a comparative study of three model-based algorithms for...

Applied Energy 164 (2016) 387–399

Contents lists available at ScienceDirect

Applied Energy

journal homepage: www.elsevier .com/locate /apenergy

A comparative study of three model-based algorithms for estimatingstate-of-charge of lithium-ion batteries under a new combined dynamicloading profile

http://dx.doi.org/10.1016/j.apenergy.2015.11.0720306-2619/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected] (F. Yang), [email protected].

edu (Y. Xing), [email protected] (D. Wang), [email protected](K.-L. Tsui).

Fangfang Yang a,⇑, Yinjiao Xing b, Dong Wang a, Kwok-Leung Tsui aaDepartment of Systems Engineering and Engineering Management, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, ChinabCenter for Advanced Life Cycle Engineering, University of Maryland, College Park, MD 20742, USA

h i g h l i g h t s

� Three different model-based filtering algorithms for SOC estimation are compared.� A combined dynamic loading profile is proposed to evaluate the three algorithms.� Robustness against uncertainty of initial states of SOC estimators are investigated.� Battery capacity degradation is considered in SOC estimation.

a r t i c l e i n f o

Article history:Received 4 May 2015Received in revised form 14 October 2015Accepted 26 November 2015

Keywords:State of chargeLithium-ion batteriesExtended Kalman filterUnscented Kalman filterParticle filterDegradation

a b s t r a c t

Accurate state-of-charge (SOC) estimation is critical for the safety and reliability of battery managementsystems in electric vehicles. Because SOC cannot be directly measured and SOC estimation is affected bymany factors, such as ambient temperature, battery aging, and current rate, a robust SOC estimationapproach is necessary to be developed so as to deal with time-varying and nonlinear battery systems.In this paper, three popular model-based filtering algorithms, including extended Kalman filter,unscented Kalman filter, and particle filter, are respectively used to estimate SOC and their performancesregarding to tracking accuracy, computation time, robustness against uncertainty of initial values of SOC,and battery degradation, are compared. To evaluate the performances of these algorithms, a new com-bined dynamic loading profile composed of the dynamic stress test, the federal urban driving scheduleand the US06 is proposed. The comparison results showed that the unscented Kalman filter is the mostrobust to different initial values of SOC, while the particle filter owns the fastest convergence abilitywhen an initial guess of SOC is far from a true initial SOC.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

With the increasing development of electric vehicles (EVs),lithium-ion batteries are gradually becoming dominant in energystorage systems due to their advantages, such as high energy andpower density, long lifespan [1]. To guarantee safe and reliable bat-tery operation, a battery management system (BMS) is required tomonitor and control lithium-ion batteries so as to provide a longerlifetime of a battery [2]. State of charge (SOC) estimation is one ofthe main concerns in the BMS. The SOC quantifies remaining

charge of a battery at the current cycle and indicates how longthe battery will sustain before the battery is recharged [3]. It canbe regarded as a ‘‘Gas Gauge” or ‘‘Fuel Gauge” function by analogyto a fuel tank in a car [4]. A precise automotive fuel gauge willrelieve drivers’ anxious about an unexpected fuel range. In addi-tion, accurate estimation of SOC is strongly helpful to determinethe end of charge and discharge. And it will effectively keep abattery operating within desired operation limits and slow downbattery failures caused by over-charging and over-discharging.However, SOC cannot be directly measured. Even though SOCcan be estimated from some measurable parameters, such ascurrent and voltage, an explicit relationship is not concluded. Inother words, voltage and current can only be used to provide arough indication of SOC. To achieve a higher SOC estimationaccuracy, other factors in operation conditions, such as ambient

http://crossmark.crossref.org/dialog/?doi=10.1016/j.apenergy.2015.11.072&domain=pdfhttp://dx.doi.org/10.1016/j.apenergy.2015.11.072mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.apenergy.2015.11.072http://www.sciencedirect.com/science/journal/03062619http://www.elsevier.com/locate/apenergy

388 F. Yang et al. / Applied Energy 164 (2016) 387–399

temperature, battery voltage and temperature, charge anddischarge rate, self-discharge rate should be taken into considera-tion [5].

Currently, existing SOC estimation algorithms can be dividedinto non-model based approaches mainly including Ampere-Hourintegral, open-circuit voltage (OCV), machine learning methodsand model-based approaches. By directly accumulating batterycurrent over time, Ampere-Hour integral methods can give approx-imate SOC estimation [6–8]. However, errors caused by inaccurateinitialization of SOC, low precision of current sensor, and dis-cretization of sample time are inevitable and difficult to be reducedbecause of open-loop estimation [9]. In practice, an OCV-basedmethod is often adopted to estimate an initial SOC via the mono-tonic relationship between OCV and SOC [5]. Machine learningmethods, such as artificial neural networks [10], fuzzy logic [11],and support vector machine [12], regard a battery as a black boxand they are able to model the nonlinear relationship betweeninputs and outputs on the basis of large quantities of training dataavailable [1,13]. Because these methods lack of specifications oflithium-ion batteries, estimation accuracy of these methodsstrongly depends on quantity and quality of training data. Besides,these methods are time-consuming.

In opposition to the aforementioned non-model basedapproaches, model-based SOC estimation approaches featured bya closed-loop are able to self-correct and overcome unexpecteddisturbances. A battery dynamic behavior can be described eitherby an electrochemical model [14,15] or by an equivalent circuitmodel. The design of observers for SOC estimation can be con-ducted by using Kalman filter family, sliding observer [16,17] andH-infinity observer [18–20], etc. [21,22]. Among all the designedobservers, a Kalman filter family takes up a large percentage dueto its advantage in finding an optimal solution for a linear Gaussiansystem. Variants of the Kalman filter emerge for a non-linear bat-tery system. Extended Kalman filter (EKF) was introduced to esti-mate SOC of a lithium-ion polymer battery pack by Plett [23–25].Later, he implemented and tested two sigma-point Kalman filters(SPKFs), including the unscented Kalman filter (UKF) and the cen-tral difference Kalman filter, on a battery pack based on a fourth-generation prototype lithium-ion polymer battery because thesetwo SPKFs did not require a Jacobian matrix, compared with theEKF [26,27]. Subsequently, the methodologies to enhance the Kal-man filter family’s performance on SOC estimation emerge, such asdual EKF [28], adaptive EKF (AEKF) [29], iteratively EKF [30], adap-tive UKF (AUKF) [31,32], square-root UKF [33], square-root spher-ical UKF [34], strong tracking SPKF [35], and adaptive cubatureKalman filter [36]. Meanwhile, some efforts have been made tocompare the performance of these model-based estimationapproaches. Sun et al. [32] compared AUKF with AEKF, EKF, andUKF and showed the AUKF has a superior performance with alow computational load and a better accuracy of SOC. Li et al.[37] compared three model-based filtering algorithms, includingthe Luenberger observer, EKF, and SPKF, and concluded that theclassical Luenberger observer relies mostly on the accuracy of thebattery model and is less accurate, while the SPKF provides betterSOC estimation results in the most cases. Tian et al. [38] comparedthe performance of AUKF against an adaptive slide mode observerin terms of convergence ability, tracking accuracy, and estimationrobustness and the AUKF was shown to have better tracking accu-racy and convergence ability in the comparison results. Other sim-ilar work can be found in [39,40].

Although the Kalman filter family yields satisfying results, itrequires the noise in the system to follow Gaussian distribution.Particle filter (PF) is free of this constraint and it can be appliedto non-linear and non-Gaussian systems [41]. The PF has beenwidely applied in object tracking and navigation, machine vision,and automatic control, whereas it is rarely exploited until the

recent years in SOC estimation. In 2011, Gao et al. [42] used thePF with the combined model to estimate the SOC of a lithium-ion battery and showed the proposed method is effective andefficient. In 2013, Schwunk et al. [43] used the PF for SOC andstate-of-health (SOH) estimation of lithium-ion batteries. Otherrelated works based on PF for SOC estimation of lithium-ion batter-ies can be found in [44–46]. In our paper, to further explore itspotential application to SOC estimation, PF is investigated and iscompared with UKF and EKF.

