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Robust Estimation of Lithium Battery State of Charge with Random Missing Current Measurement Data

Authors:
Xi Li
Xi Li
Xi Li
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Zongsheng Zheng at Sichuan University
Zongsheng Zheng
Jinhao Meng at Xi'an Jiaotong University
Jinhao Meng
Qinling Wang at Southeast University
Qinling Wang
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Abstract and Figures

The precise estimation of the state of charge (SOC) in lithium batteries is crucial for enhancing their operational lifespan. To address the issue of reduced accuracy in SOC estimation caused by the random missing values of lithium battery current measurements, a joint estimation method which combines recursive least squares with missing input data (MIDRLS) and the unscented Kalman filter (UKF) algorithm is proposed, called the MIDRLS-UKF algorithm. Firstly, the equivalent circuit model of a Thevenin battery is formulated. Then, the current imputation model is designed to interpolate the missing data, based on which the MIDRLS algorithm is derived by solving the unbiased estimation of the gradient of the objective function, thus realizing the online high-precision identification of the circuit model parameters. Furthermore, the proposed algorithm is combined with the UKF algorithm to facilitate the online precise estimation of SOC. The simulation results indicate a marked decrease in the SOC estimation error when employing the proposed joint algorithm, as opposed to the conventional forgetting factor recursive least squares (FFRLS) algorithm combined with the UKF joint estimation algorithm, which verifies the precision and effectiveness of the proposed joint algorithm.
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Electronics2024,13,4436.https://doi.org/10.3390/electronics13224436www.mdpi.com/journal/electronics
Article
RobustEstimationofLithiumBaeryStateofChargewith
RandomMissingCurrentMeasurementData
XiLi
1
,ZongshengZheng
1,
*,JinhaoMeng
2
andQinlingWang
3
1
CollegeofElectricalEngineering,SichuanUniversity,Chengdu610065,China;
sugar76@stu.scu.edu.cn
2
SchoolofElectricalEngineering,Xi’anJiaotongUniversity,Xi’an710049,China;jinhao@xjtu.edu.cn
3
SchoolofElectricalEngineering,SoutheastUniversity,Nanjing210096,China;qlingwang@seu.edu.cn
*Correspondence:zongshengzheng@scu.edu.cn;Te l.:+86-15281063498
Abstract:Thepreciseestimationofthestateofcharge(SOC)inlithiumbaeriesiscrucialfor
enhancingtheiroperationallifespan.ToaddresstheissueofreducedaccuracyinSOCestimation
causedbytherandommissingvaluesoflithiumbaerycurrentmeasurements,ajointestimation
methodwhichcombinesrecursiveleastsquareswithmissinginputdata(MIDRLS)andthe
unscentedKalmanfilter(UKF)algorithmisproposed,calledtheMIDRLS-UKFalgorithm.Firstly,
theequivalentcircuitmodelofaTheveninbaeryisformulated.Then,thecurrentimputation
modelisdesignedtointerpolatethemissingdata,basedonwhichtheMIDRLSalgorithmisderived
bysolvingtheunbiasedestimationofthegradientoftheobjectivefunction,thusrealizingtheonline
high-precisionidentificationofthecircuitmodelparameters.Furthermore,theproposedalgorithm
iscombinedwiththeUKFalgorithmtofacilitatetheonlinepreciseestimationofSOC.The
simulationresultsindicateamarkeddecreaseintheSOCestimationerrorwhenemployingthe
proposedjointalgorithm,asopposedtotheconventionalforgeingfactorrecursiveleastsquares
(FFRLS)algorithmcombinedwiththeUKFjointestimationalgorithm,whichverifiestheprecision
andeffectivenessoftheproposedjointalgorithm.
Keywords:lithiumbaery;stateofchargeestimation;randommissingcurrentdata;model
parameteridentification;unscentedKalmanfiltering
1.Introduction
Overthepastfewyears,inordertopromotesustainableenergydevelopmentand
thedevelopmentofagreeneconomy,theelectricvehicleindustry,whichhasapositive
impactonreducingurbancarbonemissions,hasdevelopedrapidly[1].Lithiumbaeries
havebecomeawidelyusedbaerytechnologyintheemergingelectricvehiclesector[2]
duetotheirelevatedenergyandpowerdensities,extendedservicelife,lowself-discharge
andsuperiorenergyconversionefficiency[3].Theseaributesrenderlithiumbaeriesan
idealchoiceforpoweringthenextgenerationofelectricvehicles,aligningwiththeglobal
shifttowardscleanerandmoresustainabletransportationoptions.Itisthereforeessential
tomonitorbaerystatususinganeffectiveandprecisebaerymanagementsystem(BMS)
[4].AmongthevariousfunctionalitiesofaBMS,theestimationofthestateofcharge(SOC)
standsoutasparticularlycrucial.TheaccuratecompletionofSOCestimationcanshow
theremainingcapacityofthebaery,whichisessentialforpreventingoverchargingand
overdischargingscenarios.Bydoingso,itensuresthereliableoperationofelectric
vehicles,optimizesthebaery’sperformance,andsignificantlyextendsitsoperational
lifespan[5].
SOCestimationmethodscanbebroadlycategorizedintofourgroups:methods
basedonthephysicalcharacteristicsofthebaery,methodsbasedonbaerymodels,
data-drivenapproachesandfusionmethods[6].Amongthem,themethodsbasedonthe
physicalcharacteristicsofbaeries,includingtheopen-circuitvoltagemethod,the
Citation:Li,X.;Zheng,Z.;Meng,J.;
Wang,Q.RobustEstimationof
LithiumBaeryStateofChargewith
RandomMissingCurrent
MeasurementData.Electronics2024,
13,4436.hps://doi.org/
10.3390/electronics13224436
AcademicEditor:FabioCorti
Received:2October2024
Revised:6November2024
Accepted:11November2024
Published:12November2024
Copyright:©2024bytheauthors.
Submiedforpossibleopenaccess
publicationunderthetermsand
conditionsoftheCreativeCommons
Aribution(CCBY)license
(hps://creativecommons.org/license
s/by/4.0/).
Electronics2024,13,44362of15
