Research on simulation calculation on short-circuit electrodynamics force of power transformer winding

Abstract

The safety and reliable operation of power grid is directly related to the ability of power transformer to withstand short-circuit, therefore, it is a problem to be solved to improve the ability of large power transformer windings to withstand short-circuit. Taking a three-phase five-limb power transformer as an example, the transient electromagnetic field, short-circuit electrodynamics force of windings and mechanical strength of coils are analyzed in depth. Firstly, the three-dimensional finite element model of the prototype is established, and the magnetic flux density distribution of the three-dimensional transient electromagnetic field of transformer under short-circuit operation and the axial and radial static force magnitude of the winding are calculated by using the field-circuit coupling method, and the distribution law can be obtained. At the same time, the mechanical strength of power transformer winding in its height direction is discussed, and the modal vibration mathematical model of transformer low-voltage winding in Z-axis direction is established. The displacement change and resonance frequency of the winding wire cake in the axial direction caused by short-circuit are calculated, and the short-circuit electrodynamics force of the winding is also checked. The research in this paper provides a theoretical basis for strengthening the design of short-circuit withstanding capacity of windings, and has a certain theoretical and engineering application value.

Keywords

DC power transformer field-circuit coupling short-circuit electrodynamics force winding mechanical strength

1. Introduction

Power transformer plays an important role in energy conversion and transmission in power grid, and whether the power grid can operate safely and reliably is directly related to its ability to withstand short-circuit. However, when the transformer is subjected to short-circuit during its operation, accidents caused by the insufficient mechanical strength of windings are common. Therefore, it is a problem to be solved to improve the ability of large power transformer windings to withstand short-circuit [1–3].

The numerical calculation method to analyze the electromagnetic field of large power transformer has been applied in practical engineering by the efforts of scholars. It has developed rapidly in use. Scholars have changed from only calculating linear problems to solving non-linear areas, from the original planar two-dimensional model analysis to the solution of the three-dimensional solid model, from only dealing with a single electromagnetic field problem to the multi-physical field coupling analysis, and the computing range and the solving ability have been improved significantly [4–6].

In the early stage, the static force of transformer winding only calculates the sum of axial force and radial force [7–10]. In the mid-term, scholars began to consider that the wire cakes cannot be treated as a whole, and they need to be decomposed into the superposition of the wire cakes, and ignoring the friction between the wire cakes [11,12]. Later, in order to make the calculation more accurate, scholars established a three-dimensional transformer model on the basis of two-dimensional research. According to the static strength inspection, even if the power transformer meets the assembly requirements, the winding will collapse and damage when the system is short-circuited. The main reason is that the axial mechanical strength of the winding is not enough. In recent years, people gradually begin to shift the research focus from static short-circuit force to dynamic force. The dynamic problems of windings can still be calculated by dividing them into axial and radial directions [13–15].

It can be seen from the above that scholars have achieved some scientific research results on the distribution of magnetic field inside transformer, the anti-short-circuit ability of coil and the axial mechanical strength of winding. However, most scholars use the two-dimensional finite element method to analyze the internal electric field or magnetic field of transformer in the field of engineering electromagnetic field. Although the two-dimensional finite element method can effectively deal with the non-linear problem, there are some errors and difficulties in the shape and distribution of complex three-dimensional magnetic field. A fast and effective method for calculating the electrodynamics force of windings, i.e. field-circuit coupling method, is proposed in this paper. This method assigns the degree of freedom to each node vector of transformer winding. Compared with the traditional calculation method, the vector of the whole model is continuous, which is beneficial to the modeling of singular points. At the same time, the field-circuit coupling method can reduce the dimension of solving the problem. When solving three-dimensional complex electromagnetic field, it is more comprehensive and accurate [16–18].

