A DISTURBED NASH GAME APPROACH FOR GAS NETWORK OPTIMIZATION Gerhard Jank ∗ Teresa Paula Azevedo-Perdico´ ulis ∗∗ ∗ Lehrstuhl II f¨ ur Mathematik, RWTH Aachen Templergraben 55, D-52056 Aachen, Germany e-mail: jank@math2.rwth-aachen.de ∗∗ ISR & UTAD P-5000-591 Vila Real, Portugal e-mail : tazevedo@utad.pt Abstract: In this paper, we propose to model the optimization of gas networks as a disturbed linear-quadratic game, where each player chooses his strategy according to a modified Nash equilibrium game under open-loop information structure. Conditions for the existence and uniqueness of such an equilibrium are related to the solution of certain Riccati difference equations and a boundary value problem. For simplicity sake, the two- player case is considered. The sought equilibrium is thus calculated through a solution of a boundary value problem. Finally, a numerical example is used to illustrate the presented algorithm. A simple network, where the players are two controllable elements relevant to the network dynamics, is used to illustrate the calculation procedure. Copyright c  2006 IFAC Keywords: Linear quadratic games, gas network optimisation, Nash games, Riccati difference equations. 1. INTRODUCTION As natural gas is becoming increasingly important in modern life, its transmission and distribution through ever expanding pipeline networks is dependent on effi- cient control and management. However, the problem characteristics, namely, large dimension, nonlinear- ity, geographical dispersion, and transient properties of the gas behaviour, make the design of efficient algorithms for optimisation of gas networks crucial. Gas networks are composed of geographically dis- persed controllable elements communicating between themselves through pipelines; network controllable el- ements are supplying sources, compressor stations, valves, gas holders and regulators. The network main optimising objective is to meet consumer demands at the lowest cost. In this work, the gas network dynamics is modelled as a disturbed linear-quadratic Nash game. Gas net- work transient dynamics have been seen before as a noncooperative Nash-game in (Azevedo-Perdico´ ulis, 1998; Ramchandani, 1993), where the players are the network controllable elements. In a two-level itera- tion scheme, similar to the economic tˆ atonnement, the players appoint their best settings and then interact to check for network feasibility. The devolved degree of network unfeasibility informs the players about the ’quality‘ of their settings. The whole process is re- peated until an equilibrium point is approached. Also, a stochastic version of the Stackelberg-Nash equilib- rium has been proposed in (Wolf and Smeers, 1991). In (Azevedo-Perdico´ ulis and Jank, 2004), the optimi- sation problem of gas networks is formulated as a 2- player open-loop Nash game with quadratic perfor- mance criteria and a linear difference equation ex- pressing network topology and transients as a con- straint. For the resulting linear-quadratic game, suffi- cient conditions exist which guarantee the existence of an unique equilibrium, and a solution can be found for the controllable input of every player in terms of solvability of certain symmetric and nonsymmetric Riccati equations (Jank and Abou-Kandil, 2003). International Journal of Tomography & Statistics Fall 2007, Vol. 7, No. F07; Int. J. Tomogr. Stat.; 43-48 ISSN 0972-9976; Copyright © 2007 by IJTS, ISDER