- Email: [email protected]
Assessment of ITER PF coil quality from magnetic measurements
Fusion Engineering and Design 85 (2010) 718–723
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Assessment of ITER PF coil quality from magnetic measurements V.M. Amoskov a,∗ , A.V. Belov a , V.A. Belyakov a , V.A. Chuyanov b , S.A. Egorov a , A.A. Firsov a , Yu.V. Gribov b , V.G. Ivkin a , V.P. Kukhtin a , E.A. Lamzin a , A.A. Lancetov a , M.S. Larionov a , N.A. Maximenkova a , I.Yu. Rodin a , S.E. Sytchevsky a a b
D.V. Efremov Scientific Research Institute of Electrophysical Apparatus, St. Petersburg, Russia ITER Organization, CS 90 046, Building 523, 13067 Saint Paul-lez-Durance Cedex, France
a r t i c l e
i n f o
Article history: Received 3 July 2009 Received in revised form 11 April 2010 Accepted 12 April 2010 Available online 5 May 2010
a b s t r a c t A feasibility has been demonstrated for numerical reconstruction on the base of magnetic measurements for geometrical displacements or deformations occurred in the manufacture and assembly of magnet coils. For validation of the proposed approach the test results of reconstruction of possible misalignments and deviations of the ITER PF1 coil are presented. © 2010 Elsevier B.V. All rights reserved.
Keywords: ITER PF coil quality Manufacture and assembly tolerances Error field Magnetic measurements Numerical reconstruction
1. Introduction Small perturbations of the axial symmetry of a tokamak magnetic field are usually referred to as “error fields” [1]. Main sources of the field asymmetry in tokamaks are possible misalignments and deviations occurred in manufacturing and assembly of the coils for the magnet system [2]. For the ITER machine a severe limitation is adopted on the allowable level of field non-axisymmetry. The error fields above this limit can cause plasma instability resulting in plasma disruption [2]. The error fields are, however, about threefold higher with regard to the accepted manufacture tolerances [3]. To suppress the ITER error fields below the threshold level, a set of correction coils is utilized [2,3]. The aim of the study was to demonstrate a feasibility of numerical reconstruction of coil positioning displacements and deformations occurred in manufacturing and assembly of the coils on the base of magnetic measurements. The ITER magnet system includes toroidal field (TF) and poloidal field (PF) coils and sections of the central solenoid (CS) as shown in Fig. 1. Only the PF1 coil has been considered in this study as a subject of RF responsibility.
∗ Corresponding author. Tel.: +7 812 4627782. E-mail address: [email protected] (V.M. Amoskov). 0920-3796/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fusengdes.2010.04.027
Fig. 2 illustrates the PF1 coil designed as a superconducting solenoid 9 m in diameter and 1 m in height, the winding being made of NbTi conductor. Table 1 summarizes main geometrical parameters of the PF1 coil in the cylindrical coordinate system related to the ITER centre. Standard displacements of the PF1 coil related to the accepted manufacture/assembly tolerances [4] include: 1. Linear shift along the X and Y axes, ıx = ±2 mm, ıy = ±2 mm (see Fig. 3). 2. Tilt about the horizontal plane, ız = ±1 mm; at low tilt angles ω = ωx , ωy the coil centreline is tilted as ız = ωR, where R is the coil radius (see Fig. 4). 3. In-plane ellipticity, x = ±3 mm, y = ±3 mm, that is a deformation of the coil shape in the plane XY defined in terms of two elliptic half-axes x = R + x, y = R + y, and an angle of rotation ˛ around Z-axis (−180◦ < ˛ < 180◦ ). Fig. 5 illustrates ellipticity at x > 0, y = 0, ˛ = 0. 4. Vertical displacement of the coil centreline, or so called warping, characterized by the amplitude z and angle ˛z, where z = ±1 mm corresponds to the maximum deviation z(ϕ) of the coil centreline, −180◦ < ˛z < 180◦ . The centreline shift z(ϕ) in the vertical direction along the toroidal angle ϕ is described by the sine-function: z = 0.5 × z × sin(ϕ + ˛z) (see Fig. 6). The shifts and tilts are resulted from geometrical displacements of the coil from its ideal position during assembly. The ellipticity
V.M. Amoskov et al. / Fusion Engineering and Design 85 (2010) 718–723
719
Fig. 1. ITER magnet system cross-section.
