Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009 WeB09.2 Design and Modeling of IER Plasma Magnetic Control System in Plasma Current Ramp-Up Phase on DINA Code Yuri V. Mitrishkin, Alexander Y. Korostelev, Vladimir N. Dokuka, and Rustam R. Khayrutdinov Abstract he paper presents cascade tracking system for plasma current and shape in plasma current ramp-up stage. he internal system cascade for tracking currents in magnetic coils was designed by means of full dynamic decoupling. For the external tracking cascade of plasma shape and current design the pseudo-inverse decoupling techniques was used together with original diagonal PII-controller containing parallel connection of proportional, integral, and doubleintegral units in each channel. he 2nd order model for plasma vertical instability was obtained by identification methodology on plasma-physics DINA code and used in scalar loop design with P-controller for plasma vertical speed suppression. he basic system idea is to transfer from tracking mode of plasma ramp-up phase to quasi-stationary regime without change of controller structure. he system was modeled on nonlinear DINA code with the usage of IER data of reference scenario 2. I. INRODUCION HE necessary condition for controlled fusion is longterm confinement of high-temperature plasma in magnetic field. his condition is satisfied in tokamaks (toroidal chambers with magnetic coils) [1] by means of usage of feedback magnetic control systems. his article is focused on development of IER (www.iter.org) plasma shape and current cascade control system using channel decoupling technique on linearized plant model. he control goal is to achieve as small as possible tracking errors of plasma shape and current during transition from plasma current ramp-up phase to quasi stationary mode without changing controller structure. In doing so, the second order model of plasma vertical speed was obtained on nonlinear plasma-physics DINA code [2] by identification procedure which made possible to choose justified proportional gain of scalar loop of plasma vertical speed suppression and estimate its stability margins. Plasma multivariable magnetic control problem is known to be solved by wide variety of different approaches, such as H theory [3, 4], linear-quadratic control [5], model Manuscript received February 26, 2009. his work was supported in part by Russian Foundation of Basic Research, grant No 05-08-00265-a. Y. V. Mitrishkin is with Bauman Moscow State echnical University, Moscow, Russia (e-mail: y_mitrishkin@hotmail.com) A. Y. Korostelev is with Bauman Moscow State echnical University, Moscow, Russia (e-mail: akorostel@gmail.com) V. N. Dokuka is with roitsk Institute for Innovation & Fusion Research (e-mail: v.dokuka@mail.ru) R. R. Khayrutdinov is with roitsk Institute for Innovation & Fusion Research (e-mail: khayrutd@mail.ru) predictive control [6]. Channel decoupling approach has been applied to tokamak magnetic control systems for a long time. Particularly it is used in magnetic control system of Joint European orus (JE), Abington, UK [7] though only at flat-top phase of plasma discharge. he key difference and novelty of the method proposed is its application at plasma current ramp-up stage. Besides, there are some other differences in the application of channel decoupling. here is dedicated plasma current control circuit in JE tokamak while in IER plasma current and shape control is supposed to be performed by common circuits simultaneously. he plant linear model for JE control system design has plasma current as an entry of its state vector, while in the model used in this paper plasma current is an output variable of the plant model. herefore plasma current control is performed by the outer cascade loop of control system, and not by the inner loop as in JE. he plasma shape in JE is characterized by 32 parameters, which is more than the number of control circuits, equal to 8. here is an opposite situation in IER tokamak, which has control circuits, while plasma shape is characterized by 6 parameters. In the controller design linear plant model DINA-L was used which was obtained by linearization of nonlinear dynamic model, numerically implemented by DINA code [2] for IER conditions [6]. In [9] H and decoupling principles with diagonal PI-controllers were used for design and modeling on only DINA-L models of closed-loop systems in tracking mode of plasma shape and current. In the work given we designed new multivariable decoupling PII-controller and simulated it in the closed-loop on nonlinear DINA code as the fullest