research article estimation of state of charge for lithium ...an accurate estimation of the state of...

Research ArticleEstimation of State of Charge for Lithium-Ion Battery Based onFinite Difference Extended Kalman Filter

Ze Cheng Jikao Lv Yanli Liu and Zhihao Yan

School of Electrical Engineering amp Automation of Tianjin University Tianjin 300072 China

Correspondence should be addressed to Jikao Lv lvjikao163com

Received 4 December 2013 Revised 24 February 2014 Accepted 10 March 2014 Published 3 April 2014

Academic Editor Gongnan Xie

Copyright copy 2014 Ze Cheng et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An accurate estimation of the state of charge (SOC) of the battery is of great significance for safe and efficient energy utilization ofelectric vehicles Given the nonlinear dynamic system of the lithium-ion battery the parameters of the second-order RC equivalentcircuit model were calibrated and optimized using a nonlinear least squares algorithm in the Simulink parameter estimationtoolbox A comparisonwasmade between this finite difference extendedKalman filter (FDEKF) and the standard extendedKalmanfilter in the SOC estimation The results show that the model can essentially predict the dynamic voltage behavior of the lithium-ion battery and the FDEKF algorithm can maintain good accuracy in the estimation process and has strong robustness againstmodeling error

1 Introduction

In the context of countries vigorously promoting energyconservation and low carbon economy to solve energy crisisand mitigate global warming the solar photovoltaic powergeneration is emerging as the technology of choice forenergy-saving and environmentally sustainable transporta-tion It is suggested that the storage battery is second onlyto the photovoltaic modules as the most important part ofsolar photovoltaic system thus its performance will directlyaffect the operational state and reliability of the system Ithighlights the need to quickly and accurately estimate thestate of charge (SOC) of the battery A battery managementsystem is required to ensure safe and reliable operation ofthe battery One of its basic functions is to measure the SOCwhich indicates the remaining charge of the battery so thatthe driver can be reminded to charge the battery prior to itsdepletion

SOC is usually estimated indirectly by some measurablequantities [1] In recent years many methods have beenproposed to improve the SOC estimation Among themthe ampere-hour integral (coulomb counting) method isthe most simple and convenient one [2] but it requires aprior knowledge of initial SOC and suffers from accumulated

errors from noise andmeasurementThe open circuit voltage(OCV) method is sufficiently accurate because there is aone-to-one correspondence between OCV and SOC but itneeds a long rest time and thus cannot be used in real timeapplications [3] A number of intelligent approaches havebeen developed in an attempt to achieve a more accurateSOC estimation such as the neural network method andKalman filter (KF) method [4] The neural network methodcan provide an accurate SOC estimation given an appropri-ate training dataset However the training process is verycomputationally intensive and at risk of over fitting and themodel performance strongly relies on the amount and qualityof the training data which could limit its application rangeThe KF method uses sample data (current voltage and tem-perature) to recursively calculate theminimummean squarederror estimate of true SOC which can solve the problem ofuncertain initial SOC and cumulative error However it isonly suitable for linear system Given the nonlinear nature ofthe dynamics of electrochemical cells a linearization processis usually used to approximate the nonlinear system by alinear time varying systemThe extendedKalman filter (EKF)is a nonlinear extension of the conventional KF which hasbeen developed particularly for systems having nonlineardynamic models using Taylor series expansions [5ndash7] The

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 348537 10 pageshttpdxdoiorg1011552014348537

2 Journal of Applied Mathematics

divergence of EKF is primarily due to the linearization errorfor ignoring high-order terms and the accuracy of EKF-based SOC estimation is sensitive to the precision of thebattery model

We introduced an alternative nonlinear Kalman filteringtechnique known as finite difference extended Kalman filter(FDEKF) in this study and used the finite difference methodinstead of the Taylor series expansions to estimate thecovariance matrix It has theoretical advantages that manifestthemselves in more accurate predictions and also strongrobustness against modeling uncertainty by making full useof the error information generated by model linearization

The remainder of this paper is arranged as followsSection 2 describes a model structure and discusses how theparameters of this model can be automatically estimatedusing the Simulink parameter estimation Section 3 briefs theEKF and deduces the FDEKF algorithm Section 4 shows thesimulation results of the proposed algorithm to the exper-imental data Section 5 shows the experimental verificationof the proposed algorithm on a test bench and Section 6concludes the paper

2 Modeling for the LiFePO4 Battery andParameter Identification Method

21 Lithium-Ion Battery Model The commonly used batterymodels include the electrochemicalmodel and equivalent cir-cuit model (ECM) [8] The electrochemical models describethe electrochemical reactions in the electrodes and electrolytein a mathematical way which can achieve high accuracyHowever they typically deploy partial differential equationswith a large number of unknown parameters Due to theintensive computation involved these models are often usedfor battery performance analysis and battery design ECMshave been developed especially for the purpose of vehiclepower management control and battery management systemdevelopment and have less parameters such as the Rintmodel [9] Thevenin model [10] and PNGV model [11]Literature [12] gives a comparative study of these ECMs forlithium-ion batteries Due to the complicated polarizationcharacteristics of battery it is suggested that models withmore parallel RC networks connected in series should havea much higher accuracy [13] Clearly the higher the orderis the more complicated the model becomes Thereforeconsidering the details in the model is a tradeoff betweenaccuracy and complexity In this paper a second-order RCmodel is considered as shown in Figure 1

Here 119880oc is the open circuit voltage 119880119905is the terminal

voltage of the battery 119868119905is the outflow current119877

0is the ohmic

resistance of the connectors electrodes and electrolyte andthe two sets of parallel resistor-capacitor elements connectedin series 119877

119890 119862119890 and 119877

119889 119862119889are the mass transport effects

and double-layer effects respectively Commonly the timeconstants of the two dynamics differ by at least an order ofmagnitude

22 Parameter Identification The model parameters needto be accurately estimated from the test data to establish

Re Rd

It

Ce Cd

R0

UocUt+

+

minus

minus

Figure 1 Equivalent circuit of second-order RC model

a battery model with good performance In this paper themodel is parameterized using a semiautomatic process thatcan satisfy the constraints on the optimized parameters Thisprocess uses a number of measured data sets under a varietyof conditions The parameters are optimized by minimizingthe error between measured and simulated results using thenonlinear least squares algorithm in the Simulink parameterestimation toolbox The battery model can be established inSimulink based on the second-order RC model as shown inFigure 2

3 SOC Estimation

31 EKF-Based SOC Estimation In order to use EKF meth-ods for battery SOC estimation the cell should bemodeled ina discrete-time state-space form Specifically we model thenonlinear battery system by a state equation and an outputequation below

119909119896+1

= 119891 (119909119896 119906119896) + 119908119896

119910119896= 119892 (119909

119896 119906119896) + V119896

(1)

where 119909119896is the system state vector at discrete-time index 119896

vector 119906119896is the measured system input at time 119896 and 119908

119896is

unmeasured ldquoprocess noiserdquo that affects the system state Theoutput of the system is 119910

119896and V119896is measurement noise 119891(sdot sdot)

and 119892(sdot sdot) are (possibly nonlinear) functions determined bythe particular cell model used

At each time step 119891(119909119896 119906119896) and 119892(119909

119896 119906119896) are linearized

by a first-order Taylo -series expansion The model can berewritten as

119909119896+1

= 119860119896119909119896+ 119861119896119906119896+ 119908119896

119910119896= 119862119896119909119896+ 119863119896119906119896+ V119896

(2)

where

119860119896=119889[119891(119909

119896 119906119896)]

119889[119909119896]

10038161003816100381610038161003816100381610038161003816119909119896119906119896

119861119896=119889 [119891 (119909

119896 119906119896)]

119889 [119906119896]

100381610038161003816100381610038161003816100381610038161003816119909119896119906119896

Journal of Applied Mathematics 3

S PS

S PS

2

1 1Current

Voltage Terminator

Solverconfiguration

Voltagesentor

Batterycurrent

Lithium-ioncell

SOC Simscope

R_table_1Temp+minus

SOC

Simscope

C_table_1Temp

+

minus

SOC Simscope

R_table_1Temp+minus

SOC

Simscope

C_table_1Temp

+minus

SOC Simscope

R_table_1Temp+

+

minus

minus

SOC

Simscopem_table_

1Temp

+

minus

PS S

Terminator

1

2

SOC

R0

R2R1

C2Em C1

+

+

minus

minus

uarr V

f(x) = 0

Vt

gt

Figure 2 Second-order RC model in Simulink

119862119896=119889 [119892 (119909

119896 119906119896)]

119889 [119909119896]

100381610038161003816100381610038161003816100381610038161003816119909119896119906119896

119863119896=119889 [119866 (119909

119896 119906119896)]

119889 [119906119896]

100381610038161003816100381610038161003816100381610038161003816119909119896119906119896

(3)

The state of the ECM in Figure 1 is set to be SOC 119880119890and

119880119889are the voltage across the RC network the input of the

model is the current 119868119905 and the output is the terminal voltage

119880119905 Thus the dynamic ECM could be described in state space

as

(

SOC (119896 + 1)119880119890(119896 + 1)

