The Application of a Numerical Model of a Proton Exchange Membrane Fuel Cell to the Estimations of Some Cell Parameters

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1 The Application of a Numerical Model of a Proton Exchange Membrane Fuel Cell to the Estimations of Some Cell Parameters Ágnes Havasi 1, Róbert Horváth 2 and Tamás Szabó 3 1 Eötvös Loránd University, Pázmány P. s. 1, H-1117 Budapest, Hungary 2 Budapest University of Technology and Economics, Egry J. u. 1, H-1111, Budapest, Hungary 3 Basque Center for Applied Mathematics, Alameda de Mazarredo 14, Bilbao, Bizkaia, Basque-Country, Spain Abstract. The functioning and the achievable power of a proton exchange membrane fuel cell (PEMFC) are determined by several parameters simultaneously. Part of these cannot be measured directly, and can only be calculated from approximate formulas. Our aim is to study the real values of these parameters and their behavior as a function of the cell conditions. In order to give reliable estimations for the parameters, we first set up an adequate finite difference numerical solution of the mathematical model of the fuel cell. Then the values of the unknown parameters are calculated by fitting the model results to measurements. For the solution of the parameter fitting problem we test three methods: the Levenberg- Marquardt method, the trust region method and the simulated annealing method, among which the simulated annealing method turns to be the most efficient one in our case. Performing several numerical tests we can give good estimations to the values of the unknown parameters and we can analyze the behavior of them with respect to the cell conditions. 1 Introduction A fuel cell is an electrochemical conversion device [10, 11]. It produces electricity from fuel (e.g., hydrogen) on the negative electrode and an oxidant (e.g., oxygen) on the positive electrode, which react in the presence of an electrolyte (e.g., Nafion). In the archetypal hydrogen-oxygen proton exchange membrane fuel cell (PEMFC), the electrolyte is a proton-conducting, electrically insulating polymer membrane. The electrons and protons of the hydrogen are separated in a catalytic reaction, and the electrons are forced to travel through a circuit, hence producing electrical power (Figure 1). It is desirable to construct fuel cells with as high as possible achievable power. There are several physical parameters that describe the behaviour of the cell. Some of these parameters can be measured directly, while another part of them cannot be measured after the construction of the Membrane-Electrode Assembly (MEA). In our case these unknown parameters are the effective conductivity of

2 Fig. 1. Schematic view of a PEMFC. the solution phase, the exchange current density of the cathode and the limiting current of the cathode. Our main goal is to calculate the real values of the unknown parameters and to give their behavior against temperature, pressure and Nafion content. To this aim we construct a numerical model of the fuel cell based on a simplified PEMFC model [4]. This model focuses on the positive electrode (cathode) where the major losses take place and writes the basic physical and chemical phenomena of the cathode into the form of mathematical equations. The efficiency of the numerical model is crucial both from the accuracy and speed point of view because we set the three unknown parameters in such a way that the numerical solution, obtained by our model, is as close to the measurements as possible. For the solution of this so-called parameter estimation problem we test three different methods. (1) The Levenberg Marquardt method [6, 8], which is a clever combination of the well-known Newton and steepest descent methods, (2) The trust region method [9], which modifies a so-called trusted region around the current estimations using trial steps and (3) the simulated annealing method [3], which is assumed to find not only local optimum points but also global ones. The structure of the paper is as follows. In Section 2 the mathematical model of the PEMFC and the parameters used in the model are introduced. Then we give the detailed numerical discretization of the model. In Section 3 we present the parameter estimation problem, and introduce the Levenberg Marquardt, the trust region and the simulated annealing methods used in parameter fitting. In the last section, the numerical results are listed and the results of the parameter fittings are discussed. The dependence of the studied parameters on temperature, pressure and Nafion content is investigated.

