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

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

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

4. Poloidal crossection Poloidal field coils Vacuum vessel
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5. 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
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

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

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

8. General form of the control problem.
The coefficients of system (2) remain continuous on half-intervals
tm corresponds electron-cyclotron (ECR) pre-ionization time,
tp breakdown time
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9. 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.
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

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

11. The structural parametric optimization of transient processes
Let us investigate the control object with presence of a constantly applied disturbance
(3)
f(t) is a disturbance vector that satisfies following equation at the moment t (disturbance ensemble)
(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

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

13. 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
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

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

15. The software of real-time plasma shape control system
Software graphic interface. Discharge information
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16. 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
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

17. 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.
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

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

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

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

22. Construction of Geometrical Model of Tokamak
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.
Poloidal section of the ITER.
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

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

24. Calculation of Matrices of Inductivities and Resistances of the Circuits
(1)
(2)
(3)
(4)
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)
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

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

26. Equilibrium Database Computation
Computation of database of plasma equilibriums can be broken into 3 stages:
Calculation of base equilibrium.
Computation of equilibriums for deviations of currents in active and passive circuits.
Computation of equilibriums for deviations of plasma parameters (plasma current, poloidal beta, plasma internal inductance).
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27. 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.
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

28. Equilibrium Database Computation
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.
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

29. 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.
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

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

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

33. Linear Model Construction
Computation of matrix elements is given by the following formulas:
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34. Linear Model Construction Workflow
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.
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35. CONSTRUCTION AND COMPARISON OF LINEAR MODELS OF VARIOUS ORDERS FOR ITER
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

36. 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.
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37. POLOIDAL SECTION OF THE ITER
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38. 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.
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

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

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

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

42. 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.
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

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

44. 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.
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

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

46. 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 )
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

47. Linear Models for Gutta: Results
Deviations of singular characteristics for different models from singular characteristic of the base model in the working frequency range
Linear Models for Gutta: Results
Reaction of different models on the perturbations
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48. 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.
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008

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

50. Linear Models for Gutta: Results
Deviations of singular characteristics for different models from singular characteristic of the base model in the working frequency range
Linear Models for Gutta: Results
Amplitude-frequency responses from input #2 of the system to the output that corresponds to the plasma current.
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51. Thank You
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