DTIC ADA583295: Online Parameterization of Lumped Thermal Dynamics in Cylindrical Lithium Ion Batteries for Core Temperature Estimation and Health Monitoring

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IEEETRANSACTIONSONCONTROLSYSTEMTECHNOLOGY,VOL.X,NO.X,XX2012 1
Online Parameterization of Lumped Thermal
Dynamics in Cylindrical Lithium Ion Batteries for
Core Temperature Estimation and Health
Monitoring
Xinfan Lin, Hector E. Perez, Jason B. Siegel, Anna G. Stefanopoulou, Fellow, IEEE,
Yonghua Li, R. Dyche Anderson, Yi Ding and Matthew P. Castanier
Abstract—Lithium ion batteries should always be prevented approximation is computationally efficient, it might lead to
fromoverheatingandhencethermalmonitoringisindispensable. over-simplification since the temperature in the battery core
Since only the surface temperature of the battery can be mea-
can be much higher than in the surface [8].
sured,athermalmodelisneededtoestimatethecoretemperature
Lumped thermal models capturing both the surface and
of the battery, which can be higher and more critical. In this
paper, an online parameter identification scheme is designed the core temperatures of the cell have also been studied in
for a cylindrical lithium ion battery. An adaptive observer [8] and [9]. Such simplified models are efficient for onboard
of the core temperature is then designed based on the on- applicationduetotheirlimitednumberofstates.Theaccuracy
line parameterization methodology and the surface temperature
of the model parameters is of great importance since it
measurement. A battery thermal model with constant internal
determines the accuracy of the core temperature estimation.
resistance is explored first. The identification algorithm and
the adaptive observer is validated with experiments on a 2:3Ah Modelparameterscanbeapproximatedbasedonthegeometry
26650lithiumironphosphate/graphitebattery.Themethodology of the battery and the volume averaging physical properties
is later extended to address temperature dependent internal of its components [9], but such approximation may not be
resistance with non-uniform forgetting factors. The capability of
accurate due to the complicated layered structure of the cell
the methodology to track the long term variation of the internal
and the interfaces between the layers. The parameters can
resistance is beneficial for battery health monitoring.
also be determined by fitting the model to the data obtained
IndexTerms—Lithiumionbattery,coretemperature,adaptive
from experiments [8], involving designed input excitation and
estimation, state of health.
measurementofthebatterycoretemperature.Thislaboratory-
oriented parameterization is invaluable for determining the
I. INTRODUCTION initialvaluesofparameters.However,someoftheparameters,
LITHIUM ion batteries have been widely considered as such as the internal resistance, may change over the battery
an energy storage device for hybrid electric vehicles lifetime due to degradation. In this case, parameter mismatch
(HEV), plug-in hybrid electric vehicles (PHEV) and battery leads to inaccurate temperature estimation, and thus identifi-
electric vehicles (BEV). Thermal management in vehicular cation of present value of the parameters is needed.
applicationsisacriticalissueforlithiumionbatteriesbecause A recursive parameter identification scheme is designed
of their narrow operating temperature range. An accurate in this paper to automatically identify the thermal model
predictionofthebatterytemperatureisthekeytomaintaining parameters based on the signals commonly measured in a
safety, performance, and longevity of these Li-ion batteries. vehicle battery management system. The algorithm is simple
Existing high fidelity thermal models can predict the de- enough to run on a typical automotive onboard controller. An
tailed temperature distribution throughout the cell [1], [2], adaptive observer is then designed for the core temperature
[3], [4]. However, these models are not suitable for onboard estimation. A lumped battery model with constant internal
applicationduetotheirhighcomputationalintensity.Reduced resistanceisinvestigatedfirst,wheretheleastsquarealgorithm
ordermodelstypicallyuseonesingletemperature,thebulk(or issufficientforidentification.Inreality,theinternalresistance
average)temperature,tocapturethelumpedthermaldynamics ofthebatterycanbetemperatureand/orstateofcharge(SOC)
ofthecell[4],[5],[6],[7].Eventhoughthesingletemperature dependent [5], [10], [11] and hence time-varying. The pure
least square algorithm may cause errors to the identification
X.Lin,H.Perez,J.SiegelandA.StefanopoulouarewiththeDepartment
if the actual parameters are non-constant. Non-uniform for-
of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109,
USA e-mail: (xfl[email protected], [email protected], [email protected] getting factors are augmented to the least square algorithm to
[email protected]). address the issue of time-varying internal resistance.
Y.LiandR.D.AndersonarewiththeVehicleandBatteryControlsDepart-
Apart from the short-term variability due to conditions
ment,ResearchandAdvancedEngineering,FordMotorCompany,Dearborn,
MI48121,USAe-mail:([email protected]@ford.com). such as temperature, the internal resistance of the lithium ion
Y. Ding and M. Castanier are with the U.S. Army Tank Automotive Re- battery may also increase over lifetime due to degradation.
search,Development,andEngineeringCenter(TARDEC),Warren,MI48397,
Thisisbecausethesolidelectrolyteinterphase(SEI)maygrow
USAe-mail:([email protected]@mail.mil).
