Research ArticleA State of Charge Estimation Method for Lithium-Ion BatteryUsing PID Compensator-Based Adaptive Extended Kalman Filter

Zheng Liu 1 Yuan Qiu1 Chunshan Yang1 Jianbo Ji1 and Zhenhua Zhao 2

1School of Electronic Information and Automation Guilin University of Aerospace Technology Guilin 541004 China2School of Foreign Language and International Business Guilin University of Aerospace Technology Guilin 541004 China

Correspondence should be addressed to Zheng Liu lzzly365163com and Zhenhua Zhao zhenhuazhaoguateducn

Received 1 January 2021 Revised 16 January 2021 Accepted 8 February 2021 Published 22 February 2021

Academic Editor Carlos Aguilar-Ibanez

Copyright copy 2021 Zheng Liu et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

With the widespread application of electric vehicles the study of the power lithium-ion battery (LIB) has broad prospects andgreat academic significance +e state of charge (SOC) is one of the key parts in battery management system (BMS) which is usedto provide guarantee for the safe and efficient operation of LIB To obtain the reliable SOC estimation result under the influence ofsimple model and measurement noise a novel estimation method with adaptive feedback compensator is presented in this paper+e simplified dynamic external electrical characteristic of LIB is represented by the one-order+evenin equivalent circuit model(ECM) and then the ECM parameters are identified by the forgetting factor recursive least squares method (FFRLS) Fully takinginto account the feedback effect of terminal voltage innovation the combination of adaptive extended Kalman filter (AEKF) andinnovation vector-based proportional-integral-derivative (PID) feedback is proposed to estimate the LIB SOC +e commonsingle proportional feedback of Kalman filter (KF) is replaced by the innovation vector-based PID feedback which means that themultiple prior terminal voltage innovation is used in the measurement correction step of KF+e results reveal that the AEKFwithPID feedback compensation strategy can improve the SOC estimation performance compared with the common AEKF and itreveals good robust capability and rapid convergence speed for initial SOC errors +e maximum absolute error and averageabsolute error for SOC estimation are close to 4 and 26 respectively

1 Introduction

Since the urgent demand for a decrease in global petroleumconsumption and environmental depredation the explo-ration of renewable energy has become a global consensusOver the past decade the application of new energy electricvehicles (EVs) has become a hotspot of the motor industrybecause of its merit of zero emission energy source di-versification and clean energy [1] Due to the significantadvantage of high energy density low self-discharge highvoltage long cyclic life and desirable safety the lithium-ionbattery (LIB) has become the main energy source of EVs Inorder to ensure the safe and effective charging or dischargingof LIB in EVs the battery management system (BMS) is welldesigned to manage the LIB [2 3] +e state of charge (SOC)which used residual capacity to qualify LIB performance isone of the most core functionalities that need to be strictly

managed by the BMS under the actual complicated con-ditions Due to the negative impact of imprecise measure-ment and open-loop control the SOC cannot be observeddirectly by vehicle-mounted transducer and Ampere-hourcounting method +e SOC of LIB is essentially dependenton various SOC estimation methods based on its relevantcharacteristics such as charge-discharge current tempera-ture and terminal voltage [4 5]

In the past years numerous researchers have put forwardrelevant methods to increase the performance of SOC es-timation In general these methods can be classified intothree categories open-loop based method model-free basedmethod and closed-loop model-based method +e open-loop based method includes Ampere-hour counting (Ah-counting) [6] and open-circuit voltage (OCV) lookup table[7] +e Ah-counting is the most widely used method withsummation of current which is simple and easy for

HindawiComplexityVolume 2021 Article ID 6665509 14 pageshttpsdoiorg10115520216665509

implementation but it is subjected to relevant errors such asinitial error cumulative error and truncation error +eOCVmethod uses nonlinear fitting function to represent therelationship between OCV and SOC directly [8] Howeveruntil the long rest time the OCV cannot completely equalthe terminal voltage and the error sensitivity of OCV easilycauses an increase of SOC estimation error due to the localflat platform effect of the OCV-SOC fitting curve [9] Be-yond that the model-free-based method is developed as ablack box and the nonlinear relationship between param-eters and state is established by learning the train data suchas terminal voltage charge-discharge current temperatureand cycles Several model-free-based methods includingneural network (NN) [10 11] and fuzzy logic [12 13] aredeveloped for SOC estimation with positive results Al-though these methods can reduce BMSrsquos dependency on theaccuracy of measurement data the BMS is subjected to itsprocessing power storage capacity and operational costAccording to the above methodsrsquo drawbacks the closed-loopmodel-based method is proposed to address this concern[14] In the model-based methods the appropriate modelchoice is the precondition of SOC estimation in BMSCompared with the complex electrochemical model [15] theequivalent circuit model (ECM) only needs a simple circuitnetwork to represent the external characteristics of LIB[16 17] In [18] the first-order ECM is used to model thedynamic response of LIB and the ECM parameters areidentified by recursive least squares (RLS) with biascompensation

Various forms of Kalman filters (KF) are widely appliedfor ECM-based state estimation by regarding the SOC asone of the state observers [19 20] Since the basic KF isessentially unsuitable for LIB with a strong nonlinearfeature some general improved KF are used for SOC es-timation such as extended Kalman filter (EKF) [21] un-scented Kalman filter (UKF) [22] cubature Kalman filter(CKF) [23] and H-Infinite filter (HIF) [17] +e EKF withproportional-integral regulation is used for SOC estimationbased on ECM with a resistance-capacitance network [24]+e major disadvantage of EKF is that only one-orderTaylor series expansion is used to approximate the non-linear state of LIB and the higher-order expansion termsare ignored which inevitably brings about linearizationerror [25] +e UKF is developed for SOC estimation of LIBby using the unscented transformation (UT) to replace themultiorder Taylor series expansion [26] Because the UT isused for approximating the SOC distribution characteristicwithout the Jacobian matrix the SOC estimation resultshows that the UKF has better performance than the EKF inrobustness and precision +e CKF with radial-sphericalcubature rule is adopted to estimate LIB SOC and theestimation results show that the CKF is better than EKF andUKF in terms of accuracy and stability [23 27] Althoughthe EKF is weaker than UKF and CKF in estimationprecision and filter convergence the symmetric nonneg-ative definition of prior information matrix in UKF andCKF is not guaranteed all the time In addition severalhybrid methods such as NN based adaptive CKF [33] andEKF [34] are proposed to estimate battery SOC For the

potential impact of inaccuracy or unknown noise the KFwith adaptive strategy are proposed to improve the SOCestimation accuracy and robustness [28 29] In summarythe basic idea of the KF based SOC estimation methods is touse prior data such as charge-discharge current and ter-minal voltage to obtain the optimal KF gain and thus getthe estimation of SOC +e KF gain multiplying the singleinnovation can be considered as a single feedback unit ofwhich the input and output are the terminal voltage errorand measurement correction respectively +us the singlefeedback unit could be considered as a feedback controllerof KF which contains only a proportional regulatingfunction +e purpose of such a controller is to force theestimated terminal voltage to approach the measuredterminal voltage which eventually forces the state of theECM convergence to the true state of the LIB

+e main contribution of this paper includes the fol-lowing on the basis of the original KF the integral anddifferential terms of voltage deviation are introduced intothe single feedback unit of KF to describe the richer his-torical data and trends of voltage deviation +e singleproportional feedback in KF is replaced by the proportional-integral-differential feedback which can be referred to as aPID feedback controller in KF +at is for the batterysystem the differential and integral item of terminal voltageerror deduced from the innovation vector is added to themeasurement correction step of posterior estimation whichis only a proportional term in the original KF +e simu-lation result validates that the proposed SOC estimatorestablished by the combination of AEKF-PID and one-order+evenin battery model could adapt itself to the nonlinearcharacteristics of the LIB system Compared with the AEKFthe proposed AEKF-PID method has better performance inrobustness and precision

+e rest of the paper is arranged as follows +e simpleone-order ECM is used to describe the LIB model and theRLS method with forgetting factor is used for parametersidentification in Section 2+e combination of PID feedbackunit based on innovation vector theory and AEKF based onnoise covariance matching is introduced in Section 3Section 4 shows the simulation and experimental resultsunder various conditions Finally the main conclusions andsubsequent plan are listed in Section 5

2 Battery Equivalent Circuit Modeling

21 Basic ECM Compared with the electrochemical modelthe ECM intuitively reflects the relationship between LIBrsquosinput and output which is conducive to battery charac-teristics analysis and model parameter identification [30]+e higher the order of the RC network in the ECM is thecloser the battery characteristics described by the ECM are tothe actual battery characteristics However the complexityof the model also increases with the increase of order whichis not suitable for the real-time calculation of electric vehicleBMS Taking into account the complexity of the ECMengineering application and other factors the paper takesthe first-order +evenin ECM as the simplified batterymathematical model as shown in Figure 1

2 Complexity

According to the one-order ECM its mathematicalmodel can be expressed as

_Vpt minusVpt

RpCp

+It

Cp

Voutt Voct minus Vpt minus R0It

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(1)

where Voc shows the OCV R0 is the ohmic resistance RPCp

includes the polarization resistance RP and polarizationcapacitance Cp Vp represents the polarization voltage onRPCp I is the chargedischarge current and Vout representsthe terminal voltage

Rewrite equation (1) as frequency domain expressionusing Laplace transform the symbol Vp of polarizationvoltage can be eliminated

Vout(s) Voc(s) minus R0 +Rp

1 + RpCps1113888 1113889I(s) (2)

+e corresponding transfer function G(s) of ECM can beformulated as

G(s) Vout(s) minus Voc(s)

I(s) minus R0 +

Rp

1 + RpCps1113888 1113889 (3)

+e sdomain can be mapped to the z domain by bilineartransformation methods (2Ts)((1 minus zminus 1)(1 + zminus 1))

G zminus 1

1113872 1113873 Vout z

minus 11113872 1113873 minus Voc z

minus 11113872 1113873

I zminus 1

1113872 1113873

a2 + a3zminus 1

1 + a1zminus 1 (4)

+e discrete form of equation (1) can be represented as

Voutk minus Vock a1 Voutkminus 1 minus Vockminus 11113872 1113873 + a2Ik + a3Ikminus 1

(5)

Since the OCV has a correlation with SOC and tem-perature the OCV is usually expressed asVock f(soc temp) SOC can be considered as constantunder certain conditions including one second samplingtime and constant temperature

dVock

dk

zVock

zSOCzSOC

zk+

zVock

ztempztemp

zk 0 (6)

+at is ΔVock Vock minus Vockminus 1 0+en equation (4) can be rewritten as

Voutk a1Voutkminus 1 + a2Ik + a3Ikminus 1 + 1 minus a1( 1113857Vock (7)

+e model parameters R0 Rp and Cp can be obtainedfrom a1 a2 and a3

R0 a3 minus a2( 1113857

1 + a1( 1113857

Rp 2 a3 + a1a2( 1113857

a21 minus 11113872 1113873

Cp minus a1 + 1( 1113857

2

4 a3 + a1a2( 1113857

(8)

22 Model Parameters Identification +e LIB modelingmethod based on ECM can provide concise and effectivesupport for subsequent state estimation An effective pa-rameter identification method is a necessary condition forachieving this goal As the LIB model parameters are notconstant when it is charged or discharged the offlineidentification of model parameters may mismatch the realparameters According to this problem the model param-eters are updated online under discharge operation of LIB

RLS algorithm based on minimum sum-squared errortheory is a commonly used model parameter identificationmethod widely applied in tracking of time-varying pa-rameters [31] To suppress the effect of data saturation in theidentification of time-varying parameters with the RLSmethod the RLS with forgetting factor (FFRLS) algorithmcan be used in battery time-varying parameter identification[32]

+e battery parameter identification is to determinecoefficient a1 a2 and a3 values based on the current andvoltage measurements

+e form of least squares can be expressed as

yk φTk θk + ξk (9)

where φk is the current and voltage measurements and θk isthe parameters to be identified

Set the objective function

J 1113944L

k1λLminus k

yk minus φTk

1113954θk1113960 11139612 (10)

where 1113954θ is the identification value of θ λ is forgetting factor0lt λle 1 and the FFRLS degenerates to RLS when λ 1

+e objective function is close to the minimum valuethrough the recursive calculation

R0 RP

CP

VocVout

Vp

I

++

ndashndash

+

ndash

Figure 1 One-order ECM

Complexity 3

ek yk minus φTk

1113954θkminus 1

Kk Pkminus 1φ

Tk

λ + φTk Pkminus 1φk

Pk Pkminus 1 minus Kkφ

Tk Pkminus 1

λ

1113954θk 1113954θkminus 1 + Kkek

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

+e correspondingmeasurements and parameters of LIBare as follows

φk Voutkminus 1 Ik Ikminus 1 11113858 1113859

θk a1 a2 a3 1 minus a1( 1113857Vock1113858 1113859T

⎧⎨

⎩ (12)

+e coefficients a1 a2 a3 and OCV are identified re-cursively by the FFRLS method in equation (11) then theparameters R0 RP and Cp can be solved reversely fromcoefficients a1 a2 and a3

3 SOC Estimation Using PID-Based AEKF

31 Basic AEKF As a linear stochastic system with whitenoises the corresponding state estimation can be calculatedby the EKF

xk Akxkminus 1 + Bkukminus 1 + wkminus 1

yk Ckxk + Dkuk + vk1113896 (13)

where xk represents the system state matrix Ak Bk Ck andDk are the dynamic coefficients of the state function andobservation function respectively yk is the output matrixuk is the input vector wk and vk are the state Gaussian whitenoise and measurement Gaussian white noise respectively+e iterative process of EKF is as follows

(i) Initialization

1113954x0 P0 Q0 R01113864 1113865 (14)

(ii) Prior estimation

1113954xk|kminus 1 Ak1113954xkminus 1|kminus 1 + Bkukminus 1

Pk|kminus 1 AkPkminus 1|kminus 1ATk + Qkminus 1

⎧⎨

⎩ (15)

(iii) Measurement correction

ek yk minus Ck1113954xk|kminus 1 minus Dkuk

Kk Pk|kminus 1CTk CkPk|kminus 1C

Tk + Rk1113872 1113873

minus 1

1113954xk|k 1113954xk|kminus 1 + Kkek

Pk|k I minus KkCk( 1113857Pk|kminus 1

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(16)

Since the noise statistics of the dynamic model are time-varying it is necessary to construct an adaptive EKF to adaptto the noise statistics In this paper the state noise varianceQk and the measurement noise variance Rk based on the

maximum likelihood (ML) criterion can be online updatedto reflect the changes of the state noise characteristics andmeasurement noise characteristics [33] so as to ensure thatthe KF can better adapt to the changes of the noise statisticalcharacteristics

+e variance matrix of the innovation sequence ek can becalculated from equation (16)

Fk Rk + CkPk|kminus 1CTk (17)

+e equivalent value of the innovation sequence ek basedon the moving window method with size L is used tosubstitute Qk and Rk can be obtained as

Rk 1L

1113944

k

nkminus L+1eke

Tk minus CkPk|kminus 1C

Tk (18)

From the definition of KF

KkCkPk|kminus 1 Kk

1L

1113944

k

nkminus L+1eke

Tk

⎛⎝ ⎞⎠KTk KkP

minus 1k|kminus 1K

Tk

(19)

Similarly we can get KkCkPk|kminus 1 from equation (16)

KkCkPk|kminus 1 Pk|kminus 1 minus Pk|k AkPkminus 1|kminus 1ATk minus Pk + Qkminus 1

(20)

Equation (20) can be substituted into equation (19)

Qk KkFkKTk + Pk|kminus 1 minus Pk (21)

Ignore the variation of the state variance matrix

Qk KkFkKTk (22)

32 Basic PIDController +e PID controller is composed ofproportional coefficient integral coefficient and differentialcoefficient Its relationship between input and output incontinuous time domain is

y(t) kP e(t) +1

TI

1113946 e(t)dt + TD

de(t)

dt1113890 1113891 (23)

Adopt the digital PID in computer control system

y(k) kP e(k) +1

TI

1113944

k

i0Te(k) + TD

e(k) minus e(k minus 1)

T⎡⎣ ⎤⎦

(24)

One of the most common digital PID is incremental PIDby suppressing integral saturation

Δy(k) kP ek minus ekminus 1( 11138571113858

+ kIek + kD ek minus 2ekminus 1 + ekminus 2( 11138571113859(25)

33 PID Feedback Unit in Improved KF To obtain the re-liable SOC estimation result on the basis of uncertainty ofmodel and measurement noise many estimation methods

4 Complexity

with KF are presented Since the loss of correction data in KFmay be due to the fact that only single proportional inno-vation feedback is used in KF the innovation vector-basedPID feedback controller is introduced into measurementcorrection step of KF in this paper +is paper puts forwardan improved AEKF based on PID controller with self-tuningPID parameters kP kI and kD Due to the fact that moreterminal voltage innovation can be described by the PID unitin measurement correction step of improved AEKF thebattery state can be predicted more accurately by the im-proved AEKF

+e PID feedback unit deduced from innovation vectoris first introduced +e single feedback unit in KF issubstituted by the PID unit which referred to the PIDfeedback strategy in the KF structure as shown in Figure 2

+e state measurement update 1113954xk|k from common KFcan be expressed by letting ek to be single innovation inequation (16)

In order to get the equivalent form of the incrementalPID in equation (25) the measurement correction by singleinnovation ek in equation (16) can be transformed as

1113954xk|k 1113954xk|kminus 1 + KPk ek minus ekminus 1( 1113857 + KIkek

+ KDk ek minus 2ekminus 1 + ekminus 2( 1113857(26)

+e structure diagram of the improved method bycombined KF and PID feedback unit is demonstrated inFigure 3 +e core idea is to expand the single innovation toan innovation vector +at is ekis extended into

ek ekminus 1 ekminus 21113858 1113859T To ensure the consistency of data di-

mension the following improvements are made gain kk isextended into kk kkminus 1 kkminus 21113858 1113859 and measurement output zk

is extended into zk zkminus 1 zkminus 21113858 1113859T

Meanwhile a posteriori estimation 1113954xk|k in the mea-surement correction step needs to be rewritten with mod-ified parameters including kk kkminus 1 kkminus 21113858 1113859

zk zkminus 1 zkminus 21113858 1113859T and ek ekminus 1 ekminus 21113858 1113859

T

1113954xk|k 1113954xk|kminus 1 + minus 2kkminus 2 minus kkminus 1( 1113857 ek minus ekminus 1( 1113857

+ kk + kkminus 1 + kkminus 2( 1113857ek + kkminus 2 ek minus 2ekminus 1 + ekminus 2( 1113857

(27)

Consequently the regulation parameters kP kI and kD

of PID unit in the measurement correction step can besolved by equations (26) and (27)

kPk minus 2kkminus 2 minus kkminus 1

kIk kk + kkminus 1 + kkminus 2

kDk kkminus 2

(28)

From equations (26) and (28) we can get the PIDfeedback unit by setting innovation vector and the PIDcoefficients kP kI and kD can be solved by the expressions ofgains kk kkminus 1 and kkminus 2 in three sampling points +ree in-novation data at times k k-1 and k-2 are used whenupdating the state at time k +e main reason for increasingthe robustness of the improved algorithm is to reuse in-novative data to update the state at adjacent times In orderto improve the performance of SOC estimation the

combination of PID feedback and AEKF is introduced inSection 34 In addition the covariance of noise wk and vk isupdated based on the principle of covariance matchingrespectively

+e PID-based KF introduces more than one prior in-novation which greatly enhances the effect of feedbackcompensation in KF Experiments show that the innovationvector-based KF has better robustness than single-innova-tion-based KF in strong nonlinear systems [9] Although thecalculation load of PID-based KF is more than that of thesingle-innovation-based KF the increased amount of cal-culation is acceptable relative to the improvement inaccuracy

34 SOC Estimation Based on AEKF-PID +e discreteequation of the LIB model in equation (1) can be rewritten asfollows

SOCk

Upk

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1 0

0 eminus Δtτ1( )

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

SOCkminus 1

Upkminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

minusηΔtC

1113874 1113875

Rp 1 minus eminus (Δtτ)

1113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ikminus 1 + wkminus 1

Uk UOC SOCk( 1113857 minus Upk minus IkR0 + vk

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

By selecting x [SOC Up]T as the state vector choosingIk and Uk as the input current and output terminal voltagerespectively the state equation and measurement equationof ECM in a discrete-time representation are as follows

xk Akxkminus 1 + BkIkminus 1 + wkminus 1

yk Ckxk + DkIkminus 1 + vk1113896 (30)

where Akminus 1 (zfzx)|xxk|kminus 1

1 00 e

minus (Δtτ1)1113888 1113889 Bkminus 1 minus1113858

((ηΔt)C)Rp(1 minus eminus (Δtτ)

)] Ck (zhzx)|xxk|kminus 1 [(dUOC

(SOCk|kminus 1))(dS OCk|kminus 1) minus 1] and Dk minus R0 we assumethat η 1 under arbitrary charge-discharge condition

+e detailed process of the AEKF-PID method is asfollows

Step 1 Initialization

(a) SOC and Up initialized state

xkminus 1|kminus 1 SOCkminus 1|kminus 1upkminus 1|kminus 11113872 1113873 (31)

(b) State error initialized covariance

Pkminus 1|kminus 1 E xkminus 1|kminus 1 minus 1113954xk|kminus 11113872 1113873 xkminus 1|kminus 1 minus 1113954xkminus 1|kminus 11113872 1113873T

1113876 1113877

(32)

(c) Noise initialized covariance Q0 R0

Step 2 State prediction

(a) State prior estimation

1113954xk|kminus 1 Ak|kminus 11113954xkminus 1|kminus 1 + Bkminus 1Ikminus 1 (33)

Complexity 5

(b) State error covariance prior estimation

Pk|kminus 1 Ak|kminus 1Pkminus 1|kminus 1ATk|kminus 1 + Qkminus 1 (34)

Step 3 Measurement correction

(a) Gain matrix update

Kk Pk|kminus 1CTk CkPk|kminus 1C

Tk + Rk1113872 1113873

minus 1 (35)

(b) State measurement update by PID feedback unit

1113954xk|k 1113954xk|kminus 1 + kPk ek minus ekminus 1( 1113857

+ kIkek + kDk ek minus 2ekminus 1 + ekminus 2( 1113857(36)

(c) State error covariance update

Pk|k I minus KkCk( 1113857Pk|kminus 1 (37)

(d) Adaptive noise covariance matching

Fk 1L

1113944

k

nkminus L+1eke

Tk

Rk Fk minus CkPk|kminus 1CTk

Qk KkFkKTk

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(38)

In summary according to the state equation andmeasurement equation of ECM in equations(29)ndash(30) and the detailed process in equations(31)ndash(38) the flowchart of the proposed SOC es-timation by AEKF-PID algorithm is shown inFigure 4

Actual battery

Single feedback

Measured voltage

Calculatedvoltage

Current

Current

+ndash

Battery model

(a)

Actual battery

PID feedback

Measured voltage

Calculatedvoltage

Current

Current

+ndash

Battery model

(b)

Figure 2 Block diagram of KF and improved KF (a) KF with single feedback (b) Improved KF with PID feedback

Bk

Bk

+

Ak

Ck+

+

+ +

+ Ck

Dk

Ak

+

xkndash1|kndash1

xk|k xk|kndash1yk

Dk

+

ndash

Zndash2

Kp

Ki

Kd

+

+ +

+

uk

uk

wk vk

xk

xk ndash 1

yk

LIB model

Actual LIB

PID

+

+

+

Residual

Zndash1

Zndash1

Zndash1

Figure 3 +e schematic of PID-based KF

6 Complexity

4 Results and Discussion

41 ExperimentalConfigurations A LIB test platform shownin Figure 5 is established to verify the effect of the ECM andmethod It consists of several major components (1) alithium-ion ferrous phosphate battery is used as the ex-perimental objects of which the nominal capacity is 20Ahand nominal voltage is 24V (2) a power battery test system(Arbin EVTS) with control accuracy which is less thanplusmn01 FSR is used for LIB chargingdischarging undervarious working condition (3) a programmable temperaturechamber is applied to control ambient temperature (4) ahost computer with MITS Pro v70 and a computerworkstation with MATLAB R2012a are used for data ac-quisition and data simulation in the experimentsrespectively

+e LIB is fully charged by constant current-constantvoltage (CC-CV) procedure after standing for two hoursthen three working conditions including 10A constantcurrent dynamic stress test (DST) and federal urban drivingschedule (FUDS) are used to test the proposed method+ree working conditions mentioned in this paper arecarried out at 25 degrees Celsius +e discharge current andterminal voltage of DSTand FUDS are shown in Figure 6 Inaddition the evaluation indicator such as maximum abso-lute error (MAE) and average absolute error (AAE) is ap-plied to the assessment of estimation algorithms

+e OCV-SOC function describing the relationshipbetween the open-circuit voltage and SOC can be fitted byOCV feature test data +e LIB is fully charged by the CC-

CV procedure after standing for two hours the measuredterminal voltage can be used as the OCV of 100 SOC +etest steps are as follows the LIB is discharged to 98 SOCwith 10A (05 C) constant current after standing for twohours the measured terminal voltage can be used as theOCV of 98 SOC Repeating the above steps 50 times eachmeasured terminal voltage can be obtained as the OCV ofevery 2 SOC+e eighth-order polynomial fitting functionto describe the mathematical relationship of OCV-SOC canbe shown in Figure 7

42 Parameters Identification +e ECM parameters iden-tification results are shown in Figures 8 and 9 the identi-fication result of ohmic resistance R0 polarization resistanceRp and polarization capacitance Cp can converge to a steadystate quickly from the unreliable initialization value undertwo cycles

R0Rp andCp always have a small range of fluctuation inFigures 8(a) and 9(a) which is directly related to the fluc-tuations of the coefficients a1 b1 and b2 +eoretically thedynamic changes of R0 Rp and Cp are inconsistent but thefixed forgetting factors of FFRLS have the same weight onthe three parameters which has a certain effect on thefluctuation of identification results

+e terminal voltage is identified in each sampling pe-riod based on R0 Rp and Cp As we can see fromFigures 8(b) and 9(b) the identification results of the ter-minal voltage can track the measured value of the terminalvoltage stably under two cycles +e maximum error of theterminal voltage identification is only 01923V and 01702Vrelative to the terminal voltage range from 23V to 29Vrespectively

43 Comparison of the SOC Estimation

431 Analysis of SOC Estimation under 10A DischargeBased on the identified ECM parameters the SOC estimationunder the 10A discharge cycle is presented in Figure 10 Asobserved the SOC reference with Arbin EVTS is formed witha black line the red line and blue line represent the SOC

Measurement dataand

model parameters

Initial conditionsequations (31) and (32)

State prior estimationequation (33)

Gain matrix updateequation (35)

State error covariance prior estimationequation (34)

PID coefficient update with gainequation (28)

State estimation measurement update by PID unitequation (36)

State error covariance updateequation (37)

Figure 4 +e flowchart of AEKF-PID estimator for LIB SOC

Arbin EVTS Temperature chamber

Computer

Figure 5 Battery test platform

Complexity 7

estimation with single AEKF and AEKF-PID respectivelyFrom the estimation results of two methods shown inFigure 10(a) these two SOC estimations can both track theSOC reference in a short period of time no matter AEKF orAEKF-PID Although the two SOC estimation results bothconverge to the SOC reference the SOC estimation error byAEKF-PID is less than that by AEKF as shown inFigure 10(b) +e errors of the two methods slowly increasewith an increase in time +e AAE and MAE of AEKF-PIDare 02266 and 05119 respectively and the AAE andMAE of AEKF are 08606 and 12478 respectively Itfollows that the improved AEKF-PID method shows betterresults in SOC estimation than single AEKF

432 Analysis of SOC Estimation under Two Cycles Inorder to further validate the accuracy of the proposedmethod the DST and FUDS cycles are implemented to

OCV

(V)

02 04 06 08 10SOC

22

23

24

25

26

27

28

29

Figure 7 OCV-SOC fitting curve

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(a)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

23

24

25

26

27

28

29

times104

(b)

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(c)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

22

23

24

25

26

27

28

29

times104

(d)

Figure 6 (a) Current under DST cycle (b) Terminal voltage under DST cycle (c) Current under FUDS cycle (d) Terminal voltage underFUDS cycle

8 Complexity

simulate the EV operationmode+e SOC estimation resultswith two methods under the DST and FUDS cycles areshown in Figures 11 and 12 respectively As observed theSOC reference by Arbin EVTS is formed with a black linethe red line and blue line represent the SOC estimation withsingle AEKF and AEKF-PID respectively From the esti-mation results of two methods shown in Figures 11(a) and12(a) two SOC estimations can both track the SOC refer-ence in a short period of time no matter AEKF or AEKF-PID +e SOC estimation errors with two algorithms underthe DST and FUDS cycles are shown in Figures 11(b) and12(b) respectivelyWe can see that the SOC estimation errorby the AEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 2 under the FUDS cycle meanwhilethe maximum SOC error by AEKF-PID is only 17141+eMAE and AAE of two methods under two cycles are listed inTable 1 According to the above analysis the combination ofthe PID feedback unit with AEKF has advantage over singleAEKF for SOC estimation

44 Analysis on Robustness of SOC Estimation +e highprecise SOC estimation results can be obtained by theproposed method on the assumption that the sampled ex-perimental data by Arbin EVTS is reliable However theexperiment data from the laboratory is not completely equalto actual data in EV operating mode Since the measurementdata by universal transducer is hard to avoid noise inter-ference from such as drift current and diffusion current theSOC estimation method with certain anti-interferencequality is important for BMS To further verify the anti-interference performance of the AEKF-PID a sequence ofnoise with feature of random normal distribution is added tothe operation current under DST and FUDS cycles re-spectively +e mean value of the noise is set as 0 and itsstandard deviation is set as 2 +e SOC estimation resultswith Gaussian noise of current under two conditions aredisplayed in Figures 13 and 14 +e two colored lines withblue and red are used to represent the SOC estimation byAEKF-PID and AEKF respectively +e MAE and AAE ofSOC estimation with Gaussian noise under two cycles are

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

010002000

0005

01

004005006

times104

times104

times104

(a)

05 1 15 2 25 3 350Time (s)

Term

inal

vol

tage

erro

r (V

)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 8 Model parameters identification results under DST (a) R0 Rp Cp (b) terminal voltage error

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

002004006

0005

01

0500

1000

times104

times104

times104

(a)

Term

inal

vol

tage

erro

r (V

)

05 1 15 2 25 3 350Time (s)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 9 Model parameters identification results under FUDS (a) R0 Rp Cp (b) terminal voltage error

Complexity 9

listed in Table 2 As we can see from Figures 13 and 14 dueto the introduction of current noise in two cycles thefluctuation of two corresponding SOC estimation errorsincreased We can see that the SOC estimation error by theAEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 28 under the FUDS cyclemeanwhile the maximum SOC error by AEKF-PID isclose to 2+e MAE and AAE of two methods under twocycles are listed in Table 2 According to the analysisabove the combination of PID feedback unit with AEKFhas advantage over single AEKF in robustness of SOCestimation

To further verify the anti-interference performance ofthe AEKF-PID a sequence of noise with feature of ran-dom non-Gaussian distribution is added to the operationcurrent +e SOC estimation result with non-Gaussiannoise is displayed in Figure 15 We can see that the twoSOC estimation errors with non-Gaussian noise increasecompared to that with Gaussian noise +e results inTable 3 show that the MAE of AEKF is greater than 55and the MAE of AEKF-PID is close to 4 At the end ofdischarge the SOC error of AEKF increases graduallymeanwhile the SOC error of AEKF-PID reaches 3 +eresults show that the basic AEKF-based SOC estimationmethod has better performance in Gaussian noise than

SOC

ReferenceAEKFAEKFndashPID

2500 3000 3500 4000082084086088

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

055

06

065

07

075

08

085

09

095

1

(a)

AEKFAEKFndashPID

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

SOC

erro

r (

)

ndash5ndash4ndash3ndash2ndash1

012345

(b)

Figure 10 Comparison of SOC estimation under 10A discharge (a) SOC estimation result (b) SOC estimation error

ReferenceAEKFAEKFndashPID

SOC

8500 9000 9500

076077078

1 15 2 25 3 3505Time (s)

