Refine
Fast and Reliable Transient Simulation and Continuous Optimization of Large-Scale Gas Networks
(2021)
We are concerned with the simulation and optimization of large-scale gas pipeline
systems in an error-controlled environment. The gas flow dynamics is locally
approximated by sufficiently accurate physical models taken from a hierarchy of
decreasing complexity and varying over time. Feasible work regions of compressor
stations consisting of several turbo compressors are included by semiconvex
approximations of aggregated characteristic fields. A discrete adjoint approach
within a first-discretize-then-optimize strategy is proposed and a sequential
quadratic programming with an active set strategy is applied to solve the
nonlinear constrained optimization problems resulting from a validation of nominations.
The method proposed here accelerates the computation of near-term forecasts of
sudden changes in the gas management and allows for an economic control of intra-day
gas flow schedules in large networks. Case studies for real gas pipeline systems
show the remarkable performance of the new method.
With this overview we want to provide a compilation of different models for the description of gas flow in networks in order to facilitate the introduction to the topic. Special attention is paid to the hierarchical structure inherent to the modeling, and the detailed description of individual components such as valves and compressors. Also included are network model classes based on purely algebraic relations, and energy-based port-Hamiltonian models. A short overview of basic numerical methods and concepts for the treatment of hyperbolic balance equations is also given. We do not claim completeness and refer in many places to the existing literature.
The idea of a model catalog came to us in the context of the application for the CRC/Transregio 154 ``Mathematical modeling, simulation and optimization using the example of gas networks''. The present
English translation is an extension from
[P. Domschke, B. Hiller, J. Lang, and C. Tischendorf. Modellierung von Gasnetzwerken: Eine Übersicht. Preprint, TRR 154, 2017]. At this point we would like to thank the DFG for its support.
Mit dieser Übersicht wollen wir eine Zusammenstellung von unterschiedlichen Modellen zur Beschreibung des Gasflusses in Netzwerken bereitstellen, um den Einstieg in das Thema zu erleichtern. Besonderes Augenmerk wird dabei auf die der Modellierung inneliegende hierarchische Struktur und die detaillierte Beschreibung einzelner Bauteile wie Ventile und Kompressoren
gelegt. Daneben finden sich ebenfalls Netzmodellklassen, die auf rein algebraische Relationen aufbauen. Am Ende geben wir einen kurzen Überblick über grundlegende numerische Verfahren und Konzepte zur Behandlung von hyperbolischen Bilanzgleichungen. Wir erheben keinen Anspruch auf Vollständigkeit und verweisen an vielen Stellen auf die bestehende Literatur.
Die Idee eines Modellkataloges ist uns im Rahmen der Antragstellung zum SFB/Transregio 154 „Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerken“ gekommen. Wir möchten an dieser Stelle die Förderung durch die DFG dankend erwähnen.
Adaptive Refinement Strategies for the Simulation of Gas Flow in Networks using a Model Hierarchy
(2017)
A model hierarchy that is based on the one-dimensional isothermal Euler equations of fluid dynamics is used for the simulation and optimisation of gas flow through a pipeline network. Adaptive refinement strategies have the aim of bringing the simulation error below a prescribed tolerance while keeping the computational costs low. While spatial and temporal stepsize adaptivity is well studied in the literature, model adaptivity is a new field of research. The problem of finding an optimal refinement strategy that combines these three types of adaptivity is a generalisation of the unbounded knapsack problem. A refinement strategy that is currently used in gas flow simulation software is compared to two novel greedy-like strategies. Both a theoretical experiment and a realistic gas flow simulation show that the novel strategies significantly outperform the current refinement strategy with respect to the computational cost incurred.
This paper is concerned with coupling conditions at junctions for transport
models which differ in their fidelity to describe transient flow in gas pipelines.
It also includes the integration of compressors between two pipes with possibly
different models. A hierarchy of three one-dimensional gas transport models is
built through the 3 × 3 polytropic Euler equations, the 2 × 2 isentropic Euler
equations and a simplified version of it for small velocities. To ensure entropy
preservation, we make use of the novel entropy-preserving coupling conditions
recently proposed by Lang and Mindt [Netw. Heterog. Media, 13:177-190,
2018] and require the equality of the total enthalpy at the junction and that
the specific entropy for pipes with outgoing flow equals the convex combination
of all entropies that belong to pipes with incoming flow. We prove the existence
and uniqueness of solutions to generalised Riemann problems at a junction in
the neighbourhood of constant coupling functions and stationary states which
belong to the subsonic region. This provides the basis for the well-posedness of
certain Cauchy problems for initial data with sufficiently small total variation.
With this overview we want to provide a compilation of different models for
the description of gas flow in networks in order to facilitate the introduction
to the topic. Special attention is paid to the hierarchical structure inherent
to the modeling, and the detailed description of individual components such
as valves and compressors. Also included are network model classes based
on purely algebraic relations, and energy-based port-Hamiltonian models. A
short overview of basic numerical methods and concepts for the treatment
of hyperbolic balance equations is also given. We do not claim completeness
and refer in many places to the existing literature.
In this paper, we are concerned with the quantification of
uncertainties that arise from intra-day oscillations in the
demand for natural gas transported through large-scale networks. The
short-term transient dynamics of the gas flow is modelled by a hierarchy
of hyperbolic systems of balance laws based on the isentropic Euler equations.
We extend a novel adaptive strategy for solving elliptic PDEs with random data,
recently proposed and analysed by Lang, Scheichl, and Silvester
[J. Comput. Phys., 419:109692, 2020], to uncertain gas transport problems.
Sample-dependent adaptive meshes and a model refinement in the physical space
is combined with adaptive anisotropic sparse Smolyak grids in the stochastic
space. A single-level approach which balances the discretization errors of
the physical and stochastic approximations and
a multilevel approach which additionally minimizes the computational costs
are considered. Two examples taken from a public gas library demonstrate the
reliability of the error control of expectations calculated from
random quantities of interest, and the further use of
stochastic interpolants to, e.g., approximate probability density functions
of minimum and maximum pressure values at the exits of the network.