Load Controlled Noise of Power Transformers: 3D Modelling of Interior

and Exterior Sound Pressure Field

M U S T A F A K A V A S O G L U

Master of Science Thesis Stockholm, Sweden 2010

Load Controlled Noise of Power Transformers: 3D Modelling of Interior

and Exterior Sound Pressure Field

M U S T A F A K A V A S O G L U

Master’s Thesis in Numerical Analysis (30 ECTS credits) at the Scientific Computing International Master Program Royal Institute of Technology year 2010 Supervisor at CSC was Jesper Oppelstrup Examiner was Michael Hanke TRITA-CSC-E 2010:095 ISRN-KTH/CSC/E--10/095--SE ISSN-1653-5715 Royal Institute of Technology School of Computer Science and Communication KTH CSC SE-100 44 Stockholm, Sweden URL: www.csc.kth.se

Abstract

Noise emissions are of ever greater concern to trans-former manufacturers. Understanding the sources of thenoise and prediction as well as reduction of sound emis-sion is of increasing interest. In this paper we study load-controlled noise. Empirical formulas based on the trans-former’s rated power of transformers are used to predictload-controlled noise, but they are of little use for new de-signs for which measurements are not available. Accuratenumerical simulations are needed for prediction. In thiswork an efficient new 2D-3D computational model of theload-controlled acoustic noise of oil insulated power trans-formers is presented. Generation and propagation of trans-former noise is a complex interaction of the electromagneticfield, the mechanical displacement field, and the acousticpressure field. This simulation is based on a COMSOLMul-tiphysics Finite Element Model and allows accurate fullycoupled calculations of the fields A 2D axisymmetric mag-netic simulation for the windings on one transformer coreleg is done to model the stray fields and Lorenz forces; theseexcite an axisymmetric, detailed mechanical winding modelto produce the displacement field. The axisymmetric dis-placement is mapped onto the transformer core, tank andexterior geometry with 120 degree phase shift between thethree windings to simulate a three phase power transformer.The model is applied to an existing transformer model forwhich ABB has supplied data and results of earlier com-puter models, and the sensitivity to numerous parametersis assessed. It is found that accurate mechanical modelingof the clamping of the windings to the core is a pressingneed to improve simulation accuracy.

Referat

Last-inducerat buller hostrefas-transformatorer: 3D-modellering av

inre och yttre ljudtryck.Buller blir alltmer viktigt för transformatortillverkare

när kunderna ställer strängare miljöcertifieringskrav. Först-åelse för bullerkällorna och prediktion liksom reduktion avutstrålad ljudeffekt är därför intressant. Här studerar vilastinducerat buller. Prediktion av last-inducerat buller görsidag med empiriska formler baserade på transformatornsmärkeffekt, men de är till föga hjälp vid ny-konstruktioner.Noggranna numeriska simuleringar behövs. I detta arbetepresenteras en effektiv 2D/3D matematisk modell av lastin-ducerat buller i en oljeisolerad tre-fas krafttransformator.Alstring och spridning av transformatorbuller är en kom-plex samverkan mellan elektromagnetiska fält, mekaniskaförskjutningar, och det akustiska tryck-fältet. Simulerin-gen baseras på en COMSOL Multiphysics finita-element-modell och ger noggranna beräkningar av fullt koppladeingående processerna. En axi-symmetrisk magnetisk mod-ell av lindningarna på ett transformatorben ger ströfältoch inducerade krafter; dessa exciterar en axisymmetriskdetaljerad mekanisk modell och ger förskjutningarna. Axi-symmetriska förskjutningar avbildas på 3D-modellen av kär-na, tank, olja och yttre, med 120 graders fasförskjutningmellan de tre lindningarna för att simulera en tre-fastrans-formator. Modellen appliceras på en befintlig transforma-tortyp för vilken ABB har resultat av mätningar och tidi-gare datormodeller, och känsligheten för de många parame-trarna beräknas. Man finner att noggrannare mekanisk mod-ellering av lindningarnas fästning mot kärnan är av vikt föratt förbättra simuleringsnogrannheten.

Contents

1 Introduction 11.1 Information about the Transformer . . . . . . . . . . . . . . . . . . . 2

2 Electromagnetic Model 52.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Short Review of Electromagnetic Fundamentals . . . . . . . . . . . . 62.3 Time Harmonic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Verification of the computer model . . . . . . . . . . . . . . . 82.3.2 Magnetic Stray Field . . . . . . . . . . . . . . . . . . . . . . . 102.3.3 Computed Magnetic Volume Forces . . . . . . . . . . . . . . 11

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Mechanical Modeling 133.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Theory Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Axisymmetric Stress Strain Relations . . . . . . . . . . . . . 143.3 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 Mechanical Model without Oil . . . . . . . . . . . . . . . . . 163.4 Mechanical Model with Oil coupling . . . . . . . . . . . . . . . . . . 183.5 Sensitivity Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5.1 Influence of Stiffness of Winding Supports . . . . . . . . . . . 203.5.2 Influence of Young’s Modulus of Insulation Paper . . . . . . . 213.5.3 Influence of Young’s Modulus of Radial Spacers . . . . . . . 213.5.4 Influence of Elastic Modulus of Pressboards . . . . . . . . . . 22

4 Acoustic Modeling 254.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Theory Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2.1 Shell elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2.2 Extrusion Coupling . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3 Quantitive Measure of Sound . . . . . . . . . . . . . . . . . . . . . . 274.3.1 Formulas for Predicting Load-Noise . . . . . . . . . . . . . . 29

4.4 2D Time Harmonic Axisymmetric Model . . . . . . . . . . . . . . . . 29

4.4.1 Observations about the 2D axisymmetric model . . . . . . . . 314.4.2 3D Acoustic Model without Oil . . . . . . . . . . . . . . . . . 33

4.5 3D Model for Oil Insulated Transformer . . . . . . . . . . . . . . . . 364.6 Investigation of material parameters on SPL . . . . . . . . . . . . . . 39

4.6.1 Influence of Tank Thickness . . . . . . . . . . . . . . . . . . . 404.6.2 Influence of Young’s Modulus of Spacers . . . . . . . . . . . . 414.6.3 Influence of Young’s Modulus of Paper . . . . . . . . . . . . . 424.6.4 Influence of Elastic Modulus of Pressboards . . . . . . . . . . 43

5 Future Work and Conclusion 45

Bibliography 47

List of Figures

1.1 Experiment type TPVR 16.000 kVA of ABB Secheron . . . . . . . . . . 3

2.1 Model Geometry and Mesh . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Surface graph for magnetic Flux density . . . . . . . . . . . . . . . . . . 92.3 Streamline plot for different tap changer positions . . . . . . . . . . . . 102.4 The blue line indicates the magnetic volume force for Low voltage wind-

ing and the red line represents the High voltage . . . . . . . . . . . . . . 12

3.1 Finite element discretization of mechanical parts of the winding . . . . . 153.2 Top view of an pressboard . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Displacement field without oil . . . . . . . . . . . . . . . . . . . . . . . . 173.4 Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5 Displacement of the mid point of the low voltage winding . . . . . . . . 193.6 Model with cooling ducts . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7 Coil cell structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.8 Effects of additional stiffness here in the x axis -4 means we are reducing

the stiffness 10−4 times. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Extrusion Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Different standards for weighting the SPL . . . . . . . . . . . . . . . . . 294.3 Initial Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 The red dashed lines means fixed boundary conditions for mechanical

part. The green line indicates the acceleration boundary condition forthe acoustic part and black line indicates the continuity boundary con-ditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.5 2D Deformed Plot in tap changer position 1 . . . . . . . . . . . . . . . . 334.6 Arrangement of the measurement points . . . . . . . . . . . . . . . . . . 344.7 SPL for the transformer without oil . . . . . . . . . . . . . . . . . . . . 354.8 Norm of the sound intensity values for air filled transformer . . . . . . . 374.9 Pressure Level inside the tank . . . . . . . . . . . . . . . . . . . . . . . . 384.10 Effects of the resonance in 102 Hz over our measurement points. . . . . 394.11 Model with cooling ducts. . . . . . . . . . . . . . . . . . . . . . . . . . . 404.12 SPL for the model with ducts . . . . . . . . . . . . . . . . . . . . . . . . 414.13 SPL for 12 measurement point with respect to transformer thickness . . 42

Chapter 1

Introduction

Transformer noise is a social problem rather than a technical problem. Audiblenoise, particularly continuously radiated discrete tones, is the type of noise that thecommunity may find unacceptable. With the growing consciousness on the ill effectsof noise pollution many users are specifying lower noise levels for transformers. Inorder to reduce noise level, it is very important to know and understand the sourcesof noise. This is the main reason why we are doing this project, prediction of thenoise and understanding the factors which contribute with the noise.

