The effect of blanket system and port plugs on the harmonic components of error field and optimizing approach investigation

The fusion plasma is sensitive to the penetration of the toroidal mode numbers n harmonic components of the error fields (EFs), and the toroidal mode components of the EFs are closely tied to the toroidal distribution of the magnetic field source. The primary source of EFs within the magnetic field sources is the current source, such as the magnet system, while the magnetic field produced by magnetized ferromagnetic materials may also generate EFs. The ferromagnetic field source, such the reduced-activation ferritic-martensitic steel, represent a smaller portion of the EFs source. However, the use of ferromagnetic materials is expected to increase significantly as tokamak research transitions from experimental devices to demonstration and commercial reactors, particularly for the structural materials of in-vessel components and port plugs. Predictably, the EFs induced by ferromagnetic materials will increase and may even surpass the design specifications. In this study, the EFs introduced by the water-cooling ceramic breeder blanket system and the vacuum vessel port plugs of China Fusion Engineering Test Reactor are comprehensively analyzed by an analytical approach and finite element numerical approach, which is based on the Fourier decomposition. The results reveal a clear linear relationship between the EFs formed by the saturated magnetization module and its rigid displacement deviating from periodic symmetry, as well as the same behavior in EFs and overall magnetization intensity. Compared to the assembly error of the blanket system, the EFs that may be introduced by the equatorial ports and plugs are substantial, including larger n⩽3 toroidal mode harmonic components that can lead to lock mode and disruption. Moreover, using the correlation between harmonic components and symmetrical periods, specific toroidal mode components can be selectively shielded.

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Introduction
The error field (EF) characterizes the non-axial symmetry of the tokamak magnetic field, which can impact the plasma through several mechanisms.The locked mode induced by EF was reported in [1][2][3][4].With the penetration of the EFs, the rotation of the plasma is continuously slowed down.When the velocity falls below a certain threshold, the local instabilities are no longer overcome by self-rotation, the tearing mode and magnetic island caused by magnetic line reconnection may occur, which may even induce plasma disruption events in severe cases [5].In addition, flow damping effects such as electromagnetic torque and neoclassical toroidal viscosity torques can also be generated by the EF permeation [6][7][8][9][10].
The quantitative evaluation of the effect of EFs on the plasma remains an open challenge.The three-mode error index (TMEI), formerly the criterion used for EFs in the International Thermonuclear Experimental Reactor (ITER), relies on the harmonic components of toroidal mode number n = 1 and poloidal mode numbers m = 1, 2, 3 to establish a simplified EFs criterion, with a primary focus on the core-resonant field while neglecting plasma response [11][12][13][14].However, the TMEI criterion proves inadequate when attempting to expand experimental data on lock modes, and it fails to yield a reliable extrapolation law [15,16].In reality, the plasma is sensitive to all toroidal harmonic components of the EFs, particularly the lower toroidal harmonic components such as n = 1, 2 . .., and it is also essential to consider the effects generated by certain poloidal harmonic components at the same toroidal mode number in EFs analysis [2-4, 10, 15, 17-19].Based on the above, the 'overlap' metrics definition in which ideal three-dimensional magnetohydrodynamics modeling is proposed and redefining the EFs correction criterion [16,[20][21][22][23].
The inherent EFs within a tokamak are closely linked to the distribution of the field source, and the magnitude of the harmonic component serves as a measure of how much the field source distribution deviates from axisymmetric periodicity.These field sources can be categorized into two types: current-based and magnetized ferromagnetic-based.The nonperiodic symmetry can be attributed to various factors, including assembly errors in field sources like magnet system and blanket system, as well as constructions that lack strict periodic symmetry, such as coil feeders and vacuum vessel port plugs.Furthermore, the toroidal harmonic component, arising from assembly errors in the field source, is both inevitable and random.Notably, the assembly errors of the poloidal and toroidal field coils (TFCs), often considered to be major contributors to EFs, have been observed at the millimeter scale in devices like ITER and JT-60SA, resulting in significant and unacceptable EFs [11,[24][25][26].In contrast, in such experimental devices, the EFs introduced by magnetized components, such as test blanket modules (TBMs) and ferromagnetic inserts, are typically tenfold or more less than those caused by the magnet system, rendering them relatively minor contributors to EFs [26,27].
However, in upcoming Demonstration Reactors and future commercial reactors, the scale of ferromagnetic components within the vacuum vessel will significantly expand.Unlike TBMs and ferromagnetic inserts, blanket modules will almost entirely encase the plasma surface.Consequently, the influence of the magnetized blanket system on the magnetic field within the plasma region cannot be underestimated, and the potential for EF effects from these magnetized ferromagnetic components may correspondingly increase.Moreover, during the actual assembly of the modular blanket system, deviations or tilting errors may occur, which could impact the toroidal and poloidal harmonic components of EFs to some extent.Additionally, it is noteworthy that due to the requirements of a neutral beam injector and plasma diagnostics, the distribution of corresponding equipment may not adhere strictly to axisymmetric patterns [28,29].Thus, another potential source of harmonic EF components arises from variations in magnetization across different components, such as blanket modules and port plugs.This study aims to address these aspects, providing guidance and data support for mitigating the EF effects of magnetized components in future fusion reactors.
In this study, based on the China Fusion Engineering Test Reactor (CFETR), the EF effects of the blanket system's assembly error and ferromagnetic components such as vacuum vessel port plugs are thoroughly investigated.The impact of the field source on the harmonic components of the EFs will also be examined using Fourier decomposition.

