1. Introduction
As high-temperature superconducting (HTS) wire manufacturing technology matures, development of HTS electrical power equipment has also accelerated [1], [2], [3], [4], [5], [6]. As a critical component in power transmission and conversion, transformers play an essential role in the power grid and transportation electrification. Comparing with conventional copper-based transformers, HTS transformers offer several benefits, including high efficiency, reduced weight, and low risk of fire hazard [7], [8], [9], [10], [11].
The feasibility of employing HTS transformer technology has been demonstrated through numerous research projects [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. However, the commercialization of HTS transformers has not been realized yet due to the significant cryogenic heat load arising from AC loss in the HTS winding, which causes a challenge for the cooling system [23]. Previously, a HTS 3-phase 4 MVA 25/1.2/0.44 kV traction transformer was fabricated by the Japanese Railway Technical Research Institute [23], [28]. To handle the 7 kW winding loss when operating at the rated current and 66 K, the equipped cooling system was larger and heavier than the transformer. Therefore, AC loss is one of the major problems in bringing HTS transformer technology to the market.
Extensive AC loss analyses on HTS transformers have been carried out numerically and experimentally. Numerical modeling utilizing the finite element method is an efficient tool for analyzing AC loss in HTS transformers [29], [30], [31], [32], [33], [34], [35], [36]. To mitigate AC loss in transformer windings, the effectiveness of exploiting flux diverters (FDs) has been verified and then widely discussed with the consideration of different shapes, positions, and materials [32], [33], [34], [35], [36], [37], [38], [39], [40]. However, these works overlooked the role of iron cores and failed to assess how iron cores affect AC loss of HTS transformer windings. Moreover, only a few experimental investigations have reported the total loss results for superconducting transformers during the short circuit test without subtracting the iron loss contribution [19], [25], [27], leaving the impact of iron cores on AC losses in transformer windings unknown.
Considering the impact of iron cores, recent research has reported that the inclusion of iron cores significantly increases AC loss in HTS coil windings [41], [42], [43], [44], [45]. The impacts of iron cores on AC loss in both single pancake coils and stacked double pancake coils (DPCs), with iron core geometries typically being cylindrical or a closed loop, have been systematically analyzed [43], [44], [45]. However, different from HTS coil windings in the earlier published works, transformer windings are broadly non-inductive [34]. A comprehensive AC loss study that investigates the effect of the iron core in transformer windings is still missing, due to the challenges of three-dimensional (3D) simulation methods and computational resources [46]. This has drawn attention to carrying out AC loss analyses of a 3-phase HTS 1 MVA transformer coupled with a three-limb iron core, which was previously designed and constructed by the Robinson Research Institute, Victoria University of Wellington [47]. To avoid the cooling penalty on iron core losses, the three-limb iron core is excluded from the cryogenic environment in the transformer design, as shown in Fig. 1. Previously, AC loss of one single phase was measured without the iron core [47], making the impact of the iron core on AC loss within the transformer winding unknown.
Fig. 1. The prototype of the 3-phase HTS 1 MVA 11 kV/415 V transformer: (a) the layout of the HTS transformer, (b) low voltage (LV) winding, and (c) high voltage (HV) winding. |
To address the research gap mentioned above, this work presents the results of 3D AC loss simulations for a 3-phase HTS 1 MVA transformer integrated with a three-limb iron core (see Fig. 2) utilizing the 3D T-A homogenization method. The validity of the modelling method was proven through a comparison between the simulated results in a single-phase transformer winding without the iron core and the corresponding experimental data. To explore AC loss behaviors of HTS transformer windings incorporated with a three-limb iron core, the following simulation tasks have been carried out:
Fig. 2. Schematic of the HTS 1 MVA transformer consisting of 3-phase HV and LV windings and a three-limb iron core. |
(1) Simulations of AC losses for both the stand-alone LV winding and the 3-phase HTS 1 MVA transformer integrated with the iron core
(2) Investigation of iron core materials with varying saturation magnetic fields and their impact on AC losses
(3) Influence of flux diverters with various geometries on AC losses
(4) Evaluation of the influence of turn spacing in the LV winding, wound using Roebel cables, on AC losses in transformer windings.
The simulated results for the stand-alone LV winding and transformer windings are explained by comparing their normalized current density as well as perpendicular magnetic field distributions in various configurations.
2. Numerical method
2.1. T-A Formulation and homogenization in 3D
A 3D model is essential for representing the actual structure of the HTS 3-phase transformer when a three-limb iron core is included. Moreover, the AC loss calculation of this transformer is challenged by the large number of turns in transformer windings (approximately three thousand in total), the highly non-linear characteristics of the voltage-current relationship in superconductors, and the computational complexity introduced by the iron core itself. The 3D T-A homogenization method, which simplifies HTS wires into thin strips and equates the coils to bulks, is a promising candidate for this task [46]. This numerical method has been applied for simulating HTS coil losses [45], [46], [48]. However, it has not yet been validated with a practical device.
