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 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
Poloidal crossection
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
Dynamic model of a poloidal circuit system
 dI (t )
1
1

L
U
(
t
)

L
R
(
t
)
I
(
t
)
 dt

 dU (t )  C 1 I (t )
 dt
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.
 dx(t , x, y, u )
 f1 (t , y )  f 2 (t , x, u )

dt

 dy (t , x)  f (t , x)
3
 dx
The coefficients of system (2) remain continuous on half-intervals [t k , t k 1 ), k  0,..., p  1
I (tk , yk , u )  g1 ( x(t m ))  g 2 ( x(t p ), y (t p ))
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
x  Ax  Bu
(1)
y  Cx,
• The control object is completed with a regulator of a
decreased dimension with the following structure
z  Ac z  Bc y (2)
u  Cc z,
• The control object closure by the
gained regulator
Control object
y
u
Shape 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
The structural parametric optimization of transient
processes
• Let us investigate the control object with presence of
a constantly applied disturbance
 x   A
 z    B C
   c
yCx
uKz
BC c   x 
 M f (t )



Ac   z 
 A BCc 
P( p )  

B
C
A
c 
 c
(3)
p  { pi }  { Ac , Bc , Cc }
• f(t) is a disturbance vector that satisfies following
equation at the moment
t
(disturbance
ensemble)
t
*
(4)
x0 G1 x0   f * ( ) G2 ( ) f ( ) d 1.
t0
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
• let us introduce a performance functional
r
N

2
2
I ( p)    sup ( yi (t ))   sup ui (t ) dt
i 1 ( x0 , f f )St

t0  i 1 ( x0 , f f )St
t
(5)
• the gradient of the functional by parameters p  { pi }
T

I
P * 
 dt .
  2 tr D
pi
pi 

0
(6)
• where
(7)
1
D  P( p ) D  DP * ( p )  MG2 M *
D(t 0 )  G11 .
  P  P*  (C *C  K * K ), (T )  0

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
VD3
R1
SA1
C1
VD4
L1
+
• There were developed
different versions of the realtime 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
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
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
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
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
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
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
Calculation of Matrices of Inductivities and
Resistances of the Circuits
L
1
ds  Mds  (1)
s 2 s s
M
1
ds Mds(2)
s1 s2 s1 s2
M
0
4
 cos 
l  l 
dl dl 
(3)
r
R   0  2 
1
r ds (4)
S 2 S
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) on Spherical Tori and the 14th International Workshop on Spherical
The 4th IAEA
Technical Meeting
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 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
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
Equilibrium Database Computation
• Grad-Shafranov equation:
 8 2

rj inside S p


   c 2 L
 8  rI i r  ri  z  z i  outside S p ,
 c i1
• Boundary conditions:
  0 when r or z   and when r  0
• 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
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
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
Linear Model Construction
Linear model has the following form:
d
 d
(I )  M 
( )  RI  U
L
dt
 dt


 g  CI  F
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
• The A, B, C, D representation of control object
equation:
 x  Ax  BU  AE

 g  Cx  DU  ( CE  F )
where A  ( L ) 1 R , B  ( L ) 1 , D – zero matrix,
E  ( L ) 1 M  , x  I  E
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
• Computation of matrix elements is given
by the following formulas:
L3  L1  L2

 g i


 I j

 g i
I I0  

 I j

 


 I j

 
I I0  

 I j
ext
L1 
I
M pl 

 


 I pl
pl
I pl
I I0
I I0
L2 
 pl
I
I  I0
I I0



 I pl
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 Workflow
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.
Similar computations are performed for all other parameters of the 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
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
Singular Plot
0
10
-1
Maximum Singular Value
10
103
93
83
73
63
53
43
33
23
21
18
16
13
-2
10
-3
10
-4
10
-5
10
-6
10
0
10
1
10
2
10
3
4
10
10
Frequency (rad/s)
5
10
6
10
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
Maximum Absolute Deviation
0.08
0.07
Absolute Deviation
0.06
0.05
0.04
0.03
0.02
0.01
0
103
93
83
73
63
53
43 33
Model Order
23
21
18
16
13
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
Linear Models for Gutta: Results
• Results of the simulation in the MATLABSimulink framework also show consistency of
considered models.
– In the simulation the perturbations of types and l i
dropswere used.
 x  Ax  BU  AE

 g  Cx  DU  ( CE  F )
 - vector of perturbations (components that
li
 constant non-zero
correspond to and
are
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
Deviations of singular characteristics for different models from singular characteristic of the base model in the working frequency range
Linear Models for Gutta: Results
-5
8.5
x 10
8
Plasma Current (mA)
7.5
13
16
18
21
23
33
43
53
63
73
83
93
103
7
6.5
6
5.5
5
0
0.005
0.01
0.015
Time (s)
0.02
0.025
0.03
Reaction of different models on the perturbations
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
Deviations of singular characteristics for different models from singular characteristic of the base model in the working frequency range
Linear Models for Gutta: Results
Bode Diagram
From: In(1)
-60
-80
To: Out(13)
Magnitude (dB)
-100
-120
-140
-160
-180
-200
0
10
1
10
2
10
3
10
4
10
5
10
6
10
Frequency (rad/sec)
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
Deviations of singular characteristics for different models from singular characteristic of the base model in the working frequency range
Linear Models for Gutta: Results
Bode Diagram
From: In(2)
-60
-70
13
33
-90
To: Out(13)
Magnitude (dB)
-80
-100
103
16
-110
-120
-130
-140
0
10
1
10
2
10
3
10
4
10
5
10
6
10
Frequency (rad/sec)
Amplitude-frequency responses from input #2 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
Thank You
The 4th IAEA Technical Meeting on Spherical Tori and the 14th International Workshop on Spherical
Torus, ENEA, Frascati, Roma, Italy, October, 7-10, 2008