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Optimal transport on gas networks

Ariane Fazeny, Martin Burger, Jan-Frederik Pietschmann

TL;DR

The paper addresses transport dynamics on gas networks by modelling the network as a metric graph with edges carrying isothermal Euler dynamics and vertices enforcing Kirchhoff-type couplings and boundary flows. It develops a generalized dynamic $p$-Wasserstein framework on graphs, enabling both balanced and unbalanced transport with vertex storage and boundary data, and derives $p$-Wasserstein gradient flows, recovering the ISO3 gas-network model at $p=3$. A primal–dual numerical scheme is extended to graphs with and without vertex dynamics, and the approach is illustrated through geodesic branching and time-dependent inflow/outflow examples. This framework provides a rigorous, computationally tractable way to analyze and optimize gas transport on networks, with potential applications to mixed natural gas–hydrogen systems and storage-aware network design.

Abstract

This paper models gas networks as metric graphs, with isothermal Euler equations at the edges, Kirchhoff's law at interior vertices and time-(in)dependent boundary conditions at boundary vertices. For this setup, a generalized $p$-Wasserstein metric in a dynamic formulation is introduced and utilized to derive $p$-Wasserstein gradient flows, specifically focusing on the non-standard case $p = 3$.

Optimal transport on gas networks

TL;DR

The paper addresses transport dynamics on gas networks by modelling the network as a metric graph with edges carrying isothermal Euler dynamics and vertices enforcing Kirchhoff-type couplings and boundary flows. It develops a generalized dynamic -Wasserstein framework on graphs, enabling both balanced and unbalanced transport with vertex storage and boundary data, and derives -Wasserstein gradient flows, recovering the ISO3 gas-network model at . A primal–dual numerical scheme is extended to graphs with and without vertex dynamics, and the approach is illustrated through geodesic branching and time-dependent inflow/outflow examples. This framework provides a rigorous, computationally tractable way to analyze and optimize gas transport on networks, with potential applications to mixed natural gas–hydrogen systems and storage-aware network design.

Abstract

This paper models gas networks as metric graphs, with isothermal Euler equations at the edges, Kirchhoff's law at interior vertices and time-(in)dependent boundary conditions at boundary vertices. For this setup, a generalized -Wasserstein metric in a dynamic formulation is introduced and utilized to derive -Wasserstein gradient flows, specifically focusing on the non-standard case .
Paper Structure (27 sections, 6 theorems, 143 equations, 6 figures, 1 algorithm)

This paper contains 27 sections, 6 theorems, 143 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Related work
  4. Modelling gas networks as metric graphs
  5. Gas flow in single pipes
  6. Gas flow at interior vertices
  7. Gas flow at boundary vertices
  8. Transport type problems on metric graphs
  9. Measure spaces on graphs
  10. Formulation of the optimal transport problems
  11. Feasibility of the optimal transport problem
  12. Strong and weak solutions of the continuity equation constraints
  13. $p$-Wasserstein distance as cost of transport
  14. General $p$-Wasserstein distance
  15. $p$-Wasserstein distance on networks
  16. ...and 12 more sections

Key Result

Proposition 2.10

The source vertex mass density $S_{\nu} \left(t\right)$ of a source vertex $\nu \in \partial^+ \mathcal{V}$ at any time $t \in \left[0, T\right]$ can be calculated by for time-dependent boundary conditions, or with $s_{\nu}$ instead of $s_{\nu}^G$ for time-independent boundary conditions.

Figures (6)

  • Figure 1: Sketch of the graph used in the first example for branching geodesics. Here, no in- or outflux via the boundary is assumed (i.e. $\partial^+ \mathcal{V} = \partial^- \mathcal{V} = \emptyset$).
  • Figure 2: Branching geodesic without vertex dynamic: snapshots of the dynamics of the densities $\rho_e$ at different times.
  • Figure 3: Branching geodesic with vertex dynamic: snapshots of the dynamics of the densities $\rho_e$ and $\gamma_{\nu}$ at different times.
  • Figure 4: Sketch of the graph used in the second example. We set $\partial^+ \mathcal{V} = \{\nu_1\}$, $\partial^- \mathcal{V} = \{\nu_3,\,\nu_4\}$ and $\mathring{\mathcal{V}} = \{\nu_2\}$.
  • Figure 5: Snapshots of the dynamics of the densities $\rho_e$ and $\gamma_{\nu}$ with symmetric boundary conditions \ref{['eq:bc_in_out_sym']} at different times.
  • ...and 1 more figures

Theorems & Definitions (32)

  • Remark 2.1: Assumptions for the graph
  • Example 2.2: Gas network as oriented graph
  • Remark 2.3: Modelling three-dimensional pipes as one-dimensional edges
  • Definition 2.5: Time-dependent boundary conditions
  • Definition 2.7: Time-independent boundary conditions
  • Remark 2.9: Non-positivity of $s_{\nu}$ and non-negativity of $d_{\nu}$
  • Proposition 2.10: Source vertex mass density
  • proof
  • Definition 3.1: Optimal transport problem
  • Remark 3.2: Extension with boundary conditions
  • ...and 22 more