IEEE AUTOTESTCON 2008 Salt Lake City, UT, 8-11 September 2008

Automotive Battery Management Systems Bharath Pattipati1, Krishna Pattipati1, Jon P. Christopherson2, Setu Madhavi Namburu3, Danil V. Prokhorov3, and Liu Qiao3 1

Department of Electrical and Computer Engineering, University of Connecticut, 371 Fairfield Road, U-2157, Storrs, CT 06269, USA Email: {bharath, krishna}@engr.uconn.edu 2 Idaho National Laboratory, Idaho Falls, Idaho 83415, USA Email: [email protected] 3 Toyota Technical Center USA 1555 Woodridge, RR #7 Ann Arbor, MI 48105, USA Email: {setumadhavi.namburu, danil.prokhorov, liu.qiao}@tema.toyota.com

Abstract – Battery management system (BMS) is an integral part of an automobile. It protects the battery from damage, predicts battery life and maintains the battery in an operational condition. The BMS performs these tasks by integrating one or more of the functions, such as protecting the cell, controlling the charge, determining the state of charge (SOC), the state of health (SOH), and the remaining useful life (RUL) of the battery, cell balancing, as well as monitoring and storing historical data. In this paper, we propose a BMS that estimates three critical characteristics of the battery (SOC, SOH, and RUL) using a data-driven approach. Our estimation procedure is based on an equivalent circuit battery model consisting of resistors, capacitor, and Warburg impedance. The resistors usually characterize the self-discharge and internal resistance of the battery, the capacitor generally represents the charge stored in the battery, and the Warburg impedance represents the diffusion phenomenon. We investigate the use of support vector machines to predict the capacity fade and power fade, which characterize the SOH of a battery, as well as estimate the SOC of the battery. The circuit parameters are estimated from Electrochemical Impedance Spectroscopy (EIS) test data using nonlinear least squares estimation techniques. Predictions of Remaining Useful Life (RUL) of the battery are obtained by support vector regression of the power fade and capacity fade estimates. Keywords – Battery Management System (BMS), State of Charge (SOC), State of Health (SOH), Power Fade, Capacity Fade, Remaining Useful Life (RUL), Support Vector Machines (SVM).

I. INTRODUCTION Battery technology has come a long way since the invention of the first voltaic cell in the 1800s. Because of the increased interest in hybrid vehicles, a Battery Management System (BMS) has become one of the chief components in an automobile. The goals of BMS are to maximize both the runtime per discharge cycle, as well as the number of life cycles attainable for the life of the battery [1]. Automotive battery management is very demanding, because it has to work in real-time in rapidly varying charge-discharge conditions as the vehicle accelerates and brakes, as well as work in a harsh

978-1-4244-2226-5/08/$25.00 ©2008 IEEE

and uncontrolled environment. In addition, it must interface with other on-board systems, such as the engine management, climate controls, communications, and safety systems. The functions of a BMS in a hybrid electric vehicle are multifaceted. They include monitoring the conditions of individual cells which make up the battery, maintaining all the cells within their operating limits, protecting the cells from out-of-tolerance conditions, compensating for any imbalances in cell parameters within the battery chain, providing information about the State of Charge (SOC), State of Health (SOH), and Remaining Useful Life (RUL) of the battery, providing the optimum charging algorithm for charging the cells, responding to changes in the vehicle operating mode and so on. The main motivation of this paper is to develop a systematic procedure for estimating three critical characteristics of a battery, viz., State of charge (SOC), State of Health (SOH), and Remaining Useful Life (RUL). The SOC, a measure of remaining capacity in the battery, is used to ensure optimum control of the charging/discharging process. The SOH is a measure of a battery's capability to deliver its specified output. This is vital for assessing the readiness of emergency power equipment, and is an indicator of whether maintenance actions are needed. We will employ capacity fade and power fade as measures of SOH of a battery. The ability to accurately predict the RUL is the key to proactive, condition-based maintenance of batteries. Our approach to estimating the SOC, SOH and RUL is based on a modified Randles circuit model of a battery. This model consists of a high frequency resistance, a parallel RC circuit for modeling charge transfer phenomenon at medium frequencies, and Warburg impedance to model diffusion phenomenon at low frequencies. The circuit parameters are estimated from the EIS data using non-linear least squares (NLLS) estimation techniques. The temporal variations of battery resistance are modeled by an auto-regressive support vector machine. Exploiting the linear correlation between the

between the battery resistance and C1/11 capacity, the nonlinear SVM model is used to make future predictions of the battery’s SOC, as well as the capacity fade and the power fade. The estimates of capacity fade and power fade, in turn, are used to estimate the Remaining Useful Life (RUL) of the battery via a moving average SVM regression. The paper is organized as follows. Section II provides an overview of our estimation approach. Section III validates our estimation approach on second-generation lithium-ion cell (i.e. Gen 2 cell) data collected at the Idaho National Laboratory (INL). Section IV concludes with a summary and future research directions.

A. Parameter Estimation using nonlinear least squares Nyquist plots of resistance versus (negative) reactance as a function of frequency are used to depict impedance changes in the electrode-electrolyte interface using equivalent circuit models. The Nyquist curve generally shows a semi-circle of radius r governed by the charge transfer phenomenon, and a 450 slope (theoretically) where the diffusion phenomenon is predominant. These profiles tend to grow as a function of cell age and are particularly sensitive to the chosen battery chemistry, the nature of the solid electrolyte interface, and the temperature.

II. ESTIMATION APPROACH The estimation framework proposed in this paper is presented in Figure 1. Characterization testing was initiated at 250 C with C1/1 static capacity tests to establish performance parameters, such as capacity, resistance, power and energy. These tests consist of a constant-current discharge from a fully charged state using a fraction of the rated capacity defined at the 1-hour rate. After the static capacity tests, all cells were subjected to electrochemical impedance spectroscopy (EIS) test. EIS was used to determine impedance changes in the electrodeelectrolyte interface [2]. EIS measurements were conducted by discharging the cells from a fully-charged state to the specified open circuit voltage (OCV) corresponding to 60% SOC, followed by an 8- to 12- hour rest at OCV (to reach electrochemical equilibrium), and then the impedance is measured over a frequency range of 10 kHz to 0.01 Hz [3][4].

