Proceedings of the 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013)

SOC estimation of Lithium-ion battery based on Kalman filter algorithm

Yu Ding-xuan Gao Yan-xia

School of Mechatronic Engineering and Automation School of Mechatronic Engineering and Automation
Shanghai University Shanghai University
Shanghai 200072,China Shanghai 200072, China
edmend_yu@163.com gaoyanxia@shu.edu.cn

Abstract -This paper presents Extended Kalman-filter (EKF) timal estimation value can be obtained. Kalman filter algo-
algorithm which is based on a first-order Lithium-ion batteries rithm needs accurate state describe model and accurate
model. Curve fitting According to the OCV(open circuit volt- measurement values, beyond all of these, EKF has no re-
age)、SOC(state of charge) parameters measured in exper- quest on initial value.
iments, descript status equation and observation equation of
Lithium-ion battery in detail , so that it can accurately demon- II. THE BATTERY MODEL
strates the characteristics of the Lithium-ion battery. Simula- The purpose of building battery model is to establish
tion and experiment results show the feasibility and effective-
battery internal state variables, external quantitative relation
ness of the algorithm.
electrical characteristics and to set up the mathematical
Keywords-Kalman filter; Lithium-ion battery; SOC model, then based on the external variables such as battery
voltage, current, temperature to calculate internal state var-
I. INTRODUCTION iables like SOC, internal resistance and electromotive force.
There are many battery models, in order to guarantee
With the advantages of energy-saving and friendly to
the precision, in this article I chooses Thevenin model as
the environment, Hybrid Electric Vehicle and pure Electric
Lithium battery model. This kind of model has good dy-
vehicle have been developing rapidly. Proper operation of
namic performance and can simulate the dynamic charac-
these new type vehicles rely on batteries and their manage
teristics of Lithium ion battery accurately [3].
system; therefore the stability and reliability of power bat-
teries are important for energy transformation [1].
One of the most important parameter in battery man-
agement system is state of charge (SOC). State of charge
means the proportion of released energy from all battery
energy storage. Only SOC estimated precisely, energy bal-
ance can be done without damaging the batteries. However
due to battery complex internal chemical and physical reac-
tion, SOC cannot be measured directly and multiple factors
should be considered, so it is difficult to estimate SOC
Fig.1 Thevenin battery model
when battery is working.
In figure.2, I represent battery charging and discharging
Ampere-hour (Ah) method is the most commonly used
current flow; V0 represent battery terminal voltage.
estimation method. However integral calculation has accu-
Thevenin dynamic circuit model includes: 1.the battery
mulation error, with the growth of working time, accumula-
open circuit voltage Vd; 2.the battery internal resistance R,
tion error will result in SOC estimation error. Ampere-hour
including ohm resistance Ri and electrochemical polariza-
method is suitable for full charge and full discharge situa-
tion resistance Rd; 3.equivalent capacitance Cd is to describe
tion, but not for hybrid electric vehicles. Besides Am-
cell electrochemical polarization and concentration polari-
pere-hour method, other normally used SOC estimation
zation, which reflects the battery transient response of
method includes open circuit voltage method, neural net-
charge or discharge. According to the Kirchhoff's voltage
work method. Open circuit voltage is often used in battery
and current laws, and capacitance voltage variation and its
overhaul rather than state of charge estimation; neural net-
current relationship, Thevenin circuit model of the state
work models are too complicated, and in order to train the
space equation can be expressed as [4]:
model, plenty of data are needed [2].
I
The method presented in this paper is extended algo- Vb = . (1)
rithm based on Kalman filter. The core of extended Kalman Cb
filter algorithm is to propose a precise state describe model, I 1
according to the measurement data to calculate the values of Vd = − Vd + I . (2)
objects with recursive algorithm. Comparing to other algo- RdCd Cd
rithms, extended Kalman filter algorithm has its own ad- V 0 = Vb + Vd + IRi . (3)
vantages: even if there were errors in system, The relationship of OCV and SOC for Lithium ion bat-
self-correction can be down within a certain range and op- tery is not linear. Drawing the curve graph base on the

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 Proceedings of the 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013)

measurement data, an OCV function relation about SOC pansion is needed for the linearization, the model after line-
can be obtained: arization is:
Vb = k * SOC + d . (4) x = At ⋅ x + Bt ⋅ u + w . (12)
-30
y = Ct ⋅ x + Dt ⋅ u + v . (13)
-20
3.5
-10
10
Among these,
3.0
20
0 0 
∂f ( x , u)
At = x(t ), u (t ) =  
∂x 0 − I 
Voltage (V)

2.5

2.0
 RdCd 
 1 
1.5
∂f ( x , u)  kCb 
Bt = x(t ), u (t ) =  .
1.0
0 500 1000 1500 2000
∂u  1 
Capacity (mAh)  Cd 
Fig.2 OCV and SOC curve from -30C。~20C。 ∂g ( x , u )
Ct = x (t ), u (t ) = [ k 1] .
Based on the equations above, the system equations can ∂x
be derived: ∂g ( x , u )
 I  Dt = x (t ), u (t ) = Ri .
  ∂u
 SO C  kC b
 Vd  =   (5) After discretization, the model can be expressed as:
  − I Vd + 1 I  xk + 1 = A k ⋅ xk + B k ⋅ u k + w k . (14)
 R d C d C d  yk = Ck ⋅ xk + Dk ⋅ uk + vk . (15)
V 0 = kSOC + Vd + IRi + d . (6) Extended Kalman filtering algorithm calculation steps
These two equations are state equations and observation are as follows:
equation for Lithium ion battery model, discretization for
continuous equation and SOC calculation will be done in Initialization: k=0, Set initial condition x*k ， Error
next part. covariance matrix P*and noise covariance matrix Q*。
III. THE EKF ESTIMATION PRINCIPLE OF SOC calculation： k = 1, 2,3...... .
Kalman filtering method algorithm equation can be di- update state estimation：
vided into two parts [5]:
 
