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On a port-Hamiltonian formulation and structure-preserving numerical approximations for thermodynamic compressible fluid flow

Sarah-Alexa Hauschild, Nicole Marheineke

TL;DR

This work develops an infinite-dimensional port-Hamiltonian formulation for compressible non-isothermal Euler flow, enabling energy-preserving coupling in pipe networks via mass and enthalpy balance. It establishes a Stokes-Dirac structure with boundary ports and derives a structure-preserving weak formulation, including a compatible variable transformation to facilitate discretization. The authors then extend structure-preserving spatial discretization to networks and propose MOR and complexity-reduction techniques (notably compatible projection bases and empirical quadrature) that preserve mass and energy dissipation. Numerical studies on single pipes and networks illustrate the critical role of compatibility conditions and parity in MOR, and compare network-wide versus pipe-wise reduction approaches, showing the potential for accurate, efficient simulations of complex gas networks. The results provide a foundation for integrating non-isothermal gas networks with power and district-heating systems in energy-technology planning and optimization.

Abstract

The high volatility of renewable energies calls for more energy efficiency. Thus, different physical systems need to be coupled efficiently although they run on various time scales. Here, the port-Hamiltonian (pH) modeling framework comes into play as it has several advantages, e.g., physical properties are encoded in the system structure and systems running on different time scales can be coupled easily. Additionally, pH systems coupled by energy-preserving conditions are still pH. Furthermore, in the energy transition hydrogen becomes an important player and unlike in natural gas, its temperature-dependence is of importance. Thus, we introduce an infinite dimensional pH formulation of the compressible non-isothermal Euler equations to model flow with temperature-dependence. We set up the underlying Stokes-Dirac structure and deduce the boundary port variables. We introduce coupling conditions into our pH formulation, such that the whole network system is pH itself. This is achieved by using energy-preserving coupling conditions, i.e., mass conservation and equality of total enthalpy, at the coupling nodes. Furthermore, to close the system a third coupling condition is needed. Here, equality of the outgoing entropy at coupling nodes is used and included into our systems in a structure-preserving way. Following that, we adapt the structure-preserving aproximation methods from the isothermal to the non-isothermal case. Academic numerical examples will support our analytical findings.

On a port-Hamiltonian formulation and structure-preserving numerical approximations for thermodynamic compressible fluid flow

TL;DR

This work develops an infinite-dimensional port-Hamiltonian formulation for compressible non-isothermal Euler flow, enabling energy-preserving coupling in pipe networks via mass and enthalpy balance. It establishes a Stokes-Dirac structure with boundary ports and derives a structure-preserving weak formulation, including a compatible variable transformation to facilitate discretization. The authors then extend structure-preserving spatial discretization to networks and propose MOR and complexity-reduction techniques (notably compatible projection bases and empirical quadrature) that preserve mass and energy dissipation. Numerical studies on single pipes and networks illustrate the critical role of compatibility conditions and parity in MOR, and compare network-wide versus pipe-wise reduction approaches, showing the potential for accurate, efficient simulations of complex gas networks. The results provide a foundation for integrating non-isothermal gas networks with power and district-heating systems in energy-technology planning and optimization.

Abstract

The high volatility of renewable energies calls for more energy efficiency. Thus, different physical systems need to be coupled efficiently although they run on various time scales. Here, the port-Hamiltonian (pH) modeling framework comes into play as it has several advantages, e.g., physical properties are encoded in the system structure and systems running on different time scales can be coupled easily. Additionally, pH systems coupled by energy-preserving conditions are still pH. Furthermore, in the energy transition hydrogen becomes an important player and unlike in natural gas, its temperature-dependence is of importance. Thus, we introduce an infinite dimensional pH formulation of the compressible non-isothermal Euler equations to model flow with temperature-dependence. We set up the underlying Stokes-Dirac structure and deduce the boundary port variables. We introduce coupling conditions into our pH formulation, such that the whole network system is pH itself. This is achieved by using energy-preserving coupling conditions, i.e., mass conservation and equality of total enthalpy, at the coupling nodes. Furthermore, to close the system a third coupling condition is needed. Here, equality of the outgoing entropy at coupling nodes is used and included into our systems in a structure-preserving way. Following that, we adapt the structure-preserving aproximation methods from the isothermal to the non-isothermal case. Academic numerical examples will support our analytical findings.
Paper Structure (31 sections, 7 theorems, 119 equations, 13 figures, 4 tables)

This paper contains 31 sections, 7 theorems, 119 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Port-Hamiltonian Framework
  3. Port-Hamiltonian Formulation of Compressible Non-Isothermal Pipe Flow
  4. Ports and Stokes-Dirac Structure
  5. Weak Formulation and Incorporation of Boundary Conditions
  6. Port-Hamiltonian Formulation for Pipe Networks
  7. Network Toplogy and Function Spaces
  8. Coupling Conditions and the Port-Hamiltonian Framework
  9. Port-Hamiltonian Network Model
  10. Structure-Preserving Approximations
  11. Spatial Discretization
  12. Snapshot-based Model Order Reduction
  13. Computation of Projection Matrices
  14. Complexity Reduction
  15. Modification of Approximation Approach
  16. ...and 16 more sections

Key Result

Corollary 4

Assume that boundary terms vanish. For $\psi,\phi\in\mathcal{H}^1(\omega)^3$ and $\tilde{\psi},\tilde{\phi}\in\mathcal{L}^2(\omega)^3$ the operators defined in System Sys:pHDCv are skew-adjoint and self-adjoint semi-elliptic in the $\mathcal{L}^2$ inner-product, respectively.

Figures (13)

  • Figure 1: Stokes-Dirac structure
  • Figure 2: We have $\mathcal{N}=\{\nu_1,\nu_2,\nu_3,\nu_4\}$ and $\mathcal{E}=\{\omega_1,\omega_2,\omega_3\}$, respectively. The incidence mapping yields $n^{\omega_1}[\nu_1]=1$, $n^{\omega_1}[\nu_2]=-1$, $n^{\omega_2}[\nu_2]=1$,$n^{\omega_2}[\nu_3]=-1$, $n^{\omega_3}[\nu_2]=1$ and $n^{\omega_3}[\nu_4]=-1$. The set of boundary nodes is $\mathcal{N}_\partial=\{\nu_1,\nu_3,\nu_4\}$, such that $\mathcal{N}_0=\{\nu_2\}$.
  • Figure 3: Total energy and mass with (left) and without (right) compatibility conditions.
  • Figure 4: $\mathfrak{E}_{t,P}$ and $\mathfrak{E}_t$ for ROMS with (left) and without (right) compatibility conditions.
  • Figure 5: $\mathfrak{E}_{t,P}$ and $\mathfrak{E}_{t}$ for different values of $r_\uprho$.
  • ...and 8 more figures

Theorems & Definitions (21)

  • Remark 3
  • Corollary 4
  • proof
  • Definition 5
  • Theorem 7
  • Remark 8
  • Definition 9
  • Theorem 12
  • proof
  • Definition 13
  • ...and 11 more