Fig. 2. Nyquist plot.

The main difficulty of battery dynamic modeling is the diffusion phenomenon. In a previous work, Kuhn et al. proposed a modified Randles scheme [6] shown in Figure 3 as an equivalent circuit model of a battery.

Cdl Zw ( s )

RHF

i (t )

Rtc Eeq

Fig. 3. Modified Randles scheme.

Here, the Warburg impedance ZW represents the diffusion phenomenon; RHF (high-frequency resistance) denotes the

Fig. 1. SOC, SOH, and RUL estimation framework. 1

C1/1 Rate: The current corresponding to the rated capacity in ampere-hours for a 1-hour discharge. For example, if the battery’s rated 1-hour capacity is 5 Ah, then the C1/1 is 5 A [5].

electrolyte and connection resistances, and the Rtc || Cdl parallel circuit models the charge transfer phenomenon. Hence, the frequency-dependent impedance expression for the above circuit becomes: Rtc Z (ω ) = RHF + + Z w (ω ) (1) 1 + jω Rtc Cdl Typically, Warburg impedance is theoretically expressed as a non-integer function [7].

Z w ( ω ) = γω1/ 2 (1 − j )

(2)

where γ is a parameter, which depends on the electrochemical phenomenon. In this paper, we implemented nonlinear least squares (nonlinear data-fitting) using the lsqnonlin function in the Optimization Toolbox of MATLAB® to estimate the parameters of the equivalent circuit of the battery cell shown in Figure 3. The optimization uses trust region method and is based on the interior-reflective Newton method. The noninteger function presented in equation (2) was used to model the Warburg impedance. The inputs and outputs of the parameter estimation procedure are shown in Figure 4 below.

trade-off between these two terms. We should note that the larger the C, the more the error is penalized. Thus, C should be chosen with care to keep away from over fitting. We use the ε -sensitive loss function for the second term in (6) given by [8] ⎧⎪0, if y(k ) − f ( x(k ),w) ε < ε y(k ) − f ( x(k ),w) ε = ⎨ ⎪⎩ y(k ) − f ( x(k ),w) ε − ε , otherwise

(7)

which means that we tolerate errors up to ε and also that errors beyond ε have a linear effect. This error function is therefore more tolerant to noise and is thus more robust [8]. By using Lagrange multiplier techniques, the minimization of Φ ( w ) leads to the following dual optimization problem: K

max −ε α,α*

K

K K

1 ∑(α +α ) +∑(α −α ) y(k) − 2 ∑∑(α −α )(α * k

* k

k

k=1

k

k=1

* k

k

* m −αm

k=1 m=1

) K( x(k),x(m)) (8)

subject to: K

∑ (α

* k

)

− α k = 0, 0 < α k , α k* < C , k = 1,..., K

k =1

(9)

This problem is solved by quadratic programming techniques [8]. The regression takes the form:

Fig. 4. Inputs and outputs of the parameter estimation procedure.

f ( x) =

B. Support Vector Machine Regression (SVMR)

K

∑ (α k =1

* k

)

− α k K ( x(k ), x ) + b

(10)

The nonlinear mapping (or kernel function) K is used to transform the original input x to a higher dimensional

The circuit parameters vary with time. We model the temporal variations of total resistance R(k) = (RHF(k)+Rtc(k)) in week k via an auto-regressive SVMR of length L (typically, L=2). Specifically, (3) [ R(k )] = f ({R(i)}ik=−k1− L ) We also use moving average SVMRs for predicting the capacity fade from C1/1 capacities, and RUL from capacity and power fade estimates. The main objective in SVM regression is to estimate a function f ( x ) that is as close as possible to the target values

where . , . denotes dot product. In practice, various kernel functions are used, such as polynomial, radial basis functions (RBF) [8], and sigmoid functions. Here, we use the RBF kernel function because it has less numerical difficulties: ⎛ − x ( k ) − x ( m) 2 ⎞ ⎟ K ( x(k ), x(m) ) = exp ⎜ (11) ⎜ ⎟ 2σ 2 ⎝ ⎠

{ y (k )}kK=1 for the corresponding {x(k )}kK=1 , where x(k ) ∈ R L .

C. State of Charge (SOC)

In the current context, y ( k ) = R( k ) and x(k ) = [ R (k − 1), R(k − 2),..., R(k − L)]T . Similar analogy applies to capacity and RUL. The training data are arranged as (4) F = {( x(1), y (1) ) , ( x(2), y (2) ) ,..., ( x( K ), y ( K ) )} By a nonlinear mapping Φ , x(k ) is mapped into a feature space and a linear estimate is constructed in this space as

y (k ) = f ( x(k ), w) = wT Φ ( x(k ) ) + b

(5)

To determine the weight vector w and bias b , the following regularized risk function is minimized: Φ ( w) =

1 2 w +C 2

K

∑ y(k ) − f ( x(k ), w) ε ; C > 0, ε > 0

(6)

k =1

The first term in the objective function represents the model complexity and the second term represents the model accuracy. Here, C is a regularization parameter to control the

feature space Ω as K ( x(k ), x(m) ) = Φ ( x(k ) ) , Φ ( x(m) ) ,

State of Charge (SOC) is defined as the available capacity in a battery expressed as a percentage of the actual (or estimated) rated capacity. This is normally referenced to a constant-current discharge at C1/1 rate. That is, estimated capacity - capacity removed × 100 SOC (%) = estimated capacity (12) where capacity removed =

∫

t

0

I (τ )dτ . For constant-current

discharge, capacity removed = I x t, where, I = discharge current in Amperes and t = time in hours. The sum of high-frequency (or ohmic) resistance and charge-transfer resistance is correlated with the C1/1 capacities from the C1/1 static capacity test data. SVMR [8] is then used to make future predictions of the resistance and, consequently, the estimated capacity is obtained from these

correlation plots. The capacity removed depends on the battery usage.

where C(k) and P(k) are the capacity fade and power fade respectively, and RUL(k) is the remaining useful life at time k. Figure 5 shows our approach for RUL estimation.

