5- Development and Validation of a Battery Model

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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 25, NO. 3, SEPTEMBER 2010 821

Development and Validation of a Battery ModelUseful for Discharging and Charging Power

Control and Lifetime EstimationVivek Agarwal, Student Member, IEEE, Kasemsak Uthaichana, Member, IEEE, Raymond A. DeCarlo, Fellow, IEEE,

and Lefteri H. Tsoukalas, Member, IEEE

AbstractAccurate information on battery state-of-charge, ex-pected battery lifetime, and expected battery cycle life is essen-tial for many practical applications. In this paper, we develop anonchemically based partially linearized (in battery power) inputoutput battery model, initially developed for lead-acid batteries ina hybrid electric vehicle. We show that with properly tuned param-eter values, the model can be extended to different battery types,such as lithium-ion, nickel-metal hydride, and alkaline. The valida-tion results of the model against measured data in terms of power

and efficiency at different temperatures are then presented. Themodel is incorporated with the recovery effect for accurate life-time estimation. The obtained lifetime estimation results using theproposed model are similar to the ones predicted by the Rakhma-tov and Virudhula battery model on a given set of typical loads atroom temperature. A possible incorporation of the cycling effect,which determines the battery cycle life, in terms of the maximumavailable energy approximated at charge/discharge nominal powerlevel is also suggested. The usage of the proposed model is compu-tationally inexpensive, hence implementable in many applications,such as low-power system design, real-time energy management indistributed sensor network, etc.

Index TermsBattery cycle life validation, battery model, life-time estimation, recovery effect, validation.

I. INTRODUCTION

BATTERIES are widely used as a finite source of energy for

a variety of applications ranging from low-power design of

portable devices to high-power hybrid electric vehicle (HEV).

Battery performance is affected by various factors, such as op-

erating temperature, humidity, discharging/charging cycles, etc.

Battery models are essential for any battery-powered system

design that aims at extending the batterys expected life and

in battery power management. This creates a need for battery

models that capture essential application-dependent character-

Manuscript received August 15, 2008; revised June 26, 2009; acceptedJanuary 6, 2010. Date of current version August 20, 2010. The work ofK. Uthaichana was supported by grant Thailand Research Fund, Commissionon Higher Education. Paper no. TEC-00320-2008.

V. Agarwal and L. H. Tsoukalas are with the School of Nuclear Engineer-ing, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]; [email protected]).

K. Uthaichana is with the Department of Electrical Engineering, Chiang MaiUniversity, Chiang Mai 50200, Thailand (e-mail: [email protected]).

R. A. DeCarlo is with the School of Electrical and Computer Engineer-ing, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TEC.2010.2043106

istics of real batteries to estimate/predict battery behavior under

various operation conditions.

The conversion of chemical energy within the battery into

electric energy is a complex electrochemical process. The bat-

tery behavior, depending upon the application can be approxi-

mated by an empirical model or by highly accurate and complex

electrochemical model. Rao et al. [1] classified battery models

as 1) physical models; 2) empirical models; 3) abstract models;and 4) mixed models.

Physical models are also known as electrochemical mod-

els [1]. They are highly accurate and involve detailed considera-

tion of electrochemical processes, thermal dynamic process, and

the physical construction for both charging/discharging charac-

teristics [2], [3]. Since, the number of parameters to be carefully

selected in electrochemical models are large, i.e., computation-

ally expensive, the usage of physical models in practical appli-

cations is limited.

Empirical battery models are easy to configure and are repre-

sented by a simple mathematical expressions [4], [5] with less

number of parameters. The parameters are usually obtained as a

solution of a least squares problem using chargingdischargingdata, operating conditions, and physical properties. Empirical

models often fail to provide accurate representation/estimation

under varying load conditions. On the other hand, abstract mod-

els are simplified equivalent representation of electrochemi-

cal process within a battery, and their choice is usually ap-

plication specific. The equivalent representation, for example,

can be in terms of electric circuits [6] or stochastic process

[7].

Mixed models are based on high-level abstraction, avoiding

excessive details of physical laws (e.g.,electrochemicalprocess)

governing battery characteristics, which leads to the derivation

of a simplified analytical expression. The number of parametersto be selected is usually low in mixed models. Rakhmatov and

Virudhula [8] and Rong and Pedram [9] battery models are

examples of mixed models.

In [10] and[11], a nonlinearempirical lead-acidbattery model

was proposed and validated for supervisory level power flow

control of an HEV. The battery model in [10] has captured the

characteristics of discharging/charging efficiencies to estimate

the battery state-of-charge (SOC). Initially, the proposed model

assumes relatively constant operating temperature, and does not

consider the capacity fading effect. Further, the model was lin-

earized in the (control) input to make the model amenable to a

large body of control literature.

0885-8969/$26.00 2010 IEEE


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822 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 25, NO. 3, SEPTEMBER 2010

In this research, we show that with appropriately tuned pa-

rameters, the battery model in [10] can be extended to accu-

rately describe behaviors of additional battery types [alkaline,

lithium-ion (Li-ion), nickel-metal hydride (Ni-MH)]. This ex-

tension is validated against the actual discharging and charging

data. Based on the validation results, we set forth a functional

relationship that exists between the model coefficients and tem-

peratures for lead-acid and Ni-MH battery types alone. The

actual data at different temperatures for Li-ion and alkaline

battery types were not available to us. Although, Uthaichana

et al. [10] did not consider the recovery effect, by incorporating

the recovery effect, the model can be used for battery lifetime

estimation under various discharging load conditions.

A possible incorporation of the cycling effect, which deter-

mines the battery cyclelife when it undergoes repeated discharg-

ing and charging is also suggested. Specifically, the maximum

available energy in a battery is characterized as a function of

number of cycles and nominal charge/discharge power level at

a specific depth of discharge (DOD). The quality of the ap-

proximation is validated against the actual data for a Li-ionbattery. Hence, the versatility of the model to different bat-

tery types widens its utility for practical applications. For ex-

ample, in distributed sensor network application, estimation of

battery longevity, estimates of scheduled maintenance, and bat-

tery replacement are some of the key issues in such applica-

tion [12][15].

Thepaperis organizedas follows.The discussion on Rakhma-

tov and Virudhula battery model [8] and the development of

nonlinear and partially linearized battery model with recovery

is presented in Section II. The relationship between both battery

models is also discussed. The validation of the proposed model

for different battery types against the actual data measured atdifferent temperatures for both discharging and charging is il-

lustrated in Section III. Section IV presents the validation of the

proposed model with the incorporated recovery model for Li-ion

battery lifetime estimation under different constant and varying

power load conditions. The approximation and the validation of

the maximum rated energy for a Li-ion battery to incorporate the

effect of cycling on battery cycle life is presented in Section V.

Finally, conclusions are drawn in Section VI.

II. BATTERY MODELING

Given various forms of battery models, we briefly summarize

the Rakhmatov and Virudhula battery model [8] and develop apartially linearized inputoutput battery model of Uthaichana

et al. [10]. The generalization of Uthaichana et al. model [10]

to different battery types and its validation against the actual

data measured at different temperatures for both charging and

discharging are some of the main contributions of the paper. In

addition, a differential equation describing the recovery effect

is presented. Under the constant maximum energy assumption,

Uthaichana et al. model [10] with the incorporated recovery

model can be applied to solve the battery lifetime estimation

problem for nonrecharging load profiles and is presented in

Section IV. A possible relaxation on the constant maximum

energy assumption to capture the cycling effect and estimate

the battery cycle life under repeated discharging and charging

cycles are mentioned at the end of the paper.

