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State-of-charge Estimation of Lithium-ionBattery Using an Adaptive Extended KalmanFilter Based on an Improved Thevenin Model

Hongwen He, Rui Xiong, Xiaowei Zhang, Fengchun Sun, Jinxin Fan

Ahstruct—An adaptive Kalman filtcr algoiïthm is adoptcd tocst imatc thé state-of-chargc (SOC) of l i th ium-ion battcry l'or théapplication in clcctric vchicles (ËVs). Gcncrally, thé Kalman fi ltcralgorithm is sclected to estimate SOC dynamically. Howcvcr, itcasi ly causes divergence due to thé uncertainty of battery niodeland System noise. To obtain better convergent and robust rcsult,an adaptivc Kalman filter algorithm is employed which cangrcatly better thé dependcnce of'the tradit ional tllter algorithm onthé battery niodel. In this papcr, thé typical characteristics ofl i thium-ion battery are analyzcd by experimcnt such as hystérésis,polari/ .ation and coulomb cf'ficiency, etc. In addition, an improvcdThevenin battery motlel is achieved by adding an extra RCbranch to thé Thevenin niodel and model parameters arcident i l l c i l by using an extcnded Kalman filter algorithm (EKF).Further, an adaptivc extendcd Kalman tllter algorithm (AEKF) isadoptcd to SOC est imat ion of l i thium-ion battery. Final ly , théproposed inethod is e v a l u a t e d by experiments with thé FédéralUrban Driving Schedules (FUDS). The proposed SOC estimationusing A E K F is more accurate and re l iable than that using EKF,(l ie comparison shows that thé max imum of SOC estimation errordecreases Irom 14.96% to 2.54% and thé mean ot'SOC estimationerror reduces t'rom 3.19% to 1.06%.

Index Ternis—Adaptive cxtended Kalman tllter (AEKF),state-of- charge (SOC), paramcter identification, battery model,electric vehicles (EVs).

I . INTRODUCTION

Battery pack, as a key component, is crucial for llieperformance of electric vehicles (EVs) such as Ihe

economy, power performance and security. etc. Il is significantto accuratcly manage thé battery pack to extend ils lifcspan,improvc its r e l i ab i l i t y and lovver its cost. The onl inestale-of-eharge (SOC) estimation of battery pack, which is alsov i i a l for thé power d is t r ibut ion strategy of EVs [l]-[4], i s o n e o f

Manuscript received September 18, 2010; revised January 16, 2011;accepted Mardi 14, 201 i. Date of pub l i ca t ion xxxxx xx, 2011; date of currenlversion xxxx xx, 2011.This \vork vvas supporied in part by Ihe Nat ional HighTechnology Research and Development Program of China (2008AA11A124).The review of th i s paper vvas coordinated by Dr. Reza Langari.

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The authors are wilh thé Nat ional Engineering Laboratory l'or ElectricVehicles, School of Mechanical Engineering, Beijing Ins t i tu te of Technology,No.5 South Zhongguancun Street, Beijing, 100081, China (e-mails: hwhebit@bit.edu.cn; xrui_ev@ 126.coin; fotSI50@bil .edu.cn; sunfch@bit.edu.cn;j inxinfan@gmail .com).

thé main tasks of battery-management System [4]-[7J. Anumber of methods to estimate SOC hâve been proposed, eachwith its own advantages and disadvantages [8]-| 17]. Accordingto thé choice of thé battery model, some commonly usedmethods can be approximately categorized into thrce types.One is coulomb counting method, which is an open-loopalgorithm with thé flavv of big accumulation of measurementerrors due to uncertain disturbances, and diff iculty to déterminethé initial value of SOC accurately. The second is based onblack-box battery models that describe thé nonlinearrelat ionship between SOC and its inf luencing factors. whichcan often produce a good estimate of SOC due to thé powerfula b i l i t y lo approximate nonl inear function surfaces, with théflaw of heavy computation burden and bad real-timeapplicat ion. The th i rd is based on state es t imat ion techniqueswith state-space battery models, which is popular using due tothé advantages of being close-loop. on l ine and avai lable toregulate thé estimation error range dynamically.

The extended Kalman filter (EKF) algorithm can nol on ly beused to implement thé parameters identification of batterymodel but also can be employed for onl ine SOC estimation|12]-|22|. In part icular . il is suitable for dynamic d r iv ingconditions. However, it strongly dépends on thé accuracy of thébattery model as vvell as thé predetermined variables of théSystem noise such as mean value, pertinence and covariancematrix. Usually, thé improper setting of thé predeterminedvariables of thé System noise may result in remarkable errorsand divergence.

