Estimation of State of Charge, Unknown Nonlinearities, and State of Health of a Lithium-Ion Battery...

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 3, MARCH 2014 1335

Estimation of State of Charge, UnknownNonlinearities, and State of Health of a

Lithium-Ion Battery Based on aComprehensive Unobservable Model

Mehdi Gholizadeh and Farzad R. Salmasi, Senior Member, IEEE

Abstract—This paper considers the estimation of the state ofcharge and state of health for lithium-ion batteries, while aninclusive model is taken into account. The model includes twoRC subnetworks, which represent the fast and slow transientresponses of the terminal voltage. Nevertheless, the linear partof the model is unobservable. On the other hand, the nonlinearbehavior of the open-circuit voltage versus state of charge isalso included in the model. The proposed observer tackles theaforementioned problems to attain a reliable estimation of thestate of charge. Moreover, as opposed to the methods in whichthe nonlinearities or uncertainties in the model are disregarded orthose terms are discarded using a conventional sliding-mode ob-server, an analytical method is considered to estimate the additivenonlinear or uncertainty term in the model. This approach leadsto a very accurate model of the battery to be used in a batterymanagement system. Moreover, an online parameter estimationmethod is proposed to estimate the battery’s state of health. Theproposed scheme benefits from an adaptive rule for the onlineestimation of the series resistance in the lithium-ion battery basedon the accurately identified model. Experimental tests certify theperformance and feasibility of the proposed schemes.

Index Terms—Adaptive observer, estimation, lithium-ionbattery, sliding motion, state of charge, state of health (SoH).

I. INTRODUCTION

L ITHIUM batteries have been used recently in many ap-plications such as portable power products and hybrid

electric vehicles. They play a significant role in powering thesetechnologies. To obtain the usable charge of the battery, tomanage it to its optimal potential, and to extend the lifetime ofthe battery, it is required to monitor the available charge or stateof charge (SoC) of the battery more accurately. The SoC, bydefinition, is the ratio of the available capacity to its maximumcapacity when the cell is completely charged [1]. There havebeen more than a dozen of schemes presented by researchers inrecent years to improve the SoC estimation. Coulomb countingor ampere-hour counting is the most commonly used method

Manuscript received October 20, 2012; revised January 15, 2013 andFebruary 28, 2013; accepted March 26, 2013. Date of publication April 24,2013; date of current version August 23, 2013.

The authors are with the Faculty of Engineering, School of Electrical andComputer Engineering, University of Tehran, Tehran 14395, Iran (e-mail:[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2013.2259779

for SoC estimation in which the time integral of the batterycurrent is considered as a direct SoC indication [2] as follows:

SoC(t) = SoC(t0)−t∫

t0

ηI

3600Cndτ (1)

where SoC(t0) is the initial value of the SoC, Cn stands forthe nominal capacity of the battery (in ampere-hours), I is theinstantaneous current (positive for discharge and negative forcharge), and η represents the Coulomb coefficient. However,this method accumulates errors in measurements and may leadto large SoC errors in real-world applications. An improvedampere-hour counting method is reported in which the open-circuit voltage (OCV) is taken into account. The measuredOCV is used for the periodic correction of the SoC estimatedby ampere-hour counting [3]. However, OCV can only bemeasured after a long rest of the battery. Hence, this strategyis not desirable in practice. In [4], the electromotive force(EMF) voltage of the battery is used to predict SoC. The EMFis approximated based on the impedance, load current, andterminal voltage of lead–acid and lithium-ion batteries.

The Kalman filter is inherently a recursive algorithm for theestimation of internal states of a dynamic system. It has beenwidely used for estimating the OCV or other parameters ofthe battery which have direct relation with the SoC (e.g., see[5]–[7]). To take the nonlinear behavior of OCV versus SoCinto account, a nonlinear extension of this filter, known as theextended Kalman filter (EKF), is considered. EKF is anotherpopular statistical tool to extract internal battery parameterssuch as SoC [8]–[10]. Furthermore, EKF is deployed by Kimand Cho to estimate both the SoC and state of health (SoH) oflithium-ion batteries, as reported in [11] and [12]. It should benoted that assumptions on certain statistical properties, alongwith the local linearization of the state equations, are thedrawbacks of this renowned filter. Improved versions of EKFsuch as the sigma-point EKF [13] and adaptive EKF [14] arealso reported for battery applications.

