20- Modeling, Estimation, And Control Challenges for Lithium-ion Batteries

8/4/2019 20- Modeling, Estimation, And Control Challenges for Lithium-ion Batteries

1/6

Modeling, estimation, and control challenges for lithium-ion batteries

Nalin A. Chaturvedi, Reinhardt Klein,,

Jake Christensen, Jasim Ahmed and Aleksandar Kojic

Abstract Increasing demand for hybrid electric vehicles(HEV), plug-in hybrid electric vehicles (PHEV) and electricvehicles (EV) has forced battery manufacturers to considerenergy storage systems that are better than contemporary lead-acid batteries. Currently, lithium-ion (Li-ion) batteries are be-lieved to be the most promising battery system for HEV, PHEVand EV applications. However, designing a battery managementsystem for Li-ion batteries that can guarantee safe and reliableoperation is a challenge, since aging and other performancedegrading mechanisms are not sufficiently well understood. Asa first step to address these problems, we analyze an existingelectrochemical model from the literature. Our aim is to presentthis model from a systems & controls perspective, and to bringforth the research challenges involved in modeling, estimationand control of Li-ion batteries. Additionally, we present a novel

compact form of this model that can be used to study the Li-ion battery. We use this reformulated model to derive a simpleapproximated model, commonly known as the single particlemodel, and also identify the limitations of this approximation.

I. BACKGROUND AND MOTIVATION

Lithium ion (Li-ion) batteries are popular as a reliable

source of power in mobile phones, and portable electronic

devices. Li-ion batteries are favored over other battery

technologies since they provide one of the best energy-to-

weight ratios, have no memory effect, and have a slow self-

discharge. Of late, there is a push to commercialize Li-

ion batteries for use in automotive and other applications

(such as aerospace or defense) due to their high energydensity [1]. In the automotive sector, increasing demand

for hybrid electric vehicles (HEVs), plug-in hybrid electric

vehicles (PHEVs), and electric vehicles (EVs) has forced

consideration of other promising battery technologies such

as Li-ion batteries to replace existing lead-acid batteries.

Unfortunately, this replacement is challenging due to the

large power and energy demands placed on such batteries,

while guaranteeing its safe operation.

A battery typically consists of the battery itself and the

battery management system (BMS). A BMS is composed

of hardware and software system that controls charging and

discharging of the battery while guaranteeing reliable and

safe operation [2]. It also takes care of other functions such ascell balancing. The design of a sophisticated BMS becomes

critical for Li-ion batteries, since these batteries can ignite

and explode when overcharged or under abuse conditions

[2], [3]. To design and build the BMS for Li-ion batteries,

a model is required that can describe the battery dynamics.

One of the key tasks of a BMS is to observe the states of

[email protected] , Robert Bosch LLC, Res. &

Tech. Center, Palo Alto, CA 94304.Otto-von-Guericke University, Institute of Automation Engineering,

Magdeburg 39106, Germany.

the battery and track physically relevant parameters as the

battery ages.

The outline of the paper is as follows. In section II,

we intuitively explain the fundamentals of a Li-ion battery.

In section III, we present equations describing a Li-ion

cells dynamic behavior. The modeling is based on using

electrochemical principles to develop a physics-based model

in contrast to equivalent circuit models [4], [5], [6], [7], [8].

While electrochemical models have been developed earlier,

our goal in this paper is to present this model with a

perspective that appeals to people with diverse backgrounds.

In section IV, we develop a novel compact form of this

model that can be used to study the full Li-ion battery model.Previous work in this field [9], [10], [11], [12] avoids such

detailed mathematical constructions since their primary aim

is cell-optimization using numerical simulations. However,

if the intention is to build control or estimation algorithms

for BMS, then a more mathematical and a systems-and-

controls-oriented understanding of the Li-ion battery model

becomes imperative. In this paper, we fill this gap and

additionally derive an approximation of the Li-ion battery

model that is used in the literature [13], [14] in section V.

This approximate model is obtained from the new compact

form derived earlier in section IV, thus demonstrating its use.