However, several existing issues are seldom addressed in theliterature. Firstly, a battery degradation issue is seldom discussedin SOC estimation. Often, experiments are carried out on brandfresh batteries. Practicable capacity, as an indicator of batterydegradation, will decline due to irreversible physical/chemicalreactions during normal operation [47–49]. Thus, an aged batterymuch more common-seen in reality, with a capacity loss, not onlyinduces to reduction of a vehicle driving range, but may results in alarge error when estimating SOC [50]. That is the motivation tostudy the effect of the aging level of the battery on SOC estimation.Secondly, SOC can only be inferred from some measurable param-eters and thus the precise initial value of SOC is always unknown inreality. The accuracy and performance of a SOC estimator will beinfluenced by the uncertainty of initial values of SOC in twoaspects: on one hand, improper initial guesses of a certain initialSOC may require different estimation times to track true SOC; onthe other hand, lithium-ion batteries have a relatively flat OCVcurve over the SOC, especially for lithium iron phosphate (LiFePO4)batteries. Inferring SOC from the flat region will cause a larger errorcomparing with that from other relatively steep regions. However,in most of the works on SOC estimation, only some certain initialvalues of SOC were used to validate the developed algorithms.Therefore, it makes great sense to test robustness of themodel-based filtering algorithms in terms of uncertainty of initialvalues of SOC [5]. What’s more, computation time is an importantfactor to evaluation the performance of estimators as estimation ofthe current state is often required to be finished before the nextmeasurement arrives in an online estimation case.

In this paper, we compare the performances of three popularcombined model-based filtering algorithms, including EKF, UKF,and PF, for SOC estimation. First of all, we propose a new combineddynamic loading profile for the simulation of real EV drivingbehaviors. Taking battery degradation into consideration, we col-lect data from a new battery and an aged battery based on the pro-posed profile. Using data collected from the batteries, we consideruncertainty of initial values of SOC and test robustness of the threefiltering algorithms to SOC estimation in terms of various initialvalues of SOC and various initial guesses of SOC. The performancesof EKF, UKF, and PF are then compared in terms of tracking accu-racy, convergence behavior, and computation time.

The rest of this paper is organized as follows. Section 2 intro-duces the battery test bench and the combined dynamic loadingtest. The implemented procedure and the details of algorithmsare presented in Sections 3 and 4, respectively. The experimentalresults and discussions are presented in Section 5. Conclusionsare drawn at the last section.

2. Experiments

The battery test bench, which composes of a battery test system(Arbin BT2000 tester) for loading and sampling the battery, a hostcomputer with Arbin MITS Pro Software for on-line experimentcontrol and data recording, and a computer with Matlab R2012bSoftware for data analysis, is shown in Fig. 1. The cylindricalA123 18650 battery (LiFePO4), was used in the test, and the keyspecifications are shown in Table 1. Two separate test schedules

Arbin BT2000 tester

Matlab R2012a

lithium-ion battery of 18650 cylindrical type (3.6V/1.1Ah)

AC PowerSignalPower

Arbin MITS Pro Sofware (v4.27)

Database

Command

Voltage CurrentTemperature

Fig. 1. Schematic of the battery test bench.

Table 1The key specifications of the test samples.

Type Ratedcapacity

Upper/lowercutoff voltage

End ofchargecurrent

Maximum continuousdischarge current

LiFePO4 1.1 A h 3.6 V/2 V 0.05 C 30 A

F. Yang et al. / Applied Energy 164 (2016) 387–399 389

were conducted on the battery test bench at room temperature(26 �C with a tolerance of 1 �C) for model identification andmethod evaluation, respectively.

2.1. Model parameter identification test

The first test, the dynamic stress test (DST), is a 360-s testdesigned by the US Advanced Battery Consortium (USABC) tosimulate a dynamic discharging regime that represents dynamicEV battery demand [51]. The DST can be scaled down to any desiredmaximum demand as specified in the test plan. The DST was run onthe LiFePO4 batteries to identify the model parameters, then thevoltage and current were measured and recorded from fullycharged to empty on our battery test bench. The rate profile isshown in Fig. 2(a). Fig. 2(b) shows the SOC trace for the entire test.

2.2. Algorithms evaluation test

To simulate the effects of EV driving behavior on the perfor-mance of a battery, we create a new combined dynamic loadingprofile, which is a serial combination of the DST, FUDS (federalurban driving schedule), and US06 test profile. Thus, we name itthe DFU (DST, FUDS, US06) profile. The 360-s DST profile is illus-trated in Section 2.1. The 1372-s FUDS is a dynamic EV perfor-mance test based on a time-velocity profile from an automobileindustry standard vehicle [51]. It can also be scaled to any desired

maximum demand as with the DST. Finally, the US06 test is a 600-stest generated when the automobile is running the supplementalfederal test procedure. A complete 2332-s DFU current profile isshown in Fig. 3.

The DFU test schedule was repeated five times on the batterytest bench at room temperature. The sampling time for the current,voltage, and other information was set to 1 s. In the test, the DFUwas run end-to-end with no time delay between tests, from100% SOC at 3.6 V to 0% SOC at 2 V in a discharge process. A posi-tive current indicates discharging while a negative value corre-sponds to charging. The measured current, voltage profile andcumulative SOC of a new battery are shown in Fig. 4. As the figureshows, in each discharge process, the DFU profile runs more thanthree times before the battery is exhausted.

3. Battery modelling

3.1. The combined model

The combined model, a combination of several empirical mod-els, including the Shepherd, Unnewehr universal, and Nernst mod-els, is chosen to describe the battery dynamic characteristics [24].The model features its simplicity in parameter identification andgenerally outperforms any of the individual models alone [52].The combined model is formulated as

Vk ¼ K0 � RIk � K1SOCk � K2SOCk þ K3 lnðSOCkÞ þ K4 lnð1� SOCkÞ;ð1Þ

where Vk is the measured terminal voltage of the battery under anormal dynamic current load at time step k, and Ik is the dynamiccurrent at the same time; R is the internal resistance of the battery;K0, K1, K2, K3, and K4 are unknown model parameters that aredetermined by the battery, but in practice, they can be estimatedusing a system identification procedure.

0 2000 4000 60000

20

40

60

80

100

Time (s)

SO

C (%

)

SOC as a function of time

0 100 200 300-2

-1

0

1

2

3

4

Time (s)

Cur

rent

(A)

Current for one DST cycle

(b) (a)

Fig. 2. SOC versus time and rate versus time for DST battery tests. The rate for one DST cycle is shown in (a); the SOC for the entire test is shown in (b).

0 500 1000 1500 2000-3

-2

-1

0

1

2

3

4

Time (s)

Cur

rent

(A)

A Complete DFU Current Profile

US06600s

FUDS1372s

DST360s

Fig. 3. A complete DFU current profile.


3.2. Model parameter identification

To perform the online SOC estimation, we first need to identifythe model parameters, namely K0; K1, K2, K3, K4 and R. The idea isto perform a least-square regression given a set ofm battery input–output three-tuples fVk; Ik; SOCkg, according to Eq. (1).

For the kth measurement, Eq. (1) can be rewritten as:

1 1SOCk �SOCk lnðSOCkÞ lnð1� SOCkÞ �Ikh i

�

K0K1K2K3K4R

2666666664

3777777775¼ Vk:

ð2ÞCombining all of the measurements, we have

1 1SOC1 �SOC1 lnðSOC1Þ lnð1�SOC1Þ �I11 1SOC2 �SOC2 lnðSOC2Þ lnð1�SOC2Þ �I2... ..

. ... ..

. ... ..

.

1 1SOCm �SOCm lnðSOCmÞ lnð1�SOCmÞ �Im

2666664

3777775�

K0K1K2K3K4R

2666666664

3777777775¼

V1V2

..

.

Vm

266664

377775:

ð3Þ

Generally, this is an over-determined equation; let

A ¼

1 1SOC1 �SOC1 lnðSOC1Þ lnð1� SOC1Þ �I11 1SOC2 �SOC2 lnðSOC2Þ lnð1� SOC2Þ �I2... ..

. ... ..

. ... ..

.

1 1SOCm �SOCm lnðSOCmÞ lnð1� SOCmÞ �Im

2666664

3777775;

X ¼ K0 K1 K2 K3 K4 R½ �T ;Y ¼ V1 V2 � � � Vm½ �T ;

ð4Þ

the least-square solution can be found by:

X ¼ ðATAÞ�1ATY : ð5ÞFirst, we run the DST test on the LiFePO4 batteries to identify

the model parameters. The voltage and current were measuredand recorded from fully charged to empty with a sampling periodof 1 s based on our battery test bench. Then, we calculate the ‘‘true”SOC using the Coulomb counting method on the measured data.The rate profile is shown in Fig. 2(a). Fig. 2(b) shows the SOCtrajectory for the entire test.