0010203040506070809

1

times104

(a)

AEKFAEKFndashPID

SOC

erro

r (

)

50 100 150 200 250142144146148

1 15 2 25 3 3505Time (s)

ndash5ndash4ndash3ndash2ndash1

012345

times104

(b)

Figure 11 Comparison of SOC estimation under DST (a) SOC estimation result (b) SOC estimation error

10 Complexity

SOC

6000 6500 7000

081

082

083

1 15 2 25 3 3505Time (s)

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 3 3505Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 13 Comparison of SOC estimation with Gaussian noise under DST (a) SOC estimation result (b) SOC estimation error

SOC

7500 8000 8500

077078079

08

1 15 2 25 305Time (s)

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)SO

C er

ror (

)

50 100 150 200 250 300076078

08082084

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

1 15 2 25 305Time (s)

AEKFAEKFndashPID

times104

(b)

Figure 12 Comparison of SOC estimation under FUDS (a) SOC estimation result (b) SOC estimation error

Table 1 SOC estimation results

AEKF-PID AEKFMAE () 12422 (DST) 17257 (DST)MAE () 17141 (FUDS) 20093 (FUDS)AAE () 03327 (DST) 07466 (DST)AAE () 05235 (FUDS) 07345 (FUDS)

Complexity 11

Table 2 SOC estimation with Gaussian noise

AEKF-PID AEKFMAE () 17726 (DST) 19795 (DST)MAE () 19924 (FUDS) 28171 (FUDS)AAE () 05101 (DST) 10416 (DST)AAE () 05261 (FUDS) 11209 (FUDS)

1 15 2 25 305Time (s)

SOC

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

SOC

erro

r (

)

1 15 2 25 305Time (s)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 15 Comparison of SOC estimation with non-Gaussian noise (a) SOC estimation result (b) SOC estimation error

7500 8000 8500076077078079

1 15 2 25 305Time (s)

SOC

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 305Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 14 Comparison of SOC estimation with Gaussian noise under FUDS (a) SOC estimation result (b) SOC estimation error

12 Complexity

that in non-Gaussian noise whether or not the PIDfeedback is used

5 Conclusion

Based on the simple one-order ECM a novel method ofcombination of AEKF and PID feedback strategy is pro-posed for LIB SOC estimation +e parameters of the ECMare updated online using the FFRLS method to ensure LIBECM reliability and parameters integrity An improvedAEKF-PID technique is introduced to alleviate the negativeeffects resulting from the ECM error and the measurementerror via replacing the single innovation in feedback loop ofAEKF with an innovation vector-based PID module Toevaluate the performance of the proposed algorithm ex-periment and simulation results under 10A discharge DSTand FUDS cycles are adopted to validate the SOC estimation+e simulation results show that the maximum estimationerror of AEKF-PID is below 06 13 and 18 under 10Adischarge DST and FUDS cycles respectively +e SOCestimation error of AEKF-PID is still below 2 even in aGaussian noise situation +ese results suggest that theproposed AEKF-PID method can provide reliable SOCestimation with high precision and robustness Due to thefact that the local flat region in OCV-SOC fitted curve issusceptible to interference from measurement error ourfuture studies will concentrate on the optimization of OCVestimation

Data Availability

Data are available upon request to the corresponding author

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Zheng Liu proposed the original idea Zheng Liu designedthe novel algorithm Yuan Qiu Chunshan Yang and JianboJi performed and analyzed the experiments together ZhengLiu wrote the original manuscript Zheng Liu and ZhenhuaZhao revised the final manuscript

Acknowledgments

+is work was financially supported by the Guangxi NaturalScience Foundation (2018GXNSFAA281161 and2020GXNSFAA297032)

References

[1] X Hu H Yuan C Zou Z Li and L Zhang ldquoCo-estimationof state of charge and state of health for lithium-ion batteriesbased on fractional-order calculusrdquo IEEE Transactions onVehicular Technology vol 67 no 11 pp 10319ndash10329 2018

[2] X Zhou and C Feng ldquo+e impact of environmental regu-lation on fossil energy consumption in China direct andindirect effectsrdquo Journal of Cleaner Production vol 142pp 3174ndash3183 2017

[3] WWaag C Fleischer and D U Sauer ldquoCritical review of themethods for monitoring of lithium-ion batteries in electricand hybrid vehiclesrdquo Journal of Power Sources vol 258pp 321ndash339 2014

[4] R Xiong Y Zhang J Wang H He S Peng and M PechtldquoLithium-ion battery health prognosis based on a real batterymanagement system used in electric vehiclesrdquo IEEE Trans-actions on Vehicular Technology vol 68 no No 5pp 4110ndash4121 2019

[5] H Fathabadi ldquoPlug-in hybrid electric vehicles (PHEVs)replacing internal combustion engine with clean and re-newable energy based auxiliary power sourcesrdquo IEEETransactions on Power Electronics vol 33 no No 11pp 9611ndash9618 2018

[6] Y Xing W He M Pecht and K L Tsui ldquoState of chargeestimation of lithium-ion batteries using the open-circuitvoltage at various ambient temperaturesrdquo Applied Energyvol 113 pp 106ndash115 2014

[7] N Yang X Zhang and G Li ldquoState of charge estimation forpulse discharge of a LiFePO4 battery by a revised Ahcountingrdquo Electrochimica Acta vol 151 pp 63ndash71 2015

[8] K S Ng C-S Moo Y-P Chen and Y-C Hsieh ldquoEnhancedCoulomb counting method for estimating state-of-charge andstate-of-health of lithium-ion batteriesrdquo Applied Energyvol 86 no 9 pp 1506ndash1511 2009

[9] Z Liu X Dang and B Jing ldquoA novel open circuit voltagebased state of charge estimation for lithium-ion battery bymulti-innovation Kalman filterrdquo IEEE Access vol 7pp 49432ndash49447 2019

[10] J D J Rubio ldquoStability analysis of the modified Levenberg-Marquardt algorithm for the artificial neural network train-ingrdquo IEEE Transactions on Neural Networks and LearningSystems 2020

[11] G Aquino J D J Rubio J Pacheco et al ldquoNovel nonlinearhypothesis for the delta parallel robot modelingrdquo IEEE Accessvol 8 no 1 pp 46324ndash46334 2020

[12] J D J Rubio ldquoSOFMLS online self-organizing fuzzy mod-ified least-squares networkrdquo IEEE Transactions on FuzzySystems vol 17 no 6 pp 1296ndash1309 2009

[13] J A M Campantildea ldquoOn the estimation and control of non-linear systems with parametric uncertainties and noisy out-putsrdquo IEEE Access vol 6 pp 31968ndash31973 2018

[14] S Pramanik and S Anwar ldquoElectrochemical model basedcharge optimization for lithium-ion batteriesrdquo Journal ofPower Sources vol 313 pp 164ndash177 2016

[15] F Sun X Hu Y Zou and S Li ldquoAdaptive unscented Kalmanfiltering for state of charge estimation of a lithium-ion batteryfor electric vehiclesrdquo Energy vol 36 no 5 pp 3531ndash35402014

[16] Z Liu and X Dang ldquoA new method for state of charge andcapacity estimation of lithium-ion battery based on dualstrong tracking adaptive H infinity filterrdquo Mathematical

Table 3 SOC estimation with non-Gaussian noise

AEKF-PID AEKFMAE () 40590 55443AAE () 26074 32427

Complexity 13

Problems in Engineering vol 2018 Article ID 521820518 pages 2018

[17] Y Wang C Zhang and Z Chen ldquoOn-line battery state-of-charge estimation based on an integrated estimatorrdquo AppliedEnergy vol 185 pp 2026ndash2032 2017

[18] R Xiao J Shen X Li W Yan E Pan and Z ChenldquoComparisons of modeling and state of charge estimation forlithium-ion battery based on fractional order and integralorder methodsrdquo Energies vol 9 no No 3 2016

[19] Y Qiu X Li W Chen Z-m Duan and L Yu ldquoState ofcharge estimation of vanadium redox battery based on im-proved extended Kalman filterrdquo ISA Transactions vol 94pp 326ndash337 2019

[20] Y Xu M Hu A Zhou et al ldquoState of charge estimation forlithium-ion batteries based on adaptive dual Kalman filterrdquoAppliedMathematical Modelling vol 77 pp 1255ndash1272 2020

[21] Y Bi and S-Y Choe ldquoAn adaptive sigma-point Kalman filterwith state equality constraints for online state-of-charge es-timation of a Li(NiMnCo)O2Carbon battery using a re-duced-order electrochemical modelrdquo Applied Energyvol 258 p 113925 2020

[22] Z Liu X Dang B Jing and J Ji ldquoA novel model-based stateof charge estimation for lithium-ion battery using adaptiverobust iterative cubature Kalman filterrdquo Electric Power Sys-tems Research vol 177 p 105951 2019

[23] J Wei G Dong and Z Chen ldquoOn-board adaptive model forstate of charge estimation of lithium-ion batteries based onKalman filter with proportional integral-based error adjust-mentrdquo Journal of Power Sources vol 365 pp 308ndash319 2017

[24] F Yang Y Xing D Wang and K-L Tsui ldquoA comparativestudy of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dy-namic loading profilerdquo Applied Energy vol 164 pp 387ndash3992016

[25] Y Li C Wang and J Gong ldquoA multi-model probability SOCfusion estimation approach using an improved adaptiveunscented Kalman filter techniquerdquo Energy vol 141pp 1402ndash1415 2017

[26] J Linghu L Kang M Liu X Luo Y Feng and C LuldquoEstimation for state-of-charge of lithium-ion battery basedon an adaptive high-degree cubature Kalman filterrdquo Energyvol 189 Article ID 116204 2019

[27] Q Zhu M Xu W Liu and M Zheng ldquoA state of chargeestimation method for lithium-ion batteries based on frac-tional order adaptive extended Kalman filterrdquo Energyvol 187 Article ID 115880 2019

[28] Z Liu and H He ldquoSensor fault detection and isolation for alithium-ion battery pack in electric vehicles using adaptiveextended Kalman filterrdquo Applied Energy vol 185 pp 2033ndash2044 2017

[29] X Lai Y Zheng and T Sun ldquoA comparative study of dif-ferent equivalent circuit models for estimating state-of-chargeof lithium-ion batteriesrdquo Electrochimica Acta vol 259pp 566ndash577 2018

[30] Y Li J Chen and F Lan ldquoEnhanced online model identi-fication and state of charge estimation for lithium-ion batteryunder noise corrupted measurements by bias compensationrecursive least squaresrdquo Journal of Power Sources vol 456p 227984 2020

[31] V Duang H Bastawrous K ldquoLim K See P Zhang andS Dou ldquoOnline state of charge and model parameters esti-mation of the LiFePO4 battery in electric vehicles usingmultiple adaptive forgetting factors recursive least-squaresrdquoJournal of Power Sources vol 296 pp 215ndash224 2015

[32] Y Tian R Lai X Li L Xiang and J Tian ldquoA combinedmethod for state-of-charge estimation for lithium-ion bat-teries using a long short-term memory network and anadaptive cubature Kalman filterrdquo Applied Energy vol 265p 114789 2020

[33] C Chen R Xiong R Yang W Shen and F Sun ldquoState-of-charge estimation of lithium-ion battery using an improvedneural network model and extended Kalman filterrdquo Journal ofCleaner Production vol 234 pp 1153ndash1164 2019

14 Complexity

According to the one-order ECM its mathematicalmodel can be expressed as

_Vpt minusVpt

RpCp

+It

Cp

Voutt Voct minus Vpt minus R0It

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(1)

where Voc shows the OCV R0 is the ohmic resistance RPCp

includes the polarization resistance RP and polarizationcapacitance Cp Vp represents the polarization voltage onRPCp I is the chargedischarge current and Vout representsthe terminal voltage

Rewrite equation (1) as frequency domain expressionusing Laplace transform the symbol Vp of polarizationvoltage can be eliminated

Vout(s) Voc(s) minus R0 +Rp

1 + RpCps1113888 1113889I(s) (2)

+e corresponding transfer function G(s) of ECM can beformulated as

G(s) Vout(s) minus Voc(s)

I(s) minus R0 +

Rp

1 + RpCps1113888 1113889 (3)

+e sdomain can be mapped to the z domain by bilineartransformation methods (2Ts)((1 minus zminus 1)(1 + zminus 1))

G zminus 1

1113872 1113873 Vout z

minus 11113872 1113873 minus Voc z

minus 11113872 1113873

I zminus 1

1113872 1113873

a2 + a3zminus 1

1 + a1zminus 1 (4)

+e discrete form of equation (1) can be represented as

Voutk minus Vock a1 Voutkminus 1 minus Vockminus 11113872 1113873 + a2Ik + a3Ikminus 1

(5)

Since the OCV has a correlation with SOC and tem-perature the OCV is usually expressed asVock f(soc temp) SOC can be considered as constantunder certain conditions including one second samplingtime and constant temperature

dVock

dk

zVock

zSOCzSOC

zk+

zVock

ztempztemp

zk 0 (6)

+at is ΔVock Vock minus Vockminus 1 0+en equation (4) can be rewritten as

Voutk a1Voutkminus 1 + a2Ik + a3Ikminus 1 + 1 minus a1( 1113857Vock (7)

+e model parameters R0 Rp and Cp can be obtainedfrom a1 a2 and a3

R0 a3 minus a2( 1113857

1 + a1( 1113857

Rp 2 a3 + a1a2( 1113857

a21 minus 11113872 1113873

Cp minus a1 + 1( 1113857

2

4 a3 + a1a2( 1113857

(8)

22 Model Parameters Identification +e LIB modelingmethod based on ECM can provide concise and effectivesupport for subsequent state estimation An effective pa-rameter identification method is a necessary condition forachieving this goal As the LIB model parameters are notconstant when it is charged or discharged the offlineidentification of model parameters may mismatch the realparameters According to this problem the model param-eters are updated online under discharge operation of LIB

RLS algorithm based on minimum sum-squared errortheory is a commonly used model parameter identificationmethod widely applied in tracking of time-varying pa-rameters [31] To suppress the effect of data saturation in theidentification of time-varying parameters with the RLSmethod the RLS with forgetting factor (FFRLS) algorithmcan be used in battery time-varying parameter identification[32]

+e battery parameter identification is to determinecoefficient a1 a2 and a3 values based on the current andvoltage measurements

+e form of least squares can be expressed as

yk φTk θk + ξk (9)

where φk is the current and voltage measurements and θk isthe parameters to be identified

Set the objective function

J 1113944L

k1λLminus k

yk minus φTk

1113954θk1113960 11139612 (10)

where 1113954θ is the identification value of θ λ is forgetting factor0lt λle 1 and the FFRLS degenerates to RLS when λ 1

+e objective function is close to the minimum valuethrough the recursive calculation

R0 RP

CP

VocVout

Vp

I

++

ndashndash

+

ndash

Figure 1 One-order ECM

Complexity 3

ek yk minus φTk

1113954θkminus 1

Kk Pkminus 1φ

Tk

λ + φTk Pkminus 1φk

Pk Pkminus 1 minus Kkφ

Tk Pkminus 1

λ

1113954θk 1113954θkminus 1 + Kkek

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

+e correspondingmeasurements and parameters of LIBare as follows

φk Voutkminus 1 Ik Ikminus 1 11113858 1113859

θk a1 a2 a3 1 minus a1( 1113857Vock1113858 1113859T

⎧⎨

⎩ (12)

+e coefficients a1 a2 a3 and OCV are identified re-cursively by the FFRLS method in equation (11) then theparameters R0 RP and Cp can be solved reversely fromcoefficients a1 a2 and a3

3 SOC Estimation Using PID-Based AEKF

31 Basic AEKF As a linear stochastic system with whitenoises the corresponding state estimation can be calculatedby the EKF

xk Akxkminus 1 + Bkukminus 1 + wkminus 1

yk Ckxk + Dkuk + vk1113896 (13)

where xk represents the system state matrix Ak Bk Ck andDk are the dynamic coefficients of the state function andobservation function respectively yk is the output matrixuk is the input vector wk and vk are the state Gaussian whitenoise and measurement Gaussian white noise respectively+e iterative process of EKF is as follows

(i) Initialization

1113954x0 P0 Q0 R01113864 1113865 (14)

(ii) Prior estimation

1113954xk|kminus 1 Ak1113954xkminus 1|kminus 1 + Bkukminus 1

Pk|kminus 1 AkPkminus 1|kminus 1ATk + Qkminus 1

⎧⎨

⎩ (15)

(iii) Measurement correction

ek yk minus Ck1113954xk|kminus 1 minus Dkuk

Kk Pk|kminus 1CTk CkPk|kminus 1C

Tk + Rk1113872 1113873

minus 1

1113954xk|k 1113954xk|kminus 1 + Kkek

Pk|k I minus KkCk( 1113857Pk|kminus 1

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(16)

Since the noise statistics of the dynamic model are time-varying it is necessary to construct an adaptive EKF to adaptto the noise statistics In this paper the state noise varianceQk and the measurement noise variance Rk based on the

maximum likelihood (ML) criterion can be online updatedto reflect the changes of the state noise characteristics andmeasurement noise characteristics [33] so as to ensure thatthe KF can better adapt to the changes of the noise statisticalcharacteristics

+e variance matrix of the innovation sequence ek can becalculated from equation (16)

Fk Rk + CkPk|kminus 1CTk (17)

+e equivalent value of the innovation sequence ek basedon the moving window method with size L is used tosubstitute Qk and Rk can be obtained as

Rk 1L

1113944

k

nkminus L+1eke

Tk minus CkPk|kminus 1C

Tk (18)

From the definition of KF

KkCkPk|kminus 1 Kk

1L

1113944

k

nkminus L+1eke

Tk

⎛⎝ ⎞⎠KTk KkP

minus 1k|kminus 1K

Tk

(19)

Similarly we can get KkCkPk|kminus 1 from equation (16)

KkCkPk|kminus 1 Pk|kminus 1 minus Pk|k AkPkminus 1|kminus 1ATk minus Pk + Qkminus 1

(20)

Equation (20) can be substituted into equation (19)

Qk KkFkKTk + Pk|kminus 1 minus Pk (21)

Ignore the variation of the state variance matrix

Qk KkFkKTk (22)

32 Basic PIDController +e PID controller is composed ofproportional coefficient integral coefficient and differentialcoefficient Its relationship between input and output incontinuous time domain is

y(t) kP e(t) +1

TI

1113946 e(t)dt + TD

de(t)

dt1113890 1113891 (23)

Adopt the digital PID in computer control system

y(k) kP e(k) +1

TI

1113944

k

i0Te(k) + TD

e(k) minus e(k minus 1)

T⎡⎣ ⎤⎦

(24)

One of the most common digital PID is incremental PIDby suppressing integral saturation

Δy(k) kP ek minus ekminus 1( 11138571113858

+ kIek + kD ek minus 2ekminus 1 + ekminus 2( 11138571113859(25)

33 PID Feedback Unit in Improved KF To obtain the re-liable SOC estimation result on the basis of uncertainty ofmodel and measurement noise many estimation methods

4 Complexity

with KF are presented Since the loss of correction data in KFmay be due to the fact that only single proportional inno-vation feedback is used in KF the innovation vector-basedPID feedback controller is introduced into measurementcorrection step of KF in this paper +is paper puts forwardan improved AEKF based on PID controller with self-tuningPID parameters kP kI and kD Due to the fact that moreterminal voltage innovation can be described by the PID unitin measurement correction step of improved AEKF thebattery state can be predicted more accurately by the im-proved AEKF

+e PID feedback unit deduced from innovation vectoris first introduced +e single feedback unit in KF issubstituted by the PID unit which referred to the PIDfeedback strategy in the KF structure as shown in Figure 2

+e state measurement update 1113954xk|k from common KFcan be expressed by letting ek to be single innovation inequation (16)

In order to get the equivalent form of the incrementalPID in equation (25) the measurement correction by singleinnovation ek in equation (16) can be transformed as

1113954xk|k 1113954xk|kminus 1 + KPk ek minus ekminus 1( 1113857 + KIkek

+ KDk ek minus 2ekminus 1 + ekminus 2( 1113857(26)

+e structure diagram of the improved method bycombined KF and PID feedback unit is demonstrated inFigure 3 +e core idea is to expand the single innovation toan innovation vector +at is ekis extended into

ek ekminus 1 ekminus 21113858 1113859T To ensure the consistency of data di-

mension the following improvements are made gain kk isextended into kk kkminus 1 kkminus 21113858 1113859 and measurement output zk

is extended into zk zkminus 1 zkminus 21113858 1113859T

Meanwhile a posteriori estimation 1113954xk|k in the mea-surement correction step needs to be rewritten with mod-ified parameters including kk kkminus 1 kkminus 21113858 1113859

zk zkminus 1 zkminus 21113858 1113859T and ek ekminus 1 ekminus 21113858 1113859

T

1113954xk|k 1113954xk|kminus 1 + minus 2kkminus 2 minus kkminus 1( 1113857 ek minus ekminus 1( 1113857

+ kk + kkminus 1 + kkminus 2( 1113857ek + kkminus 2 ek minus 2ekminus 1 + ekminus 2( 1113857

(27)

Consequently the regulation parameters kP kI and kD

of PID unit in the measurement correction step can besolved by equations (26) and (27)

kPk minus 2kkminus 2 minus kkminus 1

kIk kk + kkminus 1 + kkminus 2

kDk kkminus 2

(28)

From equations (26) and (28) we can get the PIDfeedback unit by setting innovation vector and the PIDcoefficients kP kI and kD can be solved by the expressions ofgains kk kkminus 1 and kkminus 2 in three sampling points +ree in-novation data at times k k-1 and k-2 are used whenupdating the state at time k +e main reason for increasingthe robustness of the improved algorithm is to reuse in-novative data to update the state at adjacent times In orderto improve the performance of SOC estimation the

combination of PID feedback and AEKF is introduced inSection 34 In addition the covariance of noise wk and vk isupdated based on the principle of covariance matchingrespectively

+e PID-based KF introduces more than one prior in-novation which greatly enhances the effect of feedbackcompensation in KF Experiments show that the innovationvector-based KF has better robustness than single-innova-tion-based KF in strong nonlinear systems [9] Although thecalculation load of PID-based KF is more than that of thesingle-innovation-based KF the increased amount of cal-culation is acceptable relative to the improvement inaccuracy

34 SOC Estimation Based on AEKF-PID +e discreteequation of the LIB model in equation (1) can be rewritten asfollows

SOCk

Upk

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1 0

0 eminus Δtτ1( )

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

SOCkminus 1

Upkminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

minusηΔtC

1113874 1113875

Rp 1 minus eminus (Δtτ)

1113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ikminus 1 + wkminus 1

Uk UOC SOCk( 1113857 minus Upk minus IkR0 + vk

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

By selecting x [SOC Up]T as the state vector choosingIk and Uk as the input current and output terminal voltagerespectively the state equation and measurement equationof ECM in a discrete-time representation are as follows

xk Akxkminus 1 + BkIkminus 1 + wkminus 1

yk Ckxk + DkIkminus 1 + vk1113896 (30)

where Akminus 1 (zfzx)|xxk|kminus 1

1 00 e

minus (Δtτ1)1113888 1113889 Bkminus 1 minus1113858

((ηΔt)C)Rp(1 minus eminus (Δtτ)

)] Ck (zhzx)|xxk|kminus 1 [(dUOC

(SOCk|kminus 1))(dS OCk|kminus 1) minus 1] and Dk minus R0 we assumethat η 1 under arbitrary charge-discharge condition

+e detailed process of the AEKF-PID method is asfollows

Step 1 Initialization

(a) SOC and Up initialized state

xkminus 1|kminus 1 SOCkminus 1|kminus 1upkminus 1|kminus 11113872 1113873 (31)

(b) State error initialized covariance

Pkminus 1|kminus 1 E xkminus 1|kminus 1 minus 1113954xk|kminus 11113872 1113873 xkminus 1|kminus 1 minus 1113954xkminus 1|kminus 11113872 1113873T

1113876 1113877

(32)

(c) Noise initialized covariance Q0 R0

Step 2 State prediction

(a) State prior estimation

1113954xk|kminus 1 Ak|kminus 11113954xkminus 1|kminus 1 + Bkminus 1Ikminus 1 (33)

Complexity 5

(b) State error covariance prior estimation

Pk|kminus 1 Ak|kminus 1Pkminus 1|kminus 1ATk|kminus 1 + Qkminus 1 (34)

Step 3 Measurement correction

(a) Gain matrix update

Kk Pk|kminus 1CTk CkPk|kminus 1C

Tk + Rk1113872 1113873

minus 1 (35)

(b) State measurement update by PID feedback unit

1113954xk|k 1113954xk|kminus 1 + kPk ek minus ekminus 1( 1113857

+ kIkek + kDk ek minus 2ekminus 1 + ekminus 2( 1113857(36)

(c) State error covariance update

Pk|k I minus KkCk( 1113857Pk|kminus 1 (37)

(d) Adaptive noise covariance matching

Fk 1L

1113944

k

nkminus L+1eke

Tk

Rk Fk minus CkPk|kminus 1CTk

Qk KkFkKTk

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(38)

In summary according to the state equation andmeasurement equation of ECM in equations(29)ndash(30) and the detailed process in equations(31)ndash(38) the flowchart of the proposed SOC es-timation by AEKF-PID algorithm is shown inFigure 4

Actual battery

Single feedback

Measured voltage

Calculatedvoltage

Current

Current

+ndash

Battery model

(a)

Actual battery

PID feedback

Measured voltage

Calculatedvoltage

Current

Current

+ndash

Battery model

(b)

Figure 2 Block diagram of KF and improved KF (a) KF with single feedback (b) Improved KF with PID feedback

Bk

Bk

+

Ak

Ck+

+

+ +

+ Ck

Dk

Ak

+

xkndash1|kndash1

xk|k xk|kndash1yk

Dk

+

ndash

Zndash2

Kp

Ki

Kd

+

+ +

+

uk

uk

wk vk

xk

xk ndash 1

yk

LIB model

Actual LIB

PID

+

+

+

Residual

Zndash1

Zndash1

Zndash1

Figure 3 +e schematic of PID-based KF

6 Complexity

4 Results and Discussion

41 ExperimentalConfigurations A LIB test platform shownin Figure 5 is established to verify the effect of the ECM andmethod It consists of several major components (1) alithium-ion ferrous phosphate battery is used as the ex-perimental objects of which the nominal capacity is 20Ahand nominal voltage is 24V (2) a power battery test system(Arbin EVTS) with control accuracy which is less thanplusmn01 FSR is used for LIB chargingdischarging undervarious working condition (3) a programmable temperaturechamber is applied to control ambient temperature (4) ahost computer with MITS Pro v70 and a computerworkstation with MATLAB R2012a are used for data ac-quisition and data simulation in the experimentsrespectively

+e LIB is fully charged by constant current-constantvoltage (CC-CV) procedure after standing for two hoursthen three working conditions including 10A constantcurrent dynamic stress test (DST) and federal urban drivingschedule (FUDS) are used to test the proposed method+ree working conditions mentioned in this paper arecarried out at 25 degrees Celsius +e discharge current andterminal voltage of DSTand FUDS are shown in Figure 6 Inaddition the evaluation indicator such as maximum abso-lute error (MAE) and average absolute error (AAE) is ap-plied to the assessment of estimation algorithms

+e OCV-SOC function describing the relationshipbetween the open-circuit voltage and SOC can be fitted byOCV feature test data +e LIB is fully charged by the CC-

CV procedure after standing for two hours the measuredterminal voltage can be used as the OCV of 100 SOC +etest steps are as follows the LIB is discharged to 98 SOCwith 10A (05 C) constant current after standing for twohours the measured terminal voltage can be used as theOCV of 98 SOC Repeating the above steps 50 times eachmeasured terminal voltage can be obtained as the OCV ofevery 2 SOC+e eighth-order polynomial fitting functionto describe the mathematical relationship of OCV-SOC canbe shown in Figure 7

42 Parameters Identification +e ECM parameters iden-tification results are shown in Figures 8 and 9 the identi-fication result of ohmic resistance R0 polarization resistanceRp and polarization capacitance Cp can converge to a steadystate quickly from the unreliable initialization value undertwo cycles

R0Rp andCp always have a small range of fluctuation inFigures 8(a) and 9(a) which is directly related to the fluc-tuations of the coefficients a1 b1 and b2 +eoretically thedynamic changes of R0 Rp and Cp are inconsistent but thefixed forgetting factors of FFRLS have the same weight onthe three parameters which has a certain effect on thefluctuation of identification results

+e terminal voltage is identified in each sampling pe-riod based on R0 Rp and Cp As we can see fromFigures 8(b) and 9(b) the identification results of the ter-minal voltage can track the measured value of the terminalvoltage stably under two cycles +e maximum error of theterminal voltage identification is only 01923V and 01702Vrelative to the terminal voltage range from 23V to 29Vrespectively

43 Comparison of the SOC Estimation

431 Analysis of SOC Estimation under 10A DischargeBased on the identified ECM parameters the SOC estimationunder the 10A discharge cycle is presented in Figure 10 Asobserved the SOC reference with Arbin EVTS is formed witha black line the red line and blue line represent the SOC

Measurement dataand

model parameters

Initial conditionsequations (31) and (32)

State prior estimationequation (33)

Gain matrix updateequation (35)

State error covariance prior estimationequation (34)

PID coefficient update with gainequation (28)

State estimation measurement update by PID unitequation (36)

State error covariance updateequation (37)

Figure 4 +e flowchart of AEKF-PID estimator for LIB SOC

Arbin EVTS Temperature chamber

Computer

Figure 5 Battery test platform

Complexity 7

estimation with single AEKF and AEKF-PID respectivelyFrom the estimation results of two methods shown inFigure 10(a) these two SOC estimations can both track theSOC reference in a short period of time no matter AEKF orAEKF-PID Although the two SOC estimation results bothconverge to the SOC reference the SOC estimation error byAEKF-PID is less than that by AEKF as shown inFigure 10(b) +e errors of the two methods slowly increasewith an increase in time +e AAE and MAE of AEKF-PIDare 02266 and 05119 respectively and the AAE andMAE of AEKF are 08606 and 12478 respectively Itfollows that the improved AEKF-PID method shows betterresults in SOC estimation than single AEKF

432 Analysis of SOC Estimation under Two Cycles Inorder to further validate the accuracy of the proposedmethod the DST and FUDS cycles are implemented to

OCV

(V)

02 04 06 08 10SOC

22

23

24

25

26

27

28

29

Figure 7 OCV-SOC fitting curve

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(a)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

23

24

25

26

27

28

29

times104

(b)

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(c)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

22

23

24

25

26

27

28

29

times104

(d)

Figure 6 (a) Current under DST cycle (b) Terminal voltage under DST cycle (c) Current under FUDS cycle (d) Terminal voltage underFUDS cycle

8 Complexity

simulate the EV operationmode+e SOC estimation resultswith two methods under the DST and FUDS cycles areshown in Figures 11 and 12 respectively As observed theSOC reference by Arbin EVTS is formed with a black linethe red line and blue line represent the SOC estimation withsingle AEKF and AEKF-PID respectively From the esti-mation results of two methods shown in Figures 11(a) and12(a) two SOC estimations can both track the SOC refer-ence in a short period of time no matter AEKF or AEKF-PID +e SOC estimation errors with two algorithms underthe DST and FUDS cycles are shown in Figures 11(b) and12(b) respectivelyWe can see that the SOC estimation errorby the AEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 2 under the FUDS cycle meanwhilethe maximum SOC error by AEKF-PID is only 17141+eMAE and AAE of two methods under two cycles are listed inTable 1 According to the above analysis the combination ofthe PID feedback unit with AEKF has advantage over singleAEKF for SOC estimation

44 Analysis on Robustness of SOC Estimation +e highprecise SOC estimation results can be obtained by theproposed method on the assumption that the sampled ex-perimental data by Arbin EVTS is reliable However theexperiment data from the laboratory is not completely equalto actual data in EV operating mode Since the measurementdata by universal transducer is hard to avoid noise inter-ference from such as drift current and diffusion current theSOC estimation method with certain anti-interferencequality is important for BMS To further verify the anti-interference performance of the AEKF-PID a sequence ofnoise with feature of random normal distribution is added tothe operation current under DST and FUDS cycles re-spectively +e mean value of the noise is set as 0 and itsstandard deviation is set as 2 +e SOC estimation resultswith Gaussian noise of current under two conditions aredisplayed in Figures 13 and 14 +e two colored lines withblue and red are used to represent the SOC estimation byAEKF-PID and AEKF respectively +e MAE and AAE ofSOC estimation with Gaussian noise under two cycles are

