Optimal Control of Load Changes for Molten Carbonate Fuel Cell Systems: A Challenge in PDE Constrained Optimization

Molten carbonate fuel cells provide a promising technology for the operation of future stationary power plants. In order to enhance service life, a detailed understanding of the dynamical behavior of such fuel cell systems is necessary. In particular, fast load changes shall be simulated, respectively optimized without risking material stress due to the extreme temperature differences usually accompanying fast load changes. Fast load changes are important for daily operations in order to react on varying demands. Material stress may lead to irreparable damage of the fuel cell stack. For these contradicting goals, a family of hierarchically ordered mathematical models has been developed with the aim to simulate and optimize the temporal and spatial dynamical behavior of the gas streams, the chemical reactions, and the potential fields within the fuel cells. Altogether, the most complicated system, which is investigated in the present paper, results in a Pareto-optimal control problems with constraints in form of a huge system of 28 partial differential-algebraic (PDAE) and ordinary integro-differential-algebraic equations (OIDAE) and boundary conditions which are themselves partly given by an ordinary differential-algebraic system (ODE) of dimension 9. The partial differential equations (PDE) are of parabolic and hyperbolic type; some are degenerate. Moreover, the variables being involved in the different submodels of this fully coupled multi-physics system live on considerably different time scales. Optimal control results are presented for a compromise between sufficiently fast load changes and sufficiently small temperature differences within the cell’s solid part by means of a specially tailored formulation of a chain of optimal control problems. This procedure benefits from the different time scales of the state variables and keeps the problem manageable and computable despite its tremendous complexity and scale, although standard numerical methods are employed.