energies

Article
Analysis of Winding Vibration Characteristics of
Power Transformers Based on the
Finite-Element Method
Xiaomu Duan 1 , Tong Zhao 1, * , Jinxin Liu 2 , Li Zhang 1 and Liang Zou 1
1 School of Electrical Engineering, Shandong University, Jinan 250061, China;
201613055@mail.sdu.edu.cn (X.D.); zhleee@sdu.edu.cn (L.Z.); zouliang@sdu.edu.cn (L.Z.)
2 State Grid, Jining Power Supply Company, Jining 272000, China; ljxwillingheart@gmail.com
* Correspondence: zhaotong@sdu.edu.cn; Tel.: +86-531-8169-6129




Received: 25 August 2018; Accepted: 7 September 2018; Published: 11 September 2018


Abstract: The winding is the core component of a transformer, and the technology used to diagnose
its current state directly affects the operation and maintenance of the transformer. The mechanical
vibration characteristics of a dry-type transformer winding are studied in this paper. A short-circuit
test was performed on an SCB10-1000/10 dry-type transformer, and the vibration signal at the
surface was measured. Based on actual experimental conditions, a vibration-simulation model of
the transformer was established using COMSOL Multiphysics software. A multiphysics coupling
simulation of the circuit, magnetic field, and solid mechanics of the transformer was performed on
this model. The simulation results were compared with measured data to verify the validity of the
simulation model. The simulation model for a transformer operating under normal conditions was
then used to develop simulation models of transformer-winding looseness, winding deformation,
and winding-insulation failure, and the winding fault vibration characteristics were analyzed.
The results provide a basis for detecting and analyzing the mechanical state of transformer windings.

Keywords: power transformer winding; vibration characteristics; multiphysical field analysis;

short-circuit experiment; winding-fault characteristics

1. Introduction
The safety and reliability of power transformers, which are the core pieces of equipment in a
power grid, are important for the reliable operation of the entire power grid [1]. Foreign statistics show
that approximately 2% of transformers that run continuously for more than four years will experience
accidents of varying degrees [2]. The high failure rate of transformers has always affected the safe
and stable operation of the power grid [3], and it is not difficult to find that mechanical faults in
transformer are often due to latent issues upon reviewing historical cases of transformer accidents [4].
Transformer faults generally involve the failure of main components and accessories, with the primary
source of these faults being due to windings and core failures. In China, faults have occurred in
18 transformers in and above the 110 kV class [5]. Of these faults, 10 (55.6%) were caused by winding
issues. From 2006 to 2010, the State Grid Corporation of China (SGCC) compiled statistics on the causes
of faults in 46 transformers, of which 26 (56.5%) were caused by winding deformation [6]. In 2013,
there were five accidents in transformers of the 110 kV class and above belonging to the Guangxi
Power Supply Company of the Southern China Power Grid [7]. Of these, four cases (80%) were caused
by winding deformation.
The study of electromagnetic vibration in transformers began in the 1920s, mainly by large
power-transformer manufacturers and related research institutions. However, that work was limited

Energies 2018, 11, 2404; doi:10.3390/en11092404 www.mdpi.com/journal/energies

Energies 2018, 11, 2404 2 of 19

by the technology available at that time, when winding fault simulations were not ideal [8]. Fahnoe H.
studied the forced vibration of a transformer’s vibrating iron core under magnetostriction and
resonance at the harmonic frequency [9]. A substantial amount of simulation of the transformer was
carried out. The modal-resonance frequency of the transformer was analyzed such that the transformer
avoided resonance, but the simulation results were not verified via experimentation [10]. Foster S. L.
and others used finite element numerical analysis to calculate the electromagnetic field and structural
force field of large transformers, and obtained the vibration of the transformer core [11]. By combining
electromagnetic-field theory with the theory of structural mechanics, Yang Qingxin and other scholars
in China established a mathematical model of the electromagnetic vibration of the iron core of a
power transformer [12–14]. The model was used to simulate magnetostriction of silicon steel sheets.
On this basis, the distribution of the sound field around the core was analyzed. These researchers
paid considerable attention to the vibration of the iron core, but the vibration of the windings at
various working conditions was less important to them. Liu Dichen and other scholars established an
electromagnetic mechanical sound field finite-element model of a transformer core and its winding [15].
In ANSYS Workbench, a finite-element model of the transformer winding, iron core, and oil tank were
established. Transient electromagnetic-field analysis was used to obtain the alternating electromagnetic
force of the transformer core and winding under the effect of alternating currents. Noise distribution
was analyzed, but little attention was paid to the spectrum analysis of the windings under various
fault conditions. Ji Shengchang and other scholars discussed in detail the relationship between the
vibration of the winding, the iron core, the load current, and the no-load voltage, and proposed a
method for extracting the characteristics of the vibration signal of the transformer based on wavelet
analysis [16,17]. Through simulation and experimentation, Yu Xiaohui and others discussed the
interaction between the tightening force and the natural frequency of the winding and concluded that
the pretension of the windings can change their natural frequency [18]. A comprehensive analysis of
the research conducted by experts around the world reveals that, although many effective diagnostic
methods based on vibration signals have been proposed, there still exist problems, such as incomplete
simulations of the various types of winding faults and poor diagnostic accuracy.
In recent years, various nondestructive testing methods for transformer-winding deformation
have been developed, such as the frequency-response analysis method for comparing transformer
frequency-response changes, and vibration analysis method for judging winding state based on
the transformer-vibration signal. The principle of frequency-response analysis is to detect the
amplitude-frequency response characteristics of each winding of the transformer, and compare
the detection results horizontally or vertically. According to the difference of amplitude-frequency
response characteristics, winding faults that may occur in the transformer are comprehensively judged.
In recent years, scholars have paid more and more attention to vibration-detection transformer research.
The vibration-analysis method discriminates the winding state of the transformer by detecting the
vibration signal transmitted to the body surface [19,20]. The principle is to reflect the winding states by
detecting a change in the mechanical characteristics of the winding. The frequency-response method
has many factors that affect the test results, such as the position of the signal source, the length of test
leads, the length of the test instrument grounding wire, the position of the transformer tap changer,
and connection mode. Compared with the frequency-response method, the vibration-analysis method
has fewer factors affecting the test results. The noise of the transformer cooling system will pollute the
vibration signal. When collecting the signal, it should be as far away from the fan group as possible,
or you should take noise-reduction measures. The vibration test results of transformer-winding
deformation are affected by the vibration of the core. Power frequency 150 Hz and 250 Hz components
appear in the frequency spectrum of transformer-vibration signals under a three-phase asymmetric
operation. The severe overvoltage generated in the asymmetric phase increases the amplitude of the
resonant frequency of the core, which interferes with the test results of winding. When a short-circuit
fault occurs, the vibration of the iron core is far less than that of winding, and it can be approximated
that the detected vibration contains only vibration signals of the winding. The frequency-response
Energies 2018, 11, 2404 3 of 19

method is blackout detection, while the vibration-detection method is live detection [21,22]. It can
continuously monitor transformer-winding deformation and reflect the decline trend of transformer
short-circuit resistance after repeated short-circuit shocks, which reduces the difficulty of online
monitoring and fault diagnosis of the power transformer.
Because the vibration of the transformer is a complicated process, the interaction between the
magnetic field and the load current, as well as between the magnetostriction of the silicon steel sheet
and the structural change in the transformer, produce changes in the vibration signal in both the
time and frequency domains, thus increasing the difficulty in fault monitoring and diagnosis. In this
paper, the SCB10-1000/10 dry-type transformer is studied in detail from the perspective of simulation
modeling, fault simulation, and feature analysis to obtain state diagnostics on transformer winding via
vibration analysis. Based on the mechanical-vibration characteristics of dry-type transformer windings,
a short-circuit experiment was performed on the SCB10-1000/10 transformer, and the vibration signals
at its surface were measured. A vibration-simulation model of the SCB10-1000/10 transformer was
established using COMSOL Multiphysics 5.3, and the coupling calculations were performed with
regard to the circuit, magnetic field, and solid mechanics of the transformer, among other areas of
physics. By comparing the simulated data to the actual data of the transformer, the accuracy of
the model was proven. Using this model, faults like loosening, deformation, and loss of insulation
from the transformer windings were simulated, and the vibration characteristics of the winding
fault were subsequently analyzed. The model utilizes multiphysical field-coupling simulation of the
electromagnetic solid mechanics of dry transformer windings, which can provide a new basis for the
state simulation and fault diagnosis of transformer windings.