119880119889(119896 + 1)

) = (

1 0 0

0 119890minus119879120591119890 0

0 0 119890minus119879120591119889

) times(

SOC (119896)119880119890(119896)

119880119889(119896)

)

+(

minus120578119879

119876119899

119877119890(1 minus 119890

minus119879120591119890 )

119877119889(1 minus 119890

minus119879120591119889)

)119868119905(119896)

+ 119908119896

119880119905(119896) = 119880oc (SOC (119896)) minus 119880

119890(119896)

minus 119880119889(119896) minus 119877

0119868119905(119896) + V

119896

(4)

where 120578 is the cell coulombic efficiency 119879 is the inter-sampleperiod119876

119899is the cell capacity 119868

119905(119896) is the current at time index

119896which is negative at charge and positive at discharge 120591119890and

120591119889are the time constants of the 119877

11198621and 119877

21198622circuit and

120591119890= 119877119890119862119890 120591119889= 119877119889119862119889 Both 119908

119896and V

119896are assumed to be

mutually uncorrelatedwhiteGaussian randomprocesses andthe statistical characteristics are as follows

119864 [119908119896] = 0 Cov [119908

119896 119908119879

119895] = 119876

119896120575119896119895

119864 [V119896] = 0 Cov [V

119896 V119879119895] = 119877119896120575119896119895

Cov [119908119896 V119896] = 0

(5)

where both 119876119896and 119877

119896are positive definite symmetric matri-

ces and 120575119896is the Kronecker function The OCV corresponds

to a certain SOC which can be identified by the nonlinearcharacteristic curve 119880oc(SOC(119896)) [14] The hysteresis effectwhich is beyond the scope of this paper can cause thedischarging curve to stay below the charging curve for thesame amount of SOC

Obviously the relationship between OCV and SOC isnonlinear The EKF approach is to linearize the equations ateach sample point usingTaylor series expansionsThe specificsteps are as follows

Define 119909119896= [SOC(119896) 119880

119890(119896) 119880

119889(119896)] the superscripts

ldquo mdash rdquo and ldquordquo indicate the prior and posterior estimationrespectively We linearize the nonlinear state and outputequations around the present operating point using Taylorseries expansions and ignore the second- and higher-orderterms

119860119896=120597119891 (119909119896 119906119896)

120597119909119896

100381610038161003816100381610038161003816100381610038161003816119909119896=119909119896

= (

1 0 0

0 119890minus119879120591119890 0

0 0 119890minus119879120591119889

)

119861119896=(

minus120578119879119876119899

119877119890(1 minus 119890

minus119879120591119890)

119877119889(1 minus 119890

minus119879120591119889)

)


119862119896=120597119892 (119909119896 119906119896)

120597119909119896

100381610038161003816100381610038161003816100381610038161003816119909119896= 119909119896

= [120597119880oc120597SOC

10038161003816100381610038161003816100381610038161003816SOC=SOC(119896)minus1 minus1 ]

119863119896= minus1198770(119896)

(6)

Initialization is

1199090= [SOC (0) 0 0]

0= 119864 [(119909

0minus 1199090) (1199090minus 1199090)119879

]

(7)

Recursive calculation is

119896+1

= 119860119896119909119896+ 119861119896119868119905(119896) (8)

119896+1

= 119860119896119896119860119879

119896+ 119876119896 (9)

119870119896= 119896+1119862119879

119896[119862119896119896+1119862119879

119896+ 119877119896]minus1

(10)

119909119896+1

= 119896+1+ 119870119896[119880119905(119896) minus

119905(119896)] (11)

119896+1

= 119896+1minus 119870119896119862119896119896+1 (12)

32 FDEKF-Based SOC Estimation There are two problemsfor EKF in SOC estimation (1) when the higher-order termsof Taylor series expansions are not negligible the lineariza-tion process will cause significant errors to the system oreven make the filter unstable and divergent and (2) Jacobianmatrix needs to be calculated at every sample time therebyleading to a multiplication of the calculation amount forcomplicated system FDEKF is another iterative minimummean variance error estimator which has a higher precisionthan the first-order Taylor series expansions by applyingfinite difference method and is applicable for all nonlinearfunctions

Schei first conceived the thought of finite difference [15]It uses polynomial approximations obtained with a Sterlinginterpolation formula for the derivation of state estima-tors for nonlinear systems The estimators perform betterthan that based on Taylor approximations Nevertheless theimplementation is significantly simpler as no derivatives arerequired A nonlinear function 119910 = 119891(119909) is assumed andapproximated by the interpolation formula

119891 (119909) asymp 119891 () + 1198911015840

119863119863() (119909 minus ) (13)

1198911015840

119863119863() =

119891 ( + ℎ) minus 119891 ( minus ℎ)

2ℎ (14)

where ℎ is the interval length and119891 is assumed to be analyticthen the full Taylor series expansion of (13) is

119891 (119909) = 119891 () + 1198911015840() (119909 minus )

+ [119891(3)()

3ℎ2+119891(5)()

5ℎ4+ sdot sdot sdot ] (119909 minus )

(15)

The first two terms on the right hand side of (15) areindependent of the interval length ℎ and are recognized

as the first two terms of the Taylor series expansion of 119891The ldquoremainderrdquo term given by the difference between (15)and the first-order Taylor approximation is controlled by ℎand will in general deviate from the higher order terms ofthe Taylor series expansion In some sense certain intervallengths may make the remainder term close to the higherorder terms of the full Taylor seriesThe procedure of FDEKFalgorithm is given below

First we introduce the following four square Choleskyfactorizations

119876 = 119878119908119878119879

119908 119877 = 119878V119878

119879

V

= 119878119909119878119879

119909 = 119878

119909119878119879

119909

(16)

The factorization of the noise covariance matrices 119876 and119877 can be made in advance 119878

119909and 119878

119909are updated directly

during application of the filterThen we calculate the partial derivative of the nonlinear

function by the first-order polynomial approximation

119865119909(119896) = (119891 (119909

119896+ Δ119909119896 119906119896) minus 119891 (119909

119896minus Δ119909119896 119906119896)) 2Δ119909

119896 (17)

Define Δ119909119896= ℎ119878119909(set ℎ2 = 3)

119865119909(119896) 119878119909= 119878119909119909

=1

2ℎ119891119894(119909119896+ ℎ119878119909119895 119906119896) minus 119891119894(119909119896minus Δℎ119878

119909119895 119906119896)

(18)

Let the 119895th column of 119878119909be denoted 119878

119909119895 (19) can be

derived just as (18)

119866119909(119896) 119878119909= 119878119910 119909

=1

2ℎ119892119894(119896+ ℎ 119878119909119895 119906119896) minus 119892119894(119909119896minus Δℎ 119878

119909119895 119906119896)

(19)

(1) Estimates of the prior variance

119896= 119865119909(119909) 119896119865119879

119909(119896) + 119876

119896

= 119865119909(119896) 119878119909119878119879

119909119865119879

119909(119896) + 119878

119908119878119879

119908

= 119878119909119909119878119879

119909119909+ 119878119908119878119879

119908

(20)

(2) Estimates of the posterior gain matrix and posteriorvariance

119870119896= 119896119866119879

119909(119896) [119866

119909(119896) 119896119866119879

119909(119896) + 119877

119896]minus1

= 119878119909119878119879

119909(119878119910 119909119878minus1

119909)119879

[119878119910 119909119878119879

119910 119909+ 119878V119878119879

V ]minus1

= 119878119909119878119879

119910 119909[119878119910 119909119878119879

119910 119909+ 119878V119878119879

V ]minus1


100 200 300 400 500 6002628

33234

Volta

ge (V

)

Time (min)0 700 800

01234

Curr

ent (

A)

CurrentVoltage

Figure 3 Terminal currentvoltage curves by constant-current dis-charge pulse

119896+1

= 119896minus 119870119896119866119909(119896) 119896

= 119878119909119878119879

119909minus 119870119896119866119909(119896) 119878119909119878119879

119909

= 119878119909119878119879

119909minus 119878119909119878119879

119910 119909119870119879

119896minus 119870119896119878119910 119909119878119879

119909+ 119878119909119878119879

119910 119909119870119879

119896

= 119878119909119878119879

119909minus 119878119909119878119879

119910 119909119870119879

119896

minus 119870119896119878119910 119909119878119879

119909+ 119870119896119878119910 119909119878119879

119910 119909119870119879

119896+ 119870119896119878V119878119879

V119870119879

119896

= [ 119878119909 minus 119870119896119878119910 119909 119870119896119878V] [119878119909minus 119870119896119878119910 119909

119870119896119878V]119879

(21)

