Tanvir R. TanimDepartment of Mechanical and

Nuclear Engineering,

The Pennsylvania State University,

University Park, PA 16802

e-mail: trt140@psu.edu

Christopher D. RahnProfessor

Department of Mechanical and

Nuclear Engineering,

The Pennsylvania State University,

University Park, PA 16802

e-mail: cdrahn@psu.edu

Chao-Yang WangProfessor

William E. Diefenderfer Chair

of Mechanical Engineering,

and Director of the Electrochemical

Engine Center,

Department of Mechanical

and Nuclear Engineering,

The Pennsylvania State University,

University Park, PA 16802

e-mail: cxw31@psu.edu

A Temperature Dependent,Single Particle, LithiumIon Cell Model IncludingElectrolyte DiffusionLow-order, explicit models of lithium ion cells are critical for real-time battery manage-ment system (BMS) applications. This paper presents a seventh-order, electrolyteenhanced single particle model (ESPM) with electrolyte diffusion and temperature de-pendent parameters (ESPM-T). The impedance transfer function coefficients are explicitin terms of the model parameters, simplifying the implementation of temperature depend-ence. The ESPM-T model is compared with a commercially available finite volume basedmodel and results show accurate matching of pulse responses over a wide range of tem-perature (T) and C-rates (I). The voltage response to 30 s pulse charge–discharge currentinputs is within 5% of the commercial code for 25 �C < T < 50 �C at I � 12:5C and�10 �C < T < 50 �C at I � 1C for a graphite/nickel cobalt manganese (NCM) lithiumion cell. [DOI: 10.1115/1.4028154]

Keywords: lithium ion battery, single particle model (SPM), battery management system(BMS)

1 Introduction

Corporate average fuel economy standards of 54.5 miles pergallon will double the fuel efficiency of the U.S. fleet, save con-sumers $1.7 trillion in fuel costs, and reduce U.S. oil consumptionby 12� 109 barrels [1]. This fuel efficiency standard is driving theautomotive industry to rely more heavily on partially and fullyelectrified (battery and/or fuel cell) vehicles. Thomas [2] con-cludes that most of the vehicles in the U.S. will be hybrid electricvehicles (HEVs) and plug in hybrid electric vehicles (PHEVs) by2034 and 2045, respectively. Li-ion batteries are the leading can-didates for HEVs and PHEVs, as they offer 40–50% weight reduc-tion and 30–40% volume reduction along with superiorcoulometric and energy efficiency compared to their closest rivals,Ni-MH batteries [3]. Hybrids require advanced and real-timeBMS that ensure safe and efficient power utilization, estimatestate of charge (SOC) and state of health, and balance cell strings[3]. Accurate cell models that capture the fast cell dynamics inHEVs are critical for high performance BMS design. BMS areoften based on equivalent circuit models [4–10] that are relativelyeasy to implement but lack the important underlying physiochemi-cal processes of the cell and require empirical parametrization forprecise estimation. Physics-based electrochemical models, on theother hand, include important battery dynamics, are explicitlydependent on the physical cell parameters, and accurately predictthe cell response [11,12].

A fundamental electrochemical model [13–15] could be anaccurate and reliable candidate for model based BMS design.Computational time and expense of solving the underlying highlynonlinear and coupled partial differential equations (PDEs), how-ever, limits their use in real time BMS applications. Reduced-order models have been developed by many researchers that pre-dict the cell response with varying degrees of fidelity and modelcomplexity. Smith et al. [12,16] obtains a control-oriented, iso-thermal, reduced seventh-order model in state variable form usingresidue grouping that predicts internal battery potentials,

concentration gradients, and enables SOC estimation from exter-nal current and voltage measurements. The numerical model orderreduction process, however, does not provide explicit relation-ships between the cell internal parameters and the coefficients ofthe impedance transfer function [17]. Lee et al. [18] extend thework of Smith et al. by obtaining an analytic transfer function forelectrolyte phase potential and concentration distribution. Kleinet al. [19] propose an isothermal simplified PDE based model thatassumes constant electrolyte concentration and approximates solidphase diffusion using three equations for volume average concen-tration, particle surface concentration, and average concentration,respectively. In subsequent paper [20], temperature effect isincluded by solving the energy equation and incorporating tem-perature correction terms, resulting in good agreement with ahybrid drive cycle.

The single particle model (SPM) studied by many researchers[17,21–24] is a simplified, physics-based, fundamental model.The two cell electrodes are modeled by two spherical particlesand associated diffusion equations are solved assuming averageelectrochemical reaction rate. The electrolyte dynamics are com-pletely ignored by assuming instantaneous Li-ion transfer fromone electrode to the other, leading to significant error at higherC-rates [17]. Rahimian et al. [25] propose an improved isothermalSPM by adding polynomially approximated electrolyte dynamics.Their improved SPM reduces to 13 differential algebraic equa-tions and are solved in COMSOL Inc., Marcicki et al. [26]recently added first-order truncated electrolyte dynamics to a SPMfrom cell specific transcendental transfer function.

Most of the previously developed SPMs do not consider theelectrolyte dynamics and are isothermal. Few researchers such asRefs. [25] and [26] attempted to incorporate the electrolytedynamics into the SPM model. On the other hand, isothermalmodels can be highly inaccurate at low temperatures. A BMScapable of achieving excellent SPM model based SOC estimationat room temperature, for example, would produce erroneousresults when cold (�10 �C) starts. This paper: (i) incorporateselectrolyte dynamics into a conventional, isothermal SPM of a Li-ion cell; (ii) simulates the performance compared to a commercialcode (EC Power, AutoLion-ST

TM

); (iii) adds temperature depend-ent parameters to the model; and (iv) demonstrates the improved

Contributed by the Dynamic Systems Division of ASME for publication in theJOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedJuly 21, 2013; final manuscript received July 30, 2014; published online August 28,2014. Assoc. Editor: Jwu-Sheng Hu.

