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Computers and Chemical Engineering 64 (2014) 114123

Contents lists available at ScienceDirect

Computers and Chemical Engineering

journal homepage: www.elsevier .com/ locate /compchemeng

Adaptive gain sliding mode observer for state ofcharge estimation

based on combined battery equivalent circuit model

Xiaopeng Chen, Weixiang Shen, Zhenwei Cao, Ajay Kapoor

Faculty of Science, Engineering and Technology, SwinburneUniversity of Technology, Hawthorn, Victoria 3122,Australia

a r t i c l e i n f o

Article history:

Received 3 February 2013

Received in revised form 8 February 2014Accepted 16 February 2014

Available online 22 February 2014

Keywords:

Adaptive gain sliding mode observer

Battery management system

Combined battery equivalent circuit model

Electric vehicle

Lithium-polymer battery

State of charge

a b s t r a c t

An adaptive gain sliding mode observer (AGSMO) for battery state ofcharge (SOC) estimation based on a

combined battery equivalent circuit model (CBECM) is presented. The error convergence of the AGSMO

for the SOC estimation is proved by Lyapunov stability theory. Comparing with conventional sliding

mode observers for the SOC estimation, the AGSMO can minimise chattering levels and improve the

accuracy by adaptively adjusting switching gains to compensate modelling errors. To design the AGSMO

for the SOC estimation, the state equations ofthe CBECM are derived to capture dynamics ofa battery. A

lithium-polymer battery (LiPB) is used to conduct experiments for extracting parameters ofthe CBECM

and verifying the effectiveness ofthe proposed AGSMO for the SOC estimation.

2014 Elsevier Ltd. All rights reserved.

1. Introduction

In recent decades, the progressive increase of petrol costs and

air pollution of the exhaust fumes from petrol-driven vehicles has

stimulated a surge of research and innovation in electric vehi-

cle (EV) technologies. Lithium-ion or lithium-polymer batteries

(LiPBs) have been adopted as primary power sources in EVs due

to their merits in high power and energy densities, high operating

voltages, extremely low self-discharge rate and long cycle life in

the comparison with other types of batteries such as lead-acid or

nickel-metal hydride batteries. For the application of the batter-

ies in EVs, the state of charge (SOC) is one of the key parameters

which corresponds to the amount of residual available capacity, its

accurate indication is crucial for optimising battery energy utilisa-

tion, informing drivers the reliable EV travelling range, preventing

batteries from over-charging or over-discharging and extend-

ing battery life cycles. Unfortunately, the SOC cannot be directly

measured by a sensor as it involves in complex electrochemical

processes of a battery. An advanced algorithm is required to esti-

mate the SOC with the aids of measurable parameters of a battery

such as terminal voltage and current.

A variety of the SOC estimation techniques has been reviewed

by Piller, Perrin, and Jossen (2001) and each method has its own

Corresponding author. Tel.: +61 3 9214 5886; fax: +61 3 9214 8264.

E-mail addresses:xchen@swin.edu.au(X. Chen), wshen@swin.edu.au(W. Shen).

advantagesin certain aspects.The ampere-hour (Ah)countingis the

most applicable approach for the SOC indication in many commer-cial battery management systems (BMSs). It simply integrates the

battery charge and discharge currents over time and accumulates

errors caused by the embedded noises in current measurements.

Furthermore, this non-model and open-loop based method has

difficulty in determining the initial SOC value. An improved ver-

sion of the Ah counting has exhibited better SOC estimation results

by on-line evaluating charge and discharge efficiencies with the

recalibration of the cell capacity (Ng, Moo, Chen, & Hsieh, 2009).

Battery impedance measurement technique is also used for the

SOC estimation through injecting small ac signals with a wide

range of frequencies into a battery to detect the variationof battery

internal impedances (Rodrigues, Munichandraiah, & Shukla, 2000).

However, the measured impedances cannot completely model the

dynamics of batteries in the case of large discharge current in EVs.

Furthermore, the application of impedance spectroscopy has to be

carried out in temperature-controlled environment that requires

bulky and costly auxiliary equipment since the temperature signif-

icantly affects impedance curves.

