An Optimal Real-Time Controller for Vertical Plasma Stabilization N
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1 An optimal real-time controller for vertical plasma stabilization N. Cruz, J.-M. Moret, S. Coda, B.P. Duval, H.B. Le, A.P. Rodrigues, C.A.F. Varandas, C.M.B.A. Correia and B. Gonc¸alves Abstract—Modern Tokamaks have evolved from the initial ax- presents important advantages since it allows the creation isymmetric circular plasma shape to an elongated axisymmetric of divertor plasmas, the increase of the plasma current and plasma shape that improves the energy confinement time and the density limit as well as providing plasma stability. However, triple product, which is a generally used figure of merit for the conditions needed for fusion reactor performance. However, the an elongated plasma is unstable due to the forces that pull elongated plasma cross section introduces a vertical instability the plasma column upward or downward. The result of these that demands a real-time feedback control loop to stabilize forces is a plasma configuration that tends to be pushed up or the plasma vertical position and velocity. At the Tokamak down depending on the initial displacement disturbance. For Configuration Variable (TCV) in-vessel poloidal field coils driven example, a small displacement downwards results in the lower by fast switching power supplies are used to stabilize highly elongated plasmas. TCV plasma experiments have used a PID poloidal field coils pulling the plasma down, with increased algorithm based controller to correct the plasma vertical position. strength as the plasma gets further from the equilibrium posi- In late 2013 experiments a new optimal real-time controller was tion. To compensate this instability, feedback controllers have tested improving the stability of the plasma. been designed to correct the vertical position displacement This contribution describes the new optimal real-time con- [4][5][6]. troller developed. The choice of the model that describes the plasma response to the actuators is discussed. The high order The design of vertical stabilization feedback controllers model that is initially implemented demands the application of has been based in simple models, resulting in experimentally a mathematical order reduction and the validation of the new tuned Single Input Single Output (SISO) Proportional Integral reduced model. The lower order model is used to derive the time and Derivative (PID) regulators. This procedure requires an optimal control law. A new method for the construction of the in-depth experimental treatment that is time consuming and switching curves of a bang-bang controller is presented that is based on the state-space trajectories that optimize the time to demands a big number of experimental discharges to obtain target of the system. the necessary gains optimization. This paper presents an A closed loop controller simulation tool was developed to test alternative method to design the vertical stabilization controller different possible algorithms and the results were used to improve of a tokamak using a simple plasma model and the application the controller parameters. of optimal control theory. The final control algorithm and its implementation are de- This paper is organized as follows: Section II presents scribed and preliminary experimental results are discussed. the vertical observer developed to detect the plasma centroid Index Terms—Real-Time, Tokamak, Plasma Control, Optimal vertical position and velocity in real-time; Section III briefly Control depicts the different methods that can be used to describe a tokamak plasma; Section IV describes the state-space plasma I. INTRODUCTION model that predicts the plasma response to the actuators and the model reduction performed to permit the application of ODERN tokamak devices [1] are designed to ac- the time optimal control theory that is presented in Section commodate elongated cross-section plasmas [2][3] to M V; Section VI depicts the simulation tool that permits off line improve fusion performance. A vertically elongated plasma testing and parameter tunning of the controller; The controller Manuscript received June 16, 2014. This work was supported by EU- results and future work is presented in Section VII. RATOM and carried out within the framework of the European Fu- arXiv:1406.4436v2 [physics.plasm-ph] 18 Jun 2014 sion Development Agreement. IST activities also received financial sup- II. VERTICAL PLASMA POSITION OBSERVER port from ”Fundac¸ao˜ para a Cienciaˆ e Tecnologia” through project Pest- OE/SADG/LA0010/2013. The views and opinions expressed herein do not The vertical position observer is a linear combination of necessarily reflect those of the European Commission. This work was partly the magnetic field measured using the magnetic diagnostics. supported by the Swiss National Science Foundation. A matrix containing the contribution weight of each magnetic N. Cruz is with the Instituto de Plasmas e Fusao˜ Nuclear, Instituto Superior Tecnico,´ Universidade de Lisboa, P-1049-001 Lisboa, Portugal (Telephone: probe is calculated before each plasma discharge, taking into +351-239410108, e-mail: [email protected]). account the planned plasma parameters such as shape and J.