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THERMAL MODELING OF
ELECTRICAL UTILITY TRANSFORMERS
A Dissertation
By
Haritha V V S S
Reg. No. 200742002
Submitted in partial fulfillment of the requirements for the degree of
Master of Science (by Research)in
IT in Power Systems
Faculty Advisors
Dr. M. Ramamoorty and Dr. Amit Jain
International Institute of Information Technology
Hyderabad, India
November 2011
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INTERNATIONAL INSTITUTE OF INFORMATIONTECHNOLOGY
Hyderabad, India
Certificate
It is certified that the work contained in this thesis, titled “Thermal Modeling of Electrical Utility
Transformers ” by Haritha V V S S has been carried out under our supervision and is not
submitted elsewhere for a degree.
Dr. Amit Jain (Advisor)Dr. M. Ramamoorty (Advisor)
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Acknowledgements
I would like to thank my advisers Dr. Amit Jain and Dr. M. Ramamoorty for their guidance
and support during the entire course of my research. The regular discussions with them and
their constant feedback helped me immensely in completing this thesis work satisfactorily.
I would like to thank Vijai Electricals, Hyderabad, INDIA for their technical support at several
stages of the work.
I thank my friends in the Power Systems Research Center, for their constant encouragement
and joyous company. I would like to thank my family and friends back home. Without their
support and encouragement this thesis would not have seen the light of day.
My acknowledgment would not be complete without mentioning my friends from MS and M.
Tech (2007 batch). Their company throughout my stay in IIIT, sharing the joys during the
highs and providing comfort during the lows is unforgettable. I thank them for this humbling
experience.
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Abstract
The importance of transformers, with their role in transmission and distribution of electrical
power and with the effect of their performance on the system, is an obvious axiom in the
modern day’s power systems. In addition to their momentous share in the capital investment
of a power system, transformer outages have a considerable economic impact on the operation
of the power systems.
In the course of continuous efforts to make the existing power network smarter and efficient,
thermal modeling and monitoring of transformers has become important in the field oftransformer engineering. With all the advances in the design techniques as well as material
engineering, it is the transformer thermal limitations that decide the loading and designing of
the transformer from the purview of user as well as manufacturer. With the research in power
systems on the whole progressing towards development of a ‘smart grid’, which infers that
each of the equipment should be ‘smart’, that includes that the monitoring of each individual
equipment should be intelligent, accurate as well as fast and economical, the problem of
thermal Modeling of transformers has been gaining momentum all the more.
The maximum temperature in the transformer interior is a significant parameter governing a
transformer’s performance and life expectancy. Though the temperature rise in the
transformer interior by itself may not have immediate effects, it does trigger other undesirable
consequences like excessive deterioration of insulation, which in the long run will reduce the
life of the transformer, thus affecting the economics of the power system. Thus the possible
maximum temperature rise in the transformer for certain kind of loading needs to be estimated
so as to be able to decide on the operational conditions as well as estimate the remaining life of
the transformer and plan accordingly. In the perspective of the user, temperatures in a
transformer are important to determine the amount and duration of over load it can sustain,
and to estimate the effects on the life of the transformer by operation at various temperatures.
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For a transformer design engineer, prediction of temperatures at various points becomes
necessary to determine the amount of copper to place in the coils, leads and outlet bushings,
type of cooling and ducts, position of ducts, insulation class, design and settings of control
equipment. Apart from this, increased market competency demands for accurate
determination of the thermal profile across the transformer, which might result in a more
economical as well as efficient manufacturing.
Existing thermal models calculate the winding hotspot temperature and top oil temperature
using the lumped values of heat generation inside the transformer and the rate of heat transfer
and retention in the surrounding media that finally result in the temperature rise. The heat
generation is due to the energy losses in the transformer which are the iron losses in the core
and ohmic losses in the coils. These temperatures served as an index for the interior
temperature rise in the transformer. To calculate the hotspot temperatures, the existing models
used the lumped values of losses and lumped values of heat transfer and retention in the
different media that surrounded the heat generating elements and the loss distribution across
the transformer geometry was not calculated and used in those models. However,
advancements in computing capabilities and ever ongoing research enables better transformer
interior temperature modeling, which may be a better indicator of transformer thermal status.In the current work, the use of finite element analysis technique was made to calculate the loss
distribution across the transformer geometry, which is a different approach. With the
calculation of loss distribution across the transformer geometry, the current work proposes a
new approach for thermal model of transformer and discusses the development of this
thermal model that aims at computing the interior temperatures at different as well as desired
points across the transformer geometry. The proposed thermal model has been successfully
implemented on four real transformer data to calculate the thermal profiles of transformers
that show the real life use of proposed thermal model.
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i
Table of Contents
LIST OF FIGURES ............................................................................................................................................ iii
LIST OF TABLES ...............................................................................................................................................V
1
INTRODUCTION ........................................................................................................................................ 1
1.1 TRANSFORMER – AN OVERVIEW ............................................................................................................. 1
1.2 THESIS CONTRIBUTION ............................................................................................................................ 2
1.3 ORGANIZATION OF THESIS....................................................................................................................... 2
2
THERMAL BASICS – HEAT BUILDUP IN A TRANSFORMER...................................................... 4
2.1 HEAT GENERATION AND DISSIPATION IN A SOLID BODY ........................................................................ 4
2.1.1 Heat Dissipation............................................................................................................................ 4
2.1.2 Modes of Heat Dissipation............................................................................................................ 5
2.2
NEWTON’S LAW OF COOLING.................................................................................................................. 8
2.3 THEORY OF SOLID BODY HEATING AND COOLING ................................................................................. 9
2.3.1 Heating and Cooling Curves....................................................................................................... 11
2.4
HEAT IN A TRANSFORMER ...................................................................................................................... 12
2.4.1 Heat Generation in the Transformer ..........................................................................................12
2.4.2
Heat Dissipation in the Transformer – Cooling Arrangements................................................. 13
2.4.3 Heat Build Up in the Transformer.............................................................................................. 14
2.5 CONSEQUENCES OF EXCESSIVE HEAT BUILDUP ..................................................................................... 15
2.5.1 Arrhenius Law of Insulation Ageing........................................................................................... 16
3
TRANSFORMER THERMAL MODELING – LITERATURE SURVEY ................................. 18
3.1
TECHNIQUES TO MEASURE TRANSFORMER INTERIOR TEMPERATURES................................................ 18
3.2 IEEE FORMULAE FOR CALCULATING HOTSPOT TEMPERATURES.........................................................19
3.3 FIBER OPTIC SENSORS FOR TEMPERATURE MEASUREMENTS ............................................................... 21
3.4
THERMAL MODELS TO CALCULATE HOTSPOT TEMPERATURES ........................................................... 22
3.5 TECHNIQUES BASED ON COMPUTER BASED SIMULATIONS .................................................................. 24
3.6
TECHNIQUES BASED ON ARTIFICIAL INTELLIGENCE ............................................................................. 25
3.7 OBSERVATIONS AND COMMENTS .......................................................................................................... 26
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4 TRANSFORMER THERMAL MODELING USING LOSS DISTRIBUTION................................ 28
4.1 PROPOSED METHOD OF THERMAL MODELING...................................................................................... 28
4.2
OBTAINING THE LOSS DISTRIBUTION – FINITE ELEMENT ANALYSIS....................................................31
4.2.1 Obtaining the Flux Density Distribution by Finite Element Analysis........................................ 32
4.2.2 Obtaining the Loss Distribution..................................................................................................33
4.3 DEVELOPMENT OF THERMAL MODEL ................................................................................................... 35
4.3.1 Thermal Electrical Analogy........................................................................................................ 35
4.3.2 Electrical Equivalent Model for Thermal Behavior...................................................................36
4.3.3 Calculation of Parameters of Thermal Model............................................................................ 38
4.3.4 Modeling of Radiators.................................................................................................................39
4.3.5 Modeling the Convection in Oil .................................................................................................. 40
4.3.6
Modeling the Ambient .................................................................................................................43
4.4 OBTAINING THE THERMAL PROFILE ...................................................................................................... 44
5 IMPLEMENTATION ON DIFFERENT TRANSFORMER DESIGNS............................................ 46
5.1 15 KVA SHELL TYPE TRANSFORMER – MODEL AND RESULTS ............................................................ 46
5.2
25 KVA CORE TYPE TRANSFORMER – MODEL AND RESULTS .............................................................. 59
5.3 16 KVA SHELL TYPE TRANSFORMER – MODEL AND RESULTS ............................................................ 67
5.4 45 KVA THREE PHASE TRANSFORMER – MODEL AND RESULTS .........................................................75
5.5 DISCUSSIONS.......................................................................................................................................... 83
6
CONCLUSIONS......................................................................................................................................... 84
6.1 CONCLUSIONS ........................................................................................................................................ 84
6.2 FUTURE SCOPE OF THE WORK ...............................................................................................................87
7
APPENDIX.................................................................................................................................................. 88
7.1 INTRODUCTION TO FINITE ELEMENT ANALYSIS – NISA.......................................................................88
7.2 INTRODUCTION TO MULTISIM............................................................................................................ 89
PUBLICATIONS............................................................................................................................................... 90
REFERENCES................................................................................................................................................... 91
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iii
LIST OF FIGURES
Fig. 2.1: Heating Curves……………………………………………………………………11
Fig. 2.2: Cooling Curves……………………………………………………………………12
Fig. 2.3: Arrhenius Law of Insulation Ageing ……………………………………………… 17
Fig. 4.1: Thermal Model of a single element ……………………………………………….. 28
Fig. 4.2: Steady state thermal model showing interconnection of elements…………………. 29
Fig. 4.3: Calculation of resistances – Thermal Model for single element …………………….38
Fig. 4.4: Modeling the convection – Modified thermal model of oil element ……………….. 42
Fig. 4.5: Modeling the convection – Thermal model of oil element with diode……………..43
Fig. 5.1: Transformer 1: Geometry………………….………………….…………………. 47
Fig. 5.2: Transformer 1: Elemental Division………………….………………….…………48
Fig. 5.3: Transformer 1: FEA Implementation in NISA………………….…………………49
Fig. 5.4: Transformer 1: Flux Density distribution………………….……………………… 50
Fig. 5.5: Transformer 1: Loss Distribution………………….………………….………….. 51
Fig. 5.6: Transformer 1: Numbering of Elements………………….………………………. 52
Fig. 5.7: Transformer 1: Thermal Model ………………….………………….……………. 57
Fig. 5.8: Transformer 1: Thermal Profile………………….………………….……………. 58
Fig. 5.9: Transformer 2: Geometry………………….………………….…………………..60
Fig. 5.10: Transformer 2: Elemental Division………………….………………….……….. 61
Fig. 5.11: Transformer 2: FEA Implementation in NISA………………….………………. 62
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Fig. 5.12: Transformer 2: Flux Density distribution………………….…………………….. 63
Fig. 5.13: Transformer 2: Thermal Model ………………….………………….……………65
Fig. 5.14: Transformer 2: Thermal Profile………………….………………….……………66
Fig. 5.15: Transformer 3: Geometry………………….………………….………………….68
Fig. 5.16: Transformer 3: Elemental Division………………….………………….……….. 69
Fig. 5.17: Transformer 3: FEA Implementation in NISA………………….……………….70
Fig. 5.18: Transformer 3: Flux Density distribution………………….…………………….. 71
Fig. 5.19: Transformer 3: Thermal Model ………………….……………………………….73
Fig. 5.20: Transformer 3: Thermal Profile…………………………………………………..74
Fig. 5.21: Transformer 4: Geometry………………….………………….…………………76
Fig. 5.22: Transformer 4: Elemental Division………………….…………………………...77
Fig. 5.23: Transformer 4: FEA Implementation in NISA………………….………………..78
Fig. 5.24: Transformer 4: Flux Density distribution………………….……………………..79
Fig. 5.25: Transformer 4: Thermal Model ………………….……………………………….81
Fig. 5.26: Transformer 4: Thermal Profile………………….……………………………… 82
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LIST OF TABLES
Table 1: Thermal Electrical Analogy………………………………………………………. 35
Table 2: Dimensions of the considered Transformer 1…………………………………….. 46
Table 3: Material Properties: FEA Implementation of Transformer 1…………………….49
Table 4: Material Thermal Properties: Thermal Modeling of Transformer 1……………….. 51
Table 5: Calculation of Thermal Model Parameters for Transformer 1…………………….. 52
Table 6: Calculation of Tank to Ambient Resistances for Transformer 1……………………55
Table 7: Thermal Model Implementation-Comparison with test values for Transformer 1…. 59
Table 8: Dimensions of the considered Transformer 2…………………………………….. 59
Table 9: Material Properties: FEA Implementation of Transformer 2…………………….62
Table 10: Material Thermal Properties: Thermal Modeling of Transformer 2………………. 64
Table 11: Thermal Model Implementation-Comparison with test values for Transformer 2... 67
Table 12: Dimensions of the considered Transformer 3…………………………………….67
Table 13: Material Properties: FEA Implementation of Transformer 3…………………….70
Table 14: Material Thermal Properties: Thermal Modeling of Transformer 3……………….72
Table 15: Thermal Model Implementation-Comparison with test values for Transformer 3... 75
Table 16: Dimensions of the considered Transformer 4………………………………….... 75
Table 17: Material Properties: FEA Implementation of Transformer 4…………………….78
Table 18: Material Thermal Properties: Thermal Modeling of Transformer 4………………. 80
Table 19: Thermal Model Implementation-Comparison with test values for Transformer 4... 83
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1
Chapter 1
1
INTRODUCTION
1.1 Transformer – An Overview
A transformer is a highly efficient static machine operating on the principle of mutual
induction. It transforms ac power in a circuit from one voltage level to another through
inductively coupled electrical conductors and is available in a variety of power ratings. Thetransformer principle was revealed by Michael Faraday in 1831 and it was William Stanley in
the year 1886 that first developed a transformer for commercial use. It is the most efficient
device in the power system with values of efficiency ranging up to 99.8% [1].
