Core Stress Analysis of Amorphous Alloy Transformer for Rail Transit Under Different Working Conditions

energies

Article Core Stress Analysis of Amorphous Alloy Transformer for Rail Transit under Different Working Conditions

Jianwei Shao, Cuidong Xu and Ka Wai Eric Cheng *

Department of Electrical Engineering, Faculty of Engineering, The Hong Kong Polytechnic University, Hong Kong 999077, China; [email protected] (J.S.); [email protected] (C.X.) * Correspondence: [email protected]; Tel.: +852-27666162

Abstract: The rail transit system is a large electric vehicle system that is strongly dependent on the energy technologies of the power system. The use of new energy-saving amorphous alloy transformers can not only reduce the loss of rail transit power, but also help alleviate the power shortage situation and electromagnetic emissions. The application of the transformer in the field of rail transit is limited by the problem that amorphous alloy is prone to debris. this paper studied the stress conditions of amorphous alloy transformer cores under different working conditions and determined that the location where the core is prone to fragmentation, which is the key problem of smoothly integrating amorphous alloy distribution transformers on rail transit power supply systems. In this study, we investigate the changes in the electromagnetic field and stress of the amorphous alloy transformer core under different operating conditions. The finite element model of an amorphous alloy transformer is established and verified. The simulation results of the magnetic field and stress of the core under different working conditions are given. The no-load current and no-load loss are simulated and compared with the actual experimental data to verify practicability of amorphous alloy transformers. The biggest influence on the iron core is the overload state and the maximum value is higher than the core stress during short circuit. The core strain caused by the

side-phase short circuit is larger than the middle-phase short circuit.

Citation: Shao, J.; Xu, C.; Cheng, Keywords: amorphous alloy transformer; large electric vehicle system; magnetostriction; normal K.W.E. Core Stress Analysis of load; load imbalance; overload; short circuit; no-load current; no-load loss Amorphous Alloy Transformer for Rail Transit under Different Working Conditions. Energies 2021, 14, 164. https://doi.org/10.3390/en14010164 1. Introduction New materials are now increasingly used in the traditional power industry, bringing Received: 15 October 2020 new solutions to the problems in the power system. The rail transit system is a large Accepted: 17 December 2020 power system and the use of new energy-saving amorphous alloy transformers, not only Published: 30 December 2020 can reduce the loss of rail transit power, but also help alleviate the power shortage situation [1–3]. Amorphous alloy is increasingly used in the iron core of power transformers Publisher’s Note: MDPI stays neu- due to its excellent low loss performance [4,5]. The advantages of using amorphous fer- tral with regard to jurisdictional clai- romagnetic alloys as a replacement for grain-oriented Si-steel in power transformers are ms in published maps and institutio- widely reported owing to their low no-load core losses [6]. With the rapid development nal afﬁliations. of amorphous alloy technology, the energy-saving advantages of amorphous alloy core distribution transformers have gradually been accepted by manufacturers and users [7,8]. Due to the shortcomings of amorphous alloy strips such as high hardness, poor short-circuit Copyright: © 2020 by the authors. Li- resistance, and high magnetostriction, the application and market promotion of amorphous censee MDPI, Basel, Switzerland. alloy transformers are limited [9]. This article is an open access article Therefore, it is important to study the stress conditions of amorphous alloy trans- distributed under the terms and con- former cores under different working conditions, and determine where the core is prone ditions of the Creative Commons At- to fragmentation. This is the key problem of smoothly integrating the amorphous alloy tribution (CC BY) license (https:// distribution transformer into rail transit power supply systems [3]. Many scholars have creativecommons.org/licenses/by/ conducted research on improving the short-circuit resistance of amorphous alloys, reducing 4.0/).

Energies 2021, 14, 164. https://doi.org/10.3390/en14010164 https://www.mdpi.com/journal/energies Energies 2021, 14, 164 2 of 23

vibration and noise, and improving process measures [10]. Du et al. [11] focused on magnetostriction variation at different positions on the core surface to identify the correlations between the vibration amplitude and frequency. Liu et al. [12] presented a noise reduction measurement for amorphous alloy core distribution transformers and verification was carried out by some experiments. This paper emphasizes discussion on the effects of temperature and different materials. Liu et al. [13] developed an analytical model for copper loss calculation of Litz-wire in amorphous/nanocrystalline core-based high-frequency transformer. Haifeng et al. [14] designed a new type of clamp to withstand the huge force caused by short-circuit troubles and with the help of ANSYS software, the stress and strain of the end covers and the winding clamps were calculated, respectively, confirming the feasibility of this new structural method. Guo [15] proposes a three-dimensional buckling finite element method (FEM) to calculate the tilt limit force of the rectangular winding of an amorphous alloy transformer and experimental results of the short-circuit test on several amorphous alloy transformers verify the effectiveness of the proposed method. With a large number of amorphous alloy transformers in operation, oil-immersed amorphous alloy transformer faults caused by iron core debris of amorphous alloy are more and more common [7]. There are two sources of core debris of amorphous alloys. One is the debris generated during the manufacturing, transportation, and assembly processes before operation, and the other is the core debris generated during operation of the amorphous alloy transformer. When the transformer is in operation, due to the mechanical sensitivity of the amorphous alloy, the transformer core will be deformed because of the electromagnetic force and the hysteresis force, resulting in core debris [16]. According to [16], in an amorphous alloy transformer, the strain caused by magnetostriction is much greater than the strain caused by the magnetic field force. Different working conditions such as unbalanced load, overload, and short circuit of transformer operation will cause changes of core force. After the amorphous alloy transformer is put into operation, the core debris will affect the electromagnetic performance, resulting in excessive temperature rise of the transformer, faster deterioration of the insulation medium, insulation failure, shorter effective working time of the transformer and increase of leakage field and electromagnetic interference. Therefore, this paper focuses on the research gap in the amorphous alloy core devel- opments and studies the stress of amorphous alloy transformer cores caused by magnetostriction under different working conditions and seeks to determine where the core is prone to fragmentation. In this paper, the finite element analysis method is used to study stress conditions of amorphous alloy transformer cores under different working conditions. The finite element method [17] is an effective method to analyze the electromagnetic characteristics of amorphous alloy transformers. Bahmani [18] used the equivalent elliptic loop (EEL) method in Ansoft Maxwell 3D (Ansoft Corp., Pittsburgh, USA) to calculate the core losses of high-frequency high-power transformers and compared them with the empirical equations, verifying the practicability of the finite element method in electromagnetic field analysis. Chang et al. [19] studied the magneto mechanical effects of three-phase three-leg transformers with amorphous cores in different bending structures, where the magnetic properties of audible noises related to core vibrations are discussed. Experimental results in this paper indicate that amorphous-cored transformers with rectangular cores have higher vibration intensities. This paper is organized as follows. Section2 discusses the mathematical model of stress and strain induced by magnetostriction. Section3 establishes a simulation model based on the structure, and physical and electromagnetic parameters of the actual amorphous alloy transformer. The simulation results of the magnetic field and stress of the core of amorphous alloy transformers under different working conditions are given in Section4. The no-load current and no-load loss are simulated and compared with the actual experimental data to verify the practicability of amorphous alloy transformers in Section5. The discussion and conclusion are given in Sections6 and7, respectively. The research framework of the whole paper is shown in Figure1. Energies 2021, 14, x FOR PEER REVIEW 3 of 24

4. The no-load current and no-load loss are simulated and compared with the actual ex- Energies 2021, 14, 164 perimental data to verify the practicability of amorphous alloy transformers in Section3 of 23 5. The discussion and conclusion are given in Sections 6 and 7, respectively. The research framework of the whole paper is shown in Figure 1.

Figure 1. Research framework of the paper. Figure 1. Research framework of the paper.

