State of charge estimation of lithium-ion batteries using fractional order sliding mode observer

State of charge estimation of lithium-ion batteries using fractional
order sliding mode observer
Qishui Zhong a,n
, Fuli Zhong a,n
, Jun Cheng b
, Hui Li a
, Shouming Zhong c
a
School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu, Sichuan Province611731, PR China
b
School of Science, Hubei University for Nationalities, Enshi, Hubei Province445000, PR China
c
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan Province611731, PR China
a r t i c l e i n f o
Article history:
Received 23 February 2015
Received in revised form
22 March 2016
Accepted 17 September 2016
Available online 15 October 2016
This paper was recommended for publica-
tion by Dr. Q.-G. Wang
Keywords:
State of charge estimation
Sliding mode observer
Fractional order RC equivalent circuit model
Lithium-ion battery
a b s t r a c t
This paper presents a state of charge (SOC) estimation method based on fractional order sliding mode
observer (SMO) for lithium-ion batteries. A fractional order RC equivalent circuit model (FORCECM) is
firstly constructed to describe the charging and discharging dynamic characteristics of the battery. Then,
based on the differential equations of the FORCECM, fractional order SMOs for SOC, polarization voltage
and terminal voltage estimation are designed. After that, convergence of the proposed observers is
analyzed by Lyapunov’s stability theory method. The framework of the designed observer system is
simple and easy to implement. The SMOs can overcome the uncertainties of parameters, modeling and
measurement errors, and present good robustness. Simulation results show that the presented estima-
tion method is effective, and the designed observers have good performance.
& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Battery which is an important energy storage equipment has
been widely used in various electric vehicles (EVs), and plays an
important role in EVs [1,2]. Lithium-ion batteries are favored as a
promising power source for EVs by the researchers because of the
characteristics of high cell voltage, high specific power and long
cycle-life [3,4]. In EVs, battery management system was applied to
ensure the reliable operations of battery [5], in which state of
charge (SOC) is an important parameter [6]. SOC often suffer the
influences of random factors like driving loads, operating envir-
onment and nonlinear characteristics [7]. Poor SOC estimation
may lead to larger SOC swing, over-charging and over-discharging
causing the cycle life decline or lower efficiency, it is very sig-
nificant to estimate SOC accurately to improve power distribution
efficiency and usage life [8–10].
A number of SOC estimation methods and techniques have
been proposed in recent years, e.g. ampere–hour counting
method, artificial neural network, support vector machine tech-
nique, Kalman filter-based method and electrochemical impe-
dance spectroscopy method [11]. Ampere–hour counting method
is simple and easy to implement, but requires the prior knowledge
of initial SOC and suffers from accumulated errors [12]. Estimating
the SOC based on artificial neural networks and support vector
machine [11,13] can lead to good SOC estimation results with
appropriate training data sets. But they require a great number of
training samples to train the model. Impedance measurement is
an effective technique for SOC estimation [15,16]. In [14], an
impedance spectra-based approach to estimate SOC was pre-
sented. However, this kind of method requires a set of costly and
auxiliary equipments to carry out the impedance measurement
that is inconvenient in EVs.
The Kalman filter-based method is generally applied to esti-
mate the SOC online or offline [4,7,11,17–19]. In the research on
SOC estimation, both the linear model based and nonlinear model
based methods were applied to estimate the SOC. In order to
improve the robustness and estimation accuracy, some adaptive
Kalman filter estimation methods for SOC estimation were pro-
posed, and the performance was improved. However, these Kal-
man filter-based SOC estimation algorithms often require accurate
parameters of the model, and assume that constant values of the
process and measurement noise covariance are known.
Fractional calculus has been applied in various fields, for
example, control [20–22,25], signal processing and system mod-
eling [23,24,26,27], and some related researches such as stability
analysis of fractional order systems [28]. Recently, fractional cal-
culus was applied in state of charge estimation of battery
[12,29,30]. Ref. [12] introduced a fractional calculus method to
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ISA Transactions
http://dx.doi.org/10.1016/j.isatra.2016.09.017
0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.
n
Corresponding authors.
E-mail addresses: zhongqs@uestc.edu.cn (Q. Zhong),
zhongfulicn@163.com (F. Zhong).
ISA Transactions 66 (2017) 448–459

