Haritha Thesis

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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



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)



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.



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.



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.



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



ii

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



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



iv

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



v

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 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 14: Material Thermal Properties: Thermal Modeling of Transformer 3……………….72


Table 16: Dimensions of the considered Transformer 4………………………………….... 75


Table 18: Material Thermal Properties: Thermal Modeling of Transformer 4………………. 80




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



2

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



3

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.



4

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



5

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



6

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.



7

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]



8

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.



9

( ) ( ) 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.



10

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



11

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.



12

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



13

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



14

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



15

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



16

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.



17

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.



18

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



19

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



20

∆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



21

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.



22

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



23

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



24

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



25

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



26

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



27

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.



28

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.



29

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



30

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



31

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.



32

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



33

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



34

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



35

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



36

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.



37

• 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.



38

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.



39

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



40

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



41

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.



42

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.



43

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



44

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.



45

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.



46

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



47

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.



48

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



49

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



50

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.



51

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



52

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



53

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



54

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



55

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



56

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



57

FIG. 5.7: TRANSFORMER 1: Thermal Model



58

FIG. 5.8: TRANSFORMER 1: Thermal Profile



59

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.












60




10 Oil duct width between core central limb and LV winding 4.5 mm



13 Thickness of HV winding 36.5 mm

Table 8. Dimensions of the considered Transformer 2




61


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.



62











63




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.



64

Sl.No.


(W/mK)Specific Heat

Capacity (J/kg oC)








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.



65




66




67


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.











68





10 Oil duct width between core central limb and LV winding 3.5 mm


12 Thickness of LV winding 10.5 mm






69










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.



70











71










72

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.


(W/mK)Specific Heat

Capacity (J/kg oC)








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.



73




74




75


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


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.












76




10 Oil duct width between core central limb and LV winding 10 mm










77








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.



78











79








58x10

6

mho/m. The values of thermal conductivity as well as specific heat capacity of thedifferent materials are given in table 18.



80

Sl.No.


(W/mK)Specific Heat

Capacity (J/kg oC)





Table 18. Material Thermal Properties: Thermal Modeling for Transformer 4



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.



81




82




83

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


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.



84

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



85

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



86

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



87

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.



88

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



89

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.



90

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.



91

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92

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Electrical Utility Transformers Using Finite Element Analysis and Thermal Electrical

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Haritha Thesis

Documents

Transcript of Haritha Thesis