PSM: Lithium-Ion Battery State of Charge (SOC) and Critical SurfaceCharge (CSC) Estimation using an Electrochemical Model-driven

Extended Kalman Filter

Domenico Di Domenico, Anna Stefanopoulou and Giovanni Fiengo

Abstract— This paper presents a numerical calculation of theevolution of the spatially-resolved solid concentration in thetwo electrodes of a lithium-ion cell. The microscopic solid con-centration is driven by the macroscopic Butler Volmer currentdensity distribution which is consequently driven by the appliedcurrent through the boundary conditions. The resulting, mostly-causal, implementation of the algebraic differential equationsthat describe the battery electrochemical principles, even afterassuming fixed electrolyte concentration, is of high order andcomplexity and is denoted as the full-order model. The full-order model is compared with the results in [22] and in [11]R3C1and creates our baseline model which will be further simplifiedfor charge estimation.

We then propose a low-order extended Kalman filter (EKF)for the estimation of the average-electrode charge similarly tothe single-particle charge estimation in [18] with the followingR3C2two substantial enhancements. First we estimate the average-electrode, or single-particle, solid-electrolyte surface concen-tration, called critical surface charge (CSC) in addition tothe more-traditional bulk concentration, called state of charge(SOC). Moreover, we avoid the weakly observable conditionsassociated with estimating both electrode concentations byrecognizing that the measured cell voltage depends on thedifference, and not the absolute value, of the two electrodeopen-circuit voltages. The estimation results of the reduced,single, averaged electrode model are compared with the fullorder model simulation.

I. INTRODUCTION

Lithium-ion battery is the core of new plug-in hybrid-electrical vehicles (PHEV) as well as considered in many2nd generation hybrid electric vehicles (HEV). Micro-macroscopic battery electrochemical modeling is connectedwith the hybrid vehicle design, scale-up, optimization andcontrol issues of Hybrid-electrical vehicles (HEV), wherethe battery plays an important role in this area as high-rate transient power source. The battery models based onelectrochemistry laws [5], [25] are generally preferred tothe equivalent circuit or to other kinds of simplified models,because they also predict the physical cells limitations, whichhave a relevant effect in the automotive application, wherethe battery suffers very often the stress of very high transientloads [22].

The importance of lithium-ion battery has grown in thepast years. Based on the the third lightest element, this kind

Domenico Di Domenico and Giovanni Fiengo are with Dipartimentodi Ingegneria, Universita degli Studi del Sannio, Piazza Roma 21, 82100Benevento. E-mail: {domenico.didomenico,gifiengo}@unisannio.it

Anna Stefanopoulou is with Mechanical Engineering Dept., Univ ofMichigan, Ann Arbor MI 48109-2121. E-mail: annastef@umich.edu. Sheis funded by NSF CMS-GOALI 0625610 and CBET-GOALI 0932509.

of battery has the right characteristics to be widely used inthe hybrid vehicles, thanks to its best energy-to-weight ratios,no memory effect, and a slow loss of charge when not inuse. As the majority of advanced battery systems, lithium-ion batteries employ porous electrodes in order to increasethe active area between electrolyte and solid active material,facilitating the electrochemical reactions [2], [15], [22], [24].

The literature on electrochemical modeling of batteriesis quite extensive, including both full order and simplifiedmodels. A first electrochemical approach to porous elec- R3C1trodes modeling for battery applications was presented byNewman and Tiedemann in [12]. In the porous electrodetheory [12], the electrode is treated as a superposition oftwo continua, namely the electrolytic solution and the solidmatrix. The solid matrix is modeled as microscopic sphericalparticles, where the lithium ions diffuse and react on thespheres surface. This approach was later expanded to twocomposite electrodes and a separator, by Fuller et al. in[6]. Based on this model, Ramadass et al. [16] accountedfor the decay in capacity of the cell while Sikha et al.[21] included the change in the porosity of the electrodematerial as a function of time using a pseudo 2-dimensionalapproach. Unfortunately, due to the irregular morphology,the electrode porosity increases the model complexity. Thus, R2C1a macroscopic description of the cell is needed. A micro-macroscopic coupled model meeting this requirement, withmicroscopic and interfacial phenomena, as described in theporous electrode theory, rigorously and systematically inte-grated into a macroscopic battery model was introduced byWang et al. in [26]. It incorporates solid-state physics andinterface chemistry and can be adapted to a wide range ofactive materials and electrolyte solutions. Presented at thebeginning for Ni-MH batteries [27], it was quickly expandedto lithium-ion batteries in [22] and [7], where the thermalbehavior was also described. The full order electrochemicalmodels predict the profile of solid concentration, i.e. lithiumconcentration in the solid phase across the electrode, duringcharge and discharge, but, it is difficult to apply to a real-time on-board vehicle estimator, due to the complexity of themodel. As a consequence several approximations betweenthe input and output of the dynamical system and the batteryinput and output are introduced to reduce the system orderand complexity [2], [13] and to allow the estimation of thebattery State of Charge (SOC).

The SOC estimation and regulation is one of the mostimportant and challenging tasks for hybrid and electrical

vehicle control. In fact, when the batteries operate in arelative limited range of state of charge, high efficiency, slowaging and no damaging are expected. Several techniques havebeen proposed for the SOC estimation, like equivalent circuitmodel-based algorithms [23], [9] or black-box methods (asan example using fuzzy-logic [19]). The accuracy reachedby these SOC estimations is about 2% [14].

Recently the effort to use electrochemical models inorder to estimate the internal state of the battery has beenintensified. Starting from the model introduced by Wang, in[11] a low order state variable model practical for real timeestimation and control has been obtained. In [18], based onR3C2a simplified representation of the electrode introduced byHaran et al. in [8], a Kalman filter was designed in order toestimate an average value of the bulk solid concentration.

In this paper we employ an electrode-average concentra-tion approximation, similar but higher order than the single-particle electrode estimation of [18]. We then present anextended Kalman filter for SOC which typically relates tothe bulk battery capacity through integration of the currentdrawn although in our work it represents the average particleconcentration of the average-electrode. We also estimatethe critical surface charge (CSC) estimation, which is theelectrode-average solid concentration at the electrolyte inter-face. In the following, a general micro-macroscopic lithium-ion battery model, as it is presented in [22], is summarizedand a numerical solution is shown. Then the model issimplified in order to make it compatible with a feasiblesolid concentration estimation. Finally an extended Kalmanfilter (EKF) is designed based on the averaged model.The estimation results are compared with the high ordernumerical approximation of the micro-macroscopic modelprediction.

