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Model-Based Condition Monitoring for Lithium-ion Batteries

Kim, T.; Wang, Y.; Sahinoglu, Z.; Wada, T.; Hara, S.; Qiao, W.

TR2015-055 November 2015

AbstractCondition monitoring for batteries involves tracking changes in physical parameters and oper-ational states such as state of health (SOH) and state of charge (SOC), and is fundamentallyimportant for building high-performance and safety-critical battery systems. A model-basedcondition monitoring strategy is developed in this paper for Lithium-ion batteries on the basisof an electrical circuit model incorporating hysteresis effect. It systematically integrates 1)a fast upper-triangular and diagonal recursive least squares algorithm for parameter identi-fication of the battery model, 2) a smooth variable structure filter for the SOC estimation,and 3) a recursive total least squares algorithm for estimating the maximum capacity, whichindicates the SOH. The proposed solution enjoys advantages including high accuracy, lowcomputational cost, and simple implementation, and therefore is suitable for deployment anduse in real-time embedded battery management systems (BMSs). Simulations and experi-ments validate effectiveness of the proposed strategy.

Journal of Power Sources

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Copyright c© Mitsubishi Electric Research Laboratories, Inc., 2015201 Broadway, Cambridge, Massachusetts 02139

Model-Based Condition Monitoring for Lithium-ion Batteries✩

Taesic Kima, Yebin Wangb, Huazhen Fangc, Zafer Sahinoglub, Toshihiro Wadad, Satoshi Harad,Wei Qiaoe

aDepartment of Computer Science and Computer Engineering, University of Nebraska-Lincoln, Lincoln, NE68588-0511 USA

bMitsubishi Electric Research Laboratories, 201 Broadway,Cambridge, MA 02139, USAcDepartment of Mechanical Engineering, University of Kansas, Lawrence, KS 66045, USA

dAdvanced Technology R&D Center, Mitsubishi Electric Corporation, 8-1-1, Tsukaguchi-honmachi, Amagasaki City,661-8661, Japan

eDepartment of Electrical and Computer Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0511USA

Abstract

Condition monitoring for batteries involves tracking changes in physical parameters and opera-tional states such as state of health (SOH) and state of charge (SOC), and is fundamentally impor-tant for building high-performance and safety-critical battery systems. A model-based conditionmonitoring strategy is developed in this paper for Lithium-ion batteries on the basis of an electricalcircuit model incorporating hysteresis effect. It systematically integrates 1) a fast upper-triangularand diagonal recursive least squares algorithm for parameter identification of the battery model,2) a smooth variable structure filter for the SOC estimation,and 3) a recursive total least squaresalgorithm for estimating the maximum capacity, which indicates the SOH. The proposed solutionenjoys advantages including high accuracy, low computational cost, and simple implementation,and therefore is suitable for deployment and use in real-time embedded battery management sys-tems (BMSs). Simulations and experiments validate effectiveness of the proposed strategy.

Keywords: Lithium-ion battery monitoring, fast upper-triangular and diagonal recursive leastsquares, maximum capacity estimation, recursive total least squares, smooth variable structurefilter, state of charge, state of health

1. Introduction

Lithium-ion (Li-ion) batteries have gained widespread usein applications ranging from con-sumer electronics devices to power tools and to electric vehicles (EVs) due to their high energyand power densities and long cycle life [1]. However, effective battery monitoring and control,equivalently battery management systems (BMSs), remains aremarkable challenge and necessity,having spurred a wealth of research on their core algorithms[2]. A key mission of a BMS is to

✩Corresponding author Y. Wang. Tel: 001-617-621-7500; fax:001-617-621-7550. Email address: yebin-wang@ieee.org

Preprint submitted to Journal of Power Sources March 20, 2015

monitor the state of charge (SOC) [3, Sec. 3.2], state of health (SOH), and battery parametersincluding impedance and capacity [4]. A precise understanding of these variables is crucial for aseries of BMS tasks, e.g., charging, discharging, cell balancing, and fault prognosis and diagnos-tics, to improve operational performance, safety, reliability and lifespan of batteries. However, itcannot be obtained by direct measurement and instead, needsto be built upon estimation. Suchestimation algorithms are expected to have low complexity and high computational efficiency, aprerequisite for their online implementation on resource constrained platforms such as embeddedBMSs.

Research efforts on online battery parameter estimation have led to two families of methods:Kalman filter (KF)-based and regression-based. A variety ofKFs, including the linear KF [5],extended KF (EKF) [3], and sigma point KF (SPKF) [6] have beenused to estimate battery pa-rameters and states simultaneously. Compared to the KF-based solution, the least squares methodand its variant are more computationally competitive without compromising much accuracy, thusholding significant potential for battery model identification. See [7–11] and references thereinfor details. Recently, an upper-triangular and diagonal (UD) factorization-based RLS estimationmethod with an EF [12] was proposed to solve the digital computer implementation problem of theRLS. In additional to fast speed, this method has improved numerical stability with preservationof a positive covariance.

The SOC estimation has been a subject of intensive research,which leads to a variety of esti-mation algorithms. The easiest-to-implement ones includevoltage translation and Coulomb count-ing [13]. However, multiple issues render them unreliable,especially the sensitivity to initial SOCestimate and accumulative integration errors. The computational intelligence (CI)-based methods,e.g., artificial neural network (ANN) [14], fuzzy logic [15], and support vector machine [16], con-duct the SOC estimation through data-driven learning of thenonlinear relationship between theSOC and measured quantities such as battery voltage, current and temperature. The learning pro-cess is nonetheless adverse to real-time execution due to the high computational burden. Applyingstate filters and observers to electrochemical or electrical circuit models, model-based methodshave been attracting considerable attention as an effective means to improve the SOC estimationaccuracy. The EKF has been widely used for SOC estimation [17, 18], and its upgraded variant,the iterated EKF (IEKF), is used in [19] for simultaneous SOCand model parameter estimation.In [6, 20], the hysteresis effect inherent in batteries [21]is accounted for with the developmentof approaches based on a dual EKF and a dual sigma-point KF (SPKF), respectively. However,the IEKF, dual EKF and SPKF increases the accuracy at the expense of higher computational ef-fort. One major alternative thus is the more computationally efficient observer-based approaches,e.g., linear observer [10], sliding mode observer [9, 11], nonlinear geometric adaptive observer[22], and partial differential equation (PDE) observer [23]. A further advantage they have is theavailability of convergence analysis for the estimation error dynamics. Meanwhile, some workattempts to combine the advantages of the aforementioned methods, an example of which is theSOC estimation using neural networks and EKF in [24].

