1. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Plasma stabilization control models for tokamak Ovsyannikov A.D., Ovsyannikov D.A. Suhov E.V. Vorobyov G.M., Zavadsky S.V.

2. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Gutta tokamak Main parameters: major radius – R 16 cm, minor radius – a 8 cm, aspect ratio – A 2, vessel elongation – k 2, plasma current < 150kA, toroidal field - 1.5 T

3. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Tokamak poloidal circuits. Poloidal field coils Vacuum vessel

4. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Poloidal crossection Poloidal field coils Vacuum vessel

5. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Dynamic model of a poloidal circuit system Where I-vector of currents, U-vector of voltages, L-inductance matrix, R-resistance matrix, C-capacitance matrix

6. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Poloidal currents dynamics

7. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Model testing Calculated outer loop voltage. Horizontal axis – time in microseconds, vertical axis – voltage, 1V in point. Red line - Loop voltage on outer loop. Measured outer loop voltage. Horizontal axis - time in microseconds, vertical axis – voltage, 0.5V in point. Red line - Loop voltage on outer loop.

8. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 General form of the control problem. The coefficients of system (2) remain continuous on half-intervals t m corresponds electron-cyclotron (ECR) pre-ionization time, t p breakdown time

9. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Optimal and non optimal breakdown conditions Plasma visible light. Horizontal axis – time in microseconds, red line – plasma visible light amplitude in conventional units. Optimal breakdown conditions. Plasma visible light. Horizontal axis – time in microseconds, red line – plasma visible light amplitude in conventional units. Non-optimal breakdown conditions.

10. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 The structural parametric optimization of transient processes The equations of the control object in the state space are the following The control object closure by the gained regulator The control object is completed with a regulator of a decreased dimension with the following structure (1) (2)

11. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 f(t) is a disturbance vector that satisfies following equation at the moment t (disturbance ensemble) Let us investigate the control object with presence of a constantly applied disturbance The structural parametric optimization of transient processes (3) (4)

12. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 where let us introduce a performance functional the gradient of the functional by parameters The structural parametric optimization of transient processes (5) (6) (7)

13. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 The results of the first experiments on the plasma shape control in Gutta tokamak There were developed different versions of the real- time control system for horizontal plasma position with the use of feedback The system is constructed on the bases of the power transistor switch, PC, special software and hardware, elements of electromagnetic diagnostics

14. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 The diagnostic error signal that characterizes the horizontal shift of plasma column Special PC software and hardware Signals of executing system Control System Scheme Control poloidal field coils Diagnostic coils

15. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 The software of real-time plasma shape control system Software graphic interface. Discharge information

16. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Some software parameters program measures the error signal each 2.5 microseconds program forms a control command for the switch each 5 microseconds yellow points - control discrete moments

17. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 The results of the first experiments on the plasma shape control in Gutta tokamak Fig. a. Testing experiments without controls. Horizontal axis – time in microseconds, Vertical axis – vertical magnetic flux in conventional units. Fig. b. Testing experiments with controls. Horizontal axis – time in microseconds, Vertical axis – vertical magnetic flux in conventional units.

18. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 LINEAR CONTROL MODELS FOR GUTTA TOKAMAK

19. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 DESCRIPTION OF THE SOFTWARE PACKAGE

20. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Program Workflow Build Geometrical Model Calculate matrices of inductivities and resistances of the circuits Compute Equilibrium DatabaseConstruct Linear Model

21. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Program Workflow Build Geometrical Model Calculate matrices of inductivities and resistances of the circuits Compute Equilibrium DatabaseConstruct Linear Model

22. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Construction of Geometrical Model of Tokamak Poloidal section of the ITER. The passive structure of the tokamak has to be divided into several circuits, whose induced currents together with the current in control windings are the states in the linear model. The section of each circuit that is included in the linear model can be geometrically presented by one (active circuit) or several (walls of vacuum chamber) rectangles. The division of circuits that refer to the walls of the vacuum chamber into several rectangles allows to approximate the geometry of chamber walls more precisely. The programs for calculation of geometry make the first part of the software package that is discussed here.

23. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Program Workflow Build Geometrical Model Calculate matrices of inductivities and resistances of the circuits Compute Equilibrium DatabaseConstruct Linear Model

24. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Calculation of Matrices of Inductivities and Resistances of the Circuits Для расчета равновесных плазменных конфигураций с целью построения линейной модели необходимо знать электротехнические параметры проводящих системы полоидальных проводящих контуров токамака. Программа eltech вычисляет собственные и взаимные индуктивности обмоток, согласно формулам 1 и 2, при этом обмотки разбиваются на элементарные нити тока, как показана на рис. 123 и взаимная индуктивность нитей тока вычисляется по формуле (3) и сопротивления контуров согласно (4) (1) (2) (3) (4)

25. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Program Workflow Build Geometrical Model Calculate matrices of inductivities and resistances of the circuits Compute Equilibrium DatabaseConstruct Linear Model

26. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Equilibrium Database Computation Computation of database of plasma equilibriums can be broken into 3 stages: 1.Calculation of base equilibrium. 2.Computation of equilibriums for deviations of currents in active and passive circuits. 3.Computation of equilibriums for deviations of plasma parameters (plasma current, poloidal beta, plasma internal inductance).

27. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Equilibrium Database Computation On each stage direct problem of equilibrium has to be solved for each parameter deviation. –Let us assume that coil currents and physical parameters of plasma (currents, “beta poloidal”, etc.) are known. –It is necessary to restore the magnetic surface for plasma equilibrium.

28. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Grad-Shafranov equation: Boundary conditions: The position of plasma border is not given and is determined by the problem solution. Because of that the problem is always non-linear even in cases when the right part of the equation is linear. Equilibrium Database Computation

29. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Equilibrium Database Computation For solving direct problem of equilibrium the PET code is used. Huge number of equilibriums has to be computed. –Process of equilibriums computation is automated.

30. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Program Workflow Build Geometrical Model Calculate matrices of inductivities and resistances of the circuits Compute Equilibrium DatabaseConstruct Linear Model

31. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Linear Model Construction Linear model has the following form:

32. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 The A, B, C, D representation of control object equation: where,, D – zero matrix,, Linear Model Construction

33. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Computation of matrix elements is given by the following formulas: Linear Model Construction

34. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Similar computations are performed for all other parameters of the model. Initialization of input parameters (matrices of inductivities and resistances of the circuits, equilibrium database) For each circuit: Computation of current and magnet flux variations with respect to base equilibrium values. Program finds 2 variations of magnet flux in the database of equilibriums corresponding to variations of current in j-th contour. Program computes average of relations between flux variation and variation of current in j-th contour. Linear Model Construction Workflow

35. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 CONSTRUCTION AND COMPARISON OF LINEAR MODELS OF VARIOUS ORDERS FOR ITER

36. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Linear Models for ITER Models with various degree of division of passive elements of the tokamak were computed (for divisions into 131 and 71 circuits. For base equilibrium one of the standard ITER plasma equilibriums was selected. Positive eigenvalues of A-matrices were computed for each model as well as the corresponding eigenvectors.

37. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 POLOIDAL SECTION OF THE ITER

38. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Linear Models for ITER: Results Difference between positive eigenvalues of A-matrices for different models is less than 1%. Double-ply increase of passive contours does not result in essential increase of model parameters accuracy, so for practical computations it is enough to use the model with 71 contours.

39. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 CONSTRUCTION AND COMPARISON OF LINEAR MODELS OF VARIOUS ORDERS FOR GUTTA

40. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Gutta Tokamak Main parameters: major radius – R 16 cm, minor radius – a 8 cm, aspect ratio – A 2, vessel elongation – k 2, plasma current < 150kA, toroidal field - 1.5 T

41. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 POLOIDAL SECTION OF THE GUTTA

42. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Linear Models for Gutta For prescribed plasma parameters inverse equilibrium problem was solved using DIALEQT-C code so the currents in the control coils corresponding to reference equilibrium were computed. Models of order 103, 93, 83, 73, 63, 53, 43, 33, 23, 21, 18, 16 and 13 that were built using considered software package were compared.

43. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Linear Models for Gutta: Results Dependency of maximum singular value on frequency

44. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Linear Models for Gutta: Results As a result the model of order 33 was selected as the base model, which provides compromise between order of the model and closeness of the singular characteristic of the model to singular characteristics of the higher-order models.

45. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Linear Models for Gutta: Results Deviations of singular characteristics for different models from singular characteristic of the base model in the working frequency range

46. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Linear Models for Gutta: Results Results of the simulation in the MATLAB- Simulink framework also show consistency of considered models. –In the simulation the perturbations of types and drops were used. - vector of perturbations (components that correspond to and are constant non-zero values )

47. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Linear Models for Gutta: Results Reaction of different models on the perturbations Deviations of singular characteristics for different models from singular characteristic of the base model in the working frequency range

48. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Linear Models for Gutta: Results –Amplitude-frequency responses from two inputs of the system to the output that corresponds to the plasma current were analyzed. –Consistent behaviour between models of different order is also achived.

49. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Linear Models for Gutta: Results Amplitude-frequency responses from input #1 of the system to the output that corresponds to the plasma current. Deviations of singular characteristics for different models from singular characteristic of the base model in the working frequency range

50. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Linear Models for Gutta: Results Amplitude-frequency responses from input #2 of the system to the output that corresponds to the plasma current. Deviations of singular characteristics for different models from singular characteristic of the base model in the working frequency range

51. The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008 Thank You