ManuscriptreceivedMarch6,2012;revisedJuly31,2012. in thickness [12], [13], [14] and reduce the conductivity of
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1. REPORT DATE 2. REPORT TYPE 3. DATES COVERED
13 MAR 2012 Journal Article 10-10-2011 to 08-02-2012
4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Online Parameterization of Lumped Thermal Dynamics in Cylindrical W56H2V-04-2-0001
Lithium Ion Batteries for Core Temperature Estimation and Health
5b. GRANT NUMBER
Monitoring
5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S) 5d. PROJECT NUMBER
Xinfan Lin; Hector Perez; Jason Siegel; Yi Ding; Matthew Castanier
5e. TASK NUMBER
5f. WORK UNIT NUMBER
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT
Center For Automotive Research,3005 Boardwalk St,Ste 200 ,Ann NUMBER
; #22630
Arbor,Mi,48108
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)
U.S. Army TARDEC, 6501 East Eleven Mile Rd, Warren, Mi, TARDEC
48397-5000
11. SPONSOR/MONITOR’S REPORT
NUMBER(S)
#22630
12. DISTRIBUTION/AVAILABILITY STATEMENT
Approved for public release; distribution unlimited
13. SUPPLEMENTARY NOTES
Submitted to IEEE Transactions on Control Systems Technology
14. ABSTRACT
Lithium ion batteries should always be prevented from overheating and hence thermal monitoring is
indispensable. Since only the surface temperature of the battery can be measured, a thermal model is
needed to estimate the core temperature of the battery, which can be higher and more critical. In this
paper, an online parameter identification scheme is designed for a cylindrical lithium ion battery. An
adaptive observer of the core temperature is then designed based on the online parameterization
methodology and the surface temperature measurement. A battery thermal model with constant internal
resistance is explored first. The identification algorithm and the adaptive observer is validated with
experiments on a 2:3Ah 26650 lithium iron phosphate/graphite battery. The methodology is later extended
to address temperature dependent internal resistance with non-uniform forgetting factors. The capability
of the methodology to track the long term variation of the internal resistance is beneficial for battery
health monitoring.
15. SUBJECT TERMS
Lithium ion battery, core temperature, adaptive estimation, state of health.
16. SECURITY CLASSIFICATION OF: 17. LIMITATION 18. NUMBER 19a. NAME OF
OF ABSTRACT OF PAGES RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE Public 11
unclassified unclassified unclassified Release
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IEEETRANSACTIONSONCONTROLSYSTEMTECHNOLOGY,VOL.X,NO.X,XX2012 2
a lumped parameter aggregating the conduction and contact
thermal resistance across the compact and inhomogeneous
materials. A convection resistance R is modeled between the
u
surface and the surrounding coolant to account for convective
cooling. The value of R is a function of the coolant flow
u
rate, and in some vehicle battery systems, the coolant flow
rate is adjustable to control the battery temperature. Here, it
is modeled as a constant as if the coolant flow rate is fixed
to accommodate the maximum required cooling capacity. A
model with the more complicated varying R has also been
u
Fig.1. SingleCellRadiallyLumpedThermalModel
investigated in [19].
The rates of the temperature change of the surface and the
the SEI. Hence, the least square algorithm with non-uniform core depend on their respective lumped heat capacity. The
forgettingfactorsisalsoappliedtotrackthelongtermgrowth parameter Cc is the heat capacity of the jelly roll inside the
oftheinternalresistance.Thegrowthoftheinternalresistance cell,andCs isrelatedtotheheatcapacityofthebatterycasing.
canbeviewedasanimportantindicationofthestateofhealth The complete parameter set for this model includesCc,Cs,
(SOH) of the battery, and used as a reference for the onboard Re, Rc, and Ru, of which the values cannot be easily calcu-
battery management system to extend the life of the batteries. lated. Consider the conduction resistance Rc as an example.
Parameterization of battery model and adaptive monitoring of Theoretically, Rc can be calculated based on the conductivity
the battery voltage and SOH have been explored previously and dimensions of the wound cell electrode assembly and
in various seminal papers [15], [16], [17], but this paper is the aluminum casing. However, since the rolled electrodes
among the first ones to adaptively monitor the temperatures are composed by the cathode, anode, current collectors and
(especially the core temperature) of batteries and SOH from separator, it will be difficult to obtain an accurate value
a thermal perspective. for the overall conductivity. Moreover, Rc also includes the
contact thermal resistance between the rolled electrodes and
II. LUMPEDTHERMALMODELOFACYLINDRICAL the casing, which involves various contact properties adding
LITHIUMIONBATTERY to the complexity of the calculation.
The radial thermal dynamics of a cylindrical battery are Therefore, model identification techniques are developed in
modeledbasedontheclassicheattransferproblembyassum- the following section to obtain the lumped phenomenological
ing heat generation located at the core and zero heat flux at values of the model parameters based on measurable inputs
the center, as shown in Fig. 1. and outputs of the model.