model the constant phase element in the impedance model. Based
on the impedance model, a fractional Kalman filter was introduced
to estimate the SOC of the lithium-ion battery, and good estimate
results were achieved. In Refs. [29,30], fractional order sliding
mode observer designed method was employed to estimate the
state of charge of lithium-ion batteries based on the presented
equivalent circuit model, and the experimental results show that
the designed observers were effective and possess good
performance.
In the past, sliding mode observer (SMO) has been employed to
estimate the SOC for the battery [14,31–33]. The SMO-based SOC
estimation method can overcome the drawbacks of the conven-
tional SOC estimation methods like large cumulative errors. It is
simple and robust to modeling errors. To do the further research
on sliding mode observer for SOC estimation of batteries is
significant.
The purpose of this paper is to establish an SOC estimation
method for lithium-ion batteries which combines the advantages
of SMO with the excellent modeling ability of fractional calculus.
Firstly, the fractional calculus is employed to model the battery,
and a fractional order RC equivalent circuit model (FORCECM) is
set up to characterize the charging and discharging dynamics of
the lithium-ion battery. Then, we design fractional order SMOs to
estimate the SOC. In order to guarantee the robustness stability
and the estimation performance of the designed SMOs, the rele-
vant conditions are derived out. Finally, the experiments are car-
ried out, and the results show that our method is effective.
This paper is organized as follows. In Section 2, the basic
definitions, lemmas and theorems are introduced. In Section 3, the
fractional order RC equivalent circuit model and the dynamic
equations which are employed to describe the dynamics of the
battery are presented in detail. Design methodology of the frac-
tional order SMOs for SOC estimation is presented in Section 4.
And the results of the test experiment of designed SMOs are
shown to verify the performance of the proposed method in
Section 5, followed by conclusion in Section 6.
Notations: Rn
denotes n-dimensional Euclidean space. J Á J
denotes a 2-norm.
2. Basic definitions, theorems and lemmas
Let us introduce some definitions, lemmas and theorems that
will be used in this paper. The Riemann–Liouville definition of
α-th order fractional derivative is given by [21,23,34]
Dα
t f ðtÞ ¼
1
ΓðNÀαÞ
d
N
dt
N
Z t
0
f ðsÞ
ðtÀsÞαÀ N þ 1
ds ð1Þ
where f(t) is an integrable function, ΓðÁÞ is the Gamma function, N
is the first integer larger than α (NÀ1rαoN). The Riemann–
Liouville definition of q-th fractional integral is described as
0Iq
t f ðtÞ ¼
1
ΓðqÞ
Z t
0
f ðsÞ
ðtÀsÞ1À q
ds ð2Þ
where NÀ1rqoN.
Lemma 1 ([35,37,38]). For a non-autonomous fractional-order sys-
tem Dν
xðtÞ ¼ f ðx; tÞ in which νAð0; 1Þ and f ðx; tÞ satisfies the Lipschitz
condition with a Lipschitz constant k40, let x¼0 be an equilibrium
point. When there exists a Lyapunov candidate EðxðtÞ; tÞ satisfying
ρJxJα rEðxðtÞ; tÞrσJxJαϱ; ð3Þ
Dν
EðxðtÞ; tÞr ÀγJxJαϱ; ð4Þ
where ρ; σ; γ; α; ϱ are positive constants, then the equilibrium point is
asymptotic stable.
Lemma 2 ([36,35]). For αAC, ReðαÞ40, À1ox1 ox2 o þ1, and
1rpr1, the fractional integral x1
Iα
t f ðtÞ is bounded in Lpðx1; x2Þ
Jx1
Iα
t f ðtÞJ rβJf ðtÞJ; ð5Þ
where β ¼ ðx2 À x1ÞReðαÞ
ReðαÞj ΓðαÞj .
Lemma 3 ([35]). Consider a fractional-order nonautonomous sys-
tem Dν
t xðtÞ ¼ f ðx; tÞ, where νAð0; 1Þ, f : Ω Â ½0; þ1Š-Rn
is piece-
wise continuous in t, ΩARn
is a closed set that contains the origin
x¼0, the initial value condition is xðt0Þ. The constant x0 is an equi-
librium point of fractional dynamic system (without loss of generality,
let the equilibrium point be 0). Choose a Lyapunov function
EðtÞ ¼ 2xT
ðtÞxðtÞ. According to Leibniz's rule of differentiation, the νth-
order time derivative of E(t) can be expressed as
Dν
t EðtÞ ¼ ðDν
t xÞT
xþxT
ðDν
t xÞþ2Ψ, where Ψ ¼
P1
k ¼ 1
Γð1 þνÞðDk
t xÞT
ðDν À k
t xÞ
Γð1 þkÞΓð1À k þνÞ
.
Then, there exists a positive constant ψ1 such that
X1
k ¼ 1
Γð1þνÞðDk
t xÞT
ðDνÀk
t xÞ
Γð1þkÞΓð1ÀkþνÞ

rψ1 JxJ: ð6Þ
3. Equivalent circuit model for lithium-ion battery
The charging and discharging process of lithium-ion battery is a
complex electrochemistry reaction procedure. In this paper, the
fractional calculus is applied to describe the charging and dis-
charging dynamics. A fractional order RC equivalent circuit model
for lithium-ion battery is employed, in which a fractional order RC
loop is used to model the polarization effect, nonlinear factors, and
approximate the modeling errors. Then an SOC estimation method
and fractional order equivalent circuit model for the battery are
proposed.
The model mainly consists of a capacitance Cp which is used to
model the polarization effect, a diffusion resistance Rp, an open
circuit voltage (OCV) denoted as Voc which is related to the SOC Z,
an ohmic resistance Rt employed to model the ohmic behavior of
the battery cell, terminal voltage Vt and instantaneous current.
Others are depicted by fractional-order terms. The model used in
Ref. [33] employs a capacitance, a resistance and an uncertain term
to model the polarization effect. This uncertain term can model
the uncertainty of the parameters of a battery. As the electro-
chemical reaction in the battery is extremely complex, the model
in this paper considers the characteristics of the battery further.
Not only a resistance, a capacitance and an uncertain term, but
also a special term depicted by a fractional order model is applied
to model the polarization effect, nonlinear factors, and approx-
imate the errors caused by the model. It is named as fractional-
order element (FOE) which aims at improving the model accuracy.
The polarization capacitance is in the FOE component. The voltage
of the FOE is described as DαÀ 1
t Vp which is in the form of
fractional-order integral. When α ¼ 1, it becomes the common
used one, Vp. The diffusion resistance, unknown term ϕp and
fractional-order element component form a fractional-order RC
loop. Symbols ϕp and ϕv denote uncertainties in the battery. The
FORCECM is shown in Fig. 1.
Based on the definition of SOC for lithium-ion battery, the
mathematical expression for SOC is given by
ZðtÞ ¼ Zð0Þþ
Z t
0
IðxÞ
Cca
dx
¼ Zð0Þþ
Z t
0
ImðxÞ
CnomðTÞþΔCnomðT; tÞ
dxþ
Z t
0
ΔIðxÞ
CnomðTÞþΔCnomðT; tÞ
dx
Q. Zhong et al. / ISA Transactions 66 (2017) 448–459 449

