(SOCo=0.8)
Robust EKF(SOCo=0.7)
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5. Conclusions339
The paper presented an equivalent circuit model with two RC networks characterizing battery340
activation and concentration polarization processes. The hysteresis voltage was also included in the341
battery model to improve the accuracy of SOC determination. The paper proposed a Robust Extended342Kalman filter in which the steady error of the SOC estimation was investigated, and accounted for to343
improve the estimation accuracy. Based on the knowledge of battery characteristics, a filter gain344
coefficient was introduced to decrease the fluctuation of SOC estimation caused by terminal voltage345
fluctuation, which occurs when a standard EKF is used without the gain coefficient. Simulation results346
demonstrated the accuracy of SOC estimation with the proposed Robust EKF during both the self-347
defined HPPC cycles and DST driving cycles. The error found to be less than 5% compared to nearly348
10 % error achieved by the standard EKF.349
Acknowledgments350
The work was supported by the National High Technology Research and Development Program of351
China (No. 2011AA05A108) and National Natural Science Foundation of China (No. 71041025).352
References353
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Appendix A406
The robust state estimation combined EKF based on separate bias estimator can be derived as407
follows.408
In the absence of bias error the vector bkis zero, the estimates ofxusing EKF are given by409
1 1 1 1k k k k k x A x F u
(A.1)410
, 1 , 1 1 , 1
T
x k k x k k x kP A P A Q
(A.2)411
, , ,( )x k x k k x kP I K C P
(A.3)412
,( ( , ,0))k k x k k k k x x K y h x u
(A.4)413
with initial condition (0) (0)x xP P . x is the bias-free estimate ofx,xP is the error covariance matrix of414
x , xK is the Kalman gain matrix, xQ is the process noise covariance matrix of x ,Ris the measurement415
noise covariance matrix.416
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The bias vector estimator is given by:417
1
k kb b
(A.5)418
, , 1 , 1b k b k b k P P Q
(A.6)4191
, , , ,( )
T T T
b k b k k k b k k k x k k k K P S S P S C P C R (A.7)420
, , ,( )b k b k k b k P I K S P (A.8)421
, [ ( , ,0) ]k k b k k k k k k b b K y h x u D b
(A.9)422
where the weighted matrices U, S, and Vare defined by:423
1k k k k U A V B (A.10)424
k k k k S C U D (A.11)425
,k k x k k V U K S (A.12)426
Considering the estimates of x and b , the adjusted estimates ofxare defined by:427
k k k k x x U b (A.13)428
k k k k x x V b
(A.14)429
where the present estimatex is adjusted with the current estimate of the bias vectorb .430
2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article431
distributed under the terms and conditions of the Creative Commons Attribution license432
(http://creativecommons.org/licenses/by/3.0/).433