基于扩张卡尔曼滤波的磷酸铁锂蓄电池SOC检测，给出了电池模型和算法实现过程

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基于扩张卡尔曼滤波的磷酸铁锂蓄电池SOC检测，给出了电池模型和算法实现过程.zip （2个子文件）

基于扩张卡尔曼滤波的磷酸铁锂蓄电池SOC检测，给出了电池模型和算法实现过程

battery SOC estimation based on EKF.pdf 946KB

Matlab实现无约束条件下普列姆(Prim)算法.docx 14KB

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Journal of Power Sources 174 (2007) 9–15

Li-ion battery SOC estimation method based on the reduced

order extended Kalman ﬁltering

夽

Jaemoon Lee

∗

, Oanyong Nam, B.H. Cho

School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-744, Republic of Korea

Received 7 December 2006; received in revised form 26 March 2007; accepted 27 March 2007

Available online 18 April 2007

Abstract

The extended Kalman ﬁlter (EKF) method for SOC estimation has some problems such as the lack of an accurate model, and model errors due

to the variation in the parameters of the model due to the nonlinear behavior of a battery. To solve the aforementioned issues, this paper proposes

a reduced order EKF including the measurement noise model and data rejection. In order to do so, the model of a battery in the EKF is simpliﬁed

into the type of reduced order to decrease the calculation time. Additionally, to compensate the model errors caused by the reduced order model

and variation in parameters, a measurement noise model and data rejection are implemented because the model accuracy is critical in the EKF

algorithm in order to obtain a good estimation. Finally, the proposed algorithm is veriﬁed by short and long term experiments.

Keywords: State of charge (SOC); Extended Kalman ﬁlter (EKF); Reduced order; Li-ion battery

1. Introduction

In recent years, much research has been done to improve the

estimation of the state of charge (SOC), which has increasingly

become an important issue in electric/hybrid vehicle applica-

tions. Ah counting, which is the most common method of

estimating a battery SOC, is easy and reliable. However, it has

problems such as the initial value problem, an accumulated error

problem from incorrect measurements, and no consideration of

the current losses. Open circuit voltage (OCV) for the measure

[5,6,7]. However, it requires a sufﬁcient rest period. To deal

with these problems for Ah counting and OCV, adaptive meth-

ods, such as neural network, fuzzy logic, adaptive observer and

extended Kalman ﬁlter (EKF) have been employed, based on

Ah counting, OCV and other factors [1,4,8].Adaptive methods

require an accurate model to obtain good [3,12]. Many states and

factors are necessary to develop the accurate model of a battery.

In general, if states and input factors increase, the calculation

burden also increases. If the number of states in the EKF is n

and the dimension of the measurement vector is /, the compu-

夽

The submitted work was presented at the 4th International Energy Conver-

sion Engineering Conference and Exhibit (IECEC).

∗

Corresponding author. Tel.: +82 2 880 1785; fax: +82 2 878 1452.

E-mail address: jmoonzz@shinbiro.com (J. Lee).

tational complex is (l

∗

+ n

)) [9]. Therefore, the number of

states and input factors is limited by the processor’s calculation

capability. The battery pack system in the HEV is composed

of many series-connected cells or modules. The BMS (battery

management system) in the HEV should measure the voltages

of each cell or module, the current of the battery, and the temper-

atures in the battery pack. Because the charge/discharge current

in the HEV can be rapidly ﬂuctuated, the period of the algorithm

should be less than 100 ms. Also, SOC estimation algorithm can

be applied to each cell or module of the battery pack. Therefore,

the states in the EKF should be reduced in order to decrease the

calculation time. A trade-off between performance and compu-

tational requirements is likely to be made. Also, the parameters

of the model vary due to the conditions of the battery. Variations

in the parameters of the model cause model errors, which in turn

affect the estimation results.