Based on this, taking a three-phase five-limb power transformer as an example, the transient electromagnetic field, short-circuit electrodynamics force of windings and mechanical strength of coils are analyzed in depth. Firstly, the three-dimensional finite element model of the prototype is established, and the magnetic flux density distribution of the three-dimensional transient electromagnetic field of transformer under short-circuit operation and the axial and radial static force magnitude of the winding are calculated by using the field-circuit coupling method, and the distribution law can be obtained. At the same time, the mechanical strength of power transformer winding in its height direction is discussed, and the modal vibration mathematical model of transformer low-voltage winding in Z-axis direction is established. The displacement change and resonance frequency of the winding wire cake in the axial direction caused by short-circuit are calculated, and the short-circuit electrodynamics force of the winding is also checked. The research in this paper provides a theoretical basis for strengthening the design of short-circuit withstanding capacity of windings, and has a certain theoretical and engineering application value.

2. Electromagnetic field calculation of power transformer

When large power transformer is short-circuited, the short-circuit current flowing through the winding is 15–25 times that of normal operation [19]. Under the joint action of the impulse current and the magnetic field, the windings will be subjected to huge force, which can cause the winding to be deformed, and the deformation will destroy the insulation of the winding or reduce the mechanical strength of the winding [20,21].

2.1. Basic assumptions

Because of the complex structure and large size of the transformer, the size of the three-dimensional calculation is large. In order to reduce the amount of three-dimensional calculation, the accuracy of the calculation results is the primary factor, and the following basic assumptions are put forward [22–24].

(1) Because short-circuit electrodynamics force of transformer winding is only analyzed and calculated without considering the eddy current factors in this section, the clamping devices, fuel tank and bushing, etc. in the structural devices can be ignored when modeling.

(2) Ignore the flux caused by the eddy current in the transformer winding.

(3) Ignore the influence of the displacement current, the permeability of the ferromagnetic material tends to infinity.

(4) The permeability and conductivity of the material and structural parts in the winding area are constant.

2.2. The governing differential equations

According to the three-dimensional electromagnetic field of large power transformer with complex structure, the finite element method based on the node vector is usually used to solve. In the area of the solution, the following governing differential equations are applied to the primary and secondary area, $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}\text{The primary area }\left\{\begin{array}{@{}l@{}}{\nabla}^{2}\boldsymbol{A}_{\boldsymbol{1}}=-{\mu}\boldsymbol{J}_{\boldsymbol{1}}\quad \\ \boldsymbol{J}_{\boldsymbol{1}}={\displaystyle \frac{N_{1}i_{1}}{K_{1}S_{1}}}\quad \\ u_{1}=i_{1}R_{1{\delta}}+L_{1{\delta}}{\displaystyle \frac{di_{1}}{dt}}+e_{1}\quad \end{array}\right.\\ \quad \text{The secondary area }\left\{\begin{array}{@{}l@{}}{\nabla}^{2}\boldsymbol{A}_{2}=-{\mu}\boldsymbol{J}_{2}\quad \\ \boldsymbol{J}_{\boldsymbol{2}}={\displaystyle \frac{N_{2}i_{2}}{K_{2}S_{2}}}\quad \\ i_{2}={\displaystyle \frac{u_{2}}{R_{2}}}\quad \\ u_{2}=e_{2}-\left(i_{2}R_{2{\sigma}}+L_{2{\sigma}}{\displaystyle \frac{di_{2}}{dt}}\right)\quad \end{array}\right.\end{array} & & \displaystyle\end{eqnarray}$ (1) where N ₁, K ₁, S ₁ is the turn number, duty cycle and cross-sectional area of high voltage winding, respectively, I ₁, J ₁ is the current of the high voltage winding and the current density in the primary side, R _1δ, L _1δ is the equivalent leakage resistance and leakage inductance of the high voltage winding, N ₂, K ₂, S ₂ is the turn number, duty cycle and cross-sectional area of low voltage winding, I ₂, J ₂ is the current of the high voltage winding and the corresponding current density, R _2δ, L _2δ is the equivalent leakage resistance and leakage inductance of the low voltage winding, e ₂ is the induction electrodynamics force of low voltage winding.

