V.I. Vasiliev, Yu.A. Kostsov, K.M. Lobanov, L.P. Makarova, A.B. Mineev, D.V.Efremov Scientific Research Institute of Electrophysical Apparatus, St.-Petersburg,

V.I. Vasiliev, Yu.A. Kostsov, K.M. Lobanov, L.P. Makarova, A.B. Mineev,

D.V.Efremov Scientific Research Institute of Electrophysical Apparatus, St.-Petersburg, Russia,

V.K. Gusev, R.G. Levin, Yu.V. Petrov, N.V. Sakharov

A.F.Ioffe Physico-Technical Institute, St.-Petersburg, Russia

Plasma Shape Reconstruction on-line Algorithm in Tokamaks

JOINT MEETING OF THE 3rd IAEA TECHNICAL MEETING ON SPHERICAL TORI AND THE 11th INTERNATIONAL WORKSHOP ON SPHERICAL TORUS

St. Petersburg State University, St. Petersburg, RUSSIA 3 to 6 October 2005

Plasma Shape Reconstruction Algorithm in Tokamaks

Discussed here plasma shape reconstruction algorithm is based on using of current filaments to imitate plasma column and includes two steps.

At the first step location of the plasma current centroid is estimated roughly, using so-called plasma current density moments technique for two current filaments [1].

As results two pairs of the current filaments coordinates (r1,z1) and (r2,z2) can be calculated and they determine approximately plasma centroid location.

At the second step well known the fixed current filaments technique [2] is used to calculate plasma shape boundary.

To allocate current filaments inside the vacuum vessel (in vicinity of plasma current centroid location) two points with coordinates (r1,z1) and (r2,z2), computed at the first step, are used.

Globus-M cross-section

flux loops (21)+ two component probes (32)

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First step algorithm

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 [1]

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

In the case when plasma current distribution is approximated by 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.

dl)BrgfB()z,r(fIYl

mmnkkm

2

1kk0m

0r

f

r

1

r

f

z

ff

2

2

2

2*

0r

g

r

1

r

g

z

gg

2

2

2

2

First step algorithm (continue)

Here measured value of normal component of field induction in

the point of probe location with number “k”,

measured value of tangential component of field induction in the point of probe location with number “k”,

unknown current value in the jth current filament.

Formulas described fm and gm functions are presented below as following

f0 = 1; g0 = 0;

f1 = z; g1 = -ln(r);

f2 = r2; g2 = 2z;

f3 = r2z; g3 = -(1/2)r2 + z2;

f4 = -(1/4)r4 + r2z2; g4 = -r2z + 2/3 z3.

knB

kB

jI

First step algorithm (continue)

So, to compute coordinates of two current filaments it is necessary to 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.

Points (r1,z1) and (r2,z2) serve as the reference points to set plasma current filaments in vicinity of plasma current centroid to be used at second step.

Second step algorithm

To reconstruct plasma shape boundary well known the fixed current filament technique [2] is used. In according with selected algorithm plasma is approximated by M current filaments with given coordinates (rj, zj) located inside 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.

Unknown filament currents can be calculated by minimizing functional as follows

p l

21

N

1k

N

1m

M

1j

2J

2

mm

2kk2k

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Second step algorithm (continue)

computed value of normal component of field induction in the point of probe location with number “k”,

computed value of tangential component of field induction in the point of probe location with number “k”,

measured value of magnetic flux in the point of flux loop location with number “m”,

computed value of magnetic flux in the point of flux loop location with number “m”,

unknown current value in the jth filament,

adjustable parameter,

Here knB

kB

m

m

jI

c1, c2 relative weighs

Second step algorithm (continue)

, and parameters are computed with using theoretical models of poloidal field coils and vacuum vessel and experimentally measured coil currents and loops voltages.

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 located on the vacuum vessel shell. Interpolating experimental measured data loop voltage and current value can be computed for each finite element of vacuum vessel.

Minimizing residual functional filament currents can be calculated and then plasma column boundary can be reconstructed.

knB kB m

Application of plasma shape reconstruction technique in

Globus-M data handling

Globus-M magnetic diagnostic complex consists of:

32 two-components magnetic probes located on the vacuum vessel surface

21 full-scale flux loops located on the vacuum vessel surface

Rogovsky coils to measure PF coils currents

Rogovsky coils to measure plasma current and sum Ip+Ivv

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flux loops (21)+ two component probes (32) Globus-M cross-section

)B,B( n

Application of plasma shape reconstruction technique

in Globus-M data handling (Continue)

Preliminary adjustment of plasma shape reconstruction procedure in numerical experiments with theoretical models.

Plasma evolution discharge scenario is simulated with using dynamic PET code with deformable plasma shape model.

Normal and tangential components of magnetic field induction ( ) relative to counter line with points of probes locations are numerically calculated.

Loop voltages that are used to estimate eddy current distribution in the vacuum vessel shell are calculated.

Test results of plasma shape reconstruction with using theoretical Globus-M model in the next picture.

kkn B,B

Application of plasma shape reconstruction technique

in Globus-M data handling (Continue)

Plasma equilibrium limiter configurationin Globus-M device;t=0.025 s, max difference = 7mm, Ip = 89.95 кА

Plasma equilibrium diverter configurationin Globus-M device;t=0.050 s, , max difference = 17mm, Ip = 165.4 кА

In total Using of 6 plasma current filaments gives acceptable results.

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OH OH

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Application of plasma shape reconstruction technique

in Globus-M data handling (Continue)

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. To have better-fit results plasma column is simulated here with 9 currents filaments.

Reconstructed plasma shape boundary at limiter plasma discharge stage.1. case with c1=1, c2=0;2. case with c1=0, c2=1.

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Reconstructed plasma shape boundary at divertor plasma discharge stage.1. case with c1=1, c2=0;2. case with c1=0, c2=1.

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Conclusion

Developed here plasma shape reconstructed algorithm can be useful in time between shots to analyse output data of plasma discharges.

Proposed algorithm can be used as on-line algorithm in a feedback plasma shape control.

References

[1] Yasin I.V. PhD. Thesis, Kharkow, 1999.

[2] 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.