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performance of this ESPM-T effect. Integral method analysis(IMA) is used to solve the Li-ion diffusion equation in the electro-lyte domain. Advanced order reduction techniques of parabolicdiffusion equation such as approximate inertial manifolds and theuse of empirical eigen functions can be found in Refs. [27] and[28]. Temperature is an input to the ESPM-T model so the heattransfer dynamics are not included. On boarding a vehicle, thecell temperature can be measured. In the simulated performancecomparisons, the cell temperature is provided by AutoLion-ST.

2 Mathematical Modeling

Figure 1 shows the three domains of the pseudo 2D Li-ion cellmodel: Porous anode with spherical graphite particles, porous sep-arator, and porous cathode with spherical active material particles(e.g., lithium NCM oxide, lithium cobalt oxide (LCO), lithiummanganese oxide (LMO), or lithium iron phosphate LFP). Theelectrolyte is typically 1.2 M LiPF6 in propylene carbonate (PC)/ethylene carbonate (EC)/dimethyl carbonate (DMC) and saturatesall three domains. A microporous polymer or gel polymer separa-tor isolates the direct electron path between the positive and nega-tive electrodes, but allows Li-ions to diffuse through. Aluminum(Al) and copper (Cu) foil current collectors are attached at theends of positive and negative electrodes, respectively.

During discharge, Li-ions deintercalate from the negativeelectrode

LixC Ðdischarge

Cþ xLiþ þ xe� (1)

and intercalate into the positive electrode for LMO active materialparticles

Lið1�xÞMO2 þ xLiþ Ðdischarge

LiMO2 (2)

where M stands for a metal. The opposite reactions occur duringcharge.

2.1 Governing Equations. The electrochemical model of aLi-ion cell can be described by four governing equations: Conser-vation of Li-ion (Liþ) and conservation of charge (e�) in both thesolid and electrolyte phases. Conservation of Liþ in a single,spherical, solid phase particle is described by Fick’s law ofdiffusion

@cs

@t¼ Ds

r2

@

@rr2 @cs

@r

� �(3)

where csðx; r; tÞ : ð0;LÞ � ð0;RsÞ � Rþ ! ½0; cs;max� is the concen-tration of Liþ in the solid particle. The model parameters aredefined in Table 1. The rate at which ions exit or enter the particleequals the volumetric reaction rate at the particle surface, jLi, andzero at the particle center, written as the boundary conditions

@cs

@r

� �r¼0

¼ 0 (4)

Ds

@cs

@r

� �r¼Rs

¼ � jLi

asF(5)

where jLi > 0 for ion discharge and the interfacial surface area,as ¼ 3es=Rsð Þ. Equations (3)–(5) are applied on a continuum basisacross both electrodes. The solid phase potential depends on theparticle surface concentration, cs,e(x, t)¼ cs(x, Rs, t). Diffusion inCartesian coordinates governs the conservation of Liþ in the elec-trolyte phase

ee

@ce

@t¼ Deff

e

@2ce

@x2þ

1� t0þ

FjLi (6)

where ceðx; tÞ : ð0; LÞ � Rþ ! ½0; ce;max� is electrolyte concentra-tion and ee and De are different in each domain (anode, separator,and cathode). The Bruggeman relation Deff

e ¼ Dee1:5e accounts for

the tortuous path of Liþ transport through the porous electrodesand separator. Ensuring zero flux at the current collectors and con-tinuity of concentration and flux through the adjoining domainswithin the cell, produces the boundary conditions

@cne

@x

� �x¼0

¼ 0 (7)

Deffn

@cne

@x

� �x¼Ln

¼ Deffs

@cse

@x

� �x¼Ln

(8)

cneðLn; tÞ ¼ cs

eðLn; tÞ (9)

Deffs

@cse

@x

� �x¼LnþLs

¼ Deffp

@cpe

@x

� �x¼LnþLs

(10)

cneðLn þ Ls; tÞ ¼ cp

eðLn þ Ls; tÞ (11)

@cpe

@x

� �x¼LnþLsþLp

¼ 0 (12)

Fig. 1 Schematic diagram of a pseudo 2D Li-ion cell model

011005-2 / Vol. 137, JANUARY 2015 Transactions of the ASME

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where

ceðx; tÞ ¼cn

e x; tð Þ for x 2 ð0; LnÞcs

e x; tð Þ for x 2 ðLn;Ln þ LsÞcp

e x; tð Þ for x 2 ðLn þ Ls;LÞ

8><>:

Conservation of charge in the solid phase of each electrode is

reff @2/s

@x2¼ jLi (13)

where solid phase potential

/sðx; tÞ ¼/n

s x; tð Þ for x 2 ð0;LnÞ/p

s x; tð Þ for x 2 ðLn þ Ls;LÞ

(

The fields at the current collectors are proportional to the appliedcurrent and zero at the separator

� reffn

@/ns

@x

� �x¼0

¼ reffp

@/ps

@x

� �x¼L

¼ I

A(14)

@/ns

@x

� �x¼Ln

¼ @/ps

@x

� �x¼LnþLs

¼ 0 (15)

where I> 0 indicates discharge. The effective conductivity of thesolid phase reff can be calculated from reff ¼ res, where r is thereference conductivity of the active material. The linearized elec-trolyte phase charge conservation equation [13–15]

jeff @2/e

@x2þ jeff

d

ce;0

@2ce

@x2þ jLi ¼ 0 (16)

has boundary conditions

@/ne

@x

� �x¼0

¼ 0 (17)

jeffn

@/ne

@x

� �þ jeff

d;n

@cne

@x

� �� �x¼Ln

¼ jeffs

@/se

@x

� �þ jeff

d;s

@cse

@x

� �� �x¼Ln

(18)

/neðLn; tÞ ¼ /s

eðLn; tÞ (19)

jeffs

@/se

@x

� �þ jeff

d;s

@cse

@x

� �� �x¼LnþLs

¼ jeffp

@/pe

@x

� �þ jeff

d;p

@cpe

@x

� �� �x¼LnþLs

(20)

/seðLn þ Ls; tÞ ¼ /p

eðLn þ Ls; tÞ (21)

@/pe

@x

� �x¼LnþLsþLp

¼ 0 (22)

where

/eðx; tÞ ¼/n

e x; tð Þ for x 2 ð0;LnÞ/s

e x; tð Þ for x 2 ðLn;Ln þ LsÞ/p

e x; tð Þ for x 2 ðLn þ Ls;LÞ

8><>:

The Bruggeman relation jeff ¼ je1:5e calculates the effective ionic

conductivity of individual domain. The effective diffusionalconductivity

jeffd ¼

2RTjeff

Ft0þ � 1� �

1þ d ln f6d ln ce

� �(23)

according to concentrated solution theory. Butler–Volmer (B-V)kinetics

jLi ¼ asi0 expaaFgRT

� �� exp

acFgRT

� �� �(24)

couple the four conservation Eqs. (3), (6), (13), and (16)describing the four field variables cs;e; ce;/s, and /e.Overpotential

g ¼ /s � /e � Uðcs;eÞ (25)

drives the electrochemical reaction rate. The exchange currentdensity in Eq. (24) depends on the solid particle surface and elec-trolyte concentrations

i0ðx; tÞ ¼ kðTÞcaa

e cs;max � cs;e

� �aa cac

s;e (26)

Table 1 Model parameters of a 1.78 A h high power gr/NCM Li-ion cell [29–32]

Parameter Negative electrode Separator Positive electrode

Thickness, L (cm) 40� 10�4 25� 10�4 36.55� 10�4

Particle radius, Rs (cm) 5� 10�4 5� 10�4

Active material volume fraction, es 0.662 0.58Porosity (electrolyte phase volume fraction), ee 0.3 0.4 0.3Maximum solid phase concentration, cs,max (mol cm�3) 31.08� 10�3 51.83� 10�3

Stoichiometry at 0% SOC, x0%, y0% 0.001 0.955473Stoichiometry at 100% SOC, x100%, y100% 0.790813 0.359749Average electrolyte concentration, ce,0 (mol cm�3) 1.2� 10�3

Exchange current density, i0,ref (A cm�2) 2.8� 10�3 2.0� 10�3

Activation energy of i0, (kJ mol�1) 92 58Charge transfer coefficient, aa, ac 0.5, 0.5 0.5, 0.5Liþ transference number, tþ0 0.38Film resistance, Rf (X cm2) 0 0Solid phase Li diffusion coefficient, Ds,ref (cm2 s�1) 1.4� 10�10 2.0� 10�10

Electrode plate area, A (cm2) 1020.41Activation energy of Ds (kJ mol�1) 30 25Contact resistance, Rc (X cm2) 6

Up(y), (V) �10:72y4 þ 23:88y3 � 16:77y2 þ 2:595yþ 4:563 for y 2 ð0; 1ÞUnðxÞ; ðVÞ 0:1493þ 0:8493 expð�61:79xÞ þ 0:3824 expð�665:8xÞ � expð39:42x� 41:92Þ � 0:03131 tan�1ð25:59x� 4:099Þ� 0:009434 tan�1ð32:49x� 15:74Þ for x 2 ð0:3; 1Þ

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Finally, the cell voltage is

VðtÞ ¼ /sðL; tÞ � /sð0; tÞ �Rc

AIðtÞ (27)

Finite difference discretization of the governing Eqs. (3)–(22) canproduce hundreds of state equations, requiring expensive compu-tation for on-board estimation and control. To reduce the modelorder for real-time computations, we use efficient discretizationtechniques and retain only the most significant dynamics of thefull order model.

2.2 Low-Order ESPM Formulation. In a conventional iso-thermal SPM, current density is assumed to be uniformly distrib-uted across each electrode. Thus, all the active material particlesare in parallel and each electrode can be replaced by a singlespherical particle with radius Rs but Li-ion storage capacity isequal to the electrode storage capacity. The electrolyte dynamicsare neglected, resulting in under predicted voltage swings andtransients. The assumptions of the ESPM are: (i) infinite solidphase conductivity in the individual electrodes resulting no ohmicloss, (ii) uniform current distribution in the individual electrodes,(iii) linearized conservation equations in the electrolyte domain,and (iv) all properties are evaluated at the equilibrium point (at50% SOC). Assumptions (i), (ii), and (iv) are also used in SPMmodels.

Conservation of Li in the single electrode particle for the SPMis solved by taking the Laplace transform of the particle diffusion(Eq. (3)) and applying boundary conditions (4) and (5). The solidstate diffusion impedance transfer function at the particle surfaceof a spherical particle is

~Cs;eðsÞJLiðsÞ ¼

1

asF

Rs

Ds

� �tanhðbÞ

tanhðbÞ � b

� �(28)

where b ¼ Rs

ffiffiffiffiffiffiffiffiffiffiffiffiffis=Dsð Þ

pand the tilde indicates a small perturbation

from the equilibrium condition [33] cs;eðtÞ ¼ �cs;e þ ~cs;eðtÞ. Notethat at equilibrium, �g; �j, and �ce are zero so, tildes are unnecessary.Capital letters indicate a variable has been Laplace transformed.Conservation of charge in the electrode (Eq. (13)) is simplified byintegrating in each electrode domain and applying the boundaryconditions (14) and (15). The final transfer functions are

JLin ðsÞIðsÞ ¼

1

ALn(29)

JLip ðsÞIðsÞ ¼ �

1

ALp(30)

where the assumed uniform current distributions are defined as

JLip ðsÞ ¼ 1=Lp

� � Ð LLnþLs

JLiðx; sÞdx and JLin ðsÞ ¼ 1=Lnð Þ

Ð Ln

0JLi

ðx; sÞdx, in the positive and negative electrodes, respectively.Using Eqs. (28)–(30), the solid phase surface concentrations are

~Cps;eðsÞIðsÞ ¼ �

1

aps FALp

Rps

Dps

� �tanhðbpÞ

tanhðbpÞ � bp

� �(31)

~Cns;eðsÞIðsÞ ¼

1

ans FALn

Rns

Dns

� �tanhðbnÞ

tanhðbnÞ � bn

� �(32)

in the positive and negative electrodes, respectively. These tran-scendental transfer functions are infinitely differentiable and canbe discretized using a Pad�e approximation [34,35]. Prasad andRahn [17] experimentally validated a third-order Pad�e approxima-tion with a 10 Hz bandwidth, sufficiently high for current electricvehicle applications. The third-order Pad�e approximations ofEqs. (31) and (32) are

