Samsung Battery ROM

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Short communication

An explicit algebraic reduced order algorithm for lithium ion cellvoltage prediction

V. Senthil Kumar a, Priya Gambhire a,*, Krishnan S. Hariharan a, Ashish Khandelwal a,Subramanya Mayya Kolake a, Dukjin Oh b, Seokgwang Doo b

aComputational Simulations Group, SAIT-India Lab, TRIDIB e Bagmane Tech Park, Bangalore 560 093, Indiab Energy Storage Group, Samsung Advanced Institute of Technology, 449712, South Korea

h i g h l i g h t s

Earlier work reduced the 10 PDEs to 5 linear ODEs and nonlinear expressions. Exact solutions are derived here for a linearly varying current period. Explicit algebraic algorithm which can model any chargeerestedischarge protocol.

a r t i c l e i n f o

Article history:Received 16 July 2013Received in revised form20 September 2013Accepted 21 September 2013Available online 30 September 2013

Keywords:Lithium ion cellReduced order modelVolume averagingProle based approximationsExplicit algebraic algorithmBattery management system

a b s t r a c t

The detailed isothermal electrochemical model for a lithium ion cell has ten coupled partial differentialequations to describe the cell behavior. In an earlier publication [Journal of Power Sources, 222, 426(2013)], a reduced order model (ROM) was developed by reducing the detailed model to a set of velinear ordinary differential equations and nonlinear algebraic expressions, using uniform reaction rate,volume averaging and prole based approximations. An arbitrary current prole, involving charge, restand discharge, is broken down into constant current and linearly varying current periods. The linearlyvarying current period results are generic, since it includes the constant current period results as well.Hence, the linear ordinary differential equations in ROM are solved for a linearly varying current periodand an explicit algebraic algorithm is developed for lithium ion cell voltage prediction. While the existingbattery management system (BMS) algorithms are equivalent circuit based and ordinary differentialequations, the proposed algorithm is an explicit algebraic algorithm. These results are useful to develop aBMS algorithm for on-board applications in electric or hybrid vehicles, smart phones etc. This algorithmis simple enough for a spread-sheet implementation and is useful for rapid analysis of laboratory data.

2013 Elsevier B.V. All rights reserved.

1. Introduction

An isothermal model for lithium ion cell has ten coupled partialdifferential equations (PDEs) to describe the cell behavior [1e4],namely: two solid diffusion equations and two solid currentequations in the electrode regions; three electrolyte diffusionequations and three total current balances in the negative elec-trode, separator and positive electrode regions, respectively.Recently, a reduced order model (ROM) has been developed [4],wherein, using uniform reaction rate, volume averaging and prolebased approximations, the aforementioned ten coupled PDEs are

reduced to a set of ve linear ordinary differential equations (ODEs)and nonlinear algebraic expressions. The nonlinearity of the systemonce relegated to algebraic expressions is not computationallycumbersome. This ROM [4] shows negligible errors in cell voltageprediction at nominal currents and less than 3% error at high cur-rents for a commercial cell.

The current work derives analytical solutions for the ve linearODEs in the ROM [4], thereby developing an algebraic algorithm forcell voltage prediction in the non-phase change lithium ion cells.Any arbitrary current prole, involving charge, rest and discharge,can be expressed as a series of constant current and linearly varyingcurrent periods. The latter is more generic, since it includes theconstant current period as well. Hence, analytical solutions arederived for a linearly varying current period and the results aredeveloped into an explicit algebraic algorithm for cell voltageprediction. This algorithm is simple enough for a spread-sheet

* Corresponding author. Tel.: 91 80 4181 9999; fax: 91 80 4181 9000.E-mail addresses: [email protected], [email protected]

(P. Gambhire).

Contents lists available at ScienceDirect

Journal of Power Sources

journal homepage: www.elsevier .com/locate/ jpowsour

0378-7753/$ e see front matter 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.jpowsour.2013.09.089

Journal of Power Sources 248 (2014) 383e387


implementation and can be used in battery testing equipment forrapid analysis of laboratory data. It can also be easily developed intoa battery management system algorithm for on-board applicationsin electric or hybrid vehicles and smart phones.

