Soc Estim Ekf

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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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Energies 2012, 5 2

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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Energies 2012, 5 4

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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Energies 2012, 5 8

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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Energies 2012, 5 10

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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Energies 2012, 5 11

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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Energies 2012, 5 12

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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Energies 2012, 5 13

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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Energies 2012, 5 14

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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2. Sato, S.; Kawamura, A. A New Estimation Method of State of Charge using Terminal Voltage and356Internal Resistance for Lead Acid Battery. In Proceedings of 2002 IEEE International Conference357

on Power Conversion, Osaka, Japan, 2002.358

3. Soon, Ng.K.; Moo, C.S.; Chen, Y.P.; Hsieh, Y.C. Enhanced coulomb counting method for359estimating stat-of-charge and state-of-health of lithium-ion batteries. Applied Energy 2009, 86,360

1506-1511.361

4. Shen, W.X. State of available capacity estimation for lead-acid batteries in electric vehicles using362neural network.Energy Convers. Manage.2007, 48, 433-442.363

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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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Energies 2012, 5 16

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

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