The T-A formulation benefits from treating superconducting layers with high aspect ratios as one-dimensional thin films, thereby greatly reducing the DOF (degrees of freedom) in 3D modelling [49], [50]. The current vector potential, T, in the superconducting domain is defined to calculate the current density, J, using the following equation:
$ J=\nabla \times T $
here the current vector potential is perpendicular to the surface of the conductor. For a pancake coil centered at the origin of a 3D Cartesian coordinate, its normal vector, n, is assumed to be:
$ n=\left[\begin{array}{l} n_{x} \\ n_{y} \\ n_{z} \end{array}\right]=\left[\begin{array}{c} \frac{x}{\sqrt{x^{2}+y^{2}}} \\ \frac{y}{\sqrt{x^{2}+y^{2}}} \\ 0 \end{array}\right]$
Therefore, the current density components can be calculated as:
$ \left[\begin{array}{l} J_{x} \\ J_{y} \\ J_{z} \end{array}\right]=\left[\begin{array}{l} \frac{\partial\left(T \cdot n_{z}\right)}{\partial y}-\frac{\partial\left(T \cdot n_{y}\right)}{\partial z} \\ \frac{\partial\left(T \cdot n_{x}\right)}{\partial z}-\frac{\partial\left(T \cdot n_{z}\right)}{\partial x} \\ \frac{\partial\left(T \cdot n_{y}\right)}{\partial x}-\frac{\partial\left(T \cdot n_{x}\right)}{\partial y} \end{array}\right]$
The relationship between the electric field, E, and the current density, J, in superconductors is described through the implementation of the E-J power law in a simplified form:
$ E=E_{0} \cdot\left(\frac{J}{J_{\mathrm{c}}(B)}\right)^{n}$
here E0 = 10-4 V/m. The critical current density of HTS coated conductors, which depends on magnetic fields, is represented by Jc(B). n is the exponent of flux creep.
For the Jc(B) equation, the modified Kim model is applied [51]:
$ J_{\mathrm{c}}(B)=J_{\mathrm{c} 0} \cdot\left(1+\frac{\left|B_{\text {perp }}\right|}{B_{0}}\right)^{-\alpha}$
here Jc0, B0, and α are derived from critical current experiments for the conductor under external magnetic fields. The magnetic field component that is perpendicularly aligned to the superconducting layer is denoted as Bperp.
To solve the magnetic flux density, B, the magnetic vector potential, A, is applied to the entire model:
$ B=\nabla \times A$
Moreover, in the magnetic field module (mf.) of COMSOL, the governing equation is employed for A:
$ \nabla \times\left(\frac{1}{\mu_{0} \mu_{\mathrm{r}}} \nabla \times A\right)=J$
here the vacuum permeability is μ0, while the relative permeability of the magnetic material is μr. The µr values of iron cores and FDs in this work are derived from B-H curves of respective materials. As shown in Fig. 3, Fig. 4, “Silicon Steel GO 3423” and “Silicon Steel GO 3411” are used for iron cores, while “Alloy Powder Core Hiflux 60 mu” is employed to FDs [52], [53], [54]. The Highflux 60 mu is a powder material with a high resistivity at 77 K. Its eddy current loss is negligible at 50 Hz. As shown in Fig. 5, the hysteresis loss of this material is measured at 77 K [54].
Fig. 3. B-H curves of silicon steel GO 3423 and 3411 used in iron cores. |
Fig. 4. B-H curve of alloy powder core Hiflux 60 mu employed in flux diverters. |
Fig. 5. Measured hysteresis loss of Highflux 60 mu at 77 K. |
The homogenization method assumes that the electromagnetic characteristics of a coil wound with superconducting wires can be approximated by an anisotropic bulk [55], [56], [57], [58]. To ensure that HTS layers maintains their original transport currents, Dirichlet boundary conditions are implemented on the top and bottom surfaces of the homogenous bulk, as indicated in Fig. 6 [50]:
$ I=\left(T_{1}-T_{2}\right) \cdot d$
here T1 and T2 stand for the top and bottom surface values of T, respectively, while d is the superconducting layer thickness (here considered to be 1 μm). In the homogeneous bulk, the calculated current density defined for A formulation is:
$ J_{\text {bulk }}=J \cdot \frac{d}{t_{\mathrm{HTS}}}$
here tHTS represents the entire thickness of the HTS wire.
Fig. 6. Boundaries of the homogenous bulk. |
To further fulfill the boundary definition, the internal and external surfaces are constrained by Neumann boundary conditions [46]:
$ \frac{\partial\left(n_{x} \cdot T_{x}+n_{y} \cdot T_{y}+n_{z} \cdot T_{z}\right)}{\partial n}=0$
As shown in Fig. 1, Fig. 2, the geometry of the 1 MVA transformer exhibits symmetry with respect to the x-z and x-y planes. Therefore, magnetic insulation and perfect magnetic conductor boundary conditions are employed for the cross-sections of the x-z and x-y planes in the simulation model to simplify the model into a quarter.
