Research on 750kV Reactor Vibration Based on Electromagnetic-mechanical Field Coupling Finite Element Model

The vibration signal characteristics of a 750kV shunt reactor with high voltage level and high vibration amplitude is one of the important indexes to evaluate the operating condition of the reactor. Since the high level of safe operation with Ultra-high voltage power equipment, it is challenging to collect long-term and extensive vibration signal data under safe operating conditions. This paper establishes the electromagnetic-mechanical field coupling finite element model (FEM) of a 750kV shunt reactor under a COMSOL open-source simulation environment. The resonance effect of magnetostriction of the iron core and the Maxwell forces between the iron core discs on the reactor structure is analyzed. The vibration distribution and typical frequency characteristics of each part of the 750kV shunt reactor are analyzed to provide a reference for the research of vibration fault diagnosis of Ultra-high shunt reactor.


Introduction
In order to satisfy the demand for long-distance and large-capacity power transmission in northwest China, the number of Ultra-high voltage substations is increasing.The shunt reactor has a large vibration amplitude as critical operating equipment in large substations.It produces a high noise sound pressure level.Moreover, reactor vibration characteristics can be used to detect reactor operation status, one of the leading indicator quantities for reactor fault diagnosis.
There has been much research on the electromagnetic-mechanical vibration of reactors and transformers.In [1], the magnetostriction effect is obtained for each direction of the silicon steel sheet with different magnetic field clamping angles.In [2], the vibration displacement of the core caused by magnetostrictive forces under the no-load condition of a dry-type transformer are calculated by FEM.Core magnetostriction and Maxwell forces are the leading causes of reactor core vibration suggested in [3].In [4] and [5], it is found that Maxwell forces significantly affect the vibration of a shunt reactor without an air gap pad.In contrast, its effect on the vibration of the transformer model core can be neglected.In [6], it is inferred that the vibration frequency of the transformer core is mainly the exact times of the excitation current frequency of 50 Hz, that is, 100, 200, 300, 400, 500 Hz.The above theoretical studies analyze the causes and situations of transformer and reactor vibration and determine the analysis target of electromagnetic-mechanical coupling of reactors, which provide an essential theoretical basis for calculating electromagnetic-mechanical field coupling of Ultra-high voltage shunt reactors.
The high voltage level of the 750kV shunt reactor makes it challenging to collect a large amount of abnormal acoustic data under regular operation.Therefore, this paper establishes the FEM of a 750kV shunt reactor with the coupled electromagnetic-mechanical field, which is used to validate the model simulated reactor vibration signal and provide a reference for the study of 750kV shunt reactor vibration characteristics.

Theoretical model
Establish a coupled electromagnetic mechanical sound field model for 750kV parallel reactors.Calculate the core vibration caused by the magnetostrictive effect of the reactor core and Maxwell forces between the air gaps of the core.The voltage excitation source is applied to the shunt reactor winding.
The shunt reactor with a magnetostrictive saturation coefficient of s  is assumed to be in the excitation magnetic field with a voltage source of sin According to the principle of electromagnetic induction: In formula (1): B is the core magnetic induction intensity.N is the number of coil turns.S is the core cross-sectional area.
Since the reactor operates in the linear magnetization region, the magnetic field strength in the core is [7] :

B B B B
( 2 ) In Formula (2): μ 0 μ r is permeability.B s is the saturation magnetic flux density of iron core.H c is the coercive magnetic field strength.
The magnetostrictive strain  of silicon steel sheet is formula (3) [8] : △L is the deformation of silicon steel.
The magnetostrictive forces are equivalent to the magnetostrictive phenomenon of iron core silicon steel to facilitate the coupling calculation.The equal magnetostrictive forces are calculated by the magnetostrictive strain of the iron core [9] , as shown in Formula (4): Where σ is the magnetostrictive forces; D is the size of the elastic tensor; the elastic tensor is only related to the core material parameters.
Maxwell forces is a surface force at the interface between core air gap and core discs.It is generally calculated by Maxwell forces tensor T max [10] : ) n is the normal unit vector of the air gap surface of the iron core column.
The core vibration of a UHV shunt reactor can be expressed as the periodic vibration of the core in insulating oil under the combined action of equivalent magnetostrictive forces and Maxwell forces [11] : In formula (7): M is the mass matrix of the core; c x is the displacement of the core; C is the damping matrix, which is generally assumed to be negligible; K is the stiffness matrix of the core, which the material parameters and structural dimensions of the core can calculate.F is the external forces that cause vibration, F max is the Maxwell forces matrix, F mag is the magnetostrictive forces matrix, and t is time.

