TY - JOUR A1 - Gotzes, Claudia A1 - Heitsch, Holger A1 - Henrion, Rene A1 - Schultz, Rüdiger T1 - On the quantification of nomination feasibility in stationary gas networks with random load JF - Mathematical Methods of Operations Research N2 - The paper considers the computation of the probability of feasible load constellations in a stationary gas network with uncertain demand. More precisely, a network with a single entry and several exits with uncertain loads is studied. Feasibility of a load constellation is understood in the sense of an existing flow meeting these loads along with given pressure bounds in the pipes. In a first step, feasibility of deterministic exit loads is characterized algebraically and these general conditions are specified to networks involving at most one cycle. This prerequisite is essential for determining probabilities in a stochastic setting when exit loads are assumed to follow some (joint) Gaussian distribution when modeling uncertain customer demand. The key of our approach is the application of the spheric-radial decomposition of Gaussian random vectors coupled with Quasi Monte-Carlo sampling. This approach requires an efficient algorithmic treatment of the mentioned algebraic relations moreover depending on a scalar parameter. Numerical results are illustrated for different network examples and demonstrate a clear superiority in terms of precision over simple generic Monte-Carlo sampling. They lead to fairly accurate probability values even for moderate sample size. Y1 - 2016 U6 - https://doi.org/10.1007/s00186-016-0564-y VL - 84 IS - 2 SP - 427 EP - 457 ER - TY - INPR A1 - Gugat, Martin A1 - Schultz, Rüdiger T1 - Boundary feedback stabilization of the isothermal Euler-equations with uncertain boundary data N2 - In a gas transport system, the customer behavior is uncertain. Motivated by this situation, we consider a boundary stabilization problem for the flow through a gas pipeline, where the outflow at one end of the pipe %that is governed by the customer's behavior is uncertain. The control action is located at the other end of the pipe. The feedback law is a classical Neumann velocity feedback with a feedback parameter $k>0$. We show that as long as the $H^1$-norm of the function that describes the noise in the customer's behavior decays exponentially with a rate that is sufficiently large, the velocity of the gas can be stabilized exponentially fast in the sense that a suitably chosen Lyapunov function decays exponentially. For the exponential stability it is sufficient that the feedback parameter $k$ is sufficiently large and the stationary state to which the system is stabilized is sufficiently small. The stability result is local, that is it holds for initial states that are sufficiently close to the stationary state. This result is an example for the exponential boundary feedback stabilization of a quasilinear hyperbolic system with uncertain boundary data. The analysis is based upon the choice of a suitably Lyapunov function. The decay of this Lyapunov function implies that also the $L^2$-norm of the difference of the system state and the stationary state decays exponentially. Y1 - 2017 ER - TY - JOUR A1 - Gugat, Martin A1 - Wintergerst, David A1 - Schultz, Rüdiger ED - Iske, Armin T1 - Networks of pipelines for gas with nonconstant compressibility factor: stationary states JF - Computational and Applied Mathematics N2 - For the management of gas transportation networks, it is essential to know how the stationary states of the system are determined by the boundary data. The isothermal Euler equations are an accurate pde-model for the gas flow through each pipe. A compressibility factor is used to model the nonlinear relationship between density and pressure that occurs in real gas in contrast to ideal gas. The gas flow through the nodes is governed by algebraic node conditions that require the conservation of mass and the continuity of the pressure. We examine networks that are described by arbitrary finite graphs and show that for suitably chosen boundary data, subsonic stationary states exist and are uniquely determined by the boundary data. Our construction of the stationary states is based upon explicit representations of the stationary states on each single pipe that can easily be evaluated numerically. We also use the monotonicity properties of these states as functions of the boundary data. Y1 - 2016 U6 - https://doi.org/10.1007/s40314-016-0383-z ER - TY - INPR A1 - Gotzes, Claudia A1 - Nitsche, Sabrina A1 - Schultz, Rüdiger T1 - Probability of Feasible Loads in Passive Gas Networks with up to Three Cycles N2 - Gas networks are of growing importance for the economy and offer interesting mathematical problems at the same time. The classical linear network flow allows for approximate models that more and more have come to their limits. This has raised interest in nonlinear but, for simplicity, still steady-state models. The present paper aims at mobilizing techniques from symbolic computation and reparametrization of multivariate integrals to enable validation of stochastic nominations following Gaussian distributions