Adaptive and robust observers are also utilized for stateestimation in batteries. An H∞ observer is designed to esti-mate SoC based on a linear battery model in [15]. Li et al.used a class of piecewise linear systems as the lithium-ionbattery model and then proposed an adaptive observer basedon the model estimate SoC [16]. Artificial neural networks

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1336 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 3, MARCH 2014

(ANNs) and fuzzy logic identifiers (FLIs) are other adaptiveor intelligent alternatives for estimating SoC. In [17], fuzzylogic models were used by Sing et al. for SoC estimationand available capacity approximation based on measuring theelectrochemical impedance spectroscopy of nickel/metal bat-teries. Aside from fuzzy logic, ANNs have gained much popu-larity for estimation and prediction, specifically for direct SoCestimation [18]–[21]. In [22], a neuro-fuzzy identifier knownas locally linear model tree network is developed for SoCapproximation. Although ANNs take the nonlinearities of thebattery model into account, they suffer from the training processand the dependence to large training data sets. ANNs and FLIsare also reported as SoH estimators [23], [24]. Recently, theimpulse response of a lithium battery is used for online SoCestimation [25].

In the presence of modeling errors and uncertainties, sliding-mode observers (SMOs) have been used as a reliable and robusttool for SoC or SoH estimation [26]. Kim proposed a SMObased on a simple RC model for SoC estimation, which wasrobust to the modeling uncertainties [27]. He extended hisapproach to evaluate the lithium battery SoH by the adaptiveestimation of the series capacitance and resistance as the indica-tors of battery SoH , as reported in [28]. Recently, Einhorn et al.used the measured SoC at two states and the transferred chargebetween those states to estimate the cell capacity for lithium-ionbatteries [29].

In this paper, new schemes for estimating the SoC and SoHof lithium-ion batteries are proposed, while an inclusive modelis taken into account. The model encloses two RC networks, aseries resistor, and a nonlinear voltage source with a parallelRC circuit. Nevertheless, designating the terminal and RCsubnetwork voltages, along with the SoC, as the model dynam-ical states, the system state equation is linearly unobservable.This problem is a setback to directly develop an observer forthe system. On the other hand, the nonlinear behavior of theOCV versus SoC is also encompassed by the model. Theproposed SMO tackles the aforementioned problems. More-over, unlike the methods that disregard the nonlinearities oruncertainties in the model or dump those terms using a con-ventional SMO, an analytical method is considered to estimatethe additive nonlinear or uncertainty term in the model. Thisleads to a very accurate model of the battery for control andmanagement purposes. Finally, an online parameter estimationmethod is proposed to estimate the SoH of the battery. Thismethod uses an adaptive rule to estimate the series resistanceof the battery based the accurately identified battery model.Experiments are performed for the verification of the proposedmethods. Contributions of the scheme can be summarized asfollows:

1) SoC estimation based on an unobservable but detectablenonlinear comprehensive model for lithium batteries;

2) direct estimation of battery model nonlinearities anduncertainties to derive an inclusive model of lithiumbatteries;

3) SoH estimation for the lithium battery based on thederived inclusive model with an identified additive termwhich represents nonlinearities and uncertainties.

Fig. 1. Equivalent circuit model of the lithium-ion battery.

II. COMPREHENSIVE BATTERY MODEL

Due to the variety of applications for electrochemical bat-teries, the reported battery models cover a wide spectrum.Nevertheless, they can be classified into electrochemical, math-ematical, intelligent, and electrical models. Electrical modelsare more apt to be used for estimation and control purposes.In the proposed scheme, an inclusive verified electrical batterymodel proposed by Chen and Rincon-Mora in [30] is adopted,as shown in Fig. 1. This electrical model mimics the precisedynamic behavior of the lithium-ion battery.