To obtain more accurate and robusl SOC est imat ion, in th i spaper. an improved Thevenin batlery model is put forward,which is based on thé analysis of polarization characteristics ofa l i t h i u m - i o n battery module by expérimental results, and itsparameters are identified by EKF algorithm. In addition, anadaptive exlended Kalman filter (AEKF) algorithm is designedfor thé SOC estimation of thé lithium-ion battery module.

IL MODEL AND PARAMETERS IDENTIFICATION OFL I T H I U M - I O N BATTERY MODULE

A. Lithium-ion Batlery Modeling

Generally, équivalent circuit models are selected. whichinclude Rin t model, Thevenin model. RC model and PNGVmodel [15], [23]-|25] for battery modeling. Among them. thé

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Thevenin model is vvidely usecl to model thé l i th ium- ion battery,but it is s t i l l not accurate enough since ail of its éléments canchange depending on thé condition and state of thé battery. Inorder to improve thé model accuracy, thé polarizationcharacteristics of l i thium-ion battery is especially consideredand an improved Thevenin model, vvhich is performed byadding an extra RC branch to thé Thevenin model as shovvn inFig. 1, is proposed accordingly.

Fig. i . Schematic of thé improved Thevenin model

The improved Thevenin model includes three parts asopen-circui l voltage UQC, in ternai résistances and équivalentcapacitances. The internai résistances include ohmic résistanceRO, electrochemical polarization résistance Rpa andconcentration polarization résistance Rfc. The équivalentcapacitances, vvhich include electrochemical polarizationcapacitance Cpa and concentration polarization capacitance Cpc,are used to describe thé transicnt rcsponse during thé charging/discharging transits. /L and U\_ are charging/dischargingcurrenl and terminal voltage respectively. The electricalbehavior of thé circuit can be expressed as follows:

/KC^H/CP,( i )

Where 0"pa and are voltages on Cpaand Cpc respectively.

B. Experiments

To identify thé parameters of thé improved Thevenin model,thé battery test bench is designed. Charging and dischargingexperiments are carried ont on a l i t h i u m - i o n battery module,which is from Chinese C1TIC GUOAN MGL Company vvi thnominal voltage of 57.6 V and nominal capacity of lOOAh.

B. 1. Battery Test Bench

Fig. 2 shows thé configuration of thé battery test bench. Itskey equipment is Digatron EVT500-500, vvhich can charge ordischarge battery module vvith a maximum voltage of 500V andmax imum current of 500A, and can t imely measure thé majorparameters l ike voltage, current and température. The hostcomputer vvith thé installed software BTS-600 can program thécxperiment procédure and deal vvith real-time data acquisi t ion.In order lo reduce thé inf luence of température, thé batterymodule being tested is kept in a thermal chamber.

Thermal Chamber

Battery Modu e

AC Power

DigatronEVT500-500

Power Cable

^ Communication Cable

Fig.2. Configuration of thé battery test bench

B.2 The Open-circuit Voltage Identification Test

In order to acquire data to idenlify thé parameter ofopen-circuit voltage, a test vvas performed on thé l i th ium-ionbattery module. The test procédure is as follows: thé batterymodule is first discharged at constant current of 0.3C ampèrefrom ful ly charged state to 90% of thé nominal capacity at 20°Cin thé thermal chamber. Afterwards it is left in open-circuitcondition and thé terminal voltage is monitored simultaneously.The measured terminal voltage is considered to reach théequi l ibr ium potential after ten hours since thé change of théterminal voltage is negligible and thé battery module isassumed to reach a steady state. The battery module iscontinuously discharged by a further 10% of thé nominalcapacity at thé same current and thé equ i l i b r ium potential ismeasured after ten hours. The above procédure vvas performedrepeatedly to obtain thé e q u i l i b r i u m potentials shovvn as thédischarge curve in Fig.3. The e q u i l i b r i u m potenlials on thécharge curve in Fig.3 vvere got vvith thé s i m i l a r test procédureby charging thé battery pack at constant current of 0.3C ampèrefrom fu l ly discharged state vvith thé step of 10% of thé nominalcapacity.