Next, in section VI, a comparison between the full model

and the reduced model is presented, identifying the domainswhere the approximation holds. In section VII, we present

estimation and control issues for BMS in Li-ion batteries

and present the current status of research in BMS. Finally,

we present the current solutions to estimation and control

problems, and conclude by mentioning future work.

I I . INTERCALATION-BASED BATTERIES

The commonly available Li-ion cell is an intercalation-

type cell [15]. The term intercalation-type implies that the

electrodes have a lattice structure and charging (discharging)

the cell causes the Li ions to leave the positive (negative)

electrode and enter the lattice structure of the negative

(positive) electrode. This process of ions being moved in andout of an interstitial site in the lattice is called intercalation.

A typical Li-ion battery (Figure 1) has four main com-

ponents. A porous negative electrode in a Li-ion cell is the

negative terminal of the cell. It is usually made of graphite.

Similarly, a porous positive electrode is the positive terminal

of the cell. It can have different chemistries, but is usually a

metal oxide or a blend of multiple metal oxides. A separator

is a thin porous medium that physically insulates the negative

from the positive electrode. It is an electrical insulator

that does not allow electrons to flow between the positive

2010 American Control ConferenceMarriott Waterfront, Baltimore, MD, USAJune 30-July 02, 2010

WeC12.5

978-1-4244-7427-1/10/$26.00 2010 AACC 1997


2/6

and negative electrodes. However, being porous, it allows

ions to pass through it via the electrolyte. The electrolyte

is a concentrated solution that has charged species. These

charged species can move in response to an electrochemical

potential. The key idea behind storing energy in a Li-ion

cell is that lithium has different potentials when placed in

an interstitial site of the lattice of the positive or negative

electrodes solid active material particles (sometimes referred

to as solid particles; see Figure 1). This potential of an

electrode is usually expressed as a function of the lithium

concentration in the electrode, referred to as the open-circuit

potential (OCP) of the electrode [15].

III. MODELING APPROACH

In this section, we present electrochemical equations that

describe the physics of a Li-ion battery. In a Li-ion battery,

lithium can exist either in the dissolved state in the electrolyte

or in the solid phase in the particles (interstitial site), as

shown in Figure 1, at every point along the X-axis.

Fig. 1. Schematic of model of an intercalation cell.

In Figure 1, we assume in our model that spherical solid

particles of radius Rp (shown by red and green spheres) existat every point x along the X-axis, which correspond to thelattice sites in the electrode. These particles are immersed in

a sea of electrolyte shown by the wavy, dashed line.

The seven variables required to describe this 1D model are

the current is(x, t) in the solid particles (or solid electrode),the current ie(x, t) in the electrolyte, the electric potentials(x, t) in solid electrode, the electric potential e(x, t)in electrolyte, the surface molar flux density jn(x, t) at thesurface of the spherical particle, the concentration ce(x, t) ofthe electrolyte, and the concentration cs(x,r,t) of lithium in

the solid electrode particles at a distance r from the centerof a spherical particle located at x at time t [15].

The input to the model is the current that is applied to

the battery given by I(t), and the output of the model is thecorresponding output voltage V given by

V(t) = s(0+, t)s(0

, t), (1)

where 0+ and 0 correspond to the two ends of the batteryas shown in Figure 1.

We next present the equations that describe the behavior of

the seven variables stated above. In the ensuing sections, we

assume, without loss of generality, that the cross-sectional

area of the separator is unity.

The first two equations follow from Kirchoffs law, is +ie = I, and from Ohms law

s(x, t)

x=

ie(x, t) I(t)

, (2)

where R+ is the effective electronic conductivity of

the solid. Equation (2) has no explicit boundary conditions.However, ie = 0 at 0

+ and 0, and ie = I at L+ and L.