2. Derivation of the algebraic ROM

2.1. Solid phase Li concentration and state of charge in theelectrodes

The evolution of average solid phase Li concentration in thenegative electrode (Eq. (127) in Ref. [4]) is

dc1ndt

3hjnirn

: (1)

The average reaction rate in the above equation is related to theexternal current and the electrode design parameters as (Eq. (14) inRef. [4])

hjnit ItanFln ; (2)

where, an is the specic surface area of the active material innegative electrode given by,

an 31nrn : (3)

Using Eqs. (2) and (3) in (1) gives

dc1ndt

Itln1nF

: (4)

Dening the scaled average solid Li concentration in negativeelectrode as

C1n c1n

csn max; (5)

and the maximum specic moles of solid phase Li in the negativeelectrode as

N1n ln1ncsn max; (6)Eq. (4) is rewritten in terms of scaled quantities as

dC1ndt

ItFN1n

: (7)

Time is left unscaled since there are many different time scalesin this problem. Consider a linear change of external current from Iito If, in the time interval ti to tf. The current at any time in this in-terval is

It Ii If Ii

tf ti

t ti: (8)Using the above expression in Eq. (7) and integrating from ti to t

gives

C1nt C1nti 1

FN1n

264IiZtti

dtIf Ii

tf ti

Ztti

t tidt

375: (9)

On integrating and rearranging the above expression, the scaledaverage concentration of solid phase Li in the negative electrode atany time is

C1nt C1nti 1

FN1n

"Iit ti

12

If Ii

tf ti

t ti2#: (10)

Substituting t tf and simplifying gives the average concen-tration of solid phase Li in the negative electrode at the nal timeas

C1ntf C1nti

12

1FN1n

Ii If

tf ti

: (11)

Using the condition, Ii If I the above expression reduces tothe solution for a constant current period.

According to the electro-neutrality condition, for every neutralLi ionized in an electrode, a Li ion is neutralized in the oppositeelectrode. Thereby, the total number of neutral Li in the solid phaseof both the electrodes remains constant (Eq. (128) in Ref. [4])

ln1nc1nt lp1pc1pt ln1nc1n0 lp1pc1p0 Constant:(12)

In terms of scaled quantities, the above equation becomes

N1nC1nt N1pC1pt N1nSoCn0 N1pSoCp0; (13)

where N1p lp1pcsp max: (14)The initial condition is assumed to be a completely equilibrated

state, that is, no concentration gradients or equivalently, theaverage concentration of Li within the active material spheres isequal to the surface concentration. Hence, the initial scaled con-centrations are identical to initial electrode states of charge. Usingthe Eq. (13), the average Li concentration in the positive electrode iscomputed as

C1pt SoCp0 N1nN1p

hSoCn0 C1nt

i: (15)

The average concentration gradient within the active materialspheres in the negative electrode is scaled using maximum con-centration and particle radius as

C1rn rn

csn maxc1rn: (16)

The evolution equation for the radial concentration gradient innegative electrode is given by (Eq. (153) in Ref. [4])

dc1rndt

30D1nr2n

c1rn 452r2n

hjni: (17)

With the solid diffusion time scale in negative electrode denedas

s1n r2nD1n

: (18)

Using Eqs. (2), (8), (16) and (18), Eq. (17) gets scaled as

dC1rndt

30s1n

C1rn 152

1FN1n

"Ii

If Ii

tf ti

t ti#: (19)

Eq. (19) is a rst order differential equation which can be solvedby more than one method. M. R. Spiegel [5] has details of suchsolution methods. In the present work, the equation is solved byobtaining the general solution which is the sum of a complemen-tary function and a particular integral. The complementary func-tion obtained by solving the homogeneous ODE, is

V. Senthil Kumar et al. / Journal of Power Sources 248 (2014) 383e387384


CCF1rn an exp

30 t ti

s1n

: (20)

To obtain the particular integral, a trial function linear in time issubstituted in Eq. (19)

CPI1rn bn gnt ti: (21)Equating the coefcients of the constant and the linear terms

in time, in Eq. (21), the expressions for bn and gn are obtainedas,

gn 14

If Ii

s1n

FN1n

1tf ti

; (22)

and bn 14Iis1nFN1n

130

gns1n: (23)

The complete solution is the sum of complementary functionand particular integral

C1rnt an exp 30 t ti

s1n

bn gnt ti: (24)

