AC Loss in REBCO Coil Windings Wound With Various Cables: Effect of Current Distribution Among the Cable Strands

To construct large-scale superconducting devices, it is critical to enhance the current carrying capability of superconducting coils. One practical approach is to utilise assembled cables, composed of multiple strands, for the winding. There have only been a few investigations of the dependence of current distribution among the strands on the AC losses of the cables and coils wound with these cables. In this work, we studied three types of cables; 1) 8/2 (eight 2 mm-wide strands) Roebel cable, 8/2 Roebel; 2) two parallel stacks (TPS) which have the same geometrical dimensions as the Roebel cable, 8/2 TPS; and 3) an equivalent four-conductor stack (ES) comprising four 4 mm-wide conductors, 4/4 ES. We proposed a new numerical approach that can achieve equal current sharing and free current sharing among the strands. We examined the loss behaviour of all three types of straight cables and two coil assemblies comprising two, and eight, stacks of double pancakes (DPCs) wound with these cables, respectively. 2D FEM analysis was carried out in COMSOL Multiphysics using the $H$ – formulation. The stacks were modelled in parallel connection, with the same electric field applied to all strands, so that current is distributed between the conductors. Simulated transport AC loss results in the straight 8/2 Roebel cable, 8/2 TPS and 4/4 ES were compared with previously measured results as well as each other. The numerical AC loss results in two coil windings 2-DPC and 8-DPC wound with the 8/2 Roebel cables were compared with the results in coil windings wound with the 8/2 TPS and 4/4 ES. No transposition was introduced at the connection between double pancakes, in order that current can be shared among the strands in the 8/2 TPS and 4/4 ES. The results indicate that AC loss in the straight 8/2 TPS and 4/4 ES is larger than that in the 8/2 Roebel cable. Current is more concentrated in the outer strands for the straight 8/2 TPS and 4/4 ES than the 8/2 Roebel cable and causes greater AC loss than the 8/2 Roebel cable. The 8-DPC coil winding wound with the 8/2 Roebel cable has the smallest loss and the coil winding wound with the 8/2 TPS has the greatest loss at two current amplitudes investigated. At 113 A, the AC loss value in the 8-DPC coil winding wound with the 8/2 TPS is 2.2 times of that wound with the 8/2 Roebel cable.


