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.