The model identification results are given in Table 2. The meanabsolute error (MAE) and the root-mean-squared error (RMSE) areemployed to verify the goodness-of-fit of the model. Using theseparameters, the results of the fitting, comparing the combinedmodel battery voltage estimation with the battery’s measured volt-age, are shown in Fig. 5. It can be seen that the model output isalmost the same as the general shape of the battery response.The RMSE is around 0.02 V, which is very small compared withthe output voltage, indicating that the combined model welldescribes the dynamic relationship between terminal voltage,SOC, and the simultaneous current.

4. Algorithm implementation for online estimation

Three closed-loop filtering algorithms, including extended Kal-man filter, unscented Kalman filter, and particle filter are respec-tively developed to implement SOC estimation. Under a Bayesianframework, the implementations of these algorithms have a simi-lar prediction–correction structure. For each time step k, the meanand covariance of states are first predicted based on the states atstep k � 1; and then in the correction step, the predicted meanand covariance are corrected based on new measurementsobtained at time step k. The filtering algorithms are detailed asfollows.

0 1000 2000 3000 4000 5000 6000 7000-3

-2

-1

0

1

2

3

4

Time (s)

Cur

rent

(A)

DFU DFU DFU

0 1000 2000 3000 4000 5000 6000 70001.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

Vol

tage

(V)

0 1000 2000 3000 4000 5000 6000 70000

20

40

60

80

100

Time (s)

SO

C (%

)

(a) Measured current

(b) Measured voltage

(c) Cumulative SOC

Time (s)

Fig. 4. DFU profile at 26 C.

Table 2Model parameters and statistics for the fitted error of the model.

K0 K1 K2 K3 K4 R (X) RMSE(V)

MAE (V)

3.3821 0.0016 0.1214 0.0897 �0.0293 0.1798 0.0233 0.0127 V


4.1. Extended Kalman filter

It is well known that the Kalman filter gives the best possibleestimation for a linear system with Gaussian noise. For a non-linear Gaussian noise system, the state estimation can be per-formed by the EKF. The basic idea of the EKF is to approximatethe real system via Taylor expansion at every step. To demonstratehow the EKF works, for a general hidden Markov process:

xk ¼ fðxk�1;ukÞ þ vkzk ¼ hðxk;ukÞ þwk

; ð6Þ

where xk is the system state vector; uk is the known input vector;and zk is the measurement vector at time step k. Correspondingly,fð�Þ and hð�Þ are the state function and the measurement function,respectively, and they can be either linear or nonlinear; vk and wkare the process Gaussian noise and the measurement Gaussiannoise with zero-means and covariances Q and S, respectively.Specifically, vk � Nð0;Q Þ, and wk � Nð0; SÞ. Let n, r be the dimen-sionality of the state vector and measurement vector, respectively.Using the first-order Taylor expansion approximation, the systemis linearized as

xk � Jfðxk�1 � xak�1Þ þ fðxak�1;ukÞ þ vkzk � Jhðxk � x fk Þ þ hðx fk ;ukÞ þwk

ð7Þ

where the Jacobian matrices are defined by

JfðxÞ ¼

@f 1x1� � � @f 1xn

..

. . .. ..

.

@f nx1� � � @f nxn

26664

37775; ð8Þ

JhðxÞ ¼

@h1x1� � � @h1xn

..

. . .. ..

.

@hrx1� � � @hrxn

26664

37775: ð9Þ

The EKF prediction and update equation can be derived byapplying the same technique as used to derive the standard Kal-man filter. Further details can be found in [53]. The EKF procedureis summarized in Table 3 for the combined model.

4.2. Unscented Kalman filter

The UKF, as an alternative way to estimate the state of a nonlin-ear system, doesn’t use the first order Taylor expansion. The UKFaddresses the approximation problem of the EKF by introducingthe concept of weighted sigma points, which are deterministicallyselected from the a Gaussian approximation [54]. The sigma pointsare chosen so that their mean and covariance are exactly xak�1 andPk�1. Each sigma point is then propagated through the system statefunction to yield new sigma points. The newly estimated mean andcovariance are then computed based on their statistics. This pro-cess is called the unscented transformation. Consider the systemdescribed by Eq. (6) and let n be the dimensionality of the statevector. To apply the UKF, 2nþ 1 sigma points with weightsfxik; Wig, i = 1:2n + 1 are generated according tox0k�1 ¼ xak�1;xik�1 ¼ xak�1 þ

ffiffiffiffiffiffiffiffiffiffiffinþ kp

½ffiffiffiffiffiffiffiffiffiffiPk�1

p�i;

xiþnk�1 ¼ xak�1 �ffiffiffiffiffiffiffiffiffiffiffinþ kp

½ffiffiffiffiffiffiffiffiffiffiPk�1

p�i; i ¼ 1 : n;

ð10Þ

where ½��i takes the ith column of the matrix; k ¼ 3a2 � n is theparameter to control the spread of the sigma points around the

mean;ffiffiffiffiffiffiffiffiffiffiPk�1p ffiffiffiffiffiffiffiffiffiffi

Pk�1p T ¼ Pk�1.

Each sigma point is then propagated through the non-linearstate function:

xi;fk ¼ fðxik�1;ukÞ; i ¼ 0; . . . ;2n; ð11Þ

The mean and covariance of x fk is then computed via

x fk ¼X2ni¼0

Wmi xi;fk ; ð12Þ

0 1000 2000 3000 40002

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

Time (s)

Vol

tage

(V)

Measured VoltageEstimated Voltage

0 1000 2000 3000 4000-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Time (s)

Err

or (V

)

(a) (b)

Fig. 5. Fitted results of the combined model on the DST profile at 26 C: (a) the measured and the estimated voltage responses; (b) the error between the measured andestimated voltages.

Table 3Summary of the EKF approach for SOC estimation.

� Initialization:– Initial state: xa0 ¼ E½x0�– Covariance matrix: P0 ¼ E½ðx0 � xa0Þðx0 � xa0ÞT�� For k ¼ 1;2; . . .– Prediction:

x fk ¼ fðxak�1;ukÞP fk ¼ Jf ðxak�1ÞPk�1JTf ðxak�1Þ þ Q

– Correction:Kalman Gain:

Kk ¼ P fk JThðx fk ÞðJhðx fk ÞP fk JThðx fk Þ þ SÞ�1

State Update:

xak ¼ x fk þ Kkðzk � hðx fk ;ukÞÞCovariance Update:

Pk ¼ ðI� KkJThðx fk ÞÞP fk– k kþ 1


P fk ¼X2ni¼0

Wci ðxi;fk � x fk Þðxi;fk � x fk ÞT þ Q ; ð13Þ

where Wmi and Wci are respectively defined as:

Wm0 ¼k

kþ n ;

Wc0 ¼k

kþ nþ ð1� a2 þ bÞ;

Wmi ¼1

2ðkþ nÞ ; i ¼ 1; . . . ;2n;

Wci ¼1

2ðkþ nÞ ; i ¼ 1; . . . ;2n;

ð14Þ

where b controls the prior information of xk�1. According to Ref.[55], a and b are empirically set to default values 1 and 0, respec-tively, in this paper. In other words, the spread of the sigma pointsxik�1 and x

iþnk�1 is

ffiffiffi3p½ ffiffiffiffiffiffiffiffiffiffiPk�1p �i far from the mean vector xak�1.

Similarly, the sigma points are propagated through the mea-surement function:

zi;fk ¼ hðxi;fk ;ukÞ; i ¼ 0; . . . ;2n; ð15Þ

The mean and covariance of the z fk are then computed:

z fk ¼X2ni¼0

Wmi zi;fk ; ð16Þ

covðz fk Þ ¼X2ni¼0

Wci ðzi;fk � z fk Þðzi;fk � z fk ÞT þ S: ð17Þ

The cross-covariance of the state and measurement is

covðx fk ; z fk Þ ¼X2ni¼0

Wiðxi;fk � x fk Þðzi;fk � z fk ÞT: ð18Þ

The Kalman gain is computed as

Kk ¼ covðx fk ; z fk Þcovðz fk Þ�1: ð19Þ

Like the Kalman filter, the state estimation can be obtained by

xak ¼ x fk þ Kkðzk � z fk Þ; ð20Þand the covariance can be updated by

Pk ¼ P fk � Kkcovðz fk ÞKTk : ð21ÞThe process is summarized in Table 4.

