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State of Charge Estimation for Lithium-Ion Batteries Based on

Adaptive Dual Kalman Filter

Yidan Xu , Minghui Hu , Anjian Zhou , Yunxiao Li , Shuxian Li ,

Chunyun Fu , Changchao Gong

PII: S0307-904X(19)30541-4
DOI: https://doi.org/10.1016/j.apm.2019.09.011
Reference: APM 13015

To appear in: Applied Mathematical Modelling

Received date: 3 December 2018

Revised date: 14 August 2019
Accepted date: 3 September 2019

Please cite this article as: Yidan Xu , Minghui Hu , Anjian Zhou , Yunxiao Li , Shuxian Li ,
Chunyun Fu , Changchao Gong , State of Charge Estimation for Lithium-Ion Batter-
ies Based on Adaptive Dual Kalman Filter, Applied Mathematical Modelling (2019), doi:
https://doi.org/10.1016/j.apm.2019.09.011

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Highlights

 A novel fractional order model of lithium ion battery is established.

 The DKF and DEKF based SOC estimation methods are simulated and compared,based on the fractional

equivalent circuit model.

 A novel ADKF algorithm combining the RLS method and the DKF algorithm is established.

 The effectiveness，robustness and accuracy of the proposed SOC estimation algorithm are verified.
 State of Charge Estimation for Lithium-Ion Batteries Based on

Adaptive Dual Kalman Filter

Yidan Xu, a,b Minghui Hu, a,b,* Anjian Zhou, c Yunxiao Li, a,d

Shuxian Li, a,d Chunyun Fu , a,b Changchao Gong, a,b

a
State Key Laboratory of M echanical Transmissions, Chongqing University, Chongqing, 400044, China

b
School of Automotive Engineering, Chongqing University, Chongqing, 400044, China

c
Chongqing Changan Automobile Co., Ltd., Chongqing, 400023, China

d
Chongqing Automotive Collaborative Innovation Center, Chongqing University, Chongqing, 400044, China

Abstract: Accurate estimation of the battery state of charge (SOC) is of great significance for enhancing its

service life and safety. In this study, based on the fractional-order equivalent circuit model of lithium-ion battery,

the SOC estimation methods using dual Kalman filter (DKF) and dual extended Kalman filter (DEKF) are

simulated and compared, in terms of model accuracy and SOC estimation accuracy. Then, combining the

advantages of the DKF and DEKF algorithms, an SOC estimation algorithm based on adaptive double Kalman

filter is proposed. This algorithm uses the recursive least squares (RLS) method to update the battery model

parameters online in real time, and employs the DKF algorithm to filter the SOC twice to reduce the interferences

from the battery model error and the current measurement error. In the experimental studies, the measured SOC

values are compared with the estimated SOC values produced by the proposed algorithm. The comparison results

show that SOC estimation error of the proposed algorithm is within the range of ±0.01 under most test conditions，

and it can automatically correct SOC to true value in the presence of system errors . Thus, the validity, accuracy,

robustness and adaptability of the proposed algorithm under different operation conditions are verified.

Keywords：lithium-ion battery; fractional equivalent circuit model; SOC estimation; Kalman filter; recursive least

squares method
1. Introduction

Lithium-ion batteries have become one of the main power sources for electric vehicles (EVs) and hybrid electric

vehicles (HEVs), due to the advantages of no memory effect, high energy density and low self-discharge rate [1,2].

Reliable and efficient battery management systems (BMSs) help to make the best of lithium-ion batteries, which

makes BMS one of the focuses in EV and HEV related research. The SOC is an indicator of batteries remaining

power. Accurate SOC estimation helps protect the battery by preventing over-charge and over-discharge and

improve the battery utilization rate of the battery. Besides, it also helps increase the range of the vehicle, reduce the

requirement on the power battery, and guarantees the safe and stable operation of the vehicle [3–5]. Therefore,

SOC estimation has become one of the foci in recent BMS research.

The current SOC estimation methods can be divided into two categories: non-model based methods and model

based methods. The first category mainly includes : open circuit voltage (OCV) method [6], internal resistance

method [7], ampere hour (Ah) integral method [8], and machine learning algorithms. The second category mainly

includes：Kalman filter (KF) and its improved algorithm，particle filter (PF) [9], unscented particle filter (UPF)

[10,11].

The OCV method and internal resistance method make use of the direct mapping relationship between battery

SOC and its external static characteristic parameters (OCV or internal resistance). This relationship is employed to

establish a look-up table, based on which the SOC value is estimated by measuring these characteristic parameters

[3]. These two methods can effectively suppress the errors in SOC estimation. However, the measurement of

battery OCV requires that the electrolytes inside the battery be uniformly distributed, which needs a large amount

of time. Thus, it is difficult to accurately measure the OCV in real time [12]. Besides, the battery internal resistance

cannot be measured directly either. As a result, this SOC estimation method cannot be used independently in

practice. The Ah integral method directly calculates the SOC by integrating the battery current, based on the

definition of SOC. However, the accuracy of this method is not satisfactory due to the inaccuracy of the initial
SOC value, the current measurement error and the accumulation of the integral error [13]. To reduce the initial

SOC error, Feng et al. [14] employed the OCV approach to calculate the initial SOC value based on the Ah integral

method. Although this is an effective way to recalibrate the accumulated error, the OCV can hardly be measured in

real time when driving. So, this method still results in a large error in SOC estimation. The machine learning

algorithms include the artificial neural network (ANN) [15], Fuzzy Logic (FL) [16] and Support Vector (SVR) [17],

etc. These algorithms all rely on a large amount of training data to establish the nonlinear relationship between

battery input and output, which is highly subject to the quantity and quality of training data sets. Therefore, the

applicability and estimation accuracy of these algorithms are significantly limited.

