POD-based optimal control of current profile in tokamak plasmas via nonlinear programming

In a magnetic fusion reactor, the achievement of a certain type of plasma current profiles, which are compatible with magnetohydrodynamic (MHD) stability at high plasma pressure, is key to enabling high fusion gain and noninductive sustainment of the plasma current for steady-state operation. The evolution in time of the current profile is related to the evolution of the spatial derivative of the poloidal flux profile, which is modeled in normalized cylindrical coordinates using a partial differential equation (PDE) usually referred to as the magnetic diffusion equation. The dynamics of the plasma poloidal flux profile can be modified by three actuators: the total plasma current, the non-inductive power, and the average plasma density. These three actuators, which are constrained not only in value and rate but also in their initial and final values, are used to drive the poloidal flux profile, or equivalently the current profile, as close as possible to a desired target profile at a specific final time. To solve this constrained finite-time open-loop optimal control problem, model reduction based on proper orthogonal decomposition (POD) is combined with sequential quadratic programming (SQP) in an iterative fashion. The use of a low dimensional dynamical model reduces the computational effort, and therefore the time required to solve the optimization problem, which is critical for a potential implementation of a receding horizon control strategy.