Hans Georg Bock

Jürgen Gutekunst, Armin Nurkanovic, Ekaterina Kostina et al.

Recently there has been a lot of progress in the development of economic nonlinear model predictive control (NMPC) schemes for multistage optimal power flow (OPF) problems. However, the additional inclusion of discrete decision variables to model generator runtimes and generator startup costs can amount to large scale mixed-integer nonlinear programs (MINLPs) that are computationally very challenging. This work investigates the practical approach that replaces the nonlinear AC power flow equations by convex quadratic approximations. In combination with the discrete generator dynamics this leads to a mixed-integer quadratically constrained program (MIQCP) which is of significantly lower complexity and can be solved in reasonable time by off-the-shelf solvers such as CPLEX. We further show that simple terminal constraints are not sufficient to guarantee recursive feasibility of the NMPC scheme if constraints on generator runtime and on the number of generator startup events are present. To address this challenge we propose the use of additional time-coupled constraints and prove the resulting recursive feasibility property. Based on the assumption of periodic dissipativity of the underlying system we can prove stability of the proposed controller. To illustrate our results, we present simulations of a realistic 6-bus microgrid under different demand scenarios.

Dörte Beigel, Mario S. Mommer, Leonard Wirsching et al.

With this contribution, we shed light on the relation between the discrete adjoints of multistep backward differentiation formula (BDF) methods and the solution of the adjoint differential equation. To this end, we develop a functional-analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions. We devise a finite element Petrov-Galerkin interpretation of the BDF method together with its discrete adjoint scheme obtained by reverse internal numerical differentiation. We show how the finite element approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its asymptotic convergence in the space of normalized functions of bounded variation. We also obtain asymptotic convergence of the discrete adjoints to the classical adjoints on the inner time interval. Finally, we give numerical results for non-adaptive and fully adaptive BDF schemes. The presented framework opens the way to carry over the existing theory on global error estimation techniques from finite element methods to BDF methods.