Large rated power transformers and distribution transformers form a major part of the today’s
power system network. However, the problem in large rating transformers is heat dissipation.
More the heat is accumulated without being dissipated lesser is the life of the transformer. The
insulating oil circulating inside the transformer absorbs heat from the interior of transformerthrough conduction and this heat is dissipated to the ambient through natural means or by
cooling methods using suitable coolants. But there may be some areas inside the transformer
where the oil might not reach properly or somehow the heat might not be dissipated properly,
or heat dissipation is slow (or less) in comparison to heat generation, and hence gets
accumulated. Such heat accumulation results in high temperature, which reduces the life of the
transformer drastically.
It is very important for the transformer to be operating within the safe limits for the power
system operation to be safe and reliable. The safe operation and loading of the transformer is
decided by the thermal limits of the insulation used in the design of the transformer. For the
manufacturer, given a certain loading condition, it has to be ensured that the particular loading
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will not cause the transformer interior temperature exceed the thermal limits of insulation and
accordingly design the amount of insulation such that the worst case of operation also will not
cause the interior temperatures to exceed the thermal limits of materials. From the user side,
given a transformer, it has to be ensured that the loading on the transformer doesn’t cause the
interior temperatures go beyond the threshold limits. In either case, calculating interior
temperatures of a transformer for a particular loading condition, which is termed as thermal
modeling is important with respect to transformer.
1.2 Thesis Contribution
With the significance of thermal modeling stated, it is important to have a good thermal model
that can calculate the temperatures at desired location inside the transformer. Earlier works in
this area concentrated on developing models that calculate the winding hotspot temperature as
well as top oil temperature using the design values of transformer. While the importance of
hotspot temperature and top oil temperature in indicating the transformer thermal status is
maintained, the current work primarily focuses on developing a thermal model that can
calculate the temperature profile across the transformer geometry, thus giving a better picture
of thermal status of transformer interior. With such a model, the temperature at desiredlocation in the transformer interior can be calculated, besides obtaining the general thermal
profile. In addition, besides obtaining the maximum temperatures, the model can provide the
location of the maximum temperature. This would certainly help in better understanding of
the thermal status of transformer and would also help in the optimization of insulation ratings
used inside the transformer. With further modifications and improvement this model can also
be used for thermal monitoring of transformers.
1.3 Organization of Thesis
This thesis presents the development of an electrical equivalent model simulating the thermal
behavior of the transformer based on the thermal-electrical analogy with the application of
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Finite Element Analysis to the transformer geometry to derive the distribution of losses. The
thesis is organized in to six chapters. The first chapter gives an introduction to the problem
and explains the significance of the problem along with the contribution of the work to
research in the field of transformer thermal modeling. The second chapter provides the
thermal basics. It is very important to understand how heat is generated and dissipated and
also the different modes of heat dissipation in a solid body in general, so as to understand the
problem with respect to transformers. The axioms guiding the heating and cooling processes
in the transformer, that is, the Newton’s laws of cooling are explained. The consequences of
heat buildup and deterioration of insulation are also discussed. The third chapter gives a wide
literature survey of the previous research attempts made in this area. Various techniques are
classified broadly and presented accordingly. The fourth chapter presents the proposed
method. Obtaining the loss distribution with the application of finite element analysis
technique to the transformer geometry is discussed first followed by the development of the
thermal model. The various assumptions and modeling issues are discussed. The fifth chapter
discusses the implementation of this method on four different standard transformer designs.
An implementation of this method to a three phase transformer design is also presented along
with obtained results. Observations and conclusions from previous chapters are given in
chapter six. This chapter provides a summary of the work done for the thesis and gives the
possible scope for future research in this area.
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Chapter 2
2
THERMAL BASICS – HEAT BUILDUP IN A
TRANSFORMER
2.1 Heat generation and Dissipation in a Solid Body
This section presents a few thermal concepts related to generation and dissipation of heat. This
is very important to analyze the thermal behavior of transformers, or for that matter, anyelectrical device. Heat dissipation is a pronounced problem in transformers because
transformers are enclosed devices and because transformers have no rotating parts which
provide inherent ventilation. Any solid body with losses occurring in it generates heat
inherently, because, any form of energy loss in a body is dissipated in the form of heat. The
loss can be either magnetic loss, due to interaction of fluxes or hysteresis loss, or ohmic loss,
due to the current passing through a conductor and the resistance being offered by it or
mechanical loss, due to moving parts and the friction between the surfaces in contact.
2.1.1 Heat Dissipation
The heat generated in any body is dissipated to the surrounding media and finally to the
ambient. The usage of the term ambient medium or ambient temperature is similar to
reference point in an electrical circuit. As far as radiation is concerned, the ambient
temperature is the temperature of sky and the ground to which the heat is radiated by the hot
bodies. For the other forms of heat dissipation like conduction, natural and artificial
convections, the ambient temperature is the temperature of the bulk of the air at a distance too
remote to be affected by the thermal field of the heated body. The use of air temperature as
ambient temperature is justified only where most of the heat dissipated is by convection, as in
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the case of transformers. In any case, the heat generated in a body is dissipated to the
surrounding medium through one or more of the possible modes of heat dissipation, namely
conduction, convection and radiation.
• Conduction: Conduction can be defined as the transfer of heat because of the
temperature difference between two bodies in contact.
• Convection: Convection can be defined as the heat transfer in a gas or liquid by the
circulation of fluid currents from one region to another.
•
Radiation: Radiation of heat may be defined as the emission and propagation of energy inthe form of rays or waves.
2.1.2 Modes of Heat Dissipation
The section mathematically explains the various modes of heat dissipation.
•••• Conduction: This mode of dissipation of heat is important in the case of solid parts of
machine like copper, iron and insulation. The equation of heat flow by conductionbetween two surfaces separated by a heat conducting medium is given in equation 2.1.
( )( )1 2 2.1
conQ
Rθ
θ θ −= …
Where, Qcon = Heat dissipated by conduction, J.
θ1, θ2 = Temperatures of two bounding surfaces, oC.
R θ = Thermal resistance of the conducting medium, thermal ohm
The term thermal resistance may be defined as the resistance offered by the element for heat
flow which causes a drop of 1oC per watt of heat flow. The equation 2.1 shows the analogy
between thermal and electrical behavior of a body and permits heat conduction problems to
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be solved by methods of calculation similar to those used in electrical circuits. The thermal
resistance, like electrical resistance can be written as in equation 2.2.
( ) 2.2t t
RS s
θ
ρ
σ = = …
Where, ρ = thermal resistivity of the material, Ω (thermal)m or oC-m/W
σ = 1/ ρ = thermal conductivity, W/ oC-m
t = Length of the medium, m
S = Area of the surface separated by the medium, m2.
•••• Convection: Convection can be either natural with the heat transfer taking place by means
of natural fluid currents or forced with the heat transfer taking place by means of induced
or forced fluid currents.
a) Natural Convection: Liquid and gas particles near the heater body become lighter and
rise, giving place to cooler particles, which in turn get heated and rise. This natural process,
due to changes in fluid density, is known as natural convection. The heat dissipated per unit
surface by natural convection is given by equation 2.3.
( ) ( )1 0 2.3
n
conv cQ K θ θ = − …
Where, Qconv = Heat dissipated by convection, J.
K c = a constant depending on the shape and dimensions of hot body
n = a constant depending upon shape and dimensions of hot body; its value lies
between 1 and 1.25
θ1 = temperature of emitting surface, oC
θ0 = temperature of ambient medium, oC
Convection is more complicated phenomenon and the amount of the heat convected
depends upon many variables such as power density, temperature difference between the
heated surface and coolant, thermal conductivity of the fluid and gravitational constants.
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b) Forced Convection: In modern machines where there is too much of heat generation,
artificial circulation of the cooling medium is done to enhance the convection and hence the
increase in heat dissipation. For example, a transformer tank may be cooled by blasting air on
it or a turbo–alternator may be cooled by circulating hydrogen. This is known as cooling by
artificial convection. The problem of calculation of heat dissipation by artificial convection is
even more complex as it mainly depends upon the constructional features of the machine.
These constructional details are different for every machine and so no exact relationship can
be given for artificial convection.
•••• Radiation: The heat dissipated by radiation from a surface depends upon its temperature
and its other characteristics like color, roughness etc. For the case of a very small spherical
radiating surface inside a large and or black spherical shell, the heat radiated per unit of
surface is given by Stefan Boltzmann law given in equation 2.4.
( ) ( )8 4 4
1 05.7 10 2.4rad
Q e T T −= × − …
Where, Qrad = Heat dissipated by radiation per unit area, J per unit area.
T1, T0 = Absolute temperatures of the emitting surface and the ambient medium
respectively, K
e = Coefficient of emissivity; 1 for perfect black bodies, and is always less than unity
for others.
In general, the heat transfer in any body is by conduction and convection assisted by radiation.
The above equations help in quantizing the heat transferred by each of these modes of heat
transfer. [2]
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2.2 Newton’s Law of Cooling
Losses are produced in various parts of electrical machines due to which the machinetemperature rises. After sometime, the machine attains a steady temperature rise, at which, the
heat produced in the machine is equal to the heat leaving its surface by convection and
radiation. This is known as Newton’s law of cooling. It is another form of energy conservation
principle.
The equation 2.4 for heat dissipated by radiation can be simplified as given in equation 2.5
with suitable assumptions.
( ) 2.5rad rad Q S λ θ = …
Where, Qrad = the heat emitted out by radiation, J.
λrad = Emissivity of the radiating surface.
θ = Temperature difference between the radiating surface and the ambient, oC.
S = Area of the radiating surface, m2
If temperature rise remains within normal conventional limits for electrical machines, it may beassumed that artificial convection would not be necessary and the heat dissipated by
convection can be approximated as given in equation 2.6.
( ) 2.6conv convQ S λ θ = …
Where, Qconv = the heat dissipated by natural convection, J.
λconv = Specific heat dissipation by natural convection
θ = The temperature difference between the two media, oC.
S = Area of the surface separating the two media, m2
Therefore, total heat dissipated by radiation plus convection is as given in equation 2.7.
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( ) ( ) 2.7rad conv rad conv rad convQ Q Q S S S S λ θ λ θ λ λ θ λθ = + = + = + = …
Where, λ = λrad +λconv = Net Specific heat dissipation or emissivity.
This equation represents the Newton’s law of cooling. It should be noted that the Newton’s
law of Cooling is strictly true only for the cases where the body is acted upon by a uniform
current of air, i.e, only for the natural convection case. [2]
2.3 Theory of Solid Body Heating and Cooling
The temperature of a machine rises when it is loaded, starting from cold condition. The
temperature, in the beginning, increases at a rate determined by the power loss. As the
temperature rises, the active parts of the machines dissipate heat partly by conduction, partly
by radiation and in most cases, largely by means of air cooling. The higher the temperature
rise, the greater would be the effect of these methods of cooling. Therefore, as the temperature
rises, the rate at which the temperature rises falls off owing to better heat dissipating
conditions. The temperature of any part of a machine, not only depends on the heat produced
in it, but also on the heat produced in other parts. This is because there is always a heat flow
from one part to another. For example, the heat produced in the part of the windingembedded in the slot flows partially through the insulation to the laminations and partially to
the end windings. Thus the end windings have to transfer to the air, not only the heat
produced in them but also a part of the heat produced in the slot portion of the winding. That
is why the determination of temperatures is difficult, particularly in compactly enclosed devices
like transformers. Electrical machines are not homogenous bodies and their parts are made up
of different materials like copper, iron and insulation which have different thermal resistivities.
Due to this, it is rather difficult to calculate the temperature of a part of the machine.
However, it is worthwhile taking the theory of heating of homogeneous bodies as the basis for
analyzing the process of machine heating. The results obtained from such a theory are
applicable to a certain degree, to the different parts of the machine as a whole.
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The equation of temperature rise with time of any body is given by equation 2.8.
( )1 2.8h h
t t T T
m ie eθ θ θ
− − = − + …
Where, θ = temperature rise at any time t, oC
θm = final steady temperature rise while heating, oC
θi = initial temperature rise over ambient, oC
t = time, sec
Th = Heating time constant, sec.
The heating time constant of a machine is the index of time taken by the machine to attain
its final steady temperature rise and is given by equation 2.9.
...(2.9)h
GhT S λ
=
Where, G = Weight of the active parts of the machine, kg
h = Specific heat, J/Kg-oC
S = Cooling surface area, m2
λ = Specific heat dissipation, W/m2-oC
From the equation 2.9, it can be concluded that the time constant is inversely proportional
to λ and since the value of λ is large for well-ventilated machines, the value of their heating
time constant is less. Also, with the relation of the heating time constant with the dimensions
of the machine, it can be concluded that large sized machines have large heating time
constants. Another issue is that for the same loss, the machine would attain a higher
temperature rise if its dissipating surface is small or, if its ventilation is poor.
If the machine has started from cold conditions, then θ i= 0, which modifies equation 2.9.
1 ...(2 .10)ht T
m eθ θ = −
The cooling mechanism is also guided by a similar equation given as equation 2.11
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FIG 2.1: HEATING CURVE
. . .(2.11)c
t T
i eθ θ
=
where, Tc = Cooling time constant, sec and T c = Gh/Sλ
From equations 2.10 and 2.11, it is evident that both heating and cooling processes are
exponential in nature. Despite having similar mathematical expressions, the heating and
cooling time constants of a machine may have different values owing to the difference in
specific heat dissipation in heating and cooling conditions. The cooling time constant is
usually larger owing to poorer ventilations conditions, when the machine cools. [2]
2.3.1
Heating and Cooling Curves
The heating and cooling curves are graphical representation of equations 2.10 and 2.11 and are
illustrated in figures 2.1 and 2.2 respectively.