The SOH measures the battery aging and wear, which correspond to capacity fade and powerfade [20]. Accordingly, battery maximum capacity [17, 25],and impedance components (e.g.impedance [26], internal resistance [27], diffusion resistance [28] and a diffusion capacitance[29]) are the commonly used parameters to quantify the SOH. Some straightforward ways to infer

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SOH, e.g., evaluating the maximum capacity via a full discharge test with a small current [25]or measuring the impedance [26], are not suitable for real-time estimation. This is because themaximum capacity declines gradually due to the aging and degradation, and fluctuates accordingto temperature. A knowledge of its accurate value is indispensable for SOC estimation (espe-cially in the standalone case), health prognosis and other battery management tasks. The onlineSOH estimation has been tackled by CI-based methods, e.g., ANN [30, 31], adaptive recurrentNN [32], and structured neural network (SNN) [33], and model-based methods, such as the dualEKF [6, 17, 20, 34] and the dual sliding mode observer [35]. Additionally, analytical approaches,including the two-point (TP) of SOCs method [11] and recursive total least squares (RTLS) [36],have been developed and exploited to estimate the maximum capacity based on Coulomb counting.

This paper proposes a comprehensive strategy for online condition monitoring of Li-ion bat-teries. Its design is based on a battery model that captures both the electrical circuit characteristicsand the hysteresis. The monitoring solution consists of three interrelated algorithms for batteryparameter, SOC, and SOH estimation, respectively. Specifically, a fast UD recursive least squares(FUDRLS) method is built to identify the battery model parameters. Based on the fully identi-fied model, a smooth variable structure filter (SVSF) is designed to perform the SOC estimation.Finally, the battery’s maximum capacity is determined by a Rayleigh quotient-based RTLS algo-rithm taking the estimated SOCs and measured current as inputs. The proposed algorithms areintegrated to run in parallel but at multiple time scales to achieve the best use of computationalresources. A short time scale is used in FUDRLS and SVSF to deal with the fast time-varyingelectrical parameters and SOC, and the RTLS algorithm is executed at a longer time scale fortracking the slowly time-varying capacity. The proposed strategy is endowed with high computa-tional efficiency and accuracy, and thus is suitable for real-time embedded BMS applications. Theproposed method is validated by both simulation and experimental studies.

2. The Real-time Battery Model

The battery model should be carefully chosen to ensure high-quality state and parameter esti-mation. In particular, a balance between the fidelity and complexity of the battery model should bemade for the real-time condition monitoring in embedded BMSs. Electrical circuit battery modelsare arguably the most suitable for embedded applications due to their low complexity and the abil-ity of characterizing the current-voltage dynamics of battery cells [37]. A real-time electric circuitmodel with the hysteresis will be considered throughout thepaper. The voltage hysteresis effectbetween the charge and discharge curve widely exists in Li-ion batteries, especially the popularLiFePO4-type [21]. The SOC estimation accuracy will deteriorate ifthe battery model fails toaccount for this phenomenon. The model considered here is based on a first-order RC electricalcircuit with hysteresis, as shown in Fig. 1, which features both simplicity and effectiveness [38].

As shown in Fig. 1, the open-circuit voltage (OCV), denoted asVoc, includes two parts. Thefirst part,Vs(SOC), represents the average equilibrium OCV as a function of theSOC. Since theVs is bijective, the SOC can be inferred fromVs. The second partVh is the hysteresis voltage tocapture the hysteresis behavior of the OCV curves. The RC circuit models the current-voltagecharacteristics and the transient response of the battery cell. Particularly, the series resistance,Rs, is used to describe the charge/discharge energy loss in thecell; the charge transfer resistance,

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Rc, and double layer capacitance,Cd, are used to characterize the charge transfer and short-termdiffusion voltage,Vd, of the cell;VB represents the terminal voltage of the cell.

Fig. 2 shows two measured OCV curves and their average:Voc,c, Voc,d, andVoc,a, respectively.Particularly,Voc,c andVoc,d are obtained by slowly charging and discharging the battery, and rep-resent major upper and lower hysteresis loops, respectively. We treat the average voltageVoc,a asVs(SOC). The instantaneous open circuit voltageVoc is bounded by the major hysteresis loops. BysubtractingVh(k) from VOC, theVs(SOC) can be extracted.

We will use the following voltage hysteresis model [39]:

∂Vh

∂ t=−ρ(η iB−υSD)[Vhmax+sign(iB)Vh], (1)

whereρ is the hysteresis parameter representing the convergence rate,η the Coulomb efficiency(assumingη = 1), iB the instantaneous current applied to the battery,υ the self-discharge mul-tiplier for hysteresis expression,SD the self-discharge rate, andVhmax the maximum hysteresisvoltage. The model (1) describes the dependency of the hysteresis voltageVh on the current, self-discharge, and hysteresis boundaries. The parameterρ is chosen to minimize the voltage errorbetween theVoc−SOCcurves from simulation and experiments, respectively. Note thatρ andVhmax may depend on the SOC and the battery temperature [21, 39].