internalresistancemethodandthealternatingcurrentimpedancemethod,facechallenges
inachievingtheaccurateestimationofSOConline.Thesechallengesarisefromthelonger
experimentaltime,higherrequirementsforexperimentalconditionsandlargererrors[7].
Thedata-drivenapproachtoSOCestimationiscenteredontheutilizationofmachine
learningalgorithmstoprocessextensivedatasetsofmeasurementsamplesandtodiscern
underlyingpaernsandfeatureswithinthedata.Itharnessesthecomputational
capabilitiesofmachinelearningtoanalyzeandlearnfromthecomplexrelationships
presentinlargevolumesofbaeryperformancedata,therebyenablingtheestimationof
SOC.Therearethreepredominanttypesofdeepneuralnetworksthatarefrequently
utilizedinresearch,includingfullyconnectedneuralnetworks,recurrentneuralnetworks
andconvolutionalneuralnetworks[8].Inaddition,inthelastfewyears,avarietyofother
neuralnetworkshavebeendevelopedbyresearcherstoenhancetheprecisionand
applicabilityofSOCestimation.Forinstance,arandomsearchoptimization-basedLong
Short-TermMemory(RS-LSTM)neuralnetworkwasproposedforpreciseSOCestimation
[9],whichisbasedontheCALCEdatasetandarandomsearchalgorithmtooptimizeits
performance.Althoughthiskindofmethodhashighflexibilitywithhighestimation
accuracy,itiscontingentuponthequalityofthedataset,entailssubstantialtime
investmentsformodeltraining,andisoftencharacterizedbylimitedrobustness[10].In
termsoffusionmethods,theresearchersin[11]fusedneuralnetworksandequivalent
circuitmodels,thusachievingtheaccurateestimationofSOCoverawiderangeof
temperatureconditions.Inaddition,theresearchersin[12]achievedlightweightSOC
estimationthroughEISdataandtheequivalentcircuitmodel,whichdonotrequiretime-
consumingmodeltraining.
Consideringtheaccuracy,cost,real-timecapabilitiesandotherfactorsofvarious
estimationtechniques,thecurrentbaerymodel-basedmethodhasabeerprospectof
applicationanddevelopmentpotential[13].Themethodthatemploysanequivalent
circuitmodel haslowcomputationaldemands,makingitparticularlyamenableto
embeddedsystemsdesignedforonlineSOCestimation.Baerymodel-basedapproaches
performSOCestimationbyidentifyingtheparametersofthecircuitelementswithinthe
baery’sequivalentcircuitmodel,usuallycombinedwithanadaptivefilteringalgorithm
toemulatethebaery’sdynamicstatecharacteristics[14],suchasanextendedKalman
filter(EKF),anunscentedKalmanfilter(UKF)anditsimprovedalgorithms[15].
Furthermore,theresearchersin[16]proposedahierarchicaladaptiveextendedKalman
filter(HAEKF)algorithm,whichinvolvesthedissectionofthesecond-orderRCcircuit’s
stateequationmodelinaccordancewiththeprinciplesofhierarchicalidentification.
Currently,toenhancethetimelinessandprecisionofonlinemodelparameterestimation
andtoaddresstheissueofoutdateddataaccumulationiniterativeprocesses[17],the
forgeingfactorrecursiveleastsquare(FFRLS)algorithmisoftenusedtoidentifymodel
parameters.However,itisonlyapplicableiftheinputdataarecomplete,i.e.,thereal-time
loadcurrentmeasurementsarefullyobtained.Shouldthecurrentdetectionmoduleinthe
BMSbefaulty,oriftheconnectorislooseorenvironmentalvibrationsoccur,causingthe
currentmeasurementvaluetobemissing,thiswillleadtoalargeerrorintheresultsof
parameteridentification.TheseerrorscanfurtheraffecttheaccuracyofSOCestimation,
potentiallyleadingtoimproperbaeryusageandjeopardizingthebaery’shealth.
Toensurethehighreliabilityofonlinebaerymodelparameteridentificationdespite
thepresenceofmissingcurrentmeasurementdata,inthispaper,weinnovativelydesign
acurrentimputationmodeltointerpolatethemissingvaluesofthecurrent.Basedonthe
imputationmodel,wederivearecursiveleastsquareswithmissinginputdata(MIDRLS)
algorithmtoidentifythemodelparametersbysolvingtheunbiasedestimationofthe
gradientoftheobjectivefunction.CombiningitwiththeUKFalgorithm,weachieve
onlineSOCestimationbyusingupdatedmodelparametersandsystemstatesateach
cycle.
Theremainderofthispaperisorganizedasfollows:Section2describesthederivation
oftheMIDRLSalgorithmandthemodelparameteridentificationprocess.Section3
Electronics2024,13,44363of15
describestheSOCestimationprocessbasedontheUKFalgorithmandpresentsthe
derivationoftheMIDRLSalgorithminconjunctionwiththeUKFtoperformSOC
estimationinrealtime.InSection4,simulationworksareconductedtoevaluatethe
performanceoftheproposedalgorithm.Finally,Section5presentsthediscussionand
conclusionofthepaper.
2.BaeryModelandParameterIdentification
2.1.BaeryModel
Equivalentbaerymodelsrepresenttheinternalstateanddynamiccharacteristicsof
abaerybyforminganequivalentcircuitfromidealcircuitelements.Prevalentmodels
includetheRintmodel,Theveninmodel,DPmodel,PNGVmodel,second-orderRC
model,etc.[18].TheTheveninmodeltakesintoaccounttheelectrochemicalreaction
insidethebaery,whichcanreflectthepolarizationcharacteristicsofthebaery[19].It
canreflecttheactualworkingcharacteristicsofreallithiumbaeries,soitissuitablefor
simulatingtheperformanceoflithiumbaeries.However,itsstructureisrelativelysimple
andthemodelparametersarefixedvalues,sotheaccuracyisnottoohigh.Toprecisely
reflectthedynamiccharacteristicsoflithiumbaeries,theTheveninmodel[20]was
selectedforthestudyinthispaper,asshowninFigure1.
AsillustratedinFigure1,thereisanidealvoltagesourceocv
U,whichdenotesthe
electricalpotentialofthebaeryandhasaone-to-onecorrespondencewiththeSOCof
thebaery[21].Inaddition,itincludesabaeryequivalentinternalresistance0
R
,a
polarizationresistance1
R
andapolarizationcapacitor1
C.d
Urepresentsthebaery
terminalvoltageand0
I
representstheloadcurrent.Therefore,0
R
,1
R
and1
Carethe
parametersthatneedtobeidentified.
ThemodelequationscanbeconstructedfromKirchhoff’slaw:
11 0 1
00
///
cc
docvc
dU dt U R C I C
UU UIR


