Eugenio Schuster

Design and simulation of extremum-seeking open-loop optimal control of current profile in the DIII-D tokamak

In a magnetic fusion reactor, the achievement of a certain type of plasma current profiles, which are compatible with magnetohydrodynamic stability at high plasma pressure, is key to enable high fusion gain and non-inductive sustainment of the plasma current for steady-state operation. The approach taken toward establishing such plasma current profiles at the DIII-D tokamak is to create the desired profile during the plasma current ramp-up and early flattop phases. The evolution in time of the current profile is related to the evolution of the poloidal flux, which is modeled in normalized cylindrical coordinates using a partial differential equation usually referred to as the magnetic diffusion equation. The control problem is formulated as an open-loop, finite-time, optimal control problem for a nonlinear distributed parameter system, and is approached using extremum seeking. Simulation results, which demonstrate the accuracy of the considered model and the efficiency of the proposed controller, are presented.

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.

Development of ITER-relevant plasma control solutions at DIII-D

The requirements of the DIII-D physics program have led to the development of many operational control results with direct relevance to ITER. These include new algorithms for robust and sustained stabilization of neoclassical tearing modes with electron cyclotron current drive, model-based controllers for stabilization of the resistive wall mode in the presence of ELMs, coupled linear–nonlinear algorithms to provide good dynamic axisymmetric control while avoiding coil current limits, and adaptation of the DIII-D plasma control system (PCS) to operate next-generation superconducting tokamaks. Development of integrated plasma control (IPC), a systematic approach to modelbased design and controller verification, has enabled successful experimental application of high reliability control algorithms requiring a minimum of machine operations time for testing and tuning. The DIII-D PCS hardware and software and its versions adapted for other devices can be connected to IPC simulations to confirm control function prior to experimental use. This capability has been important in control system implementation for tokamaks under construction and is expected to be critical for ITER.

Ramp-Up-Phase Current-Profile Control of Tokamak Plasmas via Nonlinear Programming

The achievement of suitable toroidal-current-density profiles in tokamak plasmas plays an important role in enabling high fusion gain and noninductive sustainment of the plasma current for steady-state operation with improved magnetohydrodynamic stability. 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 partial differential equation (PDE) usually referred to as the magnetic flux diffusion equation. The dynamics of the plasma current density profile can be modified by the total plasma current and the power of the noninductive current drive. These two actuators, which are constrained not only in value and rate but also in their initial and final values, are used to drive the current profile as close as possible to a desired target profile at a specific final time. To solve this constrained finite-time open-loop PDE optimal control problem, model reduction based on proper orthogonal decomposition is combined with sequential quadratic programming in an iterative fashion. The use of a low-dimensional dynamical model dramatically reduces the computational effort and, therefore, the time required to solve the optimization problem, which is critical for a potential implementation of a real-time receding-horizon control strategy.

Extremum-Seeking Finite-Time Optimal Control of Plasma Current Profile at the DIII-D Tokamak

In a magnetic fusion reactor, the achievement of a certain type of plasma current profiles, which are compatible with magnetohydrodynamic (MHD) stability at high plasma pressure, is key to enable high fusion gain and noninductive sustainment of the plasma current for steady-state operation. The approach taken toward establishing such plasma current profiles at the DIII-D tokamak is to create the desired profile during the plasma current ramp-up and early flattop phases. The evolution in time of the current profile is related to the evolution of the poloidal flux, which is modeled in normalized cylindrical coordinates using a partial differential equation (PDE) usually referred to as the magnetic diffusion equation. The control problem is formulated as an open-loop, finite-time, optimal control problem for a nonlinear distributed parameter system, and is approached using extremum seeking. Simulation results, which demonstrate the accuracy of the considered model and the efficiency of the proposed controller, are presented.

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.

Mixing enhancement in 2D magnetohydrodynamic channel flow by extremum seeking boundary control

The interaction between electrically conducting fluids and magnetic fields in channel flows generates significant magnetohydrodynamics (MHD) effects, which often result in the need of higher pressure gradients to drive the fluid and lower heat transfer rates due to the laminarization of the flow. Active boundary control, either open-loop or closed-loop, can be employed to overcome this limitation. Open-loop controllers are in general more sensitive to uncertainties of the system, which may result in a poorer performance. Extremum seeking is a powerful tool to tune in real time open-loop controllers, incorporating certain degree of feedback into the control scheme. In this work we combine extremum seeking with a fixed-structure open-loop controller with the ultimate goal of enhancing mixing in a 2D MHD channel by boundary actuation. We show that by carefully tuning the extremum seeking controller the modified open-loop control scheme can be as effective as previously proposed closed-loop control schemes.

Plasma models for real-time control of advanced tokamak scenarios

Anintegratedplasmaprofilecontrolstrategy,ARTAEMIS,isbeingdevelopedforextrapolatingpresent-dayadvanced tokamak (AT) scenarios to steady-state operation. The approach is based on semi-empirical modelling and was initiallyexploredonJET(Moreauetal2008Nucl.Fusion48106001). Thispaperdealswiththegeneralapplicability of this strategy for simultaneous magnetic and kinetic control on various tokamaks. The determination of thedevicespecific, control-oriented models that are needed to compute optimal controller matrices for a given operation scenario is discussed. The methodology is generic and can be applied to any device, with different sets of heating and current drive actuators, controlled variables and profiles. The system identification algorithms take advantage of the large ratio between the magnetic and thermal diffusion time scales and have been recently applied to both JT-60U and DIII-D data. On JT-60U, an existing series of high bootstrap current (∼70%), 0.9MA non-inductive AT discharges was used. The actuators consisted of four groups of neutral beam injectors aimed at perpendicular injection (on-axis and off-axis), and co-current tangential injection (also on-axis and off-axis). On DIII-D, dedicated system identification experiments were carried out in the loop voltage (Vext) control mode (as opposed to current control) to avoid feedback in the response data from the primary circuit. The reference plasma state was that of a 0.9MA AT scenario which had been optimized to combine non-inductive current fractions near unity with 3.5 <β N < 3.9, bootstrap current fractions larger than 65% and H98(y,2) = 1.5. Actuators other than Vext were co-current,counter-currentandbalancedneutralbeaminjection,andelectroncyclotroncurrentdrive. Powerandloop voltage modulations resulted in dynamic variations of the plasma current between 0.7 and 1.2MA. It is concluded that the response of essential plasma parameter profiles to specific actuators of a given device can be satisfactorily identified from a small set of experiments. This provides, for control purposes, a readily available alternative to first-principles plasma modelling. (Some figures in this article are in colour only in the electronic version)

On Iterative Learning Control of parabolic distributed parameter systems

The Iterative Learning Control (ILC) technique is extended to distributed parameter systems governed by parabolic partial differential equations (PDEs). ILC arises as an effective method to approach constrained optimization problems in PDE systems. We discuss both P-type and D-type ILC schemes for a distributed parameter system formulated as a general linear system ∑(A,B,C,D) on a Hilbert space, in which the system operator A generates a strongly continuous semigroup. Under the assumption of identical initialization condition (IIC), conditions on the learning parameters are obtained to guarantee convergence of the P-type and D-type ILC schemes. Numerical simulations are presented for a 1D heat conduction control problem solved using ILC based on semigroup analysis. The numerical results show the effectiveness of the proposed ILC schemes.

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.