In this model, the nonlinear mapping from the battery’s stateof charge to the OCV is represented by a controlled voltagesource, Voc. Furthermore, an instantaneous terminal voltagevariation due to the load current IB is given by inserting a seriesresistance denoted by R0. Nevertheless, for a step load currentchange, the terminal voltage response is far from an ideal stepwaveform, as it shows short and long time constant dynamics.These dynamics are modeled by two series RC networks,denoted by (Rf , Cf ) and (Rs, Cs), respectively. The whole-charge capacitor is denoted by Cb, and the self-discharge energyloss due to long time storage is represented by Rsd. Owing tothe slow variation of the no-load battery’s usable capacity, theself-discharge resistance may be taken out of the picture, butin order to be more precise, it is included in the model in ourscheme.

The dynamics of the voltages across the capacitors, Vf andVs, are given by

Vf = − 1

RfCfVf +

1

CfIB

Vs = − 1

RsCsVs +

1

CsIB (2)

where IB is the battery terminal current. The voltage across thewhole-charge capacitor is VSoC , and its dynamic is given by

VSoC = − 1

RsdCbVSoC − 1

CbIB . (3)

The battery terminal voltage can be expressed as

VB = Voc −R0IB − Vf − Vs. (4)

Voc(SoC) has an almost linear behavior in the region between10% and 100% SoC and exponentially drops while the SoC isapproximately below 10%. Therefore, the OCV is related to theSoC as follows:

Voc = αVSoC + g(VSoC) (5)

where g(VSoC) stands for the nonlinear part of the OCV. Giventhe fact that the rate of change of the terminal current is

GHOLIZADEH AND SALMASI: ESTIMATION OF SoC, UNKNOWN NONLINEARITIES, AND SoH OF LI-ION BATTERY 1337

negligible in comparison with other time constants in a sam-pling period, the time derivative of the terminal voltage isdetermined to be

VB =∂Voc

∂VSoCVSoC − Vf − Vs

∂Voc

∂VSoC=α+ g(VSoC). (6)

From (3), (4), and (6), the dynamic of the terminal voltage isdetermined to be

VB =

(− α

RsdCb+

α

RsCs

)VSoC +

(1

RfCf− 1

RsCs

)Vf

− 1

RsCsVB −

(R0

RsCs+

α

Cb+

1

Cf+

1

Cs

)IB

+Φ(VSoC , IB) (7)

where Φ(VSoC , IB) stands for the nonlinearities in the terminalvoltage dynamic. Moreover, the modeling uncertainties areconsidered to be embedded into this term.

Defining ξ[VSoCVfVSVB ]T as the state vector and denoting

the battery current and terminal voltage as the system inputand output, respectively, the state and output equations of thebattery based on the adopted model are given by

ξ = Aξ +BIB +DΦ VB = Cξ (8)

where

A=

⎡⎢⎢⎢⎣− 1

RsdCb0 0 0

0 − 1RfCf

0 0

0 0 − 1RsCs

0

− αRsdCb

+ αRsCs

1RfCf

− 1RsCs

0 − 1RsCs

⎤⎥⎥⎥⎦B = [ b1 b2 b3 b4 ]

T , C=DT =[ 0 0 0 1 ]

b1 = − 1

Cb, b2 = − 1

Cf, b3 =

1

Cs,

b4 = − Ro

RsCs− α

Cb− 1

Cf− 1

Cs.

The observability matrix of the linear system is given by[C CA CA2 CA3]T . It turns out that one column in thismatrix is zero. Therefore, the battery model is unobservable.This problem is a setback to design any observer to estimatethe state vector ξ. However, since the A-matrix is stable, thebattery model is detectable. It should be noted that Φ[VSoC , IB ]is bounded with respect to the battery current. Therefore, thereexists ϕ(IB) such that

|Φ(VSoC , IB)| ≤ ϑ(IB). (9)

III. OBSERVER DESIGN FOR SoC ESTIMATION

In this section, a sliding-mode-type observer is designed forthe robust estimation of the states of the lithium-ion batterybased on the unobservable model in (8). Later, using thedesigned observer, the nonlinear part of the battery model,

Φ[VSoC , IB ] is estimated directly to attain an inclusively iden-tified battery model. The proposed observer is as follows:

˙ξ = Aξ +BIB +MeVB

+ μDρ(eVB) VB = Cξ. (10)

It should be noted that eVB= VB − VB . The observer correc-

tion function is chosen as follows:

ρ(eVB) =

eVB(t)

|eVB(t)|+ γ(t)

(11)

where γ(t) : R+ → R+ is a continuous function such that γ <

−cγ(t) and c is a positive scalar. Observer gain matrix M couldbe determined by pole assignment, but the pair (A,B) is unob-servable. Thus, it hinders us to design the observer directly. Toovercome this setback, the system dynamic is decomposed intoobservable and unobservable modes. Therefore, the observablemodes of the system can be arbitrarily placed. The decomposedsystem equations are given by[

ξoξu

]=

[Ao 00 Au

] [ξoξu

] [Bo

Bu

]IB +DΦ(VSoC , IB)

VB =C

[ξoξu

](12)

where ξu and ξo denote the observable and unobservablemodes, respectively. Moreover

Ao =

⎡⎢⎣ − 1RsdCb

0 0

0 − 1RfCf

0

− αRsdCb

+ αRsCs

1RfCf

− 1RsCs

− 1RsCs

⎤⎥⎦Au = − 1

RsCs, B0 = [ b1 b2 b4 ]

T , Bu = − 1

Cs

D = [DT0 0 ]T , C = [Co 0 ]

Co =DTo = [ 0 0 1 ] . (13)

Since the unobservable subsystem Au is stable, the systemis detectable. Also, the observer gain is considered to be M =[Mo 0]T . Therefore, the observer dynamics are decomposedas follows:[˙ξo˙ξu

]=

[Ao 00 Au

] [ξξu

]+

[Bo

Bu

]IB

[Mo

0

]eVB

+μDρ(eVB)

VB =C

[ξoξu

]. (14)

For the observable subsystem, Mo is obtained arbitrarily usingpole placement. Therefore, for a given Qo > 0, there exists asymmetric Po > 0 such that

Po(Ao −MoCo) + (Ao −MoCo)TPo = −Qo. (15)

Subtracting (14) from (12), the state estimation error system isobtained as follows:

e =

[eoeu

]=

[Ao −MoCo 0

0 Au

] [eoeu

]+DΦ(VSoC , IB)− μDρ(eVB

)

eVB=Ce = C

[eoeu

]. (16)


A conventional Lyapunov function for the error system isgiven by

V (e) = eτPe. (17)

The P_ matrix is chosen to be diag(P0, 1). Moreover, it isassumed that, for some F > 0, FC = DTP . This is a commonassumption in the sliding-mode literature (e.g., see [31]).

The determination of the derivative of the above functionalong the error trajectory in (16) yields

V (e) = eP e+ eTP e =

[eoeu

]TP

[eoeu

]TP

[eoeu

]= − eTo Qoeo + 2eTuAueu + 2D

TPe (Φ− μρ(eVB

))

= − eTo Qoeo −2

RsCs|eu|2

+ 2FeVB

(Φ− μ

eVB

|eVB|+ γ(t)

). (18)

On the other hand, we have

e2VB

|eVB|+ γ(t)

= |eVB| − γ(t) +

γ2

|eVB|+ γ

= |eVB| − δ(t).

(19)Since γ(t) is positive for every t, then

δ(t) = γ(t)− γ2

|eVB|+ γ

> 0. (20)

Consequently, it leads to the following inequality:

V (e) ≤ −λmin(Q0)‖eo‖2 −2

RsCs|eu|2 + 2Fμδ(t)

+2F (eVBΦ− μ|eVB

|) . (21)

Since γ(t) < −cγ(t), δ(t) asymptotically converges to zero.Therefore, if μ is chosen such that

μ > ϑ(IB) > |Φ(VSoC , IB)| (22)

then V (e) < 0, and consequently, the asymptotic stability ofthe error system is ensured. Thus, the estimated SoC voltageconverges to the real value.