Model based SOC estimation methods are able to handle uncertainties and disturbances by self-correction in

closed loop systems, thereby achieving higher SOC estimation accuracy. Although PF and UPF provide good

results for systems with non-Gaussian white noise, their computation loads are higher than that of KF. However,

the accuracy of KF decreases when dealing with nonlinear systems. To tackle this shortcoming, various revised KF

algorithms (such as Extend Kalman Filter (EKF) [18], double extended Kalman Filter (DEKF) [19] and double

Kalman Filter (DKF) [20]) have been extensively studied for battery SOC estimation. Urbain et al. [21] estimated

the SOC of a lithium-ion battery using the KF method, and the estimation results were verified through

experiments. Rahmoun et al. [22] used the EKF method to estimate SOC based on first-order and second-order

equivalent circuit models. It was shown that the SOC estimation results resulting from the second-order equivalent

circuit model are more accurate. However, the EKF method linearizes the nonlinear system using only the

first-order term from the Taylor series. As a result, the information contained in the higher -order terms are

neglected, which inevitably brings about estimation errors and results in deterioration of estimation accuracy. The

DKF and DEKF methods use two filters to correct the model error and the measurement error in real time [23,24],

which in turn provides superior SOC estimation results.

At the same time, Model based SOC estimation methods requires higher accuracy for the battery model. At
present, in the battery model research, the fractional-order equivalent circuit model has received extensive

attention because it can better reflect the internal reaction of the battery (solid phase diffusion, electric double layer

effect, etc.) [25]. Fractional-order calculus (FOC) that deals with non-integer integrals and derivatives was first

introduced by Leibniz in 1695 [26]. Numerous studies have demonstrated that fractional-order models can

characterize real systems with even better accuracy [27]. Multiple SOC estimation schemes based on

fractional-order models have also been researched for Lithium-ion Battery. Zou et al. [28] propose a

fractional-order model-based nonlinear estimator, utilizing a combination of Luenberger observer and sliding mode

observer (SMO). The estimator gains are designed by Lyapunov’s direct method, providing a guarantee for

stability and robustness of the error system under certain assumptions. Hu et al. [29] put forward a dual

fractional-order extended Kalman filter to realize simultaneous SOC and state of health (SOH) estimation.

However, these fractional-order models are commonly not capable of predicting battery dynamics in both the time

and frequency domains over the entire operating range.

In order to further improve the battery model and SOC estimation accuracy, by improving the second-order RC

equivalent circuit model using impedance elements with fractional-order characteristics ，a fractional-order

equivalent circuit model representing the lithium-ion battery with high precision is established in Section 2. In

Section 3, the model precision and SOC estimation precision, resulting from both the DKF and DEKF algorithms,

are compared and analyzed based on the proposed fractional-order equivalent circuit model. The results show that

both methods have their own advantages and disadvantages. By making full use of the advantages of these two

algorithms and combining the recursive least squares (RLS) which c an update model perameter online, an SOC

estimation algorithm based on adaptive DKF is proposed in Section 4. Section 5 verifies the validity, accuracy and

robustness of the proposed SOC estimation algorithm through experimental studies.

2. Fractional-order equivalent circuit model of lithium-ion battery

The existing lithium-ion battery models mainly include the electrochemical model, the black-box model, and the
equivalent circuit model (integer order equivalent circuit model and fractional equivalent circuit model) [30]. The

fractional-order equivalent circuit model uses fractional impedance elements (e.g. constant phase element (CPE)

and Wahlberg element) to describe the electrochemical processes such as charge transfer reaction, electron layer

effects, mass transfer, and diffusion of lithium ion battery with sufficiently high accuracy [31,32]. The phase shift

of a fractional-order capacitor is called a phasance , a term introduced by Jean [33]. The phasance is an important

characteristic parameter of the Nyquist plot in Fig.1. In particular, the phasance of a Warburg element represents

the slope of the low-frequency straight line, while for a CPE-resistor network, it is related to the shape of the

depressed semicircle. A thorough explanation of the phasance concept and mathematics behind it can be found in

Ref. [33].The SOC estimation algorithms proposed in this paper are based on the established fractional-order

equivalent circuit model.

Fig.1. Nyquist plot of a Lithium-ion Battery cell

2.1 Establishment of lithium-ion battery fractional-order equivalent circuit model

Typical second order RC models employ a parallel RC circuit to simulate the impedance spectra of the

intermediate frequency region. However, the solid state diffusion of lithium-ion batteries is neglected, and the ideal

capacitors cannot accurately simulate the double-layer effect. Besides, the low frequency region of the impedance

spectrum is not represented by any electronic components. Therefore, based on the electrochemical process

described by the electrochemical impedance spectroscopy of lithium-ion batteries, fractional-order impedance

elements are introduced to improve the accuracy of the 2-RC integral-order model [34]. The fractional-order

equivalent circuit model is shown in Fig. 2, where UOCV represents the OCV, Ud indicates the battery terminal

voltage which can be directly measured, I stands for the current (positive for charge and negative for discharge), R0

is the ohmic resistance of lithium-ion battery, R 1 and R 2 are the polarization resistances of lithium-ion battery,

CPE 1 and CPE 2 are the constant phase elements, and W is the Wahlberg element.

R1 R2 W
R0 + +
CPE1 CPE2
+ Ud
UOCV
-
I -

Fig. 2. Schematic of the fractional-order equivalent circuit model.

The transfer function of the model impedance is given by:

U d ( s )  U OCV ( s ) R1 R2
 1
 1
 Z W  R0
I (s) 1  R1Z CPE1 1  R2 Z CPE2
(1)
1 1 1
Z CPE1  Z CPE2  ZW 
C1 S α C1 S  WS 

where ZCPE1, ZCPE2 and ZW represent the impedances of CPE 1 , CPE2 and Wahlberg element respectively, C1 , C2 and

W are the parameters of model elements, α and β are the fractional-orders of CPE1 and CPE2 , and γ is the

fractional-order of the Warburg element. Note that CPE 1 and CPE2 represent ideal capacitors for α=β=1, or denote

resistors for α=β=0. In this paper, the lithium-ion battery is modeled based on the fractional calculus theory, as a

result, the fractional-order falls in the range of (0, 1).