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

010002000

0005

01

004005006

times104

times104

times104

(a)

05 1 15 2 25 3 350Time (s)

Term

inal

vol

tage

erro

r (V

)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 8 Model parameters identification results under DST (a) R0 Rp Cp (b) terminal voltage error

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

002004006

0005

01

0500

1000

times104

times104

times104

(a)

Term

inal

vol

tage

erro

r (V

)

05 1 15 2 25 3 350Time (s)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 9 Model parameters identification results under FUDS (a) R0 Rp Cp (b) terminal voltage error

Complexity 9

listed in Table 2 As we can see from Figures 13 and 14 dueto the introduction of current noise in two cycles thefluctuation of two corresponding SOC estimation errorsincreased We can see that the SOC estimation error by theAEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 28 under the FUDS cyclemeanwhile the maximum SOC error by AEKF-PID isclose to 2+e MAE and AAE of two methods under twocycles are listed in Table 2 According to the analysisabove the combination of PID feedback unit with AEKFhas advantage over single AEKF in robustness of SOCestimation

To further verify the anti-interference performance ofthe AEKF-PID a sequence of noise with feature of ran-dom non-Gaussian distribution is added to the operationcurrent +e SOC estimation result with non-Gaussiannoise is displayed in Figure 15 We can see that the twoSOC estimation errors with non-Gaussian noise increasecompared to that with Gaussian noise +e results inTable 3 show that the MAE of AEKF is greater than 55and the MAE of AEKF-PID is close to 4 At the end ofdischarge the SOC error of AEKF increases graduallymeanwhile the SOC error of AEKF-PID reaches 3 +eresults show that the basic AEKF-based SOC estimationmethod has better performance in Gaussian noise than

SOC

ReferenceAEKFAEKFndashPID

2500 3000 3500 4000082084086088

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

055

06

065

07

075

08

085

09

095

1

(a)

AEKFAEKFndashPID

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

SOC

erro

r (

)

ndash5ndash4ndash3ndash2ndash1

012345

(b)

Figure 10 Comparison of SOC estimation under 10A discharge (a) SOC estimation result (b) SOC estimation error

ReferenceAEKFAEKFndashPID

SOC

8500 9000 9500

076077078

1 15 2 25 3 3505Time (s)

0010203040506070809

1

times104

(a)

AEKFAEKFndashPID

SOC

erro

r (

)

50 100 150 200 250142144146148

1 15 2 25 3 3505Time (s)

ndash5ndash4ndash3ndash2ndash1

012345

times104

(b)

Figure 11 Comparison of SOC estimation under DST (a) SOC estimation result (b) SOC estimation error

10 Complexity

SOC

6000 6500 7000

081

082

083

1 15 2 25 3 3505Time (s)

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 3 3505Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 13 Comparison of SOC estimation with Gaussian noise under DST (a) SOC estimation result (b) SOC estimation error

SOC

7500 8000 8500

077078079

08

1 15 2 25 305Time (s)

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)SO

C er

ror (

)

50 100 150 200 250 300076078

08082084

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

1 15 2 25 305Time (s)

AEKFAEKFndashPID

times104

(b)

Figure 12 Comparison of SOC estimation under FUDS (a) SOC estimation result (b) SOC estimation error

Table 1 SOC estimation results

AEKF-PID AEKFMAE () 12422 (DST) 17257 (DST)MAE () 17141 (FUDS) 20093 (FUDS)AAE () 03327 (DST) 07466 (DST)AAE () 05235 (FUDS) 07345 (FUDS)

Complexity 11

Table 2 SOC estimation with Gaussian noise

AEKF-PID AEKFMAE () 17726 (DST) 19795 (DST)MAE () 19924 (FUDS) 28171 (FUDS)AAE () 05101 (DST) 10416 (DST)AAE () 05261 (FUDS) 11209 (FUDS)

1 15 2 25 305Time (s)

SOC

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

SOC

erro

r (

)

1 15 2 25 305Time (s)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 15 Comparison of SOC estimation with non-Gaussian noise (a) SOC estimation result (b) SOC estimation error

7500 8000 8500076077078079

1 15 2 25 305Time (s)

SOC

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 305Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 14 Comparison of SOC estimation with Gaussian noise under FUDS (a) SOC estimation result (b) SOC estimation error

12 Complexity

that in non-Gaussian noise whether or not the PIDfeedback is used

5 Conclusion

Based on the simple one-order ECM a novel method ofcombination of AEKF and PID feedback strategy is pro-posed for LIB SOC estimation +e parameters of the ECMare updated online using the FFRLS method to ensure LIBECM reliability and parameters integrity An improvedAEKF-PID technique is introduced to alleviate the negativeeffects resulting from the ECM error and the measurementerror via replacing the single innovation in feedback loop ofAEKF with an innovation vector-based PID module Toevaluate the performance of the proposed algorithm ex-periment and simulation results under 10A discharge DSTand FUDS cycles are adopted to validate the SOC estimation+e simulation results show that the maximum estimationerror of AEKF-PID is below 06 13 and 18 under 10Adischarge DST and FUDS cycles respectively +e SOCestimation error of AEKF-PID is still below 2 even in aGaussian noise situation +ese results suggest that theproposed AEKF-PID method can provide reliable SOCestimation with high precision and robustness Due to thefact that the local flat region in OCV-SOC fitted curve issusceptible to interference from measurement error ourfuture studies will concentrate on the optimization of OCVestimation

Data Availability

Data are available upon request to the corresponding author

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Zheng Liu proposed the original idea Zheng Liu designedthe novel algorithm Yuan Qiu Chunshan Yang and JianboJi performed and analyzed the experiments together ZhengLiu wrote the original manuscript Zheng Liu and ZhenhuaZhao revised the final manuscript

Acknowledgments

+is work was financially supported by the Guangxi NaturalScience Foundation (2018GXNSFAA281161 and2020GXNSFAA297032)

References

[1] X Hu H Yuan C Zou Z Li and L Zhang ldquoCo-estimationof state of charge and state of health for lithium-ion batteriesbased on fractional-order calculusrdquo IEEE Transactions onVehicular Technology vol 67 no 11 pp 10319ndash10329 2018

[2] X Zhou and C Feng ldquo+e impact of environmental regu-lation on fossil energy consumption in China direct andindirect effectsrdquo Journal of Cleaner Production vol 142pp 3174ndash3183 2017

[3] WWaag C Fleischer and D U Sauer ldquoCritical review of themethods for monitoring of lithium-ion batteries in electricand hybrid vehiclesrdquo Journal of Power Sources vol 258pp 321ndash339 2014

[4] R Xiong Y Zhang J Wang H He S Peng and M PechtldquoLithium-ion battery health prognosis based on a real batterymanagement system used in electric vehiclesrdquo IEEE Trans-actions on Vehicular Technology vol 68 no No 5pp 4110ndash4121 2019

[5] H Fathabadi ldquoPlug-in hybrid electric vehicles (PHEVs)replacing internal combustion engine with clean and re-newable energy based auxiliary power sourcesrdquo IEEETransactions on Power Electronics vol 33 no No 11pp 9611ndash9618 2018

[6] Y Xing W He M Pecht and K L Tsui ldquoState of chargeestimation of lithium-ion batteries using the open-circuitvoltage at various ambient temperaturesrdquo Applied Energyvol 113 pp 106ndash115 2014

[7] N Yang X Zhang and G Li ldquoState of charge estimation forpulse discharge of a LiFePO4 battery by a revised Ahcountingrdquo Electrochimica Acta vol 151 pp 63ndash71 2015

[8] K S Ng C-S Moo Y-P Chen and Y-C Hsieh ldquoEnhancedCoulomb counting method for estimating state-of-charge andstate-of-health of lithium-ion batteriesrdquo Applied Energyvol 86 no 9 pp 1506ndash1511 2009

[9] Z Liu X Dang and B Jing ldquoA novel open circuit voltagebased state of charge estimation for lithium-ion battery bymulti-innovation Kalman filterrdquo IEEE Access vol 7pp 49432ndash49447 2019

[10] J D J Rubio ldquoStability analysis of the modified Levenberg-Marquardt algorithm for the artificial neural network train-ingrdquo IEEE Transactions on Neural Networks and LearningSystems 2020

[11] G Aquino J D J Rubio J Pacheco et al ldquoNovel nonlinearhypothesis for the delta parallel robot modelingrdquo IEEE Accessvol 8 no 1 pp 46324ndash46334 2020

[12] J D J Rubio ldquoSOFMLS online self-organizing fuzzy mod-ified least-squares networkrdquo IEEE Transactions on FuzzySystems vol 17 no 6 pp 1296ndash1309 2009

[13] J A M Campantildea ldquoOn the estimation and control of non-linear systems with parametric uncertainties and noisy out-putsrdquo IEEE Access vol 6 pp 31968ndash31973 2018

[14] S Pramanik and S Anwar ldquoElectrochemical model basedcharge optimization for lithium-ion batteriesrdquo Journal ofPower Sources vol 313 pp 164ndash177 2016

[15] F Sun X Hu Y Zou and S Li ldquoAdaptive unscented Kalmanfiltering for state of charge estimation of a lithium-ion batteryfor electric vehiclesrdquo Energy vol 36 no 5 pp 3531ndash35402014

[16] Z Liu and X Dang ldquoA new method for state of charge andcapacity estimation of lithium-ion battery based on dualstrong tracking adaptive H infinity filterrdquo Mathematical

Table 3 SOC estimation with non-Gaussian noise

AEKF-PID AEKFMAE () 40590 55443AAE () 26074 32427

Complexity 13

Problems in Engineering vol 2018 Article ID 521820518 pages 2018

[17] Y Wang C Zhang and Z Chen ldquoOn-line battery state-of-charge estimation based on an integrated estimatorrdquo AppliedEnergy vol 185 pp 2026ndash2032 2017

[18] R Xiao J Shen X Li W Yan E Pan and Z ChenldquoComparisons of modeling and state of charge estimation forlithium-ion battery based on fractional order and integralorder methodsrdquo Energies vol 9 no No 3 2016

[19] Y Qiu X Li W Chen Z-m Duan and L Yu ldquoState ofcharge estimation of vanadium redox battery based on im-proved extended Kalman filterrdquo ISA Transactions vol 94pp 326ndash337 2019

[20] Y Xu M Hu A Zhou et al ldquoState of charge estimation forlithium-ion batteries based on adaptive dual Kalman filterrdquoAppliedMathematical Modelling vol 77 pp 1255ndash1272 2020

[21] Y Bi and S-Y Choe ldquoAn adaptive sigma-point Kalman filterwith state equality constraints for online state-of-charge es-timation of a Li(NiMnCo)O2Carbon battery using a re-duced-order electrochemical modelrdquo Applied Energyvol 258 p 113925 2020

[22] Z Liu X Dang B Jing and J Ji ldquoA novel model-based stateof charge estimation for lithium-ion battery using adaptiverobust iterative cubature Kalman filterrdquo Electric Power Sys-tems Research vol 177 p 105951 2019

[23] J Wei G Dong and Z Chen ldquoOn-board adaptive model forstate of charge estimation of lithium-ion batteries based onKalman filter with proportional integral-based error adjust-mentrdquo Journal of Power Sources vol 365 pp 308ndash319 2017

[24] F Yang Y Xing D Wang and K-L Tsui ldquoA comparativestudy of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dy-namic loading profilerdquo Applied Energy vol 164 pp 387ndash3992016

[25] Y Li C Wang and J Gong ldquoA multi-model probability SOCfusion estimation approach using an improved adaptiveunscented Kalman filter techniquerdquo Energy vol 141pp 1402ndash1415 2017

[26] J Linghu L Kang M Liu X Luo Y Feng and C LuldquoEstimation for state-of-charge of lithium-ion battery basedon an adaptive high-degree cubature Kalman filterrdquo Energyvol 189 Article ID 116204 2019

[27] Q Zhu M Xu W Liu and M Zheng ldquoA state of chargeestimation method for lithium-ion batteries based on frac-tional order adaptive extended Kalman filterrdquo Energyvol 187 Article ID 115880 2019

[28] Z Liu and H He ldquoSensor fault detection and isolation for alithium-ion battery pack in electric vehicles using adaptiveextended Kalman filterrdquo Applied Energy vol 185 pp 2033ndash2044 2017

[29] X Lai Y Zheng and T Sun ldquoA comparative study of dif-ferent equivalent circuit models for estimating state-of-chargeof lithium-ion batteriesrdquo Electrochimica Acta vol 259pp 566ndash577 2018

[30] Y Li J Chen and F Lan ldquoEnhanced online model identi-fication and state of charge estimation for lithium-ion batteryunder noise corrupted measurements by bias compensationrecursive least squaresrdquo Journal of Power Sources vol 456p 227984 2020

[31] V Duang H Bastawrous K ldquoLim K See P Zhang andS Dou ldquoOnline state of charge and model parameters esti-mation of the LiFePO4 battery in electric vehicles usingmultiple adaptive forgetting factors recursive least-squaresrdquoJournal of Power Sources vol 296 pp 215ndash224 2015

[32] Y Tian R Lai X Li L Xiang and J Tian ldquoA combinedmethod for state-of-charge estimation for lithium-ion bat-teries using a long short-term memory network and anadaptive cubature Kalman filterrdquo Applied Energy vol 265p 114789 2020

[33] C Chen R Xiong R Yang W Shen and F Sun ldquoState-of-charge estimation of lithium-ion battery using an improvedneural network model and extended Kalman filterrdquo Journal ofCleaner Production vol 234 pp 1153ndash1164 2019

14 Complexity

ek yk minus φTk

1113954θkminus 1

Kk Pkminus 1φ

Tk

λ + φTk Pkminus 1φk

Pk Pkminus 1 minus Kkφ

Tk Pkminus 1

λ

1113954θk 1113954θkminus 1 + Kkek

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

+e correspondingmeasurements and parameters of LIBare as follows

φk Voutkminus 1 Ik Ikminus 1 11113858 1113859

θk a1 a2 a3 1 minus a1( 1113857Vock1113858 1113859T

⎧⎨

⎩ (12)

+e coefficients a1 a2 a3 and OCV are identified re-cursively by the FFRLS method in equation (11) then theparameters R0 RP and Cp can be solved reversely fromcoefficients a1 a2 and a3

3 SOC Estimation Using PID-Based AEKF

31 Basic AEKF As a linear stochastic system with whitenoises the corresponding state estimation can be calculatedby the EKF

xk Akxkminus 1 + Bkukminus 1 + wkminus 1

yk Ckxk + Dkuk + vk1113896 (13)

where xk represents the system state matrix Ak Bk Ck andDk are the dynamic coefficients of the state function andobservation function respectively yk is the output matrixuk is the input vector wk and vk are the state Gaussian whitenoise and measurement Gaussian white noise respectively+e iterative process of EKF is as follows

(i) Initialization

1113954x0 P0 Q0 R01113864 1113865 (14)

(ii) Prior estimation

1113954xk|kminus 1 Ak1113954xkminus 1|kminus 1 + Bkukminus 1

Pk|kminus 1 AkPkminus 1|kminus 1ATk + Qkminus 1

⎧⎨

⎩ (15)

(iii) Measurement correction

ek yk minus Ck1113954xk|kminus 1 minus Dkuk

Kk Pk|kminus 1CTk CkPk|kminus 1C

Tk + Rk1113872 1113873

minus 1

1113954xk|k 1113954xk|kminus 1 + Kkek

Pk|k I minus KkCk( 1113857Pk|kminus 1

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(16)

Since the noise statistics of the dynamic model are time-varying it is necessary to construct an adaptive EKF to adaptto the noise statistics In this paper the state noise varianceQk and the measurement noise variance Rk based on the

maximum likelihood (ML) criterion can be online updatedto reflect the changes of the state noise characteristics andmeasurement noise characteristics [33] so as to ensure thatthe KF can better adapt to the changes of the noise statisticalcharacteristics

+e variance matrix of the innovation sequence ek can becalculated from equation (16)

Fk Rk + CkPk|kminus 1CTk (17)

+e equivalent value of the innovation sequence ek basedon the moving window method with size L is used tosubstitute Qk and Rk can be obtained as

Rk 1L

1113944

k

nkminus L+1eke

Tk minus CkPk|kminus 1C

Tk (18)

From the definition of KF

KkCkPk|kminus 1 Kk

1L

1113944

k

nkminus L+1eke

Tk

⎛⎝ ⎞⎠KTk KkP

minus 1k|kminus 1K

Tk

(19)

Similarly we can get KkCkPk|kminus 1 from equation (16)

KkCkPk|kminus 1 Pk|kminus 1 minus Pk|k AkPkminus 1|kminus 1ATk minus Pk + Qkminus 1

(20)

Equation (20) can be substituted into equation (19)

Qk KkFkKTk + Pk|kminus 1 minus Pk (21)

Ignore the variation of the state variance matrix

Qk KkFkKTk (22)

32 Basic PIDController +e PID controller is composed ofproportional coefficient integral coefficient and differentialcoefficient Its relationship between input and output incontinuous time domain is

y(t) kP e(t) +1

TI

1113946 e(t)dt + TD

de(t)

dt1113890 1113891 (23)

Adopt the digital PID in computer control system

y(k) kP e(k) +1

TI

1113944

k

i0Te(k) + TD

e(k) minus e(k minus 1)

T⎡⎣ ⎤⎦

(24)

One of the most common digital PID is incremental PIDby suppressing integral saturation

Δy(k) kP ek minus ekminus 1( 11138571113858

+ kIek + kD ek minus 2ekminus 1 + ekminus 2( 11138571113859(25)

33 PID Feedback Unit in Improved KF To obtain the re-liable SOC estimation result on the basis of uncertainty ofmodel and measurement noise many estimation methods

4 Complexity

with KF are presented Since the loss of correction data in KFmay be due to the fact that only single proportional inno-vation feedback is used in KF the innovation vector-basedPID feedback controller is introduced into measurementcorrection step of KF in this paper +is paper puts forwardan improved AEKF based on PID controller with self-tuningPID parameters kP kI and kD Due to the fact that moreterminal voltage innovation can be described by the PID unitin measurement correction step of improved AEKF thebattery state can be predicted more accurately by the im-proved AEKF

+e PID feedback unit deduced from innovation vectoris first introduced +e single feedback unit in KF issubstituted by the PID unit which referred to the PIDfeedback strategy in the KF structure as shown in Figure 2

+e state measurement update 1113954xk|k from common KFcan be expressed by letting ek to be single innovation inequation (16)

In order to get the equivalent form of the incrementalPID in equation (25) the measurement correction by singleinnovation ek in equation (16) can be transformed as

1113954xk|k 1113954xk|kminus 1 + KPk ek minus ekminus 1( 1113857 + KIkek

+ KDk ek minus 2ekminus 1 + ekminus 2( 1113857(26)

+e structure diagram of the improved method bycombined KF and PID feedback unit is demonstrated inFigure 3 +e core idea is to expand the single innovation toan innovation vector +at is ekis extended into

ek ekminus 1 ekminus 21113858 1113859T To ensure the consistency of data di-

mension the following improvements are made gain kk isextended into kk kkminus 1 kkminus 21113858 1113859 and measurement output zk

is extended into zk zkminus 1 zkminus 21113858 1113859T

Meanwhile a posteriori estimation 1113954xk|k in the mea-surement correction step needs to be rewritten with mod-ified parameters including kk kkminus 1 kkminus 21113858 1113859

zk zkminus 1 zkminus 21113858 1113859T and ek ekminus 1 ekminus 21113858 1113859

T

1113954xk|k 1113954xk|kminus 1 + minus 2kkminus 2 minus kkminus 1( 1113857 ek minus ekminus 1( 1113857

+ kk + kkminus 1 + kkminus 2( 1113857ek + kkminus 2 ek minus 2ekminus 1 + ekminus 2( 1113857

(27)

Consequently the regulation parameters kP kI and kD

of PID unit in the measurement correction step can besolved by equations (26) and (27)

kPk minus 2kkminus 2 minus kkminus 1

kIk kk + kkminus 1 + kkminus 2

kDk kkminus 2

(28)

From equations (26) and (28) we can get the PIDfeedback unit by setting innovation vector and the PIDcoefficients kP kI and kD can be solved by the expressions ofgains kk kkminus 1 and kkminus 2 in three sampling points +ree in-novation data at times k k-1 and k-2 are used whenupdating the state at time k +e main reason for increasingthe robustness of the improved algorithm is to reuse in-novative data to update the state at adjacent times In orderto improve the performance of SOC estimation the

combination of PID feedback and AEKF is introduced inSection 34 In addition the covariance of noise wk and vk isupdated based on the principle of covariance matchingrespectively

+e PID-based KF introduces more than one prior in-novation which greatly enhances the effect of feedbackcompensation in KF Experiments show that the innovationvector-based KF has better robustness than single-innova-tion-based KF in strong nonlinear systems [9] Although thecalculation load of PID-based KF is more than that of thesingle-innovation-based KF the increased amount of cal-culation is acceptable relative to the improvement inaccuracy

34 SOC Estimation Based on AEKF-PID +e discreteequation of the LIB model in equation (1) can be rewritten asfollows

SOCk

Upk

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1 0

0 eminus Δtτ1( )

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

SOCkminus 1

Upkminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

minusηΔtC

1113874 1113875

Rp 1 minus eminus (Δtτ)

1113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ikminus 1 + wkminus 1

Uk UOC SOCk( 1113857 minus Upk minus IkR0 + vk

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

By selecting x [SOC Up]T as the state vector choosingIk and Uk as the input current and output terminal voltagerespectively the state equation and measurement equationof ECM in a discrete-time representation are as follows

xk Akxkminus 1 + BkIkminus 1 + wkminus 1

yk Ckxk + DkIkminus 1 + vk1113896 (30)

where Akminus 1 (zfzx)|xxk|kminus 1

1 00 e

minus (Δtτ1)1113888 1113889 Bkminus 1 minus1113858

((ηΔt)C)Rp(1 minus eminus (Δtτ)

)] Ck (zhzx)|xxk|kminus 1 [(dUOC

(SOCk|kminus 1))(dS OCk|kminus 1) minus 1] and Dk minus R0 we assumethat η 1 under arbitrary charge-discharge condition

+e detailed process of the AEKF-PID method is asfollows

Step 1 Initialization

(a) SOC and Up initialized state

xkminus 1|kminus 1 SOCkminus 1|kminus 1upkminus 1|kminus 11113872 1113873 (31)

(b) State error initialized covariance

Pkminus 1|kminus 1 E xkminus 1|kminus 1 minus 1113954xk|kminus 11113872 1113873 xkminus 1|kminus 1 minus 1113954xkminus 1|kminus 11113872 1113873T

1113876 1113877

(32)

(c) Noise initialized covariance Q0 R0

Step 2 State prediction

(a) State prior estimation

1113954xk|kminus 1 Ak|kminus 11113954xkminus 1|kminus 1 + Bkminus 1Ikminus 1 (33)

Complexity 5

(b) State error covariance prior estimation

Pk|kminus 1 Ak|kminus 1Pkminus 1|kminus 1ATk|kminus 1 + Qkminus 1 (34)

Step 3 Measurement correction

(a) Gain matrix update

Kk Pk|kminus 1CTk CkPk|kminus 1C

Tk + Rk1113872 1113873

minus 1 (35)

(b) State measurement update by PID feedback unit

1113954xk|k 1113954xk|kminus 1 + kPk ek minus ekminus 1( 1113857

+ kIkek + kDk ek minus 2ekminus 1 + ekminus 2( 1113857(36)

(c) State error covariance update

Pk|k I minus KkCk( 1113857Pk|kminus 1 (37)

(d) Adaptive noise covariance matching

Fk 1L

1113944

k

nkminus L+1eke

Tk

Rk Fk minus CkPk|kminus 1CTk

Qk KkFkKTk

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(38)

In summary according to the state equation andmeasurement equation of ECM in equations(29)ndash(30) and the detailed process in equations(31)ndash(38) the flowchart of the proposed SOC es-timation by AEKF-PID algorithm is shown inFigure 4

Actual battery

Single feedback

Measured voltage

Calculatedvoltage

Current

Current

+ndash

Battery model

(a)

Actual battery

PID feedback

Measured voltage

Calculatedvoltage

Current

Current

+ndash

Battery model

(b)

Figure 2 Block diagram of KF and improved KF (a) KF with single feedback (b) Improved KF with PID feedback

Bk

Bk

+

Ak

Ck+

+

+ +

+ Ck

Dk

Ak

+

xkndash1|kndash1

xk|k xk|kndash1yk

Dk

+

ndash

Zndash2

Kp

Ki

Kd

+

+ +

+

uk

uk

wk vk

xk

xk ndash 1

yk

LIB model

Actual LIB

PID

+

+

+

Residual

Zndash1

Zndash1

Zndash1

Figure 3 +e schematic of PID-based KF

6 Complexity

4 Results and Discussion

41 ExperimentalConfigurations A LIB test platform shownin Figure 5 is established to verify the effect of the ECM andmethod It consists of several major components (1) alithium-ion ferrous phosphate battery is used as the ex-perimental objects of which the nominal capacity is 20Ahand nominal voltage is 24V (2) a power battery test system(Arbin EVTS) with control accuracy which is less thanplusmn01 FSR is used for LIB chargingdischarging undervarious working condition (3) a programmable temperaturechamber is applied to control ambient temperature (4) ahost computer with MITS Pro v70 and a computerworkstation with MATLAB R2012a are used for data ac-quisition and data simulation in the experimentsrespectively

+e LIB is fully charged by constant current-constantvoltage (CC-CV) procedure after standing for two hoursthen three working conditions including 10A constantcurrent dynamic stress test (DST) and federal urban drivingschedule (FUDS) are used to test the proposed method+ree working conditions mentioned in this paper arecarried out at 25 degrees Celsius +e discharge current andterminal voltage of DSTand FUDS are shown in Figure 6 Inaddition the evaluation indicator such as maximum abso-lute error (MAE) and average absolute error (AAE) is ap-plied to the assessment of estimation algorithms

+e OCV-SOC function describing the relationshipbetween the open-circuit voltage and SOC can be fitted byOCV feature test data +e LIB is fully charged by the CC-

CV procedure after standing for two hours the measuredterminal voltage can be used as the OCV of 100 SOC +etest steps are as follows the LIB is discharged to 98 SOCwith 10A (05 C) constant current after standing for twohours the measured terminal voltage can be used as theOCV of 98 SOC Repeating the above steps 50 times eachmeasured terminal voltage can be obtained as the OCV ofevery 2 SOC+e eighth-order polynomial fitting functionto describe the mathematical relationship of OCV-SOC canbe shown in Figure 7

42 Parameters Identification +e ECM parameters iden-tification results are shown in Figures 8 and 9 the identi-fication result of ohmic resistance R0 polarization resistanceRp and polarization capacitance Cp can converge to a steadystate quickly from the unreliable initialization value undertwo cycles

R0Rp andCp always have a small range of fluctuation inFigures 8(a) and 9(a) which is directly related to the fluc-tuations of the coefficients a1 b1 and b2 +eoretically thedynamic changes of R0 Rp and Cp are inconsistent but thefixed forgetting factors of FFRLS have the same weight onthe three parameters which has a certain effect on thefluctuation of identification results

+e terminal voltage is identified in each sampling pe-riod based on R0 Rp and Cp As we can see fromFigures 8(b) and 9(b) the identification results of the ter-minal voltage can track the measured value of the terminalvoltage stably under two cycles +e maximum error of theterminal voltage identification is only 01923V and 01702Vrelative to the terminal voltage range from 23V to 29Vrespectively

43 Comparison of the SOC Estimation

431 Analysis of SOC Estimation under 10A DischargeBased on the identified ECM parameters the SOC estimationunder the 10A discharge cycle is presented in Figure 10 Asobserved the SOC reference with Arbin EVTS is formed witha black line the red line and blue line represent the SOC

Measurement dataand

model parameters

Initial conditionsequations (31) and (32)

State prior estimationequation (33)

Gain matrix updateequation (35)

State error covariance prior estimationequation (34)

PID coefficient update with gainequation (28)

State estimation measurement update by PID unitequation (36)

State error covariance updateequation (37)

Figure 4 +e flowchart of AEKF-PID estimator for LIB SOC

Arbin EVTS Temperature chamber

Computer

Figure 5 Battery test platform

Complexity 7

estimation with single AEKF and AEKF-PID respectivelyFrom the estimation results of two methods shown inFigure 10(a) these two SOC estimations can both track theSOC reference in a short period of time no matter AEKF orAEKF-PID Although the two SOC estimation results bothconverge to the SOC reference the SOC estimation error byAEKF-PID is less than that by AEKF as shown inFigure 10(b) +e errors of the two methods slowly increasewith an increase in time +e AAE and MAE of AEKF-PIDare 02266 and 05119 respectively and the AAE andMAE of AEKF are 08606 and 12478 respectively Itfollows that the improved AEKF-PID method shows betterresults in SOC estimation than single AEKF

432 Analysis of SOC Estimation under Two Cycles Inorder to further validate the accuracy of the proposedmethod the DST and FUDS cycles are implemented to

OCV

(V)

02 04 06 08 10SOC

22

23

24

25

26

27

28

29

Figure 7 OCV-SOC fitting curve

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(a)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

23

24

25

26

27

28

29

times104

(b)

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(c)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

22

23

24

25

26

27

28

29

times104

(d)

Figure 6 (a) Current under DST cycle (b) Terminal voltage under DST cycle (c) Current under FUDS cycle (d) Terminal voltage underFUDS cycle

8 Complexity

simulate the EV operationmode+e SOC estimation resultswith two methods under the DST and FUDS cycles areshown in Figures 11 and 12 respectively As observed theSOC reference by Arbin EVTS is formed with a black linethe red line and blue line represent the SOC estimation withsingle AEKF and AEKF-PID respectively From the esti-mation results of two methods shown in Figures 11(a) and12(a) two SOC estimations can both track the SOC refer-ence in a short period of time no matter AEKF or AEKF-PID +e SOC estimation errors with two algorithms underthe DST and FUDS cycles are shown in Figures 11(b) and12(b) respectivelyWe can see that the SOC estimation errorby the AEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 2 under the FUDS cycle meanwhilethe maximum SOC error by AEKF-PID is only 17141+eMAE and AAE of two methods under two cycles are listed inTable 1 According to the above analysis the combination ofthe PID feedback unit with AEKF has advantage over singleAEKF for SOC estimation

44 Analysis on Robustness of SOC Estimation +e highprecise SOC estimation results can be obtained by theproposed method on the assumption that the sampled ex-perimental data by Arbin EVTS is reliable However theexperiment data from the laboratory is not completely equalto actual data in EV operating mode Since the measurementdata by universal transducer is hard to avoid noise inter-ference from such as drift current and diffusion current theSOC estimation method with certain anti-interferencequality is important for BMS To further verify the anti-interference performance of the AEKF-PID a sequence ofnoise with feature of random normal distribution is added tothe operation current under DST and FUDS cycles re-spectively +e mean value of the noise is set as 0 and itsstandard deviation is set as 2 +e SOC estimation resultswith Gaussian noise of current under two conditions aredisplayed in Figures 13 and 14 +e two colored lines withblue and red are used to represent the SOC estimation byAEKF-PID and AEKF respectively +e MAE and AAE ofSOC estimation with Gaussian noise under two cycles are

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

010002000

0005

01

004005006

times104

times104

times104

(a)

05 1 15 2 25 3 350Time (s)

Term

inal

vol

tage

erro

r (V

)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 8 Model parameters identification results under DST (a) R0 Rp Cp (b) terminal voltage error

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

002004006

0005

01

0500

1000

times104

times104

times104

(a)

Term

inal

vol

tage

erro

r (V

)

05 1 15 2 25 3 350Time (s)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 9 Model parameters identification results under FUDS (a) R0 Rp Cp (b) terminal voltage error

