Deciding feasibility of a booking in the European gas market on a cycle is in P for the case of passive networks

Abstract

We show that the feasibility of a booking in the European entry‐exit gas market can be decided in polynomial time on single‐cycle networks that are passive, i.e., do not contain controllable elements. The feasibility of a booking can be characterized by solving polynomially many nonlinear potential‐based flow models for computing so‐called potential‐difference maximizing load flow scenarios. We thus analyze the structure of these models and exploit both the cyclic graph structure as well as specific properties of potential‐based flows. This enables us to solve the decision variant of the nonlinear potential‐difference maximization by reducing it to a system of polynomials of constant dimension that is independent of the cycle's size. This system of fixed dimension can be handled with tools from real algebraic geometry to derive a polynomial‐time algorithm. The characterization in terms of potential‐difference maximizing load flow scenarios then leads to a polynomial‐time algorithm for deciding the feasibility of a booking. Our theoretical results extend the existing knowledge about the complexity of deciding the feasibility of bookings from trees to single‐cycle networks.

A bilevel optimization approach to decide the feasibility of bookings in the European gas market

“…It is required to develop problem-specific solution approaches, especially for the case of nonlinear gas physics. Similar to the studies in Robinius et al (2019); Labbé et al (2020) for tree-shaped and in Labbé et al (2021) for single-cycle networks, algorithms to solve the nonlinear subproblems of the characterizations presented in this paper can be beneficial. Finally, the analyses of the European gas market models studied in Böttger et al (2021); can be extended to take into account linearly modeled active elements by integrating the novel characterizations of feasible bookings presented in this paper.…”

“…In the following, we consider stationary gas flows based on the Weymouth pressure loss equation (Weymouth 1912). In line with the corresponding literature , Thürauf (2020), and Labbé et al (2021), we model gas flow physics using potential-based flows, which for active networks consist of arc flows q = (q a ) a∈A , node potentials π = (π u ) u∈V , and controls = ( a ) a∈A act . In the context of gas networks with horizontal pipes, potentials represent squared gas pressures at the nodes, i.e., π u = p 2 u for u ∈ V .…”

“…We now formalize the problem of deciding the feasibility of a booking in gas networks including compressors and control valves. We follow and extend the problem description in Labbé et al (2021), which deals with the feasibility of a booking for a single-cycle network without active elements. To this end, we consider linearly modeled active elements and stationary gas flows.…”

“…For passive tree-shaped or passive single-cycle networks, this characterization can be checked in polynomial time; see Labbé et al (2020Labbé et al ( , 2021; Robinius et al (2019). However, the problem of validating a booking on general passive networks is known to be coNP-hard (Thürauf 2020).…”

“…If these maximum pressure differences satisfy certain pressure bounds, the booking is feasible and otherwise, it is infeasible. This characterization can be used to decide the feasibility of a booking in polynomial time for passive, tree-shaped networks (Labbé et al 2020) or passive, single-cycle networks (Labbé et al 2021). However, the problem is coNPhard on passive networks in general (Thürauf 2020).…”