To solve these problems, this paper proposes a reduced order

EKF implemented by a simple battery model with a measure-

ment noise model and data rejection. Generally, the battery in

the EKF is represented by an equivalent circuit model based on

its impedance spectrum. It consists of a series resistance, double

layer and charge transfer, and diffusion [2,10,11]. This equiv-

alent circuit model is complex and nonlinear. In the equivalent

circuit model, fast dynamic components can be modeled as a

resistance and, slow dynamics components can be described as

doi:10.1016/j.jpowsour.2007.03.072

10 J. Lee et al. / Journal of Power Sources 174 (2007) 9–15

a RC ladder circuit. Fast and slow dynamics are separated by

the impedance spectrum, where the RC time constant can be

determined by the impedance spectrum data. As a result, the

battery model is simpliﬁed into an open circuit voltage, RC lad-

der, and series resistance. The inﬁnite series RC ladders of the

diffusion are reduced into one or two RC ladder circuits. The

double layer and charge transfer are represented by one resis-

tance. The reduced order model can cause an accuracy problem.

The errors caused by the model simpliﬁcation are mainly caused

by the fast dynamics of diffusion, charge transfer, and double

layer. In addition, the parameters of the equivalent circuit model

vary with the SOC, battery current and so on. Variations in the

parameters due to the conditions of the battery are an additional

cause of the model errors. In order to solve the model accu-

racy and parameter variation problems, the measurement noise

model is used. The noise measurement to compensate the model

errors is separately conducted, depending on whether the model

is comparatively correct or incorrect, because it has an inﬂuence

on deciding the Kalman ﬁlter gain. The regions are classiﬁed

by the SOC, battery current, and battery dynamics. The model

errors from the fast dynamics of the battery are decreased by

data rejection.

Several tests with a 1.3Ah 18650 type Li-ion battery are con-

ducted to verify the proposed algorithm. Short term tests show

the effectiveness of each detailed algorithm. Long term tests are

performed to guarantee the stability and reliability of the algo-

rithm. As a result of the long term tests, it is shown that the error

of SOC estimate is less than 2%.

2. Li-ion battery model

The impedance-based model is used. The equivalent circuit

model is reduced to the simple model. The EKF equations of the

simple model are also described.

2.1. Impedance-based equivalent circuit model

The electrochemical characteristics of the battery are clas-

siﬁed as follows: internal resistance, charge transfer, double

layer and diffusion. As shown in Fig. 1, they are described as a

resistance, capacitance and constant phase element (CPE) in the

impedance-based circuit model [2,10].

The elements of resistance and capacitance are easily realized

in the time domain, but the CPE must be realized by a distributed

circuit model which consists of inﬁnite RC ladder elements in

order to obtain an accurate model. Usually, the diffusion and

Fig. 1. Circuit model: L (parasitic inductance), R

(internal resistance), C

(double layer: CPE), R

(charge transfer), Z

(diffusion = Warburg impedance:

CPE).

double layer phenomena in the battery are modeled by the CPE.

Fig. 2 shows the RC ladder circuit for diffusion. This series-

connected RC ladder circuit for the CPE needs to be simpliﬁed,

because the increase of the states and parameters is the main

cause of the longer EKF calculation time. This simpliﬁcation

is rather critical in reducing the digital processor memory and

calculation time, considering the nonlinearity of each parameter

(see Fig. 2).

2.2. Model simpliﬁcation

The electrochemical equivalent circuits can describe the

behaviors of the battery and can be separated into three parts,

consisting of a series resistance, charge transfer and double layer,

and diffusion process, in terms of the frequency components.

The slow dynamics of a battery is included in a simple battery

model. The fast dynamics is compensated by the measurement

noise model and data rejection. The fast and slow dynamics of

the battery can be distinguished by the impedance spectrum. The

impedance spectra of the battery used in the experiment are plot-

ted in the Nyquist domain, as shown in Fig. 3. The frequency of

the series resistance is more than 500 Hz which indicates a very

fast dynamics. The frequency of the charge transfer and double

layer is in the range between 500 and 0.63 Hz. The calculated R

and C

are 27.6 m and 0.5693 F, whose time constant is less

than 0.02 [10]. The dynamics of R

and C

is relatively fast.