1. Predict system status, system output and error; x* k k - 1 = A* k - 1 ⋅ x* k - 1 + B* k - 1 ⋅ u* k - 1 + wk - 1 . (16)
2. Correct the current state estimate value based on sys-
tem output value. update error covariance：
When the system's state equation is nonlinear, Kalman
P* k / k - 1 = A* k - 1P* k - 1A* T k - 1 + Q* . (17)
filtering method cannot be put into use directly, system
equation discretization is required, and this kind of nonline- Kalman gain matrix：
ar equation for discretization of the Kalman filtering meth-
od is called extended Kalman filtering method (EKF). Ac- Lk = P* k / k - 1C* k T (C* kP* k / k - 1C* k T )−1 . (18)
cording to equations (5) (6), the system state equation can
be written as: update state estimation and measurement results：
  
x = [ x1 x 2]T ， x1(t ) = SOC ， x 2(t ) = Vd . (7) x* k = x* k k - 1 + Lk[ y* k − (C* kx* k k - 1 + D* kuk +vk)] .(19)
System input is defined as u(t)=I，output is defined as update results of error covariance：
y(t)=V0，then the battery model can be written as: P* k = ( I - LkC* k ) P* k / k - 1 . (20)
x = f ( x , u) + w . (8) Extended Kalman filtering method include above five
y = g ( x , u) + v . (9) basic formulas，which can calculate the system state after
Among these, put parameters into the formulas. The core of EKF was to
 u  compare the predicted value and the measurement value,
 kCb  adjusting Kalman gain according to the size of the error, the
f ( x, u ) =  . (10) gain will be used to calculate the next predicted value. The
− I x2 + 1 u  bigger the error, the greater the gain, the bigger the esti-
 RdCd Cd  mated value will be corrected; the lesser the error, the
g ( x, u ) = kx1 + x 2 + Riu + d . (11) smaller the gain, the lesser the estimated value will be cor-
In these formulas, w and v represent system noise and rected. Based on such a thought, extended Kalman filtering
measurement noise. For formulas (8) (9), Taylor series ex- method has good real time calculation performance [6].

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 Proceedings of the 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013)

IV. THE SIMULATION AND EXPERIMENT 2

SOC estimate value

A．The experiment system Ah intergal value

1.5
In order to validate the result of extended Kalman fil-
tering algorithm, the simulation results were compared with
the experimental data in the Matlab simulation. The archi- 1
tecture of a system diagram showed below，In order to Ah intergal value

SOC
obtain the parameters for calculation, it contains a current
0.5
measurement circuit, voltage measurement circuit and
low-pass filter circuit which is used to filter out signal
noise[7]: 0
SOC estimate value

-0.5
0 50 100 150 200 250 300
Sample times

Fig.4 SOC estimation value and Ah integral value

Figure 4 is constant current charging experiment wave-
form diagram, SOC estimation results and expected Ah
integral values are coincide basically, which explains that
EKF algorithm can achieve accurate SOC estimation val-
ues.

Fig V. CONCLUSION
.3 The experimental system structure diagram A new method for state of charge estimation based on
The Lithium ion battery used in experiment is 26650, extended Kalman filter algorithm has been presented in this
the capacity of single battery is 2200 mah, the highest volt- article, after obtain the relationship of Lithium ion battery
age is 3.65 V, and discharge termination minimum voltage SOC and OCV, firstly confirm the parameters of Lithium
is 2.84 V, lower than this voltage will lead to battery capac- ion battery model, and secondly plug these parameters into
ity decline seriously or even scrapped. state equation and observation equation, the current and
Lithium ion battery parameters is as follow： voltage that measured will be used in equations, and then
Ccb = 11252.4 F , Ccs = 27.31F , Ri = 0.1174Ω through the EKF equations iterative calculation and update
system status constantly, an accurate estimation value can
Rt = 0.1645Ω , k = 4.38 , d=23.74 be obtained ultimately. This kind of EKF calculation algo-
Error and noise covariance matrix selected as the fol- rithm has been validated through the simulation and ex-
lowing values： periment that the SOC estimation based on EKF algorithm
is effective and accurate. If the difference between noise
2 0 0.005 0  value in equations and the actual noise value are too big,
P=  R=
0 2 ,  0 0.0191 Q = 2 estimated value is difficult to converge to a reasonable
, range. How to estimate the influence of noise in the algo-
B．The experimental results rithm is the next step research direction.
In order to simulate Lithium ion battery charge and dis-
charge state in normal use and verify the effectiveness of REFERENCES
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Published by Atlantis Press, Paris, France.

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 Proceedings of the 2nd International Conference on Computer Science and Electronics Engineering (ICCSEE 2013)

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© the authors
2319