At the initial time, the above expression reduces to

C1rn ti an bn; (25)

giving an C1rnti bn: (26)Thus, Eqs. (22), (23) and (26) determine the three coefcients in

the complete solution of Eq. (24). For the initial condition of anequilibrated state, the initial average radial concentration gradientcan be set to zero. Knowing the average concentration and theaverage radial concentration gradient, the negative electrode SoC isobtained by scaling Eq. (149) in Ref. [4] as,

SoCnt csntcsn max C1nt 835

C1nrt 1

105Its1nFN1n

: (27)

Similarly, the average radial gradient in the positive electrode isobtained by solving the following equations (Eqs. (153) and (15) inRef. [4])

dc1rpdt

30D1pr2p

c1rp 452r2p

DjpE; (28)

whereDjpEt It

apFlp; (29)

and ap 31prp

: (30)

With the maximum solid phase Li capacity from Eq. (14) and thesolid diffusion time constant

s1p r2pD1p

; (31)

Eq. (28) gets scaled as

dC1rpdt

30s1p

C1rp 152

1N1pF

"Ii

If Ii

tf ti

t ti:#

(32)

Its complete solution is obtained as

C1rpt ap exp" 30

s1pt ti

# bp gpt ti; (33)

where gp 14

If Ii

s1p

FN1p

1tf ti

; (34)bp

14Iis1pFN1p

130

gps1p; (35)

and ap C1rpti bp: (36)Using the expressions for the average concentration and the

average radial concentration gradient, the positive electrode SoC isobtained by scaling Eq. (149) in Ref. [4] as

SoCpt csptcsp max C1pt 835

C1prt 1

105Its1pFN1p

: (37)

2.2. Electrolyte phase Li concentration in the electrodes and theseparator

The initial specic moles of Li, in the electrolyte phase of theelectrode and separator regions, are dened as

N2k lk2kc20; (38)

and the electrolyte diffusion time constants s2k l2k2kD2k

; (39)

where k n, s, p. It is to be noted that in Eq. (39), D2k is the effectivediffusivity that includes the effects of porosity and the Bruggemanfactor. Details of the scaling are given in the Notation section. Thetotal specicmoles of Li in the electrolytephase of the cell is given by

N2 N2n N2s N2p: (40)Electrolyte concentration eld is characterized by the electro-

lyte ux at the two electrodeeseparator interfaces: q2in(t) andq2ip(t). The coupled ODEs for these two internal variables in ROMare (Eqs. (83) and (84) in Refs. [4], along with the average reactionrates Eqs. (2) and (5) herein)

a11dq2indt

a12dq2ipdt

q2in 1 tItF

; (41)

and a21dq2indt

a22dq2ipdt

q2ip 1 tItF

; (42)

where the coefcients (having units of time) in these ODEs arewritten in terms of the scaled quantities as

a11 12N2nN2s

s2s s2s6 s2n

3

N2nN2

12N2nN2s

s2s s2n3

; (43)

a12 12N2nN2s

s2s s2s3 s2p

3

N2nN2

12N2nN2s

s2s; (44)

a21 12N2nN2s

s2s s2s6 s2n

3

N2pN2

; (45)

and a22 12N2nN2s

s2s s2s3 s2p

3

N2pN2

s2p3

: (46)

Rearranging Eq. (41) gives

V. Senthil Kumar et al. / Journal of Power Sources 248 (2014) 383e387 385


dq2ipdt

1a12

q2in a11a12

dq2indt

1a12

1 tF

It: (47)

Using this equation in Eq. (42) and rearranging gives

q2ip a22a12

q2in detaa12

dq2indt

a22a12

1 1 t

FIt: (48)

Differentiating the above equation with respect to time, andusing the current prole of Eq. (8) gives

dq2ipdt

a22a12

dq2indt

detaa12

d2q2indt2

a22a12

1 1 t

F

If Iitf ti

!:

(49)

Using the above equation coupled with Eq. (8) in Eq. (41) fol-lowed by simplication gives

detad2q2indt2

a11 a22dq2indt

q2in

1 tF

"Ii

If Ii

tf ti

t ti a12 a22#: (50)

This equation is a second order ODE with a time-dependentforcing function. Its complementary function is obtained from thehomogeneous part of the ODE as follows: Solving the quadraticequation

detal2 a11 a22l 1 0; (51)

gives the roots l1 and l2. Then, the complementary function is

qCF2int A expl1t B expl2t: (52)The particular integral is obtained as follows: Substituting the

linear trial function

qPI2int P Qt; (53)

in the ODE Eq. (50) and simplifying gives

a11 a22Q P Qt 1 t

F

"Ii

If Ii

tf ti

t ti a12 a22

#:

(54)Equating the coefcients of the linear and constant terms in

time gives

Q 1 tF

If Ii

tf ti

; (55)

and; P 1 tF

Ii ti a11 a12Q : (56)

The complete solution to the ODE Eq. (50) is the sum of thecomplementary function Eq. (52) and the particular integral Eq.(53)

q2int A expl1t B expl2t P Qt: (57)With P and Q determined from Eqs. (55) and (56), A and B are

determined from the initial value and the initial rate of change ofq2in, as shown below. Differentiating Eq. (57) gives

dq2indt

Al1 expl1t Bl2 expl2t Q : (58)

Using Eqs. (57) and (58) in Eq. (48) and simplifying gives

q2ip 1a12

a22 detal1A expl1t 1a12

a22

detal2B expl2t a22a12

P Qt detaa12

Q a22a12

1 1 t

FIt:

(59)

The initial values of q2in and q2ip are set to zero at the beginningof the simulation based on the assumption of equilibrated state. Atsuccessive time intervals they are obtained by their continuity intime i.e. their values at the end of the previous time interval are theinitial values for the successive time interval. Once the initial valuesof q2in and q2ip are known, the coefcients A and B can be obtainedby solving Eqs. (57) and (59).

3. Algebraic ROM algorithm

In this section an explicit algebraic reduced order algorithm isconstructed using the theoretical results collated in Section 7 of [4]and those derived in the earlier section of this work. The initial stateis assumed to be equilibrated i.e. the cell is rested until the con-centration gradients in the electrolyte and solid phases haverelaxed. The steps in the algorithm are:

1. Computation of average concentration based SoC in negativeelectrode: Eq. (10).

2. Computation of average radial gradient in negative elec-trode: Eq. (24).

3. Computation of SoC in negative electrode: Eq. (27).4. Computation of average concentration based SoC in positive

electrode using solid Li balance: Eq. (15).5. Computation of average radial gradient in positive electrode:

Eq. (33).6. Computation of SoC in positive electrode: Eq. (37).7. Computation of interfacial electrolyte uxes: Eqs. (57) and

(59).8. Computation of interfacial, average, collector-end and mid-

separator electrolyte concentrations using interfacialuxes: Eqs. (77), (78), (57), (74), (65), (60), (67), (76) of [4].

9. Computation of interfacial and collector-end electrolyte po-tentials: Eqs. (105), (106), (114) and (117) of [4].

10. Computation of collector-end solid potentials and hence thecell voltage: Eqs. (157) and (158) of [4].

This algorithm is used to compare the algebraic ROM predictionwith the MatlabeSimulink result of the original ROM of [4] for aconstant current discharge case in Fig. 1. This gure shows that theorder reduction to algebraic level is indeed exact at both nominaland high current discharge rates.

4. Conclusions

Real-time voltage prediction of lithium ion cells using theelectrochemical model is difcult owing to solving coupled partialdifferential equations. In an earlier publication [4] a reduced ordermodel (ROM) was developed by reducing the partial differential

V. Senthil Kumar et al. / Journal of Power Sources 248 (2014) 383e387386


equations to a set of ve linear ordinary differential equations andnonlinear algebraic expressions. This is achieved [4] by applyingorder reduction methodologies such as uniform reaction rateapproximation, volume averaging and prole based approxima-tions. In the present work, this model reduction approach is appliedfor any arbitrary current prole. An arbitrary current prole,involving charge, rest and discharge, is expressed as a series ofconstant current and linearly varying current periods. The latter ismore generic, since it includes the constant current period as well.The linear ordinary differential equations in ROM are solved usingthe analytical form of the linearly varying current and an explicitalgebraic algorithm is developed for lithium ion cell voltage pre-diction. Owing to the simplicity of the algorithm, it is amenable forimplementation on spreadsheets which can be used to validatelaboratory test data and can be extended to a battery managementsystem for onboard application in electric vehicles andsmartphones.

References

[1] M. Doyle, J. Newman, J. Electrochem. Soc. 143 (1996) 1890e1903.[2] V.R. Subramanian, V. Boovaragavan, V.D. Diwakar, Electrochem. Solid-State Lett.

