## 1 Introduction

The mechanical condition of windings and magnetic core of power transformers can be determined by using vibration diagnostics. This process is performed for loaded transformer or in a no-load operating mode by placing vibration accelerometers on the surface of power transformer tank [1].

In Latvian power system, there are three different variants of performing mechanical diagnostics of large power transformers: in no-load operation mode, at 50% and at 100% of full rated power. The sensors do not save harmonic analysis; they obtain and save the total values of vibrations. If within the received data, there are vibration values that exceed the provided limits: displacement – 100 μm, velocity – 20 mm/s, acceleration – 10 m/s^{2} more detailed analysis is required.

With this diagnosis methodology, it is possible to obtain robust information about the power transformer and discover, if a relatively large mechanical defect has occurred. However, there are no following actions and procedures advised in the case of increased vibration values.

Therefore, there is a necessity for a different diagnostic approach of large power transformer mechanical fault detection to improve result accuracy and reliability. The aim of this paper is to develop a simulation model for transformers with high vibration level to determine if magnetostriction process is the cause for such vibrations. Authors have presented the results of this simulation model at international symposium [2].

## 2 Magnetostriction in Magnetic Core

Magnetostriction is a property of ferromagnetic materials that changes their geometric dimensions when exposed to magnetic field [3]. It is because these ferromagnetic materials are composed of small magnetic domains that have their unique magnetic orientation, placement and size.

Generally, magnetic orientation is random throughout the ferromagnetic material, when there is no magnetic field passing through. Figure 1 a) shows this situation. The color coding corresponds to a correlation with an upward direction.

However, when an external magnetic field passes through, the magnetic domains react to it and partially rearrange their sizes [4]. With this process, the domains with matching magnetic orientation with the external magnetic field grow larger by changing the materials magnetic orientation on their individual surfaces. Thus, the magnetic domains, with individual magnetic orientation pointing against the external magnetic field become smaller [4]. This process is visualised in Figure 1 b), where areas color coded in grey, represent the fractions of the non-aligning magnetic domains with the external magnetic field that will change their individual magnetic orientation and separate from their original magnetic domains. Afterwards, these grey regions will join their more aligned neighbouring magnetic domains.

The result of this process is visualised in Figure 1 c). The magnetic domains have changed their original shape. Due to this, the shape of the entire ferromagnetic body has changed as well.

In the case of large power transformers, magnetic core is made of sheet steel, which creates a complex three-dimensional system consisting of a ferromagnetic material since the largest part of magnetic flux concentrates there [5]. This causes elastic deformations and vibrations of the magnetic core that have a sinusoidal form since the externally caused magnetic field is also sinusoidal. This process over a period can cause delamination since the elastic deformations are continuous throughout the operating period of power transformer. Furthermore, the vibration amplitudes may increase since this process can slowly reduce the mechanical resistance of the transformer magnetic core.

## 3 Magnetostriction Model Description

In order to determine these elastic deformations, it is necessary to know characteristics of the magnetic field inside magnetic core and magnetostriction curve of it. In this paper, magnetostriction curve is used as a function L = f(B) that expresses the elastic deformation, depending on magnetic induction [6]. This magnetostriction curve differs for each material since the magnetic domains in multiple ferromagnetic materials can have varying sizes and can resist to external magnetic field differently.

Since the model uses two software programs, it has two distinct subsections:

- The first consists of a three-dimensional model created in
*COMSOL*software to calculate magnetic induction values within the magnetic core, windings and the surrounding area; - The second is also a three-dimensional model created in
*Matlab*software for the calculation of elastic deformations.

The first subsection in *COMSOL* begins with the creation of model geometry and the assignment of a material to every created domain. Windings are made of copper, magnetic core from steel but the surrounding area from transformer oil. Each material used in the model has defined physical characteristics and parameters. Afterwards, it is necessary to generate a finite element mesh. Then the program calculates the values of magnetic induction in all the coordinates of finite element mesh intersections within a given time intervals in a provided duration of time. A built-in calculation system carries out the computations, which contains Maxwell equations [7].

Afterwards, the model exports obtained results to *MS Excel* in the form of a table and then imports in *Matlab* where RMS values of magnetic induction are calculated. The program creates a three-dimensional matrix that represents the geometrical structure of transformer model. Within it, it is assumed that every element is 1 cm^{3}. The model uses approximation to obtain the RMS values of magnetic induction where they are unknown. A modified three-dimensional nearest neighbour problem (NNP) algorithm carries out this approximation [8].

These results from *Matlab* subsection allow determining whether magnetostriction is the cause of these vibration epicentres.

The input parameters of the proposed magnetostriction model have the following aspects:

- The geometric dimensions of transformer tank, windings and magnetic core;
- The materials for different parts and their physical characteristics;
- Frequency of the voltage, primary voltage, secondary voltage, primary winding rated current, secondary winding rated current and the transformation coefficient;
- No-load operation mode;
- Magnetostriction curve function L = f(B) of magnetic core material.

