Non-smooth Methods for Complementarity Formulations of Switched Advection-Diffusion Processes

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Organic Rankine Cycle (ORC) processes are established in process engineering for energy recovery from an exhaust heat source. They run a cyclically operated advection-diffusion process to transfer energy from exhaust heat into a working medium using a boiler. The working medium in steam phase is expanded to harness energy, while the working medium, now in fluid phase, is fed back to the boiler by way of condenser and pump.We take interest in the temporal and spatial dynamics of the isobaric phases of an ORC under transient boundary conditions described by exhaust flow and temperature. The problem is described by an advection-diffusion process modeled by instationary partial differential-algebraic equations in 1D or 2D with distributed phase-dependent medium parameters. Due to the large scale of process models and significant perturbations of the process by external load-point changes on a time scale considerably smaller than the process' time constant, the non-smooth behavior of the process is non-periodic and shows nontrivial patterns. Optimal operation of the transient behavior is key to making the concept practically worthwhile, and is a highly challenging task for classical control concepts.The proposed research project develops theory and efficient numerical methods for optimization of advection-diffusion processes with phase changes described by instationary nonlinear PDAEs.Theory and methods shall combine the state of the art in reduced approaches for PDE-constrained optimization, a modeling approach for non-smoothness that makes use of discrete-valued controls and a partial outer convexification approach to mixed-integer optimal control combined with a decomposition approach for constraints on the non-smooth parts of the problem. In particular, we develop:- a simultaneous optimization framework is chosen in which the discretized model equations enter the optimization problem as nonlinear constraints;- discrete and non-smooth phenomena are modeled in a partial outer convexification framework. Using adaptive discretizations in space and time, this gives rise a non-convex constraint structure imposed on the simplex of indicator controls;- a decomposition approach for the resulting MIOCP into an MPVC and the use of a SUR algorithm. The approach shall be used to decouple the PDE optimization task from the combinatorial task of identifying an optimal switching structure;- a theoretical framework for the class of advection diffusion processes that relates the non-smooth result of the SUR algorithm to the to the solution of the non-smooth problem. Such relations shall be shown in the spaces of controls and of PDE states, and in terms of bounded loss of feasibility and bounded suboptimality, and shall be related to the choice of discretizations- a reduced semi-smooth Newton-type framework is used to solve the non-smooth nonlinear programming problems that result from the discretization of model equations and controls.