ArticlePDF Available

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

Authors:

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.
Content may be subject to copyright.
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
unscentedKalmanlter(UKF)algorithmisproposed,calledtheMIDRLS-UKFalgorithm.Firstly,
theequivalentcircuitmodelofaTheveninbaeryisformulated.Then,thecurrentimputation
modelisdesignedtointerpolatethemissingdata,basedonwhichtheMIDRLSalgorithmisderived
bysolvingtheunbiasedestimationofthegradientoftheobjectivefunction,thusrealizingtheonline
high-precisionidenticationofthecircuitmodelparameters.Furthermore,theproposedalgorithm
iscombinedwiththeUKFalgorithmtofacilitatetheonlinepreciseestimationofSOC.The
simulationresultsindicateamarkeddecreaseintheSOCestimationerrorwhenemployingthe
proposedjointalgorithm,asopposedtotheconventionalforgeingfactorrecursiveleastsquares
(FFRLS)algorithmcombinedwiththeUKFjointestimationalgorithm,whichveriestheprecision
andeectivenessoftheproposedjointalgorithm.
Keywords:lithiumbaery;stateofchargeestimation;randommissingcurrentdata;model
parameteridentication;unscentedKalmanltering
1.Introduction
Overthepastfewyears,inordertopromotesustainableenergydevelopmentand
thedevelopmentofagreeneconomy,theelectricvehicleindustry,whichhasapositive
impactonreducingurbancarbonemissions,hasdevelopedrapidly[1].Lithiumbaeries
havebecomeawidelyusedbaerytechnologyintheemergingelectricvehiclesector[2]
duetotheirelevatedenergyandpowerdensities,extendedservicelife,lowself-discharge
andsuperiorenergyconversioneciency[3].Theseaributesrenderlithiumbaeriesan
idealchoiceforpoweringthenextgenerationofelectricvehicles,aligningwiththeglobal
shifttowardscleanerandmoresustainabletransportationoptions.Itisthereforeessential
tomonitorbaerystatususinganeectiveandprecisebaerymanagementsystem(BMS)
[4].AmongthevariousfunctionalitiesofaBMS,theestimationofthestateofcharge(SOC)
standsoutasparticularlycrucial.TheaccuratecompletionofSOCestimationcanshow
theremainingcapacityofthebaery,whichisessentialforpreventingoverchargingand
overdischargingscenarios.Bydoingso,itensuresthereliableoperationofelectric
vehicles,optimizesthebaery’sperformance,andsignicantlyextendsitsoperational
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.Althoughthiskindofmethodhashighexibilitywithhighestimation
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,usuallycombinedwithanadaptivelteringalgorithm
toemulatethebaery’sdynamicstatecharacteristics[14],suchasanextendedKalman
lter(EKF),anunscentedKalmanlter(UKF)anditsimprovedalgorithms[15].
Furthermore,theresearchersin[16]proposedahierarchicaladaptiveextendedKalman
lter(HAEKF)algorithm,whichinvolvesthedissectionofthesecond-orderRCcircuit’s
stateequationmodelinaccordancewiththeprinciplesofhierarchicalidentication.
Currently,toenhancethetimelinessandprecisionofonlinemodelparameterestimation
andtoaddresstheissueofoutdateddataaccumulationiniterativeprocesses[17],the
forgeingfactorrecursiveleastsquare(FFRLS)algorithmisoftenusedtoidentifymodel
parameters.However,itisonlyapplicableiftheinputdataarecomplete,i.e.,thereal-time
loadcurrentmeasurementsarefullyobtained.Shouldthecurrentdetectionmoduleinthe
BMSbefaulty,oriftheconnectorislooseorenvironmentalvibrationsoccur,causingthe
currentmeasurementvaluetobemissing,thiswillleadtoalargeerrorintheresultsof
parameteridentication.TheseerrorscanfurtheraecttheaccuracyofSOCestimation,
potentiallyleadingtoimproperbaeryusageandjeopardizingthebaery’shealth.