We use (20)-(21) to replace (9) (10) and (11) whichcomprises the full FDEKF

4 Results and Discussions

To verify the effectiveness and performance of the FDEKF weapplied the identification and SOC estimation algorithms tothe experimental data obtained on the LP2770102AC lithium-ion battery This is a lithium iron phosphate battery that canbe used in portable high power devices grid stabilizationenergy storage and electric vehicles and hybrid electricvehicles Its nominal capacity is 125 Ah and nominal voltageis 33 V For the tests we used a DigatronMCT 30-05-40 cellcycler with a measurement accuracy of plusmn5mV for voltageand plusmn50mA for current The battery temperature was keptat room temperature (20 plusmn 2∘C) throughout the experiment

In this section we first estimated the values of ECMparameters using an iterative numerical optimization algo-rithm implemented by Simulink parameter estimation andthen compared the FDEKF-based and EKF-based SOC esti-mation

41 Pulse Test The battery was fully charged so that SOC= 100 and then the constant-current discharge pulse testwas performed (18min discharging and 60min resting)The discharge lasted 790min and the sample time was 1 sFigure 3 shows the terminal current and voltage of the battery

42 Initial Estimation The parameters 119864119898 1198770 119877119890 119862119890 119877119889

and 119862119889needed to be identified The initial values of model

Table 1 Parameter constraints

Parameter Initial value Minimum Maximum119864119898

33 V 2V 36V1198770

001Ω 0Ω 1Ω119877119890

0005Ω 0Ω 1Ω119862119890

0005 F 0 F 1 F119877119889

10000Ω 1Ω 100000Ω119862119889

10000 F 1 F 100000 F

parameters should be assigned before running the optimiza-tion algorithm The upper and lower bounds of the modelparameters were selected through trial and error In ourinitial attempt six parameters were estimated and theirmaximumandminimumvalueswere given a broad range Aninitial guess for each parameter was chosen based on priorexperimentation with the model Table 1 shows the param-eters estimated and their constraints

43 Model Evaluation The current and voltage data wereused as the input to the identification algorithm described inSection 2 The second-order RC model in Figure 2 was rununtil the optimization process was terminatedThemeasuredand simulated terminal voltages are shown in Figure 4(a)and the errors between them are shown in Figure 4(b) It isevident that the model can essentially predict the dynamicvoltage behavior of the lithium-ion battery fromSOC = 01 toSOC = 1 with an error of 50mV However the performanceof the model is deteriorated when the battery runs outwith an error of 150mV The trajectories of the optimizationvariables after 9 iterations are shown in Figure 4(c) In orderto guarantee the global optimal parameters we choose sumof squares for error (SSE) as the cost function and the curveis shown in Figure 4(d)

SSE = sum(119909119894minus 119909)2

(22)

We used the dynamic stress test (DST) data to validatethe accuracy of the model Single DST working conditionincludes 14 steps and it is repeated five times the process isas follows (1) 20A charge for 420 seconds (2) rest for 30seconds (3) 10 A charge for 120 seconds (4) 10 A dischargefor 120 seconds (5) 4A charge for 300 seconds (6) 20Adischarge for 120 seconds (7) 10 A charge for 120 seconds (8)4A discharge for 180 seconds (9) 40A charge for 23 seconds(10) 10 A discharge for 120 seconds (11) 4A charge for 300seconds (12) 40A discharge for 60 seconds (12) 20A chargefor 120 seconds (13) 4A discharge for 120 seconds and (14)rest for 30 seconds Figure 5 shows that themodel can quicklyand accurately track the real-time voltage of the battery withan error of less than 01 V

44 Verification of SOC Estimation Algorithm We thenapplied the experimental data obtained in Section 41 toimplement the standard EKF and the FDEKF The exper-imental SOC values were computed by the battery testing


0 100 200 300 400 500 600 700 80026272829

33132333435

Time (s)

Experimental dataEstimated data

Vt

(V)

(a) Measured and calculated battery voltage with 13C discharge

0 100 200 300 400 500 600 700 800minus003

minus0025minus002

minus0015minus001

minus00050

0005001

Time (s)

Erro

r (V

)

(b) Calculated battery voltage error with 13C discharge

05

10

012

33234

0 1 2 3 4 5 6 7 8 90

002004

0 1 2 3 4 5 6 7 8 90

0102

1 2 3 4 5 6 7 8 900

0102

Trajectories of estimated parameters

Iterations

Para

met

er v

alue

s

times104

times105

Em

(V)

C1

(F)

C2

(F)

R0

(Ω)

R1

(Ω)

R2

(Ω)

(c) Trajectories of the optimization variables

Cos

t fun

ctio

n (S

SE) Cost function

IterationsC

ost

101

100

10minus1

10minus2

0 1 2 3 4 5 6 7 8 9

(d) Cost function of the optimization variables

Figure 4 The parameters estimation results with 13C discharge

2000 4000 6000 8000 100002628

332343638

4

Time (s)

Volta

ge (V

)

0 12000minus40minus2002040

Curr

ent (

A)

True voltage dataEstimated voltage dataCurrent

(a) True data of DST and its estimated data

0 2000 4000 6000 8000 10000 12000minus006minus004minus002

0002004

Time (s)

Erro

r of v

olta

ge (V

)

(b) Estimated voltage error of DST data

Figure 5 The results of model validation with DST data

system and acted as a reference for the SOC estimates TheSOC estimation results are shown in Figure 6

Figure 6(a) shows that the two estimators can trace thereference SOC but with different accuracy It also shows thatthe maximum absolute estimation error is 7 for EKF and2 for FDEKF with an improvement of 71

To evaluate the comprehensive performance of the twofilters in the quantitative analysis we define the root mean

square error (RMSE) and single mean computation time 119905costas

RMSE = radic 1119879

119873

sum

119896=1

(119909119896minus 119909119896)2

119905cost =1

119873

119873

sum

119896=1

119905119896

cost

(23)


0 100 200 300 400 500 600 700 8000

02040608

1

Time (min)

SOC

EKFFDEKF

Ture

(a) True SOC and its experiment results with EKF and FDEKF

0 100 200 300 400 500 600 700 800minus01

minus008minus006minus004minus002

0002004006008

01

Time (min)

Erro

r of S

OC

EKFFDEKF

(b) SOC experiment errors with EKF and FDEKF

Figure 6 Comparison of SOC estimation and error curve of constant-current discharge pulse

20 40 60 80 100 120 140 160 180 2003

31323334353637

Volta

ge (V

)

Time (min)0 220

05101520253035

Curr

ent (

A)

VoltageCurrent

(a) Terminal currentvoltage curves of changing-current discharge pulse

0 20 40 60 80 100 120 140 160 180 200 220010203040506070809

1

Time (min)

SOC

EKFFDEKF

Ture

(b) SOC estimation results of changing-current discharge pulse

0 20 40 60 80 100 120 140 160 180 200 220minus014minus012

minus01minus008minus006minus004minus002

0002004006

Time (min)

Erro

r of S

OC

EKFFDEKF

(c) SOC estimation errors of changing-current discharge pulse

Figure 7 Comparison of SOC estimation and error curve of changing-current discharge pulse

Table 2 Comparison of SOC estimation

Algorithm Single mean computing time RMSEEKF 38994119890 minus 05 S 00174FDEKF 39371119890 minus 04 S 00018

where 119879 is the time step 119909119896is the estimated SOC at step 119896 119909

119896

is the true SOC at step 119896 119905119896cost is the computation time at step119896 and119873 is the sampling numberThe results in Table 2 clearlyindicate that FDEKF is superior to EKF in both estimationaccuracy and algorithm complexity

In order to confirm the robustness of FDEKF we per-formed another two tests with changing current and DSTdata using the parameters in Table 2 which incorrectly iden-tified the ECM and compared the performance of EKF andFDEKF in perturbations of the system parameters The SOCestimation results are shown in Figures 7 and 8 respectively Itshows that the EKF-based estimator has a poor performancewith a maximum absolute error of 14 while the FDEKF-based estimator can correctly trace the reference SOC witha maximum absolute error of 4 Thus there is a substantialimprovement (70) The comparison results show that theFDEKF-based method provides better performance in theSOC estimation under modeling error


0 2000 4000 6000 8000 10000 120002628

332343638

4

Time (s)

Volta

ge (V

)

minus40minus2002040

Curr

ent (

A)

VoltageCurrent

(a) Terminal currentvoltage curves of DST data

0 2000 4000 6000 8000 10000 120000

010203040506070809

1

Time (s)

SOC

FDEKFTure

EKF

(b) SOC estimation results of DST data with EKF and FDEKF

0 2000 4000 6000 8000 10000 12000minus004minus002

0002004006008

01012014

Time (s)

Erro

r of S

OC

(c) SOC estimation errors of DST data with EKF and FDEKF

Figure 8 Comparison of SOC estimation and error curve of DST data with EKF and FDEKF

