Electrochemical Modeling of a Single Li Ion Cell by Martin Sommer

8/10/2019 Electrochemical Modeling of a Single Li Ion Cell by Martin Sommer

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Institute of Electrical Measurement and Measurement Signal Processing

Electrochemical Modelingof a single LiIon cell

Martin Sommer

Konstanz, Marz 4, 2010

Martin Sommer Konstanz, Marz 4, 2010 Electrochemical Modeling

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Overview

Modeling approaches Equivalent circuit model

Electrochemical model

Mathematical Model of the electrochemical system

Numerical realization

Method

Discretization in space

Discretization in time

Simulation Results

Problems


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Equivalent Circuit Model I

Pros and Cons

+ small set of parameters

+ straight-forward implementation

+ effects can be allocated to a frequency band

limited to one operating point

many measurements required for coupling of these points

only the current input/voltage output characteristic can bemonitored/displayed


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Equivalent Circuit Model II

Abbildung 1:Simple RCmodel with additional impedance ZW for lowfrequencies.

Abbildung 2:Nyquist plot of a LiIon battery showing a characteristic halfcircle and a 45 straight line.


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Electrochemical Reactions I

Abbildung 3:Electrochemical reaction during discharge of the cell.


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Electrochemical Reactions II

Reaction in the Negative during discharge:

LixC6 C6+xLi+ +xe

Reaction in the Positive during discharge:

Li1xFePO4+xe

+xLi+

LiFePO4


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Mathematical Formulation I

Charge balance in the solid phase:

(s) = j (1)

Charge balance in the fluid phase:

((ce)e) +

2(ce)RTF

ln ce

= j (2)

ButlerVolmer equation (charge transfer):

i0 =ak

ce(cs,max cs,surf)cs,surf (3)

j=i0

eFRT

(seUOCV) e(1)F

RT (seUOCV)

(4)


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Mathematical Formulation II

Concentration in the solid phase:

cs

t =Ds

1

r2

r

r2

cs

r

(5)

Concentration in the fluid phase:

(ce)

t = (Dce) +

1 t+F

j (6)

Taking the porosity of the electrodes into account often theBruggemannrelations:

eff =brugs , eff =brug and Deff =Dbrug are used.


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Boundary Conditions

Charge balance in the solid phase:

s=i fur x= 0, x=L

Charge balance in the fluid phase:

((ce)e) +2(ce)RT

F ln ce

= 0 fur x= 0, x=L


Dscsr |r=0 = 0 , Ds

csr

r=Rs = jaF


ce= 0 fur x= 0, x=L

Additionally a reference potential has to be defined! (e.g. s(0) = 0)


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Initial Conditions


cs(r, 0) =cs,0


ce(x, 0) =ce,0


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Modifications of the Equation System I

Modification of the fluid potential after [1]:

e= e 2(ce)RT

F ln ce

therby the charge balance in the fluid results as follows

((ce)e) =j (7)

as well as

j=i0

e

FRT

(s(e+2(ce)RT

F ln ce)UOCV)

e(1)F

RT (s(e+

2(ce)RTF

ln ce)UOCV) (8)

for the ButlerVolmer equation.


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Modifications of the Equation System II

Modification of the concentration in the solid phase by avariable substitution after [2]:

v=rcs (9)

Thus the equation for the concentration in the solid phasecan now be written in the new coordinates as follows

v

t =

r

Ds

v

r

(10)

Not to forget the transformation of the boundary condition,because the equation is no longer associated to a sphere!

Due to these modification the equations can no be written as a system of linear differential equations

coupled by an algebraic one.


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The Control Volume Method I

CVM on the example of the charge transfer in the solid phase[3]:

(s) = j (11)

The integration of the equation for the 1D case yields

dsdx

e

dsdx

w

=

ew

j dx . (12)

This equation can be discretized as follows

e(s,E s,P)

(x)e

+

w(s,P s,W)

(x)w

= j x . (13)


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The Control Volume Method II

This brings us to a linear equation for each discretization point

aPs,P=aEs,E+aWs,W+b (14)

with the coefficients

aE = e(x)w

, aW = w(x)w

, aP=aE+aW and b=j x .

The resulting set of equations is symmetric showing a tridiagonalstructure. Systems of that kind can be solved by simple algorithms.


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The Control Volume Method III

Abbildung 4:Discretization of the charge balance in the solid phase after [3].

Basically this method solves bilance equations, therefore theconservation equations are always fulfilled.

You have to consider that the boundary elements represent only halfcontrol volumina!


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I i f El i l M d M Si l P i


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The Control Volume Method IV

Abbildung 5:Possibilities for the discretization of the profiles; left: stepprofile, right: piecewise linear profil after [3].

For all calculations the step profile was used. This holds for the sourceterm linearization as well.