The input to the system is set to u=I(t), i.e. the battery current. The output is y = UOCV (t)－Ud(t), namely the

difference between the battery OCV and the terminal voltage. Thus, the system model can be expressed by a

fractional calculus equation in the time domain, as given by equation (2):

WD 
 WR1C1D   WR2C2 D    WR1C1R2C2 D    y (t )
 [ R1C1D  R2C2 D   ( R0  R1  R2 )WD  R1C1R2C2 D    ( R0  R2 )WR1C1D  (2)
 ( R0  R1 )WR2C2 D    R0WR1C1R2C2 D    ]u (t )  u (t )
 γ α+γ β+γ α+β+γ α β α+γ
where parameters D , D ,D ,D ,D ,D ,D are fractional-order operators.

By definition, SOC as the system state of the fractional model is shown as follows:


D1SOC  t   u t  (3)
Cn

where Cn is the battery rated capacity, and ε is the battery coulomb efficiency (0.98 for charging and 1 for

discharging).

In this paper, the Grunwald-Letnikov definition [35] is used to approximate the fractional calculus equation of

the fractional-order equivalent circuit model. The Grunwald-Letnikov definition is given by equation (4).

t / h 
1  
Dt f (t )  lim
h 0 h
 (1)  i i
 f (t  ih) (4)
i 0  

where h is the sampling period, [t/h] is the integral part of t/h, and (𝜆) is the Newton binomial coefficient.
𝑖

In this study, the following definitions are given to simplify equation (2):

[a1 a2 a3 a]4 [         ]
[b1 b2 b3 b]4 [ W W R
1 C
1 W 2R C
2 W1 R 1C]2R 2C
[c1 c2 c3 c4 c5 c6 ]c7 [       
    ]  (5)
[d1 d 2 d3 d 4 d5 d 6 ]d7 [ R1 C1 R2 C ( 2  R0  R1 ) R2 W R1 C1 R2 C2
( R0  R2 ) W R1 C1 ( R0 R)1 W R2 C2 R0 W 1R 1C ]2R C
2

According to the Grunwald-Letnikov definition, equation (2) can be discretized as:

i j   cj 
N 4 bj a N 7 d

 aj
( 1)   y (t  iT )   j
cj
(1)i   u (t  iT )  u (t ) (6)
i  0 j 1 T i  i  0 j 1 T i 

where N is the number of historical data points involved in the calculation, and T is the sampling interval.

Then the following parameters are defined to further simplify equation (6):

4 bj  aj 
A(i )   (1)i  
aj (7)
j 1 T i 
7 d cj 
B(i )   c j (1)i   i  0,1, 2    N
j

j 1 T i 

Moreover, in the Grunwald-Letnikov definition, when T approaches 0 and N approaches infinity, the equal sign

in equation (6) holds. In all other cases, the two sides of the equation are approximately equal. However in practice,

as N becomes large, the computation load increases. Considering the accuracy requirement of the lithium-ion
battery model and the short-term memory principle, the data length can be appropriately truncated. When N=1,

equation (6) can be converted to a first-order difference equation, as given by equation (8).

A(0) y (k )  A(1) y (k  1)  u (k )  B(0)u (k )  B (1)u (k  1) (8)

Rearrangement of equation (8) leads to:

A(1) 1  B(0) B(1)

y (k )   y(k  1)  u (k )  u (k  1) (9)
A(0) A(0) A(0)

Similarly, equation (3) is discretized into:

T
SOC (k )  SOC (k  1)+ u (k ) (10)
Cn

Equations (9) and (10) constitute the fractional-order equivalent circuit model for lithium-ion batteries.

Employing the experimental current and voltage data, the model parameters can be identified us ing an

optimization algorithm. The parameters to be identified in the model are as follows:

ζ= [R 0 R1 C1 C2 W α β γ] (11)

2.2 Parameter identification of fractional-order equivalent circuit model

The battery models are normally complex, time-varying and nonlinear. They involve many parameters and some

of them cannot be directly measured. As a result, parameter identification has become an important part in the

battery modeling process. In this section, the relationship between OCV and SOC is established based on the

experimental data, and the equation parameters are obtained by MATLAB f itting. Then, using mixed-swarm-based

cooperative particle swarm optimization (MCPSO), the parameters of the fractional equivalent circuit model are

identified in the time domain.

2.2.1 Establishment of the OCV-SOC relationship

The OCV is an important part in the lithium-ion battery modeling. Due to the polarization and hysteresis effects

of lithium-ion batteries, the OCV can only be measured under sufficiently static conditions, which makes practical

measurement intractable. However, one should note that there exists a certain relationship between OCV and SOC.

Through the double-pulse discharge experiment, the OCV value under certain SOC value can be obtained, and
then the relationship between OCV and SOC can be achieved by means of data fitting.

The schematic of the battery test system used in this study is shown in Fig. 3. The test system is composed of the

high and low temperature test chamber, the battery test equipment and the host computer for human-computer

interaction. In this study, the sampling time of the battery test system is set to 0.1 s.

Calorstat

Battery

Sensor: voltage, current

TCP/IP
Charge and discharge Host computer
Data and commands
equipment

Fig. 3. Schematic of the battery test system.

The OCV values corresponding to different SOC values at 25°C are measured by the double pulse method, and

the results are as shown in Table 1.

Table 1 OCV values corresponding to different SOC values.

