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Research Papers

# Robust Real-Time Optimization of a Solid Oxide Fuel Cell Stack

[+] Author and Article Information
A. Marchetti, A. Gopalakrishnan, B. Chachuat

GIAIP-CIFASIS (CONICET,UNR,UPCAM III), 27 de Febrero 210bis, S2000EZP Rosario, ArgentinaLaboratoire d’ Automatique (LA), École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, UK

D. Bonvin1

Laboratoire d’ Automatique (LA), École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland e-mail: dominique.bonvin@epfl.ch Laboratoire d’Énergétique Industrielle (LENI), EPFL CH-1015 Lausanne, Switzerland

L. Tsikonis, A. Nakajo, Z. Wuillemin, J. Van herle

Laboratoire d’ Automatique (LA), École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland e-mail: dominique.bonvin@epfl.ch Laboratoire d’Énergétique Industrielle (LENI), EPFL CH-1015 Lausanne, Switzerland

The notation $x∈IRn$ is used to indicate that $x$ is an n-dimensional vector of real variables.

3The notation requires some explanation. Here, $εkG$ are the correction terms given by 35, and $εckG$ is the subset of $εkG$ corresponding to the inequality constraints that are controlled between the RTO iterations k and k + 1. Notice that the controlled inequality constraints might change from one iteration to the next, hence the subscript k added to c.

1

Corresponding author.

J. Fuel Cell Sci. Technol 8(5), 051001 (Jun 13, 2011) (11 pages) doi:10.1115/1.4003976 History: Received May 14, 2009; Revised January 28, 2011; Published June 13, 2011; Online June 13, 2011

## Abstract

On-line control and optimization can improve the efficiency of fuel cell systems, whilst simultaneously ensuring that the operation remains within a safe region. Also, fuel cells are subject to frequent variations in their power demand. This paper investigates the real-time optimization (RTO) of a solid oxide fuel cell (SOFC) stack. An optimization problem maximizing the efficiency subject to operating constraints is defined. Due to inevitable model inaccuracies, the open-loop implementation of optimal inputs evaluated off-line may be suboptimal, or worse, infeasible. Infeasibility can be avoided by controlling the constrained quantities. However, the constraints that determine optimal operation might switch with varying power demand, thus requiring a change in the regulator structure. In this paper, a control strategy that can handle plant-model mismatch and changing constraints in the face of varying power demand is presented and illustrated. The strategy consists in the integration of RTO and model predictive control (MPC). A lumped model of the SOFC is utilized at the RTO level. The measurements are not used to re-estimate the parameters of the SOFC model at different operating points, but to simply adapt the constraints in the optimization problem. The optimal solution generated by RTO is implemented using MPC that uses a step-response model in this case. Simulation results show that near-optimality can be obtained, and constraints are respected despite model inaccuracies and large variations in the power demand.

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## Figures

Figure 1

Schematic of the SOFC stack and furnace

Figure 2

Cell voltage and power density as a function of current density. Solid lines: n·fuel,in  = 10−3 mol/s; dot-dashed lines: n·fuel,in  = 1.2 × 10−3 mol/s.

Figure 3

Contour maps and operational constraints for the plant at steady state corresponding to different power density setpoints. White area: feasible region; dotted lines: contours of the objective function; point P: optimum for the plant.

Figure 4

Contour maps and operational constraints for the nominal model at steady-state corresponding to different power density setpoints. White area: feasible region; dotted lines: contours of the objective function; point M: optimum for the model.

Figure 5

Constraint-adaptation algorithm for real-time optimization. The subscript k represents the iteration counter. The constrained quantities H, G, Hp , and Gp correspond to steady-state operation. When implemented alone, i.e., without an additional controller, the optimal inputs uuk+1∗ are applied directly to the plant as indicated in this figure; otherwise, the process inputs are determined by the controller as detailed in Sec. 5.

Figure 6

Real-time optimization of the SOFC stack. Solid lines: Constraint-adaptation results. Dashed lines: Power density setpoint and constraint bounds. The three inputs are the flow rates n·fuel, in and n·air, in , and the current I = Aactive i.

Figure 7

Combination of constraint adaptation and constraint control

Figure 8

Robust real-time optimization of the SOFC stack (case 1). Solid lines: performance of constraint adaptation and constraint control. Dashed lines: power density setpoint and constraint bounds. The three inputs are the flow rates n·fuel, in and n·air, in , and the current I = Aactive i.

Figure 9

Robust real-time optimization of the SOFC stack (case 2). Solid lines: performance of constraint adaptation and constraint control. Dashed lines: power density setpoint and constraint bounds. The three inputs are the flow rates n·fuel, in and n·air, in , and the current I = Aactive i.

Figure 10

Location of the optimal operating point upon convergence of the RTO-MPC scheme for the two power density setpoints of 0.2 and 0.4 W/cm2

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