A discrete-time battery model, including the electrical circuit model and the hysteresis model,can be written as follows

X(k+1) =

1 0 00 γ 00 0 H

X(k)+

− ηTs

Cmax0

Rc(1− γ) 00 (H−1)sign(iB)

[

iB(k)Vhmax

],

y(k) =VB(k) =Voc(SOC(k))−Vd(k)−RsiB(k)+Vh(k),

Vs(SOC) = a0exp(−a1SOC)+a2+a3SOC−a4SOC2+a5SOC3,

(2)

whereX(k+1) =[SOC(k+1) Vd(k+1) Vh(k+1)

]Tis the state,y(k) is the measured output,

k is the time index,Cmax denotes the maximum capacity of the battery,Ts is the sampling period,γ = exp(−Ts

τ ) with τ = RcCd, H(iB) = exp(−ρ |iB|Ts), anda j for 0≤ j ≤ 5 are the coefficientsused to parameterize theVoc-SOC curve. A concise form of the dynamics (2) is given by

X(k+1) = f (X(k), iB(k)),

y(k) = h(X(k), iB(k)),

where f andh are vectors of smooth functions with appropriate dimensions. Coefficientsa j for0≤ j ≤ 5 can be extracted by pulsed current tests [37] or constant charge and discharge currenttest using a small current to minimally excite transient response of the battery cell [40]. Althoughthe temperature dependency is ignored in this paper by testing the battery under the ambient tem-perature, the proposed strategy might be extended to incorporate the thermal effects.

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3. The Proposed Strategy

The proposed condition monitoring strategy, shown in Fig. 3, consists of three parts:

1) an FUDRLS-based parameter estimator,2) an SVSF-based SOC estimator, and3) an RTLS-based SOH (i.e., capacity) estimator.

The strategy operates at different time scales, where the FUDRLS and SVSF runs at a fast speedto estimate the fast time-varying parameters and the SOC, and the RTLS runs slower to track theslowly time-varying capacity parameter. In this way will the computational resources be usedeconomically with guaranteed estimation performance.

3.1. Parameter Estimation by the FUDRLSSince battery parameters (e.g, impedance) change with the SOC, temperature, and current

rates, etc., online parameter estimation is required. Previous work [41] formulated the impedanceestimation as a least squares (LS) problem and proposed the FUDRLS algorithm to identify threeimpedance parameters:Rs, Rc, andCd. In this paper, we overcome the difficulty due to the time-varying dynamics and establish that both the impedance and the hysteresis parameter estimationcan be approximately formulated as an LS problem, and thus the FUDRLS can be readily em-ployed to estimate the hysteresis parameterVhmax.

We first present procedures for impedance estimation for completeness, then show how similaridea can be used to estimate the hysteresis parameter. To estimate the impedance, we ignore thehysteresis voltage dynamics, and assumeVoc = b1SOC+b0. The battery model (2) is reduced to

[SOC(k+1)Vd(k+1)

]= A

[SOC(k)Vd(k)

]+BiB(k),

VB(k) =C

[SOC(k)Vd(k)

]+DiB(k)+b0,

(3)

where

A=

[1 00 γ

], B=

[ −ηTsCmax

Rc(1− γ)

], C=

[b1 −1

], D =−Rs.

Taking z-transformation of (3), we have [42]

VB(z)iB(z)

=C(zI2−A)−1B+D =x3+x4z−1+x5z−2

1+x1z−1+x2z−2 , (4)

whereVB(z) =VB(z)−b0, I2 ∈ R2×2 is an identity matrix, and

x1 = γ−1, x2 = γ, x3 =−Rs,

x4 =−b1Ts

Cmax+Rc(γ−1)+Rs(γ +1),

x5 = Rc(1− γ)+ γ(b1Ts

Cmax−Rs).

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The difference equation corresponding to (4) is given by

VB(k) =−x1VB(k−1)−x2VB(k−2)+x3iB(k)+x4iB(k−1)+x5iB(k−2)+b0(1+x1+x2). (5)

Considering 1+x1+x2 = 0, (5) can be reformulated into the following regression form

VB(k)−VB(k−1) = ΦT(k)Θ, (6)

whereΘ = [x2,x3,x4,x5]T , and

ΦT(k) =[VB(k−1)−VB(k−2) iB(k) iB(k−1) iB(k−2)

].

Since the map from parametersRs,Rc,Cd and b1 to Θ is a diffeomorphism, one can uniquelydetermine the estimates of parametersRs,Rc,Cd,b1 from the estimatedΘ.

Remark 1. The approach proposed above leads to a regression model for impedance estimation.It does not require additional high-pass filtering [8], thussaving on computational cost and ad-ditional effort to develop a high-pass filter. Moreover, it brings better accuracy than the methodsassuming a constant Voc [7, 9, 12], especially when Voc is highly nonlinear with respect to theSOC. In addition, the form given in(6) only uses four parameters while the work [10, 11] utilizesfive parameters.

Remark 2. Because matrices A,B,C,D are time invariant, the z-transformation technique is ap-plicable to derive(6). Alternatively, one can perform derivation in the time domain, i.e., directlywork on the difference equation(5), and establish(6).

The derivation of (6) is performed on the basis of the second-order battery model (3) andparameterizingVoc asb0+b1SOC. Specifically, linear parameterizations of theVoc is critical to thederivation, and the second-order battery model (6) is merely to simplify the presentation. Linearparameterizations ofVoc is valid in a neighborhood ofSOCwhile the hysteresis voltagevhmaxreaches steady state, but is invalid during the transient ofVh. We propose to address this limitationby imposing a less restrictive assumption: linear parameterizations of theVs-SOC curve, whichis always valid locally. This allows us to perform parameteridentification based on the followingdynamics

X(k+1) =

1 0 00 γ 00 0 H

X(k)+

− ηTs

Cmax0

Rc(1− γ) 00 (H−1)sign(iB)

[

iB(k)Vhmax

],

y(k) = b0+b1SOC−Vd(k)−RsiB(k)+Vh(k).