The two-state approximation of the radially distributed
thermal model is defined as [9]
III. PARAMETERIZATIONMETHODOLOGY
T (cid:0)T
CcT˙c= I2Re+ s c (1a) For model identification, a parametric model
R
c
T (cid:0)T T (cid:0)T
C T˙ = f s (cid:0) s c; (1b) z=θTϕ (2)
s s
R R
u c
where the two states are the surface temperature T and is derived first by applying Laplace transformation to the
s
the core temperature T . The temperature variation along model, where z is the observation, θ is the parameter vector
c
the battery height is neglected here, assuming homogeneous and ϕis the regressor [20]. Both z and ϕshould be measured
conditions. or can be generated from measured signals.
Heatgenerationisapproximatedasaconcentratedsourceof With a parametric model, various algorithms can be chosen
Joule loss in the battery core, computed as the product of the for parameter identification, such as the gradient and the least
current,I,squaredandtheinternalresistance,Re.Theinternal squares methods. The method of least squares is preferred for
resistance Re is considered as an unknown parameter to be noise reduction [20].
identified.Thissimplificationcanleadtocycle-dependentval- The recursive least squares algorithm is applied in an on-
ues for lumped resistance Re, or even non-constant resistance line fashion, as parameters are updated continuously [20]
within a single cycle, because R can vary with conditions
e εϕ
suchastemperature,SOCanddegradation[5],[10],[11],[13]. θ˙ =P ; (3a)
In subsequent sections, first, R will be identified online as m2
e
ϕϕT
a constant under a drive cycle, and then identification of Re P˙=(cid:0)P P; (3b)
as a varying parameter will be addressed in Sec. (VII). Heat m2
generationcalculatedbasedonanequivalentcircuitmodelhas ε=z(cid:0)θTϕ; (3c)
also been used for thermal model parameterization in another m2=1+ϕTϕ; (3d)
ongoing work [18].
Heatexchangebetweenthecoreandthesurfaceismodeled wheremisanormalizationfactorthatenhancestherobustness
by heat conduction over a thermal resistance, R , which is of parameter identification.
c
IEEETRANSACTIONSONCONTROLSYSTEMTECHNOLOGY,VOL.X,NO.X,XX2012 3
Insomecases,tomaketheobservationzandtheregressors For the parametric model in Eq. (8),
ϕproper (or causal), a filter 1 will have to be designed and
Λ(s) s2T (cid:0)sT
applied. The parametric model will then become Z(s)= s s;0 (10a)
Λ(s)
z ϕ
Λ =θTΛ: (4) Φ(s)=[ I2 Tf(cid:0)Ts sTs(cid:0)Ts;0]T (10b)
Λ(s) Λ(s) Λ(s)
The convergence and robustness of the identification are θ=[α β γ]T; (10c)
guaranteediftheregressors,ϕinEq.(8),arestationarysignals
and satisfy the persistent excitation (PE) conditions [20]. The where
PEconditionsaresatisfiedifthereexistsometimeintervalT ,
0 R
and positive number α and α, such that α = e (11a)
1 0
C C R
∫ c s c
αI (cid:21)U(t)= 1 t+T0ϕ(τ)ϕT(τ)dτ(cid:21)αI 8t(cid:21)0; (5) β = 1 (11b)
1 M T0 t 0 M ( CcCsRcRu )
C +C 1
where IM is the identity matrix with the same dimension as γ=(cid:0) c s + : (11c)
C C R C R
U(t) [20]. This criteria can be used to test whether a drive c s c s u
cycle can ensure robust parameter convergence. For implementation in a practical system, the identification
algorithmisformulatedalongwithsignalszandϕinthetime
domainbasedonEq.(3),orinthediscretetimedomainbased
IV. ONLINEPARAMETERIZATIONOFTHEBATTERY
on equivalent formula. For example, z(t), whose Laplace
THERMALMODEL
transform is s2Ts(cid:0)sTs;0, can be obtained by calculating the
In this section, the parameterization scheme described pre- Λ(s)
convolution of T(t)(cid:0)T and the inverse Laplace transform
viously is applied to the cylindrical battery thermal model in s s;0
of s2 . In this way, calculation of the 2nd order derivative of
Eq. (1). A parametric model for identification can be derived Λ(s)
by taking the Laplace transformation of Eq. (1) and replacing Ts, s2Ts, which can be easily corrupted by noises, is avoided.
