animprovedrecursivetotalleastsquaresestimationof...

Research ArticleAn Improved Recursive Total Least Squares Estimation ofCapacity for Electric Vehicle Lithium-Iron Phosphate Batteries

Shaohua Wang 1 Yue Yang 2 and Konghui Guo 1

1State Key Laboratory of Automobile Simulation and Control Jilin University Changchun 130025 China2College of Transportation Jilin University Changchun 130025 China

Correspondence should be addressed to Shaohua Wang shwang13mailsjlueducn

Received 24 March 2020 Revised 15 May 2020 Accepted 23 May 2020 Published 20 June 2020

Academic Editor Yang Li

Copyright copy 2020 ShaohuaWang et alis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A batteryrsquos capacity is an important indicator of its state of health and determines the maximum cruising range of electric vehiclesIt is also a crucial piece of information for helping improve state of charge (SOC) estimation health prognosis and other relatedtasks in the battery management system (BMS) In this paper we propose an improved recursive total least squares approach toonline capacity estimation which is based on the constrained Rayleigh quotient in terms of battery capacity is approachaccounts for errors in both the SOC and accumulated current measurements not traditionally considered in the battery capacitymodel to give an unbiased estimation Moreover the forgetting factor updated by minimizing the Rayleigh quotient of thecapacity estimation model is applied to track the changes in the model and get a more precise estimation of the capacity Finallythe performance of the proposed algorithm is validated via simulation and experimental studies on lithium-iron phosphatebatteries e estimation results show that the proposed algorithm improves capacity estimation accuracy

1 Introduction

To meet the need to reduce fossil fuel dependence andemissions from traditional transportation electric vehicles(EVs) are being studied and manufactured extensivelyaround the whole world [1] Li-ion batteries play an im-portant role in the energy storage system of the EV AmongLi-ion batteries the LiFePO4 (lithium-iron phosphate)battery has gained a lot of attention owing to its nontoxicitylow cost inherent safety and higher energy density [2ndash4] Inorder to achieve optimum performance and long life frombatteries their state of health (SOH) is crucial informationin addition to their state of charge (SOC) in the batterymanagement system (BMS) [5 6] e battery capacity isconsidered as one important indicator of SOH which re-flects that maximum electrical charge can be stored into thebattery and determines the maximum cruising range for EVs[7] As the battery ages the loss of battery capacity leads to areduction in the electric range of the EVs [8] Howeverbattery capacity cannot be measured by some of the availablevehicle sensors is necessitates the development of viablestate estimation algorithms

Many studies in the literature present estimation ap-proaches for battery capacity e capacity of the LiFePO4battery will change continually according to its age usagepatterns and temperature [9] Traditionally battery capacitycould only be measured offline by depleting a fully chargedbattery with a low current rate at a specific temperatureObviously this method is not suitable for real-time appli-cations Some approaches [10 11] are based on a degra-dation model calibrated under a certain usage pattern andhave limited tracking ability in terms of vehicle workingconditions Incremental capacity analysis (ICA) and dif-ferential voltage analysis (DVA) [12ndash16] are used to describethe internal electrochemical aging mechanism during thebattery life and have recently been developed for the on-board application e drawback of these two approaches isthat the battery has to be charged or discharged with aconstant current in a wide voltage region since ICA typicallyrequires constant charging or discharging data [7] Someintelligent algorithms have been applied to build a rela-tionship between capacity and its influencing factors andthereby predict the real capacity of the batteryese includeneural network (NN) methods and machine learning

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 9359076 12 pageshttpsdoiorg10115520209359076

methods such as RBF NN [16 17] adaptive recurrent NN[18] structured NN (SNN) [19] extreme learning machine[20] adaptive multikernel relevance vector machine [21]ensemble learning with LS_SVM algorithm [22] supportvector regression [23] and others ese methods usuallyrequire a large amount of experimental data so that they cantrain a more precise network for battery capacity estimation[24ndash26] is is a difficult and time-consuming task

Least squares or Kalman filters offer the most promisingcompromise between estimation accuracy and low com-putational effort [7] e dual Kalman filter-based methodsjointly estimate the capacity and SOC together [27 28]However the employment of dual filters brings about anincreased computational burden and the multiscale problemmay cause algorithm divergence [29] Some approachesbased on coulomb counting have been researched to esti-mate the capacity in real time [30 31] In these studies theSOC estimation error is not considered which could affectthe capacity estimation accuracy Since the recursive leastsquares (RLS) do not consider the SOC error the total leastsquares (TLS) algorithm is introduced to improve the es-timation accuracy by considering the SOC estimation error[32 33] However the TLS algorithm is performed bycomputing the singular value decomposition (SVD) of themodelrsquos covariance matrix Since the multiplication opera-tions of SVD for an N by N matrix have computationalcomplexity of O(N3) it can incur high computationalcomplexity [34ndash36] In [33] a recursive approximateweighted total least squares algorithm was proposed toimprove the TLS performance In [37] another recursivetotal least squares algorithm was employed to solve thecapacity estimation In [38] the constrained Rayleighquotient was developed to find the TLS solution with thereducing computational complexity In [39] a fast recursivetotal least square algorithm was proposed to solve the TLSsolution by using the adaptation minimization of the con-strained Rayleigh quotient However the forgetting factor isfixed as a constant in the above studies It could not handlethe tradeoff between robustness and tracking capabilitywhich affects the estimation performance Specifically al-gorithms with smaller forgetting factor weigh more ontracking the time-varying capacity at the expense of moresensitivity to the current measurement and SOC estimationerror on the contrary large forgetting factor improvesrobustness but compromises the tracking capability

Based on the above algorithms we propose an approachemploying an improved recursive total least squares withvariable forgetting factor (VFF-RTLS) and a coulombcounting model considering errors in both the estimatedSOC and the current measurement e forgetting factor iscontrolled by minimizing the constrained Rayleigh quotientthrough the steepest descent method is approach willimprove capacity estimation performance e rest of thispaper is constructed as follows Section 2 describes thecapacity estimation model for the battery Section 3 presentsthe proposed VFF-RTLS approach for battery capacity es-timation Section 4 presents the simulation and experimentalstudies using the proposed algorithms and the conclusionsare summarized in Section 5

2 Capacity Estimation Model of theLiFePO4 Battery

When EVs are running in a real-life environment the SOCof their lithium-iron phosphate batteries is estimated by theunscented Kalman filter through the measured current andvoltage In order to predict the battery capacity we use thefollowing coulomb counting equation in terms of themeasured current and battery SOC

soc(k + 1) + vs(k + 1) soc(k) + vs(k) +ηΔ I(k) + wi(k)( 1113857

3600Cap

(1)

where Cap is the estimated capacity of the lithium-ironphosphate battery in ampere-hours (Ah) and I(k) is themeasured current η is coulomb efficiency with the as-sumption that η 1 Δ is the sample time wi(k) is thecurrent measurement error with standard deviation σivs(k) denotes the SOC estimation error

We convert equation (1) into an iterative form as follows

soc kLb( 1113857 minus soc kLb minus Lb + 1( 1113857 + vs(k)

η1113936

jkLb

jkLbminus Lb+1 Δ I(j) + wi(j)( 1113857

3600Cap

(2)

where Lb is the capacity update time period k 1 2 Nand vs(k) is the differential SOC error vs(k) vs(kLb) minus

vs(kLb minus Lb + 1)en equation (2) can be simplified into the following

expression

y + wi Caplowast x + vs( 1113857 (3)

where x soc(kLb) minus soc(kLb minus Lb + 1) denotes the differ-ence between the battery SOCs at the beginning and end ofthe update period y η1113936

jkLb

jkLbminus Lb+1 ΔI(j)3600 denotes thecumulated measured current within the update time periodLb and wi is the cumulated measured current errorwi η1113936

jkLb

jkLbminus Lb+1 Δwi(j)3600

3 Improved Recursive Total Least SquaresAlgorithm with Variable Forgetting Factor

31 Recursive Total Least Squares with Variable ForgettingFactor (VFF-RTLS) From the capacity model in (3) wecan see that there are errors in both the model input andoutput erefore this section proposes a constrainedRayleigh quotient-based RTLS algorithm with a variableforgetting factor for the capacity estimation of LiFePO4batteries

To facilitate the presentation of the proposed algorithmthe autocorrelation matrix of the capacity model input x(k)

is defined as follows

R E x(k)xT(k)1113960 1113961 (4)

e augmented input vector is defined asxa [x(k) y(k)] and its autocorrelation matrix can berepresented as follows

