Model-based Optimization and Feedback Control of the Current Density Profile Evolution in NSTX-U

Unlike nuclear fission in present nuclear power plants, where energy is generated by splitting heavy atoms like uranium, nuclear fusion generates energy by fusing light nuclei like hydrogen isotopes under high-temperature and high-pressure conditions, at which the reactants (hydrogen isotopes) separate from their electrons and form an ionized gas called plasma, which is considered as the fourth state of matter. Contrary to fission, fusion provides more energy density, poses almost no risk of a catastrophic nuclear accident, and produces mostly short-term, low-level radioactive waste.The main difficulty in maintaining fusion reactions is the development of a device that can confine the hot plasma for sufficiently long time while preventing it from hitting the walls of the confining device. Among several techniques, magnetic confinement appears as the most promising approach. In particular, the tokamak device is a toroidal device surrounded by large magnetic coils responsible for the magnetic fields that confine the plasma. A spherical tokamak, or a spherical torus (ST), is a variation of the conventional tokamak concept. Compared to a standard tokamak, the ST device extrapolates to a more compact, potentially lower-cost reactor with higher efficiency of confinement. Nuclear fusion research is a highly challenging, multidisciplinary field seeking contributions from both plasma physics and multiple engineering areas. As an application of plasma control engineering, this dissertation mainly explores methods to control the current density profile evolution within the National Spherical Torus eXperiment-Upgrade (NSTX-U), which is a substantial upgrade based on the NSTX device, which is located in Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ. Active control of the toroidal current density profile is among those plasma control milestones that the NSTX-U program must achieve to realize its next-step operational goals, which are characterized by high-performance, long-pulse, MHD-stable plasma operation with neutral beam heating. Therefore, the aim of this work is to develop model-based, feedforward and feedback controllers that can enable time regulation of the current density profile in NSTX-U by actuating the total plasma current, electron density, and the powers of the individual neutral beam injectors.Motivated by the coupled, nonlinear, multivariable, distributed-parameter plasma dynamics, the first step towards control design is the development of a physics-based, control-oriented model for the current profile evolution in NSTX-U in response to non-inductive current drives and heating systems. Numerical simulations of the proposed control-oriented model show qualitative agreement with the high-fidelity physics code TRANSP. The next step is to utilize the proposed control-oriented model to design an open-loop actuator trajectory optimizer. Given a desired operating state, the optimizer produces the actuator trajectories that can steer the plasma to such state. The objective of the feedforward control design is to provide a more systematic approach to advanced scenario planning in NSTX-U since the development of such scenarios is conventionally carried out experimentally by modifying the tokamak’s actuator trajectories and analyzing the resulting plasma evolution.Finally, the proposed control-oriented model is embedded in feedback control schemes based on optimal control and Model Predictive Control (MPC) approaches. Integrators are added to the standard Linear Quadratic Gaussian (LQG) and MPC formulations to provide robustness against various modeling uncertainties and external disturbances. The effectiveness of the proposed feedback controllers in regulating the current density profile in NSTX-U is demonstrated in closed-loop nonlinear simulations. Moreover, the optimal feedback control algorithm has been implemented successfully in closed-loop control simulations within TRANSP through the recently developed Expert routine

Plasma Shape and Current Density Profile Control in Advanced Tokamak Operating Scenarios