Another category of the SOC estimation methods is based on

black-box established on machine learning strategies, which

includes artificial neural networks (ANNs) (Shen, 2007; Shen, Chan,

Lo,& Chau, 2002), fuzzyneural networks (Li, Wang, Su, & Lee,2007),

adaptive fuzzy neural networks (Chau, Wu, Chan, & Shen, 2003)

and support vector machine (Hansen & Wang, 2005). These data-

oriented approaches can accurately estimate the SOC without its

http://dx.doi.org/10.1016/j.compchemeng.2014.02.015

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X. Chen et al. / Computers and Chemical Engineering 64 (2014) 114123 115

Nomenclatures

Cn nominal capacity of LiPB (Ah)

Cp polarisation capacitance (F)

eVt, eZ, eVoc, eVp estimation errors

f1, f2, f3 system uncertainty termsRi ohmic resistance ()

Rp polarisation resistance ()

Z state of chargeZ estimated state of chargeVoc open circuit voltage (V)

Voc(Z) open circuit voltage asa function ofstateof charge

Vp polarisation voltage (V)

Vp estimated polarisation voltage (V)

Vt estimated battery terminal voltage (V)Vt battery terminal voltage (V)

coulomb efficiencyi uncertainty bounds

1, 2, 3 adaptive switching gains1,

2,

3 adaptive switching gains updating laws

1, 2, 3 adaptation speed adjusting values

accurate initial state, but they require a large amount of data to

train ANNs, which leads to the large computation burden in the

BMS. Moreover, the SOC estimation results would be unpredictable

in the presence of the conditions where the current profiles in EVs

are different from those represented by the training data.

The Kalman filter (KF), as an optimal recursive estimator which

is able to estimate the states of a linear dynamic system (Ristic,

Arulampalam,& Gordon, 2004), hasbeen developedto estimate the

SOC based on linear state space battery models (Barbarisi, Vasca, &

Glielmo, 2006). For nonlinear battery models, the enhanced ver-

sions of KF have been intensively investigated to achieve better

results for on-line SOC estimation, such as extended KF (EKF) (Dai,

Wei, Sun, Wang, & Gu, 2012; Hu, Youn, & Chung, 2012; Hu, Li, &Peng, 2012), adaptive extended KF (AEKF) (Han, Kim, & Sunwoo,

2009), sigma-point KF (SKF) (Plett, 2006a,b) and unscented KF

(UKF) (He, Williard, Chen, & Pecht, 2013; Zhang & Xia, 2011). The

EKF utilises the first-order Taylor series expansion to linearise

the nonlinear function. This local linearisation can give rise to

large estimation errors when the degrees of nonlinearityin battery

models are significant and the covariance of process and measure-

ment noises is assumed to be constant. Adaptively updating the

covariance of process and measurement noises, the AEKF has been

developed to improve the online SOC estimation accuracy. Instead

of local linearisation, the SKF and the UKF use an unscented trans-

formation to approximate the probability density function of the

nonlinear systems with a set of sample points or so-called sigma

points. Essentially, all above-mentioned KF-based approaches arebased on the assumption that the covariance of measurement and

process noises described by a Gaussian probability density func-

tion has to be known a priori. Moreover, their complex matrix

operations may result in numeric instabilities.

The H observer based approach has also been proposed to

estimate the SOC without the requirement of the exact statistical

properties of the battery model (Zhang, Liu, Fang, & Wang, 2012).

This approach minimises the errors between the outputs of the

battery and its model so that the SOC estimation error is less than

a given attenuation level. However, in order to tackle modelling

errors and external disturbances, the feedback gain ofHobserver

must be obtained by solving a linear matrix inequality, which may

not provide the optimal solution for ensuring tracking error con-

vergence.

More recently, sliding mode observer (SMO) based SOC esti-

mation methods were adopted to overcome battery model

uncertainties, external disturbances and measurement noises with

sufficient large switching gains (Kim, 2006; Chen, Shen, Cao, &

Kapoor, 2012). This method relies on the exhaustive understanding

of battery dynamics for the appropriate selection of the switching

gains, which lead to the trade-off between the magnitude of chat-

tering in the SOC estimation and the convergence speed to reach

the sliding mode surface and trigger the sliding motion.

In this paper, an adaptive gain slide mode observer (AGSMO)

based on a combined battery equivalent circuit model (CBECM)

has been proposed for the SOC estimation. The main advantage

of the AGSMO is that the robust behaviour of the SOC estima-

tion is guaranteed in the presence of the modelling errors, which

are considered as the bounded uncertainties. This is achieved by

dynamically adjusting the switching gain of the SMO in response

to the tracking error while ensuring the reachability of the slid-

ing mode surface and triggering the sliding mode. Once the sliding

mode is activated, theswitching gain is self-tuned to an adequate

level to counteract the modelling errors and reduce the chattering

levels, thereby improving the SOC estimation accuracy.