I. INTRODUCTION
Large scale high temperature superconducting (HTS) power applications present promising solutions for the electrical The associate editor coordinating the review of this manuscript and approving it for publication was Su Yan .systems within the transportation and energy sectors.Thanks to the remarkable capabilities enabled by HTS technology, HTS applications encompass high efficiency and power density, compact size, and lightweight design, etc.Many R&D projects and studies have been carried out on superconducting applications in the transportation and energy sectors, including superconducting transformers [1], [2], [3], [4], power transmission cables [5], [6], rotating machines [7], [8], fault current limiters [9], [10], Superconducting Magnetic Energy Storage (SMES) [11], [12], magnets [13], [14], etc.The integration of HTS-based devices in these sectors and systems enables us to tackle decarbonation and energy challenges more effectively and efficiently.
To achieve high power rating and high-power density, it is essential for the HTS devices to be wound with superconducting cables comprised of multiple tapes to carry high current [15], [16].One example of increasing current-carrying capability is Roebel cable, which is a fully transposed cable assembled by multiple punched ReBCO strands [15], [17], [18], [19], [20].Another technique is to assemble ReBCO tapes into a simply stacked cable with or without transposition of the tapes [21], [22], [23], [24], [25].In a transposed cable, the current in each strand is equally distributed.Whereas, in a simple stack without transposition, current distributes unevenly in each tape/strand, for example, with greater current flowing in outer tapes and less current in the inner tapes [26], [27], [28].This leads to an undesirable AC loss from the cables.In a coil winding wound with a simple stack, the unequal current sharing between the tapes will be more severe due to the difference in inductance for each tape and hence leads to large AC loss in the coil winding [29].It is important to note that these losses are closely linked with the device efficiency and lead to a thermal load that must be removed by the cooling systems, which has a high cooling penalty [30].Thus, for high current AC applications, the unequal current sharing between the tapes in an assembled cable is a critical issue concerning losses that must be studied and solved.
In our previous research, we explored the impact of unequal current distribution in non-inductive bifilar coils, wound with Roebel cable and simple stacks, for superconducting fault current limiter [31].We experimentally compared the measured transport AC losses of Roebel cables and simple stacks with, and without, transposition.However, there is limited work published on the AC loss in superconducting inductive coil windings comprised of double pancake coils wound with transposed, and non-transposed, cables and the impact of unequal current sharing on AC loss in the coil windings.
In this work, three types of cables are studied; i) 8/2 (eight 2 mm-wide strands) Roebel cable, 8/2 Roebel; ii) two parallel stack which have the same geometrical dimensions as the Roebel cable, 8/2 TPS; iii) an equivalent four-conductor stack comprising four 4 mm-wide conductors, 4/4 ES. Figure 1(a) shows the schematic of the three cables.Firstly, the simulated transport AC loss results in the straight 8/2 Roebel cable were compared with measured results as well as 8/2 TPS and 4/4 ES.Next, we evaluated the loss behaviour of two coil windings comprising two, and eight stacks of double pancakes (DPCs) wound with these cables, respectively.The numerical AC loss results in the two coil windings wound with the 8/2 Roebel cables were compared with the results in the coil windings wound with 8/2 TPS and 4/4 ES.
This paper is structured as follows.Section I introduces the importance and motivation of the work.Section II explains the numerical methods and specifications of the studied cables.Section III validates the numerical model through comparison with experimental results.Section IV shows the results and the analysis of straight 8/2 Roebel cable, 8/2 TPS and 4/4 ES.Section V reports the results and the analysis on superconducting coils wound with the 8/2 Roebel, 8/2 TPS and 4/4 ES.Conclusion is presented in section VI.

II. NUMERICAL APPROACHES A. SIMULATION FOR STRAIGHT CONDUCTOR AND COILS
A two-dimensional (2D) FEM model employing Hformulation was built in COMSOL to study the AC loss characteristics of the straight 8/2 Roebel cable, 8/2 TPS, 4/4 ES, as well as stacks of coil windings wound with the three cables [32], [33], [34], [35].Two state variables were implemented, H = [H x , H y ] T , where H x and H y are magnetic fields parallel and perpendicular to the tape wide-surface.A power law relation was used to represent the highly nonlinear relationship of local electric field E and local current density J in the HTS layer: where E 0 = 1 µV/cm, n = 30 is the power index derived from V -I characteristic; J c (B) is the critical current density dependence on local magnetic field and lateral position.In this work, a modified Kim model was adopted for J c (B) profile, as expressed in (2).k, α, and B 0 are curve fitting parameters and the values used in this study are 0.3, 0.7, and 42.6 mT, respectively.
The governing equation ( 3) is derived by combining Faraday's law, Ampere's law, Ohm's law, and constitutive law.Equation ( 4) is obtained by substituting H x and H y into (3).µ 0 is the magnetic permeability of the free space.µ r is the relative magnetic permeability.Here, µ r = 1 holds true for all the cases studied in this work.
For simplification, 2D axisymmetric model was employed to simulate the electromagnetic behavior in coil windings, with two state variables H = [H r , H z ] T [36].The governing  equation is shown in (5),  [16], [17], [18].Table 1 lists the specifications of the cables.Figure 2 illustrates the modelled cross-sections of the three cables and the symmetry boundary conditions applied in COMSOL software.Figure 2(a) represents the modelling of 8/2 Roebel cable and 8/2 TPS since they share the same cross-section, while Figure 2(b) simulates the 4/4 ES.Due to the symmetry, only a quarter model is simulated to reduce computing time.To consider the symmetry, H x = 0 is applied on x-axis, and H y = 0 is applied on y-axis.
Two coil windings wound with these three cables, namely 8/2 Roebel, 8/2 TPS, and 8/2 ES, were investigated, including  a coil winding comprised of a stack of two double pancake coils (2-DPC), and eight double pancake coils (8-DPC), respectively.Table 2 lists the specifications of the coil windings wound with the three cables.
Figures 3(a)-(c) show the schematic of current distribution in a 1-DPC wound with the 8/2 Roebel, 8/2 TPS and 4/4 ES.In figure 3(a), current is equally distributed in each strand of the 8/2 Roebel cable due to its fully -transposed nature.Pointwise constraint is applied in the partial differential equation (PDE) module for each strand.In figure 3(b), free current-sharing is defined in the 8/2 TPS cable, where the strand carrying the same amount of current is denoted using the same colour.This case is similar to the ''coupled at end'' case in [25].The current-sharing pattern among all strands is repeated for each turn due to the structure of the coil winding.Similarly, in figure 3(c), free current-sharing is defined for the 4/4 ES cable.The different colour in each strand of the 4/4 ES denotes free current sharing.In a 1-DPC level, the current sharing pattern is linked and repeated for each turn for this case as well.