IV. ESTIMATION OF NONLINEARITIES

The battery model in (8) is not thoroughly identified if thenonlinear term Φ(VSoC , IB) is not reconstructed. The estima-tion of SoC based on the SMO relies only on an upper bound ofthis term. Nevertheless, in order to simulate the battery transientresponses and design charge controllers and battery manage-ment systems, the reconstruction of the nonlinear behavior ofthe battery is unavoidable. Although nonlinear approximatorssuch as neural networks or fuzzy identifiers may be utilized forthis purpose, they require a large number of training data sets.In the new scheme, we extended the results from the last sectionto analytically estimate the nonlinear term.

In the previous section, the state estimation error system hasbeen obtained by decomposing the system to observable andunobservable modes, as it appears in (12).

Defining Ao = Ao −MoCo, then, the dynamic of the ob-servable error subsystem is given by

eo = Aoeo +DoΦ(VSoC , IB)− μDoρ(eVB)

eVB=Coeo = [ 0 0 1 ] eo. (23)

Decomposing eo and matrix Ao as follows:

eo =

[eo1eVB

], Ao =

[Ao11 Ao12

Ao21 − 1RsCs

](24)

results in

eo1 = Ao11eo1 + Ao12eVB

eVB= A021eo1−

1

RsCseVB

+Φ(VSoC , IB)−μρ(eVB). (25)

For this end, it is shown that, with the proposed ob-server, the error vector asymptotically converges to zero, i.e.,limt→∞ eo(t) = 0. Consequently, eo1(t) is bounded such that‖A21eo1‖ ≤ σ. In order to show that a sliding motion takesplace in finite time, i.e., eVB

= eVB= 0, the reachability con-

dition should be verified. For this purpose, we have

eVBeVB

= eVBA21eo1 −

1

RsCse2VB

− eVB(μρ(eVB

)− φ)

(26)

with μ > ϕ(IB) > Φ(VSoC , IB) and ‖A21eo1‖ ≤ σ, and ityields to

eVBeVB

≤ σ|eVB| 1

RsCse2VB

− |eVB| (μ− γ(IB))

+eVBδ(t) ≤ −ξ|eVB

|+ eVBδ(t) (27)

in which ξ = (μ− γ(IB)− σ). Therefore, if

μ > γ(IB) + σ (28)

and considering the fact that δ(t) asymptotically convergesto zero, then we conclude that the sliding motion maintainsthereafter. Now, from (25), during the sliding motion, we have

Φ(VSoC , IB) = μρ(eVB)− A21eo1 (29)

where ρeq is known as the equivalent output injection signal[32]. Let us introduce the estimated nonlinear term as follows:

Φ(VSoC , IB) = μρ(eVB) = μ

eVB(t)

|eVB(t)|+ γ(t)

. (30)

Then, we have

|Φ− Φ|=∣∣μρeq − A21eo1 − μρ

∣∣ ≤ ∥∥A21

∥∥ ‖eo1‖+ μ|ρeq − ρ|.(31)

Since limt→∞(ρeq − ρ) = 0 and eo1 tends to zero, it followsthat

limt→∞

|Φ− Φ| = 0. (32)


V. SoH ESTIMATION BASED ON ADAPTIVE OBSERVER

There are several definitions for SoH such as the batterycell’s capability of storing energy and preserving charge forlong periods [23] or the ratio of the amount of charge thatcan be drawn from a new battery to the aged battery [24]. Thebattery cranking ability has been used as a token for batterySoH approximation by Cugnet et al. [33].

Evaluating the battery SoH , a battery management systemmay conclude about the battery’s end of life (EoF) and signalsfor the replacement or restrict the number of cycles [5]. Theincrease and decrease of series resistance and capacitance,respectively, are reported as the indications of reduced SoHby Kim in [27]. In [5], the 60% decrease of battery maximumpower compared to its initial value is reported to be an indi-cation of the battery’s EoF. Thus, battery EoF is detected oncethe series resistance of the battery reaches to 160% of its initialvalue at the same condition of temperature and SoC, i.e., [34]

SoH =R0,EoF −R0

R0,EoF −R0,new× 100% (33)

where R0, R0,EoF , and R0,new denote the current, the EoF, andthe new battery series resistance.