This rest of this paper is organised as follows. In Section 2, a

CBECM is presented to model the battery dynamic behaviour. In

Section 3, the AGSMO design methodology for estimating the SOC

is explained. Section 4 elaborates the procedures to extract battery

modelparameters. Section 5 validates the proposed AGSMO forthe

SOC estimation by experimental results and Section 6 concludes.

2. Battery modelling

A suitable battery model is essential to the development of

the model-based BMS in real EVs, which requires less computa-

tion power and fast response to ever-changing road conditions.

Many types of models are developed to capture lithium-ion bat-

tey dynamics for various purposes (Ramadesigan et al., 2012). In

general, they can be categorised into two main groups, which

are electrochemical and equivalent circuit models (He, Xiong, &Fan, 2011; Hussein & Batarseh, 2011; Hu, Youn et al., 2012; Hu,

Li et al., 2012). The electrochemical models describe the physical

phenomena which occur inside batteries such as the material and

charge transfer processes, ionic conduction, solid phase diffusion.

They utilise partial differential equations with a large number of

unknown parameters andthus a large amountof memoryrequired,

which leads to long computation time and slow response. They are

usually used for battery design and simulation and hardly suitable

for the BMS design in real EVs (Smith, Rahn, & Wang, 2010).

On the other hand, the equivalent circuit models simply consist

of resistors, capacitors and voltage sources to form a circuit net-

work, which leads to short computation time and quick response.

Furthermore, they are the circuit in nature which is easily inte-

grated into the BMS and power control in real EVs. Various batteryequivalent circuit models have been proposed to reflect dynamic

characteristics of the battery as a result of the trade-off between

modelling accuracy and complexity (Lee, Kim, Lee, & Cho, 2008; Cho

et al., 2012; Chen, Gabriel, & Mora, 2006; Abu-Sharkh & Doerffel,

2004).

In this paper, the combined battery equivalent circuit model

(CBECM) is used to represent the dynamical behaviors of LiPB,

as shown in Fig. 1. A capacitor, Cn represents the nominal capac-

ity of the battery on the left in the model. The current source, I

denotes the discharge or charge current of the LiPB and the corre-

sponding battery terminal voltage is expressed by Vt. The voltage

across the Cn as the open circuit voltage (OCV), Voc varies in the

range of the SOC,Z from 0% to 100% and it represents the SOC of

the battery quantitatively. A resistor, Ri and a parallel-connected

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116 X. Chen et al. / Computers and Chemical Engineering 64 (2014) 114123

Fig. 1. Schematic of combined battery equivalent circuit model.

network of the polarisation resistance, Rpand polarisation capaci-

tance, Cpon the right are used to characterise the ohmic resistance

and simulate the transient responses or the battery relaxation

effect, respectively. The relaxation effect is defined as the slow

convergence of the battery terminal voltage to the OCV at its equi-

librium state after hours of relaxation at the end of charging or

discharging process. It is causedby the diffusioneffect anda double-

layerchargingor dischargingeffect in thebattery(Chenetal.,2006).

The voltage-controlled voltage source, Voc(Z) is used to bridge the

nonlinear relationship between the SOC and the OCV as shown

in Fig. 2, which can be derived by fitting the experimental data

obtained from the pulse current discharge (PCD) and pulse current

charge (PCC) tests (for details see Section 4). The self-discharge

resistance is ignored in this model as a LiPB has extremely low

self-discharge rate.

The SOC describes the ratio of the remaining capacity to the

present maximum available capacity of a battery, and it can be

expressed as:

Z(t) =Z(0)

t

0

I()Cn

d (1)

whereZ(0) is the initial SOC of the battery, I() is the instantaneouscurrent and it is assumed to be positive for discharge current and

negative for charge current. The denotes the coulomb efficiencyand it can be normally taken one for discharging and less than or

close to one for charging LiPB in the wide range of the current and

the temperature.

Fig. 2. Experimental OCVSOC curves of LiPB.

According to Fig. 1, Vtcan be written as follows

Vt= Voc(Z) Vp IRi (2)

The time derivatives of polarisation voltage and the SOC yield

Vp =VpRpCp

+I

Cp(3)

Z= I

Cn

(4)

where Vp is the polarisation voltages across theCp.