(7)

Note that notationb0,b1 are abused here.Since the state matrices are current-dependent or time-varying, the model (7) does not admit

z−transformation. It is not straightforward to rewrite (7) into a linear regression form. We how-ever show that an approximate linear regression form of the model (7) can be derived, and thus

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parameter identification can be readily carried out. Noticethat the main difficulty in establishingthe linear regression form arises from the time-varyingVh-dynamics, which is fortunately inde-pendent of theVd andSOC-dynamics. This decoupling feature allows us to obtain an approximatelinear regression form.

We essentially try to obtain an approximate linear parameterization of y. We consider thefollowing system

ξ (k+1) = Hξ (k)+(H−1)sign(iB), ξ (0) = ξ0.

ConsideringVh(k) = ξ (k)Vhmax for ξ0 =Vh(0)/Vhmax, we obtain linear parameterizations ofy(k)as follows

y(k) = b0+b1SOC(k)−Vd(k)−RsiB(k)+ξ (k)Vhmax, (8)

whereVhmax is unknown. We introduce a time-varying open-loop filter to estimateξ

ξ (k+1) = Hξ (k)+(H−1)sign(iB), ξ (0) = 0.

SinceH < 1, the aforementioned time-varying open-loop filter produces an exponentially conver-gent estimate ofξ (k), i.e., ξ (k) converges toξ (k) ask→ ∞ for any boundedξ0. Combining (9)and the fact thatξ (k)→ ξ (k) ask→ ∞, we have the approximate linear parameterizations ofy(k)as follows

y(k) = b0+b1SOC(k)−Vd(k)−RsiB(k)+ ξ (k)Vhmax, (9)

from which, together with dynamics ofSOC,Vd, the approximate linear regression of (7) can beestablished. Compared to (6), the approximate linear regression has an extra parameterVhmax inΘ, and an extra signalξ (k) in Φ(k).

Given (6), the parameter vectorΘ can be estimated by a multitude of algorithms, for instancethe conventional Bierman’s UD method [12], Gentleman’s UDRLS [43], etc. The Gentleman’sUDRLS is attractive to embedded applications due to its parallel implementation and the resultantfast computational speed. The RLS-based methods can be improved by using the forgetting factor[7]. The estimation algorithm with a small forgetting factor may track time-varying parametersfairly well at the expense of increased susceptibility to the noise; while the forgetting factor islarge, the tracking ability will be poor but robust to noises. In general, the RLS technique utilizesan exponential forgetting (EF) whose forgetting rate is constant [7], [12]. The main drawback ofthe EF method is called wind-up, and it comes when a data vector is not persistently exciting [44]as well as non-optimal tracking ability and noise influence due to the constant forgetting rate [44].

The FUDRLS algorithm combines the Gentleman’s UDRLS with a variable forgetting factorto estimateΘ. Methods with variable forgetting (VF) adaptively change the forgetting rate. Themain VF mechanism is: the algorithm takes a smaller forgetting factor at the presence of largeprediction errors, and a larger forgetting factor, otherwise. In this paper, the forgetting factorλ isadjusted as follows

λ (k+1) = 1−v1(k)

N0σ20

, λmin≤ λ ≤ λmax,

v1(k+1) = δ1v1(k)+(1−δ1)e2(k),

(10)

whereδ1 is a weighting factor to be taken close to 1;v1 is time-average expressions ofe2(k) and

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v1(0) is set to beσ20 ; the parameterσ2

0 is the mean value of the prediction error variance obtainedfrom the method implemented in the FUDRLS with constant forgetting factor (e.g.,λ = 0.98),assuming that the expected noise variance is much smaller than σ2

0 ; N0 represents the memorylength (e.g.,N0 = 50 corresponding to mean forgetting factor of 0.98); λmax (e.g., 0.999) andλmin (e.g., 0.95) denote maximum and minimum forgetting factors, respectively. An intuitiveinterpretation of (10) is that the forgetting factorλ is adjusted according to the square of thetime-averaged estimation of the autocorrelation of posterior errore(k).

In the FUDRLS, the regression matrixΦT(k) is combined withv(k) = VB(k)−VB(k−1) toproduce an augmented matrix:

ΦTa (k) =

[ΦT(k) v(k)

].

The detailed FUDRLS algorithm is given in Table 1, whereδ denotes an initial covariance value(e.g., 105). For real-time implementation, the computation ofF and the triangularization can bepipelined.

In practical BMS applications, the parameter identification algorithm can be implemented insystem-on-a-chip [45]. Due to the advent of the VLSI technology, the features of parallel process-ing and pipelining implementation will be attractive to improve the computation speed and reducethe size of ICs [46]. The FUDRLS will be beneficial to the development of real BMS ICs in thissense.

3.2. SOC Estimation by the SVSF

With parameters estimated by the FUDRLS algorithm, the SVSFcan be employed to estimatethe battery SOC based on the model (2). Originally proposed in [47] and built on integration of thevariable structure theory and the sliding mode notion, the SVSF is a predictor-corrector method forstate and parameter estimation. A schematic diagram of the SVSF-based state estimation is shownin Fig. 4, where the solid line is the system state trajectory. The estimated state trajectory is forcedtowards the system state trajectory until it enters a neighborhood of the actual state trajectory,referred to as the existence subspace. The existence subspace is an invariant set because oncethe estimated state enters, it remains within the region driven by a switching gain. The SVSFdemonstrates good robustness to modeling uncertainties and noises, given that uncertainties areupper-bounded. It has been applied to estimate battery parameters and the SOC in [48], with onlysimulation results available.