the unmeasured T with measured signals I, T , and T, ByusingtheparametricmodelinEq.(8),onlythreelumped
c f s
parameters, α, βand γ, can be identified under the condition
s2Ts(cid:0)sTs;0 = CcCResRcI2+(CcCs1RcRu(Tf(cid:0)Ts)+Cs1Rus(Tf)(cid:0)Ts) of persistent input excitation[20]. Prior knowledge of two of
(cid:0) 1 (C +C )sT (cid:0)C T (cid:0)C T ; (6) thephysicalparametersmustbeassumedsoastodeterminea
CcCsRc c s s s s;0 c c;0 setofuniquesolutionfortheoriginalfivephysicalparameters,
where T and T are the initial surface and core tempera- Cc,Cs, Re, Rc, and Ru from α, βand γ. Of the five physical
s;0 c;0
tures. When the initial core temperature, T , is considered to parameters, the internal resistance R may vary due to aging
c;0 e
be the same as the initial surface temperature, T , as if the and should be identified online. The conduction resistance R
s;0 c
battery starts from thermal equilibrium, Eq. (6) becomes isdifficulttoestimateasexplainedpreviously.Theconvection
resistanceR willbeinfluencedbythecoolantflowconditions
R 1 u
s2T (cid:0)sT = e I2+ (T (cid:0)T) around the cell depending on the packaging. Therefore, it is
s s;0 f s
C C R C C R R
c s c c s c u not easy to obtain prior knowledge of those three parameters.
C +C 1
(cid:0) c s(sTs(cid:0)Ts;0)+ s(Tf(cid:0)Ts): (7) The heat capacities Cc and Cs, which depend on the thermal
C C R C R
c s c s u properties and the mass of the rolled electrode assembly and
It is assumed here that T is regulated as a steady output of thecasing,arerelativelyconstantoverlifetime.Inaddition,the
f
the air-conditioning unit and thus sT =0, giving heat capacities will only affect the speed of transient response
f
of the model without having any impact on the steady state
s2T (cid:0)sT = Re I2+ 1 (T (cid:0)T) temperatures. Consequently, the heat capacitiesCc andCs are
s s;0 f s
(CcCsRc CcCsR)cRu selected to be the presumed parameters.
(cid:0) Cc+Cs + 1 (sT (cid:0)T ): (8) With Cc and Cs presumed and α, β and γ identified, Re,
C C R C R s s;0 R and R can be obtained by solving the following set of
c s c s u c u
equations:
If T is a time-varying input to the model, sT should not
f f
be dropped. In this case, Tf can also be used as an input β(Cc+Cs)CsRu2+γCsRu+1=0 (12a)
excitationintheparametricmodel.Asecondorderfiltershould 1
be applied to the observation and the regressors in Eq. (8) to Rc= βCC R (12b)
s c u
make them proper. The filter takes the form R =αC C R : (12c)
e c s c
1 1
= ; (9) The quadratic equation for R in Eq. (12) can lead to two
Λ(s) (s+λ)(s+λ) u
1 2 solutions,buttherightonecanbedecidedbasedonthecoolant
whereλ andλ arethetimeconstantsofthefilter.Thevalues flow conditions based on [21].
1 2
ofλ andλ canbechosentofilterthenoiseswithfrequencies The least squares algorithm in Eq. (3) can then be applied
1 2
higher than the temperature dynamics. forparameteridentification.In[19]and[22],themethodology
IEEETRANSACTIONSONCONTROLSYSTEMTECHNOLOGY,VOL.X,NO.X,XX2012 4
hasbeenappliedandverifiedbysimulationwithabatteryther-
malmodelwithassumedparameters.Inthefollowingsection,
the parameterization is further validated by experiments.
V. EXPERIMENTVALIDATION
A. Experiment Set-Up and Measurements
Experiments have been conducted to validate the de-
signed parameterization scheme. A 2.3Ah A123TM 26650
LiFePO /graphite battery is cycled with a BitrodeTM cycler
4
under the control of a customized testing system by A&D
TechnologyTM. A Cincinnati Sub-ZeroTM environmental sim-
ulation chamber is used to regulate the temperature of the
coolant air flow around the battery.
T-type thermocouples are installed both on the battery Fig.3. SchematicsoftheFlowChamber
casing to measure its surface temperature, and also inside
the battery core to measure the core temperature. During the
[23], hence the UAC current cycle taken from [23] is rescaled
fabrication process of the 26650 cylindrical cell, the electrode
for the experiments. The original 20-minute cycle is repeated
assembly is wound up to form a roll, leaving a cavity in the
4 times to let the battery temperature reach periodic steady
center.Tomeasurethecoretemperature,thebatterywasdrilled
state. The resulting scaled drive cycle current is plotted in
inside an argon-filled glove box through to its central cavity,
Fig. 4. The normalized unit of C-rate is commonly used to
wherethethermocouplewasinserted,asshowninFig.2.The
describe the load applied to the battery and 1 C corresponds
battery was then sealed and taken out of the glove box for
to the magnitude of the current that depletes the battery in
experiments.
one hour (in this case 2.3 A). The negative current indicates
the discharge of the battery as the energy is drawn from the
batterytodrivethevehicle,andthepositivecurrentrepresents
the regenerative braking during which the battery is charged.