model of tokamakreactor IER. Section II provides short description of the plasma nonlinear model used for simulation. he tracking problem is stated in section III. Section IV deals with the design and study of vertical stabilization controller. he development of plasma shape and current controller is represented in section V. Simulation results of developed controller on DINA code are presented in section VI. Section VII summarizes the results obtained. II. PLASMA NONLINEAR MODEL Plasma tokamak configuration with a free boundary at given plasma current and its profile is defined by currents in poloidal active coils and passive contours, and described by nonlinear vector Kirchhoff equation [2]: 978-1-4244-3872-3/09/$25.00 2009 IEEE 1354
Ξm m Ψ ( I, Ip, ξ ) + RI = Nu, N = 0 (1) ( n m) m where Ψ is poloidal fluxes vector-function, I is current vector with n states, I p is plasma current, R is matrix of resistances, Ξ is unit matrix corresponding voltage m m vector u on active coils, dimu = m, n> m, ξ = β l, β p is a scalar proportional to plasma pressure, l i is internal plasma inductance. Plasma equilibrium in magnetic field is described by Grad-Shafranov nonlinear partial differential equation [1, 2]: 1 Ψ 2 Ψ 2 r + = r r r z 2 2 ( Ψ) d F( Ψ) 8π dp 1 r2 πc +,( r, z) Spl, c dψ cr dψ = 2 n 8π rikδ( r rk) δ( z zk), ( r, z) S pl. c k = 1 Here (r, φ, z) is a system of cylindrical coordinates, Ψ is poloidal flux function, r k, z k, I k are coordinates of outer currents, p ( Ψ ) is pressure distribution function, F ( Ψ ) is poloidal current, S pl is a region with plasma. After application of linearization procedure [8] to (1) (2) one can obtain plant linear equation around equilibrium point of plasma discharge scenario p i (2) x = Ax + Bu + Eδξ, y = Cx + Fδξ (3) where x is state vector of passive and active currents deviations, dim x = n, u is vector of voltage variations on active coils, y is output vector of plasma current variation, deviations of plasma geometric parameters, and variations of currents in active coils. One eigenvalue of A is unstable. III. PROBLEM SAEMEN OF PLASMA SHAPE AND CURREN RACKING One can divide the discharge in tokamaks into three phases depending on plasma current: ramp-up, quasistationary, ramp-down phases. o provide the discharge scenario one has to calculate in advance the currents in CS&PF coils. hese currents may be driven by preprogrammed voltages on the coils or by tracking system. Usually one changes controllers when transition from rampup phase to quasi-stationary phase because of different purposes of control. On ramp-up phase plasma current is introduced into tokamak and the basic control aim at this phase is to stabilize plasma unstable vertical position. On quasi-stationary phase one has to stabilize plasma shape and current when external disturbances occur of minor disruption type [3-7]. In Fig. 1 one can see the plasma boundary movement during the current ramp-up and magnetic surfaces at flat-top phase in IER. We suggest avoiding transition of controllers and using only one controller with the same structure to control plasma vertical speed, shape and current at both phases. In IER database we have chosen 4 scenario points specifically at.5 MA (56.21 s), 12.5 MA (63.22 s), 13.5 MA (72.55 s), 15 MA (100 s) of plasma current ramp-up. For these points 4 linear models were obtained by linearization procedure [8] on DINA code. he study of these linear models showed that they do not have big difference in time and frequency domains [10]. It made possible to develop and study tracking control system only on one linear model starting from.5 MA scenario point. Fig. 1. Plasma boundary during the current ramp-up (a) and flat-top (b) modes in IER. Numeration of Central Solenoid (CS) and Poloidal Field (PF) coils in DINA code is approved as follows: CSU3 CSU2 CSU1+CSL1 CSL2 CSL3 PF1 PF2 PF3 PF4 PF5 PF6. CSU3 & CSL1 sections of CS are connected in series. his has led to the following statement of the control problem. Develop two levels (cascade) system in which the lower level tracks the scenario currents in CS&PF coils and the upper level follows the gaps and plasma current. In doing so, one has to achieve trade-off between tracking errors of CS&PF currents and gaps. he better tracking gaps the larger tracking errors of CS&PF currents. So one needs to specify the CS&PF currents errors and then to adjust control system to get acceptable gaps errors and vice versa. his contradiction is due to the phenomenon that plasma is frozen into magnetic field. So gaps are measured and calculated from the location of magnetic surfaces which are created by plasma current and CS&PF currents to be controlled. IV. PLASMA VERICAL SABILIZAION SYSEM A. Identification of scalar closed loop system Due to plasma vertical elongation in modern tokamaks, a scalar feedback controller is to be designed to suppress 1355