2.2. Mathematical Mathematical Model Model of of Stress Stress and and Strain Strain Induced Induced by by Magnetostriction Magnetostriction AmorphousAmorphous alloy alloy iron iron core core is formedis formed by by stacking stacking amorphous amorphous alloy alloy strips, strips, and and there there is eddyis eddy current current and and magnetic magnetic flux flux in thein the iron iron core core column column and and iron iron yoke. yoke. Under Under the the action action ofof ampere ampere force, force, the the amorphous amorphous alloy alloy core core has has slight slight deformation. deformation. The The comparison comparison of of the the magnetostrictionmagnetostriction coefficient coefficient of of amorphous amorphous alloy alloy and and the the magnetostriction magnetostriction coefficient coefficient of of orientedoriented silicon silicon steel steel sheet sheet is is shown shown in in Figure Figure2[ 220 [20].]. It It can can be be seen seen from from the the figure figure that that underunder the the same same magnetic magnetic field field strength, strength, the the degree degree of of magnetostriction magnetostriction of of the the amorphous amorphous alloyalloy is is much much higher higher than than that that of of the the silicon silicon steel steel sheet. sheet. Correspondingly, Correspondingly, under under the the action action ofof the the same same magnetic magnetic field field strength, strength, the the amorphous amorphous alloy alloy core core has has a a much much larger larger heart heart shapeshape variable variable than than the the silicon silicon steel steel core. core. The The largest largest strain strain caused caused by by magnetostriction magnetostriction is is wherewhere amorphous amorphous fragments fragments are are easily easily generated. generated.

Figure 2. Magnetostrictive characteristic curve of amorphous alloy and silicon steel sheet. Energies 2021, 14, 164 4 of 23

2.1. Fully Coupled Model The relationship between the object strain and the magnetic ﬁeld intensity caused by the magnetostriction of the amorphous alloy wound core can be expressed by the magnetic pressure equation, which includes two aspects [21]:

 6 3  H  ε p = ∑ spqσq + ∑ dpm Hm p = 1, 2, ··· , 6  q=1 m=1 6 3 (1)  σ  Bn = ∑ dnqσq + ∑ µnm Hm n = 1, 2, 3  q=1 m=1

where 1, 2, ··· , 6 are strain tensors of x, y, z, xy, yz, xz; ε is strain tensor; sH is elastic constant in a constant magnetic field; σ is stress tensor; d is piezomagnetic coefficient; H is magnetic field intensity; B is magnetic induction intensity; µ is magnetic conductivity. According to the relationship between the body strain caused by the magnetostriction of the amorphous alloy coil core and the magnetic field strength, the core strain when the magnetic field acts alone is:   d11 d12 d13  d21 d22 d23     Hx 0  d31 d32 d33  ε =   Hy  (2)  d41 d42 d43    Hz  d51 d52 d53  d61 d62 d63

The amorphous alloy iron core is wound, and the internal sheer force of the iron core is very small, so the shear strain of the amorphous iron core is ignored, and the shear strain magnetostriction coefficient is defined as dij = 0 (i = 3, 4, 5). Formula (2) is simplified to:    d11 d12 d13 Hx ε =  d21 d22 d23  Hy  (3) d31 d32 d33 Hz

Because amorphous alloys are isotropic materials, the magnetostriction coefﬁcient can 0 be simpliﬁed to two: dij = d (i = j), dij = d (i 6= j). The constitutive equation of stress and strain of amorphous alloy is:  σx = (a + 2b)εx + aεy + aεz   σy = aεx + (a + 2b)εy + aεz   σ = aε + aε + (a + 2b)ε z x y z (4)  σxy = bεxy   σyz = bεyz  σzx = bεzx

Ev a = (5) (1 + v)(1 − 2v) E b = (6) 2(1 + v) where E is Young’s modulus and v is Poisson’s ratio. Energies 2021, 14, 164 5 of 23

Then:  σx vσy vσz  εx = E − E − E  σ  ε = y − vσx − vσz  y E E E  σz vσx vσy  εz = E − E − E 2(1+v)σxy (7)  εxy =  E  2(1+v)σyz  εyz =  E  2(1+v)σzx εzx = E The magnetic-mechanical coupling energy of the amorphous alloy transformer core, that is, the magneto strictive energy is: According to Formula (7), the constitutive equation of strain and stress can be obtained:

λ0 = −vλ (8)

where λ is magnetostriction. Substituting Formula (9) into Formula (8), this can be simpliﬁed as:  T 0 0  (1 − v)εx + vεy + vεz λ λ λ 0 0  vεx + (1 − v)εy + vεz   λ λ λ   Z Z    0 0  Hx T E  vεx + vεy + (1 − v)εz   λ λ λ  σ λHdV =     Hy dV (9) Ω (1 + v)(1 − 2v) Ω  (1 − 2v)εxy/2   0 0 0      Hz  (1 − 2v)εyz/2   0 0 0  (1 − 2v)εzx/2 0 0 0 Z Z T σ λHdV = λE Hxεx + Hyεy + Hzεz dxyz (10) Ω Ω The total energy of amorphous alloy distribution transformer core includes strain energy, magnetic energy, magnetostrictive energy, potential energy of external force, and potential energy of current [22]. Introducing the vector magnetic potential A, the energy functional function can be obtained as: = R 1 T H + R 1 T + R T − R I Ω 2 σ s σ dV Ω 2 lH H dV Ω σ λH dV Γ fΓldV 2 1R R 2 1 (11) − JAdV− fV ldV Ω1 Ω2

where l is the deformation of the iron core; fV and f Γ are the external volume force of the transformer core and the boundary force on the surface of the iron core, respectively; A is the introduced magnetic vector position, satisfying B = ∇ × A; J represents the external current density; Ω1 represents the analysis domain of the magnetic field and Ω2 represents the analysis domain of the mechanical field. Based on the variational principle, the energy functional is subjected to unit discretiza- tion, and the variational problem of the functional is transformed into the problem of finding the extreme value of the multivariate function. The conditions for taking the extreme value of the functional I are:  ∂I = ∑ ∂Ie = 0  ∂Aij ∂Aij e i = x y z j = ··· n ∂I ∂Ie , , ; 1, 2, , (12)  ∂d = ∑ ∂d = 0 ij e ij

The finite element equations of the overall magnetic field-mechanical field strength coupling can be obtained:

SD A J = (13) CM L fV + fΓ

where S and M are the magnetic ﬁeld stiffness matrix and the mechanical stiffness matrix, respectively; A is the electromagnetic vector potential; L is the deformation of the core, C is Energies 2021, 14, 164 6 of 23

the contribution matrix of the magnetic ﬁeld to the core deformation; D is the contribution matrix of the core deformation to the magnetic ﬁeld pair; J is the current density. According to the electromagnetic field-mechanical field strength coupling theory, the full coupling model is used to simulate the magnetostrictive deformation of amorphous alloys.

2.2. Actual Calculation Model If the full coupling model is used to simulate the magnetostrictive deformation of the amorphous alloy according to the electromagnetic field-mechanical field strength coupling theory, the calculation or the computation power will be huge, and the calculation cost will be increased. Besbes et al. [23] proposed a strong and weak coupling model based on the finite element method, and compared the two models. The analysis results show that in the coupling analysis with small deformation variables, the weak coupling has stronger con- vergence and has a negligible effect on the results [23]. This effect can also be seen in the motor modeling in hybrid flux [24]. Therefore, when analyzing the influence of magnetostrictive properties on the deformation of amorphous alloy cores, the simplification of the electromagnetic field-mechanical field coupling theory only considers the magnetostrictive effect and does not consider its inverse effect. In this paper, the indirect coupling method is used to analyze the magnetostriction phenomenon. A mathematical model describing the material’s magnetostrictive properties is established based on the piezomagnetic equation, and then the model is indirectly coupled to the magnetic field finite element calculation. When calculating the magnetostriction in this paper, the influence of applied stress is not considered. Based on the above analysis and taking into account the isotropic properties of amorphous alloys, Formula (1) can be simplified as:

ε = dH (14) B = µH

In order to further derive the relationship between magnetostriction and magnetic field strength, Formula (14) with ε and B is combined and the strain in the formula is changed to magnetostriction. d λ = × B = k × B (15) µ In the formula, k is the magnetostriction coefficient. By interpolating the relationship between the magnetostriction of the amorphous alloy and the magnetic field strength in Figure2, the magnetostriction can be obtained with the help of the magnetic field distribution in the core.