The third equation relating e and ie is given by

e(x, t)

x=

ie(x, t)

+2RT

F

1 t0c

1 +

d ln fd ln ce

(x, t)

ln ce(x, t)

x, (3)

where F is Faradays constant, R is the universal gasconstant, T is the temperature and f is the mean molaractivity coefficient in the electrolyte. Also, is the ionicconductivity of the electrolyte, and t0c is the transferencenumber with respect to the solvent velocity. The boundary

condition of (3) is e(0+) = 0.The fourth equation describing the relationship between

ce and ie is given by

ce(x, t)

t=

x

De

ce(x, t)

x

+

1

F e

(t0aie(x, t))

x, (4)

where De is the diffusion coefficient, e is the volumefraction of the electrolyte and t0a is the transference numberfor the anion. The boundary conditions for the above equa-

tion are that the fluxes of the ions are zero at the current

collectors. Thus,

ce

xx=0

=ce

xx=0+

= 0.

The fifth equation relating cs and jn is described bydiffusion as

cs(x,r,t)

t=

1

r2

r

Dsr

2 cs(x,r,t)

r

, (5)

where r is the radial dimension of the particles in the activematerial. The boundary and initial conditions for (5) are

given by

csr

r=0

= 0,csr

r=Rp

= 1

Dsjn, cs(x,r, 0) = c

0s.

The sixth equation relating ie to jn is

ie(x, t)

x= aF jn(x, t), (6)

where a = 3sRp

is a constant. The boundary condition is

ie = I at L+ and L and ie = 0 at 0+ and 0.The seventh and the final equation relating jn to ce, cs,

s and e is the Butler-Volmer equation [16], [17]

jn =i0F

exp

aFRT

s exp

cFRT

s

, (7)

where a and c are transport coefficients, i0 is the exchangecurrent density, and s represents the overpotential. The

1998


3/6

overpotential s represents the change in potential that acharged species would go through as it passes through the

spherical particle into the electrolyte, and is given as

s(x) = s(x)e(x) U(css(x)) F Rfjn(x), (8)

where for any given t, css(x) cs(x, Rp). Note that theequation for the molar flux density jn is algebraic. Thus,all equations need to be solved together to satisfy the

algebraic constraint at every time t. The exchange currentdensity i0 in (8) is given by i0 = reff

ce(x, t)(cs,max

css(x, t))acss(x, t)c , where reff and cs,max are constants.

In summary, given that is + ie = I, the six equations (2),(3), (4), (5), (6) and (7) are solved for with applied current Ias the input and the output given by the voltage V as definedin (1).

IV. MODEL REDUCTION

In section III, we presented the mathematical equations

describing a Li-ion battery. In this section, we analyze these

equations and present a novel compact structure for the Li-

ion battery model. This reformulation involves changing only

the structure and hence, no information of the model is lostin the process.

A. Mathematical notation

Let Cr(a, b) denote the function-space of all real-valued,r-times continuously differentiable functions with domain(a, b) R. By abuse of notation, we include 1 value forr. A function in C1(a, b) is discontinuous, but its integralexists and is continuous. More precisely, this function-space

is the Sobolev space H1(R) [18]. Note that C(R) C(R) . . . C0(R) C1(R), where C(R) represents

the space of analytic functions.

We next introduce the concept of a function-space map-

ping. In this article, we define a function-mapFas a mappingthat takes an element of Cr(a, b) to Cq(a, b) where r and qare some integers. In other words, F : Cr(a, b) Cq(a, b) isa function-map. We also introduce the notion ofrestriction of

a function by means of an example. Suppose g(x, t) R is afunction, that is, g : RR R. Then, the t-restriction of thefunction g is defined as gt(x) g(x, t). Hence, gt : R Rdenotes the value of g for some fixed t.