A. Rakhmatov and Virudhula Battery Model

Rakhmatov and Virudhula [8] developed an analytical ex-

pression to estimate battery lifetime for various time-varying

loads by taking into account the changes in the concentration ofthe electroactive species inside the battery. The model is based

on a 1-D diffusion process of the concentration of the species,

where the concentration of the species at time t, and at distancex from the electrode is denoted by C(x, t). The battery life-time L is defined as the time at which the concentration at theelectrode surface C(0, t) drops from the initial concentration ofthe species C, to a specific threshold Ccutoff. The cutoff con-centration Ccutoff depending on the battery type and size, is theconcentration level below which no further power can be drawn.

Rakhmatov and Virudhula obtained an analytical expression for

the concentration behavior by defining two partial differential

equations based on Ficks law as follows:

J(x, t) = D C(x, t)x

(1)

C(x, t)

x= D

2 C(x, t)

2 x(2)

where 1) J(x, t) denotes the flux of the species at time t [0, L] and at distance x [0, w] from the electrode, in whichw is the length of the battery and 2) D denotes the diffusioncoefficient. The boundary conditions at the electrode surface

x = 0 based on the Faradays law, and at the other electrodex = w based on constant concentration, are expressed in terms

of the concentration gradient as follows:i(t)

nF A= D

C(x, t)

x

x= 0

(3)

0 = DC(x, t)

x

x= w

(4)

where F is Faradays constant, n is the number of electrons in-volved in the electrochemical reaction at the electrode surface,

A is the surface area of the electrode, and i(t) is the dischargingcurrent and represents the load on the battery. Throughout this

paper, the electrode refers to the one that receives the elec-

troactive species, and the other electrode refers to the one that

generates the species.

It can be shown that an analytical solution for (1) and (2), at

the electrode surface (x = 0) with the boundary conditions (3)and (4) is as follows:

C(0, t) = C 1nFAw

t0

i(t)d

+ lim0

2

m =1

t0

i(t)e 2 D ( t ) m 2

w 2 d

. (5)

The details on the derivation of (5) are presented in [16]. An

observation can be made from (5) that the concentration of

the species at the electrode surface decreases with time due to


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AGARWAL et al.: DEVELOPMENT AND VALIDATION OF A BATTERY MODEL 823

the usage (i(t) > 0). Once the concentration C(0, t) drops toCcutoff, we obtain the measure of battery lifetime L.

In certain applications, it is important to obtain an estimate

of the amount of energy consumed, and consequently, the re-

maining battery stored energy; this then allows estimates of the

battery lifetime given nominal loading profiles. To make the

model more amendable to various applications, (5) has been

expressed in terms of battery capacity as follows:

=

L0

i(t)d + 2

m = 1

L0

i(t)e2 m 2 (L) d. (6)

The first term in (6) is the total charge drawn by the load

and battery losses. The second term of (6) is the unavail-

able charge left in the battery due the nonuniform distribu-

tion of the electroactive species during discharging. Here, the

two battery parameters and are introduced. The parameter = nFAwC(1

Ccutoff/C

) denotes the battery capacity,

i.e., the estimated amount of charge used during the battery life-time and =

D/w denotes the discharge time constant. The

values of both parameters depend on the battery type. Note that

the nonuniform distribution of the electroactive species in the

second term of (6) results in lower concentration at the electrode

surface C(0, t) than at the other electrode C(w, t). If a rest pe-riod is introduced, the nonuniformity decreases over time and

more charge are available at the electrode surface, which is a part

of the recovery effect. The equation capturing this phenomenon

is described in [17].

In general, the load i(t) is unknown and nonconstant. Evenif known, the nonlinear and random behavior would make the

usage of (6) numerically difficult. Nevertheless, if i(t) has anaverage value over small intervals of time, the time-varyingdischarge rate can be approximated by piecewise constant loads,

i.e., i(t) is approximated in the time interval [0, T] by N equalsize staircase basis functions as follows:

i(t) N1k = 0

Ik

U(t tk ) U(t tk +1 )

(7)

where U(t) is the unit step function.Given a set ofM constant experimental current loads Ii , i =

1, 2, . . . , M and corresponding set ofMlifetimes Li , the battery

parameters and are selected as a least squares solutionminimizing

Mi= 1 |Ii Ii |2 , where Ii = Ii (, ) is an estimate

of the experimental current load based on the most recent values

of and within the iteration process.Given the estimated battery parameters and N-step staircase

approximation of the load [see (7)], the battery lifetime compu-

tation involves two steps. The first step is to find the subinterval

(Tk = tk +1 tk , k = 1, 2, . . . , N 1), such that the concen-tration at the electrode surface C(0, tk +1 ) is below Ccutoff attk + 1 , i.e., find the subinterval Tk , such that L [tk , tk + 1 ].The second step is to determine the smallest t within the subin-terval Tk , such that C(0, tk +1 ) Ccutoff. The smallest t canbe found using the modified secant method, as in [18].

B. Development of Nonlinear and Partially Linearized

Battery Model

As mentioned earlier, the motivation to solve the power man-

agement problem of an HEV led to the development of par-

tially linearized (control-oriented) battery model developed as

follows:

Here, normalized battery energy, denoted by Wba t (t), is de-fined as the ratio of the instantaneous stored charge to the max-

imum stored charge, i.e., Wba t (t) = Wba t (t)/Wm ax

ba t , where

Wba t (t) is the instantaneous stored battery energy and Wm axba t isthe maximum rated storage energy of the battery. As developed

in [19], Wba t (t) approximates the battery SOC under the rea-sonable assumption of a relatively constant open-circuit battery

voltage during operation.

To achievea dynamic model formeasuring theSOC of thebat-

tery, we differentiate the normalized stored energy Wba t (t) =Wba t (t)/W

ma xba t with respect to time to obtain a relationship to

the discharging and charging power of the battery

Wba t = Wba t (t)

Wma xba t= 1

Wma xba tba t (Wba t , Pba t , v)Pba t (t)

(8)

where 1) v is the battery mode of operation, where v = 0 meansthe battery is discharging and v = 1 means the battery is charg-ing; 2) ba t (Wba t , Pba t , v) is a generic efficiency for discharg-ing and charging; and 3) Pba t (t) is the discharging (+)/charging() battery power flow (input) as expressed in (9). The nega-tive sign in (8) indicates decreasing Wba t (t) during discharging,while (8) becomes positive during charging, thereby indicating

increasing Wba t (t)

Pba t (t) => 0, discharging

< 0, charging. (9)

