Receding-horizon optimal control of the current profile evolution during the ramp-up phase of a tokamak discharge

The control of the toroidal current density spatial profile in tokamak plasmas will be absolutely critical in future commercial-grade reactors to enable high fusion gain, non-inductive sustainment of the plasma current for steady-state operation, and magnetohydrodynamic (MHD) instability-free performance. 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 control objective during the ramp-up phase is to drive an arbitrary initial profile to approximately match, in a short time windows during the early flattop phase, a predefined target profile that will be maintained during the subsequent phases of the discharge. Thus, such a matching problem can be treated as an optimal control problem for a PDE system. A distinctive characteristic of the current profile control problem in tokamaks is that it admits interior, boundary and diffusivity actuation. A receding-horizon control scheme is proposed in this work to exploit this unique characteristic and to solve the associated open-loop finite-time optimal control problem using different optimization techniques. The efficiency of the proposed scheme is shown in simulations.

[1]  S. Dubljevic,et al.  Predictive control of parabolic PDEs with state and control constraints , 2004 .

[2]  Alfredo Pironti,et al.  A model-based technique for integrated real-time profile control in the JET tokamak , 2004 .

[3]  L. V. Willigenburg,et al.  The significance of crop co-states for receding horizon optimal control of greenhouse climate , 2002 .

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

[5]  T. Osborne,et al.  Emerging Applications in Tokamak Plasma Control Emerging Applications in Tokamak Plasma Control CONTROL SOLUTIONS FOR NEXT-GENERATION TOKAMAKS , 2006 .

[6]  Miroslav Krstic,et al.  Backstepping boundary control for first order hyperbolic PDEs and application to systems with actuator and sensor delays , 2007, 2007 46th IEEE Conference on Decision and Control.

[7]  Wook Hyun Kwon,et al.  Receding horizon guidance laws for constrained missiles with autopilot lags , 2001 .

[8]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

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

[10]  C. Gormezano,et al.  A review of internal transport barrier physics for steady-state operation of tokamaks , 2004 .

[11]  H. Michalska,et al.  Receding horizon control of nonlinear systems , 1988, Proceedings of the 28th IEEE Conference on Decision and Control,.

[12]  Malur K. Sundareshan,et al.  Design of Decentralized Observation Schemes for Large-Scale Interconnected Systems: Some New Results , 1989, 1989 American Control Conference.

[13]  A. Pironti,et al.  Fusion, tokamaks, and plasma control: an introduction and tutorial , 2005, IEEE Control Systems.

[14]  S. Dubljevic,et al.  Predictive control of parabolic PDEs with state and control constraints , 2006, Proceedings of the 2004 American Control Conference.

[15]  A. A. Patwardhan,et al.  Nonlinear model-predictive control of distributed-parameter systems , 1992 .

[16]  Warren D. Seider,et al.  Model-predictive control of the Czochralski crystallization process. Part I. Conduction-dominated melt , 1997 .

[17]  Romeo Ortega,et al.  Passivity of Nonlinear Incremental Systems: Application to PI Stabilization of Nonlinear RLC Circuits , 2006, CDC.

[18]  M. Krstić,et al.  Real-Time Optimization by Extremum-Seeking Control , 2003 .

[19]  J. Richalet,et al.  Industrial applications of model based predictive control , 1993, Autom..

[20]  Frank Allgöwer,et al.  An Introduction to Nonlinear Model Predictive Control , 2002 .

[21]  Barry Lennox,et al.  Infinite horizon model predictive control with no terminal constraint , 2004, Comput. Chem. Eng..

[22]  Martin Berzins,et al.  A Method for the Spatial Discretization of Parabolic Equations in One Space Variable , 1990, SIAM J. Sci. Comput..

[23]  P. C. de Vries,et al.  Real-time control of the q-profile in JET for steady state advanced tokamak operation , 2003 .

[24]  Didier Mazon,et al.  Feedback control of the safety factor profile evolution during formation of an advanced tokamak discharge , 2006 .

[25]  X. Litaudon,et al.  Feedback control of the current profile on Tore Supra , 1997 .

[26]  Yasushi Hada,et al.  Constrained Model Predictive Control , 2006 .

[27]  M. Sueoka,et al.  Recent RF Experiments and Application of RF Waves to Real-Time Control of Safety Factor Profile in JT-60U , 2005 .

[28]  Francesco Borrelli,et al.  Event-based receding horizon control for two-stage multi-product production plants , 2007 .

[29]  Jean Biston,et al.  Modeling of a distributed parameter process with a variable boundary: application to its control , 1994 .

[30]  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 .

[31]  Didier Mazon,et al.  Feedback control of the lower hybrid power deposition profile on Tore Supra , 2007 .

[32]  C. D. Challis,et al.  The use of internal transport barriers in tokamak plasmas , 2004 .

[33]  Panagiotis D. Christofides,et al.  Nonlinear and Robust Control of Pde Systems , 2001 .

[34]  T. S. Taylor,et al.  Physics of advanced tokamaks , 1997 .

[35]  Leonidas G. Bleris,et al.  Reduced order distributed boundary control of thermal transients in microsystems , 2005, IEEE Transactions on Control Systems Technology.

[36]  Gary J. Balas,et al.  Receding horizon control of an F-16 aircraft: A comparative study , 2006 .

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

[38]  David Q. Mayne,et al.  Constrained model predictive control: Stability and optimality , 2000, Autom..

[39]  M. Krstić,et al.  Boundary Control of PDEs , 2008 .

[40]  M. Ariola,et al.  The Role of Controls in Nuclear Fusion , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.