a review on existing modeling methodologies for pem ...

A REVIEW ON EXISTING MODELING METHODOLOGIES FOR PEM FUEL CELL SYSTEMS D. HISSEL1, Ch. TURPIN2 S. ASTIER2, L. BOULON1, A. BOUSCAYROL11, Y. BULTEL4, D. CANDUSSO9, S. CAUX2, S. CHUPIN6, T. COLINART6, G. COQUERY10, B. DAVAT3, A. DE BERNARDINIS10, J. DESEURE4, S. DIDIERJEAN6, J. DILLET6, F. DRUART4, M. FADEL2, G. FONTES2, B. FRANCOIS11, J.C. GRANDIDIER8, F. HAREL9, M. HILAIRET14, M. HINAJE3, S. JEMEI1, O. LOTTIN6, L. MADIER8, G. MARANZANA6, S. MARTEMANIOV7, D. NGUYEN3, R. ORTEGA13, R. OUTBIB12, M.C. PERA1, S. RAEL3, N. RETIERE5, D. RIU5, S. SAILLER4,5, R. TALJ13,14, T. ZHOU11 1

FEMTO-ST, UMR CNRS 6174, FCLAB, Rue Thierry Mieg, 90010 Belfort Cedex, France LAPLACE, UMR CNRS 5213, Univ. Toulouse, 2 rue Camichel, 31071 Toulouse cedex 7, France 3 GREEN UMR CNRS 7037, INPL, Vandoeuvre-lès-Nancy, France 4 LEPMI UMR CNRS 5631, INPG UJF, ENSEEG, BP 75, 38402 Saint Martin d’Hères, France 5 G2ELAB, UMR CNRS 5529, INPG UJF, ENSIEG, BP 46, 38402 Saint Martin d’Hères, France 6 LEMTA, UMR CNRS 7563, Nancy-Université, 54504 Vandoeuvre Cedex, France 7 LET, UMR CNRS 6608, ESIP – University of Poitiers and ENSMA, 86022 Poitiers, France 8 LMPM, UMR CNRS 6617, ENSMA, BP 40109, 86962 Futuroscope Cedex, France 9 INRETS LTN, FCLAB, Rue Thierry Mieg, 90010 Belfort Cedex, France 10 INRETS LTN, 2 avenue du Général Malleret-Joinville, 94114 Arcueil Cedex, France 11 L2EP Lille, 59655 Villeneuve d’Ascq, France 12 LSIS UMR CNRS 6168, Avenue Escadrille Normandie-Niemen, 13397 Marseille Cedex 20, France 13 L2S UMR CNRS 8506, SUPELEC, F-91192 Gif-Sur-Yvette, France 14 LGEP UMR CNRS 8507, SUPELEC, F-91192 Gif-Sur-Yvette, France 2

ABSTRACT This paper is a common review research paper written in the framework of the “Energy program” of the French National Centre for Scientific Research (CNRS). In this program, funding has been provided to a project aiming to federate all the French competencies around fuel cell systems. Thus, as a first achievement, a review on existing modelling methodologies for PEM fuel cell systems is here proposed.

1. INTRODUCTION In the framework of the “Energy program” of the French National Centre for Scientific Research (CNRS), funding has been provided to a project aiming to federate all the French competencies around fuel cell systems. A working group on this topic has also been set up. Nevertheless and before all, the first question is what is a “fuel cell system”? Figure 1 presents a general scheme of such a generic fuel cell system. As it can be seen, a fuel cell stack needs a lot of ancillaries to be operated. The fuel must firstly be produced and/or stored. Then, it is finally processed (mostly in terms of pressure, hydration and flow regulation) before entering the fuel cell stack. The oxidant must also be processed in the same way. For both fuel and oxidant gases, the water produced by the FC stack can be removed from the exhaust gases to be re-used in the

hydration of incoming gases. Then, as the electrochemical reaction is exothermal and as the FC stack must be operated in a dedicated temperature range, thermal management is essential. Moreover, the gas supplying and the stack thermal management are strongly coupled with the gases hydration level control. Finally, electrical power conditioning (in association or not with an energy storage device) and overall control of the whole system are other important subsystems. Different major research issues around such a fuel cell system have been identified by the working group: multiphysics modelling of these systems, diagnosis of these systems, and hybridization of these systems. Among them, this paper will focus on the modelling topic and propose a review on existing modelling methodologies for PEM fuel cell systems. These methodologies will be described in this paper considering the system boundary: the single cell, the stack or the whole system (i.e. the fuel cell stack and the different ancillaries). They will also be sorted by their level of required physical knowledge, by their finality, and by their ability to be implemented in real-time systems. Thus, the first part of this paper will be devoted to the description of PEM FC stack modelling approaches. The second part will then be focused of PEM FC systems modelling methodologies. Finally, the last part will conclude through an objective evaluation of the advantages and drawbacks of the described modelling approaches.

Heat exchanger

Control Supervision

Thermal power out

Thermal management

Fuel Storage

Fuel Processing

Water management

Oxidant in

Power conditioning

Fuel Cell Stack

Oxidant Processing

Electrical power out

Energy storage

Exhaust gases processing

Figure 1. Fuel cell system scheme.

2. PEM FUEL CELL STACK MODELING APPROACHES This first part of the paper is devoted to the detailed description of some PEM fuel cell stack modeling approaches. Those can be divided in two main families: the mechanistic (or theoretical) models and

the empirical ones. Of course, this separation can be considered as too binary, so-called semi-empirical models also exist but we have chosen here to limit the granularity. 2.1 Mechanistic models A mechanistic model is based on a set of electrochemical, thermodynamic, thermal, electrical and fluidic relationships. Those are based on phenomenological equations like the Butler-Volmer equation for FC voltage, the Stefan-Maxwell equation for transport phenomena, and the Nernst-Plank equation for species transport [1]. 2.1.1 General considerations In mechanistic modeling, differential and algebraic equations are based on the physics and electrochemistry governing the phenomena internal to the cell and the stack. Equations describe electrochemical reaction, mass and charge transfer. Indeed, the need of precise water management, the dehydration of membrane, the complex electrode kinetics, the mass transport and the slow rate of oxygen reduction are the most significant limiting factors on the fuel cell performances. Several domains are defined to describe the complex structure of fuel cells. Actually, Gas Diffusion Electrode with Diffusion Layer and Active Layer, Electrolyte, Gas channel can be distinguished. Different levels of complexity are proposed from the single-cell one-dimension modeling to stack or 3-dimensions modeling. Resolutions of complex models lead to expensive calculation. Then, the models propose hypothesis to focus on one mechanism or a limiting case. These models make it possible to describe quantitatively the reaction mechanisms, the polarization curves and impedance spectra of the fuel cell. For dynamic modeling, mass accumulation is integrated in mass balance in order to take into consideration the transient processes of membrane hydration, water accumulation and gas transport in channel and Gas Diffusion Electrode. As mass accumulation, electrode capacitance influences transient response. In most of existing dynamic models, capacitance is included at electrode interfaces but can also be distributed along the active layer thickness. As steady state models, dynamic models include electrochemical reaction kinetic. Some multi-scale models have been elaborated [2-3] to predict the dynamic and steady state behaviors in active layer. Mechanistic approach enables the simulation of transient response with large voltage variation. The model of [4] includes the full description of water and proton co-transport in the three-dimensional MEA, thus it could provide detailed information of water behaviors in this vital component, such as water accumulation. It is found that membrane hydration occurs over a period of 10s, the gas transport of 10–100 ms, with the double-layer discharging being negligibly fast. The two dimensions model of [5] incorporates water transport in membrane and gas distribution in channel and Gas Diffusion Layer in order to study transient response of fuel cell integrated in systems. Malfunction in the air supply is approached. In [6], the authors study the dynamic behavior of PEMFC stack operating in dead-end mode to explain the different performances between a “fresh” and “aged” stack. Simulations indicated that the liquid water accumulation is at the origin of the performances decrease with ageing, due to its effect on decreasing the actual GDL porosity that in turn cause the starving of the active layer with oxygen. The response time corresponding to time needed to charge the double-layer capacitance is about 40 ms. Mechanistic models are also used to elaborate equivalent circuit model [2-3,7]. In this model, the fuel cell is represented by an equivalent circuit, whose components are identified with the experimental technique of electrochemical impedance spectroscopy (EIS). Those dynamics models can describe small voltage variations. Those studies show that the impedance of a fuel cell is a powerful tool in order to characterize the intra-electrode processes occurring in gas diffusion electrodes. To reduce parameters involved in EIS models, in [8], the authors introduce non-integer derivatives to model diffusion phenomena. The dynamic study of a fuel cell appears to be of great interest to provide detailed understanding of transient behavior and is of extreme importance for the control strategy and system management in power generation systems, especially when there are power injections into the network. Those models

established a useful basis for understanding the hydratation in the fuel cell Membrane Electrode Assembly (MEA), the mass and charge transfer in the Gas Diffusion Electrodes and understanding time responses or limiting phenomenon. They are used for a quantitative analysis. Those models are also modular and easily allow the testing of various technological solutions. Indeed, model’s parameters depend on fuel cells conception (geometry, porosity, size) and operating conditions (temperature, pressure, gas inlet). Nevertheless, these models are complex to take into account all the physical and electrochemical phenomena: numerous parameters and expansive time calculations. Electrochemical impedance spectroscopy is an experimental technique widely used for the transient analysis of PEFCs. Analyze and fitting EIS model obtained from mechanistic approach with experimental results permit an access to those internal parameters. Mechanist models should thus be used to design and conception step, when time simulation is not a limiting factor.