SOC 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

OCV (V) 3.466 3.544 3.605 3.6385 3.673 3.733 3.828 3.9205 4.0275

An empirical formula [36] is used to fit the relationship between SOC and OCV, as given by equation (12):

1
U OCV (SOC )  C0  C1SOC  C2  C3 ln(SOC )  C4 ln(1  SOC )
SOC (12)

The resulting fitting curve is shown in Fig. 4, and the fitted coefficients are shown in Table 2. It is known from

Fig. 4 that the experimental data is very close to the curve fitting, and the calculated RMSE is 9.5 mV. Model

errors outside the SOC range are not taken into account, as the operational design of the battery pack is limited to

10%-90% SOC due to the impact of over-charge and over-discharge on battery life.
 4.2
Experimental data
4.1
Curve fitting
4

OCV (V) 3.9

3.8

3.7

3.6

3.5

3.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
SOC

Fig. 4. The OCV-SOC fitting curve.

Table 2 Fitted parameters for the OCV-SOC curve.

C0 C1 C2 C3 C4

2.9510 0.9981 -0.0515 -0.3994 -0.0856

2.2.2 Model parameter identification and model accuracy verification

In this paper, the MCPSO algorithm is used to identify the parameters of the fractional-order equivalent circuit

model. The model parameters to be identified are ζ= [R 0 R1 C1 C2 W α β γ], so the particle dimens ion

is set to 9 and each particle vector in the particle group is a model parameter vector. The fractional-order

equivalent circuit model is employed to calculate the battery terminal voltage, aiming to minimize the errors

between the terminal voltage values resulting from the model and the experiment.

In this study, the A123 battery was tested at 25 ℃ ambient temperature, using the Federal Urban Driving

Schedule (FUDS) test driving cycles with an SOC range of 0.9-0.65 for model parameter identification. The

current change under this operation condition is shown in Fig. 6. The current data are used as the input to the

model for parameter identification, and the resulting model parameter identification outcomes are obtained, as

shown in Table 3.
 Table 3 Model parameter identification results.

R0 R1 C1 R2 C2 W α β γ
0.0122 0.0084 19.7144 17.8081 51.1027 155.7854 0.8387 0.2128 0.1666

Start t =0

Initialize the probe s ubgroup and The mining subgroups are initialized
calculate the fitness to obtain the best i n t he a c t iv e a re a o f t he p ro be
a nd w or s t p os it i on. Calculate the subgroup and the fitness is calculated
active region of the probe subgroup to obtain the best and worst position

N Termination condition Y End

Whether synergistic
time is achieved
t = t+1
N Y
Mining learn
probe>mining
from probe
Y N
Probe learn from mining

N
Speed<Threshold
Update the mining subgroup and
Y calculate the fitness to obtain the
Initialization speed best and worst location
range

Update the probe subgroups and calculate

the fitness to get the best and worst position

Fig. 5. Flow chart of model parameter identification based on MCPSO.

50
Current (A)

-50

-100
0 200 400 600 800 1000 1200
Time (s)

Fig. 6. Current resulting from the FUDS test cycle with an SOC range of 0.9-0.65.

Partial verification results of the fractional-order equivalent circuit model under the above operating condition

are shown in Fig. 7. The error between the measured voltage and the model output voltage is shown in Fig. 8, and
the relative error is given in Fig. 9. It is shown in Fig. 7 that the measured voltage is very close to the model output

voltage, and the calculated Root Mean Square Error (RMSE) is 7.69 mV. Fig. 8 illustrates that the absolute error

lies within the range of ±10 mV at large, and the error exceeds the ±10 mV boundaries only when the charging

and discharging current changes significantly. It can be seen from Fig. 9 that the average relative error is less than

0.1%, which verifies the high accuracy of the proposed model.

4.1 Model output

Measured data
4
Voltage (V)

3.9

3.8
4

3.98
3.7
3.96
3.6 3.94
0 200 400 600 800 1000 1200
340 345 350
Time (s)

Fig. 7. Comparison of model output voltage and measured voltage.

0.02
0.01
Error (V)

0
-0.01
-0.02
-0.03
0 200 400 600 800 1000 1200
Time (s)

Fig. 8. Error between model output voltage and measured voltage.

Relative error Average value
0.4
Relative error (%)

0.3

0.2

0.1

0
0 200 400 600 800 1000 1200
Time (s)

Fig. 9. Relative error between model output voltage and measured voltage.

3. SOC estimation based on DEKF and DKF

The KF can correct the initial error of the system and effectively suppress the influence of system noise and

measurement noise. Therefore, the KF has been widely employed among in various lithium-ion battery SOC
estimation algorithms under complex operating conditions, the SOC estimation algorithm based on KF is favored

by many experts and scholars. However, the accuracy of KF-based algorithms largely depend on the battery model

accuracy, and their estimation accuracy deteriorates when dealing with highly nonlinear dynamic models (e.g. a

battery system). To tackle these issues, a series of modified KF based SOC estimation algorithms have been

proposed in the literature, such as the SOC estimation methods based on DEKF and DKF. In this section, the KF

and EKF theories are first introduced. Next, the DEKF and DKF methods are employed to estimate SOC based on

the fractional-order equivalent circuit model. Then, comparative analyses on the SOC estimation precisions and the

model output voltage errors are carried out.

3.1 KF and EKF

The KF algorithm, a time domain filtering method, uses state space equations to describe the system. The core

of KF is a recursive linear minimum variance algorithm, which employs the observation to correct the estimates

and drives the state estimates towards the real values recursively [37]. However, the KF algorithm is only effective

for linear systems. The fractional-order model of lithium-ion batteries considered in this paper is a highly nonlinear

model, which defies the direct usage of the KF algorithm. In this case, the observation equation in the KF

algorithm needs to be expanded through Taylor series, and only the first order term is retained for linearization.

This filtering algorithm that linearizes the observation equation is known as the EKF algorithm.