Complexity 9

listed in Table 2 As we can see from Figures 13 and 14 dueto the introduction of current noise in two cycles thefluctuation of two corresponding SOC estimation errorsincreased We can see that the SOC estimation error by theAEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 28 under the FUDS cyclemeanwhile the maximum SOC error by AEKF-PID isclose to 2+e MAE and AAE of two methods under twocycles are listed in Table 2 According to the analysisabove the combination of PID feedback unit with AEKFhas advantage over single AEKF in robustness of SOCestimation

To further verify the anti-interference performance ofthe AEKF-PID a sequence of noise with feature of ran-dom non-Gaussian distribution is added to the operationcurrent +e SOC estimation result with non-Gaussiannoise is displayed in Figure 15 We can see that the twoSOC estimation errors with non-Gaussian noise increasecompared to that with Gaussian noise +e results inTable 3 show that the MAE of AEKF is greater than 55and the MAE of AEKF-PID is close to 4 At the end ofdischarge the SOC error of AEKF increases graduallymeanwhile the SOC error of AEKF-PID reaches 3 +eresults show that the basic AEKF-based SOC estimationmethod has better performance in Gaussian noise than

SOC

ReferenceAEKFAEKFndashPID

2500 3000 3500 4000082084086088

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

055

06

065

07

075

08

085

09

095

1

(a)

AEKFAEKFndashPID

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

SOC

erro

r (

)

ndash5ndash4ndash3ndash2ndash1

012345

(b)

Figure 10 Comparison of SOC estimation under 10A discharge (a) SOC estimation result (b) SOC estimation error

ReferenceAEKFAEKFndashPID

SOC

8500 9000 9500

076077078

1 15 2 25 3 3505Time (s)

0010203040506070809

1

times104

(a)

AEKFAEKFndashPID

SOC

erro

r (

)

50 100 150 200 250142144146148

1 15 2 25 3 3505Time (s)

ndash5ndash4ndash3ndash2ndash1

012345

times104

(b)

Figure 11 Comparison of SOC estimation under DST (a) SOC estimation result (b) SOC estimation error

10 Complexity

SOC

6000 6500 7000

081

082

083

1 15 2 25 3 3505Time (s)

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 3 3505Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 13 Comparison of SOC estimation with Gaussian noise under DST (a) SOC estimation result (b) SOC estimation error

SOC

7500 8000 8500

077078079

08

1 15 2 25 305Time (s)

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)SO

C er

ror (

)

50 100 150 200 250 300076078

08082084

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

1 15 2 25 305Time (s)

AEKFAEKFndashPID

times104

(b)

Figure 12 Comparison of SOC estimation under FUDS (a) SOC estimation result (b) SOC estimation error

Table 1 SOC estimation results

AEKF-PID AEKFMAE () 12422 (DST) 17257 (DST)MAE () 17141 (FUDS) 20093 (FUDS)AAE () 03327 (DST) 07466 (DST)AAE () 05235 (FUDS) 07345 (FUDS)

Complexity 11

Table 2 SOC estimation with Gaussian noise

AEKF-PID AEKFMAE () 17726 (DST) 19795 (DST)MAE () 19924 (FUDS) 28171 (FUDS)AAE () 05101 (DST) 10416 (DST)AAE () 05261 (FUDS) 11209 (FUDS)

1 15 2 25 305Time (s)

SOC

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

SOC

erro

r (

)

1 15 2 25 305Time (s)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 15 Comparison of SOC estimation with non-Gaussian noise (a) SOC estimation result (b) SOC estimation error

7500 8000 8500076077078079

1 15 2 25 305Time (s)

SOC

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 305Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 14 Comparison of SOC estimation with Gaussian noise under FUDS (a) SOC estimation result (b) SOC estimation error

12 Complexity

that in non-Gaussian noise whether or not the PIDfeedback is used

5 Conclusion

Based on the simple one-order ECM a novel method ofcombination of AEKF and PID feedback strategy is pro-posed for LIB SOC estimation +e parameters of the ECMare updated online using the FFRLS method to ensure LIBECM reliability and parameters integrity An improvedAEKF-PID technique is introduced to alleviate the negativeeffects resulting from the ECM error and the measurementerror via replacing the single innovation in feedback loop ofAEKF with an innovation vector-based PID module Toevaluate the performance of the proposed algorithm ex-periment and simulation results under 10A discharge DSTand FUDS cycles are adopted to validate the SOC estimation+e simulation results show that the maximum estimationerror of AEKF-PID is below 06 13 and 18 under 10Adischarge DST and FUDS cycles respectively +e SOCestimation error of AEKF-PID is still below 2 even in aGaussian noise situation +ese results suggest that theproposed AEKF-PID method can provide reliable SOCestimation with high precision and robustness Due to thefact that the local flat region in OCV-SOC fitted curve issusceptible to interference from measurement error ourfuture studies will concentrate on the optimization of OCVestimation

Data Availability

Data are available upon request to the corresponding author

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Zheng Liu proposed the original idea Zheng Liu designedthe novel algorithm Yuan Qiu Chunshan Yang and JianboJi performed and analyzed the experiments together ZhengLiu wrote the original manuscript Zheng Liu and ZhenhuaZhao revised the final manuscript

Acknowledgments

+is work was financially supported by the Guangxi NaturalScience Foundation (2018GXNSFAA281161 and2020GXNSFAA297032)

References

[1] X Hu H Yuan C Zou Z Li and L Zhang ldquoCo-estimationof state of charge and state of health for lithium-ion batteriesbased on fractional-order calculusrdquo IEEE Transactions onVehicular Technology vol 67 no 11 pp 10319ndash10329 2018

[2] X Zhou and C Feng ldquo+e impact of environmental regu-lation on fossil energy consumption in China direct andindirect effectsrdquo Journal of Cleaner Production vol 142pp 3174ndash3183 2017

[3] WWaag C Fleischer and D U Sauer ldquoCritical review of themethods for monitoring of lithium-ion batteries in electricand hybrid vehiclesrdquo Journal of Power Sources vol 258pp 321ndash339 2014

[4] R Xiong Y Zhang J Wang H He S Peng and M PechtldquoLithium-ion battery health prognosis based on a real batterymanagement system used in electric vehiclesrdquo IEEE Trans-actions on Vehicular Technology vol 68 no No 5pp 4110ndash4121 2019

[5] H Fathabadi ldquoPlug-in hybrid electric vehicles (PHEVs)replacing internal combustion engine with clean and re-newable energy based auxiliary power sourcesrdquo IEEETransactions on Power Electronics vol 33 no No 11pp 9611ndash9618 2018

[6] Y Xing W He M Pecht and K L Tsui ldquoState of chargeestimation of lithium-ion batteries using the open-circuitvoltage at various ambient temperaturesrdquo Applied Energyvol 113 pp 106ndash115 2014

[7] N Yang X Zhang and G Li ldquoState of charge estimation forpulse discharge of a LiFePO4 battery by a revised Ahcountingrdquo Electrochimica Acta vol 151 pp 63ndash71 2015

[8] K S Ng C-S Moo Y-P Chen and Y-C Hsieh ldquoEnhancedCoulomb counting method for estimating state-of-charge andstate-of-health of lithium-ion batteriesrdquo Applied Energyvol 86 no 9 pp 1506ndash1511 2009

[9] Z Liu X Dang and B Jing ldquoA novel open circuit voltagebased state of charge estimation for lithium-ion battery bymulti-innovation Kalman filterrdquo IEEE Access vol 7pp 49432ndash49447 2019

[10] J D J Rubio ldquoStability analysis of the modified Levenberg-Marquardt algorithm for the artificial neural network train-ingrdquo IEEE Transactions on Neural Networks and LearningSystems 2020

[11] G Aquino J D J Rubio J Pacheco et al ldquoNovel nonlinearhypothesis for the delta parallel robot modelingrdquo IEEE Accessvol 8 no 1 pp 46324ndash46334 2020

[12] J D J Rubio ldquoSOFMLS online self-organizing fuzzy mod-ified least-squares networkrdquo IEEE Transactions on FuzzySystems vol 17 no 6 pp 1296ndash1309 2009

[13] J A M Campantildea ldquoOn the estimation and control of non-linear systems with parametric uncertainties and noisy out-putsrdquo IEEE Access vol 6 pp 31968ndash31973 2018

[14] S Pramanik and S Anwar ldquoElectrochemical model basedcharge optimization for lithium-ion batteriesrdquo Journal ofPower Sources vol 313 pp 164ndash177 2016

[15] F Sun X Hu Y Zou and S Li ldquoAdaptive unscented Kalmanfiltering for state of charge estimation of a lithium-ion batteryfor electric vehiclesrdquo Energy vol 36 no 5 pp 3531ndash35402014

[16] Z Liu and X Dang ldquoA new method for state of charge andcapacity estimation of lithium-ion battery based on dualstrong tracking adaptive H infinity filterrdquo Mathematical

Table 3 SOC estimation with non-Gaussian noise

AEKF-PID AEKFMAE () 40590 55443AAE () 26074 32427

Complexity 13

Problems in Engineering vol 2018 Article ID 521820518 pages 2018

[17] Y Wang C Zhang and Z Chen ldquoOn-line battery state-of-charge estimation based on an integrated estimatorrdquo AppliedEnergy vol 185 pp 2026ndash2032 2017

[18] R Xiao J Shen X Li W Yan E Pan and Z ChenldquoComparisons of modeling and state of charge estimation forlithium-ion battery based on fractional order and integralorder methodsrdquo Energies vol 9 no No 3 2016

[19] Y Qiu X Li W Chen Z-m Duan and L Yu ldquoState ofcharge estimation of vanadium redox battery based on im-proved extended Kalman filterrdquo ISA Transactions vol 94pp 326ndash337 2019

[20] Y Xu M Hu A Zhou et al ldquoState of charge estimation forlithium-ion batteries based on adaptive dual Kalman filterrdquoAppliedMathematical Modelling vol 77 pp 1255ndash1272 2020

[21] Y Bi and S-Y Choe ldquoAn adaptive sigma-point Kalman filterwith state equality constraints for online state-of-charge es-timation of a Li(NiMnCo)O2Carbon battery using a re-duced-order electrochemical modelrdquo Applied Energyvol 258 p 113925 2020

[22] Z Liu X Dang B Jing and J Ji ldquoA novel model-based stateof charge estimation for lithium-ion battery using adaptiverobust iterative cubature Kalman filterrdquo Electric Power Sys-tems Research vol 177 p 105951 2019

[23] J Wei G Dong and Z Chen ldquoOn-board adaptive model forstate of charge estimation of lithium-ion batteries based onKalman filter with proportional integral-based error adjust-mentrdquo Journal of Power Sources vol 365 pp 308ndash319 2017

[24] F Yang Y Xing D Wang and K-L Tsui ldquoA comparativestudy of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dy-namic loading profilerdquo Applied Energy vol 164 pp 387ndash3992016

[25] Y Li C Wang and J Gong ldquoA multi-model probability SOCfusion estimation approach using an improved adaptiveunscented Kalman filter techniquerdquo Energy vol 141pp 1402ndash1415 2017

[26] J Linghu L Kang M Liu X Luo Y Feng and C LuldquoEstimation for state-of-charge of lithium-ion battery basedon an adaptive high-degree cubature Kalman filterrdquo Energyvol 189 Article ID 116204 2019

[27] Q Zhu M Xu W Liu and M Zheng ldquoA state of chargeestimation method for lithium-ion batteries based on frac-tional order adaptive extended Kalman filterrdquo Energyvol 187 Article ID 115880 2019

[28] Z Liu and H He ldquoSensor fault detection and isolation for alithium-ion battery pack in electric vehicles using adaptiveextended Kalman filterrdquo Applied Energy vol 185 pp 2033ndash2044 2017

[29] X Lai Y Zheng and T Sun ldquoA comparative study of dif-ferent equivalent circuit models for estimating state-of-chargeof lithium-ion batteriesrdquo Electrochimica Acta vol 259pp 566ndash577 2018

[30] Y Li J Chen and F Lan ldquoEnhanced online model identi-fication and state of charge estimation for lithium-ion batteryunder noise corrupted measurements by bias compensationrecursive least squaresrdquo Journal of Power Sources vol 456p 227984 2020

[31] V Duang H Bastawrous K ldquoLim K See P Zhang andS Dou ldquoOnline state of charge and model parameters esti-mation of the LiFePO4 battery in electric vehicles usingmultiple adaptive forgetting factors recursive least-squaresrdquoJournal of Power Sources vol 296 pp 215ndash224 2015

[32] Y Tian R Lai X Li L Xiang and J Tian ldquoA combinedmethod for state-of-charge estimation for lithium-ion bat-teries using a long short-term memory network and anadaptive cubature Kalman filterrdquo Applied Energy vol 265p 114789 2020

[33] C Chen R Xiong R Yang W Shen and F Sun ldquoState-of-charge estimation of lithium-ion battery using an improvedneural network model and extended Kalman filterrdquo Journal ofCleaner Production vol 234 pp 1153ndash1164 2019

14 Complexity

with KF are presented Since the loss of correction data in KFmay be due to the fact that only single proportional inno-vation feedback is used in KF the innovation vector-basedPID feedback controller is introduced into measurementcorrection step of KF in this paper +is paper puts forwardan improved AEKF based on PID controller with self-tuningPID parameters kP kI and kD Due to the fact that moreterminal voltage innovation can be described by the PID unitin measurement correction step of improved AEKF thebattery state can be predicted more accurately by the im-proved AEKF

+e PID feedback unit deduced from innovation vectoris first introduced +e single feedback unit in KF issubstituted by the PID unit which referred to the PIDfeedback strategy in the KF structure as shown in Figure 2

+e state measurement update 1113954xk|k from common KFcan be expressed by letting ek to be single innovation inequation (16)

In order to get the equivalent form of the incrementalPID in equation (25) the measurement correction by singleinnovation ek in equation (16) can be transformed as

1113954xk|k 1113954xk|kminus 1 + KPk ek minus ekminus 1( 1113857 + KIkek

+ KDk ek minus 2ekminus 1 + ekminus 2( 1113857(26)

+e structure diagram of the improved method bycombined KF and PID feedback unit is demonstrated inFigure 3 +e core idea is to expand the single innovation toan innovation vector +at is ekis extended into

ek ekminus 1 ekminus 21113858 1113859T To ensure the consistency of data di-

mension the following improvements are made gain kk isextended into kk kkminus 1 kkminus 21113858 1113859 and measurement output zk

is extended into zk zkminus 1 zkminus 21113858 1113859T

Meanwhile a posteriori estimation 1113954xk|k in the mea-surement correction step needs to be rewritten with mod-ified parameters including kk kkminus 1 kkminus 21113858 1113859

zk zkminus 1 zkminus 21113858 1113859T and ek ekminus 1 ekminus 21113858 1113859

T

1113954xk|k 1113954xk|kminus 1 + minus 2kkminus 2 minus kkminus 1( 1113857 ek minus ekminus 1( 1113857

+ kk + kkminus 1 + kkminus 2( 1113857ek + kkminus 2 ek minus 2ekminus 1 + ekminus 2( 1113857

(27)

Consequently the regulation parameters kP kI and kD

of PID unit in the measurement correction step can besolved by equations (26) and (27)

kPk minus 2kkminus 2 minus kkminus 1

kIk kk + kkminus 1 + kkminus 2

kDk kkminus 2

(28)

From equations (26) and (28) we can get the PIDfeedback unit by setting innovation vector and the PIDcoefficients kP kI and kD can be solved by the expressions ofgains kk kkminus 1 and kkminus 2 in three sampling points +ree in-novation data at times k k-1 and k-2 are used whenupdating the state at time k +e main reason for increasingthe robustness of the improved algorithm is to reuse in-novative data to update the state at adjacent times In orderto improve the performance of SOC estimation the

combination of PID feedback and AEKF is introduced inSection 34 In addition the covariance of noise wk and vk isupdated based on the principle of covariance matchingrespectively

+e PID-based KF introduces more than one prior in-novation which greatly enhances the effect of feedbackcompensation in KF Experiments show that the innovationvector-based KF has better robustness than single-innova-tion-based KF in strong nonlinear systems [9] Although thecalculation load of PID-based KF is more than that of thesingle-innovation-based KF the increased amount of cal-culation is acceptable relative to the improvement inaccuracy

34 SOC Estimation Based on AEKF-PID +e discreteequation of the LIB model in equation (1) can be rewritten asfollows

SOCk

Upk

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1 0

0 eminus Δtτ1( )

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

SOCkminus 1

Upkminus 1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ +

minusηΔtC

1113874 1113875

Rp 1 minus eminus (Δtτ)

1113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Ikminus 1 + wkminus 1

Uk UOC SOCk( 1113857 minus Upk minus IkR0 + vk

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

By selecting x [SOC Up]T as the state vector choosingIk and Uk as the input current and output terminal voltagerespectively the state equation and measurement equationof ECM in a discrete-time representation are as follows

xk Akxkminus 1 + BkIkminus 1 + wkminus 1

yk Ckxk + DkIkminus 1 + vk1113896 (30)

where Akminus 1 (zfzx)|xxk|kminus 1

1 00 e

minus (Δtτ1)1113888 1113889 Bkminus 1 minus1113858

((ηΔt)C)Rp(1 minus eminus (Δtτ)

)] Ck (zhzx)|xxk|kminus 1 [(dUOC

(SOCk|kminus 1))(dS OCk|kminus 1) minus 1] and Dk minus R0 we assumethat η 1 under arbitrary charge-discharge condition

+e detailed process of the AEKF-PID method is asfollows

Step 1 Initialization

(a) SOC and Up initialized state

xkminus 1|kminus 1 SOCkminus 1|kminus 1upkminus 1|kminus 11113872 1113873 (31)

(b) State error initialized covariance

Pkminus 1|kminus 1 E xkminus 1|kminus 1 minus 1113954xk|kminus 11113872 1113873 xkminus 1|kminus 1 minus 1113954xkminus 1|kminus 11113872 1113873T

1113876 1113877

(32)

(c) Noise initialized covariance Q0 R0

Step 2 State prediction

(a) State prior estimation

1113954xk|kminus 1 Ak|kminus 11113954xkminus 1|kminus 1 + Bkminus 1Ikminus 1 (33)

Complexity 5

(b) State error covariance prior estimation

Pk|kminus 1 Ak|kminus 1Pkminus 1|kminus 1ATk|kminus 1 + Qkminus 1 (34)

Step 3 Measurement correction

(a) Gain matrix update

Kk Pk|kminus 1CTk CkPk|kminus 1C

Tk + Rk1113872 1113873

minus 1 (35)

(b) State measurement update by PID feedback unit

1113954xk|k 1113954xk|kminus 1 + kPk ek minus ekminus 1( 1113857

+ kIkek + kDk ek minus 2ekminus 1 + ekminus 2( 1113857(36)

(c) State error covariance update

Pk|k I minus KkCk( 1113857Pk|kminus 1 (37)

(d) Adaptive noise covariance matching

Fk 1L

1113944

k

nkminus L+1eke

Tk

Rk Fk minus CkPk|kminus 1CTk

Qk KkFkKTk

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(38)

In summary according to the state equation andmeasurement equation of ECM in equations(29)ndash(30) and the detailed process in equations(31)ndash(38) the flowchart of the proposed SOC es-timation by AEKF-PID algorithm is shown inFigure 4

Actual battery

Single feedback

Measured voltage

Calculatedvoltage

Current

Current

+ndash

Battery model

(a)

Actual battery

PID feedback

Measured voltage

Calculatedvoltage

Current

Current

+ndash

Battery model

(b)

Figure 2 Block diagram of KF and improved KF (a) KF with single feedback (b) Improved KF with PID feedback

Bk

Bk

+

Ak

Ck+

+

+ +

+ Ck

Dk

Ak

+

xkndash1|kndash1

xk|k xk|kndash1yk

Dk

+

ndash

Zndash2

Kp

Ki

Kd

+

+ +

+

uk

uk

wk vk

xk

xk ndash 1

yk

LIB model

Actual LIB

PID

+

+

+

Residual

Zndash1

Zndash1

Zndash1

Figure 3 +e schematic of PID-based KF

6 Complexity

4 Results and Discussion

41 ExperimentalConfigurations A LIB test platform shownin Figure 5 is established to verify the effect of the ECM andmethod It consists of several major components (1) alithium-ion ferrous phosphate battery is used as the ex-perimental objects of which the nominal capacity is 20Ahand nominal voltage is 24V (2) a power battery test system(Arbin EVTS) with control accuracy which is less thanplusmn01 FSR is used for LIB chargingdischarging undervarious working condition (3) a programmable temperaturechamber is applied to control ambient temperature (4) ahost computer with MITS Pro v70 and a computerworkstation with MATLAB R2012a are used for data ac-quisition and data simulation in the experimentsrespectively

+e LIB is fully charged by constant current-constantvoltage (CC-CV) procedure after standing for two hoursthen three working conditions including 10A constantcurrent dynamic stress test (DST) and federal urban drivingschedule (FUDS) are used to test the proposed method+ree working conditions mentioned in this paper arecarried out at 25 degrees Celsius +e discharge current andterminal voltage of DSTand FUDS are shown in Figure 6 Inaddition the evaluation indicator such as maximum abso-lute error (MAE) and average absolute error (AAE) is ap-plied to the assessment of estimation algorithms

+e OCV-SOC function describing the relationshipbetween the open-circuit voltage and SOC can be fitted byOCV feature test data +e LIB is fully charged by the CC-

CV procedure after standing for two hours the measuredterminal voltage can be used as the OCV of 100 SOC +etest steps are as follows the LIB is discharged to 98 SOCwith 10A (05 C) constant current after standing for twohours the measured terminal voltage can be used as theOCV of 98 SOC Repeating the above steps 50 times eachmeasured terminal voltage can be obtained as the OCV ofevery 2 SOC+e eighth-order polynomial fitting functionto describe the mathematical relationship of OCV-SOC canbe shown in Figure 7

42 Parameters Identification +e ECM parameters iden-tification results are shown in Figures 8 and 9 the identi-fication result of ohmic resistance R0 polarization resistanceRp and polarization capacitance Cp can converge to a steadystate quickly from the unreliable initialization value undertwo cycles

R0Rp andCp always have a small range of fluctuation inFigures 8(a) and 9(a) which is directly related to the fluc-tuations of the coefficients a1 b1 and b2 +eoretically thedynamic changes of R0 Rp and Cp are inconsistent but thefixed forgetting factors of FFRLS have the same weight onthe three parameters which has a certain effect on thefluctuation of identification results

+e terminal voltage is identified in each sampling pe-riod based on R0 Rp and Cp As we can see fromFigures 8(b) and 9(b) the identification results of the ter-minal voltage can track the measured value of the terminalvoltage stably under two cycles +e maximum error of theterminal voltage identification is only 01923V and 01702Vrelative to the terminal voltage range from 23V to 29Vrespectively

43 Comparison of the SOC Estimation

431 Analysis of SOC Estimation under 10A DischargeBased on the identified ECM parameters the SOC estimationunder the 10A discharge cycle is presented in Figure 10 Asobserved the SOC reference with Arbin EVTS is formed witha black line the red line and blue line represent the SOC

Measurement dataand

model parameters

Initial conditionsequations (31) and (32)

State prior estimationequation (33)

Gain matrix updateequation (35)

State error covariance prior estimationequation (34)

PID coefficient update with gainequation (28)

State estimation measurement update by PID unitequation (36)

State error covariance updateequation (37)

Figure 4 +e flowchart of AEKF-PID estimator for LIB SOC

Arbin EVTS Temperature chamber

Computer

Figure 5 Battery test platform

Complexity 7

estimation with single AEKF and AEKF-PID respectivelyFrom the estimation results of two methods shown inFigure 10(a) these two SOC estimations can both track theSOC reference in a short period of time no matter AEKF orAEKF-PID Although the two SOC estimation results bothconverge to the SOC reference the SOC estimation error byAEKF-PID is less than that by AEKF as shown inFigure 10(b) +e errors of the two methods slowly increasewith an increase in time +e AAE and MAE of AEKF-PIDare 02266 and 05119 respectively and the AAE andMAE of AEKF are 08606 and 12478 respectively Itfollows that the improved AEKF-PID method shows betterresults in SOC estimation than single AEKF

432 Analysis of SOC Estimation under Two Cycles Inorder to further validate the accuracy of the proposedmethod the DST and FUDS cycles are implemented to

OCV

(V)

02 04 06 08 10SOC

22

23

24

25

26

27

28

29

Figure 7 OCV-SOC fitting curve

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(a)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

23

24

25

26

27

28

29

times104

(b)

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(c)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

22

23

24

25

26

27

28

29

times104

(d)

Figure 6 (a) Current under DST cycle (b) Terminal voltage under DST cycle (c) Current under FUDS cycle (d) Terminal voltage underFUDS cycle

8 Complexity

simulate the EV operationmode+e SOC estimation resultswith two methods under the DST and FUDS cycles areshown in Figures 11 and 12 respectively As observed theSOC reference by Arbin EVTS is formed with a black linethe red line and blue line represent the SOC estimation withsingle AEKF and AEKF-PID respectively From the esti-mation results of two methods shown in Figures 11(a) and12(a) two SOC estimations can both track the SOC refer-ence in a short period of time no matter AEKF or AEKF-PID +e SOC estimation errors with two algorithms underthe DST and FUDS cycles are shown in Figures 11(b) and12(b) respectivelyWe can see that the SOC estimation errorby the AEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 2 under the FUDS cycle meanwhilethe maximum SOC error by AEKF-PID is only 17141+eMAE and AAE of two methods under two cycles are listed inTable 1 According to the above analysis the combination ofthe PID feedback unit with AEKF has advantage over singleAEKF for SOC estimation

44 Analysis on Robustness of SOC Estimation +e highprecise SOC estimation results can be obtained by theproposed method on the assumption that the sampled ex-perimental data by Arbin EVTS is reliable However theexperiment data from the laboratory is not completely equalto actual data in EV operating mode Since the measurementdata by universal transducer is hard to avoid noise inter-ference from such as drift current and diffusion current theSOC estimation method with certain anti-interferencequality is important for BMS To further verify the anti-interference performance of the AEKF-PID a sequence ofnoise with feature of random normal distribution is added tothe operation current under DST and FUDS cycles re-spectively +e mean value of the noise is set as 0 and itsstandard deviation is set as 2 +e SOC estimation resultswith Gaussian noise of current under two conditions aredisplayed in Figures 13 and 14 +e two colored lines withblue and red are used to represent the SOC estimation byAEKF-PID and AEKF respectively +e MAE and AAE ofSOC estimation with Gaussian noise under two cycles are

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

010002000

0005

01

004005006

times104

times104

times104

(a)

05 1 15 2 25 3 350Time (s)

Term

inal

vol

tage

erro

r (V

)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 8 Model parameters identification results under DST (a) R0 Rp Cp (b) terminal voltage error

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

002004006

0005

01

0500

1000

times104

times104

times104

(a)

Term

inal

vol

tage

erro

r (V

)

05 1 15 2 25 3 350Time (s)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 9 Model parameters identification results under FUDS (a) R0 Rp Cp (b) terminal voltage error

Complexity 9

listed in Table 2 As we can see from Figures 13 and 14 dueto the introduction of current noise in two cycles thefluctuation of two corresponding SOC estimation errorsincreased We can see that the SOC estimation error by theAEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 28 under the FUDS cyclemeanwhile the maximum SOC error by AEKF-PID isclose to 2+e MAE and AAE of two methods under twocycles are listed in Table 2 According to the analysisabove the combination of PID feedback unit with AEKFhas advantage over single AEKF in robustness of SOCestimation

To further verify the anti-interference performance ofthe AEKF-PID a sequence of noise with feature of ran-dom non-Gaussian distribution is added to the operationcurrent +e SOC estimation result with non-Gaussiannoise is displayed in Figure 15 We can see that the twoSOC estimation errors with non-Gaussian noise increasecompared to that with Gaussian noise +e results inTable 3 show that the MAE of AEKF is greater than 55and the MAE of AEKF-PID is close to 4 At the end ofdischarge the SOC error of AEKF increases graduallymeanwhile the SOC error of AEKF-PID reaches 3 +eresults show that the basic AEKF-based SOC estimationmethod has better performance in Gaussian noise than

SOC

ReferenceAEKFAEKFndashPID

2500 3000 3500 4000082084086088

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

055

06

065

07

075

08

085

09

095

1

(a)

AEKFAEKFndashPID

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

SOC

erro

r (

)

ndash5ndash4ndash3ndash2ndash1

012345

(b)

Figure 10 Comparison of SOC estimation under 10A discharge (a) SOC estimation result (b) SOC estimation error

ReferenceAEKFAEKFndashPID

SOC

8500 9000 9500

076077078

1 15 2 25 3 3505Time (s)

0010203040506070809

1

times104

(a)

AEKFAEKFndashPID

SOC

erro

r (

)

50 100 150 200 250142144146148

1 15 2 25 3 3505Time (s)

ndash5ndash4ndash3ndash2ndash1

012345

times104

(b)

Figure 11 Comparison of SOC estimation under DST (a) SOC estimation result (b) SOC estimation error

10 Complexity

SOC

6000 6500 7000

081

082

083

1 15 2 25 3 3505Time (s)

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 3 3505Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 13 Comparison of SOC estimation with Gaussian noise under DST (a) SOC estimation result (b) SOC estimation error

SOC

7500 8000 8500

077078079

08

1 15 2 25 305Time (s)

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)SO

C er

ror (

)

50 100 150 200 250 300076078

08082084

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

1 15 2 25 305Time (s)

AEKFAEKFndashPID

times104

(b)

Figure 12 Comparison of SOC estimation under FUDS (a) SOC estimation result (b) SOC estimation error

Table 1 SOC estimation results

AEKF-PID AEKFMAE () 12422 (DST) 17257 (DST)MAE () 17141 (FUDS) 20093 (FUDS)AAE () 03327 (DST) 07466 (DST)AAE () 05235 (FUDS) 07345 (FUDS)

Complexity 11

Table 2 SOC estimation with Gaussian noise

AEKF-PID AEKFMAE () 17726 (DST) 19795 (DST)MAE () 19924 (FUDS) 28171 (FUDS)AAE () 05101 (DST) 10416 (DST)AAE () 05261 (FUDS) 11209 (FUDS)

1 15 2 25 305Time (s)

SOC

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

SOC

erro

r (

)

1 15 2 25 305Time (s)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 15 Comparison of SOC estimation with non-Gaussian noise (a) SOC estimation result (b) SOC estimation error

7500 8000 8500076077078079

1 15 2 25 305Time (s)

SOC

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 305Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 14 Comparison of SOC estimation with Gaussian noise under FUDS (a) SOC estimation result (b) SOC estimation error

12 Complexity

that in non-Gaussian noise whether or not the PIDfeedback is used

5 Conclusion

Based on the simple one-order ECM a novel method ofcombination of AEKF and PID feedback strategy is pro-posed for LIB SOC estimation +e parameters of the ECMare updated online using the FFRLS method to ensure LIBECM reliability and parameters integrity An improvedAEKF-PID technique is introduced to alleviate the negativeeffects resulting from the ECM error and the measurementerror via replacing the single innovation in feedback loop ofAEKF with an innovation vector-based PID module Toevaluate the performance of the proposed algorithm ex-periment and simulation results under 10A discharge DSTand FUDS cycles are adopted to validate the SOC estimation+e simulation results show that the maximum estimationerror of AEKF-PID is below 06 13 and 18 under 10Adischarge DST and FUDS cycles respectively +e SOCestimation error of AEKF-PID is still below 2 even in aGaussian noise situation +ese results suggest that theproposed AEKF-PID method can provide reliable SOCestimation with high precision and robustness Due to thefact that the local flat region in OCV-SOC fitted curve issusceptible to interference from measurement error ourfuture studies will concentrate on the optimization of OCVestimation

Data Availability

Data are available upon request to the corresponding author

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Zheng Liu proposed the original idea Zheng Liu designedthe novel algorithm Yuan Qiu Chunshan Yang and JianboJi performed and analyzed the experiments together ZhengLiu wrote the original manuscript Zheng Liu and ZhenhuaZhao revised the final manuscript

Acknowledgments

+is work was financially supported by the Guangxi NaturalScience Foundation (2018GXNSFAA281161 and2020GXNSFAA297032)