2.1.2. Steady-state 1D model As one of the most important phenomenon reducing the fuel cell performances is linked to the hydration of the electrolyte membrane, an iterative procedure to compute the coupled diffusion equations of the reactants and the water transport in the membrane can be considered. That way, the membrane resistance behaviour can be predicted for various relative humidity of the inlet gas. The water transport is always a balance between at least two competing diffusion mechanisms. One is due to the proton displacement from anode to cathode that drags some water molecules with them (called electro-osmotic drag). The other mechanism is back diffusion of water from cathode to anode. This water flux results from the water concentration gradient created in the membrane by the electroosmotic drag and the water produced by the redox reaction at the cathode. The model presented in this section is a one-dimensional steady-state single-cell model with two differential equation systems coupled at the anode and the cathode that represent the Stefan-Maxwell diffusion of a mixture gas. The model regions consist of a membrane sandwiched between two gas diffusion layers. In the diffusion layers, dry hydrogen and humidified air travel to the catalytic sites, where the redox reaction takes place, producing water, the governing equations are: k yia ,c N aj ,c − y aj ,c N ia ,c dy ia ,c = dz cDijeff i =1,i ≠ j

∑

(1)

(where y i is the molar fraction of species i, Dijeff is the effective diffusion coefficient (m2.s-1), N i is the flux density of species i (mol.m-2.s-1) The gas consumptions are the following (nitrogen takes no part in the redox reaction; hence its flux is equal to zero N N 2 = 0 ): j j j cell , N O2 = cell , N Hpro2O = cell 2F 4F 2F (where j cell is the current density (A.m-2)) N H2 =

(2)

The phenomenological model of the membrane described two water distributions diffusive created by the concentration gradient of water and the electro-osmotic drag [9]:

ρ dry dλ jcell − D mH 2O (λ ) (3) F EW dz (where nd is the number of H2O molecules dragged per migration of H+ ion as a function of the water 2 .5 λ . However, Gore® membranes drag coefficient was recently measured to be content n d = 22 N mH 2O = n d

approximately unity [10-11], λ is the water content, ρ dry is the membrane density (kg.m-3), EW is the equivalent weight (kg.mol-1)) At the membrane/anode interface, the continuity of the water flux density leads to: N aH 2O = N mH 2O

(4)

At the membrane/cathode interface, equation (5) can be obtained: N cH 2O = N mH 2O + N pro H 2O

(5)

Solving the above coupled equations, the water content λ is found and thus the membrane resistance Rm can be computed as follow [12-13]: Lm

Rm =

∫ 0

1

(0.5139λ ( z ) − 0.326 ).e

⎛ 1 1⎞ 1268 ⎜ − ⎟ ⎝ 303 T ⎠

2 dz (Ω.m )

(6)

As an example, the influence of n d on the membrane resistance can then be shown (Fig.2). When the incoming anode gas is dry (RHa=0%) and the cathode inlet gas is humidified (RHc=62%), a water concentration gradient is established across the membrane. Migration of water by electro-osmotic drag from anode side to the cathode side is less than back diffusion water. Thus, increasing the current (so, more water is produced by the electrochemical reaction) leads to a decrease of the membrane resistance as shown in Fig. 2. 30

Rmstack (nd=0,9)

Stack membrane resistance (mOhm)

28

Rmstack (nd=0,4)

26 24

Rmstack (nd=0,7)

22

Rmstack RHc=62% nd=2,5/22λ

20 18 16 14 12 10 0

10

20 30 current (A)

40

50

Figure 2. Influence of the drag coefficient on the stack membrane resistance. Such a one dimensional steady-state model allows studying the variation of the membrane resistance while considering variations of the electro-osmotic drag coefficient.

2.1.3. FC dynamic models For the dynamic and small signal modeling, some assumptions have to be considered. In this section, the air-breathing PEMFC system is firstly considered as isothermal and isobaric. Both these approximations appear to be valid since these conditions are normally achieved in a small single-cell experimental investigation. The total pressure at the anode and cathode compartment and within the gas diffusion electrodes is considered as constant. Moreover ionic ohmic drop in the active layer and the electronic ohmic drop in the current collectors can be neglected owing to the high electronic and ionic conductivities, and thus lead to the absence of voltage drop. It is also supposed that the gas permeation in the membrane is negligible.

When there is no mass transport limitation, the redox reaction is simply represented by an equivalent electrical circuit of parallel RC cells. However when there are considerable variations of the interfacial concentrations on electrodes, the redox reaction is represented by an equivalent circuit of parallel ZfC [2]. The faradic impedance Zf is then composed of two terms: a charge transfer resistance Rt and an impedance of diffusion species on both electrodes. This impedance of diffusion is called Warburg impedance and is represented by ZW [14]. The impedance of an electrode then corresponds to the parallel combination of the faradic impedance Zf and the double layer capacitance (Cdl) to account for the dynamics of the changing concentration in the gas backing layer and the charge stored in the interfacial capacitance. The expression of the electrode impedance is thus given by:

Z(ω) =

Z f (ω) 1 + iωC dl Z f (ω)

(7)

Finally, the total impedance of the fuel cell is composed of two impedances, one impedance for each electrode (anode and cathode), in series with the internal resistance Rm linked to the membrane. The total impedance ZTotal of the fuel cell is then given by: Z Total ( ω ) = Z a ( ω ) + Rm + Z c ( ω ) (8) Around a stationary operating point, we obtain the charge transfer resistance Rtk by differentiating the equation (8) with respect to the over potential (η):

Rtk =

1 ∂j k

1

=

∂ηk

γ k jok

2.3 exp ( bk

2 .3 ηk bk

) (1 −

j jlk

(9)

)

while the analytical impedance of mass diffusion (i.e. Warburg impedance) is expressed as [2,14]: ∂j k tanh( sτ k ) ∂C k δ Z Wk ( s ) = (10) ∂j k ne F Dkeff sτ k ∂η k where s = i ⋅ ω is the Laplace operator, and τ k = δ 2 / Dkeff , the time constant of diffusion. The equation (10) can be simplified by introducing the non-linear term Ak(j) which depends on the current : tanh( sτ k ) Z Wk ( s ) = Ak ( j ) sτ k (11) δ 1 where: Ak ( j ) = eff 2.3 j (1 − ) C k* ne F Dk bk jlk

The effective double layer capacitance C dleff is defined as C dleff = γ ⋅ C dl , where γ represents the roughness factor whose value is of the order of 100. Non-integer derivatives are introduced in the equation (11) using a Taylor approximation of the tanh function [15]: tanh( sτ ) sinh( sτ ) 1 sτ 1 1 = ⋅ ≈ ⋅ ≈ (12) s τ sτ cosh( sτ ) sτ 1 + sτ 1 + sτ 2 This limited expansion however remains valid at high frequencies; indeed, using the approximation of tanh function for high frequencies :

tanh( sτ) sτ

=

sinh( sτ ) cosh( sτ )

⋅

1

≈

e sτ

s τ ∞ e sτ

⋅

1

≈

1

≈

1

s τ ∞ sτ ∞ 1 + sτ

(13)

By using the approximation, we can simplify the analytical expression of the Warburg impedance given by the equation as follows:

ZWk ( s ) =

Ak ( j )

(14)

1 + sτ k

The half-order fractional model of the faradic impedance is then given by:

Z fk ( s ) = Rtk + Z Wk ( s ) = Rtk +

Ak ( j ) 1 + sτ k

(15)

The total impedance of the electrode is then given by:

Z electrode ( s ) =

1 1

Z fk ( s )

+ sCdleff

(16)

In Fig. 3, the various models of Warburg impedance, presented in the previous section, are compared in frequency domain, i.e. • Analytical Warburg impedance given by equation (11). • Classical model using 20 RC-cells in series. • Half-order fractional model given by equation (14).

Figure 3. Various approaches for the modelling of Warburg impedance. These curves have been simulated for A(j) = 1.0 Ω ⋅ rad 1 / 2 ⋅ s −1 / 2 and τ = δ 2 D = 1s . Thus, it can be noted that the difference between the fractional Warburg impedance and the analytical Warburg impedance is very small over a wide range of frequencies. The small difference that exists between the two models can be related to the 2nd order expansion of the tangent hyperbolic function. This limited expansion however remains valid at high frequencies. It is also worth mentioning that the fractional Warburg impedance represents resistive behavior at low frequency, which is easily observable by a horizontal line (Fig. 3a) and is characterized by a zero phase (Fig. 3b). Moreover, asymptotically, the Bode plot of fractional model is a straight line having a slope of -10dB per decade; while its phase is constant and equal to -45°. The asymptotic behavior of Warburg impedance is then totally taken into account by the fractional model contrary to the classical one, which behaves like a capacitance for high frequencies. The number of parameters describing each model can also be highlighted: 2 for fractional model and 40 for the classical R-C one! This property, which leads to system order reduction, allows decreasing the simulation times of a PEMFC system. Experimental identifications can then be realized in order to validate the half-order model on a 500 W PEM fuel cell. The identification of the impedance spectra obtained by Electrochemical Impedance Spectroscopy (EIS) provided a set of model parameters whose relevance must be checked. A convenient way to do this consists in comparing experimental and simulated Nyquist plots of the fuel cell at a given current density. Fig. 4 displays the Nyquist plots versus its fractional model fitted for the current densities of 0.068 and 0.070 A cm-2 respectively. Then, it can be observed that the results of modeling are here in good agreements with measurements.

Figure 4. Experimental and simulated Nyquist plots. 2.1.4. Pseudo 2D model In this part, the system under consideration is a unit-cell composed of the Membrane Electrodes Assembly (MEA) surrounded by the bipolar plates. A better understanding of the transport phenomena at stake in this system is expected to help in developing more efficient electrochemical converter that would lead to major gains in performances of the whole fuel cell system. However, it includes a wide variety of transport processes occurring in all of the fuel cell components, and building an integrated model of the most significant of them remains a challenging issue [16]: the bipolar plates, backing layers, electrodes and membrane are made of materials of very different size and structure, with a broad spectrum of characteristic lengths. On top of that, the difficulties in modelling the operation of PEMFC come from the complexity of the geometry of the flows of hydrogen and air that are threedimensional as much as from the condensation of liquid water. To be useful for diagnostic or transport applications, a dynamic approach is also required [17].

A two dimensional numerical model is built in order to predict water management and the polarization curve of a cell [18]. Local transport phenomena in the MEA are assumed one dimensional in the through-plane direction and changes in gas composition due to the electrochemical reactions consumption are taken into account in the gas channels. The molar changes along the channels are calculated by mass balance. This allows emphasizing the non-uniformity of water content (in gas or liquid phase), of the fluxes and as a consequence of the current density. The calculated current distribution is here confirmed by the experimental results obtained on a fuel cell specially developed for comparison with the numerical data [19]. At the local level, the model takes into account: - Liquid water and proton transports through the membrane: the so-called Schroeder’s paradox [20] is taken into account considering a linear law for the water content of the membrane when it is in contact with a mixture of saturated vapor and liquid water. Transport properties in ionomers and the couplings between them are not always well understood, even in usual fuel cell components like Nafion membranes [21-22], which complex structure is still intensely studied [23]. Owing to the lack of knowledge of sorption properties, an analysis of coupled mass and charge transfer in the pores of a Nafion membrane have been realized starting from Poisson-Boltzmann theory. Thanks to a homogenisation technique, macroscopic transport coefficients are calculated and the ionic conductivity variations with the membrane water content is deduced and used in the model [24]. - Two phase flows of water and gas diffusion in the Gas Diffusion Layers (GDL). The StefanMaxwell equations are used to calculate the transport by molecular diffusion of hydrogen and vapor at the anode side and of moist air at the cathode side. The liquid water is driven by capillary action and its flow is given by the Darcy’s law developed in the case of two phase flows in hydrophobic porous media. By considering a uniform gas pressure, the liquid water saturation is obtained [25].

