Deciding Robust Feasibility and Infeasibility Using a Set Containment Approach: An Application to Stationary Passive Gas Network Operations

Abstract

In this paper we study feasibility and infeasibility of nonlinear two-stage fully adjustable robust feasibility problems with an empty first stage. This is equivalent to deciding whether the uncertainty set is contained within the projection of the feasible region onto the uncertaintyspace. Moreover, the considered sets are assumed to be described by polynomials. For answering this question, two very general approaches using methods from polynomial optimization are presented -one for showing feasibility and one for showing infeasibility. The developed methods are approximated through sum of squares polynomials and solved using semidefinite programs. Deciding robust feasibility and infeasibility is important for gas network operations, which is a nonconvex feasibility problem where the feasible set is described by a composition of polynomials with the absolute value function. Concerning the gas network problem, different topologies are considered. It is shown that a tree structured network can be decided exactly using linear programming. Furthermore, a method is presented to reduce a tree network with one additional arc to a single cycle network. In this case, the problem can be decided by eliminating the absolute value functions and solving the resulting linearly many polynomial optimization problems. Lastly, the effectivity of the methods is tested on a variety of small cyclic networks. It turns out that for instances where robust feasibility or infeasibility can be decided successfully, level 2 or level 3 of the Lasserre relaxation hierarchy typically is sufficient.

Structural properties of feasible bookings in the European entry–exit gas market system

“…We now analyze the feasibility of nominations and bookings w.r.t. (3). Each nomination of the set of nominations N is feasible, which can be shown by a direct proof or by Theorem 7.1 in [34].…”

“…and c I (x, s; ) is empty. Moreover, we set u = σ u = 0 for all inner nodes u ∈ V 0 in Constraint (3). We now analyze the feasibility of nominations and bookings w.r.t.…”

“…We set the pressure bounds to [2,2] for node s 1 and to [1,2] for the remaining nodes. Furthermore, the lower and upper bounds for the compression ratio are given by [ε − , ε + ] = [2,3] and the pressure drop coefficient Λ a equals 0.5 for every arc a ∈ A \ A cs . We neglect flow bounds in this example.…”

“…We again consider the network G = (V, A) of the last example. We modify the lower and upper pressure bounds to [0.875, 2] for v 1 and to [1,3] for v 2 and t 1 . Furthermore, the lower and upper bounds for the compression ratio are set to [1,3].…”

Decomposable robust two‐stage optimization: An application to gas network operations under uncertainty

“…Since global optimal solutions are required to ensure robust feasibility of the obtained results, we develop piecewise‐linear relaxations of which can then be used in a mixed‐integer linear program (MIP). Of course other relaxations like linear (see Appendix A) or semidefinite relaxations arising from polynomial programming are also conceivable; however, we restrict ourselves to piecewise‐linear relaxations as a priori error bounds can be computed. In the remainder of this section, we drop the arc‐specific indices of .…”