D. State of Health (SOH) SOH is the ability of a cell to store energy, source and sink high currents, and retain charge over extended periods, relative to its initial or nominal capabilities [9]. It is a figure of merit that describes the degree of degradation of a battery, and gives a quantitative measure that replaces fuzzy statements, such as ‘fresh’, ‘aged’, ‘old’, and ‘worn out’ [10]. In this paper, we characterize the SOH of a battery by its power fade and capacity fade. Power Fade: The loss of cell power due to an increase in cell impedance during aging is known as power fade. In this paper, we determine the power fade of the battery from the EIS test. The actual power from the EIS test is calculated as:

V2 (13) R where V is the voltage (5mV for EIS test) and R is the total resistance, (RHF+Rtc), obtained from EIS data using nonlinear least squares. Hence, power fade is computed as follows: ⎛ Power (k ) ⎞ R (0) Power Fade = 1- ⎜ (14) ⎟ = 1− R (k ) ⎝ Power (0) ⎠ P=

where Power(0) is the power at the beginning-of-life (BOL) and Power(k ) is the power at the desired time (week). Capacity Fade: The gradual loss of capacity of a secondary battery with cycling is known as capacity fade [5]. Capacity fade is also the percent loss in C1/1 discharge capacity [3]. ⎛ Capacity(k) ⎞ Capacity Fade (% ) = ⎜ 1(15a) ⎟×100 ⎝ Capacity(0) ⎠ where Capacity (0) is the BOL Capacity, and Capacity (k ) is the Capacity at the desired time (week), k. Given that the capacity is approximately linearly related to the total resistance, i.e., C ( k ) = β − α ( R( k ) − R(0)) , it is evident that Capacity Fade (% ) =

α ( R( k ) − R( 0 )) ×100 β

(15b)

Fig. 5. RUL estimation framework.

III.

The validation data used in this paper had been collected as part of the Advanced Technology Development (ATD) program, where performance testing was conducted on second-generation lithium-ion cells (i.e., Gen 2 cells). The 18650-size Gen 2 cells, with baseline and variant chemistry, were cycle-life tested at Idaho National Laboratory (INL); only baseline cells data was used in this paper. Life testing was interrupted every 4 weeks for reference performance tests (RPTs) consisting of C1/1 static capacity test, a lowcurrent hybrid pulse power characterization (L-HPPC) test, and electrochemical impedance spectroscopy (EIS) test [4]. The Baseline cell degradation generally increased with increasing test temperature, SOC, and test time. A. Parameter Estimation Results Estimates of RHF, Rtc, Cdl, and Warburg parameter, γ , for the different battery cells are obtained using nonlinear least squares, as a function of time (weeks). The time-dependent behavior of (RHF+Rtc) will be used to estimate the degradation of battery cells. The degradation of some of the battery cells as a function of time (weeks) is shown below (Cdl and Warburg parameter, γ , are not shown here since they show negligible change over the ageing process, and are not used in our estimation framework). TOTAL RESISTANCE

The capacity fade of a battery is estimated by predicting the C1/1 capacities using an auto-regressive SVMR: (16) [C (k )] = g ({C (i)}ik=−k1− L )

E. Remaining Useful Life (RUL) In this paper, we predict the RUL of the battery using a moving average SVMR for different thresholds on capacity fade and power fade. Formally, at week k, we estimate RUL(k ) = h({P(i ), C (i )}ik=1 ) (17)

0.055 CELL 8 CELL 9 CELL 11 CELL 14 CELL 15

0.05

0.045

RHF + Rtc

Once the capacity is estimated, the capacity fade is computed from Eq. (15a). Alternately, one can compute the capacity fade from Eq. (15b) by appealing to the linear correlation between the capacity and battery resistance.

RESULTS AND DISCUSSION

0.04

0.035

0.03

0.025

0.02

0

20

40

60 80 Time (weeks)

100

120

Fig. 6. Total resistance as a function of weeks.

140

The battery’s internal impedance parameter identified from the EIS test chosen for correlation analysis is the total internal resistance of the battery (i.e., R = RHF + Rtc ). Figure 6 shows the variation of resistance as a function of weeks. The SVMR model with L=2 for the total resistance parameter R showed excellent R2 fit values of 97% for as many as 15-20 week-ahead predictions. This implies that EIS tests can be done as part of routine vehicle maintenance (e.g., oil changes). The C1/1 discharge capacities were extracted from the C1/1 static capacity test data. Figure 7 shows the average C1/1 discharge capacity for different Baseline cells that were aged at 60% SOC.

B. State of Charge (SOC) Results The resistance predictions are used to obtain the corresponding C1/1 discharge capacity from the linear correlation plots in Figure 8. Subsequently, the SOC is obtained using equation (12). Figure 9 shows the variation of the SOC for a constant-current discharge of baseline cell 8 from beginning-of-life (BOL) to end-of-life (EOL). SOC of CELL 8 100 4th week 8th week 12th week 16th week 20th week 24th week 28th week 32nd week 36th week 40th week 44th week 48th week 52nd week 56th week 60th week 64th week 68th week

90 80 70

Average C1/1 Discharge Capacity 1

CELL 9

0.95

CELL 11

SOC (%)

60

CELL 8

50

BOL

CELL 14 0.9

40

CELL 15

EOL 30

Capacity (Amp-hr)

0.85

20 0.8

10 0.75

0

0

0.1

0.2

0.3

0.4

0.5 0.6 Time (hours)

0.7

0.8

0.9

1

0.7

Fig. 9. Percentage SOC as a function of time for Baseline Cell 8.

0.65

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100104108112116120124128132136140144 Test Time (weeks)

Fig. 7. Average C1/1 discharge capacity for Baseline cycle-life cell groups.

