Vibration and noise mechanism of a 110 kV transformer under DC bias based on finite element method

0 Introduction

A 110 kV transformer is a crucial piece of equipment in industrial and civil transmission engineering,and its safe and stable operation largely determines the reliability and power quality of the power system [1,2].Its working voltage is relatively high,accompanied by a large number of nonlinear components producing DC and harmonic components,making the vibration and noise characteristics of 110 kV transformers complex and variable,resulting in strong vibration and noise radiation during operation.This causes noise pollution and seriously affects the physical and mental health of workers.Long-term vibration can also cause mechanical fatigue in components such as cores and fixtures,leading to loosening of cores and windings,further exacerbating vibration,ultimately causing irreversible mechanical damage,and even affecting the operational safety of the equipment [3].The 110 kV substations are often located closer to urban areas,which makes their vibration and noise issues a concern for highvoltage transmission equipment.As is well known,the vibration noise of transformers is mainly generated by the magnetostrictive force of the core and the electromagnetic force of the winding.Recently,several theoretical and experimental studies have been conducted on the noise mechanism of transformers,and the contribution distribution of the vibration noise sources of transformers has been obtained,promoting the research progress of vibration noise problems [4].However,the problem of vibration and noise in the operation of 110 kV transformers has not been resolved,and the vibration and noise caused by DC bias is still one of the main problems currently affecting the environmental protection and equipment operation safety of transformers [5,6].The finite element simulation platform and powerful computing power of modern computers enable analyzing the internal characteristics of power equipment with a large capacity and complex testing and operating environments [7,8].

However,in existing transformer simulation studies,researchers have often focused on the electromagnetic characteristic changes of transformers caused by DC bias issues,and research on the internal vibration and noise issues of transformers during DC bias is insufficient [9].Yao proposed a simplified method to analyze the threedimensional nonlinear transient eddy current field of transformer windings under a DC bias by establishing a quarter finite element model and analyzing the nonlinear effect of the DC bias on online transformers.However,the modeling process ignored the reasonable simplification of the iron core [10].Zhang used the finite element method to design a multi-field coupling model of a converter transformer considering DC bias,analyzed the electromagnetic characteristics and reactive power loss characteristics of the converter transformer,and proposed and verified a method for calculating reactive power loss [11].Zeng used a two-dimensional transient field circuit directly coupled with the finite element method to quantitatively analyze the effect of DC bias on the excitation current of converter transformers.When the DC bias component increased from 0 to 2 A,the iron core loss increases by 0.5% [12].However,they did not fully consider the impact of DC bias on the vibration characteristics of transformer components [13].Liu considered the coupling effect between the electric,magnetic,and force fields,and analyzed the changes in the magnetic field and mechanical parameters of the core and winding of the converter transformer during the process of increasing the DC bias coefficient from 0 to 1.5.As the DC content increased,the magnetic flux density and stress of the core and winding increased,and the overall vibration of the converter transformer intensified.However,further quantitative analyses of vibration indicators and indepth research on the distribution characteristics of noise caused by DC bias are lacking [14].However,there are certain deficiencies in the design of the physical model of transformers in the above-mentioned studies,particularly in the winding mode,distribution of support buffer blocks,and differences between various components of the winding,which are crucial for the performance of winding vibration characteristics [15-17].

In this study,the structure of the iron core and winding components of a 110 kV transformer are optimized based on the design and operating parameters provided by the transformer manufacturer.A multi-field coupling model of the electricity,magnetism,force,and sound of the transformer was designed.The finite element method was used to comprehensively and quantitatively analyze the impact of the DC bias on the electromagnetic,vibration,and noise characteristics of the transformer.The coupling relationship between the electromagnetic,mechanical,and acoustic characteristics of the 110 kV transformer during the DC bias process was elucidated.

