# Optimal Nodal Control Of Networked Systems Of Conservation Laws

### From DFG-SPP 1253

## Project leaders

FB Mathematik

TU Kaiserslautern

Tel: 0631 205-3996

Fax: 0631 205-3052

Email: klar@itwm.fhg.de

Homepage:

Lehrstuhl für Angewandte Mathematik II

Uni Erlangen

Tel: 09131 8527509

Fax: 09131 85-28126

Email: leugering@am.uni-erlangen.de

Homepage:

## DFG funded assistants

Dr. Nils Bräutigam (Uni Erlangen)

nils.braeutigam@am.uni-erlangen.de

Veronika Schleper (TU Kaiserslautern)

sachers@mathematik.uni-kl.de

## assistant

Prof. Dr. Michael Herty (RWTH Aachen)

www.opt.rwth-aachen.de

## Description

The objective of this proposal is the investigation of controllability and optimality issues for networks of gas and water flow. The gained insight is then applied to derive adapted numerical methods for simulation and optimization of real-world gas and water networks.

Modelling of physical phenomena by conservation laws is nowadays common to many disciplines and provides a successful tool to analyze and predict behavior of many processes and we are particularly interested in gas flows in pipeline systems and flows in water channels. Especially the gas optimization is a very important industrial problem and has been investigated over several years but for reduced and simplified models only. The situation is similar in the case of water networks. Our starting point for both physical processes is a nonlinear system of conservation laws in one space dimension modelling the flow on each arc of a possibly large water or gas network. Coupling conditions determined by physical considerations are formulate in terms of Riemann problems to allow for discontinuous solutions. Furthermore, controls are applied at nodes of arbitrary degree. Hereby controlling a node of degree one corresponds to boundary controls, e.g. prescribing demand and supply behavior; of degree two to steering of compressors (resp. pumps); of degree larger than two to open auxiliary pipes (resp. channels) or valves. Finally, we deal with an initial-boundary value problems for a coupled system of conservation laws where structural information is available by the geometry of the network and due to the coupling conditions. In a first step we will carry out an analysis of the network problem for the coupled hyperbolic equation. Currently, there are no publications for general possibly discontinuous solutions to the systems for water and gas flow in networks.

However, in the context of traffic problems there exist some very recent results and partly the applied techniques can be used to analyze the gas and water network case. The key to existence is a local and nodal-based discussion and will be further exploited to derive controllability, sensitivity and optimality conditions for node control problems of water and gas networks. Those controls in particular allow discontinuous solutions which are an important complement to the existing results about classical solutions. Second, we investigate equilibrium states near a vertex. We derive conditions at the nodes which improve the regularity of the weak solutions, i.e., in particular conditions which imply weak to be strong solutions. Due to the local structure of the applied techniques it is expected to obtain results for a single vertex first. The extension to arbitrary networks will depend on suitable a priori estimates on the solution similarly to preliminary work done by one of the applicants. Third, we develop a numerical tool which benefits from the insights of the previous results. This is achieved by extending a recent collaboration of both applicants. The applied techniques proved a natural setting for methods dealing with local Riemann problems, i.e., Front- or Wave-Tracking methods. This has already been applied successfully for simulations and optimization of traffic flow networks by one of the applicants and the preliminary work shows that even control problems can be efficiently handled by this approach.

The list of references (c.f. application) shows that there exists knowledge to deal with possible questions concerning control, modelling or numerics which might occur during the analysis of the network problem. Furthermore, preliminary work has been completed successfully. Due to the similar structure of the governing equations the program outlined can be carried out for the isothermal Euler equations (gas networks) as well as for St. Venant equations (water networks).

The proposed research will also provide strong support to the larger picture of optimizing and controlling infrastructures based on both discrete decisions and controls acting on the physical system.