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FIG 2.2: COOLING CURVE
2.4 Heat in a transformer
A transformer has to be continuously in operation. By virtue of its construction, it has no
moving parts and is a closely packed and sealed device. As a result, the dissipation of the
generated heat inside the transformer is a rather difficult problem compared with any of the
other machines.
2.4.1 Heat Generation in the Transformer
The process of energy transfer in the case of transformers involves currents in the conductors
and fluxes in the ferromagnetic parts. Thus, there are I2R losses in the windings and core losses
in the ferromagnetic cores. The core losses include the hysteresis losses which are due to the
magnetic inertia of the core material and eddy current losses, which are due to the circulating
currents developed in the flux carrying parts of the transformer, mainly the core. The losses
taking place in the transformer cores and windings, during conversion of energy from one
voltage to another voltage level, are converted into thermal energy and cause heating of the
corresponding transformer parts. In addition to this, losses occur in tank walls and end plates
on account of leakage flux. All these losses appear as heat and the temperature of every
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affected part of the machine rises above the temperature of the ambient medium, which is
normally the surrounding air. The heat generated inside the transformer must be dissipated
without allowing the windings to reach a temperature, which will cause excessive deterioration
of the insulation. So, care should be taken that the heat generated in a machine is properly
dissipated without being accumulated. The heat generated in any equipment is dissipated into
its surroundings by convection and conduction assisted by radiation.
2.4.2 Heat Dissipation in the Transformer – Cooling Arrangements
The heat generated in various parts of the transformer is dissipated to the ambient in various
stages. The heat generation, majorly occurs in the core and coils and it is this heat that has to
be transferred to the external ambient in the first place. The heat transfer within the core and
coils and from the core and coils to the oil is by conduction. The heat transfer within the oil is
by convection and the heat transfer between the hot oil and the heat exchanger is through
conduction. When the insulating oil is involved in the heat transfer, since perfect contact with
the heated surface is rare, the heat transfer is mainly dependent upon the fluid flow conditions,
i.e., whether the flow is stream line or turbulent and upon the condition of the surface. The
thermal conductivity of the coolant is much smaller than that of the metals. Apart from the
heat being dissipated by means of heat exchanger surfaces, the transformer tank also partly
dissipates the heat generated by radiation. Radiation in transformers, does not normally occur
by itself and in almost every case, it is accompanied by convection. Since dull metallic paints
cause more radiation, all the electrical machines are painted with dull metallic paints usually
grey in color in order to have large heat dissipation from radiation and thus reduce
temperature rise.
The transformer oil, in the process of heat transfer gets heated up and its temperature rises
which is detrimental to its operation. Through external means, this heat in the oil should be
dissipated and then cooled oil should be circulated back into the transformer. Here, no
moisture, or gases should enter the transformer. This makes the cooling in the transformer a
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major issue. Different cooling methods, with suitable heat exchangers are used depending
upon the quantity of heat to be handled.
•••• Oil Natural Air Natural (ONAN) Cooling: Heat transfer in oil occurs because of the
natural thermal head generated due to convection in the oil. The hot oil at the top of the
tank is circulated back to the bottom of the tank through tubes, passing through which the
oil gets cooled because of cooler ambient around the tubes. It can be enhanced by the
usage of radiators and fins.
•••• Oil Natural Air Forced (ONAF) Cooling: For larger transformers, where the amount
of heat carried by the oil is huge, the heat exchange between the hot oil and cooler ambient
via the radiator tubes can be augmented by means of an air blast achieved through the
usage of fans. They blow air through the hollow spaces, drive the hot air out and suck
cooler oil in and thus cause better heat exchange between oil and external air.
•••• Oil Natural Water Forced (ONWF) Cooling: The radiator tubes can be cooled by
water instead of using an air blast as water is more effective in transferring heat, but only
when a natural water head is already available. This method proves cheaper and efficient
for transformers at hydro power stations
•••• Forced and Directed Oil Cooling: In large transformers, the natural circulation of oil is
insufficient for cooling the transformer and forced circulation is employed. Also, guiding
vanes are used to direct the oil flow in the cooling ducts in paths that ensure quicker and
efficient heat transfer from the coils to the oil.
The choice of cooling system is made depending on the loading of the transformer.
Accordingly, a transformer is given different ratings with different cooling methods.
2.4.3
Heat Build Up in the Transformer
With all the losses being generated in the transformer and the cooling system efficiently
transferring this heat to external ambient, the problem of heat buildup in the transformer still
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remains. A major reason for this is that with all the guiding vanes and cooling ducts, there
might be still some areas in the transformer where the coolant might not reach and the heat
might not be dissipated and hence gets accumulated. Apart from this, the transformer may be
operating at worse conditions of operations with load and frequency fluctuations, which might
cause increased losses that the cooling system may not dissipate effectively. It is a very usual
condition that the transformer is over loaded frequently for shorter durations or continuously
for longer durations. The losses, based on which the cooling system performance is
anticipated, may be practically different from the calculated values. Iron loss may change
because of change in grain orientation due to punching, clamping of laminations and the
pressures during these actions. Due to all or any of the above reasons, heat does get built up in
the transformer despite the existence of the cooling system.
On the top of all these, ambient temperatures do decide the rate of heat dissipation, since heat
dissipation is linearly proportional to the temperature difference between the transformer and
external ambient. Hence, it is obvious that the transformer cannot be loaded as much on
hotter days as on cooler days and on hotter days even lesser temperature rise of the
transformer oil may not be properly dissipated and might result in heat accumulation.
2.5
Consequences of Excessive heat buildup
The excess heat build-up results in undesirable consequences as given.
• Deterioration of winding Insulations which may elicit winding short circuits
• Deterioration of Insulating oil, which might reduce the quality of insulation
• It may affect the chemical properties of oil causing its dissociation and generation of
moisture and gases
•
This may increase the pressure in the tank, and in the worst case, this might causethe explosion of the tank.
• Chance of fire hazards
• This may change the thermal and electrical properties of the windings and core
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Though most of the consequences seem to have very little possibility of occurrence, they
cannot be neglected because, transformers are usually installed in remote locations unlike
other machines, where frequent maintenance is not always possible. Also, regular monitoring
of the device is not possible. The above consequences are not sudden in nature; they are
gradual, cumulative and related to each other. Also, it is difficult to identify any of these
happening unless an apparent damage takes place. So, if one of the above consequences
happens, it eventually causes the others to happen, and result in a permanent and irreparable
damage. [3]
2.5.1 Arrhenius Law of Insulation Ageing
In all the consequences that follow the heat buildup in the transformer, the major one is the
insulation deterioration. It is the first thing that happens with the excessive heat buildup and it
is the one that triggers other undesirable consequences/damages. Also, insulation is very
costly and is a major contributor in the cost of the transformer. The insulation of the
transformer tends to age and deteriorate when heated. The higher is the temperature, the
faster is the insulation deterioration.
During periods of subnormal operating temperature, the loss of life of the insulation will beless than normal. But when, the operating temperatures are greater than normal, the loss of
life will be higher than normal. Consequently, the transformer may be safely operated for a
time at above normal temperatures provided the loss of insulation life during this period is
adequately compensated for, by operation for a sufficiently long time at temperatures lower
than normal. This deterioration of life of insulation with temperature is mathematically given
by Arrhenius law of Insulation Ageing, which is a non linear relation, given in equation 2.12.
. . .(2.12)
B A T L ife e
+ =
Where A and B are constants, derived by experiment and T is the absolute temperature.
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FIG 2.3: ARRHENIUS LAW OF INSUALTION AGEING
In the range of 80oC to 120oC, which is the usual winding hot spot temperatures, this law
can be expressed in a more convenient form called Montsinger relation as given in equation
2.13.
...(2.13) p
Life e θ −=
Where p is a constant and θ is the temperature in oC. Practical observations and
investigations reveal that between 80 oC to 120 oC the rate of loss of life due to ageing of
transformer insulation is doubled for every 6 oC rise in temperature [2].
Figure 2.3 gives the graphical representation of Arrhenius law of Insulation ageing.
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Chapter 3
3
TRANSFORMER THERMAL MODELING –
LITERATURE SURVEY
3.1 Techniques to Measure Transformer Interior Temperatures
There are broadly three ways to measure transformer interior temperatures, namely, the
usage of empirical formulae (IEEE and IEC standard formulations), direct measurementusing fiber optic sensors and usage of suitable computer based mathematical thermal
models. Empirical formulae to calculate maximum interior temperatures assume the heating
process of oil and winding to be similar to the charging and discharging process of a
capacitor. Using the measurements obtained from the transformer’s heat run test, the
formulae calculate Hot Spot Temperature (HST) and Top Oil Temperature (TOT) at rated
load and predict the value of HST and TOT for any loading condition and see if the
transformer thermal limits are observed, thus ensuring the safe feasibility of such a loading
condition. While IEEE formulae give precise formula to calculate the value of HST, IEC
standards stipulate a factor to be multiplied with the measured average temperature rise over
TOT to calculate the winding hottest spot rise. On the other hand, direct measurement
involves embedding the fiber optic sensors at various locations inside the transformer and
then by running the transformer at desired loading conditions, the value of HST is directly
recorded from the sensor measurements. The HST is checked with the thermal limits of the
transformer to see if the particular loading condition is safe for the transformer.
Both of these methods have their own limitations. Empirical formulae are derived based on
certain assumptions which are made to generalize the formulae to suit to any transformer.
The other side of the coin is that, since the transformer design, construction and loading
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conditions change the transformer thermal response in a non linear way, generalizing the
formulae makes the formulae unspecific for a transformer and hence the transformer
thermal response is poorly tracked. Moreover, including too many constraints and
coefficients make the formulae too complex to be solved. However, they are cost efficient
techniques. In contrast, direct measurement techniques using sensors are quite accurate but
costly. The sensors themselves are costly and embedding them inside the transformer for
testing and then removing them after the testing is done are cost involving processes.
Moreover, placement locations for sensors also play their part as even after placing many
sensors, one may miss the hottest location. So, it can be understood that both the
conventional methods are not the optimal answers to the problem of transformer thermal
monitoring. Constant improvement of technology in the field of computer science and its
applications in wide range of research areas enables the problem to be solved with the
application of suitable software techniques. So, software based simulations and modeling
prove to be cost effective as well as efficient techniques to answer this problem.
3.2 IEEE Formulae for Calculating Hotspot Temperatures
The IEEE formulae give the possible value of hot spot temperature inside the transformerusing the value of top oil temperature measured from the heat run test so as to guide the
user (or the designer) to decide the safe loading conditions of the transformer. Applications
of loads in excess of nameplate rating involve some degree of risk. IEEE formulae are
designed to identify these risks and to establish limitations and guidelines, the application of
which will minimize the risks to an acceptable level [4]. The basic equation for the
calculation of the hottest-spot temperature is as given in equation 3.1.
...(3.1) H A TO H θ θ θ θ = + ∆ + ∆
Where, θH is the winding hottest spot temperature, °C
θA is average ambient temperature during the load cycle to be studied, °C
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∆θTO is the top-oil rise over ambient temperature, °C
∆θH is the winding hottest-spot rise over top-oil temperature, °C.
The top oil temperature is given by the equation 3.2.
...(3.2)TO A TO
θ θ θ = + ∆
The top-oil temperature rise at a time after a step load change is given by the following
exponential expression containing an oil time constant.
( ) 0
1
, , ,1 ...(3.3)T
TO TO U TO I TO I e
τ θ θ θ θ −
∆ = ∆ − ∆ − + ∆
Where, ∆θTO,U is the ultimate top oil temperature rise over the ambient for load L, °C
∆θTO,I is the initial top oil temperature rise over the ambient, °C
τT0 is the oil thermal time constant, sec.
The winding hot spot rise over the top oil temperature ∆θH is calculated using similar
exponential expression as in equation 3.3 while the value of rated hotspot temperature rise
above top oil temperature is given by equation 3.4.
, / , , ...(3.4) H R H A R TO Rθ θ θ ∆ = ∆ − ∆
Where, ∆θH/A,R is the rated hot spot temperature rise above ambient, oC
∆θTO,R is the rated top oil temperature rise above ambient, oC
∆θH/A,R can be measured by embedded detectors or is usually supplied by the manufacturer
on test report and ∆θTO,R is usually supplied by the manufacturer on test report [4].
While the IEEE formulae seem to give a general and simple formula that can be applied to
any transformer, on the other hand, it can be seen that the constants used in the formulae
are not that easily available. The accuracy of the testing procedure to determine the various
constants used in the formulae also affect the accuracy of the final estimated hot spot
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temperature value. A more accurate but iterative method is also given in the IEEE Loading
guide, but that is too elaborative and time consuming and requires lot number of
measurement data [5]. Also the IEEE model, while it accounts for the thermodynamic effect
of load on Hot Spot Temperature (HST), it does not accurately account for the effects of
Top Oil Temperature (TOT) variations on HST. From the formulae, it can be seen that if
TOT changes instantaneously, then HST will also change instantaneously.
Thermodynamically, HST cannot change instantaneously even if TOT changes
instantaneously and there must be a time lag due to the winding time constant. It has also
been noticed that the top-oil temperature time constant is shorter than the time constant
suggested by the present loading guide. These are the noticeable drawbacks of the above
discussed IEEE standard [1], [6], [7].