3.2.1. The SVSFThe dynamics of the SVSF are given by

Xk+1|k = f (Xk|k, iB(k)),

yk+1|k =CSVSFXk+1|k,(11)

whereXK+1|k is the predicted state,Xk|k is the state estimate at timek, yk+1|k is the predictedmeasurement, andCSVSFis the linearized measurement matrix given by

CSVSF=∂h(X, iB)

∂X= diag

[∂Vs(SOC)

∂SOC −1 1].

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Defining the innovation asez,k+1|k = yk+1−CSVSFXk+1|k,

the SVSF gain is calculated as follows

KSVSF,k+1 =C−1SVSF(|ez,k+1|k|+ γ|ez,k|k|)◦sat(ez,k+1|k,Ψ), (12)

whereez,k|k is a posteriori measurement error;Ψ is the smoothing boundary layer widths;γ ∈ (0,1)is the SVSF convergence rate;◦ is the Schur product. To ensure the numerical stability, thecomponents ofCSVSFshould not take singular values. This can be accomplished byusing a simpleif statement with a very small threshold (i.e., 10−10) or calculating the pseudoinverseC−1

SVSFwitha small damping parameterω (e.g., 10−8) as the following

C−1SVSF=CT

SVSF(CSVSFCTSVSF+ωI3)

−1. (13)

The corrected (or posteriori) state estimates are computedas follows

Xk+1|k+1 = Xk+1|k+KSVSF,k+1.

3.3. SOH Estimation by the RTLS

As the maximum capacity is a key factor for the battery’s health, this paper considers thefollowing quantity as a measure of the SOH

SOH(n) =Cmax

Cmaxnew, (14)

wheren is the maximum capacity estimation algorithm update index,andCmaxnew is the maximumcapacity of a new battery cell. Such an SOH represents the capacity degradation of the cell.Also, it is clear that an accurateCmax is prerequisite for Coulomb-counting-based SOC estimationalgorithms to provide a good estimation of the SOC.

In [11, 49], the maximum capacity is simply calculated as follows

Cmax=TsΣk2

k=k1

η iB(k)3600

SOC(k2)−SOC(k1), (15)

wherek1 andk2 are time indices. Rearrangement of (15) gives the followinglinear regression form

z=Cmaxu,

whereu= SOC(k2)−SOC(k1) and

z= TsΣk2k=k1

η iB(k)3600

.

Under certain conditions onz andu, an unbiased estimation ofCmax can be achieved by solvingan LS problem. The total least squares (TLS) problem was proposed to alleviate the limitation of

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the LS formulation by performing orthogonal regression [50].The TLS problem is generally solved by using singular value decomposition (SVD) algo-

rithms [50] which incur high computational complexity, andthus are not suitable for embeddedapplications [51]. In this paper, a fast RTLS algorithm is applied for maximum capacity estima-tion. The estimated maximum capacity will be consequently used for the SOH estimation using(14). The proposed RTLS algorithm is based on the constrained Rayleigh quotient, which can runin real time and enjoys fast convergence [52]. Compared to the TLS, the proposed RTLS algo-rithm entails much lower computational load, and the estimation accuracy is comparable to theTLS algorithm.

To facilitate the presentation of the proposed algorithm, it is firstly assumed that the noisyoutput and input are given by

z(n)−∆z︸ ︷︷ ︸z(n)

=Cmax(u(n)−∆u)︸ ︷︷ ︸u(n)

, (16)

whereu(n) andz(n) are the true input and output, respectively; ˜un andzn are the noisy input andoutput, respectively; the output error∆z is assumed zero-mean Gaussian with known variance ofσ2

z ; the SOC estimation error∆u is assumed zero-mean Gaussian with known variance ofσ2u . The

autocorrelation matrix of the noisy input is defined as:

Ru(n) = E[u(n)uT(n)] = Ru(n)+σ2u I ,

whereRu(n) =E[u(n)uT(n)]. Define the augmented data ¯x(n) = [u(n), z(n)]T . The autocorrelationmatrix of x(n) can be expressed as

Rx = E[x(n)xT(n)] =

[Ru(n) b(n)bT(n) c(n)

]

whereb(n) = E[u(n)zT(n)] andc(n) = E[z(n)zT(n)]. Whenn is sufficiently large, the stochasticquantitiesR(n),b(n), andc(n) can be expressed as follows [52]

Ru(n) = µRu(n−1)+ u(n)uT(n),

b(n) = µb(n−1)+ u(n)yT(n),

c(n) = µc(n−1)+ y(n)yT(n),

whereµ is the forgetting factor.The maximum capacity estimation on the basis of (16) is performed by minimizing the follow-

ing constrained Rayleigh quotient

J(Cmax) =qTRxqqTDq

=RuC2

max−2bCmax+cC2

max+β(17)

where the eigenvectorq= [Cmax,−1]T , andD = diag(1,β ) is a diagonal matrix withβ = σ2z/σ2

u .

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If the eigenvector vectorq∗ which minimizesJ(Cmax) corresponds to the smallest eigenvalue ofRx, thenq∗ is the unbiased TLS solution [53].

To avoid solving the constrained Rayleigh quotient minimization problem at each step, theCmax is assumed to be updated as follows

Cmax(n) =Cmax(n−1)+α(n)u(n), (18)

whereα(n) is chosen to minimize (17) in the direction of ˜u(n), i.e.,

∂J(Cmax(n−1)+α(n)u(n))∂α(n)

=c1α2(n)+c2α(n)+c3

d(α(n))= 0. (19)

where

c1 = 2u3(n)b(n),

c2 = 2u2(n)[2b(n)Cmax(n−1)+βR(n)−c(n)],

c3 = 2u(n)[b(n)C2max(n−1)− (βR(n)+c(n))Cmax(n−1)+βb(n)].