II. BATTERY GENERAL FEATURE

A battery is composed of three parts: the two electrodesand the separator. Referring to a porous battery, eachelectrode consists of a solid matrix inside an electrolytesolution. In particular, for a Lithium-ion battery, the negativeelectrode is composed of carbon and the positive electrode isa metal oxide. In the separator the lithium-salt electrolyte isheld in an organic solvent, such as LiPF6, LiBF4 or LiClO4,or in a solid polymer composite such as polyethylene oxideor poly-acrylonitrile (lithium-ion polymer battery). Boththe solvent and the polymer conduct the ion but they areelectronic insulators. As shown in Figure 1, at the negativeelectrode, the solid active material particles of lithium ofthe LixC6, diffuse until the electrolyte-solid interface wherethe chemical reaction occurs, transferring the lithium ionto the solution and the electrons to the collector [22]. Theproduced electrolyte material travels through the solutionto the positive electrode, where, at the interface with solidmaterial, they react and insert into the metal oxide solidparticles. The chemical reactions are:

x=0

Currentcollector

Currentcollector

Negative electrode Separator Positive electrode

Solid Material

Electrolyte

Unitary Volume

r

Solid Electrolyte interfacecs(x,r,t)

cse(x,t)

Li+

e

e

xx= sp x=Lx= n

cs(x,r,t)

r

cse(x,t)

Fig. 1. Schematic macroscopic (x-direction) cell model with coupledmicroscopic (r-direction) solid diffusion model.

negative electrode reactionLixC6 � C6 + x Li+ + x e−

positive electrode reactionLiy−xMn2O4 + x Li+ + x e− � LiyMn2O4

net reactionLiy−xMn2O4 + Lix C6 � LiyMn2O4 + C6

where the two ways of the reaction refer to the batterydischarge reaction (right arrow) and to the battery chargereaction (left arrow).

The phenomena occurring at the microscale can havestrong implications for the cell discharge and charge duringpulse operations. However, due to the complexity of solid-electrolyte interface morphology, it is problematic to solvethe equations on a microscopic scale [26]. So, in orderto mathematically model the battery, both macroscopic andmicroscopic physics have to be considered. The equationsdescribe the battery system with four quantities, i.e. solid andelectrolyte concentrations (cs, ce) and solid and electrolytepotentials (φs, φe). The complete set of equation for themicro-macroscopic model is [7], [22]

−→∇xκeff−→∇xφe +−→∇xκeff

D

−→∇x ln ce = −jLi (1)

−→∇xσeff−→∇xφs = jLi (2)

describing the potentials distribution and

∂εece

∂t=

−→∇x

(Deff

e

−→∇xce

)+

1 − t0

FjLi (3)

∂cs

∂t=

−→∇r(Ds−→∇rcs) (4)

describing the concentrations diffusion.In order to simplify the model we assume a constant R3C3

electrolyte concentration ce. This simplification can alsobe found in [18] and is justified due to the insignificant(<5%) variations observed in the electrolyte concentration

in the detailed model [22]. Then the set of equations canbe simplified by neglecting the equation (3) and simplifyingthe equation (1). Furthermore, introducing the solid currentis and the electrolyte current ie the 2th order PDE systemfor the potentials can be rewritten as

ie(x) = −κeff−→∇xφe (5)

is(x) = −σeff−→∇xφs (6)

−→∇xie(x) = jLi (7)

−→∇xis(x) = −jLi (8)

with the Butler-Volmer current density

jLi(x) = asj0

[exp

(αaF

RTη

)− exp

(− αcF

RTη

)](9)

where R and F are the universal gas and the Faraday’s con-stants, T is the absolute temperature, η is the overpotential,obtained as

η = φs − φe − U(cse), (10)

and U is the open circuit voltage, function of the solidconcentration at the electrolyte interface indicated withcse(x, t) = cs(x, Rs, t). The value of the concentration at theinterface solid-electrolyte has a particular relevance duringthe cell pulse operations and it is called critical surfacecharge, defined asR3C6

CSC(t) =θ − θ0%

θ100% − θ0%, (11)

where θ = cse/cs,max is the normalized solid-electrolyteconcentration. Its reference value, θ100%, can be defined foreach electrode finding the concentration corresponding to afull charge battery, i.e. 100% of the state of charge. Thus,the 0% reference stoichiometry can be derived from θ 100%

by subtracting the battery capacity Q, with the appropriateconversion:

θ0% = θ100% − Q

δ

(1

AFεcs,max

)(12)

where δ, ε and cs,max have appropriate values for negativeand positive electrode. The open circuit voltage for thenegative electrode, denoted with the subscript n, is calculatedusing the empirical correlation introduced in [4]

Un(θn) = 8.0029 + 5.0647θn − 12.578θ0.5n

− 8.6322× 10−4θ−1n + 2.1765× 10−5θ3/2

n

− 0.46016 exp[15.0(0.06− θn)]− 0.55364 exp[−2.4326(θn − 0.92)] .

(13)

For the positive electrode, denoted with p, the result obtainedfrom [22] has been adopted

Up(θp) = 85.681θ6p − 357.70θ5

p + 613.89θ4p

− 555.65θ3p + 281.06θ2

p − 76.648θp

+ 13.1983− 0.30987 exp(5.657θ115

p

).

(14)

Electrodes Separator

ee

e

e

idxdk

dxdi 0

Solid material

ee

e

ss

s

Lie

Lis

idxdk

idxd

jdxdi

jdxdi

se

sees

cas

Li

cjjcU

RTF

RTFjaj

00

0 expexprc

rrcD

tc ss

ss 2

2

2

Negative electrode Separator Positive electrode

nx0x spsepnx Lx

effx

s

s

AI

x

AIi

0

)0(

s

e

0

0)(

nx

s

ns

x

i

s

e

sepene

effx

e

ne

AkI

x

AIi

n

,,

)(

e

effx

e

spe

AkI

x

AIi

sp

)(

ee

s

effLx

s

s

AI

x

AILi )(

s

e

0

0)0(

0x

e

e

x

i

0)0(s

AIdxxj

n Li0

)(

0

0)(

spx

s

sps

x

i

0

0)(

Lx

e

e

x

Li

AIdxxj

LLi

sp

)(

sepepe ,,effx

e

spe

AkI

x

AIi

sep

)(effx

e

ne

AkI

x

AIi

n

)(

sRr

00r

src

0r

Faj

rcD

s

Li

Rr

ss

s

Fig. 2. Schematic representation of the set of equations and of theboundary conditions for the potentials and solid concentration. The current Iis assumed to be positive (battery discharge) and provide boundary condition(boxes with dotted line) and the non-local constraints (boxes with solid line)governing the Butler-Volmer current density.