3.3 Fiber Optic Sensors for Temperature Measurements
Fiber optic sensors use optical fiber as the sensing element. Optical fibers can be used as
sensors to measure strain, temperature, pressure and other quantities by modifying a fiber so
that the quantity to be measured modulates the intensity, phase, polarization, wave length or
transit time of light in the fiber. Temperature can be measured by an optical fiber usingits evanescent loss that varies with temperature. Thus they were an answer to the increased
need of accuracy in temperature measurements and fiber optic sensors for effective
measurement of high temperatures were designed. However, in the case of transformers,
survival of these sensors from the interior stresses was a problem due to their fragility in the
initial days. In response to this important need, fiber optic sensors have significantly
improved to the point that direct measurement of winding temperature is now becoming the
preferred method to measure the Hot Spot Temperature (HST) than using standard
empirical formulae. Compatibility of fragile fiber optic sensor with transformer factory
environment which had been a problem in the past, is now resolved with sturdy fiber jackets,
proper spooling of sensor during factory work and simplified through-wall connections.
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But the fiber optic sensors are costly and so this has effect on the final cost of transformer.
However, the demand for quality and accuracy of measurements is more prominent in the
market, which indicates that the measurement of temperatures through methodologies that
have possibilities of giving better quality results are being looked forward than using only the
present IEEE and IEC standards in the industry.
3.4 Thermal Models to Calculate Hotspot Temperatures
References [5] – [24] present the various attempts made to develop the thermal model for a
transformer based on thermal electrical analogy that can calculate the hot spot temperatures
better than what the IEEE thermal model does. A basic thermal model that uses thermal
electrical analogy, which states that the thermal parameters of heat flow, temperature, heat
storage and heat dissipation are analogous respectively to the electrical parameters of current,
potential difference (voltage), capacitance and resistance, is used as a basis to build lumped
parameter models that simulate the thermal behavior of the transformer that is exponential
[9], [10]. One model represents the winding to oil heat transfer and the other represents oil
to external air heat transfer. All the heat generated inside the transformer due to the losses is
represented by current sources and all the heat storage inside the transformer is representedby a single capacitance, claiming that the single capacitance as a lump represents the total
heat storage inside the transformer. The resistance connected in each of the models
represents the heat dissipation from winding to oil or oil to external ambient as the case may
be. This resistance is modeled to be non linear because of the fact that heat transfer from the
winding to transformer oil as well as from the transformer oil to external ambient depends
upon whether the transformer oil is directed (or forced) or natural and whether the external
air is natural or forced respectively.
The required parameters like rated load top oil temperature rise over the ambient, the top oil
time constant and the value of exponent that determines the nonlinearity of the resistance
were found from the technique of minimization of the integral-squared-error over one day
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of test data. Such a method of determination of thermal model parameters would be quite
practical for an on-line parameter determination calculation that automatically and
continually estimates the key thermal model parameters. A sudden change in the value of
one of the parameters such as the rated top oil rise could indicate that the fans or pumps
have failed. A gradual increase in the calculated rated top oil rise might indicate that the
radiators are becoming increasingly fouled in some way. Therefore this method helps to
make the transformer monitoring online [10].
As this model doesn’t consider the oil viscosity as well as temperature dependence on the
transformer oil thermal parameters, ref. [6] presents an improvisation with the inclusion of a
nonlinear thermal resistance in the model, which takes into account changes in the
transformer oil thermal characteristics and viscosity with temperature using an
experimentally determined constant. Ref. [7] presents a further improvisation of this model
which includes the effect of temperature dependence of the losses generated inside the
transformer, using a loss correction factor as well as the specific design of the transformer
windings and their influence on the oil circulation and the temperature gradient at the top of
the winding stack. Also, the oil viscosity changes with temperature are also modeled. The
method of calculating the transformer time constants and the changes in their values is
refined. Ref. [11] suggests the simultaneous solving of the two exponential equations derived
from the winding to oil and oil to ambient models. An online transformer monitoring
system based on this model is presented in ref. [12]. Ref [13] gives similar online transformer
monitoring model, but, instead of including the effect of external cooling on the transformer
by means of non linearity in the resistor, it has been separately dealt as an equation.
All the models discussed hitherto, use top oil temperature as reference to calculate the
hotspot temperature. Instead, since bottom oil temperature which is the temperature at the
winding oil ducts just below the winding, might properly track the temperature of oil
adjacent to the winding, especially in the case of transient over loads, usage of bottom oil
temperature to calculate the hotspot temperatures might improve the accuracy of
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calculations [8], [14]-[16]. Since measurement of bottom oil temperature is considered a
difficult task, an alternative method for calculation of bottom oil temperature without
disturbing the oil flow has also been explained in the [14], [15].
Apart from these models, modifications in the existing IEEE loading guide by including the
temperature dependence of losses, viscosity dependence on heat transfer and oil temperature
as well as non linearity of temperature distribution along the winding height is suggested in
[14]. Ref [5] gives a modified version of the IEEE Hot Spot Temperature (HST) model to
account for the effects of dynamic variations of the top-oil temperature (TOT) on HST.
Attempts have also been made to improve the method of measuring various parameters
such that the non-linearity is understood and measured properly. While ref. [17] presents a
method for accurate calculation of eddy current loss using a two dimensional finite element
formulation, ref. [18], [19] present an attempt to apply Finite Element Analysis to the
accurate calculation of losses in the windings, so that the resulting losses can be used in
conventional models to calculate the hot spot temperatures. Ref [20], [21] present two
dimensional models aiming at accurate calculation of the magnetic properties of the
magnetic material, flux densities and the grain orientation which would help in saving energy
as well as avoiding unnecessary heat generation in the core material.
3.5 Techniques Based on Computer Based Simulations
With the advent of computers in every domain of engineering sciences and with the
development of computer simulations, solving of complex problems has become easy along
with increased accuracy and controllability over the solution process. In the field of power
system engineering, complex multi variable problems which demanded either a compromise
in the solution accuracy or solution speed or the cost of solution have been properly cateredby these computer simulations. In the field of thermal modeling of transformers to calculate
the interior temperatures of the transformers, some work has been done to simulate the
transformer interiors using standard software packages. A technique used in these software
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packages for such problems is Finite Element Analysis (FEA). In the recent past, some work
has been done in using the computer simulations and FEM packages in the transformer
analysis and are presented in [25] – [28].
Ref. [25] describes a computer model, which can predict hot-spot temperatures for different
types of cooling regimes and transformer winding geometries coded using FORTRAN. Ref.
[19] presents a model that uses Finite Element Analysis for the accurate determination of
stray losses stating the fact that underestimation of losses inside the transformer particularly
the stray loss is one of the possible reasons for the hotspot inside the transformer. Ref [26]
presents a 3 dimensional model that aims at accurate estimation of the stray losses in a
power transformer using an Integral Equation Method (IEM) and Finite Element Method
(FEM) owing to the fact that stray losses in a large rated transformer can be around 20% to
25% of the total losses.
3.6 Techniques Based on Artificial Intelligence
The rationalism and reasoning which have been human assets are now being attempted to be
imbibed to the machines with the progressing development in the field of computation. A
few recent research attempts to answer the problem of transformer thermal modeling
through the use of artificial intelligence methods of fuzzy logic, genetic algorithms and
neural networks, are discussed in [29] – [32]. Ref [29] presents an equivalent heat circuit
based thermal model of an oil-immersed power transformer and a methodology for model
construction using intelligent learning, based on real-world data. A genetic algorithm is used
as a search method, based on a few on-site measurements, to determine the thermal model
parameters. The proposed thermal model can continuously calculate temperatures of the
main parts of an ONAN or OFAF cooled power transformer under various ambient andload conditions. Ref [30] presents a further simplified thermal-electric analogous thermal
model of an oil-immersed power transformer rooted on the principles of heat exchange and
electric circuit laws. Ref. [31] gives the application of neural networks to the transformer
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analysis as well as fault diagnosis and ref. [32] gives the application of fuzzy logic algorithms
as well as expert systems.
3.7 Observations and Comments
Economic and operation motivations have been the reason for the research for accurate
thermal modeling of transformers and the possible approaches are to measure the hot-spot
temperature using the sensors or to calculate it using standard empirical formulae or a thermal
model. In order to ensure that the conventional IEEE and IEC standards cope up with the
improvements in the design technology as well as to improve their accuracy in the modeling of
overloads etc. they are constantly being revised with more accurate and advanced models
aiming at a better representation of oil temperature inside the winding, considering variations
in the winding resistance, oil viscosity and oil inertia. Direct measurement of winding
temperature with fiber optic sensor provides a good alternative, but they are costly.
This makes evident the fact that neither the IEEE standards nor the fiber optic sensors could
be an optimal solution to address the problem of transformer thermal modeling and
transformer interior temperature calculations, to be specific, the hot spot temperature
calculations. With the improved capabilities of computer automations, constant efforts are
being made to derive software based thermal models for accurate measurement of hot spot
temperatures. Improvisation is needed not only in the model for measurement of temperatures
but also in the methods of accurate determination of eddy current losses, stray losses, cooling
mechanism etc.
A thermal model for a transformer can be created to deliver either the temperature distribution
across the transformer geometry, or the characteristic temperatures (HST, TOT) that indicate
the thermal status of the transformer interior. The current market competency demands more
research insight into the problem of transformer thermal modeling and calls for even more
precise, intelligent, accurate and above all economic solutions. The past research focused on
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the development of lumped parameter thermal models that calculated only the HST and TOT
and there is no evident attempt on obtaining a temperature distribution profile across the
geometry of transformer. An attempt has been made in this work to calculate the temperature
distribution across the transformer geometry which can give a better understanding of the
thermal conditions inside the transformer.
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FIG 4.1: THERMAL MODEL OF A SINGLE ELEMENT
Chapter 4
4
TRANSFORMER THERMAL MODELING USING LOSS
DISTRIBUTION
4.1 Proposed Method of Thermal Modeling
The proposed method is based on thermal electrical analogy and the basic guideline is the fact
that the losses in the transformer are distributed across the geometry and not concentrated at asingle point and hence the model uses distributed values of losses instead of lumping them.
The principle is that each point-element in the transformer generates heat because of the loss
in it. The heat transferred to that element or from that element depends upon the temperature
of neighboring elements. The element stores a little amount of heat, which is the cause of
temperature rise of that element and dissipates the rest into surrounding medium as long as its
temperature is greater than that of the surrounding medium. The temperature of each point-
element, therefore, depends on heat generation in that element and also the temperature of the
surrounding elements. Thus, each point element is considered to have a loss causing heat
generation in it, a heat conductor dissipating heat to neighboring elements and a heat retaining
element that causes heat storage inside it.
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FIG 4.2: STEADY STATE THERMAL MODEL SHOWING INTERCONNECTION OF ELEMENTS
As stated, there exists an analogy between the thermal and electrical parameters and the
thermal parameters of heat flow, temperature, heat storage and heat dissipation are
analogous respectively to the electrical parameters of current, potential difference (voltage),
capacitance and resistance. Based on this analogy, the behavior of every point element in the
transformer can be represented in the form of an electrical circuit as shown in fig. 4.1. The
dissipation of heat to neighboring elements in all the directions is modeled making use of two
resistors representing the heat flow in either direction. If the loading on the transformer is
steady with no major variations, the temperature rise would follow an exponential pattern and
gets constant in the steady state. In the case of such a condition, the capacitance can be
ignored and the model can be simulated. In this way, every point element can be modeled and
can be connected to neighboring elements modeled similarly as shown in fig. 4.2.
Implementation wise, since modeling of each point element is difficult practically, the
transformer geometry is divided into finite number of sections or elements (as referred to
hereafter) and each element is modeled accordingly as given in fig. 4.1. Each element can be a
part of core, winding, tank or oil, geometry wise with dimensions different from the
surrounding elements and hence the value of parameters of the model changes for each
element. The division of the geometry into elements can be based on the change in the loss
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concentration in the part of the geometry and smaller the size of the each element, higher is
the complexity in the model and finer is the thermal profile obtained. The current source
represents the heat generated in the element because of electrical loss occurring in it. If the
element is a part of core, the hysteresis loss and eddy current loss summed up forms the value
of current source in the model of that element and if the element is a part of winding, the
ohmic loss occurring in that part of winding is used as the value of current source in the model
of that element. Oil elements will not have current sources as there are no losses occurring in
the oil. The resistors used in the model to represent the heat dissipation of an element to
neighboring elements are calculated using the values of thermal conductance of the material,
which is the property of the material. The capacitor value, which represents the heat storage
inside the transformer, if used, is calculated using the value of specific heat capacity of the
material of the element.
As straightforward as the model might look, the complexity lies in determining the loss in each
element. Though it is an obvious fact that the loss is not uniformly distributed across the
geometry of the heat generating elements (core and windings), determination of the loss
distribution pattern and the gross loss in each of element division is not easy. Copper loss is
more or less uniformly distributed and hence the loss in each element division of the winding
geometry can be proportionally a fraction of the total ohmic loss in the winding. The problem
of obtaining the loss distribution across the core is a more complex issue and has to be
addressed using high end computational techniques. In the current work, the technique of
Finite Element Analysis (FEA) is applied to obtain the flux distribution, which is non uniform
across the core and calculate the core losses from the values of flux density grossed across
each element division.
Each element is thus modeled accordingly and the individually modeled elements are
interconnected to form the total geometry of the transformer. The electrical mesh circuit, thus
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formed is the thermal model of transformer and solving the model gives the temperature at
each node, thus giving the thermal profile for the transformer geometry.
