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Dynamic Model of a Polymer Electrolyte Fuel Cell Power Device Elie Laffly Marie-Cécile Péra Daniel Hissel Laboratoire de recherche en Electronique, Electrotechnique et Systèmes Laboratoire de recherche en Electronique, Electrotechnique et Systèmes Laboratoire de recherche en Electronique, Electrotechnique et Systèmes L2ES UFC/UTBM, EA3898 rue Thierry-Mieg L2ES UFC/UTBM, EA3898 rue Thierry-Mieg L2ES UFC/UTBM, EA3898 rue Thierry-Mieg 90010 Belfort Cedex 90010 Belfort Cedex 90010 Belfort Cedex FRANCE FRANCE FRANCE [email protected] [email protected] [email protected] Abstract – This paper proposes a system oriented dynamic model of a Polymer Electrolyte Fuel Cell based on equivalent electrical impedances and on steady state response. Polarisation curve and Electrochemical Impedance Spectra (EIS) provide information about respectively static and dynamic parameters of the model. The validation of the proposed model is based on experimental responses to a square pulsed current solicitation. I. INTRODUCTION During the last decade, the problem of energy became a real issue because of the rarefaction of oil stocks and greenhouse effect. Especially for transport applications, the limitation of the carbon dioxide emissions leads to find new powertrain technologies or to improve the current ones. The Fuel Cell (FC) is one of those promising technologies. It allows the production of electrical energy with hydrogen and air supply. Simulation is a milestone to avoid costly prototype realization. Several models have been published at different scales. Microscopic scale allows understanding of electrochemical and fluidic phenomena accurately and is useful to improve stack design [1]. Macroscopic scale is needed for system design. Furthermore, dynamic response has to be taken into account for transportation applications where fast power solicitation changes are imposed. This paper proposes a system oriented model of a F.C. based on equivalent electrical impedance circuits. The section II presents the principle of the equivalent circuit and the electrochemical baselines. The section III describes the experimental measurements consisting in static polarization curve and impedance spectra. Sections IV and V concern the parameter identification and the experimental validation. A. Fuel Cell Description The considered fuel cell in this study is a polymer electrolyte one (PEFC). Several cells are stacked in series and each cell is composed of two electrodes (anode and cathode) and one electrolyte (in this case a proton exchange membrane). The electrolyte spares the fuel (the hydrogen supplied at the anode) and the oxidant (oxygen of air supplied at the cathode). Gas diffusion layers (or backing layer) and bipolar plates allow the diffusion of both hydrogen and oxygen, and the mechanical standing of this elementary assembly (see Fig. 1). Fig. 1. PEFC composition. Fig. 2. PEFC principle. The electrochemical reactions in a PEFC are (Fig. 2): At the anode: the oxidation of hydrogen 2 H ⎯→ ⎯⎯ 2 + H + 2 e , (1) At the cathode: the reduction of oxygen 2 O + 4 + H + 4 e ⎯→ ⎯⎯ 2 O H 2 , (2) The global reaction is then: 2 2 H + 2 O ⎯→ ⎯⎯ 2 O H 2 , (3) Membrane Electrode Gas diffusion layer Bipolar plate 466 1-4244-0136-4/06/$20.00 '2006 IEEE

[IEEE IECON 2006 - 32nd Annual Conference on IEEE Industrial Electronics - Paris, France (2006.11.6-2006.11.10)] IECON 2006 - 32nd Annual Conference on IEEE Industrial Electronics

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Page 1: [IEEE IECON 2006 - 32nd Annual Conference on IEEE Industrial Electronics - Paris, France (2006.11.6-2006.11.10)] IECON 2006 - 32nd Annual Conference on IEEE Industrial Electronics

Dynamic Model of a Polymer Electrolyte Fuel Cell Power Device

Elie Laffly Marie-Cécile Péra Daniel Hissel Laboratoire de recherche en Electronique,

Electrotechnique et Systèmes Laboratoire de recherche en Electronique,

Electrotechnique et Systèmes Laboratoire de recherche en Electronique,

Electrotechnique et Systèmes L2ES UFC/UTBM, EA3898

rue Thierry-Mieg L2ES UFC/UTBM, EA3898

rue Thierry-Mieg L2ES UFC/UTBM, EA3898

rue Thierry-Mieg 90010 Belfort Cedex 90010 Belfort Cedex 90010 Belfort Cedex

FRANCE FRANCE FRANCE [email protected] [email protected] [email protected]

Abstract – This paper proposes a system oriented dynamic model of a Polymer Electrolyte Fuel Cell based on equivalent electrical impedances and on steady state response. Polarisation curve and Electrochemical Impedance Spectra (EIS) provide information about respectively static and dynamic parameters of the model. The validation of the proposed model is based on experimental responses to a square pulsed current solicitation.