¼ Zð0Þþ
Z t
0
ImðxÞ
CnomðTÞ
dxþ
Z t
0
ÀΔCnomðT; tÞImðxÞ
C2
nomðTÞþCnomðTÞΔCnomðT; tÞ
dx
þ
Z t
0
ΔIðxÞ
CnomðTÞ
dxþ
Z t
0
ÀΔCnomðT; tÞΔIðxÞ
C2
nomðTÞþCnomðTÞΔCnomðT; tÞ
dx
¼ Zð0Þþ
Z t
0
ImðxÞ
CnomðTÞ
dxþ
Z t
0
ΔIðxÞ
CnomðTÞ
dxþgΔcðtÞþσ1ðtÞ; ð7Þ
where Cca is the capacity of battery, CnomðTÞ denotes the history
measured capacity when temperature is T, ΔCnomðT; tÞ is the
uncertainty of capacity at time t with temperature T, ImðtÞ and ΔIðtÞ
are the measured instantaneous current and measured noise of
current respectively, gΔcðtÞ ¼
R t
0
ÀΔCnomðT;tÞImðxÞ
C2
nomðTÞ þ CnomðTÞΔCnomðT;tÞ
dx denotes the
effects caused by the measured noise and/or uncertainty of the
capacity, and σ1ðtÞ ¼
R t
0
ÀΔCnomðT;tÞΔIðxÞ
C2
nomðTÞ þCnomðTÞΔCnomðT;tÞ
dx is the effects term
that results from the measurement error of current and the
uncertainty of capacity.
From the expression (7), one can calculate the derivative with
respect to time as follows:
dZðtÞ
dt
¼
d
dt
Z t
0
ImðxÞ
CnomðTÞ
dxþ
d
dt
Z t
0
ΔIðxÞ
CnomðTÞ
dxþ
d
dt
gΔcðtÞþ
d
dt
σ1ðtÞ
¼
IðtÞ
CnomðTÞ
þ _gΔcðtÞþ _σ1ðtÞ: ð8Þ
Since some system parameters may vary with the temperature
when the power source battery working in vehicles, temperature T
is considered herein.
The mathematical relationship of terminal voltage can be
shown as:
VtðtÞ ¼ VocðZÞþIðtÞRt þDαÀ1
VpðtÞ
¼ VocðZÞþIðtÞ RtðT0ÞþΔRtðTÞþΔRtt
À Á
þDαÀ 1
VpðtÞ
¼ VocðZÞþImðtÞ RtðT0ÞþΔRtðTÞ
À Á
þDαÀ 1
VpðtÞþImðtÞΔRtt þΔIðtÞRt;
ð9Þ
where RtðT0Þ denotes the ohmic resistance with respect to T0, ΔRtðTÞ
is the changes of ohmic resistance related with temperature T, ΔRtt is
the effects of measured error and uncertainty varies with time, the
VocðZÞ is with respect to SOC, I(t) presents the instantaneous current,
and VpðtÞ is the polarization voltage, αA½0; 1Š.
From (8) and (9), one can obtain
_ZðtÞ ¼
IðtÞ
CnomðTÞ
þ _gΔcðtÞþ _σ1ðtÞ
¼
VtðtÞÀVocðZÞÀDαÀ 1
VpðtÞ
CnomðTÞRt
þ _gΔcðtÞþ _σ1ðtÞ
¼
VtðtÞÀVocðZÞÀDαÀ 1
VpðtÞ
CnomðTÞRtðTÞ
þ _gΔcðtÞþ _σ1ðtÞþmΔðtÞ
¼ φ1VtðtÞÀφ1VocðZÞÀφ1DαÀ 1
VpðtÞþ _gΔcðtÞþ _σ1ðtÞþmΔðtÞ;
ð10Þ
where φ1 ¼ 1
CnomðTÞRt ðTÞ
, φ2 ¼ À ΔRtt
CnomðTÞ R
2
t ðTÞ þΔRtt Rt ðTÞ
À Á, mΔðtÞ ¼
ÀΔRtt Vt ðtÞ ÀVocðZÞ ÀDα À 1
VpðtÞð Þ
CnomðTÞ R
2
t ðTÞ þΔRtt Rt ðTÞ
À Á is the uncertainty caused by the unknown
changes of ohmic resistance Rt, and RtðTÞ ¼ RtðT0ÞþΔRtðTÞ.
For the diffusion resistance Rp and capacitance Cp, considering
the effects of temperature and unknown factors, one can get Rp ¼
RpðT0ÞþΔRpðTÞþΔRpt and Cp ¼ CpðT0ÞþΔCpðTÞþΔCpt, where
RpðT0Þ and CpðT0Þ are measured diffusion resistance and capaci-
tance respectively when temperature is T0. Both ΔRpðTÞ and ΔCpðTÞ
are the changes of diffusion resistance and capacitance caused by
temperature. While ΔRpt and ΔCpt denote the uncertainty of dif-
fusion resistance and capacitance caused by instantaneous
unknown factors.
In the model, polarization voltage due to the instantaneous
current is presented as:
Dα
VpðtÞ ¼ À
DαÀ 1
VpðtÞ
CpRp
þ
IðtÞ
Cp
¼ Àθ1DαÀ 1
VpðtÞþθ3IðtÞÀθ2DαÀ 1
VpðtÞþθ4IðtÞ
¼ Àθ1DαÀ 1
VpðtÞþθ3ImðtÞÀθ2DαÀ 1
VpðtÞþθ3ΔIðtÞþθ4IðtÞ
¼ À
1
CpðTÞRpðTÞ
DαÀ1
VpðtÞþ
1
CpðTÞ
ImðtÞÀθ2DαÀ 1
VpðtÞ
þθ3ΔIðtÞþθ4IðtÞ; ð11Þ
where θ1 ¼ 1
CpðTÞRpðTÞ
, θ2 ¼
À ΔCpt Rp ÀCpðTÞΔRpt
C
2
pt ðTÞR
2
pðTÞ þ ΔCpt RpCpðTÞRpðTÞþ C
2
pðTÞΔRpt RpðTÞ
,
θ3 ¼ 1
CpðTÞ
, θ4 ¼
À ΔCpt
C
2
pðTÞ þΔCpt CpðTÞ
, RpðTÞ ¼ RpðT0ÞþΔRpðTÞ and
Cp ¼ CpðT0ÞþΔCpðTÞ. The term Àθ2DαÀ 1
VpðtÞ denotes the uncer-
tainty results from the uncertainties of diffusion resistance and
polarization capacitance. While θ3ΔIðtÞ and θ4IðtÞ are the uncer-
tainty caused by measurement error of current and polarization
capacitance, respectively.
From (9), one can get the derivative of terminal voltage with
respect to time as:
d
dt
VtðtÞ ¼
d
dt
VocðZÞþ
d
dt
IðtÞRtð ÞþDα
VpðtÞ
¼
d
dt
VocðZÞþ
d
dt
ImðtÞ RtðT0ÞþΔRtðTÞ
À Á
þDα
VpðtÞ
þ
d
dt
ΔIðtÞ RtðT0ÞþΔRtðTÞ
À ÁÀ Á
þ
d
dt
IðtÞΔRt
À Á
ð12Þ
Considering the high capacitance, the time derivative of term-
inal voltage Vt with respect to current is negligible if a fast sam-
pling time is obtained. From (11) and (12), one has
_V tðtÞ ¼
d
dt
VocðZÞþ
d
dt
ðImðtÞ RtðT0ÞþΔRtðTÞ
À Á
ÞþDα
VpðtÞ
þ
d
dt
ΔIðtÞ RtðT0ÞþΔRtðTÞ
À ÁÀ Á
þ
d
dt
IðtÞΔRt
À Á
¼ η_Z þΔ_RttIðtÞþDα
VpðtÞ
¼ η_Z þΔ_RttIðtÞÀθ1DαÀ 1
VpðtÞþθ3IðtÞÀθ2DαÀ 1
VpðtÞþθ4IðtÞ
¼
ηIðtÞ
CnomðTÞ
þη_gΔcðtÞþη _σ1ðtÞþΔ_RttIðtÞÀθ1DαÀ 1
VpðtÞþθ3IðtÞ
Àθ2DαÀ 1
VpðtÞþθ4IðtÞ
¼
η
CnomðTÞ
þθ3