4.2 Obtaining the Loss Distribution – Finite Element Analysis
The ohmic loss in the winding can be grossed as the product of the square of the current
passing through the winding with the resistance offered by the winding (Copper) and the loss
in each element division of the winding can be proportionally a fraction of the ohmic loss in
the entire winding. The core loss in the transformer is the sum of hysteresis loss as well as
eddy current loss. The hysteresis loss is because of the magnetic inertia of the magnetic
material, or in technical terms, due to the remnant magnetization left over in the magnetic
dipoles during alternate cycles of magnetization and demagnetization caused by sinusoidal
alternating current. Similarly, as the magnetic material also would have got certain inherent
electrical conductivity, circulating alternating currents are generated in the magnetic material
due to alternating magnetic flux in it. As less as the electrical conductivity of the magnetic
material may be, still these circulating currents termed as eddy currents would exist and the
ohmic losses due to these currents are termed as eddy current losses.
For a given transformer core material, given frequency of excitation, the core loss is dependent
on the flux density in the core cross section defined mathematically. So, if the flux density is
known, the core loss can be determined. Hence, if the flux density distribution across the core
as well as in each element division of its geometry is known, the core loss can be determined
by means of mathematical substitutions and calculations. It is here that the Finite Element
Method (FEM) is made use of. The Finite Element Analysis (FEA), sometimes referred to as
finite element method (FEM), is a computational technique used to obtain approximate
solutions of boundary value problems in engineering.
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4.2.1 Obtaining the Flux Density Distribution by Finite Element Analysis
The concept of FEM deals with applying the differential equations over smaller sub domainsof the entire large domain area, and then building up the solution of next layer of sub domains
using the already calculated values of field variables of the neighboring domains as boundary
conditions and this is extended to the current problem. Flux density in the core is associated
with the development of magnetic field in and around the core due to the alternating current
generating an alternating electric field. The values of flux densities at various points in the
transformer are governed by Maxwell’s differential equations given in equations 4.1 to 4.3. In
addition, equations 4.4 to 4.6 give the necessary constitutive relations. These equations give the
value of flux density in terms of current densities. Depending on the value of current densities
in the element and the value of boundary conditions on the element, the FEA tool calculates
the flux densities in each element. The boundary conditions for the elements on the outer end
of the geometry would be the user defined limiting value of the field variable on the boundary
of the geometry and here assumed to be the value of field variable (magnetic field) being zero.
(Dirichlet boundary condition)
...(4.1) B
E t
∂∇ × = −
∂
...(4.2) D
H J t
∂∇ × = +
∂
...(4.3) D ρ ∇ • =
...(4.4) D E ε =
...(4.5) B H µ =
...(4.6) J E σ =
Where, E is the electric field strength, V/m
H is the magnetic field strength, A/m
D is the electric flux density, C/m2
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B is the magnetic flux density, Wb/m2
J is the electric current density, A/m2
ρ is the electric charge density, C/m3
ε is the permittivity, F/m
σ is the conductivity, mho/m
µ is the permeability, H/m
Since the transformer geometry is divided into fixed number of elements to be modeled
individually, the same elemental division is followed in the FEA implementation also, so that
the tool directly gives the averaged flux density in each element. Implementation wise, the
transformer geometry is divided into elements, and the excitation and boundary conditions,
which are respectively the current densities in the windings and the values of magnetic field at
the boundaries of geometry are given. The tool implements the Maxwell’s equations starting
from the outer boundary and works towards the inner elements and finally calculates the flux
densities in all the elements.
4.2.2 Obtaining the Loss Distribution
The loss in the transformer is majorly in core and windings. The losses in the other parts are
negligible and can be ignored with no loss of accuracy of the results.
• Loss distribution in the windings: The losses in the windings are the ohmic losses due
to current flowing in the conductors. As superior as the electrical conductivity of the copper
may be, still, there would be good amount of ohmic loss in the windings owing to the high
values of currents flowing through the windings. The general expression for the calculation of
ohmic loss is given in equation 4.7.
( )2
...(4.7)Cu
lW J A
Aσ
= ×
Where, W c u is the copper loss occurring in the element, Watts
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J is the current density in the element, A/m2
A is the surface area through which the current flows, m2
σ is the electrical conductivity of the material, mho/m
l is the length of the path the current has to travel, m
The winding geometry is divided into elements and the ohmic loss in each element is to be
calculated. The current density will have a uniform distribution (this can be ensured with the
implementation of FEA also) and hence, the ohmic loss will have a uniform distribution
across the winding geometry. Hence the ohmic loss in each element can be calculated
directly from equation 4.7. Since the current density remains the same through entire
winding, substituting the value of A from the dimensions of each element gives the ohmic
loss in each element. This value of loss would be used for the current source in the thermal
model for each element.
• Loss distribution in the core: The finite element analysis of the transformer geometry is
performed and the flux density distribution across the transformer geometry as well as the
average flux density across each element is obtained. Now, the core losses are to be calculated
using the flux density values. The mathematical expressions for the core losses, which
comprise of hysteresis and eddy current loss are given in the equations 4.8 and 4.9.
1.6 / ...(4.8)h h mW K B f Watts kg=
2 2 2 2 / ...(4.9)e e f mW K K B f t n Watts kg=
Where, W h and W e are hysteresis and eddy current loss respectively, Watts
K e and K h are material constants that are found out experimentally
f is the frequency of the alternating flux, Hz
Bm is the maximum value of operating flux density, Wb/m2
K f is the form factor of the ac wave form
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t is the thickness of each lamination of the core, m
n is the number of laminations in the core.
The sum of W h and W e give the value of core losses. Since the flux density across each
element is obtained from the FEA implementation, the values of core losses can be
calculated from the mathematical formulae given in equations 4.8 and 4.9. Though the flux is
supposed to be limited to the magnetic material, i.e., the core itself, there would be some
leakage flux existing in the tank, coils as well as very feebly in the rest parts. Though the
FEA gives the values of this leakage flux also, however, the losses contributed by these
leakage fluxes are very negligible and can be ignored. The value of core losses in each
element, thus calculated, will serve as the value of current source in the thermal model for
the elements.
4.3 Development of Thermal Model
The principle behind the thermal model is explained in section 4.1 and the development of
thermal model for the entire transformer is on the same lines. The section elaborates the
thermal modeling process.
4.3.1 Thermal Electrical Analogy
The analogy between the thermal and electrical parameters is given in table 1.
Sl. No Element Thermal parameter Electrical Parameter
1 Through Variable Heat transfer rate, q Watts/s Current, i ampere
2 Across Variable Temperature, θ oC Voltage, v volts
3 Dissipation element Thermal Resistance R t h,
oC/Watt Electrical Resistance, R el Ω
4 Storage element Thermal Capacitance C th J/oC Electrical Capacitance, C el Farads
Table 1. Thermal Electrical Analogy
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The development of thermal model and solving of it to obtain the thermal profile is based on
the analogy between the thermal and electrical parameters of a material and the similarity
between the development of electrical potential in the conductor with the flow of charge and
the temperature rise in the body because of the flow of heat. The two important thermal
parameters of a substance or a material are its specific heat capacity and its thermal
conductivity. These are general properties which can be made specific to a specific volume of
the material in which case they become thermal capacitance and thermal resistance of that
particular volume of the material. This holds good for metals, but for fluids (transformer oil),
additional modeling considerations and coefficients are needed to model the heat transfer. This
is because, besides being guided by the thermal conductance and the temperature difference,
the heat transfer in liquids is affected majorly by heat convection and fluid viscosity.
4.3.2 Electrical Equivalent Model for Thermal Behavior
The basic model for each element, whatever the division of element might be, is explained in
the section 4.1. All the individually modeled elements are to be connected together to form an
electrical mesh circuit which is the thermal model for the entire transformer geometry. This
connection is needed because, the temperatures and the dissipation of heat in each element are
not independent and these depend on the temperature and heat dissipation of all the
neighboring elements. But while integrating individual elements geometrically to form the
thermal model, few modeling constraints would have to be observed.
• The loss generated in each of these elements is independent of the neighboring blocks. So
the current source of every element, which is the heat generated in that element must be
separately connected to the ambient.
• The oil sections will not have source since there are no losses in the oil sections. In the
transformer, oil functions as insulating as well as cooling liquid and is used as a medium of
heat transfer. It has no effect on heat generation.
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• The outer tank section also dissipates the heat to the ambient, which is modeled by
thermal resistances connected between each element on the tank boundary to the ambient. In
this way, ambient is modeled separately.
• Ambient serves the purpose of ground in conventional electrical circuits. But the
difference here is that in conventional electrical circuits, ground is at zero potential, whereas
ambient in this electrical equivalent for simulating the thermal model is the surrounding
atmosphere which is not at zero temperature. The ambient temperature is in the range of 20oC
or 30 oC depending on the surrounding atmosphere temperature. It can be as high as 45 oC - 50
oC and as low as 5 oC - 10 oC. Hence the temperature values obtained from the thermal model
is the temperature rise of the corresponding locations above the ambient and not the
temperatures themselves.
• Symmetry is observed in the circuit for heat dissipation. The heat generating elements in
the middle of geometry dissipate in both the directions, while the elements on either side
dissipate in the respective direction only. This does not restrict in any way the heat flow in any
direction; it is just that the circuit is made symmetrical.
•
Since the model is two dimensional, in a way, we are studying the heat dissipation in two
directions, while the losses that are generated in the transformer dissipate in all the three
directions equally. So, since we limit the model to two-directional (X-Y) heat dissipation, the
heat to be dissipated must also be proportionally reduced, i.e., made two thirds of the total
heat generation.
• In the figure 4.1, there are two resistors. The resistor in the horizontal direction represents
the heat flow along X-direction and the resistor in the vertical direction represents the heat
flow in the Y-direction.
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FIG 4.3: CALCULATION OF RESISTANCES – THERMAL MODEL FOR SINGLE ELEMENT
4.3.3 Calculation of Parameters of Thermal Model
•
Calculation of Sources: For the core elements, the value of core loss in each element willform the current source and for the winding elements, the value of ohmic loss will be the value
of current source. For the other elements, there will be no current sources as there is no heat
generation in the other parts of the transformer. The leakage fluxes and the loss due to them
may help improve the accuracy and help in transformer design analysis, but the contribution to
heat generation by them is not so significant.
• Calculation of Resistances: The thermal resistance of an element represents the
opposition offered by the element to the flow of heat through it. This resistance is offered in
both X and Y directions for either directions of heat flow and hence the thermal resistance of
each element is represented by two resistances corresponding to the opposition offered in each
direction of heat flow. The resistance along the X direction represents the opposition offered
by the element to the heat transfer in horizontal direction to neighboring elements. Similarly
the resistance placed in the Y-direction is the opposition offered to the heat flow in vertical
direction. The thermal resistance of each element depends on the material property of thermal
conductivity (Watts/mK), specific to the material of the element. Consider a single element as
shown in the figure 4.3.
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If x and y are the dimensions of the element in X and Y directions respectively, z is the Z-
direction depth and ρ is the thermal conductivity of the material of the element, the thermal
resistance offered by the element to the heat flow in either direction is given by the equations
4.10 and 4.11.
...(4.10)horizontal
x R
yz ρ =
...(4.11)vertical
y R
xz ρ =
• Calculation of Capacitance: The capacitance can be included in the thermal modeling
circuit, if the transient thermal performance is of interest or if the load is so non uniform that
the thermal transients might affect the transformer. In any case, the method to calculate the
thermal capacitance values is presented. The thermal capacitance of an element represents its
capacity to store the heat, which causes a rise in its temperature. This is determined by the
material property of specific heat capacity. The thermal capacitance of each element is
calculated by multiplying the value of specific heat capacity of the material of the element with
the mass of the block. So, if an element of dimensions x, y and z in X, Y and Z directions
respectively as in figure 4.3 is considered, whose specific heat capacity is s J/kg
0
C and density isd kg/m3, then the value of thermal capacitance of the material is given by equation 4.12.
.( ). ...(4.12)C s xyz d =
4.3.4 Modeling of Radiators
Radiators are heat exchangers used to transfer thermal energy from one medium to another
for the purpose of cooling and heating. In the transformer, radiators help draw out heat from
the hot oil in the top and re-circulate the cooled oil back into the tank from the bottom. They
are commonly used for ONAN, ONAF/ONAN and OFAF/ONAF/ONAN types of
cooling for slightly larger transformers, while for smaller transformers, corrugations on the
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tank surface itself serve as radiators. Radiators are tubes or fins through which the oil from the
top of the tank is circulated through, to the bottom of tank and while the oil is circulated the
heat from the oil is drawn out. They have heat exchanging elements joined to top and bottom
headers which are connected to the transformer tank by welding and consist of previously
rolled and pressed thin steel sheets to form a number of channels or flutes through which the
oil flows. The surface area available for heat dissipation is increased by the use of radiators. As
the oil passes downwards, due to natural circulation heat is carried away by the surrounding
atmospheric air. This cooling of oil is augmented by blowing air onto the radiator tubes and
this is what happens in forced air cooling.
While modeling, the effect of radiator is modeled by means of thermal conductance pathprovided from the top and bottom oil to the ambient, which is represented by means of
resistors connected to the top and bottom oil on either side of the tank. The dissipation occurs
through the radiator surface and hence the thermal conductivity of the radiator metal (same as
the tank metal) is to be used in the calculation of radiator resistance. The diameter of the
radiator tube would be the length of the heat transfer path and the surface area of the radiators
would be the surface area of dissipation
4.3.5
Modeling the Convection in Oil
The heat transfer process in the transformer is by conduction, convection and sometimes by
radiation. The heat transfer within inside the core and windings is by conduction, which is a
linear process depending on the thermal conductivity of the material as well as the temperature
difference between the surfaces in contact for heat conduction. The heat transfer from the
core and windings to the oil is also conduction since they are designed to be in good contact
and there exists a temperature difference between them. But when it comes to heat transfer
within the oil (heat gets transmitted within the oil to reach the outside cooling system from the
transformer interiors) the process is not entirely linear. Due to fluid currents in the oil, the heat
transfer within the oil is due to convection as well as conduction. Convection is a complicated
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phenomenon depending upon many variables such as density and viscosity of the liquid,
temperature difference between the heated surface and coolant, thermal resistivity of the fluid
and gravitational constants. However, it is to be noted that the convection takes place only in
the oil and only in the vertical direction (Y-direction) and hence doesn’t affect the heat transfer
in horizontal direction (X-direction).
