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A data-driven multi-scale extended Kalman ﬁltering based parameter

and state estimation approach of lithium-ion polymer battery in electric

vehicles

Rui Xiong

a,b,

⇑

, Fengchun Sun

, Zheng Chen

, Hongwen He

National Engineering Laboratory for Electric Vehicles, School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China

DOE GATE Center for Electric Drive Transportation, Department of Electrical and Computer Engineering, University of Michigan, Dearborn, MI 48128, USA

highlights



A data-driven multi-scale extended Kalman ﬁltering is developed for battery system.



A lumped parameter battery model against different aging levels has been proposed.



The proposed approach has less computation efﬁciency but higher estimation accuracy.



The proposed approach can estimate battery parameter, capacity and SoC concurrently.



The robustness of the proposed approach against different aging levels is evaluated.

article info

Article history:

Received 28 April 2013

Received in revised form 19 July 2013

Accepted 27 July 2013

Available online 23 August 2013

Keywords:

Electric

vehicles

Lithium-ion polymer

battery

Data-driven

State-of-Charge

Battery capacity

Multi-scale

abstract

Accurate estimations of battery parameter and state play an important role in promoting the commer-

cialization of electric vehicles. This paper tries to make three contributions to the existing literatures

through advanced time scale separation algorithm. (1) A lumped parameter battery model was improved

for achieving accurate voltage estimate against different battery aging levels through an electrochemical

equation, which has enhanced the relationship of battery voltage to its State-of-Charge (SoC) and capac-

ity. (2) A multi-scale extended Kalman ﬁltering was proposed and employed to execute the online mea-

sured data driven-based battery parameter and SoC estimation with dual time scales in regarding that the

slow-varying characteristic on battery parameter and fast-varying characteristic on battery SoC, thus the

battery parameter was estimated with macro scale and battery SoC was estimated with micro scale. (3)

The accurate estimate of battery capacity and SoC were obtained in real-time through a data-driven

multi-scale extended Kalman ﬁltering algorithm. Experimental results on various degradation states of

lithium-ion polymer battery cells further veriﬁed the feasibility of the proposed approach.

1. Introduction

To address the two urgent goals nowadays of protecting the

environment and achieving

energy sustainability, it is of strategic

signiﬁcance on a global scale to replace oil-dependent vehicles with

electric vehicles. Battery, as an important on-board electric energy

storage, has been widely used in various electric vehicles. However,

to satisfy the operation voltage and traction power requirements of

electric vehicles, battery packs have to be made up of hundreds

of cells connected in series or parallel to overcome the limitations

of low energy density, low cell capacity and low cell voltage. But

how to avoid the adverse effect of cell inconsistency on battery pack

performanc

e and

prolong the service life of both the pack and the

cells are posing tremendous technological challenges to battery

State-of-Charge (SoC) and capacity estimation techniques. Its

accurate estimation is not only beneﬁcial for the efﬁcient vehicular

energy management, but also for the diagnosis and prognosis of the

battery behavior.

A wide variety of SoC estimation methods have been put forward

to improve battery SoC determination [1–16], each one has its own

advantage. The most commonly used methods fall into two major

categories: the lumped parameter battery model including equiva-

lent circuit models [1–9] and electrochemical model [10] based

SoC estimation method and the ‘‘black box’’ based methods, such

as artiﬁcial neural networks based methods [11–13], fuzzy logic

based methods [14,15] and support vector regression (SVR) based

http://dx.doi.org/10.1016/j.apenergy.2013.07.061

⇑

Corresponding author at: National Engineering Laboratory for Electric Vehicles,

School of Mechanical Engineering, Beijing Institute of Technology, No. 5 South

Zhongguancun Street, Haidian District, Beijing 100081, China. Tel./fax: +86 10 6891

4842.

E-mail addresses: rxiong6@gmail.com, rxiong@ieee.org (R. Xiong).

