Abstract— A port-Hamiltonian model is derived for tokamak plasma current proﬁle dynamics. Magnetohydrodynamics and energy balance equations are ﬁrst written in port-Hamiltonian formalism using Stokes-Dirac structures in 3D forms and a speciﬁc interconnection structure for the magneto-hydrodynamical coupling. A balance preserving reduction scheme is applied to the 3D models, assuming classical cylindrical approximations and neglecting the diamagnetic effects. .
I. I NTRODUCTION A plasma is a gas in which an important fraction of the particles is ionized, so that the electrons and ions are separately free. A tokamak is a facility (whose main ideas came in the 1950’s from a group of Russian scientists including Andrei Sakharov and Igor Tamm) constructed with the shape of a torus (or dough-nut) in which a plasma is magnetically conﬁned and heated in order to produce nuclear fusion reactions. The magnetic conﬁnement of the plasma particles in the vacuum vessel torus is obtained through the combination of toroidal and poloidal ﬁelds produced by external coils (see ﬁg.I.1) with the additional ﬁeld produced by the electrical current ﬂowing along the plasma ring. This plasma current is generally ﬁrstly generated by induction (the plasma ring can then be considered as the secondary loop of a transformer whose primary loop is the ohmic ﬁeld coil (see ﬁg.I.1). It allows to heat up the plasma, which behaves as a resistive conductor. However, ohmic heating and current drive do not allow to reach the adequate plasma temperature and duration required for future fusion reactors. Indeed the plasma resistivity decreases with temperature and technology limits the ohmic ﬁeld coil current. Non inductive heating and current drive methods were thus developed to take it over, namely high power microwave or fast neutral beams injection. A complete review of tokamak’s principles and technologies may be found in the classical Wesson’s monograph . Tokamaks have been proved to be the most promising approach to obtaining energy production from nuclear fusion. The importance of tokamaks for the future of nuclear fusion is demonstrated by the decision to build a new experimental facility, called ITER, as a joint effort by most of the industrialized countries in the world (see http://www.iter.org/). The main objective of this project is to obtain the necessary plasma temperature and density proﬁles for the self-sustained fusion reactions and, doing so, prove the feasibility of fusion power electricity production at the industrial scale. Control problems of tokamak plasma aim at many different objectives , , . In particular, four classes of control problems have been extensively investigated: • the vertical stabilization of the plasma
Simple tokamak with the electrical solenoids
controlling the overall shape of the plasma controlling the magneto-hydrodynamic (MHD) instabilities • controlling the current, temperature and pressure density proﬁles In this and subsequent works we aim at developing a structured model useful to handle the last two problems. Readers interested in the vertical stabilization and shape problems are referred to . The mean value of the plasma current may be controlled using the central magnetic coil. This deﬁnes a boundary control problem. The current density proﬁles, which deﬁnes security factor proﬁles  necessary to avoid MHD instabilities and to obtain a satisfying conﬁnement for the plasmas, may only be controlled using distributed non inductive source term. Moreover modelling and optimizing the bootstrap current effect , as well as a better understanding of all thermal phenomena in the tokamak , are key issues for the success of the ITER project. Both require an explicit representation of...