Figure 8 shows the high degree of linear correlation between the C1/1 discharge capacity and the internal resistance parameter R for different cells. For the linear model C (k ) = β − α ( R(k ) − R (0)) , typical values for

α and β for the various cells were α ∈ [9.52, 11.03] and β = C ( 0 ) ∈ [0.93,1] .

C. State of Health (SOH) Results The rate of capacity loss (capacity fade) generally increases with increasing temperature and SOC. The SVMR model with L=2 for the C1/1 capacities showed excellent R2 fit values of 95% for as many as 10 week-ahead predictions. Figure 10 shows the capacity fade of different baselinechemistry cells obtained using equation (16). The power fade of different cells predicted using the approach described in section 2 is shown in Figure 11. CAPACITY FADE OF DIFFERENT CELLS 35

Linear Correlation between Resistance and Capacity 1

Capacity (Amp-hr)

0.9

0.85

8

30

9 11

8 9 11 14 15

25 14

CAPACITY FADE (%)

CELL fit 1 CELL fit 2 CELL fit 3 CELL fit 4 CELL fit 5

0.95

CELL CELL CELL CELL CELL

15

0.8

0.75

20

15

10 0.7

5 0.65

0.6

0 0.025

0.03

0.035 0.04 Resistance (RHF + Rtc )

0.045

0.05

0.055

Fig. 8. Correlation between resistance and battery capacity.

0

20

40

60 80 Time (weeks)

100

120

Fig. 10. Capacity fade prediction of different cells.

140

POWER FADE of CELLS 8,9,11,14 and 15 0.7

0.6 Cell 14

Power Fade

0.5

Cell 15

Cell 11

Cell 9

0.4

0.3

Cell 8

0.2

0.1

0

0

20

40

60 80 100 No. of Estimation Points

120

140

Fig. 11. Power fade estimation for different cells.

D. Remaining Useful Life (RUL) Results

RUL (weeks)

Figure 12 shows the RUL for an end-of-life (EOL) criterion of approximately 23% power fade and 30% capacity fade, at which time the battery is assumed to have failed. Remaining Useful Life of Cell 15 72 68 R-square = 0.9947 64 MSE = 2.2892 60 56 52 48 44 40 36 32 28 24 20 16 12 8 4 0 -4 68 72 76 80 84 88 92 96 100 104 108 112 116120 124 128 132 136 140 144 Time (weeks)

Fig. 12. RUL prediction.

As seen in the Figure, the RUL of the battery decreases with time and the EOL criterion is based on the applicationdependent capacity fade and power fade. The SVMR can be trained for various thresholds of capacity and power fade thresholds, and consequently, the RUL can be obtained for a specific context-dependent threshold. IV.

CONCLUSIONS AND FUTURE WORK

Next generation battery management systems will feature online tracking and monitoring of the pivotal battery characteristics, such as the SOC, SOH, and RUL, to facilitate efficient diagnostic and prognostic maintenance of batteries. Here, we proposed a framework for estimating and predicting

these salient battery performance measures. Two commonly used tests were used for our analysis, viz., electrochemical impedance spectroscopy (EIS) test and the C1/1 static capacity test. Our approach to estimating the SOC, SOH, and RUL is based on a modified Randles circuit model of a battery. This model consists of a high frequency resistance, a parallel RC circuit for modeling charge transfer phenomenon at medium frequencies, and Warburg impedance to model diffusion phenomenon at low frequencies. The circuit parameters are estimated from the EIS data using non-linear least squares (NLLS) estimation techniques. We proposed an autoregressive SVMR using an ε -sensitive loss function for predicting the resistance. The C1/1 capacity was obtained from the C1/1 static capacity test. Hence, the SOC can be obtained using the resistance-capacity mapping or from the C1/1 prediction via an autoregressive SVMR. Consequently, a framework was proposed to predict the capacity fade and power fade, which characterize the SOH of a battery. The RUL predictions were made by setting different applicationdependent thresholds on capacity fade and power fade. Our future work will focus on the following areas: 1. Parametric modeling of specific frequency areas of the Nyquist plot to address ill-conditioned NLLS, and 2. Low-current hybrid pulse power characterization (LHPPC) test analysis [4]. ACKNOWLEDGMENT This work was supported by the Toyota Technical Center. Any opinions expressed in this paper are solely those of the authors and do not represent those of the sponsor. We would also like to gratefully acknowledge technical discussions with William Morrison of Qualtech Systems, Inc. REFERENCES [1]

E. Bordeaux,“Portable power management – a holistic perspective”, Intersil Corporation, 2002. [2] J. P. Christopherson, C. D. Ho, C. G. Motloch, D. Howell, and H. L. Hess,“Effects of reference performance testing during aging using commercial lithium-ion cells”, Journal of The Electrochemical Society 153, issue 7, A1406-A1416, 2006. [3] PNGV Battery Test Manual, Rev.3, DOE/ID-10597, February 2001. [4] FreedomCAR Battery Test Manual for Power-Assist Hybrid Electric Vehicles, DOE/ID-11069, October 2003. [5] D.L. Linden and T.B. Reddy,“Handbook of Batteries”, 3rd edition, McGraw Hill, New York, 2002. [6] E. Kuhn, C. Forgez, G. Friedrich,“Modeling diffusive phenomena using non-integer derivatives: application NiMH batteries”, The European Physical Journal Applied Physics, 25, 183-190, 2004. [7] D. Landolt,“Corrosion et chimie de surfaces des métaux”, Traité des matériaux, Presses Polytechniques et Universitaires Romandes, 1993. [8] C.M. Bishop,“Pattern Recognition and Machine Learning”, Springer, 2006. [9] B.S. Bhangu, P. Bentley, D.A. Stone, and C.M. Bingham,“Nonlinear observers for predicting state-of-charge and state-of-health of lead-acid batteries for hybrid-electric-vehicles”, IEEE transactions on vehicular technology, 54(3), 783-794, May 2005. [10] E. Meissner, G. Richter,“The challenge to the automotive battery industry: the battery has become an increasingly integrated component within the vehicle electric power system”, Journal of Power Sources, 144, 438-460, 2005.

