Download Plasma shape reconstruction on-line algorithm in tokamaks

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Plasma Shape Reconstruction on-line Algorithm in Tokamaks
V.I. Vasiliev 1), Yu.A. Kostsov 1), K.M. Lobanov 1), L.P. Makarova 1), A.B. Mineev 1),
V.K. Gusev 2), R.G. Levin 2), Yu.V. Petrov 2), N.V. Sakharov 2)
1) D.V. Efremov Scientific Research Institute of Electrophysical Apparatus, St-Petersburg,
189631, Russia.
2) Ioffe Physico-Technical Institute, St-Petersburg, 189631, Russia.
e-mail contact of main author: [email protected]
Abstract. Plasma shape reconstruction on-line algorithm is necessary to design plasma position and shape
control system in tokamaks. Algorithm aimed on solving this problem is discussed here. An example of
applying of this algorithm is demonstrated with using experimental data of Globus-M discharge #10292. The
results of this analysis are reported.
1. Introduction
The main aim of magnetic diagnostic in tokamak devices is reconstruction of plasma
boundary both for limiter and diverter plasmas. In the present article a numerical algorithm,
which permits to calculate plasma column boundary in Globus-M tokamak, is described. The
magnetic diagnostic system of Globus-M tokamak consists of 32 magnetic two-component
probes located along vacuum vessel cross section boundary, which can measure normal and
tangential components of magnetic induction, and 21 magnetic flux loops located both inside
and outside of the vacuum vessel. The magnetic flux loops are used to measure loop voltage
distribution close to vacuum vessel surface and these experimental data permit then to
evaluate eddy current distribution in the vacuum vessel shell. Algorithm allowing to
reconstruct plasma shape boundary with using mentioned above experimental data will be
described below.
Fig.1. Globus-M cross-section
 flux loops
+ two component probes
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2. Problem formulation
Cross-section of Globus-M machine is shown in Fig.1. Distribution of the magnetic probes
and flux loops positions are presented here. Magnetic probes are marked with bold plus sign
and flux loops are marked with bold dot sign respectively. Magnetic two component probes
k
k
measure normal and tangential components of magnetic field induction ( B n , B  ) in points of
probes locations. Flux loops measure loop voltages that are used to estimate eddy current
distribution in the vacuum vessel shell. These computed eddy currents are used by plasma
reconstruction procedure. To reconstruct plasma shape boundary in tokamak Globus-M well
known fixed filament current technique (for example, [1]) is used. In according with selected
algorithm plasma is approximated by M current filaments with given coordinates (Rj, Zj)
located inside plasma region. Unknown filament currents can be calculated by minimizing
functional  as follows
Np

  B
  c1
k 1
k
n
Nl
M
2
2
2
k
k

 B n    B k  B     c2  m   m    I 2J .
 
 