The normalized solid-electrolyte at the negative and positiveelectrode are here indicated with, respectively, θn and θp.The coefficient j0 in (9) also exhibits a modest dependenceon the solid and electrolyte concentration, according to

j0 = (ce)αa(cs,max − cse)αa(cse)αc . (15)

Finally, the cell potential which is typically measured iscomputed as

V = φs(x = L) − φs(x = 0) − RfI (16)

where Rf is the film resistance on the electrodes surface. InFigure 2 the model equations and their boundary conditionsfor the x-domain and r-domain are shown. Note that thetemperature spatial and temporal gradients are neglected inthis work.

For completeness the battery model parameters are sum-marized in Table II and more details on the model and itsparameters can be found in [22], [27].

III. NUMERICAL SOLUTION

In order to solve the battery Partial Differential Equations(PDE) model, the boundary conditions are necessary. At the

boundary with the current collectors of the negative andpositive electrode, i.e. at x = 0 and at x = L (see Figure 1),it is

ie(x = 0) = 0 (17)

ie(x = L) = 0 (18)

is(x = 0) = IA (19)

is(x = L) = IA (20)

because at the collector there is not electrolyte flux andthe solid current, i.e. the spatial gradient of solid potential,correspond to the applied current density.

At x = δn and at x = δsp, i.e. the separator-electrodeinterfaces, the boundary conditions are

is(x = δn) = 0 (21)

is(x = δsp) = 0 (22)

because no charge passes through the separator. Assuming a1-dimensional model, the sum of the (7) and (8) give

d (is + ie)dx

= 0, (23)

that implies that the “total current” (is + ie) has to beconstant along the cell. Furthermore, this constant value isknown, because the boundary conditions (17) and (19) givethe constant value

is + ie =I

A, (24)

which translates, considering (21) and (22), to

ie(x = δn) = IA (25)

ie(x = δsp) = IA . (26)

It can also be shown, integrating (7) (or equivalently (8))between x = 0 and x = δn, that the condition (24) isequivalent to ∫ δn

0

jLi(x′)dx′ =I

A(27)

as expected because the total current generated by themicroscopic chemical reactions is the integral along the cellof the microscopic current density flux. The same conditioncan be found for the positive electrode∫ L

δsp

jLi(x′)dx′ = − I

A. (28)

By the way, the (5) - (8) are two 2nd order PDE’s withtwo set of boundary conditions defining the slopes at theboundary. Hence, the distribution shape of the potentials canbe found but not its absolute value. To provide the completesolution for φe(0 < x < δn) and φs(0 < x < δn), we firstassume φs(x = 0) = 0 and then we find φe(x = 0) by usingthe constrain given in the equation (27). Similarly, to findthe potentials φe(δsp < x < L) and φs(δsp < x < L) weapply the corresponding constrain for the positive electrode,i.e. equation (28).

Regarding the solid active material, the boundary condi-tions connect the concentration with the chemistry of the

reactions through the surface microscopic current density.So, at r = 0 is

∂cs(x, t)∂r

∣∣∣∣∣r=0

= 0 ∀x (29)

and at r = Rs the boundary conditions are

Ds∂cs(x, t)

∂r

∣∣∣∣∣r=Rs

= − jLi(x, t)aspF

(30)

for positive electrode’s particles

Ds∂cs(x, t)

∂r

∣∣∣∣∣r=Rs

= − jLi(x, t)asnF

(31)

for negative electrode’s particles.Finally, in Figure 2 the complete set of equations is

provided and a graphical representation of the boundaryconditions for x-domain and r-domain is shown.

The lithium-ion PDE system, with its initial and boundaryconditions, was simulated and numerically solved in theMatlab/Simulink environment by using the finite differencemethod. To reproduce the spatial dependence of the modelvariables, the currents (is and ie), the potential (φs and φe)and the Butler-Volmer current jLi were spatially discretizedand the differential equations (5) - (8), with the (9), weresolved using the finite difference method. Referring to thenegative electrode, let Nx the number of discretization steps.The algebraic variables, depending on time, are φsl

(t),φel

(t), isl(t), iel

(t) and jLil

(t), and the algebraic system, tobe solved each simulation step, is

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

isl= isl−1 − jLi

l−1 ∗ Δx

iel= iel−1 + jLi

l−1 ∗ Δx

Φsl= Φsl−1 −

1σeff

isl−1Δx

Φel= Φel−1 −

1keff

iel−1Δx

ηl= Φsl

− Φel− U(cse)

jLil

= asj0(cse)

[exp

(αaF

RTη

l

)− exp

(− αcF

RTη

l

)](32)

with l = 1, ...., Nx and Δx = δn

Nx−1 .Critical are the boundary condition at l = 1. Equations

(17) and (19) give an initial input to solve recursivelyequation (32) with respect to currents is and ie, but to solvethe whole system the values of the potentials at x = 0 areneeded. As discussed above, we fix φs(x = 0) = 0.

With this assumption, the problem is how to find the R3C4correct electrolyte potential value at the collector, i.e. φe(x =0), satisfying one of the equivalent conditions (21) and (25)or (27). The solution has been obtained by numericallyminimizing |iS(δn)| via a Matlab/Simulink optimization tool.Note that if the dependence of the Butler-Volmer current onthe solid concentration is not neglected, the model needs theoptimization of the parameter φe(x = 0) at each simulationstep.

The input of the algebraic recursive system (32) is thebattery current I , while the solid concentration can beconsidered as a set of parameters that, as we are going toshow, can be derived starting from the Butler-Volmer current.

The dependence of the solid concentration cs(r, x, t) onx can be described by applying the same discretizationprocedure, that turns into csl

(r, t), with l = 1, ...., Nx. Foreach l, the partial differential equation (4) governs the r-dependent csl

behavior. For a state space formulation ofthe solid concentration diffusion equations, let the batterycurrent I be the model input which governs the boundaryconditions of (5) - (8) as shown in Fig. 2. For each x-dimension discretization step, an ODE is obtained by usingthe finite difference method for the r variable [20]. Thisresults in 2Nx different systems (Nx for the negative and Nx

for the positive electrode), each of (Mr − 1)-order, drivenby a time-dependent nonlinear function of the battery inputI through the Butler-Volmer current. Making explicit the r-derivative in equation (4), it is

∂cs

∂t= Ds

(∂2cs

∂r2+

2r

∂cs

∂r

). (33)

By dividing the solid material sphere radius in Mr − 1intervals, with Mr large enough, of size Δr = Rs

Mr−1 , thefirst and the second derivative can be approximated with thefinite difference. So the partial differential equation becomes

dcsq

dt= Ds

[1

Δ2r

(cs(q+1) + cs(q−1)) −2

Δ2r

csq

+1

2Δr

1r

q

(cs(q+1) − cs(q−1) )] (34)

with q = 1, ...., Mr − 1 and rq = qΔr. The boundary condi-tions can be rewritten linearly, approximating the dependenceof cs on r at center and on the border of the solid materialsphere

cs0 = cs1

csMr= cs(Mr−1) − Δr

jLil

FasDs.