Each distortion of the coil shape and position is an independent random value. Let a magnet field generated by an undistorted coil be the ideal field. Then a field generated by a distorted coil is the distorted field, and the discrepancy between the ideal and distorted fields is the field distortion. In order to asses the resultant effect of various small geometrical distortions on the field, we can apply the superposition principle, assuming the coil distortions to be additive. Thus, the resultant
Fig. 2. PF1 coil layout.
and warping are associated with the coil distortion while manufactured. Note that the electromagnetic effect of coil positioning errors is always defined as an error in the coil current centreline. In intended magnetic measurements, all types of possible coil positioning errors should be taken into account.
Table 1 PF1 coil parameters. R, m
Z, m
R, m
Z, m
NPF1
3.943
7.557
0.968
0.976
256
R, Z: the central coordinates of the PF1 coil cross-section; R: the radial size; Z: the vertical size; NPF1 : the number of turns.
Fig. 3. Coil shift along the X-axis.
720
V.M. Amoskov et al. / Fusion Engineering and Design 85 (2010) 718–723
Diagnostics and evaluation of geometrical distortions may be obtained by solving the inverse magnetostatics problem. According to Hadamard’s classification [5], the problem is mathematically ill-conditioned and, therefore, its solution requires highly accurate magnetic measurements. Two points are essential for effective magnetic measurements:
Fig. 4. Coil tilt about the horizontal plane (rotation around the Y-axis).
• selection of types of sensors capable of field measurement to desired accuracy; • selection of observation points over typical regions, the number of observation points should be as low as possible to reduce measured data without loss in solution accuracy. The described approach is applicable to in-plant assessment of the coil quality during fabrication. Unwanted magnetic field generated by plant equipment and constructions may be detected by field mapping and then taken into account. With the use of specific field sensors, the same approach may be used for assessment of a coil field under cooldown conditions. A successful numerical reconstruction of coil distortions necessitates:
Fig. 5. Centrelines of a round coil and elliptic coil (viewed from above).
effect of all coil distortions can be defined to sufficient accuracy as a sum of their individual error fields. In the study, we took the PF1 coil parameters tabulated above as the ideal case, structure elements and current leads were ignored to keep axisymmetry for the undistorted coil. These components may be then included in a detailed calculation model. Assessment of the coil field quality as one of the main criteria for quality of manufacturing the entire coil or its components (in particular, pancakes) imply two steps: 1. room-temperature magnetic measurements over typical regions, using allowable coil current at room-temperature; 2. comparison of the measured data with a field map calculated for the ideal coil. A discrepancy in the results demonstrates quality of the coil manufacture.
• development of a numerical algorithm in order to provide evaluation of possible geometrical distortions of the coil from magnetic measurements with existing commercial field sensors in practice accurate to 10−3 –10−4 ; • development of a mathematical model of the coil to perform 3D field simulation in a form of maps for different field components. The model should allow a detailed field description, with respect to various coil components (current leads, buses, joints) capable to affect the field axisymmetry. The model should also take into account the effect of ferromagnetic surroundings. 2. Numerical experiment for evaluation of coil distortions The calculation model was developed for a cylindrical coil with the constant cross-section as described in Table 1. To perform field simulation, we used the code KLONDIKE [6] that enables analysis of precision current systems on the basis of solving magnetostatic problems in the integral formulation, using analytical solutions for a set of typical current system elements. The distorted field was determined as a superposition of individual error fields caused by geometrical coil distortions. 2.1. Calculation model of PF1 coil The calculation model of the coil is shown in Fig. 7. The coil is divided into 600 sections in the toroidal direction. The calculated results were obtained for the total coil current 156.25 A × 256 turns = 40 kA turns, taking into account the design limit of about (150–160) A/turn for a “warm” coil. 2.2. Observation points The field components were calculated at 400 points for each poloidal contour taken 0.1 m apart of a coil cross-section (see Fig. 8). The number of poloidal contours is 200. The total number of observation points was 200 × 400 = 80,000. 2.3. Computational method and results of numerical reconstruction
Fig. 6. Coil warping.
Simulation of a coil field includes two stages:
V.M. Amoskov et al. / Fusion Engineering and Design 85 (2010) 718–723
721
To assess the impact of random uncertainty of measurements , the sensors positions were assumed to be precisely specified on the measurement contour. The distorted field was added with random noise uniformly distributed within ±. Field components were treated as random values described in terms of the equiprobability distribution, dispersion 2 , and the distribution centroid coinciding with appropriate field component. For each field component 10 stochastic events were simulated at every observation point. In the analysis, the following coil distortions within the accepted tolerances (see Section 1) were taken:
Fig. 7. Calculation model of PF1 coil.