2. Study of the Mechanical Vibration Characteristics of Transformer Windings

2.1. Vibration-Signal Conduction Process and Winding Electrodynamic Analysis of Dry-Type Transformers
This paper focuses on dry-type transformers. To understand the mechanism behind the
mechanical vibration of transformer windings, a short-circuit experiment of the SCB10-1000/10
dry-type transformer was conducted, and the vibration signal at the surface was measured.
The vibration of power transformers during operation is complicated and influenced by many factors,
but there are two main phenomena: the vibration caused by the electric force on the winding and the
vibration caused by the Lorentz force and the magnetostrictive force on the silicon steel sheet [23].
Figure 1 shows the conduction process of a vibration wave for a dry-type transformer. The vibration
caused by the winding and the iron core is transferred to the surface of the fixed clamp of the
transformer through the rigid component that connects the two. A dry-type transformer consists of
layer-type windings, which cause vibration from the effect of the electrical power. These windings pass
through the rigid connecting component to the fixed-clamp surface. The iron core of a dry-type
transformer is subjected to magnetostrictive force and the action of the Lorentz force, which is
transmitted to the surface of the fixed clamp of the transformer by a support unit, such as the cushion
block or the fastening bolt [24].
When the load current of the dry-type transformer is loaded, leakage of the magnetic field occurs
in its vicinity, which produces electrical power and causes mechanical vibration of the transformer
winding. This vibration is transferred through the connecting component to the surface of the
transformer clamp. When the transformer is in a steady state, the load current inside the winding can
be found as follows:
it = I cos ωt (1)

In Equation (1), the current effective value is presented, where ω represents the current
angular frequency.
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Figure1.1.Dry-type
Figure Dry-typetransformer-vibration
transformer-vibrationtransmission
transmissionroute.
route.

The vibration of the transformer body is mainly caused by core vibration, which, in turn, is caused
The vibration of the transformer body is mainly caused by core vibration, which, in turn, is
by the magnetostriction of the silicon steel sheet and winding vibration resulting from load current.
caused by the magnetostriction of the silicon steel sheet and winding vibration resulting from load
The vibration of the core is caused by the magnetostriction of the silicon steel sheet in a strong magnetic
current. The vibration of the core is caused by the magnetostriction of the silicon steel sheet in a strong
field [25]. The amplitude of the vibration is directly proportional to the square of the excitation voltage,
magnetic field [25]. The amplitude of the vibration is directly proportional to the square of the
and the fundamental frequency is two times greater than the voltage frequency. The vibration of the
excitation voltage, and the fundamental frequency is two times greater than the voltage frequency.
winding is caused by the electromagnetic force produced by the current in the winding. The amplitude
The vibration of the winding is caused by the electromagnetic force produced by the current in the
of the vibration is proportional to the square of the winding current, and the basic frequency is two
winding. The amplitude of the vibration is proportional to the square of the winding current, and the
times greater than the current frequency. In the short-circuit test of the transformer winding, due
basic frequency is two times greater than the current frequency. In the short-circuit test of the
to low excitation voltage, the vibration of the winding is far greater than the vibration of the core.
transformer winding, due to low excitation voltage, the vibration of the winding is far greater than
Therefore, the detected vibration signals can be approximated as containing only the vibration signals
the vibration of the core. Therefore, the detected vibration signals can be approximated as containing
of the winding.
only the vibration signals of the winding.
The leakage of the magnetic field around the winding of the transformer is a function that changes
The leakage of the magnetic field around the winding of the transformer is a function that
with time. When the winding generates a change in position, the distribution of the leakage of the
changes with time. When the winding generates a change in position, the distribution of the leakage
magnetic field around the winding also changes [26]. To calculate the force on a single conductor,
of the magnetic field around the winding also changes [26]. To calculate the force on a single
the discrete magnetic field value is fitted to a continuous distribution function. By the Biot–Savart Law,
conductor, the discrete magnetic field value is fitted to a continuous distribution function. By the
magnetic flux density B at a certain point on the windings can be expressed as follows:
Biot–Savart Law, magnetic flux density B at a certain point on the windings can be expressed as
follows: → u0
Z
dl 0 × r0
Bt = it (2)
 4πu0 l 0 dlr'2× r 0
Bt =
At a given point in space, all quantities except
it
4π li' are  (2)
r 2 constant. Thus, in the calculation of the
t
static electromagnetic
At a given point field, the magnetic-flux
in space, leakage
all quantities except density
it are Bt ofThus,
constant. the winding can be simplified
in the calculation to
of the static
the following:
electromagnetic field, the magnetic-flux leakage density Bt of the winding can be simplified to the
→ →
following: B t = k I cos ωt (3)
→  
where k is the proportionality constant between kI cos ωt
Bt =magnetic-flux density and load current. (3)
 magnetic-field leakage induced by the load current flowing through the transformer
The axial
where kBztis
winding is , and
the radial electromagnetic
proportionality force
constant Fx is induced
between by the action
magnetic-flux of the
density andload
loadcurrent. Similarly,
current.
the magnetic leakage field
The axial magnetic-field B , induced by radial induction, can induce axial electromagnetic
xt leakage induced by the load current flowing through the transformer force Fz
through
winding theisload
Bzt, current.
and radial Theelectromagnetic
axial force and radial
force force
Fx is of the conductor
induced by the can be calculated
action of the loadfrom the
current.
electric-force equation as follows:
Similarly, the magnetic leakage field Bxt, induced by radial induction, can induce axial
electromagnetic force Fz through the loadFcurrent. x = it BztThe
2πRaxial force and radial force of the conductor
(4)
can be calculated from the electric-force equation Fz = it Bas
xt 2πR
follows:
F x = it B zt 2π R
(4)
F z = it B xt 2π R
Energies 2018, 11, 2404 5 of 19

By vector calculation, the axial and radial electric forces are integrated to simplify the electric
power of the winding: p
F = Fx2 + Fz2 = it Bt 2πR
= I cos ωt · kI cos ωt · 2πR (5)
= 2πRkI 2 ( 21 + 12 cos 2ωt)
where it is the load current in the winding, ω represents the angular frequency of the power, and R
represents the transformer-winding radius. Equation (5) shows that the magnitude of the electric
force on the transformer winding is proportional to the square of the load current flowing through
the transformer winding, and the fundamental frequency of the vibration signal is twice the power
frequency of the power grid.

2.2. Transformer Short-Circuit Experiment

Based on the analysis above, when the secondary winding of the transformer is short-circuited,
the vibration of the body is mainly caused by winding vibration. To eliminate the disturbance caused
by vibration of the iron core and analyze only the vibration characteristics of the transformer winding,
a short-circuit experiment was performed on an SCB10-1000/10 transformer, which is a 10 kV SCB11
epoxy resin-cast dry-type transformer.
The parts with strong vibration signal are more likely to have fault accidents occur [27]. High
signal-to-noise ratio can also be obtained by selecting a strong vibration area of the transformer
winding [28]. According to the model studied in this paper, the vibration signal near the winding
is strong. In order to understand the vibration condition of the winding, the inner side of the upper
winding is the first choice, that is, the position selected in this experiment. During the experiment,
Energies 2018, 11, x FOR PEER REVIEW 6 of 19
the intensity of the vibration signal is tested in different areas. The results show that the vibration
signal near the winding is tested. Considering the difficulty of installing the vibration sensor and
the intensity of the vibration signal, the testUpoint = a ×near
S1 × Sthe
2 B-phase winding, the test point near (6) the
B-phase winding, and the test point on the inner and upper side of the C-phase winding are selected.
The sensitivity S1 of the vibration acceleration sensor was 2 pC/ms−2, and the sensitivity S2 of the
Figure
charge2amplifier
shows the location
was of the
656 V/pC. vibration-acceleration
The sensor relative
mechanical-vibration-signal diagramto the transformer
in Figure 3 can for the field
be obtained
experiment. The
via transformation. key technical parameters are shown in Table 1.