For a linear discrete system, the standard KF recursive algorithm can be expressed as follows:

 x(k k  1)  Ak x(k  1 k  1)  Bk u (k )

 P(k k  1)  Ak P(k  1 k  1) Ak  Q
T


 Kg (k )  P(k k  1)Ck [Ck P(k k  1)Ck  Rv ]
T T
(13)
 x(k k )  x(k k  1)  Kg (k )[ y (k )  (C x(k k  1)  D u (k ))]
 k k

 P(k k )  [ I  Kg (k )Ck ]P(k k  1)



where x(k/k －1) and x(k/k) denote the prior and posterior estimates of the states at time k, P(k/k －1) and P(k/k)

are the prior and posterior estimates of the error covariance matrix at time k, Kg(k) represents the Kalman

filtering gain at time k, I stands for the identity matrix, x(k), u(k) and y(k) are state variables, input variables and
output variables of the system at time k respectively, A k, B k, Ck and Dk are the input matrix, control input matrix,

observation matrix and output value transmission matrix at time k respectively, Qω represents the covariance of the

process noise ωk at time k, and R v denotes the covariance of measured noise v k at time k.

The KF algorithm is mainly composed of two stages – prediction and update. The prediction stage is represented

by the first two equations in (13). The posterior state x(k －1/k －1) at time k-1 and the corresponding error

covariance P(k －1/k －1) are employed to solve the state x(k/k －1) and error covariance P(k/k －1) at time k. The

update stage is represented by the last three equations in (13). The experimental measurement y (k) is employed to

update the prior estimate, thereby achieving the posterior estimates x(k/k) and P(k/k) at time k. Then in the

following time steps, the above two states are repeated recursively to calculate the optimal estimates at each time

step.

For a discrete nonlinear system, the state equation and the observation equation are given by

 xk 1  f ( xk , uk )  wk
 (14)
 yk  g ( xk , uk )  vk

where f(xk, uk) is the state transition function, and g(xk, uk) is the measurement function.

When applying the EKF algorithm to a nonlinear system, the above state transition function and measurement

function are expanded using Taylor series, and only the first order terms are retained to achieve linearization. The

linearized equations are given in equation (15):

 xk 1  Ek xk  [ f ( xˆk , uk )  Ek xˆk ]  k
 (15)
 yk  Fk xk  [ g ( xˆk , uk )  Fk xˆk ]  vk

f ( xk , uk ) g ( xk , uk )
where Ek   , Fk   , and 𝑥̂𝑘 represents the estimated system state at time k.
xk xk  xk xk xk  xk

Applying the KF algorithm to equation (15) leads to the following expressions for the EKF algorithm:
  xˆk k 1  f ( xˆk 1 k 1 , uk 1 )

 Pk k 1  Ek 1 Pk 1 k 1 Ek 1  Q
T


 Kg k  Pk k 1 Fk ( Fk Pk k 1 Fk  Rv )
T T
(16)
ˆ
 xk k  xˆk k 1  Kg k ( yk  g ( xˆk k 1 , uk ))
 P  [ I  Kg F ]P
 kk k k k k 1

Although the EKF algorithm adopts approximation to linearize nonlinear systems, which inevitably brings about

some extent of model errors, yet it provides the advantages of straightforward and fast implementation. Therefore,

in practice, the EKF algorithm has been extensively employed to tackle the estimation problems for nonlinear

models.

3.2 SOC estimation based on DEKF

The EKF based SOC estimation algorithm estimates the SOC, using the error between the measured voltage and

the output voltage from the model. The effectiveness of this method largely relies on the battery model accuracy.

The parameters of the fractional-order equivalent circuit model are obtained by offline identification. However, in

practice, the battery parameters are varying in the working process, so the fixed parameters will inevitably impair

the model accuracy. However, the DEKF based SOC estimation algorithm employs two independent EKF to

estimate the battery model parameters (i.e. R 0 in this paper) and battery SOC individually. The DEKF algorithm

jointly estimates the battery SOC and updates the model parameters, which not only increases the battery model

accuracy but also enhances the SOC estimation accuracy. The proposed DEKF based SOC estimation algorithm is

schematically shown in Fig. 10.

 Input I Lithium-ion Output Ud Input SOC Output OCV
battery SOC-OCV curve

EKF + EKF
Error Error + Model
-
Model - output
U OCV
output d
Fractional SOC Fractional R0
order model order model

(a) SOC correction by primary filtering (b) R0 correction by secondary filtering

Input I Lithium-ion Output OCV

SOC-OCV curve
battery Output Ud
EKF
Error + EKF Model
Error + output
-
Model -
U OCV
output d
Fractional SOC
order model Fractional R0
order model

(c) DEKF based SOC estimation

Fig. 10. Schematic of the DEKF based SOC estimation algorithm.

Based on the lithium ion battery model, the fractional differential equations for the CPE and resistance parallel

circuits can be expressed by equations (17) and (18):

1 1
DVCPE1  t    VCPE1  t   I  t  (17)
R1C1 C1
1 1
D VCPE2  t    VCPE2  t   I t  (18)
R2C2 C2

The fractional differential equation for the Wahlberg element W is given by:

1
D VW  t    I t  (19)
W

By definition, the system state SOC in a fractional-order equivalent circuit model can be expressed as:


D1SOC  t    I t  (20)
Cn

The state variable of the system is selected as x(t)=[VCP E1 (t) VCP E2(t) VW (t) SOC(t)]. The input to the

system is u=I(t) (i.e. the battery current) and the output from the system is y=Ud (t) (i.e. the battery terminal

voltage).

Then, the pseudo-system state equation of the fractional-order equivalent circuit model is established, as given

by equation (21):
  D N x(t  1)  Ax(t )  Bu (t )
 (21)
 y(t )  f [ x(t )]  Cx(t )  Du (t )

where f [x(t)] represents the relationship between OCV and SOC. The definitions of A, B, C and D are given in

equation (22):

 1 
 1  C 
 R C 0 0 0  1 
 1 1   1 
 1  C 
A 0  0 0 B   2  C   1  1  1 0 D  R0 (22)
 R2 C2   1 
 0   
0 0 0  W 
 
 0 0 0 0   
 C 
 n

Using the definition of fractional calculus and EKF principle discrete equation (21), the result as shown

in equation (23).

 xk 1  Ak xk  Bk uk
 (23)
 yk  Ck xk  Dk uk

where the definitions of A k, B k, Ck and Dk are given by:

 T 
 
 1   C1 
  T 0 0 0
R1C1  T 
   
 1 
Ak   0  T 0 0  Bk   C2 
R2 C2  T 
    (24)
 0 0  0  W 
   T 
 0 0 0 1
 
 Cn 
df ( SOCk )
Ck  [1  1  1 ] Dk  R0
dSOC

So the detailed steps of the DEKF based SOC estimation algorithm are as follows:

Step 1: Determine the initial SOC value -- Based on the SOC-OCV curve obtained in Section 2.2.1, compute the

initial SOC value by substituting the OCV in.