References

[1] X Hu H Yuan C Zou Z Li and L Zhang ldquoCo-estimationof state of charge and state of health for lithium-ion batteriesbased on fractional-order calculusrdquo IEEE Transactions onVehicular Technology vol 67 no 11 pp 10319ndash10329 2018

[2] X Zhou and C Feng ldquo+e impact of environmental regu-lation on fossil energy consumption in China direct andindirect effectsrdquo Journal of Cleaner Production vol 142pp 3174ndash3183 2017

[3] WWaag C Fleischer and D U Sauer ldquoCritical review of themethods for monitoring of lithium-ion batteries in electricand hybrid vehiclesrdquo Journal of Power Sources vol 258pp 321ndash339 2014

[4] R Xiong Y Zhang J Wang H He S Peng and M PechtldquoLithium-ion battery health prognosis based on a real batterymanagement system used in electric vehiclesrdquo IEEE Trans-actions on Vehicular Technology vol 68 no No 5pp 4110ndash4121 2019

[5] H Fathabadi ldquoPlug-in hybrid electric vehicles (PHEVs)replacing internal combustion engine with clean and re-newable energy based auxiliary power sourcesrdquo IEEETransactions on Power Electronics vol 33 no No 11pp 9611ndash9618 2018

[6] Y Xing W He M Pecht and K L Tsui ldquoState of chargeestimation of lithium-ion batteries using the open-circuitvoltage at various ambient temperaturesrdquo Applied Energyvol 113 pp 106ndash115 2014

[7] N Yang X Zhang and G Li ldquoState of charge estimation forpulse discharge of a LiFePO4 battery by a revised Ahcountingrdquo Electrochimica Acta vol 151 pp 63ndash71 2015

[8] K S Ng C-S Moo Y-P Chen and Y-C Hsieh ldquoEnhancedCoulomb counting method for estimating state-of-charge andstate-of-health of lithium-ion batteriesrdquo Applied Energyvol 86 no 9 pp 1506ndash1511 2009

[9] Z Liu X Dang and B Jing ldquoA novel open circuit voltagebased state of charge estimation for lithium-ion battery bymulti-innovation Kalman filterrdquo IEEE Access vol 7pp 49432ndash49447 2019

[10] J D J Rubio ldquoStability analysis of the modified Levenberg-Marquardt algorithm for the artificial neural network train-ingrdquo IEEE Transactions on Neural Networks and LearningSystems 2020

[11] G Aquino J D J Rubio J Pacheco et al ldquoNovel nonlinearhypothesis for the delta parallel robot modelingrdquo IEEE Accessvol 8 no 1 pp 46324ndash46334 2020

[12] J D J Rubio ldquoSOFMLS online self-organizing fuzzy mod-ified least-squares networkrdquo IEEE Transactions on FuzzySystems vol 17 no 6 pp 1296ndash1309 2009

[13] J A M Campantildea ldquoOn the estimation and control of non-linear systems with parametric uncertainties and noisy out-putsrdquo IEEE Access vol 6 pp 31968ndash31973 2018

[14] S Pramanik and S Anwar ldquoElectrochemical model basedcharge optimization for lithium-ion batteriesrdquo Journal ofPower Sources vol 313 pp 164ndash177 2016

[15] F Sun X Hu Y Zou and S Li ldquoAdaptive unscented Kalmanfiltering for state of charge estimation of a lithium-ion batteryfor electric vehiclesrdquo Energy vol 36 no 5 pp 3531ndash35402014

[16] Z Liu and X Dang ldquoA new method for state of charge andcapacity estimation of lithium-ion battery based on dualstrong tracking adaptive H infinity filterrdquo Mathematical

Table 3 SOC estimation with non-Gaussian noise

AEKF-PID AEKFMAE () 40590 55443AAE () 26074 32427

Complexity 13

Problems in Engineering vol 2018 Article ID 521820518 pages 2018

[17] Y Wang C Zhang and Z Chen ldquoOn-line battery state-of-charge estimation based on an integrated estimatorrdquo AppliedEnergy vol 185 pp 2026ndash2032 2017

[18] R Xiao J Shen X Li W Yan E Pan and Z ChenldquoComparisons of modeling and state of charge estimation forlithium-ion battery based on fractional order and integralorder methodsrdquo Energies vol 9 no No 3 2016

[19] Y Qiu X Li W Chen Z-m Duan and L Yu ldquoState ofcharge estimation of vanadium redox battery based on im-proved extended Kalman filterrdquo ISA Transactions vol 94pp 326ndash337 2019

[20] Y Xu M Hu A Zhou et al ldquoState of charge estimation forlithium-ion batteries based on adaptive dual Kalman filterrdquoAppliedMathematical Modelling vol 77 pp 1255ndash1272 2020

[21] Y Bi and S-Y Choe ldquoAn adaptive sigma-point Kalman filterwith state equality constraints for online state-of-charge es-timation of a Li(NiMnCo)O2Carbon battery using a re-duced-order electrochemical modelrdquo Applied Energyvol 258 p 113925 2020

[22] Z Liu X Dang B Jing and J Ji ldquoA novel model-based stateof charge estimation for lithium-ion battery using adaptiverobust iterative cubature Kalman filterrdquo Electric Power Sys-tems Research vol 177 p 105951 2019

[23] J Wei G Dong and Z Chen ldquoOn-board adaptive model forstate of charge estimation of lithium-ion batteries based onKalman filter with proportional integral-based error adjust-mentrdquo Journal of Power Sources vol 365 pp 308ndash319 2017

[24] F Yang Y Xing D Wang and K-L Tsui ldquoA comparativestudy of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dy-namic loading profilerdquo Applied Energy vol 164 pp 387ndash3992016

[25] Y Li C Wang and J Gong ldquoA multi-model probability SOCfusion estimation approach using an improved adaptiveunscented Kalman filter techniquerdquo Energy vol 141pp 1402ndash1415 2017

[26] J Linghu L Kang M Liu X Luo Y Feng and C LuldquoEstimation for state-of-charge of lithium-ion battery basedon an adaptive high-degree cubature Kalman filterrdquo Energyvol 189 Article ID 116204 2019

[27] Q Zhu M Xu W Liu and M Zheng ldquoA state of chargeestimation method for lithium-ion batteries based on frac-tional order adaptive extended Kalman filterrdquo Energyvol 187 Article ID 115880 2019

[28] Z Liu and H He ldquoSensor fault detection and isolation for alithium-ion battery pack in electric vehicles using adaptiveextended Kalman filterrdquo Applied Energy vol 185 pp 2033ndash2044 2017

[29] X Lai Y Zheng and T Sun ldquoA comparative study of dif-ferent equivalent circuit models for estimating state-of-chargeof lithium-ion batteriesrdquo Electrochimica Acta vol 259pp 566ndash577 2018

[30] Y Li J Chen and F Lan ldquoEnhanced online model identi-fication and state of charge estimation for lithium-ion batteryunder noise corrupted measurements by bias compensationrecursive least squaresrdquo Journal of Power Sources vol 456p 227984 2020

[31] V Duang H Bastawrous K ldquoLim K See P Zhang andS Dou ldquoOnline state of charge and model parameters esti-mation of the LiFePO4 battery in electric vehicles usingmultiple adaptive forgetting factors recursive least-squaresrdquoJournal of Power Sources vol 296 pp 215ndash224 2015

[32] Y Tian R Lai X Li L Xiang and J Tian ldquoA combinedmethod for state-of-charge estimation for lithium-ion bat-teries using a long short-term memory network and anadaptive cubature Kalman filterrdquo Applied Energy vol 265p 114789 2020

[33] C Chen R Xiong R Yang W Shen and F Sun ldquoState-of-charge estimation of lithium-ion battery using an improvedneural network model and extended Kalman filterrdquo Journal ofCleaner Production vol 234 pp 1153ndash1164 2019

14 Complexity

(b) State error covariance prior estimation

Pk|kminus 1 Ak|kminus 1Pkminus 1|kminus 1ATk|kminus 1 + Qkminus 1 (34)

Step 3 Measurement correction

(a) Gain matrix update

Kk Pk|kminus 1CTk CkPk|kminus 1C

Tk + Rk1113872 1113873

minus 1 (35)

(b) State measurement update by PID feedback unit

1113954xk|k 1113954xk|kminus 1 + kPk ek minus ekminus 1( 1113857

+ kIkek + kDk ek minus 2ekminus 1 + ekminus 2( 1113857(36)

(c) State error covariance update

Pk|k I minus KkCk( 1113857Pk|kminus 1 (37)

(d) Adaptive noise covariance matching

Fk 1L

1113944

k

nkminus L+1eke

Tk

Rk Fk minus CkPk|kminus 1CTk

Qk KkFkKTk

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(38)

In summary according to the state equation andmeasurement equation of ECM in equations(29)ndash(30) and the detailed process in equations(31)ndash(38) the flowchart of the proposed SOC es-timation by AEKF-PID algorithm is shown inFigure 4

Actual battery

Single feedback

Measured voltage

Calculatedvoltage

Current

Current

+ndash

Battery model

(a)

Actual battery

PID feedback

Measured voltage

Calculatedvoltage

Current

Current

+ndash

Battery model

(b)

Figure 2 Block diagram of KF and improved KF (a) KF with single feedback (b) Improved KF with PID feedback

Bk

Bk

+

Ak

Ck+

+

+ +

+ Ck

Dk

Ak

+

xkndash1|kndash1

xk|k xk|kndash1yk

Dk

+

ndash

Zndash2

Kp

Ki

Kd

+

+ +

+

uk

uk

wk vk

xk

xk ndash 1

yk

LIB model

Actual LIB

PID

+

+

+

Residual

Zndash1

Zndash1

Zndash1

Figure 3 +e schematic of PID-based KF

6 Complexity

4 Results and Discussion

41 ExperimentalConfigurations A LIB test platform shownin Figure 5 is established to verify the effect of the ECM andmethod It consists of several major components (1) alithium-ion ferrous phosphate battery is used as the ex-perimental objects of which the nominal capacity is 20Ahand nominal voltage is 24V (2) a power battery test system(Arbin EVTS) with control accuracy which is less thanplusmn01 FSR is used for LIB chargingdischarging undervarious working condition (3) a programmable temperaturechamber is applied to control ambient temperature (4) ahost computer with MITS Pro v70 and a computerworkstation with MATLAB R2012a are used for data ac-quisition and data simulation in the experimentsrespectively

+e LIB is fully charged by constant current-constantvoltage (CC-CV) procedure after standing for two hoursthen three working conditions including 10A constantcurrent dynamic stress test (DST) and federal urban drivingschedule (FUDS) are used to test the proposed method+ree working conditions mentioned in this paper arecarried out at 25 degrees Celsius +e discharge current andterminal voltage of DSTand FUDS are shown in Figure 6 Inaddition the evaluation indicator such as maximum abso-lute error (MAE) and average absolute error (AAE) is ap-plied to the assessment of estimation algorithms

+e OCV-SOC function describing the relationshipbetween the open-circuit voltage and SOC can be fitted byOCV feature test data +e LIB is fully charged by the CC-

CV procedure after standing for two hours the measuredterminal voltage can be used as the OCV of 100 SOC +etest steps are as follows the LIB is discharged to 98 SOCwith 10A (05 C) constant current after standing for twohours the measured terminal voltage can be used as theOCV of 98 SOC Repeating the above steps 50 times eachmeasured terminal voltage can be obtained as the OCV ofevery 2 SOC+e eighth-order polynomial fitting functionto describe the mathematical relationship of OCV-SOC canbe shown in Figure 7

42 Parameters Identification +e ECM parameters iden-tification results are shown in Figures 8 and 9 the identi-fication result of ohmic resistance R0 polarization resistanceRp and polarization capacitance Cp can converge to a steadystate quickly from the unreliable initialization value undertwo cycles

R0Rp andCp always have a small range of fluctuation inFigures 8(a) and 9(a) which is directly related to the fluc-tuations of the coefficients a1 b1 and b2 +eoretically thedynamic changes of R0 Rp and Cp are inconsistent but thefixed forgetting factors of FFRLS have the same weight onthe three parameters which has a certain effect on thefluctuation of identification results

+e terminal voltage is identified in each sampling pe-riod based on R0 Rp and Cp As we can see fromFigures 8(b) and 9(b) the identification results of the ter-minal voltage can track the measured value of the terminalvoltage stably under two cycles +e maximum error of theterminal voltage identification is only 01923V and 01702Vrelative to the terminal voltage range from 23V to 29Vrespectively

43 Comparison of the SOC Estimation

431 Analysis of SOC Estimation under 10A DischargeBased on the identified ECM parameters the SOC estimationunder the 10A discharge cycle is presented in Figure 10 Asobserved the SOC reference with Arbin EVTS is formed witha black line the red line and blue line represent the SOC

Measurement dataand

model parameters

Initial conditionsequations (31) and (32)

State prior estimationequation (33)

Gain matrix updateequation (35)

State error covariance prior estimationequation (34)

PID coefficient update with gainequation (28)

State estimation measurement update by PID unitequation (36)

State error covariance updateequation (37)

Figure 4 +e flowchart of AEKF-PID estimator for LIB SOC

Arbin EVTS Temperature chamber

Computer

Figure 5 Battery test platform

Complexity 7

estimation with single AEKF and AEKF-PID respectivelyFrom the estimation results of two methods shown inFigure 10(a) these two SOC estimations can both track theSOC reference in a short period of time no matter AEKF orAEKF-PID Although the two SOC estimation results bothconverge to the SOC reference the SOC estimation error byAEKF-PID is less than that by AEKF as shown inFigure 10(b) +e errors of the two methods slowly increasewith an increase in time +e AAE and MAE of AEKF-PIDare 02266 and 05119 respectively and the AAE andMAE of AEKF are 08606 and 12478 respectively Itfollows that the improved AEKF-PID method shows betterresults in SOC estimation than single AEKF

432 Analysis of SOC Estimation under Two Cycles Inorder to further validate the accuracy of the proposedmethod the DST and FUDS cycles are implemented to

OCV

(V)

02 04 06 08 10SOC

22

23

24

25

26

27

28

29

Figure 7 OCV-SOC fitting curve

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(a)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

23

24

25

26

27

28

29

times104

(b)

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(c)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

22

23

24

25

26

27

28

29

times104

(d)

Figure 6 (a) Current under DST cycle (b) Terminal voltage under DST cycle (c) Current under FUDS cycle (d) Terminal voltage underFUDS cycle

8 Complexity

simulate the EV operationmode+e SOC estimation resultswith two methods under the DST and FUDS cycles areshown in Figures 11 and 12 respectively As observed theSOC reference by Arbin EVTS is formed with a black linethe red line and blue line represent the SOC estimation withsingle AEKF and AEKF-PID respectively From the esti-mation results of two methods shown in Figures 11(a) and12(a) two SOC estimations can both track the SOC refer-ence in a short period of time no matter AEKF or AEKF-PID +e SOC estimation errors with two algorithms underthe DST and FUDS cycles are shown in Figures 11(b) and12(b) respectivelyWe can see that the SOC estimation errorby the AEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 2 under the FUDS cycle meanwhilethe maximum SOC error by AEKF-PID is only 17141+eMAE and AAE of two methods under two cycles are listed inTable 1 According to the above analysis the combination ofthe PID feedback unit with AEKF has advantage over singleAEKF for SOC estimation

44 Analysis on Robustness of SOC Estimation +e highprecise SOC estimation results can be obtained by theproposed method on the assumption that the sampled ex-perimental data by Arbin EVTS is reliable However theexperiment data from the laboratory is not completely equalto actual data in EV operating mode Since the measurementdata by universal transducer is hard to avoid noise inter-ference from such as drift current and diffusion current theSOC estimation method with certain anti-interferencequality is important for BMS To further verify the anti-interference performance of the AEKF-PID a sequence ofnoise with feature of random normal distribution is added tothe operation current under DST and FUDS cycles re-spectively +e mean value of the noise is set as 0 and itsstandard deviation is set as 2 +e SOC estimation resultswith Gaussian noise of current under two conditions aredisplayed in Figures 13 and 14 +e two colored lines withblue and red are used to represent the SOC estimation byAEKF-PID and AEKF respectively +e MAE and AAE ofSOC estimation with Gaussian noise under two cycles are

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

010002000

0005

01

004005006

times104

times104

times104

(a)

05 1 15 2 25 3 350Time (s)

Term

inal

vol

tage

erro

r (V

)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 8 Model parameters identification results under DST (a) R0 Rp Cp (b) terminal voltage error

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

002004006

0005

01

0500

1000

times104

times104

times104

(a)

Term

inal

vol

tage

erro

r (V

)

05 1 15 2 25 3 350Time (s)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 9 Model parameters identification results under FUDS (a) R0 Rp Cp (b) terminal voltage error

Complexity 9

listed in Table 2 As we can see from Figures 13 and 14 dueto the introduction of current noise in two cycles thefluctuation of two corresponding SOC estimation errorsincreased We can see that the SOC estimation error by theAEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 28 under the FUDS cyclemeanwhile the maximum SOC error by AEKF-PID isclose to 2+e MAE and AAE of two methods under twocycles are listed in Table 2 According to the analysisabove the combination of PID feedback unit with AEKFhas advantage over single AEKF in robustness of SOCestimation

To further verify the anti-interference performance ofthe AEKF-PID a sequence of noise with feature of ran-dom non-Gaussian distribution is added to the operationcurrent +e SOC estimation result with non-Gaussiannoise is displayed in Figure 15 We can see that the twoSOC estimation errors with non-Gaussian noise increasecompared to that with Gaussian noise +e results inTable 3 show that the MAE of AEKF is greater than 55and the MAE of AEKF-PID is close to 4 At the end ofdischarge the SOC error of AEKF increases graduallymeanwhile the SOC error of AEKF-PID reaches 3 +eresults show that the basic AEKF-based SOC estimationmethod has better performance in Gaussian noise than

SOC

ReferenceAEKFAEKFndashPID

2500 3000 3500 4000082084086088

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

055

06

065

07

075

08

085

09

095

1

(a)

AEKFAEKFndashPID

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

SOC

erro

r (

)

ndash5ndash4ndash3ndash2ndash1

012345

(b)

Figure 10 Comparison of SOC estimation under 10A discharge (a) SOC estimation result (b) SOC estimation error

ReferenceAEKFAEKFndashPID

SOC

8500 9000 9500

076077078

1 15 2 25 3 3505Time (s)

0010203040506070809

1

times104

(a)

AEKFAEKFndashPID

SOC

erro

r (

)

50 100 150 200 250142144146148

1 15 2 25 3 3505Time (s)

ndash5ndash4ndash3ndash2ndash1

012345

times104

(b)

Figure 11 Comparison of SOC estimation under DST (a) SOC estimation result (b) SOC estimation error

10 Complexity

SOC

6000 6500 7000

081

082

083

1 15 2 25 3 3505Time (s)

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 3 3505Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 13 Comparison of SOC estimation with Gaussian noise under DST (a) SOC estimation result (b) SOC estimation error

SOC

7500 8000 8500

077078079

08

1 15 2 25 305Time (s)

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)SO

C er

ror (

)

50 100 150 200 250 300076078

08082084

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

1 15 2 25 305Time (s)

AEKFAEKFndashPID

times104

(b)

Figure 12 Comparison of SOC estimation under FUDS (a) SOC estimation result (b) SOC estimation error

Table 1 SOC estimation results

AEKF-PID AEKFMAE () 12422 (DST) 17257 (DST)MAE () 17141 (FUDS) 20093 (FUDS)AAE () 03327 (DST) 07466 (DST)AAE () 05235 (FUDS) 07345 (FUDS)

Complexity 11

Table 2 SOC estimation with Gaussian noise

AEKF-PID AEKFMAE () 17726 (DST) 19795 (DST)MAE () 19924 (FUDS) 28171 (FUDS)AAE () 05101 (DST) 10416 (DST)AAE () 05261 (FUDS) 11209 (FUDS)

1 15 2 25 305Time (s)

SOC

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

SOC

erro

r (

)

1 15 2 25 305Time (s)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 15 Comparison of SOC estimation with non-Gaussian noise (a) SOC estimation result (b) SOC estimation error

7500 8000 8500076077078079

1 15 2 25 305Time (s)

SOC

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 305Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 14 Comparison of SOC estimation with Gaussian noise under FUDS (a) SOC estimation result (b) SOC estimation error

12 Complexity

that in non-Gaussian noise whether or not the PIDfeedback is used

5 Conclusion

Based on the simple one-order ECM a novel method ofcombination of AEKF and PID feedback strategy is pro-posed for LIB SOC estimation +e parameters of the ECMare updated online using the FFRLS method to ensure LIBECM reliability and parameters integrity An improvedAEKF-PID technique is introduced to alleviate the negativeeffects resulting from the ECM error and the measurementerror via replacing the single innovation in feedback loop ofAEKF with an innovation vector-based PID module Toevaluate the performance of the proposed algorithm ex-periment and simulation results under 10A discharge DSTand FUDS cycles are adopted to validate the SOC estimation+e simulation results show that the maximum estimationerror of AEKF-PID is below 06 13 and 18 under 10Adischarge DST and FUDS cycles respectively +e SOCestimation error of AEKF-PID is still below 2 even in aGaussian noise situation +ese results suggest that theproposed AEKF-PID method can provide reliable SOCestimation with high precision and robustness Due to thefact that the local flat region in OCV-SOC fitted curve issusceptible to interference from measurement error ourfuture studies will concentrate on the optimization of OCVestimation

Data Availability

Data are available upon request to the corresponding author

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Zheng Liu proposed the original idea Zheng Liu designedthe novel algorithm Yuan Qiu Chunshan Yang and JianboJi performed and analyzed the experiments together ZhengLiu wrote the original manuscript Zheng Liu and ZhenhuaZhao revised the final manuscript

Acknowledgments

+is work was financially supported by the Guangxi NaturalScience Foundation (2018GXNSFAA281161 and2020GXNSFAA297032)

References

[1] X Hu H Yuan C Zou Z Li and L Zhang ldquoCo-estimationof state of charge and state of health for lithium-ion batteriesbased on fractional-order calculusrdquo IEEE Transactions onVehicular Technology vol 67 no 11 pp 10319ndash10329 2018

[2] X Zhou and C Feng ldquo+e impact of environmental regu-lation on fossil energy consumption in China direct andindirect effectsrdquo Journal of Cleaner Production vol 142pp 3174ndash3183 2017

[3] WWaag C Fleischer and D U Sauer ldquoCritical review of themethods for monitoring of lithium-ion batteries in electricand hybrid vehiclesrdquo Journal of Power Sources vol 258pp 321ndash339 2014

[4] R Xiong Y Zhang J Wang H He S Peng and M PechtldquoLithium-ion battery health prognosis based on a real batterymanagement system used in electric vehiclesrdquo IEEE Trans-actions on Vehicular Technology vol 68 no No 5pp 4110ndash4121 2019

[5] H Fathabadi ldquoPlug-in hybrid electric vehicles (PHEVs)replacing internal combustion engine with clean and re-newable energy based auxiliary power sourcesrdquo IEEETransactions on Power Electronics vol 33 no No 11pp 9611ndash9618 2018

[6] Y Xing W He M Pecht and K L Tsui ldquoState of chargeestimation of lithium-ion batteries using the open-circuitvoltage at various ambient temperaturesrdquo Applied Energyvol 113 pp 106ndash115 2014

[7] N Yang X Zhang and G Li ldquoState of charge estimation forpulse discharge of a LiFePO4 battery by a revised Ahcountingrdquo Electrochimica Acta vol 151 pp 63ndash71 2015

[8] K S Ng C-S Moo Y-P Chen and Y-C Hsieh ldquoEnhancedCoulomb counting method for estimating state-of-charge andstate-of-health of lithium-ion batteriesrdquo Applied Energyvol 86 no 9 pp 1506ndash1511 2009

[9] Z Liu X Dang and B Jing ldquoA novel open circuit voltagebased state of charge estimation for lithium-ion battery bymulti-innovation Kalman filterrdquo IEEE Access vol 7pp 49432ndash49447 2019

[10] J D J Rubio ldquoStability analysis of the modified Levenberg-Marquardt algorithm for the artificial neural network train-ingrdquo IEEE Transactions on Neural Networks and LearningSystems 2020

[11] G Aquino J D J Rubio J Pacheco et al ldquoNovel nonlinearhypothesis for the delta parallel robot modelingrdquo IEEE Accessvol 8 no 1 pp 46324ndash46334 2020

[12] J D J Rubio ldquoSOFMLS online self-organizing fuzzy mod-ified least-squares networkrdquo IEEE Transactions on FuzzySystems vol 17 no 6 pp 1296ndash1309 2009

[13] J A M Campantildea ldquoOn the estimation and control of non-linear systems with parametric uncertainties and noisy out-putsrdquo IEEE Access vol 6 pp 31968ndash31973 2018

[14] S Pramanik and S Anwar ldquoElectrochemical model basedcharge optimization for lithium-ion batteriesrdquo Journal ofPower Sources vol 313 pp 164ndash177 2016

[15] F Sun X Hu Y Zou and S Li ldquoAdaptive unscented Kalmanfiltering for state of charge estimation of a lithium-ion batteryfor electric vehiclesrdquo Energy vol 36 no 5 pp 3531ndash35402014

[16] Z Liu and X Dang ldquoA new method for state of charge andcapacity estimation of lithium-ion battery based on dualstrong tracking adaptive H infinity filterrdquo Mathematical

Table 3 SOC estimation with non-Gaussian noise

AEKF-PID AEKFMAE () 40590 55443AAE () 26074 32427

Complexity 13

Problems in Engineering vol 2018 Article ID 521820518 pages 2018

[17] Y Wang C Zhang and Z Chen ldquoOn-line battery state-of-charge estimation based on an integrated estimatorrdquo AppliedEnergy vol 185 pp 2026ndash2032 2017

[18] R Xiao J Shen X Li W Yan E Pan and Z ChenldquoComparisons of modeling and state of charge estimation forlithium-ion battery based on fractional order and integralorder methodsrdquo Energies vol 9 no No 3 2016

[19] Y Qiu X Li W Chen Z-m Duan and L Yu ldquoState ofcharge estimation of vanadium redox battery based on im-proved extended Kalman filterrdquo ISA Transactions vol 94pp 326ndash337 2019

[20] Y Xu M Hu A Zhou et al ldquoState of charge estimation forlithium-ion batteries based on adaptive dual Kalman filterrdquoAppliedMathematical Modelling vol 77 pp 1255ndash1272 2020

[21] Y Bi and S-Y Choe ldquoAn adaptive sigma-point Kalman filterwith state equality constraints for online state-of-charge es-timation of a Li(NiMnCo)O2Carbon battery using a re-duced-order electrochemical modelrdquo Applied Energyvol 258 p 113925 2020

[22] Z Liu X Dang B Jing and J Ji ldquoA novel model-based stateof charge estimation for lithium-ion battery using adaptiverobust iterative cubature Kalman filterrdquo Electric Power Sys-tems Research vol 177 p 105951 2019

[23] J Wei G Dong and Z Chen ldquoOn-board adaptive model forstate of charge estimation of lithium-ion batteries based onKalman filter with proportional integral-based error adjust-mentrdquo Journal of Power Sources vol 365 pp 308ndash319 2017

[24] F Yang Y Xing D Wang and K-L Tsui ldquoA comparativestudy of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dy-namic loading profilerdquo Applied Energy vol 164 pp 387ndash3992016

[25] Y Li C Wang and J Gong ldquoA multi-model probability SOCfusion estimation approach using an improved adaptiveunscented Kalman filter techniquerdquo Energy vol 141pp 1402ndash1415 2017

[26] J Linghu L Kang M Liu X Luo Y Feng and C LuldquoEstimation for state-of-charge of lithium-ion battery basedon an adaptive high-degree cubature Kalman filterrdquo Energyvol 189 Article ID 116204 2019

[27] Q Zhu M Xu W Liu and M Zheng ldquoA state of chargeestimation method for lithium-ion batteries based on frac-tional order adaptive extended Kalman filterrdquo Energyvol 187 Article ID 115880 2019

[28] Z Liu and H He ldquoSensor fault detection and isolation for alithium-ion battery pack in electric vehicles using adaptiveextended Kalman filterrdquo Applied Energy vol 185 pp 2033ndash2044 2017

[29] X Lai Y Zheng and T Sun ldquoA comparative study of dif-ferent equivalent circuit models for estimating state-of-chargeof lithium-ion batteriesrdquo Electrochimica Acta vol 259pp 566ndash577 2018

[30] Y Li J Chen and F Lan ldquoEnhanced online model identi-fication and state of charge estimation for lithium-ion batteryunder noise corrupted measurements by bias compensationrecursive least squaresrdquo Journal of Power Sources vol 456p 227984 2020

[31] V Duang H Bastawrous K ldquoLim K See P Zhang andS Dou ldquoOnline state of charge and model parameters esti-mation of the LiFePO4 battery in electric vehicles usingmultiple adaptive forgetting factors recursive least-squaresrdquoJournal of Power Sources vol 296 pp 215ndash224 2015

[32] Y Tian R Lai X Li L Xiang and J Tian ldquoA combinedmethod for state-of-charge estimation for lithium-ion bat-teries using a long short-term memory network and anadaptive cubature Kalman filterrdquo Applied Energy vol 265p 114789 2020

[33] C Chen R Xiong R Yang W Shen and F Sun ldquoState-of-charge estimation of lithium-ion battery using an improvedneural network model and extended Kalman filterrdquo Journal ofCleaner Production vol 234 pp 1153ndash1164 2019

14 Complexity

4 Results and Discussion

41 ExperimentalConfigurations A LIB test platform shownin Figure 5 is established to verify the effect of the ECM andmethod It consists of several major components (1) alithium-ion ferrous phosphate battery is used as the ex-perimental objects of which the nominal capacity is 20Ahand nominal voltage is 24V (2) a power battery test system(Arbin EVTS) with control accuracy which is less thanplusmn01 FSR is used for LIB chargingdischarging undervarious working condition (3) a programmable temperaturechamber is applied to control ambient temperature (4) ahost computer with MITS Pro v70 and a computerworkstation with MATLAB R2012a are used for data ac-quisition and data simulation in the experimentsrespectively

+e LIB is fully charged by constant current-constantvoltage (CC-CV) procedure after standing for two hoursthen three working conditions including 10A constantcurrent dynamic stress test (DST) and federal urban drivingschedule (FUDS) are used to test the proposed method+ree working conditions mentioned in this paper arecarried out at 25 degrees Celsius +e discharge current andterminal voltage of DSTand FUDS are shown in Figure 6 Inaddition the evaluation indicator such as maximum abso-lute error (MAE) and average absolute error (AAE) is ap-plied to the assessment of estimation algorithms

+e OCV-SOC function describing the relationshipbetween the open-circuit voltage and SOC can be fitted byOCV feature test data +e LIB is fully charged by the CC-

CV procedure after standing for two hours the measuredterminal voltage can be used as the OCV of 100 SOC +etest steps are as follows the LIB is discharged to 98 SOCwith 10A (05 C) constant current after standing for twohours the measured terminal voltage can be used as theOCV of 98 SOC Repeating the above steps 50 times eachmeasured terminal voltage can be obtained as the OCV ofevery 2 SOC+e eighth-order polynomial fitting functionto describe the mathematical relationship of OCV-SOC canbe shown in Figure 7

42 Parameters Identification +e ECM parameters iden-tification results are shown in Figures 8 and 9 the identi-fication result of ohmic resistance R0 polarization resistanceRp and polarization capacitance Cp can converge to a steadystate quickly from the unreliable initialization value undertwo cycles

R0Rp andCp always have a small range of fluctuation inFigures 8(a) and 9(a) which is directly related to the fluc-tuations of the coefficients a1 b1 and b2 +eoretically thedynamic changes of R0 Rp and Cp are inconsistent but thefixed forgetting factors of FFRLS have the same weight onthe three parameters which has a certain effect on thefluctuation of identification results

+e terminal voltage is identified in each sampling pe-riod based on R0 Rp and Cp As we can see fromFigures 8(b) and 9(b) the identification results of the ter-minal voltage can track the measured value of the terminalvoltage stably under two cycles +e maximum error of theterminal voltage identification is only 01923V and 01702Vrelative to the terminal voltage range from 23V to 29Vrespectively