AT90CAN128

SPI

LTC6803-4

Voltage

Cells

CANTMS320F2812

USB

CH240128B

Current

ACS750SCA-050Computer

Figure 9 Test bench

5 Experiments and Online Test

51 Test Bench The test bench is shown in Figure 9which consists of a DigatronMCT 30-05-40 cell cycler avoltage acquisition unit LTC6803-4 a high speed MCUAT90CAN128 a Hall current sensor ACS750SCA-050 and

a core data processor TMS320F2812 The Digatron MCT 30-05-40 can chargedischarge five battery packs according tothe designed program with a maximum voltage of 30V andmaximum chargedischarge current of 40A and its voltagemeasurement accuracy is plusmn5mV and current measurementaccuracy is plusmn50mA LTC6803-4 can measure up to 12 series


Table 3 Comparison of the simulated and experimental SOC results

Comparison point True SOC Simulation error of SOC Experimental error of SOCEKF FDEKF EKF FDEKF

First Loop end point 0172 08 04 17 09Second Loop end point 0352 09 03 26 11Third Loop end point 0532 10 04 40 13Forth Loop end point 0712 14 05 37 13Fifth Loop end point 0893 15 07 49 15Average error mdash 112 046 338 122

middot middot middot middot middot middotmiddot middot middot

TMS320F2812 Computer (Labview)USB

MonitorsMonitorsMonitorsMonitorsMonitorsMonitors

Cellpack

CellpackCell

packCellpack

Cellpack

Cellpack middot middot middot

middot middot middot

middot middot middotmiddot middot middot

AT90CAN128

AT90CAN128

AT90CAN128

BMC1 BMC2 BMCn

CAN bus

Figure 10 The extended structure diagram

connected battery cells at the same time AT90CAN128 is alow-power CMOS 8-bit microcontroller based on the AVRenhanced RISC structure ACS750SCA-050 can convert thechargedischarge current into 0ndash25 V voltage signal and itserror is less than 1 TMS320F2812 can receive the voltagedata of each cell from the AT90CAN128 with CAN commu-nication and thenuses the FDEKFalgorithm to calculate SOCin real time which is displayed on the computer screen Thistest bench can be extended as shown in Figure 10

We tested the single battery under DST working condi-tion on this test bench using the EKF and FDEKF algorithmsTable 3 shows that both simulation and experimental errorsincrease over timeThis is probably because the ECM param-eters are estimated offline and all of them are single constantand no adjustments are made for the changes of battery SOC

6 Conclusions

In this study we proposed a robust and powerful real-time SOC estimator for the lithium-ion batteries and theparameters of the second-order ECM were estimated usingthe nonlinear least squares algorithmThis new linearizationtechnique for SOCestimation is known as the finite difference

extended Kalman filter Compared to the EKF method theFDEKF method is able to track the real-time SOC morequickly and accurately with the accurate model When themodel parameters change it also has stronger robustnessagainst modeling uncertainties and maintains good accuracyin the estimation process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] L Lu H Xuebing J Li et al ldquoA review on the key issues forlithium-ion battery management in electric vehiclesrdquo Journal ofPower Sources vol 226 pp 272ndash288 2013

[2] K S Ng C-S Moo Y-P Chen and Y-C Hsieh ldquoEnhancedcoulomb counting method for estimating state-of-charge andstate-of-health of lithium-ion batteriesrdquoApplied Energy vol 86no 9 pp 1506ndash1511 2009

[3] H He X Zhang R Xiong Y Xu and H Guo ldquoOnline model-based estimation of state-of-charge and open-circuit voltage of


lithium-ion batteries in electric vehiclesrdquo Energy vol 39 no 1pp 310ndash318 2012

[4] G Plett ldquoLipb dynamic cell models for kalman-filter soc esti-mationrdquo in Proceedings of the 19th International Battery Hybridand Fuel Electric Vehicle Symposium and Exhibition pp 1ndash122002

[5] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packs part 1 BackgroundrdquoJournal of Power Sources vol 134 no 2 pp 252ndash261 2004

[6] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-basedHEVbattery packs part 2Modeling andidentificationrdquo Journal of Power Sources vol 134 no 2 pp 262ndash276 2004

[7] G L Plett ldquoExtended Kalman filtering for battery managementsystems of LiPB-based HEV battery packs part 3 State andparameter estimationrdquo Journal of Power Sources vol 134 no 2pp 277ndash292 2004

[8] M Marco K Lidiya L Bettina et al ldquoStudy of the localSOC distribution in a lithium-ion battery by physical and elec-trochemical modeling and simulationrdquo Applied MathematicalModelling vol 37 no 4 pp 2016ndash2027 2013

[9] V H Johnson ldquoBattery performance models in ADVISORrdquoJournal of Power Sources vol 110 no 2 pp 321ndash329 2002

[10] H He R Xiong X Zhang F Sun and J Fan ldquoState-of-charge estimation of the lithium-ion battery using an adaptiveextended Kalman filter based on an improvedTheveninmodelrdquoIEEE Transactions on Vehicular Technology vol 60 no 4 pp1461ndash1469 2011

[11] W Gao M Jiang and Y Hou ldquoResearch on PNGV modelparameter identification of LiFePO4 Li-ion battery based onFMRLSrdquo in Proceedings of the 6th IEEE Conference on IndustrialElectronics and Applications (ICIEA rsquo11) pp 2294ndash2297 June2011

[12] X Hu S Li and H Peng ldquoA comparative study of equivalentcircuitmodels for Li-ion batteriesrdquo Journal of Power Sources vol198 pp 359ndash367 2012

[13] Y-H Chiang W-Y Sean and J-C Ke ldquoOnline estimationof internal resistance and open-circuit voltage of lithium-ionbatteries in electric vehiclesrdquo Journal of Power Sources vol 196no 8 pp 3921ndash3932 2011

[14] S Abu-Sharkh and D Doerffel ldquoRapid test and non-linearmodel characterisation of solid-state lithium-ion batteriesrdquoJournal of Power Sources vol 130 no 1-2 pp 266ndash274 2004

[15] T S Schei ldquoA finite-difference method for linearization innonlinear estimation algorithmsrdquoAutomatica vol 33 no 11 pp2053ndash2058 1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


divergence of EKF is primarily due to the linearization errorfor ignoring high-order terms and the accuracy of EKF-based SOC estimation is sensitive to the precision of thebattery model

We introduced an alternative nonlinear Kalman filteringtechnique known as finite difference extended Kalman filter(FDEKF) in this study and used the finite difference methodinstead of the Taylor series expansions to estimate thecovariance matrix It has theoretical advantages that manifestthemselves in more accurate predictions and also strongrobustness against modeling uncertainty by making full useof the error information generated by model linearization

The remainder of this paper is arranged as followsSection 2 describes a model structure and discusses how theparameters of this model can be automatically estimatedusing the Simulink parameter estimation Section 3 briefs theEKF and deduces the FDEKF algorithm Section 4 shows thesimulation results of the proposed algorithm to the exper-imental data Section 5 shows the experimental verificationof the proposed algorithm on a test bench and Section 6concludes the paper

2 Modeling for the LiFePO4 Battery andParameter Identification Method

21 Lithium-Ion Battery Model The commonly used batterymodels include the electrochemicalmodel and equivalent cir-cuit model (ECM) [8] The electrochemical models describethe electrochemical reactions in the electrodes and electrolytein a mathematical way which can achieve high accuracyHowever they typically deploy partial differential equationswith a large number of unknown parameters Due to theintensive computation involved these models are often usedfor battery performance analysis and battery design ECMshave been developed especially for the purpose of vehiclepower management control and battery management systemdevelopment and have less parameters such as the Rintmodel [9] Thevenin model [10] and PNGV model [11]Literature [12] gives a comparative study of these ECMs forlithium-ion batteries Due to the complicated polarizationcharacteristics of battery it is suggested that models withmore parallel RC networks connected in series should havea much higher accuracy [13] Clearly the higher the orderis the more complicated the model becomes Thereforeconsidering the details in the model is a tradeoff betweenaccuracy and complexity In this paper a second-order RCmodel is considered as shown in Figure 1

Here 119880oc is the open circuit voltage 119880119905is the terminal

voltage of the battery 119868119905is the outflow current119877

0is the ohmic

resistance of the connectors electrodes and electrolyte andthe two sets of parallel resistor-capacitor elements connectedin series 119877

119890 119862119890 and 119877

119889 119862119889are the mass transport effects

and double-layer effects respectively Commonly the timeconstants of the two dynamics differ by at least an order ofmagnitude

22 Parameter Identification The model parameters needto be accurately estimated from the test data to establish

Re Rd

It

Ce Cd

R0

UocUt+

+

minus

minus

Figure 1 Equivalent circuit of second-order RC model

a battery model with good performance In this paper themodel is parameterized using a semiautomatic process thatcan satisfy the constraints on the optimized parameters Thisprocess uses a number of measured data sets under a varietyof conditions The parameters are optimized by minimizingthe error between measured and simulated results using thenonlinear least squares algorithm in the Simulink parameterestimation toolbox The battery model can be established inSimulink based on the second-order RC model as shown inFigure 2

3 SOC Estimation

31 EKF-Based SOC Estimation In order to use EKF meth-ods for battery SOC estimation the cell should bemodeled ina discrete-time state-space form Specifically we model thenonlinear battery system by a state equation and an outputequation below