~Cps;eðsÞIðsÞ ¼

211

aps FARp

s Lp

s2 þ 60Dps

aps FA Rp

s½ �3Lp

sþ495 Dp

s

2ap

s FA Rps½ �5Lp

" #

s3 þ 189Dps

Rps½ �2

s2 þ3465 Dp

s

2Rp

s½ �4s

(33)

~Cns;eðsÞIðsÞ ¼ �

211

ans FARn

s Lns2 þ 60Dn

s

ans FA Rn

s

3Ln

sþ495 Dn

s

2an

s FA Rns

5Ln

" #

s3 þ 189Dns

Rns

2 s2 þ3465 Dn

s

2Rn

s

4 s

(34)

The linearized B-V Eq. (24) is

gðsÞJLiðsÞ ¼

Rct

as

(35)

where the charge transfer resistance, Rct ¼ RT=i0ðaa þ acÞFð Þ.Combining Eqs. (29), (30), and (35),

gpðsÞIðsÞ ¼ �

Rpct

aps

1

ALp

(36)

gnðsÞIðsÞ ¼

Rnct

ans

1

ALn(37)

Combining Eqs. (25) and (27) and linearizing around an equilib-rium produces the voltage deviation

~VðtÞ ¼ gpðL; tÞ � gnð0; tÞ þ /eðL; tÞ � /eð0; tÞ þ@Up

@cs;e~cp

s;eðL; tÞ

� @Un

@cs;e~cn

s;eð0; tÞ �Rc

AIðtÞ (38)

where Rc is the contact resistance and the negative terminal isassigned as ground.

Taking the Laplace transform of Eq. (38) produces the imped-ance transfer function

~VðsÞIðsÞ ¼

gpðL; sÞIðsÞ �

gnð0; sÞIðsÞ þ

D/eðL; sÞIðsÞ þ @Up

@cs;e

~Cps;eðL; sÞIðsÞ

� @Un

@cs;e

~Cns;eð0; sÞIðsÞ � Rc

A(39)

The open circuit potential (OCP) slopes @Up=@cs;e

� �and

@Un=@cs;e

� �can be evaluated at any SOC from the empirically

measured OCP functions for the cathode and anode, respectively.In Eq. (39), the voltage associated with the electrolyte dynamicsD/eðL; sÞ=IðsÞ½ � is calculated from the linearized Li-ion species

conservation equation Eq. (16) assuming the reaction rate is equalto the average volumetric reaction rates in Eqs. (29) and (30).IMA [3,24,36–38] is used to solve the Li-ion conservation Eq. (6)across the three domains of the cell. Substituting Eqs. (29) and(30) into Eq. (6) for the anode, separator and cathode,

ee;nsCneðsÞ � Dn

@2CneðsÞ

@x2� b1IðsÞ ¼ 0 for x 2 ð0; LnÞ (40)

ee;ssCseðsÞ � Ds

@2CseðsÞ

@x2¼ 0 for x 2 ðLn; Ln þ LsÞ (41)

ee;psCpeðsÞ � Dp

@2CpeðsÞ

@x2þ b2IðsÞ ¼ 0 for x 2 ðLn þ Ls;LÞ

(42)

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where b1 ¼ 1� t0þ=FALn

� �; b2 ¼ 1� t0þ=FALp

� �, and the super-

script “eff” on diffusivity has been removed for simplicity. In theminimal IMA, the quadratic Li-ion concentration profiles in theindividual domains are

Cneðx; sÞ ¼ c0;nðsÞ þ c1;nðsÞxþ c2;nðsÞx2 for x 2 ð0; LnÞ (43)

Cseðx; sÞ ¼ c0;sðsÞ þ c1;sðsÞxþ c2;sðsÞx2 for x 2 ðLn;Ln þ LsÞ

(44)

Cpeðx; sÞ ¼ c0;pðsÞ þ c1;pðsÞxþ c2;pðsÞx2 for x 2 ðLn þ Ls;LÞ

(45)

where c0;n; c1;n;…; c2;p are the coefficients of the quadratic con-centration distributions in x. Note that Eqs. (43)–(45) are third-order in time/Laplace domain. These quadratic distributions aresubstituted into Eqs. (40)–(42), integrated and applied the associ-ated boundary conditions (7)–(12). Solving the nine equations forthe nine unknown coefficients in Eqs. (43)–(45) yields the transferfunctions

CneðsÞ

IðsÞ ¼p2;nðxÞs2 þ p1;nðxÞsþ p0;n

s q3;ns2 þ q2;nsþ q1;n

� � (46)

CseðsÞ

IðsÞ ¼p2;sðxÞs2 þ p1;sðxÞsþ p0;s

s q3;ns2 þ q2;nsþ q1;n

� � (47)

CpeðsÞ

IðsÞ ¼p2;pðxÞs2 þ p1;pðxÞsþ p0;p

s q3;ns2 þ q2;nsþ q1;n

� � (48)

where the coefficients p0;n; p1;n;…; q3;n are given in Appendix A.Now substituting Eqs. (29) and (30) into Eq. (16), the electrolytecharge conservation equations in the three domains of the cell are

jn@2/n

eðsÞ@x2

þ jd;n@2Cn

eðsÞ@x2

þ b3I ¼ 0 (49)

js

@2/seðsÞ

@x2þ jd;s

@2CseðsÞ

@x2¼ 0 (50)

jp

@2/peðsÞ

@x2þ jd;p

@2CpeðsÞ

@x2� b4I ¼ 0 (51)

where b3 ¼ 1=ALnð Þ; b4 ¼ 1=ALp

� �;jd;n ¼ jeff

d;n=ce;0

� �;jd;s

¼ jeffd;s=ce;0

� �; jd;p ¼ jeff

d;p=ce;0

� �, and the conductivities are

assumed constant in each domain (anode, cathode, and separator).Equations (49)–(51) and their associated boundary conditions,Eqs. (17)–(22), are singular due to the zero flux at x¼ 0 andx¼ Lnþ Lsþ Lp. This situation can be avoided by defining elec-trolyte voltage difference relative to /eð0; sÞ [3,11,16]. We definethe electrolyte phase voltage differences