The AC losses for HTS windings are estimated in units of (J/m/cycle) and Watt (W) as:
$ Q=2 \frac{d}{t_{\mathrm{HTS}} \cdot L} \int_{\frac{T}{2}}^{T} \iiint_{V} E \cdot J \mathrm{~d} V \mathrm{~d} t$
$ P=Q \cdot f \cdot L$
where tHTS represents the overall thickness of the HTS wire, f is the operating frequency, T stands for the corresponding period of one cycle, and L is defined as the wire consumption of the simulated windings. In this work, the operating frequency of the 3-phase 1 MVA transformer is 50 Hz. At this frequency, the T-A formulation, which adopts the thin film approximation, is capable of accurately simulating the AC loss of HTS coil windings [59], [60].
2.2. Model validation
The 3D T-A homogenization method is verified through a comparison of the calculated AC loss of the single-phase transformer winding while excluding the iron core with the measured data [47] and simulated values from our previous work [35]. As presented in Fig. 1 (b), the solenoid LV winding per phase is a 20-turn single-layer winding, made of 15/5 Roebel cable. The HV winding, which is depicted in Fig. 1 (c), consists of 24 DPCs in a single phase. These DPCs are wound with YBCO 4-mm wide coated conductor, and each of them has 381/4 turns. Modelling parameters for a single phase in the 1 MVA transformer can be found in Table 1 [32].
Table 1. Modelling parameters for a single phase in the 1 MVA transformer. |
| LV | HV | |
|---|---|---|
| Inner diameter of winding (mm) | 310 | 345 |
| Width of the conductor (mm) | 12.1 | 4 |
| Thickness of the conductor (mm) | 0.8 | 0.22 |
| Turn number in x direction | 20 | 48 |
| Turn number in z direction | 1 | 19 |
| Total turn number | 20 | 912 |
| Number of Roebel strand | 15 | − |
| Width of Roebel strand (mm) | 5 | − |
| Gap between Roebel stacks (mm) | 2.1 | − |
| Axial gap between turns, dts (mm) | 2.1 | 2.13 |
| Constant critical current, Ic (A) | 2226 | 118.7 |
| Amplitude of the rated current (A) | 1964 | 42.9 |
Magnetic field dependence parameters and n values for the transformer windings are given in Table 2. The values of n are set based on the measured n-Bperp curve of the HTS wire. Here, Bperp values for the transformer windings are obtained from our previous simulations of the 1 MVA transformer, conducted using the two-dimensional (2D) T-A homogenization method [35].
Table 2. Magnetic field dependence parameters and n values for the transformer windings. |
| Parameters | LV | HV |
|---|---|---|
| Jc0 (A/m2) | 3.55 × 1010 | 2.12 × 1010 |
| B0 (mT) | 149 | 149 |
| α | 0.6 | 0.6 |
| n | 19 | 17 |
Fig. 7 presents the structured meshes for a quarter model of a single-phase winding. In the LV winding, the 15/5 Roebel cable consists of fifteen 5 mm wide strands. Each strand is fully transposed and carries the same current [61], [62]. As shown in Fig. 7 (b), each Roebel turn comprises two parallel stacks of strands, which equates to a DPC in the simulation model [62], [63]. Each stack is assumed to have eight strands to avoid the modeling complexity introduced by an uneven strand in these two stacks. This assumption does not impact the loss results for the LV winding [32], [47]. Since the end discs of the windings experience a large Bperp and contribute to most of the AC loss, denser meshes are employed for these end discs, as marked in Fig. 7 (b). Regarding the LV and HV windings, their end discs are partitioned into 4 and 8 sub-blocks in the x-direction, respectively, while 12 and 8 divisions are applied in the z-direction. For the middle discs of the LV and HV windings, 3 and 6 sub-blocks are designated in the x-direction, and 8 and 6 divisions in the z-direction, respectively. Along the semi-circular circumference of the discs, 72 and 80 elements are allocated for all discs in the LV and HV windings, respectively. To reduce mesh elements without compromising the accuracy of calculations, two air domains are required. Air domain_1 is customized with a minimum element size of 1 μm, while the meshing of air domain_2 follows the normal settings. However, the solved DOF of this model still reaches 1,865,639. It is worth emphasizing that to ensure consistency in subsequent analyses and avoid any difference that comes from meshes, HTS windings utilize the same meshes in all simulation models as described above.
AC loss simulations of both the single-phase and 3-phase transformer windings are carried out when assuming the LV winding is short-circuited. The phase difference between HV and LV windings, denoted as θHV - θLV, can be obtained by [47]:
$ \theta_{\mathrm{HV}}-\theta_{\mathrm{LV}}=\arg \left(-1+\frac{\mathrm{i} R}{\omega L_{\mathrm{LV}}}\right)$
here arg is a function that refers to the argument of a complex number, i is the imaginary unit, R is the load resistance connected across the LV winding, ω is the angular frequency, and LLV stands for the self-inductance of the LV winding. Since the LV winding is in short circuit in this work (R = 0), the currents in LV and HV winding are in phase opposition.