Then,α(n) can be obtained by solving the following quadratic equationformed by the numeratorterm of (19):

c1α2(n)+c2α(n)+c3 = 0. (20)

The quadratic equation (20) has two roots, from which the solution of α(n) can be obtained asfollows

α(n) =−c2+

√c2

2−4c1c3

2c1. (21)

4. Strategy Validation

Simulation and experiments are carried out to validate the proposed condition monitoring strat-egy for a Li-ion battery cell subject to various pulsed current operations. Comparisons with exist-ing DEKF [20] methods demonstrate advantages of the proposed strategy in terms of estimationaccuracy quantified by root mean square error (RMSE) and computational cost quantified by run-ning time. Simulation and experiments are performed in MATLAB R© on a computer with 2.2GHzIntel R© CoreTMDuo 2 CPU T6600 and 64-bit OS.

4.1. Simulation Study: Non-Aging Case

For the non-aging case, simulation study assumes the battery model (2) with constant param-eters and that the battery model is subject to a current profile which is proportional to the speedprofile in the standard Urban Dynamometer Driving Schedule (UDDS). In an urban driving envi-ronment, a vehicle switches frequently between acceleration, deceleration and steady state. Thiswould lead to battery discharging profiles containing sufficient frequencies, thus bringing aboutimproved identifiability and observability of the battery model [19].

11

Table 2 lists the values of model parameters, which are basedon a polymer Li-ion batterycell [37] but with the maximum capacity scaled up to 10 Ah. Theinitial actual and estimatedstates are set, respectively, as follows:

[SOC(0),Vd(0),Vh(0)]T = [0.95,0,0]T,

[SOC(0),Vd(0),Vh(0)]T = [0.8,0,0]T.

The initial maximum capacityCmax of the estimators is set to be 6 Ah. The value ofN0 andδ1

are defined as 50 and 0.995, respectively, for the FUDRLS withthe proposed VF (λmin = 0.95and λmax = 0.995). In the SVSF, the value ofγ and Ψ are set to be 0.1 and 1. Input currentis corrupted by zero-mean Gaussian noise with varianceσ2

z = (0.01)2. In the RTLS, the SOCestimation accuracy of the SVSF is assumed 1% (i.e.,σu = 0.01), and thus an overallσ2

u is 2×(0.01)2 since two estimated SOC points are required [36]. Hence,β = (0.001)2/(0.01)2. Also,the forgetting factorµ = 0.98. The DEKF [20], which includes an EKF for SOC estimationand another EKF for estimatingRs, Rc, Cd, andCmax, is implemented to make comparison. Inthe DEKF design, the initial state covariance, process noise covariance matrix, and measurementnoise covariance matrix, are defined as diag[1,1,1], diag[0.09,0.09,0.09] and 0.25, respectively;and those of the EKF for parameter estimation are specified asdiag[10−13,10−2,5×10−2,10−4],diag[10−9,10−4,10−5,10−7] and 0.25, respectively. The parameters of the proposed algorithmsand the DEKF are selected by trial-and-error in an effort to minimize the estimation error.

Both the FUDRLS and the SVSF run at a sampling periodTs= 1 second, while the RTLS runsat a longer periodTl = 200 seconds. The DEKF however has only one sampling period:Ts = 1second. Simulation results are shown in Fig. 5. Particularly, Fig. 5(a) plots the UDDS currentprofile; Fig. 5(b) gives the corresponding voltage responseof the battery cell; Figs. 5(c)-5(e)compare the impedance estimation results; Fig. 5(f) shows the estimated SOC and the true SOCcomputed from the Coulomb counting; and Fig. 5(g) compares the estimatedCmax. One can seethat the proposed algorithms lead to at least comparable estimation accuracy as the DEKF does.Table2 3-4 compare the proposed algorithms and the DEKF using performance metrics: RMSEas a measure of estimation accuracy and simulation time as a measure of computational load.Simulation shows that the proposed strategy outperforms the DEKF in the sense of comparableestimation accuracy but lower computational cost.

4.2. Simulation Study: Aging Case

Proceeding further, we make a more compelling simulation study to verify that the proposedcondition monitoring algorithm detects effectively the aged cell condition. From Table 2, capacityfade and internal resistance deterioration are consideredas major indicators in the aging batterycell, where the trueCmax decreases from 15Ah to 12Ah andRs increase linearly over time. Fig. 6summarizes simulation results. Particularly, Fig. 6(a) gives the pulsed current cycle applied on thebattery model; Fig. 6(b) shows the cell voltage; Fig. 6(c) compares the trueRs with its estimates;and Fig. 6(d) compares the true maximum capacity with its estimates. Simulation results indicatethat the proposed method and the DEKF can track theRs and the time-varying maximum capacitywith similar accuracy.

12

4.3. Experimental Studies

The proposed condition monitoring algorithm is further validated against experimental data,which were collected from a LiMn2O4/hard-carbon battery in the Advanced Technology R&DCenter, Mitsubishi Electric Corporation. The experiment was conducted, under the ambient tem-perature 21.6◦, using a rechargeable battery test equipment produced by Fujitsu Telecom Net-works. The tuning parameters of the proposed condition monitoring algorithm are given in Table 5.In the DEKF design, the initial state covariance, process noise covariance matrix, and measure-ment noise covariance matrix, used in the EKF for SOC estimation are defined as diag[1,1,1],diag[0.16,0.16,0.16] and 0.25, respectively; and those used in the EKF for parameter estima-tion are specified as diag[10−14,10−4,10−5,10−6], diag[4×10−10,10−7,10−10,10−11] and 0.25,respectively. The true SOC trajectory is obtained using theCoulomb counting method. The pa-rameters of the OCV-SOC function of the battery cell are extracted [40]. The estimated statesX(0)and maximum capacityCmax are initialized to be[0.4,0,0]T and 5 Ah, respectively; the true statesX(0)= [0.31,0,0]T andCmax= 4.732Ah. In order to set the test battery cell with the desired initialSOC, the battery cell was first fully charged and rest for one hour. Then the cell is discharged us-ing a small current (e.g., 0.2 A) to the desired initial SOC value. The true maximum capacity wasextracted offline from full discharge test with a small current (e.g., 0.2 A) at ambient temperaturebefore testing the battery.