For transformers there are three main noise sources: [1]

• Vibration of the core, caused by magnetostrictive strain of core laminations.

• Noise generated by the operation of cooling systems like fans or oil pumps.

• Load Noise is mainly generated by the interaction of the load current in thewindings and the leakage flux produced by this current.

In this project we are mainly interested in Load Noise.As described above LoadNoise caused by winding vibrations excited by electromagnetic forces,Lorentz forces,in the windings are resulting from the interaction between the magnetic stray fieldof one current carrying winding and the total electric currents in the conductors ofthe other windings. Lorentz force is proportional to the square of the load current.This force causes winding vibrations having frequency twice the line frequency.

The acoustic pressure generated by vibration of the core and the windings istransmitted to tank surfaces through the oil medium. The tank responds to thesepressure waves depending on its natural frequencies and mode shapes. In thisproject the noise caused by the magnetic core and the equipments is not considered.

For rapidly obtaining concrete suggestions a modeling scheme for the numericalsimulation of oil insulated power transformers has been developed. This model isbased on Finite Element Method (FEM) and allows us doing efficient 3D calculation.We are using the data and measurements of experiment transformer type TPVR16.000 kVA of ABB Secheron.For FEM simulation Comsol Multiphysics software isused performing calculations and model developments which allow the precise 3D

1

calculation of the magnetic,mechanical and acoustic fields including their couplingsin a single computer run. This project consists of four parts. In the first part,an Electromagnetic model is developed to capture the Lorentz forces. The resultsare verified using TRACE software which is an in house ABB software. In thesecond part, a mechanical model for one winding is developed and coupled to theelectromagnetic model. The main aim of the mechanical model is to get the axialand the radial surface displacements for High Voltage(HV) and Low Voltage(LV)windings. In the third part axial and radial surface displacements are used asthe input values for mechanical excitation in a 3D acoustic FE model with 120degree phase shift between the three windings. The fourth and last part consists ofinspection of design parameters and their effect on the sound pressure levels.

1.1 Information about the TransformerThe measurements as well as results of the simulation are based on experiment typeTPVR 16.000 kVA of ABB Secheron with the following data.[2]

• Three phase three limb core-type transformer with tappings

• Rated power of 16.000 kVA

• Nominal voltage 52.400+ 11 × 600/11200 V

• Nominal current 159.3-179.7-206.2/825 A

• Coupling Yy0,frequency 50 Hz

• year of construction is 1978.

The transformer allows us to regulate voltage by means of the addition or re-moval of tapping turns. The measurements as well as our simulation results showthat tap changing has got significant effects. Therefore all the simulations are donefor three nominal positions:

• Tap changer position 1: The high voltage (HV)-winding and both tappingwindings are connected in series.

• Tap changer position 12: The HV winding and the coarse tapping-windingare connected in series.

• Tap changer position 23: Only HV winding is connected

Figure 1.1 illustrates the transformer studied in this project.

Figure 1.1. Experiment type TPVR 16.000 kVA of ABB Secheron

Chapter 2

Electromagnetic Model

Main aim of this part of the project is to develop and to verify 2D axisymmet-ric modeling scheme which is used for calculating the magnetic leakage field andthe resulting electromagnetic Lorentz forces. These forces will be used as couplingquantity between the mechanical field and electro-magnetic field. The equationsdescribing the magnetic system are derived from the Maxwell’s equations. In thischapter different types of analysis are done and the results are verified by TRACEsoftware. The schemes are implemented using Finite Element program Comsol Mul-tiphysics. After the verification of the model, different elements are investigated forbetter understanding of the phenomena. The following section is written to givesome background information about electromagnetic modeling like governing equa-tions as well as some modeling details. Further the simulated results are presentedand compared with the measurement results. In the last section of this chapter asummary of findings is presented.

2.1 Methodology

In this project the equations governing the magnetic field quantities are solved usingFinite Element Method. In the finite element methods, the region is subdivided intodiscrete elements, so called finite elements. Due to the almost rotational symmetryof the conductors, a two dimensional finite element method based on axisymmetryis implemented. For this purpose Azimuthal Induction Current mode from Comsolsoftware is used. This mode perfectly matches for our purpose. This predefinedmode is for axially symmetric structures with currents present only in the angulardirection. Magnetic field is formulated using the magnetic vector potential, ϕ com-ponent. First of all a time harmonic analysis is done which will be explained indetail in the following section.

5

2.2 Short Review of Electromagnetic FundamentalsThis section is just a short review about the Maxwell’s equation and the derivation ofMagnetic Vector Potential. Maxwell, based his theory on the work of Ampere,Gaussand Faraday. Maxwell’s achievement lies in the unification of the different equationsa set of partial differential equations. The electromagnetic quantities involved inMaxwell’s equations are:

Table 2.1. Notations of Electromagnetic quantities

Notation Unit and DescriptionE (V/m) electric field intensityD (As/m2) electric flux densityH (A/m) magnetic field intensityB (T) magnetic flux densityJ (A/m2) current densityP (As/m2) electric polarizationM (T) magnetizationqe (As/m3) charge density

Following material parameters are also needed for constitutive equations.

• Magnetic permeability ν(Vs/Am)

• Magnetic reluctivity υ=1/ν

• Electric permittivity ε (As/Vm)

• Electric conductivity σ (1/Ωm = S/m)

The four partial differential equations(PDEs),stated as Maxwell’s equations are:

∇×H = J + ∂D∂t

(2.1)

which states that an electric current generates a magnetic field.

∇×E = −∂B∂t

(2.2)

which states that time varying magnetic flux will generate an induced voltage in anopen conductive loop

∇ ·D = qe (2.3)

which shows clearly that the electric field is irrotational.

∇ ·B = 0 (2.4)

which states that the magnetic field is always solenoidal(closed field lines)In order to get the solution of the four partial differential equations one needs

the following constitutive equations.

J = γ(E + v ×B) (2.5)

D = εE = ε0E + P (2.6)

B = νH = ν0H + M (2.7)

The most important case for electromagnetic simulations can often referred toas the eddy current case[3]. In the quasi-static case we are going to ignore the dis-placement current density. Equation 2.4 indicates that our field is purely solenoidalso we can describe it as a curl of a vector.

B = ∇×A (2.8)

So the vector A became our magnetic vector potential. Magnetic vector potentialhas the following relation

∇×E = − ∂

∂t(∇×A) (2.9)

From this relation one can easily find that E = −∂A/∂t and then we can arrive at

∇× ν∇×A = J− γ ∂A∂t

+ σ(v×∇×A) (2.10)

2.3 Time Harmonic AnalysisOne can easily assume that all variations in time occur as sinusoidal signals. Thenour problem is considered as a time harmonic problem so we can formulate it as a sta-tionary problem with complex valued solutions. As mentioned before the softwareis solving the PDE using Finite Element Method. Figure 2.1 shows the analyzedgeometry with and without applied mesh. Some important aspects of the model.

• Magnetic vector potential formulation in the time harmonic case looks like:

(jωσ − ω2ε0)A +∇× (µ−10 ∇×A−M)− σv × (∇×A) = Je + jwP (2.11)

• Here we are not modeling the core, instead we are using electric insulationboundary conditions due to the fact that the core has a relatively large mag-netic permeability. This means that in the boundary we are assuming :

n×H = 0 (2.12)

• The nonlinear magnetization curve of iron is not taken into account since, theamplitude of the flux densities in the core is in linear region.

• Theoretically the magnetic flux density must be the same allover the windingdomain but if we put an electrical conductivity to the coil due to the fact thatwe are using an AC current we would experience a skin effect, which meanscurrent densities are concentrated close to the surface of conductor. Thereforein order to avoid the skin effect we assume 0 conductivity at the surface.

