We consider the discretization and subsequent model reduction of a system of partial differential-algebraic equations describing the propagation of pressure waves in a pipeline network. Important properties like conservation of mass, dissipation of energy, passivity, existence of steady states, and exponential stability can be preserved by an appropriate semi-
discretization in space via a mixed finite element method and also during the further dimension reduction by structure preserving Galerkin projection which is the main focus of this paper. Krylov subspace methods are employed for the construction of the reduced models and we discuss modifications needed to satisfy certain algebraic compatibility conditions; these are required to ensure the well-posedness of the reduced models and the preservation of the key properties. Our analysis is based on the underlying infinite dimensional problem and its Galerkin approximations. The proposed algorithms therefore have a direct interpretation in function spaces; in principle, they are even applicable directly to the original system of partial differential-algebraic
equations while the intermediate discretization by finite elements is only required for the actual
computations. The performance of the proposed methods is illustrated with numerical tests and the necessity for the compatibility conditions is demonstrated by examples.
Proceeding from balanced truncation-based parametric reduced order
models (BT-pROM) a matrix interpolation strategy is presented that allows the
cheap evaluation of reduced order models at new parameter sets. The method ex-
tends the framework of model order reduction (MOR) for high-order parameter-
dependent linear time invariant systems in descriptor form by Geuss (2013) by
treating not only permutations and rotations but also distortions of reduced order
basis vectors. The applicability of the interpolation strategy and different variants is
shown on BT-pROMs for gas transport in pipeline-networks
This work deals with the model order reduction (MOR) of a nonlinear-
parametric system of partial differential equations (PDEs). Applying a semidis-
cretization in space and replacing the nonlinearities by introducing new state vari-
ables, we set up quadratic-linear differential algebraic systems (QLDAE) and use a
Krylov-subspace MOR. The approach is investigated for gas pipeline modeling
Optimization and control of large transient gas networks require the fast
simulation of the underlying parametric partial differential algebraic systems. Sur-
rogate modeling techniques based on linearization around specific stationary states,
spatial semi-discretization and model order reduction allow for the set-up of para-
metric reduced order models that can act as basis sample to cover a wide parameter
range by means of matrix interpolations. However, the interpolated models are often
not stable. In this paper, we develop a stability-preserving interpolation method.
This paper provides a first contribution to port-Hamiltonian modeling of district heating networks. By introducing a model hierarchy of flow equations on the network, this work aims at a thermodynamically consistent port-Hamiltonian embedding of the partial differential-algebraic systems. We show that a spatially discretized network model describing the advection of the internal energy density with respect to an underlying incompressible stationary Euler-type hydrodynamics can be considered as a parameter-dependent finite-dimensional port-Hamiltonian system. Moreover, we present an infinite-dimensional port-Hamiltonian formulation for a compressible instationary thermodynamic fluid flow in a pipe. Based on these first promising results, we raise open questions and point out research perspectives concerning structure-preserving discretization, model reduction, and optimization.