CRC-TRR 154/1: Hierarchical PDAE-surrogate-modeling and stable PDAE-network-discretization for simulating large non-stationary gas networks (SP C02)

This subproject focuses on the development and analysis of models and methods for a stable and fast simulation of huge transient gasnetworks, which will also be used for an efficient parameter optimization and control of the network. The main aspects are the development of a numerical discretization in space and time that is adjusted to the topology of the network as well as a hierarchical modelling of several elements (pipes, compressors ect.) and subnet-structures.
For the complete network as a coupled system of nonlinear partial differential equations and algebraic equations (PDAE) we consider approximations by a spatial semi-discretization. We strive for a determination and classification of topology depending critera for the index of the time dependend differential algebraic system. Topology- and controldepending spatial discretizations will be determinded, that lead to DAEs of index 1, in order to diminish the influence of perturbations for the DAE system best possible. Moreover we want to establish a perturbationanalysis as well as existence and uniquness results for die PDAE-model. Here, the time and pressure/flow-depending control-states that can change the variable structure (dynamic as well as static) for certain points in time and for certain states of the network will be a major challange.
As a method, we focus on a Galerkin-Approach in space followed by a discretization in time of the resulting DAE with implicit or semi-implicit methods respectively, such that the algebraic constraints hold for the current point in time. Continuationmethods and space-mapping techniques are used for the initialisation to guarantee good convergence behaviour. Furthermore, to satisfy the control requirements of the systems and to enable the handling of huge networks, this subproject aims at the enhancement of the simulation speed. It is planed to detect characteristic subnetstructures and derive parameter dependent transient surrogate models with suitable error estimators by applying model order reduction methods. These input-ouput models as dynamic systems of ODEs will be coupled with die complete PDAE model in one model hierarchy.

The second phase (2018-2022) is also dedicated to the simulation of large gas networks, but with a focus on optimization of transient compressor controls, compliance with pressure and flow barriers, and overcoming simulation hurdles due to valve opening and closing. Methodologically, we take an approach that differs from the two basic concepts first-discretize-then-optimize approach and first-optimize-then-discretize approach. We follow a strategy of the form 1. discretize in space 2. optimize 3. discretize in time. We first choose an appropriate local discretization approach for the PDAE systems. In addition to the discretization approach developed in the first phase and adapted to the mesh topology, a local discretization approach via mixed finite elements is used here, which guarantees mass and energy conservation. This results in DAE systems with possibly higher index and index change. At consideration of pressure and flux barriers, these are overdetermined. Here, we now pursue the idea of solving the DAEs with a least-squares collocation method. In numerical tests, the least-squares collocation showed a surprisingly good regularizing effect on the inherent differentiation problems of DAEs for both linear and nonlinear DAEs. However, error estimates are only available for linear systems so far. These will be extended to linear systems with index changes, nonlinear systems, and overdetermined systems. Furthermore, the optimization problem of the compressor control will be solved using the adjoint calculus and solving the resulting boundary value problem by a least-squares collocation.

Principal investigators
Tischendorf, Caren Prof. Dr. (Details) (Applied Mathematics)

Participating external organisations

DFG: Sonderforschungsbereich

Duration of project
Start date: 10/2014
End date: 06/2022

Research Areas

Research Areas
Diskrete Optimierung, Kontinuierliche Optimierung, Mathematische Modellierung, Modelladaption, Numerische Analysis, Optimierung über Partielle Differentialgleichungen, Simulation, Stochastische Optimierung

Last updated on 2023-27-05 at 06:30