Advances in Electrochemical Models for Predicting the Cycling … · Advances in Electrochemical Models for Predicting the Cycling Performance of Traction Batteries: Experimental

Advances in Electrochemical Models for Predictingthe Cycling Performance of Traction Batteries:Experimental Study on Ni-MH and Simulation

J. Bernard1, A. Sciarretta2, Y. Touzani2 and V. Sauvant-Moynot1

1 Institut français du pétrole, IFP-Lyon, Rond-point de l'échangeur de Solaize, BP 3, 69360 Solaize - France2 Institut français du pétrole, IFP, 1-4 avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex - Francee-mail : [email protected] - [email protected] - [email protected] - [email protected]

Résumé — Développement de modèles électrochimiques de batteries de traction pour laprédiction de performances : étude expérimentale de batteries NiMH et simulations — Desmodèles électrochimiques fins permettant de simuler le comportement de batteries ont été développésavec succès et reportés dans la littérature. Ils constituent une alternative aux méthodes classiques pourestimer l’état de charge (SoC pour State of Charge) des batteries, cette variable étant ici un paramètreinterne du modèle physique. Cependant, pour les applications embarquées, il est nécessaire dedévelopper des modèles simplifiés sur la base de ces modèles physiques afin de diminuer le temps decalcul nécessaire à la résolution des équations. Ici, nous présenterons à titre d’exemple un modèleélectrochimique 0D avancé d’un accumulateur NiMH et sa validation. Ce modèle à paramètresconcentrés sera utilisé pour réaliser un filtre de Kalman qui permettra la prédiction de l’état de charged’un pack complet. Une étude expérimentale d’accumulateurs NiMH permettra de mieux comprendreles mécanismes physico-chimiques ayant lieu à chaque électrode et ainsi d’alimenter le modèlephysique en informations. La dernière partie de cet article sera consacrée à la validation du modèle parcomparaison à des données expérimentales obtenues sur cellule individuelle mais également sur un packbatterie NiMH commercial complet.

Abstract — Advances in Electrochemical Models for Predicting the Cycling Performance of TractionBatteries: Experimental Study on Ni-MH and Simulation — Rigorous electrochemical models tosimulate the cycling performance of batteries have been successfully developed and reported in theliterature. They constitute a very promising approach for State-of-Charge (SoC) estimation based onthe physics of the cell with regards to other methods since SoC is an internal parameter of thesephysical models. However, the computational time needed to solve electrochemical battery models foronline applications requires to develop a simplified physics-based battery model. In this work, our goalis to present and validate an advanced 0D-electrochemical model of a Ni-MH cell, as an example. Thislumped-parameter model will be used to design an extended Kalman filter to predict the SoC of a Ni-MHpack. It is presented, followed by an extensive experimental study conducted on Ni-MH cells to betterunderstand the mechanisms of physico-chemical phenomena occurring at both electrodes andsupport the model development. The last part of the paper focuses on the evaluation of the model withregards to experimental results obtained on Ni-MH sealed cells but also on the related commercialHEV battery pack.

Oil & Gas Science and Technology – Rev. IFP, Vol. 65 (2010), No. 1, pp. 55-66Copyright © 2009, Institut français du pétroleDOI: 10.2516/ogst/2009060

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INTRODUCTION

The technical and commercial success of Hybrid-ElectricVehicles (HEV) is strongly related to the effectiveness of thevehicle supervisory controller that manages the power flowsbetween the multiple energy sources. Regardless of the con-trol strategy adopted, one essential information to the super-visor is the state of the energy storage system. This quantityrelated to the concentration of reactive species inside thebattery cells is not directly measurable on board. Given thevery specific environment of traction batteries [1, 2] used inHEV possibly highly dynamic and power demanding (cur-rent usually reaches intensive C-rates in charge or dischargeup to ± 20 C), in addition covering a large temperaturerange, SoC estimation is a rather difficult task.

Numerous approaches have been studied to monitor theState-of-Charge (SoC) of a cell [3, 4]. The coulomb-count-ing procedure is not reliable for a precise SoC determina-tion in traction batteries since not all current supplied goesto charging the cell and the charge lost to any undesiredcharge reaction is not included. In order to take into accountthe typical battery usage in an automotive context, moreprecise techniques of estimation are required. In this frame-work, a clear trend in the last years is toward the use ofmathematical models of the battery systems to inspiread-hoc estimation strategies.

Among the most used method of SoC determination, theparameterization of the battery electro-motive force (emf) asa function of SoC could in principle provide information onthe SoC from an estimation or a measurement of the open-circuit voltage. In particular, the emf of Li-ion battery incombination with the coulomb-counting approach has beendemonstrated to be a good indication of SoC [5,6], since theemf-SoC dependency does not change with cycling, if theSoC is always referred to the actual battery capacity. In con-trast, there is no direct relationship between emf and SoC forNi-MH, since a typical hysteresis phenomenon is prominentin this case. Another main drawback of this estimation tech-nique is that the measurement of the emf needs long resttime, during which the battery is not yielding current –which is hardly achievable in HEV operation.