1 Finite element equivalent equation in the vibration and noise process of a 110 kV transformer

The transformer can be described using the following equation under the electromagnetic field in a normal working state:

where B is the magnetic induction intensity,H is the magnetic field strength,E is the electric field strength,D is the potential shift vector,J is current density,σ is conductivity,μ is permeability,ε is the dielectric constant,ρ is the charge density,and S is the area.The finite element dynamic equation of 110 kV transformer components during vibration can be expressed as [18]:

whereare the acceleration and node velocity vectors,respectively.Here,M is the mass matrix;C is the damping matrix;K is the stiffness matrix.Q(t) denotes the load vector.

The vibration of the iron core is composed of an electromagnetic force generated by the interaction between the silicon steel sheets caused by the gap and magnetostriction of the iron core itself.The vibration model of a silicon steel sheet under an alternating magnetic field is as follows:

where D0 is the bending stiffness of silicon steel sheet;ν is the Poisson’s ratio of the material;x and y are the transverse and longitudinal dimensions of two-dimensional silicon steel sheet respectively;ω represents the frequency;t denotes the time;lx is the elongation;E is the elastic modulus of the material;λZ is the axial magnetostriction coefficient;and V is the volume element.

The unit volume force of the magnetic field on ferromagnetic materials can be expressed as the superposition of the Lorentz force,material volume force,and material surface tension,as follows:

where f is the physical density,J is the current density,B is the magnetic flux density,and H is the magnetic field strength.μ is the permeability,τ is the volume density,and Δ is the Hamiltonian operator.

Currently,the gaps between the silicon steel sheets in the iron cores used in converter transformers are becoming increasingly smaller,and the stacking method is becoming more advanced.The Maxwell stress between the silicon steel sheets can be ignored;therefore,the main contribution of the iron core vibration is the magnetostrictive effect of ferromagnetic materials under alternating electric fields.The magnetostrictive force caused by the magnetostrictive effect in the parallel and vertical directions can be expressed as

where Fc is the magnetostrictive force and Fcmax is the amplitude of the electromagnetic force caused by magnetostriction.

The vibration of the iron core was primarily caused by the magnetostriction of the silicon steel sheets.When ferromagnetic materials are exposed to a power-frequency magnetic field,the magnetostriction frequency of the iron core is twice that of the voltage source.The core vibration acceleration ac caused by magnetostriction under the action of voltage ucos (ωt) is:

Among them,εs is the saturation magnetostriction rate of the silicon steel sheets,ω is the angular frequency of the voltage source,B is the saturation magnetic induction intensity of the iron core,L is the length of the silicon steel sheet,and A is the cross-sectional area of the silicon steel sheet.

The excitation voltage contains several high-order harmonics that distort the vibration signal of the iron core.The acceleration of the iron core vibration can be considered as a superposition of the fundamental sine voltage source and the harmonic voltage source.The acceleration of the iron core vibration under the ac of no-load voltage is:

Winding vibrations are mainly generated by the electromagnetic forces between the windings.When the load current i=I1 (1+cos (ωt)) in the winding,the winding vibration acceleration aw can be expressed as:

The vibration acceleration aw of the winding under the influence of harmonic current is:

Sound field calculation is used to solve the equation for the variation of sound pressure with time and spatial position in sound waves,including sound pressure p,particle velocity v,and density increment during sound propagation ρ.These three satisfy the following equations of motion,continuity,and state of matter [19].

Equation of motion: The relationship between sound pressure p and particle velocity v is:

Continuity equation: The relationship between particle velocity v and density increment ρ is:

where q is the volume production rate of the fluid.

Equation of State: The relationship between sound pressure p and density increment ρ is:

Among them,After obtaining the sound pressure values of the sound field in the x,y,and z directions based on the above three equations,the Helmholtz equation can be solved using the separation of variables’ method,as follows:

In the formula,k is wavenumber,andwhere c is the velocity of sound waves in a fluid;ρ0 is the fluid density;q0 is the external mass source acting on the fluid.

The calculation of the transformer radiation sound field belongs to the boundary closed external sound field problem,and the boundary conditions include a two-part closed boundary and an infinitely distant boundary.The velocity boundary condition should be satisfied on a closed boundary,that is,the structural surface vibration response results should be imported into the finite element model as the boundary condition for the acoustic analysis.The form of the acoustic finite element equation is

where pi is the node sound pressure and Vni is the node boundary condition.

