@unpublished{AignerSchaumannvonLoeperetal., author = {Aigner, Kevin-Martin and Schaumann, Peter and von Loeper, Freimut and Martin, Alexander and Schmidt, Volker and Liers, Frauke}, title = {Robust DC Optimal Power Flow with Modeling of Solar Power Supply Uncertainty via R-Vine Copulas}, abstract = {We present a robust approximation of joint chance constrained DC Optimal Power Flow in combination with a model-based prediction of uncertain power supply via R-vine copulas. It is applied to optimize the discrete curtailment of solar feed-in in an electrical distribution network and guarantees network stability under fluctuating feed-in. This is modeled by a two-stage mixed-integer stochastic optimization problem proposed by Aigner et al. (European Journal of Operational Research, (2021)). The solution approach is based on the approximation of chance constraints via robust constraints using suitable uncertainty sets. The resulting robust optimization problem has a known equivalent tractable reformulation. To compute uncertainty sets that lead to an inner approximation of the stochastic problem, an R-vine copula model is fitted to the distribution of the multi-dimensional power forecast error, i.e., the difference between the forecasted solar power and the measured feed-in at several network nodes. The uncertainty sets are determined by encompassing a sufficient number of samples drawn from the R-vine copula model. Furthermore, an enhanced algorithm is proposed to fit R-vine copulas which can be used to draw conditional samples for given solar radiation forecasts. The experimental results obtained for real-world weather and network data demonstrate the effectiveness of the combination of stochastic programming and model-based prediction of uncertainty via copulas. We improve the outcomes of previous work by showing that the resulting uncertainty sets are much smaller and lead to less conservative solutions while maintaining the same probabilistic guarantees.}, language = {en} } @article{KrugMehrmannSchmidt2019, author = {Krug, Richard and Mehrmann, Volker and Schmidt, Martin}, title = {Nonlinear Optimization of District Heating Networks}, series = {Optimization and Engineering}, journal = {Optimization and Engineering}, number = {22(2)}, pages = {783 -- 819}, year = {2019}, abstract = {We develop a complementarity-constrained nonlinear optimization model for the time-dependent control of district heating networks. The main physical aspects of water and heat flow in these networks are governed by nonlinear and hyperbolic 1d partial differential equations. In addition, a pooling-type mixing model is required at the nodes of the network to treat the mixing of different water temperatures. This mixing model can be recast using suitable complementarity constraints. The resulting problem is a mathematical program with complementarity constraints subject to nonlinear partial differential equations describing the physics. In order to obtain a tractable problem, we apply suitable discretizations in space and time, resulting in a finite-dimensional optimization problem with complementarity constraints for which we develop a suitable reformulation with improved constraint regularity. Moreover, we propose an instantaneous control approach for the discretized problem, discuss practically relevant penalty formulations, and present preprocessing techniques that are used to simplify the mixing model at the nodes of the network. Finally, we use all these techniques to solve realistic instances. Our numerical results show the applicability of our techniques in practice.}, language = {en} } @article{MehrmannSchmidtStolwijk2017, author = {Mehrmann, Volker and Schmidt, Martin and Stolwijk, Jeroen J.}, title = {Model and Discretization Error Adaptivity within Stationary Gas Transport Optimization}, series = {Vietnam Journal of Mathematics}, journal = {Vietnam Journal of Mathematics}, number = {46(4)}, pages = {779 -- 801}, year = {2017}, abstract = {The minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and isothermal gas flows. The error control is realized by switching to more detailed models or finer discretizations if necessary to guarantee that a prescribed model and discretization error tolerance is satisfied in the end. We prove convergence of the adaptively controlled optimization method and illustrate the new approach with numerical examples.}, language = {en} } @article{HauschildMarheinekeMehrmannetal.2019, author = {Hauschild, Sarah-Alexa and Marheineke, Nicole and Mehrmann, Volker and Mohring, Jan and Badlyan, Arbi Moses and Rein, Markus and Schmidt, Martin}, title = {Port-Hamiltonian modeling of district heating networks}, series = {Progress in Differential Algebraic Equations II (edited by Reis T., Grundel S., and Sch{\"o}ps S). Differential-Algebraic Equations Forum}, journal = {Progress in Differential Algebraic Equations II (edited by Reis T., Grundel S., and Sch{\"o}ps S). Differential-Algebraic Equations Forum}, pages = {17}, year = {2019}, abstract = {This paper provides a first contribution to port-Hamiltonian modeling of district heating networks. By introducing a model hierarchy of flow equations on the network, this work aims at a thermodynamically consistent port-Hamiltonian embedding of the partial differential-algebraic systems. We show that a spatially discretized network model describing the advection of the internal energy density with respect to an underlying incompressible stationary Euler-type hydrodynamics can be considered as a parameter-dependent finite-dimensional port-Hamiltonian system. Moreover, we present an infinite-dimensional port-Hamiltonian formulation for a compressible instationary thermodynamic fluid flow in a pipe. Based on these first promising results, we raise open questions and point out research perspectives concerning structure-preserving discretization, model reduction, and optimization.}, language = {en} } @unpublished{HannesVolkerRolandetal.2022, author = {Hannes, D{\"a}nschel and Volker, Mehrmann and Roland, Marius and Schmidt, Martin}, title = {Adaptive Nonlinear Optimization of District Heating Networks Based on Model and Discretization Catalogs}, pages = {29}, year = {2022}, abstract = {We propose an adaptive optimization algorithm for operating district heating networks in a stationary regime. The behavior of hot water flow in the pipe network is modeled using the incompressible Euler equations and a suitably chosen energy equation. By applying different simplifications to these equations, we derive a catalog of models. Our algorithm is based on this catalog and adaptively controls where in the network which model is used. Moreover, the granularity of the applied discretization is controlled in a similar adaptive manner. By doing so, we are able to obtain optimal solutions at low computational costs that satisfy a prescribed tolerance w.r.t. the most accurate modeling level. To adaptively control the switching between different levels and the adaptation of the discretization grids, we derive error measure formulas and a posteriori error measure estimators. Under reasonable assumptions we prove that the adaptive algorithm terminates after finitely many iterations. Our numerical results show that the algorithm is able to produce solutions for problem instances that have not been solvable before.}, language = {en} }