Methods employing electrical circuit analogy are alsowidely studied. Electrical models are electrical equivalentmodels made of a combination of voltage sources, resistorsand capacitors. Equivalent-circuit models can be further dif-ferentiated between black-box models in the form oflumped-parameter dynamic electric equivalent circuits, andgrey-box models, still in the form of equivalent electric cir-cuits, but where the circuit elements try to reproduce theinner electrochemical behavior of the cells. Many electricalmodels of batteries have been presented in the literature, forlead-acid [7], Ni-MH [8, 9] and Lithium-ion batteries [10],since they are intuitive, easy to handle and useful as batterysimulator to design hybrid architecture [11]. Those models

rely on the collection of appropriate look-up tables from celldata. This requires generating enough test points to create aseries of curves which can be broken down into intervals toestimate the curvature with a stepwise approximation. Thismethod cannot offer the best accuracy to estimate the SoCwhile dealing with spread in both battery and user behav-iour, large temperature and current range, and ageing of thecells under all realistic user conditions. In addition, equiva-lent circuit models are tracking a parameter to estimate theSoC which has often little physical significance. In conclu-sion, the results of electric equivalent circuit models arenowadays of limited value in predicting the SoC of a HEVbattery online.

Electrochemical models, as an alternative approach toequivalent circuit based models, characterise the fundamen-tal mechanisms of power generation and relate batterydesign parameters with macroscopic (e.g. battery voltageand current) and microscopic (e.g. concentration distribu-tion) information. They constitute a very promisingapproach with regards to other methods since SoC is aninternal parameter of these physical models. Sophisticatedand simpler models were presented as valuable physicalmodels to predict the SoC and other performance of bothNi-MH cells [12-16] and lithium ion cells [17-20]. However,the computational time needed to solve rigorous electro-chemical battery models for online applications requires todevelop a simplified physics-based battery model with spe-cific efforts to account for the diffusion phenomena. Thoselater are not taken into account yet in 0D-model, whichalters their predictability in power applications [21].

The objective of this paper is to present and validate anadvanced 0D-electrochemical model of a Ni-MH cell.Attempts are made to extend the model to describe a batterypack as well. This lumped-parameter model will be used infuture work to design an extended Kalman filter to predictthe SoC of a Ni-MH cell. The second part of the paper isdedicated to the development of an advanced lumped-para-meter electrochemical model in the AMESim simulationenvironment, with specific efforts to account for the diffu-sion phenomena. An extensive experimental study con-ducted on sealed Ni-MH cells to better understand the mech-anisms of physico-chemical phenomena occurring at bothelectrodes, and support the model development is presentedin the first part of the paper. Electrochemical ImpedanceSpectroscopy (EIS) with a 3-electrode set-up has been usedfor this purpose over a large temperature range (– 20°C to35°C), pointing out the large contribution of diffusion phe-nomena. The last part focuses on the comparison betweenexperimental and simulation results, looking at charge anddischarge curves obtained on Ni-MH cells but also on EISspectra thanks to an original co-simulation model(AMESim/Simulink). The discussion is completed withregards to experimental data collected on a commercialHEV pack as well.

56


1 EXPERIMENTAL

The experimental work was performed on a commercialHEV Nickel/metal hydride battery pack (168 cells, 202 Vnominal, 6.5 Ah). Sealed Ni-MH cells (1.2 V nominal, 6.5 Ah)were extracted from the battery pack to conduct charge/discharge experiments. A sealed nickel/metal hydride batterycell is composed of four regions (Fig. 1a): a negative elec-trode (Metal Hydride MH), a positive electrode (porousnickel oxyhydroxide NiOOH), a separator (porous nylonmaterial, typically) and a gas reservoir. Both electrodes areelectronic conducting porous materials with constant porosityflooded with the electrolyte (concentrated KOH aqueoussolution) which fully impregnates the separator as well. Eachcell was characterised in terms of capacity and physical para-meters such as electrode and separator nature, thicknessesand electrolyte concentration to provide related parametersfor the simulation and try to duplicate exactly the runs withthe model. However, other microscopic geometrical parame-ters were selected according to the literature [13, 22]. Underthis hypothesis, nickel oxide electrode consists of compositecylindrical needles with a substrate inside. The MH electrode(LaNi5) consists of spherical particles (powders with uniformsize) of the metal-hydride material only, without anydeposited metal layer on their surface.

Charge/discharge cycling on a cell was performed with aVMP3 multi-channel potentiostat from Biologic ScienceInstruments. Tests were conducted in a climatic chamberallowing measurements from –40°C to 180°C. Each channel

is equipped with the Electrochemical Impedance Spectroscopy(EIS) option allowing independent EIS measurements on eachchannel. EIS consists of the application of a current/voltageexcitation waveform to the system and the monitoring of thesystem’s voltage/current response. In this work, all EIS mea-surements were done in a galvanostatic mode. The currentamplitude was adapted to obtain a potential amplituderesponse under 10 mV cc. This amplitude and also the station-arity of the response signal were controlled using an oscillo-scope. Impedance diagrams were plotted between 10 kHz and50 mHz.