The electrodes of fuel cells are considered as interfaces in the model and the total pressure and the temperature are assumed uniform in the whole fuel-cell. Continuity conditions are used at the interfaces with the gas channels and with the electrodes. Kinetic losses and concentration overpotentials are taken into account in the active layers through the Butler-Volmer equation. However, since the electrodes are considered only as interfaces, macroscopic mass transport limitations are attributed to the GDL: - The gas diffusion coefficients decrease because of the reduction of the gas phase volume in the case of liquid water occurrence. - The geometrical active surface is reduced by a factor depending on the liquid water saturation at the GDL-electrode interface. Microscopic flooding inside the nanostructure of the electrodes needs elaborate models including a detailed description of the catalyst layers. The introduction in the model of the limiting current density allows taking into account the effect of liquid layer at the catalyst particles surface. The simulated current density profile and its variations with the stoichiometry coefficient of the air are presented in figure 5 and show the same tendencies as the experimental observations. Thermal transfer simulation at the local level [26] shows that temperature gradients can be very important through the MEA. Although it can strongly influence water transport and above all liquid water appearance, heat transfer has not yet been included in the model presented in this section. The numerical effective drag coefficient and polarisation curve are compared with experimental data [27]. It validates partially the numerical model and emphasis the need to develop more elaborate approaches including detailed description of the catalyst layers. Since the transport of liquid water in the GDL is a critical point for avoiding flooding, it would be important to reach a good understanding and more accurate data of the water sorption/desorption at the membrane-GDL interfaces. Finally, the development of fuel cells depends also on fundamental issues still linked to transport phenomena, but more specifically to interface mass and charges transfers. These issues will be addressed in the near future. -2

10000

i [A.m ]

8000 6000 4000

λ Air = 2 λ Air = 6 λ Air = 20

2000 0 0

0.2

rel. channel coordinate (-) 0.4 0.6 0.8

1

Figure 5. Influence of the stoichiometry on the local current density distribution.

Figure 6. Polarisation curve – Numerical and experimental results.

Figure 7. Experimental and numerical variations with the current density of the net drag coefficient.

2.1.5. 2D linear elastic-plastic model of a FC stack Numerical modelling of PEM FC can be focused on the developing of mathematical tools allowing prediction of the electrical performance and the life-time of fuel cells and stacks. This is multiphysical problem which should correctly take into account coupled electrical, heat and mass transfer processes in different components of fuel cell stacks. The main trends in developing of fuel cell technologies deals with elaboration of advanced components (catalysts, GDL, bipolar plates), optimisation of running parameters, improvement of stack design and assembly procedure. For example, it was possible to reach [28-29] electrical performances around 1 W/cm2 with rather low Pt loading (about 0.2 – 0.3 mg/cm2/electrode) by optimising methods of MEA fabrication, assembly procedure and running parameters. Actually, this optimisation is based mainly on fuel cell tests and the main goal of numerical modelling is to support this experimental procedure.

Starting from the classical works [30-31] up to this moment [32-35] the main attention in mathematical modelling of PEM FC is given for heat and water management. All the above mentioned papers suppose that the mechanical properties of different cell components are known a priori and can be introduced in the model as some given parameters. From the other hand, the mechanical stresses arising in stacks during assembly procedure and due to swelling and heat effects in running system change physical properties of cell components. These mechanical phenomena can influence significantly on the electrical performance and the life-time of fuel cells. Some attempts in modelling of the mechanical behaviour of fuel cells have been done recently [36-38], but the problem is still open. The additional physical factors (mechanical stresses and deformations) complicate the problem, so the first step developed in this section is related to modelling of stresses arising in a fuel cell during the assembly procedure [39]. A linear elastic-plastic 2D model of fuel cell with hardening has been developed for this purpose. The model simulates mechanical behavior of the main components of real fuel cell (the membrane, the gas diffusion layers, the graphite plates, and the seal joints) and clamping elements (the steel plates, the bolts, the nuts). The stress and plastic deformation in MEA have been calculated using ABAQUS code with respect to the realistic clamping conditions. The examples of the results (for the applied mechanical load 1MPa) are presented in the local (Fig. 8) and on the global (Fig. 9) scales. The first one corresponds to the single tooth/channel structure. On the scale of a single tooth/channel structure the stress distribution in the membrane is not uniform and the stresses have a periodic character (Fig. 8). The stress evolutions along the membrane have been obtained on the scale of the entire cell as well (Fig 9). In particular, a zone with a strong heterogeneous stresses in the membrane under the junction seal joint/graphite plate has been observed and quantified. This heterogeneity is caused mainly by the structural features and by the difference between the stiffness of the seal joint and the gas diffusion layer. If the applied mechanical load corresponds to 1MPa, the plasticity does not arise in the membrane. Calculations show that the membrane reaches plasticity

under the torque which is equal 15.8Nm. The plasticity arises primarily in the zone of the maximal stresses, i.e. under the junction seal joint/graphite plate.

Figure 8. Distribution of normal (a) and shear (b) stresses at the centre (1) and at the edge (2) region of membrane

Figure 9. Stress distribution in the entire fuel cell: I - normal stress , II – shear stress, III – Misses stress. The further development is related to the prediction of the mechanical effects arising in running fuel cell and theirs impact on the fuel cell performance. For this aim two models for calculation of the temperature field in running fuel cell have been developed. The first one is based on the ABAQUS code and introduces the humidity as a variable parameter. The more detailed model is running under COMSOL and takes into account some specific effects, for example anisotropy of the thermal conductivity of gas diffusion layer. The first results show that arising in MEA stresses reach important peak values in transient regime during humidification step. These stresses are 3 times more important than applied during assembly procedure mechanical load can cause plastical deformation of the membrane.

2.2 Semi-empiric and empiric models As presented in the last section, fuel cell stacks and systems appear as inherently multi-physic and multi-scale objects. To understand their physical behaviours and to improve their performances, various skills and knowledge are needed: chemistry, electrochemistry, fluid mechanics, thermal, electrical and mechanical engineering… Advances in FC system developments are obtained by conducting a variety of investigations ranging from fundamental domains and material fields (with for instance, the development of new catalysts and new electrolytes) to more application oriented works as the optimisation of FC balance-of-plants to fulfil final operating conditions and requirements (e.g. load current cycles linked with dynamical mission profiles for vehicles). FC performances estimated at different scale levels (i.e. materials, components, single cells, FC stacks, and complete FC systems) are generally highly dependent on different physical phenomena from mixed domains. A high number of input factors that contribute to the FC final output voltage could be mentioned. These factors influencing FC performances would be related with the properties and sizes of FC materials and components (platinum loading of electrodes, membrane material and thickness, channel design of gas

distributor plates…), with operating variables (load current, stack and reactant temperatures, pressures, flow and humidity rates…) and technological solutions available for the ancillaries (scroll or screw type compressors, reactant humidification sections based on gas bubbling or enthalpy wheels…). On the whole, FC power generators are difficult to model due to their complex non linear multivariate natures. Moreover, these models require a good knowledge of the process’ parameters and in most cases, these parameters are difficult to determine for fuel cell stacks or systems. Thus, in addition to the classical mechanistic modelling approach (described in the last part of this paper), some semiempirical or empirical modelling approach also exist for the fuel cell stacks. 2.2.1. DoE modelling approach To design or to characterise FC stacks or systems, the approach that consists in one-factor-at-a-time experimentations is really time-consuming and thus quite limited. It is neither appropriate for in-lab testing of large power stacks nor for FC durability tests, which imply large reactant consumptions and therefore higher financial costs. Modelling the FC system electrical or thermal power outputs by considering the input factors without proper statistical methods during the test analysis stages is not effective since strong interactions between parameters can exist. So, as FC experiments are generally long and expensive, as complex interrelations between physical parameters exist in FC gen sets, the tests have to be carefully implemented and analysed. This can motivate Design of Experiment (DoE) approaches rather than (or complementary to) first-principles / mechanistic models [40].

Many aspects and tools of the DoE methodology can be of great benefit for various scientific and technological purposes: development of FC materials, components and ancillaries, analysis and improvement of single cell and FC stack performances, evaluation and development of complete FC systems (enhanced design, improved operating conditions, more efficient diagnosis of FC state-of health…). The DoE methodology dates actually from the beginning of the last century with the work of R.A. Fisher (1925). Many other scientists have contributed to the development of the DoE methodology, among them: G.E.P Box, R.C. Bose, G. Taguchi, F. Yates [41]. The objective of the DoE method is to increase the productivity of the experimental process, especially by minimising the number of test runs and by maximising the accuracy of the results. The method allows the significant factors affecting the studied process to be determined and it can also highlight some possible interactions between the various factors. It is a structured, efficient procedure to plan some experiments and to obtain some data, which can be analysed in order to yield valid and objective conclusions about the studied product or process. As already suggested, a large number of experimental tests is generally needed to correctly determine the performances of FC stacks or to identify the parameters of FC physical models. The choice of an experimental design depends on the objectives of the experiment to be conducted and on the number of factors to be investigated: • Comparative objective: the primary goal of the experimental design is to conclude on one apriori important factor (in the presence of and/or in spite of other factors). Here, the question of interest is whether or not that particular factor is significant [42]. • Screening objective: in this case, the purpose of the experiment series is to select or screen out the few important main effects from the many less important ones [43]. Once the key factors are identified by the screening, the Response Surface Method can be used. • Response Surface Method (RSM) objective: the experiment is designed to allow the estimation of factor interactions and even quadratic effects. Therefore it gives an idea about the (local) shape of the investigated response surface. The RSM designs are used to find improved or optimal process settings, troubleshoot process problems and weak points, and to make a product or a process more robust against external and non-controllable influences [44]. • Optimal fitting of a regression model objective: if the experimental response is modelled as a mathematical function (either known or empirical) of a few continuous factors, then the model parameters have to be properly estimated using a multi-linear regression design. Once a suitable approximation for the true functional relationship between the independent variables and the surface response is found, the optimisation of the response variable can be made. A growing number of FC studies involving DoE can now be found in the literature. The DoE methodology has the potential to allow efficient test definitions for fast and well-organised characterisations of FC systems. DoE techniques can be first applied to organise the tests in the

experimental characterisation stages but also in the test analysis and modelling phases in particular when time-consuming simulations due to model complexity are encountered. The DoE methodology with its statistical techniques is then well-suited to analyse the tests conducted on FC system since it offers a wide range of practical tools (e.g. ANalysis Of Variance - ANOVA), graphical representations and techniques (e.g. graphs of the average effects and interactions) that can be really suitable mediums for FC experimenters and developers. In addition, some statistical / numerical relations can be used to predict the behaviour of the investigated systems as a function of various operating parameters. Then, some control strategies can be developed to optimise relevant criteria like FC voltage, fuel consumption, maximal electrical power or even stack lifetime [45-46]. Some design or operating parameters can also be selected so that the FC system or the FC device will work well under a variety of conditions, so that robust performances will be achieved [44]. Furthermore, the statistic-based models can be proposed in pre-stages of physical models. With the proposed approach, some mechanistic models can be developed in a more efficient way since valid assumptions can be made regarding the different influences of the various investigated parameters. The DoE exercise should certainly be considered as an intermediate and necessary stage between the pure experimentation processes and the next physical modelling phases. 2.2.2. ANN modelling approach However, in addition to this DoE approach, it is also possible to get a behavioral modeling of fuel cell stacks and still to avoid identifying all the parameters by using so-called "black box" models. These models are based on a set of easily measurable inputs like temperature, pressures, or current and are able to predict the output voltage of the fuel cell stack. Today, static and also dynamic models of fuel cell stacks or systems based on Artificial Neural Networks (ANN) can be found in the literature [47].