Applied Energy 113 (2014) 463–476

Contents lists available at ScienceDirect

Applied Energy

journal homepage: www.elsevier.com/locate/apenergy

method [16]. The authors in [1] presented a comparative study of the

equivalent circuit model-based SoC estimation approaches algo-

rithms including Luenberger observer, extended Kalman ﬁltering

(EKF) and sigma point Kalman ﬁltering (SPKF) to monitor the SoC

of a LiFePO

lithium-ion battery (LiB) cell, and the results showed

that the SPKF was an optimal choice to estimate dynamic SoC behav-

ior. The authors in [2–4] presented an equivalent circuit battery

model-based method for real time battery cell SoC estimation using

linear parameter varying (LPV) system and reduced order EKF tech-

niques. Xiong et al. [5,6] proposed an equivalent circuit model based

SoC estimation method using adaptive Extended Kalman Filter

(AEKF) to estimate battery SoC through the measurements of battery

current and voltage. The authors in [10] proposed a reduced order

electrochemical model to estimate internal battery potentials, con-

centration gradients, and SoC from external current and voltage

measurements. The authors in [11] presented a novel approach

using adaptive artiﬁcial neural network and neuron-controller for

online cell SoC determination. The authors in [16] presented v-Sup-

port Vector Regression algorithm based method to estimate the SoC.

A common drawback of the above SoC estimation methods is

that

the model

parameters are identiﬁed with ofﬂine data or the

training data for ‘‘black-box’’ models is built by previously mea-

surement; as a result, the battery model parameters variances fol-

lowing with its degradation and varied operation conditions are

ignored. Thus, the reliability and applicable of these SoC estimators

were not sufﬁciently discussed. In order to overcome these draw-

backs, online parameter identiﬁcation methods were proposed to

track the real-time behavior of the battery. The authors in

[17,18] used the method of recursive least square with an optimal

forgetting factor to carry out the online battery parameters identi-

ﬁcation and state estimation. However, both the model parameters

and capacity are important battery parameters, while the above

method fails to estimate the battery capacity and model parame-

ters concurrently. The reliable capacity estimate is indispensable

for an accurate SoC estimate, and which is of paramount impor-

tance for battery State-of-Health (SoH) indication.

A number of research methods have been proposed for estimat-

ing

the cell

capacity and then to calculate the SoH with the estimated

capacity taking the SoH is the ratio of estimated capacity over its

nominal value, most of them are carried out with lumped parameter

models [5,19–26]. The authors in [19] presented a neural-network

model online estimation method for SoH of valve-regulated lead

acid batteries on the basis of the relationship between the estimated

SoC and the battery open circuit voltage. The authors in [20] pre-

sented a probabilistic neural network (PNN) based SoH estimation

method for LiB, where the PNN was trained using 100 pieces of bat-

teries. The authors in [21] presented an experiment data based SoC

and SoH estimation method with fuzzy logic system, while the

experiment data were provided from electrochemical impedance

spectroscopy (EIS) measurements on new and aged cells. However,

for electric vehicles application, it is not easy to obtain the overall

data for training, which leads to the inaccuracy prediction in com-

plex variable practical application. On the contrary, the dynamic

battery model-based method can provide a cheap alternative in esti-

mation or it can be used along with a sensor-based data-driven

scheme to provide some redundancy. The authors in [22–26] pre-

sented battery model-based dual/joint Kalman ﬁlters method to

estimate the battery SoC and capacity concurrently. However, their

model parameters were identiﬁed by ofﬂine data and the inﬂuence

of the battery degradation or operation conditions over model

parameters are not discussed, as a result, their performance are

not veriﬁed adequately. Furthermore, the authors in [22–25] used

dual EKF or dual SPKF to execute the capacity and SoC joint estima-

tion with one time scale. However, in considering the system param-

eter inclines to change slowly over time while system state is prone

to fast variation over time, it is not an optimal choice to use the same

calculated time scale for battery parameter and state calculation; on

the contrary, it will largely increase the computational burden of

battery

management

system (BMS). Thus the time scale separation

based method, which uses macro scale to calculate the battery

parameter and uses micro scale to calculate the battery state, will

lower the computation cost of BMS. The authors in [26] used the

two time scales to estimate the battery capacity and SoC concur-

rently, and the difference of estimated SoC with dual scales was used

as an innovation to update the Kalman gain to correct the capacity.

But it is very hard to obtain the ‘‘true’’ SoC, thus it is not easy and reli-

able to obtain accurate capacity by the estimated SoC especially

when the estimated SoC is not stable.

The purpose of this paper is to establish general battery param-

eter

and state

dual estimation method using data-driven multi-

scale EKF algorithm, which is a key technique to safeguard the

optimal and safe use of battery – energy source in various electric

vehicles and promote the commercialization of electric vehicles.

The description of the research system and the data-driven mul-

ti-scale EKF algorithm are presented in Section 2. Section 3 de-

scribes the implementation ﬂowchart of the proposed approach.