Despite the nonlinearity of the OCVSOC curves as shown in

Fig. 2, there exists a piecewise linear relationship between the OCV

and the SOC in a certain range of the SOC indicated by the dots in

the curves. Therefore, the OCV is expressed as a function of the SOC

by using piecewise linearisation method

Voc(Z) = Z+ d (5)

where the values of and d are the constants in a certain range ofSOC. In this paper, there are one pair of and d in every 10% SOCand totally ten pairs for 0% to 100%. Thus, the time derivative ofVocin each 10% SOC segment is

Voc(Z) = Z (6)

Subsitituting Eq. (4) into Eq. (6) gives

Voc(Z) =

I

Cn

(7)

Therate of changeof thecurrent during charging or discharging,

Icanbe negligible dueto thethe fast sampling interval as explained

as follows (Chen et al., 2012; Chiang, Sean, & Ke,2011). For instance,

5 A h LiPB has been conducted discharge at the current of 1Cn(5A), the variation of SOC in 1s sampling period with respect to

timeas given in Eq. (4) is dZ/dt=5/(53600) =0.00028, namely

dZ/dt0. It shows that the current within 1s has an insignificant

impactof theSOC,thusthe current isassumedto beconstantin each

sampling period (e.g., one second in this paper), namely dI/dt0.

Therefore, the time derivative ofVtin Eq. (2) with the substitutionsof Eqs. (3)(7) gives

Vt=

I

Cn

+

VpRpCp

I

Cp(8)

Solving I in Eq. (2) and substituting it into Eq. (4) as well as

rearranging Eqs. (3)(5) result in the state-space equations of the

CBECM as

Vt = a1Vt+ a1Voc(Z) b1I

Z = a2Vt a2Voc(Z) + a2Vp

Vp = a1Vp + b2I

(9)

where a1 =1/(RpCp), a2 =1/(RiCn), b1 =/Cn +Ri/(RpCp)+ 1 /Cp andb

2= 1/C

p.

There are two main causes for modelling errors by using the

CBECM to represent battery behaviours. Firstly, the circuit parame-

ters of theCBECM equationsare taken as theconstantvalues, butin

fact they are varying with the battery SOC. Secondly, the OCVSOC

curve is piecewise linearised. These modelling errors represented

by the uncertainty terms,fare added to Eq. (9) as

Vt = a1Vt+ a1Voc(Z) b1I+f1

Z = a2Vt a2Voc(Z) + a2Vp +f2

Vp = a1Vp + b2I+f3

(10)

where f1, f2and f3satisfy the following bounded conditions:

fi < i where i=1, 2 and 3. Since the proposed SOC estimationapproach is applied to EV applications, EV driving schedules are

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X. Chen et al. / Computers and Chemical Engineering 64 (2014) 114123 117

Fig. 3. SMO based SOC estimation with conservative switching gains.

studied to determine average discharge rates of current profiles for

battery testing, where the average EV driving speed is equivalently

converted to average current rates of 1/3Cn1Cn. In this paper, the

average current rate of discharge current profiles up to 1.5Cn is

adopted for the battery tests to ensure that the proposed dischargecurrent profiles have included the maximum possible currents for

discharging or charging. As a result, both charge and discharge cur-

rent rates as the inputs to the model and the battery in EVs are

bounded andso arethe terminal voltages as the outputs. Therefore,

the uncertainty bound, i can be determined by using the largestmodelling errorsbetweenthe LiPB andthe CBECM underthe largest

average discharge rate of 1.5Cn.

3. Design of adaptive gain sliding mode observer for SOC

estimation

Conventional SMOs with the constant switching gains for the

SOC estimation have demonstrated the robustness to compen-

sate modelling errors and uncertainties with the properly selectedswitchinggains (Kim, 2006; Chen et al., 2012). However, the under-

estimated or overestimated switching gain has given rise to the

poor tracking performance or undesired chattering phenomena in

the SOC estimation. The SMO with a lower switching gain has no

unwanted chattering in the SOC estimation as shown in Fig. 3, but

its tracking performance under the randomly selected initial SOC

is very poor with the mean square errors (MSEs) in SOC estimation

higherthan10%. On the other hand, the SMO with a large switching

gain has a considerable chattering aroundthe true SOC as shown in

Fig. 4 and its tracking performance is robust with most of the MSEs

Fig. 4. SMO based SOC estimation with large switching gains.

bounded within 5%, but the higher magnitude of chattering ripples

blurs the SOC estimation and affects the stability of observer.