B. Reduction of PDE system

We begin by reducing the system of five PDEs and one

algebraic equation to two PDEs with time derivatives and

algebraic constraints. We focus on any one of the positive or

negative electrodes for this purpose. The spatial domain isassumed to run from 0 to L. Thus, for the positive electrodeand the negative electrode, the corresponding domains are

[0, L] and [0+, L+], respectively (see Figure 1).Consider equation (6). For a sufficiently regular I(t) R,

we have jtn(x) C1[0, L] for each time t R. Note that

It = I(t) is a scalar for each t R. Now, we can solve(6) as ie(x, t) =

x0

aF jn(, t)d + ie0(I(t)). Define thefunction-map Fie : C

1[0, L]R C0[0, L] as

Fie(g, ) (x)

x0

aF g()d + ie0(). (9)

Then it is clear that a solution of the PDE in (6) for each tis

ie(x, t) = ite(x) = Fie(j

tn, I

t) (x). (10)

Here we have absorbed the constant of integration in Fieas the boundary condition of the PDE is known. Indeed, the

constant of integration is obtained by setting ie to either zeroor I(t) at one of the boundaries (x = 0 or x = L) of the

domain. Since the value ofite(x) is known at both x = 0 andx = L, there are in fact two boundary conditions to satisfy.Depending on whether it is the positive or negative electrode

or the separator, it is

{ite(0), ite(L)} =

{0,I(t)} (electrodes),

{I(t),I(t)} (separator).(11)

For now, we choose any one of the above boundary

conditions (at x = 0 or x = L) to determine ie0 in (9).Similarly, we solve PDE in (2) for each t as

s(x, t) = ts(x) = Fs(j

tn, I

t) (x) + s0(t), (12)

where Fs : C1[0, L]R C1[0, L] is defined as

Fs(g, ) (x) 1

x0

(Fie(g, ) (w) ) dw, (13)

w is a dummy variable, and s0(t) is an integration constantthat depends on boundary condition. Note that unlike Fie , we

do not absorb the integration constant in Fs . This is because

we do not know a priori what the boundary condition is.

As we show later, the boundary condition appears implicitly

through the flux density jn(x, t) and the Butler-Volmerequation. For now, we write s0(t) as a boundary conditionthat is unknown.

Similarly, assuming a constant t0c

and performing some

manipulations, we solve for the PDE in (3) as

e(x, t) = te(x) = Fe(j

tn, c

te, I

t) (x), (14)

where Fe : C1[0, L] C0[0, L]R C1[0, L] is

Fe(g,h,) (x)

x0

Fie(g, ) (w)

(h(w))dw

+2RT

F

1 t0c

ln(fh(x)) + e0(g,h,), (15)

w is a dummy variable and f is some known functionof h or is set to a constant. Note that in this case wehave absorbed the constant of integration in Fe since the

boundary condition for this PDE is known. It is given asFe(j

tn, c

te, I

t) (0+) = 0.Summarizing until now, equations (10), (12) and (14)

imply that if jn(x, t), ce(x, t) and I(t) are given, then weobtain s(x, t), e(x, t) and ie(x, t) (and hence is(x, t)since is(x, t) + ie(x, t) = I(t) for all (x, t) [0, L]R).

Next, consider equation (7) and (8). Substituting for sand e from (13) and (15), respectively, in (8) yields

s(x, t) = Fs(jtn, I

t) (x) + ts0 Fe(j

tn, c

te, I

t) (x) U(ctss(x))Rfjtn(x)F. (16)

1999


4/6

Equation (16) suggests that we can express s as a functionof jn, ce, css, I and s0 for every x and t. Note thatts0 = s0(t) in (16) is the unknown boundary conditionor constant of integration for the s-equation, and that theexchange current density is also a function of ce and css.Thus, equation (7) can be expressed as

jn

(x, t) = jtn

(x) = Fjn

(jtn

, cte

, ctss

, It, ts0

) (x), (17)

where Fjn : C1[0, L] C0[0, L] C0[0, L] R R

C1[0, L] is

Fjn(f, g, h, , ) (x)

i0(x)

F

exp

aF

RTs(x)

exp

cF

RTs(x)

, (18)

i0(x) reff

g(x)acs,max h(x)ah(x)c , (19)

s(x) Fs(f, ) (x) Fe(f , g , ) (x)

U(h(x))Rff(x)F + . (20)