m 1 
j1


Here Bkn computed value of normal component of field induction in the point of probe
location with number “k”,
k
B  computed value of tangential component of field induction in the point of probe
location with number “k”,
k
Bn measured value of normal component of field induction in the point of probe location
with number “k”,
k
B measured value of normal component of field induction in the point of probe location
with number “k”,
I j unknown current value in the jth filament,
 adjustable parameter,
c1, c2 relative weighs
Bkn and B k magnetic field components are computed with using theoretical models of
poloidal field coils and vacuum vessel. Poloidal field coil currents are measured during
plasma discharge with Rogovsky coils and they are therefore known. Vacuum vessel current
distribution is calculated with using loop voltages measured by flux loops on the vacuum
vessel shell. Interpolating experimental measured data loop voltage can be computed for each
finite element of vacuum vessel and after then current value conducting in that one. So, Bkn
and B k components can be computed with using available experimental data.
Minimizing residual functional  filament currents can be calculated and then plasma
column boundary can be reconstructed. At the limiter discharge stage plasma column
boundary is closed and has even if one common point with limiter. At the diverter discharge
stage plasma column boundary is coincided with magnetic flux separatrix that has X-point.
It can be noted that described here procedure of plasma shape reconstruction requires
coordinates of current filaments as input data. While plasma column position is uncertain
during operational shot an algorithm is used to estimate roughly plasma column position and
then current filaments coordinates can be initialed in the plasma current centroid region. With
this aim the same plasma shape reconstruction problem is solved using so-called plasma
current density moment technique for two current filaments case.
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If to define as “measuring contour“ a curve l, which passes through magnetic probes
coordinate points, then plasma current density moments can be written as following [2]
2
Ym= 0  Ik f m (rk , zk )   (Bn f m  g m r B )dl
k 1
l
Here Ym is plasma current density moment of mth order. fm and gm are functions that
determine order of plasma current density moment and they satisfy to equations as follows
2
2
2
2
f
f 1 f
g
g 1 g
*f   2   2 
 0,
g   2   2 
 0.
z
r
r r
z
r
r r
In the case when plasma current distribution is approximated with two filament currents it is
required fm and gm only up to 4th order to compute coordinates of two current filaments under
condition that filament currents are equal each other. Equations described fm and gm functions
are presented below as following
f0 = 1;
g0 = 0;
f1 = z;
g1 = -ln(r);
f2 = r2;
g2 = 2z;
2
f3 = r z;
g3 = -(1/2)r2 + z2;
4
2 2
f4 = -(1/4)r + r z ;
g4 = -r2z + 2/3 z3.
So, to compute coordinates of two current filaments it is necessary calculate plasma current
moments with using experimental data and with using current filaments approach and equate
each other. Solving algebraic equation system with four unknown (r1,z1) and (r2,z2), current
filaments coordinates can be found.
Found thus points (r1,z1) and (r2,z2) serve reference points to set plasma current filaments in
vicinity of plasma current centroid region. In the described here procedure current filaments
are uniformly placed along ellipse with focal points (r1,z1) and (r2,z2). The value of ellipse
minor semi-axis is adjusted to have more acceptable results. Preliminary adjustment of this
procedure is carried out in numerical experiments with theoretical models.
3. Test results of plasma shape reconstruction with using theoretical Globus-M model
To verify developed numerical code based on described above algorithm, theoretical data of
computed dynamic plasma discharge scenario were used. These data were calculated with
using dynamic PET code, which can simulate plasma discharge evolution with taking into
account plasma shape deformation. Bkn and B k magnetic induction components in 32
uniformly distributed points along “measuring contour” closed to vacuum vessel boundary
were computed. These data serve as input data for plasma shape reconstruction procedure.
Besides loop voltage values in the same points were computed too. These data were used to
estimate eddy currents conducting in the vacuum vessel shell, which are required in plasma
shape reconstruction process. Below results of numerical code verification are shown in Figs.
2 and 3 as examples. In Fig.2 computed and reconstructed plasma shape boundaries are
presented. Maximal difference between them is estimated of 7 mm. This plasma equilibrium
state corresponds to plasma discharge limiter stage. The same results are presented in Fig.3
but in the case of diverter configuration, which corresponds to diverter discharge stage.
Maximal difference between theoretical and reconstructed separatrices is estimated as 17 mm.
Through present analysis loop voltage along vacuum vessel boundary is approximated with
spline which is constructed with using computed signals. Comparative analysis shows that
obtained results of plasma shape reconstruction are acceptable and developed plasma shape
reconstruction procedure can be used in practice. The model with 6 fixed plasma current
filaments permits to obtain acceptable results.
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Fig.2 Plasma eqilibrium limiter configuration in Globus-M device.
t=0.025 s, 32 Bn.B - magnetic probes, max difference = 7mm, Ip = 89.95 kА.
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Fig.3 Plasma eqilibrium diverter configuration in Globus-M device.
t=0.050 s, 32 Bn.B - magnetic probes, max difference = 17mm, Ip = 216.54 кА.
4. Results of plasma shape reconstruction in Globus-M experiments.
To demonstrate possibilities of the developed numerical plasma shape reconstruction code
shot #10292 was taken as an example. In Figs. 4 and 5 plasma shape reconstruction results are
presented. Fig. 4 corresponds to the limiter equilibrium configuration in time moment
132.257 ms and plasma current about of 168.5 kA. Diverter plasma shape configuration is
shown in Fig. 5, time moment 153.758 ms and plasma current 152.3 kA. To have better-fit
results plasma column is simulated here with 9 currents filaments.
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Fig. 4. Reconstructed plasma shape boundary at limiter plasma discharge stage.
1
2
Fig. 5. Reconstructed plasma shape boundary at diverter plasma discharge stage.
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Conclusion
 Developed here plasma shape reconstructed algorithm can be useful in time
between shots to analysis output data of plasma discharges.
 Proposed algorithm can be used as on-line algorithm in a feedback plasma shape
control.
References
[1] Ogata A., Aicawa H., Suzuki Y., “Accuracy of plasma displacement measurements in a
tokamak using magnetic probes,” Jap. J. Appl. Phys., no. 16,1, pp. 185-188, 1977.
[2] Yasin I.V. PhD. Thesis, Kharkow, 1989.