In order to accurately estimate the SOC, an AGSMO based on an

equivalent control concept is proposed as follows

Vt = a1Vt+ a1Voc(Soc) b1I+ 1sgn(eVt)

Z = a2Vt a2Voc( Z) + a2Vp + 2sgn(eVoc)

Vp = a1Vp + b2I+ 3sgn(eVp)

(11)

where Vt, Zand Vpare the estimatedVt, Zand Vp, respectively, the

1, 2and 3 are the adaptive switching gains which are adaptedaccording to the following updating laws:

1 = 1 |eVt| ,

2 = 2 |eVoc|and

3 = 3

eVp (12)where the terms 1, 2 and 3 are positive constants that should

be chosen suitably small so that they can ensure the adaptation

speed of the switching gains for state errrors convergence while

preventing the corresponding i from becoming too large andguaranteeing suitable bounded magnitude of the switching gains.

Accordingly, the battery terminal voltage and state estimation

errors are defined as

eVt = Vt Vt

eVoc = Voc(Z) Voc( Z) = (ZZ) = eZ

eVp = Vp Vp

(13)

Thus, by substracting Eq. (11) from Eq. (10), the error dynamics

of battery terminal voltage and other states are expressed as

eVt = a1eVt+ a1eVoc+f1 1sgn(eVt)

eZ = a2eVt a2eZ+ a2eVp +f2 2sgn(eZ)

eVp = a1eVp +f3 3sgn(eVp)

(14)

where sgn() is the signum function

sgn(eVt) =

+1, eVt> 0

1, eVt< 0

Itcanbe seenfrom Eq. (14) that if the switching gain 1is prop-erly adjusted so that a sliding mode motion can be induced on the

terminal voltage errorstate in Eq. (14). The asymptotic convergence

of the terminal voltage error can be proved by Lyapunov stability

theory via choosing the candidate of Lyapunov function as follows

V1 =1

2(e2Vt+

11

21) (15)

where 1 = 1 1and since the SOC operation range of the LiPB

is varying from 0 to 1, as can been seen in Fig. 2, the SOC esti-

mation error is bounded as |ez| < 1 and the Voc is also bounded as

|eVoc| =Voc(Z) Voc( Z) = (ZZ) < || |eZ|< || , and the time

derivative of the candidate of Lyapunov function V1results in

V1 = eVteVt+ 11 1

1 = eVt[a1eVt+ a1eVoc+f1 1sgn(eVt)]

+11 (1 1)

1 = [a1e

2Vt+ a1eVteVoc+ eVtf1

1 |eVt|]

+ (1 1) |eVt| = a1e2Vt+ a1eVteVoc+ eVtf1

1 |eVt|< a1 |eVt| |eVoc| + |eVt|f1

1 |eVt| = |eVt| (a1 |eVoc| +f11) (16)

There exists an unknown finite non-negative switching gain

1such that 1> a1 |eVoc| + 1, leading to V1< 0, which satisfies

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118 X. Chen et al. / Computers and Chemical Engineering 64 (2014) 114123

Fig. 5. Testing platform of LiPB.

the second method of Lyapunov stability theory. Thus, the termi-

nal voltage error as the sliding variable in Eq. (14) asymptotically

converges to zero as time tends to infinity. In other words, the

sliding surface is reached during the sliding motion as the slid-

ing variable is equal to zero, where the sliding surface is defined as

S= {eVt= 0}. Once the sliding surface is reached, the sliding modewould be induced to ensure eVt(t) = eVt(t) = 0 and the influence

of the bounded uncertainty term f1 is compensated. Accordingto the equivalent control concept (Edwards & Spurgeon, 1994), the

unmeasurable SOC error can be derived by solving Eq. (14) in terms

of1sgn(eVt) after inserting zeros foreVtand eVt

eZ=

1a1

sgn(eVt)

eq

(17)

Similarly, the SOC error in Eq. (14) asymptotically converges to

zero as time tends to infinity. After eZ= 0 and eZ= 0, the influence

of the bounded uncertainty term f2 is compensated. Finally, theVperror equation can be derived from Eq. (14) as follows

eVp =

2a2

sgn

1a1

sgn(eVt)

eq

eq

(18)

Eqs. (17) and (18) are substituted into Eq. (14), a set of the

AGSMO equations based on the equivalent control concept is

obtainedVt= a1Vt+ a1Voc( Z) b1I+ 1sgn(eVt)

Z= a2Vt a2Voc( Z)+ a2Vp + 2sgn

1a1

sgn(eVt)

eq

Vp=a1Vp + b2I+ 3sgn

2a2

sgn

1a1

sgn(eVt)

eq

eq

(19)

From Eq. (19), it is worth mentioning that the proposed AGSMO

has no requirement for the detailed knowledge of modelling errorsas long as they are bounded whereas the KF based approaches

require the covariance values of the process and measurement

noises. These covariance values are determined either by the

time-consuming trial-and-error method or by the recursive iden-

tification method. Since a priori knowledge of the noise statistical

properties is normally unknown, the former may cause the large

SOC estimation errorwith slowconvergence if inappropriatevalues

of the noise covariance were used and the latter increases compu-

tational complexity.