At first, the FUDRLS is executed for 30 seconds to estimate parameters, and then the SVSFstarts estimating the SOC. Both the FUDRLS and the SVSF have the same sampling periodTs= 1second, while the RTLS has a distinctive sampling periodTl = 20 seconds. On the other hand,the DEKF runs at the sampling periodTs = 1 second. Estimation results are shown in Fig. 7.Particularly, Fig. 7(a) shows the high pulse current cycle (iB = 10C) applied on the battery; Fig.7(b) gives the measured cell voltage; Figs. 7(c)-7(e) show the impedance estimation results; Fig.7(f) compares the estimated SOCs; and Fig. 7(g) compares theCmax estimates. One can observe:the proposed algorithms yield accurate SOC estimation; forthe maximum capacity estimation,the proposed algorithms converge to the true value, albeit the DEKF does not; and the DEKFprovides more consistent estimation of impedance parameters than the proposed algorithms. Table6 summarizes simulation time and estimation accuracy of both condition monitoring algorithms.Experimental results validate that the proposed algorithms can provide reliable SOC and SOHestimation at fairly low computational cost, and thus can besuitable for real-time embedded BMSsfor various applications.

5. Conclusions

Motivated to address the challenges arsing in the deployment and use of Li-ion batteries, thispaper has proposed a novel model-based condition monitoring strategy for real-time impedance,SOC, and maximum capacity/SOH estimation. A set of interdependent algorithms have beenconstructed and validated by both simulation and experimental studies. Owing to its low complex-ity, easy implementation, and high accuracy, the proposed strategy will be particularly suitablefor real-time embedded BMSs strongly demanded in applications such as EVs and PHEVs. Inaddition, the proposed strategy can be extended to build promising solutions to SOP and SOF es-timation, battery prognosis and fault diagnosis. In the future work, the thermal and aging effects

13

will be incorporated, and adaptive condition monitoring will be investigated. Another future ef-fort will be to develop capacity estimation approaches robust to colored noises, which will findimportant application in EVs and PHEVs.

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List of Tables

1 The FUDRLS algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Simulated battery model parameters . . . . . . . . . . . . . . . . . . .. . . . . 203 Simulation comparison of RMSE for impedance estimation . .. . . . . . . . . . 204 Simulation comparison of RMSE and simulation time . . . . . . .. . . . . . . . 205 Tuning parameters of the proposed algorithms . . . . . . . . . . .. . . . . . . . 206 Experimental comparison of RMSE and simulation time . . . . .. . . . . . . . 20

17

List of Figures

1 The first-order RC model with hysteresis . . . . . . . . . . . . . . . .. . . . . . 212 OCV-SOC curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 The proposed model-based condition monitoring method . . .. . . . . . . . . . 224 The SVSF estimation concept [47] . . . . . . . . . . . . . . . . . . . . . .. . . 225 Comparison of true and estimated impedance, SOC, and maximum capacity of the

battery model from the proposed condition monitoring algorithm and the DEKF:(a) input current profile; (b) cell voltage; (c)Rs and its estimates; (d)Rc and itsestimates; (e)Cd and its estimates; (f) SOC and its estimates; and (g)Cmax and itsestimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6 Comparison of true and estimated internal resistance and maximum capacity ofthe aged battery model from the proposed condition monitoring algorithm and theDEKF:(a) input current profile; (b) cell voltage; (c)Rs and its estimates; and (d)Cmax and its estimates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7 Estimated impedance, SOC, and maximum capacity from the proposed conditionmonitoring algorithm and a DEKF on the experimental data:(a) input current pro-file; (b) battery cell voltage response;(c)Rs estimates; (d)Rc estimates; (e)Cd

estimates; (f) SOC and its estimates; and (g)Cmax and its estimates. . . . . . . . 25

18

1: algorithm initialization: setk= 0, Θ = Θ0, andP0 = δ I5 =U0D0UT0 where

U0 =

1 0 0 0 θ1

0 1 0 0 θ2

0 0 1 0 θ3

0 0 0 1 θ4

0 0 0 0 −1

, D0 = δ

1 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 0

.

2: repeat3: k← k+14: read new dataVB(k) andiB(k)5: computef =UT

0 Φa(k)6: initialize r(0) = λ7: for h= 1 to 5do8: compute the parameters Gentleman’s transformation

r(h) = r(h−1)+D0(h) f 2(h)

D(h) = D0(h)r(h−1)/(λ r(h))

α(h) =− f (h)

β (h) = D0(h) f (h)/r(h)

K(h) = β (h)

9: end for10: for j = 2 to 5do11: compute the Gentleman’s transformation12: for i = 1 to j−1 do13: compute the Gentleman’s transformation

U(i, j) =U0(i, j)+α( j)K(i)

K(i) = K(i)+β ( j)U(i, j)

14: end for15: end for16: update parameter estimateΘ andU0,D0

Θ = [U(1,5),U(2,5),U(3,5),U(4,5)]T

U0 =U, D0 = D

17: mapΘ to Rs, Rd,Cd, b1

18: check whether estimated parameters are within the predefined range of values19: update the internal parameters20: until parameter estimation task ends

Table 1: The FUDRLS algorithm.