2 Mathematical Problems in Engineering

Ra

E xa(k)x

aprime(k)1113960 1113961 R b

b c1113890 1113891 (5)

where b E[x(k)y(k)] and c E[y(k)y(k)]e stochastic quantities Ra can be computed via an it-

eration formula with a forgetting factor μ (0lt μle 1) as follows

Ra(k)

R(k) b(k)

bT(k) c(k)1113890 1113891 μR

a(k minus 1) + x

a(k)x

aprime(k)

(6)

enR(k) b(k) and c(k) can be expressed as follows [38]

R(k) μR(k minus 1) + x(k)xT

(k) (7)

b(k) μb(k minus 1) + x(k)y(k) (8)

c(k) μc(k minus 1) + y(k)yT(k) (9)

e VFF-RTLS performs the orthogonal regressionwhichminimizes the sum of the squared orthogonal distancesfrom the data points to the fitting line [40] It makes verticaland horizontal corrections on both dependent and inde-pendent variables contain noise However the traditional RLSonly makes vertical corrections on the dependent variablecontaining noise Interested readers are referred to [39ndash42]and references therein for details In other words the pro-posed method is to solve the following optimization problem

Cap wi vs1113864 1113865 argmin wi vs1113858 1113859

F (10)

where middot F denotes the Frobenius normIt has been shown in [40] that optimization problem (10)

is equivalent to minimizing the Rayleigh quotient F(q)

F(q) qTψTψq

qTDq (11)

where q is the eigenvector associated with the smallest ei-genvalue of the symmetric positive-definite matrix ψTψ inthe TLS solution and D is a symmetric matrix

For the capacity estimation a constrained Rayleighquotient is used as a cost function where qT and ψTψ inequation (11) are replaced with [Cap minus 1] and Ra

respectively e capacity estimation problem therefore hasthe following constrained Rayleigh quotient cost function

J(Cap) [Cap minus 1]Ra[Cap minus 1]T

[Cap minus 1]D[Cap minus 1]T (12)

where Cap represents the estimated capacity of the LiFePO4battery and D diag(1 β) is a weighting matrix withβ σ2xσ

2y where σx is the standard deviation of the dif-

ferential SOC error vs and σy is the standard deviation of thecumulated current error wi

According to reference [42] the estimated capacity of thebattery can be updated as follows

Cap(k) Cap(k minus 1) + α(k)x(k) (13)

where α(k) can be determined by minimizing J(Cap(k))which can also be rewritten as minα(k)J(Cap(k minus 1) +

α(k)x(k))To find minα(k)J(Cap(k minus 1) + α(k)x(k)) let the gra-

dient of J(Cap(k minus 1) + α(k)x(k)) with respect to α(k) beequal to zero en we obtain α(k) by solving the followingequation

z[J(Cap(k minus 1) + α(k)x(k))]

zα(k) 0 (14)

In order to calculate α(k) (14) can be rewritten as

Aα2(k) + Bα(k) + C 0 (15)

epolynomial coefficients of (16) are derived as follows

A KT

(k)x(k)1113960 1113961[Cap(k minus 1)x(k)]

minus x(k)2

KT(k)Cap(k minus 1) minus x

T(k)y(k)1113960 1113961

(16)

B KT(k)x(k)1113960 1113961 β + Cap(k minus 1)

21113960 1113961

minus λ0(k) x(k)2

1113960 1113961(17)

C KT(k)Cap(k minus 1) minus x

T(k)b(k)1113960 1113961

lowast β + Cap(k minus 1)2

1113960 1113961 minus λ0(k)[Cap(k minus 1)x(k)](18)

where

K(k) R(k)x(k) (19)

λ0(k) μ(k minus 1)λ(k minus 1) β + Cap(k minus 1)2

1113960 1113961 +[Cap(k minus 1)x(k) minus y(k)]2 (20)

λ(k) λ0(k) + 2α(k) KT(k)Cap(k minus 1) minus x(k)b(k)1113858 1113859 + α2(k)x(k)K(k)1113966 1113967

β + Cap(k)21113960 1113961 (21)

us we can obtain the solution α(k) of equation (15)which is given by the following [38]

α(k) minus B +

B2 minus 4AC

radic

2A (22)

en the variable forgetting factor μ is introduced toimprove the estimation performance In order to adjust μ inthe above algorithm the steepest descent method is appliedto minimize the Rayleigh quotient J(Cap(k)) with respectto μ as follows

Mathematical Problems in Engineering 3

μ(k) μ(k minus 1) + c1zJ(Cap(k))

zμ (23)

where c1 is a tuning parameter

Using equations (7) and (9) we can show that equation(12) is equivalent to

J(Cap(k)) Cap(k)T μR(k minus 1) + x(k)x(k)T

1113960 1113961Cap(k) + μc(k minus 1) + y(k)y(k)

β + Cap(k)TCap(k) (24)

and its derivative is easily computed as follows

zJ(Cap(k))

zμCap(k)TR(k minus 1)Cap(k) + c(k minus 1)

β + CapTCap (25)

us we obtain the updating equation for the forgettingfactor

μ(k) μ(k minus 1) + c1Cap(k)TR(k minus 1)Cap(k) + c(k minus 1)

β + Cap(k)TCap(k)

(26)

Finally the details of the improved RTLS algorithm aresummarized in Algorithm 1 In this paper λ R b and c areinitialized to zero e tuning parameter c1 is set to 0001e forgetting factor μ is initialized to 099

32 Battery State of Charge (SOC) Estimation Battery SOCfor the capacity estimationmodel in this paper is obtained byusing an unscented Kalman based filter A second-orderequivalent circuit model is employed to estimate SOC estate-space equation for the model is defined as follows

soc(t + 1)

Up1(t + 1)

Up2(t + 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1 0 0

0 αp1 0

0 0 αp2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

soc(t)

Up1(t)

Up2(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+

ηΔ3600Cap

Rp1 1 minus αp11113872 1113873

Rp2 1 minus αp21113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

I(t)

V(t) Uocv(soc(t)) minus Up1(t) minus Up2(t) minus I(t)R0

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

where R0 is the internal resistance Rp1 and Rp2 are thepolarization resistance Cp1 and Cp2 are the polarizationcapacitance Cap is the estimated capacity I is the loadingcurrent (positive for discharging and negative for charging)and Uocv Up1 Up2 and V denote the open circuit voltage(OCV) polarization voltage and terminal voltage respec-tively Also Δ is the sampling time η is the coulomb effi-ciency with the assumption that η 1 αp1

exp(minus (ΔRp1Cp1)) and αp2 exp(minus (ΔRp2Cp2)) Uocv canbe described as a function of estimated SOC e rela-tionship between Uocv and SOC is defined as follows

Uocv(soc(t)) u0 minusl1

soc(t)minus l2soc(t) + l3 ln(soc(t))

+ l4 ln(1 minus soc(t))

(28)

e OCV parameters u0 l1 l2 l3 and l4 are usuallyobtained through pulse experiments

e square root spherical unscented Kalman filter (SR-SUKF) is proposed for estimating the battery SOC SOC andpolarization voltage Up1 Up2 are chosen as the state variablein the state-space model e input and output of the modelare the measured current I and terminal voltage Vrespectively

e detailed estimation process can be explained asfollows

(1) Initialization1113954x

+0 E x0( 1113857

S+0 chol E x0 minus 1113954x

+0( 1113857 x0 minus 1113954x

+0( 1113857

T1113876 11138771113874 1113875

(29)

where 1113954x+0 is the initial state estimation and S+

0 is thesquare root of the error covariance matrixe weight vector W is defined as follows

W0 isin [0 1)

Wi 1 minus W0

n + 1 i 1 n + 1

(30)

e sigma vector is initialized as follows

χ(1)0 0

χ(1)1

minus 12W1

1113968

χ(1)2

12W1

1113968

(31)

with j 2 n and the sigma vector is iterativelycalculated as follows

χ(j)i

χ(jminus 1)0

0⎡⎢⎣ ⎤⎥⎦ i 0

χ(jminus 1)i

minus 1j(j + 1)W1

1113968

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ i 1 j

0jminus 1

jj(j + 1)W1

1113968

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ i j + 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)


en the sigma points are given by

χtminus 1 1113954xtminus 1 + Stminus 1χ(n)i (33)

(2) Time update

χt|tminus 1 f χtminus 1( 1113857 (34)

1113954xt|tminus 1 1113944n+1

i0Wiχ

(i)t|tminus 1 (35)

St|tminus 1 qr W1 χ1 n+1t | tminus 1 minus 1113954xt | tminus 11113872 1113873Qtminus 1