Thedischargeloadisfairlyevenlydistributedbetween1Cand
7 C, except at around -8 C which indicates rapid acceleration.
The charge load is mostly below 7C and occasionally reaches
above 10 C during drastic braking. The SOC evolution under
this cycle is also plotted in Fig. 4, showing a decrease from
about 50% to roughly 35%.
ate) 20
C r
Fig. 2. Instrumentation of the Battery Core Temperature (left: drill press nt I ( 0
e
setupofthebattery;right:installationofthethermocouples) urr−20
C 1000 1500 2000 2500 3000 3500 4000 4500 5000
time (s)
Inside the thermal chamber, the battery was placed in a %)
e (20
designed flow chamber as shown in Fig. 3, where a fan was ag
nt10
mounted at one end to regulate the air flow around the cell. Perce 0
The speed of the fan is controlled by Pulse Width Modulation −10 −5 0 5 10 15
Current (C−rate)
(PWM) signals to change the air flow rate. The flow chamber 55
is used to emulate the pack air cooling conditions where the %)50
C (45
coolantflowrateisadjustable(likein[19]).AT-typethermo- SO40
couple is placed near the battery inside the flow chamber to 35
1000 1500 2000 2500 3000 3500 4000 4500 5000
measure the air flow temperature T . time (s)
f
B. Persistent Excitation of Input Signals Fig.4. ScaledUACCurrentExcitation(top:currentsintimeseries;middle:
histogramofthecurrents;bottom:SOCvariationunderthecycle)
A driving cycle, the Urban Assault Cycle (UAC) [23], is
appliedasthecurrentexcitationtothebatteryingalvanostatic The temperature of the thermal chamber is controlled at
mode. The UAC is originally a velocity cycle for military 26 oC. The resulting battery surface temperature T and air
s
vehicles. The current profile for a battery pack of a hybrid flow temperature T are measured and recorded by the data
f
military vehicle under UAC is derived in [23] by applying a acquisition system. The measured T and T under the scaled
s f
certain power management strategy. The type of battery used UAC cycles are plotted in Fig. 5, which along with I are then
intheexperiment(LiFePO 26650)isdifferentfromtheonein used for parameter identification.
4
IEEETRANSACTIONSONCONTROLSYSTEMTECHNOLOGY,VOL.X,NO.X,XX2012 5
x 10−4
32 3.5
oT (C)s2380 λmin2.35
26
2
1000 1500 2000 2500 3000 3500 4000 4500 5000 1000 1200 1400 1600 1800 2000 2200
27 0.09
26.5 0.08
oT (C)f 26 λmax0.07
25.5 0.06
25 0.05
1000 1500 2000 2500 3000 3500 4000 4500 5000 1000 1200 1400 1600 1800 2000 2200
time (s) time (s)
Fig. 5. Measured Ts and Tf under Scaled UAC Cycle (top: surface Fig.7. EvolutionoftheEigenvaluesofU(t)inSteadyState(Top:smallest
temperatureTs;bottom:flowtemperatureTf) eigenvalue;bottom:largesteigenvalue.)
TABLEI
INITIALGUESSANDIDENTIFICATIONRESULTSOF
The criteria in Eq. (5) is then applied to check if the
PARAMETERS
UAC cycle satisfies the PE condition, which requires the
regressors to be stationary signals first. As can be seen in Parameters Ru(KW(cid:0)1) Re(mΩ) Rc(KW(cid:0)1)
Fig. 5, the surface temperature Ts will vary periodically after InitialGuess 1:5 30 0:5
the battery finishes the warm-up phase at about 1000 second. IDResults 3:03 11:4 1:83
Consequently, the regressors, which include I2, T (cid:0)T, and
f s
sT, will become stationary signals, as shown in Fig. 6. The
s
U(t) matrix can then be calculated to check the persistent
the regressors satisfy the conditions of persistent excitation.
excitation conditions. It is noted that the measurements taken Furthermore, α are related to the speed of the convergence
0
during the warm-up period can also be used for identification,
for parameter identification. Specifically, when the gradient
even though they are not stationary signals [19]. method is used, 2α(cid:0)1 is the upper limits of the time constant
0
of the parameter convergence [20], which would be
1 τ(cid:20)8333s (13)
Λ
2φ I/10.5 in this case. Based on Eq. (13), the 90% settling time for the
convergence under the gradient search algorithm is expected
0
1000 1500 2000 2500 3000 3500 4000 4500 5000
to be less than 19186s. It is noted that 19186s is a rather
0
conservative estimation of the convergence time, in real ap-
φΛ/2−0.05 plication,theconvergenceisusuallyacceleratedbyincreasing
the adaptive gain [20], [24].