19

Table 2: Simulated battery model parameters

Cmax 10Ah Cd 4000F Rs 0.06ohm Rd 0.02ohmVhmax 0.01V ρ 2.47e-4 a0 -0.852 a1 63.867

a2 3.692 a3 0.559 a4 0.51 a5 0.508

Table 3: Simulation comparison of RMSE for impedance estimation

DEKF FUDRLSRs (ohm) 7.4814e-4 2.9532e-4Rc (ohm) 8.7396e-4 3.7908e-4Cd (F) 359.85 178.06

Table 4: Simulation comparison of RMSE and simulation time

FUDRLS SVSF RTLS DEKFEstimation Impedance SOC Capacity Impedance SOC Capacity

Accuracy (RMSE) In Table 3 0.0243 2.0989 In Table 3 0.0269 2.4685Simulation Time (s) 0.8933 5.6676 0.0034 13.0559

Table 5: Tuning parameters of the proposed algorithms

λmin 0.95 λmax 0.995 N0 50δ1 0.995 γ 0.1 Ψ 1υ 0.01 σ2

u 2(0.02)2 σ2z (0.01)2

µ 0.98 Ts 1 Tl 20

Table 6: Experimental comparison of RMSE and simulation time

FUDRLS SVSF RTLS DEKFEstimation Impedance SOC Capacity Impedance SOC Capacity

Accuracy (RMSE) N/A 0.0171 0.1617 N/A 0.0220 0.2065Simulation Time (s) 0.1474 0.9070 0.0061 2.0795

20

Vs(SOC)

+VB(k)

Rc

Cd

Rs

+Vh(k)

iB(k)

+

-

Voc

Vd(k)

Figure 1: The first-order RC model with hysteresis

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

3

3.2

3.4

3.6

3.8

4

4.2

SOC

Vo

c (V

)

Voc,c

Voc,a

Voc,d

Upper major loop

Lower major loop

Trajectory of V

oc

Figure 2: OCV-SOC curves

21

SOC Estimator

by SVSF

VB(k)

Parameter

Estimator by

FUDRLSSOH

Estimator

by RTLSCmax(n)

SOH(k)

iB(k)

SOC(k)

Cmax(n)

Parameter(k) iB (k)

VB (k)

iB(k)

Figure 3: The proposed model-based condition monitoring method

Time

Amplitude

Time

Initial Estimate

Estimated State Trajectory

Existence Subspace

System State Trajectory

Figure 4: The SVSF estimation concept [47]

22

0 200 400 600 800 1000 12000

2

4

6

8

10

12

Time (seconds)

i B(A

)

(a)

0 2000 4000 6000 8000 100002.8

3

3.2

3.4

3.6

3.8

4

4.2

Time (seconds)

Vo

ltag

e (V

)

(b)

0 2000 4000 6000 8000 100000.04

0.05

0.06

0.07

0.08

Time (seconds)

Rs (

oh

m)

TrueDEKFFUDRLS

(c)

0 2000 4000 6000 8000 100000

0.01

0.02

0.03

0.04

Time (seconds)

Rc (

oh

m)

TrueDEKFFUDRLS

(d)

0 2000 4000 6000 8000 100002000

3000

4000

5000

6000

Time (seconds)

Cd (

F)

TrueDEKFFUDRLS

(e)

0 2000 4000 6000 8000 10000

0.2

0.4

0.6

0.8

Time (seconds)

SO

C

TrueDEKFSVSF

(f)

0 10 20 30 40 5014

15

16

17

18

19

20

Algorithm update index

Max

imu

m c

apac

ity

(Ah

)

TrueDEKFRTLS

(g)

Figure 5: Comparison of true and estimated impedance, SOC, and maximum capacity of the battery model fromthe proposed condition monitoring algorithm and the DEKF: (a) input current profile; (b) cell voltage; (c)Rs and itsestimates; (d)Rc and its estimates; (e)Cd and its estimates; (f) SOC and its estimates; and (g)Cmax and its estimates.23

0 200 400 600 800 1000

−5

0

5

Time (seconds)

i B(A

)

(a)

0 2 4 6 8 10

x 104

3

3.5

4

4.5

Time (seconds)

Vol

tage

(V

)

(b)

0 2 4 6 8 10

x 104

0.02

0.03

0.04

0.05

0.06

0.07

Time (seconds)

Rs (

ohm

)

TrueDEKFFUDRLS

(c)

100 200 300 400 50010

12

14

16

18

Algorithm update index

Max

imum

cap

acity

(A

h)

TrueDEKFRTLS

(d)

Figure 6: Comparison of true and estimated internal resistance and maximum capacity of the aged battery model fromthe proposed condition monitoring algorithm and the DEKF:(a) input current profile; (b) cell voltage; (c)Rs and itsestimates; and (d)Cmax and its estimates.

24

0 500 1000 1500 2000−50

0

50

Time (seconds)

i B(A

)

(a)

0 500 1000 1500 20003

3.2

3.4

3.6

3.8

4

4.2

Time (seconds)

Vol

tage

(V

)

(b)

0 500 1000 1500 20000

1

2

3

4x 10

−3

Time (seconds)

Rs (

ohm

)

DEKF

FUDRLS

(c)

0 500 1000 1500 20000

2

4

6

8x 10

−4

Time (seconds)

Rc (

ohm

)

DEKF

FUDRLS

(d)

0 500 1000 1500 20000

1

2

3

4

5

6x 10

4

Time (seconds)

Cd (

F)

DEKF

FUDRLS

(e)

0 500 1000 1500 2000

0.3

0.4

0.5

0.6

0.7

0.8

Time (seconds)

SO

C

Coulomb countingDEKFSVSF

(f)

0 10 20 30 40 50 60 70 80 90

4

4.5

5

5.5

Algorithm update index

Max

imum

cap

acity

(A

h)

TrueDEKFRTLS

(g)

Figure 7: Estimated impedance, SOC, and maximum capacity from the proposed condition monitoring algorithm anda DEKF on the experimental data:(a) input current profile; (b) battery cell voltage response;(c)Rs estimates; (d)Rc

estimates; (e)Cd estimates; (f) SOC and its estimates; and (g)Cmax and its estimates.

25