A control-oriented model of the current profile in tokamak plasma

Abstract

This paper proposes a control-oriented approach to the tokamak plasma current profile dynamics. It is established based on a consistent set of simplified relationships, in particular for the microwave current drive sources, rather than exact physical modelling. Assuming that a proper model for advanced control schemes can be established using the socalled cylindrical approximation and neglecting the diamagnetic effects, we propose a model that focuses on the flux diffusion (from which the current profile is inferred). Its inputs are some real-time measurements available on modern tokamaks and the effects of some major actuators, such as the magnetic coils, Lower Hybrid (LHCD), Electron and Ion Cyclotron Frequency (ECCD and ICRH) systems, are particularly taken into account. More precisely, the non-inductive current profile sources are modelled as 3-parameters functions of the control inputs derived either from approximate theoretical formulae for the ECCD and bootstrap terms or from experimental scaling laws specifically developed from Hard X-ray Tore Supra data for the LHCD influence. The use of scaling laws in this model reflects the fact that the operation of future reactors will certainly depend upon a great number of scaling laws and specific engineering parameters. The discretisation issues are also specifically addressed, to ensure the robustness with respect to discretisation errors and the efficiency (in terms of computation time) of the associated algorithm. This model is compared with experimental results and the CRONOS solver for Tore Supra Tokamak.

Dynamic optimization of open-loop input signals for ramp-up current profiles in tokamak plasmas

“…From the viewpoint of distributed parameter systems governed by PDEs, several advanced control and optimization approaches have been discussed based on new developments in internal diagnostics and plasma actuation. Simplified distributed parameter system models have been studied in [18] and [27] toward current profile control at DIII-D and Tore Supra, respectively. In order to achieve an optimal current evolution in the plasma discharges of the ramp-up phase, various dynamic optimization techniques are applied to obtain numerical solutions in an open loop fashion, such as [7], [8], [19], [28].…”

“…In the papers by Witrant [27] and Ou [18], the mathematical formulation for the distributed parameter system framework is discussed, for both the Tore Supra and the DIII-D experimental tokamaks. In these papers, only the magnetic flux PDE is considered and scaling laws are used to account (in an approximate fashion) for the effects of the other PDEs.…”

“…In these papers, only the magnetic flux PDE is considered and scaling laws are used to account (in an approximate fashion) for the effects of the other PDEs. In the present work, using the framework proposed by Witrant [27] and Ou [18], we consider the problem of obtaining a specific desired output current profile at the end of the current ramp-up phase, as shown in Figure 2. This problem can be formulated as a finite-time dynamic optimization problem for the magnetic flux diffusion PDE system, which is a type of a bilinear infinite-dimensional system, i.e., the diffusivity control term appears in the second-order elliptic operator.…”

“…Here, σ k 1 is an approximation of the derivative ofn(t) on [γ k−1 , γ k ), σ k 2 is an approximation of the derivative of I(t) on [γ k−1 , γ k ), and σ k 3 is an approximation of the derivative of P (t) on [γ k−1 , γ k ). We can express (27) as follows:…”

Real-time physics-model-based simulation of the current density profile in tokamak plasmas

“…The basic idea underlying this paper was suggested earlier in work by Witrant et al [8], which presents a control-oriented current density profile model which would be suitable for real-time implementation. However, to our knowledge, such a model had until now never actually been implemented in a tokamak real-time system.…”