43 Comparison of the SOC Estimation

431 Analysis of SOC Estimation under 10A DischargeBased on the identified ECM parameters the SOC estimationunder the 10A discharge cycle is presented in Figure 10 Asobserved the SOC reference with Arbin EVTS is formed witha black line the red line and blue line represent the SOC

Measurement dataand

model parameters

Initial conditionsequations (31) and (32)

State prior estimationequation (33)

Gain matrix updateequation (35)

State error covariance prior estimationequation (34)

PID coefficient update with gainequation (28)

State estimation measurement update by PID unitequation (36)

State error covariance updateequation (37)

Figure 4 +e flowchart of AEKF-PID estimator for LIB SOC

Arbin EVTS Temperature chamber

Computer

Figure 5 Battery test platform

Complexity 7

estimation with single AEKF and AEKF-PID respectivelyFrom the estimation results of two methods shown inFigure 10(a) these two SOC estimations can both track theSOC reference in a short period of time no matter AEKF orAEKF-PID Although the two SOC estimation results bothconverge to the SOC reference the SOC estimation error byAEKF-PID is less than that by AEKF as shown inFigure 10(b) +e errors of the two methods slowly increasewith an increase in time +e AAE and MAE of AEKF-PIDare 02266 and 05119 respectively and the AAE andMAE of AEKF are 08606 and 12478 respectively Itfollows that the improved AEKF-PID method shows betterresults in SOC estimation than single AEKF

432 Analysis of SOC Estimation under Two Cycles Inorder to further validate the accuracy of the proposedmethod the DST and FUDS cycles are implemented to

OCV

(V)

02 04 06 08 10SOC

22

23

24

25

26

27

28

29

Figure 7 OCV-SOC fitting curve

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(a)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

23

24

25

26

27

28

29

times104

(b)

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(c)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

22

23

24

25

26

27

28

29

times104

(d)

Figure 6 (a) Current under DST cycle (b) Terminal voltage under DST cycle (c) Current under FUDS cycle (d) Terminal voltage underFUDS cycle

8 Complexity

simulate the EV operationmode+e SOC estimation resultswith two methods under the DST and FUDS cycles areshown in Figures 11 and 12 respectively As observed theSOC reference by Arbin EVTS is formed with a black linethe red line and blue line represent the SOC estimation withsingle AEKF and AEKF-PID respectively From the esti-mation results of two methods shown in Figures 11(a) and12(a) two SOC estimations can both track the SOC refer-ence in a short period of time no matter AEKF or AEKF-PID +e SOC estimation errors with two algorithms underthe DST and FUDS cycles are shown in Figures 11(b) and12(b) respectivelyWe can see that the SOC estimation errorby the AEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 2 under the FUDS cycle meanwhilethe maximum SOC error by AEKF-PID is only 17141+eMAE and AAE of two methods under two cycles are listed inTable 1 According to the above analysis the combination ofthe PID feedback unit with AEKF has advantage over singleAEKF for SOC estimation

44 Analysis on Robustness of SOC Estimation +e highprecise SOC estimation results can be obtained by theproposed method on the assumption that the sampled ex-perimental data by Arbin EVTS is reliable However theexperiment data from the laboratory is not completely equalto actual data in EV operating mode Since the measurementdata by universal transducer is hard to avoid noise inter-ference from such as drift current and diffusion current theSOC estimation method with certain anti-interferencequality is important for BMS To further verify the anti-interference performance of the AEKF-PID a sequence ofnoise with feature of random normal distribution is added tothe operation current under DST and FUDS cycles re-spectively +e mean value of the noise is set as 0 and itsstandard deviation is set as 2 +e SOC estimation resultswith Gaussian noise of current under two conditions aredisplayed in Figures 13 and 14 +e two colored lines withblue and red are used to represent the SOC estimation byAEKF-PID and AEKF respectively +e MAE and AAE ofSOC estimation with Gaussian noise under two cycles are

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

010002000

0005

01

004005006

times104

times104

times104

(a)

05 1 15 2 25 3 350Time (s)

Term

inal

vol

tage

erro

r (V

)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 8 Model parameters identification results under DST (a) R0 Rp Cp (b) terminal voltage error

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

002004006

0005

01

0500

1000

times104

times104

times104

(a)

Term

inal

vol

tage

erro

r (V

)

05 1 15 2 25 3 350Time (s)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 9 Model parameters identification results under FUDS (a) R0 Rp Cp (b) terminal voltage error

Complexity 9

listed in Table 2 As we can see from Figures 13 and 14 dueto the introduction of current noise in two cycles thefluctuation of two corresponding SOC estimation errorsincreased We can see that the SOC estimation error by theAEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 28 under the FUDS cyclemeanwhile the maximum SOC error by AEKF-PID isclose to 2+e MAE and AAE of two methods under twocycles are listed in Table 2 According to the analysisabove the combination of PID feedback unit with AEKFhas advantage over single AEKF in robustness of SOCestimation

To further verify the anti-interference performance ofthe AEKF-PID a sequence of noise with feature of ran-dom non-Gaussian distribution is added to the operationcurrent +e SOC estimation result with non-Gaussiannoise is displayed in Figure 15 We can see that the twoSOC estimation errors with non-Gaussian noise increasecompared to that with Gaussian noise +e results inTable 3 show that the MAE of AEKF is greater than 55and the MAE of AEKF-PID is close to 4 At the end ofdischarge the SOC error of AEKF increases graduallymeanwhile the SOC error of AEKF-PID reaches 3 +eresults show that the basic AEKF-based SOC estimationmethod has better performance in Gaussian noise than

SOC

ReferenceAEKFAEKFndashPID

2500 3000 3500 4000082084086088

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

055

06

065

07

075

08

085

09

095

1

(a)

AEKFAEKFndashPID

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

SOC

erro

r (

)

ndash5ndash4ndash3ndash2ndash1

012345

(b)

Figure 10 Comparison of SOC estimation under 10A discharge (a) SOC estimation result (b) SOC estimation error

ReferenceAEKFAEKFndashPID

SOC

8500 9000 9500

076077078

1 15 2 25 3 3505Time (s)

0010203040506070809

1

times104

(a)

AEKFAEKFndashPID

SOC

erro

r (

)

50 100 150 200 250142144146148

1 15 2 25 3 3505Time (s)

ndash5ndash4ndash3ndash2ndash1

012345

times104

(b)

Figure 11 Comparison of SOC estimation under DST (a) SOC estimation result (b) SOC estimation error

10 Complexity

SOC

6000 6500 7000

081

082

083

1 15 2 25 3 3505Time (s)

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 3 3505Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 13 Comparison of SOC estimation with Gaussian noise under DST (a) SOC estimation result (b) SOC estimation error

SOC

7500 8000 8500

077078079

08

1 15 2 25 305Time (s)

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)SO

C er

ror (

)

50 100 150 200 250 300076078

08082084

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

1 15 2 25 305Time (s)

AEKFAEKFndashPID

times104

(b)

Figure 12 Comparison of SOC estimation under FUDS (a) SOC estimation result (b) SOC estimation error

Table 1 SOC estimation results

AEKF-PID AEKFMAE () 12422 (DST) 17257 (DST)MAE () 17141 (FUDS) 20093 (FUDS)AAE () 03327 (DST) 07466 (DST)AAE () 05235 (FUDS) 07345 (FUDS)

Complexity 11

Table 2 SOC estimation with Gaussian noise

AEKF-PID AEKFMAE () 17726 (DST) 19795 (DST)MAE () 19924 (FUDS) 28171 (FUDS)AAE () 05101 (DST) 10416 (DST)AAE () 05261 (FUDS) 11209 (FUDS)

1 15 2 25 305Time (s)

SOC

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

SOC

erro

r (

)

1 15 2 25 305Time (s)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 15 Comparison of SOC estimation with non-Gaussian noise (a) SOC estimation result (b) SOC estimation error

7500 8000 8500076077078079

1 15 2 25 305Time (s)

SOC

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 305Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 14 Comparison of SOC estimation with Gaussian noise under FUDS (a) SOC estimation result (b) SOC estimation error

12 Complexity

that in non-Gaussian noise whether or not the PIDfeedback is used

5 Conclusion

Based on the simple one-order ECM a novel method ofcombination of AEKF and PID feedback strategy is pro-posed for LIB SOC estimation +e parameters of the ECMare updated online using the FFRLS method to ensure LIBECM reliability and parameters integrity An improvedAEKF-PID technique is introduced to alleviate the negativeeffects resulting from the ECM error and the measurementerror via replacing the single innovation in feedback loop ofAEKF with an innovation vector-based PID module Toevaluate the performance of the proposed algorithm ex-periment and simulation results under 10A discharge DSTand FUDS cycles are adopted to validate the SOC estimation+e simulation results show that the maximum estimationerror of AEKF-PID is below 06 13 and 18 under 10Adischarge DST and FUDS cycles respectively +e SOCestimation error of AEKF-PID is still below 2 even in aGaussian noise situation +ese results suggest that theproposed AEKF-PID method can provide reliable SOCestimation with high precision and robustness Due to thefact that the local flat region in OCV-SOC fitted curve issusceptible to interference from measurement error ourfuture studies will concentrate on the optimization of OCVestimation

Data Availability

Data are available upon request to the corresponding author

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Zheng Liu proposed the original idea Zheng Liu designedthe novel algorithm Yuan Qiu Chunshan Yang and JianboJi performed and analyzed the experiments together ZhengLiu wrote the original manuscript Zheng Liu and ZhenhuaZhao revised the final manuscript

Acknowledgments

+is work was financially supported by the Guangxi NaturalScience Foundation (2018GXNSFAA281161 and2020GXNSFAA297032)

References

[1] X Hu H Yuan C Zou Z Li and L Zhang ldquoCo-estimationof state of charge and state of health for lithium-ion batteriesbased on fractional-order calculusrdquo IEEE Transactions onVehicular Technology vol 67 no 11 pp 10319ndash10329 2018

[2] X Zhou and C Feng ldquo+e impact of environmental regu-lation on fossil energy consumption in China direct andindirect effectsrdquo Journal of Cleaner Production vol 142pp 3174ndash3183 2017

[3] WWaag C Fleischer and D U Sauer ldquoCritical review of themethods for monitoring of lithium-ion batteries in electricand hybrid vehiclesrdquo Journal of Power Sources vol 258pp 321ndash339 2014

[4] R Xiong Y Zhang J Wang H He S Peng and M PechtldquoLithium-ion battery health prognosis based on a real batterymanagement system used in electric vehiclesrdquo IEEE Trans-actions on Vehicular Technology vol 68 no No 5pp 4110ndash4121 2019

[5] H Fathabadi ldquoPlug-in hybrid electric vehicles (PHEVs)replacing internal combustion engine with clean and re-newable energy based auxiliary power sourcesrdquo IEEETransactions on Power Electronics vol 33 no No 11pp 9611ndash9618 2018

[6] Y Xing W He M Pecht and K L Tsui ldquoState of chargeestimation of lithium-ion batteries using the open-circuitvoltage at various ambient temperaturesrdquo Applied Energyvol 113 pp 106ndash115 2014

[7] N Yang X Zhang and G Li ldquoState of charge estimation forpulse discharge of a LiFePO4 battery by a revised Ahcountingrdquo Electrochimica Acta vol 151 pp 63ndash71 2015

[8] K S Ng C-S Moo Y-P Chen and Y-C Hsieh ldquoEnhancedCoulomb counting method for estimating state-of-charge andstate-of-health of lithium-ion batteriesrdquo Applied Energyvol 86 no 9 pp 1506ndash1511 2009

[9] Z Liu X Dang and B Jing ldquoA novel open circuit voltagebased state of charge estimation for lithium-ion battery bymulti-innovation Kalman filterrdquo IEEE Access vol 7pp 49432ndash49447 2019

[10] J D J Rubio ldquoStability analysis of the modified Levenberg-Marquardt algorithm for the artificial neural network train-ingrdquo IEEE Transactions on Neural Networks and LearningSystems 2020

[11] G Aquino J D J Rubio J Pacheco et al ldquoNovel nonlinearhypothesis for the delta parallel robot modelingrdquo IEEE Accessvol 8 no 1 pp 46324ndash46334 2020

[12] J D J Rubio ldquoSOFMLS online self-organizing fuzzy mod-ified least-squares networkrdquo IEEE Transactions on FuzzySystems vol 17 no 6 pp 1296ndash1309 2009

[13] J A M Campantildea ldquoOn the estimation and control of non-linear systems with parametric uncertainties and noisy out-putsrdquo IEEE Access vol 6 pp 31968ndash31973 2018

[14] S Pramanik and S Anwar ldquoElectrochemical model basedcharge optimization for lithium-ion batteriesrdquo Journal ofPower Sources vol 313 pp 164ndash177 2016

[15] F Sun X Hu Y Zou and S Li ldquoAdaptive unscented Kalmanfiltering for state of charge estimation of a lithium-ion batteryfor electric vehiclesrdquo Energy vol 36 no 5 pp 3531ndash35402014

[16] Z Liu and X Dang ldquoA new method for state of charge andcapacity estimation of lithium-ion battery based on dualstrong tracking adaptive H infinity filterrdquo Mathematical

Table 3 SOC estimation with non-Gaussian noise

AEKF-PID AEKFMAE () 40590 55443AAE () 26074 32427

Complexity 13

Problems in Engineering vol 2018 Article ID 521820518 pages 2018

[17] Y Wang C Zhang and Z Chen ldquoOn-line battery state-of-charge estimation based on an integrated estimatorrdquo AppliedEnergy vol 185 pp 2026ndash2032 2017

[18] R Xiao J Shen X Li W Yan E Pan and Z ChenldquoComparisons of modeling and state of charge estimation forlithium-ion battery based on fractional order and integralorder methodsrdquo Energies vol 9 no No 3 2016

[19] Y Qiu X Li W Chen Z-m Duan and L Yu ldquoState ofcharge estimation of vanadium redox battery based on im-proved extended Kalman filterrdquo ISA Transactions vol 94pp 326ndash337 2019

[20] Y Xu M Hu A Zhou et al ldquoState of charge estimation forlithium-ion batteries based on adaptive dual Kalman filterrdquoAppliedMathematical Modelling vol 77 pp 1255ndash1272 2020

[21] Y Bi and S-Y Choe ldquoAn adaptive sigma-point Kalman filterwith state equality constraints for online state-of-charge es-timation of a Li(NiMnCo)O2Carbon battery using a re-duced-order electrochemical modelrdquo Applied Energyvol 258 p 113925 2020

[22] Z Liu X Dang B Jing and J Ji ldquoA novel model-based stateof charge estimation for lithium-ion battery using adaptiverobust iterative cubature Kalman filterrdquo Electric Power Sys-tems Research vol 177 p 105951 2019

[23] J Wei G Dong and Z Chen ldquoOn-board adaptive model forstate of charge estimation of lithium-ion batteries based onKalman filter with proportional integral-based error adjust-mentrdquo Journal of Power Sources vol 365 pp 308ndash319 2017

[24] F Yang Y Xing D Wang and K-L Tsui ldquoA comparativestudy of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dy-namic loading profilerdquo Applied Energy vol 164 pp 387ndash3992016

[25] Y Li C Wang and J Gong ldquoA multi-model probability SOCfusion estimation approach using an improved adaptiveunscented Kalman filter techniquerdquo Energy vol 141pp 1402ndash1415 2017

[26] J Linghu L Kang M Liu X Luo Y Feng and C LuldquoEstimation for state-of-charge of lithium-ion battery basedon an adaptive high-degree cubature Kalman filterrdquo Energyvol 189 Article ID 116204 2019

[27] Q Zhu M Xu W Liu and M Zheng ldquoA state of chargeestimation method for lithium-ion batteries based on frac-tional order adaptive extended Kalman filterrdquo Energyvol 187 Article ID 115880 2019

[28] Z Liu and H He ldquoSensor fault detection and isolation for alithium-ion battery pack in electric vehicles using adaptiveextended Kalman filterrdquo Applied Energy vol 185 pp 2033ndash2044 2017

[29] X Lai Y Zheng and T Sun ldquoA comparative study of dif-ferent equivalent circuit models for estimating state-of-chargeof lithium-ion batteriesrdquo Electrochimica Acta vol 259pp 566ndash577 2018

[30] Y Li J Chen and F Lan ldquoEnhanced online model identi-fication and state of charge estimation for lithium-ion batteryunder noise corrupted measurements by bias compensationrecursive least squaresrdquo Journal of Power Sources vol 456p 227984 2020

[31] V Duang H Bastawrous K ldquoLim K See P Zhang andS Dou ldquoOnline state of charge and model parameters esti-mation of the LiFePO4 battery in electric vehicles usingmultiple adaptive forgetting factors recursive least-squaresrdquoJournal of Power Sources vol 296 pp 215ndash224 2015

[32] Y Tian R Lai X Li L Xiang and J Tian ldquoA combinedmethod for state-of-charge estimation for lithium-ion bat-teries using a long short-term memory network and anadaptive cubature Kalman filterrdquo Applied Energy vol 265p 114789 2020

[33] C Chen R Xiong R Yang W Shen and F Sun ldquoState-of-charge estimation of lithium-ion battery using an improvedneural network model and extended Kalman filterrdquo Journal ofCleaner Production vol 234 pp 1153ndash1164 2019

14 Complexity

estimation with single AEKF and AEKF-PID respectivelyFrom the estimation results of two methods shown inFigure 10(a) these two SOC estimations can both track theSOC reference in a short period of time no matter AEKF orAEKF-PID Although the two SOC estimation results bothconverge to the SOC reference the SOC estimation error byAEKF-PID is less than that by AEKF as shown inFigure 10(b) +e errors of the two methods slowly increasewith an increase in time +e AAE and MAE of AEKF-PIDare 02266 and 05119 respectively and the AAE andMAE of AEKF are 08606 and 12478 respectively Itfollows that the improved AEKF-PID method shows betterresults in SOC estimation than single AEKF

432 Analysis of SOC Estimation under Two Cycles Inorder to further validate the accuracy of the proposedmethod the DST and FUDS cycles are implemented to

OCV

(V)

02 04 06 08 10SOC

22

23

24

25

26

27

28

29

Figure 7 OCV-SOC fitting curve

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(a)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

23

24

25

26

27

28

29

times104

(b)

Curr

ent (

A)

05 1 15 2 25 3 350Time (s)

ndash10

ndash5

0

5

10

15

20

times104

(c)

Term

inal

vol

tage

(V)

05 1 15 2 25 3 350Time (s)

22

23

24

25

26

27

28

29

times104

(d)

Figure 6 (a) Current under DST cycle (b) Terminal voltage under DST cycle (c) Current under FUDS cycle (d) Terminal voltage underFUDS cycle

8 Complexity

simulate the EV operationmode+e SOC estimation resultswith two methods under the DST and FUDS cycles areshown in Figures 11 and 12 respectively As observed theSOC reference by Arbin EVTS is formed with a black linethe red line and blue line represent the SOC estimation withsingle AEKF and AEKF-PID respectively From the esti-mation results of two methods shown in Figures 11(a) and12(a) two SOC estimations can both track the SOC refer-ence in a short period of time no matter AEKF or AEKF-PID +e SOC estimation errors with two algorithms underthe DST and FUDS cycles are shown in Figures 11(b) and12(b) respectivelyWe can see that the SOC estimation errorby the AEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 2 under the FUDS cycle meanwhilethe maximum SOC error by AEKF-PID is only 17141+eMAE and AAE of two methods under two cycles are listed inTable 1 According to the above analysis the combination ofthe PID feedback unit with AEKF has advantage over singleAEKF for SOC estimation

44 Analysis on Robustness of SOC Estimation +e highprecise SOC estimation results can be obtained by theproposed method on the assumption that the sampled ex-perimental data by Arbin EVTS is reliable However theexperiment data from the laboratory is not completely equalto actual data in EV operating mode Since the measurementdata by universal transducer is hard to avoid noise inter-ference from such as drift current and diffusion current theSOC estimation method with certain anti-interferencequality is important for BMS To further verify the anti-interference performance of the AEKF-PID a sequence ofnoise with feature of random normal distribution is added tothe operation current under DST and FUDS cycles re-spectively +e mean value of the noise is set as 0 and itsstandard deviation is set as 2 +e SOC estimation resultswith Gaussian noise of current under two conditions aredisplayed in Figures 13 and 14 +e two colored lines withblue and red are used to represent the SOC estimation byAEKF-PID and AEKF respectively +e MAE and AAE ofSOC estimation with Gaussian noise under two cycles are

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

010002000

0005

01

004005006

times104

times104

times104

(a)

05 1 15 2 25 3 350Time (s)

Term

inal

vol

tage

erro

r (V

)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 8 Model parameters identification results under DST (a) R0 Rp Cp (b) terminal voltage error

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

002004006

0005

01

0500

1000

times104

times104

times104

(a)

Term

inal

vol

tage

erro

r (V

)

05 1 15 2 25 3 350Time (s)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 9 Model parameters identification results under FUDS (a) R0 Rp Cp (b) terminal voltage error

Complexity 9

listed in Table 2 As we can see from Figures 13 and 14 dueto the introduction of current noise in two cycles thefluctuation of two corresponding SOC estimation errorsincreased We can see that the SOC estimation error by theAEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 28 under the FUDS cyclemeanwhile the maximum SOC error by AEKF-PID isclose to 2+e MAE and AAE of two methods under twocycles are listed in Table 2 According to the analysisabove the combination of PID feedback unit with AEKFhas advantage over single AEKF in robustness of SOCestimation

To further verify the anti-interference performance ofthe AEKF-PID a sequence of noise with feature of ran-dom non-Gaussian distribution is added to the operationcurrent +e SOC estimation result with non-Gaussiannoise is displayed in Figure 15 We can see that the twoSOC estimation errors with non-Gaussian noise increasecompared to that with Gaussian noise +e results inTable 3 show that the MAE of AEKF is greater than 55and the MAE of AEKF-PID is close to 4 At the end ofdischarge the SOC error of AEKF increases graduallymeanwhile the SOC error of AEKF-PID reaches 3 +eresults show that the basic AEKF-based SOC estimationmethod has better performance in Gaussian noise than

SOC

ReferenceAEKFAEKFndashPID

2500 3000 3500 4000082084086088

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

055

06

065

07

075

08

085

09

095

1

(a)

AEKFAEKFndashPID

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

SOC

erro

r (

)

ndash5ndash4ndash3ndash2ndash1

012345

(b)

Figure 10 Comparison of SOC estimation under 10A discharge (a) SOC estimation result (b) SOC estimation error

ReferenceAEKFAEKFndashPID

SOC

8500 9000 9500

076077078

1 15 2 25 3 3505Time (s)

0010203040506070809

1

times104

(a)

AEKFAEKFndashPID

SOC

erro

r (

)

50 100 150 200 250142144146148

1 15 2 25 3 3505Time (s)

ndash5ndash4ndash3ndash2ndash1

012345

times104

(b)

Figure 11 Comparison of SOC estimation under DST (a) SOC estimation result (b) SOC estimation error

10 Complexity

SOC

6000 6500 7000

081

082

083

1 15 2 25 3 3505Time (s)

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 3 3505Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 13 Comparison of SOC estimation with Gaussian noise under DST (a) SOC estimation result (b) SOC estimation error

SOC

7500 8000 8500

077078079

08

1 15 2 25 305Time (s)

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)SO

C er

ror (

)

50 100 150 200 250 300076078

08082084

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

1 15 2 25 305Time (s)

AEKFAEKFndashPID

times104

(b)

Figure 12 Comparison of SOC estimation under FUDS (a) SOC estimation result (b) SOC estimation error

Table 1 SOC estimation results

AEKF-PID AEKFMAE () 12422 (DST) 17257 (DST)MAE () 17141 (FUDS) 20093 (FUDS)AAE () 03327 (DST) 07466 (DST)AAE () 05235 (FUDS) 07345 (FUDS)

Complexity 11

Table 2 SOC estimation with Gaussian noise

AEKF-PID AEKFMAE () 17726 (DST) 19795 (DST)MAE () 19924 (FUDS) 28171 (FUDS)AAE () 05101 (DST) 10416 (DST)AAE () 05261 (FUDS) 11209 (FUDS)

1 15 2 25 305Time (s)

SOC

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

SOC

erro

r (

)

1 15 2 25 305Time (s)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 15 Comparison of SOC estimation with non-Gaussian noise (a) SOC estimation result (b) SOC estimation error

7500 8000 8500076077078079

1 15 2 25 305Time (s)

SOC

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 305Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 14 Comparison of SOC estimation with Gaussian noise under FUDS (a) SOC estimation result (b) SOC estimation error

12 Complexity

that in non-Gaussian noise whether or not the PIDfeedback is used

5 Conclusion

Based on the simple one-order ECM a novel method ofcombination of AEKF and PID feedback strategy is pro-posed for LIB SOC estimation +e parameters of the ECMare updated online using the FFRLS method to ensure LIBECM reliability and parameters integrity An improvedAEKF-PID technique is introduced to alleviate the negativeeffects resulting from the ECM error and the measurementerror via replacing the single innovation in feedback loop ofAEKF with an innovation vector-based PID module Toevaluate the performance of the proposed algorithm ex-periment and simulation results under 10A discharge DSTand FUDS cycles are adopted to validate the SOC estimation+e simulation results show that the maximum estimationerror of AEKF-PID is below 06 13 and 18 under 10Adischarge DST and FUDS cycles respectively +e SOCestimation error of AEKF-PID is still below 2 even in aGaussian noise situation +ese results suggest that theproposed AEKF-PID method can provide reliable SOCestimation with high precision and robustness Due to thefact that the local flat region in OCV-SOC fitted curve issusceptible to interference from measurement error ourfuture studies will concentrate on the optimization of OCVestimation

Data Availability

Data are available upon request to the corresponding author

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Zheng Liu proposed the original idea Zheng Liu designedthe novel algorithm Yuan Qiu Chunshan Yang and JianboJi performed and analyzed the experiments together ZhengLiu wrote the original manuscript Zheng Liu and ZhenhuaZhao revised the final manuscript

Acknowledgments

+is work was financially supported by the Guangxi NaturalScience Foundation (2018GXNSFAA281161 and2020GXNSFAA297032)

References

[1] X Hu H Yuan C Zou Z Li and L Zhang ldquoCo-estimationof state of charge and state of health for lithium-ion batteriesbased on fractional-order calculusrdquo IEEE Transactions onVehicular Technology vol 67 no 11 pp 10319ndash10329 2018

[2] X Zhou and C Feng ldquo+e impact of environmental regu-lation on fossil energy consumption in China direct andindirect effectsrdquo Journal of Cleaner Production vol 142pp 3174ndash3183 2017

[3] WWaag C Fleischer and D U Sauer ldquoCritical review of themethods for monitoring of lithium-ion batteries in electricand hybrid vehiclesrdquo Journal of Power Sources vol 258pp 321ndash339 2014

[4] R Xiong Y Zhang J Wang H He S Peng and M PechtldquoLithium-ion battery health prognosis based on a real batterymanagement system used in electric vehiclesrdquo IEEE Trans-actions on Vehicular Technology vol 68 no No 5pp 4110ndash4121 2019

[5] H Fathabadi ldquoPlug-in hybrid electric vehicles (PHEVs)replacing internal combustion engine with clean and re-newable energy based auxiliary power sourcesrdquo IEEETransactions on Power Electronics vol 33 no No 11pp 9611ndash9618 2018

[6] Y Xing W He M Pecht and K L Tsui ldquoState of chargeestimation of lithium-ion batteries using the open-circuitvoltage at various ambient temperaturesrdquo Applied Energyvol 113 pp 106ndash115 2014

[7] N Yang X Zhang and G Li ldquoState of charge estimation forpulse discharge of a LiFePO4 battery by a revised Ahcountingrdquo Electrochimica Acta vol 151 pp 63ndash71 2015

[8] K S Ng C-S Moo Y-P Chen and Y-C Hsieh ldquoEnhancedCoulomb counting method for estimating state-of-charge andstate-of-health of lithium-ion batteriesrdquo Applied Energyvol 86 no 9 pp 1506ndash1511 2009

[9] Z Liu X Dang and B Jing ldquoA novel open circuit voltagebased state of charge estimation for lithium-ion battery bymulti-innovation Kalman filterrdquo IEEE Access vol 7pp 49432ndash49447 2019

[10] J D J Rubio ldquoStability analysis of the modified Levenberg-Marquardt algorithm for the artificial neural network train-ingrdquo IEEE Transactions on Neural Networks and LearningSystems 2020

[11] G Aquino J D J Rubio J Pacheco et al ldquoNovel nonlinearhypothesis for the delta parallel robot modelingrdquo IEEE Accessvol 8 no 1 pp 46324ndash46334 2020

[12] J D J Rubio ldquoSOFMLS online self-organizing fuzzy mod-ified least-squares networkrdquo IEEE Transactions on FuzzySystems vol 17 no 6 pp 1296ndash1309 2009

[13] J A M Campantildea ldquoOn the estimation and control of non-linear systems with parametric uncertainties and noisy out-putsrdquo IEEE Access vol 6 pp 31968ndash31973 2018

[14] S Pramanik and S Anwar ldquoElectrochemical model basedcharge optimization for lithium-ion batteriesrdquo Journal ofPower Sources vol 313 pp 164ndash177 2016

[15] F Sun X Hu Y Zou and S Li ldquoAdaptive unscented Kalmanfiltering for state of charge estimation of a lithium-ion batteryfor electric vehiclesrdquo Energy vol 36 no 5 pp 3531ndash35402014

[16] Z Liu and X Dang ldquoA new method for state of charge andcapacity estimation of lithium-ion battery based on dualstrong tracking adaptive H infinity filterrdquo Mathematical

Table 3 SOC estimation with non-Gaussian noise

AEKF-PID AEKFMAE () 40590 55443AAE () 26074 32427

Complexity 13

Problems in Engineering vol 2018 Article ID 521820518 pages 2018

[17] Y Wang C Zhang and Z Chen ldquoOn-line battery state-of-charge estimation based on an integrated estimatorrdquo AppliedEnergy vol 185 pp 2026ndash2032 2017

[18] R Xiao J Shen X Li W Yan E Pan and Z ChenldquoComparisons of modeling and state of charge estimation forlithium-ion battery based on fractional order and integralorder methodsrdquo Energies vol 9 no No 3 2016

[19] Y Qiu X Li W Chen Z-m Duan and L Yu ldquoState ofcharge estimation of vanadium redox battery based on im-proved extended Kalman filterrdquo ISA Transactions vol 94pp 326ndash337 2019

[20] Y Xu M Hu A Zhou et al ldquoState of charge estimation forlithium-ion batteries based on adaptive dual Kalman filterrdquoAppliedMathematical Modelling vol 77 pp 1255ndash1272 2020

[21] Y Bi and S-Y Choe ldquoAn adaptive sigma-point Kalman filterwith state equality constraints for online state-of-charge es-timation of a Li(NiMnCo)O2Carbon battery using a re-duced-order electrochemical modelrdquo Applied Energyvol 258 p 113925 2020

[22] Z Liu X Dang B Jing and J Ji ldquoA novel model-based stateof charge estimation for lithium-ion battery using adaptiverobust iterative cubature Kalman filterrdquo Electric Power Sys-tems Research vol 177 p 105951 2019

[23] J Wei G Dong and Z Chen ldquoOn-board adaptive model forstate of charge estimation of lithium-ion batteries based onKalman filter with proportional integral-based error adjust-mentrdquo Journal of Power Sources vol 365 pp 308ndash319 2017

[24] F Yang Y Xing D Wang and K-L Tsui ldquoA comparativestudy of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dy-namic loading profilerdquo Applied Energy vol 164 pp 387ndash3992016

[25] Y Li C Wang and J Gong ldquoA multi-model probability SOCfusion estimation approach using an improved adaptiveunscented Kalman filter techniquerdquo Energy vol 141pp 1402ndash1415 2017

[26] J Linghu L Kang M Liu X Luo Y Feng and C LuldquoEstimation for state-of-charge of lithium-ion battery basedon an adaptive high-degree cubature Kalman filterrdquo Energyvol 189 Article ID 116204 2019

[27] Q Zhu M Xu W Liu and M Zheng ldquoA state of chargeestimation method for lithium-ion batteries based on frac-tional order adaptive extended Kalman filterrdquo Energyvol 187 Article ID 115880 2019