Toensurethehighreliabilityofonlinebaerymodelparameteridenticationdespite
thepresenceofmissingcurrentmeasurementdata,inthispaper,weinnovativelydesign
acurrentimputationmodeltointerpolatethemissingvaluesofthecurrent.Basedonthe
imputationmodel,wederivearecursiveleastsquareswithmissinginputdata(MIDRLS)
algorithmtoidentifythemodelparametersbysolvingtheunbiasedestimationofthe
gradientoftheobjectivefunction.CombiningitwiththeUKFalgorithm,weachieve
onlineSOCestimationbyusingupdatedmodelparametersandsystemstatesateach
cycle.
Theremainderofthispaperisorganizedasfollows:Section2describesthederivation
oftheMIDRLSalgorithmandthemodelparameteridenticationprocess.Section3
Electronics2024,13,44363of15
describestheSOCestimationprocessbasedontheUKFalgorithmandpresentsthe
derivationoftheMIDRLSalgorithminconjunctionwiththeUKFtoperformSOC
estimationinrealtime.InSection4,simulationworksareconductedtoevaluatethe
performanceoftheproposedalgorithm.Finally,Section5presentsthediscussionand
conclusionofthepaper.
2.BaeryModelandParameterIdentication
2.1.BaeryModel
Equivalentbaerymodelsrepresenttheinternalstateanddynamiccharacteristicsof
abaerybyforminganequivalentcircuitfromidealcircuitelements.Prevalentmodels
includetheRintmodel,Theveninmodel,DPmodel,PNGVmodel,second-orderRC
model,etc.[18].TheTheveninmodeltakesintoaccounttheelectrochemicalreaction
insidethebaery,whichcanreectthepolarizationcharacteristicsofthebaery[19].It
canreecttheactualworkingcharacteristicsofreallithiumbaeries,soitissuitablefor
simulatingtheperformanceoflithiumbaeries.However,itsstructureisrelativelysimple
andthemodelparametersarexedvalues,sotheaccuracyisnottoohigh.Toprecisely
reectthedynamiccharacteristicsoflithiumbaeries,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
parametersthatneedtobeidentied.
ThemodelequationscanbeconstructedfromKirchho’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
Careinuencedbyfactorssuch
asenvironmentalchanges,theoperationalconditionsofthebaeryandtheagingprocess
ofthebaery[22].ThispaperemploystheproposedMIDRLSalgorithmformodel
parameteridentication.Itprovidesreal-timeidenticationoftheparametersevenwhen
theloadcurrentmeasurementsarerandomlylost,thusavoidingundesirableeectson
theSOCestimationresults.
Electronics2024,13,44364of15
2.2.TheProposedMIDRLSAlgorithm
Toenablethecapabilitytoaccuratelyidentifythemodelparametersdespitethe
presenceofmissingmodelinputcurrentdata,anewparameteridenticationalgorithm
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
parametervectortobeidentied.
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
Algorithm1ThealgorithmowoftheproposedMIDRLS
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
Theforgeingfactorallowstheweightofpreviousdatainparameteridentication
tobecontinuouslyreducedoverlongperiodsoftimeinsteadofaccumulatingovertime.
Constructingtheunbiasedestimationreducestheerrorinidenticationresultscausedby
therandomlossofcurrentmeasurements.
2.3.ParameterIdenticationoftheBaeryModel
ToidentifytheparametersofthebaerymodelthroughtheMIDRLSalgorithm,after
rewritingEquation(1)indiscreteformandleing,,

kdkocvk
EU U ,onecanobtainthe
dierentialequationsforthemodel
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,
thecoecientvector12 3
() [ ]wn

  canbeidentiedrecursivelythroughthe
MIDRLSalgorithmandthebaerymodelparametersatmomentkcanbecalculated
fromEquation(11).

Electronics2024,13,44366of15
3.SOCEstimationBasedontheJointMIDRLSUKFAlgorithm
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
 istheCoulombiceciencywhichdescribestheratioofdischargecapacity
(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.