1 KLONDIKE computation of components of the distorted field in the global coordinate system; 2 computation of field maps for the normal and tangential components of the distorted field in the local coordinates x , y , z related to the observation points in order to simulate field measured with a 3-axis field sensor. To allow for uncertainty of measurements due to random errors, the mean-square error or standard deviation = 0.01–0.1% is introduced for calculated field components. Geometrical coil distortions leading to error fields can be determined using the regularization procedure [5] to minimize the following functional
2
2
M ˛ (Z, U) = (AZ − U) + ˛Z , where Z = {Zj } is the solution vector, A = {aij } is the i × j influence matrix with respect to the distortions, U = {ui } is the related error field vector, = {i }, = {j } are the vectors of weight functions, and ˛ is the regularization parameter. Distorted fields have been calculated at all observation points so as to imitate magnetic measurements. For numerical reconstruction of coil distortions we used only data calculated at the points located on a circle passing through point #50, see Fig. 8. Thus, we have limited our analysis to 200 observations.
Fig. 8. Observation points taken on a poloidal contour.
Fig. 9. Normal error field Bn distributed along a circle passing through point #50 (as indicated in Fig. 8) at coil shift ıx = 1 mm. Bn = 34.15 Gs, 40 kA turns total coil current.
Fig. 10. Tangential error field B 1 distributed along a circle passing through point #50 (as indicated in Fig. 8) at coil shift ıx = 1 mm. B 1 = 0 Gs, 40 kA turns total coil current.
722
V.M. Amoskov et al. / Fusion Engineering and Design 85 (2010) 718–723
Fig. 12. Random noise with a standard deviation of 10−3 . Fig. 11. Tangential error field B 2 distributed along a circle passing through point #50 (as indicated in Fig. 8) at coil shift ıx = 1 mm. B 2 = −121.91 Gs, 40 kA turns total coil current.
1. 2. 3. 4.
the linear shift along the X-axis, ıx = 1 mm; the tilt about the horizontal plane, ωy = 0.01◦ ; the ellipticity x = 1 mm; the warping z = 1 mm.
To give an example, Figs. 9–11 illustrate field distortions associated with linear shift of PF1 coil along the X-axis.
Fig. 12 presents an artificial random noise = 10−3 . According to Figs. 9–11, a ratio of error field B associated with the coil distortions to the total coil field is (B /B) ≤ 1.3%. This value is rather substantial and B can be determined to a desired accuracy from magnetic measurements with the use of existing field sensors. The value B is then used in the simulations for reverse reconstruction of coil distortions. The results of the reconstruction are listed in Tables 2–5. They demonstrate a feasibility of the proposed procedure for assessment of ITER coil quality.
Table 2 Numerical reconstruction of PF1 coil distortions at displacements ␦x = 1 mm, ωy = 0.01◦ , x = 1 mm, z = 1 mm. Uncertainty of measurement = 5 × 10−3 . Event #
Shift, x = 1 mm
Tilt, ωy = 0.01
Ellipticity, x = 1 mm
Warping, z = 1 mm
1 2 3 4 5 6 7 8 9 10
1.0849 0.8207 0.9342 1.1680 0.9085 0.9821 0.9272 1.6484 1.0215 0.8471
0.0071 0.0130 0.0170 −0.0019 0.0183 0.0194 0.0167 0.0085 0.0152 0.0226
0.7192 1.1005 1.0940 0.9822 1.1911 0.8628 1.2993 0.7582 0.5345 1.0820
1.5240 1.8559 −0.4849 0.9126 0.6608 2.1186 1.0400 −0.0462 3.3672 2.8090
Average Reconstruction error
1.0343 3.4%
0.0136 36%
0.9624 3.8%
1.3757 37%
Table 3 Numerical reconstruction of PF1 coil distortions at displacements ıx = 1 mm, ωy = 0.01◦ , x = 1 mm, z = 1 mm. Uncertainty of measurement = 10−3 . Event #
Shift, x = 1 mm
Tilt, ωy = 0.01
