Start-up phase plasma discharge design of a tokamak via control parameterization method

Abstract

The tokamak start-up is a very important phase during the process to obtain a suitable equalizing plasma, and its governing model can be described as a set of nonlinear ordinary differential equations (ODEs). In this paper, we first estimate the parameters in the original model and set up an accurate model to express how the variables change during the start-up phase, especially how the plasma current changes with respect to time and the loop voltage. Then, we apply the control parameterization method to obtain an approximate optimal parameters selection problem for the loop voltage design to achieve a desired plasma current target. Computational optimal control techniques such as the variational method and the costate method are employed to solve the problem, respectively. Finally, numerical simulations are performed and the results obtained via different methods are compared. Our numerical parameterization method and optimization procedure turn out to be effective.

Dynamic optimization of trajectory for ramp‐up current profile in tokamak plasma

“…In [8] and [20], using the Galerkin method, a finite-dimensional ordinary differential equation (ODE) model based on the original magnetic diffusion PDE is obtained, then the optimization of open-loop actuator trajectories for the tokamak plasma profile control is solved by the nonlinear optimization algorithm. There are also several advanced control and optimization approaches being investigated (e.g., [3,4,6,9,22,23]) due to advances in internal diagnostics and plasma actuation.…”

“…where N denotes the number of discrete points in space within the interval ρ ∈ [0, 1] for the normalized radius and ε is a given small positive constant. Note that in order to establish the appropriate current density profile for a quasi-static operation in the flat-top phase (as shown in Figure 2), we add the second term in (9) to penalize the terminal growth rate in the end of the ramp-up phase. By introducing the penalty term, we expect to maintain the achieved profile in the end of the ramp-up phase rather than just let it fly away in the beginning of the flat-top phase.…”

“…Actually, we can solve such problems using the SQP method whose advantage is that the local extremal solution can be obtained with a local quadratic convergence. To solve Problem P N using the SQP method, the key idea is to evaluate the values of cost functional (9) and its gradients with respect to the decision parameters n(t i ), I(t i ), P (t i ), i = 1, 2, . .…”

“…Step 4. Evaluate the gradients of cost functional (9) with respect to parameters (n (m) (t k ), I (m) (t k ), P (m) (t k )), using finite difference approximation.…”

“…, t f . Note that the cost functional (9) contains the terms ι( ρi , T ) and ∂ψ( ρi,T ) ∂t . We use the forward finite difference method to approximate ι( ρi , T ) in the domain ρi ∈ (0, 1):…”