III. MODEL VERIFICATION
Figure 4 shows the simulated transport AC loss in the straight 8/2 Roebel cable at 59 Hz, plotted as a function of the Roebel cable current and compared with the measured AC losses of the cable [36].In the figure, the theoretical transport AC losses estimated by Norris-ellipse (N-e) and Norris-strip    The simulated AC losses fall between N-e and N-s in the low current region and agrees well with N-s at high current amplitudes.The simulated losses of the 8/2 Roebel cable have a good agreement with the measured ones although slightly smaller.
Figure 5 shows the simulated AC loss in a 2-DPC coil winding compared with measured results.The modelling method for the coil is identical to what has been presented in Chapter II.The 2-DPC coil winding was wound with 4 mmwide SuperPower wires.The measurement was carried out at 24.63 Hz.Calculation reasonably reproduces the measured values, and the results validate the simulation method for coil windings, which has been used to produce the results for 2-DPC and 8-DPC wound with the three cables in Chapter V.   the same current for the whole current range in the 8/2 Roebel cable.In the 8/2 TPS, with increasing I t, turn , current in both the 'inner conductor' and 'outer conductor' increases linearly only until I t, turn /I c0, cable reaches 0.5.The 'inner conductor' in the 8/2 TPS carries much less current than the 'outer conductor' and the difference between the current values is the largest when I t, turn /I c0, cable is around 0.5.The current in the 'outer conductor' in the 8/2 TPS saturates from I t, turn /I c0, cable approximately 0.7 and becomes slightly smaller with increasing I t, turn .On the other hand, current in the 'inner conductor' in the 8/2 TPS becomes larger more rapidly from I t, turn /I c0, cable > 0.5.This clearly indicates current redistribution between the two conductors through the terminals of the 8/2 TPS and the behaviour could be explained using a circuit model described in figure 7.At I t, turn /I c0, cable = 0.  6) and ( 7), we get the ratio of I 1 /I 2,