I. INTRODUCTION

During the last decade, the problem of energy became a real issue because of the rarefaction of oil stocks and greenhouse effect. Especially for transport applications, the limitation of the carbon dioxide emissions leads to find new powertrain technologies or to improve the current ones.

The Fuel Cell (FC) is one of those promising technologies. It allows the production of electrical energy with hydrogen and air supply. Simulation is a milestone to avoid costly prototype realization. Several models have been published at different scales. Microscopic scale allows understanding of electrochemical and fluidic phenomena accurately and is useful to improve stack design [1]. Macroscopic scale is needed for system design. Furthermore, dynamic response has to be taken into account for transportation applications where fast power solicitation changes are imposed.

This paper proposes a system oriented model of a F.C. based on equivalent electrical impedance circuits. The section II presents the principle of the equivalent circuit and the electrochemical baselines. The section III describes the experimental measurements consisting in static polarization curve and impedance spectra. Sections IV and V concern the parameter identification and the experimental validation. A. Fuel Cell Description

The considered fuel cell in this study is a polymer electrolyte one (PEFC). Several cells are stacked in series and each cell is composed of two electrodes (anode and cathode) and one electrolyte (in this case a proton exchange membrane). The electrolyte spares the fuel (the hydrogen supplied at the anode) and the oxidant (oxygen of air supplied at the cathode). Gas diffusion layers (or backing layer) and bipolar plates allow the diffusion of both hydrogen and oxygen, and the mechanical standing of this elementary assembly (see Fig. 1).

Fig. 1. PEFC composition.

Fig. 2. PEFC principle.

The electrochemical reactions in a PEFC are (Fig. 2):

At the anode: the oxidation of hydrogen

2H ⎯→⎯⎯⎯← 2 +H + 2 −e , (1)

At the cathode: the reduction of oxygen

2O + 4 +H + 4 −e ⎯→⎯⎯⎯← 2 OH 2 , (2)

The global reaction is then:

2 2H + 2O ⎯→⎯⎯⎯← 2 OH 2 , (3)

Membrane

Electrode

Gas diffusion layer

Bipolar plate

4661-4244-0136-4/06/$20.00 '2006 IEEE

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Fig. 3. PEFC global system description ([2], p 3-17).

B. Fuel Cell System Description

The F.C. is embedded in a complete system. The aim of this system is to supply the reactants at controlled temperatures, humidities, flows and pressures in the F.C. and also to control the stack temperature (see Fig. 3).

1) Air supply: The F.C. is not feed with pure oxygen. The gas entering the cathode is air and water. Water amount and temperature are very important for the humidity of the membrane and the efficiency of the system. The produced water (equation 3) is generally not sufficient to keep the membrane humid. That is the reason why air is preheated and humidified before entering the FC. As oxygen from air is used, it has to be pressured by a compressor or a fan in order to balance the pressure drop in the circuit and supply an oxygen partial pressure high enough to activate the reaction at the cathode. 2) Hydrogen supply: Hydrogen can be produced as it is consumed along (through a reformer for instance) or provided from a pressured tank and expanded. It can be also humidified or kept dry if the membrane is thin enough to allow easy water exchange. 3) Temperature management: As already said, temperature is a critical parameter in a F.C. system and has a great influence on the stack performance. It has to remain constant during a measurement to have reliable and repeatable data. So, the stack is water cooled.

II. DYNAMIC MODEL OF THE STACK A. Dynamic model

The chosen dynamic model is based on an equivalent

electrical impedances circuit, as shown in Fig. 4(a). Each component is related to electrochemical or fluidic phenomena. Some of them are classical electrical component (resistance and capacity) even if their value depends on operating conditions and the operating current. Some are typical of electrochemical devices and have unusual impedances like Warburg impedance.

The circuit is composed of the following impedances:

1) : Charge transfer resistance at the anode. taR2) : Charge transfer resistance at the cathode. tcR3) : Warburg Impedance modelling diffusion and

convection of the oxidant at the anode (proton). OaWδ

4) : Warburg Impedance modelling diffusion and convection of the reductant at the anode (dihydrogen).