Note that (17) is an algebraic equation that has to holdfor every x and time t. Given the electrolyte concentrationcte(x), the surface concentration of the solid particle c

tss(x) =

cs(x, Rp, t), and the current It, we need to find jtn(x) andts0 that satisfy (17). However, there are two unknowns

jn(x, t) and s0(t) and only one equation (17). To solvefor jn and s0 together, we now use the second boundarycondition on ie. Suppose to derive Fie in (9), we usedie(0, t) = 0. Then, the other boundary condition on ie at theseparator-electrode interface is ie(L, t) = I(t). Thus, from(10) it follows that

ie(L, t) = ite(L) = Fie(j

tn, I

t) (L) = I(t), (21)

which is an algebraic constraint on jn(, t). Then, (17) and(21) are solved together to obtain the current density jn(x, t)and the boundary condition on s given by s0(t), for agiven electrolyte concentration ce(x, t), surface concentra-tion of the electrode css(x, t) and current I(t).

The full Li-ion battery model is given by (2), (3), (4), (5),

(6) and (7). In this section, we have reduced these equations

to solving the dynamical equations (involving time) given by

(4) and (5) while satisfying the algebraic constraints in space

given by (17) and (21) at all times t.

V. SIMPLEST DISCRETIZATION OF THE MODEL

In the last section, we studied the equations for the model

given by PDEs (2), (3), (4), (5), (6) and (7), and reduced

them to a compact form given by (4) and (5), and the

algebraic constraint (17) and (21). In general, the equations

obtained from discretization of (17) and (21) along the

spatial dimension cannot be solved analytically. However,

it is possible to solve these equations analytically for the

case where the coarsest discretization is chosen. This yields

a model related to the single particle model (SPM) [13], [14].

A. Assumptions involving coarse discretization

Suppose we choose the lowest order of discretization for

the X-domain. Then, we have one node for the positiveelectrode and one node for the negative electrode (and a

node at the separator; see Figure 2) with quantities at this

node representing their average over the whole electrode. Let

us consider the positive electrode. Assume that cex

0 andcet 0. This approximation holds if we assume that I is

small or is large. Then ce(x, t) c0e R

+ is a constant.

Also, (4) yields that ie(x, t) = cnst(t), yielding that within adomain (positive electrode, negative electrode, or separator),

ie remains a constant or that it does not vary in x. Then wecan express ie for the entire electrode by one value in eachof the electrode in the cell.

Fig. 2. Schematic showing variables for the approximate model.

B. Solution for the coarse discretization

Since only one node exists in the electrode, we express

the corresponding variables as scalar functions of time de-

noted as j+n (t), i+e (t), +s (t), +e (t), c+e (t), c+s (r, t), andsimilarly for the negative electrode, as shown in Figure 2.

The function-maps in this case are easily solved as follows.

From (9) and (10), we obtain

0 = ie(0+, t) = i+e (t) = Fie(j

tn, I

t) (0+)

00

a+F j+,tn d + ie0(It) = ie0(I

t).

Thus, ie0(It) = 0 is obtained from the boundary condition

that ie(0+, t) = i+e (t) = 0. Next, substituting this boundarycondition in Fie and solving (21) implies that

I(t) = i

sep

e (t) = ie(L

+

, t) =Fie(j

t

n, I

t

) (L

+

)

=

L+0

a+F j+,tn d = j+,tn L

+a+F,

where isepe (t) is the current in the separator and hence,

j+n (t) = j+,tn =

I(t)

F a+L+. (22)

Next, it follows from (12) and (13) that

+s (t) =1

00

i+e (t) I(t)

dw + +s0(t) =

+s0

(t). (23)

2000


5/6

Similarly, it can be shown that +e (t) = 0 since +e0

(t) =0 from the boundary condition that e = 0 at the currentcollector of the positive electrode.

Finally, applying the last algebraic constraint (17) to (22),

and choosing a = c =12 , it follows that

I(t)

2a+L+= reff

c0ec+ss(t)(c

+s,max c

+ss(t)) sinh(

+), (24)

+ = F2RT

+s0(t) U

+(c+ss(t)) +R+f I(t)a+L+

, (25)

where U+() is the OCP of the positive electrode and isusually known from experiments.