• Induced voltage is calculated by following formula

uind = −NcdΦdt

(2.13)

In the axisymmetric case we can calculate magnetic flux density by averagingover the whole area which means to compute the integral over the volume andnormalize it at the cross section Γc of the coil

B = 2πNc

Γc

∫rAµdr (2.14)

• Due to the influence of the tap changer position, the simulation defined abovehave been performed for all possible cases

Figure 2.1. Model Geometry and Mesh

2.3.1 Verification of the computer modelThe verification of the model described above, has been performed by comparing theresults of TRACE software as well as analytical formulas. In table 2.2 tap changerposition means we are just giving the current to the defined windings and gettingback the induced current from Low Voltage winding.

Table 2.2. Comparison of Simulation Results

Tap Changer Current in LV Current in specified windingsVolt Volt

Trace 23 -813.30 206.19Simulation 23 -813.305 206.19Trace 12 -823.72 179.72Simulation 12 -823.715 179.72Trace 1 -824.39 159.27Simulation 1 -824.38 159.27

Flux Density.jpg

Figure 2.2. Surface graph for magnetic Flux density

It can be seen from table that our results perfectly match with the resultsderived from the TRACE software. One of the important result is the currentdensities between the windings. These results will be used for calculation of theLorentz forces. The Figure 2.2 is a 2D surface plot of magnetic flux density. Itcan be seen the highest magnetic field is between the High Voltage(HV) and LowVoltage(LV) windings. The results of the magnetic field can be easily estimated bythe analytical formula prepared by W.Rogoswski.[4]

BL = 4π10−7IN√

2L

(2.15)

Figure 2.3. Streamline plot for different tap changer positions

and

L = b

1− a1+a2+σbπ

(2.16)

where I denotes the RMS current and N the number of turns in LV-winding. Theconstants a1,a2,σ,b represent the width of the LV,width of the HV, the distancebetween windings and height of the windings respectively. Using (2.15) one caneasily found 162 mT.

2.3.2 Magnetic Stray Field

In the first step the magnetic stray field is calculated inside the power transformerwindow. First of all the LV winding was short circuited and HV winding wasloaded with the predefined current. The Figure 2.3 shows the magnetic flux densityby stream lines for all 3 cases. The following important aspects are observed.

• In the area between the HV and LV windings the leakage flux is uniform inthe direction parallel to the axis of the coils. The r component is negligibleBL = 163mT

• Tap changer position has a significant effect on the magnetic field densityinside the HV winding which can be seen from Figure 2.3. Left streamlineplot illustrates the case when we apply the current to HV and CT and FTwindings. The middle plot shows the case for HV CT windings and the rightplot illustrates when we are just exciting the HV windings.

• Tap changer position does not affect the maximum values of the magneticfield of radial components near the upper and lower edges of windings.

2.3.3 Computed Magnetic Volume ForcesIn this step the magnetic volume forces are calculated. Then volume forces are reallyimportant for us because we will use these forces for coupling electromagnetic fieldwith mechanical field. In the case of 2D axisymmetric electromagnetic problems,the magnetic volume forces can be calculated by the following equations

B = ∇×A = ∂Az∂z

ey −∂Az∂ez

(2.17)

fv = J ×B = −BzJxey +ByJxez (2.18)

The equations above mean that the leakage field, which was calculated in the pre-vious section and the winding currents will act as excitation forces to the windings.The magnetic volume force is radially inwards for LV winding and outwards forthe other windings. Figure 2.4 shows the calculated magnetic volume forces alongthe outside radius of the LV winding at r = 329 mm and the outside of the HV-windings(r=431mm) for the case, where we apply a voltage to HV-winding,CoarseTapping(CV)-winding and Fine Tapping(FV)-winding. Some important notes are :

• Due to the decreasing magnetic flux density near the edges the radial volumeforces are decreasing.

• The tap changer position has a remarkable effect on norm of the volume forces.The case where we apply voltage on HV,CT, and FT windings we are gettingthe maximum value of 271KN/m3 and for the case when we apply voltage tothe HV and CV winding we get the max value of 210KN/m3

2.4 DiscussionThe presented simulations have been compared with TRACE software, measure-ment results as well as analytical formulas. Some of the important conclusionsare:

Force.jpgFigure 2.4. The blue line indicates the magnetic volume force for Low voltagewinding and the red line represents the High voltage

• The axial magnetic flux densities and the radial magnetic forces are dominant.

• Axial volume forces are mostly dominant at the edges of the windings.

• LV and HV windings move in opposite direction

• Tap changer position has a significant effect on all discussed results.

Chapter 3

Mechanical Modeling

3.1 IntroductionIn this chapter the mechanical parts of the winding will be modeled. Due to the ro-tational symmetry of the windings,a two dimensional axisymmetric model suffices inthis step. The main aim of the mechanical modeling is getting the correct displace-ment field for High Voltage(HV) and Low Voltage(LV) windings, these displacementfields will later be used in the acoustic model. After making an introduction aboutthe mechanical modeling and the electro-magnetic coupling, some results will bepresented which are important for the acoustic modeling.

3.2 Theory BackgroundIn the mechanical model we are solving basically the Navier’s equations for thedynamical behavior of mechanical systems which will read as:

fv + OTσ = ρa (3.1)

where ρ denotes the density of the medium and a is the acceleration of the bodyO is the differential operator which takes the form :

O =

∂∂x 0 0 0 ∂

∂z∂∂y

0 ∂∂y 0 ∂

∂z 0 ∂∂x

0 0 ∂∂z

∂∂y

∂∂x 0

T

(3.2)

The main coupling between the mechanical and the magnetic field is Lorentzforces, which is the magnetic volume force.The magnetic volume forces arise due tothe interaction of the magnetic field of the current carrying windings and the totalelectric currents in the conductive parts of the windings. The Lorentz force can befound by the following formula:

fv = J×B =(−γ ∂A

∂t− γ∇V + γν × (∇×A)

)× (∇×A) (3.3)

13

where J denotes the total electric current density ν is the velocity of the conductivemoving body,γ is the electrical conductivity, A is the magnetic vector potential.

3.2.1 Axisymmetric Stress Strain Relations

In a cylindrical coordinate system, the displacement would components read as:

• ur displacement in radial direction(r-direction)

• uz displacement in axial direction(z-direction)

• uθ displacement in circumferential direction(θ-direction)

As we are discussing the axisymmetric case the mechanical displacements donot depend on the θ coordinate. Thus the stress strain relation for isotropic casereads as:

urruzzuθθurz

=

λL + 2µL λL λL 0

λL λL + 2µL λL 0λL λL λL + 2µL 00 0 0 λL

(3.4)

In equation 3.4 λL and µL are Lame parameters which are defined by:

λL = υpEm(1 + υp)(1− 2υp)

(3.5)

µL = Em2 ∗ (1 + υp)

(3.6)

where Em is elastic modulus and υp is Poisson’s ratio.

3.3 Model DescriptionThe finite element discretization of the mechanical part of the winding is shown infigure 3.1. It is important to note that in this model nonlinear effect of the materialsare ignored. One should consider following aspects to make a precise simulation:

• In reality all the windings are made of a combination of copper and insulationpaper. In the presented model we use homogenization method in order toobtain the mechanical parameters of windings. These parameters should beaccurate enough to capture material properties of both paper and copper.Homogenization method is really important for reducing the computing load.

• All the windings are modeled as anisotropic material.

Figure 3.1. Finite element discretization of mechanical parts of the winding

• Winding clampings are modeled as an imaginary material which supports anadditional stiffness to the pressboards. In this simulation a stiffness K=85kN/mmhas been used.Using the following equation one can calculate the equivalentYoung’s modulus.

E = k(1− ν2)LA

= 85 · 103 × 20×(1− 0.332)

π ×((0.235 + 0.25)2 − 0.2352

) = 2.679 · 106(Pa) (3.7)

• All the pressboards have a spoke like shape. In reality it is necessary to makethe oil flow inside windings. Due to the reason that different pressboards havedifferent contact areas, they have different material properties.

Table 3.1. Pressboard material property

Types Young’s Modulus Poisson’s Ratio DensityPressboard (LV) 7.9 · 108 0.33 400Pressboard (HV) 5.847 · 108 0.33 400Pressboard (CT) 5.048 · 108 0.33 400Pressboard (FT) 4.871 · 108 0.33 400

• All the windings have different copper paper ratios. All the windings havedifferent densities considering their total mass and total volume.