In the proposed model [47], inputs of the dynamic model are the requested current, the operating temperature, the incoming air hygrometry level, and the air and hydrogen incoming flows. The output is the obtained voltage. In order to give more relevance to the time dependence of an output to itself, the self feedback loops were designed to provide different time states of the output. The general structure of the dynamic model of the PEM FC stack is given in Fig. 10. Taking into account the whole possible frequency range for the current in the considered fuel cell system, the proposed neural network is composed of four modular networks. The first one, which computes the low frequency signals, the second which computes intermediate frequency signals, the third one is dedicated to high frequency signals and the last one is used for static mode. The three dynamic modular networks have the same inputs, outputs and architecture and have to be trained separately. Then, to test the network, it has to be switched on to the needed modular networks. Afterward, to have a single and efficient NN model able to operate at any frequency, the Discrete Fourier Transformation (DFT) is used to perform an input preprocessing by analysing the frequency spectrum of the current. Once all the sine curves are composed (as obtained by the DFT), the modular sub-networks corresponding to the frequencies are sought. Then, each voltage response provided by a sub-network is collected and all voltage responses are summed to obtain the final voltage response of the fuel cell stack. Experimental results are compared to those obtained thanks to the ANN dynamic model of the fuel cell stack in Fig. 11. The maximum difference between experimental points and calculated ones is lower than 2.9% which allows us to conclude on the good performance of the considered modeling methodology. Nevertheless, the main drawbacks of this modeling approach are the huge number of experimental tests that are required to perform a well-suited model and the loose of any causality in the final model, reducing the scope of interest of this kind of model.

ANN1 Low frequency

∆t = 100 ms

-1

-n

Z to Z

ANN 2 Intermediate frequency

∆t = 1 ms -1

-n

Z to Z

ANN 3 High frequency

∆t = 25 µs -1

-n

Z to Z

ANN4 Static mode

Figure 10. General structure of the dynamic ANN of a PEM FC stack. 25

UNN=f(t), FSA=2, FSC=5, T°=50°C, H=100% Umeas=f(t), FSA=2, FSC=5, T°=50°C, H=100% I=f(t), FSA=2, FSC=5, T°=50°C, H=100%

U (Volts), I (Amps)

20

15

10

5

0 0

0.05

0.1 t (s)

0.15

0.2

Figure 11. Experimental and simulation results (requested current and resulting voltage). 2.2.3 Modelling approach based on electrical analogies One step forward can be found the electrical approach. Unlike in the DoE and the ANN modeling approach, certain knowledge of the behavior of the stack is needed a priori but none or few internal parameters of the stack are used to tune the parameters. This approach can then be applied to any commercial stack without the help of the know-how of the supplier. The basic idea is to find a common way to represent the different physical laws involved in the stack in the fluidic, electrical and, if needed, thermal model, and implement it easily in the same environment [48].

Each component of an hydraulic or pneumatic circuit is represented by its electrical equivalent in an electric circuit. Following Bernoulli continuity law, mass flow and energy are conserved in an ideal pipe with an incompressible fluid; however a real pipe would introduce energy losses that are related to not conservative pipes and fluid interactions. Fluid compressibility will add stocking capacity to the pipe, which can actually store potential energy in the form of static differential pressure. Making pressure an image of potential internal energy in a gas, the pressure can be represented in an electrical circuit by an electric potential difference V. In the same way, mass flow and charge flow can be considered as equivalent expressions in either system. Following the previous directives, a quadri-pole model for the different fluid circuit components can be built. Pipes and tanks containing input and output nozzles can be modeled by a T equivalent quadri-pole circuit consisting in a capacitive first order filter as shown in Fig. 12. Since the fluid inertia for fuel cell normal operation conditions is considered relatively small, gas momentum can be neglected. Otherwise, an inductance representing kinetic energy should be included in the quadri-pole equivalent model. For this model construction, pressures are taken as relative pressures rather than absolute pressures and atmospheric pressure conditions are assumed as ground potential. Fuel cell channels can actually be modeled by series of pipes that connected together will lead to a T equivalent circuit with equivalent resistances and capacities, product of channel geometry and distribution in the fuel cell stack. In order to calculate internal partial pressure evolution, gases in cathode and anode channels have to be decoupled to their elemental components. Each specie is considered as a single gas, and physical interactions between different species is neglected, giving as result three different representations, for either cathode or anode channels. Global pressure and flow are obtained by applying superposition properties to the different models developed, according to the different gas species present in the system. In the case of hydrogen and oxygen, gas consumption due to the electrochemical reaction takes place. The gas consumption rate is a linear function of stacks current following the Faraday law. This gas consumption will be modeled in its electrical equivalent circuit by a current source. Anode and cathode channels can be treated separately for this two species, since no interaction between channels is considered. As a first approach, the nitrogen diffusion coefficient can be assumed constant for the range of water activity present in normal operating conditions of the fuel cell stack. The nitrogen diffusion constant is modelled by a resistance joining both anode and cathode models. Both water produced and electro osmotic water flows are dependent of stack current, and are modeled by current sources acting on corresponding channels. Water diffusion is modeled by a resistance linking the cathode and anode circuits. As in the case of nitrogen, water diffusion coefficient and electro osmotic drag coefficients, are dependent on membrane water content, which is usually estimated by water activity on either channel. The system is no longer linear, when water partial pressure arrives to saturation level. At this point, water partial pressure does no longer evolve and liquid water is formed, either as suspended particles or as liquid spots over channel surfaces, implying the insertion of a non linear component into the electrical equivalent model. A Zener diode in parallel with the channel capacitance in the water equivalent circuit will limit vapor partial voltage pressure to the stated saturation voltage image (Fig. 13). The voltage is then computed according to the Nernst equation. The Fig. 14 shows the validation of the model on a 20 cell PEFC stack. The thermal behavior can be simulated through the nodal approach which uses the same kind of analogy for the heat flow (as a current) and the temperature (as a potential) (Fig. 15). A similar approach has been used for SOFC as well [49].

Figure 12. Pneumatic component and electric equivalance

Figure 13. Water electric circuit

Figure 14. Voltage simulation for current step profile and experimental validation

Temperature potentiel

j

Heat flow Node of system Thermal resistance towards the others nodes, G i

Heat source, Q Heat capacity related to material, C

Figure 15. Principal of thermal nodal modelling 2.2.4 Modelling approach based on energy analogies The previous approach proposed to model fluidic and thermal phenomena with a fuel cell stack by electrical analogies. This kind of approach was in fact extended to a great number of fields of physics through an energy approach carried by the Bond Graph modelling in the sixties [68][69].

The Bond Graph is an explicit graphical unified formalism where the energy exchanges within an energy system are described by bonds which represent the power exchanges (see Fig. 16 as an example). Two variables, effort and flow, are associated with each bond. These two variables have different interpretations in the different fields of physics: voltage-current, chemical potential-molar flow, pressure-volume flow… The product of these two variables is the transferred power. The bond is arbitrarily oriented by a half arrow which indicates the positive power flow orientation. Only a limited number of elements are necessary to describe the majority of energy systems: dissipative RS element, potential energy storage C element and kinetic energy storage I element (not

used here). The connections between these elements are implemented through junctions. There are two kinds of junctions: 1-junctions and 0-junctions which respectively correspond to series connections and parallel connections in electricity field. They express in fact the generalized Kirchoff’s laws. Other elements, like the transformer TF element, are used to pass from a field of physics to another. Causality is a fundamental concept in Bond Graphs: it defines the cause-effect relations. The causal bar indicates the effort direction. A C element always enforces the effort, an element I the flow. The R element adapts itself in function of the remainder of the system. Causal rules enforce that only one connected element can fix the flow through a 1-junction or only one connected element can fix the effort at a 0-junction. For instance, a C element connected to a 0-junction fixes the effort at the junction; the rest of the system fixes the flows through this junction. For the Bond Graph model proposed in Fig. 16, the fuel cell stack was modelled by its “equivalent mean cell” (arithmetic average of the stack voltage in steady state) which allows a rather light implementation. That assumed that the unbalances are weak between the cells. With the same will of simplification, the effects of each electrode were combined in the proposed model by considering an elementary cell. But a version with dissociated electrodes was also developed [70]. Firstly, the measurements are indeed generally achieved only at the terminals of the cell. Secondly, the laws for the phenomena are identical for their literal expression for each electrode. This concept was extended to the case of a fuel cell stack. The fuel cell was thus considered from a macroscopic point of view, even if the contribution of different physicochemical phenomena were modelled. The hydrogen and oxygen tanks are seen here as a single source of effort Se which enforces the pressure to the fuel cell. The involved molar flow has thus no real sense in practice. The multiport C element models the global potential chemical power carried by both reactants. Be careful, it is not really a storage phenomenon as usually for a C element. At this stage, a first exchange, the entropic losses, occurs with the thermal model (which enforces the temperature) when the fuel cell delivers electrical current. Afterwards, the theoretical reversible conversion of this chemical energy into the electrical energy is modelled by a TF element whose ratio is 1/2F. The obtained theoretical electrical power is thus progressively degraded by all the considered irreversible physicochemical phenomena with their associated dynamics: the activation phenomena (Butler Volmer’s law), the double layer phenomena, the diffusion phenomena (response to a Fick’s law) through the gas diffusion layer and the active layer, and the ohmic phenomena (all the involved charge conductions). Each one of these phenomena is modelled by an RS element which exchanges thermal power with the thermal model (which enforces the temperature) when the fuel cell delivers electrical current. All the considered dynamics are modelled by C elements which represents storage phenomena. It is obvious for the double layer phenomena which is an electrostatic storage. For the diffusion phenomena, it can be interpreted as an image of species’ local tanks in the respective considered layers. The output of the model is the available electrical power which is “amplified” by a TF element whose ratio is the cell number. The flow source Sf models the electrical consumption; it enforces the current to the fuel cell stack. The thermal model is here expressed in pseudo Bond Graph because the thermal flow is a power. Thermal conduction and convection phenomena are modelled by R elements which are here non dissipative. The thermal inertia of the stack is modelled by a C element. The TF element whose ratio is the cell number emulates the thermal power generated by the whole fuel cell stack. In Fig. 17 is proposed the “same” model but drawn with electrical circuit formalism [71][73] which can be seen as a particular case of the Bond Graph formalism. Be careful, this model is not an impedance model: it is really a large signal circuit model. Both models give identical performances (an example is given in Fig. 18) if temperature and pressure are assumed constant. The reversible potential models directly the consequence of the energy transformation which is not explicit any more. The use of current sources piloted by voltages makes it possible to translate the causality of phenomena, causality which is clearly expressed in Bond Graph formalism.

Thermal model TM5

T(K) ηdiffAL .I (W)

C

Diffusion losses AL

ηdiffGDL

T(K) T∆S.Jf (W) P (Pa) TF

D stack (m3.s-1)

-∆G -

C

D cell (m3.s-1)

E rev If

TF

Jf

1/2F

ηact

1

T(K)

RS

Activation losses

If

T(K) (W)

RS

Thermal model TM4

Diffusion ηdiffGDL .If losses GDL

If

Thermal model TM3

η act .If (W)

Eth - η

conversion Chemical energy ↔ Electrical energy

Potential thermochemical power transported

Ideal “tank”

ηdiff

0

1

0

Thermal flow

Se

C

RS

ηdiffAL

Thermal model TM1

P (Pa)

Diffusion

Dynamics diffusion GDL

Dynamics diffusion AL

Eth - η

0

I

Eth- η- ηohm = U

1

cell

Ustack (V)

TF

I

I

Thermal model

Sf

Electrical source

Thermal power of the stack (W)

T(K)

Reaction heat TM1

Thermal power of the cell (W)

TF

C

Activation losses TM3

RS

Activation + Double layer phenomena

Diffusion losses TM4 TM5

T(K)

Double layer capacitor

Ohmic losses TM2

Ohmic losses

Charge conduction

T(K) ηohm .I (W) Thermal model TM2

c

Figure 16. Large signal and dynamic BOND GRAPH model of a PEM fuel cell (combined electrodes) Activation losses and double layer

Diffusion losses

η diff

η diff

GDL

Reversible potential

nact

AL

Erev

nohm I

Rohm

C diff GDL Cdl

C diff AL

-

Ohmic losses

+ Ucell

Figure 17. Large signal and dynamic circuit model of a PEM fuel cell (combined electrodes) P

gaz

= 2bars, T = 65oC

P

0.95

0.95

Expérimental Modèle

0.9

gaz

P

= 2bars, f=513mHz

gaz

= 2bars, T = 65oC

1 Expérimental Modèle

0.9

Expérimental Modèle

0.85

0.85

0.9

0.8

0.8

0.7 0.65

0.75

tension [V]

tension [V]

tension [V]

0.8 0.75

0.7

0.65

0.6

0.6

0.55

0.55

0.5

0.5

0.7

0.6

0.5

0.45

0

2

4

6

8

courant [A]

10

12

14

16

0.45

0

2

4

6

8

courant [A]

10

12

14

16

0.4

0

2

4

6

8

10

12

14

courant [A]

Figure 18. Comparison model/experiment for a single PEM fuel cell (2 bars, 65°C) for three low frequency large amplitude (0 to nominal current) current sweeps (the model parameter values are identical in the three cases)

16

3. PEM FUEL CELL SYSTEM MODELING APPROACHES The second part of this review paper will be devoted to the modeling of PEM FC systems, i.e. the FC stack itself and the different ancillaries required for the operation of the FC stack (air compressor, electrical power converter, cooling subsystem, gas supply and processing subsystem …). In this part, a distinguish will be done between the analytical modeling approaches devoted to FC systems and those based on graphical modeling formalisms. 3.1 Analytical modeling approaches 3.1.1. Modeling the fuel cell system for control purposes The global scheme of a fuel cell system as already been given on figure 1. This scheme can be modified into a realization scheme, taking into account the required specifications of the considered FC system. Fig. 19 presents such a realization scheme. As shown in this last figure, three control loops have to be considered in this FC system; namely: the air supply circuit (controlled by a motorcompressor group and a backpressure valve), the hydrogen supply circuit (here controlled by a passive pressure reducer valve) and the water cooling circuit (controlled by a pump and two bypassing valves). Thus, these loops can be modelled and regulated in an independent way. All dynamics (power profile, fluid, thermal, electric) must be scaled in time and controlled not only to track power but also to maintain safety and allowing easier global behaviour computation. This global behaviour must allow extracting efficiency behaviour requested by most of optimisation algorithms to manage energy consumption in systems using it. E lectric Circuit M otor/C om pressor

P o w er D em a nd

A ir E n try

H2 E vaporator/ C on den ser V en tilator

P reducer

out air P Regulator

T em perature con trol valves

cathode anode

an od

circulation out H2

Separator h ydrog en

Pum p

air, hum id air T an k

liquid water coolin g/eatin g

EA

W ater excess

E lectric circuit

Figure 19. An example of PEM fuel cell system. The hydrogen loop The hydrogen loop consists here of a hydrogen tank, a pressure valve, the anode compartment and an atmospheric release valve. This last valve is required to regularly purge nitrogen gas and water vapor that have migrated from the cathode into the anode compartment, reducing the accessibility of hydrogen onto the catalyst sites. The pressure valve reduces the pressure from the high-pressure H2 storage cylinder (i.e. the H2 tank) to the desired working pressure. This loop consists of a simple mechanical pressure regulator that is supposed to achieve the desired absolute pressure under all flow conditions. If the dynamic of the primary hydrogen supply is smaller (e.g. using hydrogen produced from reformed hydrocarbons), a modification of the control model should be made to take this possible slower dynamic into account.

The air loop To deliver the requested power at the desired output voltage, the air circuit must supply the fuel cell stack with a sufficient flow rate of air at a fixed target pressure. The air circuit is here based on a motor-compressor group and a valve, providing two control parameters – the speed of the synchronous machine that drives the compressor (used to regulate the air flow) and the set point of the valve (used to regulate the air pressure). The air flow control allows furnishing oxygen to the FC stack and thus also providing the requested electrical power to the load. A model describing species balance, pressure and flow is a way to extract parameters to describe the FC voltage versus current behaviour. Considering the air compressor, using state space representation, using the data sheet representing the compressor map in linearizing/decoupling control scheme allows fixing pressure and following flow references. The cooling loop The last, but not least, important loop in the FCS is the water loop, which is used to control the fuel cell temperature. The cooling system can be separated into 4 blocks. In fact, there are two thermal elements that contribute to the thermal model (i.e. the heat exchanger and the FC stack itself). There are two inputs by means of which this loop can be controlled: i.e. the valves S1 and S2 leading to two water flow rates. There are also two disturbance terms which are the thermal heat from the electrochemical losses within the cell (Qelec), and the ambient temperature (Tamb). FCS Behavior For an efficiency point of view, the compressor state and the voltage provided by the fuel cell are the most important variables. These values allow computing the FC efficiency and computing the power needed by the compressor Pcomp : Pcomp

Cp ⋅ Te = η m ⋅η c

⎛ ⎜ ⎛ p cathode ⎜ ⎜⎜ ⎜ ⎝ p atm ⎝

⎞ ⎟⎟ ⎠

γ −1 γ

⎞ ⎟ − 1⎟ × Q air ⎟ ⎠

(17)

where Qair is the air flow compressor outlet kg.s-1 – linked to the FC requested current, γ is the polytropic coefficient, p x are the pressure values, Te is the air temperature and η x are the efficiencies. Evaluation of the FC stack output voltage expression has here been done considering a mechanistic model. Some parameters specific to each FC stack have thus to be known (construction and scale factors, double layer capacitance, global internal resistance…). For purposes of parametric identification, a set of static tests conducted can be done on a real FC stack. FCS Chopper In the power converter used to link the FC stack to the DC bus there are classical losses due to conduction and commutation losses in switches. Depending on the switches characteristic available through constructor data sheets, it is possible to identify it. Of course the power losses depend also on the average current. 100 90 80

rendement (%)

70 60 50 40 30 20 10 0

0

50

100

150

200 250 Puissance (kW)

300

350

Figure 20: losses in boost chopper

400

FCS global efficiency The losses in the other FC ancillaries (mainly compressor) can be taken into account in the fuel cell behaviour and current computation. The losses in the inverter can be computed based on the data sheets and average current and voltage knowledge. In the same way losses and global efficiency for the electrical energy storage devices can also be done (here for example buck/boost chopper and supercacitors efficiency curve).

Figure 21. FCS and Storage Element efficiencies depending on delivered power. With such accurate data and modelling, optimization algorithms have accurate data to manage global energy consumption optimization, in a non-time consuming way. 3.1.2. Modeling of the air supply system of PEM fuel cells for control purposes As already said, a proton exchange membrane fuel cell system is composed of four main subsystems: the hydrogen subsystem, the air subsystem, the humidifier and the cooling system. On the one hand, knowing that the degree of humidity and the temperature can not change rapidly, the control problem of these two subsystems can be decoupled from the rest of the system. On the other hand, the hydrogen subsystem is mostly controlled by a (fast) electrical valve, while the air subsystem is mostly controlled with a (slow) mechanical device (e.g., an electrical motor and a compressor), suggesting another time-scale decomposition. So, this section is dedicated to the air supply subsystem of the fuel cell, whose dynamic behavior is described by a highly uncertain nonlinear model.

Mathematical model In the interesting work [50], it is shown that, for the purposes of air supply control, it is reasonable to reduce the classical 9 -th order model of [51] to the 4 -th order dynamics1 x& = f ( x) + g u u + gξ ξ (18) with the state space vector x = [ x1 , x2 , x3 , x4 ]Τ , where x1 and x2 are the oxygen and nitrogen partial pressures in the cathode, respectively, x3 is the angular speed of the compressor and x4 is the air

pressure in the supply manifold, ξ is the stack current, which is traditionally considered as a measurable disturbance to the system, and u that represents the compressor motor voltage, is the control input. The components of the vector f ( x) are

c3 x1ψ ( x1 , x2 ) , c4 x1 + c5 x2 + c6 c x ψ ( x1 , x2 ) f 2 ( x) = c8 (− x1 − x2 + x4 − c2 ) − 3 2 , c4 x1 + c5 x2 + c6

f1 ( x) = c1 (− x1 − x2 + x4 − c2 ) −

1

(19) (20)

To keep the physical meaning of all the signals and constants we have decided to avoid normalization and rescaling, alas, at the prize of a more cluttered notation.