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I tit t f El t i l M t d M t Si l P i


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The Control Volume Method V

Time discretization on the example of the modified concentration

equation in the solid phase (fully implicit scheme):

v

t =

r

Ds

v

r

(15)

(v1P v0P)t

r=

Ds,e(v1E v1P)(r)e

Ds,w(v1P v1W)(r)w

(16)

with the new coefficients for the equation system

aE = Ds,e

(r)w , aW = Ds,w

(r)e , a0P= rt, aP=aE+aW+a0P

and b=a0Pv0P .

This discretization has the advantage that its always stable!


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The Finite Difference Method

Discretization on the example of the concentration in the solidphase after [4]:

cs

t =Ds1

r2

r

r2cs

r

(17)

cs,P

t =

Ds,e

r(r)e

r2er2p

(cs,E cs,P) Ds,w

r(r)w

r2wr2p

(cs,P cs,W) (18)

This discretization leads to an asymmtric system matrix!

Martin Sommer Konstanz, Marz 4, 2010 Electrochemical Modeling18 / 31



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Implementation in Matlab R I

Description of the whole system as a pseudo2D model:

Abbildung 6:Discretization of the equations in pseudo2D coordinates.

Discretization in x: Nn = 5, Nsep= 3, Np= 7, scale = 1.3

Discretization in r: Np n= 50, Np p= 5, scale = 1




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Implementation in Matlab R II

Time discretization with variable time steps:

For the reduction of computing time the following algorithm after[5] with an error measurement and a reference gwas used.

if < 12 g

accept solution + t= t 1,5elseif 12 g < g

accept solution + t= t

elseif g 2g

accept solution + t= t 0,5elseif >2g

decline solution + t= t 0,5

end




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Implementation in Matlab R III

The error measurement is determined as follows

=vf vc2

vf2, (19)

with vf being calculated using t

2

and vcusing tfor the step size.

For the diffusion equations in the solid phased

g = 1e2 was used.

For the concentration in the liquid phaseg = 1e

3 was used.

The step size can vary in the interval1 ms


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g g

Implementation in Matlab R IV

Abbildung 7:Error due to nonlinearity in tafter[6].

Strong nonlinearities can lead to errors using a coarse step size!




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g g

Implementation in Matlab R V

Calculation of the potentials using an extended Newtonmethod

regarding to [7]:

Abbildung 8:Current distribution between the solid and the liquid phase forthe calculation of both potentials in the negative electrode.




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Implementation in Matlab R VI

For the negtive electrode the potential e(0) is varied until the cur-rent is transferred from to solid to the liquid phase or vice versa.

i=

jdx=

k

jkxk (20)

For the positive electrode the reaction runs into the opposite direc-tion. The only difference is now that e(Ln+Lsep) is known ands(Ln+Lsep) has to be determined.

stopping criteria:(i

k

jkxk) 1e6, maxiter= 500, smin= 1 pV, smax p= 50 mV,

smax n= 100 mV .




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Simulation Results I




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Simulation Results II




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Problems

Dealing with source term in potential equations (iterativemethods)

Discretization dependencies (not known)

Low i0 causes problems

Number of parameters

Errors

Physics Mathematical Formulation

Realization




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Thank You For Your Attention!




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Literatur I

J. Wu, J.Xu, H. Zou: On The Well Posedness Of A MathematicalModel For LithiumIon Battery Systems, Methods andApplications of Analysis, vol. 13, no. 3, 2006, S. 275298

A. J. Bard, L.R. Faulkner: Electrochemical Methods:

Fundamentals and Applications, John Wiley & Sons, 2nd edition,2001, S. 165

S. V. Patankar: Numerical Heat Transfer and Fluid Flow, series incomputational methods in mechanics and thermal sciences, 1980

L. Cai, R. E. White: Reduction of Model Order Based on ProperOrthogonal Decomposition for LithiumIon Battery Simulations,Journal of The Electrochemical Society, vol. 156, no. 3, 2009,S. 154161




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Literatur II

S. E. Minkoff, N. M. Kridler: A comparison of adaptive timestepping methods for coupled flow and deformatio modeling,Applied Mathematical Modeling, vol. 30, no. 9, 2006, S. 993-1009

E. Kreyszig: Advanced Egeneering Mathematics, John Wiley &Sons, 7th edition, 1993, S. 1035 ff.

B. Schweighofer: Simulation of the Dynamic Behavior of aLeadAcid Battery, Dissertation an der TUGraz, 2007



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The author wishes to thank the COMET K2 Forschungsfrderungs-Programm of

the Austrian Federal Ministry for Transport, Innovation and Technology (BMVIT), theAustrian Federal Ministry of Economics and Labour (BMWA), sterreichischeForschungsfrderungsgesellschaft mbH (FFG), Das Land Steiermark, andSteirische Wirtschaftsfrderung (SFG) for their financial support.Additionally, we would like to thank the supporting companies and project partners

as well as

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Electrochemical Modeling of a Single Li Ion Cell by Martin Sommer