3. Modeling 3.1. Transformer The research object of this paper is the SBH15-M 10 kV oil-immersed amorphous alloy transformer [8,9,17], as this type is the most commonly used in China, and its outline diagram is shown in Figure3. The rated capacity is 315 kVA, the rated voltage of high- voltage winding is 10 kV, and the rated voltage of low-voltage winding is 0.4 kV. Table1 shows its nameplate data. Energies 2021, 14, x FOR PEER REVIEW 8 of 24

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Figure 3. SBH15-M 10 kV Amorphous Alloy Distribution Transformer.

Table 1. SBH15-M 10 kV amorphous distribution transformer nameplate data.

Model Num- Phase Num- Rated Voltage Join Rated Capacity Frequency Rated Current Tapping ber ber (kV) Groups (kVA) (Hz) (A) Range 10 × (1 ± 2 × SBH15-M 3 10/0.4 Dyn11 315 50 18.2/454.7 2.5) Figure Figure3. SBH15-M 3. SBH15-M 10 kV Amorphous 10 kV Amorphous Alloy Distribution Alloy Distribution Transformer. Transformer. 3.2. Core TableTable 1. 1. SBH15-MSBH15-MThe core10 10 kV kV of amorphous amorphous SBH15-M distribution distribution 10 kV amorphous transformer transformer allo nameplate nameplatey distribution data. data. transformer is made of FA24S07-86 amorphous alloy produced by Antai Nanrui Amorphous Technology Co., Model Num- Phase Num- Rated VoltageRated VoltageJoin Rated CapacityRated Capacity FrequencyFrequency RatedRated Current Current Tapping Model Number Phase Number Join Groups Tapping Range ber ber (kV)Ltd. (kV)The coreGroups structure has(kVA) three (kVA) phases, four(Hz) frames,(Hz) and five(A)(A) columns (FigureRange 4) and the SBH15-M 3core 10/0.4 structure parameters Dyn11 are shown 315 in Table 2. 50 Core material 18.2/454.7 properties10 × 10 (1 ×can ±(1 2± ×be2 × seen2.5) in SBH15-M 3 10/0.4 Dyn11 315 50 18.2/454.7 Table 3. 2.5) 3.2. CoreFigure 5 shows the magnetization curve of the amorphous alloy, and Figure 6 shows 3.2. Corethe iron loss curve of the amorphous alloy. The red lines in the two figures are the data The core of SBH15-M 10 kV amorphous alloy distribution transformer is made of entered by the authors, and the black lines are the curves fitted for the core material by TheFA24S07-86 core of SBH15-M amorphous 10 alloykV amorphous produced by allo Antaiy distribution Nanrui Amorphous transformer Technology is made Co.,of Ltd. ANSYS according to the data entered by the user. In order to accurately simulate the loss FA24S07-86The core amorphous structure hasalloy three produced phases, by four Antai frames, Nanrui and ﬁveAmorphous columns Technology (Figure4) and Co., the core characteristics of amorphous alloy cores, the thickness of the material is set to 24 m, and Ltd. Thestructure core structure parameters has three are shown phases, in Tablefour frames,2. Core and material five columns properties (Figure can be 4) seen and in the Table 3. the lamination factor is 0.86. core structure parameters are shown in Table 2. Core material properties can be seen in Table 3. Figure 5 shows the magnetization curve of the amorphous alloy, and Figure 6 shows the iron loss curve of the amorphous alloy. The red lines in the two figures are the data entered by the authors, and the black lines are the curves fitted for the core material by ANSYS according to the data entered by the user. In order to accurately simulate the loss characteristics of amorphous alloy cores, the thickness of the material is set to 24 m, and the lamination factor is 0.86.

FigureFigure 4. 4.Amorphous Amorphous alloy alloy core core structure structure diagram. diagram.

Table 2. Core structure parameters.

Inner Frame Inner Frame Thickness Parameters Width (mm) Length (mm) Height (mm) (mm) Big frame iron core 185 130 86 142.2 Small frame iron core 185 70 86 142.2 Figure 4. Amorphous alloy core structure diagram.

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Table 3. Core material properties. Energies 2021, 14, x FOR PEER REVIEW 9 of 24

Saturated Young’s Energies 2021, 14, x FOR PEER REVIEWLamination Resistivity Density Poisson Vickers Thickness9 of 24 Performance Magnetic Modulus Coefficient (µΩm) (g/cm3) Ratio Degree (HV) (m) Induction (T) (GPa) Table 2. Core structure parameters. Size 1.56 0.86 1.3 7.18- 110 0.3 900 24 Parameters InnerTable Frame 2. Core Length structure (mm) parameters. Inner Frame Height (mm) Thickness (mm) Width (mm) BigParameters frame iron core Inner Frame 185 Length (mm) Inner Frame 130 Height (mm) Thickness 86 (mm) Width 142.2 (mm) SmallBig frame frame iron iron core core Figure 185 1855 shows the magnetization 130 curve 70 of the amorphous 86 86 alloy, and Figure 142.2 142.26 shows Small frame iron core the iron loss 185 curve of the amorphous alloy.70 The red lines in the 86 two figures 142.2 are the data enteredTable 3. byCore the material authors, properties. and the black lines are the curves fitted for the core material by ANSYS according to the data entered by the user. In order to accurately simulate the loss Perfor- Saturated Magnetic TableLamination 3. Core Co-materialResistivity properties. Density Young’s Modu- Poisson Vickers De- Thickness characteristics of amorphous alloy3 cores, the thickness of the material is set to 24 m, and Perfor-mance SaturatedInduction Magnetic (T) Laminationefficient Co- Resistivity(μΩm) Density(g/cm ) Young’slus (GPa) Modu- PoissonRatio Vickersgree (HV) De- Thickness(m) the lamination factor is 0.86. manceSize Induction1.56 (T) efficient 0.86 (μΩ1.3m) (g/cm7.18-3) lus (GPa)110 Ratio0.3 gree900 (HV) (m)24 Size 1.56 0.86 1.3 7.18- 110 0.3 900 24

Figure 5. Magnetization curve of amorphous alloy. Figure 5. Magnetization curve of amorphous alloy. Figure 5. Magnetization curve of amorphous alloy.

Figure 6. Iron loss curve of amorphous alloy.

FigureFigure 6.6. IronIron loss curve curve of of amorphous amorphous alloy. alloy. 3.3. Winding 3.3.3.3. WindingWindingSince the core cross section of the amorphous alloy transformer is rectangular, in order SincetoSince make the the core core structure cross cross section sectioncompact, of ofthe the the amorphou windin amorphousg sinterface alloy alloy transformer of transformerthe amorphous is rectangular, is rectangular,alloy isin alsoor- in orderderrectangular. to to make make theThe the structure high-voltage structure compact, compact, winding the the windin of winding theg SBH15-M interface interface of10 the ofkV the amorphousamorphous amorphous alloydistribution alloy is also is also rectangular.rectangular.transformer The is a high-voltagethree-layer high-voltage cylindrical winding winding ofstructure, of the the SBH15-M SBH15-M and the 10 low-voltage 10kV kV amorphous amorphous winding distribution distribution is a foil transformertransformerwinding. Since is a thethree-layer amorphous cylindrical cylindrical alloy is structure, structure,particularly and and sensitivethe the low-voltage low-voltage to mechanical winding winding stress, is a is foilthe a foil winding.winding isSince not attachedthe amorphous to the core alloy during is particularly assembly. sensitive In order to preventmechanical a major stress, impact the windingon the iron is not core attached from the to shortthe core circuit during of th assembly.e winding, In there order is to a preventcertain gap a major between impact the on the iron core from the short circuit of the winding, there is a certain gap between the