Geometric model
The geometric model of CFETR is constructed as a periodic symmetric model, featuring 16 sectors.This model comprises the magnet system, consisting of TFCs and poloidal field coils, the blanket system, which includes water-cooled ceramic breeder (WCCB) blankets, and the equatorial port assembly, encompassing equatorial ports and port plugs.Figure 1 illustrates the CFETR model.
All coils are represented as solid excitation structures, simplifying their complex wound structure and shell.In each sector, there are two groups of inboard (IB) blankets and three groups of outboard (OB) blankets.These blankets are secured at back supporting structure (BSS).Each blanket module retains the primary ferromagnetic components, including the first wall (FW) and shell, while the finer structures such as flow channels within the FW and shell, as well as cooling tube assemblies, are compressed.The compressed ferromagnetic structures are filled with equivalent-volume ferromagnetic modules, with detailed magnetization parameters and volumes outlined in section 2.3.Given that the focus is primarily on the positional distribution of the main ferromagnetic components rather than the impact of their internal structures on EFs, the simplified approach mentioned above is deemed suitable.This simplified approach is also extended to the modeling of equatorial ports and port plugs.

Analytical approach
The chosen material for in-vessel components, such as blankets, is reduced-activation ferritic-martensitic (RAFM) steel.When ferromagnetic RAFM steel is exposed to a highintensity magnetic field, it becomes magnetized, generating a magnetization field.This magnetization effect of ferromagnetic materials can be likened to the field produced by what is known as the 'bound current.'This bound current exists on the surface of the material structure, and the surface current density of the bound current ⃗ J s is where ⃗ M is the magnetization of the material and ⃗ n is the normal vector of the surface of the structure.The magnetic field intensity in the vacuum vessel of the tokamak is strong enough to saturate magnetized RAFM steel (figure 2(a)), so ⃗ M( ⃗ H) can be replaced by saturation magnetization M s .By doing this, the J s from the magnetic field could be decoupled, and the M s is a constant in the quasi-static field: Moreover, since the toroidal field created by the TFC is much stronger than the magnetic field in other directions, the bound current on the blanket surfaces can be simplified to poloidal coils, and there is no bound current distribution at either end of the coil.As seen in figure 2(b), the integral of the toroidal height h tor of the bound current surface density along the coil determines the ampere-turns of the coil.By the way, the approach of equivalent bound current to the coil is also applicable to other components of the magnetic field.According to Biot-Savart's law, the magnetic field distribution of the computed field is obtained: where ⃗ B is the magnetic induction intensity, µ o is the vacuum permeability, d ⃗ l is the linear microelement of the current, ⃗ e is the unit direction vector from spatial coordinates to the current microelement, and r 2 is the square of the coordinate vector.