Figure 3.2. Top view of an pressboard

Table 3.2. Density of different windings

Windings Mass Volume Density(kg) (m3) (kg/m3)

LV 722.33 0.140 5165HV 760 0.185 4103.7CT 91.67 1.46 · 10−2 6271.7FT 74.67 1.7 · 10−2 4394.3

3.3.1 Mechanical Model without OilFor understanding the effects of insulation oil in the transformer two models aredeveloped,one mechanical model with oil and a mechanical model without oil. Themodel without oil is a good way to understand the effects of the different materialparameters on the displacement field of the windings as well as the effects of the tapchanger position. After presenting the result of this magneto-mechanical model, inthe chapter for acoustic we will investigate the effects of the material parameterson the sound pressure level.

In Figure 3.3 the total displacement field is represented for tap changer position1. Our main interest are HV and LV windings. In the acoustic model only thedisplacements of the HV and LV windings will be used as an input for the 3Dmodel. Some important notes about the results are:

• It can be seen from the Figure 3.3, the biggest displacement is in HV winding.In Figure 2.4 we present that the magnetic volume forces acting on the HV andthe LV windings are nearly at same rate. However the radial displacementsare three times different. The reason for this big difference is because theratio of paper in the HV winding is higher than that of the LV winding.This ratio makes the HV winding much weaker. There is a big difference inYoung’s modulus values of paper and copper.Therefore we can conclude thatthe biggest reason of displacement is the insulation paper in the windings.

• The average value of the total displacement in the HV winding is 1.510−6 m

Figure 3.3. Displacement field without oil

where as in the LV winding the average becomes 4.6610−6m.

• Due to the direction of the Lorentz force HV winding and LV winding movesto the opposite directions.

• Tap changer position has a significant effect on the displacement field of theFine tapping(FT) and Coarse tapping(CT) windings. The Lorentz force actingon the HV and the LV windings are almost independent of the tap-changerposition.The effects of the tap changer position can be summarized by thefollowing table.

Table 3.3. Effects of tap changer position on the radial displacement in m

Windings Tap-changer position Tap-changer position Tap-changer position1 12 23

LV 1.3e-6 1.3e-6 1.3e-6HV 4.5e-6 4.5e-6 4.4e-6CT 1.5e-6 4.8e-7 2.34e-8FT 7e-7 8.9e-9 1.09e-8

Figure 3.4. Frequency Response

• From Figure 3.4 one can see the effects of the frequency sweep.The figureshows the total displacement for the HV winding. The most important thingis that we capture the first 3 eigenfrequencies this way, the eigenfrequencieshappens to be 75 Hz and 135 Hz and 190 Hz.

• If we make an eigenvalue analysis, the first three eigenvalues become 75,135,190Hz which perfectly matches with the measurement results presented in thesecond progress report written by Prof. Lerch.

3.4 Mechanical Model with Oil couplingFor cooling the mechanical parts of the transformer as well as preventing the oxi-dation,mineral oil is widely used in the power transformers. In reality oil must gothrough all the mechanical structures for picking up the heat while it is in contactwith conductors and carry the heat out to the tank surface by self convection. Oilis also needed for enhancing the dielectric strength of the winding. For this purposethe windings got radial ducts so that oil can easily go through the windings. The ra-dial ducts are also needed for increasing the contact area. In our mechanical modelfor decreasing the computational costs we are using a homogenization method. Sowe are removing the ducts but keeping their effect on the mechanical system. In thisapproach a model without ducts is designed. This model uses pressure acousticsmodule for fluid mechanic coupling. In the solid-fluid interface one has to considerthe fact that the ambient fluid causes a mechanical stress σn on the surface. This

stress acts like a pressure load on the solid. The coupling condition is:

σn = −np (3.8)

Here n is the normal component of the solid and p is the pressure in Pa. If we areusing this method for coupling the oil with the structure we experience resonancefrequency of around 102 Hz. This resonance frequency can be seen from the Fig-ure 3.5. This resonance frequency has a significant effect on the Sound PressureLevel(SPL) as well as displacement field. The biggest problem with the resonancefrequency is that its too close to 100 Hz which is our primary frequency of interest.By using this model we experience a big local pressure between the HV winding andthe LV winding. Pressure field can be seen from the Figure 4.9 this pressure dif-ference has also got an empowering effect on the displacement field of the windingswhen we are using Fluid Load condition for coupling the system which depends onpressure value.

Figure 3.5. Displacement of the mid point of the low voltage winding

For overcoming this problem we create a more detailed model. In this new modelwe are introducing some of the cooling ducts for getting more homogenized pressurelevel inside the tank. The geometry of the new model can be seen from Figure 3.6.The newly introduced ducts have a height of 4mm. In the model we put more ductsin the middle area where we are experiencing the bigger pressure difference. Thepressure is a topic of the Acoustic model so they are presented in the chapter 4.The effect of the opened cooling ducts can be seen from Figure 4.11. The maxvalue of the pressure level decreases from 3.105 Pa to 4000 Pa and the resonance

is disappearing. The future models are based on this latest model. This model isalready coupled with the pressure acoustics so more detail will be presented in thenext chapter.

Figure 3.6. Model with cooling ducts

3.5 Sensitivity TestsHere some sensitivity test are done and their influence over the displacement field isinvestigated. In the next chapter the same analysis is done for understanding theireffect over the overall sound pressure level. To make the analysis more understand-able while we are talking about the paper or spacer we are referring the insulationpaper and axial spacers in the cell structure which is used in the homogenizationmethod. The coil structure can be briefly described using Figure 3.7. In this figurethe white areas represent the air, the black domains are insulation paper,the lightgray area represent the radial spacers and the dark gray area shows the copper.

3.5.1 Influence of Stiffness of Winding Supports

In this step stiffness of the winding and core supports on the displacement fieldhave been investigated. As mentioned before the stiffness has been modeled as animaginary material which can support 85 kN/mm and is located at the top andbottom of the pressboards. If one makes a parameter sweep on additional stiffnessone can see big effects on the axial acceleration on the windings. In Figure 3.8 to

Figure 3.7. Coil cell structure

the x axis represents the change in the additional stiffness for example, -4 meanswe are reducing the additional stiffness 10−4 times. As can be seen from Figure 3.8when we increase the additional stiffness we can have a big drop in the axial windingaccelerations. After increasing the stiffness 100 times more the accelerations becamenearly stabilized and further increase does not effect the accelerations.

3.5.2 Influence of Young’s Modulus of Insulation Paper

In this step the influence of the axial spacers over radial and axial accelerationfield is investigated. The effects of the Sound pressure level is an issue of the nextchapter. In this analysis we are using the boundary integral of the outer surfaceof the windings. Than we will weight the results by their length to get an averagevalue. For HV winding we will use the vertical line located at r=0.431m and for LVwinding r=0.329m respectively. The results of the sensitivity test are summarizedin the table 3.4.

3.5.3 Influence of Young’s Modulus of Radial Spacers

In this step the influence of the radial spacers over radial and axial accelerationfield is investigated. Like the previous step we are using the boundary integral ofthe outer surface of the windings. Then we will weight the results by their lengthto get an average value. Similar to the last step, for HV winding we will use the

Figure 3.8. Effects of additional stiffness here in the x axis -4 means we are reducingthe stiffness 10−4 times.

Table 3.4. Acceleration field with respect to Young’s modulus of Insulation Paper

Rate of Change in Young’s Radial HV Radial LV Axial HV Axial LVModulus of Insulation Paper Acceleration Acceleration Acceleration Acceleration60% 2.86 0.86 0.11 0.1470% 2.16 0.67 0.11 0.1480% 2.42 0.74 0.11 0.1490% 2.27 0.69 0.11 0.14110% 2.04 0.63 0.10 0.14120% 1.95 0.60 0.10 0.14

vertical line located at r=0.431m and for LV winding r=0.329m respectively. Theresults of the sensitivity tests are summarized in the following table.