(a) (b)
vibration-acceleration sensor;
Figure 2. (a) Position of vibration-acceleration sensor; (b)
(b) field
field experiment.
experiment.

When the transformer is short-circuit tested, the secondary low-voltage side is short-circuited,
and a three-phase voltage is applied to the primary high-voltage side such that the load current in the
winding attains its rated value. The vibration signals were measured using a vibration acceleration
sensor (YD70C) (Xieli Science and Technology, Qinhuangdao, China), a charge amplifier (DHF-10),
and a Tek oscilloscope.
Energies 2018, 11, x FOR PEER REVIEW 6 of 19

U = a × S1 × S 2 (6)
Energies 2018, 11, 2404 6 of 19
The sensitivity S1 of the vibration acceleration sensor was 2 pC/ms−2, and the sensitivity S2 of the
charge amplifier was 656 V/pC. The mechanical-vibration-signal diagram in Figure 3 can be obtained
via transformation.
Table 1. Main technical parameters of the SCB10-1000/10 dry-type transformer.

Main Technical Indicators Parameter

Phase number Three-phase
Rated frequency 50 Hz
Rated capacity 1000 kVA
Rated voltage 10,000/400 V
Rated current 57.74/1443.38
Connection mode Dyn11
Cooling mode AN
Short-circuit impedance (%) 5.91

Figure 3 shows a set of typical mechanical-vibration signals measured during the experiment.
The ordinate axis in this picture represents vibration acceleration a. The relationship between vibration
acceleration a and output voltage U of the charge amplifier is as follows:
(a) (b)
U = a × S1 × S2 (6)
Figure 2. (a) Position of vibration-acceleration sensor; (b) field experiment.

Figure 3.
Figure 3. A set of typical mechanical-vibration
mechanical-vibration signals.
signals.

The sensitivity S1 of the vibration acceleration sensor was 2 pC/ms −2 , and the sensitivity S of the
3. Multiphysical Field-Coupling Model for Winding Vibration of Dry-Type Transformer 2
charge amplifier was 656 V/pC. The mechanical-vibration-signal diagram in Figure 3 can be obtained
In this paper, COMSOL Multiphysics software was used to simulate the vibration of the SCB10-
via transformation.
1000/10 dry-type transformer winding, and coupling simulation analysis of the magnetic field and
theMultiphysical
3. solid mechanics was performed.
Field-Coupling Model for Winding Vibration of Dry-Type Transformer
In this paper,
3.1. Establishment andCOMSOL Multiphysics
Mesh Generation software was used
of a Vibration-Simulation tofor
Model simulate the Transformer
a Dry-Type vibration of the
SCB10-1000/10 dry-type transformer winding, and coupling simulation analysis of the magnetic
The the
field and structure of the transformer
solid mechanics is complex and includes a variety of components such as the
was performed.
winding, iron core, and cooling device. For the purpose of the simulation, the transformer is
simplified
3.1. and treated
Establishment as Generation
and Mesh an ideal model in which the internal
of a Vibration-Simulation cooling
Model for device
a Dry-Type and supporting
Transformer
fastening components are neglected. The finite-element geometric model was set up based on the
The structure of the transformer is complex and includes a variety of components such as
actual structural parameters of the SCB10-1000/10 dry-type transformer (Xuzhou Debon Electric
the winding, iron core, and cooling device. For the purpose of the simulation, the transformer
Equipment Co., Ltd., Xuzhou, China), and a fixed constraint was applied at both ends of the winding
is simplified and treated as an ideal model in which the internal cooling device and supporting
to simulate clamping. To simulate the electrical insulation of the actual transformer, three-phase high-
fastening components are neglected. The finite-element geometric model was set up based on the
and low-voltage winding turns and layers were used, which allowed the electromagnetic field to be
actual structural parameters of the SCB10-1000/10 dry-type transformer (Xuzhou Debon Electric
solved. The solid mechanics model was set up as a fully coupled solution to understand the
Equipment Co., Ltd., Xuzhou, China), and a fixed constraint was applied at both ends of the winding
connection between the differential equation of motion and the differential equation of the
to simulate clamping. To simulate the electrical insulation of the actual transformer, three-phase high-
electromagnetic field. The simulation model is shown in Figure 4a. To improve efficiency, the
and low-voltage winding turns and layers were used, which allowed the electromagnetic field to be
solved. The solid mechanics model was set up as a fully coupled solution to understand the connection
between the differential equation of motion and the differential equation of the electromagnetic field.
Energies 2018, 11, 2404 7 of 19

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11, xx FOR
FOR PEER
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The simulation model is shown in Figure 4a. To improve efficiency, the geometric transformer model
was simplified
geometric
geometric by considering
transformer
transformer model it symmetric.
model was
was simplifiedFigure
simplified 4b illustrates
by considering
by considering the simplified
itit symmetric.
symmetric. geometric
Figure
Figure modelthe
4b illustrates
4b illustrates of
the
the body. geometric
simplified
simplified geometric model
model of
of the
the body.
body.

(a)
(a) (b)
(b)
Figure 4.
Figure (a) Integral
4. (a) Integral model
model of
of transformer;
transformer; (b)
(b) simplified transformer.
simplified model of transformer.

The geometric
The geometricmodel
geometric modelof
model ofofthe
thetransformer
the transformer
transformer is is
is meshed
meshed
meshed byby
by means
means
means of aaof
of a free
free
free tetrahedron
tetrahedron
tetrahedron network,
network,
network, and
and
and
the the mesh
the mesh
mesh modelmodel
model shown
shown shown
in in Figure
in Figure
Figure 5a can5a
5a can becan
be be obtained.
obtained.
obtained. Thequality
The mesh
The mesh mesh quality
quality is shown
is shown is in
shown
in in5b,
Figure
Figure Figure
5b, 5b,
and the
and the
and the
closer closer
the valuethe
is value
to 1, the is to
higher1, the
the higher
mesh the
quality mesh
is. quality
The is.
winding The
and winding
core and
structural
closer the value is to 1, the higher the mesh quality is. The winding and core structural parameters of core structural
parameters of
parameters of the
the SCB10-1000/10
the SCB10-1000/10
SCB10-1000/10 dry-type
dry-type transformer dry-type
transformer are transformer
are shown
shown in are shown
in Tables
Tables 22 and in
and 3.3. Tables 2 and 3.

(a)
(a) (b)
(b)
Figure 5.
Figure
Figure 5. (a)
(a) Mesh-generation results
(a) Mesh-generation
Mesh-generation results of
results of geometric
geometric model;
model; (b)
(b) quality
quality diagram
diagram of
of grid
grid division.
division.