The application of EIS with a 3-electrode set-up to Ni-MHcell in function of current, temperature and during representa-tive charge cycles will help to determine electrochemicalphenomena occurring at both electrodes [23]. To build a3-electrode set-up, a hole was made in the upper part of thecell and half of the gas reservoir was filled with an electrolyticsolution (KOH 6 mol/L). A platinum wire was then insertedin the hole and immersed in the electrolyte, a part of the wirestanding higher than the top of the cell to allow an externalconnection (Fig. 1b). The platinum wire immersed in a KOHsolution with constant concentration constitutes a quasi-refer-ence electrode. The stability of its potential versus a commer-cial Hg/HgO was verified in isothermal conditions. Finally,a resin was used to fix the platinum wire and ensure that thecell was sealed.

Different connection types were used:– the complete cell was studied with a 2-electrode configu-

ration: the Working Electrode (WE) gives the potential ofpositive electrode, the Reference Electrode (RE) and theCounter Electrode (CE) are connected to the negative;

– the positive electrode was investigated with a 3-electrodeconfiguration: WE to the positive, RE to the platinum wireand CE to the negative;

– the negative electrode was investigated with a 3-electrodeconfiguration as well: WE to the positive, RE to the platinumwire and CE to the positive.Charge/discharge cycles following the Hybrid Pulse

Power Characterisation test [24] (HPPC) were also appliedon the full battery pack which contains 168 sealed Ni-MHcells connected in series. A Digatron 500 A/500 V test benchwas used for this purpose at 20°C. The HPPC test is pro-posed in the FreedomCar US Department of Energy programto demonstrate the discharge pulse and regenerative pulsepower capabilities of a HEV battery at various Depth ofDischarge (DoD) values for both the Minimum andMaximum Power-Assist goals.

2 FORMULATION OF THE 0D-MODEL

A Ni-MH cell is a dual-intercalation electrochemical systemin which proton insertion in the positive solid nickel electrodeand hydrogen de-insertion in the negative metal/hydride

J Bernard et al. / Advances in Electrochemical Models for Predicting the Cycling Performance of Traction Batteries: Experimental Study on Ni-MH and Simulation

57

Gas reservoir Gas

Electrolyte

reservoir

Platinumwire

a) b)

WE RE CE WE RE CE

Positiveelectrode

Separator Separator

Positiveelectrode

Negativeelectrode

Negativeelectrode

Figure 1

Schematic view of a commercial NiMH cell a) beforemodification and b) after insertion of the quasi-referenceelectrode.



electrode occur during discharge and vice versa duringcharge. Electrochemical reactions taking place at the elec-trode/electrolyte interface (assuming discharge) are:

NiOOH(s) + H2O(l) + e- ↔ Ni(OH)2(s) + OH-(aq) (1)

1/2 O2 + H2O + 2e- ↔ 2OH- (2)

at the positive electrode and:

M(s) + H2O(l) + e- ↔ MH(s) + OH-(aq) (3)

1/2 O2 + H2O + 2e- ↔ 2OH- (4)

at the negative electrode. Side Equations (2) and (4) constitute the oxygen recirculation

inside the cell. The oxygen is transported from one electrodeto the other via the liquid electrolyte and, after exceeding itssolubility limit in the electrolyte, via a gas phase. Thus aNi-MH cell is a three-phase system, with one gas phasebesides the solid matrices and the liquid electrolyte. Theaccumulation of oxygen in the gas phase increases the cellpressure in the gas reservoir above the cell (constant volume).

The modelling of such a system can be very complex ifthe spatial distribution of all the species and phases are rep-resented, since the relevant mass and energy conservationlaws have to be applied locally for a three-dimensionaldomain. Several techniques have been proposed in order toreduce the complexity of this approach. The pseudo-two-dimensional method [16, 25] considers spatial variations ofconcentrations, potentials, etc., along a main spatial x-direc-tion and two fictitious y-directions. The main x-axis encom-passes the two porous electrodes and the separator and it isbounded by the battery terminal plates. The two y-axes rep-resent diffusion within fictitious solid particles in each elec-trode. These fictitious particles have a spherical geometry inthe MH electrode and a cylindrical geometry in the nickelelectrode. A main simplification of this approach (diffusionlength approach) consists in neglecting the diffusion alongthe y-directions, except for a localized concentration discon-tinuity between the bulk of the fictitious particle and itsinterface with the electrolyte. Computationally, the modelthus reduces to a one-dimensional (1D) domain [13].

Even if simplified to a single dimensional domain, distributed-parameter electrochemical battery models are of limitedvalue in predicting the SoC of a battery on-board due to theexcessive computational time required. Nevertheless, besidesbeing useful for physical understanding and diagnostics, theycan be used to assess the prediction capability of a simplerbut computationally efficient approach, the so called lumped-parameter electrochemical model.