Emulation model
The research object is a 750kV B-phase BKD-70000/800 single-column shunt reactor, and its specific parameters of the shunt reactor are shown in Tab 1.The silicon steel sheet model is 30P100.
Tab The model built in this paper does not consider the influence of the rolling direction of the silicon steel sheet, which regards the reactor core model as an isotropic material.The B-H curve of silicon steel is shown in  The air gap material is generally a marble gasket with high hardness.The relative permeability of the marble gasket is much smaller than that of a silicon steel sheet.The magnetic field in the reactor is almost all passed through the silicon steel sheet.
Based on the above conditions, the FEM of the reactor is established.The process of simulation analysis is shown in The magnetostrictive effect of the ferromagnetic material of the reactor core and the Maxwell forces between the core discs causes the core vibration.The bottom edge fixed constraint is used to simulate the reactor base set.
The reactor model winding is set to a uniform multi-turn.The core magnetic field is simulated to excite the winding with a 50Hz rated sinusoidal alternating voltage of 2 times the voltage amplitude.The calculation step is set to T/20s, and the transient magnetic field is solved for the reactor with a total solution time of 0.1s.Fig. 3 FEM analysis process [12] Fig. 4 Reactor FEM Adjust the winding parameters so the reactor winding current is close to the rated value, as is shown in Fig 5, which proves that the reactor impedance value is relative to the actual value.Fig 6 is the magnetic flux density diagram of the reactor.It can be seen that the magnetic flux of the reactor mainly passes through the air gap between the core column and the core discs.Since the air is a non-ideal magnetic insulation material, a small amount of magnetic flux passes through the air outside the core, similar to the actual situation.There is a small amount of magnetic flux leakage between the core column and the winding.

Calculation and analysis
The magnetic flux density distribution of the core at 40ms, 42.5ms, 45ms, and 55ms is shown in Fig 7.
The magnetic flux density is higher at the corners of the core.The frequency domain results for the six measuring points are obtained separately.And calculate the 100Hz fundamental frequency proportion of acceleration at six Points.The expression of the fundamental frequency proportion is [13] : Where A is the acceleration amplitude of the corresponding frequency component, A 2 is the energy of the corresponding frequency component, focusing on frequencies within 800Hz of the observed frequency.

Conclusion
(1) Magnetostriction and Maxwell forces can cause the vibration of a 750kV shunt reactor.Since the magnetic flux is not saturated, the Maxwell forces is the main reason for the vibration.
(2) The simulation results show that the primary frequency of the 750kV shunt reactor vibration signal is 100Hz and 100Hz multiplies.Under normal conditions, each position's 100Hz fundamental frequency of vibration content is over 90%.The highest movement amplitude of the 750kV shunt reactor is at the magnetic shunt.
(3) The finite element simulation model of 750kV shunt reactor has high reliability of vibration signal.It can be used for 750kV shunt reactor vibration-related studies when field acquisition conditions are unavailable, or there are no abnormal working conditions.

Fig 1 .
The internal structure of the reactor includes winding, yoke, core discs, magnetic shunt, and air gap material, as shown in Fig 2.

Fig. 2
Fig 1. B-H curve of silicon steel

Fig 3 .
The reactor geometry model and mesh sectioning are shown in Fig 4.

Fig 5 .
Fig 5. Voltage and current waveform of reactor Fig 6.Magnetic flux density of reactor

Fig 7 .
Fig 7. Magnetic flux density distribution of iron core at different times

Fig 10 . 6 Fig. 11
Fig 10 shows that the reactor movement frequency is twice as high as the magnetic flux density frequency.The displacement of core measuring point 1 is the largest and the distortion is the most serious.Point 2 and 3 also have some distortion, but not as much as point 1.The core displacement of measuring point 4, 5 and 6 are shown inFig 11:

Fig 12 .
Fig 12. Acceleration decomposition analysis of Point 1 Fig 12 shows that the reactor core vibration frequency is even multiples of the power supply excitation frequency of 50 Hz, which is 100 Hz multiple frequency.The main frequency range of vibration is mainly concentrated within 800Hz.The frequency domain results for the six measuring points are obtained separately.And calculate the 100Hz fundamental frequency proportion of acceleration at six Points.The expression of the fundamental frequency proportion is[13] : 1. Parameter table of reactor The 100Hz fundamental frequency proportion of 6 Points is shown in Tab 2.It can be seen from theTable that except for Point 1, where the distortion rate is high, the percentage of 100Hz in all other points is more than 90%.It proves that the fundamental frequency of 750kV shunt reactor vibration under regular operation is 100Hz, and its mean value is more than 90%.2023 5th International Conference on Energy Systems and Electrical Power Journal of Physics: Conference Series 2584 (2023) 012114 IOP Publishing doi:10.1088/1742-6596/2584/1/012114