[28] Z Liu and H He ldquoSensor fault detection and isolation for alithium-ion battery pack in electric vehicles using adaptiveextended Kalman filterrdquo Applied Energy vol 185 pp 2033ndash2044 2017

[29] X Lai Y Zheng and T Sun ldquoA comparative study of dif-ferent equivalent circuit models for estimating state-of-chargeof lithium-ion batteriesrdquo Electrochimica Acta vol 259pp 566ndash577 2018

[30] Y Li J Chen and F Lan ldquoEnhanced online model identi-fication and state of charge estimation for lithium-ion batteryunder noise corrupted measurements by bias compensationrecursive least squaresrdquo Journal of Power Sources vol 456p 227984 2020

[31] V Duang H Bastawrous K ldquoLim K See P Zhang andS Dou ldquoOnline state of charge and model parameters esti-mation of the LiFePO4 battery in electric vehicles usingmultiple adaptive forgetting factors recursive least-squaresrdquoJournal of Power Sources vol 296 pp 215ndash224 2015

[32] Y Tian R Lai X Li L Xiang and J Tian ldquoA combinedmethod for state-of-charge estimation for lithium-ion bat-teries using a long short-term memory network and anadaptive cubature Kalman filterrdquo Applied Energy vol 265p 114789 2020

[33] C Chen R Xiong R Yang W Shen and F Sun ldquoState-of-charge estimation of lithium-ion battery using an improvedneural network model and extended Kalman filterrdquo Journal ofCleaner Production vol 234 pp 1153ndash1164 2019

14 Complexity

simulate the EV operationmode+e SOC estimation resultswith two methods under the DST and FUDS cycles areshown in Figures 11 and 12 respectively As observed theSOC reference by Arbin EVTS is formed with a black linethe red line and blue line represent the SOC estimation withsingle AEKF and AEKF-PID respectively From the esti-mation results of two methods shown in Figures 11(a) and12(a) two SOC estimations can both track the SOC refer-ence in a short period of time no matter AEKF or AEKF-PID +e SOC estimation errors with two algorithms underthe DST and FUDS cycles are shown in Figures 11(b) and12(b) respectivelyWe can see that the SOC estimation errorby the AEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 2 under the FUDS cycle meanwhilethe maximum SOC error by AEKF-PID is only 17141+eMAE and AAE of two methods under two cycles are listed inTable 1 According to the above analysis the combination ofthe PID feedback unit with AEKF has advantage over singleAEKF for SOC estimation

44 Analysis on Robustness of SOC Estimation +e highprecise SOC estimation results can be obtained by theproposed method on the assumption that the sampled ex-perimental data by Arbin EVTS is reliable However theexperiment data from the laboratory is not completely equalto actual data in EV operating mode Since the measurementdata by universal transducer is hard to avoid noise inter-ference from such as drift current and diffusion current theSOC estimation method with certain anti-interferencequality is important for BMS To further verify the anti-interference performance of the AEKF-PID a sequence ofnoise with feature of random normal distribution is added tothe operation current under DST and FUDS cycles re-spectively +e mean value of the noise is set as 0 and itsstandard deviation is set as 2 +e SOC estimation resultswith Gaussian noise of current under two conditions aredisplayed in Figures 13 and 14 +e two colored lines withblue and red are used to represent the SOC estimation byAEKF-PID and AEKF respectively +e MAE and AAE ofSOC estimation with Gaussian noise under two cycles are

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

010002000

0005

01

004005006

times104

times104

times104

(a)

05 1 15 2 25 3 350Time (s)

Term

inal

vol

tage

erro

r (V

)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 8 Model parameters identification results under DST (a) R0 Rp Cp (b) terminal voltage error

R0 (Ω

)Rp

(Ω)

Cp (F

)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

05 1 15 2 25 3 350Time (s)

002004006

0005

01

0500

1000

times104

times104

times104

(a)

Term

inal

vol

tage

erro

r (V

)

05 1 15 2 25 3 350Time (s)

ndash02

ndash015

ndash01

ndash005

0

005

01

015

times104

(b)

Figure 9 Model parameters identification results under FUDS (a) R0 Rp Cp (b) terminal voltage error

Complexity 9

listed in Table 2 As we can see from Figures 13 and 14 dueto the introduction of current noise in two cycles thefluctuation of two corresponding SOC estimation errorsincreased We can see that the SOC estimation error by theAEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 28 under the FUDS cyclemeanwhile the maximum SOC error by AEKF-PID isclose to 2+e MAE and AAE of two methods under twocycles are listed in Table 2 According to the analysisabove the combination of PID feedback unit with AEKFhas advantage over single AEKF in robustness of SOCestimation

To further verify the anti-interference performance ofthe AEKF-PID a sequence of noise with feature of ran-dom non-Gaussian distribution is added to the operationcurrent +e SOC estimation result with non-Gaussiannoise is displayed in Figure 15 We can see that the twoSOC estimation errors with non-Gaussian noise increasecompared to that with Gaussian noise +e results inTable 3 show that the MAE of AEKF is greater than 55and the MAE of AEKF-PID is close to 4 At the end ofdischarge the SOC error of AEKF increases graduallymeanwhile the SOC error of AEKF-PID reaches 3 +eresults show that the basic AEKF-based SOC estimationmethod has better performance in Gaussian noise than

SOC

ReferenceAEKFAEKFndashPID

2500 3000 3500 4000082084086088

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

055

06

065

07

075

08

085

09

095

1

(a)

AEKFAEKFndashPID

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

SOC

erro

r (

)

ndash5ndash4ndash3ndash2ndash1

012345

(b)

Figure 10 Comparison of SOC estimation under 10A discharge (a) SOC estimation result (b) SOC estimation error

ReferenceAEKFAEKFndashPID

SOC

8500 9000 9500

076077078

1 15 2 25 3 3505Time (s)

0010203040506070809

1

times104

(a)

AEKFAEKFndashPID

SOC

erro

r (

)

50 100 150 200 250142144146148

1 15 2 25 3 3505Time (s)

ndash5ndash4ndash3ndash2ndash1

012345

times104

(b)

Figure 11 Comparison of SOC estimation under DST (a) SOC estimation result (b) SOC estimation error

10 Complexity

SOC

6000 6500 7000

081

082

083

1 15 2 25 3 3505Time (s)

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 3 3505Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 13 Comparison of SOC estimation with Gaussian noise under DST (a) SOC estimation result (b) SOC estimation error

SOC

7500 8000 8500

077078079

08

1 15 2 25 305Time (s)

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)SO

C er

ror (

)

50 100 150 200 250 300076078

08082084

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

1 15 2 25 305Time (s)

AEKFAEKFndashPID

times104

(b)

Figure 12 Comparison of SOC estimation under FUDS (a) SOC estimation result (b) SOC estimation error

Table 1 SOC estimation results

AEKF-PID AEKFMAE () 12422 (DST) 17257 (DST)MAE () 17141 (FUDS) 20093 (FUDS)AAE () 03327 (DST) 07466 (DST)AAE () 05235 (FUDS) 07345 (FUDS)

Complexity 11

Table 2 SOC estimation with Gaussian noise

AEKF-PID AEKFMAE () 17726 (DST) 19795 (DST)MAE () 19924 (FUDS) 28171 (FUDS)AAE () 05101 (DST) 10416 (DST)AAE () 05261 (FUDS) 11209 (FUDS)

1 15 2 25 305Time (s)

SOC

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

SOC

erro

r (

)

1 15 2 25 305Time (s)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 15 Comparison of SOC estimation with non-Gaussian noise (a) SOC estimation result (b) SOC estimation error

7500 8000 8500076077078079

1 15 2 25 305Time (s)

SOC

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 305Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 14 Comparison of SOC estimation with Gaussian noise under FUDS (a) SOC estimation result (b) SOC estimation error

12 Complexity

that in non-Gaussian noise whether or not the PIDfeedback is used

5 Conclusion

Based on the simple one-order ECM a novel method ofcombination of AEKF and PID feedback strategy is pro-posed for LIB SOC estimation +e parameters of the ECMare updated online using the FFRLS method to ensure LIBECM reliability and parameters integrity An improvedAEKF-PID technique is introduced to alleviate the negativeeffects resulting from the ECM error and the measurementerror via replacing the single innovation in feedback loop ofAEKF with an innovation vector-based PID module Toevaluate the performance of the proposed algorithm ex-periment and simulation results under 10A discharge DSTand FUDS cycles are adopted to validate the SOC estimation+e simulation results show that the maximum estimationerror of AEKF-PID is below 06 13 and 18 under 10Adischarge DST and FUDS cycles respectively +e SOCestimation error of AEKF-PID is still below 2 even in aGaussian noise situation +ese results suggest that theproposed AEKF-PID method can provide reliable SOCestimation with high precision and robustness Due to thefact that the local flat region in OCV-SOC fitted curve issusceptible to interference from measurement error ourfuture studies will concentrate on the optimization of OCVestimation

Data Availability

Data are available upon request to the corresponding author

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Zheng Liu proposed the original idea Zheng Liu designedthe novel algorithm Yuan Qiu Chunshan Yang and JianboJi performed and analyzed the experiments together ZhengLiu wrote the original manuscript Zheng Liu and ZhenhuaZhao revised the final manuscript

Acknowledgments

+is work was financially supported by the Guangxi NaturalScience Foundation (2018GXNSFAA281161 and2020GXNSFAA297032)

References

[1] X Hu H Yuan C Zou Z Li and L Zhang ldquoCo-estimationof state of charge and state of health for lithium-ion batteriesbased on fractional-order calculusrdquo IEEE Transactions onVehicular Technology vol 67 no 11 pp 10319ndash10329 2018

[2] X Zhou and C Feng ldquo+e impact of environmental regu-lation on fossil energy consumption in China direct andindirect effectsrdquo Journal of Cleaner Production vol 142pp 3174ndash3183 2017

[3] WWaag C Fleischer and D U Sauer ldquoCritical review of themethods for monitoring of lithium-ion batteries in electricand hybrid vehiclesrdquo Journal of Power Sources vol 258pp 321ndash339 2014

[4] R Xiong Y Zhang J Wang H He S Peng and M PechtldquoLithium-ion battery health prognosis based on a real batterymanagement system used in electric vehiclesrdquo IEEE Trans-actions on Vehicular Technology vol 68 no No 5pp 4110ndash4121 2019

[5] H Fathabadi ldquoPlug-in hybrid electric vehicles (PHEVs)replacing internal combustion engine with clean and re-newable energy based auxiliary power sourcesrdquo IEEETransactions on Power Electronics vol 33 no No 11pp 9611ndash9618 2018

[6] Y Xing W He M Pecht and K L Tsui ldquoState of chargeestimation of lithium-ion batteries using the open-circuitvoltage at various ambient temperaturesrdquo Applied Energyvol 113 pp 106ndash115 2014

[7] N Yang X Zhang and G Li ldquoState of charge estimation forpulse discharge of a LiFePO4 battery by a revised Ahcountingrdquo Electrochimica Acta vol 151 pp 63ndash71 2015

[8] K S Ng C-S Moo Y-P Chen and Y-C Hsieh ldquoEnhancedCoulomb counting method for estimating state-of-charge andstate-of-health of lithium-ion batteriesrdquo Applied Energyvol 86 no 9 pp 1506ndash1511 2009

[9] Z Liu X Dang and B Jing ldquoA novel open circuit voltagebased state of charge estimation for lithium-ion battery bymulti-innovation Kalman filterrdquo IEEE Access vol 7pp 49432ndash49447 2019

[10] J D J Rubio ldquoStability analysis of the modified Levenberg-Marquardt algorithm for the artificial neural network train-ingrdquo IEEE Transactions on Neural Networks and LearningSystems 2020

[11] G Aquino J D J Rubio J Pacheco et al ldquoNovel nonlinearhypothesis for the delta parallel robot modelingrdquo IEEE Accessvol 8 no 1 pp 46324ndash46334 2020

[12] J D J Rubio ldquoSOFMLS online self-organizing fuzzy mod-ified least-squares networkrdquo IEEE Transactions on FuzzySystems vol 17 no 6 pp 1296ndash1309 2009

[13] J A M Campantildea ldquoOn the estimation and control of non-linear systems with parametric uncertainties and noisy out-putsrdquo IEEE Access vol 6 pp 31968ndash31973 2018

[14] S Pramanik and S Anwar ldquoElectrochemical model basedcharge optimization for lithium-ion batteriesrdquo Journal ofPower Sources vol 313 pp 164ndash177 2016

[15] F Sun X Hu Y Zou and S Li ldquoAdaptive unscented Kalmanfiltering for state of charge estimation of a lithium-ion batteryfor electric vehiclesrdquo Energy vol 36 no 5 pp 3531ndash35402014

[16] Z Liu and X Dang ldquoA new method for state of charge andcapacity estimation of lithium-ion battery based on dualstrong tracking adaptive H infinity filterrdquo Mathematical

Table 3 SOC estimation with non-Gaussian noise

AEKF-PID AEKFMAE () 40590 55443AAE () 26074 32427

Complexity 13

Problems in Engineering vol 2018 Article ID 521820518 pages 2018

[17] Y Wang C Zhang and Z Chen ldquoOn-line battery state-of-charge estimation based on an integrated estimatorrdquo AppliedEnergy vol 185 pp 2026ndash2032 2017

[18] R Xiao J Shen X Li W Yan E Pan and Z ChenldquoComparisons of modeling and state of charge estimation forlithium-ion battery based on fractional order and integralorder methodsrdquo Energies vol 9 no No 3 2016

[19] Y Qiu X Li W Chen Z-m Duan and L Yu ldquoState ofcharge estimation of vanadium redox battery based on im-proved extended Kalman filterrdquo ISA Transactions vol 94pp 326ndash337 2019

[20] Y Xu M Hu A Zhou et al ldquoState of charge estimation forlithium-ion batteries based on adaptive dual Kalman filterrdquoAppliedMathematical Modelling vol 77 pp 1255ndash1272 2020

[21] Y Bi and S-Y Choe ldquoAn adaptive sigma-point Kalman filterwith state equality constraints for online state-of-charge es-timation of a Li(NiMnCo)O2Carbon battery using a re-duced-order electrochemical modelrdquo Applied Energyvol 258 p 113925 2020

[22] Z Liu X Dang B Jing and J Ji ldquoA novel model-based stateof charge estimation for lithium-ion battery using adaptiverobust iterative cubature Kalman filterrdquo Electric Power Sys-tems Research vol 177 p 105951 2019

[23] J Wei G Dong and Z Chen ldquoOn-board adaptive model forstate of charge estimation of lithium-ion batteries based onKalman filter with proportional integral-based error adjust-mentrdquo Journal of Power Sources vol 365 pp 308ndash319 2017

[24] F Yang Y Xing D Wang and K-L Tsui ldquoA comparativestudy of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dy-namic loading profilerdquo Applied Energy vol 164 pp 387ndash3992016

[25] Y Li C Wang and J Gong ldquoA multi-model probability SOCfusion estimation approach using an improved adaptiveunscented Kalman filter techniquerdquo Energy vol 141pp 1402ndash1415 2017

[26] J Linghu L Kang M Liu X Luo Y Feng and C LuldquoEstimation for state-of-charge of lithium-ion battery basedon an adaptive high-degree cubature Kalman filterrdquo Energyvol 189 Article ID 116204 2019

[27] Q Zhu M Xu W Liu and M Zheng ldquoA state of chargeestimation method for lithium-ion batteries based on frac-tional order adaptive extended Kalman filterrdquo Energyvol 187 Article ID 115880 2019

[28] Z Liu and H He ldquoSensor fault detection and isolation for alithium-ion battery pack in electric vehicles using adaptiveextended Kalman filterrdquo Applied Energy vol 185 pp 2033ndash2044 2017

[29] X Lai Y Zheng and T Sun ldquoA comparative study of dif-ferent equivalent circuit models for estimating state-of-chargeof lithium-ion batteriesrdquo Electrochimica Acta vol 259pp 566ndash577 2018

[30] Y Li J Chen and F Lan ldquoEnhanced online model identi-fication and state of charge estimation for lithium-ion batteryunder noise corrupted measurements by bias compensationrecursive least squaresrdquo Journal of Power Sources vol 456p 227984 2020

[31] V Duang H Bastawrous K ldquoLim K See P Zhang andS Dou ldquoOnline state of charge and model parameters esti-mation of the LiFePO4 battery in electric vehicles usingmultiple adaptive forgetting factors recursive least-squaresrdquoJournal of Power Sources vol 296 pp 215ndash224 2015

[32] Y Tian R Lai X Li L Xiang and J Tian ldquoA combinedmethod for state-of-charge estimation for lithium-ion bat-teries using a long short-term memory network and anadaptive cubature Kalman filterrdquo Applied Energy vol 265p 114789 2020

[33] C Chen R Xiong R Yang W Shen and F Sun ldquoState-of-charge estimation of lithium-ion battery using an improvedneural network model and extended Kalman filterrdquo Journal ofCleaner Production vol 234 pp 1153ndash1164 2019

14 Complexity

listed in Table 2 As we can see from Figures 13 and 14 dueto the introduction of current noise in two cycles thefluctuation of two corresponding SOC estimation errorsincreased We can see that the SOC estimation error by theAEKF method is larger than that of the AEKF-PIDmethod +e results show that the maximum SOC error ofAEKF is greater than 28 under the FUDS cyclemeanwhile the maximum SOC error by AEKF-PID isclose to 2+e MAE and AAE of two methods under twocycles are listed in Table 2 According to the analysisabove the combination of PID feedback unit with AEKFhas advantage over single AEKF in robustness of SOCestimation

To further verify the anti-interference performance ofthe AEKF-PID a sequence of noise with feature of ran-dom non-Gaussian distribution is added to the operationcurrent +e SOC estimation result with non-Gaussiannoise is displayed in Figure 15 We can see that the twoSOC estimation errors with non-Gaussian noise increasecompared to that with Gaussian noise +e results inTable 3 show that the MAE of AEKF is greater than 55and the MAE of AEKF-PID is close to 4 At the end ofdischarge the SOC error of AEKF increases graduallymeanwhile the SOC error of AEKF-PID reaches 3 +eresults show that the basic AEKF-based SOC estimationmethod has better performance in Gaussian noise than

SOC

ReferenceAEKFAEKFndashPID

2500 3000 3500 4000082084086088

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

055

06

065

07

075

08

085

09

095

1

(a)

AEKFAEKFndashPID

2000 3000 4000 5000 6000 7000 8000 90001000Time (s)

SOC

erro

r (

)

ndash5ndash4ndash3ndash2ndash1

012345

(b)

Figure 10 Comparison of SOC estimation under 10A discharge (a) SOC estimation result (b) SOC estimation error

ReferenceAEKFAEKFndashPID

SOC

8500 9000 9500

076077078

1 15 2 25 3 3505Time (s)

0010203040506070809

1

times104

(a)

AEKFAEKFndashPID

SOC

erro

r (

)

50 100 150 200 250142144146148

1 15 2 25 3 3505Time (s)

ndash5ndash4ndash3ndash2ndash1

012345

times104

(b)

Figure 11 Comparison of SOC estimation under DST (a) SOC estimation result (b) SOC estimation error

10 Complexity

SOC

6000 6500 7000

081

082

083

1 15 2 25 3 3505Time (s)

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 3 3505Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 13 Comparison of SOC estimation with Gaussian noise under DST (a) SOC estimation result (b) SOC estimation error

SOC

7500 8000 8500

077078079

08

1 15 2 25 305Time (s)

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)SO

C er

ror (

)

50 100 150 200 250 300076078

08082084

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

1 15 2 25 305Time (s)

AEKFAEKFndashPID

times104

(b)

Figure 12 Comparison of SOC estimation under FUDS (a) SOC estimation result (b) SOC estimation error

Table 1 SOC estimation results

AEKF-PID AEKFMAE () 12422 (DST) 17257 (DST)MAE () 17141 (FUDS) 20093 (FUDS)AAE () 03327 (DST) 07466 (DST)AAE () 05235 (FUDS) 07345 (FUDS)

Complexity 11

Table 2 SOC estimation with Gaussian noise

AEKF-PID AEKFMAE () 17726 (DST) 19795 (DST)MAE () 19924 (FUDS) 28171 (FUDS)AAE () 05101 (DST) 10416 (DST)AAE () 05261 (FUDS) 11209 (FUDS)

1 15 2 25 305Time (s)

SOC

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

SOC

erro

r (

)

1 15 2 25 305Time (s)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 15 Comparison of SOC estimation with non-Gaussian noise (a) SOC estimation result (b) SOC estimation error

7500 8000 8500076077078079

1 15 2 25 305Time (s)

SOC

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 305Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 14 Comparison of SOC estimation with Gaussian noise under FUDS (a) SOC estimation result (b) SOC estimation error

12 Complexity

that in non-Gaussian noise whether or not the PIDfeedback is used

5 Conclusion

Based on the simple one-order ECM a novel method ofcombination of AEKF and PID feedback strategy is pro-posed for LIB SOC estimation +e parameters of the ECMare updated online using the FFRLS method to ensure LIBECM reliability and parameters integrity An improvedAEKF-PID technique is introduced to alleviate the negativeeffects resulting from the ECM error and the measurementerror via replacing the single innovation in feedback loop ofAEKF with an innovation vector-based PID module Toevaluate the performance of the proposed algorithm ex-periment and simulation results under 10A discharge DSTand FUDS cycles are adopted to validate the SOC estimation+e simulation results show that the maximum estimationerror of AEKF-PID is below 06 13 and 18 under 10Adischarge DST and FUDS cycles respectively +e SOCestimation error of AEKF-PID is still below 2 even in aGaussian noise situation +ese results suggest that theproposed AEKF-PID method can provide reliable SOCestimation with high precision and robustness Due to thefact that the local flat region in OCV-SOC fitted curve issusceptible to interference from measurement error ourfuture studies will concentrate on the optimization of OCVestimation

Data Availability

Data are available upon request to the corresponding author

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Zheng Liu proposed the original idea Zheng Liu designedthe novel algorithm Yuan Qiu Chunshan Yang and JianboJi performed and analyzed the experiments together ZhengLiu wrote the original manuscript Zheng Liu and ZhenhuaZhao revised the final manuscript

Acknowledgments

+is work was financially supported by the Guangxi NaturalScience Foundation (2018GXNSFAA281161 and2020GXNSFAA297032)

References

[1] X Hu H Yuan C Zou Z Li and L Zhang ldquoCo-estimationof state of charge and state of health for lithium-ion batteriesbased on fractional-order calculusrdquo IEEE Transactions onVehicular Technology vol 67 no 11 pp 10319ndash10329 2018

[2] X Zhou and C Feng ldquo+e impact of environmental regu-lation on fossil energy consumption in China direct andindirect effectsrdquo Journal of Cleaner Production vol 142pp 3174ndash3183 2017

[3] WWaag C Fleischer and D U Sauer ldquoCritical review of themethods for monitoring of lithium-ion batteries in electricand hybrid vehiclesrdquo Journal of Power Sources vol 258pp 321ndash339 2014

[4] R Xiong Y Zhang J Wang H He S Peng and M PechtldquoLithium-ion battery health prognosis based on a real batterymanagement system used in electric vehiclesrdquo IEEE Trans-actions on Vehicular Technology vol 68 no No 5pp 4110ndash4121 2019

[5] H Fathabadi ldquoPlug-in hybrid electric vehicles (PHEVs)replacing internal combustion engine with clean and re-newable energy based auxiliary power sourcesrdquo IEEETransactions on Power Electronics vol 33 no No 11pp 9611ndash9618 2018

[6] Y Xing W He M Pecht and K L Tsui ldquoState of chargeestimation of lithium-ion batteries using the open-circuitvoltage at various ambient temperaturesrdquo Applied Energyvol 113 pp 106ndash115 2014

[7] N Yang X Zhang and G Li ldquoState of charge estimation forpulse discharge of a LiFePO4 battery by a revised Ahcountingrdquo Electrochimica Acta vol 151 pp 63ndash71 2015

[8] K S Ng C-S Moo Y-P Chen and Y-C Hsieh ldquoEnhancedCoulomb counting method for estimating state-of-charge andstate-of-health of lithium-ion batteriesrdquo Applied Energyvol 86 no 9 pp 1506ndash1511 2009

[9] Z Liu X Dang and B Jing ldquoA novel open circuit voltagebased state of charge estimation for lithium-ion battery bymulti-innovation Kalman filterrdquo IEEE Access vol 7pp 49432ndash49447 2019

[10] J D J Rubio ldquoStability analysis of the modified Levenberg-Marquardt algorithm for the artificial neural network train-ingrdquo IEEE Transactions on Neural Networks and LearningSystems 2020

[11] G Aquino J D J Rubio J Pacheco et al ldquoNovel nonlinearhypothesis for the delta parallel robot modelingrdquo IEEE Accessvol 8 no 1 pp 46324ndash46334 2020

[12] J D J Rubio ldquoSOFMLS online self-organizing fuzzy mod-ified least-squares networkrdquo IEEE Transactions on FuzzySystems vol 17 no 6 pp 1296ndash1309 2009

[13] J A M Campantildea ldquoOn the estimation and control of non-linear systems with parametric uncertainties and noisy out-putsrdquo IEEE Access vol 6 pp 31968ndash31973 2018

[14] S Pramanik and S Anwar ldquoElectrochemical model basedcharge optimization for lithium-ion batteriesrdquo Journal ofPower Sources vol 313 pp 164ndash177 2016

[15] F Sun X Hu Y Zou and S Li ldquoAdaptive unscented Kalmanfiltering for state of charge estimation of a lithium-ion batteryfor electric vehiclesrdquo Energy vol 36 no 5 pp 3531ndash35402014

[16] Z Liu and X Dang ldquoA new method for state of charge andcapacity estimation of lithium-ion battery based on dualstrong tracking adaptive H infinity filterrdquo Mathematical

Table 3 SOC estimation with non-Gaussian noise

AEKF-PID AEKFMAE () 40590 55443AAE () 26074 32427

Complexity 13

Problems in Engineering vol 2018 Article ID 521820518 pages 2018

[17] Y Wang C Zhang and Z Chen ldquoOn-line battery state-of-charge estimation based on an integrated estimatorrdquo AppliedEnergy vol 185 pp 2026ndash2032 2017

[18] R Xiao J Shen X Li W Yan E Pan and Z ChenldquoComparisons of modeling and state of charge estimation forlithium-ion battery based on fractional order and integralorder methodsrdquo Energies vol 9 no No 3 2016

[19] Y Qiu X Li W Chen Z-m Duan and L Yu ldquoState ofcharge estimation of vanadium redox battery based on im-proved extended Kalman filterrdquo ISA Transactions vol 94pp 326ndash337 2019

[20] Y Xu M Hu A Zhou et al ldquoState of charge estimation forlithium-ion batteries based on adaptive dual Kalman filterrdquoAppliedMathematical Modelling vol 77 pp 1255ndash1272 2020

[21] Y Bi and S-Y Choe ldquoAn adaptive sigma-point Kalman filterwith state equality constraints for online state-of-charge es-timation of a Li(NiMnCo)O2Carbon battery using a re-duced-order electrochemical modelrdquo Applied Energyvol 258 p 113925 2020

[22] Z Liu X Dang B Jing and J Ji ldquoA novel model-based stateof charge estimation for lithium-ion battery using adaptiverobust iterative cubature Kalman filterrdquo Electric Power Sys-tems Research vol 177 p 105951 2019

[23] J Wei G Dong and Z Chen ldquoOn-board adaptive model forstate of charge estimation of lithium-ion batteries based onKalman filter with proportional integral-based error adjust-mentrdquo Journal of Power Sources vol 365 pp 308ndash319 2017

[24] F Yang Y Xing D Wang and K-L Tsui ldquoA comparativestudy of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dy-namic loading profilerdquo Applied Energy vol 164 pp 387ndash3992016

[25] Y Li C Wang and J Gong ldquoA multi-model probability SOCfusion estimation approach using an improved adaptiveunscented Kalman filter techniquerdquo Energy vol 141pp 1402ndash1415 2017

[26] J Linghu L Kang M Liu X Luo Y Feng and C LuldquoEstimation for state-of-charge of lithium-ion battery basedon an adaptive high-degree cubature Kalman filterrdquo Energyvol 189 Article ID 116204 2019

[27] Q Zhu M Xu W Liu and M Zheng ldquoA state of chargeestimation method for lithium-ion batteries based on frac-tional order adaptive extended Kalman filterrdquo Energyvol 187 Article ID 115880 2019

[28] Z Liu and H He ldquoSensor fault detection and isolation for alithium-ion battery pack in electric vehicles using adaptiveextended Kalman filterrdquo Applied Energy vol 185 pp 2033ndash2044 2017

[29] X Lai Y Zheng and T Sun ldquoA comparative study of dif-ferent equivalent circuit models for estimating state-of-chargeof lithium-ion batteriesrdquo Electrochimica Acta vol 259pp 566ndash577 2018

[30] Y Li J Chen and F Lan ldquoEnhanced online model identi-fication and state of charge estimation for lithium-ion batteryunder noise corrupted measurements by bias compensationrecursive least squaresrdquo Journal of Power Sources vol 456p 227984 2020

[31] V Duang H Bastawrous K ldquoLim K See P Zhang andS Dou ldquoOnline state of charge and model parameters esti-mation of the LiFePO4 battery in electric vehicles usingmultiple adaptive forgetting factors recursive least-squaresrdquoJournal of Power Sources vol 296 pp 215ndash224 2015

[32] Y Tian R Lai X Li L Xiang and J Tian ldquoA combinedmethod for state-of-charge estimation for lithium-ion bat-teries using a long short-term memory network and anadaptive cubature Kalman filterrdquo Applied Energy vol 265p 114789 2020

[33] C Chen R Xiong R Yang W Shen and F Sun ldquoState-of-charge estimation of lithium-ion battery using an improvedneural network model and extended Kalman filterrdquo Journal ofCleaner Production vol 234 pp 1153ndash1164 2019

14 Complexity

SOC

6000 6500 7000

081

082

083

1 15 2 25 3 3505Time (s)

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 3 3505Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 13 Comparison of SOC estimation with Gaussian noise under DST (a) SOC estimation result (b) SOC estimation error

SOC

7500 8000 8500

077078079

08

1 15 2 25 305Time (s)

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)SO

C er

ror (

)

50 100 150 200 250 300076078

08082084

ndash5

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

5

1 15 2 25 305Time (s)

AEKFAEKFndashPID

times104

(b)

Figure 12 Comparison of SOC estimation under FUDS (a) SOC estimation result (b) SOC estimation error

Table 1 SOC estimation results

AEKF-PID AEKFMAE () 12422 (DST) 17257 (DST)MAE () 17141 (FUDS) 20093 (FUDS)AAE () 03327 (DST) 07466 (DST)AAE () 05235 (FUDS) 07345 (FUDS)

Complexity 11

Table 2 SOC estimation with Gaussian noise

AEKF-PID AEKFMAE () 17726 (DST) 19795 (DST)MAE () 19924 (FUDS) 28171 (FUDS)AAE () 05101 (DST) 10416 (DST)AAE () 05261 (FUDS) 11209 (FUDS)

1 15 2 25 305Time (s)

SOC

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

SOC

erro

r (

)

1 15 2 25 305Time (s)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 15 Comparison of SOC estimation with non-Gaussian noise (a) SOC estimation result (b) SOC estimation error

7500 8000 8500076077078079

1 15 2 25 305Time (s)

SOC

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 305Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 14 Comparison of SOC estimation with Gaussian noise under FUDS (a) SOC estimation result (b) SOC estimation error

12 Complexity

that in non-Gaussian noise whether or not the PIDfeedback is used

5 Conclusion

Based on the simple one-order ECM a novel method ofcombination of AEKF and PID feedback strategy is pro-posed for LIB SOC estimation +e parameters of the ECMare updated online using the FFRLS method to ensure LIBECM reliability and parameters integrity An improvedAEKF-PID technique is introduced to alleviate the negativeeffects resulting from the ECM error and the measurementerror via replacing the single innovation in feedback loop ofAEKF with an innovation vector-based PID module Toevaluate the performance of the proposed algorithm ex-periment and simulation results under 10A discharge DSTand FUDS cycles are adopted to validate the SOC estimation+e simulation results show that the maximum estimationerror of AEKF-PID is below 06 13 and 18 under 10Adischarge DST and FUDS cycles respectively +e SOCestimation error of AEKF-PID is still below 2 even in aGaussian noise situation +ese results suggest that theproposed AEKF-PID method can provide reliable SOCestimation with high precision and robustness Due to thefact that the local flat region in OCV-SOC fitted curve issusceptible to interference from measurement error ourfuture studies will concentrate on the optimization of OCVestimation