To evaluate the proposed approach, four different health status of

lithium-ion polymer battery (LiPB) cells are used to carry out the

veriﬁcation are shown in Section 4. The experiment, simulation re-

sults and evaluation of the proposed method are reported in Sec-

tion 5 before conclusions are drawn in Section 6.

2. Data-driven multi-scale extended Kalman ﬁltering

To make the discussion more convenient, ﬁrstly the section con-

structs

a very

general framework for discrete-time lumped dy-

namic system with dual scales. Afterwards, based on the review

of dual EKF, the implementation process of online measured data

driven based multi-scale EKF is built.

2.1. System description

In regarding that the slow-varying characteristic on battery

paramete

r and

fast-varying characteristic on battery state, we

use multi-scale method to construct the discrete time state-space

equation, where the system parameter and system state are pre-

dicted with the macro and micro scale separately. To be more spe-

ciﬁc, we consider the problem of learning both the hidden states

and parameters h of a very general framework for discrete-time

nonlinear dynamical system as:

k;lþ1

¼ Fð

k;l

; h

; u

k;l

Þþ

k;l

; h

kþ1

¼ h

k;l

¼ Gð

k;l

; h

; u

k;l

Þþ

k;l

(

ð1Þ

where

k,l

is the system state matrix at the time t

k,l

= t

k,0

+l T

(1 6 l 6 L), herein the T is a ﬁxed sampling interval between two adja-

cent measurement points, k and l being the two time-scales indices

for system parameter with macro scale and system state with micro

scale respectively; u

k,l

is the exogenous input matrix at time t

k,l

; y

k,l

the system observation (or measurement) matrix at time t

k,l

;

k,l

and

are the process noise matrix for state and model parameter respec-

tively,

k,l

is the measurement noise matrix. Note that L represents the

level of time-scale separation and that

k,0

k1,L

. h

is the parame-

ters matrix under the kth macro scale, and h

= h

k,0:L1

. With the de-

ﬁned system, we aim at estimating both the system state

and

model parameter h from the noisy observations Y.

2.2. Review of dual EKF method

The EKF provides an efﬁcient approach for generating approxi-

mate maximum-likelihood

estimates of the state of a discrete-time

nonlinear dynamical system. However, the prediction precision of

464 R. Xiong et al. / Applied Energy 113 (2014) 463–476

the EKF largely depends on the predetermined parameter values

used in the system model. Therefore, the dual estimation for the bat-

tery parameter and state is very important. Furthermore, to make

the estimation accuracy of the researched system state more reli-

able, the system parameter is continuously requiring periodic cali-

bration, otherwise the estimation accuracy may be reduced

greatly and even leads to invalid. In other words, when the accurate

system model is not available, a dual estimation approach is re-

quired to achieve reliable and accurate system parameter and state

dual estimation. In this case, the dual EKF, in which both the state of

the dynamical system and parameter are estimated simultaneously,

is proposed to achieve a better prediction precision [25–28].

In this section, we ﬁrstly review the dual EKF algorithm, which

combines

the Kalman

state and parameter ﬁlter. It is noted that its

goal is to predict both the state and model parameter from only

noisy observations. Essentially, two EKF are run concurrently. At

each calculation time, an EKF state ﬁlter-EKF

estimates the state

using the present model estimate h

, while the EKF parameter ﬁl-

ter-EKF

estimates the parameters using the present state estimate

. The system is shown schematically in Fig. 1.

Since the dual EKF does not take into account the time-scale

separation, we use

the notations

to replace

k,l

. In this case,

the measurement is replaced with Y

. It is noted that the system

state matrix



and parameter matrix



are the priori estimates

before the measurement Y

is taken into account; the system state

matrix

and parameter matrix

are the posteriori estimates after

the measurement y

is taken into account.

From Fig. 1, we

ﬁnd that the dual EKF algorithm consists of two

EKF that run concurrently. The top EKF generates state estimate,

and requires



for its time update, thus the ﬁrst step of the dual

EKF operation is time-update for the system parameter, afterwards

the system state is estimated with the



. The bottom EKF gener-

ates parameter estimate, and requires

for the measurement up-

date. The detail numerical formulation of is presented in [28].

2.3. Data-driven multi-scale extended Kalman ﬁltering

To reduce the computation cost of the BMS, a multi-scale EKF

algorithm is proposed

and expected to employ the macro scale to

identify the system parameter of slowly time-varying and employ

the micro scale to estimate the system state of fast time-varying,

and provide more stable estimates of system parameter and state.