119909119896+1

= 119891 (119909119896 119906119896) + 119908119896

119910119896= 119892 (119909

119896 119906119896) + V119896

(1)

where 119909119896is the system state vector at discrete-time index 119896

vector 119906119896is the measured system input at time 119896 and 119908

119896is

unmeasured ldquoprocess noiserdquo that affects the system state Theoutput of the system is 119910

119896and V119896is measurement noise 119891(sdot sdot)

and 119892(sdot sdot) are (possibly nonlinear) functions determined bythe particular cell model used

At each time step 119891(119909119896 119906119896) and 119892(119909

119896 119906119896) are linearized

by a first-order Taylo -series expansion The model can berewritten as

119909119896+1

= 119860119896119909119896+ 119861119896119906119896+ 119908119896

119910119896= 119862119896119909119896+ 119863119896119906119896+ V119896

(2)

where

119860119896=119889[119891(119909

119896 119906119896)]

119889[119909119896]

10038161003816100381610038161003816100381610038161003816119909119896119906119896

119861119896=119889 [119891 (119909

119896 119906119896)]

119889 [119906119896]

100381610038161003816100381610038161003816100381610038161003816119909119896119906119896


S PS

S PS

2

1 1Current

Voltage Terminator

Solverconfiguration

Voltagesentor

Batterycurrent

Lithium-ioncell

SOC Simscope

R_table_1Temp+minus

SOC

Simscope

C_table_1Temp

+

minus

SOC Simscope

R_table_1Temp+minus

SOC

Simscope

C_table_1Temp

+minus

SOC Simscope

R_table_1Temp+

+

minus

minus

SOC

Simscopem_table_

1Temp

+

minus

PS S

Terminator

1

2

SOC

R0

R2R1

C2Em C1

+

+

minus

minus

uarr V

f(x) = 0

Vt

gt


119862119896=119889 [119892 (119909

119896 119906119896)]

119889 [119909119896]

100381610038161003816100381610038161003816100381610038161003816119909119896119906119896

119863119896=119889 [119866 (119909

119896 119906119896)]

119889 [119906119896]

100381610038161003816100381610038161003816100381610038161003816119909119896119906119896

(3)





as

(

SOC (119896 + 1)119880119890(119896 + 1)

119880119889(119896 + 1)

) = (

1 0 0

0 119890minus119879120591119890 0

0 0 119890minus119879120591119889

) times(

SOC (119896)119880119890(119896)

119880119889(119896)

)

+(

minus120578119879

119876119899

119877119890(1 minus 119890

minus119879120591119890 )

119877119889(1 minus 119890

minus119879120591119889)

)119868119905(119896)

+ 119908119896

119880119905(119896) = 119880oc (SOC (119896)) minus 119880

119890(119896)

minus 119880119889(119896) minus 119877

0119868119905(119896) + V

119896

(4)






11198621and 119877

21198622circuit and

120591119890= 119877119890119862119890 120591119889= 119877119889119862119889 Both 119908

119896and V



119864 [119908119896] = 0 Cov [119908

119896 119908119879

119895] = 119876

119896120575119896119895

119864 [V119896] = 0 Cov [V

119896 V119879119895] = 119877119896120575119896119895

Cov [119908119896 V119896] = 0

(5)

where both 119876119896and 119877





Define 119909119896= [SOC(119896) 119880

119890(119896) 119880



119860119896=120597119891 (119909119896 119906119896)

120597119909119896

100381610038161003816100381610038161003816100381610038161003816119909119896=119909119896

= (

1 0 0

0 119890minus119879120591119890 0

0 0 119890minus119879120591119889

)

119861119896=(

minus120578119879119876119899

119877119890(1 minus 119890

minus119879120591119890)

119877119889(1 minus 119890

minus119879120591119889)

)


119862119896=120597119892 (119909119896 119906119896)

120597119909119896

100381610038161003816100381610038161003816100381610038161003816119909119896= 119909119896

= [120597119880oc120597SOC


119863119896= minus1198770(119896)

(6)

Initialization is

1199090= [SOC (0) 0 0]

0= 119864 [(119909

0minus 1199090) (1199090minus 1199090)119879

]

(7)


119896+1

= 119860119896119909119896+ 119861119896119868119905(119896) (8)

119896+1

= 119860119896119896119860119879

119896+ 119876119896 (9)

119870119896= 119896+1119862119879

119896[119862119896119896+1119862119879

119896+ 119877119896]minus1

(10)

119909119896+1

= 119896+1+ 119870119896[119880119905(119896) minus

119905(119896)] (11)

119896+1

= 119896+1minus 119870119896119862119896119896+1 (12)



119891 (119909) asymp 119891 () + 1198911015840

119863119863() (119909 minus ) (13)

1198911015840

119863119863() =

119891 ( + ℎ) minus 119891 ( minus ℎ)

2ℎ (14)


119891 (119909) = 119891 () + 1198911015840() (119909 minus )

+ [119891(3)()

3ℎ2+119891(5)()


(15)




119876 = 119878119908119878119879

119908 119877 = 119878V119878

119879

V

= 119878119909119878119879

119909 = 119878

119909119878119879

119909

(16)


119909and 119878




119865119909(119896) = (119891 (119909

119896+ Δ119909119896 119906119896) minus 119891 (119909

119896minus Δ119909119896 119906119896)) 2Δ119909

119896 (17)

Define Δ119909119896= ℎ119878119909(set ℎ2 = 3)

119865119909(119896) 119878119909= 119878119909119909

=1

2ℎ119891119894(119909119896+ ℎ119878119909119895 119906119896) minus 119891119894(119909119896minus Δℎ119878

119909119895 119906119896)

(18)


119909119895 (19) can be


119866119909(119896) 119878119909= 119878119910 119909

=1

2ℎ119892119894(119896+ ℎ 119878119909119895 119906119896) minus 119892119894(119909119896minus Δℎ 119878

119909119895 119906119896)

(19)


119896= 119865119909(119909) 119896119865119879

119909(119896) + 119876

119896

= 119865119909(119896) 119878119909119878119879

119909119865119879

119909(119896) + 119878

119908119878119879

119908

= 119878119909119909119878119879

119909119909+ 119878119908119878119879

119908

(20)


119870119896= 119896119866119879

119909(119896) [119866

119909(119896) 119896119866119879

119909(119896) + 119877

119896]minus1

= 119878119909119878119879

119909(119878119910 119909119878minus1

119909)119879

[119878119910 119909119878119879

119910 119909+ 119878V119878119879

V ]minus1

= 119878119909119878119879

119910 119909[119878119910 119909119878119879

119910 119909+ 119878V119878119879

V ]minus1


100 200 300 400 500 6002628

33234

Volta

ge (V

)

Time (min)0 700 800

01234

Curr

ent (

A)

CurrentVoltage


119896+1

= 119896minus 119870119896119866119909(119896) 119896

= 119878119909119878119879

119909minus 119870119896119866119909(119896) 119878119909119878119879

119909

= 119878119909119878119879

119909minus 119878119909119878119879

119910 119909119870119879

119896minus 119870119896119878119910 119909119878119879

119909+ 119878119909119878119879

119910 119909119870119879

119896

= 119878119909119878119879

119909minus 119878119909119878119879

119910 119909119870119879

119896

minus 119870119896119878119910 119909119878119879

119909+ 119870119896119878119910 119909119878119879

119910 119909119870119879

119896+ 119870119896119878V119878119879

V119870119879

119896

= [ 119878119909 minus 119870119896119878119910 119909 119870119896119878V] [119878119909minus 119870119896119878119910 119909

119870119896119878V]119879

(21)










33 V 2V 36V1198770

001Ω 0Ω 1Ω119877119890

0005Ω 0Ω 1Ω119862119890

0005 F 0 F 1 F119877119889

10000Ω 1Ω 100000Ω119862119889

10000 F 1 F 100000 F



SSE = sum(119909119894minus 119909)2

(22)




0 100 200 300 400 500 600 700 80026272829

33132333435

Time (s)


Vt

(V)


0 100 200 300 400 500 600 700 800minus003

minus0025minus002

minus0015minus001

minus00050

0005001

Time (s)

Erro

r (V

)


05

10

012

33234

0 1 2 3 4 5 6 7 8 90

002004

0 1 2 3 4 5 6 7 8 90

0102

1 2 3 4 5 6 7 8 900

0102


Iterations

Para

met

er v

alue

s

times104

times105

Em

(V)

C1

(F)

C2

(F)

R0

(Ω)

R1

(Ω)

R2

(Ω)


Cos

t fun

ctio

n (S

SE) Cost function

IterationsC

ost

101

100

10minus1

10minus2

0 1 2 3 4 5 6 7 8 9



2000 4000 6000 8000 100002628

332343638

4

Time (s)

Volta

ge (V

)

0 12000minus40minus2002040

Curr

ent (

A)




0002004

Time (s)

Erro

r of v

olta

ge (V

)








119873

sum

119896=1

(119909119896minus 119909119896)2

119905cost =1

119873

119873

sum

119896=1

119905119896

cost

(23)