(35)

The discretized PDE system at the l location along thex-dimension forms the l-state cs = (cs1 , cs2 , ....csMr−1)T

showing explicitly the dependency on the Butler-Volmercurrent jLi

l⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

csq =(

q + 1q

)α1csq+1 − 2α1csq +

(q − 1

q

)α1csq−1

csMr−1 =(

Mr − 2Mr − 1

)α1csMr−2 −

(Mr − 2Mr − 1

)α1csMr−1

−(

Mr

Mr − 1

)α2j

Lil

(36)with q = 1, . . . , Mr − 2, l = 1, . . . , Nx, α1 = Ds/Δ2

r

and α2 = 1FasΔr

and jLil (cse, I) = Ul(cse, I), where the

dependence on the input I of the Butler-Volmer current,through the (32), is explicitly highlighted. It follows

cs = Acs + bUl (37)

where A and b can be determined from (36) and the U l

can be considered as a set of Nx parameters, each of themappearing in one of the respective Nx state-space systems,and has to be derived from (5) - (8). The output of the systemis the value of the solid concentration on the sphere radius,that can be rewritten

cse = csMr−1 −α2

α1Ul(cs, I). (38)

These equations show that solid concentrations and poten-tials are related through the Butler-Volmer equation throughthe boundary condition. Furthermore, along each electrode,as the boundary condition, i.e. the input U l, depends on xthe system output cse is also function of x.

For the positive electrode the same procedure was applied.Also in that case, one needs to fix the boundary conditionsat the collector, so the optimization tool was used to find thevalue of φs(L) that satisfies the condition on the solid currentat x = L. On the contrary, at the separator, the discretizationcan be avoided and the analytical solutions can be obtained.For δn < x < δsp, the current of the solid material is zerohence the electrolyte current has to be

ie =I

A(39)

to satisfy the (24). Thus, using (25) and (26), it is

φe(x) = φe(δn) − I

Akeffx. (40)

Equations (39) and (40) were used in the simulation.

0 10 20 30 40 50 60−50

0

50

Cur

rent

[A]

0 10 20 30 40 50 603.4

3.5

3.6

3.7

3.8

3.9

Time [s]

Cel

l vol

tage

[Vol

t]

30% SOC50% SOC70% SOC

Fig. 3. Battery voltage prediction. SOC initial conditions used in the modelvary from 30% to 70% of the total charge.

IV. SIMULATION OF FULL ORDER MODEL

The micro-macroscopic battery model is difficult to vali-date. We followed the results presented in [22] where themodel was validated with a current profile according toFreedom CAR test procedure, an U.S. Department of En-ergy program for the zero-emission vehicle and technologyresearch, as reported in [1]. The test procedures, shown inthe top plot of Figure 3, consist of a 30 A battery dischargefor 18 s, open-circuit relaxation for 32 s, 22.5 A charge for

10 s followed by open-circuit relaxation. Figure 3 is inspiredby [22] and try to reproduce the numerical solution used tovalidated the model. In [22] the battery voltage prediction,for several battery SOC initial conditions, is shown. Theinitial SOC can be related whit the initial conditions on thesolid concentration by introducing the stoichiometry ratioθb = csb

cs,max, where csb is the average bulk concentration

that can be obtained as

csb =1

Rs

∫ Rs

0

cs(r, t)dr. (41)

Then, the state of charge of the battery is, with a goodapproximation, linearly varying with θb between the twostoichiometry values θ0% and θ100%

SOC(t) =θb − θ0%

θ100% − θ0%. (42)

We here assume, according with [22], a single cell witha nominal capacity Q = 6 Ah and, for simplicity, we setuniform initial conditions on the solid concentration.

Figures 4 and 5 show the solid and electrolyte potentialand the solid material concentration during the battery dis-charge and charge. The Figures refer to the 50% SOC line inFigure 3. In detail, the discharge refers to a battery currentI = 30 A after 2 seconds (i.e. after 4 s of whole test) and thecharge refers to a current I = −25 A after 3 seconds (i.e.after 55 s of the whole test). Note that the potential profilesduring the discharge agree with the slope imposed by theboundary conditions.

Figures 4 and 5 point out that the solid concentration isa function of the position along the electrodes. The criticalvalues for electrodes depletion and saturation effects arethe lowest one for the ion-production electrode and thehighest one for the ion-insertion electrode. Because of thesolid concentration profile during charge and discharge, asshown in Figures 4 and 5), the critical points are x = δn

and x = δsp both during discharge and during the charge.

V. AVERAGE MODEL

A model simplification can be provided neglecting thesolid concentration distribution along the electrode and con-sidering the material diffusion inside a representative solidmaterial particle for each electrode. This approximation leadsto an average value of the solid concentration that can berelated with the definition of battery state of charge and ofcritical surface concentration. Although this simplified modelresults in a heavy loss of information, it can be useful incontrol and estimation applications and still maintains a smallconnection with the physical phenomena and dimensions [3].

In accordance with the average solid concentration, thespatial dependence of the Butler-Volmer current is ignoredand a constant value jLi is considered which satisfies thespatial integral (27), giving for the negative electrode (andcan be reproduced accordingly in the positive electrode,considering (28))∫ δn

0

jLi(x′)dx′ =I

A= jLi

n δn (43)

where δn is the negative electrode thickness, as shown inFigure 1.

The battery voltage (16) using (10) can be rewritten as

V (t) = η(L, t) − η(0, t) + (φe(L, t) − φe(0, t))+ (Up(cse(L, t)) − Un(cse(0, t))) − RfI

(44)

and using the average values at the negative and the posi-tive electrode instead of the boundary values the followingrelation is obtained

V (t) = ηp − ηn +(φe,p − φe,n

)+ (Up(cse,p) − Un(cse,n)) − RfI.