Ellipticity, x = 1 mm
Warping, z = 1 mm
1 2 3 4 5 6 7 8 9 10
0.8774 0.9897 1.0889 1.0006 0.9932 0.9888 0.9677 0.9766 1.0457 0.9701
0.0125 0.0088 0.0079 0.0094 0.0110 0.0097 0.0128 0.0149 0.0087 0.0096
1.1403 0.9480 0.9465 1.0258 0.9706 1.0394 0.8946 0.9327 0.9837 1.1259
0.9748 0.7829 0.6687 0.3672 1.1882 0.2637 1.3787 0.5081 1.1866 1.4345
Average Reconstruction error
0.9899 1.0%
0.0106 6.0%
1.0008 0.1%
0.8753 12.5%
V.M. Amoskov et al. / Fusion Engineering and Design 85 (2010) 718–723
723
Table 4 Numerical reconstruction of PF1 coil distortions at displacements ıx = 1 mm, ωy = 0.01◦ , x = 1 mm, z = 1 mm. Uncertainty of measurement = 5 × 10−4 . Event #
Shift, x = 1mm
Tilt, ωy = 0.01
Ellipticity, x = 1 mm
Warping, z = 1 mm
1 2 3 4 5 6 7 8 9 10
1.0201 1.0305 0.9936 1.0209 1.0219 1.0053 1.0261 1.0207 0.9859 0.9940
0.0102 0.0090 0.0106 0.0102 0.0098 0.0095 0.0084 0.0092 0.0102 0.0096
1.0117 0.9934 0.9608 1.0024 0.9994 0.9632 1.0326 0.9954 0.9869 0.9942
0.9299 0.9175 1.0856 0.8843 0.8282 0.7743 0.9913 1.0703 1.0112 1.2836
Average Reconstruction error
1.0119 1.2%
0.0097 3.0%
0.9940 0.6%
0.9776 2.2%
Table 5 Numerical reconstruction of PF1 coil distortions at displacements ıx = 1 mm, ωy = 0.01◦ , x = 1 mm, z = 1 mm. Uncertainty of measurement = 10−4 . Event #
Shift, x = 1 mm
Tilt, ωy = 0.01
Ellipticity, x = 1 mm
Warping, z = 1 mm
1 2 3 4 5 6 7 8 9 10
1.0065 0.9933 0.9933 0.9946 1.0087 1.0039 0.9937 1.0068 0.9931 0.9976
0.0101 0.0099 0.0099 0.0103 0.0097 0.0098 0.0101 0.0098 0.0096 0.0101
0.9982 1.0003 1.0003 1.0008 0.9966 0.9912 0.9942 1.0000 0.9999 0.9992
1.0362 0.9943 0.9943 1.0421 0.9558 1.0488 0.9820 1.0063 0.9637 1.0299
Average Reconstruction error
0.9992 0.1%
0.00998 0.2%
0.9981 0.2%
1.0053 0.5%
3. Conclusion
References
The reconstruction error, with respect to the introduced mean-square error = 10−3 –10−4 , is within a tolerable range of ∼(10–0.5)%. With increasing number of simulated random events N, the reconstruction error, which is proportional to N−1/2 , is expected to be decreased. It should be noted that some errors of sensor positioning and orientation can be treated as random if a proper measurement procedure is utilized. Then it is possible to apply statistical laws to their analysis. Also, numerical reconstruction of coil distortions can be improved with the increase in observation points and optimization of computation algorithm, particularly, by properly selected weight functions. Without any restriction the proposed approach can be applied both to PF and TF coil quality.
[1] R.J. La Haye, R. Fitzpatrick, T.C. Hender, A.W. Morris, J.T. Scoville, T.N. Todd, Critical error fields for locked mode instability in tokamaks, Phys. Fluids 4 (1992) 2098. [2] ITER Technical Basis, ITER EDA Documentation Series No. 24, International Atomic Energy Agency, Vienna, 2002. [3] V. Amoskov, A. Belov, V. Belyakov, O. Filatov, Yu. Gribov, E. Lamzin, N. Maximenkova, B. Mingalev, S. Sytchevsky, Statistical analysis of expected error fields in tokamaks and their correction, Plasma Devices Oper. 13 (2005) 87–103. [4] N. Mitchell, J. Knaster, Contributions to Plasma Error Fields from the CS, PF and TF Coils, ITER D 23DVQU v1.3, 2 September 2006. [5] A. Tikhonov, V. Arsenin, Methods of Solving Ill-Conditioned Problems, Nauka Publisher, Moscow, 1986. [6] V.Amoskov, A.Belov, V.Belyakov, T.Belyakova, E.Gapionok, D.Garkusha et.al., KLONDIKE 1.0: computer code for 3D simulation of magnet systems of complex geometry with retentive and non-retentive materials and current carrying components. RF Computer Program Register, Registration Certificate № 2003612487, Moscow, November 12, 2003.
