James Buchanan

The tokamak is a world-leading concept for producing sustainable energy via magnetically confined nuclear fusion. Identifying where to position the magnets within a tokamak, specifically the poloidal field (PF) coils, is a design problem which requires balancing a number of competing economical, physical, and engineering objectives and constraints. In this paper, we show that multi-objective Bayesian optimization (BO), an iterative optimization technique utilizing probabilistic machine learning models, can effectively explore this complex design space and return several optimal PF coilsets. These solutions span the Pareto front, a subset of the objective space that optimally satisfies the specified objective functions. We outline an easy-to-use BO framework and demonstrate that it outperforms alternative optimization techniques while using significantly fewer computational resources. Our results show that BO is a promising technique for fusion design problems that rely on computationally demanding high-fidelity simulations.

A key aspect in the modelling of magnetohydrodynamic (MHD) equilibria in tokamak devices is having access to fast, accurate, and stable numerical simulation methods. There is an increasing demand for reliable methods that can be used to develop traditional or machine learning-based shape control feedback systems, optimise scenario designs, and integrate with other plasma edge or transport modelling codes. To handle such applications, these codes need to be flexible and, more importantly, they need to have been validated against both analytically known and real-world tokamak equilibria to ensure they are consistent and credible. In this paper, we are interested in solving the static forward Grad–Shafranov (GS) problem for free–boundary MHD equilibria. Our focus is on the validation of the static forward solver in the Python-based equilibrium code FreeGSNKE by solving equilibria from magnetics-only EFIT ++ reconstructions of MAST-U shots. In addition, we also validate FreeGSNKE against equilibria simulated using the well-established MATLAB-based equilibrium code Fiesta. To do this, we develop a computational pipeline that allows one to load the same (a)symmetric MAST-U machine description into each solver, specify the required inputs (active/passive conductor currents, plasma profiles and coefficients, etc.) from EFIT++, and solve the GS equation for all available time slices across a shot. For a number of different MAST-U shots, we demonstrate that both FreeGSNKE and Fiesta can successfully reproduce various poloidal flux quantities and shape targets (e.g. midplane radii, magnetic axes, separatrices, X-points, and strikepoints) in agreement with EFIT++ calculations to a very high degree of accuracy. We also provide public access to the code/data required to load the MAST-U machine description in FreeGSNKE/Fiesta and reproduce the equilibria in the shots shown.

We present a Python-based numerical solver for the two-dimensional dynamic plasma equilibrium problem. We model the time evolution of toroidally symmetric free-boundary tokamak plasma equilibria in the presence of the non-linear magnetohydrodynamic coupling with both currents in the “active” poloidal field coils, with assigned applied voltages, and eddy currents in the tokamak passive structures. FreeGSNKE (FreeGS Newton–Krylov Evolutive) builds and expands on the framework provided by the Python package FreeGS (Free boundary Grad–Shafranov). FreeGS solves the static free-boundary Grad–Shafranov (GS) problem, discretized in space using finite differences, by means of Picard iterations. FreeGSNKE introduces: (i) a solver for the static free-boundary GS problem based on the Newton–Krylov (NK) method, with improved stability and convergence properties; (ii) a solver for the linearized dynamic plasma equilibrium problem; and (iii) a solver for the non-linear dynamic problem, based on the NK method. We propose a novel “staggered” solution strategy for the non-linear problem, in which we make use of a set of equivalent formulations of the non-linear dynamic problem we derive. The alternation of NK solution steps in the currents and in the plasma flux lends this strategy an increased resilience to co-linearity and stagnation problems, resulting in favorable convergence properties. FreeGSNKE can be used for any user-defined tokamak geometry and coil configuration. FreeGSNKE's flexibility and ease of use make it a suitably robust control-oriented simulator of plasma magnetic equilibria. FreeGSNKE is entirely written in Python and easily interfaced with Python libraries, which facilitates machine learning based approaches to plasma control.

The optimization of scenarios and design of real-time-control in tokamaks, especially for machines still in design phase, requires a comprehensive exploration of solutions to the Grad–Shafranov (GS) equation over a high-dimensional space of plasma and coil parameters. Emulators can bypass the numerical issues in the GS equation, if a large enough library of equilibria is available. We train an ensemble of neural networks to emulate the typical shape-control targets (separatrix at midplane, X-points, divertor strike point, flux expansion, and poloidal beta) as a function of plasma parameters and active coil currents for the range of plasma configurations relevant to spherical tokamaks with a super-X divertor, with percent-level accuracy. This allows a quick calculation of the classical-control shape matrices, potentially allowing real-time calculation at any point in a shot with submillisecond latency. We devise a hyperparameter sampler to select the optimal network architectures and quantify uncertainties on the model predictions. To generate the relevant training set, we devise a Markov-chain Monte Carlo algorithm to produce large libraries of forward Grad–Shafranov solutions without the need for user intervention. The algorithm promotes equilibria with desirable properties, while avoiding parameter combinations resulting in problematic profiles or numerical issues in the integration of the GS equation.

Spherical tokamaks have many desirable features that make them an attractive choice for a future fusion power plant. Power-plant viability is intrinsically related to plasma heat and particle confinement, and this is often determined by the level of micro-instability-driven turbulence. Accurate calculation of the properties of turbulent microinstabilities is therefore critical for tokamak design; however, the evaluation of these properties is computationally expensive. The considerable number of geometric and thermodynamic parameters and the high resolutions required to accurately resolve these instabilities make repeated use of direct numerical simulations in integrated modeling workflows extremely computationally challenging and create the need for fast, accurate, reduced-order models. This paper outlines the development of a data-driven reduced-order model, often termed a surrogate model for the properties of micro-tearing modes (MTMs) across a spherical tokamak reactor-relevant parameter space utilizing Gaussian process regression and classification, techniques from machine learning. These two components are used in an active learning loop to maximize the efficiency of data acquisition, thus minimizing computational cost. The high-fidelity gyrokinetic code GS2 is used to calculate the linear properties of the MTMs: the mode growth rate, frequency, and normalized electron heat flux, and core components of a quasi-linear transport model. Data cross-validation and direct validation on unseen data are used to ascertain the performance of the resulting surrogate models.