2 Electrical magnetic force acoustic model of a 110 kV transformer

According to the structure and working characteristics of the 110 kV transformer in operation provided by Shandong Electric Power Equipment Co.,Ltd.,as listed in Table 1,a physical model of the 110 kV transformer was established using a finite element simulation platform.The sound field domain is equivalent to a hemisphere with a diameter of 16 m.The core and winding parts were selected as the solid mechanics domain,and the hemisphere air domain was selected as the sound field domain to calculate the interface coupling between the solid mechanics and sound field.A simplified sound-field calculation model is shown in Fig.1.The grid was refined and meshed,the research was set as coil geometry analysis and transient,and geometric nonlinearity was applied.After completing the model structure and solver configuration,the settlement of the electromagnetic and mechanical properties of the 110 kV transformer was determined,and the calculation results were used as boundary conditions for the external acoustic calculations of the model.

Table 1 Operating parameters of the 110 kV transformer

Fig.1 Multi-field coupling model of the 110 kV transformer

Fig.2 Flowchart of multi-field coupled finite element solution

Considering the time cost,the computational complexity and convergence speed of the model were optimized,and certain details were simplified without affecting the vibration characteristics.The material parameters of each part of the model are shown in Table 2.

Table 2 Model parameter configuration

To clarify the contribution of each component to the harmonic resonance effect of the transformer in more detail,the modal vibration analysis of the 110 kV transformer was divided into three parts: core,winding,and oil tank before conducting the active coupling solution.Only solid mechanics were considered,and the characteristic frequency of the transformer was solved in a passive state.The first four distinct natural frequencies of the transformer as a whole,core,and winding are listed in Table 3,and the corresponding modal vibration mode distributions are shown in Fig.3.The performance of each mode is staggered by the vibration frequency corresponding to the added harmonic excitation.

Table 3 Natural frequency distribution of transformer components and the overall system

Table 4 Electromagnetic and vibration noise characteristic parameters

Fig.3 Modal distribution of the whole,core,and winding of the first four stages of transformers

The harmonics that invade the 110 kV transformer are injected into the winding through field circuit coupling.To make the applied harmonics more accurate,all excitation sources were converted into parallel current sources,as shown in Fig.4.We introduce the transformer DC bias coefficient K dc=I dc/ I0.In the above equation,Idc is the amplitude of the DC current and I0 is the amplitude of the no-load current.

Fig.4 Coupling circuit: (a) Field circuit coupling principle;(b) DC circuit diagram;and (c) Overall structure of winding

3 Results and discussion

3.1 Electromagnetic characteristic analysis

Figure 5 shows the effect of DC bias on the excitation characteristics of a 110 kV transformer.The bias voltage leads to a continuous increase in the positive half-cycle area of the core excitation,whereas it continues to decrease in the negative half-cycle area gradually presenting a sharp peak waveform that is no longer symmetrical.Fast Fourier decomposition was performed on the time-domain data of the excitation current to observe the changes in the harmonic composition during the DC bias,as shown in Fig.6.According to the distribution data displayed by the red column,it is not difficult to find that before applying the DC component,the harmonic components generated during the excitation process were basically odd harmonics.When Kdc reached 0.85,a large range of even components was generated during the excitation process,which reflects the acceleration of the saturation process of the silicon steel sheet under the action of the DC component,and the working point of the transformer rapidly moves up on the hysteresis loop.According to the modal distribution data in Table 3,an increase in even harmonic components will increase the possibility of resonance effects with the fourth natural frequency of the core and the second natural frequency of the winding,which is one of the reasons for the intensification of the vibration group noise.As Kdc increased,the magnetic flux density exhibited an upward trend.When Kdc increased to 0.85,the magnetic flux of the core gradually saturated,and the increase in magnetic flux density slowed.The electromagnetic and vibration properties of the 110 kV transformer under various DC components are listed in Table 2.