In order to model the convection, the effect of convection is to be understood. Convection in
the transformer oil has effect heat transfer in a manner such that the heat tends to rise up and
thus top portion of oil is at a relatively higher temperature than the bottom portion. As the oil
gets heated, the molecules become lighter and hence rise up giving place to cooler and hence
heavier oil molecules which, by gravity come down. Hence, within the oil, the heat transfer in
the upward direction is more than the heat transfer in the downward direction and hence the
top oil is always at a higher temperature than the bottom oil. The amount of convection fluid
currents also depends upon the viscosity of the fluids, here, the oil. More the viscosity of the
oil, lesser is the convection. The viscosity of a fluid is in turn dependent on the temperature of
the fluid and this makes convection a complex process to explain mathematically. Hence the
increase in the heat transfer due to convection is to be observed practically and a multiplicative
factor to the vertical resistance in the thermal model of oil elements is to be decided to
simulate this increase in heat transfer.
As oil is a viscous liquid, there cannot be too much convection as in the case with non viscous
liquids. The convection, as observed experimentally could increase the heat transfer by 100%
and not much more than that in any case and hence the heat transfer within the oil in the
vertical direction now becomes twice of the original value. Hence, the resistances in the
vertical direction (Y-direction) of oil blocks are modified, i.e., halved to include this convection
effect. Thus, an increase in heat transfer is achieved by reducing the resistance which accounts
for increased heat transfer due to convection.
Therefore, if x , y and z are the dimensions of an oil block with ρ as the thermal conductivity,
the horizontal resistance R h and vertical resistance R v would be as in equations 4.13 and 4.14.
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FIG 4.4: MODELING THE CONVECTION – MODIFIED THERMAL MODEL OF OIL ELEMENTS
...(4.13)h
x R
yz ρ =
...(4.14)2
v
y R
xz ρ =
The tendency is of the hotter oil to go up against gravity, leaving the cooler oil at the bottom
of the tank is to be simulated. So, in other words, it can be stated that the convection in oil
doesn’t allow heat flow downwards as far as convection effect is concerned. It is not that the
bottom portion of oil does not have any temperature rise, but relatively, the heat concentration
is towards the upper portion of oil. This phenomenon also takes place only in the vertical
direction and is modeled by partly obstructing the oil flow in downward direction. This is
achieved by using diodes in the thermal model, because the resistors will not simulate this
behavior and this obstruction has to be partial only. In order to ensure the convenience of
modeling, the oil blocks are modified as given in figure 4.4. Here R h represents the horizontal
direction thermal resistance while R v represents the vertical direction thermal resistance after
the multiplicative factor accounting for the increase in the heat transfer because of convection
fluid currents being incorporated.
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FIG 4.5: MODELING THE CONVECTION –THERMAL MODEL OF OIL ELEMENTS WITH DIODE
Since the two vertical resistors equivalent to one single vertical direction thermal resistance
stand parallel to each other, their value has to be double the value of a single vertical resistor
that can equivalently replace them. In order to present partial obstruction to the downward
flow of heat, thus modeling convection, a diode is included in one of the arms of the model
shown in fig. 4.4. Diode, by nature, allows only unidirectional flow of charge, which by
analogy, means heat and hence the downward flow of heat is obstructed. The other arm does
not contain any diode, thus allowing the bidirectional flow of heat. In this way, the summed
effect of viscosity as well as fluid convection is pronounced in the model. A single oil element
modeled in this way is shown in fig. 4.5.
All the oil blocks are modeled in the same way. This totally would present the summed effect
of viscosity as well as convection.
4.3.6 Modeling the Ambient
The heat transfer from the surface of the tank to ambient is both by convection (air currents)
and radiation (tank radiates heat to ambient). In transformers, radiation does not normallyoccur by itself and in almost every case, it is accompanied by convection. The surface of the
tank radiates heat to the ambient and acts like a heat sink dissipating the heat to the open
ambient. With the completion of modeling of all the blocks and the total thermal model being
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assembled, the ambient has got to be modeled. The heat dissipation from the tank surface to
the ambient, i.e., the heat sink behavior of the tank surface is modeled by resistors connecting
the elements on the outer surface of the tank elements to ground (ambient).
The resistance offered by heat sink is given in terms of ohm-m2. So, for each element on the
outer surface of the tank to which the resistor modeling the heat dissipation from the tank is
connected to, if ρ ohm-m2 is the heat sink resistance and A is the area of surface through
which it dissipates heat to the ambient in m2, ( A= lz , where l and z are the dimensions of the
element to which the resistance is connected to; l can be either x or y depending on the
location of the element), then the resistance offered by that element of heat sink is given by
equation 4.15.
...(4.15) R ohms ohms A lz
ρ ρ = =
While calculating the ambient resistances to be connected to each element of the tank, if the
tank element is situated on the vertical section of the tank, the dimension l would be the Y-
dimension, where as if it is on the horizontal portion of the tank geometry, then l would be the
X-dimension of the element. Accordingly, since, the corner blocks have two dissipating
surfaces, there will be resistances on both X and Y direction. Thus, the values of ambientresistors can be determined.
4.4 Obtaining the Thermal Profile
The entire transformer geometry thus modeled as an electrical mesh circuit consisting of
current sources and resistors (and capacitors) has to be solved for potentials at each node,
which, by analogy are the temperature rises of the corresponding points on the transformer
geometry above the ambient. Summing the ambient temperature value to each of these valuesof temperature rise, the value of temperatures at each corresponding location on the
transformer can be obtained.
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Since the transformer thermal model would be too big to be solved by hand, suitable software
can be made use of to solve the electrical circuit to yield potentials at each node. MULTISIM
is the software package used in the current work.
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Chapter 5
5
IMPLEMENTATION ON DIFFERENT TRANSFORMER
DESIGNS
The modeling, explained theoretically in chapter 4 is discussed with its practical
implementation on four transformer designs, which would explain the model better. Every
step that has been discussed in the chapter 4 is elaborated in the section 5.1 (the
implementation of the model on the first transformer design) and the implementation resultson three more designs are presented in sections 5.2, 5.3 and 5.4.
5.1 15 KVA Shell type Transformer – Model and Results
The considered transformer is 15 KVA, 11KV/250V, single phase shell type (3-limb two
winding) transformer with the winding wound on the central limb. The current density in the
LV and HV winding is 2A/mm2 and 0.95A/mm2 respectively. The dimensions of this
transformer (mentioned as transformer 1 now onward in this thesis work) are given in table 2.
Sl. No. Description Dimension
1 Winding Stack Height (LV) 230 mm
2 Winding Stack Height (HV) 230 mm
3 Core Limb width (Diameter of core limb) 90 mm
4 Width of the top yoke and bottom base of the core 90 mm
5Space between the limbs (measured between the centre of a
limb and centre of adjacent limb)190 mm
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FIG 5.1: TRANSFORMER 1: GEOMETRY
6 Height of each limb (without the top and bottom yokes) 240 mm
7 Tank wall thickness 5 mm
8 Tank length (Outer) 540 mm
9 Tank height (Outer) 580 mm
10 Oil duct width between core central limb and LV winding 10 mm
11 Oil duct width between LV and HV Coils 10 mm
12 Thickness of LV winding 15 mm
13 Thickness of HV winding 30 mm
Table 2. Dimensions of the considered transformer 1
The dimensions of the transformer are represented pictorially in figure 5.1.
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FIG 5.2: TRANSFORMER 1: ELEMENTAL DIVISION
The considered geometry here represents the two dimensional cross section of the transformer
in the XY plane. As the Z-direction depth would be needed for calculations and taking the Z
direction depth as unity, which means 1 meter, is not a likely case, a better consideration of the
Z-direction depth of the considered plane is maintained that is same as the core thickness. So,
the plane that is being modeled through the thermal modeling process is that of core thickness
transformer cross section in the XY plane. The transformer geometry is divided into finite
number of elements and this virtual elemental division with the dimensions is as shown in
figure 5.2.
On performing the finite element analysis on the transformer using the FEA Software toolfollowing the same elemental division shown in fig. 5.2, the flux density distribution across the
transformer geometry is obtained. Figure 5.3 shows the transformer modeled in the FEA tool
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NISA and figure 5.4 shows the flux density distribution obtained after performing Finite
Element Analysis on the considered transformer geometry.
FIG. 5.3: TRANSFORMER 1: FEA Implementation in NISA
The material properties used in the analysis in the implementation are as given in table 3.
Sl. No. MATERIAL MUXX(1/µoµr ) SIXX(1/ρ)
1 Windings (Copper) 795800 58000000
2 Transformer oil 795800 0
3 Core (CRGO steel) 400 4000000
4 Tank (Structural steel) 800 4000000
Table 3. Material Properties: FEA Implementation of Transformer 1
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FIG. 5.4: TRANSFORMER 1: Flux density distribution
From the flux density distribution and the values of flux density grossed over each element, as
generated by the tool, the values of core losses are calculated in each element division of the
core as explained in the equations 4.8 and 4.9. The values of K h and K e of the core are 0.0062
and 3.48 respectively. The value of form factor K f is taken as 1.1 while the thickness of
lamination t was 0.27mm. The loss in the each element division of the winding is calculated
from the ohmic loss calculations given in equation 4.7. The electrical conductivity of copper is
58x106 mho/m. The loss distribution across the considered transformer geometry is as shown
in figure 5.5.
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FIG. 5.5: TRANSFORMER 1: Loss Distribution
The values given are the losses in the corresponding blocks in rate of heat transfer W/sec.
These values of losses are used for current sources in the model. In the calculation of the
parameters of the thermal model the values of thermal conductivity as well as specific heat
capacity of the different materials used in the transformer, as given in table 4, are used.
Sl.No.
Material DescriptionThermal conductivity
(W/mK)Specific Heat
Capacity (J/kg oC)
1 CRGO steel Core 26 450
2 Copper Windings 400 386
3 Mineral Oil Transformer Insulating oil 0.72 2060
4 Structural Steel Tank 45 400
Table 4. Material Thermal Properties: Thermal Modeling of Transformer 1
Using the values of thermal conductivity as well as specific heat capacity, the values of
resistances as well as capacitances to be used for modeling of each element are calculated. The
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value of z wherever required is used as 0.09m and x and y would be the geometrical
dimensions of the element in the XY plane. The modeling discussed in the current work aims
at steady state temperature profile for a uniform loading on the transformer and so the
capacitors are eliminated from the thermal model. The numbering of elements for reference is
as shown in figure 5.6 and the corresponding calculation of resistances is shown in Table 5.
FIG. 5.6: TRANSFORMER 1: Numbering of Elements
Sl. no. MATERIAL ELEMENTS R H(Ω) R V (Ω)Current
Source (A)
1 LV Winding 1, 2, 3, 4 0.004 0.213 6.6
2 HV Winding 5, 6, 7, 8 0.0073 0.107 3
3 83, 95, 108, 120 0.43 0.43 0.601
4 84, 94, 109, 119 0.167 1.1 0.434
5 85, 93, 110, 118 0.143 1.29 0.376
6
Core outer limbs
86, 92, 111, 117 0.048 3.87 0.1255
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7 87, 91, 112, 116 0.072 2.58 0.187
8 88, 90, 113, 115 0.048 3.87 0.1258
9 89, 114 0.43 0.43 1.141
10 96, 98, 105, 107 0.774 0.24 0.61
11
Core Outer Limbs
99, 101, 102, 104 0.337 0.55 1.39
12 97, 106 0.774 0.24 2.172
13Core inner limbs
100, 103 0.337 0.55 5.07
14Oil between core
and LV21, 22, 27, 28 1.62 212.4
15Oil between LV
and HV20, 23, 26, 29 1.62 212.4
16 9, 18, 31, 40 12.96 26.4
17 10, 17, 32, 39 11.124 30.84
18 11, 16, 33, 38 3.7 92.4
19 12, 15, 34, 37 5.55 61.68
20 13, 14, 35, 36 3.7 92.4
21
Inner Oil sections
19, 24, 25, 30 1.4 15.18
22 41, 59, 62, 80 18.51 18.51
23 42, 58, 63, 79 10.2 33.312
24 43, 57, 64, 78 18.51 18.51
25 44, 56, 65, 77 33.33 10.3
26
Outer oil section
45, 55, 66, 76 12.96 26.4
0
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27 46, 54, 67, 75 11.124 30.84
28 47, 53, 68, 74 3.7 92.4
29 48, 52, 69, 73 5.55 61.68
30 49, 51, 70, 72 3.7 92.4
31 50, 71 33.24 10.2
32
Outer Oil Section
60, 61, 81, 82 8.1 33.33
33 121, 143, 146, 168 0.043 4.3
34 122, 142, 147, 167 0.024 7.74
35 123, 141, 148, 166 0.215 0.86
36 124, 140, 149, 165 0.43 0.43
37 125, 139, 150, 164 0.86 0.215
38
Tank
126, 138, 151, 163 7.74 0.024
39 127, 137, 152, 162 3.01 0.06
40 128, 136, 153, 161 2.58 0.07241 129, 135, 154, 160 0.86 0.215
42 130, 134, 155, 159 1.29 0.143
43 131, 133, 156, 158 0.86 0.215
44 132, 157 3.87 0.048
45
Tank
144, 145, 169, 170 0.02 9.9
0
Table 5. Calculation of Thermal Model Parameters for Transformer 1
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Now, the tank to ambient heat dissipation and the heat sink behavior of tank surface is to be
modeled. The heat sink resistivity of the transformer tank is taken as 0.05 ohm /m2. And the
correspondingly calculated value of resistance connected to each element is as given in table 6.
