Open Access

Holger Heitsch

• The question for the capacity of a given gas network, i.e., determining the maximal amount of gas that can be transported by a given network, appears as an essential question that network operators and political administrations are regularly faced with. In that context we present a novel mathematical approach in order to assist gas network operators in managing increasing uncertainty with respect to customers gas nominations and in exposing free network capacities while reliability of transmission and supply is taken into account. The approach is based on the rigorous examination of optimization problems with nonlinear probabilistic constraints. As consequence we deal with solving a problem belonging to the class of probabilistic/robust optimization problems, which can be formulated with some joint probabilistic constraint over an infinite system of random inequalities. We will show that the inequality system can be reduced to a finite one in the situation of considering a tree network topology. A detailed study of the problem of maximizing bookable capacities in a stationary gas network is presented that comes up with an algebraic model involving Kirchhoff's first and second laws. The focus will be on both the theoretical and numerical side. The analytical part consists in introducing and validating a generalized version of the known rank two constraint qualification implying the differentiability of the considered capacity problem. The results are important in order to solve the capacity problem numerically, where function and gradient evaluations of the probabilistic constraints are performed by an approach using spheric-radial decomposition applicable for multivariate Gaussian random variables and more general distributions.