RaWδ

5) : Warburg Impedance modelling diffusion and convection of the oxidant at the cathode (O

OcWδ2-).

6) : Warburg Impedance modelling diffusion and convection of the reductant at the cathode (dioxygen).

RcWδ

7) : Double-layer capacitance at the anode. It is due to accumulation of charges at the anode/electrolyte interface.

dcaC

8) : Double-layer capacitance at the cathode. It is due to accumulation of charges at the cathode/electrolyte interface.

dccC

9) : Membrane resistance. mR10) : Connector inductance. LThe symbol represents an impedance of diffusion

convection which form is given in (4). iWδ

( ) ( )s

sRsW

i

iii

⋅⋅=

τ

τδ

tanh, (4)

Transfer resistances at the anode and cathode model the

transfer of charges: i.e. electrons at the electrodes and ions diffusion through the backing layer and through the membrane.

Warburg impedances model the diffusion and convection of species in the backing layers.

Double layer capacitances exist at the interface between the electrolyte and electrodes. It is due to the separation between ions in the membrane and ions stuck on electrodes.

The membrane resistance exists because of the low or high

ionic conductance of the electrolyte which depends mainly on its water content.

Finally, the connection of the FC leads to an inductive behaviour of the complete system when high frequency solicitation occurs, so an inductance is added to the model.

taR

mR

tcR

L

OaWδ RaWδ

dccC

OcWδ RcWδ

dcaC

Fig. 4(a). Complete model of the P.E.M.F.C.

dcaC dccCmR L

OcWδtcRtaRFig. 4(b). Simplified model of the P.E.M.F.C.

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B. Simplification and normalisation of the model Due to their negligible values, the following impedances

are withdrawn from the model: OaWδ

RaWδ

RcWδ

Only one Warburg Impedance modelling diffusion and convection remains at the cathode: . As shown in Fig. 4(b), the model becomes simpler.

OcWδ

Then, impedances, voltages and current are normalised in

order to be independent from the number and the active surface of the stack. All values are related to a 1 cell. 2cm

N : Number of cells. ( )1

S : Surface of the membrane. ( )2cm ( )ti : Stack FC current. ( )A( )tV : Stack FC voltage. ( )V

( )tj : Current density. ( )2−⋅ cmA ( )tu : Voltage per cell. ( )V

( )sZ : Normalised impedance. ( )2cm⋅Ω ( )sZmes : Total measured impedance. ( )Ω

( ) ( )Stitj = , ( ) ( )

NtVtu = , ( ) ( )

NSsZsZ mes ⋅= , (5), (6), (7)

The Nyquist diagram of this simplified electrical

impedance circuit is given in Fig. 5. The pulsations 1ω 2ω and 3ω are defined as:

dcata CR ⋅=

11ω ,

dcctc CR ⋅=

12ω ,

Ocτω 54.2

3 = , (8)

TABLE I

MODEL PARAMETERS AND NORMALISATION GAIN

PARAMETERS NORMALISATION

GAIN NGNORMALISED

UNITS

taR NS / ( )2cm 2cm⋅Ω

tcR NS / ( )2cm 2cm⋅Ω

OcRδ NS / ( )2cm 2cm⋅Ω

Ocτ 1 ( )1 s

dcaC SN / ( )2−cm 2/ cmF

dccC SN / ( )2−cm 2/ cmF

mR NS / ( )2cm 2cm⋅Ω

L NS / ( )2cm 2cmH ⋅

Fig. 5. Nyquist diagram of the equivalent electrical circuit.

III. METHODOLOGY OF MEASUREMENT A. Considered Fuel Cell Stack

Measurements have been performed on a FC supplied by the Zentrum für Sonnenenergie – und Wasserstoff-Forschung (ZSW). It is a 3 cell stack and 100 cm² membrane surface Measurements are performed at constant temperature of 58°C. Anodic and cathodic stoichiometry factors are kept constant at the values 2 and 5 resp.. B. Polarisation curve

A polarization curve is a plot of the steady state voltage as a function of current. On Fig. 6, it is related to one cell and to the current density according to the normalisation. The temperature and stoichiometry factors are constant during this measurement. It allows plotting of the power delivered by the stack as a function of current density (Fig. 7).

Fig. 6. Polarisation curve of the P.E.M.F.C.

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Fig. 7. Power as a function of current density.

Fig. 8. Electrochemical impedance spectroscopy principle.

C. Electrochemical Impedance Spectroscopy

A PEFC is a non-linear system. The first assumed hypothesis is the linearity of the FC around a given point of current. The system is then modelled with linearized electrical impedances.