-M. Moret, S. Coda, B. P. Duval and H. B. Le are with the Centre position. The contribution of each probe to the observer has de Recherches en Physique des Plasmas, Ecole´ Polytechnique Fed´ erale´ de Lausanne (EPFL), CH-1015 Lausanne, Switzerland. in account the pre-planned plasma parameters, because the A. P. Rodrigues, C. A. F. Varandas and B.S. Gonc¸alves are with the Instituto probes closer to the plasma are more efficient estimating its de Plasmas e Fusao˜ Nuclear, Instituto Superior Tecnico,´ Universidade de position and will be given more weight in the observer. Lisboa, P-1049-001 Lisboa, Portugal. C. M. B. A. Correia is with the Centro de Instrumentac¸ao,˜ Departamento A set of coefficients are calculated to define the observer de F´ısica, Universidade de Coimbra, P-3004-516 Coimbra, Portugal from a finite element set of plasma current filaments, using Green’s functions, thus it is possible to calculate the magnetic geometry. This code is presently used to make transport field produced in the probes. The matrix is built with the set simulations of tokamak and stellarator plasmas. The first of probes that are going to be used in the measurements and version of ASTRA was implemented at the Kurchatov inverted to obtain the observer coefficients [7][8]. Institute in Moscow, but an international community The following equation relates the magnetic field measure- continues to develop the code and new features are added ments with the currents in the plasma: to its functionality regularly. • TSC (Tokamak Simulation Code) was originally devel- b − B :I = B :I m mc c mx x (1) oped by S. C. Jardin at Plasma Physics Laboratory, with bm the vector of measured quantities in the magnetic Princeton University for free boundary 2D transport [12]. • probes, Bmc the matrix of the Green’s functions between the EFIT (Equilibrium FITting) is a code developed to per- coils and the magnetic probes, Ic the coil currents vector, Bmx form magnetic and kinetic-magnetic analysis for Doublet- the matrix with the Green’s functions transforming the current III, at General Atomics. EFIT takes the measurements in the plasma filaments into the magnetic field measured by from plasma diagnostics and calculates relevant plasma the probes and Ix the currents in the toroidal filaments. properties such as geometry, stored energy and current From the inversion of the equation, the currents can be profiles. Although it is a very fast computational code, it obtained by: lacks the accuracy of other more computational intensive algorithms [13]. −1 T T Ix = A (Bmx:bm − Bmx:Bmc:Ic) (2) • FBT (Free Boundary Tokamak) is a code originally where A = BT :B . Equation (2) gives the current in the developed by F. Hofmann at Centre de Recherches en mx mx ´ toroidal filaments as a linear combination of the magnetic field Physique des Plasmas, Ecole Polytechnique Fed´ erale´ de measured by each probe (first term) with the correction of the Lausanne. FBT allows the computation of arbitrarily coil current influence in the measurements. shaped tokamak equilibrium specially dedicated to highly The observer is then given by: shaped and elongated plasmas [14]. • PROTEUS is a nonlinear tokamak simulation code that 0 zIp = (zx − za):Ix (3) solves the Grad-Shafranov equation by an iterative finite element method. This code is used to simulate the evo- with z the vertical position of the plasma centre, z the x lution of a tokamak plasma for a fixed plasma current position of the filament of the plasma column and z the a [15]. reference position of the plasma axis. • CREATE-L is a linearized plasma equilibrium response The plasma velocity observer (d(zI )=dt) uses the same p model in view of the current, position and shape control method and because the time derivative of I has a slow vari- c of plasmas in tokamaks [16][17]. The origin of this code’s ation compared to vertical position growth rate, the equations name is the consortium where it was originally developed, can be reduced to: the Consorzio di Ricerca per l’ Energia e le Applicazioni dIx −1 T dbm Tecnologiche dell’Elettromagnetismo (CREATE). = A :Bmx: (4) dt dt • DINA is a tokamak plasma axisymmetric, time- d(zI ) dI dependent, resistive MHD simulation code and a free p = (z0 − z ): x (5) dt x a dt boundary equilibrium solver developed at the RRC Kur- The coil currents correction is added to the reference signal, chatov and TRINITI institutes in Moscow [18]. which makes the output error signal completely consistent. • RZIP is a rigid plasma model that predicts the plasma current, as well as the radial and the vertical plasma positions, used at Centre de Recherches en Physique des III. PLASMA DESCRIPTION Plasmas, Lausanne [19]. The modeling of a tokamak plasma demands complex mathematical calculation, in depth physical knowledge and Some of these codes are accurate for plasma simulation and computational power for numerical calculation during simu- reconstruction but due to its complex structure are not suitable lation phase.