Fig. 7. Structured meshes for a quarter model of the single-phase transformer windings (upper half): (a) meshes for the model, (b) meshes for the transformer windings, presented in an enlarged view. |
Fig. 8 compares the simulated results in a single-phase transformer winding without the iron core using both 3D and 2D T-A homogenization methods with its experimental results [47]. When f = 50 Hz, the experimental AC loss value is 107 W at the rated current, while the simulated values are 84.9 W and 82.2 W in 3D and 2D models, respectively. The difference between the experimental data and the simulated results may arise from numerous factors, such as slight differences in coil winding geometries compared to the design parameters including horizontal and vertical turn gaps, non-uniform critical current along the conductor length, and non-uniform lateral critical current density distributions in the conductors. The loss results simulated using the 3D T-A homogenization agree better with experiments than those of the 2D method. Due to the high computational resource demands, the high-performance computing cluster, Rāpoi, is the main platform for running the simulations. With 2 AMD EPYC 7742 64-Core processors and 128 GB RAM, the computation time for the single-phase model at the rated current still amounted to 42.8 h.
Fig. 8. Comparison of the simulated results in a single-phase transformer (TX) winding without the iron core with its experimental data (f = 50 Hz). WOIC stands for without the iron core. |
2.3. Model description of the 3-phase transformer with a three-limb iron core
Fig. 9 presents the structured mesh for the 3-phase transformer model with a three-limb iron core. Even with the high-performance computing cluster, treating all the 3-phase transformer windings as superconductors poses a considerable challenge to computer performance. Therefore, copper windings are used to generate equivalent magnetic fields to mitigate the computational burden caused by the highly non-linear E-J relationships of superconductors [64], [65]. In the simulation model, when one phase is superconducting, the other two phases are composed of copper. For instance, phase B features HTS windings, whereas copper is utilized for phases A and C. The meshes for HTS windings remain consistent as described in the model validation section. For each disc in the copper LV and HV windings, the following subdivisions are applied: 2 sub-blocks along the x-direction for the LV winding and 4 sub-blocks for the LV winding, 4 divisions in the z-direction for both LV and HV windings, and 36 elements along the circumference of the semicircle for the LV winding, with 40 elements for the HV winding. Even so, the DOF of this model increases to 3,010,621, and the computational time at the rated current is 62.5 h.
Fig. 9. Structured meshes for the 3-phase 1 MVA transformer with the three-limb iron core. |
Fig. 10 plots the currents in each phase of HV and LV windings under the rated condition at f = 50 Hz. To avoid the convergence issue, the initial values of the currents applied to the HTS windings, taking phase B as an example, are set to zero.
Fig. 10. Rated currents in each phase of the transformer (f = 50 Hz): (a) applied currents in HV windings, (b) applied currents in LV windings. |
3. Results and discussions
Table 3. Design parameters of the three-limb iron core. |
| Iron core | |
|---|---|
| Type | Three-phase three-limbs |
| Diameter (mm) | 225 |
| Window height (mm) | 910 |
| Center distance between limbs (mm) | 590 |
| Effective sectional area (cm2) | 350.6 |
| Material | Grain-oriented silicon steel |
| Flux density (T) | 1.54 |
3.1. Stand-alone LV winding coupled with a three-limb iron core
To explore how the iron core affects AC losses in both inductive and non-inductive windings, AC losses of the stand-alone LV winding with/without an iron core (WIC/WOIC) are first simulated at various operating currents. This stand-alone LV winding, which has the same electrical and geometrical configurations as the LV winding in the 3-phase transformer, serves as a reference for the inductive winding [34].
Fig. 11 compares simulated AC losses of the stand-alone LV winding in the cases of WIC and WOIC at different current amplitudes when f = 50 Hz. The results show that the AC loss in the stand-alone LV winding increases significantly with the inclusion of the iron core. Moreover, the difference in loss values between the cases of WIC and WOIC grows as the current amplitude increases. At It, peak = 1964 A, the loss of the stand-alone LV winding in the case of WIC is 354.1 W, compared to 215.5 W of WOIC. The AC loss value at Irated is increased by 48.7% owing to the inclusion of the iron core.
Fig. 11. Simulated AC losses of the stand-alone LV winding with/without the iron core (f = 50 Hz). WIC stands for with the iron core. |
Fig. 12 shows the AC loss in each disc of the stand-alone LV winding WIC/WOIC when f = 50 Hz and Irated = 1964 A. As presented in Fig. 7 (b), the discs are numbered in sequence from the upper end to the center of the winding. In both cases, the AC loss in the end discs is much larger than in the discs near the center. Compared to the case of WOIC, an increase in AC loss is observed in most discs when the iron core is present, apart from the central discs numbered from 17 to 20. E.g., the loss value of disc 1, denoted as the end disc, is increased from 31.4 W to 43.5 W with the presence of the iron core.
