A structured control model for the thermo-magneto-hydrodynamics of plasmas in tokamaks

Abstract

International audienceA thermo-magneto-hydrodynamics port-Hamiltonian model is derived for the plasmas in tokamaks. Electromagnetic field and material domain balance equations are expressed in covariant forms, together with the magneto-hydrodynamics interconnection structure connecting them together. The balance equations for the entropy, mass and momentum, as well as closure equations in the material domain, are derived from the Boltzmann equation (kinetic theory). The Gibbs-Duhem equation is used to compute the irreversible entropy source term and to define the interdomain R-field of the model. All derived interdomain couplings in the material domain are represented using Dirac and Stokes-Dirac structures and the resistivity R-field structure. The complete model is summarized in a Bond Graph

Numerical Approximation of Port-Hamiltonian Systems for Hyperbolic or Parabolic PDEs with Boundary Control

“…We start by selecting the computational mesh which is generated with Gmsh 4 and saved as a.xml file. Here the parameter rfn_num corresponds to a mesh refinement parameter.…”

Distributed and backstepping boundary controls for port-Hamiltonian systems with symmetries

“…Tokamak plasma dynamics may be described using the port-Hamiltonian formulation. The mass, entropy, momentum and electromagnetic balance equations were first introduced in [20] in order to produce such a port Hamiltonian model for the electromagnetic domain and then developed in [19,23] for the full thermomagneto-hydrodynamics model for the plasma. Classical axi-symmetry and quasistatic equilibrium assumptions (see [24]) may be used together with a symplectic reduction method to obtain a reduced 1D Tokamak control model which preserves not only the energetic properties but also the port-Hamiltonian structure (namely a Stokes-Dirac structure) from the original system.…”

“…In this paper, the infinite-dimensional PHS case is investigated. The reduction approach is applied to the 3D model derived from the mass, momentum, entropy and electromagnetic balance equations [19], assuming axisymmetry and quasi-static hydrodynamic equilibrium. These two assumptions are equivalent to assuming a toroidal symmetry.…”

A partitioned finite element method for power-preserving discretization of open systems of conservation laws

“…Classical academic examples such as the transmission line, the shallow water or the beam equations have been investigated in the port-Hamiltonian framework (Duindam et al, 2009). Besides, two-and three-dimensional problems have been recently considered (Trenchant et al, 2017;Vu et al, 2016;Wu et al, 2015). In many of these examples, e.g.…”