Numerical approach
Based on commercial finite element software ANSYS, the numerical approach is used to analyze the static field.The magnetic field is regarded as a quasi-static field ∂ ⃗ B/∂t = 0, and the magnetic field is solely determined by a constant excitation current.The differential equation of the quasi-static magnetic field is obtained by removing the excitation current from the estimated field, and adding the magnetic scalar potential ϕ m when no current exists: where ⃗ H is the magnetic field intensity of the field, µ is the permeability.2(c).The surfaces of the plasma boundary components ψ are built to make sure that the field data used to calculate EFs is accessible at the nodes on the surfaces, and the mesh around the surfaces is encrypted to strive for higher calculation accuracy.Besides, the mesh geometry of the model is periodic symmetric to avoid mesh effects on low toroidal modes harmonic components.
The relationship between magnetization and magnetic induction intensity of RAFM steel is nonlinear, so the corresponding magnetization to magnetic induction intensity is determined by numerical table interpolation.In addition, the magnetization modified by the geometric factor G will be applied to the blanket module (equation ( 6)) [30,31],

EFs analysis
The B ⊥ (θ, φ ) normal to the plasma boundary component ψ b is used for Fourier decomposition of EFs analysis, and the B ⊥ could be get by the corresponding field ⃗ B (θ, φ ) and unit vector normal to the plasma boundary ⃗ e ⊥ (θ, φ ), where the θ and φ is the poloidal coordinate and toroidal coordinate, respectively.Next, four components, m,n , and B S,S m,n , are obtained as the following equations ( 8)- (11), and the harmonic component b m,n is obtained as equation (12) shows, where the m and n is the poloidal mode number and toroidal mode number.Moreover, the dimensionless parameter is more suitable for discussing the b m,n .Here, a dimensionless parameter b * m,n is For the convenience of expression, the dimensionless parameter b * m,n will be expressed by b m,n in the following.

The EFs introduced by assembly error
The primary source of assembly errors in the WCCB blanket system of CFETR is believed to be the connections between the blanket module and the BSS, as well as between the BSS and the vacuum vessel [32].These errors primarily manifest as rigid body displacements in the radial direction (d 1 at the principal coordinates), normal direction (d 2 ), and deflection angle (θ) at the connection surface between the BSS and the blankets (see figure 1(a)).These deviations can be integrated into the displacement projected in the normal direction of the plasma scrape-off layer (SOL).The b m,n introduced by the rigid body displacements of the blanket modules within one single segment were calculated.These assembly errors may occur at the connection points between the BSS and the vacuum vessel.The displacements, denoted as ∆d, range from 0 to 20 mm, spaced at 5 mm intervals, representing the straight-line distance from the blankets to the SOL.As indicated in figure 3  Besides, according to the results (table 1 and figure 4), the displacement of the magnetized blanket modules is linearly related to the b m,n , and the b m,n can be determined through linear superposition, which involves adding the contributions from each source term: where and C 2 α2 m,n is the linear coefficient, α 1 and β 1 is the number and deviation type (referring different displacement directions like the d 1 and d 2 shown in figure 1(a)) of field sources for ∆d, respectively, and the same as α 2 for ∆M.The ∆d is the displacement deviating from the symmetry state, and ∆M ≡ |M 1 − M 2 | is the magnetization difference of the assembly.So for analytical calculations, it is reasonable to treat the bound current of the saturated magnetized blanket as a constant, independent of the spatial magnetic field intensity.
To be expected, there are some disparity between the numerical results and the analytical results, with the relative error expected to be larger in the numerical results.To minimize the influence of the element on the numerical results, the element meshing is strictly periodic symmetric, which theoretically avoids the influence of the element on the modes that  n < 15.However, due to the non-circular cross-section of the tokamak, it is impossible to achieve full symmetry in the poloidal direction during element meshing.This introduces errors in the poloidal harmonic components, contributing to the larger numerical results in b m,n .Furthermore, the coefficient C 1 remains nearly identical for similar structures such as IBS and IBC, as well as OBS and OBC.It is closely related to the geometric parameters of the structure.Additionally, since C 1 is proportional to the surface area of the blankets, IB blankets have larger C 1 values compared to OB blankets.
In addition to the rigid body displacement of the vacuum vessel and BSS, assembly errors also occur at the connection points between blanket modules and BSS.These errors are independent of each other at each assembly point for blanket modules and BSS.To systematically assess the EFs introduced by this assembly error in the blanket system [26,[33][34][35], the Monte Carlo method is employed.The assembly error is assumed to follow a normal distribution, and the b m,n values for over 2000 instances introduced by various displacements are calculated using the analytical approach.As shown in figure 5 and table 2 the probability density distribution of the EFs exhibits a positive bias, with the b m,n values roughly on the order of 1 × 10 −5 .This is slightly higher than the b m,n values introduced by the assembly error of single-segment blanket modules.