Thiscanbefurthersimpliedandexpressedasafunctionalrelationship:
11
(,)
(, )
kkkk
kkkk
x
gx u w
yhxu v



. (15)
3.2.SOCEstimationBasedontheUKFAlgorithm
TheKalmanlteralgorithmisanalgorithmfortheoptimalstateestimationofalinear
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.ImplementationFlowoftheMIDRLSUKFAlgorithm
Thevaluesofthemodelparameters0
R
,1
R
and1
Careaectedbyanumberof
factors,includingtheexternalenvironmentandtheinternalstateofthebaery.The
MIDRLSalgorithmmodulecanaccuratelyidentifytheparameters0
R
,1
R
and1
Ceven
whencharging/dischargingcurrentsareincompletelymeasured.Theidentiedvaluesare
passedtotheUKFalgorithmmodule,whichparticipatesintheformationofthefunctional
relationshipsofthebaerystateandmeasurementequations.
Thus,theUKFalgorithmmoduleobtainsanestimationoftheSOCandreturnsthe
corresponding,ocv k
U totheMIDRLSalgorithmmodulefromthefunction ()
k
f
SOC ,
whichthencalculatesitandproceedstothenextmomentofparameteridentication.As
aresult,theMIDRLSalgorithmandtheUKFalgorithmjointlyrealizethebaerySOC
estimation,withtherealizationowstructurediagramdisplayedinFigure2.
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-timeidentication
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,itcanbeseenthatbasedontheidentied
modelparameters,theterminalvoltagecanbeprojectedusingtheUKFalgorithm.Ascan
beseeninFigure5,fortheMIDRLSalgorithm,initialdeviationsinterminalvoltage
predictionsoccurduetotheerraticuctuationsintheparameterestimates.However,the
accuracyimprovesovertime.Moreover,ingeneral,comparedtotheFFRLS-UKF
algorithm,theterminalvoltageestimationresultsbasedontheMIDRLS-UKFalgorithm
aremuchclosertotheactualmeasuredvalues.
AcomparisonofthenalSOCestimationresultsisshowninFigure6.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.Modelparameteridenticationresults:(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.Theyaredened,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-
UKFaresignicantlylowercomparedtothoseoftheFFRLS-UKF,whichveriesthe
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
rectiesinitialvaluediscrepancies,achievinghighlyaccurateSOCestimationwith
remarkablestability.Thisfurtherveriestheexceptionalapplicability,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,
stabilityandapplicabilityoftheproposedalgorithminthispaperaresignicantly
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.
ConictsofInterest:Theauthorsdeclarethatthisresearchwasconductedintheabsenceofany
commercialornancialrelationshipsthatcouldbeconstruedaspotentialconictsofinterest.
Abbreviations
SOCstateofcharge
UKFunscentedKalmanlter
EKFextendedKalmanlter
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 parametervectortobeidentied
forgeingfactor
()
nobjectivefunctionoftheFFRLSalgorithm
ktimestepindex
0
Tsamplingtime
0
Qmaximumcapacityofthebaery
Coulombiceciency
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
References
1. Liu,C.;Liu,W.;Wang ,L.;Hu,G.;Ma,L.;Ren,B.Anewmethodofmodelingandstateofchargeestimationofthebaery.J.
PowerSources2016,320,1–12.
2. Tong,Y.;Zheng,Z.;Fan,W.;Liu,Z.ImprovedunscentedKalmanlterforstateofchargeestimationoflithium-ionbaerywith
one-steprandomlymeasurementlossandinaccuratenoisecovariancematrices.Digit.SignalProcess.2022,131,103780.
3. Zubi,G.;Dufo-López,R.;Carvalho,M.;Pasaoglu,G.Thelithium-ionbaery:Stateoftheartandfutureperspectives.Renew.
Sustain.EnergyRev.2018,89,292–308.
4. Singh,S.;More,V.;Batheri,R.DrivingElectricVehi clesIntotheFutureWithBaeryManagementSystems.IEEEEng.Manag.
Rev.2022,50,157–161.
5. Charkhgard,M.;Farrokhi,M.State-of-ChargeEstimationforLithium-IonBaeriesUsingNeuralNetworksandEKF.IEEE
Trans.Ind.Electron.2010,57,4178–4187.