The algorithm of the multi-scale EKF for the system described in

Eq. (1) is

summarized in Table

Where h

is the initial system parameter and

is its guess va-

lue;

0,0

is the initial system state and

0;0

is its guess value. The

algorithm is initialized by providing the model parameters and

states to the best guesses based on the prior information, the

covariance matrices are also initialized based on the prior informa-

tion, which is shown in Eq. (2).

Additionally, the computation of C

involves a total derivative of

the measurement function with respect to the parameters as Eq.

(10). In considering that the system state maybe the function of

system parameter, we should decompose the total derivative into

partial derivatives by the following recursive equations.

dGð

k;0

; h; u



h¼



@Gð

k;0

;



; u



@Gð

k;0

;



; u

k;0



k;0



ð11Þ

Then,

k;0



dFð

k1;L1

;



k1;L1



@Fð

k1;L1

;



k1;L1



@Fð

k1;L1

;



k1;L1



ð12Þ

k1;L1



k1;L1

þ K

k1;L1

ðY

k1;L1

 Gð



k1;L1

;



; u

k1;L1

ÞÞÞ ð13Þ



ðK

k1;L1

Þ¼Y

k1;L1



ð14Þ



k1;L1

Gð



k1;L1

;



; u

k1;L1



¼ K

k1;L1

dGð



k1;L1

;



; u

k1;L1



k1;L1



k1;L1

;



; u

k1;L1



ð15Þ

Fig. 1. The dual extended Kalman ﬁlters.

R. Xiong et al. / Applied Energy 113 (2014) 463–476

465

Table 1

Algorithm of multi-scale extended Kalman ﬁltering.

Initialization

¼ E ½h

;

¼ E ðh



Þðh



;

0;0

¼ E ½

0;0

;

0;0

¼ E ½ð

0;0



0;0

Þð

0;0



0;0

ð2Þ

For k

{1, , 1},calculation:

Step 1: Time-update equations for parameters ﬁlter



with macro scale



k1

;

¼

k1

ð3Þ

For l

{1, ..., L}, calculate the state ﬁlter at each micro scale

Step 2: Time-update equations for state



k;l

with micro scale



k1;l

¼ Fð

k1;l1

;



; u

k;l1

Þ;

k1;l

¼A

k1;l1

ð4Þ

Step 3: Measurement-update equations for state

k;l

with micro scale

k1;l

ðC

k1;l

ðC

k1;l



k1;l

þ K

k1;l

½Y

k;l

 Gð



k1;l

;



; u

k1;l

Þ

k1;l

¼ðI  K

k1;l



ð5Þ

Step 4: For time-series calculation l =1:L (l ? L)



k1;L

¼ Fð

k1;L1

;



; u

k1;L1

Þ;

k1;L

¼A

k1;L1

k1;L

k1;l

ðC

k1;L

½C

k1;L

ðC

k1;L



k1;L



k1;L

þ K

k1;L

½Y

k;L

 Gð



k1;L

;



; u

k1;L

Þ

k1;L

¼ðI  K

k1;L



ð6Þ

Step 5: Time scale transform

k;0

k1;L

;

k;0

k1;L

; y

k;0

¼ y

k1;L

; u

k;0

¼ u

k1;L

ð7Þ

For k

{1, ... , 1}, calculation:

Step 6: Measurement-update equations for state ﬁlter

with macro scale

ðC

½C

ðC





þ K

½Y

k;0

 Gð

k;0

;



; u

k;0

Þ

¼ðI  K



ð8Þ

where

k1;l1

@Fð

;



; u

k1;l1

:; C

k1;l

@Gð

;



; u

k1;l



k1;l

ð9Þ

dGð

k;0

; h; u

k;0

h¼



ð10Þ

466 R. Xiong et al. / Applied Energy 113 (2014) 463–476

3. Application to battery system

Section 3.1 presents the discrete-

time cell dynamic model used

in this study. Section 3.2 presents the implementation ﬂowchart of

the proposed approach.

3.1. Lumped parameter battery model

In order to execute the parameter and system estimation with

multi-scale

EKF algorithm,

we need a ‘‘discrete-time cell dynamic

model’’ that relates the battery parameter and SoC to its voltage.