IðtÞÀθ1 VtðtÞÀVocðZÞÀIðtÞRtð Þþm2ðtÞ
¼ Àθ1VtðtÞþθ1VocðZÞþImðtÞ

θ1RtðT0Þþθ1ΔRtðTÞ
þ
η
CnomðTÞ
þθ3

þm2ðtÞþθ1ΔRttImðtÞ
þΔIðtÞ θ1Rt þ
η
CnomðTÞ
þθ3

; ð13Þ
where m2ðtÞ ¼ η_gΔcðtÞþη _σ1ðtÞþΔ_RttIðtÞÀθ2DαÀ 1
VpðtÞþθ4IðtÞ,
θ1ΔRttImðtÞ and θ1RtΔIðtÞþ ηΔIðtÞ
CnomðTÞþθ3ΔIðtÞ denote the
uncertainty terms.
When α ¼ 1, the dynamic system equations for SOC, polariza-
tion voltage and terminal voltage can be respectively shown as:
_ZðtÞ ¼ φ1VtðtÞÀφ1VocðZÞÀφ1VpðtÞþ ~ϕzðtÞ;
Fig. 1. The fractional order RC equivalent circuit model (FORCECM).
Q. Zhong et al. / ISA Transactions 66 (2017) 448–459450

_V pðtÞ ¼ Àθ1VpðtÞþθ3ImðtÞþ ~ϕpðtÞ;
_V tðtÞ ¼ Àθ1VtðtÞþθ1VocðZÞþImðtÞ θ1RtðT0Þþθ1ΔRtðTÞþ
η
CnomðTÞ
þθ3

þ ~ϕvðtÞ;
where ~ϕpðtÞ ¼ Àθ2VpðtÞþθ3ΔIðtÞþθ4IðtÞ, ~ϕzðtÞ ¼ _gΔcðtÞþ _σ1ðtÞþ
~mΔðtÞ, ~mΔðtÞ ¼
ÀΔRtt Vt ðtÞ À VocðZÞ ÀVpðtÞð Þ
CnomðTÞ R
2
t ðTÞ þ ΔRtt Rt ðTÞ
À Á , ~m2ðtÞ ¼ η_gΔcðtÞþη _σ1ðtÞþ
Δ_RttIðtÞÀθ2VpðtÞþθ4IðtÞ, and ~ϕvðtÞ ¼ ~m2ðtÞþθ1ΔRttImðtÞþΔIðtÞ
θ1Rt þ η
CnomðTÞ þθ3

. Then one can ﬁnd that, without the uncer-
tainty terms ~ϕzðtÞ, ~ϕpðtÞ and ~ϕvðtÞ, the above special case integer
order model shares the similar form of the one in Ref. [31].
Compared with the integer order model, the presented FORCECM
provides more parameter options for approximating the battery
dynamic system, which helps to model the dynamics of battery
accurately.
4. Fractional order observer design for SOC, polarization
voltage estimation
The state of the battery can be estimated by using observer
generally since the observability matrix of the system always has
full rank [33]. Let evðtÞ ¼ VtðtÞÀ ^V tðtÞ, ð ^V t; ^ZÞ be the estimations of
ðVt; ZÞ. For the dynamic system equation of terminal voltage
_V tðtÞ ¼ Àθ1VtðtÞþθ1VocðZÞþImðtÞ θ1RtðT0Þþθ1ΔRtðTÞþ
η
CnomðTÞ
þθ3

þϕvðtÞ;
ð14Þ
where ϕvðtÞ ¼ m2ðtÞþθ1ΔRttImðtÞþθ1RtΔIðtÞþ ηΔIðtÞ
CnomðTÞ þθ3ΔIðtÞ.
Then the output of the SMO for terminal voltage is given by
_^V tðtÞ ¼ Àθ1
^V tðtÞþθ1Vocð ^ZÞþθ13ImðtÞþLvwðVtðtÞÀ ^V tðtÞÞ; ð15Þ
where θ13 ¼ θ1RtðT0Þþθ1ΔRtðTÞþ η
CnomðTÞ þθ3, Lv is the constant that
will be designed. wðÞ is a switch control function which is a
smooth monotone increasing function (wðxÞ ¼ χi when x4εiu, 0
rwðxÞrχi while 0rxrεie, Àχi rwðxÞo0 when Àεie rxo0,
wðxÞo Àχi when xo Àεie, where εie 40 and 0oχi o1). Espe-
cially, one can choose the tanhðÞ to act as wðÞ.
From (14) and (15), one has the following error dynamic
equation:
_evðtÞ ¼ Àθ1evðtÞþθ1ðVocðZÞÀVocð ^ZÞÞþϕvðtÞÀLvwðevðtÞÞ: ð16Þ
For the convergence analysis of the error ev, one can choose a
Lyapunov candidate function as EvðtÞ ¼ 1=2e2
v , and its derivative
with respect to time is shown as
_EvðtÞ ¼ evðtÞ_evðtÞ ¼ Àθ1e2
v ðtÞþθ1evðtÞðVocðZÞÀVocð ^ZÞÞþevðtÞϕvðtÞ
ÀLvevðtÞwðevðtÞÞ:
Select Lv 4 θ1ðVocðZÞÀVocð ^ZÞÞþϕvðtÞ