Table
Table 2.Main
Table2.
2. Mainparameters
Main parametersof
parameters ofthe
of theSCB10-1000/10
the dry-type
SCB10-1000/10 dry-type
SCB10-1000/10 transformer
dry-type transformer coil
transformer coil structure.
coil structure.
structure.
Coil Parameter Type Size Size
(mm) Coil Parameter Type Size (mm)
Coil Parameter Type Size Coil Parameter Type Size (mm)
Coil Parameter
Internal Type
diameter 286 Coil Parameter
Internal Type
diameter Size (mm)
252
(mm)
(mm)
External diameter 369 External diameter 280
High pressure
Internal
Internal diameter
Heightdiameter
286
286
405
Internal diameter
Low pressure Internal diameter
Height
252446
252
External
External
Turn diameter
diameter
number 369
369
1125 External
External diameter
diameter
Turn number 28045
280
Low
Low
High pressure
High pressure Type
Height
Height Layer405
type
405 Type
Height
Height Layer
446 type
446
pressure
pressure
Turn number
Turn number 1125
1125 Turn number
Turn number 45
45
Type
Table
Type Layer type
3. Main parameters
Layer type Type
of the transformer core structure.
Type Layer type
Layer type

Parameter Type Parameter Parameter Type Size (mm)

Table 3.
Table 3. Main
Main parameters
parameters of
of the
the transformer
transformer core
core structure.
structure.
Structure Three-phase three-column General length 990
Joint method of side column Standard oblique connection Total height Size
Size
900
Parameter Type
Parameter Type Parameter
Parameter Parameter
Parameter Type
Type
Silicon steel sheet material 35Q165 Thickness (mm)
196
(mm)
Core column radius (mm)
Structure
Structure Three-phase90
Three-phase three-column
three-column Upper- and lower-yoke
General
General lengthheight
length 180
990
990
Joint method
Joint method of of side
side column
column Standard oblique
Standard oblique connection
connection Total height
Total height 900
900
Silicon steel sheet material
Silicon steel sheet material 35Q165
35Q165 Thickness
Thickness 196
196
Core column
Core column radius
radius (mm)
(mm) 90
90 Upper- and
Upper- and lower-yoke
lower-yoke height
height 180
180
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3.2.
3.2.Simulation
SimulationModel
Modelofof
ofElectromagnetic
ElectromagneticField
Field
3.2. Simulation Model Electromagnetic Field
Figure
Figure 666shows
shows the circuit
circuit diagramofofthe thetransformer.
transformer. In In the diagram, an AC voltage source
Figure showsthethe circuitdiagram
diagram of the transformer. the diagram,
In the an AC
diagram, an voltage source
AC voltage with
source
with
a 50 a
Hz50 Hz frequency
frequency was was applied
applied to theto the three-phase
three-phase high-voltage
high-voltage side,side,
and and
the the low-voltage
low-voltage side side
was
with a 50 Hz frequency was applied to the three-phase high-voltage side, and the low-voltage side
was short-circuited.
short-circuited. A 50
A 50AHz Hz AC
ACAC voltage
voltage to
to to the
thethe rated
rated current
current was applied to the high-voltage winding
was short-circuited. 50 Hz voltage rated currentwas
wasapplied
appliedtotothe
thehigh-voltage
high-voltage winding
by an external voltage source. The Thethree-phase
three-phaseinduction
inductioncurrent
currentof the low-voltage winding was
by an external voltage source. The three-phase induction current of the low-voltage winding was
obtained
obtainedby byelectromagnetic-coupling
electromagnetic-couplingcalculation,
calculation,as asshown
shownin inFigure
Figure7.7.AsAsaaresult,
result,the
theamplitude
amplitudeofof
the three-phase induction current of the low-voltage winding was 2041 A, and its effective value was
the three-phase induction current of the low-voltage winding was 2041 A, and its effective value was
1443.38
1443.38 A. The simulation results are consistent with the rated current of the transformer,as
A. The simulation results are consistent with the rated current of the transformer, asshown
shown
ininTable
Table1.1.

power
power
supply
supply high voltage
high voltage low voltage
low voltage
winding
AC

AC

winding winding
winding
Resistor 1 Resistor 2
Resistor 1 Resistor 2

Figure 6. External-circuit equivalent diagram.

Figure 6. External-circuit equivalent diagram.

Transformerlow-voltage
Figure7.7.Transformer
Figure low-voltagecoil
coilthree-phase
three-phaseinduction
inductioncurrent.
current.
Figure 7. Transformer low-voltage coil three-phase induction current.
The current obtained from the circuit module is used as the excitation source of the magnetic-field
The current obtained from the circuit module is used as the excitation source of the magnetic-
The
model, andcurrent obtained
current densityfrom
J wasthe circuitinto
brought module is used as thedifferential
the magnetic-field excitation source of as
equation thefollows:
magnetic-
field model, and current density J was brought into the magnetic-field differential equation as
field model, and current density J was brought into the magnetic-field differential equation as
follows:
follows: µ0−1 µr−1 ∇2 A = J e (7)
−1 −1 2 e
μμ0−1μμr −∇
1 2A = J e
= J space and has a value of 4π *10−7 H/m,(7)
0 r ∇of
In Equation (7), µ0 represents the permeability Afree (7)
R is
the relative permeability, A is the vector magnetic potential, and J e is the current density. Additionally,
In Equation (7), μ0 represents the permeability of free space and has a value of 4π *10 −7H/m, R is −7
In Equation (7), μ0 represents the permeability of free space and has a value of 4π *10 H/m, R is
the relative
the followingpermeability,
relationship exists
A is inthe
thevector
electromagnetic-field model:and Je is the current density.
magnetic potential,
the relative permeability, A is the vector magnetic potential, and Je is the current density.
Additionally, the following relationship exists in the electromagnetic-field model:
Additionally, the following relationshipBexists = µ0 µinr H the=electromagnetic-field
∇×A model: (8)
B = μ0 μr H = ∇ × A (8)
B = μ0 μr H = ∇ × A (8)
where B represents the magnetic-flux density, and H represents the magnetic-field intensity. Figure 8
where B represents the magnetic-flux density, and H represents the magnetic-field intensity. Figure
where
shows B represents
the H–B curve theofmagnetic-flux
the transformer density,
when and H represents
it is in operation. the
Themagnetic-field
curve in Figureintensity. Figure
8 also provides
8 shows the H–B curve of the transformer when it is in operation. The curve in Figure 8 also provides
8information
shows the H–B curve of the transformer when it is in operation. The curve in Figure 8 also
on the material properties of the dry-type transformer core, which agrees with the actual provides
information on the material properties of the dry-type transformer core, which agrees with the actual
information on the material
material properties properties of the dry-type transformer core, which agrees with the actual
of the transformer.
material properties of the transformer.
material properties of the transformer.
The magnetic-flux density distribution around the winding and the core of the dry-type
The magnetic-flux density distribution around the winding and the core of the dry-type
The magnetic-flux
transformer density distribution
during a short-circuit around
test was obtained by the winding
coupling and the core of themodel.
the electromagnetic-field dry-type
transformer during a short-circuit test was obtained by coupling the electromagnetic-field model.
transformer during a short-circuit test was obtained by coupling the electromagnetic-field model.
 As seen from Figure 9, the maximum flux-leakage density of the dry-type transformer winding
was 0.08 T, and the maximum flux density of the core was 1.75 T. The results of the simulation are in
line with the output parameters of the SCB10-1000/10 dry-type transformer. The results verify the
Energies 2018, 11, x FOR PEER REVIEW 9 of 19
correctness
Energies of2404
2018, 11, the model, demonstrating that the model can be further used for the simulation of
9 of 19
various faults in the windings.
As seen from Figure 9, the maximum flux-leakage density of the dry-type transformer winding
was 0.08 T, and the maximum flux density of the core was 1.75 T. The results of the simulation are in
line with the output parameters of the SCB10-1000/10 dry-type transformer. The results verify the
correctness of the model, demonstrating that the model can be further used for the simulation of
various faults in the windings.

Figure 8.
Figure 8. H–B
H–B curve.
curve.

As seen from Figure 9, the maximum flux-leakage density of the dry-type transformer winding
was 0.08 T, and the maximum flux density of the core was 1.75 T. The results of the simulation are
in line with the output parameters of the SCB10-1000/10 dry-type transformer. The results verify
Figure 8. H–B curve.
the correctness of the model, demonstrating that the model can be further used for the simulation of
various faults in the windings.

(a) (b)
Figure 9. (a) Flux-density distribution of the transformer winding; (b) Flux-density distribution of the
transformer iron core.