(45)

Using the microscopic current average values and imposingthe boundary conditions and the continuity at the interfaces,the solutions of equations (5) - (8) are, for the negativeelectrode

φe(x) = φe(0) − I

2Akeffδnx2 (46)

φs(x) = − I

Aσeff

(x − x2

2δn

)(47)

for the separator

φe(x) = φe(0) − I

2Akeffδn − I

Akeff(x − δn), (48)

which correspond with (40) with value of φe(δn) computedby (46), and for the positive electrode

φe(x) = φe(0) − I

2Akeffδn − I

Akeffδsep+

I

2Akeffδp(x − δsp)2 − I

Akeff(x − δsp)

(49)

φs(x) = φs(L) − I

Aσeff

((x − δsp) − 1

2δn(x − δsp)2

)(50)

where δsep = δsp − δn is the separator thickness. Theapproximate solutions (46) and (49) lead to

φe,p−φe,n = φe(L)−φe(0) = − I

2A

(δn

keff+ 2

δsep

keff+

δp

keff

).

(51)Furthermore, considering

jLin =

I

Aδn= asj0

[exp

(αaF

RTηn)

)− exp

(− αcF

RTηn

)](52)

jLip = − I

Aδp= asj0

[exp

(αaF

RTηp)

)−exp

(−αcF

RTηp

)](53)

ηn and ηp can be estimated as

ηn =RT

αaFln(ξn +

√ξ2n + 1

)(54)

ηp =RT

αaFln(ξp +

√ξ2p + 1

)(55)

where

ξn =jLin

2asj0and ξp =

jLip

2asj0. (56)

Finally, the battery voltage (45) can be written as a func-tion of battery current and of the average solid concentration

V (t) =RT

αaFln

ξn +√

ξ2n + 1

ξp +√

ξ2p + 1

+ φe,p − φe,n

+ (Up(cse,p) − Un(cse,n)) − RfI.

(57)

Thus, the averaging Butler-Volmer current leads to con-sider a representative solid material particle somewhere alongthe negative and positive electrode. This simplified modelhas similarities with the “single particle” model introducedin [8]. In [8] the diffusion dynamics are approximates witha 1st order ordinary differential equation (ODE) whereasin the electrode-average model a (Mr − 1)-order ODE isused. Furthermore, in the electrode-average model the cellR3C2voltage depends, through Un and Up, on the solid-electrolyteconcentration instead of the average single particle bulkconcentration.

Figures 6 - 10 demonstrate the performance of the reducedorder model driven by the (Mr−1)-order ordinary differentialequation (37) comparing the simulated signal with the full-order model given by (37) resolved at Nx locations acrossthe x-direction. In particular, Figure 6 shows a good cellvoltage prediction at different discharge rates from 10 Ato 300 A. The depletion and saturation effects, respectivelyfor negative and positive electrode, are evident for higherdischarge rate, where a sudden drop of initial SOC and cellvoltage can be noticed. It is to be highlighted that the averagevoltage response reproduces the full order model numericalsolution with comparable precision for different dischargerates and SOC levels. Figures 7 show a good battery voltageprediction during a FreedomCAR test procedure, with amaximum error of 0.3 mV for the considered charge anddischarge rate. Figure 8 shows the temporal distribution ofsolid concentration, highlighting a good agreement betweenthe distributed value of the solid concentration and thepredicted average. Figure 9 shows the error between theaverage solid concentration and the solid concentration atthe electrodes depletion/saturation effects critical points, i.e.the electrodes-separator interfaces, as it is predicted by thefull order model. Figure 10 shows the spatial distributionof electrodes solid surface concentration at various timesduring a 30 A discharge (constant for the average model)and the error introduced by averaging the spatial distributionwith the average solid concentration value. Finally, Figure11 highlights, particularly for the negative electrode, that thesolid material depletion is deeper during the pulse opera-tions, implying an unavoidable degradation of the averagemodel performance. The predicted gradient in solid surfaceR3C8concentration throughout the electode here, is smaller thatthe ones shown in [11], but as far as we know, nobody hasreally shown experimentally what the exact spatio-temporalprofile of the Li-ion concentrations are.

VI. KALMAN FILTER STATE OF CHARGE ESTIMATION

In most cases the battery voltage is measured, and alongthe known input (current demanded) one needs to estimate

the battery SOC and CSC. The physical quantity related tothe battery charge is the solid concentration at the electrodes.As the solid concentration is a function of the position alongthe electrodes, the critical values for the state of chargeestimation, able to predict correctly the electrodes deple-tion and saturation effects, are the lowest one for the ion-production electrodeand the highest one for the ion-insertionelectrode. Unfortunately, a spatially resolved concentrationestimation cannot be probably realized in real-time due to theiterative process involved in the 300-state numerical solutionfot the full-order model. A solid concentration estimationthat predicts the critical concentration levels would havebeen more elegant. Unfortunately, the key feature of SOCestimation relies on the Butler-Volmer current values at theelectrode borders. The microscopic current is not measurable,so it should be derived from the battery current by solvingthe PDEs (5)-(8) or the discretized equivalent (30) and (35)2Nx times, which is similarly intractable. Instead, we explorethe estimation using the averaged model of Section V.

Specifically, the average dynamical system describes thediffusion effects into two solid material particles, one forthe positive electrode and one for the negative electrode,and allows the computation of the solid concentration atthe spheres radius, which represents an average value of thesolid concentration along the electrodes. However the cellvoltage (33) depends on (Up(cse,p) − Un(cse,n)) making thedifference of the open circuit voltage observable but does notnecessarily guarantee the observability of each open circuitvoltage. Indeed, the system that includes both positive andnegative electrode concentration states is weakly observable(in the linear sense) from the output cell voltage.

It’s possible to find a relation between the positive andnegative electrode average solid concentrations which can beused for the estimation of the negative electrode concentra-tion based on the positive electrode. As it is shown below, thepositive electrode concentration states are observable fromthe output cell voltage. Then the voltage measurement is usedto compute the voltage prediction error and it is then fed backto the positive (alone) electrode concentration observer. Theenabling relation for realizing the negative electrode concen-tration as a function of the positive electrode concentrationcan be found in the SOC definition introduced in (42). Inparticular, we assume equal average solid concentration ofboth electrodes and allow the SOC estimation using a singleaveraged electrode solid concentration, even though it wouldhave been safer to estimate SOC based on the most criticallevel of the solid concentration along the electrode, whichcan be at the electrode ends and not in the middle. By assum-ing that the SOC value is equal at the two electrodes, using(42) the negative electrode concentration can be computedas function of positive electrode concentration as

cse,n = cs,max,n(θn0% +

cse,p − θp0%cs,max,p(θp100% − θp0%

)cs,max,p

(θn100% − θn0%)

)

(58)

where θn0%, θn100%, θp0% and θp100% are the reference sto-ichiometry points for the negative and the positive electrode.

