Stochastic Methods for PEMFC Parameter Identification

Fuel cells are complex multiphysical systems characterized by a stratified geometry consisting of thin layers of different materials. Their full characterization involves a number of physical parameters, which are important for quantifying and comparing the performance of the materials. The physical parameters are also used in fuel cell models, which allow the fast exploration of different operating scenarios and can be implemented in optimization procedures aimed at achieving the best structural and operating conditions. The system of equations involved in these models, e.g. the Nernst equation, the Butler-Volmer equation and the diffusion equation, are strongly non-linear; thus, the model itself is extremely sensitive to parameter variations. Careful ex-situ measurements of such parameters can be performed by means of a number of diagnostic techniques, whose results are however not completely consistent with fuel cell operation due to several effects, e.g. variable interactions and boundary effects in the thin layers of the fuel cell. Conversely, “in situ” measurements can provide meaningful operational values, but a very few techniques are available to determine a limited number of parameters.

A solution can consist in multiple parameter identification from multiple fundamental measurements, such as polarization curve and electrochemical impedance spectroscopy, performed in different conditions, e.g. at different temperature, pressure, concentration, and humidification.

If there is only one unknown parameter, the solution is easy, requiring just a statistical interpolating technique. In the case of multiple unknown parameters, which is the most frequent condition, the problem becomes increasingly difficult as the number of parameters increases as duplicity problem emerges, i.e., several groups of parameters may lead to the same performance (e.g. polarization curve).

Recently, a number of numerical tools are under development to face this kind of problems, some based on stochastic mathematical models, each presenting specific features in terms of accuracy, convergence, stability, robustness, and calculation speed [1]. Among the most promising methods, some of which have already been applied to multiple fuel cell parameter identification, are the Ishibashi–Kimura method [2], the Lambert W-function method [3], the Levenberg-Marquardt algorithm (LMA), the Gauss-Newton algorithm (GNA), the hybrid artificial bee colony algorithm [4], the trust region method, the current switching method [5], the simulated annealing method [6], and the particle swarm optimization (PSO) [7], and the differential evolution method [8].

The presentation will focus on the application of a particular class of stochastic optimization methods to the problem of multiple parameters identification. These methods have already been widely tested and applied successfully to the simultaneous search of sets of optimal parameters with different physical dimensions, making them ideal tools to cope with the present problem. Their recent relevant literature is vast. They are especially suited for multimodal and noisy functions over large search spaces, which makes them particularly appealing for the problem at hand. The main drawback of such methods is the high number of objective function evaluations needed for convergence; however, thanks to a particular formulation of the PEMFC model, such drawback can be of minor impact, i.e. the overall computational times are on the order of a few tens of seconds.

We are going to present the application of two of the most well-known metaheuristic methods, namely Particle Swarm Optimization (PSO) and Differential Evolution (DE), developed in previous work and suitably adapted. It should be noted that PSO has already been applied to parameter identification for FCs, while the application of DE is novel in this context. This is somewhat surprising since, as results will show, DE provides better results and offers higher robustness. We will present the implementation details regarding PSO and DE. The optimization algorithms have not been tuned to the specific problem and algorithm control parameters have been chosen from reasonable values suggested in literature.

References

[1]   Guarnieri M., Stella A., Trevisan F., Methodological analysis of different formulations for solving inverse electromagnetic problems”, IEEE Trans Mag, 26-2, pp. 622-625, 1990.

[2]   Ishibashi, K.I., Kimura, Y., Niwano, M., An extensively valid and stable method for derivation of all parameters of a solar cell from a single current–voltage characteristic. Journal of Applied Physics 103. 2008.

[3]   Jain, A., Kapoor, A., A new approach to study organic solar cell using Lambert W-function. Solar Energy Materials and Solar Cells 86, 197–205, 2005.

[4]   Zhang W., Wang N., Yang S., Hybrid artificial bee colony algorithm for parameter estimation of proton exchange membrane fuel cell, Int. J. Hydrogen Energy 38, 5796-5806, 2013.

[5]   Chang W.-Y., Estimating equivalent circuit parameters of proton exchange membrane fuel cell using the current change method, Electrical Power and Energy Systems 53, 584–591, 2013.

[6]   Outeiro MT, Chibante R, Carvalho AS, De Almeida AT. A new parameter extraction method for accurate modeling of PEM fuel cells. Int J. Energy Res 33, 978-88, 2009.

[7]   Alotto P., Guarnieri M., Stochastic Methods for Parameter Estimation of Multiphysics Models of Fuel Cells, IEEE Trans Mag, in press.

[8]   Qin A.K., Huang V.L., Suganthan P.N., Differential Evolution Algorithm With Strategy Adaptation for Global Numerical Optimization, IEEE Trans Evol Comp, 13-2, 398-417, 2009.