in passive gas networks with more than one cycle. KW - Mathematical Gas Network Models, Stochastic Nomination Validation, Parametric Models Y1 - 2017 ER - TY - JOUR A1 - Schultz, Rüdiger A1 - Wollenberg, Tobias T1 - Unit commitment under uncertainty in AC transmission systems via risk averse semidefinite stochastic programs JF - RAIRO-Operations Research N2 - This paper addresses unit commitment under uncertainty of load and power infeed from renewables in alternating current (AC) power systems. Beside traditional unit-commitment constraints, the physics of power flow are included. To gain globally optimal solutions a recent semidefinite programming approach is used, which leads us to risk averse two-stage stochastic mixed integer semidefinite programs for which a decomposition algorithm is presented. KW - Stochastic Programming KW - Semidefinite Programming KW - AC Power Flow Y1 - 2017 U6 - https://doi.org/https://doi.org/10.1051/ro/2016031 VL - 51 IS - 2 SP - 391 EP - 416 ER - TY - INPR A1 - Schmidt, Martin A1 - Hiller, Benjamin A1 - Koch, Thorsten A1 - Pfetsch, Marc A1 - Geißler, Björn A1 - Henrion, René A1 - Joormann, Imke A1 - Martin, Alexander A1 - Morsi, Antonio A1 - Römisch, Werner A1 - Schewe, Lars A1 - Schultz, Rüdiger A1 - Steinbach, Marc C. T1 - Capacity Evaluation for Large-Scale Gas Networks N2 - Natural gas is important for the energy turnaround in many countries like in Germany, where it serves as a "bridging energy" towards a fossil-free energy supply in the future. About 20% of the total German energy demand is provided by natural gas, which is transported through a complex pipeline network with a total length of about 30000 km and the efficient use of the given transport infrastructure for natural gas is of political, economic, and societal importance. As a consequence of the liberalization of the European gas market in the last decades, gas trading and transport have been decoupled. This has led to new challenges for gas transport companies, and mathematical optimization is perfectly suited for tackling many of these challenges. However, the underlying mathematical problems are by far too hard to be solved by today's general-purpose software so that novel mathematical theory and algorithms are needed. The industrial research project "ForNe: Research Cooperation Network Optimization" has been initiated and funded by Open Grid Europe in 2009 and brought together experts in mathematical optimization from seven German universities and research institutes, which cover almost the entire range of mathematical optimization: integer and nonlinear optimization as well as optimization under uncertainty. The mathematical research results have been put together in a software package that has been delivered to Open Grid Europe at the end of the project. Moreover, the research is still continuing - e.g., in the Collaborative Research Center/Transregio 154 "Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks" funded by the German Research Foundation. Y1 - 2019 ER - TY - JOUR A1 - Adelhütte, Dennis A1 - Aßmann, Denis A1 - Gonzàlez Grandòn, Tatiana A1 - Gugat, Martin A1 - Heitsch, Holger A1 - Liers, Frauke A1 - Henrion, René A1 - Nitsche, Sabrina A1 - Schultz, Rüdiger A1 - Stingl, Michael A1 - Wintergerst, David T1 - Joint model of probabilistic/robust (probust) constraints applied to gas network optimization N2 - 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 deal with a class of joint probabilistic/robust constraints as its appears in optimization problems under uncertainty. 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 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. KW - robust optimization KW - chance constraints KW - optimal control KW - spheric-radial decomposition Y1 - 2017 U6 - https://doi.org/10.1007/s10013-020-00434-y ER - TY - JOUR A1 - Gugat, Martin A1 - Schultz, Rüdiger A1 - Schuster, Michael T1 - Convexity and Starshapedness of Feasible Sets in Stationary Flow Networks JF - Networks and Heterogeneous Media N2 - We deal with a stationary model for flow through a network. The flows are determined by the values at the boundary nodes of the network that we call the loads of the network. In the applications , the feasible loads must satisfy some box constraints. We analyze the structure of the set of feasible loads. Our analysis is motivated by gas pipeline flows, where the box constraints are pressure bounds. We present sufficient conditions to show, that the feasible set is star-shaped with respect to special points. For stronger conditions, we prove the convexity of the set of feasible loads. All the results are given for active and passive networks, i.e. networks with and without inner control. This analysis is motivated by the aim to use the spheric-radial decomposition for stochastic boundary data in this model. This paper can be used for simplifying the algorithmic use of the spheric-radial decomposition. KW - Validation of nominations, Stationary states, Isothermal Euler equations, Gas networks, Convexity, Star-Shapedness, Spheric-radial decomposition Y1 - 2020 U6 - https://doi.org/10.3934/nhm.2020008 SP - 171 EP - 195 ER -