Hence, introducing the state vector x =(cs,p1, cs,p2, ....., cs,p(Mr−1))T , the dynamical systemis

x = Apx(t) + Bpu(t) with u = jLip and y = V (x, u)

(59)where the matrices Ap and Bp are obtained from (36) withreference to the positive electrode. For a linear state-spaceformulation, the linearized battery voltage results in an outputmatrix C = ∂V/∂x which is a row matrix with zeros in itsfirst Mr − 2 elements and the last non-zero term being

∂V

∂cs,p(Mr−1)=

∂Up

∂cs,p(Mr−1)− ∂Un

∂cse,n

∂cse,n

∂cs,p(Mr−1)(60)

due to the fact that the battery potential V is only a functionof the solid concentration at interface. This output matrix Cleads to a strongly observable system.

The non linear system observability was also studied. The(59) leads to a (Mr−1)-dimensional codistribution H of theobservation space H , which imply that the system is stronglylocally observable ∀cse,p �= 0 [10], [17].

Based on the average model developed in the previoussection, a Kalman filter can be designed, according to

˙x = Apx + Bu + Ke(y − y)y = V (x, u)

(61)

where x and y are respectively the estimated state and output,V is the output nonlinear function in (57), Ap, Bp are thematrices describing the dynamical system defined in (59),C defined in (60) and Ke is the Kalman gain, obtained asfollows

Ke = PCR−1 (62)

where P is the solution of the Riccati equation

P = ApP + PATp − PCR−1CT P + Q

P (0) = P0,(63)

and Q and R are weight matrices appropriately tuned.Note that one of the main advantage of the filter proposed

in this paper is that the bulk and the interface solid con-centration are estimated, allowing to estimation of both theCSC and the SOC.

Figures 12-17 highlight the filter performance. In partic-ular the Figures 16 and 17 show the simulation results forsome reasonable choice of Q and R when band-limited whitenoise is added to the battery current and voltage in order tosimulate the sensors noise. More noisy (and less expensiveR1C2sensors) will require more filtering which will slow down theestimation rate and future work on an experimental batteryneeds to address these issues. The 4th order Kalman filterR3C9estimation results are compared with the full 300th ordermodel. The error in the initial condition, close to 10%, isfully and quickly recovered, showing that the filter is ableto estimate the correct value of the battery state of chargeeven if its open loop model prediction was 10% wrong. The

step in current demand results in a step in solid concentrationwhich is again estimated by the filter, as shown in a zoom ofthe same figure. In particular, the Figure 13 shows explicitlythe error between the estimated solid concentration andthe solid concentration depletion/saturations critical values,which are the interesting quantities to estimate for a refinedestimation. It is shown that the error is less than ±0.4% and±3.0% in the positive and negative, respectively, electrodes.The negative electrode has higher prediction errors than thepositive electrode due to the lack of direct observability fromthe voltage prediction error. In Figure 14 the estimation ofthe CSC and SOC is compared with coulomb counting (CC)that evaluates the battery charge with the integral of batterycurrent. As it is shown, the CC is unable to recover the errorin the initial conditions. Figure 14 also highlights that thediscontinuity in the CSC during the current steps, which isslowly recovered due to the diffusion into the particle, is onlypartially observable through the SOC estimation.

Furthermore, the reduced order model based Kalman filterprovides also a good estimation for the single electrode opencircuit voltage even though just the open circuit voltage U p−Un is observable. This accuracy is accomplished because themeasured output V is not very sensitive to Un, so the correctvalue of the battery voltage can be predicted uniquely by thepositive electrode solid concentration, as confirmed by theresults shown in Figure 15.

Figures 16 and 17 show that the filter performance arebarely affected by the measurement imprecision. Introducingthe sensors noise into the simulation, in fact, increasesthe estimation convergence time and reduce the precision.However the performance is still satisfying. R3C9

VII. CONCLUSION

An isothermal numerical calculation of the solid concen-tration evolution in the two electrodes of a lithium-ion batterycell driven by a spatially distributed Butler-Volmer currentwas presented. The numerical solution is implemented inMatlab/Simulink using the finite difference method. Theisothermal electrochemical model of the lithium-ion batterycell was then used to derive an averaged model couplingthe average microscopic solid material concentration withthe average values of the chemical potentials, electrolyteconcentration and current density. The full and the reducedorder models were then compared and simulation resultswere shown and discussed. Finally, an EKF, based on theelectrode-average model, was designed for the CSC and SOCestimation. Using a typical discharging followed by chargingtest the estimation results show a ±0.4% and ±3.0% errorin predicting the critical solid concentrations in the positiveand negative electrode, respectively.

GLOSSARY

REFERENCES

[1] Freedomcar battery test manual for power-assist hybrid elecric veicles.DOE/ID-11069, 2003.

[2] O. Barbarisi, F. Vasca, and L. Glielmo. State of charge kalman filterestimator for automotive batteries. Control Engineering Practice,14:267 – 275, 2006.

TABLE I

LITHIUM-ION MODEL NOMENCLATURE.

Symbol Name Unit

ie electrolyte current density A cm−2

is solid current density A cm−2

φe electrolyte potential Vφs solid potential Vce electrolyte concentration mol cm−3

cs solid concentration mol cm−3

cse solid concentration at electrolyte interface mol cm−3

jLi Butler-Volmer current density A cm−3

θn normalized solid concentration at anode -θp normalized solid concentration at cathode -U open circuit voltage VUn anode open circuit voltage VUp cathode open circuit voltage Vη overpotential VF Faraday’s number C mol−1

I battery current AR gas constant J K−1 mol−1

T temperature K

[3] D. Di Domenico, G. Fiengo, and A. Stefanopoulou. Lithium-ionbattery state of charge estimation with a kalman filter based on aelectrochemical model. Proceedings of 2008 IEEE Conference onControl Applications, 1:702 – 707, 2008.

[4] M. Doyle and Y. Fuentes. Computer simulations of a lithium-ionpolymer battery and implications for higher capacity next-generationbattery designs. J. Electrochem. Soc., 150:A706–A713, 2003.

[5] M. Doyle, T.F. Fuller, and J. Newman. Modeling of galvanostaticcharge and discharge of the lithium/polymer/insertion cell. J. Elec-trochem. Soc., 140, 1993.