4.3. Particle filter

State estimation of the system described in Eq. (6) can be car-ried out in two steps using Bayesian estimation [56,57]:

� Prediction:

pðxkjz1:k�1Þ ¼Z

pðxkjxk�1Þpðxk�1jz1:k�1Þdxk�1; ð22Þ

� Update:

pðxkjz1:kÞ ¼ pðzkjxkÞpðxkjz1:k�1ÞZpðzkjxkÞpðxkjz1:k�1Þdxk

: ð23Þ

However, this relation is deceptively simple because theanalytical form of the posterior distribution cannot usually bedetermined. To approximate the posterior distribution, the MonteCarlo method uses a large number of weighted particles torepresent the distribution. In our case, we first draw Ns random

particles fxi0gNsi¼1 from pðx0Þ. The weight of each particle is set to

1=Ns and their sum is equal to 1. Denote fxikgNsi¼1 and fxikg

Nsi¼1 be

kth system state and their associated weights for Ns random

Table 4Summary of the UKF approach for SOC estimation.

� Initialization:– Initial state: xa0 ¼ E½x0�– Covariance matrix: P0 ¼ E½ðx0 � xa0Þðx0 � xa0ÞT�� For k ¼ 1;2; � � �– Prediction� Generate sigma points via Eq. (10)� Propagate the sigma points via Eq. (11)� Calculate the mean and covariance of the predicted state via

Equations (12) and (13)

– Correction� Propagate the sigma points via Eq. (15)� Calculate the mean and covariance of the measurement via Eqs.

(16) and (17)� Calculate the cross-covariance of the state and measurement via

Eq. (18)� Compute the Kalman gain via Eq. (19)� Update the posterior mean of the state by Eq. (20)� Update the posterior covariance of the state by Eq. (21)

– k kþ 1

Table 5Summary of the PF approach for SOC estimation.

� Initialization:– Draw Ns samples, xi0 � pðx0Þ; i ¼ 1 : Ns– Assign weight, xi0 ¼ 1=Ns ; i ¼ 1 : Ns� For k = 1, 2, ...– Prediction� Draw new samples, xik � pðxkjxik�1Þ; i ¼ 1 : Ns

– Correction� Update weights, according to Eq. (27)� Normalize weights, according to Eq. (25)� Update the state via Eq. (26)

– Resampling if degeneracy is detected� Calculate Neff using Eq. (28)� If Neff < NT , then, resample particles

– k kþ 1


particles, respectively. The posterior distribution cycle can then beapproximated by:

pðxkjz1:kÞ �XNsi¼1

~xikdðxk � xikÞ; ð24Þ

where dð�Þ is the Dirac delta function, and ~xik is the normalizedweight, i.e.

Fig. 6. Block diagram of the S

~xik ¼xikPNsi¼1xik

: ð25Þ

The real state is then approximated by

xk ¼Z 1�1

xkpðxkjz1:kÞdxk �XNsi¼1

~xikxik: ð26Þ

To calculate the weight of each particle, we use the sequentialimportance sampling as proposed in [57], where,

~xik / ~xik�1pðzkjxikÞpðxikjxik�1Þpðxikjxi0:k�1; z1:kÞ

; ð27Þ

where pðxikjxi0:k�1; z1:kÞ ¼ pðxikjxi0:k�1; zkÞ is the importance distribu-tion and its popular choice is a prior distribution, namely,pðxikjxi0:k�1; zkÞ ¼ pðxikjxik�1Þ. In real applications, it is necessary toconsider degeneracy or sample impoverishment, in which few par-ticles account for almost all of the weights. This phenomenon iscommonly avoided by using resampling whenever the number ofeffective particles Neff falls below a certain threshold NT . The basicidea behind resampling is to eliminate particles with small weightsand thus focus on those with large weights. The main resamplingmethods, including stratified resampling and systematic resam-pling, are compared in [58,59]. Hol et al. [59] pointed out that sys-tematic resampling is preferable due to its quality and simplicity.Neff can be approximated [56] by:

N̂eff ¼ 1PNsi¼1ðxikÞ

2 : ð28Þ

The PF procedures for SOC estimation is summarized in Table 5.

4.4. SOC estimation using each of the above three algorithms

In our case, the state vector is x ¼ SOC. The state function in Eq.(29) follows the Coulomb counting methodmentioned in Section 1.The terminal voltage of the battery is the measured vector z ¼ V .

The state function:

SOCk ¼ SOCk�1 � DtCn Ik þ vk; ð29Þ

The measurement function:

Vk ¼ K0 � RIk � K1SOCk � K2SOCk þ K3 lnðSOCkÞþ K4 lnð1� SOCkÞ þwk; ð30Þ

where Ik is the current as the input uk at time step k; Dt is the sam-pling interval, which is 1 s according to the sampling rate; Cn is the

OC estimation procedure.

(b)

0 2000 4000 60000

20

40

60

80

100SOC Estimation (100% && 100%)

Time (s)

SO

C E

stim

atio

n (%

)

True SOCSOC EKFSOC UKFSOC PF

0 2000 4000 6000-4

-3

-2

-1

0

1

2

3

4Estimation Error (100% && 100%)

Time (s)

Est

imat

ion

Err

or (%

)

Error by EKFError by UKFError by PF

(a)

Fig. 7. Results of SOC estimation using the three algorithms with an initial SOC = 100%, a guess = 100%: (a) SOC tracking; (b) estimation error.

Table 6RMSE (%) comparison of the EKF, the UKF and the PF in the DFU tests to estimate SOC.

Cycle 1 Cycle 2 Cycle 3 Cycle 4 Cycle 5 Mean

Correctly initialized with an initial SOC = 100%, a guess = 100%EKF 0.74 1.18 1.25 1.16 1.09 1.084UKF 0.28 0.45 0.5 0.46 0.45 0.428PF 1.6 2.52 2.0 1.41 1.91 1.888Incorrectly initialized with an initial SOC = 80%, a guess = 50%EKF 3.72 3.17 3.0 3.28 3.22 3.278UKF 2.85 2.29 2.11 2.39 2.33 2.394PF 1.95 1.31 2.93 1.98 2.57 2.148


rated capacity of the test samples and it is equal to 1.1 A h; and R,K0, K1, K2, K3, and K4 are unknown parameters with their initialvalue identified by least-square regression using data collectedfrom DST profile, as shown in Table 2. Here Q and S are set to10�5 and 10�2, respectively, by considering the scales and uncer-tainties of the state and the terminal voltages.

According to the proposed state-space model in Eqs. (29) and(30), a block diagram illustrating the process of SOC estimationbased on these algorithms is shown in Fig. 6, in which the modelparameters are updated using the same technique as for modelparameter identification.

0 1000 2000 3000 4000 50000

20

40

60

80

100SOC estimation (80% && 50%)

Time (s)

SO

C E

stim

atio

n (%

)


Est

imat

ion

Err

or (%

)

(a)

Fig. 8. Results of SOC estimation using the three algorithms with an init

5. Experimental results and discussion of SOC estimation

5.1. Experiments on fully charged battery

Fig. 7 demonstrates the SOC estimation on a fully charged newbattery. The three aforementioned algorithms are used with theDFU loading profile. As the battery is fully charged, it makes senseto assume that the initial SOC is 100%. The tracking results areshown in Fig. 7(a). The estimation error (calculated as estimatedvalue minus true value) is plotted in more detail in Fig. 7(b). Theestimation error for all three algorithms starts at zero, becausethe initial guess is correct. Over time, the error of the UKF slowlydiverges from zero, with the maximum absolute error less than1%. The error for the PF is a little larger and more fluctuating. Nev-ertheless, the maximum absolute error of the PF is within 4%.Table 6 summarizes the results from all five cycles of the DFU test.The mean RMSE is 1.084% for the EKF, 0.428% for the UKF, and1.888% for the PF. The UKF has the lowest RMSE, although theoverall performance of all three algorithms is satisfactory. Notethat the number of the particles Ns in the PF should be set to aproper value to make a tradeoff between a small RMSE andaffordable computation time. To determine a proper Ns,Ns = 10, 50, 100, 200, 400, respectively, were used. The RMSEs forthese situations are 0.0436, 0.0264, 0.0188, 0.0175, and 0.0169,

(b)

0 1000 2000 3000 4000 5000 6000-30

-25

-20

-15

-10

-5

0

5


time/s


ial SOC = 80%, a guess = 50%: (a) SOC tracking; (b) estimation error.

(b)

0 1000 2000 3000 40000

20

40

60

80


Time (s)

SO

C E

stim

atio

n (%

)


0 500 1000 15000

20

40

60

80


Time (s)

SO

C E

stim

atio

n (%

)


0 500 1000 1500 2000 25000

20

40

60

80


Time (s)

SO

C E

stim

atio

n (%

)


0 1000 2000 3000 4000 50000

20

40

60

80

100SOC estimation (80% && 50%)

Time (s)

SO

C E

stim

atio

n (%

)


(a)

(c) (d)

Fig. 9. Tracking results of SOC estimation using the three algorithms with SOC incorrectly initialized: (a) an initial SOC = 20%, a guess = 50%; (b) an initial SOC = 40%, aguess = 70%; (c) an initial SOC = 60%, a guess = 70%; (a) an initial SOC = 80%, a guess = 50%.