11139681113966 1113967 (36)

St|tminus 1 cholupdate St | tminus 1 χ0t | tminus 1 minus 1113954xt | tminus 1 W01113864 1113865 (37)

ζttminus 1 h χt | tminus 11113872 1113873 (38)

1113954yt 1113944n+1

i0Wiζ ittminus 1 (39)

where qr denotes the QR decomposition of thecompound matrix in (36) cholupdate denotes theCholesky update of (36)

(3) Measurement update

S1113954yt qr W1 ζ1 n+1t | tminus 1 minus 1113954yt1113872 1113873

Rt

11139681113966 1113967

S1113954yt cholupdate S1113954yttminus 1

ζ0t | tminus 1 minus 1113954yttminus 1 W01113966 1113967

Pxy 1113944n+1

i0Wi χit | tminus 1 minus 1113954xttminus 11113872 1113873 ζ it | tminus 1 minus 1113954yt1113872 1113873

T

Gt PxyST

1113954yt

1113874 1113875

S1113954yt

1113954xt 1113954xt|tminus 1 + Gt yt minus 1113954yt( 1113857

U GtS1113954yt

St cholupdate St | tminus 1 U minus 11113864 1113865

(40)

33 Coestimation Scheme with VFF-RTLS and SOCEstimation e VFF-RTLS capacity identification methodwas combined with an SR-SUKF SOC estimator to calculatethe battery capacity e coestimation scheme is shown inFigure 1

As shown in Figure 1 the variables in VFF-RTLS and SR-SUKF are initialized Each time the battery current andvoltage are measured the SR-SUKF can be conducted andbattery SOC will be updated e measured current is addedup during the capacity estimation update period e dif-ference SOC is obtained at the beginning and end of thecapacity estimation update period en the VFF-RTLS canbe conducted to identify the battery capacity e recentlyupdated capacity will be transferred to the SR-SUKF sectionto replace the previous capacity e SOC is then estimatedthrough SR-SUKF with the updated capacity measuredcurrent and voltage en the algorithm moves to the nextiteration

4 Simulation and Experimental Studies

Simulation and experimental studies are executed to validatethe proposed capacity estimation algorithm for a LiFePO4battery subject to different driving conditions e proposedestimation algorithms show an advantage in terms of esti-mation accuracy All of these tests are programmed andperformed in MATLAB software on a computer with24GHz Intel(R) Core (TM) i3 CPU M370 and 32-bit OS

41 Simulation Studies

411 Simulation Study Dynamic Stress Test (DST)Condition In this scenario the simulation study assumesthat the battery has a constant capacity without any deg-radation and that the battery is subject to a current profilethat is proportional to the power profile in the standard DSTcondition e DST condition is a battery testing programdesigned to mimic the current of an electric car when drivenon a real road e DST current profile is shown inFigure 2(a) and repeated end-to-end until the battery SOCreaches 0

e RLS TLS RTLS and VFF-RTLS run over an updateperiod of 100 seconds e update period can be determined

Initialization of Cap(0) μ(0) λ(0) c1 R(0) b(0) c(0)

For k 1 2

(1) Input the battery SOC difference x(k)

(2) Input the cumulated current y(k)

(3) Update of K(k) with (19)(4) Update of λ0(k) with (20)(5) Update of R(k) b(k) c(k) with (7)ndash(9)(6) Update of A B C with (16)ndash(18)(7) Update of α(k) with (22)(8) Update of Cap(k) with (13)(9) Update of μ(k) with (26)(10) Update of λ(k) with (21)

ALGORITHM 1 Detailed procedures of the proposed recursive total least square algorithm


according to a real application In EVs the current throughthe battery is always measured by a Hall current sensor withcertain high precision In the capacity identification modelthe current is obtained at a frequency of 1 second and theoutput y is calculated by summing the measured currentduring the estimation update time period of 100 secondse model input x is obtained by taking the differencebetween the two SOCs at the beginning and end of eachestimation update period

e initial parameters and noise properties of the pro-posed algorithms are set as follows

Noise in the current measurement is assumed to be arandom white noise with standard deviation σi 0001Aen the noise in the model output y is obtained bysumming the cumulated current noise over 100 secondswhich is expressed as σy

100

radiclowastΔlowastσi3600 Noise in the

SOC is assumed to be random white noise with standarddeviation σs 001 en noise in the model input x isderived as σx

2

radiclowastσs e tuning parameter c1 is set to

0001 and the initial forgetting factor is set to μ 099 eactual capacity of the battery is set to be 125Ah

e simulation results under the DST condition arepresented in Figure 2 Specifically Figure 2(a) plots theDST current profile for one cycle Figure 2(b) plots thewhole current profile during the entire running timeFigure 2(c) gives the battery SOC Figure 2(d) shows theestimated capacity computed by the estimation algorithmsand the actual capacity of the battery and Figure 2(e)shows the estimation error as a percentage From theestimation results we can see that the proposed VFF-RTLS algorithm outperforms the other three algorithmswith an estimation error below 1 Of the other threealgorithms the traditional RTLS is best with an esti-mation error below 4 Since the traditional RLS only

consider the error on the current measurement withoutthe SOC estimation error the result shows that RLSconverges to a biased value and has the worst performancewith a large estimation error Owing to the errors in SOCestimation RLS converges to a steady state with a lowerspeede SVD based TLS algorithm converges to a steadyvalue with the increasing data input and it is worse thanthe RTLS e RTLS with an appropriate initial state canconverge faster than the SVD based TLS Our proposedmethod has the best performance due to the variableforgetting factor

412 Simulation Study Highway Cycles In this scenariothe simulation study assumes that the battery has a constantcapacity without any degradation and that the battery issubject to a current profile proportional to the speed profilein the highway cycles e highway cycles are designed totest EVs driven on highways at a high speed In a highwaydriving environment the vehicle always runs at a relativelyhigh speede highway cycle profile is shown in Figure 3(a)and repeated end-to-end until the battery SOC reaches 0

e RLS TLS RTLS and VFF-RTLS run over a period of100 seconds In the capacity identification model the cur-rent is obtained at a frequency of 1 second e model inputx and output y are obtained in the same way as for the DSTdriving condition e initial parameters and noise prop-erties of the model inputoutput are set to be the same as forthe DST condition e initial forgetting factor is set toμ 099 In this section we use a battery with an actualcapacity of 100Ah

e simulation results under the highway cycles arepresented in Figure 3 Particularly Figure 3(a) plots thecurrent profile for one highway cycle Figure 3(b) plots the

V

R(k) = μR(k ndash 1) + xndash(k)xndashT(k)

c(k) = μc(k ndash 1) + xndash(k)yndashT(k)

b(k) = μb(k ndash 1) + xndash(k)yndash(k)

μ(k) = μ(k ndash 1) + γ1partJ(Cap(k))partμ

Measurement

I

VFF-RTLS capacity identification method

α(k) = ndashB + B2 ndash4AC 2A

Cap(k) = Cap(k ndash 1) + α(k)xndash(k)

SR-SUKF for SOC estimation

χtndash1 = xtndash1 + Stndash1χi(n)

χt|tndash1 = f (χtndash1)

xt|tndash1 =n+1

i=0Wiχ(i)

t|tndash1

n+1

i=0Pxy = wi(χit|tndash1 ndash xt|tndash1)(ζit|tndash1 ndash yt)T

Gt = (Pxy ST) Syt yt

xt = xt|tndash1 + Gt(yt ndash yt)

Cap

SOC

Figure 1 Diagram of the coestimation scheme with VFF-RTLS and SOC estimation


whole current profile during the entire running timeFigure 3(c) gives the battery SOC Figure 3(d) shows theestimated capacity computed using the estimation

algorithms and the actual capacity of the battery andFigure 3(e) shows the estimation error as a percentage Inthis working condition the results have shown a similar

200 400 600 800 1000 1200 1400 1600 18000Time (s)

ndash50ndash40ndash30ndash20ndash10

01020304050

Curr

ent a

mpl

itude

(A)

(a)

times1041 2 3 4 50

Time (s)


01020304050

Curr

ent a

mpl

itude

(A)

(b)

times1041 2 3 4 50

Time (s)

0

02

04

06

08

1

SOC

(c)

240 48080 120 160 200 280 320 360 400 440 5200 40Estimation update index

RLSTLSRTLS

VFF-RTLSActual capacity

9

10

11

12

13

14

15

16

Estim

ated

capa

city

(Ah)

(d)

100 200 300 400 5000Estimation update index

0

5

10

15

20

25

30

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)