The aim of Electrochemical Impedance Spectroscopy (EIS) is to measure those impedances. It is based on small sinusoidal current perturbations applied around a fixed steady point . The impedance information is located in the sinusoidal voltage response (Fig. 8).

( )tI∆

0I( )tU∆

( ) ( ) ( )JtsinJJtJJtj ϕ+⋅ω⋅+=∆+= 100 , (9)

( ) ( ) ( )UtUUtUUtu ϕω +⋅⋅+=∆+= sin100 , (10)

The impedance is calculated as follows: ( )sZ

( ) ( )( )sJsUsZ

∆∆

= , (11)

( )tZ Normalised impedance ( )2cm⋅Ω ( )tU∆ Sinusoidal voltage per cell ( )V

( )tJ∆ Current density solicitation ( )2−⋅ cmA IV. DYNAMIC MODEL PARAMETER IDENTIFICATION

A set of dynamic parameters is calculated for some different steady state points. A nonlinear least-squares data fitting by the Gauss-Newton method is used in order to fit Nyquist plots of impedance. Results are gathered in Table II.

Fig. 9 shows the good results obtained on the impedance identification. The Nyquist plots of experimental results and model fittings are in good agreement. It can be seen that experimental data are noised at frequencies below 1 Hz (arc on the right in the figure) as steady state operating conditions are hardly obtained on a long time duration. However, the identification method has succeed in overcoming this problem.

TABLE II MODEL NORMALISED PARAMETER VALUES.

NORMALISED VALUE AT DIFFERENT CURRENT DENSITIES

PARAMETERS j=0.10 2/ cmA

j=0.25 2/ cmA

j=0.40 2/ cmA

j=0.55 2/ cmA

taR ( )2cmm ⋅Ω 128 95 82 74

tcR ( )2cmm ⋅Ω 490 290 188 138

OcRδ ( )2cmm ⋅Ω 678 368 249 221

Ocτ ( )s 1.07 0.45 0.27 0.20

dcaC ( )2/ cmmF 4.55 6.30 6.34 5.82

dccC ( )2/ cmmF 20.7 27.8 31.6 29.6

mR ( )2cmm ⋅Ω 167 128 143 144

L ( )2cmµH ⋅ 1.22 1.24 1.73 1.73

Fig. 9. Electrochemical impedance spectroscopy experimental results and

comparison with simulation.

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V. EXPERIMENTAL RESULTS AND VALIDATION

The model is now implemented and validated with a square pulsed current solicitation. The experimental data have been acquired on a 20 cell stack based on the same technology than the one used for the parameter identification. Voltages, currents and impedances have been normalised according to Table I.

A10

In order to obtain the voltage response, the current signal

is separated in two parts: DC component and AC component (Fig. 10).

( ) ( ) ( )tSPJJtJJtj ⋅+=∆+= 100 , (12)

Where is a function having the successive values 0

and 1.

( )tSP2

10 05.0 −⋅== cmAJJ Under the simple linear stack behaviour assumption, the

voltage per cell is the addition of the dynamic and static responses and , computed from static measurements for and from the dynamic response of the impedance model for .

( )tu

0U ( )tU∆

0U( )tU∆

( ) ( )tUUtu ∆+= 0 , (13)

The static value cannot be obtained directly from the

polarisation curve. Experimentally, during the whole measurement, the fuel and air flows are kept constant and calibrated for the static current point . Then, the stochiometry factors FSA and FSC are not kept constant, as it is the case in the polarization curve used for the parameter identification, except at zero current where a minimal flow is achieved for 0.05 A/cm

0U

0I

2. FSA and FSC oscillate between the same value for the low level current period and the half value during the high level current pulse. That’s why the stack voltage at a current density of is less important than the voltage in the polarisation curve. It can be seen on Fig. 11, where the following results are reported: the experimental measurements at constant stochiometry factors (points), the simulated polarization curve (dotted line) where the limit current is taken into account and the voltage static values obtained during the square current solicitation at constant flows (continuous line). Moreover, the shift can also be partly due to the fact that the measurements are performed on two different stacks, which are not industrially produced and whose performances are not guaranteed to be exactly the same, even for similar technology. Finally, the two experiments have not been performed the same day, so ambient parameters (pressure, temperature, hygrometry…) are not strictly the same: it might have also a minor influence.

21.0 −⋅ cmA

Then, static values measured during the square solicitation have been used to correct the polarization curve.