3.3. Modeling of Solid Mechanics

(a)and Analysis of Winding Vibration (b)
In the 9.
Figure
Figure solid-mechanics
9. (a)
(a) Flux-density model,
Flux-density distributionthe of
distribution vibration
of the characteristics
the transformer
transformer winding;of
winding; (b)the
(b) dry-type distribution
Flux-density
Flux-density transformer ofwinding
distributionof the
the
musttransformer
be coupled.
transformer The
iron
iron mass inertia, elasticity, and damping of the winding are the key factors affecting
core.
core.
its vibration. Equation (9) provides the differential equations of motion for the solid mechanics.
3.3.
3.3. Modeling
Modeling of of Solid
Solid Mechanics
Mechanics andand Analysis
Analysis2of of Winding
WindingVibration
Vibration
d z dz
In M + C + k i z = f (t ) of (9)
In the
the solid-mechanics
solid-mechanics model, model, thethevibration
i
dt 2
vibration icharacteristics
characteristics
dt ofthe
thedry-type
dry-typetransformer
transformerwinding
winding
must
mustbe becoupled.
coupled. The The mass
mass inertia,
inertia, elasticity,
elasticity,and anddamping
dampingof ofthe
thewinding
windingare arethe
the key
key factors
factors affecting
affecting
its In Equation
vibration. Equation(9), M(9)
i represents
provides thedifferential
the mass matrix; Ci represents
equations of motionthefor
damping
the coefficient
solid mechanics. matrix,
its vibration. Equation (9) provides the differential equations of motion for the solid mechanics.
[ ]
ki 0,1 represents the stiffness coefficient2 matrix,
d zd 2 z dzdz
z represents the deformation (displacement) of the
dz M M
i + +
C C
i ++kki zi z== ff((tt)) 2 (9)
(9)
dtof the winding, d z represents the deformation
dti 2 2 i
dt
winding, represents the deformation dtvelocity
dt
In Equation (9), Mi represents the mass matrix; Ci represents the dt2 damping coefficient matrix,
In Equation (9), M i represents the mass matrix; Ci represents the damping coefficient matrix,
kacceleration of the winding, and f(t) represents the magnitude of the force on the winding. Equation
i [0, 1] represents the stiffness coefficient matrix, z represents the deformation (displacement) of
k
the[ ]
(5) 0,1
for electrodynamic
represents
i winding,
dt
dz forcethe
represents is incorporated
the stiffness coefficient matrix,
deformation into the
velocity differential
z represents
of the winding, equation
the
dt2
d2 z of Motion
deformation (9). deformation
(displacement)
represents the of the
accelerationdzof the winding, and f(t) represents the magnitude of the force
d2z on the winding. Equation (5)
winding, represents the deformation velocity of the winding, 2 represents the deformation
dt
for electrodynamic force is incorporated into the differential equation dtof Motion (9).
acceleration of the winding, and f(t) represents the magnitude of the force on the winding. Equation
(5) for electrodynamic force is incorporated into the differential equation of Motion (9).
Energies 2018, 11, 2404 10 of 19

2
l Mi ddt2z + Ci dz
dt + k i z (10)
= 21 BI 2 + 12 BI 2 cos 2ωt + Mi g
To converge to a solution, it is necessary to impose fixed constraints on both ends of the
transformer winding.
In Equation (10), mass matrix Mi , damping coefficient Ci , and stiffness coefficient ki are all constant,
which makes it a constant coefficient differential equation. The solution of the equation is composed of
the general solution and the special solution. For the homogeneous part,

d2 z dz
Mi + Ci + k i z = 0 (11)
dt2 dt
the general solution is as follows:
Ct
− 2Mi
z0 = Ye i sin(v0 t + θ ) (12)

In Equation (12), Y and θ are constants whose values are determined by the initial conditions,
and v0 represents the natural frequency of the vibration of the transformer winding, which is expressed
as follows: s
Ki C 2
v0 = −( i ) (13)
Mi 2Mi
There are two special solutions for Equation (10):

0.5BI 2 + Mi g
z1 = =D (14)
Ki

z2 = G cos(2ωt + ψ) (15)
BI 2
G= q
2
;
In the equation: (Ki −4Mi ω 2 ) +4Ci2 ω 2 .
tan ψ = − K −2C iω
2
i 4Mi ω
The total displacement of the transformer winding at any time can be expressed as follows in
Equation (16):
Ct
− 2Mi
z = z0 + z1 + z2 = Ye i sin(v0 t + θ ) + D + G cos(2ωt + ψ) (16)

where t, the total displacement of the winding, and the displacement velocity are zero. Integral
constants Y and θ can then be obtained.
By quadratic derivation of the total displacement equation of the transformer winding,
the vibration acceleration a of the transformer winding at any time t can be obtained as in Equation (17):
Ci t
a = −v02 Ye− 2M sin(v0 t + θ ) (17)
−4ω 2 G sin(2ωt + ψ)

From the solution above, it can be seen that the vibration characteristics of a given transformer’s
windings are mainly related to the elastic coefficient and the winding geometry, that is, when elastic
coefficient K and the geometric structure of the transformer winding are changed, the vibration
acceleration of the transformer-winding surface changes accordingly. Table 4 shows the main material
properties of the winding and the core in the solid mechanics model [29].
Energies 2018, 11, x FOR PEER REVIEW 11 of 19

acceleration of the transformer-winding surface changes accordingly. Table 4 shows the main
material properties of the winding and the core in the solid mechanics model [29].
Energies 2018, 11, 2404 11 of 19
Table 4. The main material properties of the solid mechanics model.

Structure Table 4. The

Modulus of main material(Pa)
Elasticity properties ofDensity
the solid mechanics
(g/cm3) model.Poisson’s Coefficient
Winding Structure 1.16 ×of1011
Modulus Elasticity (Pa) Density (g/cm 3.2
3) Poisson’s Coefficient 0.32
Iron core Winding 2 ×1.16
1011
× 1011 3.2 7.6 0.32 0.24
Iron core 2 × 1011 7.6 0.24

The amplitude and axial forces on the transformer winding are obtained through a simulation
of the solidThe amplitude as
mechanics, anddemonstrated
axial forces on the transformer
in Figure winding
10. The are obtained
legend shows thethrough a simulation
magnitude of
of stress, and
the solid mechanics, as demonstrated in Figure 10. The legend shows the magnitude
the red arrows indicate the direction of force. As seen in Figure 10a, the external high-voltage of stress, and the
red arrows indicate the direction of force. As seen in Figure 10a, the external high-voltage windings
windings receive outward traction, while the inner low-voltage windings are pushed inward; the
receive outward traction, while the inner low-voltage windings are pushed inward; the force of the
force of thewinding
outer outer winding is obviously
is obviously lower
lower than that than
of the that
inner of theFigure
winding. inner10b
winding. Figure
shows that the two10b shows
ends of that
the twotheends of the winding
transformer transformer winding
are subjected to are
inwardsubjected topressure.
extrusion inward Therefore,
extrusionthe
pressure. Therefore, the
above simulation
above results
simulation results
are in line withare
the in lineforce
actual withonthetheactual force windings.
transformer on the transformer windings.

(a)

(b)
FigureFigure
10. (a)10. (a) Radial-stress
Radial-stress distributiondiagram
distribution diagram of
of transformer
transformerwindings; (b) axial-stress
windings; distribution
(b) axial-stress distribution
diagram of transformer
diagram of transformer winding. winding.

To further verify the accuracy of the above model, the vibration signals measured by the
To further verify
transformer the test
short-circuit accuracy of the above
were compared with themodel,
vibrationthe vibration
signals from thesignals measured
simulation. Figure 11 by the
transformer
shows short-circuit
the layout of test were compared
the actual withmeasurement
vibration-signal the vibrationpoints
signalsandfrom the simulation. Figure
the simulation-model
11 shows the layout
measurement of the
points. Theactual vibration-signal
locations in the simulationmeasurement points
model are consistent and
with thethe simulation-model
vibration signal
measured in the experiment.
measurement points. The locations in the simulation model are consistent with the vibration signal
measured in the experiment.
Figure 12 shows the calculated results of vibration at the placement of the acceleration vibration
sensor and the measured results. Because the current passing through the winding in the short-circuit
experiment is rated current, the simulation-calculation conditions are close to the actual measurement
conditions. As shown in Figure 12, the simulation results are in good agreement with the measured
 Energies 2018, 11, x FOR PEER REVIEW 12 of 19

results, which further illustrates the validity of the finite-element model and the correctness of the
calculation
Energies results.
2018, 11, 2404 In addition, the vibration waveform of the test point appears in the shape12of the
of 19
top cusp, which may be due to the material characteristics [30].