¼ θ1ηez þϕvðtÞ

, then the
sign of ev and _ev has opposite sign, and _EvðtÞ ¼ evðtÞ_evðtÞo0. Thus
when the sliding mode is activated, one can ﬁnd that ev tends to a
small value and _ev tends to zero.
Based on the equivalent control technology, when ev and _ev are
0, the error plant in the sliding mode behaves as if LvwðevðtÞÞ is
replaced by Lv wðevðtÞÞ½ Šeq. When evðtÞ ¼ _evðtÞ ¼ 0 and the sliding
mode surface is achieved, the ϕvðtÞ will vanish, one has the fol-
lowing equation:
VocðZðtÞÞÀVocð ^ZðtÞÞ ¼
Lv wðevðtÞÞ½ Šeq
θ1
: ð17Þ
As the OCV is monotonically increasing with the SOC, the term
VocðZðtÞÞÀVocð ^ZðtÞÞ can be thought of as piecewise linear in
ZðtÞÀ ^ZðtÞ, there is VocðZðtÞÞÀVocð ^ZðtÞÞ % ηZðtÞÀη^ZðtÞ, where η is
the piecewise linear gain. From (10), one has
ezðtÞ ¼
ηθ1
; ð18Þ
where ezðtÞ ¼ ZðtÞÀ ^ZðtÞ is the SOC estimation error.
Based on the dynamic equation for SOC we have
_ZðtÞ ¼ φ1VtðtÞÀφ1VocðZÞÀφ1DαÀ1
VpðtÞþϕzðtÞ; ð19Þ
where ϕzðtÞ ¼ _gΔcðtÞþ _σ1ðtÞþmΔðtÞ, the observer for SOC estima-
tion is designed as:
_^Z ðtÞ ¼ φ1
^V tðtÞÀφ1Vocð ^ZÞÀφ1DαÀ1 ^V pðtÞþhzðZðtÞÀ ^ZðtÞÞþLzwðZðtÞÀ ^ZðtÞÞ;
ð20Þ
where ^ZðtÞ and ^V pðtÞ are the estimates for ZðtÞ and VpðtÞ, respec-
tively. hz and Lz are constants. From (19) and (20), one has an error
dynamic equation as follows:
_ezðtÞ ¼ φ1evðtÞÀφ1 VocðZÞÀVocð ^ZÞ

Àφ1 DαÀ1
VpðtÞÀDαÀ 1 ^V pðtÞ

þϕzðtÞÀhzezðtÞÀLzwðezðtÞÞ: ð21Þ
Select a Lyapunov function EzðtÞ ¼ 1=2e2
z , and calculate its
derivative with respect to time, one has
_EzðtÞ ¼ ezðtÞ_ezðtÞ
¼ φ1ezðtÞevðtÞÀφ1ezðtÞ VocðZÞÀVocð ^ZÞ

Àφ1ezðtÞ DαÀ1
VpðtÞÀDαÀ1 ^V pðtÞ

þezðtÞϕzðtÞÀhzezðtÞezðtÞÀLzezðtÞwðezðtÞÞ
¼ Àðφ1ηþhzÞe2
z ðtÞþezðtÞ φ1evðtÞÀφ1DαÀ1
epðtÞþϕzðtÞ

ÀLzezðtÞwðezðtÞÞ;
where epðtÞ ¼ VpðtÞÀ ^V pðtÞ denotes the estimation error of polar-
ization voltage.
When hz Z Àφ1η and Lz 4 φ1evðtÞÀφ1DαÀ1
epðtÞþϕzðtÞ

, there
is _EzðtÞo0, which ensures that the estimation error ezðtÞ tends to a
very small value around zero. While the ezðtÞ and _ezðtÞ reach zero,
the error plant in the sliding mode behaves as if hzezðtÞþLzwðezðtÞÞ
is replaced by hz ezðtÞ½ Šeq þLz wðezðtÞÞ½ Šeq. When ezðtÞ ¼ _ezðtÞ ¼ 0 and
the sliding mode surface is achieved, the ϕzðtÞ will vanish, one has
0 ¼ φ1evðtÞÀφ1DαÀ 1
epðtÞÀLzwðezðtÞÞÀhzezðtÞ, then according to
equivalent control theory method, the following equation can be
obtained:
DαÀ1
epðtÞ ¼ evðtÞÀ
Lz
φ1
wðezðtÞÞ½ Šeq À
hz ezðtÞ½ Šeq
φ1
¼ evðtÞÀ
Lz
φ1
w
ηθ1
!
eq
À
hzLv wðevðtÞÞ½ Šeq
φ1ηθ1
: ð22Þ
From (20) and (18), one can rewrite the observer for SOC as:
_^Z ðtÞ ¼ φ1
^V tðtÞÀφ1Vocð ^ZÞÀφ1DαÀ 1 ^V pðtÞþ
hzLv wðevðtÞÞ½ Šeq
ηθ1
þLzw
ηθ1

: ð23Þ
Based on the dynamic system equation for the polarization
voltage which is rewritten as:
Dα
VpðtÞ ¼ Àθ1DαÀ1
VpðtÞþθ3ImðtÞþϕpðtÞ; ð24Þ
where ϕpðtÞ ¼ Àθ2DαÀ 1
VpðtÞþθ3ΔIðtÞþθ4IðtÞ, the fractional order
sliding mode observer for estimating the state parameter of
polarization voltage are given by
Dα ^V pðtÞ ¼ Àθ1DαÀ1 ^V pðtÞþθ3ImðtÞþhpðVpðtÞÀ ^V pðtÞÞþLpw VpðtÞÀ ^V pðtÞ

;
ð25Þ
where hp and Lp are constants which will be designed.
From (24) and (25), one has the error dynamic system for
polarization voltage as
Dα
epðtÞ ¼ Àθ1DαÀ1
epðtÞþϕpðtÞÀhpepðtÞÀLpwðepðtÞÞ: ð26Þ

To analyze the convergence of this error dynamic system,
choose a Lyapunov function EpðtÞ ¼ 2e2
pðtÞ, and calculate its frac-
tional order derivative with respect to time as:
Dα
EpðtÞ ¼ Dα
epðtÞ
À ÁT
epðtÞþeT
pðtÞ Dα
epðtÞ
À Á
þ2Ψ
¼ 2Ψ þ2eT
pðtÞðÀθ1DαÀ 1
epðtÞþϕpðtÞÀLpwðepðtÞÞÞÀ2hpe2
pðtÞ
r2ψ ep