4. Battery model parameters determination

The component values of the CBECM in Fig. 1 are obtained

from the transient-response method via an experiment of the PCD

profile at room temperature. A LiPB is used in the test and it has

a nominal capacity of 5.0Ah and a nominal voltage of 3.8V. The

dimension of the cell is 135mm50mm9mm and the weight

of the cell is 130 g. A battery testing platform as illustrated in Fig. 5

is constructed to perform the experiments, and it consists of pro-

grammable power supply (Sorensen DLM50-60), electronic load(Prodigit 3320) and switches safety box. The testing platform can

control charging/discharging battery, sample experimental data

and store the data into the PC via a graphic user interface program

designed by using the LABVIEW software.

The nonlinear relationship between battery OCV and SOC has

been identified by performing PCD and PCC tests on the LiPB. For

discharge test, the PCD is comprised of a sequence of pulse current

with 6-min discharge and 1-h rest to allow the battery to return

to its equilibrium state before running the next cycle as shown in

Fig. 6. The discharge current of 5.0 A is used, which corresponds to

1Cnrate. For the fully charged LiPB (Z= 100%), each pulse discharge

approximates10% of nominal capacity equivalent to 10%of theSOC

reduction, andthe procedureof the pulse discharge andrecovery is

repeated until the battery is fully discharged to the cut-off voltage

of 2.7 V (Z=0%). For charge test, the PCC is similar tothe PCD test, a

fully discharged LiPB has been charged from 0% to 10% SOC at the

Fig. 6. Pulse current discharge and correspondingterminal voltage.

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X. Chen et al. / Computers and Chemical Engineering 64 (2014) 114123 119

Table 1

Parameters of CBECM and fitting errorsextracted from thePCD profile.

No. of set p (s) Ri(m) Rp (m) Cp (kF) MSEs

1 48.52 102.3 5.5 8.822 1.2321%

2 83.94 102.7 7.4 11.343 1.4641%

3 319.2 103.1 8.5 37.553 5.8081%

4 157.5 103.2 5.0 31.50 3.0625%

5 45.09 103.2 4.1 10.998 0.8281%

6 71.08 103.3 5.1 13.937 1.1664%

7 107.9 103.6 4.6 23.457 1.6384%8 121.8 103.7 4.1 29.71 2.1316%

9 252.3 103.5 6.5 38.554 6.4009%

10 436.3 105.8 20.4 21.387 0.4624%

recommended 0.5Cnrate, followed by 1-h rest, and this process is

repeated until the battery reaches 100% SOC. Fig. 2 shows the mea-

sured OCVat differentSOCs during dischargeand charge processes.

It canbe observed that theOCV of charging process is alwayshigher

thanthat of the dischargingprocess,which accounts for a hysteresis

phenomenon between two OCV-SOC curves during the discharge

and charge, respectively. In fact, the hysteresis effect is correlated

with the relaxation effect due to lithium ion diffusion inside the

LiPB and the level of hysteresis is decreasing with the longer rest

period. Forthe consideration of hysteresis effect, theOCV as a func-tion of SOC is defined as the average OCV values between charging

and discharging curves as shown in the blue dashed line in Fig. 2.

The circuit model parameters are extracted based on the PCD as

shown in Fig. 6. The corresponding terminal voltage response after

each 10% SOC discharge is also illustrated in Fig. 6. It can be seen

that totally ten sets of transient response in terminal voltage have

been generated to determine circuit parameters corresponding to

each discharge pulse and thus the ten sets of circuit parameters

are identified. Table 1 summarises those parameters and the cor-

responding fitting errors represented by MSEs. Since the ninth set

of circuit parameters causes the largest model error, it is used to

determine the parameters in Eq. (9).

The transient response of theterminalvoltage at theninth pulse

current dischargeindicated by a redcircle in Fig.6 is used to extract

the circuit parameters, where the circled section is magnified in

Fig.7. It canbe seen that when thebatterystops discharging theter-

minal voltage has a steep rise as the voltage drop across the ohmic

Fig. 7. Transientresponseof LiPB and circuit model at thecircled PCD.

resistance Ri disappears immediately, so the ohmic resistance can

be calculated by

Ri =VtI

(20)

whereVtis thechangeof thevoltage acrossRiat theinstant whenthe discharge current Idisappears.