0.4 0.5 0.6

State of Charge

Fig.3. Equi l ib r ium potential and open-circuit voltage curves of thé bauerymodule

I t can be found that thé e q u i l i b r i u m potential lias différentvalues after charging and discharging respectively at thé sameSOC. The e q u i l i b r i u m potential is higher at thé chargingprocess than that at thé discharging process, which indicatesthat thé e q u i l i b r i u m potential dépends on thé previoustreatment of thé battery module and thé hystérésis occursdur ing thé charging/discharging. In this paper, thé hystérésis is

ignored and thé open-circuit voltage Uoc is defined as thé

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average of thé e q u i l i b r i u m potenlials of charging anddischarging to s i m p l i f y thé model.

The open-circuit voltage Uoc measured as thé fonction of théSOC, vvhich is shovvn in (2), can be obtained by thé curvefiti ing vv i th thé quadratic f i t t ing method.

L'oc(SOC) = -13xSOC 2 +22xSOC + 57

B.3. H PPC Test

(2)

In order to acquire data to ident i fy thé model parameters, aHybr id Puise Power Characterization (HPPC) test wasconducted on thé l i t h i u m - i o n batlery module al 10% SOCinterva is slarting from 0.9 to 0.1. During thé test, thétempérature of battery module was kept at 20°C in thé thermalchamber. Fig.4 shows thé terminal voltage profile of thé testedbattery module during a HPPC test at SOC=0.1.

O 1 ;

60.5

> 60

|59.5O> 59

58.5

y/cWge .(/.. 3 ri

/ / 2 -4. Open circui t [/, • . _ï _~v~... . . ' 'dischan

Voltagedrop

t^\ — __-58 — - '-- .1 i i

10 20 30lime (sec)

50 60

Fig.4. Batlery module voltage profile du r ing HPPC test at SOC=0.1

As shovvn in Fig.4, a voltage drop appears vvhen a puisedischarge/charge current is loaded, vvhich can be expressed as:

When discharge

When charginge

Then, thé ohmic résistance RQ can be given by:

(3)

(4)

The ohmic résistance R0 is found to be independent on thébattery load current. and is a function of SOC shown in Fie.5.

0.022S-

0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.S 0.9State of Charge

Fi».5 Ohmic résistance characteristic curve o f l h e battery module

C. Parameters Identification

C. I. The Extended Kalman Fi/ter A Igorithm

The Kalman fil ter is a set of mathematical équations thatprovides an efficient recursive means to estimate thé state of a

process, in a vvay that minimizes thé mean of thé squared error.The fi lier has bcen applied extensively to thé field of linearestimation inc lud ing state estimation, parameters estimationand dual estimation. The EK.F Algor i thm is a nonlinear versionof thé Kalman filter that linearizes about thé current mean andcovariance of thé state, it can be described as follows for acontinuous-t ime System with discrete-time measurements

x = /(x,u, w./)

w - (0,Q) ( 3 >

vk - (0 ,R k )

Where u is thé System input ; w is process noise, il iscontinuous-time Gaussian white noise with zéro mean andcovariance of Q; \ is measurement noise, it is discrete-timeGaussian white noise with zéro mean and covariance of Rt.The non-linear function /?(•) relates thé state at thé previoustime step k-\o thé state at thé current t ime step k,

The filter step for optimal state and parameter estimation canbe summarised as follows.

Step / : In i t i a l i s a t i on .

The i n i t i a l estimation of x0 before any measurement ismodelled as a Gaussian random vector wi th mean of X0 andcovariance of P0, vvhich is expressed by:

il = £[x(1]„ „ T (6)

Step //: Time update (from time (k - 1)+ to t ime (k) ).

In th is step, an estimate of thé state at thé current t imestep (k)~ is produced based on thé state est imate and ilscovariance from thé previous time step (k - \)+ .

(7)x = /(x,u, w, / )

P = AP + PA1 +Q

A is thé Jacobian matrix of partial derivatives of /(•) with

respect to x.

A,.., =-[ml <5xy|(8)

In th is step, thé intégration is processed w i t h xt. = xj ,

andPt. = P/., . At thé end of thé intégration, we hâve \ = x j

and Pj = Pj . Wi th référence to (7), thé estimate state and ils

covariance propagates from t ime (k - \ to lime (k)~ based on

thé previous values. System dynamic, thé control input and thé

errors of thé actual System.