Energies 2021, 14, 164 9 of 23

Energies 2021, 14, x FOR PEER REVIEWwinding. Since the amorphous alloy is particularly sensitive to mechanical stress,10 the of 24 Energies 2021, 14, x FOR PEER REVIEW 10 of 24 winding is not attached to the core during assembly. In order to prevent a major impact on the iron core from the short circuit of the winding, there is a certain gap between the winding and the iron core, and the gap is ﬁlled by the laminate, which can minimize the winding and the iron core, and the gap is filledlled byby thethe laminate,laminate, whichwhich cancan minimizeminimize thethe impact of the deformation of the iron core when a short circuit occurs. impactimpact ofof thethe deformationdeformation ofof thethe ironiron corecore whenwhen aa shortshort circuitcircuit occurs.occurs. TheThe connection connection groupgroup ofof thethe winding is is Dyn1 Dyn11.1. As As shown shown in inFigure Figure 7,7 the, the third third har- harmonicThe canconnection form a loopgroup in of the the high-voltage winding is sideDyn1 delta1. As winding, shown in so Figure that there 7, the is third no third harmonic can form a loop in the high-voltage side delta winding, so that there is no third harmonicharmonic voltage voltage component component in thein the secondary secondary side side voltage. voltage. The harmonicThe harmonic causes causes extra extra AC lossharmonic [25] and voltage therefore component harmonics in shouldthe secondary be minimized. side voltage. The harmonic causes extra AC loss [25] and therefore harmonics should be minimized.

ABC ABC

A A

XY Z XY Z abc a abc a b b

c c B B C C xyz xyz

Figure 7. Winding connection diagram. Figure 7. Winding connection diagram. Winding parameters are shown in Table4. Winding parameters are shown in Table 4 Table 4. SBH15-M 10 kV amorphous distribution transformer winding parameters. Table 4. SBH15-M 10 kV amorphous distribution transformertransformer windingwinding parameters.parameters. Winding High-Voltage Winding Low-Voltage Winding Winding High-Voltage Winding Low-Voltage Winding TurnsTurns 818 818 18 22 SectionalSectional area area (mm (mm2))) 3.943.943.94 152 152 ResistanceResistance ((ΩΩ))) 5.23995.2399 5.2399 0.002614 0.002614 Rated current (A) 18.2 454.7 Rated current (A) 18.2 454.7 Material red copper red copper Material red copper red copper WindingWinding connectionconnection mode mode D D Y WindingSinusoidal connection voltage (kV) mode D 10 0.4 Y Sinusoidal voltage (kV) 10 0.4 3.4. Meshing 3.4. Meshing In this paper, a meshing method combining manual and adaptive methods is used. In this paper, a meshing method combining manual and adaptive methods is used. A manual meshing method is used for the core and windings, and only the maximum A manual meshing method is used for the core and windings, and only the maximum distancedistance of of the the meshing meshing unit unit is is set set (37.42 (37.42 mm mm for for core core and and 41.2 41.2 mm mm for for winding). winding). The The meshingdistance result of the is meshing shown in unit Figure is 8set. (37.42 mm for core and 41.2 mm for winding). The meshing result is shown in Figure 8.

Figure 8. Core and winding meshing results. FigureFigure 8. 8.Core Core and and winding winding meshing meshing results. results.

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3.5. Electromagnetic-Circuit Coupling 3.5. Electromagnetic-Circuit Coupling According to the connection group of transformers, the Dyn11 type is adopted, and a Accordingtriangular connection to the connection method group is adopted of transformers, at the power the Dyn11supply typeside. isThe adopted, terminal and on a the triangularpower supply connection side is method set through is adopted Maxwell at thefield power coupling supply and side.connected The terminal to the transformer on the powerwinding supply in sidethe simulation is set through model Maxwell during ﬁeld editing. coupling The and primary connected resistance to the in transformer the excitation windingcircuit in is the 5.2399 simulation Ω. To prevent model duringthe software editing. from The iter primaryation error resistance by using in the a Norton excitation equiv- circuit is 5.2399 Ω. To prevent the software from iteration error by using a Norton equivalent alent circuit in the calculation, a very small resistance of 1 mΩ is added to the power sup- circuit in the calculation, a very small resistance of 1 mΩ is added to the power supply. The ply. The excitation circuit model corresponding to the simulation model is shown in Fig- excitation circuit model corresponding to the simulation model is shown in Figure9. ure 9.

+ 8164.97V LabelID=V52 1mOhm 5.2399ohm LA1 R46 R49

+ 8164.97V LabelID=V51 1mOhm 5.2399ohm LB1 R45 R48

+ 8164.97V LabelID=V50 1mOhm 5.2399ohm LC1 0 R44 R47

FigureFigure 9. No-load 9. No-load driving driving circuit. circuit.

AsAs secondary secondary side, side, active active power power output output by three-phaseby three-phase transformer: transformer:

P = PPPPPa +=++Pb + Pc (16)(16) abc Secondary phase voltage: Secondary phase voltage:

U = 231V =Ua = Ub = Uc (17) === UUUU231V= abc (17) The phase current is Ia, Ib, Ic. Then active power per phase:

The phase current is Ia, Ib, Ic. Then= active power per phase:  Pa Ua Ia cos β Pb = Ub Ib cos β (18)  P =PUIU=I cos βcos β c  aaa=c c β PUIbbbcos (18) If circuit is a pure resistive load, then cos=β = 1. β PUIccccos Load factor is: P If circuit is a pure resistive load,η = then× cos100%β = 1. (19) 315 Load factor is: 1. Normal load Considering that amorphous alloy transformers are mostly used in the distribution network, the load rate is generallyP 30~60%. In this study, a 1 Ω load being added (η = 50%), the transformerη =× windings100% and the load are connected to the(19) three-phase power supply. 315 2. Load imbalance Phase A and Phase B add 1 Ω load, and Phase C adds 0.7 Ω load. 1. Normal load Pa = 53.3 kVA, Pb = 53.3 kVA, Pc = 76.2 kVA. With the gradual decrease of phase C Consideringresistance, that phase amorphous C active power alloy transformers gradually increases, are mostly and used the loadin the becomes distribution more net- Ω work,unbalanced. the load Whenrate is thegenerally resistance 30~60%. drops In to this 0, a study, short circuit a 1 occurs.load being added (η = 50%), 3. theOverload transformer 0.3 Ωwindingsloads have and the been load added are connected to simulate to thethe overloadthree-phase situation power supply. of the transformer during operation (η = 169%).

Energies 2021, 14, x FOR PEER REVIEW 12 of 24

2. Load imbalance Phase A and Phase B add 1 Ω load, and Phase C adds 0.7 Ω load. Pa = 53.3 kVA, Pb = 53.3 kVA, Pc = 76.2 kVA. With the gradual decrease of phase C resistance, phase C active power gradually increases, and the load becomes more unbalanced. When the resistance drops to 0, a short circuit occurs. Energies 2021, 14, 164 3. Overload 11 of 23 0.3 Ω loads have been added to simulate the overload situation of the transformer during operation (η = 169%). 4.4. ShortShort circuit circuit Setting the three-phase loads to 1 Ω, 1 × 10−9 Ω, and 1 Ω, respectively, a Settingsingle-phase the three-phase short circuit loads occurs.to 1 Ω, The1 × 10 very−9 Ω small, and resistance1Ω, respectively, of phase a Bsingle-phase is equivalent short to circuitshort-circuit. occurs. The very small resistance of phase B is equivalent to short-circuit.