Figure 2 Estimated capacity from the VFF-RTLS and the other three algorithms under the DSTcondition (a) DSTcurrent profile for onecycle (b) whole current profile (c) input SOC with noise (d) estimated capacity (e) estimation error as a percentage


phenomenon as in the DST condition From the estimationresults we can see that the proposed VFF-RTLS algorithmoutperforms the other three algorithms with an estimation

error below 01 Of the other three algorithms the RTLS isthe best with an estimation error below 04e RLS is theworst with a large capacity estimation error

100 200 300 400 500 600 700 8000Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80Cu

rren

t am

plitu

de (A

)

(a)

times10405 1 15 2 250

Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80

Curr

ent a

mpl

itude

(A)

(b)

05 1 15 2 250Time (s)

ndash02

0

02

04

06

08

1

12

SOC

times104

(c)

50 100 150 2000Estimation update index

98

985

99

995

100

1005

101

1015

102

Estim

ated

capa

city

(Ah)

RLSTLSRTLS


(d)


0

05

1

15

2

25

3

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)

Figure 3 Estimated capacity from the VFF-RTLS and the other three algorithms under the highway cycles (a) highway current profile forone cycle (b) whole current profile (c) input SOC with noise (d) estimated capacity (e) estimation error as a percentage


413 Simulation Study Battery with Degradation BehaviorIn this section a simulation study is carried out to verify thatthe proposed capacity estimation algorithm can effectivelytrack a variable battery capacity e battery is assumed tohave a variable capacity to mimic degradation behavior andthe battery SOC is in a range from 90ndash10 e capacitydecreases from 100Ah to 90Ah e model input x issimulated by a uniform random value from [minus 08 +08] andthen the model output y is simulated by multiplying theactual capacity by the model input x e initial parametersand the noise properties for the model inputoutput are setto be the same as for the highway cyclese initial forgettingfactor is set to μ 099

e simulation results are presented in Figure 4 Spe-cifically Figure 4(a) shows the estimated capacity computedby the proposed algorithms and the actual capacity of thebattery and Figure 4(b) shows the estimation error of thealgorithms as a percentage From the estimation results wecan see that the proposed VFF-RTLS algorithm outperformsthe other three algorithms with an estimation error below05 Of the other three algorithms the RTLS algorithm is thebest with an estimation error below 1 Both the RLS and TLSperform poorly with large estimation errors Without con-sidering the error in SOC estimation the RLS algorithm doesnot track the varying battery capacity And the SVD basedTLS can not track the capacity precisely and the error growswith the increasing data input e proposed method (VFF-RTLS) in this paper can track the varying battery capacitybetter than the RTLS because of the variable forgetting factorwhich improves the traditional RTLS performance

42 Experimental Study To validate the proposed algorithmagainst experimental data an experiment was conducted ona test bench where a lithium-iron phosphate battery of 28Ahwas kept in a temperature-controlled box with ambienttemperature 252degC BTS-5V-200A battery test equipment

was used to chargedischarge the battery with a maximumvoltage of 5V and maximum chargedischarge current of200A It has high accuracy in voltage and current mea-surement with plusmn01 of FS A BTS 75X test softwaresystem was installed on a laptop computer which generatedthe test current profile for the battery en the experimentbegan with the battery fully chargede current and voltagedata were measured with a frequency of 1 second e truebattery capacity was extracted offline from a full dischargetest with a small current e battery parameters wereidentified offline using constant-current pulse charge-dis-charge experiments

e SOC estimation model parameters obtained were asfollows R0 11mΩ Rp1 19mΩ Rp2 36mΩCp1 58605KF and Cp2 9142KF

e OCV parameters identified offline were as followsu0 338 l1 minus 303e minus 5 l2 01327 l3 00778 andl4 minus 000284

e capacity estimation is performed as in Section 33e initial forgetting factor is set to μ 099e experimentresults are presented in Figure 5 Specifically Figure 5(a)plots the current profile for one cycle Figure 5(b) plots thewhole current profile over the whole running timeFigure 5(c) gives the estimated battery SOC Figure 5(d)shows the estimated capacity computed by the estimationalgorithms and the actual capacity of the battery andFigure 5(e) shows the estimation error as a percentage

In the experimental studies we find a similar phe-nomenon as in the simulation studies From the estimationresults we can see that the proposed VFF-RTLS algorithmoutperforms the other three algorithms with an estimationerror below 01 Owing to the variable forgetting factor theVFF-RTLS has a better performance than the conventionalRTLS Of the other three algorithms the RTLS algorithm isbest with an estimation error below 03 e RLS has theworst performance with a large estimation error


90

92

94

96

98

100

102Es

timat

ed ca

paci

ty (A

h)

RLSTLSRTLS


(a)

0123456789

10

Estim

atio

n er

ror (

)


RLSTLS

RTLSVFF-RTLS

(b)

Figure 4 Estimated capacity from the VFF-RTLS and the other three algorithms under the degradation condition (a) estimated capacity(b) estimation error as a percentage


200 400 600 800 1000 1200 14000Time (s)

ndash20

ndash10

0

10

20

30Cu

rren

t am

plitu

de (A

)

(a)

times104

ndash20

ndash10

0

10

20

30

Curr

ent a

mpl

itude

(A)

1 2 3 4 5 60Time (s)

(b)

times1041 2 3 4 50 6

Time (s)

0

02

04

06

08

1

SOC

(c)

27272274276278

28282284286288

29

Estim

ated

capa

city

(Ah)

80 160 240 320 400 480 560 6400Estiamtion update index

RLSTLSRTLS


(d)

100 200 300 400 500 6000Estimation update index

005

115

225

335

445

5

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)

Figure 5 Estimated capacity from the VFF-RTLS and the other three algorithms using experimental data (a) experimental current profilefor one cycle (b) whole current profile (c) estimated SOC (d) estimated capacity (e) estimation error as a percentage


5 Conclusion

In order to obtain a capacity estimation with high accuracythe improved RTLS algorithm with a variable forgettingfactor (VFF-RTLS) is adopted in this paper It is based onconstrained Rayleigh quotient to consider the errors in thecapacity model inputoutput e variable forgetting factoris developed to further improve the estimation accuracyCompared with other algorithms through the simulationand experimental studies the results have shown that theproposed VFF-RTLS has better estimation performance inreal time In future work the thermal effects on the capacityestimation will be studied and more realistic experimentswill be designed to validate the proposed algorithm

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is project was partly supported by the National NaturalScience Foundation Joint Fund Project of China (U1664257)and partly supported by Guangxi Scientific Research andTechnology Development Plan Project (14122002-4)

References

[1] M A Hannan M M Hoque A Mohamed and A AyobldquoReview of energy storage systems for electric vehicle appli-cations issues and challengesrdquo Renewable and SustainableEnergy Reviews vol 69 no 5 pp 771ndash789 2017

[2] K Striebel A Guerfi J Shim M Armand M Gauthier andK Zaghib ldquoLiFePO4gelnatural graphite cells for the BATTprogramrdquo Journal of Power Sources vol 119ndash121 no 6pp 951ndash954 2002

[3] L Zheng L Zhang J Zhu G Wang and J Jiang ldquoCo-es-timation of state-of-charge capacity and resistance for lith-ium-ion batteries based on a high-fidelity electrochemicalmodelrdquo Applied Energy vol 180 pp 424ndash434 2016

[4] M Dubarry and B Y Liaw ldquoIdentify capacity fadingmechanism in a commercial LiFePO4 cellrdquo Journal of PowerSources vol 194 no 1 pp 541ndash549 2009

[5] L Lu X Han J Li J Hua and M Ouyang ldquoA review on thekey issues for lithium-ion battery management in electricvehiclesrdquo Journal of Power Sources vol 226 no 3 pp 272ndash288 2013

[6] X Feng J Li M Ouyang L Lu J Li and X He ldquoUsingprobability density function to evaluate the state of health oflithium-ion batteriesrdquo Journal of Power Sources vol 232no 18 pp 209ndash218 2013

[7] A Farmann W Waag A Marongiu and D U SauerldquoCritical review of on-board capacity estimation techniquesfor lithium-ion batteries in electric and hybrid electric ve-hiclesrdquo Journal of Power Sources vol 281 pp 114ndash130 2015

[8] X Han M Ouyang L Lu J Li Y Zheng and Z Li ldquoAcomparative study of commercial lithium ion battery cycle life

in electrical vehicle aging mechanism identificationrdquo Journalof Power Sources vol 251 no 2 pp 38ndash54 2014