IV. RESULTS ANALYSIS: STRAIGHT ROEBEL CABLE VS. STACKS
When I is small, R 1 and R 2 are nearly zero; Figures 8(a) and (c) compare the current distribution J /J c in the 8/2 Roebel cable and the 8/2 TPS at I t /I c0 = 0.8, respectively.As shown in figure 8(a), we observe nearly the same pattern of J /J c distribution in ''Outer conductor'' and ''Inner conductor'' at I t /I c0 = 0.8, and both conductors are not yet fully penetrated, that both conductors have a portion of sub-critical area.In contrast, the pattern of J /J c distribution in figure 8(c) is remarkably different.|J /J c | value of the ''Outer conductor'' in the 8/2 TPS is greater than 1 across the conductor width whereas there is 1/3 of the conductor width has less than 1 for the ''Inner conductor''.
Loss power density distribution in the 8/2 Roebel cable and the 8/2 TPS at I t /I c0 = 0.8 are plotted and compared in figures 8(b) and (d).Both ''Outer conductor'' and ''Inner conductor'' have a portion of the conductor width that does not generate AC loss, as seen in figure 8(b), and this is aligned with the fact that both conductors have an area where | J/J c | < 1.However, in figure 8(d), ''Outer conductor'' in the 8/2 TPS generates AC loss across the conductor width, and ''Inner conductor'' has a big area that does not generate AC loss at all due to the low J /J c values in the area.This shows that each strand in the 8/2 Roebel cable behaves broadly equally in generating AC loss; whereas each strand in non-transposed cables behaves unequally in loss generation due to unequal current-sharing.
Figure 9 compares the transport AC loss values in the straight 8/2 Roebel cable, 8/2 TPS and 4/4 ES at 59 Hz.It is observed that for a given cable current, 4/4 ES has the greatest loss while Roebel cable has the smallest loss among the three cable types due to the equal current sharing as explained in figure 6 and figure 8.The difference in the AC loss values between the cables becomes greater with increasing the amplitude of the cable current.
At I t, peak = 151 A and 302 A, AC loss in the 8/2 TPS is 1.06 times and 1.14 times of that in Roebel, respectively.AC loss values in the 8/2 TPS is smaller than 4/4 ES, due to  the existence of the horizontal gap [26], [37].At I t, peak = 151 A and 302 A, AC loss in 4/4 ES is 1.47 times and 1.76 times of that in the 8/2 Roebel cable, respectively.

V. RESULTS ANALYSIS: 2-DPC COIL WINDING A. AC LOSS IN 2-DPC COIL WINDING
Figure 10 presents a comparison of the simulated AC loss values in the 2-DPC coil windings wound with the 8/2 Roebel cable, 8/2 TPS, and 4/4 ES and plotted as a function of the coil current amplitude.There is nearly no difference in AC loss values in the coil windings wound with different cables when the coil current is less than 60 A. The difference in the AC loss values between the coil windings wound with the 8/2 Robel cable and those wound with the 8/2 TPS and 4/4 ES becomes greater at high current amplitudes.We attribute the lower AC loss of the Roebel coil winding to the equal current sharing capability.In contrast, the 2-DPC coil winding wound with the 8/2 TPS generates the highest AC loss for all current amplitudes.The AC loss value in the 2-DPC winding wound with the 8/2 TPS is 2.3 times of that wound with the 8/2 Roebel cable at 151 A.
Figures 11 shows the normalised current density distribution, J /J c in the 2-DPC coil windings wound with the 8/2 Roebel cable and 8/2 TPS, respectively, at I t = 113.4 A. As a result of the equal current sharing between strands, the 2-DPC coil winding wound with the 8/2 Roebel cable is equivalent to a 4-DPC coil winding wound with 2 mm strands.In Figure 11(a), we could observe shielding currents (magnetization currents) in the two end coil discs that shield the radial magnetic field component in the end part of the coil winding [19], [36].The fully penetrated region where |J /J c | > 1 is the greatest for the outermost disc and the smallest for the innermost disc.There is un-penetrated or subcritical region even in the outermost disc.In contrast, current in the end pancake coil of the 8/2 TPS is mostly concentrated in the outermost disc and there is almost no shielding current.The induced shielding current for the bottom half of the pancake coil (blue colour) is almost the same as the transport current (red colour), which implies the net current for the bottom half of the PC is almost zero.The inner PC shows similar behaviour as the upper PC although its upper half of the PC does not contain any shielding current.These observations clearly indicate highly unequal current distribution of the coil winding wound with the 8/2 TPS.This behaviour causes high AC loss and hence the 8/2 TPS is not a favourable option for high-current applications.Figure 16 compares the AC loss values in the 8-DPC coil windings wound with the 8/2 Roebel, 8/2 TPS and 4/4 ES at I t = 56.7A and 113.4 A. The 8-DPC coil winding wound with the 8/2 Roebel cable has the smallest loss and the coil winding wound with the 8/2 TPS has the greatest loss at both current amplitudes.At 113 A, loss value in the 8-DPC coil winding wound with the 8/2 TPS is 2.2 times of that wound with the 8/2 Roebel cable.