0 100 200 300 400 500 600 700 8000

02040608

1

Time (min)

SOC

EKFFDEKF

Ture


0 100 200 300 400 500 600 700 800minus01


0002004006008

01

Time (min)

Erro

r of S

OC

EKFFDEKF



20 40 60 80 100 120 140 160 180 2003

31323334353637

Volta

ge (V

)

Time (min)0 220

05101520253035

Curr

ent (

A)

VoltageCurrent


0 20 40 60 80 100 120 140 160 180 200 220010203040506070809

1

Time (min)

SOC

EKFFDEKF

Ture


0 20 40 60 80 100 120 140 160 180 200 220minus014minus012


0002004006

Time (min)

Erro

r of S

OC

EKFFDEKF






119896




0 2000 4000 6000 8000 10000 120002628

332343638

4

Time (s)

Volta

ge (V

)

minus40minus2002040

Curr

ent (

A)

VoltageCurrent


0 2000 4000 6000 8000 10000 120000

010203040506070809

1

Time (s)

SOC

FDEKFTure

EKF


0 2000 4000 6000 8000 10000 12000minus004minus002

0002004006008

01012014

Time (s)

Erro

r of S

OC



AT90CAN128

SPI

LTC6803-4

Voltage

Cells

CANTMS320F2812

USB

CH240128B

Current


Figure 9 Test bench











Cellpack

CellpackCell

packCellpack

Cellpack




AT90CAN128

AT90CAN128

AT90CAN128

BMC1 BMC2 BMCn

CAN bus




6 Conclusions





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










S PS

S PS

2

1 1Current

Voltage Terminator

Solverconfiguration

Voltagesentor

Batterycurrent

Lithium-ioncell

SOC Simscope

R_table_1Temp+minus

SOC

Simscope

C_table_1Temp

+

minus

SOC Simscope

R_table_1Temp+minus

SOC

Simscope

C_table_1Temp

+minus

SOC Simscope

R_table_1Temp+

+

minus

minus

SOC

Simscopem_table_

1Temp

+

minus

PS S

Terminator

1

2

SOC

R0

R2R1

C2Em C1

+

+

minus

minus

uarr V

f(x) = 0

Vt

gt


119862119896=119889 [119892 (119909

119896 119906119896)]

119889 [119909119896]

100381610038161003816100381610038161003816100381610038161003816119909119896119906119896

119863119896=119889 [119866 (119909

119896 119906119896)]

119889 [119906119896]

100381610038161003816100381610038161003816100381610038161003816119909119896119906119896

(3)





as

(

SOC (119896 + 1)119880119890(119896 + 1)

119880119889(119896 + 1)

) = (

1 0 0

0 119890minus119879120591119890 0

0 0 119890minus119879120591119889

) times(

SOC (119896)119880119890(119896)

119880119889(119896)

)

+(

minus120578119879

119876119899

119877119890(1 minus 119890

minus119879120591119890 )

119877119889(1 minus 119890

minus119879120591119889)

)119868119905(119896)

+ 119908119896

119880119905(119896) = 119880oc (SOC (119896)) minus 119880

119890(119896)

minus 119880119889(119896) minus 119877

0119868119905(119896) + V

119896

(4)






11198621and 119877

21198622circuit and

120591119890= 119877119890119862119890 120591119889= 119877119889119862119889 Both 119908

119896and V



119864 [119908119896] = 0 Cov [119908

119896 119908119879

119895] = 119876

119896120575119896119895

119864 [V119896] = 0 Cov [V

119896 V119879119895] = 119877119896120575119896119895

Cov [119908119896 V119896] = 0

(5)

where both 119876119896and 119877





Define 119909119896= [SOC(119896) 119880

119890(119896) 119880



119860119896=120597119891 (119909119896 119906119896)

120597119909119896

100381610038161003816100381610038161003816100381610038161003816119909119896=119909119896

= (

1 0 0

0 119890minus119879120591119890 0

0 0 119890minus119879120591119889

)

119861119896=(

minus120578119879119876119899

119877119890(1 minus 119890

minus119879120591119890)

119877119889(1 minus 119890

minus119879120591119889)

)


119862119896=120597119892 (119909119896 119906119896)

120597119909119896

100381610038161003816100381610038161003816100381610038161003816119909119896= 119909119896

= [120597119880oc120597SOC


119863119896= minus1198770(119896)

(6)

Initialization is

1199090= [SOC (0) 0 0]

0= 119864 [(119909

0minus 1199090) (1199090minus 1199090)119879

]

(7)


119896+1

= 119860119896119909119896+ 119861119896119868119905(119896) (8)

119896+1

= 119860119896119896119860119879

119896+ 119876119896 (9)

119870119896= 119896+1119862119879

119896[119862119896119896+1119862119879

119896+ 119877119896]minus1

(10)

119909119896+1

= 119896+1+ 119870119896[119880119905(119896) minus

119905(119896)] (11)

119896+1

= 119896+1minus 119870119896119862119896119896+1 (12)



119891 (119909) asymp 119891 () + 1198911015840

119863119863() (119909 minus ) (13)

1198911015840

119863119863() =

119891 ( + ℎ) minus 119891 ( minus ℎ)

2ℎ (14)


119891 (119909) = 119891 () + 1198911015840() (119909 minus )

+ [119891(3)()

3ℎ2+119891(5)()


(15)




119876 = 119878119908119878119879

119908 119877 = 119878V119878

119879

V

= 119878119909119878119879

119909 = 119878

119909119878119879

119909

(16)


119909and 119878




119865119909(119896) = (119891 (119909

119896+ Δ119909119896 119906119896) minus 119891 (119909

119896minus Δ119909119896 119906119896)) 2Δ119909

119896 (17)

Define Δ119909119896= ℎ119878119909(set ℎ2 = 3)

119865119909(119896) 119878119909= 119878119909119909

=1

2ℎ119891119894(119909119896+ ℎ119878119909119895 119906119896) minus 119891119894(119909119896minus Δℎ119878

119909119895 119906119896)

(18)


119909119895 (19) can be


119866119909(119896) 119878119909= 119878119910 119909

=1

2ℎ119892119894(119896+ ℎ 119878119909119895 119906119896) minus 119892119894(119909119896minus Δℎ 119878

119909119895 119906119896)

(19)


119896= 119865119909(119909) 119896119865119879

119909(119896) + 119876

119896

= 119865119909(119896) 119878119909119878119879

119909119865119879

119909(119896) + 119878

119908119878119879

119908

= 119878119909119909119878119879

119909119909+ 119878119908119878119879

119908

(20)


119870119896= 119896119866119879

119909(119896) [119866

119909(119896) 119896119866119879

119909(119896) + 119877

119896]minus1

= 119878119909119878119879

119909(119878119910 119909119878minus1

119909)119879

[119878119910 119909119878119879

119910 119909+ 119878V119878119879

V ]minus1

= 119878119909119878119879

119910 119909[119878119910 119909119878119879

119910 119909+ 119878V119878119879

V ]minus1


100 200 300 400 500 6002628

33234

Volta

ge (V

)

Time (min)0 700 800

01234

Curr

ent (

A)

CurrentVoltage


119896+1

= 119896minus 119870119896119866119909(119896) 119896

= 119878119909119878119879

119909minus 119870119896119866119909(119896) 119878119909119878119879

119909

= 119878119909119878119879

119909minus 119878119909119878119879

119910 119909119870119879

119896minus 119870119896119878119910 119909119878119879

119909+ 119878119909119878119879

119910 119909119870119879

119896

= 119878119909119878119879

119909minus 119878119909119878119879

119910 119909119870119879

119896

minus 119870119896119878119910 119909119878119879

119909+ 119870119896119878119910 119909119878119879

119910 119909119870119879

119896+ 119870119896119878V119878119879

V119870119879

119896

= [ 119878119909 minus 119870119896119878119910 119909 119870119896119878V] [119878119909minus 119870119896119878119910 119909

119870119896119878V]119879

(21)










33 V 2V 36V1198770

001Ω 0Ω 1Ω119877119890

0005Ω 0Ω 1Ω119862119890

0005 F 0 F 1 F119877119889

10000Ω 1Ω 100000Ω119862119889

10000 F 1 F 100000 F



SSE = sum(119909119894minus 119909)2

(22)




0 100 200 300 400 500 600 700 80026272829

33132333435

Time (s)


Vt

(V)


0 100 200 300 400 500 600 700 800minus003

minus0025minus002

minus0015minus001

minus00050

0005001

Time (s)

Erro

r (V

)


05

10

012

33234

0 1 2 3 4 5 6 7 8 90

002004

0 1 2 3 4 5 6 7 8 90

0102

1 2 3 4 5 6 7 8 900

0102


Iterations

Para

met

er v

alue

s

times104

times105

Em

(V)

C1

(F)

C2

(F)

R0

(Ω)

R1

(Ω)

R2

(Ω)