Fig. 12. Loss in each disc of the stand-alone LV winding WIC/WOIC (f = 50 Hz, Irated = 1964 A). |
Fig. 13 depicts Bperp distributions and magnetic flux lines of the end six discs in the stand-alone LV winding WIC/WOIC at f = 50 Hz, Irated = 1964 A, and t = 3/4 T. Comparing Fig. 13 (a) and (b), the end four discs are exposed to a significantly larger Bperp when the stand-alone LV winding in the case of WIC, compared to the case of WOIC. Furthermore, the area occupied by a large Bperp is larger in all six discs in the case of WIC. With the incorporated iron core, the magnetic flux lines become more perpendicular to the HTS tape surface in the surrounding area, leading to the AC loss increase in these discs. Fig. 13 provides explanations for the loss results presented in Fig. 12.
Fig. 13. Bperp distributions and magnetic flux lines of the end six discs in the stand-alone LV winding WIC/WOIC. (f = 50 Hz, Irated = 1964 A, t = 3/4 T): (a) WIC, and (b) WOIC. The red dashed rectangles highlight the area with a large perpendicular magnetic field. |
Fig. 14 plots J/Jc distributions of the end six discs in the stand-alone LV winding WIC/WOIC at f = 50 Hz, Irated = 1964 A, and t = 3/4 T. AC loss is resulted from the high current density area where |J/Jc|>1 due to magnetic field penetration. In both cases, currents with a reverse direction flow in all discs to shield the Bperp. As shown in Fig. 14 (a), when the stand-alone LV winding WIC, the end six discs exhibit a larger region where |J/Jc|>1. Furthermore, the area filled with shielding current in these discs is also broader compared to the case of WOIC. As presented in Fig. 13, Fig. 14, the area filled with a large Bperp is equivalent to the |J/Jc|>1 area. Fig. 14 further supports the results presented in Fig. 12. Hence, in the later section, the AC loss characteristics are only described by plotting Bperp distributions.
Fig. 14. J/Jc distributions of the end six discs in the stand-alone LV winding WIC/WOIC (f = 50 Hz, Irated = 1964 A, t = 3/4 T): (a) WIC, and (b) WOIC. |
3.2. 3-ph ase HTS 1 MVA transformer coupled with a three-limb iron core
Fig. 15 presents AC losses of phase B in the cases of WIC/WOIC at various current amplitudes when f = 50 Hz. Unexpectedly, in the case of WIC, there is only a minor increase in the loss of phase B under various current amplitudes compared to the case of WOIC. With an increase in current, the difference in AC loss values between the cases of WIC and WOIC grows. At Irated, 85.9 W of AC loss is obtained in the case of WIC, while it is 84.9 W in the case of WOIC. The presence of the iron core only results in a 1 W loss increase in the transformer winding of phase B, which differs from the results derived from the stand-alone LV winding.
Fig. 15. AC losses of phase B WIC/WOIC (f = 50 Hz). |
For phases A and C, their loss behaviors are expected to be similar to phase B, as each phase is independently coupled with a single limb of the core, and the variation of magnetic flux in the single limb is equivalent within one cycle. Although the difference in AC loss values between configurations of WIC and WOIC for phases A and C may vary from the results obtained for phase B, the minor difference observed in phase B could still be used to predict the AC losses in phases A and C. Therefore, later discussions focus only on phase B to assess the impact of the iron core on the AC loss of the 1 MVA transformer.
Fig. 16 shows the loss value of each disc within the HV and LV windings of phase B WIC/WOIC at Irated and f = 50 Hz. As for phase B WOIC, AC loss in the HV winding is 26.8 W, while 58.1 W in the LV winding. For the case of WIC, the loss in the HV winding decreases to 25.5 W, while the loss in the LV winding increases to 60.4 W. In either case, the LV winding loss dominates the total loss of phase B. As plotted in Fig. 16 (a) and (b), the difference in loss values caused by the iron core in each disc is only noticeable in discs 1 and 2 for both the HV and LV windings. Regarding the end disc in the HV winding, disc 1, its loss value decreases from 6 W to 5.5 W after integrating with the iron core. On the contrary, the loss value of the end disc in the LV winding increases from 7.6 W to 8.3 W in the case of WIC. Fig. 16 reveals that the iron core only influences the loss values of the discs near the ends of windings for transformer windings.