plasma vertical instability. he plasma mathematical model is needed to design stabilizing controller systematically. It has to be useful to get simple low order linear model of plasma vertical speed. Such a simple model can be obtained by solving a system identification problem with data produced in numerical experiment on complex nonlinear plasma model realized by DINA code. o perform the numerical experiment the plasma nonlinear model was pre-stabilized by proportional (P) controller. he resulting closed loop system was driven by the periodic binary signal at input r (Fig. 2). he numerical experiment was carried out for 4 selected scenario points and 4 distinct datasets were produced as a result. identified models of about 0.14%. his means that plasma in tokamak combined together with actuator and differentiating filter may be represented by linear second order model with one stable ( 1 > 0 ) and one unstable poles ( 2 < 0 ) in (5) having acceptable identification accuracy. Fig. 2. Control system block-diagram for identification of plasma vertical speed model. he identification was done by estimation of parameters of closed-loop transfer function from r to dz dt : ( s) K ( s 1)( s 1) Φ = Φ Φ1 + Φ2 +. (4) he estimation procedure consisted in minimization of mean square error between output signals of original and identified (4) systems on the sets of registered sampled input and output signals by parameters change of linear model (4). B. Identification of scalar open loop system From known transfer functions of the closed loop system (4) and P-controller the unstable open loop transfer function was obtained which includes actuator, plasma itself, and differential filter. he estimated parameters K, 1 and 2 of the open-loop transfer function from u vs to dz dt are listed in able 1. ( ) = ( + 1)( + 1) W s K s s 1 2 (5) ABLE 1 OPEN LOOP ESIMAED PARAMEERS & CLOSED LOOP SABILIY MARGINS Scenario time, s Plasma current, MA K, m/s/v 1, s 2, s Gain marg in, db Phase margin, deg 56.2.5 0.086 0.032-0.12-5.56 34.9 63.2 12.5 0.082 0.039-0.12-5.13 29.6 72.6 13.5 0.077 0.041-0.12-4.55 28.0 100.0 15.0 0.070 0.038-0.12-3.69 28.8 he simulation results of the closed loop system (Fig. 3) produced difference between outputs of original and Fig. 3. Closed loop response to rectangular periodic signal at original (nonlinear) and identified (linear) plant models. C. Proportional controller Given four linear 2 nd order models of plasma vertical speed at different scenario points the P-controller was selected which moves the closed loop system poles to the left on the complex plane to provide acceptable closed loop system stability and performance. Fig. 4. Nyquist diagrams of open loop system with proportional controller and identified models (5) at four points of IER scenario. he stability margins of the vertical stabilization system with selected P-controller for 4 scenario points are presented in able 1. Four Nyquist diagrams of the identified systems with selected controller are shown in Fig. 4. V. CASCADE DECOUPLING SYSEM DESIGN A. Decoupling and tracking of scenario CS&PF currents After adding scalar stabilizing feedback for plasma vertical speed the next step in designing plasma shape & current control system is to develop multivariable tracking loop of scenario currents in CS&PF coils as system internal cascade. In this case the square sub-plant has outputs as current variations in CS&PF coils and voltages on these coils (Fig. 5) as manipulated variables. Active coils, passive structures and plasma filament in tokamak are strongly magnetically coupled so the easiest way to control CS&PF 1356
currents independently is to decouple them in square plant. Fig. 5. Coil currents tracking control system. MC: Main Converter, VSC: Vertical Stabilization Converter. At this design stage the DINA-L plasma multivariable linearized model of IER was used. All inputs and outputs of the model are deviations from their equilibrium values at the scenario point about which the model was linearized. Linear coupled plant model is represented in state space form: x = Ax + Bu, y = Cx (6) 127 where x is state vector specifically variations of currents in active and passive structures, u is control voltages variations vector (inputs) and y is vector of measured CS&PF currents variations (outputs). he form of proposed control law with decoupling channels is u = Kspr K fby where r is the vector of desired (reference) CS&PF currents. his control law results in the closed loop system as x =Λ x+ BK r (7) where is to choose matrices Λ= A BK fbc. he basic idea of the proposed law sp K fb and K sp so that Λ becomes desired diagonal matrix. If matrices B and C are square (that means the system (6) has equal numbers of inputs, outputs, and states) and non-singular then desired requirements are met by choosing fb ( ) 1 1 1 1 K = B A Λ C, K = B Λ C. (8) he above conditions can be satisfied by reduction of given linearized model (6) of 127-order up to states. he additive error introduced by reduction W Wred W is about 0.1% where W and W red are original and reduced open system transfer matrices respectively. hus the closed loop system of internal cascade with K fb and K sp chosen by (8) will have identity matrix gain Ξ at zero frequency when x 1 = 0 in (7): y0 = Cx0 = C Λ BKspr 0 =Ξ r 0. All diagonal elements of Λ are chosen to be equal to 1.0, and multivariable closed loop system becomes separated sp (decoupled) into independent 1 st order units with the same time constant of 1.0 s. B. Decoupling and tracking of plasma shape and current o track plasma shape and current the outer cascade controller was designed (Fig. 6). he controlled outputs are the variation of plasma current and deviations of 6 gaps between the plasma surface and the first wall of tokamak. he manipulated variables of this controller are the inputs of the CS&PF coils current decoupling controller, so overall system has a cascade structure (Fig. 6). Fig. 6. Plasma shape and current tracking loop in overall cascade control system structure. Design approach of the outer loop controller is as follows. In linear model the deviations of the gaps and variation of plasma current may be approximated by linear combination of variations of CS&PF coils currents. It follows from the analysis of matrix C of the linearized plasma model (3). he first states of this model are variations of CS&PF coils currents while other states are variations of currents in passive structures. he entries of matrix C corresponding to the first states are several orders larger than other entries of C. It means that in linearized model (3) variations of gaps δ g and plasma current δ I pl depend mainly on variations of CS&PF coils currents δ Icoils (disturbances of β p and l i were not taken into account in this study). his dependence of variations may be approximated by the equation δg δi 7 pl C0δI coils where C 0 is a sub-matrix of C. Since number of CS&PF coil currents (eleven) is more than number of controlled variables (seven) and sub-matrix C 0 has full row rank then it is possible to set gaps and plasma current variations to any desired values by properly adjusting CS&PF coils currents. Moreover, infinite number of combinations (vectors from ) of CS&PF coils currents exists which give the same values of gaps and plasma current. he logical way of choosing among this infinity a proper CS&PF currents set is to select only one vector with least vector norm in. his could be done by choosing δi C δg δi coils ref = 0 ref pl ref where C C ( C C ) 1 0 0 0 0 (9) = is Moore-Penrose pseudo-inverse of matrix C 0 []. In order to incorporate pseudodecoupling feedback into system the multivariable diagonal 1357
(7 7) controller consisting of seven similar independent scalar PI-controllers was added (Fig. 6). However, preliminary simulations showed that the controller leads to noticeable non-zero tracking errors. In order to eliminate such errors new PII controller was used with transfer function PII ( ) 1 = + 2 / + 3/ W s diag k k s k s 2 which gave astatism of 2nd order in multivariable feedback. he PIIcontroller W PII with coefficients k 1, k 2, k 3 tuned by trialand-error per se on DINA code led to high tracking accuracy. VI. DECOUPLING SYSEM SIMULAION ON DINA CODE he simulation was done on DINA code by feeding preprogrammed gaps and plasma current deviations to the input (setpoint) of the control system (Fig. 6) to provide the transition from the plasma current ramp-up to flat-top stage of plasma discharge. he simulation time range was 70 s starting at time point 56.21 s of IER scenario 2. Scenario CS&PF coils voltages were added between actuators and plant (Fig. 6) as it is shown on the scheme given in IER echnical Basis document (www.iter.org). Although more realistic way would be to feed scenario voltages to the inputs of main converters we preferred to follow IER document approach in this study. Fig. 7 shows tracking of plasma current by the system. he maximum tracking error is about 60 ka which is less than 0.4%. racking of gaps is shown in Fig. 8 with maximum error of about 3 cm. Plasma vertical position displacement and speed are shown in Fig. 9. he currents and voltages variations in CS&PF coils are shown in Fig. 10,. It is possible to significantly increase CS&PF coils currents tracking accuracy, but at the price of larger gaps tracking errors. herefore, the trade-off between CS&PF coils currents tracking and plasma shape and current tracking was found by properly tuning PII controllers of the outer loop. he maximum relative errors of 25.8% for CS&PF coil currents and 6.3% for gaps were obtained. he relative errors of vector variables listed above (-entries vector of CS&PF coils currents and 6-entries vector of gaps) were calculated as ratio of Euclidian norms of error and scenario vectors of given variable. Fig. 8. racking of 6 gaps and tracking gaps errors. Fig. 7. racking of plasma current and tracking error. 1358
Fig. 9. Vertical position displacement and vertical speed of plasma center. VII. CONCLUSIONS he paper shows the results of multivariable control system design for tracking of plasma current, gaps between some magnetic surface/separatrix and the first wall, and scenario currents in CS&PF coils of IER at transition mode from plasma current ramp-up stage to flat-top phase. he resulting system is based on decoupling techniques and has cascade structure: the inner loop for tracking of CS&PF coils currents and the outer loop for tracking of plasma current and shape. he 2nd order astatism was used in the multivariable feedback to achieve high level of tracking accuracy by new diagonal PII controller. For plasma vertical stabilization the systematic choice of the P-controller was done by stability analysis of 2nd order linear model of feedback system obtained by identification procedure on the data of numerical experiment with nonlinear plasma-physics DINA code. his made possible to estimate realistic amplitude and phase stability margins of scalar closed-loop system of plasma vertical stabilization. he system designed with scalar and multivariable decoupling loops was simulated on nonlinear DINA code and showed acceptable stability and tracking accuracy. he future development of the system for earlier part of plasma current ramp-up phase is supposed to be carried on to cover the whole plasma discharge. Fig. 10. CS&PF current variations. Fig.. CS&PF voltage variations. REFERENCES [1] J. Wesson, okamaks (2nd ed.). Clarendon Press, Oxford, 1997. [2] R. R. Khayrutdinov and V. E. Lukash. Studies of Plasma Equilibrium and ransport in a okamak Fusion Device with the Inverse-Variable echnique. Journal Comp. Physics, 109, pp. 193 201, 1993. [3] Y. V. Mitrishkin, V. N. Dokuka, R. R. Khayrutdinov, A. V. Kadurin. Plasma magnetic robust control in tokamak-reactor, Proc. of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA, pp. 2207-2212, 2006, ISBN: 1-4244-0171-2. [4] M. Ariola, A. Pironty. Magnetic Control of okamak Plasmas. Springer-Verlag, 2008. [5] V. Belyakov, A. Kavin, V. Kharitonov, B. Misenov, Y. Mitrishkin et al. Linear Quadratic Gaussian Controller Design for Plasma Current, Position and Shape Control System in IER, Fusion Engineering and Design, vol. 45, pp. 55-64, 1999. [6] Y. Mitrishkin, A. Korostelev. System with Predictive Model for Plasma Shape and Current Control in okamak. Control Sciences, no.5, pp. 22-34, 2008 (In Russian). [7] M. Ariola and A. Pironti. Plasma shape control for the JE tokamak. IEEE Control Systems Magazine, vol. 25, no.5, pp. 65-75, 2005. [8] Y. V. Mitrishkin, V. N. Dokuka, and R. R. Khayrutdinov Linearization of IER Plasma Equilibrium Model on DINA Code. Proc. of the 32nd EPS Plasma Physics Conference, arragona, Spain, ID P5.080, 2005. [9] Y. V. Mitrishkin, A. Korostelev, I. Sushin, R. R. Khayrutdinov, V. N. Dokuka. Plasma Shape and Current racking Control System for okamak. Proc. of the 13 th IFAC Symposium on Information Control Problems in Manufacturing, Moscow, Russia, pp. 2133-2138, 2009. [10] Y. Mitrishkin, A. Korostelev, N. Kartsev, R. Khayrutdinov, V. Dokuka, A. Kadurin, A. Vertinskiy, I. Sushin. Synthesis and Modeling of Plasma Vertical Speed, Shape, and Current Profile Control Systems in okamak. Proc. of International Workshop Control for Nuclear Fusion, Eindhoven University of echnology, he Netherlands, May 7-8, 2008. Available: www.wtb.tue.nl/cnf/program.php. [] A. Albert. Regression and the Moore-Penrose pseudo inverse. Academic Press, NY, 1972. 1359