6. Qin,P.;Zhao,L.ANovelTransferLearning-BasedCellSOCOnlineEstimationMethodforaBaeryPackinComplex
ApplicationConditions.IEEETrans.Ind.Electron.2023,71,1606–1615.
7. Dubarry,M.;Svoboda,V.;Hwu,R.;Liaw,B.Y.Capacitylossinrechargeablelithiumcellsduringcyclelifetesting:The
importanceofdeterminingstate-of-charge.J.PowerSources2007,174,1121–1125.
8. Tian,J.;Chen,C.;Shen,W.;Sun,F.;Xiong,R.Deeplearningframeworkforlithium-ionbaerystateofchargeestimation:Recent
advancesandfutureperspectives.EnergyStorageMater.2023,61,102883.
9. Chai,X.;Li,S.;Liang,F.AnovelbaerySOCestimationmethodbasedonrandomsearchoptimizedLSTMneuralnetwork.
Energy2024,306,132583.
10. Zhou,X.;Wan g, Y.;Shi,Y.;Jiang,Q.;Zhou,C.;Zheng,Z.DeepReinforcementLearning-BasedOptimalPMUPlacement
ConsideringtheDegreeofPowerSystemObservability.IEEETrans.Ind.Inform.2024,20,8949–8960.
11. Tan g,A.;Huang,Y.;Liu,S.;Yu,Q.;Shen,W.;Xiong,R.Anovellithium-ionbaerystateofchargeestimationmethodbasedon
thefusionofneuralnetworkandequivalentcircuitmodels.Appl.Energy2023,348,121578.
12. Buchicchio,E.;DeAngelis,A.;Santoni,F.;Carbone,P.;Bianconi,F.;Smeraldi,F.BaerySOCestimationfromEISdatabased
onmachinelearningandequivalentcircuitmodel.Energy2023,283,128461.
13. Yang , H.;Sun,X.;An,Y.;Zhang,X.;We i, T.;Ma,Y.Onlineparametersidenticationandstateofchargeestimationforlithium-
ioncapacitorbasedonimprovedCubatureKalmanlter.J.EnergyStorage2019,24,100810.
14. Li,X.;Wan g, Z.;Zhang,L.Co-estimationofcapacityandstate-of-chargeforlithium-ionbaeriesinelectricvehicles.Energy
2019,174,33–44.
15. Shrivastava,P.;KokSoon,T.;BinIdris,M.Y.I.;Mekhilef,S.;Adnan,S.B.R.S.CombinedStateofChargeandStateofEnergy
EstimationofLithium-IonBaeryUsingDualForgeingFactor-BasedAdaptiveExtendedKalmanFilterforElectricVehi cle
Applications.IEEETrans.Veh .Te chnol .2021,70,1200–1215.
16. Wang,D.;Ya n g,Y. ;Gu,T.AhierarchicaladaptiveextendedKalmanlteralgorithmforlithium-ionbaerystateofcharge
estimation.J.EnergyStorage2023,62,106831.
17. Zhang,S.;Guo,X.;Zhang,X.Animprovedadaptiveunscentedkalmanlteringforstateofchargeonlineestimationoflithium-
ionbaery.J.EnergyStorage2020,32,101980.
18. Shrivastava,P.;Soon,T.K.;Idris,M.Y.I.B.;Mekhilef,S.Overviewofmodel-basedonlinestate-of-chargeestimationusingKalman
lterfamilyforlithium-ionbaeries.Renew.Sustain.EnergyRev.2019,113,109233.
19. Putra,W.S.;Dewangga,B.R.;Cahyadi,A.;Wahyunggoro,O.CurrentestimationusingTheveninbaerymodel.InProceedings
oftheJointInternationalConferenceonElectricVehic ul arTech no log yandIndustrial,Mechanical,ElectricalandChemical
Engineering(ICEVT&IMECE),Surakarta,Indonesia,4–5November2015;pp.5–9.