Energies 2018, 11, x FOR PEER REVIEW 12 of 19

results, which further illustrates the validity of the finite-element model and the correctness of the
calculation results. In addition, the vibration waveform of the test point appears in the shape of the
top cusp, which may be due to the material characteristics [30].

(a) (b)
Figure11.
Figure 11.(a)
(a)Experiment
Experiment locations
locations of measurement
of measurement points;
points; (b) Simulation
(b) Simulation locations
locations of measurement
of measurement points.
points.
Figure 12 shows the calculated results of vibration at the placement of the acceleration vibration
sensor and the measured results. Because the current passing through the winding in the short-circuit
experiment is rated current, the simulation-calculation conditions are close to the actual measurement
conditions. As shown in Figure 12, the simulation results are in good agreement with the measured
results, which further illustrates
(a) the validity of the finite-element (b)model and the correctness of the
calculation results. In addition, the vibration waveform of the test point appears in the shape of the
Figure 11. (a) Experiment locations of measurement points; (b) Simulation locations of measurement
top cusp, which may be due to the material characteristics [30].
points.

Figure 12. Comparison of vibration signals at each test point.

4. Simulation and Analysis of the Mechanical Faults of Transformer Windings

4.1. Simulation Geometric Model of the Mechanical Faults of Transformer Winding

Models were simulated for winding-insulation loss, winding looseness, and winding
deformation, with the number of coil turns held constant. The nonlinear characteristics of insulating
pads and clamps in transformer
Figure
Figure 12. windingsof
12. Comparison
Comparison cause
of theirsignals
vibration
vibration elastic
signals at modulus
at each
eachtest to vary with pretightening
testpoint.
point.
force. That is to say, the elastic coefficient of insulating pads on both sides of windings is related to
4.
4. Simulation
Simulation and
and Analysis
Analysis of the
the Mechanical
ofassembly
Mechanical Faults ofofTransformer Windings
the degree of tightening after [31]. In Faults
this paper, Transformer
as shown inWindings
Figure 13a, the core and clip
were reasonably simplified. Pretightening force was reduced
4.1. Simulation Geometric Model of the Mechanical Faults of Transformer Winding by changing the elastic coefficient of the
4.1. Simulation Geometric Model of the Mechanical Faults of Transformer Winding
B-phase winding material, and the loosening of the B-phase winding was simulated by slightly
Modelsthe
extending were simulated
length for winding-insulation
of the B-phase loss, winding looseness, and winding deformation,
Models were simulated for winding. Transformer-winding
winding-insulation loss, windingdeformation
looseness, is usually
and caused
winding by
with the number
the electric force of coil turns held constant. The nonlinear characteristics of insulating pads and
deformation, with formed
the numberby the short-circuit
of coil turns heldcurrent
constant. andTheaxial magnetic
nonlinear field. For the
characteristics winding-
of insulating
clamps in transformer
deformation faults windingsin
simulated cause their13b,
Figure elastic
the modulus to vary
high-voltage with pretightening
windings were force.toThat
subjected is
radial
pads and clamps in transformer windings cause their elastic modulus to vary with pretightening
toelectrodynamic
say, the elastic force
coefficient of
ofelastic insulating
outward pads
expansion, on both sides
resultingpads of
in the windings is related to the degree of
force. That is to say, the coefficient of insulating on windings
both sidesbulging.
of windings The islow-voltage
related to
tightening
winding after
was assembly
twisted by [31].
the In this
internal paper,
radial as shown
electric in
force.Figure
In the13a, the
model core
studiedand inclip were
this reasonably
paper, in order
the degree of tightening after assembly [31]. In this paper, as shown in Figure 13a, the core and clip
simplified.
to study Pretightening
the axial and force
radialwas reduced by changing
displacement of winding the after
elastica coefficient of the B-phasefault
winding-deformation winding
more
were reasonably simplified. Pretightening force was reduced by changing the elastic coefficient of the
material, andcircular
intuitively, the loosening
windingof the changed
was B-phase winding
into an was simulated bywith
slightly extending the length
B-phase winding material, and the loosening of octagonal
the B-phase cylinder
winding obvious
was edgesby
simulated andslightly
angles
oftothe B-phase
simulate the winding.
bulging Transformer-winding
condition deformation is As
usually caused by the electric forceof
extending the length of the B-phaseafter winding
winding. deformation.
Transformer-winding shown in Figure
deformation 13c,
is usually the effect
caused by
formed by the short-circuit current and axial magnetic field. For the winding-deformation faults
the electric force formed by the short-circuit current and axial magnetic field. For the winding-
simulated in Figure 13b, the high-voltage windings were subjected to radial electrodynamic force of
deformation faults simulated in Figure 13b, the high-voltage windings were subjected to radial
electrodynamic force of outward expansion, resulting in the windings bulging. The low-voltage
winding was twisted by the internal radial electric force. In the model studied in this paper, in order
to study the axial and radial displacement of winding after a winding-deformation fault more
intuitively, circular winding was changed into an octagonal cylinder with obvious edges and angles
 Energies 2018, 11, x FOR PEER REVIEW 13 of 19
Energies 2018, 11, 2404 13 of 19
the winding insulation was simulated by reducing the number of B-phase winding coils. During the
simulation, the other conditions were not changed.
outward expansion, resulting in the windings bulging. The low-voltage winding was twisted by the
internal radial electric force. In the model studied in this paper, in order to study the axial and radial
displacement of winding after a winding-deformation fault more intuitively, circular winding was
changed into an octagonal cylinder with obvious edges and angles to simulate the bulging condition
Energies 2018, 11, x FOR PEER REVIEW 13 of 19
after winding deformation. As shown in Figure 13c, the effect of the winding insulation was simulated
by
thereducing
windingthe numberwas
insulation of B-phase winding
simulated coils. the
by reducing During the of
number simulation, the other
B-phase winding conditions
coils. were
During the
not changed.
simulation, the other conditions were not changed.

(a) (b) (c)

Figure 13. (a) Simulation model of winding loosening; (b) simulation model of winding-insulation
failure; (c) simulation model of winding deformation.

4.2. Analysis of the Simulation Results of Three Typical Faults

As shown in Figures 14 and 15, the total displacement of normal windings and the total
displacement of windings under the effect of insulation shedding, winding loosening, and winding
deformation were (a) compared for a time period of 0.005 (b) s. (c)
Figure13.14(a)shows
Figure that the total displacement
Simulation of the transformer
winding loosening; windings was smallest under
Figure 13. (a) Simulation model of winding loosening;(b) (b)simulation
simulationmodel
modelofofwinding-insulation
winding-insulation
normal conditions,
failure;(c) while
(c)simulation the
simulation model winding
model of was
of winding more prone
winding deformation. to displacement in the event of a fault. Total
failure; deformation.
displacement was greatest in the case of winding deformation. The results validate the cumulative
characteristics
4.2.
4.2.Analysis
Analysis of ofSimulation
of the
the winding mechanical
Simulation faults,
Results of Three
Three that is,
Typical
Typical small mechanical faults increase force on the
Faults
Faults
winding, which makes it relatively easy for greater mechanical faults to occur. Therefore,
As
As shown in
in Figures
shownattention Figures 14 and 15, the the total displacement
displacement ofnormal normalwindings
windingsand andthethetotal
total
considerable should be paid to thetotal
mechanical state ofof the winding. If a small mechanical
displacement
displacement of
of windings
windings under the effect
effect of
of insulation
insulation shedding,
shedding, winding
winding loosening,
loosening, and
and
failure occurs, it should be dealt with in a timely manner; if it is not, it becomes relatively easy for a winding
winding
deformation
deformation were
serious accidentwereto compared
occur. for
compared for a time period of 0.005
0.005 s.s.
Figure 14 shows that the total displacement of the transformer windings was smallest under
normal conditions, while the winding was more prone to displacement in the event of a fault. Total
displacement was greatest in the case of winding deformation. The results validate the cumulative
characteristics of winding mechanical faults, that is, small mechanical faults increase force on the
winding, which makes it relatively easy for greater mechanical faults to occur. Therefore,
considerable attention should be paid to the mechanical state of the winding. If a small mechanical
failure occurs, it should be dealt with in a timely manner; if it is not, it becomes relatively easy for a
serious accident to occur.