Fig. 16. Loss value of each disc within the HV and LV windings of phase B WIC/WOIC at Irated (f = 50 Hz): (a) HV winding, and (b) LV winding. |
Fig. 17 presents Bperp distributions and magnetic flux lines for phase B WIC/WOIC at Irated, f = 50 Hz, and t = 3/4 T. Similar Bperp distributions for phase B with and without the iron core in Fig. 17 (a) and (b) prove that the difference in AC loss between the cases of WIC and WOC for the transformer windings is minor. As shown in Fig. 17 (a), the iron core with a high relative permeability attracts magnetic flux lines from adjacent windings. When making a comparison between Fig. 17 (a) and (b), the LV winding is positioned next to the iron core, causing the core to draw more of the flux lines perpendicular to the LV winding, resulting in the increased AC loss value for the LV winding. On the other hand, the HV winding is situated slightly farther from the iron limb of phase B. As shown in Fig. 10, the HV and LV windings carry currents in opposite directions. Consequently, a stronger magnetic flux density is generated between the HV and LV windings, effectively shielding the influence from the iron core for the HV winding. However, the HV winding of phase B is influenced by the windings of Phase C. This results in flux lines aligning more parallel to the HV winding in phase B, leading to a minor decrease in the AC loss value. In both Fig. 17 (a) and (b), only the end discs remain exposed to a larger Bperp, particularly the discs at the end of the upper-half transformer windings. Fig. 17 further supports the results presented in Fig. 16.
Fig. 17. Bperp distributions and magnetic flux lines for phase B WIC/WOIC at Irated (f = 50 Hz, t = 3/4 T). |
3.3. AC loss dependence on different saturation magnetic flux densities of the iron materials
Iron core materials featuring varying saturation magnetic flux densities are taken into account to analyze their impact on AC loss of both the stand-alone LV winding and transformer windings. Fig. 3 plots the B-H curves of the iron materials. The saturation magnetic flux density of ‘Silicon Steel GO 3423′ is approximately 1.5 T, denoted as low saturation, while 1.9 T of ‘Silicon Steel GO 3411′ is referred as high saturation.
Fig. 18 presents a comparison of the loss results between the stand-alone LV winding and phase B with different iron cores at various currents when f = 50 Hz. Incorporating a high saturation field iron core results in greater loss values for both the stand-alone LV winding and phase B of the 1 MVA transformer, compared with using an iron core with a low saturation field. This difference can be attributed to the variation in the relative permeability between the two iron materials. It is worth emphasizing that, at all current amplitudes, the magnetic flux densities within the cores are below their saturation limits. When It,peak of the LV winding is 1964 A, the loss in the stand-alone LV winding is 365.2 W with the high saturation field iron core, while it is 354.1 W with the low saturation field iron core. However, the increase in the loss value of phase B when using the high saturation iron core is only 0.003 W. Compared with the non-inductive transformer winding, the saturation magnetic flux densities of iron cores have a more pronounced impact on the inductive stand-alone winding.
Fig. 18. Comparisons of loss results between the stand-alone LV winding and phase B with different iron cores at various currents (f = 50 Hz). |
Fig. 19 shows magnetic flux density distributions and flux lines of the stand-alone LV winding with various iron cores at f = 50 Hz, Irated = 1964 A, and t = 3/4 T. In both cases, neither of the iron cores is in the saturated state. However, the iron core with a high saturation magnetic flux density more effectively attracts flux lines from the stand-alone LV winding, resulting in a broader region filled with a stronger magnetic flux density in the core. Consequently, the high saturation iron core, with a higher relative permeability, leads to a larger increase in loss for the stand-alone LV winding than the case with the low saturation iron core.
Fig. 19. Magnetic flux density distributions and flux lines of the stand-alone LV winding with different iron cores (f = 50 Hz, Irated = 1964 A, t = 3/4): (a) low saturation, and (b) high saturation. |
3.4. Stand-alone LV winding and 3-phase 1 MVA transformer coupled with the iron core and flux diverters
Fig. 20 illustrates the positions of FDs attached to the ends of the HV and LV windings for phase B. The detailed parameters of two flux diverter combinations, labeled as FDs_C1 and FDs_C2, are provided in Table 4. The dimensions of FDs for the HV winding are denoted as WHV and HHV, whereas WLV and HLV are used for the LV winding. The We, HV and We, LV indicate the overhang distance to the inner radius of HV and LV windings, respectively. The gHV and gLV represent the gaps between the end of the HV or LV winding and FDs. In Combination 1, square flux diverters are employed for the HV winding. As the LV winding loss dominated the total loss in phase B and We, LV has a large influence on loss, a rectangular-shaped FD with a wider We, LV is considered in combination 2 [33]. For the stand-alone LV winding, the FD parameters in Table 4 are used and we recap the FDs for the LV winding in Table 5. The material used for the FDs is ‘Alloy Powder Core Hiflux 60 mu’, which can be referred as high flux 60μ [52], [54]. Fig. 4 plots the B-H curve of this material. It is worthwhile mentioning that the eddy current loss in FDs is negligible [53], [54].