Cos

t fun

ctio

n (S

SE) Cost function

IterationsC

ost

101

100

10minus1

10minus2

0 1 2 3 4 5 6 7 8 9



2000 4000 6000 8000 100002628

332343638

4

Time (s)

Volta

ge (V

)

0 12000minus40minus2002040

Curr

ent (

A)




0002004

Time (s)

Erro

r of v

olta

ge (V

)








119873

sum

119896=1

(119909119896minus 119909119896)2

119905cost =1

119873

119873

sum

119896=1

119905119896

cost

(23)


0 100 200 300 400 500 600 700 8000

02040608

1

Time (min)

SOC

EKFFDEKF

Ture


0 100 200 300 400 500 600 700 800minus01


0002004006008

01

Time (min)

Erro

r of S

OC

EKFFDEKF



20 40 60 80 100 120 140 160 180 2003

31323334353637

Volta

ge (V

)

Time (min)0 220

05101520253035

Curr

ent (

A)

VoltageCurrent


0 20 40 60 80 100 120 140 160 180 200 220010203040506070809

1

Time (min)

SOC

EKFFDEKF

Ture


0 20 40 60 80 100 120 140 160 180 200 220minus014minus012


0002004006

Time (min)

Erro

r of S

OC

EKFFDEKF






119896




0 2000 4000 6000 8000 10000 120002628

332343638

4

Time (s)

Volta

ge (V

)

minus40minus2002040

Curr

ent (

A)

VoltageCurrent


0 2000 4000 6000 8000 10000 120000

010203040506070809

1

Time (s)

SOC

FDEKFTure

EKF


0 2000 4000 6000 8000 10000 12000minus004minus002

0002004006008

01012014

Time (s)

Erro

r of S

OC



AT90CAN128

SPI

LTC6803-4

Voltage

Cells

CANTMS320F2812

USB

CH240128B

Current


Figure 9 Test bench











Cellpack

CellpackCell

packCellpack

Cellpack




AT90CAN128

AT90CAN128

AT90CAN128

BMC1 BMC2 BMCn

CAN bus




6 Conclusions





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119862119896=120597119892 (119909119896 119906119896)

120597119909119896

100381610038161003816100381610038161003816100381610038161003816119909119896= 119909119896

= [120597119880oc120597SOC


119863119896= minus1198770(119896)

(6)

Initialization is

1199090= [SOC (0) 0 0]

0= 119864 [(119909

0minus 1199090) (1199090minus 1199090)119879

]

(7)


119896+1

= 119860119896119909119896+ 119861119896119868119905(119896) (8)

119896+1

= 119860119896119896119860119879

119896+ 119876119896 (9)

119870119896= 119896+1119862119879

119896[119862119896119896+1119862119879

119896+ 119877119896]minus1

(10)

119909119896+1

= 119896+1+ 119870119896[119880119905(119896) minus

119905(119896)] (11)

119896+1

= 119896+1minus 119870119896119862119896119896+1 (12)



119891 (119909) asymp 119891 () + 1198911015840

119863119863() (119909 minus ) (13)

1198911015840

119863119863() =

119891 ( + ℎ) minus 119891 ( minus ℎ)

2ℎ (14)


119891 (119909) = 119891 () + 1198911015840() (119909 minus )

+ [119891(3)()

3ℎ2+119891(5)()


(15)




119876 = 119878119908119878119879

119908 119877 = 119878V119878

119879

V

= 119878119909119878119879

119909 = 119878

119909119878119879

119909

(16)


119909and 119878




119865119909(119896) = (119891 (119909

119896+ Δ119909119896 119906119896) minus 119891 (119909

119896minus Δ119909119896 119906119896)) 2Δ119909

119896 (17)

Define Δ119909119896= ℎ119878119909(set ℎ2 = 3)

119865119909(119896) 119878119909= 119878119909119909

=1

2ℎ119891119894(119909119896+ ℎ119878119909119895 119906119896) minus 119891119894(119909119896minus Δℎ119878

119909119895 119906119896)

(18)


119909119895 (19) can be


119866119909(119896) 119878119909= 119878119910 119909

=1

2ℎ119892119894(119896+ ℎ 119878119909119895 119906119896) minus 119892119894(119909119896minus Δℎ 119878

119909119895 119906119896)

(19)


119896= 119865119909(119909) 119896119865119879

119909(119896) + 119876

119896

= 119865119909(119896) 119878119909119878119879

119909119865119879

119909(119896) + 119878

119908119878119879

119908

= 119878119909119909119878119879

119909119909+ 119878119908119878119879

119908

(20)


119870119896= 119896119866119879

119909(119896) [119866

119909(119896) 119896119866119879

119909(119896) + 119877

119896]minus1

= 119878119909119878119879

119909(119878119910 119909119878minus1

119909)119879

[119878119910 119909119878119879

119910 119909+ 119878V119878119879

V ]minus1

= 119878119909119878119879

119910 119909[119878119910 119909119878119879

119910 119909+ 119878V119878119879

V ]minus1


100 200 300 400 500 6002628

33234

Volta

ge (V

)

Time (min)0 700 800

01234

Curr

ent (

A)

CurrentVoltage


119896+1

= 119896minus 119870119896119866119909(119896) 119896

= 119878119909119878119879

119909minus 119870119896119866119909(119896) 119878119909119878119879

119909

= 119878119909119878119879

119909minus 119878119909119878119879

119910 119909119870119879

119896minus 119870119896119878119910 119909119878119879

119909+ 119878119909119878119879

119910 119909119870119879

119896

= 119878119909119878119879

119909minus 119878119909119878119879

119910 119909119870119879

119896

minus 119870119896119878119910 119909119878119879

119909+ 119870119896119878119910 119909119878119879

119910 119909119870119879

119896+ 119870119896119878V119878119879

V119870119879

119896

= [ 119878119909 minus 119870119896119878119910 119909 119870119896119878V] [119878119909minus 119870119896119878119910 119909

119870119896119878V]119879

(21)










33 V 2V 36V1198770

001Ω 0Ω 1Ω119877119890

0005Ω 0Ω 1Ω119862119890

0005 F 0 F 1 F119877119889

10000Ω 1Ω 100000Ω119862119889

10000 F 1 F 100000 F



SSE = sum(119909119894minus 119909)2

(22)




0 100 200 300 400 500 600 700 80026272829

33132333435

Time (s)


Vt

(V)


0 100 200 300 400 500 600 700 800minus003

minus0025minus002

minus0015minus001

minus00050

0005001

Time (s)

Erro

r (V

)


05

10

012

33234

0 1 2 3 4 5 6 7 8 90

002004

0 1 2 3 4 5 6 7 8 90

0102

1 2 3 4 5 6 7 8 900

0102


Iterations

Para

met

er v

alue

s

times104

times105

Em

(V)

C1

(F)

C2

(F)

R0

(Ω)

R1

(Ω)

R2

(Ω)


Cos

t fun

ctio

n (S

SE) Cost function

IterationsC

ost

101

100

10minus1

10minus2

0 1 2 3 4 5 6 7 8 9



2000 4000 6000 8000 100002628

332343638

4

Time (s)

Volta

ge (V

)

0 12000minus40minus2002040

Curr

ent (

A)




0002004

Time (s)

Erro

r of v

olta

ge (V

)








119873

sum

119896=1

(119909119896minus 119909119896)2

119905cost =1

119873

119873

sum

119896=1

119905119896

cost

(23)


0 100 200 300 400 500 600 700 8000

02040608

1

Time (min)

SOC

EKFFDEKF

Ture


0 100 200 300 400 500 600 700 800minus01


0002004006008

01

Time (min)

Erro

r of S

OC

EKFFDEKF



20 40 60 80 100 120 140 160 180 2003

31323334353637

Volta

ge (V

)

Time (min)0 220

05101520253035

Curr

ent (

A)

VoltageCurrent


0 20 40 60 80 100 120 140 160 180 200 220010203040506070809

1

Time (min)

SOC

EKFFDEKF

Ture


0 20 40 60 80 100 120 140 160 180 200 220minus014minus012


0002004006

Time (min)

Erro

r of S

OC

EKFFDEKF






119896




0 2000 4000 6000 8000 10000 120002628

332343638

4

Time (s)

Volta

ge (V

)

minus40minus2002040

Curr

ent (

A)

VoltageCurrent


0 2000 4000 6000 8000 10000 120000

010203040506070809

1

Time (s)

SOC

FDEKFTure

EKF


0 2000 4000 6000 8000 10000 12000minus004minus002

0002004006008

01012014

Time (s)

Erro

r of S

OC



AT90CAN128

SPI

LTC6803-4

Voltage

Cells

CANTMS320F2812

USB

CH240128B

Current


Figure 9 Test bench











Cellpack

CellpackCell

packCellpack

Cellpack




AT90CAN128

AT90CAN128

AT90CAN128

BMC1 BMC2 BMCn

CAN bus




6 Conclusions





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










100 200 300 400 500 6002628

33234

Volta

ge (V

)

Time (min)0 700 800

01234

Curr

ent (

A)

