David A. Humphreys

Optimal Tracking Control of Current Profile in Tokamaks

Setting up a suitable current spatial profile in tokamak plasmas has been demonstrated to be a key condition for one possible advanced scenario with improved confinement and possible steady-state operation. Experiments at the DIII-D tokamak focus on creating the desired current profile during the plasma current ramp-up and early flattop phases with the aim of maintaining this target profile during the subsequent phases of the discharge. The evolution in time of the current profile is related to the evolution of the poloidal magnetic flux, which is modeled in normalized cylindrical coordinates using a parabolic partial differential equation usually referred to as the magnetic diffusion equation. We propose a framework to solve a finite-time, optimal tracking control problem for the current profile evolution via diffusivity, interior, and boundary actuation during the ramp-up and early flattop phases of the discharge. The proposed approach is based on reduced order modeling via proper orthogonal decomposition and successive optimal control computation for a bilinear system. Simulation results illustrate the performance of the proposed controller.

Plasma vertical stabilization with actuation constraints in the DIII-D tokamak

In the advanced tokamak (AT) operating mode of the DIII-D tokamak, an integrated multivariable controller takes into account highly coupled influences of plasma equilibrium shape, profile, and stability control. Time-scale separation in the system allows a multi-loop design: the inner loop closed by the nominal vertical controller designed to control a linear exponentially unstable plant and the outer loop closed by the nominal shape controller designed to control a linear stabilized plant. Due to actuator constraints, the nominal vertical controller fails to stabilize the vertical position of the plasma inside the tokamak when large or fast disturbances are present or when the references coming from the shape controller change suddenly. Anti-windup synthesis is proposed in this paper to find a nonlinear modification of the nominal vertical controller that prevents vertical instability and undesirable oscillations but leaves the inner loop unmodified when there is no input saturation.

First-principles-driven model-based current profile control for the DIII-D tokamak via LQI optimal control

In tokamak fusion plasmas, control of the spatial distribution profile of the toroidal plasma current plays an important role in realizing certain advanced operating scenarios. These scenarios, characterized by improved confinement, magnetohydrodynamic stability, and a high fraction of non-inductively driven plasma current, could enable steady-state reactor operation with high fusion gain. Current profile control experiments at the DIII-D tokamak focus on using a combination of feedforward and feedback control to achieve a targeted current profile during the ramp-up and early flat-top phases of the shot and then to actively maintain this profile during the rest of the discharge. The dynamic evolution of the current profile is nonlinearly coupled with several plasma parameters, motivating the design of model-based control algorithms that can exploit knowledge of the system to achieve desired performance. In this work, we use a first-principles-driven, control-oriented model of the current profile evolution in low confinement mode (L-mode) discharges in DIII-D to design a feedback control law for regulating the profile around a desired trajectory. The model combines the magnetic diffusion equations with empirical correlations for the electron temperature, resistivity, and non-inductive current drive. To improve tracking performance of the system, a nonlinear input transformation is combined with a linear-quadratic-integral (LQI) optimal controller designed to minimize a weighted combination of the tracking error and controller effort. The resulting control law utilizes the total plasma current, total external heating power, and line averaged plasma density as actuators. A simulation study was used to test the controllers performance and ensure correct implementation in the DIII-D plasma control system prior to experimental testing. Experimental results are presented that show the first-principles-driven model-based control schemes successful rejection of input disturbances and perturbed initial conditions, as well as target trajectory tracking.

Toroidal current profile control during low confinement mode plasma discharges in DIII-D via first-principles-driven model-based robust control synthesis