[6] T.F. Fuller, M.Doyle, and J. Newman. Simulation and optimization ofthe dual lithium ion insertion cell. J. Electrochem. Soc., 141, 1994.

[7] W.B. Gu and C.Y. Wang. Thermal and electrochemical coupledmodeling of a lithium-ion cell. Proceedings of the ECS, 99, 2000.

[8] B.S. Haran, B.N. Popov, and R.E. White. Determination of the hydro-gen diffusion coefficient in metal hydrides by impedance spectroscopy.Journal of Power Sources, 75:56–63, 1998.

[9] B. Hariprakasha, S. K. Marthaa, Arthi Jaikumara, and A. K.Shukla. On-line monitoring of leadacid batteries by galvanostatic non-destructive technique. Journal of Power Sources, 137, 2004.

[10] R. Hermann and A. J. Krener. Nonlinear controllability and observ-ability. IEEE Trans. Aut. Contr., pages 728–740, 1977.

[11] C.Y. Wang K. Smith, C.D. Rahn. Control oriented 1d electrochemicalmodel of lithium ion battery. Energy Conversion and Management,48:25652578, 2007.

[12] J. Newman and W. Tiedemann. Porous-electrode theory with batteryapplications. AIChE Journal, 21, 1975.

[13] B. Paxton and J. Newmann. Modeling of nickel metal hydride. Journalof Electrochemical Society, 144, 1997.

[14] V. Pop, H. J. Bergveld, J. H. G. Op het Veld, P. P. L. Regtien,D. Danilov, and P. H. L. Notten. Modeling battery behavior foraccurate state-of-charge indication. J. Electrochem. Soc., 153, 2006.

[15] J.A. Prins-Jansen, J.D. Fehribach, K. Hemmes, and J.H.W. de Wita.A three-phase homogeneous model for porous electrodes in molten-carbonate fuel cells. J. Electrochem. Soc., 143, 1996.

[16] P. Ramadass, B. Haran, P.M. Gomadam, R.White, and B.N. Popov.Development of first principles capacity fade model for li-ion cells. J.Electrochem. Soc., 151, 2004.

[17] W. Respondek. Geometry of Static and Dynamic Feedback. inMathematical Control Theory, Lectures given at the Summer Schoolson Mathematical Control Theory Trieste, Italy, 2001.

[18] R.E. White S. Santhanagopalan. Online estimation of the state ofcharge of a lithium ion cell. J. Power Sources, 161:13461355, 2006.

[19] A.J. Salkind, C. Fennie, P. Singh, T. Atwater, and D.E. Reisner.Determination of state-of-charge and state-of-health of batts. by fuzzylogic methodology. J. Power Sources, 80:293–300, 1999.

[20] W.E. Schiesser. The Numerical Method of Lines: Integration of PartialDifferential Equations. Elsevier Science & Technology, 1991.

[21] G. Sikha, B.N. Popov, and R.E. White. Effect of porosity on thecapacity fade of a lithium-ion battery. J. Electrochem. Soc., 151, 2004.

[22] K. Smith and C.Y. Wang. Solid-state diffusion limitations on pulseoperation of a lithium-ion cell for hybrid electric vehicles. Journal ofPower Sources, 161:628–639, 2006.

[23] M. Verbrugge and E. Tate. Adaptive state of charge algorithm fornickel metal hydride batteries including hysteresis phenomena. Journalof Power Sources, 126, 2004.

[24] M.W. Verbrugge and B.J. Koch. Electrochemical analysis of lithiatedgraphite anodes. J. Electrochem. Soc., 150, 2003.

[25] P. De Vidts, J. Delgado, and R.E. White. Mathematical modeling forthe discharge of a metal hydride electrode. J. Electrochem. Soc., 142,1995.

[26] C.Y. Wang, W.B. Gu, and B.Y. Liaw. Micro-macroscopic coupledmodeling of batteries and fuel cells. part i: Model development. J.Electrochem. Soc., 145, 1998.

[27] C.Y. Wang, W.B. Gu, and B.Y. Liaw. Micro-macroscopic coupledmodeling of batteries and fuel cells. part ii: Application to ni-cd andni-mh cells. J. Electrochem. Soc., 145, 1998.

0 2 4−20

−10

0x 10

−6 Negative electrode φs

x [µm]solid

pot

entia

l [V

olt]

8 9 10 113.551

3.5511

3.5512

Positive electrode φs

x [µm]solid

pot

entia

l [V

olt]

0 2 4 6 8 10

−0.1425

−0.142

−0.1415

−0.141

−0.1405

−0.14φ

e

x [µm]

elec

trol

yte

pote

ntia

l [V

olt]

0 2 46.12

6.14

6.16x 10

−3 negative electrode solid concentration

x [µm]

c se,n

8 9 10 110.0164

0.0165

0.0165positive electrode solid concentration

x [µm]

c se,p

Fig. 4. Numerical solution during discharge referring to a battery currentI = 30 A after 2 seconds (i.e. after 4 s of the Freedom CAR test).The solid concentration critical point for electrodes depletion/saturation arehighlighted with a circle.

0 2 4−1

012

x 10−5 Negative electrode φ

s

x [µm]Sol

id p

oten

tial [

Vol

t]

8 9 10 113.7206

3.7207

3.7208

Positive electrode φs

x [µm]solid

pot

entia

l [V

olt]

0 2 4 6 8 10

−0.0794

−0.0792

−0.079

−0.0788

−0.0786

−0.0784

φe

x [µm]

elec

trol

yte

pote

ntia

l [V

olt]

0 2 47.58

7.6

7.62x 10

−3 Negative electrode solid concentratione

x [µm]

c se,n

8 9 10 110.0151

0.0152

Positive electrode solid concentration

x [µm]

c se,p

Fig. 5. Numerical solution during charge referring to a current I =−25 A after 3 seconds (i.e. after 55 s of the Freedom CAR test). Thesolid concentration critical point for electrodes depletion/saturation arehighlighted with a circle.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

1

1.5

2

2.5

3

3.5

4

1−SOC

volta

ge [V

]

full orderaverage

10A

50A100A

200A

300A

Fig. 6. Voltage response of average versus full order model for differentconstant current from 10 A to 300 A.

0 10 20 30 40 50 60−50

0

50

curr

ent [

A]

0 10 20 30 40 50 603.4

3.6

3.8

volta

ge [V

]

full order average

0 10 20 30 40 50 60−5

0

5x 10

−4

Time

erro

r [V

]

Fig. 7. Average versus full order model.