Step 2: Initialize the state -- Set the error covariance, the process noise variance and the measurement noise

variance for parameters R 0 and SOC.

Step 3: SOC estimation update -- According to equations (16), (23) and (24), estimate and update SOC based on
the EKF algorithm, using the error between the model output voltage and the experimental measured voltage. Use

the SOC-OCV characteristic curve to calculate the OCV.

Step 4: R 0 estimation update. According to equations (16), (23) and (24), estimate and update R 0 based on the

EKF algorithm, using the error between the model output OCV and the OCV obtained from the SOC-OCV

characteristic curve.

Step 5: Repeat steps 3 and 4 to acquire SOC estimate at each moment, and update in real time the main

parameter R 0 of the fractional-order model.

3.3 SOC estimation based on DKF

The DKF based SOC estimation is a combination of the EKF algorithm and the Ah integral method. Another

Kalman filter is constructed on the basis of these two algorithms, which not only overcomes the estimation error

interference of the Ah integral method but also suppresses the estimation error interference of the EKF. The Ah

integral method is affected by the accuracy of current measurement, and the error mainly results from the

accumulation of current error and interference. On the other hand, the EKF algorithm is dependent on the battery

model accuracy, and the error mainly originates from the battery model error. The DKF algorithm greatly reduces

the adverse influences from these two kinds of error interferences, and provides more stable and accurate SOC

estimation. A schematic of the DKF based SOC estimation is shown in Fig. 11.
 Input I Lithium-ion Output Ud
battery Input I Lithium-ion
battery Output
EKF +
Error EKF Ud
- Error +
Model
Ud - Model
output Ud
Fractional SOC output
order model Fractional
(a) Primary filtering corrects the SOC order model
Output
SOC
Input SOC Output SOC KF Model
EKF Error +
- output
SOC
KF Model
Error +
output
- SOC Ah integral SOC
method
Ah integral SOC
method (c) The DKF estimates SOC
(b) Secondary filtering corrects the SOC

Fig. 11. Schematic of the DKF based SOC estimation.

The detailed process of the DKF based SOC estimation algorithm is as follows:

Step 1: Determine the initial SOC value -- According to the SOC-OCV curve obtained in Section 2.2.1, calculate

the initial SOC value by substituting OCV in.

Step 2: Initialize the state -- Set the error covariance, the process noise variance and the measurement noise

variance for SOC.

Step 3: Primary SOC filtering. According to equations (16), (23) and (24), estimate and update SOC based on

the EKF algorithm, using the error between the model output voltage and the experimental measured voltage. This

process overcomes the interference of battery model error.

Step 4: Secondary SOC filtering. According to equations (13), (23) and (24), estimate and update SOC based on

the KF algorithm, using the error between the SOC obtained by the Ah integral method and the one obtained by the

EKF algorithm. This process overcomes the interference of the accumulated current measurement error resulting

from the Ah integral method.

Step 5: Repeat steps 3 and 4 to acquire the SOC estimate at each time step.

3.4 Accuracy comparison between the two algorithms

 For comparison purposes, the same fractional-order equivalent circuit model and initial SOC value are adopted

for both algorithms. The input to the model is the same current condition (FUDS condition), and then the two

algorithms are compared and analyzed in terms of the SOC estimation accuracy and the model accuracy.

4.2
DEKF algorithm
4 DKF algorithm
measured value
Voltage (V)

3.8

3.6
4.05

4
3.4 3.95

3.9
0 500 1000 1500 2000 2500 3000
3.85
200 220 240 260 280 300 Time (s)
(a)Model output voltage comparison based on DEKF and DKF

0.02 DEKF algorithm DEKF algorithm

Error (V)

-0.02
0 500 1000 1500 2000 2500 3000
Time (s)
(b)Model output voltage error comparison based on DEKF and DKF

Fig. 12. Output voltage comparison between the DKEF and DKF based algorithms.

The model output voltages and the errors resulting from these two algorithms are shown in Fig. 12, and the

estimated SOC values and the errors are given in Fig. 13.

As shown in Figs. 12 and 13, the DEKF based algorithm updates in real time the main parameter R 0 of the

fractional-order equivalent circuit model, which in turn makes the model output voltage well fit the measured

voltage. The error is within the range of ± 10 mV, and is smaller than that resulting from the DKF based

algorithm. However, since the DKF based algorithm suppresses both the current measurement error and the battery

model error, the resulting SOC estimation outcomes are better fitted to the real SOC curve, compared with the

results of the DEKF based algorithm. The error is within the range of ± 0.01, which shows an improved

accuracy of SOC estimation. Based on the above results, the following conclusions can be drawn: the DEKF based

algorithm improves the accuracy of the battery model and better describes the dynamic characteristics of
lithium-ion batteries, while the DKF algorithm improves the accuracy of SOC estimation, both of which have

advantages and disadvantages.

1
DEKF estimate
0.8 DKF estimate
ture value
SOC

0.6
0.58

0.4 0.57

0.56
0.2
0 0.55 500 1000 1500 2000 2500 3000
1600 1620 1640 1660 1680 1700
Time (s)
(a) SOC estimation comparison based on DEKF and DKF based algorithms.

0.02 DEKF algorithm DKF algorithm

SOC error

0.01
0
-0.01
-0.02
0 500 1000 1500 2000 2500 3000
Time (s)
(b) SOC estimation error comparison based on DEKF and DKF based algorithms.