Table 7RMSE (%) of SOC estimation by the EKF, the UKF and the PF when DFU was run attemperature 26 C.

TrueinitialSOC (%)

EKF UKF PFInitial SOC Guess(%)

Initial SOC Guess(%)

Initial SOC Guess (%)

30 50 70 30 50 70 30 50 70

20 3.01 3.23 7.23 2.47 2.85 3.95 1.01 0.91 1.1240 3.49 2.49 3.31 3.58 2.37 1.51 2.89 1.96 1.8860 10.41 7.12 3.49 3.11 4.67 2.11 16.25 14.53 16.4080 6.16 3.28 2.51 3.76 2.39 1.86 1.86 2.15 2.27


respectively. It was observed that the estimated RMSEs do notchange significantly after Ns P 100 in our case study. Therefore,Ns ¼ 100 was used for the rest of this paper unless statedotherwise.

5.2. Experiments with unknown initial state

In real applications, it is impossible to measure an initial SOCbefore use; that is, the initial SOC is unknown. A relatively closeestimation still cannot be made if we only know the open circuitvoltage, because it remains steady for a wide range of SOC [5].Fig. 8 shows the experimental results with an initial SOC of 80%

and a SOC guess of 50%. The PF converges faster than the otheralgorithms.

Table 6 summarizes the RMSE results for the five cycles. TheRMSEs are slightly larger than those in the case of correct SOCinitialization. In both cases, the UKF outperforms the EKF. TheRMSEs of the UKF are 60.5% and 27% smaller than those of theEKF for correct and incorrect initialization, respectively. Moreover,the UKF shows faster convergence than the EKF in Fig. 8. The resultis consistent with the fact that the EKF uses the first-order Taylorexpansion to linearize the system which may introduce largeerrors and uncertainties in calculating the posterior mean andcovariance [26], while the UKF tries to approximate the first-order and second-order moments directly via the unscentedtransformation.

To further explore how these algorithms perform, we conductexperiments with various initial values of SOC (20%, 40%, 60%and 80%) and various initial guesses (30%, 50%, 80%). Fig. 9 showspart of the tracking results.

In all cases, all of the tracking trajectories eventually convergeto the true SOC. The RMSEs for all of the experiments are tabulatedin Table 7 for comparison. As there is little variation in perfor-mance over different cycles, as shown in Table 6, the mean RMSEover five cycles is used henceforward unless stated otherwise.Overall the tracking result is satisfactory, as indicated by theRMSEs.

Two points should be noted from Table 7. Firstly, for the PF,when the true initial SOC is 20%, 40%, and 80%, respectively, the

(b)

0 200 400 600 800 1000-30

-25

-20

-15

-10

-5

0

5


time/s

Est

imat

ion

Err

or (%

)


1000 2000 3000 4000 5000 6000-4

-3

-2

-1

0

1

2

3


time/s

Est

imat

ion

Err

or (%

)


(a)

Fig. 10. Comparison of the estimation error using the three algorithms with an initial SOC = 80%, a guess = 50%: (a) estimated before 1000 s; (b) estimated from 1000 s everafter.

Table 8RMSE (%) of selected data range, initial SOC = 80%, guess = 50%.

[1s,5888s]

[1001s,5888s]

[2001s,5888s]

[3001s,5888s]

[4001s,5888s]

EKF 3.17 1.95 1.51 1.29 1.35UKF 2.29 1.35 1.05 1.02 1.12PF 2.07 1.40 1.34 1.50 1.19


RMSE is almost the smallest, but becomes rather large when thetrue initial SOC is 60%. This is in contrast to the fluctuating perfor-mance of the PF in Fig. 7. The reason for the difference is that the PFconverges fastest to the true SOC at the beginning. For the othertwo algorithms, the error at this stage is so large that the overallRMSE is increased. To validate this claim, we amplify the detailof Fig. 8(b) in Fig. 10 and recalculate the RMSE, but this time thedata points at the beginning are neglected. Table 8 shows thatwhen estimated from 1000 s ever after, the UKF has the smallestRMSE, which verifies the fluctuating performance of the PF. Underdifferent initial values of SOC and varying initial guesses, the UKFprovides more stable and accurate estimation results than theother two algorithms.

Second, the estimation error tends to be greater when the trueinitial SOC is 60% for all three algorithms. This can be explained bythe fairly flat phase between 45% SOC and 65% SOC of the OCV ver-sus the SOC curve [24]. A typical OCV–SOC curve for the battery is

0 0.2 0.4 0.6 0.8 12

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

SOC

OC

V (V

)

(a)

Fig. 11. OCV–SOC curve for new battery at 26 C, (a) betw

shown in Fig. 11. This flat region is a common phenomenon forlithium-ion batteries, especially for LiFePO4 batteries [5]. Whenthe SOC is in this stage, changes in the SOC barely effect the OCVvalue, making the SOC near-unobservable. For the PF, the effectis much more severe, because (a) the weight of all particles isalmost the same, and (b) the non-linear state transfer functionmakes the particle distribution unpredictable.

5.3. Experiments with varying state of health

Normally, the capacity of lithium-ion battery degrades as thenumber of charging cycles increases. The SOH of a battery isdefined as the ratio of the battery’s actual capacity Cbatt over itsrated capacity Cn [43]:

SOH ¼ CbattCn

: ð31Þ

The battery’s actual capacity is determined by capacity testsbefore profiles are run. For a new battery, the actual capacity isthe same as the rated capacity, hence the SOH of a new batteryis 1. To compare the robustness of the three algorithms againstvarying SOH, we perform the DFU profile on an aged battery ofthe same kind, with an SOH of 0.87. The measured current, voltageprofile and cumulative SOC at room temperature are shown in

(b)

0.3 0.4 0.5 0.6 0.7 0.8

3.24

3.25

3.26

3.27

3.28

3.29

3.3

3.31

SOC

OC

V (V

)

een 0% and 100% SOC; (b) between 30% and 80% SOC.

0 1000 2000 3000 4000 5000 6000-3

-2

-1

0

1

2

3

4

Time (s)

Cur

rent

(A)

DFUDFUDFU

0 1000 2000 3000 4000 5000 60002

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

Time (s)

Vol

tage

(V)

0 1000 2000 3000 4000 5000 60000

10

20

30

40

50

60

70

80

90

100

Time (s)

SO

C (%

)

(a) Measured current

(b) Measured voltage

(c) Cumulative SOC

Fig. 12. DFU profile for an aged battery at temperature 26 C.


Fig. 12. We can see from the figure that in each discharge process,the DFU profile runs less than three times before the battery isexhausted.

For the aged battery, we use Eq. (32) instead of Eq. (29) as thestate function to perform the SOC estimation.

State function:

SOCk ¼ SOCk�1 � DtSOH � Cn Ik þ vk ð32Þ

The initial parameters of the combined model use the resultsfrom Table 2 (so the parameters are the same for the new andthe aged battery). Fig. 13 shows the SOC estimation results forthe three algorithms on the aged battery. Fig. 13(a) and (b) showthe results with the SOC state correctly initialized to 100%, whileFig. 13(c) and (d) show the results with the SOC state incorrectlyinitialized to 50% instead of 80%. We can see that, the three algo-rithms provide satisfactory SOC estimations for the aged battery.When correctly initialized, the performance of the PF fluctuatesand is the worst among all the three, with maximum absolute esti-mation error within 3%. When incorrectly initialized, the PF showsthe fastest convergence. The UKF performs better than the EKF inboth cases. The reasons for these phenomena are discussed inSection 5.2.

Fig. 14 summarizes the RMSE results of the DFU test on new andaged batteries for comparison. We note that the results are verysimilar. There are two possible reasons for this result: first, thecombined model works for aged batteries; and second, the rela-tionship between the OCV and the SOC remains almost the samewhile the SOH ranges from 0.8 to 1.0.