Fig. 10. Separation of static and dynamic behaviour of the F.C.

Fig. 11. Voltage shift between the polarisation curve at constant

stochiometry factors (experimental: points, simulation: dotted line) and the static voltage response at constant reactant flows calibrated at 0.05 A/cm2

(continuous line)

Fig. 12(a). Square pulsed current solicitation.

Fig. 12(b). Voltage responses (measure and simulation).

Fig. 12(a) and Fig. 12(b) show the good agreement

between experimental results and the simulation. The decomposition into the DC and the AC components and the adaptation of a small signal model to large current solicitation are validated. It proves the static and dynamic robustness of the model. Moreover, it shows that the dynamic

+

-

( )tj

0I

( )tI∆

0U

( )tU∆

470

Page 6: [IEEE IECON 2006 - 32nd Annual Conference on IEEE Industrial Electronics - Paris, France (2006.11.6-2006.11.10)] IECON 2006 - 32nd Annual Conference on IEEE Industrial Electronics

model can be extrapolated from a short stack (3 cells) to a longer one (20 cells), even if the gas distribution among the cells is less uniform in the second case.

VI. CONCLUSION

The FC technology as already been approved as a solution for future no emission transport applications. But the feasibility of a large production of FC powered vehicles needs a good knowledge of FC static and dynamic behaviour at a macroscopic level.

This paper proposed a static and dynamic model based on electrochemical impedance spectroscopy and polarisation curve. The macroscopic approach which has been developed uses only few parameters, compared to the high complexity of the phenomena in a fuel cell. Furthermore, the electrical impedances calculated from the measurements are directly linked to the physical aspects of the FC (transfer phenomena, diffusion and convection, double-layer capacitance…). These parameters can be related to physical component characteristic (membrane resistance, water evacuation in the gas diffusion layer…). Then, it will allow the studying and taking into account the influence of ageing in the model. As a matter of fact, the identification of parameter variation function with time will help us to identify and to quantify different ageing phenomena.

V. REFERENCES [1] M. Ceraolo, C. Miulli and A. Pozio, Modelling static

and dynamic behaviour of proton exchange membrane fuel cells on the basis of electrochemical description, Journal of Power Sources 113, 2003, pp. 131-144.

[2] EG&G Technical Services, Inc, Fuel Cell Handbook

(Seventh Edition), U.S. Department of Energy Office of Fossil Energy, National Energy Technology Laboratory: 2004.

[3] J.-P. Diard, B Le Gorrec, C. Montella, C. Poinsignon

and G. Vitter, Impedance measurements of polymer electrolyte membrane fuel cells running on constant load, Journal of Power Sources 74, 1998, pp. 244-245.

[4] C. Wang, M. H Nehrir and S. R. Shaw, Dynamic models

and model validation for PEM fuel cells using electrical circuits, IEEE transactions on energy conversion, Vol. 20, no. 2, June 2006, pp. 442-451.

[5] J. Garnier, M.-C Pera, D. Hissel, A. De Bernardinis, J.-

M Kauffmann and G. Coquery, Dynamic behaviour of a proton exchange membrane fuel cell under transportation cycle load, Industrial Electronics, 2004 IEEE International Symposium on, Vol. 1, May 2004, pp. 329-333.

[6] W. Friede, S. Rael and B. Davat, Mathematical model and characterisation of the transient behaviour of a PEM fuel cell, Power Electronics, IEEE Transactions on, Vol. 19, Issue 5, Sept 2004, pp. 1234-1241.

[7] A.Rowe and Xianguo Li, Mathematical modelling of

proton exchange membrane fuel cells, Journal of Power sources, Vol. 102, 2001, pp. 82-96.

[8] J. Garnier, M.-C Pera, D. Hissel, F. Harel, D. Candusso,

N. Glandut, J.-P. Diard, A. De Bernardinis, J.-M Kauffmann and G. Coquery, Dynamic PEM fuel cell modelling for automotive applications, Vehicular Technology Conference, 2003. VTC 2003-Fall, 2003 IEEE 58th, Vol. 5, Oct 2003, pp. 3284-3288.

[9] C. Brunetto, G. Tina, G. Squadrito and A. Moschetto,

PEMFC diagnostics and modelling by electrochemical impedance spectroscopy, Electrotechnical Conference, 2004, MELECON 2004, Proceedings of the 12th IEEE Mediterranean, Vol. 3, May 2004, pp. 1045-1050.

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