The lumped-parameter description is based on the assumptionof considering the concentrations of the five species ashomogeneous within the battery, i.e., the planar electrodeapproximation [14, 15]. The reduction of the number of statevariables is usually achieved with a simplification that con-sists of evaluating the concentration of NiOOH as a function

of nickel hydroxide concentration. Moreover, the concentra-tion of the OH- ions in the electrolyte is taken as a constant.Consequently, the model state equations are given by themass balance equations of the three remaining species. Themass flows involving the three species are related to the cur-rent densities of the main and side reactions shown above.The four current densities are calculated using the Butler-Volmer equations as a function of the species concentrations,the potential differences at the solid-liquid interface on theelectrodes, and the equilibrium potentials (open-circuit poten-tials) of the reactions. Current balance equations provide themissing relationships between the reaction current densities,the potential differences, the cell voltage, and the cell current.Moreover, the equilibrium potential of the nickel reaction canbe calculated in such a way to account for the typical hysteresiseffect of Ni-MH batteries.

A comparison between 1D and 0D electrochemical modelstaken from the literature for a Ni-MH battery cell has beenrecently presented, including predictions of the terminalvoltage along a given current profile [21].

Although the main cell dynamics are qualitatively reproducedby the 0D model, nevertheless the comparison has shownthat, contrarily to what often stated in the literature, the 0Dapproach introduces substantial quantitative differences withrespect to its 1D counterpart. These differences are mainlydue to the fact that literature 0D models renounce represent-ing the diffusion processes. Indeed, literature 0D models onlyinclude three state variables for a single cell, while a typical1D model requires more than thirty dynamic variables.

2.1 Assumptions

An improved electrochemical lumped-parameter model for asealed Ni-MH cell is proposed in order to better approach theoutcome of 1D models, while keeping the same level of com-plexity of standard 0D models. The main assumptions arelisted below:– the concentration of active species is homogeneous in

each of the cell regions, i.e., spatial gradients areneglected. Thus the concentrations of active species,namely hydrogen in metal hydride material, hydrogen inthe Ni(OH)2 material), oxygen dissolved in the liquidphase, gaseous oxygen and electrolyte are only functionsof time;

– a double-layer capacitance is introduced in the chargebalance equation to account for the electrolyte ionicspecies accumulation at both electrode interfaces due tothe diffusion gradient;

– the diffusion into a single solid particle is accounted forby an equation based on the superposition integral [12].Then the bulk concentration is distinguished from theinterfacial concentration for each species, both beingfunctions of time;

58


– liquid/gas interfacial equilibrium for the oxygen specieslinks the dissolved oxygen concentration to the gaseousoxygen pressure via the Henry law, while the ideal gaslaw applies in the reservoir;

– potential hysteresis is considered;– thermal effects are adequately taken into account by intro-

ducing an energy balance equation for the whole cell.Heat production due to irreversibility in the reactions, heatlosses through the cell external surface, as well as energyaccumulation in the cell are the terms of such equation.Moreover, cell temperature affects the reaction kinetics.

2.2 Equations

The kinetics of reactions (1-4) are derived from the generalButler-Volmer equations with respect to a specified referencestate:

(5)

(6)

(7)

(8)

where cn is the proton interfacial concentration in nickelhydroxide, ce is the constant concentration of KOH elec-trolyte representing the concentration of OH- ions, c0 is theconcentration of dissolved oxygen, cm is the interfacial con-centration of hydrogen in metal hydride material and μ itsstoichiometric coefficient. The latter is variously calculatedin the literature, and corresponds to 1/6 for LaNi5H6 [13]. In

J t Jc

cee

e ref

K t4 4 0

2

1 5 4( ) ,,

. ( )=⎛

⎝⎜⎜

⎞

⎠⎟⎟

η −−⎧⎨⎪

⎩⎪

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎫

⎬⎪

⎭

−c t

ce

ref

K t0

0

1

20 5 4

( )

,

. ( )η

⎪⎪−

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟exp ,E

R T Ta 4

0

1 1

J t Jc

c

c

cm

m ref

e

e ref

3 3 0( ) ,, ,

=⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

μ

ee eK t K t0 5 0 53 3. ( ) . ( )η η−⎫⎬⎪

⎭⎪

⎧⎨⎪

⎩⎪

−

× −⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟exp ,E

R T Ta 3

0

1 1

J t Jc

cee

e ref

K t2 2 0

2

1 5 2( ) ,,

. ( )=⎛

⎝⎜⎜

⎞

⎠⎟⎟ −η

⎧⎧⎨⎪

⎩⎪

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎫

⎬⎪

⎭⎪

−c t

ce

ref

K t0

0

1

20 5 2

( )

,

. ( )η eexp ,E

R T Ta 2

0

1 1−

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

J t Jc t

c

c

cn

n ref

e

e ref

1 1 0( )( )

,, ,

=⎛

⎝⎜⎜

⎞

⎠⎟⎟⎛⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎧⎨⎪

⎩⎪−

−

e

c c t

c

K t

n n

n

0 5 1. ( )

,max

,max

( )

η

−−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎫⎬⎪

⎭⎪−−

ce

E

R Tn ref

K t a

,

. ( ) ,exp0 5 111η 11

0T

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟

the above equations, η’s with various superscripts stand forthe surface overpotentials:

(9)

where ΔΦpos and ΔΦneg are the potential differences at thesolid-liquid interface on the positive and negative electrodes,respectively. The latter terms are conveniently parameterisedas a function of the species concentrations and the tempera-ture. The typical hysteresis behavior of Ni-MH batteries canbe conveniently simulated by differentiating Ueq,ref,1 betweencharging and discharging and relaxing the switching with anexponential term:

(10)

For example, the values used in Reference [15] areUeq,ref,1,d = 0.427 for discharging and Ueq,ref,1,c = 0.527 forcharging.