To compare the computation time, all the algorithms are imple-mented using MATLAB version 2012a and tested on a computerwith a 2.2-GHz processor, 8 GB of memory, and a 64-bit operatingsystem. The computation time is started right before the predictionstep of the first state and ended right after the correction step ofthe last state. Fig. 15 summarizes the computation time for theexamples in Fig. 14. We note that the computation time of theUKF is the smallest, and the EKF is of the same order. The PF takesabout five times as much computation time as the EKF whenNs ¼ 100. When Ns ¼ 1000, the associated calculation times ofthe PF are 71.2 s, 62.4 s, 56.5 s, 49.5 s, respectively. Moreover, thePF of Ns ¼ 1000 uses almost ten times as much computation timeas that of Ns ¼ 100. The observation indicates that the computationtime of the PF is proportional to the number of samples used.

6. Conclusion and future work

This study compared several algorithms, including the EKF, theUKF, and the PF, for estimating state of charge of cylindrical-type18650 (LiFePO4) batteries by using the proposed combineddynamic loading profile, which was composed of the dynamicstress test, the federal urban driving schedule, and the US06. Inmost of the instances, the three algorithms produced the satisfyingSOC estimates as well as the small RMSEs (less than 4%) when thedifferent initial guesses of SOC were considered. Our findings weresummarized as follows.

(1) The UKF was more accurate than the EKF and the PF whenthe SOC was correctly initialized. The EKF underperformedbecause of the first-order Taylor expansion.

(2) All of the algorithms were robust to the uncertainty of initialvalues of SOC. The UKF was more robust than the EKF andthe PF. When the SOC was estimated from the relatively flatphase of the OCV-SOC curve (SOC ranged from 45% to 65%),the estimation errors tend to be increased. Moreover, the PFperformed worse.

(3) The PF showed the fastest convergence ability than the UKFand the EKF at the beginning of the SOC estimation, in whichthe initial guess of SOC was far from the true SOC.

(4) The EKF and the UKF were more computationally efficientthan the PF.

(5) All the three algorithms were able to estimate the SOC of theaged battery.

(b)

(c) (d)

0 1000 2000 3000 4000 5000 60000

20

40

60

80


Time (s)

SO

C E

stim

atio

n (%

)


0 1000 2000 3000 4000 5000 6000-3

-2

-1

0

1

2

3


Time (s)

Est

imat

ion

Erro

r (%

)


0 1000 2000 3000 4000 50000

20

40

60

80


Time (s)

SO

C E

stim

atio

n (%

)


0 1000 2000 3000 4000 5000-25

-20

-15

-10

-5

0

5


Time (s)

Est

imat

ion

Erro

r (%

)


(a)

Fig. 13. SOC estimation results for the aged battery: EKF vs. UKF vs. PF with (a and b) correct initialization, an initial SOC = 100%, a guess = 100%; and (c and d) incorrectinitialization, an initial SOC = 80%, a guess = 50%.

1.080.43

1.89

0.950.37

1.67

3.28

2.39 2.15

3.84

2.752.22

0

1

2

3

4

5

EKF UKF PFCorrectly ini�alized (new cell) Correctly ini�alized (aged cell)

Incorrectly ini�alized (new cell) Incorrectly ini�alized (aged cell)

Fig. 14. RMSE (%) of SOC estimation on new and aged batteries.

1.57 1.48

7.35

1.40 1.31

6.57

1.27 1.17

5.93

1.13 1.08

5.33

0.00

2.00

4.00

6.00

8.00

EKF UKF PF

Correctly ini�alized (new cell) Correctly ini�alized (aged cell)

Incorrectly ini�alized (new cell) Incorrectly ini�alized (aged cell)

Fig. 15. Computational time (unit: s) of SOC estimation on new and aged batteries.


As the UKF was the most stable and the PF was the fastestconvergent at the beginning of the SOC estimation, a combinedalgorithm that utilizes the PF for initialization and then the UKFfor the accurate estimation may provide a higher SOC estimationaccuracy in our future work.

As degradation is inevitable in real applications, this studyinvestigated the influence of degradation on SOC estimation. Dueto the lab conditions and limited time, instead of waiting untilthe battery degrades, our experiments used a different battery.While this is one possible solution, for more persuasive evidence,impeccable and time-consuming experiments will need to beperformed.

Acknowledgments

This research work was partly supported by General ResearchFund (Project No. CityU 11216014) and National Natural ScienceFoundation of China (Project No. 11471275).

References

[1] Lu L, Han X, Li J, Hua J, Ouyang M. A review on the key issues for lithium-ionbattery management in electric vehicles. J Power Sources 2013;226:272–88.

[2] Xing Y, Ma EW, Tsui KL, Pecht M. Battery management systems in electric andhybrid vehicles. Energies 2011;4:1840–57.

[3] Chen X, Shen W, Cao Z, Kapoor A, Hijazin I. Adaptive gain sliding modeobserver for state of charge estimation based on combined battery equivalent

http://refhub.elsevier.com/S0306-2619(15)01522-6/h0005http://refhub.elsevier.com/S0306-2619(15)01522-6/h0005http://refhub.elsevier.com/S0306-2619(15)01522-6/h0010http://refhub.elsevier.com/S0306-2619(15)01522-6/h0010


circuit model in electric vehicles. In: Industrial Electronics and Applications(ICIEA), 2013 8th IEEE Conference on: IEEE; 2013. p. 601–6.

[4] Chen XP, Shen WX, Cao ZW, Kapoor A, Hijazin I. Adaptive gain sliding modeobserver for state of charge estimation based on combined battery equivalentcircuit model in electric vehicles. C Ind Elect Appl 2013:601–6.

[5] Xing Y, He W, Pecht M, Tsui KL. State of charge estimation of lithium-ionbatteries using the open-circuit voltage at various ambient temperatures. ApplEnergy 2014;113:106–15.

[6] Ng KS, Moo C-S, Chen Y-P, Hsieh Y-C. Enhanced coulomb counting method forestimating state-of-charge and state-of-health of lithium-ion batteries. ApplEnergy 2009;86:1506–11.

[7] Liu T-H, Chen D-F, Fang C-C. Design and implementation of a battery chargerwith a state-of-charge estimator. Int J Electron 2000;87:211–26.

[8] Aylor JH, Thieme A, Johnso B. A battery state-of-charge indicator for electricwheelchairs. Ind Electron IEEE Trans 1992;39:398–409.

[9] Hu X, Li S, Peng H, Sun F. Robustness analysis of State-of-Charge estimationmethods for two types of Li-ion batteries. J Power Sources 2012;217:209–19.

[10] Kang L, Zhao X, Ma J. A new neural network model for the state-of-chargeestimation in the battery degradation process. Appl Energy 2014;121:20–7.

[11] Salkind AJ, Fennie C, Singh P, Atwater T, Reisner DE. Determination of state-of-charge and state-of-health of batteries by fuzzy logic methodology. J PowerSources 1999;80:293–300.

[12] Hu X, Sun F. Fuzzy clustering based multi-model support vector regressionstate of charge estimator for lithium-ion battery of electric vehicle. In:Intelligent Human-Machine Systems and Cybernetics, 2009 IHMSC’09International Conference on: IEEE; 2009. p. 392–6.

[13] Restaino R, Zamboni W. Comparing particle filter and extended kalman filterfor battery State-Of-Charge estimation. In: IECON 2012–38th annualconference on IEEE industrial electronics society: IEEE; 2012. p. 4018–23.

[14] Dey S, Ayalew B. nonlinear observer designs for state-of-charge estimation oflithium-ion batteries. In: American control conference; 2014. p. 248–53.

[15] Domenico DD, Fiengo G, Stefanopoulou A. Lithium-ion battery state of chargeestimation with a Kalman Filter based on a electrochemical model. In: IEEEinternational conference on control applications; 2008. p. 702–7.

[16] Kim I-S. The novel state of charge estimation method for lithium battery usingsliding mode observer. J Power Sources 2006;163:584–90.

[17] Kim D, Koo K, Jeong JJ, Goh T, Kim SW. Second-order discrete-time slidingmode observer for state of charge determination based on a dynamicresistance Li-ion battery model. Energies 2013;6:5538–51.

[18] Zhang F, Liu GJ, Fang LJ, Wang HG. Estimation of battery state of charge with H-infinity observer: applied to a robot for inspecting power transmission lines.IEEE Trans Ind Electron 2012;59:1086–95.

[19] Li X, Jiang JC, Zhang CP, Zhang WG, Sun BX. Effects analysis of modelparameters uncertainties on battery SOC estimation using H-infinity observer.In: Proc Ieee int symp; 2014. p. 1647–53.

[20] Yan JY, Xu GQ, Xu YS, Xie BL. Battery state-of-charge estimation based on H(infinity) filter for hybrid electric vehicle. In: 2008 10th Internationalconference on control automation robotics & vision, vols. 1–4. Icarv; 2008. p.464–9.

