# Control of System Dynamics in Gas and Water Networks

Lehrstuhl für Angewandte Mathematik II
Universität Erlangen-Nürnberg

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Email: gugat@am.uni-erlangen.de

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Lehrstuhl C für Mathematik
RWTH Aachen

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Homepage:

Lehrstuhl für Angewandte Mathematik II
Uni Erlangen

Tel: 09131 85-67135
Fax: 09131 85-67134

## DFG funded assistants

Markus Dick (Uni Erlangen)
markus.dick@am2.am.uni-erlangen.de

## Description

We are interested in gas and water transportation networks where the dynamics is governed by hyperbolic balance laws. Typically, the transient dynamics inside a pipe or channel $j$ is described by an spatial one--dimensional system of balance laws for which as a prototype one considers the $p-$system.

Typically $\rho_j$ being either the density in case gas flow or the water height in case of flow in open channels and $q_j$ the mass flux. A simple model for the pressure term in gas networks is $p(\rho)=a^2 \rho$ where as in the case of water networks we have the typical $p(\rho)= g/2 \rho^2.$ The main difference is the source term: The major physical effect in gas networks is pipe--wall friction where as in water networks also the gravity force due to the slope of the pipe is essential. Both terms can be included in $g(x, \rho,\rho u).$ In order to pose a complete problem we need to pose additional algebraic coupling conditions at the vertices. These induce (nonlinear) boundary conditions for the $p-$system and couple the dynamics on the connected pipes. Depending on the type of the intersection (pipe--to--pipe, compressor, valve, gate, $\dots$), different conditions are imposed, in general, they can be written as

$$Psi ( (\rho_{i})_{i\in\delta^\pm}, (\rho_{i} u_{i})_{i\in\delta^\pm} , U(t) ) = 0,$$

where $U(t)$ is a possible control with the interpretation of either a compressor power or the height of an underflow gate or the position of a valve.

Three main problems can be identified as tasks for gas and water net providers: {\em Under minimal costs they should (1.) satisfy the costumer's demands (pressure, water height), (2.) avoid the occurence of shock waves (in order to prevent pipe breakes) and (3.) provide a gas (resp. water) mixture of a certain quality}. The latter problem is due to the fact the providers procure gas (water) from several sources with different quality bu need to provide customers with a desired composition.

Currently, there exist results in the following directions. At first, existing commercial tools as well as most parts of the literature use only simplified models such as quasi--static and heuristic approximations , linearizations or coarse grid approximations to treat problem (1). However, these results are too simple since several crucial effects of the system dynamics, such as the occurrence of shock waves or the finite speed of wave propagation, are lost. Furthermore, the construction of controls or stabilization cannot be discussed in this context.

Second, in the previous funding period of the DFG SPP1253 within the project 'Optimal Nodal Control of Networked Hyperbolic Equations' the problem (1) has been studied in the case of weak solutions including shock waves. Several results concerning (1) could be achieved and have been published in a series of papers. In particular, in well--posedness of the problem under the expected assumption of a small $TV$--bound of the initial data could be established. The result is a well--posedeness and regularity result for coupled nonlinear system of balance laws and valid for the $p-$system under a subsonic condition and rather weak assumptions on the coupling function $\Psi.$ The obtained solutions do only posses BV--regularity in space, but may well include discontinuous solutions (shock waves). However, a result on the dependence of the solution on the control $U(t)$ could be established. This in turn has been used to prove existence of weak entropic and possibly discontinuous optimal controls and states. These results partly solve problem (1) in the sense of an optimal control problem, but do not yet extend to controllability or stabilitzation questions.

Last, there has been intense research concerning controllability, i.e., solving problem (1) and (2) using classical solutions. The existence of classical solutions for quasilinear hyperbolic systems on a given finite time interval (so-called semi-global solutions) has been investigated by Li Ta-Tsien and his group. For the case without source terms, also classical solutions on networks have been considered. However, for the system the influence of the source term is essential and, for example, the form of the equilibria is determined by the source term: Due to the source term, for each fixed non-zero flow rate at most one constant equilibrium exists. The existence of classical semi-global solutions for systems with source term has been analysed only recently, and the corresponding case of networked systems with source-terms has not yet been analyzed. We have a similar situation for problem (1) concerning stabilization of classical solutions at equilibria. There is some material for the case without source terms, also for networks where feedback laws are constructed using strict Lyapunov functions, but the case of systems with source terms is still open.