3 2 The PEMFC model and its numerical solution The fuel cell model applied is based on Litster s potential summation algorithm [7] and Kulikovsky s diffusion kinetic approximation [5] and the Weber membrane conductivity model [12]. Additionally, the double layer capacity is taken into account to simulate the time-dependent (transient) curves as well. The cell potential according to Litster can be calculated from the equation E cell (t) = E OC (t) V (t) η a (t) W mem κ mem I(t), (1) where E OC is the open circuit potential, V is the potential loss at the cathode, W mem is the thickness of the membrane, κ mem is the membrane conductivity and η a is the potential loss at the anode, which is the solution of the equation I(t) = i a 0 [ ( ) α a exp a F RT ηa ( )] exp αa c F RT ηa. (2) (The description and the units of the parameters used in the model are given in Table 1.) Our aim is to calculate the cell potential as a function of the load current. The potential loss at the cathode is V (t) = ϕ 1 (t, L) ϕ 2 (t, 0). If we choose ϕ 2 (t, 0) = 0, then by definition of the overpotential we obtain where ϕ 2 (t, L) = ϕ 2 (t, 0) + V (t) = ϕ 1 (t, L) = η(t, L) + ϕ 2 (t, L), (3) L 0 ( σ ) eff 1 s η(t, s) + I(t) ds. (4) κ eff + σ eff κ eff + σ eff Here the overpotential η of the cathode is computed from the solution u(t, x) of the basic cathode equation 1 t u(t, x) = x S(x) ( I(t) 1 ac dl K + σ eff x u(t, x)) (5) + S(x) x (σ eff x u(t, x)) i 0 g(u(t, x)). ac dl C dl K The function u is related to η by the transformation u(t, x) = η(t, x)/k with K = RT/(αF ), and usually depends on space and time. The notation S(x) stands for the expression S(x) = κ eff /(κ eff + σ eff ). Equation (5) is supplemented with an initial condition u(0, x) = 0, x (0, L) (6) where L is the thickness of the cathode, and the Neumann type boundary condition 1 x u(t, L) = I(t), t (0, T ). (7) σ eff (L)K

4 Symbol Description Unit a(x) Specific interfacial area cm 1 C dl (x) Double-layer capacitance F/cm 2 E cell (t) Cell potential V E OC Open circuit potential V F Faraday constant (96487) C/mol I(t) Total cell current density A/cm 2 i 0(x) Exchange current density at the cathode A/cm 2 i a 0 Exchange current density at the anode A/cm 2 j D (x) Limiting current at the cathode A/cm 2 L Thickness of the cathode cm R Universal gas constant (8.3144) J/molK T Cell temperature K V (t) Potential loss at the cathode V Wmem Membrane thickness cm α Transfer coefficient in the cathode Anodic transfer coefficient at the anode α a a α a c Cathodic transfer coefficient at the anode η(t, x) Overpotential at the cathode V η a (t) Overpotential at the anode V ϕ 1 (t, x) Solid phase potential V ϕ 2 (t, x) Solution phase potential V κ eff (x) Effective solution phase conductivity S/cm σ eff (x) Effective solid phase conductivity S/cm κmem Membrane conductivity S/cm Table 1. The parameters in the PEMFC model. The function g on the right-hand side of (5) describes the kinetics of the oxygen reduction reaction. In our case diffusion kinetics was used, where g(u(t, x)) = j D exp(u(t, x)) j D exp( u(t, x)). (8) i 0 exp(u(t, x)) + j D i 0 exp( u(t, x)) + j D The above initial-boundary value problem is solved numerically, by using an implicit-explicit Euler method. The method is explicit in the source term, so iterations for solving nonlinear problems can be avoided. At the same time, the spatial derivative is approximated using an implicit scheme, which maintains the stability of the time stepping. This approach provides a good balance between accuracy and relatively low computational costs. There is a possibility to enhance the method with Richardson extrapolation to accelerate the convergence. The Newton Raphson method is used for solving Eq. (2) of the anode [1]. 3 The parameter estimation problem In the PEMFC model, introduced in the previous section, little is known about the following parameters:

5 κ eff - the effective conductivity of the solution phase; i 0 - the exchange current density of the cathode; j D - the limiting current at the cathode, involved by the function g on the right-hand side of (5) in case of diffusion kinetics (see (8)). In [4], the following values, calculated on the base of [5] and [7], are given: κ eff = Scm 1, i 0 = Acm 2 and j D = 1.05 Acm 2. These values can only be considered as rough estimations, since they have been calculated from simplified formulas and for given cell conditions. The dependence of these parameters on the cell conditions (such as temperature, pressure and Nafion content) are not given by these formulas. Our aim is to specify the same parameters in another way, where measurements, taken under different conditions and our numerical model are applied together. We consider the stationary case, i.e., we assume that the quantities in the equation do not depend on time. During the measurements, the cell potential E cell, derived from the unknown function u, is measured for different constant values of the current density I. The three parameters are to be set such that the E cell I curve calculated by the model best fits the measured E cell I curve. The mathematical formulation of the problem is as follows. Let us introduce the following notations: w := [κ eff, j D, i 0 ] T - the column vector of the three parameters to be estimated, M: the number of measurements, I i : the current density in the ith measurement, i = 1, 2,..., M, Vi meas : the cell potential measured at the constant current density I i, V (I i : w) - the cell potential calculated by the model with parameter vector w for current density I i, r i (w) := V (I i : w) Vi meas - the ith remainder, r(w) := [r 1 (w), r 2 (w),..., r M (w)] T - vector of remainders. Our aim is to minimize the scalar function f : IR 3 IR, f(w) = 1 2 rt (w)r(w) (9) which is called objective function, and measures the distance between the measurements and model results. So, the task is to find a minimum point of a multivariable scalar function. This is a nonlinear optimization problem, for which the application of the Levenberg Marquardt (LM) method is advocated in [2] for a PMEFC cathode model. The Levenberg-Marquardt method can be generally applied to minimization problems where a local minimum point of a function f : IR n IR is to be found. For the objective function f, the following properties are required: There exists a point w and a positive constant δ such that f(w ) f(w) for all w B δ (w ), where B δ (w ) denotes the open ball with radius δ centered at w,