VI. CONCLUSION
A new numerical approach using the widely used Hformulation in COMSOL Multiphysics is both proposed and demonstrated that can achieve equal current sharing and free current sharing among multiple strands both in straight cables and coil windings.This was done using 2D FEM on three cable case studies, i) 8/2 (eight 2 mm-wide strands) Roebel cable, 8/2 Roebel ii) two parallel stacks (8/2 TPS) which have the same geometrical dimensions as the 8/2 Roebel cable, 8/2 TPS iii) an equivalent four-conductor stack comprising of four 4 mm-wide conductors, 4/4 ES, as well as the windings wound by them, namely the two, and eight stacks of double pancakes (DPCs) wound with these cables.The numerical model was validated by comparing the simulated transport AC loss results in the straight 8/2 Roebel cable with previously measured results.
An electric circuit has been used to explain the current sharing among different strands in the straight 8/2 Roebel cable and 8/2 TPS.The analytical formula can explain the unequal current-sharing among different strands, consistent with the results from the numerical model.
On the straight cable level, we observe nearly the same pattern of J /J c distribution in ''Outer conductor'' and ''Inner conductor'' at I t /I c0 = 0.8 in Roebel cable, and both conductors are not fully penetrated having a portion of sub-critical area.In contrast, the pattern of J /J c distribution in TPS is remarkably different.|J/J c | value of the ''Outer conductor'' in TPS is greater than 1 across the conductor width whereas there is 1/3 of the conductor width has less than 1 for the ''Inner conductor''.In terms of AC losses, it is observed that for a given cable current, the straight 4/4 ES has the greatest loss while the 8/2 Roebel cable has the smallest loss among the three cable types due to the equal current sharing behaviour.The difference in the AC loss values between the cables becomes greater with increasing the amplitude of the cable current.
On the coil winding level, transport current flows at the outer edge of each disc for the 8-DPC winding wound with the 8/2 Reobel cable, while magnetization current is observed in each disc excluding the central disc.The transport current is completely concentrated in the outer disc of each PC in the 8-DPC coil winding wound with the 8/2 TPS, and the inner disc of each PC carries no net current similar to the 2-DPC coil winding case.The result shows that strong unequal current-sharing occurs in the entire 8-DPC coil winding wound with the 8/2 TPS and this, in turn, leads to high loss generation.AC loss value in the end disc wound with the 8/2 TPS is nearly 5 times of that wound with the 8/2 Roebel cable.A zig-zag shaped loss distribution was found for the coil winding wound with the 8/2 TPS due to strong unequal current distribution in the 8/2 TPS strands: high AC loss values correspond to high current concentration and low loss values correspond to low net current flow.

FIGURE 1 .
FIGURE 1.(a) Schematic of the three studied cables (b) Cross-section of the three cables.

FIGURE 2 .
FIGURE 2. Schematic of the numerical model for the three studied cable (a) 8/2 Roebel cable and 8/2 TPS (b) 4/4 ES (only a quarter model was simulated considering symmetry).

FIGURE 3 .
FIGURE 3. Schematic of a 1-DPC wound with the three cables (a) wound with the 8/2 Roebel cable, where equal current flowing in each strand (b) wound with 8/2 TPS, which has the same geometry as the 8/2 Roebel cable, but allowing free current distribution among strands (c) wound with 4/4 ES, which has the identical valid conductor cross-section area but allowing free current distribution among strands.
(N-s) models using the measured cable I c value 164 A are plotted together.The modelling method of the 8/2 Roebel is identical to what we described in Chapter II.

FIGURE 4 .
FIGURE 4. Comparison of measured and simulated AC loss values in the 8/2 Roebel cable.