⎡⎛ x ⎞ c12 ⎤ ⎢⎜⎜ 4 ⎟⎟ − 1⎥ h3 ( x3 , x4 ), (21) ⎢⎣⎝ c11 ⎠ ⎥⎦ ⎡ ⎡⎛ x ⎞ c12 ⎤ ⎤ ⎢ f 4 ( x) = c14 1 + c15 ⎢⎜⎜ 4 ⎟⎟ − 1⎥ ⎥[h3 ( x3 , x4 ) − c16 (− x1 − x2 + x4 − c2 )]. (22) c11 ⎠ ⎥ ⎢ ⎢ ⎥ ⎝ ⎣ ⎦⎦ ⎣ where all the constants ci , i = 1, K,24 are positive, and depend on the physical parameters of the fuel c f 3 ( x) = −c9 x3 − 10 x3

cell.2 The input vectors are g u = c13e3 and gξ = −c7 e1 , with ei the i -th element of the Euclidean

basis. The function ψ ( x1 , x2 )

is the total flow rate at the cathode exit, defined by

⎧ c11 ⎪c17 ( x1 + x 2 + c 2 )⎛⎜ ⎜ x + x +c ⎪ 2 2 ⎝ 1 ⎪ ψ ( x1 , x 2 ) = ⎨ ⎪ ⎪c 20 ( x1 + x 2 + c 2 ), ⎪ ⎩

c

⎛ ⎞ 18 c11 ⎟⎟ 1 − ⎜⎜ x + ⎝ 1 x2 + c2 ⎠

c

⎞ 12 ⎟⎟ , ⎠

for

for

c11 > c19 x1 + x 2 + c 2

(23)

c11 ≤ c19 x1 + x 2 + c 2

The vector of measurable outputs is y = [ y1 , y 2 , y 3 ] , where y1 = h1 ( x1 , x2 ) is the fuel cell voltage Τ

(also known as polarization characteristic), y2 = x4 and y3 = h3 ( x3 , x4 ) is the compressor flow map. See [51],[53] for the description of h1 ( x1 , x2 ) and h3 ( x3 , x4 ) . The performance variables for the fuel cell system are T

⎡ ⎤ c z = [z1 , z 2 ] = ⎢ y1ξ − c 21u (u − c 22 x3 ), 23 ( x 4 − x1 − x 2 − c 2 )⎥ , c 24 ⎣ ⎦ where z1 is the net power delivered to the load and z 2 is the oxygen excess ratio. Further details of T

this model, including the assumptions that have been made to validate the reduction from the model of nine states described in [51], can be found in [50]. Model reduction The special dependance of the system dynamics on the coordinates x1 , x2 motivates the (partial)

change of coordinates χ := x1 + x2 + c2 . Furthermore, we observe that in a wide range of the operating region the following approximation holds: Assumption A1 c4 x1 + c5 x2 + c6 = κ ( x1 + x2 + c2 ) , for some positive constant κ . After some calculations for the dynamics of χ , we noticed that the system dynamics depend only on

χ , and not on x1 or x2 ; the system can be described by a third order model in the state ( χ , x3 , x4 ) . A further simplification is obtained observing that ψ ( χ ) can be suitably approximated, in a wide range of the operation regime, by a linear function in χ . Thus, we make the following: Assumption A2 ψ ( χ ) = c20 χ , for some positive constant c20 . Under this new assumption, the dynamical equation of χ becomes a linear function χ& = − µ1 χ + µ 2 x4 + µ3 − µ 4ξ , where

µ1 = c1 + c8 +

c3c20

κ

, µ 2 = c1 + c8 , µ3 =

c2 c3c20

κ

, µ 4 = c7 .

To prove the validity of these two reduction hypotheses, simulations have been done to compare the response of the original system and the reduced one, when changing the perturbation ξ in a large Check [51] and [52] for the definitions of the constants ci , i = 1, K ,24 , that appear in the model of the fuel--

2

cell system (1) and for the numerical values and other details.

domain. It has been noticed that the relative error does not exceed 1.5% in open loop simulations. 3.2 Graphical modeling approaches 3.2.1. Energetic Macroscopic Representation As presented in the previous section, a fuel cell system contains several components which have to supply several functions. Furthermore, the thermal, hydraulic, pneumatic and electrical phenomena are strongly coupled and present different time response constants. Then a graphical approach is a powerful tool to handle the complexity linked to the multi-physics and multi-scale nature of fuel cell systems. The Energetic Macroscopic Representation which has been initially developed for electromechanical systems [54] can be successfully applied in the fuel cell field [55,56]. It organizes the model around the action reaction principle, i.e. an action on an element leads to a reaction of the element on its environment. This basis leads to express clearly the causality between the different components or phenomena taking place in the system. The coupling is respected but the cause and consequence question cannot be avoided. When it is possible, the action and reaction pair is preferably chosen to 1

The graphical description of the EMR includes three element types: the sources, the converters and the accumulation blocks. Converters can be mono or multi physics and can couple many subsystems. The relation between their inputs and their outputs are instantaneous. The accumulation elements represent the parts of the systems which store energy. Consequently, the input-output relation is time dependent and induces delays. The way the component is modeled inside the block can change from one block to another and the scale of the description as well as long as the general rules concerning the inputs and the outputs are respected. The stack is divided into three major parts for three physical fields (Fig. 22). Firstly, a fluidic part provides the gas supply. It is composed of two lines (one for the hydrogen supply and the second for the air supply) modeled with an electric analogy. Secondly, an electrochemical part represents reactions that occur within the stack. Thirdly, an electric part modeled the capacitive effects of a Fuel Cell (FC) due to the charge double layer phenomenon [56, 57, 58]. This air supply subsystem is made up of a compression head (here a vane supercharger) driven by an electric motor (here a DC motor). Previous studies [59] show that the dynamic of the compressor is provided by the DC motor. Consequently, the compression head is modeled in a quasi-static way (17). The EMR formalism represents the compression head (Fig. 23) with a multi-physics coupling between two domains (fluidic and mechanic). It is interesting to focus on the fluidic inputs and outputs. On the air inlet, the pressure is imposed by the atmosphere (atmospheric pressure). On the outlet, it is imposed by the stack. The air flow is imposed by the rotation speed of the vane supercharger. The compression head is not an accumulation element and thus, there is no causality problem: the inputs and outputs are fixed by the environment of the compression head (the atmosphere, the DC motor and the stack). The DC motor is supplied through a chopper (mono physical converter) which enables the control of the rotation speed. The EMR of the motor consists of two accumulation elements (the electrical windings and the mechanical shaft, modeled as first order systems) and an electromechanical converter. The power electronic converter allows the coupling between the FC stack and the load, a DC bus for instance. In many cases, the fuel cell and the DC bus present very different voltage levels. Thus, a multi stage converter is required. The architecture presented here is a DC/DC full bridge converter [60, 61] (Fig. 24). It is designed around a High Frequency (HF) transformer. The DC/AC conversion is realized by an inverter on the FC side and the AC/DC conversion by a rectifier on the DC bus side. A LC filter reduces the current ripples on the rectifier output. The power converter losses are not modeled. In fact, the switches and transformer efficiency is very high (up to 90%) compared to the fuel cell efficiency. The behavior of these devices is approached by using mean values models. For example, the inverter is modeled as (2). minv is the modulation function imposed by the control part: the switch functions are not detailed. The EMR of this converter is directly deduced from the

architecture (Fig. 25).

Figure 22. EMR of the fuel cell stack

Figure 23. EMR of the fuel cell compressor

Figure 24. Electric scheme of the DC/DC full bridge converter

Figure 25. EMR of the DC/DC full bridge converter

3.2.2. Causal Ordering Graph Another approach for the graphical modeling of a fuel cell system lies on the so-called Causal Ordering Graph (COG). The Causal Ordering Graph (COG) is a graphical representation of mathematical equations based on integral causality, which is used to model systems and to design controls [62-64]. Balloons with equation numbers represent modeling relations. A static relation is depicted as a balloon with a bi-directional arrow (Fig. 26a). Physically, it has an external causality orientation. Either of x and y can be externally set as input and the other becomes output. Ra : y (t ) = Ra ( x(t )) . The task of the control function is to tune the input in order to make equal the output to a reference value yref. The exact value xreg of the input is calculated by inverting the modeling equation Ra, which is called Rac : x reg (t ) = Rac ( y ref (t )) .

A dynamic relation is depicted as a balloon with a unidirectional arrow (Fig. 26b). A dynamic relation is usually modeled by a differential equation, dy(t)/dt=ax(t)+b. It has an internal causality orientation,

since the output y is always obtained by integration of the input x. Rb : y (t ) = Rb ( x(t ), t ) . To control the output, the pure inversion of the equation Rb introduces instabilities due to the differential terms. An alternative method is the feedback control or closed loop control. It consists in calculating the error between the sensed value of the output and the wished reference value. Hence, a corrector Cm calculate ) the input variable xreg in order to minimize this error: Rbc : x reg (t ) = C m ( y ref (t ) − y (t )) . x

Ra

xreg

Model Control R ac

y

y ref

(a) static relation

x

xreg

Rb

Model Control Rbc

y

) y yref

(b) dynamic relation

Figure 26. COG of static and dynamic relations with their control schemes.

Modeling of a PEM fuel cell system by COG In order to simplify the modeling, only power conditioning system, fuel cell stack and gas handling system are considered (Fig. 27). We omit the thermal and humidity supervisory systems by assuming that the temperature and the humidity are well controlled. Here, the load is represented by a current source. The COG of the system modeling is shown in Fig. 28 and the mathematical equations are listed in Tables I, II and III. Power Conditioning Unit Modeling: Since fuel cells work always as a dc current source, a dc-dc (boost) converter is used, including a dc-link capacitor, a (boost) chopper and a chock filter (Table I). Fuel Cell Stack Modeling: The voltage across the fuel cell stack is calculated as a function of the stack current, reactant partial pressures, the fuel cell temperature and the membrane humidity by using a combination of physical and empirical relationships [65]. Since the temperature and the humidity are assumed as constant parameters, only electrical modeling, electrochemical modeling and hydraulic modeling are considered (Table II). Gas Handling System Modeling: The gas handling system regulates the gas input and output flow rates to control the pressure conditions of the stack. According to the fuel cell type (atmospheric fuel cell or pressurized fuel cell), the gas handling principles can be very different. Here, we list universal modeling equations (Table III). Advantage of Modeling by COG Advantages - Clear view of the causality: The causalities are clearly shown with bidirectional arrows for static relations and unidirectional arrows for dynamic relations. The state variables, which can be easily found with dynamic relations, are usually the main concerns for control design, such as fuel cell current ifc (with equation R4), sufficient reactant quantities PH2 and PO2 (with equations R8 and R9). - Easy design of the control system: By implying inverting rules, all control functions are easily established. The control inputs can be easily found to control the state variables, such as the duty ratio d, the H2 input valve αH2 and the air compressor supply power Ps, and βH2 βO2 in certain cases [66]. Disadvantages For large scale systems, too many equations lead to too many balloons. It will then be very difficult to organize the different relations in order to give a clear view of the system.