3.5.4 Influence of Elastic Modulus of Pressboards

With using same methodology we investigate the influence of elastic modulus ofpressboards over axial and radial acceleration fields. Elastic modulus of pressboardsgot a big influence on second and third eigenfrequencies[5]. We summarize ourresults using table 3.6. As the table shows that 10% of change in the elastic modulushas a small influence over the acceleration field. The one of the interesting result is,though we are increasing the elastic modulus of pressboards there is a increase inthe axial acceleration of the Low Voltage winding. We must note that the change

Table 3.5. Acceleration field with respect to Young’s modulus of radial spacers

Rate of Change in Youngs Radial HV Radial LV Axial HV Axial LVModulus of Radial Spacers Acceleration Acceleration Acceleration Acceleration60% 2.83 0.88 0.12 0.1470% 2.60 0.79 012 0.1480% 2.43 0.73 0.11 0.1490% 2.28 0.685 0.11 0.14110% 2.05 0.60 0.10 0.14120% 1.97 0.57 0.10 0.14

Table 3.6. Acceleration field with respect to Elastic Modulus of Pressboards

Rate of Change in Elastic Radial HV Radial LV Axial HV Axial LVModulus of Pressboards Acceleration Acceleration Acceleration Acceleration60% 2.21 0.660 0.13 0.1070% 2.20 0.653 0.12 0.1180% 2.18 0.651 0.12 0.1290% 2.17 0.647 0.11 0.14110% 2.15 0.640 0.10 0.14120% 2.14 0.637 0.10 0.14

of rate is too small. The phenomena may need further research.

Chapter 4

Acoustic Modeling

4.1 Introduction

The main aim of this project is to get a clue about sound pressure level outside thetank. Acoustic phenomena in the transformer depend highly on the electromagneticand mechanical models,which includes all the couplings between the electromagneticand the mechanical fields. This situation makes the problem nearly impossible tosolve analytically. There are some standard formulas that we present in section4.2.1. These formulas are for predicting the load noise, which primarily dependonly on the rated power of the transformer. These formulas are the state of artto predicting the load noise. However, the main disadvantage of these predictionformulas is that accurate parameters on the load-controlled noise are not available.So one needs a good numerical acoustic model, which contains all the fields andtheir couplings. In order to simulate three windings with 120 degree phase shift a2D axisymmetric model is not adequate. For this reason an accurate 3D model isdeveloped. The input of this model is the axial and the radial displacements of theLow Voltage and High Voltage windings derived form 2D axisymmetric model. Thelink between the 2D axisymmetric results to 3D geometry is a mapping operationcalled extrusion coupling. To clearly present all the ideas about acoustic model,first theoretical background about the pressure acoustics is presented. After theexplanation of governing equations 2D axisymmetric model is shown and in the lastsection a 3D acoustic model is presented.

4.2 Theory Background

In fluids such as air and water, sound waves propagate as disturbances in the ambientpressure level. The pressure changes in the media create a repulsive force for theparticles of the fluid. These particles move forwards and backwards in the directionof propagation and produce local compression and expansions. Sound waves in alossless medium are governed by the following equation for the acoustic pressure,p(with SI unit Pa)

25

∇2p− 1c2∂2p

∂t2= 0 (4.1)

In this formula c is the speed of sound. For our simulations we use time harmoniccase, in which the pressure varies with time as

p(x, t) = p(x)ejωt (4.2)

where ω =2πf is the angular frequency and f(Hz) is the frequency of the system.Then our wave equation becomes

∇2p− 1c2ω

2p = 0 (4.3)

The other important condition is the coupling of the mechanical field with theacoustic field. The mechanical vibrations will generate acoustic waves, which itselfact as a surface pressure load on the vibrating structure. The main coupling happensin solid-fluid interface. Rules of continuity requires that the normal component ofthe mechanical surface acceleration of the solid must match with the accelerationof the fluid .

n∂2u∂t2

= −n.a[acoustic] (4.4)

Here n is the outward pointing unit normal vector seen from inside the solid domainand a is acceleration of the fluid particles. As mentioned above ambient fluid causesa mechanical stress on the surface. The stress is equal to:

−np = σn (4.5)

In the wave equation we are using acoustic pressure. The relation between acousticpressure and accelerations can be shown by following formula

n∂2u∂t2

= −1ρ

∂p

∂n(4.6)

4.2.1 Shell elementsA shell is a thin-walled structure in 3D where you can assume a simple form forthe displacements variation through the thickness. We are using shell elements fortank modeling in the 3D simulations for reducing computational effort. Due tothe fact that tank width is 10 mm and dimensions are 3.4m×2.97m×2.1m solidelements causes unnecessary amount of elements.(nearly 700.000 mesh elements).Ifwe use shell elements for defining the same tank we just need 47.150 elements. InComsol Multiphysics shells can be defined as boundary elements. Shell elementsare described by their thickness and by their material properties.The element usedfor the shell application mode is of Mindlin-Reissner type which can be used forstrain-displacement relations to obtain bending and transverse shear strain. Eventhough shell elements greatly reduce the computing effort, one must consider twoimportant aspects of the shells

• Preconditioned Krylov subspace methods have proved to be efficient in solvinglarge, sparse linear systems in many areas of scientific computing. The successof these methods in many cases is due to the existence of good preconditioningtechniques[6].In the analysis of thin shells the situation is not so transparent.It is well known that the stiffness matrices generated by the FE discretizationof thin shells are very ill-conditioned. Thus, many preconditioning techniquesfail to converge or they converge too slowly to be competitive with directsolvers.[7]. So one can not use solvers like GMRES or Conjugate gradient.For this reason we can only use direct solvers like UMFPACK and Pardiso inComsol.

• The second aspect,which is written in the Comsol Multipyhsics handbook forstructural mechanics is shell elements in Comsol Multiphysics have limitedmultiphysic capabilities so one should really know how to handle them. Inour case Comsol does not let us have big pressure differences on the two sideof a boundary so we are using two acoustic mod to overcome this problem.One acoustic mode is for the subdomain inside the tank and the other one isfor air domain outside the tank.

4.2.2 Extrusion Coupling

For using the 2D-axisymmetric results in the 3D model we are using mapping vari-ables in Comsol, so called Extrusion Coupling Variables. In Comsol we can selectlinear or general transformation of the extrusion coupling variables. In presentedmodel we are using general transformation due to the fact that the destination,which is a 3D model, has more space dimensions.To map the solution from a 2Daxially symmetric model onto the original 3D domain, we use the source transfor-mation r, z and destination transformation sqrt(x2 +y2), z. The most importantthing here is that one must use the extrusion couplings for the bound-aries not for the sub domain. In the presented model if one use the extrusioncouplings for the sub domain the software creates the mesh for 3D model for thedisplacement values for the hollow cylinders but when solving the PDE we are justusing the boundary values for the displacement so software removes the meshes inthe cylinders causing errors. For overcoming this problem one must extrude theboundaries to the surfaces.

4.3 Quantitive Measure of Sound

Sound is characterized by its pressure amplitude and its frequency spectra. Thesmallest pressure that a average human ear can detect is 20µPa. This is a referencelevel mainly used for the calculation of the acoustic quantity called the decibel(dB).The threshold of pain is about 200 Pa which corresponds to a large interval ofamplitudes. This is also one of the advantages of using decibel scale, which is also

Figure 4.1. Extrusion Coupling

a logarithmic one. The sound pressure level Lp(SPL) is defined by

Lp = 20log10prmspref

(4.7)

where pref = 20µPa. The other important quantity is the Sound Intensity, whichis defined as the rate of energy flow per unit area. The usual context is the noisemeasurement of sound intensity in the air at a listener’s location. For instantaneousacoustic pressure pinst(t) and particle velocity v(t) the average acoustic intensityduring time T is given by

I = 1T

∫ T

0pinst(t)v(t)dt (4.8)

In the industrial applications the sound intensity level is preferable over soundpressure levels [8]:

• an intensity meter responds only to the propagating part of the sound field andignores any non propagating part,for example , standing waves and reflection;

• The intensity method reduces the influence of external sound sources, as longas their sound level is constant

In general low frequency and high frequency sounds are perceived to be not asloud as mid-frequency sounds and the effect is more pronounced at low pressurelevels.[9].Therefore Sound pressure level(SPL) meters uses weighting filters,whichreduce the contribution. International standards like IEC 60076-10 for determina-tion of sound levels, uses A-weighted sound pressure levels. One can simply use

Figure 4.2. Different standards for weighting the SPL

the Figure 4.2 to convert the sound pressure level to the A-weighted sound pressurelevel. In our model mechanical excitation is two times higher than the line frequencyso its 100 Hz. For converting the SPL to A-weighted SPL one need to subtract 20dB.