þ2eT
epðtÞþϕpðtÞÀLpwðepðtÞÞÞÀ2hpe2
pðtÞ;
ð27Þ
where Ψ ¼
P1
k ¼ 1
Γðαþ1ÞðDk
t epðtÞÞT
ðDα À k
t epðtÞÞ
Γðk þ1ÞΓðαþ1 ÀkÞ
. According to a Lemma 2,
one has DαÀ 1
epðtÞ

rκ epðtÞ

. From Lemma 3, one gets the rela-
tionship Ψ

¼
P1
k ¼ 1
Γðαþ 1ÞðDk
t epðtÞÞT
ðDα À k
t epðtÞÞ
Γðk þ 1ÞΓðαþ 1À kÞ

rψ epðtÞ

. As a result,
for (27), when there exist hp Z0 and LpwðepðtÞÞ

4
ψ þθ1κ ep

þ ϕp

, there is Dα
EpðtÞr2ψ ep

þ 2eT
epðtÞ
þ ϕpðtÞÀLpwðepðtÞÞÞÀ2hpe2
pðtÞo0. From Lemma 1, one can con-
clude that epðtÞ tends to a very small value around 0.
From (22) and (25), the fractional order sliding mode observer
for polarization voltage can be rewritten as
Dα ^V pðtÞ ¼ Àθ1DαÀ 1 ^V pðtÞþθ3ImðtÞþhpDβ
evðtÞÀ
hpLz
φ1
Dβ
w
ηθ1
!
eq
À
hphzLvDβ
wðevðtÞÞ½ Šeq
φ1ηθ1
þLpw Dβ
evðtÞÀ
Lz
φ1
Dβ
w
ηθ1
!
eq
À
hzLvDβ
wðevðtÞÞ½ Šeq
φ1ηθ1

; ð28Þ
where β ¼ 1Àα.
5. Simulation results
After the fractional order RC equivalent circuit model of a
lithium-ion battery is established and the fractional order sliding
mode observers for the SOC, terminal voltage and polarization
voltage estimation are designed, the validation experiment of the
SOC estimation with the proposed method for lithium-ion bat-
teries is carried out to analyze the performance. Based on the
simulation platform Matlab/simulink, the fractional order RC
equivalent circuit model and the designed fractional order sliding
mode observers are established firstly. Then, the hybrid charging
and discharging, pulse discharging, pulse charging, and other
multiple groups of simulation experiments are conducted. To
investigate the accuracy and online estimation performance of our
method, the SOC estimation in the simulated city driving condi-
tions is done with the simulation platform. As the driving process
often contains acceleration, speed reduction, idling speed, con-
stant speed condition, etc., the current in the simulated city
driving conditions is applied to drive the FORCECM and fractional
order SMOs to investigate the online SOC estimation.
In the hybrid charging and discharging tests and simulated city
driving conditions simulation tests, the designed fractional order
SMOs are used to estimate the SOC, and the tracking performance
is analyzed at the same time.
The main objective of the experiment is to validate the effec-
tiveness of the proposed method, therefore, some parameters of
FORCECM of a lithium-ion battery are assumed. The assumed
profile of OCV versus SOC of the lithium-ion battery is shown in
Fig. 2, and some values of dots of the piecewise linearization
profiles of the curve of OCV versus SOC are listed in Table 1.
Parameters of the FORCECM are as follows: Cnom ¼ 6 A h,
Rt ¼ 0:01 Ω, Rp ¼ 0:007 Ω, Cp ¼ 1:4 Â 104
F, ϕz ¼ 0:0015 sin ðtÞ,
ϕp ¼ 0:005 sin ðtÞ, and ϕv ¼ 0:0001 sin ðtÞ. And the parameters for
the designed observers are Cnom ¼ 6 A h, Rt ¼ 0:0095 Ω,
Rp ¼ 0:00735 Ω, Cp ¼ 1:47 Â 104
F, M1 ¼ 6, M2 ¼ 0:008, and
M3 ¼ 10, where M1, M2 and M3 denote the sliding mode gains Lv, Lz
and Lp, respectively. hz and hp are set to 0. The tanh ðÞ is selected
to act as the switch control function wðÞ in observers. For the tests
pulse discharging, hybrid complex pulse, complex charging–dis-
charging current case and pulse charging, most parameters are the
same as those set above. The different ones are ϕp ¼ 0:01,
ϕv ¼ 0:001, M1 ¼ 6, M2 ¼ 0:004, M3 ¼ 2.
Initial states of the variables in the tests are listed in Table 2, in
which AddTest1, AddTest2 and AddTest3 present the pulse dis-
charging, hybrid complex pulse, complex charging–discharging
current case tests respectively.
The FORCECM of the lithium-ion battery is employed to simulate
the charging, discharging, and hybrid charging–discharging tests,
and the terminal voltage and current are generated in the simula-
tion experiment. The output terminal voltage and current are fed
into the designed fractional order SMOs, and then the fractional
order SMOs calculate and output the estimated SOC, terminal vol-
tage and the other estimated variable. The tracking ability, estima-
tion performance of the fractional order SMOs are analyzed by
comparing the output estimated state variables (terminal voltage,
SOC, etc.) with the true ones, respectively. Then the effectiveness
and accuracy of the proposed method are well analyzed.
Figs. 3–35 present the experiment results. Among them,
Figs. 4–7 are the estimation results of the hybrid charging and
discharging test, the profiles including the current, SOC, terminal
voltage and SOC estimation errors; Figs. 9–12 are the results of the
SOC estimation test in simulated city driving conditions. Fig. 3 is
the hybrid charging and discharging current in the test, and Fig. 8
shows the curve of current of the test in the simulated city driving
conditions. While Figs. 13, 18, 23 and 28 present the currents in
the hybrid complex pulse, pulse charging, pulse discharging and
complex charging–discharging current case tests separately.
Figs. 4 and 5 indicate that the output SOC of the fractional order
SMO of SOC can track the true SOC quickly, and the output
terminal voltage of the relative SMO can also track the real one in a
fast rate accurately. The estimation errors are large at the begin-
ning stage, because the setting initial states of the SMOs are quite
different from the true ones. But they can tend to the true ones
quickly and accurately, which means that the designed SMOs
possess good performance. Similar conclusions are obtained
through analyzing the results in Figs. 9 and 10. Therefore, the
proposed SOC estimation method in this paper is effective.
Results depicted in Figs. 6 and 11 are the SOC estimation errors.
And Figs. 7 and 12 show the SOC estimation relative errors. The error
ranges which are calculated after 1000 s in the estimating process are
listed in Table 3. From the experimental results presented in Table 3
0 0.2 0.4 0.6 0.8 1
3.4
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
SOC
OpenCircuitVoltage(V)
Fig. 2. The curve of open circuit voltage versus SOC of a lithium-ion battery.

