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Research Papers

Surrogate Modeling for Spatially Distributed Fuel Cell Models With Applications to Uncertainty Quantification

[+] Author and Article Information
A. A. Shah

School of Engineering,
University of Warwick,
Coventry CV4 7AL, UK
e-mail: Akeel.Shah@warwick.ac.uk

Manuscript received August 29, 2016; final manuscript received March 26, 2017; published online May 30, 2017. Assoc. Editor: Jan Van herle.

J. Electrochem. En. Conv. Stor. 14(1), 011006 (May 30, 2017) (15 pages) Paper No: JEECS-16-1115; doi: 10.1115/1.4036491 History: Received August 29, 2016; Revised March 26, 2017

Detailed physics-based computer models of fuel cells can be computationally prohibitive for applications such as optimization and uncertainty quantification. Such applications can require a very high number of runs in order to extract reliable results. Approximate models based on spatial homogeneity or data-driven techniques can serve as surrogates when scalar quantities such as the cell voltage are of interest. When more detailed information is required, e.g., the potential or temperature field, computationally inexpensive surrogate models are difficult to construct. In this paper, we use dimensionality reduction to develop a surrogate model approach for high-fidelity fuel cell codes in cases where the target is a field. A detailed 3D model of a high-temperature polymer electrolyte membrane (PEM) fuel cell is used to test the approach. We develop a framework for using such surrogate models to quantify the uncertainty in a scalar/functional output, using the field output results. We propose a number of alternative methods including a semi-analytical approach requiring only limited computational resources.

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Figures

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Fig. 1

A schematic of the fuel cell (single channel with symmetry) on which the mathematical model is based

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Fig. 2

A slice of the fuel cell geometry in the x–z plane (see Fig. 1)

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Fig. 3

Profiles of the z component ie,z1 (A m−2) of the ionic current density vector ie=−σe∇ϕe in the x–y plane located at z = z1 in the cathode CL in Fig. 2: (a) Vcell = 0.8902 V, εe = 0.2398, T = 423.05 K, ωH2O,in,c=0.1238; (b) Vcell = 0.4433 V, εe = 0.388, T = 395.70 K, ωH2O,in,c=0.2808; (c) Vcell = 0.7925 V, εe = 0.3570, T = 392.57 K, ωH2O,in,c=0.3277; and (d) Vcell = 0.6519 V, εe = 0.3008, T = 376.95 K, ωH2O,in,c=0.7461

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Fig. 4

Boxplots of the relative square error in predicting ie,z1 against the 160 test points as the number of training points m and the number of principal components r in the approximation yr are increased: (a) m = 20 and (b) m = 40

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Fig. 5

Predicted profiles compared to test examples of the z component ie,z1 (A m−2) in the x–y plane located at z = z1: (a) Vcell = 0.6519 V, εe = 0.3008, T = 376.95 K, ωH2O,in,c=0.0746; and (c) Vcell = 0.7203 V, εe = 0.2906, T = 392.19 K, ωH2O,in,c=0.0547. Figures (b) and (d) are the corresponding predictions, respectively. In both cases, m = 40 and r = 6.

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Fig. 6

Boxplots of the relative square error in predicting ωO2 against the 160 test points as the number of training points m and the number of principal components r in the approximation yr are increased: (a) m = 20 and (b) m = 40

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Fig. 7

A comparison of the prediction of ωO2 (b) against the test case (a). The input parameters are Vcell = 0.4332 V, ϵe = 0.3883, T = 395.70 K, and ωH2O,in,c=0.2808. The value of ωO2 in the cathode channel, GDL, and CL is shown in five x–z planes along the length of the cell. In both cases, m = 40 and r = 6.

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Fig. 8

An example of the prediction of ωO2 (b) against the test case (a) with a large relative error of 1.0108 × 10−4. The input parameters are Vcell = 0.8004 V, ϵe = 0.2039, T = 409.76 K, and ωH2O,in,c=0.1262. The value of ωO2 in the cathode channel, GDL, and CL is shown in five x–z planes along the length of the cell. In both cases, m = 40 and r = 6.

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Fig. 9

Boxplots of the relative square error in predicting cH2O (mol m−3) in the cathode (channel, GDL, and CL) against the 160 test points as the number of training points m and the number of principal components r in the approximation yr are increased: (a) m = 20 and (b) m = 40

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Fig. 10

An example of the prediction of cH2O (mol m−3) in the cathode (channel, GDL, and CL), shown in (b), against the test case shown in (a). The relative error in this case is close to the median in the r = 6 box plot in Fig. 9(b). The input parameters are Vcell = 0.4488 V, εe = 0.2945, T = 361.33 K, and ωH2O,in,c=0.3465. The value of cH2O in the cathode channel, GDL, and CL is shown in five x–z planes along the length of the cell. In both cases, m = 40 and r = 6.

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Fig. 11

Histograms of the mean Mn and variance Vn, n = 1,…, Nr (see Algorithm 2)

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