4.4. Verification Verification of of Transformer Transformer Model Model InIn order order to to verify verify the the correctness correctness of of the the finite finite element element model model of of amorphous amorphous alloy alloy transformerstransformers established established in thisin this paper, paper, the simulationthe simulation data aredata compared are compared with experimental with experi- data.mental No-load data. No-load current andcurrent no-load and lossno-load are selectedloss are forselected verification, for verification, so that the so electro-that the magneticelectromagnetic characteristics characteristics and the and loss the characteristics loss characteristics of the of transformer the transformer can be can verified. be veri- Sincefied. theSince secondary the secondary winding winding of the transformerof the transformer is unloaded, is unloaded, the current the current of the secondary of the sec- windingondary canwinding be set can to zero,be set which to zero, not which only not simulates only simulates the actual the unloaded actual unloaded condition condition of the transformerof the transformer accurately, accurately, but also but makes also the makes excitation the excitation circuit simpler. circuit simpler. 4.1. No-Load Current 4.1. No-Load Current Circuit excitation (Figure9) is connected with transformer model to realize joint Circuit excitation (Figure 9) is connected with transformer model to realize joint sim- simulation of electromagnetic field and circuit. The simulated no-load current is shown in ulation of electromagnetic field and circuit. The simulated no-load current is shown in Figure 10. Figure 10.

FigureFigure 10. 10.No-load No-load current. current.

FromFrom the the simulation simulation results, results, it it can can be be seen seen that that the the maximum maximum excitation excitation phase phase current current ofof the the transformer transformer cancan reachreach 41.84441.844 A, and th thee peak peak value value of of rated rated phase phase current current on on the theprimary primary side side of the of the10 kV 10 amorphous kV amorphous alloy alloy distribution distribution transformer transformer of SBH15-M of SBH15-M type is type14.681 is 14.681 A. The A. maximum The maximum excitation excitation phase phasecurrent current can reach can 2.81 reach times 2.81 the times rated the current, rated current,which is which consistent is consistent with the with fact the that fact when that the when secondary the secondary winding winding is unloaded, is unloaded, the pri- themary primary magnetic magnetic potential potential will willhave have a large a large amplitude amplitude at the at thebeginning. beginning. At Atthis this time, time, the theamorphous amorphous alloy alloy core core is in is the in the saturation saturation region, region, the thepermeability permeability is very is very small, small, and andthere there will be a high unloaded current. After a period of time, the no-load current tends to stabilize, taking phase B current, as shown in Figure 11. It can be seen that the amplitude of the transformer tends to be stable after a period of time, but the current waveform is not completely sine wave, which is due to the hysteresis characteristic of the transformer core. When the transformer is unloaded, the current is almost completely used for excitation, and the inﬂuence of hysteresis characteristic will appear, causing current ﬂuctuation. The peak value of no-load phase current at steady state is 82.295 mA, the percentage of no-load current is 82.295/14.681. EnergiesEnergies 20212021,, 1414,, xx FORFOR PEERPEER REVIEWREVIEW 1313 ofof 2424

willwill bebe aa highhigh unloadedunloaded current.current. AfterAfter aa periodperiod ofof time,time, thethe no-loadno-load currentcurrent tendstends toto stabi-stabilize,lize, takingtaking phasephase BB current,current, asas shownshown inin FigureFigure 11.11. ItIt cancan bebe seenseen thatthat thethe amplitudeamplitude ofof thethe transformertransformer tendstends toto bebe stablestable afterafter aa periodperiod ofof time,time, butbut thethe currentcurrent waveformwaveform isis notnot com-com- pletelypletely sinesine wave,wave, whichwhich isis duedue toto thethe hysterhysteresisesis characteristiccharacteristic ofof thethe transformertransformer core.core. WhenWhen thethe transformertransformer isis unloaded,unloaded, thethe currentcurrent isis almostalmost completelycompletely usedused forfor excitation,excitation, Energies 2021, 14, 164 andand thethe influenceinfluence ofof hysteresishysteresis characteristiccharacteristic willwill appear,appear, causingcausing currentcurrent fluctuation.fluctuation.12 of TheThe 23 peakpeak valuevalue ofof no-loadno-load phasephase currentcurrent atat steadysteady statestate isis 82.29582.295 mA,mA, thethe percentagepercentage ofof no-no- loadload currentcurrent isis 82.295/14,681.82.295/14,681.

FigureFigureFigure 11. 11.11.Steady-state Steady-stateSteady-state no-load no-loadno-load current. current.current. 4.2. No-Load Loss 4.2.4.2. No-LoadNo-Load LossLoss Circuit excitation (Figure9) is connected with transformer model to realize joint CircuitCircuit excitationexcitation (Figure(Figure 9)9) isis connectedconnected withwith transformertransformer modelmodel toto realizerealize jointjoint sim-sim- simulation of circuit and electromagnetic ﬁeld. The simulated no-load loss is shown in ulationulation ofof circuitcircuit andand electromagneticelectromagnetic field.field. ThThee simulatedsimulated no-loadno-load lossloss isis shownshown inin FigureFigure Figure 12. 12.12.

Figure 12. No-load loss. FigureFigure 12. 12.No-load No-load loss. loss.

FromFromFrom the thethe simulation simulationsimulation results, results,results, it itit can cancan be bebe seen seenseen that thatthat the thethe average averageaverage no-load no-loadno-load loss lossloss of ofof the thethe trans- trans-trans- formerformerformer is isis 176.2439 176.2439176.2439 W WW when whenwhen it is itit stable, isis stable,stable, and andand the thethe actual actualactual product productproduct field fieldfield test no-loadtesttest no-loadno-load loss data lossloss aredatadata 170 W, with a difference of 3.67%, which again shows good agreement with the simulation (provided by Shandong Zhixin Intelligent Equipment Co. Ltd.). The finite element model of the amorphous alloy transformer established in Section3 has a no-load current and no-load loss which are similar to the field test data of the product when the secondary winding is in no-load operation, and the data error is relatively small. Therefore, it can be verified that the model established in this paper is practical and it is feasible to ignore the influence of transformer oil in electromagnetic field simulation. The paper provides a means to minimize the core loss for high power transformers and the associated harmonics studied in the core analysis. It notes the application for Energies 2021, 14, x FOR PEER REVIEW 14 of 24

are 170 W, with a difference of 3.67%, which again shows good agreement with the simulation (provided by Shandong Zhixin Intelligent Equipment Co. Ltd). The finite element model of the amorphous alloy transformer established in Section 3 has a no-load current and no-load loss which are similar to the field test data of the product when the secondary winding is in no-load operation, and the data error is relatively small. Therefore, it can be verified that the model established in this paper is practical and it is feasible to ignore the influence of transformer oil in electromagnetic field simulation. Energies 2021, 14, 164 13 of 23 The paper provides a means to minimize the core loss for high power transformers and the associated harmonics studied in the core analysis. It notes the application for rail systems and other power distribution. It is expected that using the proposed study, elec- railtromagnetic systems and emission otherpower and loss distribution. can be improved. It is expected that using the proposed study, electromagnetic emission and loss can be improved. 5. Simulation Analysis of Iron Core Electromagnetic Field and Strain under Different 5.Working Simulation Conditions Analysis of Iron Core Electromagnetic Field and Strain under Different Working Conditions 5.1. Normal Load 5.1. Normal Load TheThe excitation excitation circuit circuit for forelectromagnetic electromagnetic field ﬁeld simulation simulation is shown is shown in Figure in Figure 13, 13 and, and the themagnetic magnetic field ﬁeld distribution distribution in the in thecore core of the of amorphous the amorphous alloy alloy transformer transformer during during nor- normal operation can be obtained [16]. mal operation can be obtained [16].

+ 8164.97V LabelID=V8 1mOhm 5.2399ohm LA1 LA2 0.002614ohm 1ohm R17 R11 R56 R14

0.002614ohm 1ohm

+ 8164.97V LabelID=V9 R57 R15 1mOhm 5.2399ohm LB1 LB2 R18 R12 0.002614ohm 1ohm

+ 8164.97V LabelID=V10 R58 R16 1mOhm 5.2399ohm LC1 LC2 R19 R13 0

FigureFigure 13. 13.Excitation Excitation circuit circuit during during normal normal operation. operation.