D/neðx; sÞ ¼ /n

eðx; sÞ � /neð0; sÞ for x 2 ð0;LnÞ

D/seðx; sÞ ¼ /s

eðx; sÞ � /neð0; sÞ for x 2 ðLn;Ln þ LsÞ

D/peðx; sÞ ¼ /p

eðx; sÞ � /neð0; sÞ for x 2 ðLn þ Ls; LÞ

At this point of the derivation the electrolyte concentration profilein each individual domain of the cell is known and expressed asEqs. (46)–(48). Integrating Eqs. (49)–(51) with respect to x in theindividual cell domains and subtracting /n

eð0; sÞ gives

jnD/ne þ jd;n Cn

eðx; sÞ � Cneð0; sÞ

þ b3Ix2

2¼ C1nx (52)

jsD/se þ jd;sC

seðx; sÞ �

jsjd;n

jnCn

eð0; sÞ ¼ C1sxþ C2n (53)

jpD/pe þ jd;pCp

eðx; sÞ �jpjd;n

jnCn

eð0; sÞ �b4Ix2

2¼ C1pxþ C2p

(54)

where C1;n;…;C2;p are constants of integration. Equations(52)–(54) are solved analytically using the associated boundaryconditions (18)–(22). After further simplification, the electrolytephase potential difference is

D/eðL; sÞIðsÞ ¼ R2s2 þ R1sþ R0

L2s2 þ L1sþ L0

(55)

The coefficients R0;…;L2 are listed in Appendix A. Equations(33), (34), (36), (37), and (55) are substituted in Eq. (39) to deducethe final ESPM impedance transfer function

~VðsÞIðsÞ ¼ K þ K1 þ K2

s

þ b00s6 þ b01s5 þ b02s4 þ b03s3 þ b04s2 þ b05sþ b06

s6 þ a01s5 þ a02s4 þ a03s3 þ a04s2 þ a05sþ a06

(56)

where the coefficients a01;…; a06; b00;…; b06;K;K1; andK2 areexplicitly given in terms of the model parameters in Appendix B.

2.3 ESPM With Temperature Effect (ESPM-T). The trans-fer function (Eq. (56)) is converted to a state space realizationusing observer canonical form [39] which produces seven ordi-nary differential equations. For isothermal ESPM simulation at aparticular temperature all the coefficients are constants and lsim inMATLAB simulates the test response. For nonisothermal case(ESPM-T) the coefficients are updated with cell temperatureobtained from AutoLion-ST output at every time step and simu-lated using ode45.

For the gr/NCM chemistry simulated in this paper, most signifi-cant temperature dependent parameters are solid phase diffusioncoefficient, exchange current density, electrolyte diffusion coeffi-cient, electrolyte ionic conductivity, and electrolyte diffusionalionic conductivity. Arrhenius equation

w ¼ wref expEact;w

R

1

Tref

� 1

T

� �� �(57)

is used to calculate solid particle diffusion coefficient andexchange current density. The temperature dependent property we:g:;Dp

s ;Dns ; and i0

� �depends on the reference value and the acti-

vation energy Eact;w. The empirical correlations for electrolyteproperties are extracted from Valøen and Reimers [40]

De Tð Þ ¼ 10

� 4:43þ 54

T � 229þ ce;0

� �þ 0:22ce;0

" #(58)

jðTÞ ¼ ce;0½ð�10:5þ 0:074T � 6:96� 10�5T2Þþ ce;0ð0:668� 0:0178T � 2:8� 10�5T2Þþ c2

e;0ð0:494� 8:86� 10�4T2Þ�2 (59)

jd;i ¼2RTjeff

i

Ftþ0 � 1� �

1þ d ln f6d ln ce

� �¼ � 2RTjeff

i

Ftþ0 � 1� �

� Tð Þ

(60)

where jeffi ¼ j Tð Þe1:5

i and subscript i denotes individual domain.The empirical correlation

� Tð Þ¼ 0:601�0:24c12

e;0þ0:982 1�0:0052ðT�293Þ½ �c32

e;0 (61)

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3 Results and Discussion

3.1 Comparison of SPM and ESPM With AutoLion-STUnder Isothermal Conditions. In this work, EC Power’sAutoLion-ST is considered to be the truth model and used to com-pare the accuracy of reduced-order SPM, ESPM, and ESPM-T.AutoLion-ST is a pseudo 2D, fully nonlinear, finite volume basedmodel in MATLAB/SIMULINK. It uses robust numerical algorithms tosimulate electrochemical and thermal interactions of Li-ion bat-teries over a wide range of operating conditions [41]. Figures2(a)–2(c) compare the SPM, ESPM and AutoLion-ST voltageresponses at 25 �C and 50% initial SOC corresponding to the pulsecurrent input in Fig. 2(e). The zoomed in Fig. 2(a) shows that theSPM response deviates significantly from the ESPM andAutoLion-ST voltage responses, even at lower C-rates. TheESPM voltage response, however, closely matches AutoLion-ST,including 20 C–10 s pulses (see zoomed view in Fig. 2(b)).

Figure 2(d) shows the time response of the electrolyte potentialD/eðtÞ ¼ /eðL; tÞ � /eð0; tÞ is strongly C-rate dependent. This

internal variable can not be predicted by ECMs. The ESPM, how-ever, is based on the fundamental equations of the cell and inter-nal variables such as D/eðtÞ can be predicted. The ESPM predictselectrolyte potential difference accurately at low C-rates but hasalmost 50% (314 mV) error at high C-rates. The quadratic concen-tration distributions and constant current distributions contributeinto this error at higher C-rates. Although D/eðtÞ is underpre-dicted by the ESPM, it is a significant improvement over SPMwhich neglects electrolyte dynamics entirely.