FIGURE 5 .
FIGURE 5.The simulated results for a 2-DPC coil winding agree with the measured AC loss results.

Figure 6
Figure 6 shows the simulated current distributions in the 'inner conductor' and 'outer conductor' of the 8/2 Roebel cable and 8/2 TPS defined in figure 1 at various I t, turn /I c0, cable values when ωt = 3π/2, where I t, turn is the total current amplitude for the 8/2 Roebel cable and 8/2 TPS.It is worth noting that 'inner conductor' and 'outer conductor' could represent the other three pairs of conductors in the 8/2 Roebel cable and 8/2 TPS due to the symmetry.The 'inner conductor' and 'outer conductor' carry exactly

FIGURE 7 .
FIGURE 7. Schematic of magnetic field coupling among strands (only half of the cable is considered).
3 and 0.7, the current amplitudes of the 'surface conductor' and 'inner conductor' in the TPS are 4.97 A, 23.40 A, and 23.8 A, 42.38 A, respectively.Considering symmetry, the circuit illustrates half cable, i. e. 4 strands in the 8/2 Roebel cable and 8/2 TPS: the 'outer conductor' and 'inner conductor' in the upper right half; the 'outer conductor' and 'inner conductor' in the bottom right half which carry current with the same direction as their upper counterpart conductors.Branch 1 refers to the upper 'outer conductor', branch 2 refers to the upper 'inner conductor', branch 3 refers to the bottom 'inner conductor', branch 4 refers to the bottom 'outer conductor'.Considering current direction in each conductor, the equations for the circuit for the 'outer conductor' and 'inner conductor' can be written as follows, V = (R 1 + jωL 1 ) I 1 + jωM 12 I 2 + jωM 13 I 3 + jωM 14 I 4 (6) V = (R 2 + jωL 2 ) I 2 + jωM 21 I 1 + jωM 23 I 3 + jωM 24 I 4 (7) where R 1 and R 2 are resistance values defined derived from the E-J relationship of the conductors; L 1 , L 2 are self-inductance values of the two conductors, I 1 , I 2 , I 3 , and I 4 are current values in each branch; I is total current for the two branches; and M 12 , M 13 , M 14 , M 21 , M 23 , M 24 are the mutual inductance of branches defined as in figure 7. Due to the symmetry in the cable level, I 1 + I 2 = I 3 + I 4 ; I 1 = I 4 and I 2 = I 3 ; M 13 = M 24 .In addition, L 1 = L 2 = L due to the same geometry and material of the conductors.Substituting the equivalent parameters in equations (

Figures 12
Figures 12 plots and compares the radial magnetic field component, B r , distribution and the flux streamlines around the 2-DPC coil windings wound with the 8/2 Roebel cable and 8/2 TPS, respectively, at I t = 113.4 A. The area filled with large B r in the outermost disc of the 2 DPC coil winding wound with the 8/2 TPS is much greater than that of the 8/2 Roebel cable, while the area filled with large B r in the second disc of the 2-DPC coil winding is smaller than that of the 8/2

Figures 14 (
Figures 14(a) and (b) show the current density distribution in the 8-DPC coil windings wound with the 8/2 Roebel cable and 8/2 TPS, respectively, at I t = 113.4 A. As shown in

figure 15 (
figure 15(a), transport current flows at the outer edge of each disc for the 8-DPC winding wound with the 8/2 Reobel cable, while magnetization current is observed in each disc excluding the central disc due to symmetry to shield the radial magnetic field component in the end part of the coil winding, similar to figure 11(a).Unlike the 8-DPC coil winding wound with the 8/2 Roebel cable, the transport current is completely concentrated in the outer disc of each PC in the 8-DPC coil winding wound with the 8/2 TPS, and the inner disc of each PC carries no net current similar to the 2-DPC coil winding

TABLE 1 .
Specifications of different cables.

TABLE 2 .
Specifications of modelled coil windings.