FigureFigure 13 13 shows shows the the magnetic magnetic induction induction intensity intensity vector vector distribution distribution when when the the three- three- phasephase magnetic magnetic fluxes fluxes of of A, A, B, B, and and C C reach reach their their maximums. maximums. ItIt can can be be seen seen from from Figure Figure 14 14 that that the the three-phase three-phase magnetic magnetic flux flux lags lags by by 2/3 2/3 when when it it reachesreaches the the maximum. maximum. The The magnetic magnetic induction induction intensity intensity of of the the transformer transformer core core column column and iron yoke is about 1.34 T, basically working in the linear working area of the magneti- and iron yoke is about 1.34 T, basically working in the linear working area of the magnet- zation curve, which is the normal working range of the transformer. The magnetic field ization curve, which is the normal working range of the transformer. The magnetic field simulation results show that the transformer’s magnetic field changes sinusoidally under simulation results show that the transformer’s magnetic field changes sinusoidally under the excitation of a three-phase voltage source. The test maximum magnetic induction inten- the excitation of a three-phase voltage source. The test maximum magnetic induction in- sity of the actual product is 1.348 T (provided by Shandong Zhixin Intelligent Equipment tensity of the actual product is 1.348 T (provided by Shandong Zhixin Intelligent Equip- Co. Ltd.). Thus, this model can accurately simulate the magnetic field distribution of the ment Co. Ltd). Thus, this model can accurately simulate the magnetic field distribution SBH15-M 10 kV amorphous alloy distribution transformer. of the SBH15-M 10 kV amorphous alloy distribution transformer. Figure 15 shows the strain cloud diagram of the amorphous alloy core caused by magnetostriction when the three-phase magnetic flux reaches the maximum in the normal operation of the amorphous alloy transformer. It can be seen that the strain caused by magnetostriction is mainly concentrated at the corner where the iron core column and the iron yoke intersect, which is consistent with the distribution of the magnetic induction intensity in Figure 15, and the maximum strain is 2.3376 × 10−5 m/m.

Energies 2021, 14, x FOR PEER REVIEW 15 of 24

Energies 2021, 14, 164 14 of 23

Energies 2021, 14, x FOR PEER REVIEW 15 of 24

(a) ωt = π/2

(b) ωt = 7π/6

(c) ωt = 11π/6 Figure 15 shows the strain cloud diagram of the amorphous alloy core caused by Figure 14. Magnetic induction intensity vector at different moments during normal operation. magnetostrictionFigure 14.whenMagnetic the three-phase induction magnet intensityic flux vector reaches at differentthe maximum moments in the during normal normal operation. operation of the amorphous alloy transformer. Figure 15 shows the strain cloud diagram of the amorphous alloy core caused by magnetostriction when the three-phase magnetic flux reaches the maximum in the normal operation of the amorphous alloy transformer.

(a) ωt = π/2

Energies 2021, 14, x FOR PEER REVIEW 16 of 24

(b) ωt = 7π/6

(c) ωt = 11π/6 Figure 15. Strain of transformer core caused by magnetostriction during normal operation. Figure 15. Strain of transformer core caused by magnetostriction during normal operation. It can be seen that the strain caused by magnetostriction is mainly concentrated at the corner where the iron core column and the iron yoke intersect, which is consistent with the distribution of the magnetic induction intensity in Figure 15, and the maximum strain is 2.3376 × 10−5 m/m.

5.2. Load Imbalance The excitation circuit for electromagnetic field simulation is shown in Figure 16.

+ 8164.97V LabelID=V8 1mOhm 5ohm LA1 LA2 1ohm R17 R11 R14

LB2 1ohm

+ 8164.97V LabelID=V9 R15 1mOhm 5ohm LB1 R18 R12 LC2 0.7ohm

+ 8164.97V LabelID=V10 R16 1mOhm 5ohm LC1 R19 R13 0

Figure 16. Excitation circuit during load imbalance.

The magnetic field distribution of the transformer core model is shown in Figure 17 when the load is unbalanced.

(a) ωt = π/2

Energies 2021, 14, x FOR PEER REVIEW 16 of 24

ω π (c) t = 11 /6 (c) ωt = 11π/6 Figure 15. Strain of transformer core caused by magnetostriction during normal operation. Figure 15. Strain of transformer core caused by magnetostriction during normal operation. It can be seen that the strain caused by magnetostriction is mainly concentrated at the Energies 2021, 14, 164 Itcorner can be where seen that the the iron strain core caused column by and ma gnetostrictionthe iron yoke isintersect, mainly concentratedwhich is consistent at the with15 of 23 cornerthe where distribution the iron of core the columnmagnetic and induction the iron intensity yoke intersect, in Figure which 15, and is consistent the maximum with strain the distributionis 2.3376 × of10 the−5 m/m. magnetic induction intensity in Figure 15, and the maximum strain is 2.3376 × 10−5 m/m. 5.2.5.2. LoadLoad ImbalanceImbalance 5.2. Load ImbalanceTheThe excitationexcitation circuit for for electromagnetic electromagnetic field ﬁeld simulation simulation is isshown shown in Figure in Figure 16. 16. The excitation circuit for electromagnetic field simulation is shown in Figure 16.

+ 8164.97V LabelID=V8

+ 8164.97V LabelID=V8 1mOhm 5ohm LA1 LA2 1ohm 1mOhm R17 5ohm R11 LA1 LA2 1ohm R14 R17 R11 R14

LB2 1ohm

+ 8164.97V LabelID=V9 LB2 1ohm R15

+ 8164.97V LabelID=V9 1mOhm 5ohm LB1 R15 1mOhm R18 5ohm R12 LB1 R18 R12 LC2 0.7ohm

+ 8164.97V LabelID=V10 LC2 0.7ohm R16

+ 8164.97V LabelID=V10 1mOhm 5ohm LC1 R16 0 1mOhm R19 5ohm R13 LC1 R19 R13 0 0 0 Figure 16. Excitation circuit during load imbalance. Figure 16. Excitation circuit during load imbalance. Figure 16. Excitation circuit during load imbalance. TheThe magneticmagnetic field ﬁeld distribution distribution of of the the tran transformersformer core core model model is shown is shown in Figure in Figure 17 17 The magnetic field distribution of the transformer core model is shown in Figure 17 whenwhen thethe loadload is unbalanced. when the load is unbalanced.

Energies 2021, 14, x FOR PEER REVIEW 17 of 24 ω π (a) t = /2 (a) ωt = π/2

(b) ωt = 7π/6

(c) ωt = 11π/6 Figure 17. Magnetic induction intensity vector at different moments during load imbalance. Figure 17. Magnetic induction intensity vector at different moments during load imbalance. Figure 18 shows the strain cloud diagram of the amorphous alloy core caused by magnetostriction when the load is unbalanced.

(a) ωt = π/2

(b) ωt = 7π/6

Energies 2021, 14, x FOR PEER REVIEW 17 of 24

(b) ωt = 7π/6

Energies 2021, 14, 164 16 of 23 (c) ωt = 11π/6 Figure 17. Magnetic induction intensity vector at different moments during load imbalance. Figure 18 shows the strain cloud diagram of the amorphous alloy core caused by Figuremagnetostriction 18 shows the strain when cloud the load diagram is unbalanced. of the amorphous alloy core caused by magnetostriction when the load is unbalanced.

(a) ωt = π/2

(b) ωt = 7π/6

Figure 18. StrainFigure of 18. transformerStrain of transformer core caused core by ma causedgnetostriction by magnetostriction during load during imbalance. load imbalance.

Further reducing the resistance of the C, the stress situation in a more unbalanced state is explored as shown in Table5.

Table 5. Maximum values of strain in different unbalanced states.