Conservation equations in the lumped-parameter approachare derived from traditional transport equations modelledusing concentration solution theory [26]. The basis of con-centrated solution theory is that the gradient in the electro-chemical potential is the driving force for mass transfer. Themass concentration of species i in a general form is written as:

(11)

where ci is the concentration of the i-th species, Ni its fluxdensity (mol/(cm2.s)), and Ri its generation rate (mol/(cm3.s)).

Applying the general volume-averages conservationequation of species and making use of the interfacial balancecondition to active species in the finite-volume approachyield:

(12)

for the electrolyte concentration, which is constant in the0D-approximation:

(13)

(14)

where r is the radius of the MH particles (mass conservationwritten in a cylindrical geometry) and y(1) is the thickness

dc

dt y

J

Fn = −

1

1

1

( )

dc

dt r

J

Fm = −

2 3

εeedc

dt= 0

ε( )ki

i i

c

tN R

∂∂

= −∇ +

dU

dtU U k I Ieq ref

eq ref eq ref, ,

, , , , ,( ). ( ).11 1= −∞

kk Ik I

k I

UU

c

d

eq ref

eq ref

( ),

,

, , ,

, , ,

=>

<

⎧⎨⎩

=∞

0

0

1

1 cc

eq ref d

I

U I

,

,, , ,

>

<

⎧⎨⎪

⎩⎪

0

01

η

η1 1

2

( ) ( ) ( )

( ) ( ),t t U t

t t Upos eq ref

pos e

= −

= −

ΔΦ

ΔΦ qq ref

neg eq ref

t

t t U t

t

,

,

( )

( ) ( ) ( )

( )

2

3 3

4

η

η

= −

=

ΔΦ

ΔΦΦneg eq reft U t( ) ( ),− 4


59



of active materials in the nickel electrode surrounding thesubstrate along the central axis of the cylinders.

To take into account in a lumped-parameter way the diffusionin the solid phase, the concentrations calculated withEquations (13-14) can be distinguished from the concentra-tions used in the Butler-Volmer equations. Inspired by thediffusion length approach where the diffusion rate is constantalong a diffusion length lse across the characteristic geometry[13], the rules to differentiate the average concentration fromthe interfacial concentration are:

(15)

(16)

where Dh is the diffusion coefficient of hydrogen, ci is theaverage concentration of species i and ci with bar is the inter-facial concentration, which is also used in the Butler-VolmerEquations (5) to (8).

The concentration of the oxygen in the gaseous phase isaveraged in (17) as in literature Reference [13], while theconcentration of oxygen in the liquid phase is assumed tovary in a quasistatic way, thus:

(17)

(18)

with the latter being used in the Butler-Volmer Equations (6),(8).

The way proposed to include the effect of the double-layercapacitance on the electrode kinetics is as follows. Equation(5) is substituted by (19):

(19)

where f(·) is the right-hand side of Equation (5). Similarly,(7) is substituted by (20):

(20)

where g(·) is the right-hand side of Equation (7). The twocapacitances per surface unit C(1) and C(3) might be differentin principle. The model is completed by a voltage balanceacross the cell including the Ohmic resistance (mainly of theelectrolyte), Rint:

(21)

Thus, the system is described by the five non-linear differentialEquations (13), (14), (17), (19), and (20) in the five

V t t t R I tpos neg( ) ( ) ( ) ( )int= − +ΔΦ ΔΦ

J g c Cd

dti3 3 33= +( , ) ( )η

η

J f c Cd

dti1 1 11= +( , ) ( )η

η

c t c tFK

a J t k

k

a

k

tran

0 0

1 2

3

1

4

1

0 2( )

( )

( )

( ) ( )

( )

= +

=

=

JJ t k4 3( ) =

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

d

dtp t

Rv

V

A l a J t A l ab

g

01 1 1 2 3 3( )

( )( ) ( ) ( ) ( ) ( ) (= −+ 33 4) ( )J t

F

c cl J

FDm mse

h

= − 3

c cl J

FDn nse

h

= − 1

unknowns cm, cn, p0, ΔΩpos and ΔΩneg. Algebraic equationslike (18) and (5-8) introducing new intermediate variablescomplete the model.

Temperature can be calculated from an energy balance ofthe whole cell. The heat flux generated by the cell is given by:

(22)

where the term Ueq,ref,z – V can be associated to the irre-versible losses for each reaction z, while the reversible gener-ation term T × dUeq,ref,z/dT is associated to entropy variations.On the other hand, the heat flux to the ambient is given byNewton’s law of cooling:

(23)

where h is a transfer coefficient related to convection andradiation, Ta is the ambient temperature, and Acell is the cellsurface. The cell energy conservation is thus written as:

(24)

where Cp is the specific thermal capacity of the cell and Mcell

its mass. Note that certain cell parameters, such as internalresistance, diffusion coefficients, etc., are usually dependenton the temperature.