When the winding of the transformer is connected with the external circuit, the magnetic field is an unknown, the source of the voltage is excitation, the magnetism is inducted by electricity, and the distribution of magnetic field and the stress of the winding are explored. The key to solve the problem of coupling field is to calculate the equation of the finite element and the circuit equation synchronously, and the coupling equation of point vector bit can be calculated by $\begin{eqnarray}\displaystyle \left[\begin{array}{@{}cc@{}}\boldsymbol{0} & \boldsymbol{0}\\ \boldsymbol{C}^{\boldsymbol{i}\boldsymbol{A}} & \boldsymbol{0}\end{array}\right]\left\{\begin{array}{@{}c@{}}\boldsymbol{A}\\ \boldsymbol{ 0 }\end{array}\right\}+\left[\begin{array}{@{}cc@{}}\boldsymbol{K}^{\boldsymbol{A}\boldsymbol{A}} & \boldsymbol{K}^{\boldsymbol{A}\boldsymbol{i}}\\ \boldsymbol{0} & \boldsymbol{K}^{\boldsymbol{i}\boldsymbol{i}}\end{array}\right]\left\{\begin{array}{@{}c@{}}\boldsymbol{A}\\ \boldsymbol{ i }\end{array}\right\}=\left\{\begin{array}{@{}c@{}}\boldsymbol{0}\\ \boldsymbol{V }_{\boldsymbol{0 }}\end{array}\right\} & & \displaystyle\end{eqnarray}$ (2) where A is magnetic vector potential of the node, i is current density of the node, K ^{A

A} is the coefficient of position stiffness, K ^{A

i} is the bit current coupling matrix, K ^{i

i} is the resistance matrix, C ^{i

A} is the induction damping matrix, V ₀ is the applied voltage matrix.

By solving the above formula, the current, magnetic potential and induced electromotive force of primary and secondary coils at different times can be obtained. In addition, in the calculation and analysis, it is necessary to set the first kind of boundary conditions on the surface of the oil tank in parallel, while the second kind of boundary conditions should set the voltage load of the primary side to make it equal to the rated voltage.

2.3. Solutions and analysis

The power transformer structure has certain complexity, so in order to improve the computation efficiency, three-dimensional finite element model of three-phase five-limb transformer is simplified, when calculating its transient magnetic field distribution. The upper and lower yokes and the adjacent yokes are respectively equivalent to cuboids, and the core and the winding are equivalent to the cylinder. In the subsequent finite element calculation, the corresponding material property is given respectively, and the simulation physical model of the transformer is similar to that of Fig. 1.

Fig. 1.

The 3D modal graph of the eddy current loss in power transformer.

The model of the prototype transformer is SSZ10 (S—“three” phases; S—“three” coil; Z—on load voltage regulation; 10–Type 10-(transformer performance level code)), and the relevant parameters of three-phase five-column power transformer prototype are as follows:

Table 1

Basic parameters of the prototype

Parameter	Value	Parameter	Value
Model	SSZ10-180000/220	Rated capacity (kVA)	180000
Vector group	YNyn0d11	Frequency (Hz)	50
Core diameter (mm)	880	Core height (mm)	3140
Tank length (mm)	8080	Tank height (mm)	3310
Tank width (mm)	2530	Core window height (mm)	2140
Low voltage winding internal diameter (mm)	478	Low voltage winding external diameter (mm)	507
High voltage winding inner diameter (mm)	736	High voltage winding outer diameter (mm)	851
Low voltage winding turns	48	High voltage winding turns	1100
Filling ratio of winding coil	0.5	Relative permeability of air	1
Relative permeability of core	2000	Relative permeability of winding	0.99
The voltage ratings of the high voltage and low voltage windings	20

The magnetic flux density distribution of short-circuit winding is calculated and analyzed by the three-dimensional field circuit coupling method. The magnetic flux density distribution of low voltage winding short-circuit is shown in Fig. 2, and the axial magnetic flux density distribution and radial distribution of low voltage winding short-circuit are shown in Fig. 3.