Step ///: Measurement update

In this step, thé measurement information at t ime k isprocessed to refine thé estimate of xk to reach a more accurateslale estimate. And thé observed vector y^ is used to correct thé

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state es t imat ion and covariance estimation. The resu l t ingestimate of xk is denoted as xk , and ils covariance is denotedas P k + . The measurement update of thé state estimation andestimation error covariance is performed according to théIb l l owing équations:

(9)

Where K^ is Kalman gain matr ix; Ck is thé par t ia l derivatives of/?(•) vvi lh respect x , which is given by:

It should be noted that Xj. and x^ are both estimations o f the

saine vectorxA . . However, x~ is thé estimate of \ before thé

measurement y t is taken into account, which is called priori

estimate, and \l is thé estimate of \ after thé measurement

yt is considered. which is called posteriori estimate [18].

C.2. Parameters Identification with EKF

This section présents thc implementation of EKF fores t imat ing states and parameters of thé improved batterymodel.

The te rmina l voltage (7, can be rearranged based on (1) as:

UL=Uoc+(-\1

On thé assumption that d(7oc / dt ~ 0 ,taking time derivative

of UL gives:

1

'R "*

u+\L [RfaCfa

1 ' 1c, cj"

1«pcCpcJ

1_/?PCCJ

Translbrm (12 ) inlo thé form of(5):

/(x,U) = iy; /, /3 /, /5 /„ /7ry t = / , (x k ) = |0 0 Uu 0 0 0 0|T

Where/(/ =1,2,3,4,5,6,7) and x are defined as follows:

^ = - A 1 A 4 A ' 5 + A V ,

^ = - . X 2 A 6 A 7 + A ? W

/4 =/,=/«=/,= 0

The System matr ix A is calculated as:

(14)

=«•>

«110

«31

0

0

0

0

0

«22

0

0

0

0

0

0

0

«33

0

0

0

0

«14

0

«34

0

0

0

0

«15

0

«35

0

0

0

0

0

«26

«36

0

0

0

0

0 "

«27

«37

0

0

0

0

(16)

Where in matr ix \,an = -x4x^ ,

= ~X2X6 + » > «3l = -V5 ~ 'VV7

« = ' V ~ U

«22 ~ *VY7 ' «26 ~ X1X

«33 = --VV7 ' «34 = 'VV1

-R0x7n + Uocx7, o37 = -

The observe matrix C can be derived from (14):

C = [0 0 1 0 0 0 0]

, - ROXGII -11

(17)

The identif icat ion results of thé battery model parameterswith EKF are shown as Fig.6.

(a) O.I2r—=

i 0.06" 0.040.0200 10 20 30 40 50 60

time(sec)(c)

„

0 10 20 30 40 50 60 0 10 20 30 40 50 60

(0200.5 [

'^, 200 j-

0 10 20 30 40 50 60timc(sec)

Fig.6. Est imat ion results o f t h e battery model parameters with EKF (SOC=0.1)

The est imation of polar izat ion voltages as a fonction of t imeare shown in Fig.6(a) and Fig.6(b) respectively. In Fig.6(a), itcan be found that thé voltage of thé battery activationpolarizat ion reaches steady-state w i th in 0.2 second, whichindicates thé time constant o f t h e activation polarization RCbranch is smal l . However, it takes a much longer time for thévoltage o f t h e battery concentration polarization to obtain thésteady state, shown in Fig.6(b). The curves of polar izat ionrésistances and polarization capacilors with t ime are shown in

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Fig.6(c). Fig.6(d), Fig.6(e) and Fig.6(f) respectively. whichindicale thé estimated parameters quickly converge and keepconstant w i th in 60 seconds. I t is verified that thé parametersidentification using EKF algorithm is stable. Fig.6(g) shows théidentif icat ion error is small and demonstrates good agreementbetvveen resul ts obtained us ing thé voltage model-based EKFestimation and experiments.

Table I shows thé identif ication results w i t h i n thé SOC rangefrom 0.1 to 1.0.

TABLE I MODEL PARAMETERS LIST IDENTIFIED OF THE BATTERY MODULE

SOC

fip(n)c,»(F)M")Cjsifj

soc«,«,(0)

C',»(F)S,*(n)rpc(F)

0.10.001

200.01

0.110000

0.60.001

204.20.097

9903

0.20.001

2000.1

10000

0.70.001

2 0 1 . 10.09419921

0.30.001

199.99

0.0987

9909

0.80.001

200.2

0.099610210

0.40.001

2020099

10010

0.90001

200.10.0999

9666

0.50.001

202.2

0.0994

10009

1.00.001

200.50.1025

9987

D. Model vérification

To evaluate thé va l id i ty of thé improved Thevenin model, alesting cycle ofHPPC shown in Fig.7 is adopted as thé input forboth thé l i t h i u m - i o n battery module and thé battery modulemodel. The in i t i a l SOC is 1.0. The parameters of thé batterymodule model as a function of SOC are updated via l inearlookup table and extrapolation. Fig.8 présents thé comparisonot ' the expérimental data w i th thé model s imula t ion data, whichclearly shows thé coïncidence of two cases. (The / L i s positivein discharging process and négative in charging process).