and the profiles of the estimation errors, it can be found based on the
conditions we set in the simulation above, the SOC estimation error is
limited in a range of ½À0:01; 0:01Š, and the relative error range can be
limited to a range ½À2:5%; 2:5%Š. These results show that the
observers designed in this paper have good accuracy. The method
presented in this paper is effective for SOC estimation.
In order to analyze the effects of the observer gain on the
estimation accuracy further and try to search proper ranges of
observer gains, multiple simulation experiments are implemented
by using the charging and discharging current used in the simu-
lated city driving conditions test, with various observer gains
ðM1; M2; M3Þ. In the first group simulation experiment, we set
M2 ¼ 0:008, M3 ¼ 10, and conduct simulation with various M1,
they are 0.006, 0.06, 0.6, 1.8, 3.6, 4.8, 6, 8, 10, 14, 18, 24, 30, 36, 45,
85, 200, 2000, 20,000, 200,000. In the second group experiment,
fixing M1 ¼ 6 and M3 ¼ 10, we conduct SOC estimation with dif-
ferent M2, they are 8eÀ6, 8eÀ5, 8eÀ4, 8eÀ3, 0.016, 0.08, 0.8, 2, 4,
6, 8 and 10 separately. When it comes to the third group experi-
ment, we fix M1 ¼ 6, M2 ¼ 0:008, and implement the SOC esti-
mation with the proposed observers with various M3 shown as
0.001, 0.01, 0.1, 1, 10, 20, 30, 40, 50, 60, 70 and 80. Due to the limit
of the pages, only some results are listed herein. The experimental
results are shown in Figs. 33–35.
Table 1
Some dots of the piecewise linearization profiles of the curve of OCV versus SOC.
SOC 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
OCV(V) 3.48 3.54 3.59 3.63 3.662 3.688 3.718 3.763 3.828 3.918 4.028
η 0.6 0.5 0.4 0.32 0.26 0.3 0.45 0.65 0.9 1.1 1.1
Table 2
Initial values of variables in the tests.
Variables Simulated city
driving condi-
tions test
Charging and
discharging test
AddTest1,
AddTest2,
AddTest3
Pulse
charging
Z 0.95 0.8 0.95 0.3
bZ 0.6 0.6 0.6 0.2
Vt (V) 4.2 4.2 4.2 3.7
bV t (V) 3 3 3 3
Vp (V) 0 0 0 0
bV p (V) 0 0 0 0
0 1000 2000 3000 4000 5000
−2
−1
0
1
2
Time (s)
I(A)
Fig. 3. Current curve of the hybrid charging and discharging test.
0 1000 2000 3000 4000 5000 6000
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
SOC
True SOC
Estimated SOC
Fig. 4. SOC estimation result of hybrid charging and discharging test.
0 1000 2000 3000 4000 5000 6000
3
3.5
4
4.5
Time (s)
Terminalvoltage(V)
True terminal voltage
Estimated terminal voltage
Fig. 5. Terminal voltage estimation result of hybrid charging and discharging test.
0 1000 2000 3000 4000 5000 6000
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
Time (s)
SOCerror(1)
SOC estimation error
Fig. 6. SOC estimation error of hybrid charging and discharging test.

0 1000 2000 3000 4000 5000 6000
−4
−3
−2
−1
0
1
2
3
4
Time (s)
SOCrelativeerror(%)
SOC estimation relative error
Fig. 7. SOC estimation relative error of hybrid charging and discharging test.
0 500 1000 1500 2000 2500
−12
−10
−8
−6
−4
−2
0
2
4
Time (s)
I(A)
Current
Fig. 8. Current curve of SOC estimation test in the simulated city driving
conditions.
0 500 1000 1500 2000 2500
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
SOC
True SOC
Estimated SOC
Fig. 9. SOC estimation result of the test in the simulated city driving conditions.
0 500 1000 1500 2000 2500
3
3.5
4
4.5
Time (s)
Terminalvoltage(V)
Fig. 10. Terminal voltage estimation result of the test in the simulated city driving
conditions.
0 500 1000 1500 2000 2500
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
Time (s)
SOCerror(1)
Fig. 11. SOC estimation error of the test in the simulated city driving conditions.
0 500 1000 1500 2000 2500
−4
−3
−2
−1
0
1
2
3
4
Time (s)
SOCrelativeerror(%)
Fig. 12. SOC estimation relative error of the test in the simulated city driving
conditions.

As shown in Fig. 33, for the setting parameters in the simulation
tests, the range of observer gain, M1, can be selected as about 0.06–10
for obtaining smaller error range. As presented in Fig. 34, for the model
in the simulation tests, the range of observer gain, M2, can be selected
as about 8eÀ6–0.016 to achieve smaller error range. From the results
in Fig. 35, for the model in the simulation tests, the range of observer
0 1000 2000 3000 4000 5000
−7
−6
−5
−4
−3
−2
−1
0
1
2
Time (s)
I(A)
Current
Fig. 13. Current of the pulse discharging test.
0 1000 2000 3000 4000 5000
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
SOC
True SOC
Estimated SOC
Fig. 14. Estimation result of SOC in the pulse discharging test.
0 1000 2000 3000 4000 5000
3
3.5
4
4.5
Time (s)
Terminalvoltage(V)
Fig. 15. Estimation result of Vt in the pulse discharging test.
0 1000 2000 3000 4000 5000
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Time (s)
SOCerror(1)
Fig. 16. SOC estimation errors in the pulse discharging test.
0 1000 2000 3000 4000 5000
−10
−8
−6
−4
−2
0
2
4
6
8
10
Time (s)
SOCrelativeerror(%)
Fig. 17. SOC estimation relative errors in the pulse discharging test.
0 1000 2000 3000 4000 5000
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time (s)
I(A)
Current
Fig. 18. Current of the pulse charging test.