 (1)
ocv
U
d
U
0
R
1
R
1
C
0
I
c
U
Figure1.Theveninbaerymodel.
Thevaluesofthemodelparameters0
R
,1
R
and1
Careinfluencedbyfactorssuch
asenvironmentalchanges,theoperationalconditionsofthebaeryandtheagingprocess
ofthebaery[22].ThispaperemploystheproposedMIDRLSalgorithmformodel
parameteridentification.Itprovidesreal-timeidentificationoftheparametersevenwhen
theloadcurrentmeasurementsarerandomlylost,thusavoidingundesirableeffectson
theSOCestimationresults.
Electronics2024,13,44364of15
2.2.TheProposedMIDRLSAlgorithm
Toenablethecapabilitytoaccuratelyidentifythemodelparametersdespitethe
presenceofmissingmodelinputcurrentdata,anewparameteridentificationalgorithm
isproposedinthispaper.
Itispositedthattheacquiredinputdatawithrandomlosscanbemodeledas
() ()()
ic
x
ngnxn, (2)
where()
g
n isarandomvariableobeyingBernoulli’sindependenthomogeneous
distribution,whichisindependentof()
x
n.Theprobabilityof()
ntaking1or0is
p
or1p,respectively.
Weinterpolatedthemissingdata,i.e.,themissingdatawereresettotheconstant
timesofthedataavailableatthepreviousmoment.Theprocesscanbedescribedas
() ()() (1 ())( 1)xn gnxn gn xn