One can tune the SPM contact resistance to account for theunmodeled electrolyte dynamics. At higher C-rates, however, theSPM voltage response overshoots the AutoLion-ST voltageresponse due to the neglect of electrolyte diffusion dynamics [17].Although ESPM underpredicts D/eðtÞ at higher C-rates, theESPM’s voltage accurately matches the AutoLion-ST’s voltage(see Fig. 2(b)). Overprediction of overpotentials resulting fromthe linearization of the B-V kinetic equation compensates for theunderpredicted D/eðtÞ. The maximum voltage error between theESPM and AutoLion-ST is 3% (62 mV) for the 20 C–10 s pulse.

Fig. 3 Voltage response of AutoLion-ST and ESPM at 25 �C and 50% initial SOC:(a) magnified voltage during 20 C pulse (left box in Fig. 3(c)), (b) magnified voltageduring 15 C pulse (right box in Fig. 3(c)), (c) voltage response, (d) SOC, and (e)pulse current input

Fig. 2 Voltage response of AutoLion-ST, ESPM, and SPM at 25 �C and 50% initialSOC: (a) magnified voltage during 8.5 C discharge pulse (left box in Fig. 2(c)), (b)magnified voltage during 20 C discharge pulse (right box in Fig. 2(c)), (c) voltageresponse, (d) electrolyte potential difference, and (e) pulse current input

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3.2 Performance of the ESPM Away From the LinearizedPoint. To test the validity of the ESPM away from the lineariza-tion SOC, Fig. 3 simulates a hybrid current cycle operating in alarger 35–65% SOC window. Figure 3(c) compares the voltageresponses of the ESPM and AutoLion-ST for the current inputshown in Fig. 3(e). Figure 3(d) shows the corresponding SOCswing. Worst case scenarios at the highest and lowest SOCs arepresented in the zoomed in Figs. 3(a) and 3(b), with 20 C–10 s and15 C–15 s pulses, respectively. The voltage error increases withdistance from the linearization point and increasing C-rate. Theincreased error is caused due the B-V linearization and constantproperties assumptions (at 50% SOC) in the ESPM. Nevertheless,the maximum error between the ESPM and AutoLion-ST voltageresponses remains less than 4.3% (137 mV) during the entiresimulation.

3.3 Comparison of ESPM and ESPM-T With AutoLion-ST Under Nonisothermal Conditions. Figure 4 shows the influ-ence of cell initial temperature on the voltage response bycomparing the simulation results from ESPM, ESPM-T, andAutoLion-ST. The cell temperature, as predicted by AutoLion-ST

starts at 10 �C and increases under adiabatic conditions to a tem-perature limit of 25 �C over the course of the simulation. TheAutoLion-ST model includes a cooling system that prevents thecell temperature from exceeding 25 �C. The ESPM is isothermalat 25 �C and the ESPM-T temperature dependent physiochemicalproperties are updated using the AutoLion-ST output temperatureas one could use a temperature sensor on-board a vehicle. ESPMvoltage response does not match the AutoLion-ST results at lowtemperature, even at low C-rates. The highly distributed currentalong each electrode due to sluggish reaction kinetics, reducedelectrolyte diffusivity, and ionic conductivity may attribute thevoltage difference at low temperatures. As the cell temperatureapproaches 25 �C, ESPM and ESPM-T produce identical voltageresponses that closely match AutoLion-ST voltage response. Themagnified Figs. 4(a) and 4(b) at 2.5 C and 20 C, respectively,show the relative agreement of these simulation methods. Overall,ESPM-T matches AutoLion-ST to within 3% (62 mV) for theentire 10 �C–25 �C temperature range and up to 20 C rates.

3.4 Operating Range of ESPM-T. Figure 5 shows that theaccuracy of the ESPM-T model depends on both current rate and

Fig. 4 Voltage response of AutoLion-ST, ESPM, and ESPM-T from 10 �C initial tem-perature and 50% initial SOC: (a) magnified voltage during 2.5 C discharge pulse(left box in Fig. 4(c)), (b) magnified voltage during 20 C discharge pulse (right boxin Fig. 4(c)), (c) voltage response, (d) cell temperature, and (e) pulse current input

Fig. 5 Voltage response of AutoLion-ST and ESPM-T from 0 �C and 50% initialSOC: (a) voltage response, (b) cell temperature, and (c) pulse current input

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temperature. In this simulation, the cell starts at 0 �C and, underadiabatic conditions, the temperature grows to 25 �C. Low temper-ature and low C-rate (t< 200 s) produces modest errors. ModerateC-rates at low temperatures produce large errors and high C-ratesat higher temperatures produce the least errors.

To identify the working range of ESPM-T, we separate the C-rate and temperature effects by simulating constant hybrid pulsecycles at different temperatures. The temperature range is from�10 �C to 50 �C for 1 C, 2 C, 5 C, 7.5 C, 10 C, and 12.5 C constanthybrid 30 s pulse cycles. Figure 6 shows an example case of adia-batic simulation of 7.5 C–30 s pulse charge–discharge cycle. Asthe cell temperature increases with time, the initial voltage errorbetween the AutoLion-ST and ESPM-T diminishes.

Figure 7 summarizes the maximum voltage error relative toAutoLion-ST at different C-rates versus cell temperature.Each curve is for a different C-rate pulse cycle as shown inFig. 6(c). As the cell warms up due to cycling, the temperatureincreases and the maximum voltage error for the correspondingpulse is evaluated. The error eventually decreases with increasingtemperature and decreasing C-rate, although the error increasesslightly for T> 25 �C due to slight asymmetry between chargeand discharge voltage errors. To maintain errors less than 1%requires T> 17 �C and I< 5 C. If 5% errors are acceptable then12.5 C is possible for T> 25 �C and 1 C for T>�10 �C. The soliddot in Fig. 7 corresponds to the solid dot in Fig. 6(a).

Computation time of a 20 min simulation at 1 Hz sampling rateon an Intel

VR

CoreTM

2 Quad 2.4 GHz desktop computer ofAutoLion-ST and other reduced-order models are shown inTable 2. ESPM and ESPM-T models are, respectively, 11.9 and 5times faster than AutoLion-ST in this case.