Fig. 13. SOC estimation comparison between the DEKF and DKF based algorithms.

4. The adaptive DKF based SOC estimation

The lithium-ion battery model parameters of lithium-ion battery are affected by SOC, ambient temperature,

battery aging and other factors, which are closely related to the battery working state of the battery. During battery

charging and discharging cycle, the SOC value and battery internal resistance keep varying all the time. However,

the parameter identification method introduced in Section 2.2 can only obtain the initial values of model

parameters offline, and is not able to identify and update model parameters online in real time. Therefore, in order

to improve the accuracy of lithium-ion battery model, the adaptive online real-time parameter updating is required.

Besides, the DKF is employed to reduce the adverse impac ts of current measurement error and battery model error

on SOC estimation accuracy. In this paper, an SOC estimation method based on adaptive DKF is proposed,

utilizing the advantages of the two algorithms introduced in section 3. This method uses the RLS to update the

main parameter R 0 of the model in real time, and employs the DKF algorithm to estimate SOC.

4.1 Principle of recursive least squares identification

 The least squares (LS) method is a simple and easy-to-implement parameter identification approach. It is able to

identify model parameters online and make corrections in real time. The RLS is an improved LS algorithm. The

principle of RLS identification is : After each new observation is acquired, the new observation data is used to

correct the previous parameter estimation results, thereby achieving new estimation values. By performing this

process recursively, real-time updating of model parameters can be achieved [38]. The RLS method not only

reduces the computation and storage loads, but also fulfills online real-time parameters identification. Therefore,

the parameters of the fractional-order equivalent circuit model are updated using the RLS method in this paper.

Based on equation 8, the input u(k) and output y(k) of the system can be extended to n dimensions. The resulting

matrix equation is as follows:

Y   T  e (25)

where Y, Φ, δ, e are given by:

 Ak (1) 1  Bk (0) Bk (1) 

 A (0) Ak (0) Ak (0) 
  y (k  1) u (k ) u (k  1)   k 
  y (k )   Ak 1 (1) 1  Bk 1 (0) Bk 1 (1) 
u (k  1) u (k )  
     Ak 1 (0) Ak 1 (0) Ak 1 (0) 
  (26)
   
  y (k  n) u (k  n) u (k  n  1)   
 Ak  n (1) 1  Bk  n (0) Bk  n (1) 
 A (0) Ak  n (0) 
 k n Ak  n (0)
Y  [ y (k  1) y (k  2) y (k  n)] e  [e(k  1) e(k  2) e(k  n)]

The obtained parameter estimation results at time k-1 and k are:

ˆ (k  1)  (Tk 1 k 1 )1 Tk 1Yk 1

(27)
ˆ (k )  (Tk  k )1 Tk Yk

At the k-th recursion, the parameter estimate ˆ (k  1) , the observation vector  (k 1) and the actual

measurement y(k) have been obtained. Thus, the new parameter estimate ˆ (k ) can be obtained using the

following equation:

 K (k )  P(k  1) (k  1)[1   T (k  1) P(k  1) ( k  1)]1


 P(k)  [I K (k ) (k  1)]P(k  1)
T
(28)
ˆ ˆ ˆ
 (k)   (k  1)  K (k )[ y (k )   (k  1) (k  1)]
T
where K(k) is the gain vector, and P(k) = (Tk  k )1 .

The detailed algorithm of RLS is shown in Fig. 14.

Start

Generate input data：current

u and output voltage y

Calculate given initial value P(0)

according to parameter ˆ (0)
identified in section 2.2.

Calculate and update the updated

P(k), k (k) and δ (k)

Check the N
termination condition

Y
End

Fig. 14. Flow chart of the model parameter update algorithm.

4.2 The adaptive DKF based SOC estimation

The schematic of the adaptive DKF based SOC estimation algorithm proposed in this paper is shown in Fig. 15.

The detailed steps of this algorithm are as follows:

 Initial SOC value
(k time)
Current at time k + 1 Estimation error P0/0
Estimate SOC at time k+1
Predicted estimation
Update model error Pk/k-1 Corrected Primary
Estimate OCV at time k+1 estimation filtering
Parameter parameter by RLS of SOC
error Pk
updating
Model outputs Ud at time k+1 Filtering gain Kk

Measured Ud at
time k+1 Terminal voltage error Modify SOC at time k+1

Estimate SOC by Ah Second modification of

SOC error
integral method the SOC at time k+1
Estimation error Secondary
P
0/0 Predicted estimation filtering
Filtering gain K
k Output SOC
error P
k/k-1 of SOC
at time k+1
Corrected
estimation error P
k

Fig. 15. Schematic of the adaptive DKF based SOC estimation.

Step 1: Determine the initial state of the battery. The initial parameters of the battery model are determined by

the parameters of the battery model established in section 2.2. Based on the SOC-OCV curve, find the initial SOC

value using the corresponding OCV.

Step 2: Initialize the LS method and filter state. Select the initial value P (0) for the LS method, the SOC error

covariance, the process noise variance of SOC estimation and the measurement noise variance of SOC estimation.

Step 3: Update the model parameters. Based on equation (28), employ the RLS method to update the model

parameters in real time.

Step 4: Primary SOC filtering. Based on equations (16), (23) and (24), employ the EKF algorithm to estimate

and update SOC, using the error between the model output voltage and the experimental measured voltage. This

process suppresses the influence of battery model error on SOC estimation.

Step 5: Secondary SOC filtering. Based on equations (13), (23) and (24), employ the KF algorithm to estimate

and update SOC, via the error between the SOC obtained from the Ah integral method and the SOC resulting from

the EKF algorithm. This process suppresses the influence of the accumulated current measurement error in the Ah

integral method on SOC estimation.

 Step 6: Repeat steps 3 and 5 to update the model parameters in real time according to the battery operating

condition, and obtain the SOC estimate at each time step.