The link between species concentration and SoC is givenas a double integral equation whose finite-volume counter-part is:

(25)

3 RESULTS

3.1 EIS

Figure 2 shows the impedance responses of the entire cell(2-electrode measurement) and of both electrodes (3-elec-trode measurement) of a commercial Ni-MH cell modifiedwith a reference electrode as describe previously (SoC = 50%).To check the validity of the measurement the sum of the half-cell impedance results and the two-electrode measurementswere compared.

For the entire cell a single semi-circle in the mediumfrequency range and a straight line in the low frequencyrange can be distinguished. According to Figure 2, the semi-circle is due to the contribution of the negative electrode. Thestarting point of this semi-circle on the abscissa axis fits theinternal resistance of the battery (or RHF for high frequencyresistance). The semi-circle shape is typical of a chargetransfer resistance in parallel with the double layer capaci-tance. The beginning of the line in low frequencies is prin-cipally controlled by the positive electrode. This line

qc c

cn n

n

= −−,max

,max

M CdT t

dtt tcell p gen tra

( )( ) ( )= −ϕ ϕ

ϕtra cell at hA T t T( ) ( ( ) )= −

ϕgen z eq ref zeq ref zt J t U t T t

dU t( ) ( ) ( ) ( )

( ), ,

, ,= −ddT

A

V t I tz

z

⎛

⎝⎜

⎞

⎠⎟

−

∑ ( )

( ) ( )

60


reflects the diffusion processes that can be modeled by aWarburg impedance with a 45° slope.

To extract parameters of use in the model, EIS diagramscan be fitted with electrical equivalent circuits. For the

negative electrode, a classical Randles circuit can be used(Fig. 3). The Randles circuit is composed of a double layercapacitance (Cdl) in parallel with a charge transfer resistance(Rt) and a Warburg impedance (Wd). This circuit is com-pleted with a high frequency resistance representing theinternal resistance of the cell and a parallel RL circuit attrib-uted to the inductive effects due to the electrode geometryand the cell connections. The EIS diagram fitting using para-meters given in Table 1 is depicted in Figure 4. The doublelayer capacitance of the negative electrode is estimatedaround 45 F.

The shape of the positive EIS diagram (Fig. 5) is verydifferent from the negative one. A classical Randles circuit ora circuit with a limited number of parameters cannot simulateits impedance properly. To estimate the double layer capaci-tance of the positive electrode, a diagram at colder tempera-ture was used for which a semi-circle appears. The doublelayer capacitance of the positive electrode is estimatedaround 450 F in that condition, and this value is supposed tobe independent of the temperature.

Evolutions of the EIS diagrams of both positive andnegative electrode are shown in Figures 6 and 7. The shapeof the EIS diagram of the positive electrode changes very fewbetween SoC = 30% and 90%. For SoC = 30%, the low fre-quency part of the diagram becomes a vertical capacitiveline. The diameter of the medium frequency semi-circle of thenegative electrode EIS diagram decreases between SoC = 90%and 40%. A second semi-circle appears when the element iscompletely discharged. For both evolutions presented inFigures 6 and 7, the study of the evolution of the RHF is not


61

50 mHz

0.00300.00200.00100

0

50 mHz

50 mHz

2.3 Hz 2.3 Hz

52 Hz83 Hz

–Im

(Z)/

(Ohm

s)

Re(Z)/(Ohms)

-0.0005

0.0005

0.0010

0.0015

Figure 2

EIS diagrams of a NiMH commercial cell modified with areference electrode at SoC = 50%, 20°C. Positive (reddiamonds), negative (blue circles), entire cell (blackrectangles), positive + negative (green triangles).

0.00200.00150.00100

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

88 Hz

3.8 Hz

0.16 Hz

50 mHz

–Im

(Z)/

(Ohm

s)

Re(Z)/(Ohms)

Figure 4

EIS diagram of a negative electrode (line + diamonds) atSoC = 50%, 20°C and EIS diagram of the equivalent circuit(red circles).

L

RHF

RL

Cdl

WdRt

j2πft

j2πftthRfZ

d

dd

)

)

()( =Warburg :

Figure 3

Equivalent circuit for the negative electrode of the NiMH celland expression of the Warburg impedance for semi-infinitediffusion.

TABLE 1

Parameters of the equivalent circuit of the negative electrode

RHF 0.977 × 10-3 Ω

Cdl 45.73 F

Rt 0.946 × 10-3 Ω

Rd 1.234 × 10-3 Ω

td 81.14 s

L 0.127 × 10-6 H

RL 1.391 × 10-3 Ω



possible because the connections between the potentiostatand the cell must be changed between each measurement.The lack of reproducibility induced by resistances of thecell connections disturbs the measurement of very low RHFresistance.