Automotive Battery Management Systems Bharath Pattipati1, Krishna Pattipati1, Jon P. Christopherson2, Setu Madhavi Namburu3, Danil V. Prokhorov3, and Liu Qiao3 1

Department of Electrical and Computer Engineering, University of Connecticut, 371 Fairfield Road, U-2157, Storrs, CT 06269, USA Email: {bharath, krishna}@engr.uconn.edu 2 Idaho National Laboratory, Idaho Falls, Idaho 83415, USA Email: [email protected] 3 Toyota Technical Center USA 1555 Woodridge, RR #7 Ann Arbor, MI 48105, USA Email: {setumadhavi.namburu, danil.prokhorov, liu.qiao}@tema.toyota.com

Abstract – Battery management system (BMS) is an integral part of an automobile. It protects the battery from damage, predicts battery life and maintains the battery in an operational condition. The BMS performs these tasks by integrating one or more of the functions, such as protecting the cell, controlling the charge, determining the state of charge (SOC), the state of health (SOH), and the remaining useful life (RUL) of the battery, cell balancing, as well as monitoring and storing historical data. In this paper, we propose a BMS that estimates three critical characteristics of the battery (SOC, SOH, and RUL) using a data-driven approach. Our estimation procedure is based on an equivalent circuit battery model consisting of resistors, capacitor, and Warburg impedance. The resistors usually characterize the self-discharge and internal resistance of the battery, the capacitor generally represents the charge stored in the battery, and the Warburg impedance represents the diffusion phenomenon. We investigate the use of support vector machines to predict the capacity fade and power fade, which characterize the SOH of a battery, as well as estimate the SOC of the battery. The circuit parameters are estimated from Electrochemical Impedance Spectroscopy (EIS) test data using nonlinear least squares estimation techniques. Predictions of Remaining Useful Life (RUL) of the battery are obtained by support vector regression of the power fade and capacity fade estimates. Keywords – Battery Management System (BMS), State of Charge (SOC), State of Health (SOH), Power Fade, Capacity Fade, Remaining Useful Life (RUL), Support Vector Machines (SVM).

I. INTRODUCTION Battery technology has come a long way since the invention of the first voltaic cell in the 1800s. Because of the increased interest in hybrid vehicles, a Battery Management System (BMS) has become one of the chief components in an automobile. The goals of BMS are to maximize both the runtime per discharge cycle, as well as the number of life cycles attainable for the life of the battery [1]. Automotive battery management is very demanding, because it has to work in real-time in rapidly varying charge-discharge conditions as the vehicle accelerates and brakes, as well as work in a harsh

978-1-4244-2226-5/08/$25.00 ©2008 IEEE

and uncontrolled environment. In addition, it must interface with other on-board systems, such as the engine management, climate controls, communications, and safety systems. The functions of a BMS in a hybrid electric vehicle are multifaceted. They include monitoring the conditions of individual cells which make up the battery, maintaining all the cells within their operating limits, protecting the cells from out-of-tolerance conditions, compensating for any imbalances in cell parameters within the battery chain, providing information about the State of Charge (SOC), State of Health (SOH), and Remaining Useful Life (RUL) of the battery, providing the optimum charging algorithm for charging the cells, responding to changes in the vehicle operating mode and so on. The main motivation of this paper is to develop a systematic procedure for estimating three critical characteristics of a battery, viz., State of charge (SOC), State of Health (SOH), and Remaining Useful Life (RUL). The SOC, a measure of remaining capacity in the battery, is used to ensure optimum control of the charging/discharging process. The SOH is a measure of a battery's capability to deliver its specified output. This is vital for assessing the readiness of emergency power equipment, and is an indicator of whether maintenance actions are needed. We will employ capacity fade and power fade as measures of SOH of a battery. The ability to accurately predict the RUL is the key to proactive, condition-based maintenance of batteries. Our approach to estimating the SOC, SOH and RUL is based on a modified Randles circuit model of a battery. This model consists of a high frequency resistance, a parallel RC circuit for modeling charge transfer phenomenon at medium frequencies, and Warburg impedance to model diffusion phenomenon at low frequencies. The circuit parameters are estimated from the EIS data using non-linear least squares (NLLS) estimation techniques. The temporal variations of battery resistance are modeled by an auto-regressive support vector machine. Exploiting the linear correlation between the

between the battery resistance and C1/11 capacity, the nonlinear SVM model is used to make future predictions of the battery’s SOC, as well as the capacity fade and the power fade. The estimates of capacity fade and power fade, in turn, are used to estimate the Remaining Useful Life (RUL) of the battery via a moving average SVM regression. The paper is organized as follows. Section II provides an overview of our estimation approach. Section III validates our estimation approach on second-generation lithium-ion cell (i.e. Gen 2 cell) data collected at the Idaho National Laboratory (INL). Section IV concludes with a summary and future research directions.

A. Parameter Estimation using nonlinear least squares Nyquist plots of resistance versus (negative) reactance as a function of frequency are used to depict impedance changes in the electrode-electrolyte interface using equivalent circuit models. The Nyquist curve generally shows a semi-circle of radius r governed by the charge transfer phenomenon, and a 450 slope (theoretically) where the diffusion phenomenon is predominant. These profiles tend to grow as a function of cell age and are particularly sensitive to the chosen battery chemistry, the nature of the solid electrolyte interface, and the temperature.

II. ESTIMATION APPROACH The estimation framework proposed in this paper is presented in Figure 1. Characterization testing was initiated at 250 C with C1/1 static capacity tests to establish performance parameters, such as capacity, resistance, power and energy. These tests consist of a constant-current discharge from a fully charged state using a fraction of the rated capacity defined at the 1-hour rate. After the static capacity tests, all cells were subjected to electrochemical impedance spectroscopy (EIS) test. EIS was used to determine impedance changes in the electrodeelectrolyte interface [2]. EIS measurements were conducted by discharging the cells from a fully-charged state to the specified open circuit voltage (OCV) corresponding to 60% SOC, followed by an 8- to 12- hour rest at OCV (to reach electrochemical equilibrium), and then the impedance is measured over a frequency range of 10 kHz to 0.01 Hz [3][4].

