1. Date of download: 7/7/2016 Copyright © ASME. All rights reserved. From: Robust Real-Time Optimization of a Solid Oxide Fuel Cell Stack J. Fuel Cell Sci. Technol. 2011;8(5):051001-051001-11. doi:10.1115/1.4003976 Schematic of the SOFC stack and furnace Figure Legend:

2. Date of download: 7/7/2016 Copyright © ASME. All rights reserved. From: Robust Real-Time Optimization of a Solid Oxide Fuel Cell Stack J. Fuel Cell Sci. Technol. 2011;8(5):051001-051001-11. doi:10.1115/1.4003976 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 Legend:

3. Date of download: 7/7/2016 Copyright © ASME. All rights reserved. From: Robust Real-Time Optimization of a Solid Oxide Fuel Cell Stack J. Fuel Cell Sci. Technol. 2011;8(5):051001-051001-11. doi:10.1115/1.4003976 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 Legend:

4. Date of download: 7/7/2016 Copyright © ASME. All rights reserved. From: Robust Real-Time Optimization of a Solid Oxide Fuel Cell Stack J. Fuel Cell Sci. Technol. 2011;8(5):051001-051001-11. doi:10.1115/1.4003976 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.. Figure Legend:

5. Date of download: 7/7/2016 Copyright © ASME. All rights reserved. From: Robust Real-Time Optimization of a Solid Oxide Fuel Cell Stack J. Fuel Cell Sci. Technol. 2011;8(5):051001-051001-11. doi:10.1115/1.4003976 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 = Aactivei. Figure Legend:

6. Date of download: 7/7/2016 Copyright © ASME. All rights reserved. From: Robust Real-Time Optimization of a Solid Oxide Fuel Cell Stack J. Fuel Cell Sci. Technol. 2011;8(5):051001-051001-11. doi:10.1115/1.4003976 Combination of constraint adaptation and constraint control Figure Legend:

7. Date of download: 7/7/2016 Copyright © ASME. All rights reserved. From: Robust Real-Time Optimization of a Solid Oxide Fuel Cell Stack J. Fuel Cell Sci. Technol. 2011;8(5):051001-051001-11. doi:10.1115/1.4003976 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 = Aactivei. Figure Legend:

8. Date of download: 7/7/2016 Copyright © ASME. All rights reserved. From: Robust Real-Time Optimization of a Solid Oxide Fuel Cell Stack J. Fuel Cell Sci. Technol. 2011;8(5):051001-051001-11. doi:10.1115/1.4003976 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 = Aactivei. Figure Legend:

9. Date of download: 7/7/2016 Copyright © ASME. All rights reserved. From: Robust Real-Time Optimization of a Solid Oxide Fuel Cell Stack J. Fuel Cell Sci. Technol. 2011;8(5):051001-051001-11. doi:10.1115/1.4003976 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 Figure Legend: