Linear and Impulse Control Systems for Plasma Unstable Vertical Position in Elongated Tokamak

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1 51st IEEE Conference on Decision and Control December 10-13, Maui, Hawaii, USA Linear and Impulse Control Systems for Plasma Unstable Vertical Position in Elongated okamak Yuri V. Mitrishkin, Semyon M. Zenckov, Nikolai M. Kartsev, Alexander A. Efremov, Vladimir N. Dokuka, and Rustam R. Khayrutdinov Abstract²he paper presents the results of design and simulation of linear and impulse control systems of a plasma unstable vertical position for three plasma linear DINA-L models of the tokamak -15. he models were obtained from the plasma-physics DINA code on the plasma current flat top of a -15 scenario and corresponded to three different locations of a horizon field coil: outside of the toroidal coil, between the toroidal coil and vacuum vessel, inside the vacuum vessel. For design of linear controllers a loop shaping approach was used. he controllers designed were simulated in the feedback systems with three types of power supply models: linear element, pulse-width modulation element, and relay element. he feedback systems were tested by Heaviside step-functions and disturbances of minor disruption type. he assessment of performances and stability margins of the systems designed and simulated was done in accordance with a multi-criterion approach. I. INRODUCION OKAMAKS are thermonuclear installations as future energy sources from nuclear fusion [1]. he suppression of plasma vertical instability by means of magnetic field leading to stabilization of plasma vertical position is a challenge in modern tokamaks with vertically elongated plasma for instance in JE (UK) [2], NSX (US) [3], GLOBUS-M (Russia) [4], IER (France) [5] and others. Vertically elongated tokamak plasma is unstable and the capabilities of control coils are naturally constrained, so WKHUH VDOZD\VDULVNRISODVPDYHUWLFDOSRVLWLRQVWDEOLOLW\ loss due to unpredictable disturbances, which can cause damage to the walls of vacuum vessel. here are different approaches to suppress plasma vertical instability [2], [3], [5], [8]. In this paper the study of the possibilities of plasma vertical position stabilization was done for the tokamak -15 announced in [9]. he -15 poloidal system contains along with Central Solenoid (CS) Manuscript received March 5, Y. V. Mitrishkin is with V.A. rapeznikov Institute of Control Sciences of the Russian Academy of Sciences, Moscow, Russia (corresponding author to provide phone: ; fax: ; y_mitrishkin@hotmail.com). S. V. Zenckov, N. M. Kartsev, and A.A. Efremov are with V.A. rapeznikov Institute of Control Sciences of the Russian Academy of Sciences, Moscow, Russia ( szenckov@gmail.com, n.kartsev@yandex.ru, alexander.efremoff@gmail.com). V. N. Dokuka and R. R. Khayrutdinov are with roitsk Institute of Innovations and hermonuclear Research, Moscow Region, Russia ( v.dokuka@mail.ru, khayrutd@mail.ru). and Poloidal Field (PF) coils an additional Horizon Field Coil (HFC) for plasma vertical stabilization. he goal of the study was to find out the functional controllability [10], [11] of the tokamak -15 plasma in respect to vertical position stabilization for various locations of the HFC: outside of the toroidal coil, between the vacuum vessel (VV) and the toroidal coil, and inside the VV. he approach for controller design used in our work is based on the following. he actuator for applying the voltage to the HFC is supposed to be an impulse power supply, specifically a voltage inverter which transforms the DC voltage to the AC voltage of the rectangular form on an inductive load. his is because the dynamics of the elongated plasma moving in the vertical direction in -15 is very fast and the control system should provide maximum speed of response to keep plasma in the VV. he impulse actuator may operate in Pulse-Width Modulation (PWM) or relay mode. he plant model is a linear one which is obtained by the linearization procedure of the non-linear plasma model realized by plasma-physics DINA code [12]. First, we design a linear controller for a linear model, simulate it on the linear plant model in a feedback system, estimate control system basic characteristics. Second, we add into the closed loop the nonlinear actuator model, simulate the feedback system one more time and make assessment of its performance issues and stability margins. Such approach is proved by the fact that in the reality of about 99% of the controllers in fusion research and about 80% of the controllers used outside fusion in industry are classical linear controllers. Section 2 provides description of the tokamak -15 plasma under control, linear DINA-L models, and the statement of the control problem. he nonlinear models of the power supply as a voltage inventor in PWM and relay modes are given in section 3. he design and simulation of a H f SISO controller for plasma vertical position when the HFC is located outside of the toroidal coil is done in section 4. he estimation and comparison of controllability regions for plasma vertical position for different locations of the HFC is done in section 5. he comparison of linear systems with H and proportional controllers in the feedback for the HFC located inside and outside of the VV is shown in section 6. he results of comparison of impulse SISO control systems for two locations of the HFC are presented in section 7. Section 8 summarizes the results obtained /12/$ IEEE 1697

2 II. SAEMEN OF HE CONROL PROBLEM A. Controlled plant he controlled plant is plasma in the -15 tokamak. All basic tokamak parameters are presented in [9]. he tokamak -15 poloidal system, the location of six measured gaps between the seperatrix and the first tokamak wall are shown in Fig. 1. Plasma equilibrium in magnetic field of a tokamak is described by the Grad-Shafranov partial-differential equation, the solution of which gives the space distribution of poloidal field inside plasma against external currents [1]. he dynamics of plasma in the tokamak is determined by variations of currents in passive structures surrounding plasma and variations of currents in control coils around the toroidal VV. hese variations are described by the Kirchhoff equation. Grad-Shafranov and Kirchhoff equations are numerically realized together with transport equations in the plasma-physics DINA code [12]. A linearized plasma model DINA-L in -15 for a plasma current flat-top phase of a plasma discharge has the following state space form: d x / dt AxBu Edw / dt, y C xf w (1) where A, B, C, D, E, F are matrixes of the model, y ª G Z G g G I º p G Icoils G R¼ is the output vector composed of the plasma vertical displacement /=, displacements of six gaps /g, the plasma current variation /I p, variations of currents /I coils in CS, PF coils, the HFC, and the plasma horizontal displacement /R. Vector u ª U º CS UPF UHFC¼ is the input U CS composed of CS coil voltages, the input U PF of PF coil voltages, and the U HFC voltage on the HFC. Vector w ª GE p Glº i¼ is the disturbance composed of drops of relative plasma pressure p and internal plasma inductance l i [1]. he matrix A has one unstable eigenvalue which determines the plasma unstable vertical mode. he model state vector Ð4 :5 is a set of current variations in passive structures and active (control) coils. he evolution of disturbance parameters is defined by the GE K GE h t t, following expressions: p md p0 disr l K l ht t G G where h(t) is the Heaviside step i md i0 disr function, t disr is a time moment when disturbance occurs, GE p and G l i are the values of p and l i drops respectively, and K md is a variable parameter. Such kind of disturbance simulates the tokamak plasma effect known as a minor disruption [1]. B. Control problem statement he primary task for a -15 plasma control system is to stabilize plasma vertical position with input voltages limited by 2 kv in magnitude. Also, it is important to keep other system parameters such as coils currents limited i.e. to make the closed loop system internally stable. a b c Fig. 1. Location of control coils in -15: (a) the HFC is outside the toroidal coil (case I), (b) the HFC is inside the toroidal coil and outside the VV (case II), (c) the HFC is inside the VV (case III). he objective of this research is to explore functional controllability of different poloidal systems of tokamak -15 with different location of the HFC as well as to make clear how different control approaches will work with two types of actuating devices: PWM and relay units. Control system operation is estimated by its performance and stability margin at minor disruptions. he last one is evaluated by maximum K md disturbance parameter, with which the closed loop system maintains its stability. III. POWER SUPPLY MODELS A. PWM unit model he PWM unit model has been created by means of MALAB SimPowerSystems oolbox [13] using PWM and H-bridge blocks. 7KHUH VQRLQIRUPDWLRQDERXW practical voltage inverter parameters yet, so the 0% duty cycle input value was set to 0 V and the 100% duty cycle input value was set to be equal to the output signal amplitude. o test the accuracy of this model the sine wave with 2.5 khz frequency and 2000 V amplitude was signaled. he PWM frequency was set to 10 khz and the output amplitude was set to 2000 V. Both input and output signals have been integrated. he results of this test are presented in Fig. 2 and 3. Fig. 2. Input sine wave signal and PWM output signal. It can be seen from Fig. 2, 3 that the PWM unit model operates correctly with sufficient accuracy. 1698

3 IV. LINEAR SISO CONROLLER FOR PLASMA VERICAL POSIION Z he robust H f open loop shaping design technique based on the normalized coprime factorization (NCF) [15] of a 1 plant model transfer function G M N was applied to synthesize a stabilizing controller in the feedback (Fig. 6). Fig. 3. Integrated input sine wave signal and PWM output signal. B. Relay unit model he relay unit model was created on the base of the experimental data for the thyristor bridge (voltage inverter) obtained in Ioffe Institite (St. Petersburg, Russia) on UMAN-3 tokamak [14]. his data includes thyristor bridge time dead zones which were measured on the voltage inverter connected to an inductive load in a self-oscillation mode with the negative proportional feedback. he following time dead zones values were obtained in [14]: in the case of positive load current+pr while switching from positive voltage E' to negative voltage F' the dead zone width equals ì > LwräO and while switching from negative voltage F' to positive voltage E' the dead zone width equals ì? LtwräO; in the case of the negative load current +Or ì > LtwräO and ì? LwräO. he relay unit was modeled in the MALAB\Simulink environment using m-functions. he relay unit testing control system is presented in Fig. 4. During the simulation the relay unit switches to negative proportional feedback after the inductive load current I reaches the reference value I 0. he results of this simulation are shown in Fig. 5. Fig. 6. Block diagram of the closed-loop system. o perform the open loop shaping the plant model transfer function Ps () given by (1) was augmented with the weighting function W s 7 ( ) 9 10 / s, which was chosen by trial and error procedure to satisfy the requirements on disturbance rejection and robustness. Fig. 7. Step response of the feedback system on 0.1 m plasma position reference. Comparison block Load current Fig. 8. HFC, PF5, PF8 currents responses. I0 Switch + - I0 - I Relay unit Inductive load -1 Fig. 4. Block-diagram of the control system for testing the relay unit. Fig. 5. Output voltage and current of the relay unit in the self-oscillation mode of the closed-loop system. he achieved robust stability margin after the synthesis done is H max where H! ' N ' M in the f perturbed plant model transfer function 1 ' ' G M N with stable unknown transfer p M N functions ' M, '. N In Fig. 7 one can see that the value Z has a stable behavior in the system with the designed feedback controller but the synthesized system is not reliable because the HFC current grows unconstrained and it may cause damage to the HFC and power supply (Fig. 8). In addition, one can see that PF5 and PF8 coil currents are increasing as well. his fact can be explained by the law of induced current. Indeed, upper HFC and PF5 sections (lower HFC and PF8 coil sections respectively) have similar electric and magnetic properties and are mounted very close to each other in the case I of the tokamak -15 poloidal system (Fig. 1,a). Because of that, the resulting horizon field is formed by the sum of HFC and PF5, PF8 coil currents which are opposite. his fact one can see in the transfer functions from U HFC to I HFC and Z: 1699

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5 ABLE I PERFORMANCE INDEXES AND SABILIY MARGINS OF LINEAR SYSEMS HFC location Index Cont. g, db 3 deg. Out-VV H Z: 0.04 I: 0.25 K = Z: 0.10 I: 0.25 In-VV H Z: I: K = Z: I: In/Out K Z: 0.1 I:0.048 VII. IMPULSE SYSEMS COMPARISON t r, s t s, s. % I ss, A he feedback systems with the impulse power supply models were tested at 0.1 m plasma vertical position step reference and minor disruption disturbance with - à Ls. he proportional controller with K = 10 5 has been chosen for testing of all systems. Simulation results for the case II of the systems step responses are given in Fig. 11 and 12. Fig. 14. Step response for the case III system with the relay unit. Simulation characteristics of the impulse systems are collected in able II. ABLE II SEP RESPONSE RESULS OF IMPULSE SYSEMS FOR CASES II, III Operating parameter Out-VV HFC (case II) In-VV HFC (case III) PWM Relay PWM Relay Overshoot, % Response time, sec Max HFC current, A Max power, MW he case III system has 4 times better overshoot than the case II system (2 times better in the case of relay unit), about 5 times better response time and about 5 times lower max HFC current and power. Fig.11. Step response of the case II system with the PWM unit. Fig. 15. Minor disruption response for the case II system with the PWM unit. Fig. 12. Step response of the case II system with the relay unit. Results for the case III systems of step responses are demonstrated in Fig. 13 and 14. Fig. 16. Minor disruption response for the case II system with the relay unit. Fig. 13. Step response for the case III system with the PWM unit. Fig. 17. Minor disruption response for the case III system with the PWM unit. 1701

6 Fig. 18. Minor disruption response for the case III system with the relay unit. Simulation results for the case II systems of the minor disruption response are presented in Fig. 15 and 16. Simulation results for the case III systems of the minor disruption response are displayed in Fig. 17 and 18. Simulation system characteristics are accumulated in able III, which also contains minor disruption stability margins K md max for all four systems. he value K md max is the maximum minor disruption coefficient, with which the system maintains its stability. It was measured with a glance WROLPLWDWLRQVRQYHUWLFDOSODVPDGLVSODFHPHQW/Z < 0.1 m and I HFC < 10 ka to hold the limitations of real experiment. ABLE III MINOR DISRUPION RESPONSE RESULS FOR CASES II, III IMPULSE SYSEMS Out-VV HFC (case Operating parameter II) In-VV HFC (case III) PWM Relay PWM Relay Max. plasma vertical displacement, mm Response time, sec Max HFC current, A Max power, MW Minor disruption stability margin K md max he case III system has higher plasma vertical displacement than the case II system, but it has 100 times better response time for the relay unit and 20 times better for the PWM unit, about 5 times lower max HFC current and power for the relay unit and less than 2 times for the PWM unit and roughly equal minor disruption stability margins. hus, the system with the HFC located inside VV has significant performance advantage over the system with HFC located outside VV at the same proportional controller. VIII. CONCLUSION Configurations of poloidal systems of modern vertically elongated tokamaks play important role in plasma confinement in magnetic field. hese configurations may be grouped into three sets: (i) PF-coils are inside toroidal coil [DIII-D, NSX (USA), CV (Switzeland)], no HFC inside VV; (ii) PF-coils are outside toroidal coils, no HFC coil inside VVs [JE (UK), ASDEX Upgrade (Germany)], (iii) superconducting PF coils outside toroidal coil, HFC is inside VV [EAS (China), IER (France), KSAR (South Korea), J-60SC (Japan)]. In the last two tokamaks two active coils are inside the VV specifically the HFC for plasma vertical position control and a Vertical Field Coil for plasma horizon control. A set of linear controllers were designed for the tokamak -15 to find out the effectiveness of the HFC located in different positions in accordance with the general picture of poloidal systems of tokamaks mentioned above. In the first case when the HFC is outside the toroidal coil it is not possible to control plasma vertical position by only the HFC because of the unrestricted HFC current rise. In the tokamak -15 the most effective for magnetic control system is the location of the HFC inside the VV. But real choice of the HFC location will be determined not only by the best plasma functional controllability but the trade-off between tokamak technological conditions of realization and exploitation. REFERENCES [1] J. A. Wesson, okamaks. Oxford: Clarendon Press, [2] A. Neto, R. Albanese, G. Ambrosino, M. Ariola et al, ³Exploitation of modularity in the JE tokamak vertical stabilization system in Proc. the 50th IEEE Conf. Decision and Control and European Control Conf., Orlando, FL, USA, 2011, pp [3] W. Shi, J. Barton, M. Alsarheed, and E. Schuster, ³Multivariable multi-model-based magnetic control system for the current ramp-up phase in the National Spherical orus Experiment (NSX) in Proc. the 50th IEEE Conf. Decision and Control and European Control Conf., Orlando, FL, USA, 2011, pp [4] E. A. Kuznetsov and Y. V. Mitrishkin. Self-Oscillation Stabilization System of Plasma Unstable Vertical Position of Spherical okamak GLOBUS-M. Moscow: V.A. rapeznikov Institute of Control Sciences, Russian Academy of Sciences, 2005 (in Russian). [5] Y.V. Mitrishkin and N. M. Kartsev, ³Hierarchical plasma shape, position, and current control system for IER in Proc. the 50th IEEE Conf. on Decision and Control and European Control Conference, Orlando, FL, USA, pp [6] D. A. Hamphreys et al, ³Experimental vertical stability studies for IER performance and design guidance. Nuclear Fusion, vol. 49, 2009, 10 pp. [7] G. Stein, ³Respect the Unstable IEEE Control Systems Magazine, vol. 23, no. 4, pp , Aug [8] Y. V. Mitrishkin and H. Kimura, ³Plasma Vertical Speed Robust Control In Fusion Energy Advanced okamak LQ Proc. 40th IEEE Conf. on Decision and Control, Florida, USA, 2001, pp [9] E. A. Azizov, V. A. Belyakov, O. G. Filatov, E. P. Velikhov, and - 15MD 7HDP³Status of project of engineering-physical tokamak in 23rd International Atomic Energy Agency (IAEA) Fusion Energy Conf., Daejon, South Korea, 2010, FP/P6-01. [10] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control (2nd ed.), Chichester: John Wiley & Sons Ltd., [11] Y.V. Mitrishkin, ³)unctional controllability of plasma shape and current linear models of tokamaks in Proc. VIII International Workshop on Stability and Oscillations of Non-linear Control Systems, V.A rapeznikov Institute of Control Sciences, Moscow, Russia, 2004, pp [12] 9(/XNDVK91'RNXND55.KD\UXWGLQRY³0$7/$%V\VWHP software-frpsxwlqj',1$frpsoh[iruvroylqjsodvpdfrqwurowdvnv Problems of atomic science and technology. Series: Nuclear Fusion, no. 1, pp , 2004 (in Russian). [13] 6LP3RZHU6\VWHPVIRUXVHZLWK6LPXOLQN8VHU V*XLGHYHUVLRQ. he MathWorks Inc., September Available at: wersys/powersys.pdf [14] Y.V. Mitrishkin, ³&omprehensive design and implementation of plasma adaptive self-oscillations and robust control systems in thermonuclear installations LQ Proc. 8 th World Multi-Conf. Systemics, Cybernetics and Informatics, Orlando, FL, USA, vol. 15, 2004, pp [15] '0F)DUODQHDQG.*ORYHU³$/RRS6KDSLQJ'HVLJQ3URFHGXUH Using H 6\QWKHVLV IEEE rans. Automatic Control. vol. 37, no. 6, 1992, pp [16] J. P. Hespanha, Linear Systems heory. New Jersey: Princeton University Press,