Fig. 2. Nyquist plot.

The main difficulty of battery dynamic modeling is the diffusion phenomenon. In a previous work, Kuhn et al. proposed a modified Randles scheme [6] shown in Figure 3 as an equivalent circuit model of a battery.

Cdl Zw ( s )

RHF

i (t )

Rtc Eeq

Fig. 3. Modified Randles scheme.

Here, the Warburg impedance ZW represents the diffusion phenomenon; RHF (high-frequency resistance) denotes the

Fig. 1. SOC, SOH, and RUL estimation framework. 1

C1/1 Rate: The current corresponding to the rated capacity in ampere-hours for a 1-hour discharge. For example, if the battery’s rated 1-hour capacity is 5 Ah, then the C1/1 is 5 A [5].

electrolyte and connection resistances, and the Rtc || Cdl parallel circuit models the charge transfer phenomenon. Hence, the frequency-dependent impedance expression for the above circuit becomes: Rtc Z (ω ) = RHF + + Z w (ω ) (1) 1 + jω Rtc Cdl Typically, Warburg impedance is theoretically expressed as a non-integer function [7].

Z w ( ω ) = γω1/ 2 (1 − j )

(2)

where γ is a parameter, which depends on the electrochemical phenomenon. In this paper, we implemented nonlinear least squares (nonlinear data-fitting) using the lsqnonlin function in the Optimization Toolbox of MATLAB® to estimate the parameters of the equivalent circuit of the battery cell shown in Figure 3. The optimization uses trust region method and is based on the interior-reflective Newton method. The noninteger function presented in equation (2) was used to model the Warburg impedance. The inputs and outputs of the parameter estimation procedure are shown in Figure 4 below.

trade-off between these two terms. We should note that the larger the C, the more the error is penalized. Thus, C should be chosen with care to keep away from over fitting. We use the ε -sensitive loss function for the second term in (6) given by [8] ⎧⎪0, if y(k ) − f ( x(k ),w) ε < ε y(k ) − f ( x(k ),w) ε = ⎨ ⎪⎩ y(k ) − f ( x(k ),w) ε − ε , otherwise

(7)

which means that we tolerate errors up to ε and also that errors beyond ε have a linear effect. This error function is therefore more tolerant to noise and is thus more robust [8]. By using Lagrange multiplier techniques, the minimization of Φ ( w ) leads to the following dual optimization problem: K

max −ε α,α*

K

K K

1 ∑(α +α ) +∑(α −α ) y(k) − 2 ∑∑(α −α )(α * k

* k

k

k=1

k

k=1

* k

k

* m −αm

k=1 m=1

) K( x(k),x(m)) (8)

subject to: K

∑ (α

* k

)

− α k = 0, 0 < α k , α k* < C , k = 1,..., K

k =1

(9)

This problem is solved by quadratic programming techniques [8]. The regression takes the form:

Fig. 4. Inputs and outputs of the parameter estimation procedure.

f ( x) =

B. Support Vector Machine Regression (SVMR)

K

∑ (α k =1

* k

)

− α k K ( x(k ), x ) + b

(10)

The nonlinear mapping (or kernel function) K is used to transform the original input x to a higher dimensional

The circuit parameters vary with time. We model the temporal variations of total resistance R(k) = (RHF(k)+Rtc(k)) in week k via an auto-regressive SVMR of length L (typically, L=2). Specifically, (3) [ R(k )] = f ({R(i)}ik=−k1− L ) We also use moving average SVMRs for predicting the capacity fade from C1/1 capacities, and RUL from capacity and power fade estimates. The main objective in SVM regression is to estimate a function f ( x ) that is as close as possible to the target values

where . , . denotes dot product. In practice, various kernel functions are used, such as polynomial, radial basis functions (RBF) [8], and sigmoid functions. Here, we use the RBF kernel function because it has less numerical difficulties: ⎛ − x ( k ) − x ( m) 2 ⎞ ⎟ K ( x(k ), x(m) ) = exp ⎜ (11) ⎜ ⎟ 2σ 2 ⎝ ⎠

{ y (k )}kK=1 for the corresponding {x(k )}kK=1 , where x(k ) ∈ R L .

C. State of Charge (SOC)

In the current context, y ( k ) = R( k ) and x(k ) = [ R (k − 1), R(k − 2),..., R(k − L)]T . Similar analogy applies to capacity and RUL. The training data are arranged as (4) F = {( x(1), y (1) ) , ( x(2), y (2) ) ,..., ( x( K ), y ( K ) )} By a nonlinear mapping Φ , x(k ) is mapped into a feature space and a linear estimate is constructed in this space as

y (k ) = f ( x(k ), w) = wT Φ ( x(k ) ) + b

(5)

To determine the weight vector w and bias b , the following regularized risk function is minimized: Φ ( w) =

1 2 w +C 2

K

∑ y(k ) − f ( x(k ), w) ε ; C > 0, ε > 0

(6)

k =1

The first term in the objective function represents the model complexity and the second term represents the model accuracy. Here, C is a regularization parameter to control the

feature space Ω as K ( x(k ), x(m) ) = Φ ( x(k ) ) , Φ ( x(m) ) ,

State of Charge (SOC) is defined as the available capacity in a battery expressed as a percentage of the actual (or estimated) rated capacity. This is normally referenced to a constant-current discharge at C1/1 rate. That is, estimated capacity - capacity removed × 100 SOC (%) = estimated capacity (12) where capacity removed =

∫

t

0

I (τ )dτ . For constant-current

discharge, capacity removed = I x t, where, I = discharge current in Amperes and t = time in hours. The sum of high-frequency (or ohmic) resistance and charge-transfer resistance is correlated with the C1/1 capacities from the C1/1 static capacity test data. SVMR [8] is then used to make future predictions of the resistance and, consequently, the estimated capacity is obtained from these

correlation plots. The capacity removed depends on the battery usage.

where C(k) and P(k) are the capacity fade and power fade respectively, and RUL(k) is the remaining useful life at time k. Figure 5 shows our approach for RUL estimation.