In general, during discharging, when 0 Pba t (t) < , where is a small threshold and usually negligible compared to the dis-

charging loads, a battery undergoes charge recovery, whichis de-

scribed in Section II-D. During discharge, the generic efficiency

ba t (Wba t , Pba t , v) = 1/0ba t (Wba t , Pba t , v) > 1, where

0ba t

is the actual discharging efficiency. A desired power output

causes the battery to discharge more rapidly than what is re-

quired to account for losses. On the other hand, during charging

ba t (Wba t , Pba t , v) = 1ba t (Wba t , Pba t , v) < 1, where

1ba t is

the actual charging efficiency, indicates that more power is

needed to overcome charging losses.Given this relationship, the approximations of the generic ef-

ficiency ba t (Wba t , Pba t , v), v {0, 1} as a function ofWba t(state) and Pba t (input) for both cases is done by interpolatinga nonlinear function against the downscaled battery efficiency

map (see Appendix B) for v {0, 1}. Intuitively observing thatgeneric efficiency curves of a battery appear to have a logarith-

mic characteristic, the following approximation was empirically

determined:

ba t (Wba t , Pba t , v)=ln(Wba t + d1 ,v ) + d2,v Pba t (t) + d3 ,v .(10)

Given appropriate coefficients, the objective of this paper is to

show the quality of this approximation for four distinct battery


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types. The coefficients dk ,v , k = 1, 2, 3 are chosen to fit thegeneric battery efficiency map. The coefficients during battery

discharging and charging are computed as a solution to a nonlin-

ear least-squares regression similar to that described earlier for

obtaining parameters of the Rakhmatov and Virudhula model.

Specifically, in each mode, the coefficients dk ,v for k = 1, 2, 3are the numerical solution to the following minimization:

mindk R

ba t (Wba t , Pba t , v)

[ln(Wba t + d1,v ) + d2 ,v Pba t (t) + d3,v ]2 . (11)This nonlinear minimization problem is solved via a subroutine

nlinfitin MATLABs statistical toolbox in this study.

As per the aforementioned development, the differential equa-

tion for Wba t (t) [see (8)] is nonlinear in the control input(Pba t (t)). To makethe model amenableto a large bodyof controlliterature, (8) is partially linearized about the mode-dependent

nominal battery operating power Pvba t,no m . The resulting lin-earized differential equation is as follows:

Wba t (t) =d2 ,v

Wm axba t

Pvba t,no m

2

ln(Wba t (t) + d1,v ) + 2d2,v Pvba t,no m + d3 ,v

Pba t (t)

Wm axba t.

(12)

The details on the methodology to obtain the partially lin-

earized model [see (12)] are presented in Appendix A.

C. Relationship Between Rakhmatov and Virudhula and Par-

tially Linearized Battery ModelIn many practical applications, the actual knowledge about

the battery SOC is essential. In Rakhmatov and Virudhula [8]

battery model the concentration of the electroactive species at

the electrode surface is given by (5). Since the specific gravity

of the electroactive species is greater than that of water, the

higher the concentration of the electrospecies is the higher the

specific gravity. When a battery is fully charged (SOC = 1),then the concentration in the electrolyte is at its maximum C

and so is the specific gravity. On the other hand, if the concen-

tration of the species is at Ccutoff (SOC = C = 0), then thespecific gravity is at minimum operating value. In fact, there

is a linear relationship between the battery SOC and the spe-cific gravity as shown in [20] under an assumption of uniform

concentration of the electroactive species at equilibrium. Based

on aforementioned discussion, there exists a relationship be-

tween the battery SOC and concentration in the Rakhmatov and

Virudhula battery model at equilibrium. Uthaichana et al. bat-

tery model [10] approximates the battery SOC as normalized

battery energy (Wba t ) under the assumption of relatively con-stant open-circuit voltage. This relationship between the con-

centration in the Rakhmatov and Virudhula battery model [8]

and Wba t in Uthaichana et al. battery model [10] is expressedas follows:

C(t) = Wba t (t)(C Ccutoff) + Ccutoff (13)

where C(t) is the concentration of the species. Note that atequilibrium, the concentration of the species is uniform, i.e.,

C(0, t) = C(w, t) = C(t).

D. Battery Recovery Model

The recovery effect is the increase in Wba t (the battery SOC)

due to the adjustment of the concentration gradient toward zerounder no load condition 0 Pba t (t) < , where is a smallthreshold and usually negligible compared to the discharging

power loads. The recovery of the Wba t , depending upon thebattery type, reaches its steady-state value after a few time con-

stants. However, under the same condition, the derivative of

Wba t (t) given by the partially linearized model (12) approxi-mately reduces to d2,v /W

ma xba t (P

vba t,no m )

2 for v = 0. This is notequivalent to the charge recovery effect. Therefore, to achieve a

better representation of the battery behavior, we incorporate the

recovery effect by augmenting an additional equation [see (14)]

to the partially linearized battery model

Wrba t (t) = Wrba t (t) + (1 + )Wba t (ti ), 0Pba t (t) < (14)

where 1) = 1/, is the recovery time constant; 2) Wrba t (t)

denotes the battery SOC during recovery; 3) the superscript rdenotes recovery; and 4) denotes the percentage of recovery,whose value depends upon battery type and is usually obtained

from experimental data [21]. In (14), Wba t (ti ) denotes the bat-tery SOC at time ti , where ti is the time instance when recoverystarts, hence the initial condition of W

rba t (t) is Wba t (ti ). The

value of is computed using the logarithmic approach (see

Appendix C).

III. PARTIALLY LINEARIZED BATTERY MODEL VALIDATIONFOR CHARGING AND DISCHARGING

In this section, we validate the partially linearized battery

model developed in Section II-B for different battery types with-

out incorporating the recovery model. Since, the selected mag-

nitude of the input power (Pba t ) during validation is greater thanthe threshold , the recovery effect is negligible. The procedureto validate the partially linearized model for each battery type

can be briefly summarized as follows: first, we obtain the battery

efficiency data. Second, we estimate the efficiency coefficients

dk, v for charging and discharging, as described in Section II-B.Finally, we evaluate the quality of the partially linearized model

without recovery effect against the actual data by computing thetwo-norm normalized error. Specifically, we compute the error

between Wba t given by (12) at a given Pvba t,no m and the actualdata for charging and discharging under various loads, SOC,

and temperatures.

The validation results during charging and discharging at

0 C, 25 C, and 50 C are shown for 17.2 Ah, 12 V lead-acidbattery type. The validation results during discharging alone are

shown for 6.5 Ah, 7.2 V Ni-MH battery type at 5 C, 20 C,30 C, and 40 C, as the internal resistance data during chargingwas not available to us. Based on the validation, we set forth

a functional relationship between the model coefficients, dk ,v ,

k = 1, 2, 3 andtemperature(T) duringcharging anddischarging


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TABLE IMODEL COEFFICIENTS AND THE APPROXIMATION ERROR FOR 17.2 Ah, 12 V

LEAD-ACID BATTERY FOR THE NONLINEAR EFFICIENCY APPROXIMATION FORBOTH DISCHARGING AND CHARGING AT DIFFERENT TEMPERATURE

for both lead-acid and Ni-MH battery types. On the other hand,

the validation results during both charging and discharging for

1.41 Wh Li-ion and during discharging alone for 2850 mAh

alkaline batteries1 at room temperature are shown.

A. Efficiency Data for Validation

To obtain the efficiency data for lead-acid, Ni-MH, Li-ion,

and alkaline batteries, we follow the procedure describedin [22].