Data Availability

Data are available upon request to the corresponding author

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Zheng Liu proposed the original idea Zheng Liu designedthe novel algorithm Yuan Qiu Chunshan Yang and JianboJi performed and analyzed the experiments together ZhengLiu wrote the original manuscript Zheng Liu and ZhenhuaZhao revised the final manuscript

Acknowledgments

+is work was financially supported by the Guangxi NaturalScience Foundation (2018GXNSFAA281161 and2020GXNSFAA297032)

References

[1] X Hu H Yuan C Zou Z Li and L Zhang ldquoCo-estimationof state of charge and state of health for lithium-ion batteriesbased on fractional-order calculusrdquo IEEE Transactions onVehicular Technology vol 67 no 11 pp 10319ndash10329 2018

[2] X Zhou and C Feng ldquo+e impact of environmental regu-lation on fossil energy consumption in China direct andindirect effectsrdquo Journal of Cleaner Production vol 142pp 3174ndash3183 2017

[3] WWaag C Fleischer and D U Sauer ldquoCritical review of themethods for monitoring of lithium-ion batteries in electricand hybrid vehiclesrdquo Journal of Power Sources vol 258pp 321ndash339 2014

[4] R Xiong Y Zhang J Wang H He S Peng and M PechtldquoLithium-ion battery health prognosis based on a real batterymanagement system used in electric vehiclesrdquo IEEE Trans-actions on Vehicular Technology vol 68 no No 5pp 4110ndash4121 2019

[5] H Fathabadi ldquoPlug-in hybrid electric vehicles (PHEVs)replacing internal combustion engine with clean and re-newable energy based auxiliary power sourcesrdquo IEEETransactions on Power Electronics vol 33 no No 11pp 9611ndash9618 2018

[6] Y Xing W He M Pecht and K L Tsui ldquoState of chargeestimation of lithium-ion batteries using the open-circuitvoltage at various ambient temperaturesrdquo Applied Energyvol 113 pp 106ndash115 2014

[7] N Yang X Zhang and G Li ldquoState of charge estimation forpulse discharge of a LiFePO4 battery by a revised Ahcountingrdquo Electrochimica Acta vol 151 pp 63ndash71 2015

[8] K S Ng C-S Moo Y-P Chen and Y-C Hsieh ldquoEnhancedCoulomb counting method for estimating state-of-charge andstate-of-health of lithium-ion batteriesrdquo Applied Energyvol 86 no 9 pp 1506ndash1511 2009

[9] Z Liu X Dang and B Jing ldquoA novel open circuit voltagebased state of charge estimation for lithium-ion battery bymulti-innovation Kalman filterrdquo IEEE Access vol 7pp 49432ndash49447 2019

[10] J D J Rubio ldquoStability analysis of the modified Levenberg-Marquardt algorithm for the artificial neural network train-ingrdquo IEEE Transactions on Neural Networks and LearningSystems 2020

[11] G Aquino J D J Rubio J Pacheco et al ldquoNovel nonlinearhypothesis for the delta parallel robot modelingrdquo IEEE Accessvol 8 no 1 pp 46324ndash46334 2020

[12] J D J Rubio ldquoSOFMLS online self-organizing fuzzy mod-ified least-squares networkrdquo IEEE Transactions on FuzzySystems vol 17 no 6 pp 1296ndash1309 2009

[13] J A M Campantildea ldquoOn the estimation and control of non-linear systems with parametric uncertainties and noisy out-putsrdquo IEEE Access vol 6 pp 31968ndash31973 2018

[14] S Pramanik and S Anwar ldquoElectrochemical model basedcharge optimization for lithium-ion batteriesrdquo Journal ofPower Sources vol 313 pp 164ndash177 2016

[15] F Sun X Hu Y Zou and S Li ldquoAdaptive unscented Kalmanfiltering for state of charge estimation of a lithium-ion batteryfor electric vehiclesrdquo Energy vol 36 no 5 pp 3531ndash35402014

[16] Z Liu and X Dang ldquoA new method for state of charge andcapacity estimation of lithium-ion battery based on dualstrong tracking adaptive H infinity filterrdquo Mathematical

Table 3 SOC estimation with non-Gaussian noise

AEKF-PID AEKFMAE () 40590 55443AAE () 26074 32427

Complexity 13

Problems in Engineering vol 2018 Article ID 521820518 pages 2018

[17] Y Wang C Zhang and Z Chen ldquoOn-line battery state-of-charge estimation based on an integrated estimatorrdquo AppliedEnergy vol 185 pp 2026ndash2032 2017

[18] R Xiao J Shen X Li W Yan E Pan and Z ChenldquoComparisons of modeling and state of charge estimation forlithium-ion battery based on fractional order and integralorder methodsrdquo Energies vol 9 no No 3 2016

[19] Y Qiu X Li W Chen Z-m Duan and L Yu ldquoState ofcharge estimation of vanadium redox battery based on im-proved extended Kalman filterrdquo ISA Transactions vol 94pp 326ndash337 2019

[20] Y Xu M Hu A Zhou et al ldquoState of charge estimation forlithium-ion batteries based on adaptive dual Kalman filterrdquoAppliedMathematical Modelling vol 77 pp 1255ndash1272 2020

[21] Y Bi and S-Y Choe ldquoAn adaptive sigma-point Kalman filterwith state equality constraints for online state-of-charge es-timation of a Li(NiMnCo)O2Carbon battery using a re-duced-order electrochemical modelrdquo Applied Energyvol 258 p 113925 2020

[22] Z Liu X Dang B Jing and J Ji ldquoA novel model-based stateof charge estimation for lithium-ion battery using adaptiverobust iterative cubature Kalman filterrdquo Electric Power Sys-tems Research vol 177 p 105951 2019

[23] J Wei G Dong and Z Chen ldquoOn-board adaptive model forstate of charge estimation of lithium-ion batteries based onKalman filter with proportional integral-based error adjust-mentrdquo Journal of Power Sources vol 365 pp 308ndash319 2017

[24] F Yang Y Xing D Wang and K-L Tsui ldquoA comparativestudy of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dy-namic loading profilerdquo Applied Energy vol 164 pp 387ndash3992016

[25] Y Li C Wang and J Gong ldquoA multi-model probability SOCfusion estimation approach using an improved adaptiveunscented Kalman filter techniquerdquo Energy vol 141pp 1402ndash1415 2017

[26] J Linghu L Kang M Liu X Luo Y Feng and C LuldquoEstimation for state-of-charge of lithium-ion battery basedon an adaptive high-degree cubature Kalman filterrdquo Energyvol 189 Article ID 116204 2019

[27] Q Zhu M Xu W Liu and M Zheng ldquoA state of chargeestimation method for lithium-ion batteries based on frac-tional order adaptive extended Kalman filterrdquo Energyvol 187 Article ID 115880 2019

[28] Z Liu and H He ldquoSensor fault detection and isolation for alithium-ion battery pack in electric vehicles using adaptiveextended Kalman filterrdquo Applied Energy vol 185 pp 2033ndash2044 2017

[29] X Lai Y Zheng and T Sun ldquoA comparative study of dif-ferent equivalent circuit models for estimating state-of-chargeof lithium-ion batteriesrdquo Electrochimica Acta vol 259pp 566ndash577 2018

[30] Y Li J Chen and F Lan ldquoEnhanced online model identi-fication and state of charge estimation for lithium-ion batteryunder noise corrupted measurements by bias compensationrecursive least squaresrdquo Journal of Power Sources vol 456p 227984 2020

[31] V Duang H Bastawrous K ldquoLim K See P Zhang andS Dou ldquoOnline state of charge and model parameters esti-mation of the LiFePO4 battery in electric vehicles usingmultiple adaptive forgetting factors recursive least-squaresrdquoJournal of Power Sources vol 296 pp 215ndash224 2015

[32] Y Tian R Lai X Li L Xiang and J Tian ldquoA combinedmethod for state-of-charge estimation for lithium-ion bat-teries using a long short-term memory network and anadaptive cubature Kalman filterrdquo Applied Energy vol 265p 114789 2020

[33] C Chen R Xiong R Yang W Shen and F Sun ldquoState-of-charge estimation of lithium-ion battery using an improvedneural network model and extended Kalman filterrdquo Journal ofCleaner Production vol 234 pp 1153ndash1164 2019

14 Complexity

Table 2 SOC estimation with Gaussian noise

AEKF-PID AEKFMAE () 17726 (DST) 19795 (DST)MAE () 19924 (FUDS) 28171 (FUDS)AAE () 05101 (DST) 10416 (DST)AAE () 05261 (FUDS) 11209 (FUDS)

1 15 2 25 305Time (s)

SOC

0

02

04

06

08

1

ReferenceAEKFAEKFndashPID

times104

(a)

SOC

erro

r (

)

1 15 2 25 305Time (s)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 15 Comparison of SOC estimation with non-Gaussian noise (a) SOC estimation result (b) SOC estimation error

7500 8000 8500076077078079

1 15 2 25 305Time (s)

SOC

0

01

02

03

04

05

06

07

08

09

1

ReferenceAEKFAEKFndashPID

times104

(a)

1 15 2 25 305Time (s)

SOC

erro

r (

)

ndash5

0

5

AEKFAEKFndashPID

times104

(b)

Figure 14 Comparison of SOC estimation with Gaussian noise under FUDS (a) SOC estimation result (b) SOC estimation error

12 Complexity

that in non-Gaussian noise whether or not the PIDfeedback is used

5 Conclusion

Based on the simple one-order ECM a novel method ofcombination of AEKF and PID feedback strategy is pro-posed for LIB SOC estimation +e parameters of the ECMare updated online using the FFRLS method to ensure LIBECM reliability and parameters integrity An improvedAEKF-PID technique is introduced to alleviate the negativeeffects resulting from the ECM error and the measurementerror via replacing the single innovation in feedback loop ofAEKF with an innovation vector-based PID module Toevaluate the performance of the proposed algorithm ex-periment and simulation results under 10A discharge DSTand FUDS cycles are adopted to validate the SOC estimation+e simulation results show that the maximum estimationerror of AEKF-PID is below 06 13 and 18 under 10Adischarge DST and FUDS cycles respectively +e SOCestimation error of AEKF-PID is still below 2 even in aGaussian noise situation +ese results suggest that theproposed AEKF-PID method can provide reliable SOCestimation with high precision and robustness Due to thefact that the local flat region in OCV-SOC fitted curve issusceptible to interference from measurement error ourfuture studies will concentrate on the optimization of OCVestimation

Data Availability

Data are available upon request to the corresponding author

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Zheng Liu proposed the original idea Zheng Liu designedthe novel algorithm Yuan Qiu Chunshan Yang and JianboJi performed and analyzed the experiments together ZhengLiu wrote the original manuscript Zheng Liu and ZhenhuaZhao revised the final manuscript

Acknowledgments

+is work was financially supported by the Guangxi NaturalScience Foundation (2018GXNSFAA281161 and2020GXNSFAA297032)

References

[1] X Hu H Yuan C Zou Z Li and L Zhang ldquoCo-estimationof state of charge and state of health for lithium-ion batteriesbased on fractional-order calculusrdquo IEEE Transactions onVehicular Technology vol 67 no 11 pp 10319ndash10329 2018

[2] X Zhou and C Feng ldquo+e impact of environmental regu-lation on fossil energy consumption in China direct andindirect effectsrdquo Journal of Cleaner Production vol 142pp 3174ndash3183 2017

[3] WWaag C Fleischer and D U Sauer ldquoCritical review of themethods for monitoring of lithium-ion batteries in electricand hybrid vehiclesrdquo Journal of Power Sources vol 258pp 321ndash339 2014

[4] R Xiong Y Zhang J Wang H He S Peng and M PechtldquoLithium-ion battery health prognosis based on a real batterymanagement system used in electric vehiclesrdquo IEEE Trans-actions on Vehicular Technology vol 68 no No 5pp 4110ndash4121 2019

[5] H Fathabadi ldquoPlug-in hybrid electric vehicles (PHEVs)replacing internal combustion engine with clean and re-newable energy based auxiliary power sourcesrdquo IEEETransactions on Power Electronics vol 33 no No 11pp 9611ndash9618 2018

[6] Y Xing W He M Pecht and K L Tsui ldquoState of chargeestimation of lithium-ion batteries using the open-circuitvoltage at various ambient temperaturesrdquo Applied Energyvol 113 pp 106ndash115 2014

[7] N Yang X Zhang and G Li ldquoState of charge estimation forpulse discharge of a LiFePO4 battery by a revised Ahcountingrdquo Electrochimica Acta vol 151 pp 63ndash71 2015

[8] K S Ng C-S Moo Y-P Chen and Y-C Hsieh ldquoEnhancedCoulomb counting method for estimating state-of-charge andstate-of-health of lithium-ion batteriesrdquo Applied Energyvol 86 no 9 pp 1506ndash1511 2009

[9] Z Liu X Dang and B Jing ldquoA novel open circuit voltagebased state of charge estimation for lithium-ion battery bymulti-innovation Kalman filterrdquo IEEE Access vol 7pp 49432ndash49447 2019

[10] J D J Rubio ldquoStability analysis of the modified Levenberg-Marquardt algorithm for the artificial neural network train-ingrdquo IEEE Transactions on Neural Networks and LearningSystems 2020

[11] G Aquino J D J Rubio J Pacheco et al ldquoNovel nonlinearhypothesis for the delta parallel robot modelingrdquo IEEE Accessvol 8 no 1 pp 46324ndash46334 2020

[12] J D J Rubio ldquoSOFMLS online self-organizing fuzzy mod-ified least-squares networkrdquo IEEE Transactions on FuzzySystems vol 17 no 6 pp 1296ndash1309 2009

[13] J A M Campantildea ldquoOn the estimation and control of non-linear systems with parametric uncertainties and noisy out-putsrdquo IEEE Access vol 6 pp 31968ndash31973 2018

[14] S Pramanik and S Anwar ldquoElectrochemical model basedcharge optimization for lithium-ion batteriesrdquo Journal ofPower Sources vol 313 pp 164ndash177 2016

[15] F Sun X Hu Y Zou and S Li ldquoAdaptive unscented Kalmanfiltering for state of charge estimation of a lithium-ion batteryfor electric vehiclesrdquo Energy vol 36 no 5 pp 3531ndash35402014

[16] Z Liu and X Dang ldquoA new method for state of charge andcapacity estimation of lithium-ion battery based on dualstrong tracking adaptive H infinity filterrdquo Mathematical

Table 3 SOC estimation with non-Gaussian noise

AEKF-PID AEKFMAE () 40590 55443AAE () 26074 32427

Complexity 13

Problems in Engineering vol 2018 Article ID 521820518 pages 2018

[17] Y Wang C Zhang and Z Chen ldquoOn-line battery state-of-charge estimation based on an integrated estimatorrdquo AppliedEnergy vol 185 pp 2026ndash2032 2017

[18] R Xiao J Shen X Li W Yan E Pan and Z ChenldquoComparisons of modeling and state of charge estimation forlithium-ion battery based on fractional order and integralorder methodsrdquo Energies vol 9 no No 3 2016

[19] Y Qiu X Li W Chen Z-m Duan and L Yu ldquoState ofcharge estimation of vanadium redox battery based on im-proved extended Kalman filterrdquo ISA Transactions vol 94pp 326ndash337 2019

[20] Y Xu M Hu A Zhou et al ldquoState of charge estimation forlithium-ion batteries based on adaptive dual Kalman filterrdquoAppliedMathematical Modelling vol 77 pp 1255ndash1272 2020

[21] Y Bi and S-Y Choe ldquoAn adaptive sigma-point Kalman filterwith state equality constraints for online state-of-charge es-timation of a Li(NiMnCo)O2Carbon battery using a re-duced-order electrochemical modelrdquo Applied Energyvol 258 p 113925 2020

[22] Z Liu X Dang B Jing and J Ji ldquoA novel model-based stateof charge estimation for lithium-ion battery using adaptiverobust iterative cubature Kalman filterrdquo Electric Power Sys-tems Research vol 177 p 105951 2019

[23] J Wei G Dong and Z Chen ldquoOn-board adaptive model forstate of charge estimation of lithium-ion batteries based onKalman filter with proportional integral-based error adjust-mentrdquo Journal of Power Sources vol 365 pp 308ndash319 2017

[24] F Yang Y Xing D Wang and K-L Tsui ldquoA comparativestudy of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dy-namic loading profilerdquo Applied Energy vol 164 pp 387ndash3992016

[25] Y Li C Wang and J Gong ldquoA multi-model probability SOCfusion estimation approach using an improved adaptiveunscented Kalman filter techniquerdquo Energy vol 141pp 1402ndash1415 2017

[26] J Linghu L Kang M Liu X Luo Y Feng and C LuldquoEstimation for state-of-charge of lithium-ion battery basedon an adaptive high-degree cubature Kalman filterrdquo Energyvol 189 Article ID 116204 2019

[27] Q Zhu M Xu W Liu and M Zheng ldquoA state of chargeestimation method for lithium-ion batteries based on frac-tional order adaptive extended Kalman filterrdquo Energyvol 187 Article ID 115880 2019

[28] Z Liu and H He ldquoSensor fault detection and isolation for alithium-ion battery pack in electric vehicles using adaptiveextended Kalman filterrdquo Applied Energy vol 185 pp 2033ndash2044 2017

[29] X Lai Y Zheng and T Sun ldquoA comparative study of dif-ferent equivalent circuit models for estimating state-of-chargeof lithium-ion batteriesrdquo Electrochimica Acta vol 259pp 566ndash577 2018

[30] Y Li J Chen and F Lan ldquoEnhanced online model identi-fication and state of charge estimation for lithium-ion batteryunder noise corrupted measurements by bias compensationrecursive least squaresrdquo Journal of Power Sources vol 456p 227984 2020

[31] V Duang H Bastawrous K ldquoLim K See P Zhang andS Dou ldquoOnline state of charge and model parameters esti-mation of the LiFePO4 battery in electric vehicles usingmultiple adaptive forgetting factors recursive least-squaresrdquoJournal of Power Sources vol 296 pp 215ndash224 2015

[32] Y Tian R Lai X Li L Xiang and J Tian ldquoA combinedmethod for state-of-charge estimation for lithium-ion bat-teries using a long short-term memory network and anadaptive cubature Kalman filterrdquo Applied Energy vol 265p 114789 2020

[33] C Chen R Xiong R Yang W Shen and F Sun ldquoState-of-charge estimation of lithium-ion battery using an improvedneural network model and extended Kalman filterrdquo Journal ofCleaner Production vol 234 pp 1153ndash1164 2019

14 Complexity

that in non-Gaussian noise whether or not the PIDfeedback is used

5 Conclusion

Based on the simple one-order ECM a novel method ofcombination of AEKF and PID feedback strategy is pro-posed for LIB SOC estimation +e parameters of the ECMare updated online using the FFRLS method to ensure LIBECM reliability and parameters integrity An improvedAEKF-PID technique is introduced to alleviate the negativeeffects resulting from the ECM error and the measurementerror via replacing the single innovation in feedback loop ofAEKF with an innovation vector-based PID module Toevaluate the performance of the proposed algorithm ex-periment and simulation results under 10A discharge DSTand FUDS cycles are adopted to validate the SOC estimation+e simulation results show that the maximum estimationerror of AEKF-PID is below 06 13 and 18 under 10Adischarge DST and FUDS cycles respectively +e SOCestimation error of AEKF-PID is still below 2 even in aGaussian noise situation +ese results suggest that theproposed AEKF-PID method can provide reliable SOCestimation with high precision and robustness Due to thefact that the local flat region in OCV-SOC fitted curve issusceptible to interference from measurement error ourfuture studies will concentrate on the optimization of OCVestimation

Data Availability

Data are available upon request to the corresponding author

Conflicts of Interest

+e authors declare no conflicts of interest

Authorsrsquo Contributions

Zheng Liu proposed the original idea Zheng Liu designedthe novel algorithm Yuan Qiu Chunshan Yang and JianboJi performed and analyzed the experiments together ZhengLiu wrote the original manuscript Zheng Liu and ZhenhuaZhao revised the final manuscript

Acknowledgments

+is work was financially supported by the Guangxi NaturalScience Foundation (2018GXNSFAA281161 and2020GXNSFAA297032)

References

[1] X Hu H Yuan C Zou Z Li and L Zhang ldquoCo-estimationof state of charge and state of health for lithium-ion batteriesbased on fractional-order calculusrdquo IEEE Transactions onVehicular Technology vol 67 no 11 pp 10319ndash10329 2018

[2] X Zhou and C Feng ldquo+e impact of environmental regu-lation on fossil energy consumption in China direct andindirect effectsrdquo Journal of Cleaner Production vol 142pp 3174ndash3183 2017

[3] WWaag C Fleischer and D U Sauer ldquoCritical review of themethods for monitoring of lithium-ion batteries in electricand hybrid vehiclesrdquo Journal of Power Sources vol 258pp 321ndash339 2014

[4] R Xiong Y Zhang J Wang H He S Peng and M PechtldquoLithium-ion battery health prognosis based on a real batterymanagement system used in electric vehiclesrdquo IEEE Trans-actions on Vehicular Technology vol 68 no No 5pp 4110ndash4121 2019

[5] H Fathabadi ldquoPlug-in hybrid electric vehicles (PHEVs)replacing internal combustion engine with clean and re-newable energy based auxiliary power sourcesrdquo IEEETransactions on Power Electronics vol 33 no No 11pp 9611ndash9618 2018

[6] Y Xing W He M Pecht and K L Tsui ldquoState of chargeestimation of lithium-ion batteries using the open-circuitvoltage at various ambient temperaturesrdquo Applied Energyvol 113 pp 106ndash115 2014

[7] N Yang X Zhang and G Li ldquoState of charge estimation forpulse discharge of a LiFePO4 battery by a revised Ahcountingrdquo Electrochimica Acta vol 151 pp 63ndash71 2015

[8] K S Ng C-S Moo Y-P Chen and Y-C Hsieh ldquoEnhancedCoulomb counting method for estimating state-of-charge andstate-of-health of lithium-ion batteriesrdquo Applied Energyvol 86 no 9 pp 1506ndash1511 2009

[9] Z Liu X Dang and B Jing ldquoA novel open circuit voltagebased state of charge estimation for lithium-ion battery bymulti-innovation Kalman filterrdquo IEEE Access vol 7pp 49432ndash49447 2019

[10] J D J Rubio ldquoStability analysis of the modified Levenberg-Marquardt algorithm for the artificial neural network train-ingrdquo IEEE Transactions on Neural Networks and LearningSystems 2020

[11] G Aquino J D J Rubio J Pacheco et al ldquoNovel nonlinearhypothesis for the delta parallel robot modelingrdquo IEEE Accessvol 8 no 1 pp 46324ndash46334 2020

[12] J D J Rubio ldquoSOFMLS online self-organizing fuzzy mod-ified least-squares networkrdquo IEEE Transactions on FuzzySystems vol 17 no 6 pp 1296ndash1309 2009

[13] J A M Campantildea ldquoOn the estimation and control of non-linear systems with parametric uncertainties and noisy out-putsrdquo IEEE Access vol 6 pp 31968ndash31973 2018

[14] S Pramanik and S Anwar ldquoElectrochemical model basedcharge optimization for lithium-ion batteriesrdquo Journal ofPower Sources vol 313 pp 164ndash177 2016

[15] F Sun X Hu Y Zou and S Li ldquoAdaptive unscented Kalmanfiltering for state of charge estimation of a lithium-ion batteryfor electric vehiclesrdquo Energy vol 36 no 5 pp 3531ndash35402014

[16] Z Liu and X Dang ldquoA new method for state of charge andcapacity estimation of lithium-ion battery based on dualstrong tracking adaptive H infinity filterrdquo Mathematical

Table 3 SOC estimation with non-Gaussian noise

AEKF-PID AEKFMAE () 40590 55443AAE () 26074 32427

Complexity 13

Problems in Engineering vol 2018 Article ID 521820518 pages 2018

[17] Y Wang C Zhang and Z Chen ldquoOn-line battery state-of-charge estimation based on an integrated estimatorrdquo AppliedEnergy vol 185 pp 2026ndash2032 2017

[18] R Xiao J Shen X Li W Yan E Pan and Z ChenldquoComparisons of modeling and state of charge estimation forlithium-ion battery based on fractional order and integralorder methodsrdquo Energies vol 9 no No 3 2016

[19] Y Qiu X Li W Chen Z-m Duan and L Yu ldquoState ofcharge estimation of vanadium redox battery based on im-proved extended Kalman filterrdquo ISA Transactions vol 94pp 326ndash337 2019

[20] Y Xu M Hu A Zhou et al ldquoState of charge estimation forlithium-ion batteries based on adaptive dual Kalman filterrdquoAppliedMathematical Modelling vol 77 pp 1255ndash1272 2020

[21] Y Bi and S-Y Choe ldquoAn adaptive sigma-point Kalman filterwith state equality constraints for online state-of-charge es-timation of a Li(NiMnCo)O2Carbon battery using a re-duced-order electrochemical modelrdquo Applied Energyvol 258 p 113925 2020

[22] Z Liu X Dang B Jing and J Ji ldquoA novel model-based stateof charge estimation for lithium-ion battery using adaptiverobust iterative cubature Kalman filterrdquo Electric Power Sys-tems Research vol 177 p 105951 2019

[23] J Wei G Dong and Z Chen ldquoOn-board adaptive model forstate of charge estimation of lithium-ion batteries based onKalman filter with proportional integral-based error adjust-mentrdquo Journal of Power Sources vol 365 pp 308ndash319 2017

[24] F Yang Y Xing D Wang and K-L Tsui ldquoA comparativestudy of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dy-namic loading profilerdquo Applied Energy vol 164 pp 387ndash3992016

[25] Y Li C Wang and J Gong ldquoA multi-model probability SOCfusion estimation approach using an improved adaptiveunscented Kalman filter techniquerdquo Energy vol 141pp 1402ndash1415 2017

[26] J Linghu L Kang M Liu X Luo Y Feng and C LuldquoEstimation for state-of-charge of lithium-ion battery basedon an adaptive high-degree cubature Kalman filterrdquo Energyvol 189 Article ID 116204 2019

[27] Q Zhu M Xu W Liu and M Zheng ldquoA state of chargeestimation method for lithium-ion batteries based on frac-tional order adaptive extended Kalman filterrdquo Energyvol 187 Article ID 115880 2019

[28] Z Liu and H He ldquoSensor fault detection and isolation for alithium-ion battery pack in electric vehicles using adaptiveextended Kalman filterrdquo Applied Energy vol 185 pp 2033ndash2044 2017

[29] X Lai Y Zheng and T Sun ldquoA comparative study of dif-ferent equivalent circuit models for estimating state-of-chargeof lithium-ion batteriesrdquo Electrochimica Acta vol 259pp 566ndash577 2018

[30] Y Li J Chen and F Lan ldquoEnhanced online model identi-fication and state of charge estimation for lithium-ion batteryunder noise corrupted measurements by bias compensationrecursive least squaresrdquo Journal of Power Sources vol 456p 227984 2020

[31] V Duang H Bastawrous K ldquoLim K See P Zhang andS Dou ldquoOnline state of charge and model parameters esti-mation of the LiFePO4 battery in electric vehicles usingmultiple adaptive forgetting factors recursive least-squaresrdquoJournal of Power Sources vol 296 pp 215ndash224 2015

[32] Y Tian R Lai X Li L Xiang and J Tian ldquoA combinedmethod for state-of-charge estimation for lithium-ion bat-teries using a long short-term memory network and anadaptive cubature Kalman filterrdquo Applied Energy vol 265p 114789 2020

[33] C Chen R Xiong R Yang W Shen and F Sun ldquoState-of-charge estimation of lithium-ion battery using an improvedneural network model and extended Kalman filterrdquo Journal ofCleaner Production vol 234 pp 1153ndash1164 2019

14 Complexity

Problems in Engineering vol 2018 Article ID 521820518 pages 2018

[17] Y Wang C Zhang and Z Chen ldquoOn-line battery state-of-charge estimation based on an integrated estimatorrdquo AppliedEnergy vol 185 pp 2026ndash2032 2017

[18] R Xiao J Shen X Li W Yan E Pan and Z ChenldquoComparisons of modeling and state of charge estimation forlithium-ion battery based on fractional order and integralorder methodsrdquo Energies vol 9 no No 3 2016

[19] Y Qiu X Li W Chen Z-m Duan and L Yu ldquoState ofcharge estimation of vanadium redox battery based on im-proved extended Kalman filterrdquo ISA Transactions vol 94pp 326ndash337 2019

[20] Y Xu M Hu A Zhou et al ldquoState of charge estimation forlithium-ion batteries based on adaptive dual Kalman filterrdquoAppliedMathematical Modelling vol 77 pp 1255ndash1272 2020

[21] Y Bi and S-Y Choe ldquoAn adaptive sigma-point Kalman filterwith state equality constraints for online state-of-charge es-timation of a Li(NiMnCo)O2Carbon battery using a re-duced-order electrochemical modelrdquo Applied Energyvol 258 p 113925 2020

[22] Z Liu X Dang B Jing and J Ji ldquoA novel model-based stateof charge estimation for lithium-ion battery using adaptiverobust iterative cubature Kalman filterrdquo Electric Power Sys-tems Research vol 177 p 105951 2019

[23] J Wei G Dong and Z Chen ldquoOn-board adaptive model forstate of charge estimation of lithium-ion batteries based onKalman filter with proportional integral-based error adjust-mentrdquo Journal of Power Sources vol 365 pp 308ndash319 2017

[24] F Yang Y Xing D Wang and K-L Tsui ldquoA comparativestudy of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dy-namic loading profilerdquo Applied Energy vol 164 pp 387ndash3992016

[25] Y Li C Wang and J Gong ldquoA multi-model probability SOCfusion estimation approach using an improved adaptiveunscented Kalman filter techniquerdquo Energy vol 141pp 1402ndash1415 2017

[26] J Linghu L Kang M Liu X Luo Y Feng and C LuldquoEstimation for state-of-charge of lithium-ion battery basedon an adaptive high-degree cubature Kalman filterrdquo Energyvol 189 Article ID 116204 2019

[27] Q Zhu M Xu W Liu and M Zheng ldquoA state of chargeestimation method for lithium-ion batteries based on frac-tional order adaptive extended Kalman filterrdquo Energyvol 187 Article ID 115880 2019

[28] Z Liu and H He ldquoSensor fault detection and isolation for alithium-ion battery pack in electric vehicles using adaptiveextended Kalman filterrdquo Applied Energy vol 185 pp 2033ndash2044 2017

[29] X Lai Y Zheng and T Sun ldquoA comparative study of dif-ferent equivalent circuit models for estimating state-of-chargeof lithium-ion batteriesrdquo Electrochimica Acta vol 259pp 566ndash577 2018

[30] Y Li J Chen and F Lan ldquoEnhanced online model identi-fication and state of charge estimation for lithium-ion batteryunder noise corrupted measurements by bias compensationrecursive least squaresrdquo Journal of Power Sources vol 456p 227984 2020

[31] V Duang H Bastawrous K ldquoLim K See P Zhang andS Dou ldquoOnline state of charge and model parameters esti-mation of the LiFePO4 battery in electric vehicles usingmultiple adaptive forgetting factors recursive least-squaresrdquoJournal of Power Sources vol 296 pp 215ndash224 2015

[32] Y Tian R Lai X Li L Xiang and J Tian ldquoA combinedmethod for state-of-charge estimation for lithium-ion bat-teries using a long short-term memory network and anadaptive cubature Kalman filterrdquo Applied Energy vol 265p 114789 2020

[33] C Chen R Xiong R Yang W Shen and F Sun ldquoState-of-charge estimation of lithium-ion battery using an improvedneural network model and extended Kalman filterrdquo Journal ofCleaner Production vol 234 pp 1153ndash1164 2019

14 Complexity