D. State of Health (SOH) SOH is the ability of a cell to store energy, source and sink high currents, and retain charge over extended periods, relative to its initial or nominal capabilities [9]. It is a figure of merit that describes the degree of degradation of a battery, and gives a quantitative measure that replaces fuzzy statements, such as ‘fresh’, ‘aged’, ‘old’, and ‘worn out’ [10]. In this paper, we characterize the SOH of a battery by its power fade and capacity fade. Power Fade: The loss of cell power due to an increase in cell impedance during aging is known as power fade. In this paper, we determine the power fade of the battery from the EIS test. The actual power from the EIS test is calculated as:

V2 (13) R where V is the voltage (5mV for EIS test) and R is the total resistance, (RHF+Rtc), obtained from EIS data using nonlinear least squares. Hence, power fade is computed as follows: ⎛ Power (k ) ⎞ R (0) Power Fade = 1- ⎜ (14) ⎟ = 1− R (k ) ⎝ Power (0) ⎠ P=

where Power(0) is the power at the beginning-of-life (BOL) and Power(k ) is the power at the desired time (week). Capacity Fade: The gradual loss of capacity of a secondary battery with cycling is known as capacity fade [5]. Capacity fade is also the percent loss in C1/1 discharge capacity [3]. ⎛ Capacity(k) ⎞ Capacity Fade (% ) = ⎜ 1(15a) ⎟×100 ⎝ Capacity(0) ⎠ where Capacity (0) is the BOL Capacity, and Capacity (k ) is the Capacity at the desired time (week), k. Given that the capacity is approximately linearly related to the total resistance, i.e., C ( k ) = β − α ( R( k ) − R(0)) , it is evident that Capacity Fade (% ) =

α ( R( k ) − R( 0 )) ×100 β

(15b)

Fig. 5. RUL estimation framework.

III.

The validation data used in this paper had been collected as part of the Advanced Technology Development (ATD) program, where performance testing was conducted on second-generation lithium-ion cells (i.e., Gen 2 cells). The 18650-size Gen 2 cells, with baseline and variant chemistry, were cycle-life tested at Idaho National Laboratory (INL); only baseline cells data was used in this paper. Life testing was interrupted every 4 weeks for reference performance tests (RPTs) consisting of C1/1 static capacity test, a lowcurrent hybrid pulse power characterization (L-HPPC) test, and electrochemical impedance spectroscopy (EIS) test [4]. The Baseline cell degradation generally increased with increasing test temperature, SOC, and test time. A. Parameter Estimation Results Estimates of RHF, Rtc, Cdl, and Warburg parameter, γ , for the different battery cells are obtained using nonlinear least squares, as a function of time (weeks). The time-dependent behavior of (RHF+Rtc) will be used to estimate the degradation of battery cells. The degradation of some of the battery cells as a function of time (weeks) is shown below (Cdl and Warburg parameter, γ , are not shown here since they show negligible change over the ageing process, and are not used in our estimation framework). TOTAL RESISTANCE

The capacity fade of a battery is estimated by predicting the C1/1 capacities using an auto-regressive SVMR: (16) [C (k )] = g ({C (i)}ik=−k1− L )

E. Remaining Useful Life (RUL) In this paper, we predict the RUL of the battery using a moving average SVMR for different thresholds on capacity fade and power fade. Formally, at week k, we estimate RUL(k ) = h({P(i ), C (i )}ik=1 ) (17)

0.055 CELL 8 CELL 9 CELL 11 CELL 14 CELL 15

0.05

0.045

RHF + Rtc

Once the capacity is estimated, the capacity fade is computed from Eq. (15a). Alternately, one can compute the capacity fade from Eq. (15b) by appealing to the linear correlation between the capacity and battery resistance.

RESULTS AND DISCUSSION

0.04

0.035

0.03

0.025

0.02

0

20

40

60 80 Time (weeks)

100

120

Fig. 6. Total resistance as a function of weeks.

140

The battery’s internal impedance parameter identified from the EIS test chosen for correlation analysis is the total internal resistance of the battery (i.e., R = RHF + Rtc ). Figure 6 shows the variation of resistance as a function of weeks. The SVMR model with L=2 for the total resistance parameter R showed excellent R2 fit values of 97% for as many as 15-20 week-ahead predictions. This implies that EIS tests can be done as part of routine vehicle maintenance (e.g., oil changes). The C1/1 discharge capacities were extracted from the C1/1 static capacity test data. Figure 7 shows the average C1/1 discharge capacity for different Baseline cells that were aged at 60% SOC.

B. State of Charge (SOC) Results The resistance predictions are used to obtain the corresponding C1/1 discharge capacity from the linear correlation plots in Figure 8. Subsequently, the SOC is obtained using equation (12). Figure 9 shows the variation of the SOC for a constant-current discharge of baseline cell 8 from beginning-of-life (BOL) to end-of-life (EOL). SOC of CELL 8 100 4th week 8th week 12th week 16th week 20th week 24th week 28th week 32nd week 36th week 40th week 44th week 48th week 52nd week 56th week 60th week 64th week 68th week

90 80 70

Average C1/1 Discharge Capacity 1

CELL 9

0.95

CELL 11

SOC (%)

60

CELL 8

50

BOL

CELL 14 0.9

40

CELL 15

EOL 30

Capacity (Amp-hr)

0.85

20 0.8

10 0.75

0

0

0.1

0.2

0.3

0.4

0.5 0.6 Time (hours)

0.7

0.8

0.9

1

0.7

Fig. 9. Percentage SOC as a function of time for Baseline Cell 8.

0.65

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100104108112116120124128132136140144 Test Time (weeks)

Fig. 7. Average C1/1 discharge capacity for Baseline cycle-life cell groups.