4 Conclusion

The traditional SPM neglects electrolyte diffusion only pro-vides satisfactory performance over narrow C-rate and tempera-ture ranges. Using only seven states, the ESPM developed inthis paper includes an IMA model of electrolyte diffusion andmatches the AutoLion-ST pulse response up to 20 C at room tem-perature with 3% (62 mV) error. The ESPM linearized at 50%SOC, has slightly higher voltage error (4.3%/137 mV) forwider SOC swing (35–65%). The ESPM-T model updatesthe ESPM parameters with temperature, maintaining the voltageresponse to pulse charge–discharge current inputs to within 5%of the AutoLion-ST for 25 �C < T < 50 �C at 12.5 C and�10 �C < T < 50 �C at 1 C.

Acknowledgment

The authors greatly appreciate EC Power LLC. for providingAutoLion-ST

TM

software.

Fig. 6 Voltage response of AutoLion-ST and ESPM-T from 10 �C and 50% initialSOC: (a) voltage response, (b) cell temperature, and (c) 7.5C–30 s hybrid pulsecharge– discharge cycle

Fig. 7 ESPM-T error relative to AutoLion-ST for different C-ratepulse cycles (see Fig. 6(c)) versus cell temperature. The soliddot corresponds to the dot in Fig. 6(a).

Table 2 Computation time of different Li-ion cell models

Li-ion cell model Solver Computation time (s)

AutoLion-STTM Nonlinear solver 12.5ESPM-T ode45 2.5ESPM lsim 1.05SPM lsim 1

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Appendix A: Submodel Coefficients