In order for ITER to be capable of operating in advanced tokamak operating regimes, characterized by a high fusion gain, good plasma confinement, magnetohydrodynamic stability and a non-inductively driven plasma current, for extended periods of time, several challenging plasma control problems still need to be solved. Setting up a suitable toroidal current density profile in the tokamak is key for one possible advanced operating scenario characterized by non-inductive sustainment of the plasma current. At the DIII-D tokamak, the goal is to create the desired current profile during the ramp-up and early flat-top phases of the plasma discharge and then actively maintain this target profile for the remainder of the discharge. The evolution in time of the toroidal current profile in tokamaks is related to the evolution of the poloidal magnetic flux profile, which is modelled in normalized cylindrical coordinates using a first-principles, nonlinear, dynamic partial differential equation (PDE) referred to as the magnetic diffusion equation. The magnetic diffusion equation is combined with empirical correlations developed from physical observations and experimental data from DIII-D for the electron temperature, the plasma resistivity and the non-inductive current drive to develop a simplified, control-oriented, nonlinear, dynamic PDE model of the poloidal flux profile evolution valid for low confinement mode discharges. In this work, we synthesize a robust feedback controller to reject disturbances and track a desired reference trajectory of the poloidal magnetic flux gradient profile by employing the control-oriented model of the system. A singular value decomposition of the static gain matrix of the plant model is utilized to identify the most relevant control channels and is combined with the dynamic response of system around a given operating trajectory to design the feedback controller. A general framework for real-time feedforward + feedback control of magnetic and kinetic plasma profiles was implemented in the DIII-D Plasma Control System and was used to demonstrate the ability of the feedback controller to control the toroidal current profile evolution in the DIII-D tokamak. These experiments constitute the first time ever a first-principles-driven, model-based, closed-loop magnetic profile controller was successfully implemented and tested in a tokamak device.

Magnetohydrodynamic mode identification from magnetic probe signals via a matched filter method

A matched filter analysis has been developed to identify the amplitude and phase of magnetohydrodynamic modes in DIII-D tokamak plasmas using magnetic probe signals (δB p ). As opposed to conventional Fourier spatial analysis of toroidally spaced probes, this analysis includes data from both toroidally and poloidally spaced magnetic probe arrays. Using additional probes both improves the statistics of the analysis and more importantly incorporates poloidal information into the mode analysis. The matched filter is a numeric filter that matches signals from the magnetic probes with numerically predicted signals for the mode. The numerical predictions are developed using EFIT equilibrium reconstruction data as input to the stability code GATO and the vacuum field code VACUUM. Changes is the plasma equilibrium that occur on the same time scale as the mode are taken into account by modeling simple matched filter vectors corresponding to changes in total plasma current, plus vertical and horizontal plasma shifts. The matched filter method works well when there is good understanding of a mode and good modeling of its structure. Matched filter analysis results for a fast growing ideal kink mode, where equilibrium change effects are minimal, show the effectiveness of this method. A slow growing resistive-wall mode (RWM) is also analyzed using the matched filter method. The method gives good results for identifying the amplitude and phase of the RWM but the simple equilibrium vectors are insufficient for complete elimination of equilibrium changes on this time scale. An analysis of the computational requirements of the scheme indicates that real-time application of the matched filter for RWM identification will be possible.

Backstepping Control of the Toroidal Plasma Current Profile in the DIII-D Tokamak

One of the most promising devices for realizing power production through nuclear fusion is the tokamak. To maximize performance, it is preferable that tokamak reactors achieve advanced operating scenarios characterized by good plasma confinement, improved magnetohydrodynamic stability, and a largely noninductively driven plasma current. Such scenarios could enable steady-state reactor operation with high fusion gain, the ratio of produced fusion power to the external power provided through the plasma boundary. For certain advanced scenarios, control of the spatial profile of the plasma current will be essential. The complexity of the current profile dynamics, arising due to nonlinearities and couplings with many other plasma parameters, motivates the use of model-based control algorithms that can account for the system dynamics. A first-principles-driven, control-oriented model of the current profile evolution in low-confinement mode (L-mode) discharges in the DIII-D tokamak is employed to address the problem of regulating the current profile evolution around desired trajectories. In the primarily inductive L-mode discharges considered in this paper, the boundary condition, which is dependent on the total plasma current, has the largest influence on the current profile dynamics, motivating the design of a boundary feedback control law to improve the system performance. The backstepping control design technique provides a systematic method to obtain a boundary feedback law through the transformation of a spatially discretized version of the original system into an asymptotically stable target system with desirable properties. Through a nonlinear transformation of the available physical actuators, the resulting control scheme produces references for the total plasma current, total power, and line averaged density, which are tracked by existing dedicated control loops. Adaptiveness is added to the control scheme to improve upon the backstepping controllers disturbance rejection and tracking capability. Prior to experimental testing, a Simserver simulation was carried out to study the controllers performance and ensure proper implementation in the DIII-D Plasma Control System. An experimental test was performed on DIII-D to test the ability of the controller to reject input disturbances and perturbations in initial conditions and to demonstrate the feasibility of the proposed control approach.

