Control of ramp-up current profile dynamics in tokamak plasmas via the minimal-surface theory

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 control-oriented 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.