Fig. 2.

The magnetic flux density distribution of low voltage winding in short-circuit runtime.

Fig. 3.

Contrast figure of magnetic flux density distribution.

It can be seen from Fig. 2, it is known that the maximum magnetic flux density of the low voltage winding is 2.565 T when the transformer is short-circuited. According to Fig. 3(a), when the transformer is short-circuited, the axial magnetic flux density distribution of the winding is still at the height of 0.6 m and 1.2 m of the winding, and the magnetic flux density reaches the maximum and then gradually decreases to the two ends. When the transformer is short circuited, the magnetic flux density of the winding in the middle is up to 2.285 T. According to Fig. 3(b), for radial magnetic flux density distribution, the magnetic flux density distribution at 0.985 m is close to zero during no-load operation, and then increases gradually to both sides, and reaches the maximum at the end of winding, the maximum values are 1.241t and −1.086 T, respectively.

3. Calculation of static force of winding short-circuit of power transformer

When the transformer is impacted by the short-circuit current, the transformer will be disturbed, which can cause local overheating in the transformer and can also produce large electrodynamics force. Therefore, when the structure of the transformer is designed, it is necessary to calculate the electrodynamics force of the transformer when the transformer is short-circuited.

3.1. Calculation of short-circuit current of transformer

When a transformer is short-circuited, there will be a transient process when the transformer starts short-circuit to short-circuit stability. During this process, there will be an impulse current, the current gradually attenuates with the short-circuit time until the steady state. The stability strength of the transformer is determined by the sudden current, the thermal stability of the transformer is determined by the steady-state short-circuit current and its duration [25,26]. The sudden current can be expressed by $\begin{eqnarray}\displaystyle i_{dm}={\displaystyle \frac{U_{{\varphi}}}{Z}}\sin \left({\pi}-{\displaystyle \frac{{\pi}}{2}}\right)+{\displaystyle \frac{U_{{\varphi}}}{Z}}e^{\frac{-R}{L}0.01}={\displaystyle \frac{U_{{\varphi}}}{Z}}\left[e^{-\frac{R}{L}0.01}+1\right]=\sqrt{2}K_{d}I_{1d} & & \displaystyle\end{eqnarray}$ (3) where K _d is the amplitude ratio of the steady-state current of the maximum short-circuit current, small capacity transformer K _d = 1.2 ∼ 1.3, large capacity transformer K _d = 1.5 ∼ 1.8, Z is the sum of Z ₁ and Z ₂ for transformer impedance.

The short circuit current is extracted by the field circuit coupling method and is shown in Fig. 4.

Fig. 4.

Relationship between short-circuit current and time in winding.

It can be seen from Fig. 4, the maximum value of sudden current is 6871 A, and is basically the same as the short-circuit current calculated by the scalar method, which reduces the error for the short-circuit electrodynamics force calculation.

3.2. Calculation of static force of power transformer winding

According to the experience of the domestic and foreign scholars on the short-circuit test of transformer winding and the experience of engineering practice for a long time, when the transformer is short-circuited, the main reason for the collapse and damage of the windings is the axial and radial forces acting on the winding [27,28].

The short-circuit electrodynamics force is constantly changing according to the complex law, the size of the short-circuit electrodynamics force on the winding is determined by the magnetic flux density in the magnetic field and the current flowing through the conductor. When the transformer runs in the rated operation condition, the short-circuit electrodynamics force on the winding does not threaten its safe operation, but if the transformer is short circuited suddenly, the value of the short-circuit current is dozens of times of the rated value. Then the electromagnetic force will suddenly become lager, and the relay protection device is too late to make a protective action in a very short period of time. So many of the components of the transformer are damaged. When the low voltage winding is short circuited by three-phase, and the calculation of the distribution of the short-circuit radial force is shown in Fig. 5.

