The paper considers the computation of the probability of feasible load constellations in a stationary gas
network with uncertain demand. More precisely, a network with a single entry and several
exits with uncertain loads is studied. Feasibility of a load constellation is understood in the sense of
an existing flow meeting these loads along with given pressure bounds in the pipes.
In a first step, feasibility of deterministic exit loads is characterized algebraically and these general
conditions are specified to networks involving at most one cycle.
This prerequisite is essential for determining probabilities in a stochastic setting when exit loads
are assumed to follow some (joint) Gaussian distribution when modeling uncertain customer demand.
The key of our approach is the application of the spheric-radial decomposition of Gaussian random
vectors coupled with Quasi Monte-Carlo sampling. This approach requires an efficient algorithmic
treatment of the mentioned algebraic relations moreover depending on a scalar parameter. Numerical
results are illustrated for different network examples and demonstrate a clear superiority in terms of
precision over simple generic Monte-Carlo sampling. They lead to fairly accurate probability values
even for moderate sample size.
Gas networks are of growing importance for the economy and offer interesting mathematical problems at the same time. The classical linear network flow allows for approximate models that more and more have come to their limits. This has raised interest in nonlinear but, for simplicity, still steady-state models. The present paper aims at mobilizing techniques from symbolic computation and reparametrization of multivariate integrals to enable validation of stochastic nominations following Gaussian distributions in passive gas networks with more than one cycle.