6 f is sufficiently smooth, f(w ) = 0, the Hessian matrix 2 f(w ) is strictly positive definite. The Newton method is known to converge quickly to the solution of a nonlinear equation when the initial iterate is close to the minimum point. The steepest descent method shows slower convergence in the vicinity of the solution than Newton s method, however, from a distant initial iterate it can get quickly to the vicinity of the solution. The Levenberg Marquardt method [6, 8], introducing a weighting coefficient, combines the Newton method and the steepest descent method as applied to the solution of the equation f = 0. Denote by w c the current estimation to w, and by w + the estimation obtained in the next iteration step. Let the weighting coefficient, the so-called Levenberg-Marquardt parameter, denoted by λ. Then the LM method iterates as follows: 1. We choose a stating estimation w c and an initial value of λ, say, λ := We calculate the next estimation w + with the formula w + = w c ( 2 f(w c ) + λdiag( 2 f(w c ))) 1 f(w c ). (10) 3. If f(w + ) f(w c ), then we keep w c and increase λ, e.g., by multiplying it by 10 (closer to the steepest descent method). If f(w + ) < f(w c ), then we accept the new iteration w +, and decrease λ, e.g., divide it by 10 (closer to the Newton method). 4. We check a stopping condition (is f(w + ) or/and w + w c sufficiently small). If it is fulfilled then we stop the procedure and if it is not then we go back to step 2 with w c = w +. Now we apply the LM method in the PEMFC model. Taking into account the special form of the objective function (9), in our case the gradient vector f(w) and the Hessian matrix 2 f(w) can be given as follows. Denote by J the Jacobian matrix [ ] ri (w) J(w) = w j M 3 of f at w. The gradient vector and the Hessian matrix can be expressed in terms of J as f(w) = J T (w)r(w) IR 3, and 2 f(w) = J T (w)j(w) + M r i (w) 2 r i (w) IR 3 3. i=1 In the latter expression we omit the sum on the right-hand side, and use the approximate equality 2 f(w) J T (w)j(w).

7 Then the application of the LM method yields the iteration w + = w c (J T (w c )J(w c ) + λdiag(j T (w c )J(w c )) 1 J T (w c )r(w c ). Here the computation of the Jacobian matrix J is crucial. In the literature [2] two basic approaches are recommended: 1. Finite difference method The elements of J are approximated by some finite difference method, e.g., by the forward differences: J ij = r i(..., w cj + w j,...) r i (..., w cj,...) w j, i = 1,..., M, j = 1, 2, 3, where w j is a small increment in the current value of the jth parameter. This method is easy to apply, however, it only gives approximate elements due to the truncation error. The smaller the increment, the smaller the truncation error, however, with a decrease of w j the rounding error increases. 2. Sensitivity approach Differential equations are derived for the derivatives in J, which are then solved numerically. In principle, this method should give more exact results, however, it is rather complicated to implement. Moreover, in [2] the application of forward differences proved to be competitive with the sensitivity approach. In the present phase of the experiments we decided to apply the first approach, i.e., approximating the elements of J by forward differences. Note that computing each element of J in this manner requires a further model run (with the modified parameters). Therefore this method is rather time-consuming. 4 Parameter estimation results In the numerical experiments the current density was studied as a function of the cell potential. The measurements were done at three temperatures (40 o C, 60 o C and 80 o C) and two pressures (1 bar and 2 bar). We studied catalysts with different Nafion contents (14, 30 and 50%) in the fuel cell. The parameters κ eff, j D and i 0 were fitted to the measurements by using the Levenberg Marquardt method. The starting values for the iteration were those obtained by manual fitting. For comparison we also show the numerical solution obtained by manual fitting. In Fig. 2 a typical example from the results of the fittings is given (80 o C, 1 bar pressure, 50% Nafion content). The curves show the cell potential versus the current density. One can see clearly that the LM curve is much closer to the measurements than the curve obtained by manual fitting. We were interested to see whether the parameter estimation works better if the starting values of the three parameters are somewhat modified. Therefore we ran the model with halved/doubled values of the initial parameter triplet.

