@phdthesis{Schuster2021, author = {Michael Schuster}, title = {Nodal Control and Probabilistic Constrained Optimization using the Example of Gas Networks}, type = {doctoralthesis}, year = {2021}, abstract = {In this thesis we analyze stationary and dynamic gas flow with uncertain boundary data in networks of pipelines. The gas flow in pipeline networks is modeled by the isothermal Euler equations. The uncertain boundary data is modeled by probability distributions, they represent the a priori unknown gas demand of the consumers. The aim of this work is the analysis of optimization problems with probabilistic constraints in the context of gas transport. For computing the probability that an uncertain gas demand is feasible we use both, a kernel density estimator approach and the spheric radial decomposition. Feasible in this context means, that the demanded gas can be transported through the network, s.t. bounds for the pressure at the nodes are satisfied. Moreover we discuss advantages and disadvantages of both methods. In the stationary case we extend our model by compressor control and bounds for the pressure at the entry nodes, and we also compute the probability for an uncertain gas demand to be feasible. In the dynamic setting the uncertain gas demand is time dependent, which is modeled by randomized Fourier series. Further we analyze certain optimization problems with probabilistic constraints, in which the probabilistic constraints are approximated by the kernel density estimator approach. On the one hand we show the existence of optimal solutions for both, the exact and the approximated problems, and on the other hand we show that if the approximation is sufficiently accurate, then the optimal solutions of the approximated problems are close to the solutions of the exact problems. With the approximation of the probabilistic constraints via the kernel density estimator we are able to compute derivatives of the approximated optimization constraints, which allows us to derive necessary optimality conditions for the approximated optimization problems with probabilistic constraints.}, url = {https://opus4.kobv.de/opus4-trr154/410}, type = {Preprint}, institution = {TRR 154} }