−0.1
1000 1500 2000 2500 3000 3500 4000 4500 5000
0.5
C. Results and Discussion
Λ
φ/3 0 The measured signals I, Ts and Tf in Fig. 4 and Fig. 5 are
used for recursive least squares parameterization. The three
−0.5
1000 1500 2000 2500 3000 3500 4000 4500 5000
time (s) parameters to be identified, Ru, Re and Rc, are initialized with
the initial guess values in Table I. For the heat capacity, the
Fig.6. EvolutionofRegressorsϕinPeriodicSteadyState single Cp in [8] is split into Cc and Cs here representing
the battery core and surface heat capacities respectively. The
Since the current input consists of repeated UAC cycles heat capacity of the battery core, C , is assumed to be 67
c
(each lasting for 1200s), the values of U(t) only need to be JK(cid:0)1,slightlysmallerthanC in[8].Theheatcapacityofthe
p
calculated over a time interval T =1200s for 1000s(cid:20)t (cid:20) battery surface, C , is assumed to be 4:5 JK(cid:0)1 based on the
0 s
2200s.Itisnotedthatinthiscase,U(t)isnotadiagonalmatrix, dimensions of the aluminum casing of the 26650 battery and
and thus its eigenvalues are calculated to check the persistent the specific heat capacity of aluminum.
excitationconditions.Thesmallestandthelargesteigenvalues The results of the recursive identification are plotted in
ofU(t),λ andλ ,areplottedinFig.7.Itcanbeconcluded Fig. 8. It is noted that the identification procedures are started
max min
fromFig.7thatα inEq.(5)canbefoundas0:086s(cid:0)1,which after the first 1000 seconds when the temperature enters peri-
1
isthemaximumofλ (t),andα as2:4(cid:2)10(cid:0)4 s(cid:0)1,whichis odic steady state. It can be seen that starting at some random
max 0
theminimumofλ (t). Consequently,undertheUACcycle, initialvalues,the3parametersconvergetothevalueslistedin
min
IEEETRANSACTIONSONCONTROLSYSTEMTECHNOLOGY,VOL.X,NO.X,XX2012 6
TABLEII
TableI.TheupperplotinFig.8showstheconvergenceofthe
COMPARISONOFTHEIDENTIFIEDPARAMTERSTO[8]
lumped parameters α, β and γin Eq. (8), and the lower plot
shows the convergence of the physical parameters R , R and
u c Parameters Value Equivalencein[8] Value
Re, which are obtained by solving Eq. (12). It is noted that Rc(KW(cid:0)1) 1:83 Rin(KW(cid:0)1) 3:2(cid:24)3:4
the convergence time is within the range (less than 19186 s) Ru(KW(cid:0)1) 3:03 Rout(KW(cid:0)1) 8:4(cid:24)9:1
discussed in Sec. (V-B), which is strictly speaking only valid Cc(JK(cid:0)1) 67 Cp(JK(cid:0)1) 73(cid:24)78
for the gradient method. The convergence rate is accelerated Cs(JK(cid:0)1) 4:5 - -
herebyincreasingtheinitialadaptivegainP [24],[25],which
0
is the initial value of P(t) in Eq. (3).
core temperatures in real time without actually measuring it
(as in the lab set-up).
s 6
er α The identification results are also compared to those in [8],
amet 4 βγ where thermal parameters of the same battery are identified
Par 2 based on the measurement of both surface and the core
d
pe 0 temperatures under designed current inputs. In [8], the battery
m
u ismodeledwithasingledynamicstate(thecoretemperature),
L−2
1000 1500 2000 2500 3000 3500 4000 4500 5000 and the surface temperature is related to the core temperature
with an algebraic equation by assuming the surface heat
50
capacitytobezero.In[8],theheatgenerationispre-calculated
ers40 Ru (0.1KW−1) by resistive heat dissipation (due to ohmic voltage drop) plus
met30 entropic heat, and in this work, the entropic heat is ignored
a20 Rc (0.1KW−1)
Par Re (mΩ) and the heat generation is accounted for by multiplying the
10
current square with an identified parameter R . It is noted that
e
10000 1500 2000 2500 3000 3500 4000 4500 5000 the entropic heat is generally small comparing to the resistive
time (s)
heat, especially in the middle SOC range here as shown in
Fig. 4.
Fig. 8. On-line Parameter Identification Results (top: convergence of the
Table II summarizes the comparison between the thermal
lumpedparameters;convergenceoftheoriginalparameters)
parametersidentifiedin[8]andinthispaper.Itcanbeseenthat
Forvalidationpurpose,theidentifiedparametersareapplied the identified value of the conduction resistance (Rc) between
to Eq. (1) to estimate both the battery surface temperature T the core and the surface is smaller than that in [8]. This
s
and the core temperature T . The estimation is then compared is probably because the surface temperature in this work is
c
withthemeasurement,asplottedinFig.9.Theestimatedsur- measured at the aluminum casing instead of at the outside
papercover(asin[8]),whichindicatesbetterheatconduction.
The identified convection resistance between the surface and
32
the coolant R is significantly smaller than that in [8], which
u
can be explained by the fact that during the experiment, the
30
C)
oT (s28 Measured Ts awirhiflchowenihsacnocnesstathnetlycobnlvoewctnivientcooothliengflothwrocuhgahmtbheercboyoltahnetfaairn.