[9] I Fernandez C Calvillo A Sanchez-Miralles and J BoalldquoCapacity fade and aging models for electric batteries andoptimal charging strategy for electric vehiclesrdquo Energyvol 60 pp 35ndash43 2013

[10] P Ramadass B Haran R White and B N Popov ldquoMath-ematical modeling of the capacity fade of Li-ion cellsrdquo Journalof Power Sources vol 123 no 2 pp 230ndash240 2003

[11] G Ning and B N Popov ldquoCycle life modeling of lithium-ionbatteriesrdquo Journal of gte Electrochemical Society vol 151no 10 pp 1584ndash1591 2004

[12] C Weng Y Cui J Sun and H Peng ldquoOn-board state ofhealth monitoring of lithium-ion batteries using incrementalcapacity analysis with support vector regressionrdquo Journal ofPower Sources vol 235 no 4 pp 36ndash44 2013

[13] M Berecibar M Garmendia I Gandiaga J Crego andI Villarreal ldquoState of health estimation algorithm of LiFePO4battery packs based on differential voltage curves for batterymanagement system applicationrdquo Energy vol 103 pp 784ndash796 2016

[14] C Weng X Feng J Sun and H Peng ldquoState-of-healthmonitoring of lithium-ion battery modules and packs viaincremental capacity peak trackingrdquo Applied Energy vol 180pp 360ndash368 2016

[15] L Wang C Pan L Liu Y Cheng and X Zhao ldquoOn-boardstate of health estimation of LiFePO4 battery pack throughdifferential voltage analysisrdquo Applied Energy vol 168pp 465ndash472 2016

[16] I Bloom A N Jansen D P Abraham et al ldquoDifferentialvoltage analyses of high-power lithium-ion cellsrdquo Journal ofPower Sources vol 139 no 1-2 pp 295ndash303 2005

[17] X Chen W Shen M Dai Z Cao J Jin and A KapoorldquoRobust adaptive sliding-mode observer using RBF neuralnetwork for lithium-ion battery state of charge estimation inelectric vehiclesrdquo IEEE Transactions on Vehicular Technologyvol 65 no 4 pp 1936ndash1947 2015

[18] A Eddahech O Briat N Bertrand J-Y Deletage andJ-M Vinassa ldquoBehavior and state-of-health monitoring of Li-ion batteries using impedance spectroscopy and recurrentneural networksrdquo International Journal of Electrical Power ampEnergy Systems vol 42 no 1 pp 487ndash494 2012

[19] D Andre A Nuhic T Soczka-Guth and D U SauerldquoComparative study of a structured neural network and anextended Kalman filter for state of health determination oflithium-ion batteries in hybrid electricvehiclesrdquo EngineeringApplications of Artificial Intelligence vol 26 no 3 pp 951ndash961 2013

[20] H Pan L Zhiqiang B Fu HWei L Chen andG UniversityldquoOnline estimation of lithium-ion batteryrsquos state of healthusing extreme learning machinerdquo Automotive Engineeringvol 39 no 12 pp 1375ndash1381 2017

[21] Y Zhang and B Guo ldquoOnline capacity estimation of lithium-ion batteries based on novel feature extraction and adaptivemulti-kernel relevance vector machinerdquo Energies vol 8no 11 pp 12439ndash12457 2015

[22] Y Peng S Lu W Xie D Liu and H Liao ldquoLithium-ionbattery remaining useful life estimation based on ensemblelearning with LS-SVM algorithmrdquo Advances in BatteryManufacturing Service and Management Systems vol 9pp 217ndash232 2016

[23] J Wei G Dong and Z Chen ldquoRemaining useful life pre-diction and state of health diagnosis for lithium-ion batteriesusing particle filter and support vector regressionrdquo IEEE


Transactions on Industrial Electronics vol 65 no 7pp 5634ndash5643 2018

[24] H Dai G Zhao M Lin J Wu and G Zheng ldquoA novelestimation method for the state of health of lithium-ionbattery using prior knowledge-based neural network andMarkov chainrdquo IEEE Transactions on Industrial Electronicsvol 66 no 10 pp 7706ndash7716 2018

[25] A A Hussein ldquoCapacity fade estimation in electric vehicle Li-ion batteries using artificial neural networksrdquo IEEE Trans-actions on Industry Applications vol 51 no 3 pp 2321ndash23302015

[26] W Ji C Zhang and Z Chen ldquoAn online method for lithium-ion battery remaining useful life estimation using importancesampling and neural networksrdquo Applied Energy vol 173pp 134ndash140 2016

[27] C Hu B D Youn T Kim and J Chung ldquoOnline estimationof lithium-ion battery state-of-charge and capacity with amultiscale filtering techniquerdquo Transition vol 1 no 1 2011

[28] J Kim S Lee and B H Cho ldquoComplementary cooperationalgorithm based on DEKF combined with pattern recognitionfor SOCcapacity estimation and SOH predictionrdquo IEEETransactions on Power Electronics vol 27 no 1 pp 436ndash4512011

[29] X Li Z Wang and L Zhang ldquoCo-estimation of capacity andstate-of-charge for lithium-ion batteries in electric vehiclesrdquoEnergy vol 174 pp 33ndash44 2019

[30] X Tang X Mao J Lin and B Koch ldquoCapacity estimation forLi-ion batteriesrdquo in Proceedings of the 2011 American ControlConference pp 947ndash952 San Francisco CA USA 2011

[31] P Shen M Ouyang L Lu J Li and X Feng ldquoe co-es-timation of state of charge state of health and state offunction for lithium-ion batteries in electric vehiclesrdquo IEEETransactions on Vehicular Technology vol 67 no 1 pp 92ndash103 2018

[32] B Balasingam G V Avvari B Pattipati K R Pattipati andY Bar-Shalom ldquoA robust approach to battery fuel gaugingpart II real time capacity estimationrdquo Journal of PowerSources vol 269 pp 949ndash961 2014

[33] G L Plett ldquoRecursive approximate weighted total leastsquares estimation of battery cell total capacityrdquo Journal ofPower Sources vol 196 no 4 pp 2319ndash2331 2011

[34] G H Golub and C F Van Loan ldquoAn analysis of the total leastsquares problemrdquo SIAM Journal on Numerical Analysisvol 17 no 6 pp 883ndash893 1980

[35] G H Golub and C F Van Loan Matrix Computations JohnWiley and Sons Hoboken NJ USA 3rd edition 1996

[36] I Markovsky and S Van Huffel ldquoOverview of total least-squares methodsrdquo Signal Processing vol 87 no 10pp 2283ndash2302 2007

[37] T Kim Y Wang H Fang et al ldquoModel-based conditionmonitoring for lithium-ion batteriesrdquo Journal of PowerSources vol 295 pp 16ndash27 2015

[38] C E Davila ldquoRecursive total least squares algorithms foradaptive filteringrdquo in Proceedings of the International Con-ference on Acoustics Speech and Signal Processing vol 3pp 1853ndash1856 Toronto Canada May 1991

[39] D-Z Feng X-D Zhang D-X Chang and W X Zheng ldquoAfast recursive total least squares algorithm for adaptive FIRfilteringrdquo IEEE Transactions on Signal Processing vol 52no 10 pp 2729ndash2737 2004

[40] S Sagara and K Wada ldquoOn-line modified least-squares pa-rameter estimation of linear discrete dynamic systemsrdquo In-ternational Journal of Control vol 25 no 3 pp 329ndash3431977

[41] C E Davila ldquoAn efficient recursive total least squares algo-rithm for FIR adaptive filteringrdquo IEEE Transactions on SignalProcessing vol 42 no 2 pp 268ndash280 1994

[42] M Levin ldquoEstimation of a system pulse transfer function inthe presence of noiserdquo IEEE Transactions on AutomaticControl vol 9 no 3 pp 229ndash235 1964


methods such as RBF NN [16 17] adaptive recurrent NN[18] structured NN (SNN) [19] extreme learning machine[20] adaptive multikernel relevance vector machine [21]ensemble learning with LS_SVM algorithm [22] supportvector regression [23] and others ese methods usuallyrequire a large amount of experimental data so that they cantrain a more precise network for battery capacity estimation[24ndash26] is is a difficult and time-consuming task

