Non-linear model-based optimization of actuator trajectories for tokamak plasma profile control

A computational method is presented to determine the tokamak actuator time evolution (trajectories) required to optimally reach a given point in the tokamak operating space while satisfying a set of constraints. Usually, trajectories of plasma auxiliary heating, current drive and plasma current required during the transient phases of a tokamak shot to reach a desired shape of the plasma temperature and safety factor (q) profiles are determined by trial-and-error by physics operators. In this paper, these trajectories are calculated by solving a non-linear, constrained, finite-time optimal control problem.The optimization problem contains a physics model of the non-linear plasma profile dynamics, a cost function to be minimized, and a set of constraints on the actuators and plasma quantities. The method is tested by optimizing the trajectories of Ip, heating and current drive power to obtain a typical hybrid plasma q profile at the end of the current ramp-up phase, while minimizing both the Ohmic flux swing and the distance from a stationary condition, and requiring q > 1 and edge Vloop > 0 at all times. The optimized trajectories feature an Ip overshoot similar to that used in existing experiments, and are shown to perform significantly better than a set of non-optimized trajectories, allowing stationary profiles to be obtained at the beginning of the flat-top phase. Additional information is obtained, including the parameter sensitivity of the optimal solution, a linear model describing the linearized dynamics of the profiles around the optimal trajectory, as well as a classification of the actuator trajectories based on the critical constraint which limits their value at a given time. This provides a solid basis for subsequent closed-loop feedback controller design. The tools presented in this paper could be useful to improve existing tokamak operational scenarios, to prepare operation of future machines and optimize their design.

[1]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[2]  N. R. Sauthoff,et al.  Studies of internal disruptions and m=1 oscillations in tokamak discharges with soft--X-ray techniques , 1974 .

[3]  F. Hinton,et al.  Theory of plasma transport in toroidal confinement systems , 1976 .

[4]  D. Cacuci.  Sensitivity theory for nonlinear systems. I. Nonlinear functional analysis approach , 1981 .

[5]  Joan M. Ogden,et al.  Baldur: A one-dimensional plasma transport code , 1988 .

[6]  B. P. Duval,et al.  Creation and control of variably shaped plasmas in TCV , 1994 .

[7]  A. Bondeson,et al.  The CHEASE code for toroidal MHD equilibria , 1996 .

[8]  O. Sauter,et al.  Neoclassical conductivity and bootstrap current formulas for general axisymmetric equilibria and arbitrary collisionality regime , 1999 .

[9]  O. Sauter,et al.  Long-pulse improved central electron confinement in the TCV tokamak with electron cyclotron heating and current drive. , 2001, Physical review letters.

[10]  O. Sauter,et al.  Control of neoclassical tearing modes by sawtooth control. , 2002, Physical review letters.

[11]  O. Gruber,et al.  Progress towards steady-state advanced scenarios in ASDEX Upgrade , 2002 .

[12]  O. Sauter,et al.  Erratum: “Neoclassical conductivity and bootstrap current formulas for general axisymmetric equilibria and arbitrary collisionality regime” [Phys. Plasmas 6, 2834 (1999)] , 2002 .

[13]  R. Prater,et al.  Heating and current drive by electron cyclotron waves , 2003 .

[14]  M. R. Wade,et al.  High performance stationary discharges in the DIII-D tokamak , 2004 .

[15]  A Bottino,et al.  Inductive current density perturbations to probe electron internal transport barriers in tokamaks. , 2005, Physical review letters.

[16]  J. Garcia,et al.  Importance of electron cyclotron wave energy transport in ITER , 2005 .

[17]  J. B. Lister,et al.  Safety factor profile requirements for electron ITB formation in TCV , 2005 .

[18]  Inward thermodiffusive particle pinch in electron internal transport barriers in TCV , 2006 .

[19]  Ilya Grigorenko.  Optimal control and forecasting of complex dynamical systems , 2006 .

[20]  E. Joffrin,et al.  A control-oriented model of the current profile in tokamak plasma , 2007 .

[21]  Olivier Sauter,et al.  The Physics of Electron Internal Transport Barriers in the TCV Tokamak , 2007 .

[22]  D. A. Humphreys,et al.  Towards model-based current profile control at DIII-D , 2007 .

[23]  A. Polevoi,et al.  Chapter 1: Overview and summary , 2007 .

[24]  M. Walker,et al.  Design and simulation of extremum-seeking open-loop optimal control of current profile in the DIII-D tokamak , 2008 .

[25]  C. Kessel,et al.  Predictions of H-mode performance in ITER , 2008 .

[26]  Jet Efda Contributors,et al.  A two-time-scale dynamic-model approach for magnetic and kinetic profile control in advanced tokamak scenarios on JET , 2008 .

[27]  O. Sauter,et al.  Current density evolution in electron internal transport barrier discharges in TCV , 2008 .

[28]  T. Fujita.  Tokamak equilibria with nearly zero central current: the current hole , 2010 .

[29]  E. Schuster,et al.  Ramp-Up-Phase Current-Profile Control of Tokamak Plasmas via Nonlinear Programming , 2010, IEEE Transactions on Plasma Science.

[30]  E. Joffrin,et al.  The CRONOS suite of codes for integrated tokamak modelling , 2010 .

[31]  Yongsheng Ou,et al.  Robust Control Design for the Poloidal Magnetic Flux Profile Evolution in the Presence of Model Uncertainties , 2010, IEEE Transactions on Plasma Science.

[32]  Eugenio Schuster,et al.  Optimal Tracking Control of Current Profile in Tokamaks , 2011, IEEE Transactions on Control Systems Technology.

[33]  Sylvain Brémond,et al.  Validation of plasma current profile model predictive control in tokamaks via simulations , 2011 .

[34]  R. J. La Haye,et al.  Plasma models for real-time control of advanced tokamak scenarios , 2011 .

[35]  F. Felici,et al.  Real-time physics-model-based simulation of the current density profile in tokamak plasmas , 2011 .

[36]  F L Tabarés,et al.  Tritium inventory control during ITER operation under carbon plasma-facing components by nitrogen-based plasma chemistry: a review , 2013 .