Fig. 5.

Radial electrodynamics force of low voltage winding.

It can be seen from Fig. 5, the force of the low voltage winding is negative in the radial direction, which can illustrate the force of winding is inward compressibility force. The magnitude of the radial force changes between −18159 and 20139 N, and the maximum electromagnetic force is in the middle part of the winding, and the value is 20139 N.

The axial force distribution of the windings is shown in Fig. 6.

Fig. 6.

Axial electrodynamics force of low voltage winding.

It can be seen from Fig. 6, the electromagnetic force on the axial side is symmetrical, and the maximum electromagnetic force is near the end of the winding, the maximum axial force on the winding is 8081 N. When the value is larger than the critical value of the axial instability of the winding, the winding would be damaged. The overall force of the low voltage winding is shown in Fig. 7.

Fig. 7.

The axial and radial electrodynamics force distribution of low voltage winding.

It can be seen from Fig. 7 that the axial force on the low voltage winding is larger at the two ends, and the middle part is not affected by the short-circuit electrodynamics force. With the continuous change of the winding height, the axial force has also corresponding changes. For the radial force, it can be found that the magnitude of the force is negative. It shows that the winding of the power transformer is subjected to the internal compression force on the radial direction, and the force at the middle part is the largest, and then gradually decreases to the both ends.

3.3. Calibration of static force of power transformer winding

In general, the inclined collapse under the joint action of the axial and radial short-circuit electrodynamics force a is called winding instability. Therefore, the winding mechanical force should be checked to determine whether the magnitude of the force has an impact on the winding damage.

The axial electrodynamics force is checked as follows: $\begin{eqnarray}\displaystyle F_{t}=\left[k_{1}E_{0}{\displaystyle \frac{nbh^{2}}{D_{m}}}+k_{2}{\displaystyle \frac{nczb^{3}r}{h}}\right]k_{3}k_{4} & & \displaystyle\end{eqnarray}$ (4) where E ₀ is the elastic modulus of the copper conductor; n is the number of winding wires; b is the radial dimension of the winding conductor; h is the axial height dimension of winding conductor; D _m is the average diameter of winding; c is the width of the cushion block; z is the number of cushion blocks in the circumferential direction; r is the distortion term coefficient, which is 0.85; k ₁ is the coefficient of layered superposition term, which is 0.5; k ₂ is the coefficient of layered superposition term, which is 22; k ₃ is the coefficient related to the hardness of copper material, which is 1.0; k ₄ is the coefficient related to the dynamic tilt, which is 1.2.

The radial electrodynamics force is checked as follows: $\begin{eqnarray}\displaystyle F_{B}={\displaystyle \frac{12.25\times 10^{8}\left({\displaystyle \frac{x}{2}}\right)^{y}n^{1.5}b_{1}^{3}t(m^{2}-1)}{12\times r_{1}^{3}}} & & \displaystyle\end{eqnarray}$ (5) where x is the number of conductors in transposition conductor; y is the empirical value, which is taken as 1.1 in this paper; b ₁ is the radial thickness of each conductor; t is the axial height of each conductor; m is half of the effective stay in the inner winding; r ₁ is the average radius of the winding.

According to the above simulation results of short-circuit electrodynamics force, the maximum value of electromagnetic force in the axial direction of winding is 8081 N, which may lead to winding instability, so it needs to be checked. The critical force of winding in this area can be calculated as F = 12408 N by formula ((4)). Compared with the simulation results, it is found that the critical force value of winding in this area is larger than the maximum value of resultant force calculated by software simulation, and there is a certain safety margin. Therefore, it can be concluded that the low-voltage winding of this model has enough mechanical strength in its height direction, which will not cause axial collapse damage of the winding.

Similarly, for the axial force of low-voltage winding, according to formula (5), the critical force of radial instability of low-voltage winding is F = 121.23 kN/m, and the maximum value of radial compression force greater than that of low-voltage winding is 20.139 kN/m. Therefore, the transformer will not have low-voltage winding bending, insulation damage and radial instability.