1007550250

-25

-50-75

1-1 PPC

4000 6000t i ine (sec)

SOOO 10000

Fig.7. A tes t ing cycle for e v a l u a t i n g thé va l id i ty of thé battery module mode

— Model Estimation |i--•Expérimental Data1

"o 2060" 4000 6000time(sec)

8000 10000

Fig.8. Comparison between voltage response o f t h e model and experiment

Fig.9 shows thé voltage error curves between thé s imula t iondata and thé expérimental data. Table II gives thé statistic resuito f the voltage errors.

3

2>

oI 0

4000 6000time(sec)

8000" 10000

Fig.9. Voltage Error curves between thé s imula t ion and thé experimenial data

Table 11 shows thé error variance is 0.0708V2 and thé errormean is around 0.0337V, thé maximum error rate is 4.15%.which vérifies thé parameters identif ication algorithm and théimproved Thevenin model are accurate and reasonable. Theerrors are higher in Fig.9 than thé results shown in Fig.6(g)because thé SOC range is from 1.0 to 0.1. while thé Fig.6(g) isonly for a single SOC point.

TABLE II STATISTIC LIST OF THE VOLTAGE ERRORS

Index Maximum Mean Variance Error rate

Value 2.79V 0.0337V 0.0708V2

I I I . SOC ESTIMATION USING ADAPTIVE EXTENDEDKALMAN FILTER

//. SOC Définition

SOC is a relative quant i ty that desciïbes thé ratio of théremaining capacity to thé nominal capacity of a battery. it isgiven by:

soc = soc,,-- (18)

Where SOCo is thé i n i t i a l value of SOC; ;; is thé coulombicefficiency, C^ts thé nominal capacity.

Transform (18) to discrète form. and use s as théabbreviation of SOC.

Where Al is thé discrète time step, s/, and s^.\e thé SOC valueal k and /t-lstep,i'i,,k-|is thé bus current at k-\step.

Due to thé polarization o f t h e battery module, power lossesobviously occur in thé charging/discharging process and thécoulombic efficiency conséquent!}' needs to be considered. Theexpérimental data o f t h e coulombic efficiency uncler différentcharging/discharging current for thé l i t h i u m - i o n batlery moduleis shown in Table I I I .

TABLE I I ICOULOMBIC EFFICIENCY LIST OF THE LITHIUM-ION BATTERY MODULE

Currenl(A)Discharging

coulombic efficiency

Chargingcoulombic efficiency

C/3 0.5C 1C 1.5C 2C 3C

100 99.3 98.5 9S.05 97.4 95.16

100 98.5 97.4 96.0 94.0

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B. SOC Estimation

B. I. The Adaplive Extended Kalman Filter

When using Kalman f i l te r to implement SOC estimation,al though sometimes thé estimation error covariance matrixcontinues to decrease, thé actual error s t i l l may be larger thanils theoret ical ly value, and thé actual error w i l l extendconstantly vv i th thé data increase, this phenomenon is oftendefined as divergence. Divergence w i l l cause faillites inKalman filter. at least w i l l decrease ils accuracy. One purposeof thé adaptive filter is to judge whether thé dynamic of thélargel lias changed dur ing doing recursive filter based onmeasurement data. 1 f yes. thé adaptive filter must further décidewhether to treat th is changes as a random noise and classifythem to thé model noise or to modify thé pr imary model so thatil coulcl adapt thé dynamic of thé target. Another purpose of théadaptive filter is to estimate and modify thé statistics of thénoise or gain matr ix of thé filter continuously employing théi n f o r m a t i o n brought by observation data to improve théaccuracy of thé filter when thé System noise variance and théobservation noise variance is unknown [22]. Sage-Husaadaptive f ï l te r a lgor i thm treat thé target dynamic as a randomdis tu rb and classify it as a model noise, il is easy to beimplemented and applied for t ime-varying Systems vvith thédefects of poor real-time calculat ion [26]. To overcome thédrawback, \ve introducc a filter divergence judgment condition,if thé f i l le r ing resuit excceds thé presetting divergencecondi t ion, thé noise estimator w i l l be starled to do adaptivefilter calculat ing. Thus, it not only avoids filter divergenceeffectively and improves thé s t ab i l i t y of filter calculation, butalso improve thé real- t ime a b i l i t y of thé filter.