10 (2007) A255eA260.[3] V.R. Subramanian, V.D. Diwakar, D. Tapriyal, J. Electrochem. Soc. 152 (2005)

A2002eA2008.[4] V. Senthil Kumar, J. Power Sources 222 (2013) 426e441.

[5] M.R. Spiegel, Theory and Problems of Advanced Mathematics for Engineers andScientists, Schaums Outline Series, McGraw-Hill Inc., 1971 edition, 24thprinting 1996.

Notation

Constants & variablesR: Gas constant, 8.314 J mol1 K1

F: Faradays constant, 96487 C mol1.T: Temperature, 298 K in this work.I (t): External current density at time t, A m2.Ii, If: External current densities at times ti and tf in a linearly varying current pulse,

A m2.t: Electrolyte transference number, dimensionless.brug: Bruggeman factor, used to calculate effective porous media properties,

dimensionless.ln,ls,lp: Thickness of negative electrode, separator and positive electrode, m.

L ln ls lp is total cell thickness.rn, rp: Radii of active material spheres in the negative and positive electrodes, m.1k,2k,fk: Volume fractions of active material, electrolyte and ller in electrode or

separator domains, dimensionless.1k 2k fk 1. Separator has no activematerial. Hence 1s 0.

ak 31k=rk: Specic surface area of active materials in n and p electrodes, m1.jk: Local surface reaction rate in k n, p electrodes, mol m2s1.jk0 kkcsk max csk0:5c0:5sk c0:52 : Over-potential independent rate pre-factor,

mol m2 s1.kk: Surface reaction rate constant in k n, p electrodes, (mol m2 s1)(mol m3)1.5.c1, c2: Solid phase concentration in active material spheres and electrolyte concen-

tration, mol m3.csk, csk max: Surface solid phase concentrations, and their maximum value, mol m3.C1k c1k=csk max: Scaled average solid phase concentrations in k n, p electrodes,

dimensionless.C1rk rkc1rk=cskmax: Scaled average radial gradient in k n, p electrodes,

dimensionless.c2ik,q2ik: Interfacial electrolyte concentraions and uxes, mol m3, mol m2 s1.D2: Electrolyte diffusivity (material property), m2 s1.D2k D2 brug2k : Effective electrolyte diffusivity in the porous k n,s,p domains.D1k: Solid diffusivity of k n,p electrode material spheres, m2 s1.k2: Electrolyte conductivity (material property), S m1.k2k k2 brug2k : Effective electrolyte conductivity in porous k n, s, p domains.N1k: Maximum specic moles of solid Li in k n,p electrodes, mol m2.N2k: Initial specic moles of electrolyte Li in k n, s, p regions, mol m2.N2 N2n N2s N2p: Total molar capacity of electrolyte Li in the cell, mol m2.sk: Electrical conductivity of k n, p electrode materials (material property), S m1s1k sk brug1k : Effective electrical conductivity in the porous k n, p domains,

S m1.F1, F2: Solid and liquid potential, V.s1k: Solid diffusion time constants in k n, p electrodes.s2k: Electrolyte diffusion time constants in k n, s, p domains.Uk: Open circuit voltage of k n, p electrodes, V.SoCk csk=cs max k: State of charge of k n, p electrodes, dimensionless.

Subscripts and operators:1, 2, f: Solid active material, liquid electrolyte phases and solid llers (including

additives)k n, s, p: Negative electrode, separator, and positive electrode.in: Separator interface of the negative electrode.ip: Separator interface of the positive electrode.CF: Complementary function of an ordinary differential equation.PI: Particular integral of an ordinary differential equation.0: Initial condition.r: Radial gradient.h$i: Volume average, dened in the three domains, n, s, p.:: Volume average within an active material sphere in n or p electrode.

2.9

3.1

3.3

3.5

3.7

3.9

4.1

0 500 1000 1500 2000 2500 3000 3500

Vo

ltag

e (V

)

Time (s)

1C Excel1C Simulink2C Excel2C Simulink5C Excel5C Simulink10C Excel10C Simulink

Fig. 1. Comparison of Excel implementation of algebraic ROM with the Simulinkimplementation of ODEs based ROM for constant current discharge at variousdischarge rates.

V. Senthil Kumar et al. / Journal of Power Sources 248 (2014) 383e387 387

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