, (3)
where()
x
n
representstheimputeddata.
TheobjectivefunctionoftheFFRLSalgorithmistheweightederrorsumofsquares[23]:
2
0
( ) () () ( )
n
ni T
i
J
ndiiwn

x (4)
where
istheforgeingfactor,()di istheoutputdataatmomentiand()wn isthe
parametervectortobeidentified.
Topreventtheparameterestimationupdatesequencefromfailingtoconvergetothe
objectivefunction’sminimumduetothemissingdata,weusedtheimputeddatasequence
{()}
x
ntoconstructanunbiasedestimationofthegradientoftheobjectivefunction.
Considering
0
( ) 2 ()[ () () ( )]
n
ni T
i
J
nxipdiiwn

x, (5)
weneedtoobtainanexpressionfor()
J
nthrough[()]
J
n.Aftersomemathematical
derivation,oneobtains
,
0
2
0
,1 1,1
2
,1,1
[ ()] 2 [ () (()) () ]
( ) 2 [ (1 ) ( ) ( ( 1))
(1 )( (1 ) ) ( )
(1 ) (
n
ni
ii
i
n
ni
i
ii i i
ii i i
Jn pdi xi wnR
pJn p pdi xi
pY pR wn
pdiagR R







 




,1
)()].
ii
Ywn
 (6)
where,(() ())

T
ii
R
xix iand,1 ,1 1,


ii ii i i
YRR
.
Leing() () (1 ) ( 1)

xi xi pxi
 and() () ( 1)x i xi xi


,respectively,andin
combinationwithEquation(6),onecanobtain
2
0
0
2
() - [ ( () () ()) ()]
+(1 ) [ ( ( ) ( )) ( )]}
n
ni T
i
n
ni T
i
J
npdixiwnxi
p
p
diag x i x i w n


 (7)
Whensolvingfor()wn ,theupdateprocessoftheMIDRLSalgorithmcanbeobtained
byrecursionandmatrixinversionlemmas,aspresentedinAlgorithm1.

Electronics2024,13,44365of15
Algorithm1ThealgorithmflowoftheproposedMIDRLS
Input:thedatasequencewithrandommissing{()}
ic
x
n
,
theobtainedoutputsequence
{()}dn ,0
,(0,1)
,[0,1]p.
Initialize:(0) 0w,(0)
M
I
,
isapreysmallnumber,theimputeddata
sequence{()}
x
n
.
Update:-1 -1 1
() (1)( (1))
nn
nMnIXMn X


 () ( ()) ( 1) ( ()) ( 1)()()
p
wn I n wn I n M n d nx n


 -1
() [ ( 1) () ( 1)]Mn Mn nMn


where() () (1 )( 1)xn xn pxn


 () () ( 1)x n xn xn


 () () (1 ) ( () ())
TT
n
X
xnx n pdiagx nx n


EndFor
Theforgeingfactorallowstheweightofpreviousdatainparameteridentification
tobecontinuouslyreducedoverlongperiodsoftimeinsteadofaccumulatingovertime.
Constructingtheunbiasedestimationreducestheerrorinidentificationresultscausedby
therandomlossofcurrentmeasurements.
2.3.ParameterIdentificationoftheBaeryModel
ToidentifytheparametersofthebaerymodelthroughtheMIDRLSalgorithm,after
rewritingEquation(1)indiscreteformandleing,,

kdkocvk
EU U ,onecanobtainthe
differentialequationsforthemodel
11 20, 30,1kk k k
E
EII


  (8)
,,0,0,dk ocvk k ck
UU IRU
,(9)
where
1
20
310
(1)
R
R
R




. (10)
0
Tisthesamplingtime,011
exp( / )TRC
.
Therefore,thebaerymodelparameterscanbeobtainedas
12
01231
1101123
()/(1)
(1 ) / l n( ) ( )
R
R
CT





. (11)
()k
dk E istheoutputdataatmomentk,0,
()
ic k
x
kI
 istheinputdatawith
randomlossand10, 0,1
() [ ]
T
kkk
xk E I I


istheimputedinputvectoratmomentk.Thus,
thecoefficientvector12 3
() [ ]wn

  canbeidentifiedrecursivelythroughthe
MIDRLSalgorithmandthebaerymodelparametersatmomentkcanbecalculated
fromEquation(11).