CurrentVoltage


119896+1

= 119896minus 119870119896119866119909(119896) 119896

= 119878119909119878119879

119909minus 119870119896119866119909(119896) 119878119909119878119879

119909

= 119878119909119878119879

119909minus 119878119909119878119879

119910 119909119870119879

119896minus 119870119896119878119910 119909119878119879

119909+ 119878119909119878119879

119910 119909119870119879

119896

= 119878119909119878119879

119909minus 119878119909119878119879

119910 119909119870119879

119896

minus 119870119896119878119910 119909119878119879

119909+ 119870119896119878119910 119909119878119879

119910 119909119870119879

119896+ 119870119896119878V119878119879

V119870119879

119896

= [ 119878119909 minus 119870119896119878119910 119909 119870119896119878V] [119878119909minus 119870119896119878119910 119909

119870119896119878V]119879

(21)










33 V 2V 36V1198770

001Ω 0Ω 1Ω119877119890

0005Ω 0Ω 1Ω119862119890

0005 F 0 F 1 F119877119889

10000Ω 1Ω 100000Ω119862119889

10000 F 1 F 100000 F



SSE = sum(119909119894minus 119909)2

(22)




0 100 200 300 400 500 600 700 80026272829

33132333435

Time (s)


Vt

(V)


0 100 200 300 400 500 600 700 800minus003

minus0025minus002

minus0015minus001

minus00050

0005001

Time (s)

Erro

r (V

)


05

10

012

33234

0 1 2 3 4 5 6 7 8 90

002004

0 1 2 3 4 5 6 7 8 90

0102

1 2 3 4 5 6 7 8 900

0102


Iterations

Para

met

er v

alue

s

times104

times105

Em

(V)

C1

(F)

C2

(F)

R0

(Ω)

R1

(Ω)

R2

(Ω)


Cos

t fun

ctio

n (S

SE) Cost function

IterationsC

ost

101

100

10minus1

10minus2

0 1 2 3 4 5 6 7 8 9



2000 4000 6000 8000 100002628

332343638

4

Time (s)

Volta

ge (V

)

0 12000minus40minus2002040

Curr

ent (

A)




0002004

Time (s)

Erro

r of v

olta

ge (V

)








119873

sum

119896=1

(119909119896minus 119909119896)2

119905cost =1

119873

119873

sum

119896=1

119905119896

cost

(23)


0 100 200 300 400 500 600 700 8000

02040608

1

Time (min)

SOC

EKFFDEKF

Ture


0 100 200 300 400 500 600 700 800minus01


0002004006008

01

Time (min)

Erro

r of S

OC

EKFFDEKF



20 40 60 80 100 120 140 160 180 2003

31323334353637

Volta

ge (V

)

Time (min)0 220

05101520253035

Curr

ent (

A)

VoltageCurrent


0 20 40 60 80 100 120 140 160 180 200 220010203040506070809

1

Time (min)

SOC

EKFFDEKF

Ture


0 20 40 60 80 100 120 140 160 180 200 220minus014minus012


0002004006

Time (min)

Erro

r of S

OC

EKFFDEKF






119896




0 2000 4000 6000 8000 10000 120002628

332343638

4

Time (s)

Volta

ge (V

)

minus40minus2002040

Curr

ent (

A)

VoltageCurrent


0 2000 4000 6000 8000 10000 120000

010203040506070809

1

Time (s)

SOC

FDEKFTure

EKF


0 2000 4000 6000 8000 10000 12000minus004minus002

0002004006008

01012014

Time (s)

Erro

r of S

OC



AT90CAN128

SPI

LTC6803-4

Voltage

Cells

CANTMS320F2812

USB

CH240128B

Current


Figure 9 Test bench











Cellpack

CellpackCell

packCellpack

Cellpack




AT90CAN128

AT90CAN128

AT90CAN128

BMC1 BMC2 BMCn

CAN bus




6 Conclusions





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 80026272829

33132333435

Time (s)


Vt

(V)


0 100 200 300 400 500 600 700 800minus003

minus0025minus002

minus0015minus001

minus00050

0005001

Time (s)

Erro

r (V

)


05

10

012

33234

0 1 2 3 4 5 6 7 8 90

002004

0 1 2 3 4 5 6 7 8 90

0102

1 2 3 4 5 6 7 8 900

0102


Iterations

Para

met

er v

alue

s

times104

times105

Em

(V)

C1

(F)

C2

(F)

R0

(Ω)

R1

(Ω)

R2

(Ω)


Cos

t fun

ctio

n (S

SE) Cost function

IterationsC

ost

101

100

10minus1

10minus2

0 1 2 3 4 5 6 7 8 9



2000 4000 6000 8000 100002628

332343638

4

Time (s)

Volta

ge (V

)

0 12000minus40minus2002040

Curr

ent (

A)




0002004

Time (s)

Erro

r of v

olta

ge (V

)








119873

sum

119896=1

(119909119896minus 119909119896)2

119905cost =1

119873

119873

sum

119896=1

119905119896

cost

(23)


0 100 200 300 400 500 600 700 8000

02040608

1

Time (min)

SOC

EKFFDEKF

Ture


0 100 200 300 400 500 600 700 800minus01


0002004006008

01

Time (min)

Erro

r of S

OC

EKFFDEKF



20 40 60 80 100 120 140 160 180 2003

31323334353637

Volta

ge (V

)

Time (min)0 220

05101520253035

Curr

ent (

A)

VoltageCurrent


0 20 40 60 80 100 120 140 160 180 200 220010203040506070809

1

Time (min)

SOC

EKFFDEKF

Ture


0 20 40 60 80 100 120 140 160 180 200 220minus014minus012


0002004006

Time (min)

Erro

r of S

OC

EKFFDEKF






119896




0 2000 4000 6000 8000 10000 120002628

332343638

4

Time (s)

Volta

ge (V

)

minus40minus2002040

Curr

ent (

A)

VoltageCurrent


0 2000 4000 6000 8000 10000 120000

010203040506070809

1

Time (s)

SOC

FDEKFTure

EKF


0 2000 4000 6000 8000 10000 12000minus004minus002

0002004006008

01012014

Time (s)

Erro

r of S

OC



AT90CAN128

SPI

LTC6803-4

Voltage

Cells

CANTMS320F2812

USB

CH240128B

Current


Figure 9 Test bench











Cellpack

CellpackCell

packCellpack

Cellpack




AT90CAN128

AT90CAN128

AT90CAN128

BMC1 BMC2 BMCn

CAN bus




6 Conclusions





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 8000

02040608

1

Time (min)

SOC

EKFFDEKF

Ture


0 100 200 300 400 500 600 700 800minus01


0002004006008

01

Time (min)

Erro

r of S

OC

EKFFDEKF



20 40 60 80 100 120 140 160 180 2003

31323334353637

Volta

ge (V

)

Time (min)0 220

05101520253035

Curr

ent (

A)

VoltageCurrent


0 20 40 60 80 100 120 140 160 180 200 220010203040506070809

1

Time (min)

SOC

EKFFDEKF

Ture


0 20 40 60 80 100 120 140 160 180 200 220minus014minus012


0002004006

Time (min)

Erro

r of S

OC

EKFFDEKF






119896




0 2000 4000 6000 8000 10000 120002628

332343638

4

Time (s)

Volta

ge (V

)

minus40minus2002040

Curr

ent (

A)

VoltageCurrent


0 2000 4000 6000 8000 10000 120000

010203040506070809

1

Time (s)

SOC

FDEKFTure

EKF


0 2000 4000 6000 8000 10000 12000minus004minus002

0002004006008

01012014

Time (s)

Erro

r of S

OC



AT90CAN128

SPI

LTC6803-4

Voltage

Cells

CANTMS320F2812

USB

CH240128B

Current


Figure 9 Test bench











Cellpack

CellpackCell

packCellpack

Cellpack




AT90CAN128

AT90CAN128

AT90CAN128

BMC1 BMC2 BMCn

CAN bus




6 Conclusions





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2000 4000 6000 8000 10000 120002628

332343638

4

Time (s)

Volta

ge (V

)

minus40minus2002040

Curr

ent (

A)

VoltageCurrent


0 2000 4000 6000 8000 10000 120000

010203040506070809

1

Time (s)

SOC

FDEKFTure

EKF


0 2000 4000 6000 8000 10000 12000minus004minus002

0002004006008

01012014

Time (s)

Erro

r of S

OC



AT90CAN128

SPI

LTC6803-4

Voltage

Cells

CANTMS320F2812

USB

CH240128B

Current


Figure 9 Test bench











Cellpack

CellpackCell

packCellpack

Cellpack




AT90CAN128

AT90CAN128

AT90CAN128

BMC1 BMC2 BMCn

CAN bus




6 Conclusions





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Cellpack

CellpackCell

packCellpack

Cellpack




AT90CAN128

AT90CAN128

AT90CAN128

BMC1 BMC2 BMCn

CAN bus




6 Conclusions





References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article estimation of state of charge for lithium ...an accurate estimation of the state of...

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