Ben G. Penaflor

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.

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.

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.

A two-time-scale model-based combined magnetic and kinetic control system for advanced tokamak scenarios on DIII-D

System identification techniques have been successfully used to obtain linear dynamic plasma response models around a particular equilibrium in different tokamaks. This paper identifies a two-time-scale dynamic model of the rotational transform ι profile and βN in response to the electric field due to induction as well as to heating and current drive (H&CD) systems based on experimental data from DIII-D. The control goal is to regulate the plasma ι profile and βN around a particular target value. 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 solved to determine a stabilizing feedback controller that minimizes the reference tracking error and rejects external disturbances with minimal control energy. The feedback controller is augmented with an anti-windup compensator, which keeps the given controller well-behaved in the presence of magnitude constraints in the actuators and leaves the nominal closed-loop unmodified when no saturation is present. Experimental results illustrate the performance of the proposed controller, which is one of the first profile controllers integrating magnetic and kinetic variables ever implemented in DIII-D.

Optimal feedback control of the poloidal magnetic flux profile in the DIII-D tokamak based on identified plasma response models

First-principles predictive models based on flux-averaged transport equations often yield complex expressions not suitable for real-time control implementations. It is however always possible to reduce these models to forms suitable for control design while preserving the dominant physics of the system. If further model simplification is desired at the expense of less model accuracy and controller capability, data-driven modeling emerges as an alternative to first-principles modeling. System identification techniques have the potential of producing low-complexity, linear models that can capture the system dynamics around an equilibrium point. This paper focuses on the control of the poloidal magnetic flux profile evolution in response to the heating and current drive (H&CD) systems and the total plasma current. Open-loop data for model identification is collected during the plasma current flattop in a high-confinement scenario (H-mode). Using this data a linear state-space plasma response model for the poloidal magnetic flux profile dynamics around a reference profile is identified. The control goal is to use the H&CD systems and the plasma current to regulate the magnetic profile around a desired target profile in the presence of disturbances. The target profile is defined close enough to the reference profile used for system identification in order to stay within the range of validity of the identified model. An optimal state feedback controller with integral action is designed for this purpose. Experimental results showing the performance of the proposed controller implemented in the DIII-D tokamak are presented.

On-Line Calculation of Feedforward Trajectories for Tokamak Plasma Shape Control

In this paper, we report on the implementation and experimental test of an algorithm for computing feedforward coil current trajectories that can produce an approximately correct evolution of plasma shape, using a nonlinear online optimization method that maintains coil currents far from their limits. The voltage trajectory that produces these currents can be used as the feedforward component of plasma shape control when combined with a multivariable linear feedback control algorithm.

Backstepping control of the plasma current profile in the DIII-D tokamak

Control of the spatial profile of plasma current in tokamak plasmas has been demonstrated to be a key condition for achieving advanced scenarios with improved confinement and possible steady-state operation. The dynamics of the current profile are nonlinear and coupled with several other plasma parameters, motivating the design of model-based controllers that can account for these complexities. In this work, we consider a control-oriented model of the current profile evolution in DIII-D and the problem of regulating the current profile around a desired feed-forward trajectory. In open-loop, the response of the system to disturbances and perturbed initial conditions may be undesirable. To improve the performance of the system, the PDE model is discretized in space using a finite difference method and a backstepping design is applied to obtain a discrete transformation from the original system into an asymptotically stable target system with desirable properties. Through a nonlinear transformation, the resulting boundary control law utilizes the total plasma current, total power, and line averaged density as actuators. A Simserver simulation study is done to test the controllers performance and its implementation in the DIII-D plasma control system. Finally, experimental results showing the ability of the controller to reject input disturbances and perturbations in initial conditions are presented.

Experimental and Simulation Testing of Physics-model-based Safety Factor Profile and Internal Energy Feedback Controllers in DIII-D Advanced Tokamak Scenarios

Abstract Active closed-loop control of the plasma safety factor profile ( q -profile) and internal energy dynamics in nuclear fusion tokamak devices has the potential to significantly impact the success of the ITER project. These plasma properties are related to both the stability and performance of a given plasma operating scenario. In this work, we develop integrated feedback control algorithms to control the q -profile and internal energy dynamics in DIII-D advanced tokamak (high performance) scenarios. The feedback controllers are synthesized by embedding a nonlinear, physics-based, control-oriented partial differential equation model of the plasma dynamics into the control design and to be robust to uncertainties in the plasma electron density, electron temperature, and plasma resistivity profiles. The auxiliary heating and current-drive system and the total plasma current are the actuators utilized by the feedback controllers to control the plasma dynamics. Finally, the feedback controllers are tested both through simulations based on the physics-based model and experimentally in the DIII-D tokamak.

Nonlinear Physics-Model-Based Actuator Trajectory Optimization for Advanced Scenario Planning in the DIII-D Tokamak

Abstract Extensive research has been conducted to find operating scenarios that optimize the plasma performance in nuclear fusion tokamak devices with the goal of enabling the success of the ITER project. The development, or planning, of these advanced scenarios is traditionally investigated experimentally by modifying the tokamaks actuator trajectories, such as the auxiliary heating/current-drive (H&CD) scheme, and analyzing the resulting plasma evolution. In this work, a numerical optimization algorithm is developed to complement the experimental effort of advanced scenario planning in the DIII-D tokamak. Two properties related to the plasma stability and performance are the safety factor profile ( q -profile) and the normalized plasma beta ( β N ). The optimization algorithm goal is to design actuator trajectories that steer the plasma to a target q -profile and plasma β N , such that the achieved state is stationary in time, subject to the plasma dynamics (described by a physics-based, nonlinear, control-oriented partial differential equation model) and practical plasma state and actuator constraints, such as the maximum available amount of H&CD power. This defines a nonlinear, constrained optimization problem that we solve by employing sequential quadratic programming. The optimized trajectories are then tested through simulation with the physics-based model and experimentally in DIII-D.