In the following, an adaptive scheme is developed to estimatethe series resistance as an indicator of SoH based on theinclusive battery model in (8) and the results of the last sections.The proposed SoH estimator has the following structure:

˙ξ = Aξ +BIB +DΦ(VSoC , IB). (34)

It should be noted that R0 only appears in the B-matrix.Therefore, in the above observer equation, an estimatedB-matrix, i.e., B, is substituted. The estimation error dynamicsis given by

ε = ξ − ˙ξ = Aε+ (B − B)IB +D(Φ− Φ). (35)

Moreover

B − B = − R0

RsCsD (36)

where R0 = R0 − R0. The estimated resistance is denoted byR0. Based on the above dynamic, a simple adaptive observeris designed to estimate the series resistance. The followingLyapunov function:

V (ε) = −1

2εTMε+

1

2τR2

0, M > 0 (37)

yields to

˙V (ε)=−12εTNε+(B−B)TMε IB− 1

τR0

˙R0+εTMD(Φ−Φ).

(38)With the following adaptation rule:

˙R0 = τDMε (39)

we have

V (ε) = −1

2εTNε+ εTMD(Φ− Φ). (40)

Fig. 2. Schematic of the test setup for lithium-ion battery.

Therefore, from (32), it follows that ˙V (ε) ≤ 0. It should benoted that N = −(ATP + PA) is positive definite.

VI. EXPERIMENTAL VERIFICATIONS

A. Test Setup

To identify and validate the battery model and verify theproposed SoC and SoH estimation methods, a test systemwas designed. The battery studied in this test was a lithium-ion battery, which is widely used in laptop computers and hasa nominal capacity of 2400 mAh. The nominal, maximum, andcutoff voltages of the battery under study are 3.6, 4.1, and 3 V,respectively. The test system includes a programmable resistiveload consisting of a few resistances in parallel and series tocontrol the discharge current of the battery by adding or remov-ing the resistances, a 1-Ω resistance as the current sensor (thevoltage drops across this resistor indicate the battery current),an AVR microcontroller to control the load and to measurethe current and voltage signals, an LCD for displaying pur-poses, and a MAX-232 interface for transferring measured datato a PC. Two ADC channels of the microcontroller with 10-bdigital output were assigned for the current and voltage signalmeasurement in a sampling rate of 20 Hz. The test setup isdepicted in Fig. 2.

To obtain a discharge profile for the battery, the pro-grammable resistive load is applied. By programming the load,an arbitrary current profile can be generated. These characteris-tics of the battery can be used for identifying the battery modeland evaluating the proposed methods.

B. Model Validation

Based on the model identification scheme proposed bySpagnol et al. [35] and from (4) and (7), it follows that

VB = Voc −R0IB −Gm(s)IB (41)

where

Gm(s) =Rs

1 + sRsCs+

Rf

1 + sRfCf. (42)


Fig. 3. Discharge current and voltage measured for extracting OCV versusSoC.

Fig. 4. SoC calculated using the ampere-hour counting method.

In the following, schemes for the identification of Voc, R0, andGm(s) parameters are explained.

1) OCV Versus SoC Measurement: The OCV is usuallymeasured as the steady-state open-circuit terminal voltage atdifferent SoCs. However, it takes around a couple of hours toattain steady-state condition. In practice, a shorter idle time,around a quarter of an hour, is enough. In order to extract theOCV versus SoC, a discharge test is run, in which the battery isconnected to the programmable resistive load with a resistanceof 1.84 Ω and discharged from a fully charge condition by a0.93-C rate. The discharge and idle times were 120 and 720 s,respectively. The test repeated until the terminal voltage re-duced to the cutoff voltage of 3 V. The measured currentand voltage during the discharge test are shown in Fig. 3.The squares represent the values of OCV. From the dischargecurrent, the SoC of the battery can be calculated using theampere-hour counting method. The calculated SoC has beenshown in Fig. 4. Moreover, the profile of OCV versus SoC isextracted as shown Fig. 5.

2) Parameter Estimation and Model Validation: To estimatethe parameters of the model shown in Fig. 1, we use thedischarge test data and (42) in discrete form.