Least squares or Kalman filters offer the most promisingcompromise between estimation accuracy and low com-putational effort [7] e dual Kalman filter-based methodsjointly estimate the capacity and SOC together [27 28]However the employment of dual filters brings about anincreased computational burden and the multiscale problemmay cause algorithm divergence [29] Some approachesbased on coulomb counting have been researched to esti-mate the capacity in real time [30 31] In these studies theSOC estimation error is not considered which could affectthe capacity estimation accuracy Since the recursive leastsquares (RLS) do not consider the SOC error the total leastsquares (TLS) algorithm is introduced to improve the es-timation accuracy by considering the SOC estimation error[32 33] However the TLS algorithm is performed bycomputing the singular value decomposition (SVD) of themodelrsquos covariance matrix Since the multiplication opera-tions of SVD for an N by N matrix have computationalcomplexity of O(N3) it can incur high computationalcomplexity [34ndash36] In [33] a recursive approximateweighted total least squares algorithm was proposed toimprove the TLS performance In [37] another recursivetotal least squares algorithm was employed to solve thecapacity estimation In [38] the constrained Rayleighquotient was developed to find the TLS solution with thereducing computational complexity In [39] a fast recursivetotal least square algorithm was proposed to solve the TLSsolution by using the adaptation minimization of the con-strained Rayleigh quotient However the forgetting factor isfixed as a constant in the above studies It could not handlethe tradeoff between robustness and tracking capabilitywhich affects the estimation performance Specifically al-gorithms with smaller forgetting factor weigh more ontracking the time-varying capacity at the expense of moresensitivity to the current measurement and SOC estimationerror on the contrary large forgetting factor improvesrobustness but compromises the tracking capability

Based on the above algorithms we propose an approachemploying an improved recursive total least squares withvariable forgetting factor (VFF-RTLS) and a coulombcounting model considering errors in both the estimatedSOC and the current measurement e forgetting factor iscontrolled by minimizing the constrained Rayleigh quotientthrough the steepest descent method is approach willimprove capacity estimation performance e rest of thispaper is constructed as follows Section 2 describes thecapacity estimation model for the battery Section 3 presentsthe proposed VFF-RTLS approach for battery capacity es-timation Section 4 presents the simulation and experimentalstudies using the proposed algorithms and the conclusionsare summarized in Section 5

2 Capacity Estimation Model of theLiFePO4 Battery

When EVs are running in a real-life environment the SOCof their lithium-iron phosphate batteries is estimated by theunscented Kalman filter through the measured current andvoltage In order to predict the battery capacity we use thefollowing coulomb counting equation in terms of themeasured current and battery SOC

soc(k + 1) + vs(k + 1) soc(k) + vs(k) +ηΔ I(k) + wi(k)( 1113857

3600Cap

(1)

where Cap is the estimated capacity of the lithium-ironphosphate battery in ampere-hours (Ah) and I(k) is themeasured current η is coulomb efficiency with the as-sumption that η 1 Δ is the sample time wi(k) is thecurrent measurement error with standard deviation σivs(k) denotes the SOC estimation error

We convert equation (1) into an iterative form as follows

soc kLb( 1113857 minus soc kLb minus Lb + 1( 1113857 + vs(k)

η1113936

jkLb

jkLbminus Lb+1 Δ I(j) + wi(j)( 1113857

3600Cap

(2)

where Lb is the capacity update time period k 1 2 Nand vs(k) is the differential SOC error vs(k) vs(kLb) minus

vs(kLb minus Lb + 1)en equation (2) can be simplified into the following

expression

y + wi Caplowast x + vs( 1113857 (3)

where x soc(kLb) minus soc(kLb minus Lb + 1) denotes the differ-ence between the battery SOCs at the beginning and end ofthe update period y η1113936

jkLb

jkLbminus Lb+1 ΔI(j)3600 denotes thecumulated measured current within the update time periodLb and wi is the cumulated measured current errorwi η1113936

jkLb

jkLbminus Lb+1 Δwi(j)3600

3 Improved Recursive Total Least SquaresAlgorithm with Variable Forgetting Factor

31 Recursive Total Least Squares with Variable ForgettingFactor (VFF-RTLS) From the capacity model in (3) wecan see that there are errors in both the model input andoutput erefore this section proposes a constrainedRayleigh quotient-based RTLS algorithm with a variableforgetting factor for the capacity estimation of LiFePO4batteries

To facilitate the presentation of the proposed algorithmthe autocorrelation matrix of the capacity model input x(k)

is defined as follows

R E x(k)xT(k)1113960 1113961 (4)

e augmented input vector is defined asxa [x(k) y(k)] and its autocorrelation matrix can berepresented as follows


Ra

E xa(k)x

aprime(k)1113960 1113961 R b

b c1113890 1113891 (5)



Ra(k)

R(k) b(k)

bT(k) c(k)1113890 1113891 μR

a(k minus 1) + x

a(k)x

aprime(k)

(6)



(k) (7)





F (10)



F(q) qTψTψq

qTDq (11)















zα(k) 0 (14)


Aα2(k) + Bα(k) + C 0 (15)


A KT

(k)x(k)1113960 1113961[Cap(k minus 1)x(k)]

minus x(k)2


T(k)y(k)1113960 1113961

(16)


21113960 1113961

minus λ0(k) x(k)2

1113960 1113961(17)


T(k)b(k)1113960 1113961



where

K(k) R(k)x(k) (19)




β + Cap(k)21113960 1113961 (21)


α(k) minus B +

B2 minus 4AC

radic

2A (22)




zμ (23)







zJ(Cap(k))


β + CapTCap (25)



β + Cap(k)TCap(k)

(26)



soc(t + 1)

Up1(t + 1)

Up2(t + 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1 0 0

0 αp1 0

0 0 αp2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

soc(t)

Up1(t)

Up2(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+

ηΔ3600Cap

Rp1 1 minus αp11113872 1113873

Rp2 1 minus αp21113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

I(t)


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)






(28)





+0 E x0( 1113857


+0( 1113857 x0 minus 1113954x

+0( 1113857

T1113876 11138771113874 1113875

(29)



W0 isin [0 1)

Wi 1 minus W0

n + 1 i 1 n + 1

(30)


χ(1)0 0

χ(1)1

minus 12W1

1113968

χ(1)2

12W1

1113968

(31)


χ(j)i

χ(jminus 1)0

0⎡⎢⎣ ⎤⎥⎦ i 0

χ(jminus 1)i

minus 1j(j + 1)W1

1113968

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ i 1 j

0jminus 1

jj(j + 1)W1

1113968

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ i j + 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)




(2) Time update


1113954xt|tminus 1 1113944n+1

i0Wiχ

(i)t|tminus 1 (35)


11139681113966 1113967 (36)



1113954yt 1113944n+1





Rt

11139681113966 1113967



Pxy 1113944n+1


T

Gt PxyST

1113954yt

1113874 1113875

S1113954yt


U GtS1113954yt


(40)









For k 1 2









100



2








V





Measurement

I







xt|tndash1 =n+1

i=0Wiχ(i)

t|tndash1

n+1




Cap

SOC





200 400 600 800 1000 1200 1400 1600 18000Time (s)


01020304050

Curr

ent a

mpl

itude

(A)

(a)

times1041 2 3 4 50

Time (s)


01020304050

Curr

ent a

mpl

itude

(A)

(b)

times1041 2 3 4 50

Time (s)

0

02

04

06

08

1

SOC

(c)


RLSTLSRTLS


9

10

11

12

13

14

15

16

Estim

ated

capa

city

(Ah)

(d)


0

5

10

15

20

25

30

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)





100 200 300 400 500 600 700 8000Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80Cu

rren

t am

plitu

de (A

)

(a)

times10405 1 15 2 250

Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80

Curr

ent a

mpl

itude

(A)

(b)

05 1 15 2 250Time (s)

ndash02

0

02

04

06

08

1

12

SOC

times104

(c)


98

985

99

995

100

1005

101

1015

102

Estim

ated

capa

city

(Ah)

RLSTLSRTLS


(d)


0

05

1

15

2

25

3

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)












90

92

94

96

98

100

102Es

timat

ed ca

paci

ty (A

h)

RLSTLSRTLS


(a)

0123456789

10

Estim

atio

n er

ror (

)


RLSTLS

RTLSVFF-RTLS

(b)



200 400 600 800 1000 1200 14000Time (s)

ndash20

ndash10

0

10

20

30Cu

rren

t am

plitu

de (A

)

(a)

times104

ndash20

ndash10

0

10

20

30

Curr

ent a

mpl

itude

(A)

1 2 3 4 5 60Time (s)

(b)

times1041 2 3 4 50 6

Time (s)

0

02

04

06

08

1

SOC

(c)

27272274276278

28282284286288

29

Estim

ated

capa

city

(Ah)


RLSTLSRTLS


(d)


005

115

225

335

445

5

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)