4.3.1 Formulas for Predicting Load-NoiseAccording to the Standard 60076-10/IEC-2001, in order to decide magnitude of theload current sound power one can use following formula

LWA,IN = 39 + 18lg SrSp

(4.9)

where LWA,IN is the A-weighted sound power level of the transformer at ratedcurrent,rated frequency and impedance voltage; Sr is the rated power in megavoltamperes(MVA) Sp is the reference power(1 MVA)

4.4 2D Time Harmonic Axisymmetric ModelTo add the oil to the mechanical system, as well as for a better understandingof the acoustic phenomena and the coupling conditions a 2D axisymmetric modelis created. Figure 4.3 shows the geometry and Figure 4.4 shows the used bound-ary conditions in that model. This model is also used for the basis of 3D model.

Figure 4.3. Initial Geometry

The axial and radial displacements of the Low voltage(LV) winding and the Highvoltage(HV) winding will be used as input for the following acoustic simulation tocalculate the radiated transformer noise. Even though the resulting sound pressurelevels outside the tank ,in the 2D axisymmetric case,is not an important result. Theadvantage of using this model is that one can learn and experiment coupling condi-tions to get the knowledge which is important to use in 3D model. As told beforethe displacement field is our main interest in this model. Some of the importantaspects of the model are:

• Electromagnetic fields, mechanical fields and acoustic fields are coupled to-gether.This model is an add-on for the electro-mechanical model so all theboundary conditions and the initial conditions of electromagnetic and me-chanical model are same.

• The pressboards are only active in the mechanical part and are not activein the acoustic part for solving the pressure problem. The reason for thistreatment is that in reality oil can pass through everywhere but in the softwarewhen we are making an acoustic simulation we must define the subdomainsas fluid or solid domains. In this simulation we treat the subdomain forpressboards ,which are located in the top and bottom of the windings, as asolid in mechanical simulation and as fluid in acoustic simulation for gettinga more homogenized pressure in the tank.

• The top and the bottom of the winding clamping are fixed for all constraints.

They are imaginary materials which add 85 KN/mm stiffness to the press-boards.

• The tank is modeled using the material properties of the steel. Three triangu-lar mesh elements per thickness is used for solving the added stiffness problembecause of the fixed boundary conditions in the bottom.

• We are using homogenized winding model practiced by my project partnerYuanjun Zhang. One can find the parameter values in his report.All thewindings have anisotropic material parameters.[5]

• Due to the spoke like geometry of press boards we are using reduced param-eters.

• For non reflecting boundary conditions we are using an artificial boundaryconditions called Matched Boundary Condition, which allow an outgoing waveto leave the modeling domain with minimal or no reflections.

• When modeling the oil, a fluid density 841kg/m3 is used. The speed of soundin the oil is 1450 m/s.

• For the bottom surface we are using hard wall boundary condition whichmeans the normal component of the particle velocity is vanishing.

Some Notes about the Comsol Multiphysics

For making clear the model I would like to explain briefly how Comsol Multiphysicssoftware works. After creating your geometry one can specify the required physicson that geometry. Than one must select subdomains and boundary conditionswhich are active for that particular partial differential equation(PDE). When youare adding other physic modules or user defined PDE’s like we are doing, softwareis allowing us doing some tricks like activating the pressboards in the mechanicalmodel and accepting it as a solid, which we need for applying support on windings.In the acoustic PDE we are accepting these pressboards as fluids. So the softwareis always solving the PDE’s by the way you are defining them.

4.4.1 Observations about the 2D axisymmetric modelFrom this 2D axisymmetric model one can extract all the information about elec-tromagnetic and mechanic results for one winding. In this model we are mainlyinterested in acceleration field of the High Voltage(HV) and Low voltage(LV) wind-ings.For obtaining just one numeric result, line integrals along the windings areused at r=329mm for LV winding and r=431mm for HV winding. After enteringthe parameters given from Yuanjun Zhang we can make following observations.

Figure 4.4. The red dashed lines means fixed boundary conditions for mechanicalpart. The green line indicates the acceleration boundary condition for the acousticpart and black line indicates the continuity boundary conditions.

• For the model without ducts, the LV winding has an average radial accel-eration 12.6 m/s2 along the line at r=329mm and for the HV winding atr=431mm the average value is 20.6 m/s2. This result is obtained when allthree windings are active. If we change the tap changer position 13,for LVwinding the average value becomes 12.08 m/s2 and for HV winding the av-erage value becomes 19.22 m/s2. For the last tap changer position 23 for LVwinding we are getting 11.05 m/s2 and for HV winding we are obtaining 18.26m/s2 so the tap changer position has no significant effect on accelerations ofthe windings. Which means that the magnetic volume forces acting on theinnermost LV and HV winding are almost independent of the tap changerposition.

• It can be seen easily, that these accelerations are not realistic due to theresonance problem. This is why we include the ducts to our model. When weintroduce the ducts the resonance frequency near 102 Hz is disappearing and

Figure 4.5. 2D Deformed Plot in tap changer position 1

the results become much more reasonable and realistic. If we make the sameanalysis which is stated above for tap changer position 1 the average value forthe radial acceleration of the LV winding become 0.58 m/s2 and for the HVwinding average become 1.66 m/s2. So as stated in the mechanical chapterthe big difference is due to the paper copper ratios in the windings.

• Due to the direction of the Lorentz forces and the compressing effect of theedge fringing, which also compress the windings the biggest radial displace-ments happens in the middle.

• For the similar reasons, on the top and bottom of the HV and LV windings,the biggest axial displacements happen in the inner side of the HV windingand the outer side of the LV winding. Displacement field can be seen fromFigure 4.5.

• In order to capture the information about the radial and axial displacementand to use correctly this information, in the 3D model, we are using mappingtechnique.

4.4.2 3D Acoustic Model without OilIn order to taking into account the 120 degree phase shift between the three wind-ings and to see the effects, a 3D model of oil insulated three phase transformer isestablished. The input of this model is the acceleration field derived from the 2D

Figure 4.6. Arrangement of the measurement points

axisymmetric model. The main results of this model are the calculated tank surfacedisplacements as well as sound pressure level outside the tank. The software allowsus to extract many more results about the acoustic field inside and outside the tankas well as information about the tank which is modeled with shell elements. Someof the important aspects of the model are

• The 2D results are carried to the 3D geometry using extrusion operation.

• The tank is modeled using shell elements due to the memory considerations.

• The main coupling happens in the radial displacements of the windings andthe axial displacements of the clamping.

• The model uses match boundary condition which is an absorbing boundarycondition for eliminating the reflections.

• The ground is modeled using sound hard boundary condition.

• Model contains 916738 DOF and the running time is around 8 hours on Hmon-ster machine.

Sound Pressure Level

The main objective of the this master thesis is creating a model which can simulatethe Sound Pressure Level(SPL) outside the tank. In this section we are going topresent the sound pressure level outside the tank for air filled transformer. As canbe seen from Figure 4.7 there is a strong directivity in the sound pressure outsidethe tank. One can also see there is nearly a perfect symmetry due to the fact thatall the forces are acting symmetric.The international standards recommend us tomeasure the sound pressure level at the middle of the tank height and using 12measurement points. Arrangement of these points can be seen from Figure 4.6. If

Figure 4.7. SPL for the transformer without oil

we use the recommended measurement way we can summarize the sound pressurelevel with the following table.

Table 4.1. SPL Results for tap position 1 in 100Hz

Points 1 2 3 4 5 6 7 8 9 10 11 12Simulation (dB) 54 56 62 57 54 66.5 54 57 62 55.7 54 66.5

The presented results are given in decibels. One can subtract 20 dB from thenominal sound pressure level for founding the A-weighted levels. In this presentmodel we can not estimate the effect of the tap changer position because of the factthat we are not modeling the fine tapping and coarse tapping windings.