Similarly, for the negative electrode, ie0(I(t) ) = 0 isobtained from the boundary condition that ie(0

, t) =ie (t) = 0. Substituting this boundary condition in Fie andsolving (21) yields

jn (t) = j,tn =

I(t)

F aL. (26)

Equations (12) and (13) imply that

s (t) =1

0

0

ie (t) I(t)

dw + s0(t) =

s0

(t). (27)

Also (15) with the boundary condition that e = 0 atthe current collector of the positive electrode yields that

+e0(t) = 0 and hence e (t) =

1

00+

ie(x, t)dx 0.The above follows since ie(x, t) = I(t) in the separator andI(t)/ 1 from the assumption for SPM that I(t) is smallor the conductance is large. Lastly, (17), along with theassumption that a = c =

12 , yields

I(t)

2aL= reff

c0ecss(t)(c

s,max c

ss(t)) sinh(

), (28)

= F2RT

s0(t) U(css(t))R

fI(t)

aL

, (29)

where U() is the OCP of the negative electrode and isknown from experiments. We can solve (24) and (25) for

+s0(t) and (28) and (29) for s0

(t) yielding +s (t) from (23)and s (t) from (27). Note that we need to compute c

+ss(t) =

c+s (Rp, t) and css(t) = c

s (Rp, t) by solving the PDE (4)

where jn(x, t) = j+n (t) and jn(x, t) = j

n (t), respectively.

Then, the output voltage V(t) = +s0(t) s0

(t).

V I . SIMULATION RESULTS

In section IV, we presented a compact form of the model

that is subsequently used to derive an approximation of

the electrochemical model (SPM) in section V. Since SPMis an approximation, its applicability is only valid over

certain regimes. In this section, we illustrate some of these

limitations by means of simulation results. We present a brief

comparison of the full model given by (4) and (5), and the

algebraic equations (17) and (21) with SPM. In particular, we

consider cells that correspond to a high energy configuration

with applications in EVs.

To compare model performance, we compare output volt-

ages and surface concentrations computed from the two

models. The comparison of surface concentration css can be

used as an indicator of when the approximate model starts to

fail. We present a comparison between SPM and full model

for a high energy cell configuration with a nominal capacity

of 3.5 A-h. The applied currents are a constant discharge atrates of C/25, C/2, 1 C and 2 C, where 1 C corresponds to3.5 A.

0 1 2 3 42.5

3

3.5

4

4.5

Voltage[V]

C/25

0 1 2 3 42.5

3

3.5

4

4.5

C/2

0 1 2 3 42.5

3

3.5

4

4.5

Voltage[V]

Discharge Capacity [Ah]

1C

0 1 2 3 42.5

3

3.5

4

4.5


2C

Full ModelSPM

Full ModelSPM

Full ModelSPM

Full ModelSPM

Fig. 3. Comparison of the full model and the single particle model (SPM)for a high-energy cell configuration.

The corresponding voltages for the full model and the

SPM are compared in Figure 3. As seen in the figure, SPM

is accurate until C, /2 where the discharge curves are almostindistinguishable.

As seen in Figure 4, the surface concentration in an SPM

is the average surface concentration in the electrode. This

averaging follows from the fact that variables in an SPM

represent spatial average over electrodes. At higher discharge

rates of 1 C and 2 C, the uniformity in the concentration is

lost and SPM is no longer valid since concentrations cannotbe effectively represented by its spatial average due to large

variance. This failure of the approximation is noted in the

corresponding rate plots in Figure 3. In fact, even at C /2,the variance in the surface concentrations shown in Figure 4

is large suggesting that SPM may have significant errors in

its prediction of states of the full model.