During the time interval (t0 t t1), the terminal voltage

increases exponentially as it slowly converges to the OCV, namely

Voc(t1). This battery terminal voltage is driven by thedynamic char-

acteristics of the battery and can be found by setting discharge

current to zero in Eqs. (2) and (3), then solving the differential

equations gives

Vt(t) = Voc(t1) Vp exp

t

p

(21)

where p is the time constant for the polarisation voltage duringtransient response, Voc(t1) is the OCV after a full relaxation and Vpis the voltage of the polarisation capacitor and its value equals to

Voc(t0).

In order to identify model parameters, a curve fitting technique

as a nonlinear least square algorithm is applied to search for the

best fitting values which lead to the least fitting errors betweenthe measured voltages and the voltages fitted by the exponential

function, f(t) =V1 V2exp(t). The coefficients V1, V2 and aftercurving fitting can be used to determine the CBECM parameters

such as Voc(t1) =V1, Vp =Voc(t0) =V2and p = 1/.By using the time intervalt= (t1 t0) andrearranging Eq. (21),

the following equations are derived to calcualte the circuit param-

eters

Rp =Vp

(1 exp(t/p)I (22)

Cp =pRp

(23)

From Eq. (21), it can be seen that as time t increases or tends

to infinity the terminal voltage would be equal to the open cir-cuit voltage and the battery relaxation effect disappears. With

the parameters calculated by Eqs. (21)(23), the voltage transient

response represented by the fitting function is replotted in Fig. 7

in the red dash line and it can approximately match the measured

voltage.

To verifythe accuracy of the extractedparameters, the PCDpro-

fileis applied tothe CBECMwith theparameterscalculatedfromthe

ninth pulse current discharge again. Fig. 8 shows the terminal vol-

tages of the battery and the CBECM as well as their corresponding

modelling errors represented by MSEs within 0.4%.

5. Verification of AGSMO for SOC estimation

The configuration of the AGSMO for the SOC estimation is illus-trated in Fig. 9, where the properly defined current profiles are

simultaneously applied to the LiPB module, the Ah counting mod-

ule and the AGSMO module. The LiPB terminal voltage is sampled

and fed into the AGSMO to generate the voltage tracking error,

which can be used to update the corresponding swiching gains of

the AGSMO to compenstate the modelling errors. The outputof the

AGSMO module is the estimated SOC, which is concurrently com-

pared to thetrue SOCdirectlygenerated by theAh counting module

to demostrate the accuracy of the SOC estimation.

To validate the effectiveness and robustness of the AGSMO for

the SOC estimation, three types of current profiles with different

discharge rates have been conducted on the LiPB at room temper-

ature. As the battery is fully charged, the initial SOC of the LiPB is

set to 100% at the beginning of discharging process.

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120 X. Chen et al. / Computers and Chemical Engineering 64 (2014) 114123

Fig. 8. Terminal voltage of LiPB and CBECMand MSE in the PCD profile of1Cn.

The testing data are obtained by using the constant current dis-

charge (CCD) profile with discharge rates of 1/3Cn, 1Cn and 1.5Cn,

namely 1.67A, 5A, and 7.5A. The positive contants are selected

to satisfy the adaptation speed for the switching gains, they are

1 =0.5, 2 =0.3 and 3 =0.1. The initial SOC is also set to the ran-dom value away from the true value. The proposed AGSMO is used

to estimate theterminalvoltage andthe SOCfor differentdischarge

rates, as an example, the results of the CCD with 1 C discharge rate

are shown in Figs. 10 and 11, respectively. It can be seen that they

can track the true bateryterminal voltage and the true SOC with no

chattering ripples. The SOC tracking MSEs are all within the small

range of 5% after a few seconds, which shows that the proposed

SOC observer is capable of tracking the true SOC accurately in thepresence of the incorrect initial SOC. This is due to the fact that

the switching gains are adjusted to the appropriate levels as the

corresponding errors decrease.

The testing data are obtained by using the variable current dis-

charge(VCD) profilewith thesame average dischargeratesas those

of theCCD profile. The proposed AGSMO is used to estimate theter-

minal voltage andthe SOC forthe VCD profile in different discharge

rates. As an illustration, the results of terminal voltage and the SOC

of the VCD with the average discharge rate of 1Cn are shown in

Figs. 12 and 13, respectively. It can be seen that the AGSMO has

Fig. 10. Terminal voltage of LiPB and AGSMOand MSE in the CCD profile of 1Cn.

peformed great robustness and capability to track the battery ter-

minalvoltage andthetrue SOCregardless ofan incorrectinitialSOC.