Electrical DC-Link Boost Chopper Load Cdc

Fuel Cell stack

Filter Lfc

H2 O

iload

im_fc

ifc

Gas Tube

O2 loss

βO2

ifc

Pcomp

DC Cdc

um_fc

udc

Air Compressor

O2

Lfc ufc

ElectroH2 Valve Storage

H2

DC βH2

d

αH2

H2 loss

Figure 27. Simplified fuel cell system. Power Conditioning System

im_fc

Fuel Cell Stack

Gas Handling System

ufc

iload isto

udc

R1

DC-link Capacitor

R2

um_fc

R4

ifc

ufc

R5

Electrical/ Thermodynamic Modeling

Filter Lfc

im_fc

R3

ifc

qH2_in ifc

Boost Chopper

dfc

αH2

qH2

R6

R8

ifc

R7

Hydraulic Modeling

qO2

R9

qair_out

Figure 28. The COG modeling of the simplified fuel cell system.

Choke Filter

R11

Patm

βair

βH2

R12

pO2 qair_in

Elements DC-link Capacitor Boost Chopper

pH2_sto

pH2 qH2_out

Electrochemical Modeling

R10

R13

P

TABLE I. Modeling Equations for the Power Conditioning System Equations Remarks du 1 (im _ fc + isto − iload ) R1 : dc = Losses are not taken into account dt C dc R2 : um _ fc = d ⋅ udc ; R3 : im _ fc = d ⋅ i fc R4 :

di fc dt

=

1 (u fc − u m _ fc ) L fc

d is the duty cycle for average model; or the switch On/Off state for discrete model. Losses are not taken into account

TABLE II. Modeling Equations for the fuel cell stack Equations Remarks

Elements

Electrical/ Thermodynamic Modeling

R5 :ufc = Nfc[Eo(i, pH2, pO2 ) −Vact(i) −Vohm(i) −Vconc(i)]

Electrochemical Modeling

R6 : q H 2 =

Hydraulic Modeling

dp H 2 RT (q H 2 _ in − q H 2 _ out − q H 2 ) = dt V dp RT R9 : O2 = (ηO2 _ inqair_ in −ηO2 _ outqair_ out − qO2 ) dt V R8 :

N fc ⋅ i 2F

; R7 : q O 2 =

N fc ⋅ i 4F

Nfc is the number of cells in series; Eo is the thermodynamic voltage; Vact, Vohm and Vconc represent voltages due to activation loss, ohmic loss and concentration loss. See [65] for details. qH2 and qO2 are the molar rates of the H2 and O2 in the fuel cell stack. F is the Faraday constant. pH2 and pO2 are the partial pressures of H2 and O2 in the fuel stack. pH2_in and pO2_in are the input rates. pH2_out and pO2_out are the output rates. ηO2_in and ηO2_out are the molar concentration ratio of the O2 in the mixed gas.

TABLE III. Modeling Equations for the Gas Handling System Equations Remarks

Elements

H2 Handling system

Air/O2 Handling system

R11 : q H 2 _ out = β H 2

R10 : qH 2 _ in = α H 2

R12 : qair_ out = βO2 R13 :

PH 2 − Patm R H 2 _ out

PH 2 _ sto − PH 2 RH 2 _ in

PO2 /ηO2 _ out − Patm Rh _ O2

⎞ dqair_in 1⎛ kf km k2f = ⎜ Ps − pO2 ⎟ ηO2_out ⎟⎠ dt J ⎜⎝ Us

Input: The stored H2 pressure PH2_sto is usually much higher than the H2 pressure in the stack PH2, and the hydraulic resistance of tube is very small for H2 (RH2_inÆ0). So when the electro-valve is open (αH2=1), the H2 flow rate is very high (qH2_inÆ∞); when αH2=0, qH2_in=0. So we often assume that the pH2 regulation is done instantaneously in practice. Output: For atmospheric fuel cell, βH2 =1; for pressurized fuel cell, βH2 can be regulated between 0 and 1. Output: Same principle as the H2 output handling system. Input: If the fuel cell is fed by O2, ηO2=1, the same output handling system can be used as for H2. If the fuel cell is fed by air, the concentration ratio of the O2 in the mixed air is need for quantity conversion, and a (motor-driven turbo) compressor is needed to supply the air to the stack. J is compressor inertia; kf is fan coefficient, km is motor coefficient, Us is constant supply voltage, Ps is the variable supply voltage that we can control through a power converter [67].

4. CONCLUSION This paper proposed a review of various modeling methodologies for PEM fuel cells themselves and for PEM fuel cell systems including their ancillaries. Even if the presented methodologies are not exhaustive, it clearly appears that the approaches in the French community are very various and complementary. The general tendencies are: • The mechanistic models (1-D, 2-D) are used to study finely physicochemical phenomena (membrane hydration, flooding…) or to design a fuel cell (channels, mechanical constraints…) • Models based on energy analogies are used to study from a macroscopic point of view the consequences of the physicochemical phenomena (activation, diffusion, ohmic, fluidic…). • Empiric models are used to describe (even to decipher!) the very complex behavior of a fuel cell particularly concerning the impact of the operating conditions (temperature, pressure, current, gases conditioning, water management…). • Analytical models are used to develop the control process of a fuel cell system (air compressor piloting, humidifier piloting, electrical production piloting…) • Graphical modeling approaches are used in order to synthesize control laws of the fuel cell systems from the fluidics to the electricity conditioning. Finally, one can once again observe that there are as many modeling approaches as scientific objectives. All these approaches can be interesting. They should not be seen as competitive but as complementary. Sure, a so complex system such as a fuel cell system, could be better understood only by crossing several modeling approaches.

REFERENCES 1. 2. 3.

K. Haraldsson, K. Wipke, 2004, Evaluating PEM fuel cell models, J. Power Sources, vol. 126, pp. 88-97. M. Bautista, Y. Bultel, P. Ozil, Polymer Electrolyte Membrane Fuel Cell Modeling: DC and AC Solutions, Trans IChemE, Part A, July 2004, 82 (A7) 907-917. A. A. Franco, P. Schott, C. Jallut, B. Maschke, A Multi-Scale Dynamic Mechanistic Model for the Transient Analysis of PEFCs, Fuel Cells, Volume 7, Issue 2, April, 2007, Pages: 99-117.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

Y. Wang, C.Y. Wang, Transient analysis of polymer electrolyte fuel cells, Electrochim. Acta, 50 (2005) 1307-1315. M. F. Serincan, S. Yesilyurt, Transient Analysis of Proton Electrolyte Membrane Fuel Cells (PEMFC) at Start-Up and Failure, Fuel Cells, Volume 7, Issue 2, 2007, Pages: 118-127. Ph. Moçotéguy, F. Druart, Y. Bultel, S. Besse, A. Rakotondrainibe , Monodimensional modeling and experimental study of the dynamic behavior of proton exchange membrane fuel cell stack operating in dead-end mode, Journal of Power Sources, Volume 167, Issue 2, 2007, pp. 349-357. J. Deseure, Coupling RTD and EIS modelling to characterize operating non-uniformities on PEM cathodes, Journal of Power Sources, Volume 178, Issue 1, 2008, pp. 323-333. M. Usman Iftikhar, D. Riu, F. Druart, S. Rosini, Y. Bultel, N. Retière, Dynamic modeling of proton exchange membrane fuel cell using non-integer derivatives, Journal of Power Sources, Volume 160, Issue 2, 2006, Pages 1170-1182. F. Chena, M.-H. Changb, C.-F. Fang, 2007, Analysis of water transport in a five-layer model of PEMFC, J. of Power Sources, Vol. 164, pp. 649-658. F. Liu , G. Lu, C.-Y. Wang, 2007, Water transport coefficient distribution through the membrane in polymer electrolyte fuel cell, J. Membrane Science, vol. 287, pp. 126-131. X. Ye, C. Y. Wang, 2007, Measurements of water transport properties through membrane electrode assemblies. Part I membranes, J. Electrochem. Soc., B676-B682. T. E. Springer, T. A. Zawodzinski, S. Gottesfeld, 1991, Polymer Electrolyte Fuel Cell Model, J. Electrochem. Soc., Vol. 138, No. 8. S. H. Ge, B.-L. Yi, 2003, A mathematical model for PEMFC in different flow modes, J. of Power Sources, Vol. 124, pp. 1-11. S. Walkiewicz, Y. Bultel, B. Le Gorrec, J.P. Diard, Modeling impedance diagrams of Fuel Cell, in 4th International Symposium on electrocatalysis, Satellite meeting of the 53rd ISE meeting in Duesseeldorf, Abstract p.71, 22-25, 2002, Como (Italy). D. Riu, N. Retière, M. Ivanès, Induced Currents Modeling by Half-Order Systems Application to Hydro- and Turbo-Alternators, IEEE Transaction on Energy Conversion, 18(1), 2003, pp. 94-99. N. Djilali, Computational modelling of polymer electrolyte membrane (PEM) fuel cells: Challenges and opportunities, Energy, Volume 32, Issue 4, Pages 269-280, 2007. J.-P. Poirot-Crouvezier, F. Roy, GENAPAC Project: Realization of a cell fuel stack prototype dedicated to the automotive application, WHEC 16, 2006, Lyon, France. O. Lottin, B. Antoine, T. Colinart, S. Didierjean, G. Maranzana, C. Moyne, J. Ramousse, Modelling of the operation of Polymer Exchange Membrane Fuel Cells in the presence of electrodes flooding, International Journal of Thermal Sciences, Available online 21 April 2008. G. Maranzana, O. Lottin, T. Colinart, S. Chupin, S. Didierjean. A multi-instrumented polymer exchange membrane fuel cell - Observation of the in-plane non-homogeneities, Journal of Power Sources, Volume 180, Issue 2, 1 June 2008, Pages 748-754. C. Vallieres, D. Winkelmann, D. Roizard, E. Favre, Ph. Scharfer, M. Kind, On Schroeder's paradox, Journal of Membrane Science, Volume 278, Issues 1-2, Pages 357-364, 2006. K. D. Kreuer, On the complexity of proton conduction phenomena, Solid State Ionics, Volumes 136-137, Pages 149-160, 2000. O. Tatsuhiro, G. Xie, O. Gorseth, S. Kjelstrup, N. Nakamura, T. Arimura, Ion and water transport characteristics of Nafion membranes as electrolytes, Electrochimica Acta, Volume 43, Issue 24, Pages 3741-3747, 1998. G. Gebel, Structural evolution of water swollen perfluorosulfonated ionomers from dry membrane to solution, Polymer, Volume 41, Issue 15, Pages 5829-5838, 2000. T. Colinart, S. Didierjean, O. Lottin, G. Maranzana, C. Moyne. Transport in PFSA Membrane. J. of the Electrochemical Society, Vol 155, 3, B244-B257, 2008. T. Colinart O. Lottin, B. Antoine, S. Didierjean, G. Maranzana, C. Moyne, Pseudo 2D polarization model of a Polymer Exchange Membrane Fuel Cell including liquid water transport, Eindhoven 5th European Thermal-Sciences Conference, Eindhoven, the Netherlands, 2008. J. Ramousse, J.Deseure, O. Lottin, S. Didierjean, D. Maillet, Modelling of heat, mass and charge transfer in a PEMFC single cell, Journal of Power Sources, Volume 145, Issue 2, Pages 41642718, 2005.