As shown in Fig. 2, it is very important to correctly identify the

model parameter, including the open circuit voltage (OCV) which

is used to describe the voltage source, series resistance (R

) which

is used to describe the electrical resistance of various battery com-

ponents or with the accumulation and dissipation of charge in the

electrical double layer, diffusion resistance (R

Diff

) and diffusion

capacitance (C

Diff

) which consist of a RC network to describe the

mass transport effects and dynamic voltage performances.

Where i

is the load current (assumed positive for discharge,

negative for charge), U

is the terminal voltage, U

describes the

diffusion voltage over the RC network. The battery OCV is replaced

with g(z, Q) in this study, which describes the OCV under different

aging levels and SoCs. Where the z is the abbreviation for battery

SoC, Q denotes the present maximum available capacity of the LiPB

cell. It is noted that the relationship of battery SoC and capacity

with its voltage behavior has been enhanced through the g(z, Q),

which is very important for improving the capacity and SoC predic-

tion precision. The electrical behavior of the lumped parameter

university cell model can be expressed as:

¼

Diff

¼ gðz; QÞU

 i

(

ð16Þ

Herein, we use an electrochemical equation to describe the OCV,

which is from [5].

gðz; Q Þ¼K

þ K

z þ K

þ K

=z þ K

lnðzÞþK

lnð1  zÞ

ð17Þ

where K

(i =0,1,..., 6) are seven polynomial coefﬁcients to accu-

rately ﬁt the OCVs under different SoCs and aging levels.

Considering that the slow-varying behavior in model parame-

ter, we assume that the battery is a time-invariant system and

the load current keeps constant at each sampling interval. After-

wards, we can obtain the analytical solution of Eq. (16) as:

ððk þ 1ÞTÞ¼exp 

Diff



ðkTÞ

exp 

Diff



dt 

ðkTÞ

Diff

ð18Þ

Then we can calculate the diffusion voltage by:

ððk þ 1ÞTÞ¼exp 

Diff



ðkTÞ

þ R

Diff

ðkTÞ 1  exp 

Diff



ð19Þ

Afterwards we can rewrite the diffusion voltage with the sys-

tem equation in Eq. (1).

k;l

¼ exp 

Diff



 U

k;l1

þ 1  exp 

Diff



 i

k;l1

Diff

ð20Þ

where U

k;l

is the diffusion voltage U

at time t

k,l

, i

k,l1

is the load cur-

rent at time t

k,l1

On the other hand, the SoC is the equivalent of a fuel gauge for

the battery in electric vehicles. The basis for SoC state-equation is

developed as follows: If z

k,l

= SoC, we know that

k;l

¼ z

0;0



k;l

0;0

ði

ÞÞi

ð21Þ

where z

0,0

is the initial SoC, i

(

) is the cell current at time

, and

(

)) is the Coulombic efﬁciency of the battery [6]. Herein Q is

the cell maximum available capacity and not the cell nominal capac-

ity, the nominal capacity is the maximum total electrical charge that

can be stored in a cell. It is not a function of temperature and aging

level. The maximum available capacity (discharge capacity) is the

product of the nominal discharge current and discharge time with

the nominal current, thus a shorter discharge times mean a lower

discharge capacity. This indicates that the maximum available

capacity is inﬂuenced by the operation condition and aging level

of battery, thus the Q is a function of temperature and aging level.

A discrete-time approximate recurrence may then be written

as:

k;l

¼ z

k;l1



ði

k;l1

Þi

k;l1

ð22Þ

From Eqs. (16), (20), and (22), we can get:

k;l

exp 

Diff



k;l1

1 exp 

Diff



Diff



ði

k;l

ÞT

k;l1

ð23Þ

We then obtain the state transition and measurement equations as:

k;lþ1

¼ Fðv

k;l

Þ¼

exp 

Diff



k;l

1exp 

Diff



Diff



ði

k;l

ÞT

k;l

¼ Gðv

k;l

Þ¼gðz

k;l

ÞU

k;l

R

k;l

ð24Þ

where

k;l

¼ R

Diff

Q½

k;l

¼ U

k;l

¼ i

k;l

ð25Þ

where U

k;l

is the measured battery voltage at time t

k,l

3.2. Implementation ﬂowchart for advanced multi-scale approach

In accordance with the dual EKF based estimation method

shown in Fig. 1,

we intend to build a multi-scale EKF based estima-

tion method and which allows for a separation of time scale in

state and parameter estimation. To be more speciﬁc, we use macro

Fig. 2. Lumped parameter battery model.

R. Xiong et al. / Applied Energy 113 (2014) 463–476

467

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