Electronics2024,13,44366of15
3.SOCEstimationBasedontheJointMIDRLS‐UKFAlgorithm
3.1.EquationsofStateandMeasurementfortheBaeryModel
TheSOCofthelithiumbaery istypicallycalculatedastheproportionofthe
remainingcapacityrelativetothebaery’sactualmaximumcapacity[24],andtheSOC
canbeestimatedusingtheampere-timeintegrationmethod
0
0
0
1
() (0) ()
t
SOC t SOC I t dt
Q

 (12)
where(0)SOC isthestartingpointvalueofSOC,0
Qisthebaery’smaximumcapacity
and
 istheCoulombicefficiencywhichdescribestheratioofdischargecapacity
(mAh/g)tochargecapacity(mAh/g).
AfterdiscretizingEquation(12)andcombiningEquations(8)and(9),thestateand
measurementequationsofthebaerymodelcanbewrienas
1
10,1
00
(1 )
0
/01
kk kk
R
x
xIw
TQ






 
 (13)
,0,0,
()
dk k k ck k
UfSOCIRUv
, (14)
where,
(, )
T
kck k
xUSOC isthestatevariable,()
k
f
SOC  expressesthecorrespondence
betweenocv
U andk
SOC  andk
w andk
v aretheprocessnoiseandmeasurement
noiseofthesystem,respectively.
Thiscanbefurthersimplifiedandexpressedasafunctionalrelationship:
11
(,)
(, )
kkkk
kkkk
x
gx u w
yhxu v



. (15)
3.2.SOCEstimationBasedontheUKFAlgorithm
TheKalmanfilteralgorithmisanalgorithmfortheoptimalstateestimationofalinear
system,whereas()
k
f
SOC  isanonlinearfunctionalrelationshipinthemeasurement
equationofthebaerymodel.
TheEKFalgorithmlinearizesthesystemusingaTaylor expansion.TheUKF
algorithmlinearizesthenonlinearsystemthroughatracelesstransformation[25],which
approximatestheprobabilitydistributionofthesystem’sstatevariablesbyobtainingaset
ofsigmasamplingpointsaroundtheinitialestimate,thusobtainingthemeanand
varianceoftheestimatedquantity.Inthisprocess,wedonotneedtoderiveanditeratively
computetheJacobimatrix,whichimprovestheestimationaccuracyandreducesthe
computationalcomplexity[26].
ThedetailedimplementationstepsoftheUKFalgorithmareoutlinedbelow:
(1) Initialize:
0
0
00 0
0
0
ˆ
ˆˆ
()
[( )( ) ]
T
x
Px x
x
xx

 (16)
where0
ˆ
x
 istheestimatedvalueoftheinitialstateand0
P istheinitialerror
covariancematrix.
(2) Obtain21LSigmapointsattime1k:
Electronics2024,13,44367of15
,1 1
,1 1 1
,1 1 1
ˆ,0
ˆ( ( ) ) , 1,2,...,
ˆ( ( ) ) , 1,...,2
ik k
ik k k i
ik k k i L
xxi
xx LPi L
x
xLPiLL






 (17)
whereListhedimensionofthestatevariable,2()LL

,and
servesto
regulatetheproximityofthesamplingpointtothemeanvalue,generally
2
10 1

.
needstomeet0
.
Theweightingfactorsare
2
,0
1,0
1, 1, 2,3, ..., 2
2( )
m
i
c
i
mc
ii
i
L
i
L
iL
L





 (18)
where2
forGaussiandistributedvariables,m
i
istheweightedvalueofthemean
ofthesamplingpointsandc
i
istheweightedvalueoftheerrorcovariancematricesat
thesamplingpoints.
(3) Calculatetheforecastedvaluesofthemeanandcovariancematrix:
,/ 1 , 1 1
( , ), 0,1,..., 2
ik k ik k
x
gx u i L


 (19)
2
,/ 1
0
ˆ,0,1,...,2
L
m
kiikk
i
x
xi L

 (20)
2
/1 /1 /1
0
L
cT
kk i kk kk
i
PAA

 (21)
where,/ 1ik k
x and/1kk
P arethepredictedvalueofthestatevectorandtheerror
covariancematrixatthenextmomentbasedonthemoment1k,respectively,and
ˆk
x
 istheestimatedvalueofthestatevectoratmomentk.Inaddition,
/1 ,/1
ˆ


kk k ikk
Axx
.
(4) Updatethesigmasamplepoints:
,/ 1
,/ 1 / 1
,/ 1 / 1
ˆ,0
ˆ( ( ) ) , 1,2,...,
ˆ( ( ) ) , 1,..., 2
ik k k
ik k k k k i
ik k k kk i L
xxi
xxLPi L
x
xLP iL L





 (22)
(5) CalculatetheestimatedvaluesoftheoutputandtheKalmangain:
,/ 1 ,/ 1
2
,/ 1
0
()
ˆ, 0,1,..., 2
ik k ik k
L
m
kiikk
i
yhx
yyi L


 (23)
1
,,
k xyk yk
L
PP
(24)
Electronics2024,13,44368of15
2
,,/1,/1
0
2
,,/1,/1
0
ˆˆ
()( )
ˆˆ
()()
L
cT
xy k i i k k k i k k k
i
L
cT
yk i ikk k ik k k
i
P
xxyy
Pyyyy