According to international standard 60076-10, the uncorrected average A-weightedsound pressure level,LpA0, shall be calculated from the A-weighted sound pressurelevels,LpAi,measured with the test object energized by using equation 4.10

LpA0 = 10log(

N∑i=1

100.1LpAi

)(4.10)

where N is the total number of measuring positions. Using the formula 4.10 we aregetting 39.384 dB as average A-weighted sound pressure level.

Sound Intensity

Sound intensity describes the flow of acoustic energy produced by a sound source.Theunit is Wm−2. Unlike simple microphones and sound level meters, sound intensitymeasurements accurately capture only the sound produced by the source under test,eliminating interference from other sounds. The connection between sound intensityand sound pressure can be expressed as:

I = p2/ρ ∗ c (4.11)

where p is the pressure, ρ is the density of the fluid and c is the speed of sound.The dynamic range of human hearing and sound intensity spans from 10−12W/m2

to 100W/m2 so like sound pressure we need a logarithmic scale relative to thelowest human hearable sound 10−12W/m2. The Sound Intensity Level(SIL) can beexpressed as:

Li = 10log abs(In)I0

(4.12)

where I0 is 10−12W/m2

Figure 4.8 is the norm of the sound intensity for air filled transformer. If we usethe same measurement point arrangement like sound pressure level calculation wecan summarize the results with the following table

Table 4.2. Sound Intensity Quantities

Points 1 2 3 4 5 6 7 8 9 10 11 12Intensity Wm−2 8.45e-7 1.31e-6 3.4e-6 1.62e-6 0.97e-6 7.1e-6 9.6e-7 1.55e-6 3.25e-6 1.3e-6 0.82e-6 0.6e-5SIL dB 49 51.19 55.4 52.1 50 58.5 50 52 55.1 52.2 49 57.6

4.5 3D Model for Oil Insulated TransformerWhen we are adding the oil to the system and using acoustic coupling conditions weare getting a locally high pressure area in between HV and LV windings which canbe seen from Figure 4.9. My understanding in this problem is,we are using fluid load

Figure 4.8. Norm of the sound intensity values for air filled transformer

condition, which is described in equation 4.5, for coupling the oil to the mechanicalsystem. Ideally there should not be any big pressure differences inside the tank sowhen we use the fluid load it must damp the radial movement of the windings fromthe both sides. In our case we have really a big pressure difference between the rightand left side of the windings so Fluid Load conditions empower the movement fromthe outermost side of the Low voltage winding and the innermost side of the HVwinding. As discussed before the oil coupling causes a resonance in 102Hz which istoo close to the frequency of interest,100 Hz. At 102 Hz the sound pressure levelsbecome 140 dB and for 100 Hz the average value become 105 dB which is way toofar from the measurement results,88 dB.

Because the computed results so far are too far from the measurements, we refinethe model by including cooling ducts. The results can be seen form Figure 4.12, themodel with holes has a significant effect on the pressure level inside the tank. In thisapproach the average SPL value for the 3D became 94.61 dB. The biggest problemhere is the height of the ducts,which are 4 mm long. So it causes a big number ofelements in the ducts and the area near the ducts. If we further put the ducts weare experiencing a memory problem. Here the biggest problem, or the restrictingelement,is the actual choices of the solver which we are using. The presented modelconsist of coupling of three physic modes. Then we are coupling the results withanother geometry using the extrusion coupling and in the second geometry we areagain using three physic modes. Comsol has a big selection of solver like directsolvers, parallel direct solvers and iterative solvers like GMRES and Conjugate

Figure 4.9. Pressure Level inside the tank

Gradient and of course Multigrid solver. As mentioned before we are using shellelements for modeling the tank in the 3D geometry. The biggest problem is its illcondition matrix after the assembly so we can not use the GMRES or the ConjugateGradient. We are using a mapping technique from 2D to 3D when we are using theGeometric Multigrid, we are creating the coarser meshes and after the coarseningprocess so called, restriction process, the software is complaining for mismatchesbetween the source, 2D geometry, and the destination, 3D geometry. After all thesethings we just have the direct methods. Direct solvers are really robust but they areslow for big problems and requires too much memory. When we create a model withall the windings and couplings in 3D model. The memory requirement is jumping tothe 200 Gigabytes so the supercomputer which we are using do not let us have thatmuch memory. For solving the resonance problem we are adding the ducts to themechanical model and these ducts have 4 mm height and our windings are around1 meter here again we are experiencing a memory problem and the model needs 22hours to be solved. The result is we have a 10 dB more SPL level than the modelwhich is filled with air so the coupling of the oil in the mechanical model needsmore experimenting and understanding. The results can be seen from Figure 4.11.As mentioned before the overall sound pressure level(OSPL) became 94.615dB. Thefollowing table shows the SPL for our 12 measurement points.

The measurement results are around 69 dBA for overall sound pressure levelwhere for us the overall sound pressure level 74.618 dBA.

Figure 4.10. Effects of the resonance in 102 Hz over our measurement points.

Table 4.3. SPL Results for tap position 1 with ducts in 100 Hz

Points 1 2 3 4 5 6 7 8 9 10 11 12Simulation (dB) 78.63 79.84 77.93 85.36 82.39 88.5 82.60 85.24 78.12 79.78 78.78 88.38

4.6 Investigation of material parameters on SPLThe main goal of this section is the investigation of the influence of different pa-rameters on the load controlled surface vibrations and sound pressure levels andthe second aim is to find a way to improve the load controlled noise of oil insulatedtransformers. One important condition is when we are making the modifications,we are showing importance not to change the electrical behavior. The parameterinvestigation contains:

• Influence of Tank Thickness

• Influence of Additional Stiffness

• Influence of Elastic Modulus of Pressboards

• Influence of Young’s Modulus of Paper

• Influence of Young’s Modulus of Spacers

Figure 4.11. Model with cooling ducts.

Main aim for optimization is done for mechanical parameters which can easily ap-plied for the real world applications. The parameters for the sensitivity test aboutthe elastic modulus of pressboards, paper, spacers is taken from my project partnerYuanjun Zhangs paper.

4.6.1 Influence of Tank ThicknessIn our first sensitivity test we are investigating the effects of the thickness of the tank.Due to the fact that our model with oil needs 7 hours for running, this sensitivitytest is done for the case without oil which has four hours of running time. TheFigure 4.13 shows the SPL for our 12 measurement points with respect to the tankthickness. The deformation of the tank is highly dependent on the tank thicknessdue to the fact that thickness has a big effects of the eigenmodes of the tank. That’swhy, even though we are increasing the thickness, as the results show, we are notdecreasing the SPL outside the tank. Design of tank needs more eigenfrequencyanalysis for a better understanding for each thickness. In our simulations sometimeswe are hitting a critical points like the ones in 13 mm tank. In this case we aregetting SPL of 27 dB and 38.5 dB this result is due to the unique way of the totaldisplacement of the tank surface. For better understanding the simulation results,a table is done to summarize the SPL for all measurement points for different tankthickness cases.The default value is 10mm for the side walls. If we use equation 4.12for calculating the overall sound pressure level (OSPL) we are finding the minimumOSPL when we have 16 mm of tank thickness. The important result is in 15mm

Figure 4.12. SPL for the model with ducts

,with just changing 1 mm, we are getting nearly the maximum overall sound pressurelevel.

