TP C02: Hierarchische PDAE-Surrogate-Modellierung und stabile PDAE-Diskretisierung zur Simulation großer instationärer Gasnetzwerke

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 dependen 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.