Deciding Robust Feasibility and Infeasibility Using a Set Containment Approach: An Application to Stationary Passive Gas Network Operations - Aßmann, Denis; Liers, Frauke; Stingl, Michael; Vera, Juan C. (SIAM Journal on Optimization, 2018)
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 uncertainty-space. 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 (SOS) 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.<br
Keywords: Natural gas transport; Polynomial optimization; Robust optimization

Decomposable robust two-stage optimization: An application to gas network operations under uncertainty - Aßmann, Denis; Liers, Frauke; Stingl, Michael (Networks, 2019)
We study gas network problems with compressors and control valves under uncertainty that can be formulated as two-stage robust optimization problems. Uncertain data are present in the physical parameters of the pipes as well as in the overall demand. We show how to exploit the special decomposable structure of the problem to reformulate the two-stage problem as a single-stage robust optimization problem. The right-hand side of the single-stage problem can be precomputed by solving a series of optimization problems and multiple elements of the right-hand side can be combined into one optimization task. The practical feasibility and effectiveness of our approach is demonstrated with benchmarks on several gas network instances, among them a realistic model of the Greek natural gas network. Overall, aggregation and preprocessing allow us to quickly solve large gas network instances under uncertainty for the price of slightly more conservative solutions.
Keywords: decomposition; gas transport networks; mixed-integer optimization; piecewise linearization; robust optimization; two-stage optimization

Exact methods for two-stage robust optimization with applications in gas networks - Aßmann, Denis (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2019)
Today, natural gas is one of the most important sources of energy and is regarded as a key instrument for achieving the politically set climate goals. Gas-fired power plants are valued as flexible buffers to compensate for fluctuations in electricity generation from renewable energy sources at short notice. Additionally, gas network operators face new challenges as a result of the liberalization of the European gas market. In the new entry-exit model, the gas network operators have to ensure that all possible market outcomes can be transported over the network. Hence, the operation of gas networks under uncertain conditions increasingly requires new AIDS for decision-making. To this end, this thesis investigates a class of general two-stage robust optimization problems whose second-stage variables are uniquely determined by non-convex constraints. This structure occurs, e.g., in gas network operations under uncertainty. Three general solution methods are developed for this problem class. The first two approaches use ideas from polynomial optimization to decide feasibility or infeasibility of a problem variant with an empty first stage. Both procedures use polynomial formulations that are approximated by semidefinite programs using the Lasserre relaxation hierarchy. The effectiveness of the methods is investigated on cyclic gas networks. It can be observed that often a low level of the Lasserre hierarchy is sufficient to decide robust feasibility or infeasibility. The third approach is based on a transformation of the two-stage problem into a normal, single-stage optimization problem. To this end, several subproblems have to be solved whose optimal values form the right-hand side of the transformed problem. An additional aggregation step can significantly reduce the number of subproblems that have to be considered. For a practical application to real-world gas network instances, mixed-integer linear relaxations of the subproblems are developed. Finally, the performance of the approach is demonstrated by benchmarks on several gas network instances, including a realistic model of the Greek natural gas network. Overall, robust feasible solutions for large networks under uncertainty can be found within a short time.
CRIS-Team Scopus Importer:2019-11-26

GasLib - A Library of Gas Network Instances - Schmidt, Martin; Aßmann, Denis; Burlacu, Robert; Humpola, Jesco; Joormann, Imke; Kanelakis, Nikolaos; Koch, Thorsten; Oucherif, Djamal; Pfetsch, Marc E.; Schewe, Lars; Schwarz, Robert; Sirvent, Mathias (Data, 2017)
The development of mathematical simulation and optimization models and algorithms for solving gas transport problems is an active field of research. In order to test and compare these models and algorithms, gas network instances together with demand data are needed. The goal of GasLib is to provide a set of publicly available gas network instances that can be used by researchers in the field of gas transport. The advantages are that researchers save time by using these instances and that different models and algorithms can be compared on the same specified test sets. The library instances are encoded in an XML format. In this paper, we explain this format and present the instances that are available in the library.
Keywords: Gas Transport; Networks; Problem Instances; Mixed-Integer Nonlinear Optimization; GasLib

Joint model of probabilistic/robust (probust) constraints with application to gas network optimization - Adelhütte, Dennis; Aßmann, Denis; Gonzàlez Grandòn, Tatiana; Gugat, Martin; Heitsch, Holger; Liers, Frauke; Henrion, René; Nitsche, Sabrina; Schultz, Rüdiger; Stingl, Michael; Wintergerst, David (Vietnam Journal of Mathematics, 2020)
Optimization tasks under uncertain conditions abound in many real-life applications. Whereas solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually used in case solutions are sought that are feasible for all realizations of uncertainties within some pre-defined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we introduce a class of joint probabilistic/robust constraints as its appears in optimization problems under uncertainty. Focussing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound, with high probabilit
Keywords: Stabilization; Wave equation; Feedback; Robust optimization; Probabilistic constraints; Probust; Karhunen–Loève