4. Deformation analysis

In the actual process, the current in the short circuit process is continuously changing, and its axial vibration is a non-linear and complex mechanical motion. In this dynamic process, if the axial short-circuit electrodynamics force frequency acting on the winding is close to the axial natural frequency, the winding will produce resonance phenomenon, which causes the displacement between the winding cushion blocks to increase. Therefore, it is necessary to accurately calculate the natural frequency and displacement of windings.

In the analysis and study of transformer windings, the axial vibration of windings is equivalent to a spring vibration subsystem consisting of N degree of freedom distributed mass units according to the specific structure characteristics of windings. The winding is divided into several wire cakes by cross-over block and fixed between the iron yoke plates at both ends. The cushion block can be equivalent to a spring working in the compressed state, and with the end coil can be regarded as an elastic element together. The winding, as the sum of all wire cakes with concentrated mass, can be equivalent to a spring-mass-resistance under the action of axial force. The vibration model is shown in Fig. 8.

Fig. 8.

The axial vibration model of winding.

The dynamic equation of the axial vibration of the winding is as follows [29]: $\begin{eqnarray}\displaystyle M{\displaystyle \frac{d^{2}z}{dt^{2}}}+C{\displaystyle \frac{dz}{dt^{2}}}Kz=F+Mg & & \displaystyle\end{eqnarray}$ (6) where z is the displacement in Z direction, C is the sum of frictional coefficients of each wire cake in transformer oil, K is the sum of elastic coefficients of insulation gaskets between the upper and lower end coils of winding and pressure plate device, F is the resultant force acting on each unit of winding; M is the sum of mass of each wire cake.

Under the absence of external force, the influence of damped vibration of the system is neglected. Equation (6) can be simplified as $\begin{eqnarray}\displaystyle M{\displaystyle \frac{d^{2}z}{dt^{2}}}+KZ=0. & & \displaystyle\end{eqnarray}$ (7)

Equation (8) is the dynamic characteristic equation of winding vibration system. M and K are the mass matrix and elastic coefficient matrix of the system respectively. The natural frequency of winding vibration can be obtained by solving the equation.

The solution of the (7) assumed $\begin{eqnarray}\displaystyle Z=Z_{m}\sin (wt+{\varphi}) & & \displaystyle\end{eqnarray}$ (8) where Z _m is the main mode of the system.

Equation (8) is brought into (7) $\begin{eqnarray}\displaystyle (K-w^{2}M)Z=0. & & \displaystyle\end{eqnarray}$ (9)

The conditions for the equation to have non-zero solutions are as follows: $\begin{eqnarray}\displaystyle |K-w^{2}M|=0. & & \displaystyle\end{eqnarray}$ (10)

According to (10), the natural frequency of winding vibration is obtained. By expanding the above formulas, an algebraic equation of degree N about w ². For an elastic system with N degrees of freedom, its vibration system has N angular frequencies, whose value is the root of the equation.

Fig. 9.

1–4th-order mode shape of low voltage winding.

Fig. 10.

1–4th-order displacement of low voltage winding.

Table 2

Relationship between natural frequency and preload of transformer winding

Pre-tightening/Mpa	0.9	1.28	1.44	1.58	1.74
Displacement/m	0.02	0.04	0.06	0.08	0.1
Modulus of elasticity/Mpa	107.1	113.4	123.9	138.6	157.5
First order/Hz	18.17	19.04	20.51	21.84	22.13
Second order/Hz	18.193	19.075	20.879	22.212	22.571
Third-order/Hz	20.312	21.215	23.352	24.341	25.504
Fourth order/Hz	31.243	32.624	34.024	39.732	40.179
Fifth order/Hz	40.243	42.002	44.041	50.012	54.271

The vector expressions of all angular frequencies arranged in order of magnitude are as follows: $\begin{eqnarray}\displaystyle w=[w_{1}w_{2}\ldots w_{N}]. & & \displaystyle\end{eqnarray}$ (11)

The transformation analysis of (11) shows that the expression of the frequency column vector is as follows: $\begin{eqnarray}\displaystyle f={\displaystyle \frac{w}{2{\pi}}}=[f_{1}f_{2}\ldots f_{N}]^{T} & & \displaystyle\end{eqnarray}$ (12) where f ₁ is the first natural frequency, f ₂ is the second natural frequency, f _N is the N _th natural frequency.