VII. CURRENT STATUS ON BM S AND FUTURE WORK

Earlier in the paper, we mentioned that a BMS has to

perform certain tasks that are critical to the operation of

the battery. In particular for vehicle electrification, a sample

of these tasks include prediction of maximum available

power and energy, safe charging and discharging to meetregenerative braking and load bearing requirements, tracking

the state of health of the battery pack as it ages, and updating

the BMS to maintain accuracy of its tasks throughout its life.

The importance of prediction of maximum available power

and energy is self-evident since this knowledge allows the

electronic control unit (ECU) to compute the vehicles all-

battery range in miles and the power it can deliver to

accelerate, if demanded. Though ideally, it is desirable to

have the ability to charge or discharge the battery as quickly

as possible, such processes can dangerously stress the battery

2001


6/6

0 1 2 3 40

0.5

1

css

/cmax

C/25

0 1 2 3 40

0.5

1

C/2

0 1 2 3 40

0.5

1


css

/cmax

1C1C

0 1 2 3 40

0.5

1


2C2C

css

(x=0)

css

(x=L)

css

(SPM)

css

(x=0)

css

(x=L)

css

(SPM)

css(x=0)

css

(x=L)

css

(SPM)

css(x=0)

css

(x=L)

css

(SPM)

Fig. 4. Surface concentrations over the electrode computed from SPM andthe full model for a high-energy cell configuration.

and accelerate aging. Thus, a BMS has to monitor overpoten-

tials and other relevant states that indicate potential damage

to the battery [19]. Finally, as the battery ages, a BMS

needs to track model parameters to maintain accuracy of

power and energy prediction throughout the life of the battery

pack. Each of the above BMS tasks, reflects an estimation

or control problem.

We now briefly mention some of the contemporary re-

search work on design of a BMS. A large section of the

research work on batteries uses a simple equivalent circuit

model for design of the BMS [2], [6], [3], [5], [7], [8]. This

choice stems from the fact that existing BMS for portable

electronics mostly models the battery as an equivalent circuit

and hence, its use in modeling is naturally extended to Li-ion

batteries for high-energy applications.

In contrast to equivalent circuit approach, [19], [20], [21],

[22], [23] study estimation problems using other models,

including electrochemical-based models. In particular, they

use approximations of electrochemical models and other

physics-based models to improve accuracy of estimation

algorithms for BMS. With this picture in mind, we mention

the future work that needs to be addressed for design of

improved and sophisticated BMS.

Referring to our earlier discussion in this section, the

future challenges are characterization of an approximation or

reduction of the full electrochemical model given by (4) and

(5), and the algebraic equations (17) and (21) such that themodel is accurate over a large range of operation and simple

enough that it is analytically tractable. Retaining the physical

significance of the parameters is important since it helps in

characterizing aging phenomena in batteries. Next, the design

of simple algorithms for observing states of this model is an

open problem, especially when applied to the whole battery

pack and not just one cell. Finally, the estimation of all

parameters of the model to maintain accuracy of the model

and to identify age of the battery pack by tracking relevant

physical parameters is also an open problem.

REFERENCES

[1] M. Armand and J.-M. Tarascon, Building better batteries, Nature,vol. 451, pp. 652657, 2008.

[2] T. Stuart, F. Fang, X. Wang, C. Ashtiani, and A. Pesaran, A modularbattery management system for HEVs, SAE Future Car Congress,vol. 2002-01-1918, 2002.

[3] Y.-S. Lee and M.-W. Cheng, Intelligent control battery equalizationfor series connected Lithium-ion battery strings, IEEE Transactionson Industrial Electronics, vol. 52, no. 5, pp. 12971307, 2005.

[4] I. J. Ong and J. Newman, Double-layer capacitance in a dual lithiumion insertion cell, Journal of the Electrochemical Society, vol. 146,no. 12, pp. 43604365, 1999.

[5] M. W. Verbrugge and R. S. Conell, Electrochemical and thermalcharacterization of battery modules commensurate with electric vehicleintegration, Journal of the Electrochemical Society, vol. 149, no. 1,pp. A45A53, 2002.