Fig.5 Excitation current under different DC content

Fig.6 Excitation current spectrum under different DC contents

3.2 Analysis of force displacement characteristics

As described in Fig.7,the main contribution of DC bias to the stress distribution of core is in the main column,which is closely related to flow direction of the magnetic flux circuit.A 110 kV transformer is a three-phase threewinding transformer.The windings on the main and side columns have sinusoidal alternating currents with the same amplitude and a phase difference of 120°.The magnetic field generated by each winding in the core column passes through the main column.Therefore,when the DC component invaded,the increase in the magnetic flux density on the main column was the fastest,resulting in a more significant change in the stress distribution.Second,there was an increase in the number of stress points on the side columns,particularly,at the junction of the three core columns and the upper and lower yokes,which exhibited very strong stress.Overall,the maximum stress on the core increases with an increase in the DC component and reaches its maximum growth rate at Kdc 0.85,with a maximum value of 5.42×105 N/m2 increased to 9.35×105 N/m2 followed by a continued increase in stress but at a slower rate.This is because at this point,the core is near the saturation point of the magnetization curve,and the stress in the core is positively correlated with voltage U and magnetic flux density B [20].

Fig.7 Force distribution characteristics of core,under different DC bias coefficients

Ignoring the phase difference,the stress distribution characteristics of the high-and low-voltage windings in the three columns were not significantly different.Therefore,the B-phase winding was selected as an example to compare the stress distribution characteristics of the high-and lowvoltage winding coils during the DC component increase process,as shown in Fig.8.At the same time point and viewing direction,the distribution of stress points on the surface of the winding exhibited two different states.When a DC source was not set,the stress points on the front of the winding were concentrated at the central axis symmetry line,and at every 90° clockwise rotation along the z-axis,the maximum stress value appeared at 1.03×105 N/m2,as shown in Figure 8 (a).When Kdc was 0.35,0.85,and 1.25,the stress points of the winding were almost distributed at all surface finite element points within the visible range,and the range of stress intensity points gradually increased,with the maximum stress values increasing to 5.13×105,8.24 ×105,and 2.19 × 106 N/m2.

Fig.8 Force distribution characteristics of winding,under different DC bias coefficients

Figure 9 shows the deformation distribution characteristics of the core during the DC bias process.The hardness of silicon steel materials is relatively high,and the force direction of the widely used oriented silicon steel sheets is relatively stable;therefore,the deformation distribution of the core does not change significantly in morphology.However,the effect of the DC component on the amplitude of the core deformation was evident,which changed from 1.29×10-8 m (Kdc=0.00) to 3.07×10-7 m (Kdc=1.25).The DC bias had a significant effect on the distribution of winding deformation.Similarly,considering the A-phase winding as an example,the deformation characteristics of the winding during the DC bias process were compared,as shown in Fig.10.The maximum displacement of the winding,when the DC bias was not set,was 1.15×10-7 m,which is similar to the stress distribution of the winding,and the displacement points were concentrated in the middle of the winding,showing an overall inward squeezing state.When Kdc increased to 0.35,0.85,and 1.25,the maximum displacement of the winding increased to 3.37×10-7,6.26×10-7,and 8.34×10-7 m,respectively,and the distribution range of points with larger displacements increased,covering almost the entire surface of the high-voltage winding at Kdc=1.25.This was related to the frequency range of the excitation harmonics flowing through the windings.In the field-circuit coupling stage of the winding model used in this study,the winding was set as a tangled circuit connection sequence at the ends,and the middle winding was set as a continuous circuit connection sequence.Under the effect of a DC bias,the frequency range of the current harmonics on the winding changes from odd harmonics to odd and even harmonics,significantly increasing the force time on each winding and increasing the overall force surface of the winding.