(a) (b)

(a) (b)

(c) (d)

Figure 14. Axial diagram of the total displacement of windings under various working conditions.
(a) Normai working; (b) Insulation Shedding; (c) Winding loosing and (d) Winding deformation.

(c) (d)
that the force of the low-voltage coil was greater than that of the high-voltage coil. This result is due
to the opposing directions of current flow along the windings between the high- and low-voltage
sides of the transformer, as well as the mutual exclusion of the electromagnetic force in the radial
direction of the two windings, which is in accordance with the actual force. This result verifies the
Energies 2018,
accuracy of 11,
the2404
model. 14 of 19

(a) (b)

(c) (d)
Figure
Figure15.
15.Amplitude
Amplitudedirection
directiondiagram of of
diagram winding force
winding displacement
force under
displacement various
under conditions.
various (a)
conditions.
Normal working;
(a) Normal (b) Insulation
working; Shedding;
(b) Insulation (c) Winding
Shedding; loosening
(c) Winding andand
loosening (d) (d)
Winding deformation.
Winding deformation.

Figure
Figure 16
14 shows
shows athat
time-domain diagram of the
the total displacement vibration-acceleration
of the transformer windingssignal
wasatsmallest
measurement
under
point
normal 1 under normal
conditions, working
while conditions,
the winding was as well
more as intothe
prone cases of insulation
displacement shedding,
in the event winding
of a fault. Total
loosening,
displacementand was
winding deformation.
greatest in the case of winding deformation. The results validate the cumulative
characteristics of winding mechanical faults, that is, small mechanical faults increase force on the
winding, which makes it relatively easy for greater mechanical faults to occur. Therefore, considerable
attention should be paid to the mechanical state of the winding. If a small mechanical failure occurs,
it should be dealt with in a timely manner; if it is not, it becomes relatively easy for a serious accident
to occur.
Figure 15 shows that the total displacement of the low-voltage winding in the inner part of the
transformer was greater than the displacement of the lateral high-voltage winding, which indicates
that the force of the low-voltage coil was greater than that of the high-voltage coil. This result is due to
the opposing directions of current flow along the windings between the high- and low-voltage sides of
(a) (b)
the transformer, as well as the mutual exclusion of the electromagnetic force in the radial direction
of the two windings, which is in accordance with the actual force. This result verifies the accuracy of
the model.
Figure 16 shows a time-domain diagram of the vibration-acceleration signal at measurement point
1 under normal working conditions, as well as in the cases of insulation shedding, winding loosening,
and winding deformation.
As seen from Figure 16, the vibration signal at measurement point 1 under normal conditions was
more stable than the vibration signal at the time of failure, and the average amplitude of the vibration
signal was approximately 0.06 m/s2 . When insulation shedding occurred, the maximum amplitude of
the vibration-acceleration signal at measurement point 1 was 0.08 m/s2 . When the winding was loose,
the maximum amplitude of the vibration acceleration signal at measurement point 1 was 0.28 m/s2 .
When winding deformation occurred, the maximum amplitude of the vibration-acceleration signal at
measurement point 1 was 0.25 m/s2 . When a winding fault occurred, the amplitude of the vibration
signal was nonstationary. Moreover, when the winding was loose, and the winding deformation
failed, the change was more intense. When the transformer windings failed, vibration signals would
 (c) (d)
Figure 15. Amplitude direction diagram of winding force displacement under various conditions. (a)
Normal
Energies working;
2018, 11, 2404 (b) Insulation Shedding; (c) Winding loosening and (d) Winding deformation. 15 of 19

Figure 16 shows a time-domain diagram of the vibration-acceleration signal at measurement

obviously
point 1 underchange.
normal The amplitude of the vibration
working conditions, signal
as well as ininthe
thecases
fault was higher than
of insulation the amplitude
shedding, of
winding
the normal
loosening, vibration
Energies and winding
2018, 11, signal, and the
deformation.
x FOR PEER REVIEW fluctuation of the signal was intense. 15 of 19

Energies 2018, 11, x FOR PEER REVIEW 15 of 19

(a)(c) (d)(b)
Figure 16. Time-domain diagram of vibration-acceleration signals at measurement point 1 under
various working conditions: (a) normal working conditions; (b) insulation shedding; (c) winding
loosening; (d) winding deformation.

As seen from Figure 16, the vibration signal at measurement point 1 under normal conditions
was more stable than the vibration signal at the time of failure, and the average amplitude of the
vibration signal was approximately 0.06 m/s2. When insulation shedding occurred, the maximum
amplitude of the vibration-acceleration signal at measurement point 1 was 0.08 m/s2. When the
winding was loose, the maximum amplitude of the vibration acceleration signal at measurement
point 1 was 0.28 m/s2. When winding deformation occurred, the maximum amplitude of the
(c) (d)
vibration-acceleration signal at measurement point 1 was 0.25 m/s2. When a winding fault occurred,
the amplitude
Figure of the vibration
16. Time-domain
Time-domain signal
diagram
diagram ofwas
of nonstationary. Moreover,
vibration-acceleration
vibration-acceleration signals
signals when
at the winding was
at measurement
measurement pointloose,
point and
11 under
under
the winding
various
various deformation
working
working failed,
conditions:
conditions: (a)the
(a) change
normal
normal was more
working
working intense. When
conditions;
conditions; (b) the transformer
(b) insulation
insulation windings
shedding;
shedding; (c) failed,
(c) winding
vibration signals
loosening; would
(d) winding obviously change. The amplitude of the vibration signal in the fault was
deformation.
higher than the amplitude of the normal vibration signal, and the fluctuation of the signal was
intense.
Figure
As seen17from
presents
Figurea spectrum-analysis
16, the vibration signaldiagram of the vibration
at measurement signal
point at measurement
1 under point 1,
normal conditions
whenmore Figure 17 presents
the transformer a spectrum-analysis
wasvibration
working signal
under at diagram
normal of the vibration
conditions, asandsignal
wellthe at
as formeasurement
the cases point 1,
of insulation
was stable than the the time of failure, average amplitude of the
when the transformer was working under normal conditions, as well as for the cases of insulation
shedding, winding loosening, and winding deformation.
vibration signal was approximately 0.06 m/s . When insulation shedding occurred, the maximum
2
shedding, winding loosening, and winding deformation.
amplitude of the vibration-acceleration signal at measurement point 1 was 0.08 m/s2. When the
winding was loose, the maximum amplitude of the vibration acceleration signal at measurement
point 1 was 0.28 m/s2. When winding deformation occurred, the maximum amplitude of the
vibration-acceleration signal at measurement point 1 was 0.25 m/s2. When a winding fault occurred,
the amplitude of the vibration signal was nonstationary. Moreover, when the winding was loose, and
the winding deformation failed, the change was more intense. When the transformer windings failed,
vibration signals would obviously change. The amplitude of the vibration signal in the fault was
higher than the amplitude of the normal vibration signal, and the fluctuation of the signal was
intense. (a) (b)
Figure 17 presents a spectrum-analysis diagram of the vibration signal at measurement point 1,
when the transformer was working under normal conditions, as well as for the cases of insulation
shedding, winding loosening, and winding deformation.