The state équation and observation équat ion of thé discrèteSystem of interest is as fo l lowing:

Yk = C k X k + D k u k + v k

vk - ( r k , R k )

wk ~ ( i k ^ Q k )

(20)

Where X is nx I state vector; Y is mx I observe matrix; A, B, C,D and F are nxn , nx 1 , mxn , mx 1 and nxn matrix respectively;wt is process noise vv i th mean of q( and covariance of Qt ;v;. is measurement noise vvi th mean of rt and covarianceofRj .

The slatistical characteristics of thé i n i t i a l state X0 is thé sameas (6).Sage and Husa proposed thé maximum posterior sub-optimal unbiased estimator based on thé noise statist ics of théobserver Y, but thé unbiased estimalor they proposed is théar i thmet ic average, and thé weighted coefficients of thé eachitem is 1 /k+l . For thé t ime-vary ing noise, thé rôle of thé récentdata should be paid more emphasis on. Many researchers hâveadoptée! thé index weighted method [22], that is, use thé indexweighled coefficient to replace thé simple coefficient 1 /k+l .

The index weighted coefficient /^ usually is designed as:

As an example, (22) gives a calculation method of fa.

= 0, 1,2,-",*)

(21)

(22)

Where b is a forgetting factor.The improved noise estimator is recalculated as (23)~(26) by

replacing thé weighted coefficient 1/k+l vvi th thé nevv weightedcoefficient /?K.|.

*

.T K k >P k l -A k P k A[] )C T } ^

' k + , - C k X k + 1 - D k u k ] } (25)

Rk+, = d - r f k ) R k +rfk{S[Yk+lY tT+1-C t+1Pk-+lC^]} (26)/=()

Where in (23)-(26): Gk = [r'k P']"' Tk ; Error matr ix Yk+l =

k" — p" c^ IP p~ c*1 4- R r1•^k+l "" Ik+lv"k+llVk+l rk-fl^ ' t ;-r- l "' '^kJ Estimate of thé

covariance Pk + l=AkPk A k T +r k Q k ^ , Pk+1 = (I -KkT,CkT,)P l+1 .

The judgment condition for thé divergence of A E K Falgorithm is defined as:

Where r is thé adjusiable coefficient and r>=\ d i f fé ren t r isselected corresponding to différent Systems. Tr is thé trace ofthé matrix.

I f (27) is not true, thé filter may be divergent and thé priorivalues are not accurate.

B. 2. SOC Estimation Wilh A EKF

Transform ( I ) to a discrète System.

(28)

Contrast vvith (20). thé expressions of matrixes and vectorsare lisled as follows:

X =

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> VT-2010-01085.RI <

'exp(-A//( /? p aC p a))0

0

010

1

/?p c(l-exp(-A//(/?p cCp c)))

> /A/ /C N

d\

The charging/discharging current is loaded on thél i l h i u m - i o n batlery module and thé battery modelsimultaneously. Voltage Error between thé estimation and théexpérimental data is reduced by adaptively updating thé AEKFobserver gain. Then thé observer vvi th thé updated gain is usedto compensate for thé state est imation error. SOC is determinedby thé estimated voltage based on thé open-circuil voltageprofi le of thé battery module. The estimation of SOC is then fedback to update thé parameters of thé battery module model forthé next SOC estimation. And thé SOC estimation methodbased on AEKF is shovvn in Fig. 10.

Fig. 10 SOC es t imat ion method based on AEKF

C. Vérification and Evaluation

The Fédéral Urban Driv ing Schedules (FUDS) is a typicald r i v i n g cycle vvhich is often used to evaluate varions SOCes t imat ion algori thms 1 1 7|,|27]-[29J. For Digalron EVT500-500. FUDS test is an cx is t ing program and BTS-600 uses théTABLE operator lo run thé FUDS test, vvhich is normal lytenninated by a certain amount of ampère hours removed fromthé battery or if thé batlery reaches a certain voltage level. Inthis paper eight periods of FUDS are employed to verify théSOC estimation approach. The current profiles samplcd duringeight consécutive FUDS cycles are shown in Fig. 1 1.