0 10 20 30 40 50 60

0.62

0.64

0.66

0.68

0.7

0.72

0.74normalized chatode solid material concentration

solid

mat

eria

l con

cent

ratio

n

average

0 10 20 30 40 50 60

0.35

0.4

0.45

0.5normalized anode solid material concentration

Time [s]

solid

mat

eria

l con

cent

ratio

n

14.3 14.70.7165

0.7185

54.4 550.469

0.473

Fig. 8. Average versus full order battery model: negative and positiveelectrode solid material concentration. The different lines represent theconcentration values along the x-direction.

0 10 20 30 40 50 60−5

0

5

10x 10

−4 average normalized chatode solid material concentration error

erro

r

0 10 20 30 40 50 60−2

−1

0

1x 10

−3 average normalized anode solid material concentration error

Time [s]

erro

r

Fig. 9. Average versus full order battery model: error between the averagesolid concentration and the solid concentration at the electrodes-separatorinterface.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.3

0.4

0.5

0.6

0.7

0.8

norm

aliz

ed s

olid

con

cent

ratio

n

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1x 10

−3

full

orde

r vs

. ave

rage

err

or

x/L

0.7 0.8 0.9 10.726

0.727

0 0.2 0.4

0.34

0.341

0.1s 1s 5s15s

Fig. 10. Average versus full order battery model: spatial distribution ofelectrode solid surface concentration at various time (0.1 s, 1 s, 5 s and15 s) during a 30 A discharge and its error with respect to the averagemodel prediction.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x/L

norm

aliz

ed s

olid

con

cent

ratio

n

0 0.1 0.2 0.3 0.40.08

0.1

0.12

0.14

0.7 0.8 0.9 10.83

0.835

0.1s0.3s0.5s

Fig. 11. Spatial distribution of electrode solid surface concentration duringpulse operation at various time (0.1 s, 0.3 s, 0.5) during a 180 A dischargeand its error with respect to the average model prediction.

0 10 20 30 40 50 60 70

0.58

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74normalized chatode solid material concentration

Time [s]

solid

mat

eria

l con

cent

ratio

n

estimated averagefull order

15 16 17

0.718

0.72

0.722

0.724

60.95 61 61.0561.161.15

0.63

0.64

0.65

39.8 39.9

0.6747

0.6748

0.6749

61 61.1

0.646

0.648

Fig. 12. Kalman Filter: solid concentration estimation.

0 10 20 30 40 50 60−4

−2

0

2

4x 10

−3 normalized chatode solid material concentration error

erro

r

0 10 20 30 40 50 60−0.04

−0.02

0

0.02normalized anode solid material concentration error

Time [s]

erro

r

Fig. 13. Kalman Filter: error between the average solid concentrationestimation and full order model prediction at the electrolytes-separatorinterface. The transient in the EKF estimation is not shown.

0 10 20 30 40 50 60 70

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time [s]

CSCSOCCC

Fig. 14. Kalman Filter: CSC and SOC estimations, compared with thecoulomb counting SOC.

0 10 20 30 40 50 60 703.5

3.6

3.7

3.8Cell voltage

Time [s]Cel

l vol

tage

[Vol

t]

0 10 20 30 40 50 60 700.09

0.1

0.11

0.12negative electrode open circuit voltage

Un [V

olt]

full orderestimation

0 10 20 30 40 50 60 703.7

3.75

3.8

3.85positive electrode open circuit voltage

Up [V

olt]

Fig. 15. Kalman Filter: open circuit voltage estimation.

0 10 20 30 40 50 60 700.62

0.64

0.66

0.68

0.7

0.72

0.74

Time [s]

solid

mat

eria

l con

cent

ratio

n

normalized chatode solid material concentration

estimated averagefull order

18.2 18.4 18.6 18.8 190.724

0.7245

0.725

0.7255

Fig. 16. Kalman Filter: solid concentration estimation when noise is addedto the measurements.

0 10 20 30 40 50 60 70−50

0

50

Cur

rent

[A]

Battery current

0 10 20 30 40 50 60 703.5

3.55

3.6

3.65

3.7

3.75

Time [s]

Cel

l vol

tage

[Vol

t]

Cell voltage

full orderestimation

30 353.63

3.635

Fig. 17. Kalman Filter: cell voltage estimation when noise is added to themeasurements.

TABLE II

BATTERY PARAMETERS.

Parameter Negative electrode Separator Positive electrode

Thickness (cm) δn = 50 × 10−4 δsep = 25.4 × 10−4 δp = 36.4 × 10−4

Particle radius Rs (cm) 1 × 10−4 - 1 × 10−4

Active material volume fraction εs 0.580 - 0.500Electrolyte phase volume fraction (porosity) εe 0.332 0.5 0.330Conductivity of solid active materialσ (Ω−1 cm−1) 1 - 0.1

Effective conductivity of solid active material σeff = εsσ - σeff = εsσTransference number t0+ 0.363 0.363 0.363Electrolyte phase ionic conductivityκ (Ω−1 cm−1) κ = 0.0158ce exp(0.85c1.4

e ) κ = 0.0158ce exp(0.85c1.4e ) κ = 0.0158ce exp(0.85c1.4

e )

Effective electrolyte phase ionic conductivity κeff = (εe)1.5κ κeff = (εe)1.5κ κeff = (εe)1.5κ

Effective electrolyte phase diffusion conductivity κeffD = 2RTκeff

F(t0+ − 1) κeff

D = 2RTκeff

F(t0+ − 1) κeff

D = 2RTκeff

F(t0+ − 1)

Electrolyte phase diffusion coefficientDe (cm2 s−1) 2.6 × 10−6 2.6 × 10−6 2.6 × 10−6

Effective electrolyte phase diffusion coefficient Deffe = (εe)1.5De Deff

e = (εe)1.5De Deffe = (εe)1.5De

Solid phase diffusion coefficient Ds (cm2 s−1) 2.0 × 10−12 − 3.7 × 10−12

Maximum solid-phase concentrationcs,max (mol cm−3) 16.1 × 10−3 - 23.9 × 10−3

Stoichiometry at 0%θ0% 0.126 - 0.936Stoichiometry at 100%θ100% 0.676 - 0.442Average electrolyte concentration ce (mol cm−3) 1.2 × 10−3 1.2 × 10−3 1.2 × 10−3

Change transfers coefficients αa, αc 0.5,0.5 - 0.5, 0.5Active surface area per electrode unit volumeas (cm−1) asn = 3εe

Rs- asp = 3εe

Rs

Electrode plate area, A (cm2) 10452 - 10452