 (25)
wherek
L
isthegainmatrix.
(6) Updatethestatevariablesanderrorcovariancematrix:
ˆˆˆ
()
kkkkk
x
xLyy  (26)
/1 ,
T
kkk kykk
PP LPL
 (27)
where,
kdk
yU
istheactualmeasuredvalueofthebaeryterminalvoltageatthe
momentk.
Theabovestepsareiterativelyupdatedtoenablethereal-timeestimationofSOC.
3.3.ImplementationFlowoftheMIDRLS‐UKFAlgorithm
Thevaluesofthemodelparameters0
R
,1
R
and1
Careaffectedbyanumberof
factors,includingtheexternalenvironmentandtheinternalstateofthebaery.The
MIDRLSalgorithmmodulecanaccuratelyidentifytheparameters0
R
,1
R
and1
Ceven
whencharging/dischargingcurrentsareincompletelymeasured.Theidentifiedvaluesare
passedtotheUKFalgorithmmodule,whichparticipatesintheformationofthefunctional
relationshipsofthebaerystateandmeasurementequations.
Thus,theUKFalgorithmmoduleobtainsanestimationoftheSOCandreturnsthe
corresponding,ocv k
U totheMIDRLSalgorithmmodulefromthefunction ()
k
f
SOC ,
whichthencalculatesitandproceedstothenextmomentofparameteridentification.As
aresult,theMIDRLSalgorithmandtheUKFalgorithmjointlyrealizethebaerySOC
estimation,withtherealizationflowstructurediagramdisplayedinFigure2.
Measured current I
Initialization
MIDRLS for online
parameter
identification
UKF for state
estimation
Model parameters
011
RRC、、
d
SOC U、
Initialization
Obtain
ocv
UE、
Figure2.SOCestimationprocessbasedonMIDRLS-UKF.
Electronics2024,13,44369of15
4.SimulationResultsandAnalysis
Inthispaper,MATLABR2022bwasusedtocarryoutsimulationworkstoverifythe
accuracyoftheabovealgorithmicprocess.Thevoltageandcurrentdatasequenceswere
previouslymeasured.ThesubjectofourtestsistheINR18650-20Rbaery,witha2000
mAhcapacity,whichwaschargedusingaCC-CVprotocolata1Crateatatest
temperatureof25°C.Intheconstantcurrentphase,thebaerywaschargeduntilit
reachedaterminalvoltageof4.2V,afterwhichthecurrentgraduallydecreasedto0.01C
intheconstantvoltagephase.Subsequently,thebaerywasdischargedatarateofC/20
untilthevoltagewascloseto2.5V,andthenrechargedatthesamerateuntilthevoltage
wasnearly4.2Vtoobtainthedatausedinthesimulationworks.Thevalueofcurrents
appearstobezeroatrandommomentpoints,asshowninFigure3.Thecurve()
k
f
SOC
ofthefunction
ocv
USOC
 usedinthesimulationsisderivedfromtheUniversityof
MarylandexperimentaldataonOCVincrementaltestingofINR18650-20Rbaeries[27].
Tomitigatetherandomnessoftheexperimentaloutcomes,MonteCarlosimulations
wereusedinthesimulations,withatotalof200runs.Thesamplingtime01sT.Thetotal
simulationdurationwasestablishedat9434sandtheinitialvaluesofthebaery
equivalentmodelparametersweresetas-2
0=1 10R
,-3
1=1 10R
 and3
1=1 10 FC.
TheadditionalparametersofthealgorithmaresetinTabl e1.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Time/s
-3
-2
-1
0
1
2
3
4
5
6
7
8
Current/s
2000 2100 2200 2300 2400
-0.1
0
0.1
0.2
Figure3.Currentmeasurementswithrandommissingdata.
Tab le1.Parametersforsimulations.
ParametersoftheMIDRLSAlgorithmParametersoftheUKFAlgorithm
0.999
0[0 0.8]T
x
0.8p=0
1
-2
110