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Assessment of ITER PF coil quality from magnetic measurements V.M. Amoskov a,∗ , A.V. Belov a , V.A. Belyakov a , V.A. Chuyanov b , S.A. Egorov a , A.A. Firsov a , Yu.V. Gribov b , V.G. Ivkin a , V.P. Kukhtin a , E.A. Lamzin a , A.A. Lancetov a , M.S. Larionov a , N.A. Maximenkova a , I.Yu. Rodin a , S.E. Sytchevsky a a b
D.V. Efremov Scientific Research Institute of Electrophysical Apparatus, St. Petersburg, Russia ITER Organization, CS 90 046, Building 523, 13067 Saint Paul-lez-Durance Cedex, France
a r t i c l e
i n f o
Article history: Received 3 July 2009 Received in revised form 11 April 2010 Accepted 12 April 2010 Available online 5 May 2010
a b s t r a c t A feasibility has been demonstrated for numerical reconstruction on the base of magnetic measurements for geometrical displacements or deformations occurred in the manufacture and assembly of magnet coils. For validation of the proposed approach the test results of reconstruction of possible misalignments and deviations of the ITER PF1 coil are presented. © 2010 Elsevier B.V. All rights reserved.
Keywords: ITER PF coil quality Manufacture and assembly tolerances Error field Magnetic measurements Numerical reconstruction
1. Introduction Small perturbations of the axial symmetry of a tokamak magnetic field are usually referred to as “error fields” [1]. Main sources of the field asymmetry in tokamaks are possible misalignments and deviations occurred in manufacturing and assembly of the coils for the magnet system [2]. For the ITER machine a severe limitation is adopted on the allowable level of field non-axisymmetry. The error fields above this limit can cause plasma instability resulting in plasma disruption [2]. The error fields are, however, about threefold higher with regard to the accepted manufacture tolerances [3]. To suppress the ITER error fields below the threshold level, a set of correction coils is utilized [2,3]. The aim of the study was to demonstrate a feasibility of numerical reconstruction of coil positioning displacements and deformations occurred in manufacturing and assembly of the coils on the base of magnetic measurements. The ITER magnet system includes toroidal field (TF) and poloidal field (PF) coils and sections of the central solenoid (CS) as shown in Fig. 1. Only the PF1 coil has been considered in this study as a subject of RF responsibility.
∗ Corresponding author. Tel.: +7 812 4627782. E-mail address: [email protected] (V.M. Amoskov). 0920-3796/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.fusengdes.2010.04.027
Fig. 2 illustrates the PF1 coil designed as a superconducting solenoid 9 m in diameter and 1 m in height, the winding being made of NbTi conductor. Table 1 summarizes main geometrical parameters of the PF1 coil in the cylindrical coordinate system related to the ITER centre. Standard displacements of the PF1 coil related to the accepted manufacture/assembly tolerances [4] include: 1. Linear shift along the X and Y axes, ıx = ±2 mm, ıy = ±2 mm (see Fig. 3). 2. Tilt about the horizontal plane, ız = ±1 mm; at low tilt angles ω = ωx , ωy the coil centreline is tilted as ız = ωR, where R is the coil radius (see Fig. 4). 3. In-plane ellipticity, x = ±3 mm, y = ±3 mm, that is a deformation of the coil shape in the plane XY defined in terms of two elliptic half-axes x = R + x, y = R + y, and an angle of rotation ˛ around Z-axis (−180◦ < ˛ < 180◦ ). Fig. 5 illustrates ellipticity at x > 0, y = 0, ˛ = 0. 4. Vertical displacement of the coil centreline, or so called warping, characterized by the amplitude z and angle ˛z, where z = ±1 mm corresponds to the maximum deviation z(ϕ) of the coil centreline, −180◦ < ˛z < 180◦ . The centreline shift z(ϕ) in the vertical direction along the toroidal angle ϕ is described by the sine-function: z = 0.5 × z × sin(ϕ + ˛z) (see Fig. 6). The shifts and tilts are resulted from geometrical displacements of the coil from its ideal position during assembly. The ellipticity
V.M. Amoskov et al. / Fusion Engineering and Design 85 (2010) 718–723
719
Fig. 1. ITER magnet system cross-section.