[21] Xu J, Mi CC, Cao BG, Deng JJ, Chen Z, Li SQ. The state of charge estimation oflithium-ion batteries based on a proportional–integral observer. Ieee T VehTechnol 2014;63:1614–21.

[22] Chaoui H, Golbon N, Hmouz I, Souissi R, Tahar S. Lyapunov-based adaptivestate of charge and state of health estimation for lithium-ion batteries. IEEETrans Industr Electron 2014:1.

[23] Plett GL. Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs Part 1. Background. J Power Sources 2004;2004(134):252–61.

[24] Plett GL. Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 2. Modeling and identification. J Power Sources2004;134:262–76.

[25] Plett GL. Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs: Part 3. State and parameter estimation. J PowerSources 2004;134:277–92.

[26] Plett GL. Sigma-point Kalman filtering for battery management systems ofLiPB-based HEV battery packs: Part 1: introduction and state estimation. JPower Sources 2006;161:1356–68.

[27] Plett GL. Sigma-point Kalman filtering for battery management systems ofLiPB-based HEV battery packs: Part 2: simultaneous state and parameterestimation. J Power Sources 2006;161:1369–84.

[28] Lee S, Kim J, Lee J, Cho BH. State-of-charge and capacity estimation of lithium-ion battery using a new open-circuit voltage versus state-of-charge. J PowerSources 2008;185:1367–73.

[29] Xiong R, He HW, Sun FC, Zhao K. Evaluation on state of charge estimation ofbatteries with adaptive extended Kalman filter by experiment approach. Ieee TVeh Technol 2013;62:108–17.

[30] Fang H, Zhao X, Wang Y, Sahinoglu Z, Wada T, Hara S, et al. Improved adaptivestate-of-charge estimation for batteries using a multi-model approach. J PowerSources 2014;254:258–67.

[31] Sun FC, Hu XS, Zou Y, Li SG. Adaptive unscented Kalman filtering for state ofcharge estimation of a lithium-ion battery for electric vehicles. Energy2011;36:3531–40.

[32] Sun F, Hu X, Zou Y, Li S. Adaptive unscented Kalman filtering for state of chargeestimation of a lithium-ion battery for electric vehicles. Energy2011;36:3531–40.

[33] Gholizade-Narm H, Charkhgard M. Lithium-ion battery state of chargeestimation based on square-root unscented Kalman filter. IET PowerElectron 2013;6:1833–41.

[34] Aung H, Soon Low K, Ting Goh S. State-of-charge estimation of lithium-ionbattery using square root spherical unscented Kalman filter (Sqrt-UKFST) innanosatellite. Power Electron, IEEE Trans 2015;30:4774–83.

[35] Li D, Ouyang J, Li H, Wan J. State of charge estimation for LiMn2O4 powerbattery based on strong tracking sigma point Kalman filter. J Power Sources2015;279:439–49.

[36] Xia B, Wang H, Tian Y, Wang M, Sun W, Xu Z. State of charge estimation oflithium-ion batteries using an adaptive cubature Kalman filter. Energies2015;8:5916–36.

[37] Li J, Klee Barillas J, Guenther C, Danzer MA. A comparative study of state ofcharge estimation algorithms for LiFePO4 batteries used in electric vehicles. JPower Sources 2013;230:244–50.

[38] Tian Y, Xia B, Wang M, Sun W, Xu Z. Comparison study on two model-basedadaptive algorithms for SOC estimation of lithium-ion batteries in electricvehicles. Energies 2014;7:8446–64.

[39] Hu X, Sun F, Zou Y. Comparison between two model-based algorithms for Li-ion battery SOC estimation in electric vehicles. Simul Model Pract Theory2013;34:1–11.

[40] He H, Qin H, Sun X, Shui Y. Comparison study on the battery SoC estimationwith EKF and UKF algorithms. Energies 2013;6:5088–100.

[41] Gordon NJ, Salmond DJ, Smith AF. Novel approach to nonlinear/non-GaussianBayesian state estimation. In: IEE Proceedings F (Radar and Signal Processing).IET; 1993. p. 107–13.

[42] Gao M, Liu Y, He Z. Battery state of charge online estimation based on particlefilter. In: Image and signal processing (CISP), 2011 4th International Congresson. IEEE; 2011. p. 2233–6.

[43] Schwunk S, Armbruster N, Straub S, Kehl J, Vetter M. Particle filter for state ofcharge and state of health estimation for lithium–iron phosphate batteries. JPower Sources 2013;239:705–10.

[44] Wang Y, Zhang C, Chen Z. A method for joint estimation of state-of-charge andavailable energy of LiFePO4 batteries. Appl Energy 2014;135:81–7.

[45] He Y, Liu X, Zhang C, Chen Z. A new model for State-of-Charge (SOC)estimation for high-power Li-ion batteries. Appl Energy 2013;101:808–14.

[46] Liu X, Chen Z, Zhang C, Wu J. A novel temperature-compensated model forpower Li-ion batteries with dual-particle-filter state of charge estimation. ApplEnergy 2014;123:263–72.

[47] Alamgir M, Sastry AM. Efficient batteries for transportation applications. SAEtechnical paper; 2008.

[48] Miao Q, Xie L, Cui H, Liang W, Pecht M. Remaining useful life prediction oflithium-ion battery with unscented particle filter technique. MicroelectronReliab 2013;53:805–10.

[49] Wang D, Miao Q, Pecht M. Prognostics of lithium-ion batteries based onrelevance vectors and a conditional three-parameter capacity degradationmodel. J Power Sources 2013;239:253–64.

[50] Hu X, Li SE, Jia Z, Egardt B. Enhanced sample entropy-based healthmanagement of Li-ion battery for electrified vehicles. Energy 2014;64:953–60.

[51] Hunt G. USABC electric vehicle battery test procedures manual. Washington,DC, USA: United States Department of Energy; 1996.

[52] Plett G. LiPB dynamic cell models for Kalman-filter SOC estimation. In: The19th international battery, hybrid and fuel electric vehicle symposium andexhibition; 2002. p. 1–12.

[53] Terejanu GA. Extended kalman filter tutorial. Disponible: ; 2008.

[54] ArulampalamMS, Maskell S, Gordon N, Clapp T. A tutorial on particle filters foronline nonlinear/non-Gaussian Bayesian tracking. Ieee T Signal Process2002;50:174–88.

[55] Särkkä S. Bayesian filtering and smoothing. Cambridge University Press; 2013.[56] ArulampalamMS, Maskell S, Gordon N, Clapp T. A tutorial on particle filters for

online nonlinear/non-Gaussian Bayesian tracking. Signal Process, IEEE Trans2002;50:174–88.

[57] Doucet A. An introduction to sequential Monte Carlo methods. In: Doucet A, deFreitas JFG, Gordon NJ, editors. Sequential Monte Carlo methods inpractice. New York: Springer-Verlag; 2001.

[58] Douc R, Cappé O. Comparison of resampling schemes for particle filtering. In:Image and Signal Processing and Analysis, 2005 ISPA 2005 Proceedings of the4th International Symposium on: IEEE; 2005. p. 64–9.

[59] Hol JD, Schon TB, Gustafsson F. On resampling algorithms for particle filters.In: Nonlinear Statistical Signal Processing Workshop, 2006 IEEE: IEEE; 2006. p.79–82.

