Publikationen

To guarantee the safe and dependable operation of a magnetic levitation train, the distance between the magnet and the reaction rail needs to be kept within a given range. In this work, we design model predictive controllers which, in addition to complying with these constraints, provide a favorable behavior with regard to performance criteria such as travel comfort and control effort. For this purpose, we present a model of the system and the disturbances affecting it. Several results regarding the mathematical properties of this model are proven to gain insight for controller design. Finally we compare three different controllers w.r.t. performance criteria such as robustness, travel comfort, control effort, and computation time in an extensive numerical simulation study: a linear feedback controller, a model predictive control (MPC) scheme with quadratic stage costs, and the recently-proposed funnel MPC scheme. We show that the MPC closed loop complies with the constraints while also exhibiting excellent performance. Furthermore, we implement the MPC algorithms within the GRAMPC framework. This allows us to reduce the computational effort to a point at which real-time application becomes feasible.

Many real-world applications require the joint optimization of a large number of flexible devices over time. The flexibility of, e.g., multiple batteries, thermostatically controlled loads, or electric vehicles can be used to support grid operation and to reduce operation costs. Using piecewise constant power values, the flexibility of each device over d time periods can be described as a polytopic subset in power space. The aggregated flexibility is given by the Minkowski sum of these polytopes. As the computation of Minkowski sums is in general demanding, several approximations have been proposed in the literature. Yet, their application potential is often objective-dependent and limited by the curse of dimensionality. We show that up to 2d vertices of each polytope can be computed efficiently and that the convex hull of their sums provides a computationally efficient inner approximation of the Minkowski sum. Via an extensive simulation study, we illustrate that our approach outperforms ten state-of-the-art inner approximations in terms of computational complexity and accuracy for different objectives. Moreover, we propose an efficient disaggregation method applicable to any vertex-based approximation. The proposed methods provide an efficient means to aggregate and to disaggregate energy storages in quarter-hourly periods over an entire day with reasonable accuracy for aggregated cost and for peak power optimization.

The model predictive control (MPC) scheme funnel MPC enables output tracking of smooth reference signals with prescribed error bounds for nonlinear multi-input multi-output systems with stable internal dynamics. Earlier works achieved the control objective for system with relative degree restricted to one or incorporated additional feasibility constraints in the optimal control problem. Here we resolve these limitations by introducing a modified stage cost function relying on a weighted sum of the tracking error derivatives. The weights need to be sufficiently large and we state explicit lower bounds. Under these assumptions we are able to prove initial and recursive feasibility of the novel funnel MPC scheme for systems with arbitrary relative degree - without requiring any terminal conditions, a sufficiently long prediction horizon or additional output constraints.

In this paper, the application of the method of affinely adjustable robust optimization to a planning model of an energy system under uncertain parameters is presented, and the total scheduling costs in comparison with the deterministic model are evaluated. First, the basics of optimization under uncertain data are recapped, and it is described how these methods can be used in different applications for energy systems. This is followed by the methodology of adjustable robust optimization by defining the affinely adjustable robust counterpart. Finally, a numerical case study is conducted to compare the adjustable robust method with a rolling deterministic scheduling method. Both are implemented on a model of an energy system and compared with each other by simulation using real-world data. By calculating the total operating costs for both methods, it can be concluded that the adjustable robust optimization provides a significantly more cost-effective solution to the scheduling problem.

We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results.