A Riccati solution for the ideal MHD plasma response with applications to real-time stability control

Abstract

Active feedback control of ideal MHD stability in a tokamak requires rapid plasma stability analysis. Toward this end, we reformulate the dW stability method with a Hamilton-Jacobi theory, elucidating analytical and numerical features of the generic tokamak ideal MHD stability problem. The plasma response matrix is demonstrated to be the solution of an ideal MHD matrix Riccati differential equation. Since Riccati equations are prevalent in the control theory literature, such a shift in perspective brings to bear a range of numerical methods that are well-suited to the robust, fast solution of control problems. We discuss the usefulness of Riccati techniques in solving the stiff ordinary differential equations often encountered in ideal MHD stability analyses-for example, in tokamak edge and stellarator physics. We demonstrate the applicability of such methods to an existing 2D ideal MHD stability code-DCON [A. H. Glasser, Phys. Plasmas 23, 072505 (2016)]-enabling its parallel operation in near real-time, with wall-clock time (1s. Such speed may help enable active feedback ideal MHD stability control, especially in tokamak plasmas whose ideal MHD equilibria evolve with inductive timescale s տ 1s-as in ITER.

Advancing Fusion with Machine Learning Research Needs Workshop Report

“…• Ensuring research focuses on interpretations of measurements that maximize the specificity of controlrelated phenomena. For example, models or predictive algorithms that provide outputs specific to particular instabilities or plasma phenomena to be controlled are most likely to enable effective control [7,18]. • Linking derived results to specific relevant temporal and spatial scales for actuation and control that will lead to well-defined control actions • Following machine learning procedures that enable physics constrained extrapolation to different operating regimes, system conditions, or fusion devices • Developing robust ML training methods with quantified stability margins and uncertainty that would perform robustly under the dynamic nature of the fusion plasma…”

The DESC stellarator code suite. Part 2. Perturbation and continuation methods

“…This has been extended to a wide range of of codes for analysing fusion devices under small perturbations to a MHD equilibrium, such as the GPEC suite of codes (Park, Boozer & Glasser 2007; Park et al. 2009; Glasser 2016; Glasser, Kolemen & Glasser 2018) for analysing tokamak configurations under 3-D perturbations. A common feature of perturbation methods is using local approximation methods (commonly Taylor series) to examine solutions nearby to some equilibrium in an infinitesimal limit.…”

The DESC Stellarator Code Suite Part II: Perturbation and continuation methods

“…Despite this, designing and optimizing 3D magnetic equilibria that have good properties is still a computationally intensive task, for which a number of codes and software packages have been developed [4][5][6][7][8][9] Perturbation methods have been used heavily in tokamak plasma physics, primarily to analyze the stability of axisymmetric MHD equilibria by searching for a perturbation that minimizes the energy of the plasma, as in Bernstein's energy principle 10 . This has been extended to a wide range of of codes for analyzing fusion devices under small perturbations to an MHD equilibrium, such as the GPEC suite of codes [11][12][13][14] for analyzing tokamak configurations under 3D perturbations.…”