http://refhub.elsevier.com/S0306-2619(15)01522-6/h0020http://refhub.elsevier.com/S0306-2619(15)01522-6/h0020http://refhub.elsevier.com/S0306-2619(15)01522-6/h0020http://refhub.elsevier.com/S0306-2619(15)01522-6/h0025http://refhub.elsevier.com/S0306-2619(15)01522-6/h0025http://refhub.elsevier.com/S0306-2619(15)01522-6/h0025http://refhub.elsevier.com/S0306-2619(15)01522-6/h0030http://refhub.elsevier.com/S0306-2619(15)01522-6/h0030http://refhub.elsevier.com/S0306-2619(15)01522-6/h0030http://refhub.elsevier.com/S0306-2619(15)01522-6/h0035http://refhub.elsevier.com/S0306-2619(15)01522-6/h0035http://refhub.elsevier.com/S0306-2619(15)01522-6/h0040http://refhub.elsevier.com/S0306-2619(15)01522-6/h0040http://refhub.elsevier.com/S0306-2619(15)01522-6/h0045http://refhub.elsevier.com/S0306-2619(15)01522-6/h0045http://refhub.elsevier.com/S0306-2619(15)01522-6/h0050http://refhub.elsevier.com/S0306-2619(15)01522-6/h0050http://refhub.elsevier.com/S0306-2619(15)01522-6/h0055http://refhub.elsevier.com/S0306-2619(15)01522-6/h0055http://refhub.elsevier.com/S0306-2619(15)01522-6/h0055http://refhub.elsevier.com/S0306-2619(15)01522-6/h0080http://refhub.elsevier.com/S0306-2619(15)01522-6/h0080http://refhub.elsevier.com/S0306-2619(15)01522-6/h0085http://refhub.elsevier.com/S0306-2619(15)01522-6/h0085http://refhub.elsevier.com/S0306-2619(15)01522-6/h0085http://refhub.elsevier.com/S0306-2619(15)01522-6/h0090http://refhub.elsevier.com/S0306-2619(15)01522-6/h0090http://refhub.elsevier.com/S0306-2619(15)01522-6/h0090http://refhub.elsevier.com/S0306-2619(15)01522-6/h0105http://refhub.elsevier.com/S0306-2619(15)01522-6/h0105http://refhub.elsevier.com/S0306-2619(15)01522-6/h0105http://refhub.elsevier.com/S0306-2619(15)01522-6/h0110http://refhub.elsevier.com/S0306-2619(15)01522-6/h0110http://refhub.elsevier.com/S0306-2619(15)01522-6/h0110http://refhub.elsevier.com/S0306-2619(15)01522-6/h0115http://refhub.elsevier.com/S0306-2619(15)01522-6/h0115http://refhub.elsevier.com/S0306-2619(15)01522-6/h0115http://refhub.elsevier.com/S0306-2619(15)01522-6/h0120http://refhub.elsevier.com/S0306-2619(15)01522-6/h0120http://refhub.elsevier.com/S0306-2619(15)01522-6/h0120http://refhub.elsevier.com/S0306-2619(15)01522-6/h0125http://refhub.elsevier.com/S0306-2619(15)01522-6/h0125http://refhub.elsevier.com/S0306-2619(15)01522-6/h0125http://refhub.elsevier.com/S0306-2619(15)01522-6/h0130http://refhub.elsevier.com/S0306-2619(15)01522-6/h0130http://refhub.elsevier.com/S0306-2619(15)01522-6/h0130http://refhub.elsevier.com/S0306-2619(15)01522-6/h0135http://refhub.elsevier.com/S0306-2619(15)01522-6/h0135http://refhub.elsevier.com/S0306-2619(15)01522-6/h0135http://refhub.elsevier.com/S0306-2619(15)01522-6/h0140http://refhub.elsevier.com/S0306-2619(15)01522-6/h0140http://refhub.elsevier.com/S0306-2619(15)01522-6/h0140http://refhub.elsevier.com/S0306-2619(15)01522-6/h0145http://refhub.elsevier.com/S0306-2619(15)01522-6/h0145http://refhub.elsevier.com/S0306-2619(15)01522-6/h0145http://refhub.elsevier.com/S0306-2619(15)01522-6/h0150http://refhub.elsevier.com/S0306-2619(15)01522-6/h0150http://refhub.elsevier.com/S0306-2619(15)01522-6/h0150http://refhub.elsevier.com/S0306-2619(15)01522-6/h0155http://refhub.elsevier.com/S0306-2619(15)01522-6/h0155http://refhub.elsevier.com/S0306-2619(15)01522-6/h0155http://refhub.elsevier.com/S0306-2619(15)01522-6/h0160http://refhub.elsevier.com/S0306-2619(15)01522-6/h0160http://refhub.elsevier.com/S0306-2619(15)01522-6/h0160http://refhub.elsevier.com/S0306-2619(15)01522-6/h0165http://refhub.elsevier.com/S0306-2619(15)01522-6/h0165http://refhub.elsevier.com/S0306-2619(15)01522-6/h0165http://refhub.elsevier.com/S0306-2619(15)01522-6/h0170http://refhub.elsevier.com/S0306-2619(15)01522-6/h0170http://refhub.elsevier.com/S0306-2619(15)01522-6/h0170http://refhub.elsevier.com/S0306-2619(15)01522-6/h0175http://refhub.elsevier.com/S0306-2619(15)01522-6/h0175http://refhub.elsevier.com/S0306-2619(15)01522-6/h0175http://refhub.elsevier.com/S0306-2619(15)01522-6/h0175http://refhub.elsevier.com/S0306-2619(15)01522-6/h0175http://refhub.elsevier.com/S0306-2619(15)01522-6/h0180http://refhub.elsevier.com/S0306-2619(15)01522-6/h0180http://refhub.elsevier.com/S0306-2619(15)01522-6/h0180http://refhub.elsevier.com/S0306-2619(15)01522-6/h0185http://refhub.elsevier.com/S0306-2619(15)01522-6/h0185http://refhub.elsevier.com/S0306-2619(15)01522-6/h0185http://refhub.elsevier.com/S0306-2619(15)01522-6/h0185http://refhub.elsevier.com/S0306-2619(15)01522-6/h0190http://refhub.elsevier.com/S0306-2619(15)01522-6/h0190http://refhub.elsevier.com/S0306-2619(15)01522-6/h0190http://refhub.elsevier.com/S0306-2619(15)01522-6/h0195http://refhub.elsevier.com/S0306-2619(15)01522-6/h0195http://refhub.elsevier.com/S0306-2619(15)01522-6/h0195http://refhub.elsevier.com/S0306-2619(15)01522-6/h0200http://refhub.elsevier.com/S0306-2619(15)01522-6/h0200http://refhub.elsevier.com/S0306-2619(15)01522-6/h0215http://refhub.elsevier.com/S0306-2619(15)01522-6/h0215http://refhub.elsevier.com/S0306-2619(15)01522-6/h0215http://refhub.elsevier.com/S0306-2619(15)01522-6/h0220http://refhub.elsevier.com/S0306-2619(15)01522-6/h0220http://refhub.elsevier.com/S0306-2619(15)01522-6/h0220http://refhub.elsevier.com/S0306-2619(15)01522-6/h0225http://refhub.elsevier.com/S0306-2619(15)01522-6/h0225http://refhub.elsevier.com/S0306-2619(15)01522-6/h0230http://refhub.elsevier.com/S0306-2619(15)01522-6/h0230http://refhub.elsevier.com/S0306-2619(15)01522-6/h0230http://refhub.elsevier.com/S0306-2619(15)01522-6/h0240http://refhub.elsevier.com/S0306-2619(15)01522-6/h0240http://refhub.elsevier.com/S0306-2619(15)01522-6/h0240http://refhub.elsevier.com/S0306-2619(15)01522-6/h0245http://refhub.elsevier.com/S0306-2619(15)01522-6/h0245http://refhub.elsevier.com/S0306-2619(15)01522-6/h0245http://refhub.elsevier.com/S0306-2619(15)01522-6/h0250http://refhub.elsevier.com/S0306-2619(15)01522-6/h0250http://refhub.elsevier.com/S0306-2619(15)01522-6/h0250http://users%20ices%20utexas%20edu/~terejanu/files/tutorialEKFpdfhttp://users%20ices%20utexas%20edu/~terejanu/files/tutorialEKFpdfhttp://refhub.elsevier.com/S0306-2619(15)01522-6/h0270http://refhub.elsevier.com/S0306-2619(15)01522-6/h0270http://refhub.elsevier.com/S0306-2619(15)01522-6/h0270http://refhub.elsevier.com/S0306-2619(15)01522-6/h0275http://refhub.elsevier.com/S0306-2619(15)01522-6/h0280http://refhub.elsevier.com/S0306-2619(15)01522-6/h0280http://refhub.elsevier.com/S0306-2619(15)01522-6/h0280http://refhub.elsevier.com/S0306-2619(15)01522-6/h0285http://refhub.elsevier.com/S0306-2619(15)01522-6/h0285http://refhub.elsevier.com/S0306-2619(15)01522-6/h0285

A comparative study of three model-based algorithms for estimating state-of-charge of lithium-ion batteries under a new combined dynamic loading profile1 Introduction2 Experiments2.1 Model parameter identification test2.2 Algorithms evaluation test

3 Battery modelling3.1 The combined model3.2 Model parameter identification

4 Algorithm implementation for online estimation4.1 Extended Kalman filter4.2 Unscented Kalman filter4.3 Particle filter4.4 SOC estimation using each of the above three algorithms

5 Experimental results and discussion of SOC estimation5.1 Experiments on fully charged battery5.2 Experiments with unknown initial state5.3 Experiments with varying state of health

6 Conclusion and future workAcknowledgmentsReferences

a comparative study of three model-based algorithms for...

Documents