In [22], the battery is represented by a simple resistive Thevenin

equivalent circuit. The internal resistance of the battery is a

function of load rate, SOC, and temperature. The internal re-

sistance value is obtained as an average value under various

loads for each SOC at different temperature values for lead-acid

and Ni-MH batteries and for each SOC at room temperature

for Li-ion and alkaline batteries. The battery efficiency is then

obtained as the ratio of the terminal voltage to the open-circuit

voltage for discharging and vice versa for charging, where the

voltage difference is due to the losses through the internal re-

sistance. The data on the internal resistance for each SOC atdifferent temperature values for 17.2 Ah, 12 V lead-acid, and

6.5 Ah, 7.2 V Ni-MH batteries is obtained from [23] and [24],

respectively. While, the internal resistance data for each SOC at

room temperature for 1.41 Wh Li-ion, and 2850 mAh alkaline

batteries is obtained from [25].

B. Validation Results for Lead-Acid at Different Temperature

Initially, we present the discharging and charging validation

results of the partially linearized battery model without consid-

ering the recovery model for 17.2 Ah, 12 V lead-acid battery

type at 0 C, 25 C, and 50 C.In the case of a 17.2 Ah, 12 V lead-acid battery, we assume a

nominal load power of 27.5 W during discharging and 22 Wduring charging for all the temperature values. The coefficients

dk, v , k = 1, 2, 3 used to validate the linearized approximationduring discharging and charging at different temperatures are

presented in Table I along with the two-norm normalized error.

The approximation of the linearized generic battery efficiency

is accurate around the nominal input power of 27.5 W during

discharging and around the nominal input power of 22 Wduring charging. Fig. 1, whose y-axis is Pba t ba t , illustrates

1Alkaline batteries in the market are mostly nonrechargeable as the existing

rechargeable alkaline batteries are not commonly used.

the accuracy of the efficiencies from the linearized approxima-

tion against the actual data for both discharging and charging,

respectively. The approximation error increases for large excur-

sion from the nominal input power level, as observed in Fig. 1, as

expected. The partially linearized battery model, initially devel-

oped for an HEV application, showed a similar accuracy pattern

over the range of battery power in the case of a 30 of 12 Ah

12 V lead-acid battery [10], [11], [26].

From Table I, upon interpolation of the data, we see that

the variation in the model coefficients is essentially linear with

temperature and is given by a functional relationship in Table II

for both discharging and charging.

C. Validation Results for Ni-MH at Different Temperature

The validation results of the partially linearized battery model

for 6.5 Ah, 7.2 V Ni-MH battery type at 5 C, 20 C, 30 C, and40 C for discharging alone arepresented. A nominal load powerof 15 W is assumed during discharging for all the temperature

values. The approximation of the linearized battery model isaccurate around the nominal input power level of 15 W during

discharging. Fig. 2 shows a similar accuracy pattern over the

range of battery power. The coefficients used to validate the

linear approximation during discharging for 6.5 Ah, 7.2 V Ni-

MH battery at 5 C,20 C,30 C,and40 C are listed in Table IIIalong with the two-norm approximation error. From Table III,

we observe that the variation in the model coefficients during

discharging is essentially linear with temperature and is given

by a functional relationship in Table IV.

D. Validation Results for Li-ion and Alkaline Batteries

at Room Temperature

Initially, we present the discharging and charging validation

results of the partially linearized battery model for 1.41 Wh Li-

ion battery at room temperature alone, as the internal resistance

data at different temperature values was not available to us.

Later, we present the validation result during discharging for

2850 mAh alkaline battery at room temperature.

In the case of a 1.41 Wh Li-ion battery, the same valida-

tion procedure is applied. The two-norm normalized error for

discharging is 3.15% and for charging is 1.38%. The approx-

imation of the linearized battery model is accurate around the

nominal input power level of 253 mW during discharging and

250 mW during charging, as shown in Fig. 3. The approxima-tion error increases during both discharging and charging as the

load power level deviates from respective nominal power level,

but not significantly over the range, as shown in Fig. 3.

In the case of 2850 mAh alkaline battery, the same validation

procedure is applied during discharging alone, since alkaline

batteries are mostly nonrechargeable. The existing rechargeable

alkaline batteries are not commonly used. Fig. 4 shows the ac-

curacies of the linearized battery model for 2850 mAh alkaline

battery, which is linearized about the nominal input power of

125 mW. Thetwo-norm normalized error is 4.01%. From Fig. 4,

a similar accuracy pattern over the range of battery power is

observed.


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Fig. 1. Linearized approximation of 17.2 Ah 12 V lead-acid battery. (Left column) During discharging around the nominal power of 27.5 W at (a) 0 C,(b) 25 C, and (c) 50 C. (Right column) During charging around the nominal power of22 W at (d) 0 C, (e) 25 C, and (f) 50 C.

Thecoefficients used to validate the linear approximation dur-

ing discharging and charging for 1.41 Wh Li-ion and 2850 mAh

alkaline batteries during discharging alone are listed in Table V.

The linearized approximation of the battery captures the fact

that the battery efficiency decreases as SOC decreases and bat-

tery input power increases. This is consistent with the behavior

of the battery efficiency maps. In the validation of all the battery

types during discharging and charging, at different temperatures

(including the room temperature), we observed that the partially

linearized model is accurate about its nominal input power level


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Fig. 2. Linearized approximation of 6.5 Ah, 7.2 V Ni-MH battery during discharging around the nominal power of 15 W at (a) 5C, (b) 20 C, (c) 30 C, and(d) 40 C.

TABLE IIFUNCTIONAL RELATIONSHIP OF THE MODEL COEFFICIENTS WITH

TEMPERATURE DURING DISCHARGING AND CHARGING FOR 17.2 Ah, 12 VLEAD-ACID BATTERY

TABLE IIIMODEL COEFFICIENTS AND THE APPROXIMATION ERROR FOR 6.5 Ah, 7.2 V

Ni-MH BATTERY FOR THE NONLINEAR EFFICIENCY APPROXIMATION DURINGDISCHARGING AT DIFFERENT TEMPERATURES

TABLE IVFUNCTIONAL RELATIONSHIP OF THE MODEL COEFFICIENTS WITH

TEMPERATURE DURING DISCHARGING FOR 6.5 Ah, 7.2 V Ni-MH Battery

and the approximation error increases slightly as the load powerdeviates from the respective nominal input power level. The ac-

curacy of the partially linearized can be improved by selecting

different power level based on the operating load power.2

IV. LI-ION BATTERY LIFETIME ESTIMATION USING PARTIALLY

LINEARIZED MODEL WITH RECOVERY

In this section, we present the description of typical power

load usage during battery operation and numerical estimation

2A maximum of 5.24% error was observed during the validation, which isquite reasonable. However, one can create a set of range of load powers andselect an associatednominal power for eachrange to lower the deviation between

the load and nominal power, and hence to reduce the approximation error.


8/15


Fig. 3. Linearized approximation of 1.41 Wh Li-ion battery during (a) battery discharging around the nominal power of 253 mW and (b) battery charging aroundthe nominal power of250 mW.