The toroidal harmonic effect of port plugs
The vacuum vessel ports and their associated plugs are essential components designed to meet specific requirements, such as neutral beam injection (NBI) and plasma  diagnostics.They play a crucial role in shaping the toroidal period distribution characteristics of in-vessel components.Additionally, RAFM steel is used in the construction of these port plugs, serving as a candidate material to shield against extreme heat and radiation.Consequently, similar to the blanket system, the toroidal distribution characteristics of the port plugs have a significant impact on the toroidal harmonic components of the EFs.The equatorial ports of the vacuum vessel are situated closest to the plasma boundary surface, with six equatorial ports distributed across six separate sectors [28,29].Given their proximity, the EFs introduced by these equatorial ports are expected to have a more substantial impact compared to the upper and lower ports present in the sixteen sectors.Consequently, a total of four equatorial port plugs toroidal distributions and the corresponding magnetic fields were established, focusing on the detailed analysis of the toroidal harmonic components of EFs.
As depicted in figure 6, the magnetic field generated by the source exhibits a periodicity that determines the frequency domain distribution of its harmonic components.For instance, the TFCs generate a magnetic field with a period T = T 0 /16, where T 0 = 2π .Consequently, the dominant contribution to the toroidal harmonic components with n = 16 is produced by the TFCs, commonly referred to as the toroidal field ripple.By analogy, the magnetic field with a distribution period T = T 0 /p tor results from the superposition of harmonic components with n = p tor k, where k is a positive integer, and p tor is the number of periods in the toroidal direction.
The results have revealed a clear relationship between the harmonic components of EFs and the distribution of field sources.This understanding provides valuable insights into how to suppress these harmonic components through optimization of the field source distribution.For example, the harmonic components with n ̸ = 2k can be completely canceled by ensuring the field period T = π (as shown in figure 6(b)), similarly, T = π /2 period effectively reduces the components of n ̸ = 4k (figure 6(c)).In the case of CFETR ports, which are positioned as depicted in Case 4 [28], figure 6(d) illustrates that the b m,n spectrum in Case 4 comprises n = 2k and n = 8k harmonic components.Based on the previously established conclusion, the n = 8k peaks can be attributed to a locally periodically symmetrical distribution with T = T 0 /8 within the sectors where the port is located.Consequently, However, this outcome stems from considering the distribution of ports while overlooking the influence of variations in the magnetization of the plugs.In reality, the six equatorial ports of CFETR serve distinct purposes, with ports 1 and 2 dedicated to NBI and the others for plasma diagnostics [28].These differences in magnetism and magnetic field are a result of the plug designs tailored to specific functions.In essence, the ports and plugs generate a magnetic field with a period T ̸ = π .To account for this, a relative difference in the magnetization of the NBI plugs compared to the diagnostic plugs is assumed, denoted as ∆ M = | MNBI − Mdiag |/ Mdiag × 100% (∆ M is dimensionless ∆M, ∆M is defined in equation ( 14) in section 3.1).The ∆ M is incremented from 0% to 50% in 10% intervals.Here, M is defined as M ≡ ∑ i M i V i /V 0 for an overall magnetization representation, where M i and V i represent the magnetization and corresponding volume of each ferromagnetic plug material, and V 0 is the total plug volume.It is important to note that the specified magnetization difference aims to establish a correlation between the magnetization difference of the port plugs and the EFs.It does not represent an actual value for the NBI system or the diagnostic system in the engineering design.
Figures 7(a) and (b) illustrates a clear linear relationship between the difference in magnetization and b m,n , which aligns with the findings in equation ( 14) where ∆d is held constant.Notably, the trend of b m,2 shows a negative correlation with ∆ M, and the coefficient C 2 in equation ( 14) is negative.This indicates that ∆ M initiates a transformation in the model from a T = π to a T = 2π periodic distribution, causing the n = 2k harmonic components to disperse into n = k components to some extent.It is worth mentioning that the n = 8k peaks persist because ports 4, 5, and 6 maintain a local T = π /4 period distribution.
To sum up, to reduce the n harmonic components of the EFs as much as possible, the toroidal distribution period of the vacuum vessel port plugs should be as small as possible, and the ∆ M between different plugs should be reduced as much as feasible.To be specific, the ∆ M can be changed through variation of structures and materials.However, due to the requirements of thermophysical, mechanical, and radiotolerance properties of materials, the materials that plugs can choose are very limited and are ferromagnetic materials [ [36][37][38], so how to make plugs with different uses and different structures have similar magnetization field is an engineering design problem to be solved.So, when considering the local distribution of EFs, both the inner structure and material distribution of different functional plugs should be similar as much as possible.Additionally, the difference among various plug designs drives local EFs, so there are certain restrictions and upper limits in practical engineering applications for the approach of limiting the amplitude of ∆ M. Alternately, the port can be moved such that it is precisely periodic and symmetrical.For example, the locations of ports 1 and 5 can be switched in Case 4, provided that without affecting NBI and diagnostic functions.

