the 4th iaea technical meeting on spherical tori and the 14th international workshop on spherical...
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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.
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
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 fieldcoils 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
Poloidal crossection
Poloidal fieldcoils
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
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
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
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.
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.
),(),(
),,(),(),,,(
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
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.
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
,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)
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
zKu
xCy
tfMz
x
ACB
BCA
z
x
cc
c
)(
t
t
dfGfxGx0
.1)()()( 2*
01*
0
The structural parametric optimization of transient
processes
cc
c
ACB
BCApP )(
(3)
(4)
},,{}{ ccci CBApp
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
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
The structural parametric optimization of transient
processes
(5)
(6)(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
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 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 SchemeControl poloidal field coils
Diagnostic coils
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
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
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.
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
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
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
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
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.
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
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)
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)
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
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).
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.
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
,outside8
inside8
1
2
2
p
L
iiii
p
SzzrrrIc
Srjc
00 rwhenandzorrwhen
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.
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
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
FICg
UIRdt
dMI
dt
dL
)()(
Linear model has the following form:
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:
)FCE(UDCxg
AEUBAxx
where , , D – zero matrix, ,
RLA 1)( 1)( LB
MLE 1)( EIx
Linear Model Construction
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:
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
Linear Model Construction
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
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
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.
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
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.
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
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
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
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.
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
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
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.
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
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
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 )
il
)FCE(UDCxg
AEUBAxx
il
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
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
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.
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
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)
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
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
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