Fig. 4. Linearized approximation during discharging for 2850 mAh alkalinebattery around the nominal power of 125 mW.

of 1.41 Wh Li-ion battery lifetime using the partially linearized

model during discharging (v = 0) [see (12)] with the incorpo-rated recovery model [see (14)]. The actual lifetime measure-

ments on a real 1.41 Wh Li-ion battery for various typical load

usages are not available. The DUALFOIL simulator [3], which

numericallysolves set of partial differential equations governing

the chemical reaction in the rechargeable Li-ion batteries, can

be used to obtain lifetime estimates of 1.41 Wh Li-ion batteryfor a given set of typical power loads. However, we do not use

DUALFOIL simulator because it is a cumbersome processes

to appropriately choose over 50 parameters in order to obtain

accurate simulation results. Therefore, to estimate the lifetime

of 1.41 Wh Li-ion battery, we downscale the power loads used

during the actual lifetime measurement of same type 2.2 Wh

Li-ion battery in [16] and [27]. Under the downscaling assump-

tion, (see Appendix B for details), the lifetime of 2.2 Wh Li-ion

battery subject to typical power loads summarized in Tables VI

and VII and the lifetime of 1.41 Wh Li-ion battery subject to

downscaled typical power loads also summarized in Tables VI

and VII must be the same. We compare the aforementioned ac-

TABLE VMODEL COEFFICIENT FOR THE NONLINEAR EFFICIENCY APPROXIMATION FORBOTH DISCHARGING AND CHARGING FOR 1.41 Wh AND FOR DISCHARGING

ALONE FOR 2850 mAh ALKALINE BATTERIES AT ROOM TEMPERATURE

TABLE VIDESCRIPTION OF ACTUAL [16] AND DOWNSCALED POWER LOADS

OFS

UBSET

PT

tual battery lifetime with the estimate obtained using the battery

model in Section IIB and D [see (12) and (14)] and with the

estimate obtained using the Rakhmatov and Virudhula battery

model (6).


9/15


TABLE VIIDESCRIPTION OF ACTUAL [16] AND DOWNSCALED VARYING POWER LOADS OF SUBSET Pc

In general, the battery open-circuit voltage Vo c is a functionof battery SOC and operating temperature. In this paper, all

the computation assumes room temperature. Initially, Wba t (t =0) = 1 corresponds to fully charged battery with theinitial open-circuit voltage Vo c = V

o c , while Wba t (t = L) = 0 corresponds

to Vo c dropping to the cutoff voltage (Vcutoff). The value ofV

o c

and Vcutoff depends upon the battery type. So under a typicalload, battery Vo c drops to Vcutoff over time. Hence, the batterylifetime L can also be defined as the time taken by battery Vo cto drop to Vcutoff below which no further power (current) canbe drawn from the battery. This definition is consistent with the

battery lifetime definition in Section II-A, i.e., the time duration

duringwhich the concentration of the electroactivespecies drops

from its initial concentration C to a specific threshold Ccutoff.

A. Description of Typical Power Load Usage

The typical power loads (both constant and varying) used in

this paper are the average power consumed during various oper-

ating modes of the Itsy pocket computer, which is powered by

a 2.2 Wh Li-ion battery [27]. The power loads used to estimatebattery lifetime in this paper were taken from [27] in which the

power consumption of Itsy pocket computer were reported. Let

P represent a set consisting of various actual average powerloads (both constant and varying). The set P is divided into twosubsets, PT and PC, where the subset PT consists of15 constantaverage power loads and the subset PC consists of seven varyingaverage power loads. The average power loads in both subsets

are downscaled and are used to estimate the lifetime of 1.41 Wh

Li-ion battery as mentioned earlier. See Appendix B, for details

on the computation of downscaling factor and assumptions. The

actual and downscaled power loads are summarized in Tables

VI and VII were computed using average current and average

voltage of 3.75 V [27]. So the lifetimes estimated and measuredare average lifetimes. Given the upper and lower bounds on

the load values, the models discussed in Section II can be used

to estimate upper and lower bounds on the actual lifetimes.

From here onward, we will refer average power load used to

estimate battery lifetime as power load for simplicity.

In Table VI, the constant power loads (T1T10) represent

the power consumed during different operating modes of Itsy

computer. In addition, peripheral devices, such as Microdrive

hard disk [28] and a WaveLAN wireless card [29] were attached

to Itsy, and power consumptions (T11T15) were recorded for

different operating modes of Itsy and peripheral devices [16],

[27]. The following letter abbreviations are used to describe the

operation modes of Itsy and peripheral devices in Table VI [16],

[27].

1) Itsy: IIdle, MMPEG, DDictation, TTalk1, WWAV1,

and SSleepDC.

2) Microdrive: SStandby and AAccess.

3) WaveLAN: DDoze, RReceive, and TTransmit.

The power loads in Table VII represent the measured power

consumption when Itsy operation is switched between dif-

ferent modes. In Table VII, the second column represents the

actual power measurements, the third column represents the

downscaled power measurements, and the fourth column repre-

sent the time duration in which each power level was measured.

Cases C1 and C2 have the same load power levels. However, the

order in which the battery is subjected to power load is reversed.

In case C1, load power is a decreasing staircase function, except

for the last period in which the load power jumps up and remains

constant. In case C2, load power is an increasing staircase func-

tion and that remains constant at the same power value as case

C1 during the final period. Case C3 is similar to C2, except it

has a rest period of 50 min. Case C4 is a periodic repetition of

C2 for ten cycles (represented by superscript 10 in Table VII),but the time for each power level is downscaled by a factor of

ten (also represented by superscript 10 in Table VII); hence the

total load power consumption in cases C2 and C4 is identical.

Case C5 is similar to C4 except that the first two power levels in

C5 are lower than the ones in C4. The time interval column in

Table VII for cases C4 and C5 has a subscript 200, which means

that after ten load cycles, the total time duration is 200 min.

Cases C6 and C7 have different rest periods. Cases C3, C6, and

C7 are used to evaluate the recovery effect of the battery, since

they contain rest period.

The typical power loads summarized in Tables VI and VII

represent a broad range of loading condition used to exercisethe battery.

B. Numerical Estimation of Battery Lifetime

Given the partially linearized battery model during discharg-

ing (v = 0) with the incorporated recovery model [see (12) and(14)], the discharging coefficients dk ,v , k = 1, 2, 3 for down-scaled operating nominal power Pba t,no m of 181.2 mW pre-sented in Table VIII is computed, as per the procedure described

in Section II-B, and various downscaled power loads in Tables

VI and VII, a numerical lifetime estimation of 1.41 Wh Li-ion

battery is computed in MATLAB. Note that all the parame-

ters and the downscaled power load profiles used in the lifetime


10/15


TABLE VIIIMODEL COEFFICIENTS FOR THE NONLINEAR EFFICIENCY APPROXIMATION FOR

DISCHARGING (v = 0) AT THE NOMINAL POWER LEVEL OF 181.2 mW FOR1.41 Wh LI-ION BATTERY AT ROOM TEMPERATURE

TABLE IXMEASURED AND ESTIMATED LIFETIMES FOR EACH CONSTANT POWER LOAD

TABLE XMEASURED AND ESTIMATED LIFETIMES FOR EACH VARYING POWER LOAD

estimation process is for 1.41 Wh Li-ion battery and is measured

at room temperature. The downscaled nominal operating input

power Pba t,no m is 181.2 mW, which is obtained by downscalingthe average battery power flow in the Itsy pocket computer.