(c) (d)
Figure
Figure 17. 17.Spectrum
Spectrum diagram
diagramof of
vibration signalsignal
vibration at measurement point 1 under
at measurement different
point 1 underworking
different
conditions: (a) normal condition; (b) insulation shedding; (c) winding loosening;
working conditions: (a) normal condition; (b) insulation shedding; (c) winding loosening; (d) winding
deformation.
(d) winding deformation.
(a) (b)
Energies 2018, 11, 2404 16 of 19

As shown in Figure 17, the maximum frequency attained by the vibration-acceleration signal
at transformer measurement point 1 was 100 Hz under normal operating conditions, at which point
the amplitude was 0.05. Signals with frequencies of 50 Hz, 200 Hz, and 300 Hz can also be seen in
this diagram. When insulation was shed, the amplitudes of the 100 Hz, 200 Hz, and 300 Hz signals
increased. When the windings were loose, the maximum amplitude of the signal in the frequency
domain occurred at 200 Hz, and the amplitude of the signal of 300 Hz increased apparently as well.
When winding deformation occurred, the maximum amplitude of the vibration signal occurred at 300
Hz, and the amplitude of the signal spectrum at 200 Hz was greater than the amplitude at 100 Hz.
These results are consistent with the experimental results of transformer faults in Reference [32],
which verifies the accuracy of the model.
By studying the vibration characteristics of transformer windings, it is known that the
corresponding mechanical vibration will change when the mechanical state of the windings
changes. In order to make the change more intuitive and distinguish the characteristics of
different mechanical states, based on the analysis of transformer-winding vibration characteristics,
some artificial-intelligence algorithms can be used to identify fault features, such as Improved
Variational Mode Decomposition (IVMD)-Weight Divergence, which is a feature-extraction method
presented for vibration signals of transformer windings. In the process of feature extraction,
the mechanical vibration signals of transformer windings in different states are decomposed into
a series of finite-bandwidth intrinsic-mode functions (IMFs) by means of Improved Variational Mode
Decomposition; K-L divergence (K-L) between the IMF component and the original vibration signal is
calculated, and the weighting coefficients are multiplied. Weight divergence is obtained to represent
the time-frequency domain complexity of mechanical-vibration signals of transformer windings
in different states. It can effectively extract the characteristics of a mechanical-vibration signal of
transformer windings [33].
Many scholars have studied the vibration state of the transformer. Gu Hongxia of Kunming
University of Technology, through the finite-element analysis of winding-loosening faults, showed that
the natural frequency of the four orders in winding-loosening faults and the natural frequency of the
normal state of the winding transfer to a low frequency [34]. The simulation experiment shows that it
can be used. Natural frequency is used to judge the working condition of power-transformer windings.
However, this paper only focuses on the loosening fault of windings, and other faults such as winding
deformation and insulation shedding are not studied. From the point of view of acoustic measurement,
C. Bartoletti extracts the weighted values and proportions of the middle-, low-, and high-frequency
signals from the noise signals for transformer-fault diagnosis [35]. In this paper, the operation state of
the transformer is diagnosed by the sound signal. But the vibration signal is transmitted through air,
and noise reduction is needed to obtain the available vibration signal. The monitoring effect is not as
accurate as the vibration of the transformer itself. Hyun-Mo Ahn has more analysis on the vibration
and stress of short-circuit faults, but less research on the vibration caused by load changes in normal
operation [36]. Ji Shengchang of Xi’an Jiaotong University has done long-term work on the vibration
characteristics of the transformer. The radial vibration of the transformer winding, axial-vibration
characteristics of the pressure plate, and influencing factors were studied [37]. The vibration law and
propagation characteristics of transformer windings were revealed. There is no theoretical calculation
of the radial vibration of the winding, and no research on the change rule of the vibration characteristics
of the typical winding mechanical-fault state, which makes it difficult to judge the actual fault.
By setting winding looseness, winding deformation, and insulation shedding, and by means of
simulation and experiment, the frequency-spectrum characteristics of winding-vibration signals under
different conditions were analyzed. From the research, it can be seen that the transformer-vibration
model provides a reliable basis for transformer-winding state simulation and fault diagnosis, and can
be used to easily calculate the distribution of the magnetic field and the winding characteristics of the
transformer. Moreover, this model can be used to analyze vibration control during the design of a
transformer, thereby shortening the design cycle. It also provides a theoretical basis and method of
Energies 2018, 11, 2404 17 of 19

calculation for further analysis and verification of new methods of reducing electromagnetic noise
in transformers.

5. Conclusions
In this paper, the mechanical-vibration characteristics of transformer windings were studied
theoretically. Changes in the mechanical state of the windings were reflected in their vibration
signals. A simulation model of vibration was established for an SCB10-1000/10 dry-type transformer
winding using COMSOL software, and a multifield-coupling simulation of the circuit, magnetic
field, and solid mechanics was performed. The following conclusions were obtained through
simulation and experimental analysis. By changing the geometry model and power parameters,
similar conclusions can be obtained for other types of dry-type transformers, which can be extended
to the study of winding-vibration characteristics of various types of dry-type power transformers.
For the oil-immersed transformer, the simulation model was greatly changed, so it was necessary to
add an oil-tank wall to the transformer model, replace the medium with insulation oil, and consider
the vibration-signal propagation process in the insulation oil. At the same time, we also studied
the occurrence and propagation of the winding vibration of oil-immersed transformers. Because the
oil-immersed transformer was closed, the vibration-signal sensor could only be mounted on the wall of
the oil tank. When analyzing the signal received by the vibration-signal sensor, we needed to consider
the properties of the different insulating oil. Therefore, the vibration analysis in this paper was limited
to dry-type transformers.
(1) A short-circuit experiment was performed on an SCB10-1000/10 transformer, which is a
dry-type transformer. Information on the vibration acceleration at the surface of the transformer
was obtained using a vibration-acceleration sensor (YD70C), a charge amplifier (DHF-10), and a Tek
oscilloscope. Vibration-signal analysis showed that, when the secondary winding of the transformer
was short-circuited, the signal was mainly composed of a 100 Hz component and contained relatively
small components at 50 Hz, 200 Hz, and 300 Hz.
(2) Based on the vibration data obtained from the transformer short-circuit experiment,
a multiphysical field-coupling vibration-simulation model of the dry-type transformer winding is
established using the parameters of the actual transformer. The vibration-acceleration signal was
obtained from actual measurement points, which was then compared to the corresponding points in
the simulation. The similarity between the two exceeded 80%. Therefore, the model can be used to
investigate the vibration of transformer windings and possesses high value in engineering applications.
(3) Based on the simulation model, for normal working conditions, fault simulations of the
transformer windings were carried out. Specifically, winding loosening, deformation, and insulation
shedding were simulated, and the vibration characteristics of these winding faults were analyzed.
When insulation was shed, the amplitude in the frequency domain at 100 Hz, 200 Hz, and 300 Hz
increased. When the winding was loose, the maximum amplitude in the frequency domain appeared
at 200 Hz, and the amplitude at 300 Hz was also relatively larger. When winding deformation occurred,
the maximum amplitude of the vibration signal occurred at 300 Hz, and the amplitude at 200 Hz
exceeded the amplitude at 100 Hz. The simulation results are consistent with the experimental results,
which verify the accuracy of the fault model. This method can provide a new basis for simulating and
diagnosing transformer-winding faults.

Author Contributions: X.D. and J.L. designed the multiphysics field-coupling model for transformer-winding
vibration, performed the data analysis, and prepared the manuscript. L.Z. (Li Zhang) and L.Z. (Liang Zou)
assisted the project and managed to obtain the transformer-vibration signal. T.Z. led the project and research.
All authors discussed the results and approved the publication.
Funding: This research was supposed by the Key Research and Development project of Shandong Province
(2018GGX104009).
Conflicts of Interest: The authors declare no conflict of interest.
Energies 2018, 11, 2404 18 of 19

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