Figure 8 shows the high degree of linear correlation between the C1/1 discharge capacity and the internal resistance parameter R for different cells. For the linear model C (k ) = β − α ( R(k ) − R (0)) , typical values for

α and β for the various cells were α ∈ [9.52, 11.03] and β = C ( 0 ) ∈ [0.93,1] .

C. State of Health (SOH) Results The rate of capacity loss (capacity fade) generally increases with increasing temperature and SOC. The SVMR model with L=2 for the C1/1 capacities showed excellent R2 fit values of 95% for as many as 10 week-ahead predictions. Figure 10 shows the capacity fade of different baselinechemistry cells obtained using equation (16). The power fade of different cells predicted using the approach described in section 2 is shown in Figure 11. CAPACITY FADE OF DIFFERENT CELLS 35

Linear Correlation between Resistance and Capacity 1

Capacity (Amp-hr)

0.9

0.85

8

30

9 11

8 9 11 14 15

25 14

CAPACITY FADE (%)

CELL fit 1 CELL fit 2 CELL fit 3 CELL fit 4 CELL fit 5

0.95

CELL CELL CELL CELL CELL

15

0.8

0.75

20

15

10 0.7

5 0.65

0.6

0 0.025

0.03

0.035 0.04 Resistance (RHF + Rtc )

0.045

0.05

0.055

Fig. 8. Correlation between resistance and battery capacity.

0

20

40

60 80 Time (weeks)

100

120

Fig. 10. Capacity fade prediction of different cells.

140

POWER FADE of CELLS 8,9,11,14 and 15 0.7

0.6 Cell 14

Power Fade

0.5

Cell 15

Cell 11

Cell 9

0.4

0.3

Cell 8

0.2

0.1

0

0

20

40

60 80 100 No. of Estimation Points

120

140

Fig. 11. Power fade estimation for different cells.

D. Remaining Useful Life (RUL) Results

RUL (weeks)

Figure 12 shows the RUL for an end-of-life (EOL) criterion of approximately 23% power fade and 30% capacity fade, at which time the battery is assumed to have failed. Remaining Useful Life of Cell 15 72 68 R-square = 0.9947 64 MSE = 2.2892 60 56 52 48 44 40 36 32 28 24 20 16 12 8 4 0 -4 68 72 76 80 84 88 92 96 100 104 108 112 116120 124 128 132 136 140 144 Time (weeks)

Fig. 12. RUL prediction.

As seen in the Figure, the RUL of the battery decreases with time and the EOL criterion is based on the applicationdependent capacity fade and power fade. The SVMR can be trained for various thresholds of capacity and power fade thresholds, and consequently, the RUL can be obtained for a specific context-dependent threshold. IV.

CONCLUSIONS AND FUTURE WORK

Next generation battery management systems will feature online tracking and monitoring of the pivotal battery characteristics, such as the SOC, SOH, and RUL, to facilitate efficient diagnostic and prognostic maintenance of batteries. Here, we proposed a framework for estimating and predicting

these salient battery performance measures. Two commonly used tests were used for our analysis, viz., electrochemical impedance spectroscopy (EIS) test and the C1/1 static capacity test. Our approach to estimating the SOC, SOH, and RUL is based on a modified Randles circuit model of a battery. This model consists of a high frequency resistance, a parallel RC circuit for modeling charge transfer phenomenon at medium frequencies, and Warburg impedance to model diffusion phenomenon at low frequencies. The circuit parameters are estimated from the EIS data using non-linear least squares (NLLS) estimation techniques. We proposed an autoregressive SVMR using an ε -sensitive loss function for predicting the resistance. The C1/1 capacity was obtained from the C1/1 static capacity test. Hence, the SOC can be obtained using the resistance-capacity mapping or from the C1/1 prediction via an autoregressive SVMR. Consequently, a framework was proposed to predict the capacity fade and power fade, which characterize the SOH of a battery. The RUL predictions were made by setting different applicationdependent thresholds on capacity fade and power fade. Our future work will focus on the following areas: 1. Parametric modeling of specific frequency areas of the Nyquist plot to address ill-conditioned NLLS, and 2. Low-current hybrid pulse power characterization (LHPPC) test analysis [4]. ACKNOWLEDGMENT This work was supported by the Toyota Technical Center. Any opinions expressed in this paper are solely those of the authors and do not represent those of the sponsor. We would also like to gratefully acknowledge technical discussions with William Morrison of Qualtech Systems, Inc. REFERENCES [1]

E. Bordeaux,“Portable power management – a holistic perspective”, Intersil Corporation, 2002. [2] J. P. Christopherson, C. D. Ho, C. G. Motloch, D. Howell, and H. L. Hess,“Effects of reference performance testing during aging using commercial lithium-ion cells”, Journal of The Electrochemical Society 153, issue 7, A1406-A1416, 2006. [3] PNGV Battery Test Manual, Rev.3, DOE/ID-10597, February 2001. [4] FreedomCAR Battery Test Manual for Power-Assist Hybrid Electric Vehicles, DOE/ID-11069, October 2003. [5] D.L. Linden and T.B. Reddy,“Handbook of Batteries”, 3rd edition, McGraw Hill, New York, 2002. [6] E. Kuhn, C. Forgez, G. Friedrich,“Modeling diffusive phenomena using non-integer derivatives: application NiMH batteries”, The European Physical Journal Applied Physics, 25, 183-190, 2004. [7] D. Landolt,“Corrosion et chimie de surfaces des métaux”, Traité des matériaux, Presses Polytechniques et Universitaires Romandes, 1993. [8] C.M. Bishop,“Pattern Recognition and Machine Learning”, Springer, 2006. [9] B.S. Bhangu, P. Bentley, D.A. Stone, and C.M. Bingham,“Nonlinear observers for predicting state-of-charge and state-of-health of lead-acid batteries for hybrid-electric-vehicles”, IEEE transactions on vehicular technology, 54(3), 783-794, May 2005. [10] E. Meissner, G. Richter,“The challenge to the automotive battery industry: the battery has become an increasingly integrated component within the vehicle electric power system”, Journal of Power Sources, 144, 438-460, 2005.