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Humboldt-Universität zu Berlin

Dept. of Mathematics, Group Prof. Dr. Tischendorf

MatheonLogo Research Center Matheon     BMSLogo Berlin Mathematical School     ECMathLogo Einstein Center ECMath

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Stable Transient Modeling and Simulation of Flow Networks

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Background


An increasing interest in transient modeling, simulation and optimization of flow networks can be observed in diverse application fields. Future energy supplying concepts aim at better adapted scheduling of energy consumption and energy generation. It needs intelligent control systems for energy supplying and energy consuming processes.

Gas distributing and water distributing companies have to guarantee gas and water supply whenever and wherever the consumers require it. The deployment of highly energy consuming network elements as gas turbines and water pumps has to be adapted to the ongoing consumer demands. In the field of electronic circuits, modeling, simulation and optimization are well established. However, also here we are confronted with further development needs. The coupled 3D electromagnetic field and circuit simulation is a hot topic for resolving cross talking effects. Beside these engineering application fields, the modeling and simulation of flow networks becomes of interest in the understanding and control of electrophysiological processes, in particular in pediatric cardiology.

All such networks have in common that their topology can be described by incidence matrices representing the branch to node relations. Their different transient behavior is reflected by differential equations in terms of time dependent flow variables through branches and time dependent nodal potentials/pressures. Depending on the modelling level for each branch element of the network, the flow variables may also depend on space and the system includes partial differential equations. Typical examples are 1D transport equations for gas/water pipe models and 3D Maxwell equations for electromagnetic field component models.

The resulting model equations are partial differential-algebraic equations which lead to differential-algebraic equations after space discretization via finite differences or Galerkin approaches. Recent results have shown that the stability of the resulting differential-algebraic equations (DAEs) is strongly influenced by the choice of the model formulation and the choice of space discretization. Improper gauge and improper local boundary conditions may result in higher index DAE systems.

Research program for next funding period


The project aims a modeling guideline for flow networks guaranteeing stable partial differential-algebraic equation systems (PDAEs). Furthermore, prototype space and time discretizations shall be identified to ensure stable numerical solutions for such network PDAEs.

The first aim shall be achieved by an extension of the results about the stability behavior of abstract differential equations. Instead of monotonicity criteria for the whole system, monotonicity criteria for network elements shall be combined with Lipschitz coupling criteria.

The second aim shall be realized by a combination of tools for directed graphs and flow direction adapted space discretizations. Certain nodes are particular: supply nodes and demand nodes. At these nodes, the flow or the pressure/potential are prescribed. Consequently, the flow or pressure at other nodes might be fixed. For a proper space discretization we have to identify a proper direction for each branch regarding the type of boundary conditions for all nodes.

The goals of the project can be summarized as follows:

Cooperation


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