5. Accuracy verification of the adaptive DKF based SOC estimation algorithm

This section presents the experimental verification results and analyses of the adaptive DKF based SOC

estimation algorithm. In three sets of experiments, different operating conditions and different SOC initial values

are employed to validate this SOC estimation algorithm. The accuracy, robustness and adaptability to different

operating conditions are illustrated.

5.1 Accuracy verification of SOC estimation under different operating conditions

In order to verify the accuracy of the SOC estimation algorithm proposed in this paper under different operating

conditions, an experiment is carefully designed and the influence of different test conditions on SOC estimation

performance is evaluated.

In this experiment, a 10 Hz sampling frequency is used for all three test conditions – the FUDS condition,

Dynamic Stress Test (DST) condition and Hybrid Power Pulse Power Characterization ( HPPC) condition. The

SOC initial value used in this algorithm is the actual SOC initial value.

The SOC estimation results in this experiment are shown in Fig. 16, where figures (a)--(c) represent the

estimation errors under the FUDS condition, DST condition and HPPC condition, respectively. The frequency of

current change under the DST condition is close to that under the FUDS condition, and the resulting SOC errors

under both conditions fluctuate within a small range of ±0.01. The error fluctuation under the HPPC condition is

relatively larger, but still remains within a small range of ±0.03. The above experimental results demonstrate that

the proposed adaptive DKF based SOC estimation algorithm provides accurate SOC estimation results under

different operating conditions, which verifies the effectiveness of the proposed algorithm.
 0.02

SOC error
0

-0.02
0 500 1000 1500 2000 2500 3000 3500 4000
Time (s)
(a) FUDS condition
0.02
SOC error

-0.02
0 500 1000 1500 2000 2500 3000 3500 4000
Time (s)
(b) DST condition
SOC error

0.02

-0.02

0 0.5 1 1.5 2 2.5

4
Time (s) x 10
(c) HPPC condition

Fig. 16. SOC estimation results under different operating conditions

5.2 Accuracy verification of SOC estimation with system errors

The accuracy verification of SOC estimation with system errors is conducted to evaluate the SOC estimation

performance when the initial SOC is less than or greater than the actual SOC. To this end, the following two

experiments are performed in this study.

Experiment 1: SOC estimation accuracy verification experiment when the estimated initial SOC is less than the

actual SOC.

The FUDS condition with a sampling frequency of 10 Hz is used in this experimental study. The estimated

initial SOC value is set to 0.7, and the actual initial SOC value is 0.9.

The estimation results in this experiment are shown in Fig. 17. Subfigure (a) demonstrates a comparison

between the SOC value estimated by the proposed algorithm and the real SOC value. We see that the error between

the two curves decreases gradually, as the estimated SOC value converges to the actual SOC value. Subfigure (b)

plots the SOC error during the entire estimation process. It is shown that the error fluctuates within a small range

of ±0.01.
 1
ture value
estimated value

SOC
0.5 0.86

0.855

0.85

0 0.845
0 500 1000 1500 2000 2500 3000
300 310 320 330 340 350
Time (s)
(a) SOC estimation comparison curve

0.02
SOC error

-0.02

0 500 1000 1500 2000 2500 3000

Time (s)
(b) SOC estimation error

Fig. 17. Estimation results when the estimated initial SOC is less than the actual initial SOC.

Experiment 2: SOC estimation accuracy verification experiment when the estimated initial SOC is greater than

the actual initial SOC.

The step pulse condition with a sampling frequency of 10 Hz is employed in this experimental study. The

estimated initial SOC value is set to 0.9, and the actual initial SOC value is 0.5.

The estimation results of this experiment are shown in Fig. 18 The comparison results in subfigure (a)

demonstrate that the converging speed is relatively slow with a large initial error, but the estimation error at the end

of simulation is rather small. Besides, the estimation error remains within the range of ±0.01 after the error

stabilizes.

1
0.38
ture value
0.36
estimated value
SOC

0.34
0.5 600 610 620 630 640 650

0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time (s)
(a) SOC estimation comparison curve
0.02
SOC error

-0.02

200 400 600 800 1000 1200 1400 1600 1800 2000
Time (s)
(b) SOC estimation error

Fig. 18. Estimation results when the estimated initial SOC is greater than the actual initial SOC.
 The above experimental results indicate that the adaptive DKF algorithm proposed in this paper can

automatically correct SOC to true value in the presence of system errors. In other words, the proposed adaptive

DKF algorithm is able to provide accurate SOC estimation results with high robustness.

6. Conclusion

Based on the fractional-order equivalent circuit model, this paper looks into the DEKF and DKF based SOC

estimation methods, and then proposes an adaptive DKF based SOC estimation algorithm. This algorithm employs

the DKF technique to suppress the adverse influences resulting from the battery model error and the current

measurement error. In the meantime, the parameters of the battery model are updated in real time using the RLS

method to improve the battery model accuracy. The effectiveness and accuracy of the proposed SOC estimation

algorithm are verified under different operating conditions with system errors, via three groups of experimental

studies. The experimental results show that the SOC estimation errors under the DST and FUDS conditions remain

within the range of ±0.01, and the SOC estimation error under the HPPC condition is kept within the range of ±

0.03. These results indicate that the proposed algorithm can provide accurate SOC estimation and high robustness

under different operating conditions. In presence of the system errors, the proposed algorithm is able to correct the

initial SOC error and drives the estimated value towards the true value, regardless of the value of initial SOC error.

This proves that the proposed algorithm is able to provide accurate SOC estimation w ith high robustness.

Acknowledgments

This work is financially supported by the National Natural Science Foundation of the People’s Republic of

China [No.51675062], the National Key Research and Development Project [No. 2018YFB0106102], the

Fundamental Research Funds for the Central Universities [No.106112016CDJXZ338825], and the Major Program

of Chongqing Municipality [No. cstc2018jszx-cyztzxX0007].

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