A 1-D physical model of lithium-titanate battery forcontrol applications

Emil Namor, Rachid Cherkaoui, Dimitri Torregrossa, Mario PaoloneEcole Polytechnique Federale de Lausanne, Switzerland

Email: emil.namor@epfl.ch

Abstract—This document proposes a semi-physical model for alithium titanate battery. Its aim is to find a compromise betweenthe complex battery models based on PDEs and electrochemicallaws and the simple models needed in control applications. Inthis work this objective is fulfilled using a single particle model(SPM) and a series of reasonable simplifications derived fromobservation and experimental data. The parameter identificationprocedure is described and an experimental validation is carriedon.

I. INTRODUCTION

There has been a considerable effort in recent years towardsthe integration of physical models of lithium-ion batteries intheir control algorithms. The main reasons for this are:

1) the major accuracy of these models;2) their ability to estimate the internal states of the battery;3) their capacity to integrate aging phenomena within the

models and control strategies.The literature focuses on the reduced order modeling oflithium-ion batteries (LIBs) and whereas several methods havebeen described [1][2] still there are few examples of worksthat conjugate the model formulation with its parameter iden-tification and experimental validation. In the present work wepresent a model that although simpler that many presented inliterature, preserves its physical foundation and allows a plainparameter identification. The paper is structured as follows.In chapter II the model is described. The SPM on which itis based and the simplifications made are detailed. ChapterIII treats the parameter identification procedure. Chapter IVpresents the data of the experimental validation. ChapterV analyze the computational efficiency of the model andcompares it to that of a commonly used circuit model. Theconclusion sums up the document and underlines the meritswe see in this work.

II. MODEL OVERVIEW

A. The single particle model

The most widely used electrochemical model for lithium-ion batteries is the so called pseudo-2-dimensional modelfirstly presented in [3][4]. This model takes into accountthe reaction kinetics at the two electrodes, the insertion andconduction phenomena within them and the conduction anddiffusion of ions in the electrolyte through the thickness of thecell. The insertion phenomena are described through diffusionpartial differential equations along the radial dimension ofthe particles composing the electrodes, whereas the diffusion

Fig. 1. Scheme of the pseudo-2-dimensional model

and conduction in the electrolyte composing the electrodesthrough additional PDEs along an x dimension. These lastphenomena force to consider the insertion problem at everyx discretization point, hence multiplying the number of radialPDEs to consider and greatly increasing the complexity andthe computational burden of the model.

In first approximation, the inhomogeneities along the xdirection can be disregarded therefore representing the twoelectrodes through two particles of appropriate size and limit-ing the model to two diffusion PDEs, two surface reactionequations describing the reaction kinetics and a simplifiedconduction through the electrolyte law.

The reaction kinetics follows a Butler-Volmer equation thatcan be inverted as in eq. 1. Us and Ueq are respectively thepotential at the surface of the particle and the equilibriumpotential, R is the universal gas constant, T is the temperature,F is the Faraday constant, a is the area of the electrode, i0 theexchange current and Iapp the applied current. This leads to anon-linear relation between the overpotential at the surface ofthe particle due to the charge transfer process and the appliedcurrent. The equilibrium potential Ueq is defined through anempirical relation with the lithium ion stoichiometry on thissame surface.

η = Us − Ueq =2RT

Farcsinh(

Iapp2ai0

) (1)

Ueq = f(cs) (2)

The diffusion equation describes how the reacted lithiumions intercalate or de-intercalate in the active material ofthe electrodes. In the SPM, where the whole electrode isrepresented by one particle whose size is accordingly scaled,the whole current participate to the intercalation process,and the boundary condition at the surface of the particle istherefore defined as in eq. 4.

∂c

∂t=D

r

∂

∂r(1

r

∂c

∂r) (3)

With the boundary conditions:

∂c

∂r|r=0 = 0

∂c

∂r|r=R =

IappS

(4)

In eq. 4 r is the radial coordinate, R the particle radius,c the solid concentration, D the diffusion coefficient of theactive material and S the total surface (CONTROLLARE) ofthe electrode.

B. Simplifications based on physical observations

This work focuses on commercial batteries having NickelCobalt Aluminum Oxide (NCA) cathodes and lithium-titante(LTO) anodes and a nominal capacity of 30 Ah. Their specificchemistry allows some simplifications to the model describedabove. The LTO anode has a very flat open-circuit potential(OCP) curve in its functioning region [7], so its intercalationand de-intercalation processes do not affect considerably theoutput voltage. The diffusion problem related to the anodecan therefore be neglected causing little loss in accuracy anda considerable speed-up in computational time.