Electronics2024,13,443615of15
20. Kwak,M.;Lkhagvasuren,B.;Park,J.;You,J.-H.ParameterIdenticationandSOCEstimationofaBaeryUndertheHysteresis
Eect.IEEETrans.Ind.Electron.2019,67,9758–9767.
21. Pop,V.; Bergveld,H.J.;hetVel d, J.O.;Regtien,P.P.L.;Danilov,D.;Noen,P.H.L.Modelingbaerybehaviorforaccuratestate-
of-chargeindication.J.Electrochem.Soc.2006,153,A2013.
22. Lai,X.;Zheng,Y.;Sun,T.Acomparativestudyofdierentequivalentcircuitmodelsforestimatingstate-of-chargeoflithium-
ionbaeries.Electrochim.Acta2018,259,566–577.
23. Leung,S.-H.;So,C.F.Gradient-basedvariableforgeingfactorRLSalgorithmintime-varyingenvironments.IEEETrans.Signal
Process.2005,53,3141–3150.
24. Piller,S.;Perrin,M.;Jossen,A.Methodsforstate-of-chargedeterminationandtheirapplications.J.PowerSources2001,96,113–
120.
25. Hossain,M.;Haque,M.E.;Arif,M.T.Kalmanlteringtechniquesfortheonlinemodelparametersandstateofchargeestimation
oftheLi-ionbaeries:Acomparativeanalysis.J.EnergyStorage2022,51,104174.
26. Wang,K.;Feng,X.;Pang,J.;Duan,C.;Li,L.Stateofcharge(SOC)estimationoflithium-ionbaerybasedonadaptivesquare
rootunscentedkalmanlter.Int.J.Electrochem.Sci.2020,15,9499–9516.
27. Zheng,F.;Xing,Y.; Jiang,J.;Sun,B.;Kim,J.;Pecht,M.Inuenceofdierentopencircuitvoltagetestsonstateofchargeonline
estimationforlithium-ionbaeries.Appl.Energy2016,183,513–525.
Disclaimer/Publisher’sNote:Thestatements,opinionsanddatacontainedinallpublicationsaresolelythoseoftheindividual
author(s)andcontributor(s)andnotofMDPIand/ortheeditor(s).MDPIand/ortheeditor(s)disclaimresponsibilityforanyinjury
topeopleorpropertyresultingfromanyideas,methods,instructionsorproductsreferredtointhecontent.
ResearchGate has not been able to resolve any citations for this publication.
Article
A phasor measurement unit (PMU) is the core measurement component of power system monitoring and analysis. The placement of PMUs directly impacts the state estimation confidence, and hence the optimal PMU placement (OPP), that, minimizing the number of PMUs and ensuring system observability, is appealing to power engineers. However, the current observability analysis mostly relies on topological methods, which cannot reflect the influence of the operating environment. Moreover, intricate OPP models are driving higher demand for efficient solvers. Confronting these challenges, we propose a reinforcement learning graph convolutional network-deep deterministic policy gradient algorithm-based OPP strategy, which effectively captures the system graph structure and PMU state, thereby independently identifying the PMUs value. Furthermore, the degree of system observability is considered, and three observability quantitative indicators are proposed in the OPP strategy, which will enhance the confidence of the perceived state under complex operating environment. The effectiveness of the proposed strategy have been validated in multiple test systems.
Article
Accurate estimating the state of charge (SOC) can improve battery reliability, safety, and extend battery service life. The existing battery models used for SOC estimation inadequately capture the dynamic characteristics of a battery in a wide temperature over the full SOC range, leading to significant inaccuracies in SOC estimation, especially in low temperature and low SOC. A novel SOC estimation approach is developed based on a fusion of neural network model and equivalent circuit model. Firstly, the weight-SOC-temperature relationship is established by obtaining the weights of the equivalent circuit model and the neural network model offline using the standard deviation weight assignment method. Following that, an online adaptive weight correction approach is implemented to update the weight-SOC-temperature relationship. Finally, a novel multi-algorithm fusion technique is utilized to achieve SOC estimation accuracy within 1%. The results clearly demonstrate that the developed approach achieves twice the accuracy of the existing approach, highlighting its superior effectiveness.