(0) 0.001*
M
I2
-4 -4
0[1 1 0 0; 0 2 10 ]P
Figure4showstheoutcomesoftheMIDRLSalgorithmforthereal-timeidentification
forthebaerymodelparameters0
R
,1
R
 and1
C.Overall,thebaery’sinternal
resistance0
R
graduallyincreasesastheSOCdecreases,whilethepolarizationresistance
1
R
exhibitsaninitialrisefollowedbyadeclinewithdecreasingSOC.Inaddition,the
polarizationcapacitance1
Cshowsadrasticchangeintheinitialestimationprocessand
thentendstoincreaseprogressivelyalongsideareductioninSOC.Theresultsshowthat
theMIDRLSalgorithmenablesthereal-timecorrectionoftheparameterestimates,
leadingtomoreaccurateSOCstateestimationresults.
Electronics2024,13,443610of15
Subsequently,weconductedacomparativeanalysisoftheestimationoutcomes
betweentheMIDRLS-UKFandFFRLS-UKFalgorithmsunderidenticalsimulation
conditions.Accordingtotheestimationprocess,itcanbeseenthatbasedontheidentified
modelparameters,theterminalvoltagecanbeprojectedusingtheUKFalgorithm.Ascan
beseeninFigure5,fortheMIDRLSalgorithm,initialdeviationsinterminalvoltage
predictionsoccurduetotheerraticfluctuationsintheparameterestimates.However,the
accuracyimprovesovertime.Moreover,ingeneral,comparedtotheFFRLS-UKF
algorithm,theterminalvoltageestimationresultsbasedontheMIDRLS-UKFalgorithm
aremuchclosertotheactualmeasuredvalues.
AcomparisonofthefinalSOCestimationresultsisshowninFigure6.Inthepresence
ofrandommissingcurrentmeasurements,despitetheincreasedcomputationaldemands
oftheMIDRLS-UKFalgorithmovertheFFRLS-UKFalgorithm,theSOCestimationresult
curveofMIDRLS-UKFalignscloselywiththeactualSOCcurveverywell.Itcanbe
inferredthatthisisbecausethemodelparametersestimatedbytheMIDRLSalgorithm
turnedouttobeaccurate.However,theFFRLS-UKFalgorithmexhibitsinitial
discrepanciesinSOCestimation,whichareexacerbatedovertime.Thisdivergencecanbe
aributedtotheFFRLSalgorithm’sfailuretoconvergetotheminimumsteadystateerror
whenthegradientdescentisappliedinthepresenceofmissingcurrentdata.
(a)
Time/s
0.06
0
/R
02000 4000 6000 8000
0
1
/R
(b)
02000 4000 6000 8000
0
0.01
0.02
0.03
0.04
0.05
1
/CF
0
400
800
1200
02000 4000 6000 8000
(c)
Time/s
Time/s
0.04
0.02
Figure4.Modelparameteridentificationresults:(a)baeryinternalresistance0
R
;(b)polarization
resistance1
R
;(c)polarizationcapacitance1
C.
Electronics2024,13,443611of15
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Time/s
3.1
3.3
3.5
3.7
3.9
4.1
Voltage/V
Actual voltage
MIDRLS-UKF
RLS-UKF
Figure5.d
Uestimationresults.
Tomaketheresultsmoreintuitive,Figure7illustratestheSOCestimationerrorrates.
ItisevidentthattheFFRLS-UKFalgorithmexperiencesarapidincreaseinSOCestimation
error,whereastheerrorfortheproposedMIDRLS-UKFalgorithmremainsbelow0.05%,
indicatingasubstantialenhancementinaccuracy.Tofurthercomparetheestimation
accuracyofthealgorithms,wealsocalculatedtherootmeansquareerror(RMSE)andthe
maximumabsoluteerror(MAE)inTab l e2.Theyaredefined,respectively,as
2
1
1
=( )
T
ii
i
RMSE SOC SOC
T
 (28)
max
ii
MAE SOC SOC
 (29)
wherei
SOC istheactualvalueoftheSOCatmomentiandi
SOCistheestimatedvalue
oftheSOCatmomenti.AspresentedinTab le2,theMAEsandRMSEsfortheMIDRLS-
UKFaresignificantlylowercomparedtothoseoftheFFRLS-UKF,whichverifiesthe
superiorestimationaccuracyoftheproposedalgorithm.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Time/s
0
10
20
30
40
50
60
70
80
90
SOC /%
Reference
MIDRLS-UKF
RLS-UK F
Figure6.SOCestimationresults.
Electronics2024,13,443612of15
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Time / s
0
0.5
1
1.5
2
2.5
SOC estimation error / %
RLS-UKF
MIDRLS-UKF
8500 8600 8700
-0.02
0
0.02
Figure7.SOCestimationerror.
Tab le2.MSEsandMAEsofSOCestimationmethods.
AlgorithmRMSE(%)MAE(%)
MIDRLS-UKF0.43%0.81%
FFRLS-UKF8.12%14.64%
Sincetheinitialvalueofthealgorithmisoftenunknown,wetestedtheperformance
oftheproposedalgorithmagainstincorrectinitialvalues.SOCestimationwascarriedout
afterseingtheinitialvalueofSOCto0.7and0.9,respectively,andtheoutcomesare
