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Hierarchical Gas Model Coupling on Networks

Mindt, Pascal (2019):
Hierarchical Gas Model Coupling on Networks.
Darmstadt, Technische Universität, [Online-Edition: https://tuprints.ulb.tu-darmstadt.de/8710],
[Ph.D. Thesis]

Abstract

In recent years the simulation of gas flow on networks attracts increasing interest. Since natural sources of energy, like wind and solar power, might lack of continuity, some demands in energy are compensated by gas. Therefore, accurate simulations for gas transport are essential. However, a highly detailed simulation suffers from great computational costs. Consequently, it becomes natural to use models with less physical detail in pipes with lower activity, while for pipes with greater dynamics, models with higher physical detail are used.

In the analytical part of this work, we consider a network, with one single junction and a given model hierarchy. It appears the question how these models are coupled at the junction and which kind of coupling conditions have to be posed such that a resulting solution is unique and physically correct, as far as it even exists.

In order to answer the above questions, we propose mass-, energy- and entropy- preserving coupling conditions at the junction. By introducing, a so called generalized Riemann problem at the junction, i.e., piecewise constant initial data, all models are connectible to each other through the coupling conditions. Afterwards, we show well-posedness of the generalized Riemann problem, i.e., there exists a unique physically correct solution.

The well-posedness above creates a foundation for a more general setting, the so called Cauchy problem, in which initial data needs to be integrable with small total variation only. Here, well-posedness is shown as well. Based on these results, even existence of an optimal control can be proven.

In the second part of this work, we like to give some numerical illustrations, built on our analytical results.

Item Type: Ph.D. Thesis
Erschienen: 2019
Creators: Mindt, Pascal
Title: Hierarchical Gas Model Coupling on Networks
Language: English
Abstract:

In recent years the simulation of gas flow on networks attracts increasing interest. Since natural sources of energy, like wind and solar power, might lack of continuity, some demands in energy are compensated by gas. Therefore, accurate simulations for gas transport are essential. However, a highly detailed simulation suffers from great computational costs. Consequently, it becomes natural to use models with less physical detail in pipes with lower activity, while for pipes with greater dynamics, models with higher physical detail are used.

In the analytical part of this work, we consider a network, with one single junction and a given model hierarchy. It appears the question how these models are coupled at the junction and which kind of coupling conditions have to be posed such that a resulting solution is unique and physically correct, as far as it even exists.

In order to answer the above questions, we propose mass-, energy- and entropy- preserving coupling conditions at the junction. By introducing, a so called generalized Riemann problem at the junction, i.e., piecewise constant initial data, all models are connectible to each other through the coupling conditions. Afterwards, we show well-posedness of the generalized Riemann problem, i.e., there exists a unique physically correct solution.

The well-posedness above creates a foundation for a more general setting, the so called Cauchy problem, in which initial data needs to be integrable with small total variation only. Here, well-posedness is shown as well. Based on these results, even existence of an optimal control can be proven.

In the second part of this work, we like to give some numerical illustrations, built on our analytical results.

Place of Publication: Darmstadt
Divisions: 04 Department of Mathematics
04 Department of Mathematics > Numerical Analysis and Scientific Computing
04 Department of Mathematics > Numerical Analysis and Scientific Computing > Hierarchical Modelling and Model Adaptivity for Gas Flow on Networks
Date Deposited: 15 Sep 2019 19:55
Official URL: https://tuprints.ulb.tu-darmstadt.de/8710
URN: urn:nbn:de:tuda-tuprints-87104
Referees: Lang, Prof. Dr. Jens and Herty, Prof. Dr. Michael
Refereed / Verteidigung / mdl. Prüfung: 7 November 2018
Alternative Abstract:
Alternative abstract Language
In den vergangen Jahren erfuhr die Simulation von Gasfluss in Netzwerken gesteigertes Interesse. Auf Grund des schwankenden Energieflusses durch natuerliche Ressourcen, wie Wind und Solarenergie, werden einige Nachfragen durch Gas kompensiert. Folglich ist eine detaillierte Simulation des Gastransportes essentiell. Kehrseite ist jedoch der, damit verbundene, hohe Rechnungsaufwand. Um dies zu vermeiden, nutzt man Modelle mit niedrigem physikalischem Detail, bei wenig Gasdynamik im Rohr und Modelle mit hohem Detailgrad bei steigender Dynamik. Im analytischen Teil dieser Arbeit betrachten wir ein Netzwerk, versehen mit einer Modellhierarchie und einem einzigen Verzweigungspunkt. Es stellt sich nun die Frage, wie die vorhandenen Modelle an der Verzweigung zu koppeln sind und wie die Kopplungsbedingungen gestellt werden muessen, sodass eine resultierende Loesung physikalisch korrekt ist, sofern diese ueberhaupt existiert. Um diese Fragen zu beantworten, fordern wir Bedingungen an der Verzweigung zur Masse-, Energie- und Entropieerhaltung. Durch Einfuehrung eines sogenannten verallgemeinerten Riemann-Problems am Verzweigungspunkt, i.e. stueckweise konstante Anfangsdaten, lassen sich alle Modelle durch die Kopplungsbedingungen miteinander verbinden. Anschliessend zeigen wir Wohlgestelltheit des verallgemeinerten Riemann-Problems, i.e. es existiert eine physikalisch korrekte Loesung. Die Wohlgestelltheit des obigen Problems dient als Grundlage fuer ein allgemeineres Problem, dem sogenannten Cauchy-Problem. Hier muessen Anfangsdaten lediglich integrierbar und von beschraenkter totaler Variation sein. Die Wohlgestelltheit des allgemeineren Problems wird ebenfalls gezeigt. Basierend auf den obigen Resultaten, weisen wir ferner die Existenz einer optimalen Steuerung nach. Im zweiten Teil dieser Arbeit werden einige numerische Beispiele illustriert, welche auf den analytischen Ergebnissen basieren.German
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