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- Gas Dynamics (1)
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Consider a star-shaped network
of strings. Each string is governed by the wave equation.
At each boundary node of the network there is
a player that performs Dirichlet boundary control action
and in this way influences the system state.
At the central node, the states are coupled
by algebraic conditions in such a way that the energy is conserved.
We consider the corresponding antagonistic game
where each player minimizes a certain quadratic objective function
that is given by the sum of a control cost and
a tracking term for the final state.
We prove that under suitable assumptions
a unique Nash equilibrium exists
and give an explicit representation
of the equilibrium strategies.
The European gas market is governed by rules that are agreed on by the European Union. We present a mathematical market model that
takes into account this structure, where the technical system operator (TSO)
offers certain transportation capacities that can be booked and later nominated within the previously chosen bookings. The TSO also fixes booking fees and defines an operational control of the gas pipeline system in order to deliver the gas according to the nominations. Since the gas
flow is governed by a system of partial differential equations, to realize this control structure partial differential equations (PDEs) should be involved in the model.
While the four level gas market model has been discussed previously, in this
paper we take into account the
flow model by PDEs in the discussion of the model and in the reduction to a single level problem, where we also state the corresponding necessary optimality conditions.