1) Cb: The whole-charge capacitor Cb, neglecting the tem-perature and cycle dependences, is given by Cb = 3600×(Nominal Capacity)Ah. Therefore, it is determined to be8640 F.

2) Rsd: This resistor is estimated so that the SoC obtainedfrom (6) will be consistent with the SoC calculated usingthe ampere-hour counting method. Therefore, the valueof 100 Ω has been obtained for this resistor.

3) R0: As proposed in [35], to calculate the internal resis-tance, the ratio of the voltage variation to the current jump

Fig. 5. Measured OCV versus SoC.

Fig. 6. Current and voltage profiles used for R0 identification.

is used. The value of R0 at the ith point of the currentjump can be given by

R0,i =vmax,i − vmin,i

imax,i − imin,i(43)

where vmax,i, vmin,i, imax,i, and imin,i are shown inFig. 6. The mean value of R0,i at some points of thecurrent jump, i.e., R0 = Σn

1R0,i/n, gives the best ap-proximate of R0.

4) RC networks: For estimating the parameters of the RCnetworks in the electrical model, (41) is rewritten asfollows:

Voc − VB −R0IB = GmIB ≡ Veq. (44)

Assuming dIB/dt ≈ 0 over each sampling period, (44) isdiscretized as follows:

Veq[k] =

(Rf (1− e−Ts/Tpf )z−1

1− e−Ts/Tpf z−1

+Rs(1− e−Ts/Tps)z−1

1− e−Ts/Tpsz−1

)IB [k]

=

(z−1(b1 + b2z

−1)

1 + a1z−1 + a2z−2

)IB [k] (45)


TABLE IIDENTIFIED RC-NETWORK PARAMETERS

Fig. 7. Measured output voltage versus the model output.

in which TS is the sampling period and Tps and Tpf

represent the time constants for the RC subnetworks.Thus, a1, a2, b1, and b2 are estimated using the dynamicARMAX model, in which the parameters of the RCnetworks are obtained. The results of the parameter esti-mation procedure are given in Table I. With the estimatedparameters, the measured and modeled outputs have beencompared in Fig. 7. It can be seen that the maximummodeling error is at most 30 mV. For further validation,another discharge test has been applied to the battery. Inthis case, the discharge and rest intervals are 100 and700 s, respectively. In Fig. 8, the modeled voltage againstthe measured one is shown. It is right to conclude thatthere is a good match between the true voltage and theestimated one with less than 30-mV error.

C. Experimental Verification of Estimators

1) SoC Estimation: Based on the identified parameters ofthe battery model, the proposed observer is designed. Theclosed-loop pole set of the observable subsystem is chosen to be[−0.000116 − 0.3 − 3], which yields to the observer gain[0.000121 3.783 − 3.661]. Furthermore, μ = 0.5, γ(t) =0.005e−0.001t, and Q0 = diag([0.0001 0.1 0.1]). The dis-charge current profile of the battery can be seen in Fig. 9. Thebattery current decreases during discharge, while a constantresistive load was drawing the current.

In Fig. 10, the measured terminal voltage is shown andcompared to the estimated terminal voltage by the observer.From this figure, it is clear that the proposed observer demon-strates desirable performance in estimating the terminal volt-age. Despite the sudden jumps caused by the current jump, the

Fig. 8. Comparison of the model-estimated voltage to the measured voltagebased on the new current profile.

Fig. 9. Discharge current profile of the battery.

Fig. 10. Estimated terminal voltage compared to the measured profile.

estimation error tends to zero quickly. The amplitude of thesejumps depends on the assigned poles of the observer.

The SoC estimation result is given in Fig. 11. It showsthat the estimation error is less than 1.2%, when compared tothe SoC calculated from the calibrated ampere-hour countingmethod. The results verify the effectiveness of the proposedscheme for SoC estimation.


Fig. 11. Estimated SoC compared to the measured profile by precise andcalibrated ampere-hour counting.

Fig. 12. Estimation of the nonlinear part of the battery model.