To study the contribution of each electrode on the RHF ameasurement of the high frequency part of this diagram wasdone during a discharge. Figure 8 shows that the RHF of eachelectrode are not much sensitive to the SoC. The negativeRHF higher than the positive RHF shows a light decreasewhereas the positive RHF increases with increasing SoC. In

a first approximation, the RHF of the entire cell will beconsidered constant in the model.

3.2 Simulation

On the basis of the equations described above, an advancedlumped-parameter electrochemical model has been devel-oped in the AMESim simulation environment, using theAMESet programming tool. This battery model is aimed atbeing used (as a component) in the IFP simulator for hybridelectric vehicles (Fig. 9a).

62

0.00100-0.0002

0

0.0002

0.0004

0.0006

0.0008

0.0010

–Im

(Z)/

(Ohm

s)

Re(Z)/(Ohms)Figure 5

EIS diagram of a negative electrode at SoC = 50%, –20°C(blue triangles), 0°C (green diamonds), 20°C (black circles)and 35°C (red squares).

0.00200.00100

0

0.0005

0.0010

0.0015

2,4 Hz

50 mHz

50 mHz

–Im

(Z)/

(Ohm

s)

Re(Z)/(Ohms)

83 Hz

0.0005

-Im

(Z)/

Ohm

50 mHz

18 mHz

110 mHz

242 mHz

0.00100

0.0035

0.0030

0.0025

0.0020

0.0015

0.0010

0.0005

0

Re(Z)/Ohm 0.0020

Re(Z)/(Ohms)

–Im

(Z)/

(Ohm

s)

0

0.0002

0.0004

0.0006

0.0008

0

SoC = 0%

SoC = 10%

Figure 6

EIS diagram of the positive electrode of a NiMH cell atSoC = 0% (blue circles), 10% (green triangles), 30% (blacksquares) and 90% (red diamonds).

0 1.00.2 0.4 0.6 0.8

RH

F (

Ohm

s)

9.20E-04

8.20E-04

7.20E-04

6.20E-04

5.20E-04

1.02E-03

SoC

Positive

Negative

Figure 7

EIS diagram of the negative electrode of a NiMH cell atSoC = 0% (red diamonds), 20% (blue circles), 40% (greentriangles), 80% (grey crosses) and 90% (black squares).

Figure 8

Evolution of the RHF of the positive and the negativeelectrode of a Ni-MH commercial cell with the SoC, T = 20°C.


In order to apply the EIS to the battery model executed inthe AMESim platform, a special simulation technique wasalso developed. This technique consists of running simulta-neously (co-simulation) two interexchanging models inAMESim and, respectively, Matlab, platforms. The Matlabmodel introduces a modulated frequency signal in the cellcurrent during charge/discharge and it calculates the real andimaginary parts of the impedance for each frequency. In theEIS simulator the number of periods for each frequency canbe set to ensure the stability of the system. The EIS simulatoris able to reproduce the same test cases than the experimentalEIS test bench (Fig. 9b).

3.3 Model Parameter Adjustment

As explained in Section 3.1, cells were characterised in termsof capacity and physical macroscopic characteristics to providenumerical parameters to the model. Electrical characteristicssuch as cell internal resistance and double-layer capacitancefor each electrode were also measured as described inSection 2. Since the numerical values for the active materialloading were not available for the Ni-MH cell under study,some physical parameters were tuned using two experimentalmeasurements: an EIS diagram at a 60% SoC and a discharge/charge/discharge profile at C/2 rate.

The EIS diagram for the cell at 60% SoC, 20°C permitsto adjust the parameters related to the specific surface areaand the double-layer capacity per surface unit, C(1) and C(3),in such a way that the products of electrode specific surfaceareas and their respective C(i)’s equal the capacitance valuesexperimentally determined. The resulting fitted EIS diagramis shown in Figure 10. A good qualitative and quantitative

agreement between model and measurements is clearlyobservable in the frequency range corresponding to thesemi-circle.

A charge/discharge profile was used to adjust the parametersrelated to the electrode potentials and the hysteresis phenom-enon. The resulting simulated voltage curve matches veryprecisely with the experimental curve, as shown in Figure 11.

3.4 Model Evaluation

To provide experimental validation of the model developedand tuned as described above, two different dynamic profileswere used.


63

T

v_celli_cellZ_im

Z_re

S.I.E

HC

+ –

HC

T

T

HC

+ –

HC

T

a) b)

Figure 9

Simulation sketches for Ni-MH cell cycling a) and for EISb) in AMESim simulation platform.

2.5 3.02.01.5

Model

Experimental

-300-200-100

00100200300400

500600

700800900

1000

x10-6

x10-3

–Im

(Z)/

(Ohm

s)

Re(Z)/(Ohms)Figure 10

Comparison of the EIS diagram of a Ni-MH cell at a SoC of60% and 20°C and the simulated diagram.

20151050

Vol

tage

(V

)

0

1.00

1.10

1.20

1.30

1.40

1.50

Time (s x103)Figure 11

Comparison of the cell voltage of a Ni-MH cell during C/2rate charge/discharge cycles (black) and the fitted model(red).