Fig.9 Deformation distribution characteristics of core,under different DC bias coefficients

Fig.10 Deformation distribution characteristics of winding,under different DC bias coefficients

Figure 11 depicts the time-frequency domain curve of the core vibration acceleration when there was no DC and the DC bias coefficient reached 0.85.The rated extraction points of the vibration signals are shown in Figure 7 (a).As the DC bias coefficient increased from 0.0 to 0.85 and the vibration acceleration amplitude increased from 0.14 to 0.36 m/s2,a large number of “spikes”appeared in the time-domain curve of the vibration acceleration.The distribution of the time-domain signal points at the upper and lower amplitudes was uneven and mainly manifested as dense signals near the zero axis and sparse signals away from the zero axis.By comparing the frequency domain of the vibration acceleration before and after the addition of DC content,it can be observed that the harmonic content increased sharply after Kdc reached 0.85,and the main frequency shifted from 100 to 300 Hz.Simultaneously,the vibrational acceleration components at 200,350,400,and 500 Hz increased significantly.At this point,the iron core had already operated at the saturation point of the B-H curve.

Fig.11 Vibration acceleration time-frequency domain of the core at: (a) Kdc=0,and (b) Kdc=0.85

3.3 Analysis of noise characteristics

Figure 12 shows the sound field distribution characteristics of a 110 kV transformer under the influence of different DC components.Owing to the consistent distribution pattern of sound pressure and sound pressure level and the fixed conversion relationship between sound pressure and sound pressure level in amplitude,the sound pressure distribution characteristics will not be discussed separately.According to Fig.12,it can be observed that the amplitude of the sound pressure level of the 110 kV transformer increased from 58.2 to 73.6 dB after considering DC bias,with the most significant increase occurring at Kdc=0.85.Simultaneously,under the influence of the DC bias,the sound field distribution of the 110 kV transformer changed from the initial vertical layering (a) to horizontalvertical partitioning,and the greater the DC content,the more evident the partitioning area.This is due to the change in the harmonic distribution caused by the DC bias,and the increase in harmonic components makes the threedimensional distribution of noise more extensive at the finite element points in the sound field.That is,at the same finite element point and at the same time,the richness of the different dimensional noise components increases.This certainly intensifies the radiation efficiency of the noise,coupled with the contribution of the DC component to the amplitude of the excitation source,ultimately resulting in a maximum possible increase of 10 dB in noise during an actual operation of the 110 kV transformer.

Fig.12 Sound pressure level distribution of the 110 kV transformer under different DC bias coefficients

Figure 13 illustrates the time-frequency distribution of the sound pressure at 1 m in front of the converter transformer at Kdc=0 and Kdc=0.85.Comparing the two figures,it can be observed that the overall harmonic content of the noise signal increased sharply under the influence of the DC bias,and the amplitude of the sound pressure increased by approximately two times.By comparing the frequency domain distribution of sound pressure signals,it can be noted that the sound pressure increased in the frequency range of 50 Hz multiples,and the main frequency shifted from 100 to 300 Hz,reflecting the shift in the overall vibration noise radiation energy,which is closely related to the changes in the frequency domain distribution of the excitation harmonics.

Fig.13 Time frequency distribution of sound pressure at(a) Kdc=0 and (b) Kdc=0.85

4 Conclusions

The conclusions drawn from this study are as follows:

(1) The DC bias changes the harmonic frequency component of the excitation current,which is prone to generating even harmonics.

(2) The intensification of the vibration during the DC bias process is closely related to the modal vibration mode of the transformer.The harmonics generated during the DC bias process are prone to increase the probability of overlapping of the natural frequencies of the core and winding.

(3) The influence of the DC bias on the characteristics of 110 kV transformers lies in increasing the number and range of force points on the surface simultaneously,which occurs at a turning point of Kdc=0.85,after which the increase in force displacement slowed.

(4) Under the influence of the DC bias,the overall noise of the 110 kV transformer may increase by 10 dB,with the fastest increase occurring at Kd=0.85,when the noise frequency shifted to the right to 300 Hz.

(5) The research method in this study has reference value for the study of DC bias vibration characteristics of large-scale oil-filled equipment in high-voltage transmission systems,particularly,for high-voltage DC transmission equipment with serious DC bias problems.

Acknowledgement

This work was supported by the Key R&D Program of Shandong Province (2021CXGC010210).

Declaration of Competing Interest

We declare that we have no conflicts of interest.