5 Conclusion


Data Availability




Acknowledgments


References















































Ra

E xa(k)x

aprime(k)1113960 1113961 R b

b c1113890 1113891 (5)



Ra(k)

R(k) b(k)

bT(k) c(k)1113890 1113891 μR

a(k minus 1) + x

a(k)x

aprime(k)

(6)



(k) (7)





F (10)



F(q) qTψTψq

qTDq (11)















zα(k) 0 (14)


Aα2(k) + Bα(k) + C 0 (15)


A KT

(k)x(k)1113960 1113961[Cap(k minus 1)x(k)]

minus x(k)2


T(k)y(k)1113960 1113961

(16)


21113960 1113961

minus λ0(k) x(k)2

1113960 1113961(17)


T(k)b(k)1113960 1113961



where

K(k) R(k)x(k) (19)




β + Cap(k)21113960 1113961 (21)


α(k) minus B +

B2 minus 4AC

radic

2A (22)




zμ (23)







zJ(Cap(k))


β + CapTCap (25)



β + Cap(k)TCap(k)

(26)



soc(t + 1)

Up1(t + 1)

Up2(t + 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1 0 0

0 αp1 0

0 0 αp2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

soc(t)

Up1(t)

Up2(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+

ηΔ3600Cap

Rp1 1 minus αp11113872 1113873

Rp2 1 minus αp21113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

I(t)


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)






(28)





+0 E x0( 1113857


+0( 1113857 x0 minus 1113954x

+0( 1113857

T1113876 11138771113874 1113875

(29)



W0 isin [0 1)

Wi 1 minus W0

n + 1 i 1 n + 1

(30)


χ(1)0 0

χ(1)1

minus 12W1

1113968

χ(1)2

12W1

1113968

(31)


χ(j)i

χ(jminus 1)0

0⎡⎢⎣ ⎤⎥⎦ i 0

χ(jminus 1)i

minus 1j(j + 1)W1

1113968

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ i 1 j

0jminus 1

jj(j + 1)W1

1113968

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ i j + 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)




(2) Time update


1113954xt|tminus 1 1113944n+1

i0Wiχ

(i)t|tminus 1 (35)


11139681113966 1113967 (36)



1113954yt 1113944n+1





Rt

11139681113966 1113967



Pxy 1113944n+1


T

Gt PxyST

1113954yt

1113874 1113875

S1113954yt


U GtS1113954yt


(40)









For k 1 2









100



2








V





Measurement

I







xt|tndash1 =n+1

i=0Wiχ(i)

t|tndash1

n+1




Cap

SOC





200 400 600 800 1000 1200 1400 1600 18000Time (s)


01020304050

Curr

ent a

mpl

itude

(A)

(a)

times1041 2 3 4 50

Time (s)


01020304050

Curr

ent a

mpl

itude

(A)

(b)

times1041 2 3 4 50

Time (s)

0

02

04

06

08

1

SOC

(c)


RLSTLSRTLS


9

10

11

12

13

14

15

16

Estim

ated

capa

city

(Ah)

(d)


0

5

10

15

20

25

30

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)





100 200 300 400 500 600 700 8000Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80Cu

rren

t am

plitu

de (A

)

(a)

times10405 1 15 2 250

Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80

Curr

ent a

mpl

itude

(A)

(b)

05 1 15 2 250Time (s)

ndash02

0

02

04

06

08

1

12

SOC

times104

(c)


98

985

99

995

100

1005

101

1015

102

Estim

ated

capa

city

(Ah)

RLSTLSRTLS


(d)


0

05

1

15

2

25

3

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)












90

92

94

96

98

100

102Es

timat

ed ca

paci

ty (A

h)

RLSTLSRTLS


(a)

0123456789

10

Estim

atio

n er

ror (

)


RLSTLS

RTLSVFF-RTLS

(b)



200 400 600 800 1000 1200 14000Time (s)

ndash20

ndash10

0

10

20

30Cu

rren

t am

plitu

de (A

)

(a)

times104

ndash20

ndash10

0

10

20

30

Curr

ent a

mpl

itude

(A)

1 2 3 4 5 60Time (s)

(b)

times1041 2 3 4 50 6

Time (s)

0

02

04

06

08

1

SOC

(c)

27272274276278

28282284286288

29

Estim

ated

capa

city

(Ah)


RLSTLSRTLS


(d)


005

115

225

335

445

5

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)



5 Conclusion


Data Availability




Acknowledgments


References
















































zμ (23)







zJ(Cap(k))


β + CapTCap (25)



β + Cap(k)TCap(k)

(26)



soc(t + 1)

Up1(t + 1)

Up2(t + 1)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1 0 0

0 αp1 0

0 0 αp2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

soc(t)

Up1(t)

Up2(t)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+

ηΔ3600Cap

Rp1 1 minus αp11113872 1113873

Rp2 1 minus αp21113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

I(t)


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)






(28)





+0 E x0( 1113857


+0( 1113857 x0 minus 1113954x

+0( 1113857

T1113876 11138771113874 1113875

(29)



W0 isin [0 1)

Wi 1 minus W0

n + 1 i 1 n + 1

(30)


χ(1)0 0

χ(1)1

minus 12W1

1113968

χ(1)2

12W1

1113968

(31)


χ(j)i

χ(jminus 1)0

0⎡⎢⎣ ⎤⎥⎦ i 0

χ(jminus 1)i

minus 1j(j + 1)W1

1113968

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ i 1 j

0jminus 1

jj(j + 1)W1

1113968

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ i j + 1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(32)




(2) Time update


1113954xt|tminus 1 1113944n+1

i0Wiχ

(i)t|tminus 1 (35)


11139681113966 1113967 (36)



1113954yt 1113944n+1





Rt

11139681113966 1113967



Pxy 1113944n+1


T

Gt PxyST

1113954yt

1113874 1113875

S1113954yt


U GtS1113954yt


(40)









For k 1 2









100



2








V





Measurement

I







xt|tndash1 =n+1

i=0Wiχ(i)

t|tndash1

n+1




Cap

SOC





200 400 600 800 1000 1200 1400 1600 18000Time (s)


01020304050

Curr

ent a

mpl

itude

(A)

(a)

times1041 2 3 4 50

Time (s)


01020304050

Curr

ent a

mpl

itude

(A)

(b)

times1041 2 3 4 50

Time (s)

0

02

04

06

08

1

SOC

(c)


RLSTLSRTLS


9

10

11

12

13

14

15

16

Estim

ated

capa

city

(Ah)

(d)


0

5

10

15

20

25

30

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)





100 200 300 400 500 600 700 8000Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80Cu

rren

t am

plitu

de (A

)

(a)

times10405 1 15 2 250

Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80

Curr

ent a

mpl

itude

(A)

(b)

05 1 15 2 250Time (s)

ndash02

0

02

04

06

08

1

12

SOC

times104

(c)


98

985

99

995

100

1005

101

1015

102

Estim

ated

capa

city

(Ah)

RLSTLSRTLS


(d)


0

05

1

15

2

25

3

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)












90

92

94

96

98

100

102Es

timat

ed ca

paci

ty (A

h)

RLSTLSRTLS


(a)

0123456789

10

Estim

atio

n er

ror (

)


RLSTLS

RTLSVFF-RTLS

(b)



200 400 600 800 1000 1200 14000Time (s)

ndash20

ndash10

0

10

20

30Cu

rren

t am

plitu

de (A

)

(a)

times104

ndash20

ndash10

0

10

20

30

Curr

ent a

mpl

itude

(A)

1 2 3 4 5 60Time (s)

(b)

times1041 2 3 4 50 6

Time (s)

0

02

04

06

08

1

SOC

(c)

27272274276278

28282284286288

29

Estim

ated

capa

city

(Ah)


RLSTLSRTLS


(d)


005

115

225

335

445

5

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)



5 Conclusion


Data Availability




Acknowledgments


References

















































(2) Time update


1113954xt|tminus 1 1113944n+1

i0Wiχ

(i)t|tminus 1 (35)


11139681113966 1113967 (36)



1113954yt 1113944n+1





Rt

11139681113966 1113967



Pxy 1113944n+1


T

Gt PxyST

1113954yt

1113874 1113875

S1113954yt


U GtS1113954yt


(40)









For k 1 2









100



2








V





Measurement

I







xt|tndash1 =n+1

i=0Wiχ(i)

t|tndash1

n+1




Cap

SOC





200 400 600 800 1000 1200 1400 1600 18000Time (s)


01020304050

Curr

ent a

mpl

itude

(A)

(a)

times1041 2 3 4 50

Time (s)


01020304050

Curr

ent a

mpl

itude

(A)