The former was applied to a single cell after discharge to a80% SoC. It is composed of an alternation of 1 C rate chargeand discharge steps lasting 50 s each (Fig. 12). It should benoted that the “infinite” voltage values (vertical lines) visibleon the experimental curve are due to the potentiostat com-muting between two current ranges. The figure clearly showsa qualitative correspondence of the simulated and experimen-tal curves. A quantitative analysis of differences reveals thatthe discrepancy does not exceed ± 1.8% of the operatingrange of the NiMH cell under these cycling conditions.

The second validation test presented is a HPPC testperformed at 20°C for the entire battery pack. The pack con-sists of 168 serially connected cells. However, it is assumedthat cell construction, SoC, and temperature are uniformthroughout the pack. Thus no attempt was made to account forcell-to-cell differences arising from manufacturing variabilityor temperature distribution within the pack. The completeHPPC test consists with a succession of sequences as shown inFigure 13. Figure 14 gives the results for the complete HPPCtest and Figure 15 displays the result for a single profile.

64

7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2

Vol

tage

(V

)

1.30

1.32

1.34

1.36

1.38

Time (s ×103)

Figure 12

Comparison of the cell voltage of a Ni-MH cell during C ratecycling (black) with the simulation (red).

403530252015105

Vol

tage

(V

)

160

180

200

220

240

260

×103

100%

80%70% 60% 50% 40% 30%

20%10% 0%

90%

Time (s ×103)

Figure 14

Comparison of measured (black line) and simulated (reddots) Ni-MH battery pack voltage during a HPPC test withtheoretical SoCs.

11.40 11.50 11.60 11.70 11.80 11.90 12.00 12.10

Cur

rent

(A

)

100

-50

0

50

Time (s ×103)

×103

Figure 13

High Hybrid Pulse Power Characterization test profile.

19.50 19.60 19.70

12

19.80 19.90 20.00

Vol

tage

(V

)

180

190

200

210

220

230

240

250

260

Time (s ×103)

×103

Figure 15

Comparison of measured (black line)and simulated (red dots)Ni-MH battery pack voltage during a HPPC profile atSoC = 60%.


The model predictions in these highly dynamic conditionsreflect the experimental measurements quite correctly overthe nine periods of the HPPC test. However, there is anincreasing difference in the voltage at the end of the restperiod with increasing discharge/charge pulses, suggestingthat the relaxation phenomena could be better taken intoaccount in the model. Nevertheless model prediction can bestill considered as fairly good in the relaxation periods, sincethe relative error between measurements and simulationremains below ±3.8% of the operating range of the NiMHbattery pack. Notice however that discharge pulses are betterreproduced than charge pulses, for which the differencereaches 9% after the fourth pulse.

4 DISCUSSION

Generally speaking, the improved 0D model of a sealedNi-MH cell and battery pack is validated at 20°C as far asvoltage prediction is concerned. Additional work is inprogress, focusing on the thermal aspects of the modelling inconditions representative of actual operation.

It is now interesting to discuss the model predictions interms of SoC with regards to the conventional Coulombcounting method during the HPPC test. The initial state ofthe battery was set to 100% SoC. After the eight pulse wherethe SoC theoretically equals 10%, it is observed in theexperimental curve (Fig. 16) that the discharge pulse waslimited by the low-voltage limit of the battery, suggestingthat less than 10% of charge was remaining in the battery.Figure 16 compares the SoC simulated with the 0D electro-

chemical model with SoC calculated using the Coulombcounting method. Focusing on the model prediction after theeight pulse, for example, the estimated SoC value usingEquation (25) is around 5%, which is in agreement with theaforementioned observation. This is indeed a very goodresult in our scope of work where this model would be usedwith confidence as a SoC observer within a BMS system.

CONCLUSION

Electrochemical lumped-parameter models based on the cellelectrochemistry appear as very promising mathematicalmodels of the battery systems to provide reliable monitoringwithin the vehicle management system. Indeed the SoC iscalculated from the concentration of the reactants, which aredynamic states updated with mass and current balance equa-tions. The direct benefit of this approach, using a model thatincludes the SoC as a state, is that a Kalman filter or anequivalent mathematical tool can automatically give adynamic estimate of the SoC and its uncertainty. Moreover,the computational time needed to solve such simplifiedphysics-based battery models is well adapted for onlineapplications.

The improved 0D model of a sealed Ni-MH cell/batterypresented in this paper shows a good agreement with mea-surements performed for low rate solicitations on a single cellbut also for high rate solicitations like HPPC test on a entirebattery pack. Work is in progress to adjust the thermal para-meters of the model and provide a complete voltage/thermal-behaviours battery model.

Future development of the model will take into accountthe ageing phenomena occurring in the battery during serviceoperation.

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65

50 4510 15 20 25 30 35 40

SoC

0.2

0

0.4

0.6

0.8

1.0

Time (s ×103)

×103

Figure 16

Comparison of simulated SoC evolution (line) and SoCevolution obtained with coulomb counting method (dot line)during a complete HPPC test.



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Final manuscript received in August 2009Published online in November 2009

66

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