The frequency of the diffusion is less than 0.63 Hz. The equiv-

alent circuits of the diffusion are composed of series-connected

inﬁnite RC ladder circuits. The time constants of the ﬁrst and

second RC ladder are calculated as about to be 44 and 11,

respectively.

The complex equivalent circuit model can be simpliﬁed based

on the dynamics of the battery. The fast dynamics of a RC ladder

circuit can be represented as a resistance. The charge transfer

and double layer are modeled as a resistance due to its very fast

dynamics. The diffusion is represented by one or two RC ladder

Fig. 2. The inﬁnite numbers of the RC ladder model about diffusion.

J. Lee et al. / Journal of Power Sources 174 (2007) 9–15 11

Fig. 3. Nyquist plot of battery used in experiment.

circuits. The ﬁrst RC ladder is very slow and is included in the

model. The second RC ladder is used as a criterion distinguishing

the fast and slow dynamics for the diffusion. The dynamics of the

other RC ladder circuits, for the diffusion, is fast and negligible.

Finally, the proposed reduced order model of the battery is

shown in Fig. 4. The proposed model consists of the OCV, one

RC ladder for the diffusion, and, R

or R

one resistance for a

series resistance and for charge transfer and double layer. It has

two states, which are OCV and C

Diff

2.3. Reduced order extended Kalman ﬁlter using simple

model

In general, the process and measurement models used in the

EKF are as follows:

Process model : x

= f

k−1

) + g(u

k−1

) + w

k−1

∼ N(0,Q

), Measurement model :

= h

) + i(u

) + v

∼ N(0,R

) (1)

The equations that decide the Kalman gain are as follows:

= P

+ R

]

−1

(2)

ˆx

(+) = ˆx

(−) + K

− H

ˆx

(−)] (3)

Fig. 4. Simpliﬁed model: V

Diff

(diffusion), V

(internal resistance and charge

transfer).

where X

is the kth order value of X, X(−) the priori value of

X, X(+) the posteriori value of X, x the state x, w the process

noise, Q the process noise covariance, v the measurement noise,

R the measurement noise covariance, K the Kalman gain, P the

covariance matrix of the state estimation uncertainty and H is

the measurement sensitivity matrix.

As described in Fig. 4, the model has only two states. With the

states incorporated, the process model in the EKF is expressed

as follows:

dSOC

, SOC

= SOC

K−1

t

· i

K−1

(4)

Diff

−

Diff

· R

Diff

Diff K



1 −

t

Diff

· R

Diff



· V

Diff K−1

t

Diff

· i

K−1

(5)



SOC

Diff K



⎡

⎣

01−

t

Diff

· R

Diff

⎤

⎦



SOC

K−1

Diff K−1



⎡

⎢

⎣

Δt

Diff

⎤

⎥

⎦

· i

K−1

(6)

The measurement model and the terminal voltage of a battery,

are expressed by a nonlinear function as follows:

= h

(OCV,V

Diff

) − V

= OCV − V

Diff

− V

(7)

∂h

∂x

⎡

⎣

∂h

soc

(SOC)

∂SOC

0 −1

⎤

⎦

where OCV = h

soc

(SOC),h

soc

= f

−1

ocv

(8)

In this equation, the OCV-SOC table is expressed as an OCV

function.

3. Measurement noise model and data rejection

Model errors are the voltage errors between the measured

battery voltage and the model output voltage. The model errors

are caused by the model simpliﬁcation and variations in the

parameters of the model. The reduced order model involves

the slow dynamics of the battery equivalent circuit model,

but excludes the fast dynamics. The fast dynamics cause the

model errors, which lead to inaccurate estimations. The fast

dynamics are mainly composed of the charge transfer and

double layer and diffusion. When there is a step change in

the battery current, the fast dynamics of the charge transfer

and double layer and diffusion are present. The measurement

noise model and data rejection are necessary to compensate

the model errors caused by the fast dynamics. The parame-

ters of the equivalent circuit model vary with the SOCs, the

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