Joint Model of Probabilistic-Robust (Probust) Constraints Applied to Gas Network Optimization

Dennis Adelhütte,Frauke Liers,Sabrina Nitsche,David Wintergerst,Michael Stingl,Rüdiger Schultz,Tatiana Gonzàlez Grandòn,René Henrion,Martin Gugat,Denis Aßmann,Holger Heitsch

doi:10.1007/s10013-020-00434-y

Abstract

Optimization problems under uncertain conditions abound in many real-life applications. While 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 applied in case solutions are sought that are feasible for all realizations of uncertainties within some predefined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we deal with a class of joint probabilistic/robust constraints. Focusing 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 finding 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 probability.

Highlights

Decision making problems are more than often affected by parameter uncertainty
For a standard reference on optimization problems with probabilistic constraints we refer to the monograph [22]
The aim of this paper is to illustrate the application of this new class of probust constraints to optimization problems in gas transport under uncertainty in the exit and entry loads

Summary

Introduction

Decision making problems are more than often affected by parameter uncertainty. An optimization problem dealing with uncertain variables has the typical form min x g0(x). There are two main approaches for dealing with uncertainty in the constraints of an optimization problem: the first one is the use of probabilistic constraints This approach is based on the assumption that historical data is available to estimate the probability distribution of the uncertain parameter and considering it as a random vector ξ taking values in Rm. (1) may be given the form min x (2). Robust optimization is preferred in the absence of statistical data, and as a conservative approximation of the probabilistic programming setting This conservatism, may be significant up to the point of ending up at very small or even empty feasible sets possibly coming at much higher costs than under a probabilistic constraints. This trade-off propels the use of probabilistic constraints in the presence of statistical information at least in moderate dimension

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