For thé l i t h i u m - i o n battery module, thé i n i t i a l parameters foi-thé AEKF are specified as follovvs:

"I0°0 2ÔOÔ 4000 ~ 6000 8000 foOÔO~ 12000lime(sec)

Fig. 1 1 . Current prof i le sampled d u r i n g eight consécutive FUDS cycles

"148 0 O1

A , = A 0 = 0 1.001 0

0 0 1

"-0.1474 0 0

B , = B , , = 0 -0.0001 0

0 0 0.01

C , = C 0 = [ - 1 -I -4]T:

D, = D,,= [0.02225];

" 1 0 0

r= o i o0 0 I

"10 0 0

P,, = 0 10 0

0 0 0.1

"0.001 0 0

Q 0 = 0 0.001 0

0 0 0.001

The AEKF algori thm can precisely estimale thé voltage andt imely modulate thé noise signal according to thé error matr ix .The comparison between thé EKF and thé AEKF isSLimmarized in Table IV.

TABLE IVSTATISTIC LIST OF THE VOLTAGE ERRORS WITH EKF AND AEKF

EKFAEKF

Maximum1.285V

0.605 V

Mean~0~0~451 V~

0.0129V

Variance Error rate0.1204V2

0.0125V2

Fig. 12 présents thé instantaneous polarizat ion voltage curves,vvhich also represent thé dynamic performance of thé improvedThevenin model and objectively simulate thé polarizationcharacteristics of l i t h i u m - i o n battery module .

In ternis of thé AEKF, thé electrochemical polarizalionvoltage gradually increases \vhen thé open-circuil voltagedecreases under thé FUDS cycle, vvhereas thé f luc tua t ion ofconcentration polarization voltage vvil l be tardi ly slovved dovvn.Therefore il is necessary to analyze thé eleclrochemicalpolarizat ion voltage and concentration polarization voltageseparately, vvhich can greatly improve Ihe accuracy and théfeas ib i l i ty of thé proposed model and Ihe estimated approach.

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> VT-20IO-OI085 .R1 < 8

Fig. 13 and Fig. 14 show thé comparison of SOC curves and itserror curves with EKF estimation and AEKF estimation,respectively.

0.6; < r r , T-— -

— Electrochemical polarization voltage il

Concentration polarization voltage U

2000 4000 6000 8000 10000 12000time(sec)

Fig. 12. Polarizat ion voltage curvc of l i t h i u m - i o n battery' module

I 2Expérimental DataEKF algorithm

•—AEKF algorilhm j

4000 6000 SOOOtime(sec)

10000 12000

Fig 13. SOC curves w i t h EKF estimation, AEKF estimation and experiment

0.1 r . 1 -r- -r - - ; -=—ai — EKF algorithm Ir~AE_KF a lgor i thmj

-0.15„'--2000 4000 6000 8000

lime (sec)10000 12000

Fig 14 SOC es t imat ion error curves vvith EKF and AEKF

From Fig. 13 and Fig. 14, it can be seen that thé AEKFprésents thé improvements in thé tracking performance and fastconvergence vvhile thé EKF provides thé performance of théslow convergence and ins tab i l i ty which is adverse to théapplication in EVs. The delailed statistical analysis is shown inTable V.

TABLE V STATISTIC LIST OF SOC ESTIMATION ERROR

Maximum Mean

KEFAKEF 2.54%

3.19%1.06%

Variance

oTôbT~~5.75xlO'5

The AEKF a lgor i thm can compensate thé model error andthé presetting error of thé noise statistics, which has numericalstabi l i ty and high accuracy. On one hand, thé priori eiror occursin model parameters and thé noise statistics. The noise observercan estimate thé mean as well as thé covariance of dynamicnoise according to thé prédictive residual Xk - (A^X,.,,+ B k l u k l ) and measurement residual Yk , consequently thémodel and thé noise statistics error can be compensated. On thé

other hand, thé adaptive algorithm can improve thé stability andthé robustness of thé algorithm via real-time online correction.Normally, thé Kalman fi lier algorithm possesses a certains tab i l i ty , however, it easily cause thé divergence due to théerror existed in thé model and thé noise statistics, especially thénoise statistics evaluated by trial-and-error method. Fig. 13shows thé divergence took place using thé EKF algori thm at thécari 1er stage of thé SOC estimation and thé estimation errorreached to 14.96%. Nevertheless, thé remarkable merit of théAEKF algorithm is thé real-time amendment. Though thémaximum absolute error of thé AEKF algorithm is 2.54%, théabsolute error mean is only 1.06% and thé error variance is verysmall . which manifest its fast convergence, s tab i l i ty and minorerror. Thus thé proposed approach of thé SOC estimationalgorilhm is suitable for l i t h i u m - i o n power baltery module,which can quickly reduce thé model error and improve théaccuracy of thé filter.