The simulation model has the following restrictions:

- The magnetic core consists of 10 steel sheets. Each of them is 60 mm thick;
- The bracings of magnetic core are not considered;
- The terminals of primary and secondary windings are not considered;
- Interwinding insulation is not considered;
- The model functions as an enclosed magnetic system – magnetic field does not exist outside of it.

The result of this magnetostriction model is a visualisation of calculated elastic deformation values within the magnetic core of the transformer.

## 4 Research on Magnetic Core Anisotropy of Proposed Model

The created three-dimensional model in *COMSOL* software consists of domains for the magnetic core that are isotropic. Therefore, it is necessary to verify, whether anisotropy would have relative influence on the results of the magnetostriction model since it is present in magnetic cores of actual large power transformers [9].

As verification, two types of magnetic core anisotropy are used. Figures 2 and 3 show their layout.

The arrows display the direction in magnetic core rods and yokes with lesser magnetic permeability. The first case of magnetic core anisotropy has a characteristic that the anisotropy direction aligns with the direction of the transformer magnetic field. The intersections of the different domains are located at the corners between rods and yokes of transformer magnetic core.

The difference in resulting magnetic field for this layout of anisotropy shows Δ = 0.7119% absolute change. The model obtains this result by comparing the values of magnetic induction in all positions within the *Matlab* subsection three-dimensional model matrix. Equation (1) shows this calculation,

where *B _{base}* – RMS value of magnetic induction with isotropic magnetic core;

*B _{ani}* – RMS value of magnetic induction with anisotropic magnetic core;

*n* – total amount of positions in the *Matlab* subsection of the model.

However, for the second case of magnetic anisotropy, the difference of magnetic field was *Δ* = 11.90%. This is because the intersections of domains with different directions of anisotropy shifted from the corners of magnetic core towards the rods and had 90° cut angle between rod and yoke domains (see Figure 3).

Therefore, for this anisotropy type of magnetic core, the magnetic field is more concentrated near the inner corners where rods and the yoke connect and there is a winding around the rod. Figure 4 shows this result, where both the a) FEM mesh and b) instantaneous values of magnetic induction are visualised.

Increased instantaneous values of magnetic induction are visualised in red tones (see Figure 4 b)). These positions are the closest segments of the magnetic core to the windings due to the geometry of the magnetic core. Furthermore, the rapidly changing direction of the anisotropy of magnetic permeability contributes to this result since the magnetic field has least resistance in the region.

When taken in consideration that the program calculates this difference by using absolute values, the total magnetic field experiences these differences as a change in placement of the highest magnetic induction values not an overall increase or decrease. Therefore, the conclusion is that isotropic magnetic core is suitable for this magnetostriction model.

## 5 Case Study

As case study, the authors chose 5-legged large power transformer [10] with primary voltage – 13.8 kV, secondary voltage – 137 kV, primary and secondary rated power – 200 MVA for this paper. For this large power transformer vibration measurement results on tank surface are available in the study [11]. Continued research after a year showed that the values of vibration velocity exceed the given limit of 20 mm/s 2.5 times. However, the values of vibration displacement and acceleration did not exceed limits that indicate necessity for in depth analysis.

This magnetostriction model uses a method proposed by author in [12] for visualization and approximation of vibration results for areas of transformer tank surface where vibration sensors were not installed (see Figure 5). If this visualisation displays vibration amplitudes that exceed the given limits, the model begins the simulation since otherwise there is no necessity to calculate possible mechanical faults.

Such approximation allows determining the possible vibration epicentre regions and their intensity. However, it is not possible to distinguish between vibrations caused by the magnetic core and vibrations caused by transformer windings since the output of the visualisation is the result of a surface not a three-dimensional model. Therefore, there is a necessity for further diagnostics of the magnetic core, which the magnetostriction model carries out.

The first subsection of the magnetostriction model for case study has the following results. Overall, from the calculations, magnetic core has 12779 positions, primary and secondary windings have 4167 positions and transformer oil has 14181 positions, where the instantaneous values of magnetic induction were calculated using *COMSOL* software. Figure 6 shows a cross-section of instantaneous values of magnetic induction from these results.

Afterwards, the model uses the obtained values of magnetic induction to calculate the values of elastic deformation by using magnetostriction curve. Figure 7 shows the visualisation of data as the final output of the model.

## 6 Conclusions

The created magnetostriction model of transformer magnetic core yields results for the numerical values and location of elastic deformations. Based on the obtained results it can be concluded that the most intensive elastic deformations caused by magnetostriction effect for particular transformer analysed within case study are located in the middle of phase B rod. Yet, area with the highest vibration values measured on the tank surface of this transformer is located on the outer limb of phase A.

Therefore, by comparing the results of visualisation and magnetostriction model, the conclusion is that in this particular case the cause magnetostriction in magnetic core did not cause high level of vibrations in the transformer. Furthermore, it is necessary to consider the possibility of other mechanical faults such as insufficient mechanical bracing of some parts of magnetic core.

The authors plan to implement this magnetostriction model as a separate section of a more extensive diagnostics methodology of power transformers with the modelling of the electrodynamic force influence on the transformer windings.

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