Discussions and conclusions
In summary, the establishment of the analytical model has enhanced the efficiency of EFs evaluation in the early stages of engineering design and serves as a prerequisite for Monte-Carlo method analysis of EFs caused by random rigid body displacements.Through a systematic analysis using both numerical and analytical approaches, it has been determined that the b m,n introduced by the assembly error of the CFETR blanket system remains below 1.3 ×10 −5 .However, a cause for concern is the inappropriate arrangement of equatorial ports, which can introduce low n mode components (e.g.b m,1 , b m,2 , b m,3 , etc) exceeding the EFs produced by assembly errors by approximately three orders of magnitude.Given reference to criteria from similar devices like ITER, it is challenging to imagine that CFETR can tolerate such substantial EFs [16,20,23].Additionally, the equatorial port arrangement in CFETR may not effectively suppress the n = 2 harmonic components of EFs, which can lead to field penetration, locked modes, and plasma disruptions [17,18].This aspect requires careful attention in the later optimization design of the device.
The symmetrical period of the tokamak is intricately linked to its inherent harmonic components of EFs.By modifying the toroidal distribution and magnetization intensity of port plugs, it is possible to efficiently correct the impact of EFs from the equatorial ports and selectively reduce the toroidal harmonic components.To significantly reduce the n ⩽ 3 harmonic components, the minimum toroidal symmetry period of the tokamak should closely approach T = π /2.This limit period may be further reduced if higher toroidal mode harmonic components require correction.

Z. Liu et al
In addition, it is worth noting that while active correction methods such as correction coils can effectively target n = 1 and even n = 2 harmonic components, the approach of optimizing port distribution and managing magnetization differences among various port plug types offers a more direct and efficient means of reducing the inherent EFs in fusion devices.However, both of these approaches pose practical engineering challenges.Addressing differences in internal ferromagnetic structures among the plugs is one such challenge, requiring careful design and material considerations.Additionally, there is a need to balance EF reduction with the functionality of the port system, as significant changes in port locations may impact the performance of subsystems, such as NBI and diagnostic systems.Furthermore, the EF correction strategies discussed in this study are not limited to equatorial ports alone.They can be extended to address EFs in the upper and lower parts of the vacuum vessel, as well as in other components with aperiodic symmetrical distribution.
This study provides data support and methodological recommendations for the evaluation and correction of EFs in fusion reactors.Subsequent work will focus on the detailed design of system port plugs, enabling a comprehensive analysis of EFs and the exploration of appropriate optimization approaches.

Figure 1 .
Figure 1.(a) the model of CFETR and the schematic diagram of the assembly errors in the in-vessel components, (b) the model of WCCB blankets, (c) the model of equatorial ports and port plugs, where IBC, IBS, OBC, and OBS are the inboard-center, inboard-side, outboard-center, outboard-side blankets, separately, the side blankets is closed to the toroidal field coils and the center blankets is between the side blankets.

Figure 2 .
Figure 2. (a) the B(H) of the RAFM (b) Conceptual diagram of a magnetization equivalent coil (c) the finite element model of CEFTR, and the 1/16 model is shown except for the magnet system.
and table 1, the b m,n values primarily emphasize the high toroidal mode harmonic components, with the m = 1, 2, 3 poloidal mode components being more prominent compared to other poloidal modes.Except for the n = 16 components, the b m,n values remain below 1.3 ×10 −5 .

Figure 6 .
Figure 6.The harmonic components of bm,n in different cases, and corresponding conceptual diagrams of ports' distribution (top view).