Sl. No. Tank ElementTank element to ambient resistance
value (Ω)
1 121, 143, 146, 168 11.4
2 122, 142, 147, 167 6.335
3 123, 141, 148, 166 57
4 124, 140, 149, 165 114
5 125, 139, 150, 164 57
6 126, 138, 151, 163 6.335
7 127, 137, 152, 162 21.285
8 128, 136, 153, 161 19
9 129, 135, 154, 160 57
10 130, 134, 155, 159 38
11 131, 133, 156, 158 57
12 132, 157 12.66
13 144, 145, 169, 170 4.955
Table 6. Calculation of Tank to Ambient Resistances for Transformer 1
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Regarding the modeling of radiator resistances, the radiator tubes are made of pressed steel
and the cross sectional thickness is low so as to allow quicker and efficient transfer of heat to
the external ambient. The thermal conductivity of radiator tube material is taken as
0.12W/mK. The diameter of the radiator tube, which is the length of heat transfer path, is
0.01m. The surface area of the tank to which the radiators are connected is equal to the height
of the tank l z multiplied by l y, which is 0.54 x 0.09 m2. As explained in the section 4.3.4, the
value of radiator resistance would be as given in equation 5.1.
1 0.011.7 ...(5.1)
0.12 0.54 0.09rad R = × = Ω
×
This resistance has to be connected both to the top oil as well as bottom oil and hence the
value of resistance connected at each point (top oil and bottom oil) will be 3.4 Ω, as the total
resistance has to be split into two.
The different modeling constraints discussed in section 4.3.2 are taken care of and the
convection in the oil is also modeled as explained in section 4.3.5 and the final thermal model
of the transformer is constructed as shown in figure 5.7.
This electrical equivalent mesh circuit has to be solved for potentials at each node, which are
the temperature rise of the corresponding point on the transformer geometry above the
ambient. MULTISIM has been used for the purpose of solving the network. An introduction
to the software is provided in the appendix. The problem geometry is modeled in the
schematic GUI of the software and the simulations are run, which gives the value of potential
at desired node or the potential at every node as desired in our study. The values of
temperature rise thus noted and superposed on the geometry of the transformer for enhancing
the understanding is given in figure 5.8, which is the Thermal Profile across the considered
geometry of the transformer
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FIG. 5.7: TRANSFORMER 1: Thermal Model
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FIG. 5.8: TRANSFORMER 1: Thermal Profile
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The critical points have been tabulated and compared against the practical test values as given
in table 7.
Sl. No.Parameter Test data
Value from thermalprofile
1 Windingtemperature
550C 42.660C
2 Top OilTemperature
400C 31.20C
Table 7. Thermal Model Implementation - Comparison with test values for transformer 1
5.2 25 KVA Core type Transformer – Model and Results
The considered transformer is 25 KVA, 33KV/250V rating, single phase, core type (2-limb
two winding) transformer with the winding wound on the both the limbs one over the other.
The dimensions of this transformer (mentioned as transformer 2 now onward in this thesis
work) are given in table 8.
Sl. No. Description Dimension
1 Winding Stack Height (LV) 250 mm
2 Winding Stack Height (HV) 210 mm
3 Core Limb width (Diameter of core limb) 100 mm
4 Width of the top yoke and bottom base of the core 100 mm
5Space between the limbs (measured between the centre of a
limb and centre of adjacent limb)260 mm
6 Height of each limb (without the top and bottom yokes) 300 mm
7 Tank wall thickness 5 mm
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FIG 5.9: TRANSFORMER 2: GEOMETRY
8 Tank length (Outer) 612 mm
9 Tank height (Outer) 610 mm
10 Oil duct width between core central limb and LV winding 4.5 mm
11 Oil duct width between LV and HV Coils 17 mm
12 Thickness of LV winding 13 mm
13 Thickness of HV winding 36.5 mm
Table 8. Dimensions of the considered Transformer 2
The dimensions of the transformer are represented pictorially in figure 5.9.
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FIG 5.10: TRANSFORMER 2: ELEMENTAL DIVISION
The transformer geometry is divided into finite number of elements and this virtual elemental
division with the dimensions is as shown in figure 5.10 (elements are shown as same size in
figure only for the clarity purpose though in the simulation they are considered according to
the actual dimensions).
On performing the finite element analysis on the transformer using the FEA Software tool
following the same elemental division shown in fig. 5.10, the flux density distribution across
the transformer geometry is obtained. It is not necessary to limit the elements to minimum
number and if required, more number of divisions can be made. Figure 5.11 shows the
transformer modeled in the FEA tool NISA and figure 5.12 shows the flux density distribution
obtained after performing Finite Element Analysis on the considered transformer geometry.
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FIG. 5.11: TRANSFORMER 2: FEA Implementation in NISA
The material properties used in the analysis in the implementation are as given in table 9.
Sl. No. MATERIAL MUXX(1/µoµr ) SIXX(1/ρ)
1 Windings (Copper) 795800 58000000
2 Transformer oil 795800 0
3 Core (CRGO steel) 400 4000000
4 Tank (Structural steel) 800 4000000
Table 9. Material Properties: FEA Implementation of Transformer 2
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FIG. 5.12: TRANSFORMER 2: Flux density distribution
From the flux density distribution and the values of flux density grossed over each element, as
generated by the tool, the values of core losses are calculated in each element division of the
core as explained in the equations 4.8 and 4.9. The values of coefficients K h and K e of the core
are 0.005 and 2.523 respectively. The value of form factor K f is 1.1 while the thickness of
lamination t was 0.27mm. The loss in the each element division of the winding is calculatedfrom the ohmic loss calculations given in equation 4.7. The electrical conductivity of copper is
58x106 mho/m. The values of thermal conductivity as well as specific heat capacity of the
different materials are given in table 10.
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Sl.No.
Material DescriptionThermal conductivity
(W/mK)Specific Heat
Capacity (J/kg oC)
1 CRGO steel Core 26 450
2 Copper Windings 400 386
3 Mineral Oil Transformer Insulating oil 0.72 2060
4 Structural Steel Tank 45 400
Table 10. Material Thermal Properties: Thermal Modeling of Transformer 2
Using the values of thermal conductivity as well as specific heat capacity, the values of
resistances as well as capacitances to be used for modeling of each element are calculated. The
value of z wherever required is used as 0.1m. The tank to ambient heat dissipation and the heat
sink behavior of tank surface is modeled using the heat sink resistivity of the transformer tank
as 0.05 ohm /m2. The radiators are modeled and the value of each resistance to be connected
both to the top oil as well as bottom oil is calculated as 2.732 Ω. The different modeling
constraints discussed in section 4.3.2 are taken care of and the convection in the oil is also
modeled as explained in section 4.3.5 and the final thermal model of the transformer is
constructed as shown in figure 5.13 and the obtained Thermal Profile across the consideredgeometry of the transformer is given in figure 5.14.
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FIG. 5.13: TRANSFORMER 2: Thermal Model
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FIG. 5.14: TRANSFORMER 2: Thermal Profile
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The critical points have been tabulated and compared against the practical test values as given
in table 11.
Sl. No.Parameter Test data
Value from thermalprofile
1 Windingtemperature
600C 69.940C
2 Top OilTemperature
550C 47.150C
Table 11. Thermal Model Implementation - Comparison with test values for Transformer 2
5.3 16 KVA Shell type Transformer – Model and Results
The considered transformer is 16 KVA, 11KV/433V rating, single phase, shell type (3-limb
two winding) transformer with the winding wound on the both the limbs one over the other.
The current density in the LV winding is 3.071 A/mm2 and the current density in the HV
winding is 2.795 A/mm2.The dimensions of this transformer (mentioned as transformer 3 now
onward in this thesis work) are given in table 12.
Sl. No. Description Dimension
1 Winding Stack Height (LV) 163 mm
2 Winding Stack Height (HV) 135 mm
3 Core Limb width (Diameter of core limb) 84 mm
4 Width of the top yoke and bottom base of the core 84 mm
5Space between the limbs (measured between the centre of a
limb and centre of adjacent limb)365 mm
6 Height of each limb (without the top and bottom yokes) 175 mm
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FIG 5.15: TRANSFORMER 3: GEOMETRY
7 Tank wall thickness 5 mm
8 Tank length (Outer) 600 mm
9 Tank height (Outer) 530 mm
10 Oil duct width between core central limb and LV winding 3.5 mm
11 Oil duct width between LV and HV Coils 11 mm
12 Thickness of LV winding 10.5 mm
13 Thickness of HV winding 19 mm
Table 12. Dimensions of the considered Transformer 3
The dimensions of the transformer are represented pictorially in figure 5.15.
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FIG 5.16: TRANSFORMER 3: ELEMENTAL DIVISION
The transformer geometry is divided into finite number of elements and this virtual elemental
division with the dimensions is as shown in figure 5.16 (elements are shown as same size in
figure only for the clarity purpose though in the simulation they are considered according to
the actual dimensions).
On performing the finite element analysis on the transformer using the FEA Software tool
following the same elemental division shown in fig. 5.16, the flux density distribution across
the transformer geometry is obtained. It is not necessary to limit the elements to minimum
number and if required, more number of divisions can be made. Figure 5.17 shows the
transformer modeled in the FEA tool NISA and figure 5.18 shows the flux density distributionobtained after performing Finite Element Analysis on the considered transformer geometry.
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FIG. 5.17: TRANSFORMER 3: FEA Implementation in NISA
The material properties used in the analysis in the implementation are as given in table 13.
Sl. No. MATERIAL MUXX(1/µoµr ) SIXX(1/ρ)
1 Windings (Copper) 795800 58000000
2 Transformer oil 795800 0
3 Core (CRGO steel) 400 4000000
4 Tank (Structural steel) 800 4000000
Table 13. Material Properties: FEA Implementation of Transformer 3
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FIG. 5.18: TRANSFORMER 3: Flux density distribution
From the flux density distribution and the values of flux density grossed over each element, as
generated by the tool, the values of core losses are calculated in each element division of the
core as explained in the equations 4.8 and 4.9. The values of coefficients K h and K e of the core
are 0.0052 and 2.82 respectively. The value of form factor K f is 1.1 while the thickness of
lamination t was 0.27mm. The loss in the each element division of the winding is calculated
from the ohmic loss calculations given in equation 4.7. The electrical conductivity of copper is
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58x106 mho/m. The values of thermal conductivity as well as specific heat capacity of the
different materials are given in table 14.
Sl.No.
Material DescriptionThermal conductivity
(W/mK)Specific Heat
Capacity (J/kg oC)
1 CRGO steel Core 26 450
2 Copper Windings 400 386
3 Mineral Oil Transformer Insulating oil 0.72 2060
4 Structural Steel Tank 45 400
Table 14. Material Thermal Properties: Thermal Modeling of Transformer 3
Using the values of thermal conductivity as well as specific heat capacity, the values of
resistances as well as capacitances to be used for modeling of each element are calculated. The
value of z wherever required is used as 0.084m. The tank to ambient heat dissipation and the
heat sink behavior of tank surface is modeled using the heat sink resistivity of the transformer
tank as 0.05 ohm /m2. The radiators are modeled and the value of each resistance to be
connected both to the top oil as well as bottom oil is calculated as 3.8 Ω. The different
modeling constraints discussed in section 4.3.2 are taken care of and the convection in the oil
is also modeled as explained in section 4.3.5 and the final thermal model of the transformer is
constructed as shown in figure 5.19 and the obtained Thermal Profile across the considered
geometry of the transformer is given in figure 5.20.
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FIG. 5.19: TRANSFORMER 3: Thermal Model
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FIG. 5.20: TRANSFORMER 3: Thermal Profile
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The critical points have been tabulated and compared against the practical test values as given
in table 15.
Sl. No. Parameter Test data Value from thermal profile
1 Winding temperature 550C 42.380C
2 Top Oil Temperature 500C 36.760C
Table 15. Thermal Model Implementation - Comparison with test values for Transformer 3
5.4 45 KVA Three Phase Transformer – Model and Results
The considered transformer is 45 KVA, 11 KV/250 V rating, three phase, core type (3-limb
three winding) transformer with each phase of the winding wound on each of the limbs one
over the other. The current density in the LV winding is 2 A/mm2 and the current density in
the HV winding is 0.95 A/mm2.The dimensions of this transformer (mentioned as
transformer 4 now onward in this thesis work) are given in table 16.
Sl. No. Description Dimension
1 Winding Stack Height (LV) 230 mm
2 Winding Stack Height (HV) 230 mm
3 Core Limb width (Diameter of core limb) 90 mm
4 Width of the top yoke and bottom base of the core 90 mm
5Space between the limbs (measured between the centre of a
limb and centre of adjacent limb)255 mm
6 Height of each limb (without the top and bottom yokes) 240 mm
7 Tank wall thickness 5 mm
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FIG 5.21: TRANSFORMER 4: GEOMETRY
8 Tank length (Outer) 800 mm
9 Tank height (Outer) 580 mm
10 Oil duct width between core central limb and LV winding 10 mm
11 Oil duct width between LV and HV Coils 10 mm
12 Thickness of LV winding 15 mm
13 Thickness of HV winding 30 mm
Table 16. Dimensions of the considered Transformer 4
The dimensions of the transformer are represented pictorially in figure 5.21.
The transformer geometry is divided into finite number of elements and this virtual elemental
division with the dimensions is as shown in figure 5.22 (elements are shown as same size in
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FIG 5.22: TRANSFORMER 4: ELEMENTAL DIVISION
figure only for the clarity purpose though in the simulation they are considered according to
the actual dimensions).