TheSOC estimation MSEs areboundedin therangeof 5% for1090%

of SOC with minor chattering ripples. Again, this is due to the factthat the switching gains are adaptively adjusted to adequate levels

against the tracking errors calculated by Eq. (12).

Inorderto demonstratethe SOCestimationbasedon theAGSMO

superior to the conventional SMO with the consideration of hys-

teresis effect, the urban dynamometer driving schedule (UDDS)test

with corresponding current profile in Fig. 14 has been conducted

on the LiPB. The UDDS test is a typical dynamic driving cycle,which

is usually used to evaluate the vehicle performance. As shown in

Fig. 14, the UDDS cycle as speed versus time has been converted

to current versus time with respect to the capacity of 5Ah LiPB

Fig. 9. Configuration of proposed AGSMO system.

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X. Chen et al. / Computers and Chemical Engineering 64 (2014) 114123 121

Fig. 11. True and AGSMO estimated SOCand SOC MSE in the CCD profile of 1Cn.

through the EV simulation. The negative current during the decel-

eration and braking of the EVs represents regenerative energy to

charge LiPB. The current profile of the UDDS is loaded to the LiPB

until the battery reaches its cut-off voltage of 2.7 V and the results

are shown in Fig. 15. The same current profile is also loaded to the

proposed AGSMO is used to estimate the battery terminal voltage

and the SOC.As shown in Figs. 15and16, theAGSMO isable totrack

the battery terminal voltage and true SOC with incorrect initial

Fig. 12. Terminal voltage of LiPB and AGSMO and MSE in the VCD profile of 1Cn.

Fig. 13. True and AGSMOestimated SOC and SOCMSE in the VCD profile of1Cn.

values. It can also achieve fast SOC convergence with minor chat-tering ripples. The AGSMO has compared with the conventional

SMO with the constant switching gains for the SOC estimation

(Kim, 2006; Chen et al., 2012), the MSEs of the SOC estimation

for each type of current profile with different discharge rates

are summarised in Table 2. It can be seen that the MSEs of the

SOC estimation based on the AGSMO is always lower than those

based on the conventional SMO approach. Therefore, the proposed

AGSMO can provide more robustand accurate SOC estimation than

Fig. 14. One UDDS cycle with correspondingcurrent profile.

Table 2

MSEs of SOC estimation based SMO and AGSMO in different current profiles.

Current profile SMO SOC estimation

MSEs (%)

AGSMO SOC estimation

MSEs (%)

CCD 1.5Cn 8.364 3.676

CCD 1Cn 7.523 2.583

CCD 1/3Cn 6.362 2.204

VCD 1.5Cn 9.364 3.432

VCD 1Cn 8.438 2.318

VCD 1/3Cn 7.238 2.069

UDDS 8.152 2.573

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122 X. Chen et al. / Computers and Chemical Engineering 64 (2014) 114123

Fig. 15. Terminal voltage of LiPB and AGSMO and MSE in theUDDS profile.

conventional SMO-based approach. Furthermore, the proposed

AGSMO has compared with the EKF, which is one of popular KF

based approaches. Fig. 17 illustrates the EKF based SOC estimation

results in the UDDS current profile. It can be observed that the EKF

based approach has large estimation errors with slow convergence

in thecomparison with theproposedAGSMOapproachas shown in

Fig. 16, which demonstrates that the proposed AGSMO can provide

robust tracking capability against modelling errors and incorrect

initial states.

Fig. 16. True and AGSMO estimated SOC and SOC MSE in the UDDS profile.

Fig. 17. True and EKFestimated SOC and SOC MSE in theUDDS profile.

6. Conclusions

The adaptive gain sliding mode observer (AGSMO) for the SOC

estimation basedon the combined battery equivalentcircuit model

(CBEDM) has been presented. The system state equations for the

AGSMO are derived from the CBEDM using equivalent control

concepts. The LiPB is utilised to conduct the experiments. The

parameters of theCBEDMare extractedfromthe experimentaldata

under a sequence of pulse current discharge. Theexperimental data

of the LiPB in constant current and variable current discharges are

used to verify the performance and effectiveness of the proposed

observer forthe SOCestimation.It shows that theproposed AGSMO

has outperformed conventional SMO forSOC estimation of the LiPB

in terms of robust tracking capability with less chattering ripples

and high estimation accuracy.

Acknowledgment

This research work is supportedby Commonwealthof Australia,

through the Cooperative Research Centre for Advanced Automotive

Technology (AutoCRC),under the project of Electric Vehicle Control

Systems and Power Management (C2-801).

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