Each distortion of the coil shape and position is an independent random value. Let a magnet field generated by an undistorted coil be the ideal field. Then a field generated by a distorted coil is the distorted field, and the discrepancy between the ideal and distorted fields is the field distortion. In order to asses the resultant effect of various small geometrical distortions on the field, we can apply the superposition principle, assuming the coil distortions to be additive. Thus, the resultant
Fig. 2. PF1 coil layout.
and warping are associated with the coil distortion while manufactured. Note that the electromagnetic effect of coil positioning errors is always defined as an error in the coil current centreline. In intended magnetic measurements, all types of possible coil positioning errors should be taken into account.
Table 1 PF1 coil parameters. R, m
Z, m
R, m
Z, m
NPF1
3.943
7.557
0.968
0.976
256
R, Z: the central coordinates of the PF1 coil cross-section; R: the radial size; Z: the vertical size; NPF1 : the number of turns.
Fig. 3. Coil shift along the X-axis.
720
V.M. Amoskov et al. / Fusion Engineering and Design 85 (2010) 718–723
Diagnostics and evaluation of geometrical distortions may be obtained by solving the inverse magnetostatics problem. According to Hadamard’s classification [5], the problem is mathematically ill-conditioned and, therefore, its solution requires highly accurate magnetic measurements. Two points are essential for effective magnetic measurements:
Fig. 4. Coil tilt about the horizontal plane (rotation around the Y-axis).
• selection of types of sensors capable of field measurement to desired accuracy; • selection of observation points over typical regions, the number of observation points should be as low as possible to reduce measured data without loss in solution accuracy. The described approach is applicable to in-plant assessment of the coil quality during fabrication. Unwanted magnetic field generated by plant equipment and constructions may be detected by field mapping and then taken into account. With the use of specific field sensors, the same approach may be used for assessment of a coil field under cooldown conditions. A successful numerical reconstruction of coil distortions necessitates:
Fig. 5. Centrelines of a round coil and elliptic coil (viewed from above).
effect of all coil distortions can be defined to sufficient accuracy as a sum of their individual error fields. In the study, we took the PF1 coil parameters tabulated above as the ideal case, structure elements and current leads were ignored to keep axisymmetry for the undistorted coil. These components may be then included in a detailed calculation model. Assessment of the coil field quality as one of the main criteria for quality of manufacturing the entire coil or its components (in particular, pancakes) imply two steps: 1. room-temperature magnetic measurements over typical regions, using allowable coil current at room-temperature; 2. comparison of the measured data with a field map calculated for the ideal coil. A discrepancy in the results demonstrates quality of the coil manufacture.
• development of a numerical algorithm in order to provide evaluation of possible geometrical distortions of the coil from magnetic measurements with existing commercial field sensors in practice accurate to 10−3 –10−4 ; • development of a mathematical model of the coil to perform 3D field simulation in a form of maps for different field components. The model should allow a detailed field description, with respect to various coil components (current leads, buses, joints) capable to affect the field axisymmetry. The model should also take into account the effect of ferromagnetic surroundings. 2. Numerical experiment for evaluation of coil distortions The calculation model was developed for a cylindrical coil with the constant cross-section as described in Table 1. To perform field simulation, we used the code KLONDIKE [6] that enables analysis of precision current systems on the basis of solving magnetostatic problems in the integral formulation, using analytical solutions for a set of typical current system elements. The distorted field was determined as a superposition of individual error fields caused by geometrical coil distortions. 2.1. Calculation model of PF1 coil The calculation model of the coil is shown in Fig. 7. The coil is divided into 600 sections in the toroidal direction. The calculated results were obtained for the total coil current 156.25 A × 256 turns = 40 kA turns, taking into account the design limit of about (150–160) A/turn for a “warm” coil. 2.2. Observation points The field components were calculated at 400 points for each poloidal contour taken 0.1 m apart of a coil cross-section (see Fig. 8). The number of poloidal contours is 200. The total number of observation points was 200 × 400 = 80,000. 2.3. Computational method and results of numerical reconstruction
Fig. 6. Coil warping.
Simulation of a coil field includes two stages:
V.M. Amoskov et al. / Fusion Engineering and Design 85 (2010) 718–723
721
To assess the impact of random uncertainty of measurements , the sensors positions were assumed to be precisely specified on the measurement contour. The distorted field was added with random noise uniformly distributed within ±. Field components were treated as random values described in terms of the equiprobability distribution, dispersion 2 , and the distribution centroid coinciding with appropriate field component. For each field component 10 stochastic events were simulated at every observation point. In the analysis, the following coil distortions within the accepted tolerances (see Section 1) were taken:
Fig. 7. Calculation model of PF1 coil.
1 KLONDIKE computation of components of the distorted field in the global coordinate system; 2 computation of field maps for the normal and tangential components of the distorted field in the local coordinates x , y , z related to the observation points in order to simulate field measured with a 3-axis field sensor. To allow for uncertainty of measurements due to random errors, the mean-square error or standard deviation = 0.01–0.1% is introduced for calculated field components. Geometrical coil distortions leading to error fields can be determined using the regularization procedure [5] to minimize the following functional
2
2
M ˛ (Z, U) = (AZ − U) + ˛Z , where Z = {Zj } is the solution vector, A = {aij } is the i × j influence matrix with respect to the distortions, U = {ui } is the related error field vector, = {i }, = {j } are the vectors of weight functions, and ˛ is the regularization parameter. Distorted fields have been calculated at all observation points so as to imitate magnetic measurements. For numerical reconstruction of coil distortions we used only data calculated at the points located on a circle passing through point #50, see Fig. 8. Thus, we have limited our analysis to 200 observations.
Fig. 8. Observation points taken on a poloidal contour.
Fig. 9. Normal error field Bn distributed along a circle passing through point #50 (as indicated in Fig. 8) at coil shift ıx = 1 mm. Bn = 34.15 Gs, 40 kA turns total coil current.
Fig. 10. Tangential error field B 1 distributed along a circle passing through point #50 (as indicated in Fig. 8) at coil shift ıx = 1 mm. B 1 = 0 Gs, 40 kA turns total coil current.
722
V.M. Amoskov et al. / Fusion Engineering and Design 85 (2010) 718–723
Fig. 12. Random noise with a standard deviation of 10−3 . Fig. 11. Tangential error field B 2 distributed along a circle passing through point #50 (as indicated in Fig. 8) at coil shift ıx = 1 mm. B 2 = −121.91 Gs, 40 kA turns total coil current.
1. 2. 3. 4.
the linear shift along the X-axis, ıx = 1 mm; the tilt about the horizontal plane, ωy = 0.01◦ ; the ellipticity x = 1 mm; the warping z = 1 mm.
To give an example, Figs. 9–11 illustrate field distortions associated with linear shift of PF1 coil along the X-axis.
Fig. 12 presents an artificial random noise = 10−3 . According to Figs. 9–11, a ratio of error field B associated with the coil distortions to the total coil field is (B /B) ≤ 1.3%. This value is rather substantial and B can be determined to a desired accuracy from magnetic measurements with the use of existing field sensors. The value B is then used in the simulations for reverse reconstruction of coil distortions. The results of the reconstruction are listed in Tables 2–5. They demonstrate a feasibility of the proposed procedure for assessment of ITER coil quality.
Table 2 Numerical reconstruction of PF1 coil distortions at displacements ␦x = 1 mm, ωy = 0.01◦ , x = 1 mm, z = 1 mm. Uncertainty of measurement = 5 × 10−3 . Event #
Shift, x = 1 mm
Tilt, ωy = 0.01
Ellipticity, x = 1 mm
Warping, z = 1 mm
1 2 3 4 5 6 7 8 9 10
1.0849 0.8207 0.9342 1.1680 0.9085 0.9821 0.9272 1.6484 1.0215 0.8471
0.0071 0.0130 0.0170 −0.0019 0.0183 0.0194 0.0167 0.0085 0.0152 0.0226
0.7192 1.1005 1.0940 0.9822 1.1911 0.8628 1.2993 0.7582 0.5345 1.0820
1.5240 1.8559 −0.4849 0.9126 0.6608 2.1186 1.0400 −0.0462 3.3672 2.8090
Average Reconstruction error
1.0343 3.4%
0.0136 36%
0.9624 3.8%
1.3757 37%
Table 3 Numerical reconstruction of PF1 coil distortions at displacements ıx = 1 mm, ωy = 0.01◦ , x = 1 mm, z = 1 mm. Uncertainty of measurement = 10−3 . Event #
Shift, x = 1 mm
Tilt, ωy = 0.01