gain, M3, can be selected as about 0.001–10 to insure smaller error
range. However, these three observer gains cannot be too large at the
same time, for it may cause bad effect on the estimation performance.
And they cannot set to too small value in the meanwhile, otherwise, it
may cause observers' un-convergence which also lead to unsuccessful
estimation.
0 1000 2000 3000 4000 5000
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
SOC
True SOC
Estimated SOC
Fig. 19. Estimation result of SOC in the pulse charging test.
0 1000 2000 3000 4000 5000
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Time (s)
Terminalvoltage(V)
Fig. 20. Estimation result of Vt in the pulse charging test.
0 1000 2000 3000 4000 5000
−0.1
−0.05
0
0.05
0.1
0.15
Time (s)
SOCerror(1)
Fig. 21. SOC estimation errors in the pulse charging test.
0 1000 2000 3000 4000 5000
−20
−15
−10
−5
0
5
10
15
20
Time (s)
SOCrelativeerror(%)
Fig. 22. SOC estimation relative errors in the pulse charging test.
0 1000 2000 3000 4000 5000
−10
−5
0
5
10
Time (s)
I(A)
Current
Fig. 23. Current of the hybrid complex pulse charging test.
0 1000 2000 3000 4000 5000
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
SOC
True SOC
Estimated SOC
Fig. 24. Estimation result of SOC in the hybrid complex pulse test.

6. Conclusion
State of charge estimation of lithium-ion batteries using frac-
tional order sliding mode observer has been presented in this
paper. A fractional order RC equivalent circuit model is employed
to model lithium-ion battery, and fractional order SMOs are
designed to estimate the SOC. This method absorbs the beneﬁts of
0 1000 2000 3000 4000 5000
3
3.5
4
4.5
Time (s)
Terminalvoltage(V)
Fig. 25. Estimation result of Vt in the hybrid complex pulse test.
0 1000 2000 3000 4000 5000
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Time (s)
SOCerror(1)
Fig. 26. SOC estimation errors in the hybrid complex pulse test.
0 1000 2000 3000 4000 5000
−10
−8
−6
−4
−2
0
2
4
6
8
10
Time (s)
SOCrelativeerror(%)
Fig. 27. SOC estimation relative errors in the hybrid complex pulse test.
0 500 1000 1500 2000 2500
−15
−10
−5
0
5
10
Time (s)
I(A)
Current
Fig. 28. Current in the test of complex charging–discharging current case.
0 500 1000 1500 2000 2500
0
0.2
0.4
0.6
0.8
1
1.2
Time (s)
SOC
True SOC
Estimated SOC
Fig. 29. Estimation result of SOC in the complex charging–discharging current test.
0 500 1000 1500 2000 2500
3
3.5
4
4.5
Time (s)
Terminalvoltage(V)
Fig. 30. Estimation result of Vt in the complex charging–discharging current test.

SMOs in dealing with parameter uncertainties and measurement
noises, and the powerful ability for the description of different
substances of fractional order derivatives and integrals. It is simple
and able to overcome the weakness of conventional methods like
large cumulative errors with robustness to modeling errors. The
Lyapunov candidate conditions have been invoked to establish the
convergence of the presented observers. Compared with the
integer order model, the presented FORCECM provides more
parameter options for approximating the battery dynamic system.
For the case that the fractional order is 1, the FORCECM becomes
an integer RC equivalent circuit model. The designed sliding mode
observer contains fractional order derivative which is more gen-
eral. The simulation results validate the effectiveness and accuracy
of the proposed method. Under the assuming conditions in this
paper, the SOC estimation error range can be controlled in the
range of 70:01, and the relative error range can be controlled in
72:5% with proper observer gain values.
0 500 1000 1500 2000 2500
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Time (s)
SOCerror(1)
Fig. 31. SOC estimation errors in the complex charging–discharging current test.
0 500 1000 1500 2000 2500
−10
−8
−6
−4
−2
0
2
4
6
8
10
Time (s)
SOCrelativeerror(%)
Fig. 32. SOC estimation relative errors in the complex charging–discharging
current test.
10
−2
10
0
10
2
10
4
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
M1
SOCrelativeerror(%)
Minimum of SOC relative error
Maximum of SOC relative error
Fig. 33. Range of SOC estimation relative error in the tests with various M1
(M2 ¼ 0:008, M3 ¼ 10).
10
−4
10
−2
10
0
−6
−4
−2
0
2
4
6
8
M
2
SOCrelativeerror(%)
Fig. 34. Range of SOC estimation relative error in the tests with various M2 (M1 ¼ 6,
M3 ¼ 10).
10
−3
10
−2
10
−1
10
0
10
1
10
2
−3
−2
−1
0
1
2
3
M3
SOCrelativeerror(%)
Fig. 35. Range of SOC estimation relative error in the tests with various M3 (M1 ¼ 6,
M2 ¼ 0:008).

Acknowledgment
The authors would like to thank the editor and the anonymous
reviewers for their comments and suggestions. The authors also
thank Dr. Chun Yin for precious suggestions and help. This
research was supported by the China Postdoctoral Science Foun-
dation (2013T60848) and Sichuan Science and Technology Plan
(2014GZ0079).
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Table 3
SOC estimation error ranges after 1000 s in the tests.
Tests Range of estimation
error
Range of estimation
relative error (%)
Hybrid charging and dischar-
ging test
[À0.002270,
0.00222]
[À0.275, 0.256]
Simulated city driving condi-
tions test
[À0.004699,
0.005039]
[À0.7056, 0.7338]
Pulse discharging [À0.001532,
0.009404]
[À0.5884, 2.3208]
Pulse charging [À0.005286,
0.007820]
[À0.8545, 1.572]
Hybrid complex pulse [À0.002480,
0.003193]
[À0.2808, 0.3644]
Complex charging–dischar-
ging current case
[À0.001201,
0.002702]
[À0.1425, 0.3083]

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