The recovery time constant () computed using the downscaled

load data is 0.016 s1 (see Appendix C). The percentage ofrecovery is selected as 3.27% because the partially linearized

battery model during discharging without the recovery modelunderestimated the battery lifetime by 3.27% on average at nom-

inal power.

Theresults of lifetime estimation using thepartially linearized

battery model during discharging (v = 0) with the incorporatedrecovery model for downscaled power load profiles PT and PCare summarized in Tables IX and X, respectively. Based on the

downscaling assumption in Appendix C, the lifetime of 1.41 Wh

Li-ion battery computed using the downscaled power loads must

be equal to the lifetime of 2.2 Wh Li-ion battery computed (mea-

sured) using the actual power loads. Henceforth, the estimated

lifetimes of 2.2 Wh Li-ion battery using the Rakhmatov and

Virudhula battery model [8] [see (6)] for PT and PC are also

TABLE XIMEASURE OF BATTERY RECOVERY

summarized in Tables IX and X along with measured lifetimes

in the first column. The Rakhmatov and Virudhula battery model

parameters: = 33706 and = 0.75 reported in [16] is used toestimate battery lifetime. The error between the measured and

the estimated lifetime is computed using (15) and is also listed

in Tables IX and X

Error(%) =measured lifetime estimated lifetime

measured lifetime 100.

(15)

C. DiscussionIn this section, we analyze and discuss the performance of

the partially linearized model with the incorporated recovery

model under varying load conditions. A broad range of selected

load profiles used to exercise the battery model in this paper

is a real-time power consumption of Itsy computer under dif-

ferent operating modes [16], [27]. The typical loads presented

in Tables VI and VII captures both static and dynamic nature

of the power usage observed in many real-time applications.

In addition, to lifetime estimation using the partially linearized

model with the incorporated recovery model, we also compute

the percentage of recovery observed in cases C3, C6, and C7.

We define a few terms as in [8]. Let Loriginal represent the esti-mated lifetime for the power profile when no load relaxation (norest period) is exercised. When the rest period is present, let denote the rest period in a load profile and let Lunaffected denotethe expected lifetime as if no recovery took place during the rest

period of duration , i.e., Lunaffected = Loriginal + . In general,the lifetime L should be greater than Lunaffected due to recoveryeffect. The quantity ((L Lunaffected )/Loriginal ) 100% rep-resents the lifetime extension due to recovery effect. Table XI

summarizes the recovery percentage (i.e., lifetime extension)

obtained by using the partially linearized battery model along

with the incorporated recovery model with respect to Loriginal .Cases C3, C6, and C7 having a rest period of 50, 35.2, and

24.9 min, respectively, show an increase in estimated batterylifetime by 0.94%, 4.3%, and 6.7%, respectively, (with respect

to Loriginal ) due to recovery effect. These observations ascer-tain the fact that the partially linearized battery model with the

incorporated recovery model is able to capture the recovery ef-

fect. To establish the accuracy of the recovery percentage, we

compare the percentage increase in the lifetime computed using

the partially linearized model with the incorporated recovery

model and the percentage increase in the lifetime when the DU-

ALFOIL simulator is used on the same cases C3, C6, and C7.

The percentage increase in the lifetime with the DUALFOIL

reported in [8] is 1%, 1.7%, and 3.7% for C3, C6, and C7, re-

spectively. However, the percentage of recovery observed in C3,


11/15


C6, and C7 can be slightly higher than the actual recovery of real

battery. This is due to the fact that the partially linearized model

underestimates the power losses when the load power is higher

than Pba t,no m , as shown in Fig. 3. In general, the battery recov-ery has more profound effect when the power discharge rate is

high. Since the power discharge rate in C7 is higher than one in

C6 and C7 exhibits higher percentage recovery, as indicated in

Table XI.

Intuitively, a battery at a higher SOC must be able to handle

a large power load better than a battery at a lower SOC. This

assists in identifying the power load pattern that would result in

better lifetime. The cases C1 and C2 have the same power load

values except the order in which the battery is subjected to load

power is reversed (as described in Section IV-A). As expected,

the lifetime of C1 is slightly higher than the lifetime of C2. Case

C4 is a periodic repetition of an increasing staircase as in C2

for ten cycles, but the time for each power level is downscaled

by a factor of ten, while the total load power consumption in

both cases is identical. The battery in case C2 handles high-load

power at low SOC, while in case C4, the battery is subject toperiodic variation of both low- and high-load power at low SOC.

As a result, the lifetime estimated in case C4 is slightly better

than the lifetime estimated in case C2.

In the case of selected constant loads (see Table VI), the

lifetime estimation widely ranges from 2 h to 10 days. Table IX

shows the comparison of the accuracy of the lifetime estimations

between the partially linearized model and the chemical-based

Rakhmatov and Virudhula battery model [8]. The estimation

error of the partially linearized battery model is less than 10%

on average at nominal load power for the typical constant loads

listed in Table VI. We observe that the estimation error increases

as the load power level increases or decreases from the nominalinput power. This observation is consistent with the assumption

made during the linearization of nonlinear battery efficiency

[see (8)] and the validation results presented in Section III. For

example, the estimation error for 701.4 mW load power, which

is 520.2 mW more than nominal input power, is 9.8%, while

the error for 7.2 mW load power, which is 174 mW less than

nominal input power, is 7.2%. In the case of Rakhmatov and

Virudhula battery model (6), the estimation error increases as

the load on the battery decreases.

Based on the numerical results presented earlier, we observe

that the inputoutput partially linearized model with incorpo-

rated recovery model provides reasonably accurate lifetime es-

timation on average at nominal power in comparison to thelifetime estimation result obtained from the Rakhmatov and

Virudhula model [8].

V. BATTERY CYCLE LIFE VALIDATION

In general, cycling a battery at different DODs and

charge/discharge power rates affects battery cycle life as battery

ages. Data in the literature suggests that Wm axba t is a function ofnumber of cycles, DOD, and nominal charge/discharge power

level. In this section, we specifically model how the maximum

rated energy stored in a battery is affected by repeated charging

and discharging at different power levels given a specific level

of DOD. Full-discharge cycling test (equivalent to 100% DOD)

is one of many cycling tests. IEEE Standard recommends full-

discharge cycling test as it corresponds to a worst case scenario,

but there are lots of charge/discharge cycling test at different

DODs [30]. Incorporation of the DOD into the battery model is

beyond the scope of this paper.

In this paper, battery cycle life is defined as the number of

cycles (discharging than charging) the battery can withstand

before the maximum capacity of the battery reduces to 80% of

its initial ampere-hour rating at a specific level of DOD under

the assumed condition of constant temperature. Battery cycle

life represents a metric different from the chemical lifetime

definition in Section IV. For a fixed DOD, if the battery is

cycled at a rated charge/discharge power level, the battery last

for N cycle. Further, smaller rated power levels in each cycleprolongs the battery cycle life (>N), while the effect of higherpower shorten the battery cycle life (


12/15


Fig. 5. Maximum rated energy term Wm axba t approximation at the nominalcharge/discharge power level of 3780 mW for 965 mAh 4.1 V Li-ion battery.

error for charge/discharge power level closer to nominal power

level is 2.5%. As the charge/discharge power level deviatesfrom the nominal power level, the two-norm normalized error

increases, as expected.