Fig. 20. The positions and dimensions of FDs attached to the ends of the HV and LV windings. |
Table 4. FD parameters used in different combinations for the 1 MVA transformer. |
| Parameter | Combination 1 (FDs_C1) | Combination 2 (FDs_C2) |
|---|---|---|
| WHV (mm) | 8.18 | 8.18 |
| HHV (mm) | 8.18 | 8.18 |
| We, HV (mm) | 2 | 2 |
| gHV (mm) | 2 | 2 |
| WLV (mm) | 4.8 | 12.8 |
| HLV (mm) | 4.8 | 4.8 |
| We, LV (mm) | 2 | 6 |
| gLV (mm) | 2 | 2 |
Table 5. FD parameters used for the stand-alone LV winding. |
| Parameter | FDs_1 | FDs_2 |
|---|---|---|
| WLV (mm) | 4.8 | 12.8 |
| HLV (mm) | 4.8 | 4.8 |
| We, LV (mm) | 2 | 6 |
| gLV (mm) | 2 | 2 |
Table 6, Table 7 present loss values of the stand-alone LV winding and phase B in the 1 MVA transformer with the iron core and FDs at the rated current. FDs significantly reduce AC loss in the stand-alone LV winding and phase B. For the stand-alone LV winding integrated with the iron core, using FDs_1 results in a 17.7% reduction in loss, while a more substantial decrease of 40.6% is achieved with FDs_2, which has a wider WLV. The loss reduction rates for phase B with the iron core using combinations 1 and 2 are 21.8% and 32.8%, respectively. Comparing losses in the different windings of phase B with and without FDs_C2, the loss value of the HV winding is decreased from 25.5 W to 14.8 W, while the loss in the LV winding drops from 60.4 W to 46.9 W. This indicates that the geometrical design of FDs plays a significant role in reducing AC loss [33]. By applying FDs_C2 and considering a cooling penalty of 30 (requiring 30 W of energy at room temperature to extract 1 W of heat from the cryogenic environment) [22], the overall efficiency of this HTS transformer can increase from 98.5% [22] to 99%, based on Minimum Energy Performance Standard for a transformer operating at 50% of its rated load [22], [67].
Table 6. Loss values of the stand-alone LV winding with the iron core and different FDs at Irated. |
| Stand-alone LV winding | |||
|---|---|---|---|
| Cases | WIC | WIC_FDs_1 | WIC_FDs_2 |
| Values (W) | 354.1 | 296.5 | 234.5 |
Table 7. Loss values of phase B with the iron core and different FDs at Irated. |
| Phase B | |||
|---|---|---|---|
| Cases | WIC | WIC_FDs_C1 | WIC_FDs_C2 |
| Values (W) | 85.9 | 69 | 61.7 |
Fig. 21 shows Bperp distributions and magnetic flux lines of the stand-alone LV winding incorporating the iron core and different FDs at f = 50 Hz, Irated = 1964 A, and t = 3/4 T. Compared to the configuration without FDs, the use of FDs leads to a reduction in the Bperp amplitude and a reduced area filled with large Bperp, which implies the loss reduction. When the stand-alone LV winding is coupled with the iron core and FDs_2, the magnetic flux lines are more parallel to the coil surface, leading to a further shrunk area with a large Bperp, especially for the end four discs. Moreover, with wider FDs, a higher magnetic field concentration is observed within the FDs. However, the maximum magnetic flux density (Bmax) in FDs_1 and FDs_2 is 0.54 T and 0.68 T at Irated, respectively, which are still below the saturation field of 1.5 T. For the stand-alone LV winding coupled with the iron core and FDs_2, the calculated hysteresis loss of FDs_2 is 3.8 W.
Fig. 21. Bperp distributions and magnetic flux lines of the stand-alone LV winding with the iron core and different FDs (f = 50 Hz, Irated = 1964 A, t = 3/4): (a) WIC, (b) WIC_FDs_1, (c) WIC_FDs_2. The black dashed rectangles highlight the area with a large perpendicular magnetic field. |
Fig. 22 shows Bperp distributions and magnetic flux lines of phase B with the iron core and different combinations of FDs at Irated, f = 50 Hz, and t = 3/4 T. As plotted in Fig. 22 (b) and (c), the utilization of FDs greatly reduces the amplitude of Bperp in the end discs of the transformer windings. Furthermore, increasing WLV further reduces the area with large Bperp when using FDs_C2. With the combination of the iron core and FDs_C2, the Bmax in the FDs of the HV and LV windings is 0.26 T and 0.34 T, respectively. For the transformer winding coupled with the combination of the iron core and FDs_C2, the calculated hysteresis loss of FDs in the HV and LV windings is 0.87 W and 1.17 W, respectively. Even including the loss values of FDs, the overall efficiency of the HTS transformer can still be greater than 99%. The obtained results show that FDs perform effectively in reducing AC loss in both inductive and non-inductive windings coupled with the iron core.