Ellipticity, x = 1 mm
Warping, z = 1 mm
1 2 3 4 5 6 7 8 9 10
0.8774 0.9897 1.0889 1.0006 0.9932 0.9888 0.9677 0.9766 1.0457 0.9701
0.0125 0.0088 0.0079 0.0094 0.0110 0.0097 0.0128 0.0149 0.0087 0.0096
1.1403 0.9480 0.9465 1.0258 0.9706 1.0394 0.8946 0.9327 0.9837 1.1259
0.9748 0.7829 0.6687 0.3672 1.1882 0.2637 1.3787 0.5081 1.1866 1.4345
Average Reconstruction error
0.9899 1.0%
0.0106 6.0%
1.0008 0.1%
0.8753 12.5%
V.M. Amoskov et al. / Fusion Engineering and Design 85 (2010) 718–723
723
Table 4 Numerical reconstruction of PF1 coil distortions at displacements ıx = 1 mm, ωy = 0.01◦ , x = 1 mm, z = 1 mm. Uncertainty of measurement = 5 × 10−4 . Event #
Shift, x = 1mm
Tilt, ωy = 0.01
Ellipticity, x = 1 mm
Warping, z = 1 mm
1 2 3 4 5 6 7 8 9 10
1.0201 1.0305 0.9936 1.0209 1.0219 1.0053 1.0261 1.0207 0.9859 0.9940
0.0102 0.0090 0.0106 0.0102 0.0098 0.0095 0.0084 0.0092 0.0102 0.0096
1.0117 0.9934 0.9608 1.0024 0.9994 0.9632 1.0326 0.9954 0.9869 0.9942
0.9299 0.9175 1.0856 0.8843 0.8282 0.7743 0.9913 1.0703 1.0112 1.2836
Average Reconstruction error
1.0119 1.2%
0.0097 3.0%
0.9940 0.6%
0.9776 2.2%
Table 5 Numerical reconstruction of PF1 coil distortions at displacements ıx = 1 mm, ωy = 0.01◦ , x = 1 mm, z = 1 mm. Uncertainty of measurement = 10−4 . Event #
Shift, x = 1 mm
Tilt, ωy = 0.01
Ellipticity, x = 1 mm
Warping, z = 1 mm
1 2 3 4 5 6 7 8 9 10
1.0065 0.9933 0.9933 0.9946 1.0087 1.0039 0.9937 1.0068 0.9931 0.9976
0.0101 0.0099 0.0099 0.0103 0.0097 0.0098 0.0101 0.0098 0.0096 0.0101
0.9982 1.0003 1.0003 1.0008 0.9966 0.9912 0.9942 1.0000 0.9999 0.9992
1.0362 0.9943 0.9943 1.0421 0.9558 1.0488 0.9820 1.0063 0.9637 1.0299
Average Reconstruction error
0.9992 0.1%
0.00998 0.2%
0.9981 0.2%
1.0053 0.5%
3. Conclusion
References
The reconstruction error, with respect to the introduced mean-square error = 10−3 –10−4 , is within a tolerable range of ∼(10–0.5)%. With increasing number of simulated random events N, the reconstruction error, which is proportional to N−1/2 , is expected to be decreased. It should be noted that some errors of sensor positioning and orientation can be treated as random if a proper measurement procedure is utilized. Then it is possible to apply statistical laws to their analysis. Also, numerical reconstruction of coil distortions can be improved with the increase in observation points and optimization of computation algorithm, particularly, by properly selected weight functions. Without any restriction the proposed approach can be applied both to PF and TF coil quality.
[1] R.J. La Haye, R. Fitzpatrick, T.C. Hender, A.W. Morris, J.T. Scoville, T.N. Todd, Critical error fields for locked mode instability in tokamaks, Phys. Fluids 4 (1992) 2098. [2] ITER Technical Basis, ITER EDA Documentation Series No. 24, International Atomic Energy Agency, Vienna, 2002. [3] V. Amoskov, A. Belov, V. Belyakov, O. Filatov, Yu. Gribov, E. Lamzin, N. Maximenkova, B. Mingalev, S. Sytchevsky, Statistical analysis of expected error fields in tokamaks and their correction, Plasma Devices Oper. 13 (2005) 87–103. [4] N. Mitchell, J. Knaster, Contributions to Plasma Error Fields from the CS, PF and TF Coils, ITER D 23DVQU v1.3, 2 September 2006. [5] A. Tikhonov, V. Arsenin, Methods of Solving Ill-Conditioned Problems, Nauka Publisher, Moscow, 1986. [6] V.Amoskov, A.Belov, V.Belyakov, T.Belyakova, E.Gapionok, D.Garkusha et.al., KLONDIKE 1.0: computer code for 3D simulation of magnet systems of complex geometry with retentive and non-retentive materials and current carrying components. RF Computer Program Register, Registration Certificate № 2003612487, Moscow, November 12, 2003.