Phase C (Ω) 0.6 0.5 0.4 0.3 0.2 0.1 10−9 maximum values of 2.78 2.81 2.85 2.90 2.93 2.95 2.99 strain at 11π/6 (10−5)

As the load becomes more unbalanced, the strain of the iron core gradually increases. When a short circuit occurs, the strain of the iron core reaches the maximum.

5.3. Overload The excitation circuit is shown in Figure 19. The magnetic induction intensity vector distribution of the transformer core model when there is overload is shown in Figure 20. Figure 21 shows the strain cloud diagram of overload of transformer core caused by magnetostriction in the overload situation. Energies 2021, 14, x FOR PEER REVIEW 18 of 24

Energies 2021, 14, x FOR PEER REVIEW 18 of 24

Further reducing the resistance of the C, the stress situation in a more unbalanced state is exploredFurther as reducing shown in the Table resistance 5. of the C, the stress situation in a more unbalanced state is explored as shown in Table 5. Table 5. Maximum values of strain in different unbalanced states.

Table 5. MaximumPhase C values (Ω) of strain in different 0.6 unbalanced0.5 0.4 states.0.3 0.2 0.1 10−9 maximum values ofPhase strain Cat ( 11Ω)π /6 (10−5) 2.78 2.81 0.6 2.850.5 2.900.4 2.930.3 2.950.2 2.990.1 10−9 maximum values of strain at 11π/6 (10−5) 2.78 2.81 2.85 2.90 2.93 2.95 2.99 As the load becomes more unbalanced, the strain of the iron core gradually increases. When a short circuit occurs, the strain of the iron core reaches the maximum. As the load becomes more unbalanced, the strain of the iron core gradually increases. 5.3. OverloadWhen a short circuit occurs, the strain of the iron core reaches the maximum. Energies 2021, 14, 164 17 of 23 5.3.The Overload excitation circuit is shown in Figure 19. The excitation circuit is shown in Figure 19. + 8164.97V LabelID=V8 1mOhm 5ohm LA1 LA2 0.3ohm

+ 8164.97V R17 R11 R14 LabelID=V8 1mOhm 5ohm LA1 LA2 0.3ohm R17 R11 R14 LB2 0.3ohm

+ 8164.97V LabelID=V9 R15 1mOhm 5ohm LB1 R18 R12 LB2 0.3ohm + 8164.97V R15 LabelID=V9 LC2 0.3ohm + 8164.97V 1mOhm 5ohm LB1 LabelID=V10 R16 R18 R12 1mOhm 5ohm LC1 R19 R13 LC2 0 0.3ohm

+ 8164.97V LabelID=V10 R16 0 1mOhm 5ohm LC1 R19 R13 0

Figure 19.0 Excitation circuit during overload.

The magnetic induction intensity vector distribution of the transformer core model Figure 19.19.Excitation Excitation circuit circuit during during overload. overload. when there is overload is shown in Figure 20. The magnetic induction intensity vector distribution of the transformer core model when there is overload is shown in Figure 20.

(a) ωt = π/2

Energies 2021, 14, x FOR PEER REVIEW 19 of 24

(b) ωt = 7π/6

(c) ωt = 11π/6 Figure 20. Magnetic induction intensity vector at different moments during overload. Figure 20. Magnetic induction intensity vector at different moments during overload. Figure 21 shows the strain cloud diagram of overload of transformer core caused by magnetostriction in the overload situation.

(a) ωt = π/2

(b) ωt = 7π/6

We reduce the resistance of the three phases at the same time, and simulate the iron core stress under different overload conditions as shown in Table 6.

Table 6. Maximum values of strain in different overload states.

η 50 70 90 110 130 160 maximum values of strain at 11π/6 × (10−5) 2.19 2.29 2.42 2.66 2.85 3.02

Energies 2021, 14, x FOR PEER REVIEW 19 of 24

(c) ωt = 11π/6 Figure 20. Magnetic induction intensity vector at different moments during overload. Energies 2021, 14, 164 18 of 23 Figure 21 shows the strain cloud diagram of overload of transformer core caused by magnetostriction in the overload situation.

(a) ωt = π/2

(b) ωt = 7π/6

(c) ωt = 11π/6 Figure 21. Strain of transformer core caused by magnetostriction during overload. Figure 21. Strain of transformer core caused by magnetostriction during overload.

We reduceWe the reduce resistance the resistance of the three of the phases three at phases the same at the time, same and time, simulate and simulate the iron the iron core stresscore under stress different under differentoverload overload conditions conditions as shown as in shown Table in 6. Table 6.

Table 6. MaximumTable 6. Maximumvalues of strain values in of different strain in differentoverload overload states. states.

ηη 5050 70 70 9090 110110 130 130 160 160 maximum values of strain maximum values of strain at 11π/6 × (102.19−5) 2.19 2.29 2.29 2.422.42 2.662.66 2.85 2.85 3.02 3.02 at 11π/6 × (10−5)

Under normal circumstances, as the load increases, the stress of the iron core increases. When the transformer is overloaded, the core stress increases faster. The maximum value is higher than the core stress during C-phase short circuit.

5.4. Short Circuit Figure 22 shows the excitation circuit. The magnetic induction intensity vector distribution of the transformer core model when a short circuit occurs on the secondary side is shown in Figure 23. Energies 2021, 14, x FOR PEER REVIEW 20 of 24

Energies 2021, 14, x FOR PEER REVIEW 20 of 24

Under normal circumstances, as the load increases, the stress of the iron core increases. When the transformer is overloaded, the core stress increases faster. The maximum value is Underhigher normalthan the circumstances, core stress during as the C-phase load inshortcreases, circuit. the stress of the iron core increases. When the transformer is overloaded, the core stress increases faster. The maxi- 5.4. Shortmum Circuit value is higher than the core stress during C-phase short circuit.

Energies 2021, 14, 164 Figure 22 shows the excitation circuit. 19 of 23 5.4. Short Circuit

+ 8164.97VFigure 22 shows the excitation circuit. LabelID=V8 1mOhm 5ohm LA1 LA2 1ohm

+ 8164.97VR17 R11 R14 LabelID=V8 1mOhm 5ohm LA1 LA2 1ohm R17 R11 R14 LB2 0.001uOhm

+ 8164.97V LabelID=V9 R15 1mOhm 5ohm LB1 LB2 0.001uOhm

+ 8164.97VR18 R12 LabelID=V9 R15 LC2 1ohm

+ 8164.97V 1mOhm 5ohm LB1 R16 LabelID=V10 R18 R12 1mOhm 5ohm LC1 LC2 0 1ohm

+ 8164.97VR19 R13 LabelID=V10 R16 0 1mOhm 5ohm LC1 R19 R13 0

Figure 22. Excitation0 circuit during short circuit.

TheFigure Figuremagnetic 22.22.Excitation Excitation induction circuitcircuit intensity duringduring vectorshort short circuit. distributioncircuit. of the transformer core model when a short circuit occurs on the secondary side is shown in Figure 23. The magnetic induction intensity vector distribution of the transformer core model when a short circuit occurs on the secondary side is shown in Figure 23.

(a) ωt = π/2

(b) ωt = 7π/6

Figure 23. Magnetic induction intensity vector(c) ω att different= 11π/6 moments during short circuit.

Figure 24 shows the strain cloud diagram of overload of transformer core caused by magnetostriction when a short circuit occurs. The maximum value of strain at 11π/6 is

2.76 × 10−5, which is less than C-phase short circuit and overload situation. The core strain caused by the side-phase short circuit is larger than the middle-phase short circuit. Energies 2021, 14, x FOR PEER REVIEW 21 of 24

Figure 23. Magnetic induction intensity vector at different moments during short circuit.

Figure 24 shows the strain cloud diagram of overload of transformer core caused by

Energies 2021, 14, 164 magnetostriction when a short circuit occurs. The maximum value of strain at 11π/6 is20 of 23 2.76 × 10−5, which is less than C-phase short circuit and overload situation. The core strain caused by the side-phase short circuit is larger than the middle-phase short circuit.