A.1 Coefficients of the Electrolyte Concentration Profile

q1;n¼ 36DnDpD2s enLnþ esLsþ epLp

� �q2;n¼ 12DnD2

s L2pep LnenþLsesð Þþ12D2

s DpL2nenðLsesþLpepÞþ12DnDsDpL2

s es LnenþLpep

� �þ36DnDsDpenepLnLsLp

q3;n¼LnLsLpenesep 3DnDpL2s þ4D2

s LnLpþ4DsLs DnLpþDpLn

� � p0;n¼ p0;s¼ p0;p¼ 36DnDpD2

s ðLnb1�Lpb2Þ

p1;n¼ 12DnDsLnLpb1ep DsLpþ3DpLs

� �þ12DnDsL

2s es Lnb1þLpb2

� �þ 18D2

s Dpb1 LsesþLpep

� �þ6D2

s DpLpb2en

L2

n�x2� �

p2;n¼LpLses

DnLnLsb1ep 4DsLpþ3DpLs

� �þ6Dsb1ep DsLpþDpLs

� �L2

n�x2� �

þDpDsLsb2en 3x2�L2n

� �p1;s¼ 6Ds

3DnDpLnb1es L2

n�x2� �

þ6DnDpL2nb1es Ls�xð ÞþDnDpLpb2es L2

s �3x2� �

þ2DnDpLsb1es Ls�3xð Þ

þ3DnDpLnLpb2es 2x�Lnð Þ�2DpDsL2nLpb2enþ6DnDpLnLp b2enþb1ep

� �Ln� xð Þþ2DnLnLpb1ep DsLpþ3DpLs

� �p2;s¼LnLpes

3DnDpLsb1epðL2

s þ3x2Þþ6DnDp Ln� xð Þ L2s b1ep�L2

s b2en

� �þ3 L2

nþ x2� �

2DnDpLpb1ep�2DsDpLnb2en�3DnDpLsb2en

� �� 12DnDsLpb1ep LsþLnð Þþ18DnDpLnLs b2enþb1ep

� � �xþ2DpDsLnb2en 6LnxþL2

s

� �þDnb1epf4DsLpLs 3LnþLsð Þ

þ9DpL2nLsg

p1;p¼ 18D2

s LnLpðDnLpb1ep�DpLnb2enÞ�6DnDpDsLnLsðb1Lsesþ6Lpb2enÞþ18DnD2s

2b2 Ln�xð Þ LnLpenþL2

s es

� �þ2 Ls�xð Þ L2

nb2enþL2nb1epþLnLpb1esþLsLpb2ep

� �þ2LnLsb2 Lp� x

� �enþ esð ÞþLn b2enþb1ep

� �L2

nþ x2� �

þLsb2es b2enþb1ep

� �L2

s þx2� �

þLnLs Lnb2esþLsb2enþLsb1ep

� �p2;p¼LnLses

3DnLsb2enð2DsL

2n�DpLpLsÞþ2DsLnLsb2enð3DsLs�2DpLpÞ�DnDsLsb1ep 2L2

pþ3L2s

� �þ6DnDsLs 2Lpb2en�Lsb1ep

� �ðLn� xÞþ6DnDsLs 2Lnb2en�Lpb1ep

� �ðLs�xÞþð12D2

s L2nb2enþ12D2

s LnLsb2enþ12DnDsL2s b2en

�6DnDsLnLsb1epÞ Lp� x� �

þ12D2s Lnb2en LnLs� xLp

� �þðL2

nþ x2Þ 6D2s Lnb2en�3DnDsLsb1ep

� �þ6DnDsLsb2en L2

s þx2� �

A.2 Coefficients of Electrolyte Phase Potential Difference

L0 ¼ 72DnDpD2s jnjpjs Lnen þ Lses þ Lpep

� �L1 ¼ 24Dsjnjpjs

DpDsenL2

n epLp þ esLs

� �þ DnenepLnLp DsLp þ 3DpLs

� �þ DnepesLpLs DsLp þ DpLs

� �þ DnDpenesLnL2

s

L2 ¼ 2LnLpLsenesepjnjpjsð3DnDpL2

s þ 4D2s LnLp þ 4DnDsLpLs þ 4DpDsLnLsÞ

R0 ¼ 36DnDpD2s L2

nb3jpjs Lnen þ Lses þ Lpep

� �� 36DnDpD2

s L2pb4jsj

2n Lnen þ Lses þ Lpep

� �þ 36DnD2

s L2pb2jd;pjnjs Lnen þ Lsesð Þ þ 36DpD2

s L2nb1jd;njpjs Lpep þ Lses

� �þ 36D2

s LnLpjs DpLnb2enjd;njp þ DnLpb1epjd;pjn

� �þ 36DnDpDsL

2s esjd;sjnjpðb1Ln þ b2LpÞ � 72ðDnDpD2

s LpLsb4j2njp þ DnDpD2

s LnLpb4jnjpjsÞðLnen þ Lses þ LpepÞþ 72DnDpDsLnLpLsjd;sjnjpðb2en þ b1epÞ

R1 ¼ 12DpD2s L2

nenjs L2nb3jp � L2

pb4jn

� �Lpep þ Lses

� �þ 12DnD2

s L2pjs L2

nb3jpen � L2pb4jnep

� �Lpep þ Lses

� �þ 12DnDpDsL

2s esjs L2

nb3jp � L2pb4jn

� �Lnen þ Lpep

� �� 24D2

s LnL2pb4enepjp Lsjn þ Lpjs

� �DnLp þ DpLn

� �� 24DnDsLsL

2pb4epesjp Lnjs þ Lsjnð Þ DsLp þ DpLs

� �� 24DpDsLnLpLsb4enesjp Lnjs þ Lsjnð Þ DsLn þ DnLsð Þ

� 72DnDpDsLnL2pLsb4enepjp Lsjn þ Lnjsð Þ þ 12DsLnLpL2

s esjd;sjnjp DpLnb2en þ DnLpb1ep

� �þ 12D2

s L2nLsL

2pesjs b2enjnjd;p þ b1epjd;njp

� �þ 36DnDpDsLnLpenepjs L2

nb3jn � L2pb4jp

� �þ 6DnDpLnLpL3

s esjd;sjnjp b2en þ b1ep

� �þ 6DnDsLnL2

pL2s esjd;pjnjs 2b2en � b1ep

� �� 6DpDsL

2nLpL2

s b2enesjd;njpjs

R2 ¼ �LnLpLsenesep

ð�b3jpjsL

2n þ 2b4jpjsjnLnLp þ b4jsj

2nL2

p þ 2b4jpj2nLsLpÞð3DnDpL2

s þ 4D2s LnLp þ 4DnDsLpLs þ 4DpDsLnLsÞ

Journal of Dynamic Systems, Measurement, and Control JANUARY 2015, Vol. 137 / 011005-9

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Appendix B: ESPM Impedance Transfer Function

Derivation

Substituting Eqs. (33), (34), (36), (37), and (55) into Eq. (39),we obtain

~V sð ÞIðsÞ ¼ K þ R2s2 þ R1sþ R0

L2s2 þ L1sþ L0

þ a1s2 þ 60a1a2sþ 495a1a22

s3 þ 189a2s2 þ 3465a22s

þ b1s2 þ 60b1b2sþ 495b1b22

s3 þ 189b2s2 þ 3465b22s

(B1)

where

K ¼ � Rpct

aps

1

ALpþ Rn

ct

ans

1

ALnþ Rc

A

� �

Cp ¼ 21@Up

@cs;e

Cn ¼ 21@Un

@cs;e

a1 ¼Cp

aps FARp

s Lp

a2 ¼Dp

s

Rps½ �2

b1 ¼Cn

ans FARn

s Ln

b2 ¼Dn

s

Rns

2 :Simplifying Eq. (B1)

~V sð ÞIðsÞ ¼KþE0s2þE1sþE2

s2þF1sþF2

þ A2s2þA1sþA0

sðs2þB2sþB1Þþ C2s2þC1sþC0

sðs2þD2sþD1Þ(B2)

where

E0 ¼R2

L2

E1 ¼R1

L2

E2 ¼R0

L2

F1 ¼L1

L2

F2 ¼L0

L2

A0 ¼ 495a1a22

A1 ¼ 60a1a2

A2 ¼ a1

B1 ¼ 3465a22

B2 ¼ 189a2

C0 ¼ 495b1b22

C1 ¼ 60b1b2

C2 ¼ b1

D1 ¼ 3465b22

D2 ¼ 189b2

Factoring out one integrator, Eq. (B2) can be written as

~V sð ÞIðsÞ ¼ K þ K1 þ K2

sþ E0s2 þ E1sþ E2

s2 þ F1sþ F2

þ g1s3 þ g2s2 þ g3sþ g4

s4 þ h1s3 þ h2s2 þ h3sþ h4

(B3)

where

K1 ¼A0

B1

e0 ¼A1B1 � A0B2

B1

e1 ¼A2B1 � A0

B1

K2 ¼C0

D1

f0 ¼C1D1 � C0D2

D1

f1 ¼C2D1 � C0

D1

g1 ¼ e1 þ f1

g2 ¼ e0 þ e1D2 þ f0 þ f1B2

g3 ¼ e0D2 þ e1D1 þ f0B2 þ f1B1

g4 ¼ e0D1þ f0B1

h1 ¼ B2 þ D2

h2 ¼ B1 þ B2D2 þ D1

h3 ¼ B1D2 þ B2D1

h4 ¼ B1D1

After further manipulation of Eq. (B3), the final equation suitablefor state space realization can be written as follows:

~V sð ÞIðsÞ ¼ K þ K1 þ K2

s

þ b00s6 þ b01s5 þ b02s4 þ b03s3 þ b04s2 þ b05sþ b06

s6 þ a01s5 þ a02s4 þ a03s3 þ a04s2 þ a05sþ a06

(B4)

where

b00 ¼ E0

b01 ¼ E1 þ g1 þ E0h1

b02 ¼ E2 þ g2 þ E0h2 þ E1h1 þ F1g1

b03 ¼ g3 þ E0h3 þ E1h2 þ E2h1 þ F1g2 þ F2g1

b04 ¼ g4 þ E0h4 þ E1h3 þ E2h2 þ F1g3 þ F2g2

b05 ¼ E1h4 þ E2h3 þ F1g4 þ F2g3

b06 ¼ E2h4 þ F2g4

a01 ¼ F1 þ h1; a02 ¼ F2 þ h2 þ F1h1

a03 ¼ h3 þ F1h2 þ F2h1

a04 ¼ h4 þ F1h3 þ F2h2

a05 ¼ F1h4 þ F2h3

a06 ¼ F2h4

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