Fig. 22. Bperp distributions and magnetic flux lines of phase B with the iron core and different FDs at Irated (f = 50 Hz, t = 3/4): (a) WIC, (b) WIC_FDs_C1, (c) WIC_FDs_C2. The red and blue dashed rectangles highlight the area with a large perpendicular magnetic field. |
3.5. Influence of turn spacing on loss of the LV winding for the 1 MVA transformer coupled with the iron core and flux diverters
In the 3-phase 1 MVA transformer, the LV windings are made of Roebel cables. As depicted in Fig. 20, the turn spacing (dts) represents the gap between each Roebel turn. Fig. 23 shows the loss dependence of phase B in the 1 MVA transformer with the iron core and FDs_C1 on dts, with dts = 1 mm, 2.1 mm, 3 mm, and 4 mm. The half-height of the HV winding is 146 mm. With dts values set at 1 mm, 2.1 mm, 3 mm, and 4 mm, the relative height of the HV winding is 15.6 mm taller, 5.1 mm taller, 3.4 mm shorter, and 12.9 mm shorter than the LV winding, respectively. At Irated, the maximum AC loss in Phase B of the 1 MVA transformer occurs when dts = 1 mm, and the minimum AC loss is achieved when dts = 2.1 mm. This variation is attributed to the optimized cancellation of Bperp, which depends on the relative height between the LV and HV windings [32].
Fig. 23. Loss dependency of phase B on dts with the iron core and FDs_C1 at Irated (f = 50 Hz). |
Fig. 24 plots the AC loss values of each disc within the LV winding in phase B with the iron core and FDs_C1 for different dts values at Irated and f = 50 Hz. As listed in Table 1, the space between Roebel stacks is 2.1 mm. Therefore, when dts is 2.1 mm, the uniform gap between Roebel stacks and each turn in the LV winding leads to a better cancellation of Bperp, resulting in a uniform AC loss distribution from discs 3 to 20. When dts is not equal to the gap of the Roebel stacks, an uneven gap causes fluctuations in the AC loss within each disc of the LV winding. With a smaller turn spacing of dts = 1 mm, a larger Bperp is generated in the discs near the upper end due to the magnetic flux superposition with a denser arrangement of the LV winding. However, when dts = 4 mm, the enlarged turn spacing causes weakened magnetic field interactions between each turn and a reduced Bperp cancellation for the discs 9 to 20, making their AC loss larger than the cases with different turn spacings [68].
Fig. 24. AC loss values of each disc within the LV winding in phase B with the iron core and FDs_C1 for varied dts and Irated (f = 50 Hz). |
4. Conclusions
In this work, the influence of a three-limb iron core on the AC loss behaviors of a stand-alone LV winding and transformer windings in a 1 MVA HTS transformer is systematically studied for the first time by using the 3D T-A homogenization method.
A substantial loss increase is observed in the simulated results of the stand-alone LV winding with the iron core at different current amplitudes compared to the case excluding the iron core. Moreover, a 138.6 W (48.7%) increase in loss is obtained with the inclusion of the iron core at the rated current. This can be attributed to the strong penetration of Bperp in most discs of the stand-alone LV winding owing to the iron core. On the contrary, the AC loss value phase B increases by only 1 W (1.2%) at the rated current. This minor difference is owing to the strong magnetic field generated in the space between the LV and HV windings, which effectively shields the influence of the iron core.
Compared to the case of coupling with a low saturation iron core, the iron core with a high saturation magnetic flux density contributes to an 11.1 W (3%) loss increase in the stand-alone LV winding due to its higher relative permeability. However, the high saturation iron core has a negligible impact on the loss of the transformer windings.
Flux diverters are effective in reducing AC loss in the stand-alone LV winding and transformer windings, even when the iron core is coupled with the windings. Moreover, the loss reduction shows dependence on the geometry of the flux diverters. The loss reduction of 119.6 W (40.6%) for the stand-alone LV winding and 24.2 W (32.8%) for the single phase in the 1 MVA transformer is achieved when applying wider FDs for the LV winding.
Regarding the LV winding in the 1 MVA transformer with the iron core and flux diverters, an optimized cancellation of Bperp can be realized by adjusting the turn spacing between each Roebel turn, thereby minimizing the loss of the transformer.
CRediT authorship contribution statement
Yue Wu: Conceptualization, Data curation, Formal analysis, Methodology, Validation, Visualization, Writing - original draft. Shuangrong You: Software. Jin Fang: Conceptualization, Project administration, Supervision. Rodney A. Badcock: Funding acquisition, Writing - review & editing. Nicholas J. Long: Supervision, Writing - review & editing. Zhenan Jiang: Conceptualization, Formal analysis, Funding acquisition, Methodology, Project administration, Resources, Supervision, Validation, Writing - review & editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
This work was supported by the New Zealand Ministry of Business, Innovation and Employment (MBIE) Strategic Science Investment Fund “Advanced Energy Technology Platforms” under contract No. RTVU2004. Yue Wu acknowledges financial support from the Chinese Scholarship Council (CSC) and the CSC/Victoria University of Wellington Scholarship. The authors extend their appreciation to the staff and facilities of Rāpoi for their invaluable support. Additionally, the authors acknowledge Nancy Marquez of Victoria University of Wellington for her assistance with grammar checks.