The need for new sources of energy is expected to become a critical problem within the next few decades. Nuclear fusion has sufficient energy density to potentially supply the world population with its increasing energy demands. The tokamak is a magnetic confinement device used to achieve controlled fusion reactions. Experimental fusion technology has now reached a level where tokamaks are able to produce about as much energy as is expended in heating the fusion fuel. The next step towards the realization of a nuclear fusion tokamak power plant is ITER, which will be capable of exploring advanced tokamak (AT) modes, characterized by a high fusion gain and plasma stability. The extreme requirements of the advanced modes motivates researchers to improve the modeling of the plasma response as well as the design of feedback controllers. This dissertation focuses on several magnetic and kinetic control problems, including the plasma current, position and shape control, and data-driven and first-principles-driven modeling and control of plasma current density profile and the normalized plasma pressure ratio βN.The plasma is confined within the vacuum vessel by an external electromagnetic field, produced primarily by toroidal and poloidal field coils. The outermost closed plasma surface or plasma boundary is referred to as the shape of the plasma. A central characteristic of AT plasma regimes is an extreme elongated shape. The equilibrium among the electromagnetic forces acting on an elongated plasma is unstable. Moreover, the tokamak performance is improved if the plasma is located in close proximity to the torus wall, which guarantees an efficient use of available volume. As a consequence, feedback control of the plasma position and shape is necessary. In this dissertation, an H∞-based, multi-input-multi-output (MIMO) controller for the National Spherical Torus Experiment (NSTX) is developed, which is used to control the plasma position, shape, and X-point position.Setting up a suitable toroidal current profile is related to both the stability and performance of the plasma. The requirements of ITER motivate the research on plasma current profile control. Currently, physics-based control-oriented modeling techniques of the current profile evolution can be separated into two major classes: data-driven and first-principles-driven. In this dissertation, a two-timescale linear dynamic data-driven model of the rotational transform profile and βN is identified based on experimental data from the DIII-D tokamak. A mixed-sensitivity H∞ controller is developed and tested during DIII-D high-confinement (H-mode) experiments by using the heating and current drive (H&CD) systems to regulate the plasma rotational transform profile and βN around particular target values close to the reference state used for system identification. The preliminary experimental results show good progress towards routine current profile control in DIII-D. As an alternative, a nonlinear dynamic first-principles-driven model is obtained by converting the physics-based model that describes the current profile evolution in H-mode DIII-D discharges into a form suitable for control design. The obtained control-oriented model is validated by comparing the model prediction to experimental data. An H∞ control design problem is formulated to synthesize a stabilizing feedback controller, with the goal of developing a closed-loop controller to drive the current profile in DIII-D to a desirable target evolution. Simulations show that the controller is capable of regulating the system around the target rotational transform profile in the presence of disturbances. When compared to a previously designed data-driven model-based controller, the proposed first-principles-driven model-based controller shows potential for improving the control performance

Strategies for Optimal Control of the Current and Rotation Profiles in the DIII-D Tokamak

The tokamak is currently the most promising device for realizing commercially-viable fusion energy production. The device uses magnetic fields to confine a circulating ring of hydrogen in the plasma state, i.e. a cloud of hydrogen ions and electrons. When sufficiently heated the hydrogen ions can overcome the electrostatic forces and fuse together, providing an overwhelmingly abundant energy source. However, stable, high-performance operation of a tokamak requires several plasma control problems to be handled simultaneously. Moreover, the complex physics which governs the tokamak plasma evolution must be studied and understood to make correct choices in controller design. In this thesis, two key control issues are studied intensely, namely the optimization and control of the plasma current profile and control of the plasma rotation (or flow).In order to maximize performance, it is preferable that tokamaks achieve advanced scenarios (AT) characterized by good plasma confinement, improved magnetohydrodynamic stability, and a largely non-inductively driven plasma current. A key element to the development of AT scenarios is the optimization of the spatial distribution of the current profile. Also, research has shown that the plasma rotation can stabilize the tokamak plasma against degradations in the desired MHD equilibrium.In this thesis, new model-based control approaches for the current profile and rotation profile are developed to allow experimental exploration of advanced tokamak scenarios. Methods for separate control of both the current profile and rotation are developed. The advanced model-based control methods presented in this thesis have contributed to theory of tokamak profile control and in some cases they have been successfully validated experimentally in the DIII-D tokamak

Physics-guided machine learning approaches to predict stability properties of fusion plasmas