Estimated T
s
26
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
t (s)
VI. ADAPTIVEBATTERYCORETEMPERATURE
35
ESTIMATION
oT (C)c30 Measured T onIna cpolanntrtoml oapdpellictoatieosnti,maanteobthseervstearteiss ooffteanpdlaensti,gneespdebciaasleldy
c
Estimated Tc ones that are not measured, e.g. the core temperature Tc of
25
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 the battery in this case. Such model based observers can be
t (s)
categorized as either an open loop observer or a closed loop
observer. For a linear system
Fig. 9. Experimental Validation (top: estimated surface temperature Ts vs.
measured;bottom:estimatedcoretemperatureTc vs.measured)
x˙=Ax+Bu (14)
face temperatures T match the measurement exactly, since T
s s
isdirectlyusedforidentification.Itisnotedthatthemeasured where x are the states and u are the inputs, an open loop
core temperature T also agrees closely with the measured observer is simply
c
Tc (which was not used for parameterization), showing the x˙ˆ=Axˆ+Bu; (15)
capability of the parameterized model to predict the correct
battery core temperatures. Once the parameterization scheme as the estimated states xˆ are calculated by the model solely
is validated, it can be run in onboard BMS to estimate the based on the inputs u. For the battery thermal model specifi-
IEEETRANSACTIONSONCONTROLSYSTEMTECHNOLOGY,VOL.X,NO.X,XX2012 7
cally, we have
x=[T T]T (16a)
c s
u=[I2 T ]T (16b)
[ f ]
(cid:0) 1 1
A= RcCc RcCc (16c)
1 (cid:0)1( 1 + 1 )
RcC[s Cs Rc ] Ru
ReRc 0
B= Cc (16d)
0 1
RuCs
However,theestimationbysuchopenloopobservercanoften
be corrupted by unknown initial conditions, and noises in the
measurement of the inputs. To address such issues, a closed
loop observer, such as a Luenberger observer or a Kalman Fig.10. On-lineIdentificationSchemeandAdaptiveObserverStructure
filter, is often designed to estimate the states based on the
model and the feedback of some measurable outputs [26],
The data in Sec. (V) are used to test the response of the
x˙ˆ=Axˆ+Bu+L(y(cid:0)yˆ) (17a) adaptive observer, as plotted in Fig. 11. The initial estimated
y=Cx+Du (17b) temperaturesoftheadaptiveobserveraresetat30oC forboth
the surface and the core, whereas the correct value is 26 oC,
yˆ=Cxˆ+Du (17c)
and the parameters are initialized with the initial guess values
whereyarethemeasuredsystemoutputs,xˆandyˆareestimated in Table I. It can be seen from Fig. 11 that the estimated
states and output, L is the observer gain, and A, B, C and D surface temperature Ts converges to the actual values much
are model parameters. For the battery thermal model, since faster than the core temperature Tc. The reason is that the sur-
the surface temperature Ts is measurable, we have facetemperatureTs isaccessiblebytheadaptiveobserverboth
via parameter identification and closed loop error feedback,
C=[0 1] (18a) and thus the observer can adjust its estimation of T quickly
s
D= 0: (18b) basedondirectreferenceofthemeasurement.Butforthecore
temperatureT ,whichisnotmeasured,itsestimationaccuracy
c
It is noted that the difference between the measured and
depends on the accuracy of the model parameters. Therefore,
the estimated output is used as the feedback to correct the
the convergence of T to the actual values will only happen
c
estimated states. Comparing with an open loop observer, the
after the identified parameters converge to the correct model
closed loop observer can accelerate the convergence of the
parameters (at approximately 3000 seconds).
estimated states to those of the real plant under unknown
initialconditions,e.g.aLuenbergerobserver[26],oroptimize
the estimation by balancing the effect of unknown initial 32
conditions and noises, e.g. a Kalman filter [27].
30
By taking the structure of a closed loop observer, an adap- C)
tive observer is then designed based on certainty equivalence oT (s28 Measured
principle [20], start of identification Estimated
26
C T˙ˆ = I2Rˆ +Tˆs(cid:0)Tˆc +l (T (cid:0)Tˆ) (19a) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
c c e Rˆ 1 s s
c 36
C T˙ˆ =Tf(cid:0)Tˆs (cid:0)Tˆs(cid:0)Tˆc +l (T (cid:0)Tˆ); (19b) 34
s s Rˆu Rˆc 2 s s C) 32
where Tˆ and Tˆ are the estimated surface and core temper- oT (c30 Measured
s c
atures, and the observer parameters Rˆ , Rˆ and Rˆ are taken 28 Estimated
e c u
26
from the online identification results in Sec. (V). The block 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
t (s)
diagram of the adaptive observer is shown in Fig. 10. The
input current I, coolant temperature T , and the measured
f Fig. 11. Response of the Closed loop Adaptive Observer (top: adaptive
surfacecelltemperatureT arefedintotheparameteridentifier
s estimation of the surface temperature vs. measurement; bottom: adaptive
to estimate model parameters R , R and R . The adaptive estimationofthecoretemperaturevs.measurement)
u e c
observer uses the estimated parameters to estimate the core
and the surface temperatures. The estimated T is compared
s
to the measurement and the error is fed back to correct VII. PARAMETERIZATIONOFTHEBATTERYTHERMAL
the core temperature and surface temperature estimation. The MODELWITHTEMPERATUREDEPENDENTRe
estimations for both parameters and temperatures are updated For most lithium ion batteries, their internal resistance R
e
at each time step. depends on temperature and SOC, [5], [6], [11]. In general
IEEETRANSACTIONSONCONTROLSYSTEMTECHNOLOGY,VOL.X,NO.X,XX2012 8
cases, R is high when the temperatures are lowand when the Since the least squares identification algorithm in Eq. (3)
e
SOC is close to 0% or 100%. An Arrhenius function is often identifies each parameter as a constant, when R is varying,
e
used to describe the relationship between R and the battery errorswillbeobservedinR identificationasshowninFig.13.
