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8/12/2019 Adaptive Gain Sliding Mode Observer for State of Charge Estimation Based on Combined Battery Equivalent Circui
1/10
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:[email protected](X. Chen), [email protected](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
0098-1354/ 2014 Elsevier Ltd. All rights reserved.
http://localhost/var/www/apps/conversion/tmp/scratch_8/dx.doi.org/10.1016/j.compchemeng.2014.02.015http://www.sciencedirect.com/science/journal/00981354http://www.elsevier.com/locate/compchemengmailto:[email protected]:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_8/dx.doi.org/10.1016/j.compchemeng.2014.02.015http://localhost/var/www/apps/conversion/tmp/scratch_8/dx.doi.org/10.1016/j.compchemeng.2014.02.015mailto:[email protected]:[email protected]://crossmark.crossref.org/dialog/?doi=10.1016/j.compchemeng.2014.02.015&domain=pdfhttp://www.elsevier.com/locate/compchemenghttp://www.sciencedirect.com/science/journal/00981354http://localhost/var/www/apps/conversion/tmp/scratch_8/dx.doi.org/10.1016/j.compchemeng.2014.02.0158/12/2019 Adaptive Gain Sliding Mode Observer for State of Charge Estimation Based on Combined Battery Equivalent Circui
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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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