Dynamic optimization of open-loop input signals for ramp-up current profiles in tokamak plasmas

Abstract

Establishing a good current spatial profile in tokamak fusion reactors is crucial to effective steadystate operation. The evolution of the current spatial profile is related to the evolution of the poloidal magnetic flux, which can be modeled in the normalized cylindrical coordinates using a parabolic partial differential equation (PDE) called the magnetic diffusion equation. In this paper, we consider the dynamic optimization problem of attaining the best possible current spatial profile during the rampup phase of the tokamak. We first use the Galerkin method to obtain a finite-dimensional ordinary differential equation (ODE) model based on the original magnetic diffusion PDE. Then, we combine the control parameterization method with a novel time-scaling transformation to obtain an approximate optimal parameter selection problem, which can be solved using gradient-based optimization techniques such as sequential quadratic programming (SQP). This control parameterization approach involves approximating the tokamak input signals by piecewise-linear functions whose slopes and break-points are decision variables to be optimized. We show that the gradient of the objective function with respect to the decision variables can be computed by solving an auxiliary dynamic system governing the state sensitivity matrix. Finally, we conclude the paper with simulation results for an example problem based on experimental data from the DIII-D tokamak in San Diego, California.

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

“…In [25], the open-loop optimal control problem of the q-profile is solved in ramp-up tokamak plasmas using the minimal-surface theory. 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.…”

“…In this paper, we are aiming at proposing an effective framework for the solution of PDE-constrained optimization problem in tokamak plasmas. We propose a numerical solution procedure directly to solve the original PDE-constrained optimization problem rather than discretizing the original PDE model over the space into a finite-dimensional system of ODEs (e.g., [6,8,20,24]). Applying the control parameterization approach, each control function is approximated by a linear combination of temporal basis functions with the constant coefficients to be determined by numerical optimization procedures such as sequential quadratic programming (SQP) based on the gradient optimization technique.…”

“…However, there are also other methods such as adjoint method or sensitivity method to evaluate the cost functional gradients. For example, in our previous work [20], Galerkin method is used to obtain a finite-dimensional ODE model based on the original magnetic diffusion PDE, then the gradients of the cost function with respect to the decision parameters can be evaluated analytically based on the forward sensitivity analysis. The computation of the cost functional gradient using these alternative approaches such as the adjoint or sensitivity methods can be readily incorporated into current framework.…”

“…The computation of the cost functional gradient using these alternative approaches such as the adjoint or sensitivity methods can be readily incorporated into current framework. This will be our next step to integrate our previous work on different gradient computation modules [20] to unify the numerical dynamic optimization process.…”

Time Optimal Control of a Thermoelastic System

“…As far as we know, little work has been done on numerical methods for solving this problem. Our method follows the discrete-then-optimize approach [23][24][25], whereby the PDE system is first projected onto the finite-dimensional subspace to obtain a system of ordinary differential equations (ODEs), and computational optimal control techniques are applied; then the resulting problem is solved by appropriate optimization methods. Specifically, by making use of the standard finite element method (FEM), the original problem was firstly projected 2 Mathematical Problems in Engineering into a time optimal control problem governed by a system of ODEs.…”