Axisymmetric equilibrium and stability analysis in Alcator C-Mod, including effects of current profile, measurement noise and power supply saturation

Abstract

The vertical position of elongated tokamak plasmas is unstable on the time scale of the eddy currents in the axisymmetric conducting structures. In the absence of feedback control, the plasma would drift vertically and quench on the wall, a situation known as Vertical Displacement Event (VDE), with serious consequences for machine integrity. As tokamaks approach reactor regimes, VDE's cannot be tolerated: vertical feedback control must be robust against system uncertainty and the occurrence of noise and disturbances. At the same time, adaptive routines should be in place to handle unexpected events. The problem of robust control of the vertical position can be formulated in terms of identifying which variables affect vertical stability and which ones are not directly controlled/controllable; identifying the physical region of these variables, and the corresponding most unstable equilibria; and designing the control system to stabilize all equilibria with sufficient margin. The margin should be enough to allow the system to tolerate realistic scenarios of noise and disturbances. A set of metrics is introduced to characterize the problem of vertical stability: the stability margin describes the plasma-wall interaction and the open-loop growth rate; the maximum controllable displacement looks at the vertical stabilization power supplies and their ability to handle noise and off-normal events; the gain and phase margins quantify the linear stability of the feedback control loop.

(cont.) The dependence of these metrics on relevant plasma parameters is proven with analytic calculations and numerical simulations: in particular, it is shown that the stability margin is a decreasing function of the plasma internal inductance, for a given plasma elongation. An upper bound of the value of the internal inductance is derived and validated with database analysis, which describes the most unstable equilibrium for given values of the external elongation and the edge safety factor. The stability metrics are evaluated for typical and ITER-like C-Mod plasmas to give an example of the C-Mod operational space and of feasible control conditions. The vertical stabilization system should be able to tolerate realistic scenarios of noise and disturbances. The main sources of noise and pick-ups in Alcator C-Mod are identified and their effects on the measurement and control of the vertical position are evaluated. Broadband noise may affect controllability of C-Mod plasmas at limit elongations and may become an issue with high-order controllers, therefore two applications of Kalman filters are investigated. A Kalman filter is compared to a state observer based on the pseudo-inverse of the measurement matrix and proves to be a better candidate for state reconstruction for vertical stabilization, provided adequate models of the system, the inputs, the intrinsic and measurement noise and an adequate set of diagnostic measurements are available. A single-input single-output application of the filter for the vertical observer rejects high frequency noise without destabilizing high-elongation plasmas, however does not match the performance of an optimized low-pass filter.

(cont.) Aggressive control targets and large off-normal events can cause a control current to rail. The magnetic topology is consequently perturbed and the plasma might become uncontrollable. An adaptive anti-saturation control routine is demonstrated which avoids an impending saturation by interpolating in real-time to a safe equilibrium. This approach becomes necessary when poor redundancy of control coils may require mid-shot pulse rescheduling, as opposed to an adaptation in control.