4.1. Winding vibration mode

The static calculation is needed for modal analysis under preloading, the lower steel plate is fully restrained, and the upper steel plate is pre-loaded in the axial direction for analysis. Hence, according to the mathematical model of axial vibration of transformer winding, the finite element software is used to build three-dimensional solid model. The cable cake is equivalent to a solid ring. The axial frequency only need be discussed in the analysis process, only the vibration degrees DOF of Z direction of the whole winding are set, the X and Y directions DOF are constrained, and the displacement of the upper and lower pressure plates are all constrained. The natural frequency of transformer winding is analyzed by modal analysis, and the prestressing option is opened. The 1st–4th order modes of low-voltage windings with pre-tightening force of 1.2 MPa is shown in Fig. 9.

4.2. Winding displacement

When the transformer is subjected to external force, the axial displacement of the low-voltage winding is as follows:

According to Figs 9–10, it can be seen that the maximum displacement occurs in the middle of the winding when the winding resonates. The pressure of the pad in the middle of the winding is smaller, the pressure of the pad at the end of the winding is larger, and the upper and lower position of the force on the winding is basically symmetrical, which is in accordance with the actual situation.

4.3. Relation between pre-tightening force and natural frequency

The relationship between pre-tightening and natural frequency of low voltage windings is shown in Table 2.

It can be seen from Table 2, the natural frequencies of each order of windings increase with the increase of pre-tightening force, and the elastic modulus of gaskets changes with the change of pre-tightening and displacement. Therefore, this variable should be taken into account when establishing the mechanical model of low-voltage windings.

5. Conclusions

A finite element analysis method is proposed to study the force acting on transformer windings. Based on the physical model and vibration model of transformer, the following conclusions are given from the analysis:

1. The magnetic field distribution of the transformer core under three-phase short circuit conditions is analyzed by the “field-circuit” coupled finite element method. The magnetic flux density of the winding is uniform in the radial direction. The maximum magnetic flux density appears in the middle part of the winding in the axial direction, and then decreases gradually to both sides. And the relationship between the short-circuit current and time is given.

2. The electrodynamics force of the low-voltage winding in the axial direction is much smaller than that in the radial direction. At the same time, the distribution of short-circuit electrodynamics force of low-voltage winding in radial direction is that the force in the middle part of the winding is larger, the two ends are gradually decreasing, and the axial direction is exactly the opposite, and the axial direction is just the opposite. For this prototype, if the radial force is greater than 121.23 kN, it will make the low-voltage winding cake bend or warp easily, and the insulation is damaged. When the axial force is greater than 12408 N, it will cause the support failure, fall off and axial instability.

3. When the winding is deformed by external force, the coil displacement in the middle part of the winding is the most obvious. The displacement at both ends of the winding is basically symmetrical, and the displacement at the end of the winding is much smaller than that at the middle part, and the displacement value is 0.04046 m. The natural frequencies of windings under several different preloads are also calculated. It is concluded that appropriate preloads and corresponding natural frequencies should be selected as far as possible in the design of transformer structures. The main reason is that the natural frequencies of power transformer windings change with preloads. The greater the force, the higher the natural frequency of the winding. The natural frequency of the winding is ensured to avoid 50 Hz, which can the axial instability of windings be prevented from occurring due to resonance.

Footnotes

Acknowledgements

The paper was supported by the Natural Science Foundation of Heilongjiang Province, China (Grant No. LH2020E092).

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