(a) ωt = π/2

(b) ωt = 7π/6

(c) ωt = 11π/6 Figure 24. Strain of transformer core caused by magnetostriction during short circuit. Figure 24. Strain of transformer core caused by magnetostriction during short circuit.

Figures 25Figures and 26 25 show and 26the show maximum the maximum values of values magnetic of magnetic induction induction and strain and strainof of Energies 2021, 14, x FORtransformer PEER REVIEWtransformer core, respectively, core, respectively, in the four in theoperating four operating modes (Section modes (Section 3.5) when 3.5) the when three- the three-22 of 24

phase magneticphase magneticfluxes of ﬂuxesA, B, and of A, C B,reach and Ctheir reach maximums. their maximums.

1.8 π/2 1.6 7π/6 11π/6 1.4

1.2

1.0

0.8

0.6

0.4 Magnetic induction intensity (T) intensity induction Magnetic

0.2

0.0 normal load load imbalance overload short circuit Transformer operating status

FigureFigure 25. 25.Maximum Maximum values values of of magnetic magnetic induction induction in in the the four four operating operating modes. modes.

3.8 3.6 π/2 3.4 7π/6 3.2 11π/6 3.0 2.8

- m/m) 5 2.6 － 2.4 2.2 2.0

1.8 1.6 1.4 1.2 1.0 0.8 0.6 Strain of transformer core (10 0.4 0.2 0.0 normal load load imbalance overload short circuit Transformer operating status Figure 26. Maximum values of strain of transformer core in the four operating modes.

It can be seen from Figures 25 and 26 that under load imbalance, overload, and short- circuit conditions, the magnetic induction intensity of the transformer core and the strain caused by magnetostriction both increase. The biggest influence on the iron core is the overload state, and the average stress is 49.37% higher than that under normal operation. In the short-circuit state, the average stress increased by 36.93%. Load imbalance has little effect on the transformer core strain.

6. Discussion In this paper, the physical model of the SBH15-M 10 kV amorphous alloy distribution transformer is established, the model being reasonably divided, and the no-load current simulation and no-load loss simulation verifying the correctness of the model. A mathematical model of the electromagnetic field-mechanical field coupling of the amorphous alloy core is established, and the electromagnetic field distribution and the strain under the influence of the magnetostrictive characteristics of the transformer under different working conditions are analyzed. When a transformer or core with a different structure or characteristics is used, the force result will change, but the conclusion will not be affected, because the electromagnetic field law applying to the core is the same under the same working conditions.

Energies 2021, 14, x FOR PEER REVIEW 22 of 24

1.8 π/2 1.6 7π/6 11π/6 1.4

1.2

1.0

0.8

0.6

0.4 Magnetic induction intensity (T) intensity induction Magnetic

0.2

0.0 Energies 2021, 14, 164 normal load load imbalance overload short circuit 21 of 23 Transformer operating status Figure 25. Maximum values of magnetic induction in the four operating modes.

3.8 3.6 π/2 3.4 7π/6 3.2 11π/6 3.0 2.8

- m/m) 5 2.6 － 2.4 2.2 2.0

1.8 1.6 1.4 1.2 1.0 0.8 0.6 Strain of transformer core (10 0.4 0.2 0.0 normal load load imbalance overload short circuit Transformer operating status FigureFigure 26. 26.Maximum Maximum values values of of strain strain of of transformer transformer core core in in the the four four operating operating modes. modes.

ItIt can can be be seen seen from from Figures Figures 25 25 and and 26 26 that that under under load load imbalance, imbalance, overload, overload, and and shortshort- circuitcircuit conditions, conditions, the the magnetic magnetic induction induction intensity intensity of of the the transformer transformer core core and and the the strain strain causedcaused by by magnetostriction magnetostriction both both increase. increase. TheThe biggest biggest inﬂuence influence on on the the iron iron core core is is the the overloadoverload state, state, and and the the average average stress stress is is 49.37% 49.37% higher higher than than that that under under normal normal operation. operation. InIn the the short-circuit short-circuit state, state, the the average average stress stress increased increased by by 36.93%. 36.93%. Load Load imbalance imbalance has has little little effecteffect on on the the transformer transformer core core strain. strain.

6.6. Discussion Discussion In this paper, the physical model of the SBH15-M 10 kV amorphous alloy distribution In this paper, the physical model of the SBH15-M 10 kV amorphous alloy distribution transformer is established, the model being reasonably divided, and the no-load current transformer is established, the model being reasonably divided, and the no-load current simulation and no-load loss simulation verifying the correctness of the model. A mathe- simulation and no-load loss simulation verifying the correctness of the model. A mathematical model of the electromagnetic field-mechanical field coupling of the amorphous matical model of the electromagnetic field-mechanical field coupling of the amorphous alloy core is established, and the electromagnetic field distribution and the strain under the alloy core is established, and the electromagnetic field distribution and the strain under influence of the magnetostrictive characteristics of the transformer under different working the influence of the magnetostrictive characteristics of the transformer under different conditions are analyzed. working conditions are analyzed. When a transformer or core with a different structure or characteristics is used, the When a transformer or core with a different structure or characteristics is used, the force result will change, but the conclusion will not be affected, because the electromagnetic force result will change, but the conclusion will not be affected, because the electromag- field law applying to the core is the same under the same working conditions. netic field law applying to the core is the same under the same working conditions. The results of the model provide a certain basis and reference for calculating the force of the iron core under different working conditions, optimizing transformer design, and providing technical support for the integration of amorphous alloy transformers. However, the simulation model in this paper was established after proper simplification. Only the simulation research of amorphous alloy transformer cores under different working conditions was carried out, and it was not strongly combined with the experiment. In addition, this paper does not analyze the debris generated under stress. Further research can be carried out on the following aspects: 1. Combining the simulation analysis results with the experiment and testing the stress of the iron core under different working conditions; 2. Calculating the amount of amorphous debris generated under stress by simulation; 3. Studying the flow trajectory and distribution of the amorphous debris in the transformer oil after generation; 4. Studying detailed strain analysis under different overloads and load imbalance, and its effect during high-speed rail operation.

7. Conclusions The work done and the conclusions obtained are as follows: Energies 2021, 14, 164 22 of 23

1. A strong coupling theory of electromagnetic field and mechanical field for magnetostriction of amorphous alloys and an actual calculation model of indirect coupling method using interpolation are proposed. 2. The finite element model of the amorphous alloy transformer established has no- load current and no-load loss, similar to the field test data of the product when the secondary winding is in no-load operation, and the data error is relatively small. 3. The normal operation of amorphous alloy transformers is simulated, and the characteristics of electric field, magnetic field, and eddy current field are analyzed. During normal operation, the electromagnetic characteristics of the transformer conform to the actual situation, which verifies the correctness of the simulation. 4. As the load becomes more unbalanced, the strain of the iron core gradually increases. When a short circuit occurs, the strain of the iron core reaches the maximum. Under normal circumstances, as the load increases, the stress of the iron core increases. When the transformer is overloaded, the core stress increases faster and the maximum value is higher than the core stress during C-phase short circuit.

Author Contributions: Conceptualization, C.X. and K.W.E.C.; methodology, J.S.; software, J.S.; validation, J.S.; formal analysis, J.S. and C.X.; investigation, J.S.; resources, K.W.E.C. and C.X.; data curation, J.S.; writing—original draft preparation, J.S.; writing—review and editing, C.X. and K.W.E.C.; visualization, C.X.; supervision, K.W.E.C. and C.X.; project administration, C.X.; funding acquisition, K.W.E.C. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Hong Kong Branch of National Rail Transit Electrification and Automation Engineering Technology Research Centre, grant number BBVR. Acknowledgments: We would like to thank Yake Tang, Hong Huang, and Shandong Zhixin Intelli- gent Equipment Co. Ltd. for the assistance in technical support. Conflicts of Interest: The authors declare no conflict of interest.

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