Table 4.4. SPL Results with respect to tank thickness

Points 1 2 3 4 5 6 7 8 9 10 11 12 OSPLdB dB dB dB dB dB dB dB dB dB dB dB dB

7mm thickness 57.16 55.46 63.89 53.7 57.04 67.69 56.99 55.00 64.14 55.83 55.97 67.32 72.908mm thickness 55.10 51.56 61.81 54.77 56.84 67.43 57.43 54.24 61.64 52.17 53.72 66.12 71.789mm thickness 71.14 66.38 69.26 67.06 69.20 66.83 69.21 67.02 69.24 66.46 71.22 64.75 79.3610mm thickness 56.36 57.54 68.53 58.72 59.08 65.80 59.93 60.40 70.62 59.68 56.80 65.07 75.0111mm thickness 70.66 63.72 71.31 55.98 71.20 70.96 71.18 56.91 71.25 63.34 70.53 70.72 80.2412mm thickness 62.12 60.05 58.63 58.68 60.94 63.23 60.05 56.22 57.77 57.87 61.35 62.86 71.2613mm thickness 56.62 38.50 57.02 27.53 57.55 64.04 57.54 28.57 57.10 38.90 56.63 63.05 68.8314mm thickness 61.66 45.10 57.20 46.31 62.72 66.78 62.72 46.40 57.38 45.46 61.67 65.79 72.1615mm thickness 73.26 66.80 62.87 66.61 72.60 73.75 72.59 66.52 62.86 66.79 73.24 73.97 81.7516mm thickness 56.48 54.30 55.48 55.84 51.04 62.04 50.96 55.78 55.45 54.26 56.73 55.83 67.11

4.6.2 Influence of Young’s Modulus of SpacersIn this section we are investigating the effects of elastic modulus of radial spacers,which are made of paper, over sound pressure level (SPL) in oil insulated powertransformer. The elastic modulus of spacers is stated as they have a big influ-

ence in axial vibrations and also effective in changing the eigenfrequency of higheigenmodes[5]. Here in the sensitivity test we are changing the Young’s modulus ofthe radial spacers from 60% to 120% by step of 10%. After altering the effectiveelastic modulus of the HV and LV windings we are getting following results. If

Table 4.5. SPL Results with respect to Young’s modulus of paper

Points 1 2 3 4 5 6 7 8 9 10 11 12 OSPLdB dB dB dB dB dB dB dB dB dB dB dB dB

60% 80.87 82.11 80.18 87.61 84.64 90.73 84.85 87.48 80.38 82.06 81.03 90.61 96.8570% 80.21 81.44 79.52 86.95 83.97 90.07 84.18 86.82 79.72 81.38 80.37 89.95 96.1980% 79.62 80.85 78.92 86.36 83.39 89.48 83.59 86.24 79.12 80.79 79.78 89.36 95.4190% 79.13 80.34 78.43 85.87 82.89 88.98 83.11 85.74 78.63 80.28 79.29 88.86 95.11110% 78.20 79.40 77.50 84.93 81.96 88.04 82.17 84.80 77.69 79.35 78.35 87.92 94.17120% 77.81 79.01 77.11 84.54 81.57 87.65 81.78 84.41 77.3 78.95 77.95 87.55 93.78

we compare the results of the overall sound pressure level, one can easily notice anearly linear relationship 10% of change causes 0.6 dB change.

4.6.3 Influence of Young’s Modulus of PaperIn this section we are investigating the effects of elastic modulus of paper over SoundPressure Level (SPL) in oil insulated power transformers. Here from the paper

Figure 4.13. SPL for 12 measurement point with respect to transformer thickness

one should understand the paper surrounding the coil bundle. Elastic modulus ofpaper is stated as one of the most important factor which changes position of higheigenfrequency of the windings. Here in the sensitivity test we are changing theYoung’s modulus of the insulation paper from 60% to 120% by step of 10%. Afterusing the homogenization method we are getting a new effective elastic modulus forHigh Voltage and Low Voltage windings. As the table also shows there is nearly a

Table 4.6. SPL Results with respect to Young’s modulus of paper

Points 1 2 3 4 5 6 7 8 9 10 11 12 OSPLdB dB dB dB dB dB dB dB dB dB dB dB dB

60% 81.14 82.32 80.43 87.86 84.92 90.98 85.1 87.74 80.62 82.28 81.29 90.86 97.1170% 80.4 81.5 79.65 87.1 84.12 90.2 84.20 87.02 79.5 81.5 80.51 90.05 96.3180% 79.72 80.90 79.02 86.44 83.46 89.55 83.67 86.31 79.19 80.85 79.85 89.42 95.6790% 79.13 80.34 78.43 85.87 82.89 88.98 83.11 85.74 78.63 80.28 79.29 88.86 95.11110% 78.18 78.38 77.48 84.91 81.94 88.02 82.15 84.78 77.67 79.33 78.32 87.90 94.12120% 77.75 77.98 77.06 84.50 81.52 87.61 81.74 84.37 77.26 78.92 77.92 87.51 93.71

linear relationship between the Young’s modulus of the insulation paper and soundpressure level 10% of change influence approx. again 0.6 dB like the radial spacersensitivity.

4.6.4 Influence of Elastic Modulus of Pressboards, In this section we alter the elastic modulus of the pressboards and investigate theeffects on the sound pressure level outside the tank. In the paper of my projectpartner, the elastic modulus of the pressboards has the biggest influence on thesecond and the third eigenmodes.The reason is the compression and elongation ofthe clamping in the second and third eigenmodes. We use the same methadologyas previous sensitivity tests and summarize the results with following table.

Table 4.7. SPL Results with respect to Elastic Modulus of Pressboards

Points 1 2 3 4 5 6 7 8 9 10 11 12dB dB dB dB dB dB dB dB dB dB dB dB

60% 78.84 80.04 78.14 85.57 82.61 88.69 82.82 85.44 78.33 80.00 79.00 88.5670% 78.78 79.98 78.07 85.51 82.55 88.63 82.76 85.37 78.27 79.93 78.93 88.5080% 78.72 79.93 78.02 85.45 82.49 88.57 82.70 85.32 78.21 79.87 78.88 88.4490% 78.67 79.88 77.97 85.40 82.43 88.52 82.64 85.27 78.16 79.82 78.83 88.40110% 78.58 79.79 77.89 85.32 82.34 88.43 82.60 85.19 78.08 79.74 78.74 88.31120% 78.55 79.76 77.85 85.28 82.30 88.39 82.55 85.15 78.04 79.70 78.70 88.28

Elastic modulus of the pressboards has a linear but a really minor effect on thesound pressure level. 10% of change has an influence of 0.06db.

Chapter 5

Future Work and Conclusion

The beginning aim of this master thesis is trying to establish a way to predictsound pressure level outside of the tank for 2D axisymmetric case. We furthercarry to this aim to a next level and establish a 3D model for oil insulated 3Dpower transformers. In this thesis we present a new modeling scheme using anextensive multiphysics approach. In the model we successfully couple three physicmodes as well as two geometries. Models are created in a way that solutions canbe achieved with a single computer run. We also show that this model can beeasily used for optimization purposes. Several simplifications are made for reducingthe computing effort. Comsol mutipyhsics is used for modeling tool which helpsuser with its predefined physic modes. Predefined modes greatly reduce the initiallearning time of the software and help to establish complicated models. Furtherprocedures may include :

• Reducing the computing time, which is currently near 9 hours, and memoryrequirement of the model. Due to the usage of direct solvers and 3D geometrythe model still needs shared memory machines to be able to run.

• Further research about the modeling and optimizing the tank can be donewhich is an important parameter for overall sound pressure level.

• Results can be further verified with help of the measurement data. So onecan make further model modifications.

• Case of transformer in the closed room can be simulated. In the thesis we areassuming the tank is in a free field.

• Further investigation can be done about 3D mechanical winding model. Soone can successfully introduce cooling ducts and model the openings of thepressboard to understand and compare the effects with our presented model

• To understand the effects of the tap changer position one can try to modelFine Tapping (FT) and Course Tapping (CT) windings. During the thesis

45

some attempts are done but due to the memory problem even our sharedmemory machine which got 236GB memory gives a memory error.

• The mechanical vibrations which are carried by the clampings to the tank,can be investigated for understanding the effects on the sound pressure level.

Bibliography

[1] M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actua-tors, 2nd Edition,Springer, Berlin ,pp.321.

[2] Private Communication with Msc.Jan Anger.

[3] M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actua-tors, 2nd Edition,Springer, Berlin, pp.101

[4] W.Rogoswski,Die Transformatoren, Springer, pp.435

[5] Yuanjun Zhang,Numerical Investigation on Load Controlled Noise of PowerTransformers,Electromagnetic and Mechanical Analysis of Winding Struc-ture,Master thesis,KTH,2009

[6] Anne Greenboum Iterative Methods for Solving Linear System ,SIAM, page10

[7] An Assessment of Some Preconditioning Techniques In Shell Prob-lems,Communications in Numerical Methods in Engineering Volume 14 Issue10, Pages 897 - 906

[8] International Standart IEC 60076-10,Part 10 Determination of sound lev-els,pp.6

[9] Michael Talbot Smith Audio Engineer’s Reference Book, 2nd Ed 1999, , FocalPress. pp.120

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