[6] B. Schweighofer, K. M. Raab, and G. Brasseur, Modeling of highpower automotive batteries by the use of an automated test system,

IEEE Transactions on Instrumentation and Measurement, vol. 52,no. 4, pp. 10871091, 2003.

[7] M. Chen and G. A. Rincon-Mora, Accurate electrical battery modelcapable of predicting runtime and I-V performance, IEEE Transac-tions on Energy Conversion, vol. 21, no. 2, pp. 504511, 2006.

[8] M. W. Verbrugge and R. S. Conell, Electrochemical characterizationof high-power lithium ion batteries using triangular voltage and currentexcitation sources, Journal of Power Sources, vol. 174, no. 1, pp. 28,

2007.[9] J. Newman and W. Tiedemann, Porous-electrode theory with batteryapplications, AIChE J., vol. 21, no. 1, pp. 2541, 1975.

[10] M. Doyle, T. F. Fuller, and J. Newman, Modeling of galvanostaticcharge and discharge of the lithium/polymer/insertion cell, Journalof The Electrochemical Society, vol. 140, no. 6, pp. 15261533,1993. [Online]. Available: http://link.aip.org/link/?JES/140/1526/1

[11] T. F. Fuller, M. Doyle, and J. Newman, Simulation and optimizationof the dual lithium ion insertion cell, Journal of The ElectrochemicalSociety, vol. 141, pp. 110, 1994.

[12] K. Thomas, J. Newman, and R. Darling, Advances in lithium-ionbatteries: Mathematical Modeling of Lithium Batteries. New York:Springer US, 2002.

[13] G. Ning and B. N. Popov, Cycle life modeling of Lithium-ionbatteries, Journal of the Electrochemical Society, vol. 151, no. 10,pp. A1584A1591, 2004.

[14] S. Santhanagopalan, Q. Guo, P. Ramadass, and R. E. White, Review

of models for predicting the cycling performance of lithium ionbatteries, Journal of Power Sources, vol. 156, no. 2, pp. 620628,2006.

[15] N. Chaturvedi, R. Klein, J. Christensen, J. Ahmed, and A. Kojic,Algorithms for Advanced Battery Management Systems: Modeling,estimation, and control challenges for lithium-ion batteries, IEEEControl Systems Magazine, vol. 30, no. 3, 2010.

[16] A. J. Bard and L. R. Faulkner, Electrochemical Methods: fundamentalsand applications. New York: John Wiley & Sons, Inc., 2001.

[17] J. Newman and K. E. Thomas-Aleya, Electrochemical Systems. NewJersey: John Wiley & Sons, Inc., 2004.

[18] L. C. Evans, Partial Differential Equations. New Jersey: AmericanMathematical Society: Graduate Studies in Mathematics, 1998.

[19] R. Klein, N. Chaturvedi, J. Christensen, J. Ahmed, R. Findeisen, andA. Kojic, State estimation of a reduced electrochemical model of alithium-ion battery, Proceedings of the American Control Conference,2010.

[20] G. L. Plett, Extended Kalman filtering for battery managementsystems of LiPB-based HEV battery packs: Part 2. Modeling andidentification, Journal of Power Sources, vol. 134, no. 2, pp. 262276,2004.

[21] , Extended Kalman filtering for battery management systemsof LiPB-based HEV battery packs: Part 3. State and parameterestimation, Journal of Power Sources, vol. 134, no. 2, pp. 277292,2004.

[22] S. Santhanagopalan and R. E. White, Online estimation of the stateof charge of a lithium ion cell, Journal of Power Sources, vol. 161,no. 2, pp. 13461355, 2006.

[23] K. Smith, C. Rahn, and C.-Y. Wang, Control oriented 1D elec-trochemical model of lithium ion battery, Energy Conversion and

Management, vol. 48, no. 9, pp. 25652578, 2007.

2002

20- Modeling, Estimation, And Control Challenges for Lithium-ion Batteries

Documents

Transcript of 20- Modeling, Estimation, And Control Challenges for Lithium-ion Batteries