e e
(core) temperature T , as This will not only introduce errors in R estimation but might
c ( ) e
T also affect the estimation of other parameters, and eventually
ref
Re=Re;refexp T ; (20) corrupt the estimation of the core temperature Tc. To address
c
such issue, a least squares algorithm with forgetting factors is
where R is the reference resistance value at a certain
e;ref then designed to identify R as a time-varying parameter.
e
reference temperature T , and T and T are in K. It is
ref ref c
noted that the change in resistance with respect to SOC
A. Identification Design with Forgetting Factor
is negligible in the normal vehicle battery operating range
(20%(cid:0)80% SOC) and thus is not considered here. The When forgetting factors are adopted, most parts of the least
relationshipbetweenR andT describedbyEq.(20)isplotted square algorithm will be the same as Eq. (3), except that
e s
in Fig. 12, by taking R =0:091 mΩ and T =1543 K.
e;ref ref ϕ(t)ϕT(t)
P˙(t)=ηTP(t)η(cid:0)P(t) P(t); (21)
m2(t)
30
where ηis the forgetting factor matrix [20].
The least square identification algorithm tries to find the
25
optimal parameters that best fit the inputs and outputs over
Ω) 20 the whole data set. A pure least square algorithm treats
m each data point as equal, no matter if it is acquired most
R (e15 recently,orobtainedmuchearlier.However,whenaforgetting
factor is applied, the data points will be weighted differently.
10 Specifically,thenewlyacquireddataarefavoredovertheolder
ones.IntheformshowninEq.(21),theweightofthedatawill
50 10 20 30 40 50 60 70 decay exponentially with the time elapsed, and the larger the
Tc (oC) forgettingfactoris,thefasterthedecaywillbe.Consequently,
the least square algorithm can track the parameters when they
Fig.12. DependenceofRe onTc are time-varying.
As a result, in real application, R will be varying as the The least square algorithm with forgetting factors can be
e
temperature fluctuates. Such variation can not be neglected applied to the original linear parametric model in Eq. (7). Of
when the power demands are high and dramatically varying. the three lumped parameters, namely α, β, and γin Eq. (7),
Simulation is used in this section for illustration. Simulated only α is related to time varying Re, and all the others are
variation of R due to T fluctuation under a drastic current constant. Therefore, non-uniform forgetting factors should be
e c
cycleisshowninFig.13.Itcanbeseenthatthedrasticcurrent adopted with the ηmatrix designed as
 
η 0 0
1
η=0 0 0; (22)
20
e) 0 0 0
at
C−r 0 whereη istheforgettingfactorassociatedwithα(andhence
I ( 1
−20 Re).
0 1000 2000 3000 4000 5000 Simulation is conducted with η =0:25, and the results of
60 1
identification are shown in Fig. 14. It can be seen that that
oT (C)c40 tthheeriedceunrtsiifiveedleRaestcsaqnuafroelsloownlitnheeidseimntuifilactaetdiovnawryiitnhgfoRrgeetatfitnegr
200 1000 2000 3000 4000 5000 factorsisactivatedat1500s.AsshowninFig.15,theadaptive
Simulated observer, taking the structure in Eq. (19) and parameters
14 Identified
Ω) 12 identified online (now Re varying as shown in the bottom
m
R (e108 plot of Fig. 14), can estimate the battery core temperature Tc
accurately after the identified R converges to the simulated
6 e
0 1000 2000 3000 4000 5000 R at around 3700s.
t (s) e
Fig.13. ErrorsinRe EstimationwhentheTemperatureVariesSignificantly VIII. DEGRADATIONDETECTIONBYMONITORING
(top:drivecyclecurrent;middle:fluctuationofthebatterycoretemperature;
GROWTHININTERNALRESISTANCE
bottom:errorsinRe identification)
The recursive least square algorithm with forgetting factors
variation creates a 10 oC of fluctuation in the battery core can also track the long term growth of the internal resistance,
temperature T . The resulting variation of R is about 20% as which can be used as an indication for the state of health
c e
shown by the blue line in the bottom plot of Fig. 13. (SOH) of the battery.