Disruption prediction and avoidance is a critical need for next-step tokamaks such as the International Thermonuclear Experimental Reactor (ITER). The Disruption Event Characterization and Forecasting Code (DECAF) is a framework used to fully determine chains of events, such as magnetohydrodynamic (MHD) instabilities, that can lead to disruptions. In this thesis, several interpretable and physics-guided machine learning techniques (ML) to forecast the onset of resistive wall modes (RWM) in spherical tokamaks have been developed and incorporated into DECAF. The new DECAF model operates in a multi-step fashion by analysing the ideal stability properties and then by including kinetic effects on RWM stability. First, a random forest regressor (RFR) and a neural network (NN) ensemble are employed to reproduce the change in plasma potential energy without wall effects, δWno-wall, computed by the DCON ideal stability code for a large database of equilibria from the National Spherical Torus Experiment (NSTX). Moreover, outputs from the ML models are reduced and manipulated to get an estimation of the no-wall β limit, βno-wall, (where β is the ratio of plasma pressure to magnetic confinement field pressure). This exercise shows that the ML models are able to improve previous DECAF characterisation of stable and unstable equilibria and achieve accuracies within 85-88%, depending on the chosen level of interpretability. The physics guidance imposed on the NN objective function allowed for transferability outside the training domain by testing the algorithm on discharges from the Mega Ampere Spherical Tokamak (MAST). The estimated βno-wall and other important plasma characteristics, such as rotation, collisionality and low frequency MHD activity, are used as input to a customised random forest (RF) classifier to predict RWM stability for a set of human-labeled NSTX discharges. The proposed approach is real-time compatible and outperforms classical cost-sensitive methods by achieving a true positive rate (TPR) up to 90%, while also resulting in a threefold reduction in the training time. Finally, a model-agnostic method based on counterfactual explanations is developed in order to further understand the model's predictions. Good agreement is found between the model's decision and the rules imposed by physics expectation. These results also motivate the usage of counterfactuals to simulate real-time control by generating the βN levels that would keep the RWM stable

Predicting resistive wall mode stability in NSTX through balanced random forests and counterfactual explanations

Recent progress in the disruption event characterization and forecasting framework has shown that machine learning guided by physics theory can be easily implemented as a supporting tool for fast computations of ideal stability properties of spherical tokamak plasmas. In order to extend that idea, a customized random forest (RF) classifier that takes into account imbalances in the training data is hereby employed to predict resistive wall mode (RWM) stability for a set of high beta discharges from the NSTX spherical tokamak. More specifically, with this approach each tree in the forest is trained on samples that are balanced via a user-defined over/under-sampler. The proposed approach outperforms classical cost-sensitive methods for the problem at hand, in particular when used in conjunction with a random under-sampler, while also resulting in a threefold reduction in the training time. In order to further understand the model’s decisions, a diverse set of counterfactual explanations based on determinantal point processes (DPP) is generated and evaluated. Via the use of DPP, the underlying RF model infers that the presence of hypothetical magnetohydrodynamic activity would have prevented the RWM from concurrently going unstable, which is a counterfactual that is indeed expected by prior physics knowledge. Given that this result emerges from the data-driven RF classifier and the use of counterfactuals without hand-crafted embedding of prior physics intuition, it motivates the usage of counterfactuals to simulate real-time control by generating the β N levels that would have kept the RWM stable for a set of unstable discharges

Comparison of BES measurements of ion-scale turbulence with direct, gyrokinetic simulations of MAST L-mode plasmas

Observations of ion-scale (k_y*rho_i <= 1) density turbulence of relative amplitude dn_e/n_e <= 0.2% are available on the Mega Amp Spherical Tokamak (MAST) using a 2D (8 radial x 4 poloidal channel) imaging Beam Emission Spectroscopy (BES) diagnostic. Spatial and temporal characteristics of this turbulence, i.e., amplitudes, correlation times, radial and perpendicular correlation lengths and apparent phase velocities of the density contours, are determined by means of correlation analysis. For a low-density, L-mode discharge with strong equilibrium flow shear exhibiting an internal transport barrier (ITB) in the ion channel, the observed turbulence characteristics are compared with synthetic density turbulence data generated from global, non-linear, gyro-kinetic simulations using the particle-in-cell (PIC) code NEMORB. This validation exercise highlights the need to include increasingly sophisticated physics, e.g., kinetic treatment of trapped electrons, equilibrium flow shear and collisions, to reproduce most of the characteristics of the observed turbulence. Even so, significant discrepancies remain: an underprediction by the simulations of the turbulence amplituide and heat flux at plasma periphery and the finding that the correlation times of the numerically simulated turbulence are typically two orders of magnitude longer than those measured in MAST. Comparison of these correlation times with various linear timescales suggests that, while the measured turbulence is strong and may be `critically balanced', the simulated turbulence is weak.Comment: 27 pages, 11 figure