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 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.


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


Tokamak poloidal circuits.

Poloidal fieldcoils Vacuum vessel


Poloidal crossection

Poloidal fieldcoils

Vacuum vessel


Dynamic model of a poloidal circuit system

)()(

)()()()(

1

11

tICdt

tdU

tItRLtULdt

tdI

Where I-vector of currents, U-vector of voltages, L-inductance matrix, R-resistance matrix, C-capacitance matrix


Poloidal currents dynamics


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.


General form of the control problem.

),(),(

),,(),(),,,(

3

21

xtfdx

xtdy

uxtfytfdt

uyxtdx

))(),(())((),,( 21 ppm tytxgtxguyktkI

The coefficients of system (2) remain continuous on half-intervals

1,...,0),,[ 1 pktt kk

tm corresponds electron-cyclotron (ECR) pre-ionization time, tp breakdown time


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 structural parametric optimization of transient

processes • The equations of the control object in the state space are the

following

,Cxy

BuAxx

• The control object closure by the gained regulator

,zCu

yBzAz

c

cc

y

u

Control object

Shape regulator

• The control object is completed with a regulator of a decreased dimension with the following structure

(1)

(2)


• 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

zKu

xCy

tfMz

x

ACB

BCA

z

x

cc

c

)(

t

t

dfGfxGx0

.1)()()( 2*

01*

0


processes

cc

c

ACB

BCApP )(

(3)

(4)

},,{}{ ccci CBApp


• where

• let us introduce a performance functional

dttutypIt

t

r

ii

Sfx

N

ii

Sfx tftf

000 1

2

),(1

2

),()(sup))((sup)(

.20

*

dtp

PDtr

p

I T

ii

}{ ipp • the gradient of the functional by parameters

.)(

)()(1

10

*12

*

GtD

MMGpDPDpPD

0)(),( *** TKKCCPP


processes

(5)

(6)(7)


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

L1+

C1

SA1

VD3

VD4

R1


The diagnostic error signal that

characterizes the horizontal shift of plasma column

Special PC software

and hardware

Signals of executing system

Control System SchemeControl poloidal field coils

Diagnostic coils


The software of real-time plasma shape control system

Software graphic interface. Discharge information


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 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.


LINEAR CONTROL MODELS FOR GUTTA TOKAMAK


DESCRIPTION OF THE SOFTWARE PACKAGE


Program Workflow


Program Workflow


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.


Program Workflow


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)

s s

sdMsds

L2

1(1)

1 221

1

s s

sdMsdss

M (2)

l lr

ldldcosM

4

0 (3) S

dsrS

R20

12 (4)


Program Workflow


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).



• 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.


• 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.


,outside8

inside8

1

2

2

p

L

iiii

p

SzzrrrIc

Srjc

00 rwhenandzorrwhen



• 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.


Program Workflow


Linear Model Construction

FICg

UIRdt

dMI

dt

dL

)()(

Linear model has the following form:


• The A, B, C, D representation of control object equation:

)FCE(UDCxg

AEUBAxx

where , , D – zero matrix, ,

RLA 1)( 1)( LB

MLE 1)( EIx



• Computation of matrix elements is given by the following formulas:

213 LLL 02 II

pl

IL

01 IIext

IL

0IIpl

plpl I

M

jII

j II

0

plII

pl II

0

j

iII

j

i

I

g

I

g

0



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


CONSTRUCTION AND COMPARISON OF LINEAR

MODELS OF VARIOUS ORDERS FOR ITER


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.


POLOIDAL SECTION OF THE ITER


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.


CONSTRUCTION AND COMPARISON OF LINEAR

MODELS OF VARIOUS ORDERS FOR GUTTA


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


POLOIDAL SECTION OF THE GUTTA


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.


Linear Models for Gutta: Results

100

101

102

103

104

105

106

10-6

10-5

10-4

10-3

10-2

10-1

100

Frequency (rad/s)

Max

imum

Sin

gula

r V

alue

Singular Plot

10393

83

73

6353

43

3323

21

18

1613

Dependency of maximum singular value on frequency



• 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.



Deviations of singular characteristics for different models from singular characteristic of the base model in the working frequency range

103 93 83 73 63 53 43 33 23 21 18 16 130

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Model Order

Abs

olut

e D

evia

tion

Maximum Absolute Deviation



• 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 )

il

)FCE(UDCxg

AEUBAxx

il



Reaction of different models on the perturbations


0 0.005 0.01 0.015 0.02 0.025 0.035

5.5

6

6.5

7

7.5

8

8.5x 10

-5

Time (s)

Pla

sma

Cur

rent

(m

A) 13

16

18

21

2333

43

5363

73

83

93103



– 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.



Amplitude-frequency responses from input #1 of the system to the output that corresponds to the plasma current.


Bode Diagram

Frequency (rad/sec)

Mag

nitu

de (

dB)

100

101

102

103

104

105

106

-200

-180

-160

-140

-120

-100

-80

-60From: In(1)

To:

Out

(13)



Amplitude-frequency responses from input #2 of the system to the output that corresponds to the plasma current.


Bode Diagram

Frequency (rad/sec)

Mag

nitu

de (

dB)

100

101

102

103

104

105

106

-140

-130

-120

-110

-100

-90

-80

-70

-60From: In(2)

To:

Out

(13)

13

16103

33


Thank You

Documents

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