Distortion of magnetic field lines caused by radial displacements of ITER toroidal field coils

Fusion Engineering and Design 118 (2017) 64–72

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Distortion of magnetic field lines caused by radial displacements of ITER toroidal field coils V.M. Amoskov a,∗ , Y.V. Gribov b , E.A. Lamzin a , S.E. Sytchevsky a a b

D.V. Efremov Scientific Research Institute of Electrophysical Apparatus, St. Petersburg, Russia ITER Organization, Route de Vinon-sur-Verdon, CS 90 046, 13067 St Paul Lez Durance Cedex, France

a r t i c l e

i n f o

Article history: Received 15 September 2016 Received in revised form 15 March 2017 Accepted 16 March 2017

a b s t r a c t An assessment of distortions of ideal (circle) field lines caused by random radial displacements of the TF coils by |R| ≤ 5 mm has been performed from the statistical analysis assuming a uniform probability density function for displacements. © 2017 Elsevier B.V. All rights reserved.

Keywords: ITER Toroidal field coils Field lines Monte-Carlo method

1. Introduction Were the Toroidal Field (TF) of ITER tokamak [1] to coincide with an “ideal toroidal solenoid” (i.e. the number of TF coils NTFC → ∞), the magnetic field lines should be horizontal circles with the centers located on the vertical axis Z (see curve 1 in Fig. 1). The finite number NTFC = 18 of the TF coils in their nominal positions produces non-axisymmetric periodic deviations of field lines (i.e. TF ripple) with the toroidal main mode number n = 18 (see curve 2 in Fig. 1). This will cause perturbation of the plasma boundary and peaking of heat loads on the first wall due to an increase in the angle  between the first wall and the field line striking the wall (Fig. 1). The TF ripple may be partially decreased by the Ferromagnetic Inserts (FIs) [2,3]. Non-axisymmetric (random) shifts of the TF coils from their nominal positions destroy the periodical character of the TF ripple and can locally (at some toroidal angles ϕ) increase the angle  and, correspondingly, the heat loads. The paper is devoted to assessment of distortions of ideal (circular) field lines caused by random radial displacements of the TF coils on the first wall for the blanket modules (BMs) within the expected range of installation tolerances. In the general case, the field line equation has a form dx/Bx = dy/By = dz/Bz , while the line location is determined by means of integration. The study presented is focused on a simplified technique [3], due to low field distortions typical for TF coils. Dis-

∗ Corresponding author. E-mail address: [email protected] (V.M. Amoskov). http://dx.doi.org/10.1016/j.fusengdes.2017.03.091 0920-3796/© 2017 Elsevier B.V. All rights reserved.

placements of the TF coils are found in terms of deviations from an ideal (circular) field line. The results of such study will be quantitatively assessed in terms of (i) a deviation of the perturbed field lines in the direction normal to the first wall plasma facing surface, (ii) an angle between the perturbed field lines and the first wall plasma facing surface, and (iii) a horizontal shift of the “magnetic axis” of the field produced by the TF coils. 2. Statement of the problem In the study the first wall of each Blanket Module (BM) is defined as a ring given by the edges with the following coordinates: (R1 , Z1 ) for the lower edge and (R2 , Z2 ) for the upper edge. The coordinates of eighteen blanket modules edges in the Tokamak General Coordinate System (TGCS) are given in Table 1 and indicated in Fig. 2. The coordinates of a unit poloidal vector (Cr , Cz ) normal to each indicated line 1–2 can be found as Cr =



−(Z 2 − Z1 ) 2

(R2 − R1 ) + (Z2 − Z1 )

2

1/2 , Cz = 

R2 − R1 2

(R2 − R1 ) + (Z2 − Z1 )

2

1/2 .

(1)

Let us consider an ideal toroidal field line (circle (l) below) passing through a point with the coordinates (R, Z). The field B0 (R) and the current ITF in every TF coil are related to each other by Ampere’s law: 1 0



Bϕ dl = (l)

2R B0 (R) = 18 · ITF . 0

(2)

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Fig. 1. Horisontal cross-section of ITER TF coils. 1–ideal circular line, 2 – non-axisymmetric periodic (NTFC = 18) field line for nonshifted TF coils, 3–field line for the radially shifted TF coils (variant corresponding to Case 3). h, ␹, and ␨ are also indicated. Table 1 Coordinates of BM edges (R1 ,Z1 ), (R2 ,Z2 ), and middle points (ri , zi ). Blanket module

i

R1 , m

Z1 , m

R2 , m

Z2 , m

ri , m

zi , m

BM1 BM2 BM3 BM4 BM5 BM6 BM7 BM8 BM9 BM10 BM11 BM12 BM13 BM14 BM15 BM16 BM17 BM18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

4.1053 4.1053 4.1053 4.1053 4.1053 4.1053 4.1253 4.3246 4.9354 5.7534 6.5524 7.4015 7.9062 8.2697 8.3938 8.3057 7.9002 7.2824

−2.5037 −1.4914 −0.4761 0.5402 1.5566 2.5719 3.5852 4.3371 4.7196 4.5344 3.9256 3.1796 2.4647 1.6847 0.6354 −0.4200 −1.3372 −2.2544

4.1053 4.1053 4.1053 4.1053 4.1053 4.1253 4.3246 4.9354 5.7534 6.5524 7.4015 7.9062 8.2697 8.3938 8.3057 7.9002 7.2824 6.2661

−1.4914 −0.4761 0.5402 1.5566 2.5719 3.5852 4.3371 4.7196 4.5344 3.9256 3.1796 2.4647 1.6847 0.6354 −0.4200 −1.3372 −2.2544 −3.0434

4.10530 4.10530 4.10530 4.10530 4.10530 4.12030 4.22495 4.63000 5.34440 6.15290 6.97695 7.65385 8.08795 8.33175 8.34975 8.10295 7.59130 6.77425

−1.99755 −0.98375 −0.03205 1.04840 2.06425 3.07855 3.96115 4.52835 4.62700 4.23000 3.55260 2.82215 2.07470 1.16005 0.10770 −0.87860 −1.79580 −2.64890

For the nominal current ITF = −9.128 MAt the field B0 (R) is equal to

B0 =

−32.8608 [T]. R[m]

(3)

The corresponding toroidal field B0 at the radius R = 6.2 m is −5.3 T. Here the negative sign corresponds to the clockwise direction of the toroidal field if viewed from above. Deviation h(ϕ) of the perturbed field line relative to the ideal field line passing through the point with the coordinates (R, Z, ϕ0 )

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Fig. 2. ITER EM system cross-section (TF coils, Central Solenoid (CS) and Poloidal Field (PF) coils, Vacuum Vessel (VV), Ferromagnetic Inserts (FI)), plasma and first wall. Numbers indicate middle points of the first wall formed by blanket modules BM1–BM18.

in the direction normal to the first wall can be assessed using the following expressions: h(ϕ) = Cr r(ϕ) + Cz z(ϕ),

ϕ r(ϕ) = ϕ0

Br (R, Z, ϕ ) Rdϕ B0 (R)

(4)

ϕ z(ϕ) =

Bz (R, Z, ϕ ) Rdϕ , B0 (R)

(5)

(ϕ) = Cr

Br (R, Z, ϕ) Bz (R, Z, ϕ) + Cz . B0 (R) B0 (R)

where Br and Bz are the radial and vertical components of the perturbed field, respectively; (Cr , Cz ) are the coordinates of the unit vector (1). The value h(ϕ) can be interpreted as a depth of penetration of the corresponding perturbed field line into a plasma facing component. The distortion of the ideal circular field lines is studied in the terms of the maximal peak-to-peak deviation, H, of the perturbed field lines in the direction normal to the first wall

(6)

(7)

Then, the distortion of the ideal circular field lines may be studied in terms of the maximal angle, , between the perturbed field line and the first wall:  = max |(ϕ)|,

ϕ0

H = maxh(ϕ) − minh(ϕ), 0 ≤ ϕ < 360◦ .

The angle (ϕ) between the perturbed field line and the first wall at the cross-point with the coordinates (R, Z, ϕ) can be assessed as:

0 ≤ ϕ < 360◦

(8)

In the study, the dependences r(ϕ), z(ϕ), h(ϕ) and (ϕ) are calculated in the range from ϕ = ϕ0 = 0◦ to ϕ = 360◦ over a circle with the coordinates (ri , zi ) passing through the middle of the first wall of the i-th blanket module: (i)

ri =

(i)

R1 + R2 2

(i)

,

zi =

(i)

Z1 + Z2

(9)

2

Table 1 gives the mid-points (9) for all blanket modules. In Fig. 2 the point with the coordinates (ri , zi ) are marked by module numbers. The horizontal shifts x and y of the “magnetic axis” of a disturbed field line for the system of TF coils shifted in the radial

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Fig. 3. Computational model of FIs for regular VV sector. 1/18 of full model. Indicated number of ferromagnetic plates in shielding blocks corresponds to optimized FIs set.

Table 2 Maximal deviations H (6), angles  (8), and shifts of magnetic axes (10) found on circles passing though points (ri , zi ) (9) of corresponding BMs for Case 1, Case 2 and Case 3. Blanket module

B0 , T

BM1 BM2 BM3 BM4 BM5 BM6 BM7 BM8 BM9 BM10 BM11 BM12 BM13 BM14 BM15 BM16 BM17 BM18

−8.00448 −8.00448 −8.00448 −8.00448 −8.00448 −7.98503 −7.77780 −7.09737 −6.14864 −5.34070 −4.70991 −4.29337 −4.06293 −3.94405 −3.93554 −4.05541 −4.32874 −4.85084

Case 1

Case 2

Case 3

H, mm

, deg

H, mm

, deg

H, mm

, deg

, mm

0.125 0.125 0.125 0.125 0.125 0.120 0.113 0.708 3.015 5.144 7.937 12.57 15.66 15.35 12.56 8.152 4.134 1.324

0.0157 0.0158 0.0158 0.0158 0.0157 0.0150 0.0138 0.0788 0.2910 0.4311 0.5866 0.8470 0.9984 0.9504 0.7760 0.5188 0.2808 0.1008

0.125 0.125 0.125 0.125 0.125 0.120 0.117 0.711 2.183 1.757 1.022 2.760 4.301 3.801 1.813 0.521 2.043 1.286

0.0157 0.0158 0.0158 0.0158 0.0158 0.0151 0.0143 0.0791 0.2101 0.1484 0.0671 0.1804 0.2660 0.2589 0.1534 0.0705 0.1361 0.0979

6.801 6.882 6.901 6.879 6.795 6.547 5.618 2.222 6.567 10.96 14.17 19.67 23.23 23.33 20.56 15.66 11.18 7.060

0.1051 0.1056 0.1057 0.1056 0.1051 0.1022 0.0897 0.1074 0.3174 0.4769 0.6337 0.8995 1.0535 1.0066 0.8306 0.5689 0.3272 0.1393

2.286 2.351 2.370 2.348 2.280 2.155 1.961 1.697 1.410 1.173 0.952 0.764 0.651 0.604 0.623 0719 0.893 1.162

direction along the axes X and Y of the TGCS, respectively, are defined as follows:

ri y (ri , zi ) = B0 (ri )

2 Bϕ (ri , zi , ϕ) sin(ϕ)dϕ, 0

ri x (ri , zi ) = B0 (ri )

2 Bϕ (ri , zi , ϕ) cos(ϕ)dϕ, 0

where the coordinates ri , zi are given by Eq. (9); B0 (ri ) is the ideal toroidal field (3) on the circle with coordinates (ri , zi ) passing through the middle of the first wall of the i-th blanket module. Note that at x = 0 and y = 0 the magnetic axis is the vertical axis Z.

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Fig. 4. Distributions of deviation H (a), angle  (b), and horizontal shift of “magnetic axis” (c) for BM4.

Finally, the horizontal shift of the magnetic axis of the perturbed field line for the chosen first wall panel i is:



i =



2

2 ( x (ri , zi )) + y (ri , zi )

(10)

To perform field simulation, we used the code KLONDIKE [4] 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 elements. 3. Distortion of toroidal field lines caused by TF ripple The first stage of the study is the assessment of the distortion of ideal (circular) field lines caused by the TF ripple with/without FIs

[2] in a regular Vacuum Vessel (VV) sector, in the region of the first wall of the blanket modules (BM1–BM18), assuming the following two cases of magnetic field perturbations: Case 1.

TF coils only (nominal positions);

Case 2.

TF coils (nominal positions) and FIs [2];

Comparison of the field line distortions in the Case 1 with that in the Case 2 allows assessing the decrease of the TF ripple caused by FIs [2]. The simulations were performed with the use of the optimized set of FIs for a VV regular sector described in detail in [2], see also Fig. 3. A cross-section of the shielding blocks with optimized FIs in a VV regular sector is shown in Fig. 3. During operation FIs are

V.M. Amoskov et al. / Fusion Engineering and Design 118 (2017) 64–72 Table 3 Parameters of statistical distributions (N = 10,000) for innermost blanket module BM4.

69

Table 4 Parameters of statistical distributions (N = 10,000) for uppermost blanket module BM10.

Parameter

H, mm

, deg

, mm

Parameter

H, mm

, deg

, mm

Minimum values, Xmin Average values, Xav X(P = 50%) X(P = 90%) X(P = 99%) X(P = 99.7%) Maximal values, Xmax S

0.805 2.478 2.432 3.299 4.001 4.272 4.808 0.603

0.03265 0.05708 0.05669 0.06756 0.07665 0.08059 0.08576 0.00788

0.006 0.447 0.422 0.761 1.042 1.157 1.313 0.228

Minimum values, Xmin Average values, Xav X(P = 50%) X(P = 90%) X(P = 99%) X(P = 99.7%) Maximal values, Xmax S

5.010 6.004 5.923 6.900 7.710 8.069 8.841 0.642

0.43882 0.45016 0.44991 0.45512 0.45974 0.46190 0.46488 0.00374

0.003 0.224 0.211 0.380 0.520 0.578 0.656 0.114

saturated. The saturation magnetization of FIs (the steel SS-430) is assumed to be 0 Ms = 1.6 T. Table 2 summarizes the values of H and  evaluated on the circles passing through the middle points (9) of corresponding blanket modules. Due to cyclic symmetry of Case 1 and Case 2, shifts of magnetic axes are equal to zero and omitted. Column B0 contains the nominal toroidal field (3) for the middle points (9) of the first wall of blanket modules. 4. Distortion of toroidal field lines caused by random radial shifts of TF coils

Table 5 Parameters of statistical distributions (N = 10,000) for outermost blanket module BM14. Parameter

H, mm

, deg

, mm

Minimum values, Xmin Average values, Xav X(P = 50%) X(P = 90%) X(P = 99%) X(P = 99.7%) Maximal values, Xmax S

14.941 16.587 16.496 17.866 18.985 19.454 20.496 0.936

0.96137 0.97728 0.97708 0.98310 0.98822 0.99071 0.99447 0.00442

0.002 0.115 0.109 0.196 0.268 0.298 0.338 0.059

In the second stage of the study we assess the distortion of the circular circle field lines in terms of H, Eq. (6), , Eq. (8), and , Eq. (10), using the Monte Carlo approach on the base of statistical distributions of the parameters under investigation. The probability of a TF coil radial displacement, R, is expressed through the probability density function p(R). The probability of the coil radial displacement, R, in the interval R1 ≤ R ≤ R2 can be calculated as follows.



R2

P(R1 ≤ R ≤ R2 ) =

p(R)d (R) R1

In the study the radial displacements of the TF coils are assumed to be within −Rmax ≤ R ≤ Rmax with a uniform probability density function:

⎧ ⎨ p(R) = ⎩

1 , if|R| ≤ Rmax 2Rmax ,

(11)

p(R) = 0, if|R| > Rmax

with Rmax = 5 mm. In fact, 10,000 3D models of TF coils have been built in order to calculate statistical distributions for field line distortions in terms of H,  and . Every model represents a random event and implements a unique random set of radial shifts for all eighteen TF coils according to the uniform probability density function (11). The statistical analysis of the radial displacements within the specified range has been performed for 3 observation points at the middle of the BM4, BM10, and BM14 first walls. These three modules represent innermost (BM4), uppermost (BM10) and outermost (BM14) locations of blanket modules. The fields produced by each of 10,000 models have been calculated along three circles (9) with the coordinates (ri , zi ) taken from Table 1 for i = 4, 10, 14. Tables 3–5 summarize obtained statistical parameters of distributions for field line distortions related to BM4, BM10, and BM14 using the following definitions: – N is the total number of random events, with each event representing a TF coil set with a random radial displacement of each TF coil generated using the Monte Carlo method;





Fig. 5. Distribution of horizontal shifts x of “magnetic axis” for BM4. Dashed line – Gaussian distribution p( x ) =

√1

2

exp −

( x −a)2 2 2

with a = 0 and = 0.355 mm.

– xn is the randomized variable H, , or of the obtained set of the TF coils with the number “n”; n = 1. . .N; – Xav is the averaged value of H, , or for N sets of the “generated” TF coils in the randomization process, Xav =

1 N

N

xn ;

 N

1 – S is the standard deviation, S = N−1 (xn − Xav )2 ; n=1

n=1

– X(P) means that with the probability P the value X (which is H, , or ) is less than the relevant value in the row X(P). – Xmin and Xmax show the minimal and maximal values found among N events. For outer blanket modules (BM14 as an example) significantly higher deviations of the field lines are expected than that presented

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Fig. 6. The worst case with respect to angle  among 100,000 events with random radial displacements of the TF coils by |R| ≤ 5 mm. Parameters of field line perturbations along circle passing through middle point of first wall of BM4: radial Br , toroidal B␸ = B␸ – B0 , and vertical Bz components of perturbed field (a), radial r, vertical z and normal h deviation of perturbed field line (b), and angle  between perturbed field and plasma facing surface.

in Table 5, if the FIs, Test Blanket Modules and ELM coils [3] are taken into account with the maximal current and the worst toroidal phase of the current distribution among the coils. In this case, the angle  between the perturbed field line and the first wall can be up to 3◦ and H (defined as peak-to-peak deviation) reaches 40 mm. So, the contribution of TF coils radial shifts to the total effect is not essential. For the innermost blanket modules (BM4 as an example) with the probability 99% the angle  between the perturbed field line and the first wall is less than 0.077◦ , the horizontal shift of the “magnetic axis” ␨, defined by Eq. (10), is less than 1 mm, and the deviation of the field line in the direction normal to the first wall H is less than 3.5 mm. This means that for the innermost blanket modules, where the influence of FIs, Test Blanket Modules and ELM coils is weak, the radial TF coils shifts produce the main impact on distortions of field lines.

The uppermost blanket module BM10 demonstrates interjacent values as presented in Table 4. The obtained distortions of toroidal field lines are less than the ones calculated for the set of TF coils, designated in Table 2 as “Case 3” with Hmax = 6.880 mm,  max = 0.105599◦ , and max = 2.348 mm. Case 3 corresponds to a set when one half of coils (for instance, TF1· · ·TF9) is shifted by R = −5 mm, while another half of coils (TF10· · ·TF18) is shifted radially in the opposite direction by R = + 5 mm. TF coil shifts shown in Fig. 1 correspond to Case 3. From physical understanding, Case 3 should to be close to maximal field line distortions caused by TF coil radial shifts in the range of |R| ≤ 5 mm. The results obtained demonstrate that the number of events N = 10,000 is insufficient for calculation of the maximum values of field line distortions, and the data of Table 3 should be recalculated

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Fig. 7. TF coils radial displacements Case 3. Parameters of field line perturbations along circle passing through middle point of first wall of BM4: radial Br , toroidal B␸ = B␸ – B0 , and vertical Bz components of perturbed field (a), radial r, vertical z and normal h deviation of perturbed field line (b), and angle  between perturbed field and plasma facing surface.

for a wider statistics. So, we increase the number of random events from 10,000 to 100,000. Histograms in Fig. 4 show distributions of the maximal deviation H of the perturbed field lines in the direction normal to the first wall, Eq. (6), angle , Eq. (8), and horizontal shift , Eq. (10), calculated for BM4 on the base of 100,000 random events. Table 6 shows parameters of statistical distributions obtained at N = 100,000. As seen from Table 6, all integral characteristics of distributions (Xav , X(P), and S) are practically equal for both statistics (compare Table 6 with Table 3). Further increase in the number of events requires excessive computational cost. On the other hand, an estimation of a probability of large field perturbations corresponding to Case 3 may be provided as follows. Three main characteristics (H, , ) of field perturbations are far away of the Gaussian statistics. However, a series of tests (estimation of the skewness and kurtosis, ␹2 -criterion) has shown that

Table 6 Parameters of statistical distributions (N = 100,000) for innermost blanket module BM4. Parameter

H, mm

, deg

, mm

Minimum values, Xmin Average values, Xav X(P = 50%) X(P = 90%) X(P = 99%) X(P = 99.7%) Maximal values, Xmax S

0.661 2.468 2.427 3.278 3.991 4.271 5.293 0.604

0.03032 0.05705 0.05669 0.06747 0.07656 0.08028 0.08895 0.00789

0.001 0.444 0.419 0.754 1.042 1.158 1.515 0.227

distributions of the both components of ( x and y ) appear to be close to the Gaussian statistics, see Fig. 5. For BM4 a standard deviation of both components is 0.355 mm. Gaussian statistic allows us

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to estimate the probability of x (or y ) to exceed the corresponding value = 2.348 mm in Table 2, Case 3 as: 2.348 = 4 · 10−11 . P( x ≥ 2.348) = erfc √ 2 · 0.355 This value is practically unreachable. Figs. 6 and 7 additionally illustrate the field (Br , B␸ = B␸ – B0 , Bz ), deviations r, z, h, and angle  vs. the toroidal angle ϕ for the worst case among 100,000 events with respect to angle ␪ (Fig. 6) and for Case 3 (Fig. 7). 5. Conclusion The assessment of distortions of ideal (circular) toroidal field lines caused by random radial displacements of the TF coils by |R| ≤ 5 mm has been performed. A statistical analysis of 10,000 TF coil sets generated by the Monte Carlo method has been utilized assuming a uniform probability density function p(R). As shown in [3], for the outer blanket modules, the effect of Ferromagnetic Inserts, Test Blanket Modules and ELM coils with maximal currents and the worst phase leads to the angle  up to 3◦ between the perturbed field line and the first wall of the BM14, while the value of H can be up to 40 mm. The study has revealed that for the outer blanket modules deviations of the field lines caused by random radial displacements of the TF coils are significantly lower and may be ignored in the analysis. For the inner blanket modules, where the TF ripple is very low, deviations of the magnetic field lines are caused mainly by random radial displacements of the TF coils rather than the Test Blanket Modules and ELM coils. With high probability is predicted that:

1) the maximal horizontal shift of the “magnetic axis” (corresponding to the n = 1 toroidal mode of the perturbed field), is less than 1 mm, 2) the maximal value of angle  between the perturbed field line and the first wall is less than 0.08◦ . 3) the maximal peak-to-peak deviation of the perturbed field lines in the direction normal to the first wall, H, is less than 4 mm. For a wider statistics of 100,000 random events a minor difference as compared to 10,000 events has been demonstrated for the integral characteristics of distributions (i.e. average values, standard deviations, and values corresponding to probabilities P = 50%, 90%, 99%, 99.7%). This supports our conclusions. Disclaimer: The views and opinions expressed herein do not necessarily reflect those of the ITER Organization. References [1] O. Motojima, The ITER project construction status, Nucl. Fusion 55 (2015) 104023 http://iopscience.iop.org/article/10.1088/0029-5515/55/10/104023/ pdf. [2] E.A. Lamzin, V.M. Amoskov, E.I. Gapionok, Y.V. Gribov, N.A. Maximenkova, S.E. Sytchevsky, Analysis and optimization of the impact of ferromagnetic inserts on the toroidal field ripple in ITER, Appl. Supercond. IEEE Trans. 22 (2012) 4901004. [3] Y. Gribov, V. Amoskov, E. Lamzin, S. Sytchevsky, Assessment of 3D perturbation of plasma boundary and variation in field lines inclination near outboard first wall caused by non-axisymmetric magnetic field expected in ITER, Proc. 40th EPS Conference on Plasma Physics, P1 (2013), 257. [4] V.M. Amoskov, A.V. Belov, V.A. Belyakov, T.F. Belyakova, Yu. A. Gribov, V.P. Kukhtin, E.A. Lamzin, S.E. Sytchevsky, Computation technology based on KOMPOT and KLONDIKE codes for magnetostatic simulations in tokamaks, Plasma Devices Oper. 16 (2008) 89.