Control of Ramp-Up Current Profile Dynamics in Tokamak Plasmas via the Minimal-Surface Theory
C. Xu and E. Schuster
48th IEEE Conference on Decision and Control
Shanghai, China, December 16-18, 2009
Abstract
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The central task of current profile control during
the ramp-up phase of a tokamak discharge is to find the
actuator trajectories that are necessary to achieve certain
desired current profile at some time between the end of the
ramp-up phase and early stage of the flattop phase. The
magnetic diffusion partial differential equation (PDE) models
the dynamics of the poloidal magnetic flux profile, which is
closely related to the toroidal current density profile, and plays
a key role in the model-based control synthesis. Given the initial
and desired target profiles, splines are used in this work to
generate evolutionary curves connecting their boundaries at
both endpoints of the spatial domain. Then, a closed four-edge
frame (initial profile, target profile, two boundary curves) in the
three dimensional space (time, space, poloidal magnetic flux) is
obtained without knowing the transient dynamics inside. The
minimal surface theory is used in this work to define a surface
spanned by the closed four-edge frame, which represents the
desired transient dynamics for the poloidal magnetic flux.
Then, the control task becomes a trajectory tracking problem.
Once the desired transient dynamics is defined, the temporal
and spatial derivatives of the poloidal magnetic flux in the
magnetic diffusion equation can be computed, and the controloriented
PDE model can be reformulated into an algebraic
equation where the control values at each time instant represent
the to-be-determined unknown variables. Numerical simulation
results show the effectiveness of this approach. This method is
characterized by high speed computation and shows potential
for real-time implementation in a closed-loop receding-horizon
scheme, particularly for long-discharge tokamaks such as ITER.