Physics-based control-oriented modeling of the safety factor profile dynamics in high performance tokamak plasmas

The tokamak is a device that utilizes magnetic fields to confine a reactant gas to generate energy from nuclear fusion reactions. The next step towards the realization of a tokamak power plant is the ITER project, and extensive research has been conducted to find high performance operating scenarios characterized by a high fusion gain and plasma stability. A key property related to both the stability and performance of the plasma is the safety factor profile (q-profile). In this work, a general control-oriented physics-based modeling approach is developed, with emphasis on high performance scenarios, to convert the first-principles physics model that describes the q-profile evolution in the tokamak into a form suitable for control design, with the goal of developing closed-loop controllers to drive the q-profile to a desired target evolution. The DINA-CH& CRONOS and PTRANSP advanced tokamak simulation codes are used to tailor the first-principles-driven (FPD) model to the ITER and DIII-D tokamak geometries, respectively. The models prediction capabilities are illustrated by comparing the prediction to simulated data from DINA-CH&CRONOS for ITER and to experimental data for DIII-D.

On feedback stabilization of the tokamak plasma vertical instability

For more than a decade the instability intrinsic to vertically elongated tokamak plasmas has been relatively well understood. Controllers that stabilize this instability have been in routine use at experimental devices since the 1980s. However, most analysis of this instability has used low order approximations, often just one state, from which understanding of the much higher order system is extrapolated. We expand on [Walker & Humphreys (2006a). A multivariable analysis of the plasma vertical instability in tokamaks. In Proceedings of the 45th IEEE conference on decision & control (pp. 2213)] using a full multivariable model to provide a rigorous treatment of this problem, including the case in which some control coils are superconducting. The results are consistent with the physics knowledge routinely used by specialists in designing vertical stabilization. We examine two models of the tokamak-and-plasma system, one assuming the plasma has mass, the other assuming zero mass. Although the with-mass model is more correct, the massless model is most often used in control analyses. We examine multiple systems distinguished by whether there are superconducting control coils, the magnitude of the instability, and how strongly conductors are coupled. PD feedback is used as the prototype controller to study these models. Answers to stability questions depend critically on whether the plasma is assumed to have mass, but undergo only minor changes with the presence of superconducting coils. Examples show analyses using a massless plasma model can reach erroneous conclusions. Since the problem is unfamiliar to the general control community, tutorial information provides insight into the origin of the problem.

Multivariable robust control of the plasma rotational transform profile for advanced tokamak scenarios in DIII-D

The tokamak is a high order, distributed parameter, nonlinear system with a large number of instabilities. Therefore, accurate theoretical plasma models are difficult to develop. However, linear plasma response models around a particular equilibrium can be developed by using data-driven modeling techniques. This paper introduces a linear model of the rotational transform ι profile evolution based on experimental data from the DIII-D tokamak. The model represents the response of the ι profile to the electric field due to induction as well as to heating and current drive (H&CD) systems. The control goal is to use both induction and H&CD systems to regulate the plasma ι profile around a particular target profile. A singular value decomposition (SVD) of the plasma model at steady state is carried out to decouple the system and identify the most relevant control channels. A mixed sensitivity H∞ control design problem is formulated to synthesize a stabilizing feedback controller without input constraint that minimizes the reference tracking error and rejects external disturbances with minimal control energy. The feedback controller is then augmented with an anti-windup compensator, which keeps the given profile controller well-behaved in the presence of magnitude constraints in the actuators and leaves the nominal closed-loop unmodified when no saturation is present. Finally, computer simulations and experimental results illustrate the performance of the model-based profile controller.

Data-driven modeling and feedback tracking control of the toroidal rotation profile for advanced tokamak scenarios in DIII-D

First-principle predictive tokamak plasma models based on flux averaged transport equations often yield complex expressions not suitable for real time control implementations. Addition of turbulent transport phenomena further encumbers these models with transport coefficients that must be determined experimentally and the interdependences between parameters must be accounted for with ad hoc assumptions. As an alternative to first principle modeling, data-driven modeling techniques involving system identification have the potential to obtain practical, low complexity, dynamic models without the need for ad hoc assumptions. This paper considers the evolution of the toroidal rotation profile in response to the heating and current drive (H&CD) systems. Experiments are conducted during plasma current flattop, in which the actuators are modulated in open-loop to obtain data for the model identification. The rotation profile is discretized in the spatial coordinate by Galerkin projection. Then a linear state space model is generated by the prediction error method (PEM) to relate the rotation profile to the actuators according to a least squares fit. An optimal tracking controller is proposed to regulate the rotation profile to a desired reference trajectory.