On performing the finite element analysis on the transformer using the FEA Software tool
following the same elemental division shown in fig. 5.22, the flux density distribution across
the transformer geometry is obtained. It is not necessary to limit the elements to minimum
number and if required, more number of divisions can be made. Figure 5.23 shows the
transformer modeled in the FEA tool NISA and figure 5.24 shows the flux density distribution
obtained after performing Finite Element Analysis on the considered transformer geometry.
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FIG. 5.23: TRANSFORMER 4: FEA Implementation in NISA
The material properties used in the analysis in the implementation are as given in table 17.
Sl. No. MATERIAL MUXX(1/µoµr ) SIXX(1/ρ)
1 Windings (Copper) 795800 58000000
2 Transformer oil 795800 0
3 Core (CRGO steel) 400 4000000
4 Tank (Structural steel) 800 4000000
Table 17. Material Properties: FEA Implementation of Transformer 4
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FIG. 5.24: TRANSFORMER 4: Flux density distribution
From the flux density distribution and the values of flux density grossed over each element, as
generated by the tool, the values of core losses are calculated in each element division of the
core as explained in the equations 4.8 and 4.9. The values of coefficients K h and K e of the core
are 0.0062 and 3.48 respectively. The value of form factor K f is 1.1 while the thickness of
lamination t was 0.27mm. The loss in the each element division of the winding is calculated
from the ohmic loss calculations given in equation 4.7. The electrical conductivity of copper is
58x10
6
mho/m. The values of thermal conductivity as well as specific heat capacity of thedifferent materials are given in table 18.
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Sl.No.
Material DescriptionThermal conductivity
(W/mK)Specific Heat
Capacity (J/kg oC)
1 CRGO steel Core 26 450
2 Copper Windings 400 386
3 Mineral Oil Transformer Insulating oil 0.72 2060
4 Structural Steel Tank 45 400
Table 18. Material Thermal Properties: Thermal Modeling for Transformer 4
Using the values of thermal conductivity as well as specific heat capacity, the values of
resistances as well as capacitances to be used for modeling of each element are calculated. The
value of z wherever required is used as 0.09m. The tank to ambient heat dissipation and the
heat sink behavior of tank surface is modeled using the heat sink resistivity of the transformer
tank as 0.05 ohm /m2. The radiators are modeled and the value of each resistance to be
connected both to the top oil as well as bottom oil is calculated as 3.464 Ω. The different
modeling constraints discussed in section 4.3.2 are taken care of and the convection in the oil
is also modeled as explained in section 4.3.5 and the final thermal model of the transformer is
constructed as shown in figure 5.25 and the obtained Thermal Profile across the consideredgeometry of the transformer is given in figure 5.26.
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FIG. 5.25: TRANSFORMER 4: Thermal Model
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FIG. 5.26: TRANSFORMER 4: Thermal Profile
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The critical points have been tabulated as given in table 19.
Sl. No. Parameter Value from thermal profile
1 Winding temperature 42.380C
2 Top Oil Temperature 36.760C
Table 19. Thermal Model Implementation - Comparison with test values for Transformer 4
5.5 Discussions
An observation of the obtained results shows that the obtained results are near to themeasured values, which shows that the model is definitely reliable. Though there is some
deviation of the obtained results with the actual measurements, it is owed to the some
limitations of modeling and the simplifications assumed while modeling. The credibility of the
model is proved in its application to different types of transformers and yet yielding reliable
results.
An observation of winding temperatures shows that the values obtained from the thermal
model for the shell type of transformers is less than the values obtained from directmeasurements. In the case of core type transformers, the temperatures obtained from the
thermal model are greater than the direct measurement values. This difference in trend
perhaps comes because of the two dimensional modeling involved. When the model is
assumed to be in the two dimensional plane, the third dimensional width does not get
modeled, the core type design allows lesser oil to get included as the design in 2-d plane
occupies a lot of space in the plane. However, in practical implementation, the volume of oil is
maintained the same irrespective of the design for a given rating of transformer. Owing to the
limitations of the two dimensional modeling over the realistic three dimensional modeling, the
model show difference in trends in temperature results for the core type transformers and shell
type transformers.
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Chapter 6
6
CONCLUSIONS
This chapter summarizes the proposed modeling and gives the conclusions. The future
potential of the work is also discussed.
6.1 Conclusions
Transformer thermal modeling is an important aspect to be considered while aiming at the
optimization of its usage, because with all the advances in the design techniques of the
transformer, it is the thermal limitations that hinder the exploitation of the transformer’s
loadability. Particularly in a country like India, where the cost of power system infrastructure
and operation is increasing which results in the undesirable hike in the cost of electrical power,
this exploitation of available loading capability is a must. Because of the issues with
unavailability of various measurement data and the cumbersomeness of the processes, the
standard IEEE and IEC models and other basic level thermal models could not present a verysatisfactory solution to this problem and this has caused the suboptimal and uneconomical
usage of transformer loadability particularly in the case of dynamically changing loads and this
in turn results in the wastage of transformer loadability capacity.
The current work presents a thermal model, based on loss distribution, for the transformer in
the form of an electrical mesh circuit which simulates the thermal behavior of the transformer.
The thermal model discussed uses the concepts of Finite Element Analysis technique and
thermal-electrical analogy and designs the model using the distribution of losses, instead of
lumping them. The potential difference at various nodes in the circuit, when the thermal
model is solved is the temperature values measured with respect to ambient. So the output
temperatures are actually temperature rise above ambient. Therefore, depending on maximum
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limit on the temperature, one can understand that the loading on the temperature on hotter
days cannot be as high as on cooler days. The parameters of the thermal model are based on
the dimensions of the transformer and properties of the materials. So, for a transformer with
different dimensions and properties, the parameter values used in the thermal model would be
different. As our interest was more towards the steady state thermal behavior, our designed
model gives steady state response. If required, we can modify the circuit and use that model to
simulate the transient behavior too. The work has been presented in six different chapters and
this chapter summarizes some of the observations and conclusions from each of the previous
chapters.
Chapter 1 provides the basic introduction to the problem by discussing the contribution of
thesis to the problem of thermal modeling of transformers. An overview of the transformer is
provided wherein the cause and effects of the heat accumulation and temperature rise inside
the transformer is explained.
Chapter 2 presents the thermal concepts required to understand the heating process, the heat
generation and dissipation inside the transformer. The heat generation and dissipation and the
different modes of heat dissipation is presented in general for a solid body and then explained
with respect to the transformer. The various axioms that guide the heating and cooling process
are explained. Then, the process of heat accumulation in the transformer is explained with the
reasons for its happening. The consequences of heat accumulation are given with emphasis
laid on the insulation ageing and its undesirable effects.
Chapter 3 presents the literature survey of the transformer thermal modeling techniques. The
techniques of measuring transformer interior temperatures are classified into the empirical
formulae which include the IEEE formulations to calculate the winding hotspot temperatures
for a given loading, the usage of fiber optic sensors which is a costly and the mathematical
thermal models. The mathematical thermal models are further classified as basic thermal
models, thermal models that use computer based simulations, thermal models using advanced
techniques like artificial intelligence techniques. The chapter provides the concluding remarks
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discussing the drawbacks of these techniques, emphasizing the need for wider thermal profile,
thus providing a platform based on which the proposed thermal model technique was
developed.
Chapter 4 forms the crux of this thesis, where the thermal modeling of transformer using the
loss distribution has been presented. The proposed model with necessary theory has been
explained along with elaboration on various modeling issues. The technique is dealt in two
parts, namely, finding out the loss distribution in the transformer and developing the thermal
model for the transformer with the calculated loss distribution. Summarizing the methodology
of the thermal model, the losses in the core and coils for a particular loading condition are
found out. The issue of non uniform core loss distribution is handled by using Finite Element
Analysis. The values of losses along with the values of resistances and capacitance are used to
construct the thermal model for that particular loading condition, which gives the
corresponding steady state thermal profile. The model, which has been explained in theory is
explained with examples in the subsequent chapters
Chapter 5 deals with the implementation of the proposed model on four different transformer
designs. Two single phase shell type transformer designs, a single phase core type transformer
design and a three phase core type transformer designs are taken and the model is
implemented on each of them. To ease the understanding of the proposed thermal model,
presented in chapter 4, the implementation of the model on a practical transformer design is
explained in detail. The modeling constraints, the values of different parameters and properties
and their calculations are presented and the simulation results which yield the thermal profile
for the considered transformer geometry are presented. Three more different transformer
designs are taken and the modeling as well as the thermal model with the obtained thermal
profile results has been presented. A comparison with the transformer heat run test results ispresented in all the cases for the purpose of assessment of the results. The results, which are
not far away from the test cases, do give the credibility to the model and its application to
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different transformers. This model is a general model for any type of transformer and hence
can be executed to any kind of transformer construction as well as rating.
6.2 Future Scope of the Work
The work presented is a model towards development of a comprehensive thermal model to
simulate the thermal behavior of the transformer which is based on the loss distribution
across the transformer geometry. Since the quest for perfection is a never ending one, the
work leaves further scope for future research potential. A few pointers for future research
are given below.
The model can be fine tuned by increasing the number of elements and reducing the
individual element size such that the model gives the temperature at almost every minute
location on the transformer cross section. Also, the modeling of mitred construction of core,
joints and the nuts and bolts which do affect the flux path can be included in the model. The
model can be extended to study the effect of transient loads on transformer thermal status.
The current work presents the two dimensional analysis of the transformer. However,
developing a three dimensional model would be better in improving the accuracy of the
results and better modeling of the transformer geometry, and that is left to the future scope.
In addition, the application of the technique of Finite Element Analysis to the problem of
transformers has a huge potential for future research. This technique can be used for stray
loss evaluation in the transformer, leakage flux calculations, optimal design and location of
magnetic shunts to reduce the stray losses, and the design of baffles to direct oil flow etc.
Particularly the problem of design of magnetic shunts to reduce the leakage fluxes and hence
stray loss due to leakage fluxes can be properly handled with the technique of Finite Element
Analysis.
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7
APPENDIX
The appendix gives the necessary details and introduction about the software used in the
current work.
7.1 Introduction to Finite Element Analysis – NISA
NISA provides an integrated and comprehensive suite of general and special purpose
programs for computer aided engineering (CAE). The NISA Suite of FEA Software covers a
wide spectrum of engineering applications, e.g., linear and nonlinear structural and heat
transfer analysis, structural and shape optimization, electromagnetic analysis, fatigue analysis,
fluid flow analysis and printed circuit board stress and heat transfer analysis. It consists of
three phases of programming, all of which are interfaced with the parent module. They are:
• Pre Processing: Deals with the creation of model, finite element modeling and defining the
analysis and boundary conditions
• Analysis: The actual analysis chosen in the Pre processing module is applied to the
problem and the results are showed in the output file
• Post processing: The viewing of results in display and further conversions of code from
one programming format to the other etc, come under this phase of programming.
The two major analysis types in NISA/EMAG are the electric field analysis (EFIELD) and the
magnetic field analysis (MFIELD). Magnetic field analysis is used in the project, since magnetic
fields are being dealt with. NISA is used to find the flux densities across various elements in
the designed transformer geometry, which is further used to calculate the losses in various
elements and thereby proceed to the thermal model. The analysis chosen is 2D magneto
dynamic analysis (MGDN) as it deals with ac sinusoidal excitation and the output desired
being the Magnetic flux density. MGDN analysis provides for Magnetic field calculations in
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magnetic and conducting materials due to sinusoidal (ac) current excitation. The outputs
provided would be Magnetic vector potential distribution, Magnetic Flux density distribution,
Magnetic field distribution, Eddy current density distribution, Total current density
distribution, Electric field distribution due to Eddy currents, Total Electric field distribution,
Power loss density distribution, Stored magnetic energy density for each element, Total stored
magnetic energy, Total power loss [37].
7.2 Introduction to MULTISIM
A number of schematic based and SPICE based tools are available for solving simulating
electrical networks. They solve the electrical networks using circuit theorems and analyses.
SPICE based tools require programming, while schematic based tools are more user friendly.
A few important tools which are SPICE based, schematic based, and tools integrating both,
are HSPICE, PSPCIE, Tanner tool (LTSPICE), MATLAB (SIMULINK), MULTISIM etc. Of
these MULTISIM is chosen, since it is more user-friendly. MULTISIM has the ability to
calculate the potentials at different nodes of an electrical circuit. Thus, by using MULTISIM
the potentials at every node of the thermal model are being calculated, which gives the
temperature at those points on the transformer geometry using the values of losses calculatedfrom the flux densities available through simulation in NISA. The measurements of interest
are recorded by connecting the oscilloscopes (or any desired measuring instrument available
from the library) and then running the simulation.
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PUBLICATIONS
[1]
Haritha V V S S, T R Rao, Ramamoorty M, Amit Jain, “Thermal Monitoring ofElectrical Utility Transformers”, Proceedings of 7th IEEE POWERCON 2010, 24-28
October, 2010, Hangzhou, China.
[2] Haritha V V S S, T R Rao, Ramamoorty M, Amit Jain, “Thermal Modeling of
Electrical Utility Transformers Using Finite Element Analysis and Thermal Electrical
Analogy”, Proceedings of 3rd IEEE ICPS 2009, 27-29 December, 2009, IIT
Kharagpur, India.
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[33] Haritha V V S S, T R Rao, Ramamoorty M, Amit Jain, “Thermal Modeling of
Electrical Transformers”, NPSC 2010, 15-17 December, Hyderabad, India.
[34] Haritha V V S S, T R Rao, Ramamoorty M, Amit Jain, “Thermal Monitoring of
Electrical Utility Transformers”, 7th IEEE POWERCON 2010, 24-28 October,
Hangzhou, China.
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[35] Haritha V V S S, T R Rao, Ramamoorty M, Amit Jain, “Thermal Modeling of
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