8 Measurement Manual LM 1 Cell Potential / V Current / I Fig. 2. Cell potential versus current density obtained by parameter estimation ( Fitted ) and by manual fitting ( Unfitted ) in comparison with measurements (80 o C, 1 bar, 50% Nafion content). Fig. 3. Cell potential versus current density started from halved values (left panel) and doubled values (right panel) by parameter estimation ( Fitted ) and by using literature data ( Unfitted ) in comparison with measurements (80 o C, 1 bar, 50% Nafion content).

9 The results obtained for the same set of measurements as in Fig. 2 are shown in Fig. 3. Since the improved parameter values gave satisfactory results in comparison with the measurements, our model is applicable for studying the dependence of the investigated parameter values on the pressure, temperature and Nafion content. Tables 2 and 3 show the results obtained for the pressures of 1 bar and 2 bar, in both cases for different temperatures and Nafion contents. The effective conductivity K eff is expected to be of the magnitude of As one can see, the obtained values are in this range in most of the cases. It should be independent of pressure, however, the obtained dependence is negligible. It is promising that the trend of the changes of K eff are in accordance with the results of the manual fitting, not given here. The values of the exchange current density J D are everywhere in the expected interval. The manual fitting gave somewhat better results only on low pressure. The most problematic parameter during manual fitting was the limiting current I 0. The formula used in the numerical model is not able to describe reliably the temperature dependence of this quantity. The results obtained by parameter estimation is of the expected behavior and magnitude. Since the temperature dependence is an exponential function, therefore it seems correct that its increase is quicker for higher temperatures. The comparison of the two tables shows that it also increases with pressure. 14% Nafion content 30% Nafion content 50% Nafion content T ( o C) K eff J D I 0 K eff J D I 0 K eff J D I E E E E E E E E E E E E E E E E E E E E E E E E E E E-09 Table 2. The parameter values K eff, J D and I 0 obtained by parameter estimation for different temperatures T and Nafion contents at 1 bar. 14% Nafion content 30% Nafion content 50% Nafion content T ( o C) K eff J D I 0 K eff J D I 0 K eff J D I E E E E E E E E E E E E E E E E E E E E E E E E E E E-07 Table 3. The parameter values K eff, J D and I 0 obtained by parameter estimation for different temperatures T and Nafion contents at 2 bar.

10 5 Conclusion We used the Levenberg Marquardt method in the model of a PEMFC cathode to obtain improved values of three critical parameters and study their behavior in different functioning condititons. These parameters included the effective conductivity of the solution phase, the exchange current density of the cathode and the limiting current of the cathode. The results of the parameter estimation were in most cases better than those obtained by manual fitting, and they were in good agreement with the expected magnitudes and the tendencies obtained by manual fitting. Acknowledgements. The Financial support of the National Office of Research and Technology (OMFB /2008) is acknowledged. References 1. I. Faragó, F. Izsák, T. Szabó, Á. Kriston, A heterogeneous PEMFC model (to be submitted). 2. Q. Guo, V. A. Sethuraman, R. E. White, Parameter estimates for a PEMFC cathode, Journal of Electrochemical Society 151, (7), A983-A993 (2004). 3. S. Kirkpatrick, C. D. Geddat, M. P. Vecchi, Optimization by simulated annealing, Science, 220, (1983). 4. Á. Kriston, Gy. Inzelt, I. Faragó, T. Szabó, Simulation of the transient behavior of fuel cells by using operator splitting techniques for realtime applications, Computers and Chemical Engineering 34, (2010), doi: /j.compchemeng A. A. Kulikovsky, The voltage-current curve of a polymer electrolyte fuel cell: exact and fitting equations, Electrochemistry Communications 4, (2002). 6. K. Levenberg, A Method for the solution of certain non-linear problems in least squares, The Quarterly of Applied Mathematics 2, (1944). 7. S. Litster, N. Djilali, Performance analysis of microstructured fuel cells for portable devices, Electrochimica Acta 52, (11), (2007). 8. D. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, SIM Journal on Applied Mathematics II, (1963). 9. J. Nocedal, S. J. Wright, Numerical Optimization, Springer, Series in Operations Research, J. T. Pukrushpan, A. G. Stefanopoulou, H. Peng, Control of Fuel Cell Power Systems, Principles, Modeling, Analysis and Feedback Design, Springer, p. 161 (2004). 11. V. R. Subramanian, S. Devan, R. E. White, An approximate solution for a pseudocapacitor, Journal of Power Sources 135, (2004). 12. A. Z. Weber, J. Newman, Transport in Polymer-Electrolyte Membranes, Journal of the Electrochemical Society 151, A311 (2004).