IV. CONCLUSIONSFor a l i th ium- ion battery module, an improved Thevenin

model is proposed and its parameters ident i f ica t ion isperformed using EKF algori thm. The expérimental andsimulat ion results show thé maximum error of thé improvedThevenin model is vv i th in 4.15% and thé error covariance is0.0708V2. The proposed model has achieved enough accuracy.

To improve thé accuracy, re l i ab i l i ty and robustness of théSOC estimation, a SOC est imation vvi th AEKF algor i thm wasput forvvard based on thé improved Thevenin model. A fier 8FUDS cycles, thé maximum error of thé SOC est imat ion isvvi th in 2.54% and its mean is only 1.06%, which présent théestimation algorithm has higher accuracy compared with théEKF algorithm.

ACKNOWLEDGMENT

We would l ike to express our deep gratitude to Dr. ZhangCai-ping in Beijing Jiaotong University for many he lp fu ldiscussions and data support.

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Copyright (c) 201 I IEEE. Personal use is pcrmittcd. For any othcr puiposcs, Permission must be oblaincd from thc IEEE by cma i l ing pubs-permissions@iccc.org.

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Mongwen Ile received thé M.E. degree inVehicle Engineering from thé J i l i n Universi tyof Technology, Changchun, China, in 2000and Ph.D. degree in Vehicle engineering fromBeijing Ins t i tu te of Technology, Beij ing,

'•-.' •*M"! China, in 2003., ' " » : . - • • , i He is currently an associate Professer of

•fr' • : " . ' • ' . '/• Nat iona l Engineer ing Laboratory for Electric: j^pV1 'lit Vehicles and School of Mechanical

: •'.,- Engineering, Beijing Ins t i l u t e of Technology.

He has published more than 60 papers and hold 6 patents. Flis researchinterests include power battery modeling and s imula t ion, electricvehicles, design and control theory of thé hybrid power t r a in .

Dr. He received thé first prize of Scient i f icorTechnica l Awards fromBeijing I n s t i t u t e of Technology (2007).

Rui Xiong received thé M.E. degree in VehicleEngineering from Beijing Ins t i lu te ofTechnology, Beijing, China, in 2010.

Fie is currently a Ph.D candidate of VehicleEngineering at Be i j i ng Insti tule of Technologyin thé National Engineering Laboratory forElectric Vehicles. He has published 7 papers.His research mainly focuses on model ing ands i m u l a t i o n , of Power baltery and veh icu la rhybr id power System.

Xiuowci Zhang received thé M.E. degree fromBeijing I n s t i t u t e of Technology, Be i j ing ,China, in 2007.

Fie is cur rent ly a Ph.D candidate of VehicleEngineering at Beijing Inst i tu te ofTechnologyin thé National Engineering Laboratory forElectric Vehicles. Fie has published 3 papers.His research main ly focuses on opt imizat ioncontrol and H I L s imu la t ion of thé veh icu la rhybr id power train.

Fcngchun Sun received thé M.E. and Ph.D.degrees in Vehicle Engineering from BeijingInsti tute of Technology, Beijing, China, in1984 and 1989 respectively. He studied atTU-Ber l in from 1987 lo 1989 as a jo in t Ph.Dstudent.

He is currently a Professer and vicePrésident of Beijing Insti tute of Technology,and Director of Na t iona l EngineeringLaboratory for Electric Vehicles. He isconferred thé t i t l e of "Cheung Kong Scholar"

by thé Ministry of Education, China. He has published more than 150papers and hold 19 patents. His research interests include electricvehicles, electric drive System, EV démonstration and Infrastructure.

Dr. Sun received thé second prize of National Science andTechnology Progress Awards (2008), thé second prize of NationalTechnological Innovation Awards twice (2004,2009) and thé winner ofAward for I n d u s t r i a l Innova t ion from HO LEUNG HO LEEFOUNDATION (2007).

J i n X i n Fan ( S ' 1 0 - S ' l l ) received thé M.Edegree in Vehicle Engineering from A n h u iAgr icu l tu ra l Universi ty, Hefei, China, in 2004.

She is currently a Ph.D candidate of VehicleEngineering at Beij ing Ins t i tu te of Technologyin thé Nat ional Engineer ing Laboratory7 forElectric Vehicles. She studied at MichiganState University in US from 2008 to 2010 as aj o i n t Ph.D student.

Her research ma in ly focuses on permanentmagnet electric machines, cooling System and thermal analysis,especially in thé field of eleclric vehicles.

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