(b)

times1041 2 3 4 50

Time (s)

0

02

04

06

08

1

SOC

(c)


RLSTLSRTLS


9

10

11

12

13

14

15

16

Estim

ated

capa

city

(Ah)

(d)


0

5

10

15

20

25

30

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)





100 200 300 400 500 600 700 8000Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80Cu

rren

t am

plitu

de (A

)

(a)

times10405 1 15 2 250

Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80

Curr

ent a

mpl

itude

(A)

(b)

05 1 15 2 250Time (s)

ndash02

0

02

04

06

08

1

12

SOC

times104

(c)


98

985

99

995

100

1005

101

1015

102

Estim

ated

capa

city

(Ah)

RLSTLSRTLS


(d)


0

05

1

15

2

25

3

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)












90

92

94

96

98

100

102Es

timat

ed ca

paci

ty (A

h)

RLSTLSRTLS


(a)

0123456789

10

Estim

atio

n er

ror (

)


RLSTLS

RTLSVFF-RTLS

(b)



200 400 600 800 1000 1200 14000Time (s)

ndash20

ndash10

0

10

20

30Cu

rren

t am

plitu

de (A

)

(a)

times104

ndash20

ndash10

0

10

20

30

Curr

ent a

mpl

itude

(A)

1 2 3 4 5 60Time (s)

(b)

times1041 2 3 4 50 6

Time (s)

0

02

04

06

08

1

SOC

(c)

27272274276278

28282284286288

29

Estim

ated

capa

city

(Ah)


RLSTLSRTLS


(d)


005

115

225

335

445

5

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)



5 Conclusion


Data Availability




Acknowledgments


References


















































100



2








V





Measurement

I







xt|tndash1 =n+1

i=0Wiχ(i)

t|tndash1

n+1




Cap

SOC





200 400 600 800 1000 1200 1400 1600 18000Time (s)


01020304050

Curr

ent a

mpl

itude

(A)

(a)

times1041 2 3 4 50

Time (s)


01020304050

Curr

ent a

mpl

itude

(A)

(b)

times1041 2 3 4 50

Time (s)

0

02

04

06

08

1

SOC

(c)


RLSTLSRTLS


9

10

11

12

13

14

15

16

Estim

ated

capa

city

(Ah)

(d)


0

5

10

15

20

25

30

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)





100 200 300 400 500 600 700 8000Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80Cu

rren

t am

plitu

de (A

)

(a)

times10405 1 15 2 250

Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80

Curr

ent a

mpl

itude

(A)

(b)

05 1 15 2 250Time (s)

ndash02

0

02

04

06

08

1

12

SOC

times104

(c)


98

985

99

995

100

1005

101

1015

102

Estim

ated

capa

city

(Ah)

RLSTLSRTLS


(d)


0

05

1

15

2

25

3

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)












90

92

94

96

98

100

102Es

timat

ed ca

paci

ty (A

h)

RLSTLSRTLS


(a)

0123456789

10

Estim

atio

n er

ror (

)


RLSTLS

RTLSVFF-RTLS

(b)



200 400 600 800 1000 1200 14000Time (s)

ndash20

ndash10

0

10

20

30Cu

rren

t am

plitu

de (A

)

(a)

times104

ndash20

ndash10

0

10

20

30

Curr

ent a

mpl

itude

(A)

1 2 3 4 5 60Time (s)

(b)

times1041 2 3 4 50 6

Time (s)

0

02

04

06

08

1

SOC

(c)

27272274276278

28282284286288

29

Estim

ated

capa

city

(Ah)


RLSTLSRTLS


(d)


005

115

225

335

445

5

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)



5 Conclusion


Data Availability




Acknowledgments


References

















































200 400 600 800 1000 1200 1400 1600 18000Time (s)


01020304050

Curr

ent a

mpl

itude

(A)

(a)

times1041 2 3 4 50

Time (s)


01020304050

Curr

ent a

mpl

itude

(A)

(b)

times1041 2 3 4 50

Time (s)

0

02

04

06

08

1

SOC

(c)


RLSTLSRTLS


9

10

11

12

13

14

15

16

Estim

ated

capa

city

(Ah)

(d)


0

5

10

15

20

25

30

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)





100 200 300 400 500 600 700 8000Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80Cu

rren

t am

plitu

de (A

)

(a)

times10405 1 15 2 250

Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80

Curr

ent a

mpl

itude

(A)

(b)

05 1 15 2 250Time (s)

ndash02

0

02

04

06

08

1

12

SOC

times104

(c)


98

985

99

995

100

1005

101

1015

102

Estim

ated

capa

city

(Ah)

RLSTLSRTLS


(d)


0

05

1

15

2

25

3

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)












90

92

94

96

98

100

102Es

timat

ed ca

paci

ty (A

h)

RLSTLSRTLS


(a)

0123456789

10

Estim

atio

n er

ror (

)


RLSTLS

RTLSVFF-RTLS

(b)



200 400 600 800 1000 1200 14000Time (s)

ndash20

ndash10

0

10

20

30Cu

rren

t am

plitu

de (A

)

(a)

times104

ndash20

ndash10

0

10

20

30

Curr

ent a

mpl

itude

(A)

1 2 3 4 5 60Time (s)

(b)

times1041 2 3 4 50 6

Time (s)

0

02

04

06

08

1

SOC

(c)

27272274276278

28282284286288

29

Estim

ated

capa

city

(Ah)


RLSTLSRTLS


(d)


005

115

225

335

445

5

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)



5 Conclusion


Data Availability




Acknowledgments


References

















































100 200 300 400 500 600 700 8000Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80Cu

rren

t am

plitu

de (A

)

(a)

times10405 1 15 2 250

Time (s)

ndash60

ndash40

ndash20

0

20

40

60

80

Curr

ent a

mpl

itude

(A)

(b)

05 1 15 2 250Time (s)

ndash02

0

02

04

06

08

1

12

SOC

times104

(c)


98

985

99

995

100

1005

101

1015

102

Estim

ated

capa

city

(Ah)

RLSTLSRTLS


(d)


0

05

1

15

2

25

3

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)












90

92

94

96

98

100

102Es

timat

ed ca

paci

ty (A

h)

RLSTLSRTLS


(a)

0123456789

10

Estim

atio

n er

ror (

)


RLSTLS

RTLSVFF-RTLS

(b)



200 400 600 800 1000 1200 14000Time (s)

ndash20

ndash10

0

10

20

30Cu

rren

t am

plitu

de (A

)

(a)

times104

ndash20

ndash10

0

10

20

30

Curr

ent a

mpl

itude

(A)

1 2 3 4 5 60Time (s)

(b)

times1041 2 3 4 50 6

Time (s)

0

02

04

06

08

1

SOC

(c)

27272274276278

28282284286288

29

Estim

ated

capa

city

(Ah)


RLSTLSRTLS


(d)


005

115

225

335

445

5

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)



5 Conclusion


Data Availability




Acknowledgments


References
























































90

92

94

96

98

100

102Es

timat

ed ca

paci

ty (A

h)

RLSTLSRTLS


(a)

0123456789

10

Estim

atio

n er

ror (

)


RLSTLS

RTLSVFF-RTLS

(b)



200 400 600 800 1000 1200 14000Time (s)

ndash20

ndash10

0

10

20

30Cu

rren

t am

plitu

de (A

)

(a)

times104

ndash20

ndash10

0

10

20

30

Curr

ent a

mpl

itude

(A)

1 2 3 4 5 60Time (s)

(b)

times1041 2 3 4 50 6

Time (s)

0

02

04

06

08

1

SOC

(c)

27272274276278

28282284286288

29

Estim

ated

capa

city

(Ah)


RLSTLSRTLS


(d)


005

115

225

335

445

5

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)



5 Conclusion


Data Availability




Acknowledgments


References















































200 400 600 800 1000 1200 14000Time (s)

ndash20

ndash10

0

10

20

30Cu

rren

t am

plitu

de (A

)

(a)

times104

ndash20

ndash10

0

10

20

30

Curr

ent a

mpl

itude

(A)

1 2 3 4 5 60Time (s)

(b)

times1041 2 3 4 50 6

Time (s)

0

02

04

06

08

1

SOC

(c)

27272274276278

28282284286288

29

Estim

ated

capa

city

(Ah)


RLSTLSRTLS


(d)


005

115

225

335

445

5

Estim

atio

n er

ror (

)

RLSTLS

RTLSVFF-RTLS

(e)



5 Conclusion


Data Availability




Acknowledgments


References















































5 Conclusion


Data Availability




Acknowledgments


References















































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