27. C. Bonnet, S. Didierjean, N. Guillet, S. Besse, T. Colinart, P. Carré. Design of an 80 kWe PEM fuel cell system: scale up effect investigation, J. of Power Sources, 182 (2008) 441–448. 28. J.-J. Kadjo, J.-P. Garnier, J.-P. Maye, F. Relot, S. Мartemianov, Russian Journal of Electrochemistry, 42, (2006) 467-475. 29. J.-J.Kadjo, P. Brault, A. Caillard, C. Coutanceau, J.-P. Garnier, S. Martemianov, Journal of Power Sources, 172, (2007), 613-622. 30. T.E. Springer, T.A. Zawodzinski, S. Gottesfeld, Journal of the Electrochemial Society, 138, (1991), 2334-2342. 31. T.F. Fuller, J. Newman, Journal of the Electrochemial Society, 140, (1993), 1218-1225. 32. K.Z. Yao, K. Koran, K.B. McAuley, P.Oosthuizen, B. Peppley, and T. Xie. Fuel Cells, 4, (2004), 3-29. 33. A.A. Kulikovsky, Journal of Power Sources, 162, (2006), 1236-1240. 34. A.A. Kulikovsky, Electrochmistry Communications, 9, (2007), 6-12. 35. C.J.Bapat, S.T.Thynell, Journal of Heat Transfer, 129, (2007), 1109-1118. 36. Y. Tang, M.H. Santare, A.M. Karlsson, S. Cleghorn, W.B. Johnson, J. Fuel Cell Sci. Technol., 3, (2006), 119-124. 37. A. Kusoglu, A.M. Karlsson, M.H. Santare, S. Cleghorn, W.B. Johnson, Journal of Power Sources, 161, (2006) 987-996. 38. A. Kusoglu, A.M. Karlsson, M.H. Santare, S. Cleghorn, B. J. William, Journal of Power Sources, 170, (2007) 345-358. 39. D. Bograchev, M. Gueguen, J. Grandidier, S. Martemianov. Journal of Power Sources, 180, (2008) 393-401. 40. B. Wahdame, Analysis and optimisation of fuel cell operations using the experimental design methodology, Ph.D. Thesis, University of Franche-Comte - UTBM, [in French], Belfort, 2006, http://tel.archives-ouvertes.fr/tel-00163317/en/. 41. D.C. Montgomery, Design and Analysis of Experiments, 2005, John Wiley and Sons, New York. 42. B. Wahdame, D. Candusso, J.M. Kauffmann, Study of gas pressure and flow rate influences on a 500W PEM fuel cell thanks to the experimental design methodology, J. Power Sources, 2006, 156(1), 92–99. 43. B. Wahdame, D. Candusso, X. François, F. Harel, A. De Bernardinis, J.M. Kauffmann, G. Coquery, Study of a 5kW PEMFC using experimental design and statistical analysis techniques, Fuel Cells, 2007, 1, 47-62. 44. B. Wahdame, D. Candusso, X. François, F. Harel, M.C. Péra, D. Hissel, J.M. Kauffmann, Analysis of a Fuel Cell Durability Test Based on Design of Experiment Approach, IEEE Transactions on Energy Conversion, to be published. 45. B. Wahdame, D. Candusso, X. François, F. Harel, M.C. Péra, D. Hissel, J.M. Kauffmann, Comparison between two PEM fuel cell durability tests performed at constant current and under solicitations linked to transport mission profile, Int. J. Hydrogen Energy, 2007, 32(17), 45234536. 46. B. Wahdame, D. Candusso, F. Harel, X. François, M.C. Péra, D. Hissel, J.M. Kauffmann, Analysis of a PEMFC durability test under low humidity conditions and stack behaviour modelling using experimental design techniques, J. Power Sources, 2008, 182(2), 429-440. 47. S. Jemeï, D. Hissel, M.C. Péra, J.M. Kauffmann, “A new Modeling Approach of Embedded fuel cell generators based on Artificial Neural Network”, IEEE Transactions on Industrial Electronics, vol. 55, n°1, pp. 437-447, 2008. 48. A. Hernandez, D. Hissel, R. Outbib, “Electric equivalent model for a polymer electrolyte fuel celll (PEMFC)”, Electrimacs’2005 Conference, 5p., CD-ROM, Hammamet, Tunisie, 2005. 49. M. Chnani, “Transient thermal behaviour of a solid oxide fuel cell”, ASME-5th Fuel cell science, engineering and technology conference, 18-20 June 2007, New York, USA. 50. K. W. Suh, “ Modeling, Analysis and Control of Fuel Cell Hybrid Power Systems,” Department of Mechanical Engineering, The University of Michigan, PhD thesis, 2006. 51. J. T. Pukrushpan, A. G. Stefanopoulou, H. Peng, “Control of Fuel Cell Power Systems: Principles, Modeling, Analysis and Feedback design,” Springer edition, London, September 16, 2004.

52. R. Talj, R. Ortega, M. Hilairet, “ A controller tuning methodology for the air supply system of a PEM fuel cell system with guaranteed stability properties,” Internal Report of L2S, 2008. 53. M. Grujicic, K. M. Chittajallu, E. H. Law, J. T. Pukrushpan, “ Model-based control strategies in the dynamic interaction of air supply and fuel cell,” Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, Vol. 218, n°7, pp. 487-499, 2004. 54. A. Bouscayrol, M. Pietrzal-David, P. Delarue, R Pena-Eguiluz, P-E Vidal, and X. Kestelyn, “Weighted Control of Traction Drives with Parallel-Connected AC Machines,” IEEE Trans. on Industry Electronics, vol. 53, no. 6, Dec. 2006, pp. 1799–1806. 55. L. Boulon, M-C. Pera, D. Hissel, A. Bouscayrol, and Ph. Delarue, “Energetic Macroscopic Representation of a fuel cell-supercapacitor system”, In proc. IEEE-VPPC (CD-ROM), Arlington (USA), 2007. 56. J. Larminie, and A. Dicks, “Fuel cell systems explained”, John Wiley & Sons, 2003. 57. L. Boulon, M-C Pera, P. Delarue, A. Bouscayrol, D. Hissel, "Causal fuel cell system model suitable for transportation simulation applications", ASME Journal of Fuel Cell Science and Technology, to be published, 2009. 58. D. Chrenko, M.C. Pera, A. Bouscayrol, D. Hissel, “Inversion-based control of a PEM Fuel Cell system using Energetic Macroscopic Representation”, ASME Journal of Fuel Cell Science and Technology, to be published. 59. M. Tekin, D. Hissel, M-C. Pera, and J-M. Kauffmann. “Energy consumption reduction of a PEM fuel cell motor-compressor group thanks to efficient control laws”, Journal of Power Sources, vol. 156, Issue 1, May 2006, pp. 57-63. 60. H. Xu, L. Kong, and X. Wen. “Fuel cell power system and high power DC-DC converter”, Power Electronics, IEEE Transactions on vol. 19, Issue 5, September 2004, pp 1250 – 1255. 61. A. Narjiss, D. Depernet, F. Gustin, D. Hissel, and A. Berthon, “Design of a High Efficiency Fuel Cell DC/DC Converter dedicated to Transportation Applications”, ASME Journal of Fuel Cell Science and Technology, vol. 5, n°4, 041004-1/11, 2008. 62. J. P. Hautier and P. J. Barre, “The causal ordering graph – A tool for modelling and control law synthesis”, Studies in Informatics and Control Journal, vol. 13, no. 4, Dec. 2004, pp. 265-283. 63. R. Bearee, P.J. Barre and J.P. Hautier, “Control structure synthesis for electromechanical systems based on the concept of inverse model using Causal Ordering Graph,” IEEE IECON’06, Nov. 2006, pp. 5289- 5294. 64. T. Zhou, B. François, M. Lebbal and S. Lecoeuche, “Modeling and Control Design of Hydrogen Production Process by Using a Causal Ordering Graph for Wind Energy Conversion System,” IEEE ISIE’07, June 2007 pp. 3192- 3197. 65. N.A. Ahmed, “Computational modelling and polarization characteristics of proton exchange membrane fuel cell with evaluation of its interface systems,” EPE Journal, vol.18, iss.1, Mars 2008, pp.32-41. 66. B. François, D. Hissel and M.T. Iqbal, “Dynamic Modelling of a Fuel Cell and Wind Turbine DC-Linked Power System,” Electrimacs’05, Hammamet, Tunisia, April 2005, CD-ROM. 67. D. Chrenko, M.C. Pera, and D.Hissel, “Fuel Cell System Modeling and Control with Energetic Macroscopic Representation,” IEEE ISIE’07, June 2007, pp. 169-174. 68. H. Paynter, “Analysis and design of engineering systems”, MIT Press, 1961. 69. D. Karnopp, R. Rosenberg, “Systems Dynamics: A Unified Approach”, John Wiley & Sons, 1991. 70. R. Saisset, G. Fontes, C. Turpin, S. Astier, “Bond graph model of a PEM fuel cell”, Journal of Power Sources, Vol. 156, May 2006. 71. G. Fontes, “Interactions between fuel cells and power converters: influence of current harmonics on a fuel cell stack”, IEEE Transactions on Power Electronics,Vol. 22, March 2007. 72. C. Turpin, G. Fontes, S. Astier, M. Garcia Arregui, V. Phlippoteau, “A novel large signal and dynamic characterisation to parameter a novel large signal and dynamic circuit model of a PEM fuel cell”, Hydrogen & Fuel Cells 2007, Vancouver (Canada), 29th April – 2nd May 2007. 73. G. Fontes, C. Turpin, S. Astier, “A large signal dynamic circuit model of a H2/O2 PEM fuel cell: description, parameter identification and exploitation”, FDFC, Nancy (France), December 1012th, 2008.