2) Estimation of Nonlinearities: The nonlinear term Φ inthe lithium-ion battery model is estimated directly based on(30) and compared to the measured profile. The measuredprofile is obtained by subtracting the output of the linear batterymodel, i.e., (8) without the additive nonlinear term, from themeasured terminal voltage. From Fig. 12, it can be seen that theestimated nonlinear term converges to the measured profile.

3) SoH Estimation: The SoH of the battery is determinedbased on (33). All the experiments have been performed atroom temperature. The initial series resistance of the bat-tery was R0,new = 0.088 Ω. Therefore, the EoF resistance isR0,EoF = 0.1408. Moreover, the series resistance at the begin-ning of the experiment was R0 = 0.0972 Ω, and thus, the SoHof the test bench battery was determined to be 82.6%.

In order to verify the adaptive scheme for SoH estimation,it was assumed that the series resistance was unknown. Theadaptation gain was chosen to be 20. The results of R0 andSoH estimation using the proposed adaptive method have beenshown in Figs. 13–15. In Fig. 13, the adaptive model output(terminal voltage) tracks the measure value, and the estimation

Fig. 13. Adaptive model output compared to the measured terminal voltage.

Fig. 14. Estimated series resistance compared to the measured value.

Fig. 15. Estimated SoH compared to the measured value.

error tends to zero. Furthermore, Figs. 14 and 15 verify that theestimated series resistance and SoH converge to their measuredvalues based on offline verifications.

4) Updating Model Resistance Variations With Temperature:In order to consider the thermal effect on model resistances,the battery parameters are updated every 1000 s to adapt tothe thermal variations. The flowchart of the proposed updatingscheme is shown in Fig. 16. The estimated model resistancesbased on this scheme are shown in Fig. 17. Moreover, because


Fig. 16. Updating model resistances based on ARMAX model.

Fig. 17. Updated model resistance values versus temperature.

of using a SMO and the estimation of the nonlinear term,which represents the nonlinear dynamics and uncertainties, theproposed SoC observer is inherently robust to the temperature

variations. However, the state of health can be updated based onthe modified adaptation of R0.

VII. CONCLUSION

Based on an inclusive and unobservable model of the lithium-ion battery, a new scheme for the estimation of the battery SoCis presented. The model is decomposed into the observable andunobservable subsystems. An accurate model for the battery hasbeen developed by estimating the nonlinear part of the batterymodel. For this purpose, the reachability of the sliding motion isinvestigated. Using this accurate battery model, an adaptive pa-rameter estimator method has been proposed for estimating theseries resistance to monitor the battery SoH . The effectivenessof the proposed schemes has been verified experimentally on alithium-ion battery, which shows the feasibility and eligibilityof the proposed methods in providing accurate estimation of theSoC, nonlinear term in the mathematical model, and SoH ofthe battery.

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Mehdi Gholizadeh received the B.S. and M.S. de-grees in control engineering from the University ofTehran, Tehran, Iran, in 2009 and 2012, respectively.

His research interests include battery managementsystems, battery charging, estimation techniques, ro-bust adaptive observers, and self-oscillating adaptivesystems.

Farzad R. Salmasi (S’99–M’02–SM’09) receivedthe B.S. degree in electrical engineering from theSharif University of Technology, Tehran, Iran, in1994, the M.Sc. degree in electrical engineering fromthe Amirkabir University of Technology, Tehran, in1997, and the Ph.D. degree in electrical engineeringfrom Texas A&M University, College Station, TX,USA, in 2002.

From 1999 to 2002, he worked as a ResearchAssistant for the Advanced Motor Drives and Hy-brid Vehicles Laboratory, Department of Electrical

engineering, Texas A&M University. From 2002 to 2004, he was with theElectro Standards Laboratories, Cranston, RI, USA, as a Research Scientist. InSeptember 2004, he joined the School of Electrical and Computer Engineering,Faculty of Engineering, University of Tehran. He is currently the Head of theControl Engineering Department. His main research areas include the designand control of advanced motor drives, power electronic systems, hybrid electricvehicles, networked control systems, and smart grids.

Dr. Salmasi is a technical committee member of the International Associationof Science and Technology for Development on power and energy systems.

Estimation of State of Charge, Unknown Nonlinearities, and State of Health of a Lithium-Ion Battery...

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