The voltage drops that this model keep into considerationare that relative to the electrolyte conduction Rcond and thecharge-transfer resistances Rct, that can be extrapolated fromthe Butler-Volmer relation in eq. (1). Experimental observa-tions led to consider the latter as the major reason for thevoltage drop during charges and discharges. In first approx-imation, and for low currents this relation can be linearized.The charge-transfer resistance Rct can be defined through theequation:

Rct =RT

nFi0(5)

where:i0 = k

√ce(cmax − cs)cs (6)

This equation holds for low currents and reflects a Tafelbehaviour.

Fig. 2. Open circuit potential curve for the anode (a) and for the cathode (b)

C. Model implementation

The model described here above has been implemented inMatlab R©. The diffusion problem has been discretized througha forward finite difference method, as in equation []. Theparticle radius has been discretized in 10 steps, whereas thetime steps have been adapted to the different simulationsexecuted, within the interval 1s - 10s.

ct+1i − cti

∆t=( D

∆r2+

D

r∆r)ct+1

i+1−2D

∆r2ct+1i +

( D

∆r2− D

r∆r)ct+1

i−1

(7)With the BCs:

ct+11 − ct+1

−1

∆r= 0

ct+1R+1 − c

t+1R−1

∆r=IappS

(8)

The resistance is defined in the general form in eq. 9, inorder to represent the two sources of voltage drop described inchapter II.b, i.e. the conduction in electrolyte and the chargetransfer. A, B and C are the parameters to be determinedthrough the optimization procedure.

Rcond+ct = A+B√

C(cmax − cs)cs(9)

III. PARAMETER IDENTIFICATION

The literature on LIB electrochemical modeling for controlpurposes focuses so far mainly on the model formulation andis less broad, to the best of the author’s knowledge, relatively

TABLE IEXPERIMENTAL VALIDATION SUMMARY

Cycle SOC0 LSME (V) Max error (%) EOS error (V)RCC 1 0.15 0.0092 0.072 0.0039RCC 2 0.22 0.0152 0.1358 0.0028RCC 3 0.34 0.0051 0.0062 0.0015

to the parameter identification and experimental validation.[5] uses a genetic algorithm procedure to characterize awhole P2D model, seeking 88 parameters. This is a quitecomplex procedure. [6] assumes instead that the knowledgeof certain values is given and limits the optimization to asmaller number of relevant parameters. In this paper thissecond procedure is utilized. Notably, the OCP curve is foundthrough experimental and manufacturer data. The radius of theactive material particles and the geometrical data of the cell aregiven by the manufacturer. The optimization’s duty is thereforeto determine the diffusion coefficient of the cathode and thevariables given in equation [] related to the internal resistanceof the battery. The optimization is carried out by seeking theminimum of a least square value produced by experimentaland simulation data.

minDs,Rct

ε =∑

(Vsim − Vmeas)2 (10)

The problem thus expressed is then solved through theMatlab R© function GA.

IV. EXPERIMENTAL VALIDATION (TEMPORARY)

The model is then validated experimentally. A series ofcharging-discharging cycles has been carried out on a 30 Ahcell kept at the constant temperature T = 20 ◦C thanks toa climatic chamber. These cycles are composed by constant-current and constant-voltage charges and discharges of variableduration and intensity. The starting point for these cyclesis at different SOCs. The cycles follow a period of rest ofseveral hours, so that the concentration profile can be assumedconstant and the initial SOC is known through the open circuitvoltage curve, and can be inserted as an initial condition inthe simulations. A further work will concern the estimation ofthe initial SOC in a non-rest condition. The cycle RRC 1 hasthen been used to fit the model, whereas the others have beenused to estimate its goodness.

The simulated voltage satisfactorily follows the measuredone on a timescale of several hours. The tab. I summarizes thecharacteristics of the various cycles and the obtained resultsin terms of maximum and mean error.

It is worth notice that two weaknesses are for the momentevident: the spikes at the beginning of the charges and thedrift of the voltage at very high SOC. This is due to the highgradient of its curve at high SOCs and therefore to its highsensitivity with regard to the parameters.

V. COMPUTATIONAL EFFICIENCY

In this section the computational efficiency of the model iscompared to that of a circuit model.

Fig. 3. Results for the cycle RCC1. (a) applied current (b) measured (blue)and simulated (red) voltage (c) error

Fig. 4. Results for the cycle RCC2. (a) applied current (b) measured (blue)and simulated (red) voltage (c) error

Fig. 5. Results for the cycle RCC3. (a) applied current (b) measured (blue)and simulated (red) voltage (c) error

VI. CONCLUSION

A simplified single particle model has been proposed for thesimulation of a commercial lithium-titanate cell. The modelhas been parametrized through an optimization procedure andvalidated on a series of voltage profiles. A good performancehas been observed. Further work will concern the estimationof the SOC and internal states, the utilization of this model in acontrol algorithm and its improvement through the integrationof an aging model.

REFERENCES

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