Thus, from Fig. 5, we observe that the validation result for

the maximum rated energy Wm axba t is accurate about its nominalcharge/discharge power level and the error increases slightly

as the power level deviates from Pba t,no m . This validation pat-tern is consistent and similar to the pattern observed during

the validation of the linearized model for Li-ion battery type in

Section III-D. This is due to the fact that the battery efficiency

decreases with repeated discharging and charging, as the inter-

nal resistance of the battery increases, which is consistent with

the findings of Shim and Striebel [34] on a Li-ion battery.Thus, in this section, we showed that by approximating the

maximum rated energy term Wm axba t as a function of number ofcycle, nominal charge/discharge power level at a fixed DOD,

we can incorporate the cycling effect.

VI. CONCLUSION

In this paper, we have shown that the dynamic measure-

ment of the battery SOC was approximated by the partially

linearized (control-oriented) battery model for both charging

and discharging. A complete mathematical development of the

partially linearized battery model was presented in Section II-B

and in Appendix A. The model was augmented with an addi-tional differential equation to capture the recovery effect. Then,

the generalization of the scalable partially linearized model to

different battery types at different temperatures and its valida-

tion against the actual data for both charging and discharging

was demonstrated. Based on the validation results, a functional

relationship between the model coefficients and temperature

was set forth for lead-acid and Ni-MH battery types. The model

was found to be reasonably accurate around its nominal input

power level for various battery types at different temperatures.

The battery model with the incorporated recovery model was

exhaustively exercised under a variety of power loads for Li-ion

battery lifetime estimation. Actual lifetime measurements and

the estimated lifetimes using the Rakhmatov and Virudhula bat-

tery model were given for comparison. By approximating the

maximum rated energy term of the linearized model as a func-

tion of number of cycle and nominal charge/discharge power

level at a specific DOD level, the effect of repeated charging

and discharging on the overall battery cycle life was presented.

Although the developed model is linear in the input, it captures

all the essential nonlinear characteristics of the battery under

varying load conditions.

The partial linearized inputoutput battery model has been

used successfully for control-oriented power flow management

in an HEV application [11], [26]. Specifically, the model is very

useful for control when an inverter controls the currentvoltage

levels (and thus power levels) between the battery and any de-

vice, such as an electric drive, sensor, or actuator. In distributed

sensor network application, estimation of battery longevity, es-

timates of scheduled maintenance, and replacement are some

of the key issues. The usage of the model is computationally

inexpensive and can provide reasonably accurate information

on the SOC of the batteries. This information will assist in in-telligent network activity scheduling, dynamic rerouting, and

battery energy aware protocol designs in scenarios, where bat-

tery maintenance and replacement is impractical.

The accuracy of the partially linearized battery model can be

improved by selecting different nominal input powers based on

the operating load power. For example, one can create a set of

range of load powers and select an associated nominal input

power for each range to lower the deviation between load power

and nominal input power, andhence to reducethe approximation

error.

APPENDIX A

Consider a general nonlinear state equation x = f(x, u),where x is the state, and u is the (control) input. A Taylorseries expansion around the nominal operating input u is givenby

x = f(x, u)|u + f(x, u)u

|uu + H.O.T. (18)where H.O.T. means higher order terms that are assumed small

and negligible. Therefore, the linearization of (8) about a nom-

inal battery power Pvba t,no m , v {0, 1}, becomes

Wba t

=[ln(Wba t + d1,v ) + d2,v Pvba t,no m + d3 ,v ] Pv

ba t,no m

Wm axba t

+{[ln(Wba t + d1,v ) + d2,v Pvba t,no m + d3,v]Pba t}

Pba t

Pv

b a t , n o m

Pba tWm axba t

(19)

where Pba t = Pba t (t) Pvba t,no m . Now, we compute the par-tial derivative in (19) and evaluate at Pvba t,no m

{[ln(Wba t + d1 ,v ) + d2,v Pba t + d3,v ]Pba t}

Pba t

Pvb a t , n o m


13/15


= {[ln(Wba t + d1,v ) + d2,v Pba t + d3,v ] d2,v Pba t}|Pv

b a t , n o m

= [ln(Wba t + d1,v ) + 2d2,v Pvba t,no m + d3,v ]. (20)Substituting the expression in (20) into (19) results in the

following equation:

Wba t = [ln(Wba t + d1,v ) + d2 ,v Pvba t,no m + d3,v ]Pvba t,no m

Wma xba t

[ln(Wba t + d1,v ) + 2d2,v Pvba t,no m + d3,v ]Pba tWma xba t

.

(21)

For our control purposes, we want Wba t equation to dependon the actual battery input Pba t (t) rather than Pba t . For this,we substitute Pba t = Pba t Pvba t,no m into (21)

Wba t =

[ln(

Wba t + d1,v ) + d2 ,v Pv

ba t,no m+ d3,v ]

Pvba t,no m

Wma xba t

[ln(Wba t + d1 ,v ) + 2d2 ,v Pvba t,no m + d3,v ]

Pba t (t) Pv

ba t,no m

Wm axba t(22)

and arrive at the final partially linearized equation for the battery

SOC

Wba t (t) =d2,v

Wm axba t(Pvba t,no m )

2 [ln(Wba t (t) + d1,v )

+ 2d2,v Pv

ba t,no m + d3 ,v ]Pba t (t)

Wma x

ba t

. (23)

The coefficients dk ,v , k = 1, 2, 3 and v {0, 1} for differentbattery types are presented in Tables I, III, and V.

APPENDIX B

The discharging and charging efficiency of lead-acid (25 of

18 Ah 12.5 V) battery pack is obtained from [35]. In order to

obtain discharging and charging efficiency maps of lead-acid (30

of 12 Ah 12 V) battery pack used in this paper, a downscaling

is performed.

Let Wm axbat1 be the maximum rated energy of lead-acid batterypack1 [35] and Wm axbat2 be the maximum rated energy of the lead-

acid battery pack2, the one used in this paper. We define scalingfactor = Wm axbat2 /W

m axbat1 . The assumption used in downscaling

the battery efficiency map of [35] to the size used in this paper is

as follows: let Pbat1 denote the power delivered by battery pack1 and Pbat2 that of battery pack 2, whenever = Pbat2 /Pbat1 ,then for v {0, 1}

bat1 (SOC, Pbat1 , v) = bat2 (SOC, Pbat2 , v). (24)

A similar scaling procedure and assumption is adopted in down-

scaling the load powers of [16], [27] to the energy level of

Li-ion battery used in this paper: let Wm axbat1 represent the max-imum rated energy of Li-ion battery 1 (2.2 Wh) and Wm axbat2 be

the maximum rated energy of Li-ion battery 2 (1.41 Wh) usedin this paper. Let = Wm axbat2 /Wm axbat1 be a scaling factor. The

lifetime of 2.2 Wh battery is same as the lifetime of 1.41 Wh

battery, whenever the Li-ion battery 2 (1.41 Wh) is subject topower load Pbat2 = Pbat1 , where Pbat1 denote the averagepower delivered by 2.2 Wh Li-ion battery. Hence, the average

typical power load used in this study for 1.41 Wh Li-ion battery

is obtained.