depictedinFigure8.TheresultsdemonstratethattheMIDRLS-UKFalgorithmswiftly
rectifiesinitialvaluediscrepancies,achievinghighlyaccurateSOCestimationwith
remarkablestability.Thisfurtherverifiestheexceptionalapplicability,robustnessand
reliableestimationperformanceoftheproposedalgorithm.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Time / s
0
10
20
30
40
50
60
70
80
90
SOC /%
Reference
Initial value is 0.7
Initial value is 0.9
Figure8.SOCestimationresultsunderinaccurateinitialvaluesofMIDRLS-UKF.
5.Conclusions
Focusingontheproblemthatreal-timecurrentmeasurementsmayberandomly
missingduringtheSOCestimationoflithiumbaeries,thusimpairingestimation
precision,thisstudyemploystheMIDRLSalgorithmtoidentifythemodelparameters
basedontheTheveninbaerymodel.Concurrently,theUKFalgorithmisutilizedto
estimatethebaerystatevariables,thusenablingthejointestimationoftheSOC.The
estimationaccuracyisthencontrastedwiththoseoftheconventionaljointFFRLS-UKF
jointalgorithm.ThesimulationresultsverifytheaccuracyofthejointMIDRLS-UKF
algorithmproposedinthispaperforSOCpurposes.ThemaximumSOCestimationerror
oftheMIDRLS-UKFalgorithmdoesnotexceed0.05%,withtheRMSEandMAEofthe
estimationresultsbeing0.43%and0.81%,respectively,indicativeofexcellentestimation
Electronics2024,13,443613of15
accuracy.Comparedwiththeexistingestimationmethods,theestimationperformance,
stabilityandapplicabilityoftheproposedalgorithminthispaperaresignificantly
enhanced,whichfurtherimprovesthereliabilityoftheBMS.
Aut ho rContributions:Conceptualization,X.L.andZ.Z.;methodology,X.L.;software,X.L.;validation,
Z.Z.andJ.M.;formalanalysis,X.L.andQ.W.;investigation,Z.Z.andJ.M.;resources,Q.W.andZ.Z.;
datacuration,X.L.andQ.W.;writing—originaldraftpreparation,X.L.;writing—reviewandediting,
Z.Z.;visualization,X.L.;supervision,Z.ZandJ.M.;projectadministration,Z.Z.;fundingacquisition,
Z.Z.andJ.M.Allauthorshavereadandagreedtothepublishedversionofthemanuscript.
Funding:ThisresearchwaspartiallysupportedbytheNationalNaturalScienceFoundationof
China(62101362)andtheKeyResearchandDevelopmentProgramofShaanxiProvince(2024GX-
YBXM-442).
DataAvailabilityStatement:Therawdatasupportingtheconclusionsofthisarticlewillbemade
availablebytheauthorsonrequest.
Acknowledgments:Theauthorswouldliketothanktheeditorandreviewersfortheirsincere
suggestionsforimprovingthequalityofthispaper.
ConflictsofInterest:Theauthorsdeclarethatthisresearchwasconductedintheabsenceofany
commercialorfinancialrelationshipsthatcouldbeconstruedaspotentialconflictsofinterest.
Abbreviations
SOCstateofcharge
UKFunscentedKalmanfilter
EKFextendedKalmanfilter
FFRLSforgeingfactorrecursiveleastsquare
MIDRLSrecursiveleastsquarewithmissinginputdata
BMS
b
aerymanagementsystem
RMSErootmeansquareerror
MAEmaximumabsoluteerror
Symbols
ocv
Uvoltageoftheidealvoltagesource
0
R
b
aeryequivalentinternalresistance
1
R
polarizationresistance
1
Cpolarizationcapacitor
d
U
b
aeryterminalvoltage
0
I
loadcurrent
()
g
nBernoullirandomvariable
()
x
ninputdata
()
ic
x
ninputdatawithrandommissingvalues
p
probabilitythatinputdataarenotmissing
constanttimesofimputation
()di outputdata
()wn parametervectortobeidentified
forgeingfactor
()
nobjectivefunctionoftheFFRLSalgorithm
ktimestepindex
0
Tsamplingtime
0
Qmaximumcapacityofthebaery
Coulombicefficiency
k
x
statevector
k
wprocessnoise
Electronics2024,13,443614of15
k
vmeasurementnoise
k
Eterminalvoltageminusidealvoltagesourcevoltage
()
k
f
SOC functionofocv
USOC
k
ymeasurevector
0
ˆ
x
estimatedvalueoftheinitialstate
0
Pinitialerrorcovariancematrix
Ldimensionofstatevariable
m
i
weightedvalueofthemeanofthesamplingpoints
c
i
weightedvalueoftheerrorcovariancematrices
,/ 1ik k
xpredictedvalueofthestatevariableattimeinstant/1kk
/1kk
Ppredictedvalueoftheerrorcovariancematrixattimeinstant/1kk
ˆk
x
estimatedvalueofthestatevariableatmomentk
k
L
gainmatrix
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