Perturbation analysis and numerical discretisation of hyperbolic partial differential algebraic equations describing flow networks

Huck, Christoph

doi:10.18452/19596

Cited by 3 publications

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“…Second, it is well-known from DAE theory that the topology of a network directly influences how perturbations of the initial data and force terms propagate over time. The same holds true for the discretization of such a system: As shown in [62], it is necessary to discretize such a gas transport system properly to avoid unstable numerical solutions. With respect to such questions, a DAE perspective seems unavoidable.…”

Section: Relation To Other Coupled Systems and Hyperbolic Systemsmentioning

confidence: 96%

“…29 sq.]. The analysis of abstract DAEs and PDAEs, their numerical treatment, and their applications are part of current and ongoing research, see for instance [9,40,62,86,128].…”

Section: Notions Of Daesmentioning

confidence: 99%

“…In bio-mathematics, the blood flow through the cardiovascular system can be modeled by coupled systems of DAEs and PDEs, see for instance [67,87,111]. Due to the climate crisis, the interest in modeling and simulating energy transport networks is ever-increasing and the research of such and related problems has recently drawn more and more focus, see [62,67,116]. Coupled systems of DAEs and PDEs can often be found in literature as PDAEs, and we refer to the introduction in Chapter 2 for more information.…”

Section: Analysis Of a Coupled System Introduction To Coupled Systemsmentioning

confidence: 99%

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A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation

Groh

2024

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Diese Dissertation befasst sich mit der Analyse eines gekoppelten Systems der Form (Eu) ′ (t) + φ 1 (t, u(t), v(t)) = q 1 (t), v ′′ (t) − ∆v(t) + φ 2 (t, u(t), v(t)) = q 2 (t).(1.1)Es besteht aus einer semilinearen abstrakten differential-algebraischen Gleichung (DAE) und einer semilinearen hyperbolischen partiellen Differentialgleichung zweiter Ordnung. Beide Gleichungen sind durch die nichtlinearen Kopplungsoperatoren φ 1 und φ 2 miteinander gekoppelt.Gekoppelte Systeme dieser Art können als spezielle abstrakte DAEs und als Verallgemeinerungen von partiell differential-algebraischen Gleichungen aufgefasst werden. Sie spielen in vielen Anwendungen wie der Modellierung von multiphysikalischen Systemen, bei der Simulation von Schaltkreisen, oder der Optimalsteuerung von Gasnetzwerken eine Rolle.In der vorliegenden Arbeit diskutieren wir zunächst nur die abstrakte DAE und führen sogenannte Matrix-induzierte lineare Operatoren ein. Wir übertragen unter Nutzung dieser Operatoren einen Entkopplungsansatz für DAEs auf den unendlichdimensionalen Fall der vorliegenden abstrakten DAE. In Kombination mit einer neuartigen Index-1-Charakterisierung für semilineare abstrakte DAEs gelingt es uns, die inhärente gewöhnliche Differentialgleichung und die algebraischen Nebenbedingungen aus der abstrakten DAE zu extrahieren und Existenz und Eindeutigkeit von Lösungen zu zeigen. Anschließend verbinden wir die entwickelten Ansätze zur Behandlung von derartigen abstrakten DAEs mit bereits bekannten Ansätzen für die Analyse von hyperbolischen Gleichungen zweiter Ordnung, und schaffen einen einheitlichen Rahmen, in dem wir das gekoppelte System (1.1) diskutieren können. Mithilfe eines Fixpunktansatzes zeigen wir Existenz und Eindeutigkeit von lokalen und globalen Lösungen zu diesem gekoppelten System. Zu guter Letzt formulieren wir ein Optimalsteuerungsproblem, in dem das System (1.1) als Nebenbedingung auftritt. Wir zeigen die Existenz einer optimalen globalen Steuerung und einer globalen Minimalstelle. vii I wish to say thanks to Prof. Dr. Barbara Zwicknagl, my mentor within the Berlin Mathematical School. Her professional advice and her kindness helped more than once to push through difficult moments. I also wish to thank Prof. Dr. Axel Kröner for his support and motivating words. Gratitude is equally owed to Prof. Dr. Etienne Emmrich who acted as reviewer for this thesis. I would also like to thank some of the most important staff members for technology, service, and administration at the Faculty of Mathematics and Natural Sciences as well as the Department of Mathematics of the Humboldt-Universität zu Berlin. I would like to thank

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Section: Relation To Other Coupled Systems and Hyperbolic Systemsmentioning

confidence: 96%

“…29 sq.]. The analysis of abstract DAEs and PDAEs, their numerical treatment, and their applications are part of current and ongoing research, see for instance [9,40,62,86,128].…”

Section: Notions Of Daesmentioning

confidence: 99%

Section: Analysis Of a Coupled System Introduction To Coupled Systemsmentioning

confidence: 99%

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A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation

Groh

2024

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MathEnergy – Mathematical Key Technologies for Evolving Energy Grids

Clees

Baldin

Benner

et al. 2021

Mathematical Modeling, Simulation and Optimization for Power Engineering and Management

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On the numerical treatment of interlaced target values: modeling, optimization and simulation of regulating valves in gas networks

2023

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Due to the current and foreseeable shifts towards carbon dioxide neutral energy production, which will likely result in balancing fluctuating renewable energy generation by transforming power-to-gas-to-power as well as building a large-scale hydrogen transport infrastructure, the trading and transport operations of gas will become more dynamic, volatile, and hence also less predictable. Therefore, computer-aided support in terms of rapid simulation and control optimization will further broaden its importance for gas network dispatching. In this paper, we aim to contribute and openly publish two new mathematical models for regulators, also referred to as control valves, which together with compressors make up the most complex and involved types of active elements in gas network infrastructures. They provide direct control over gas networks but are in turn controlled via target values, also known as set-point values, themselves. Our models incorporate up to six dynamical target values to define desired transient states for the elements’ local vicinity within the network. That is, each pair of every two target values defines a bounding box for the inlet pressure, outlet pressure as well as the passing mass flow of gas. In the proposed models, those target values are prioritized differently and are constantly in competition with each other, which can only be resolved dynamically at run-time of either a simulation or optimization process. Besides careful derivation, we compare simulation and optimization results with predictions of the widely adopted commercial simulation tool SIMONE, serving as our substitute for actual real-world transport operations.

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Perturbation analysis and numerical discretisation of hyperbolic partial differential algebraic equations describing flow networks

Cited by 3 publications

References 0 publications

A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation

A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation

MathEnergy – Mathematical Key Technologies for Evolving Energy Grids

On the numerical treatment of interlaced target values: modeling, optimization and simulation of regulating valves in gas networks

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