APPENDIX C

In Section II-D, we discussed that batteries exhibit recovery

effect when, during which the SOC increases before, reaching a

steady-state value. The rate of increase in SOC during recovery

is approximated using a recovery time constant. In the recovery

model presented in Section II-D, the parameter [see (14)]

represents the recovery time constant. The value of depends

upon battery type and is often computed using experimental

data. The recovery time constant () of 1.41 Wh battery used in

this paper is computed by using the downscaled experimental

recovery data of same type 2.2 Wh Li-ion battery from [21]. The

actual data is monotonically increasing and reaches a steady-

state value over time. In another words, the data consists of the

transient response of batterys SOC and its steady-state value.

A simple logarithmic approach is used to compute . This

common procedure is to compute the time constant of a RC

circuit from the transient response. Consider a general state

equation of an RC circuit with zero input. The state trajectory

(due to an initial condition) is exponentially decreasing and is

expressed as follows:

x(t) = x(0)et/ (25)

where x(t) is the state at time t, x(0) is the initial value of thestate, and = RC is the time constant with R and C being the

resistance and the capacitance, respectively.Taking natural logarithmic on both side of (25), we get

ln(x(t)) = ln(x(0)) t/. (26)Equation (26) is a straight line when plotted against time t, withthe slope of1/ and the intercept ofln(x(0)). Only the slopeof the line determines the time constant.

To compute the time constant of the complete response of (14)

due to both constant input and the initial condition, we modify

the data into an exponentially decreasing response. Therefore,

we first determine the steady-state value of the response, and

then, subtract the data from the steady-state value. This leads

to an exponentially decreasing response. Then, take the naturallogarithm and perform least-square fit to the response data. The

fit will be a straight line when plotted against time t and theslope of the line is negative inverse of RCtime constant.

The procedure described earlier to compute time constant

of a simple RC circuit with nonzero input is used to compute

. The downscaled data of 2.2 Wh Li-ion battery is subtracted

from the steady-state value to obtain an exponentially decreasing

function. Let Wtrba t (t) be the difference between the steady-

state value and the complete response of the SOC, with an

initial condition of Wtrba t (0). Similar to (25), W

trba t (t) decays

exponentially to zero as per

Wtrba t = W

trba t (0)e

t/ . (27)


14/15


Again, take natural logarithmic on both side of (27) and result

in

ln(Wtrba t (t)) = ln(W

trba t (0)) t/. (28)

Thus, the function Wtrba t (t) is linear with time t and with the

slope of1/. Hence, the estimate of is 1/.

ACKNOWLEDGMENT

The authors would like to thank reviewers for their valuable

comments and suggestions. They would also like to thank Rick

Mayer (Purdue University) for his thoughtful discussions on

battery modeling.

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Vivek Agarwal (S98) received the B.E. degreefrom the University of Madras, Chennai, India, andthe M.S. degree from the University of Tennessee,Knoxville, TN, in 2001 and 2005, respectively, bothin electrical engineering. He is currently working to-ward the Ph.D. degree from the School of NuclearEngineering, Purdue University, West Lafayette, IN.His doctoral research focuses on power managementin battery powered interconnectedwireless networks.

In 2005, he was a Research Associate at theHewlett-Packard Research Laboratories, Palo Alto,

CA . His current research interests include power management in wireless sen-sor networks, battery modeling, signal and image processing, nuclear materialdetection, regulation techniques, and machine learning.

Kasemsak Uthaichana (M07) received the B.S. de-gree in electrical, computer, and systems engineeringfrom Rensselaer Polytechnic Institute, Troy, NY, in2000, and the M.S. and Ph.D. degrees in electricaland computer engineering from Purdue University,West Lafayette, IN, in 2002 and 2006, respectively.His doctoral research focused on modelingand powerflow control for parallel hybrid electric vehicles.

From 2007 to 2008, he was with Caterpillar Inc.,Peoria, IL,where he wasengagedin integrated powersystems control software for wheel loader machines.

He is currentlyin the Department of Electrical Engineering, Chiang Mai Univer-sity, Chiang Mai, Thailand. His research interests include power management inhybrid vehicles, operational control of fuel cells, battery management system,and friction estimation and compensation.

Dr. Uthaichana is an Active Member of the IEEE Control Systems Society

and the IEEE Power and Energy Society.


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Raymond A. DeCarlo (F89) He received the B.S.and M.S. degrees in electrical engineering from theUniversity of Notre Dame, Notre Dame, IN, in 1972and 1974, respectively, and the Ph.D. degree fromTexas Tech University, Lubbock, TX.

In 1977, he joined Purdue University, WestLafayette, IN as an Assistant Professor of electricalengineering, and became an Associate Professor in1982 andFullProfessor in 2005. Duringthe summersof 1985 and 1986, he was at the General Motors Re-search Laboratories. He has authored or coauthored

three books, more than 50 journal, more than 100 conference, and eight bookchapter/reprint articles. His current research interests include interdisciplinaryranging from variable structure control, hybrid optimal control and stability,hybrid electric vehicle modeling and control, biological modeling and control,as well as the control and modeling of the software test process.

Dr. DeCarlo was a Secretary Administrator of the IEEE Control SystemsSociety, and from 1986 to 1992 and from 1999 to 2003, a member of the Boardof Governors. He was an Associate Editor of Technical Notes and Correspon-dence, and Survey and Tutorial Papers, both for the IEEE TRANSACTIONS ONAUTOMATIC CONTROL. He was a Program Chairman for the 1990 IEEE Con-ference on Decision and Control (CDC), Honolulu, and a General Chairmanof the 1993 IEEE CDC, San Antonio. During 2001 and 2002, he was the VicePresident for Financial Activities for the IEEE Control System Society (CSS).He was the recipient of CSSs Distinguished Member Award in 1990, the IEEE

Third Millennium Medal in 2000, the EATON Award in Electrical and Com-puter Engineering (ECE) in 2002, the Motorola Excellence in Teaching Awardin 2006, and the award for Best Theoretical Paper in Automatica in 2008. Hewas a Chair of the Purdue University Senate and a Committee on InstitutionalCooperation (CIC) Faculty Fellow.

Lefteri H. Tsoukalas (M89) received the Ph.D.degree from the University of Illinois, Urbana-Champaign, in 1989.

He was a Professor and Former Head of theSchool of Nuclear Engineering, Purdue University,West Lafayette, IN, where he also holds a courtesyprofessorial appointment in construction engineeringand management. His current research interests in-clude in developing smart energy methodologies. Hehas authored or coauthored more than 200 researchpublications in the area including a book titled Fuzzy

and Neural Approaches in Engineering, (New York: John Wiley and Sons,1997). He was with the faculty of the University of Tennessee, Knoxville, Aris-totle University, and the University of Thessaly. He was also a Researcher andan Advisor, and a Consultant at the Japan Atomic Energy Research Institute, theMinistry of Education, ON, Canada; the International Atomic Energy Agency;the Agency for Science, Technology and Research of the Government ofSingapore; and the U.S. Department of Energy.

Dr. Tsoukalaswas the recipient of numerous awards and recognitions includ-ing Best Teacher Award at Purdue, Fellow of the American Nuclear Society,and the 2009 recipient of the Humboldt Prize.

Documents

5- Development and Validation of a Battery Model