Modeling and OpenFOAM simulation of streamers in transformer oil

Department of Physics, Chemistry and Biology

Master’s Thesis in Applied Physics

Modeling and OpenFOAM simulation of streamers

in transformer oil

Jonathan Fors

LiTH-IFM-EX--12/2696--SE

Research conducted in cooperation with

ABB Corporate Research, Västerås, Sweden

Department of Physics, Chemistry and Biology Linköping University

Master’s Thesis in Applied Physics LiTH-IFM-EX--12/2696--SE

Modeling and OpenFOAM simulation of streamers

in transformer oil

Jonathan Fors

Supervisor: Nils Lavesson

ABB Corporate Research

Weine Olovsson

ifm, Linköping University

Examiner: Peter Münger

ifm, Linköping University

Avdelning, Institution

Division, Department IFM

Department of Physics, Chemistry and Biology Linköping University

SE-581 83 Linköping, Sweden

Datum Date 2012-06-21 Språk Language  Svenska/Swedish  Engelska/English   Rapporttyp Report category  Licentiatavhandling  Examensarbete  C-uppsats  D-uppsats  Övrig rapport  

URL för elektronisk version

http://www.ifm.liu.se http://www.ep.liu.se ISBN — ISRN LiTH-IFM-EX--12/2696--SE

Serietitel och serienummer

Title of series, numbering

ISSN

—

Titel

Title

Modellering och OpenFOAM-simulering av streamers i transformatorolja Modeling and OpenFOAM simulation of streamers in transformer oil

Författare

Author

Jonathan Fors

Sammanfattning

Abstract

Electric breakdown in power transformers is preceded by pre-breakdown events such as streamers. The understanding of these phenomena is important in order to optimize liquid insulation systems. Earlier works have derived a model that describes streamers in transformer oil and utilized a finite element method to pro-duce numerical solutions. This research investigates the consequences of changing the numerical method to a finite volume-based solver implemented in OpenFOAM. Using a standardized needle-sphere geometry, a number of oil and voltage combi-nations were simulated, and the results are for the most part similar to those produced by the previous method. In cases with differing results the change is attributed to the more stable numerical performance of the OpenFOAM solver. A proof of concept for the extension of the simulation from a two-dimensional axial symmetry to three dimensions is also presented.

Nyckelord

Keywords Computational Physics, High voltage, OpenFOAM, Streamer, Electric insulation, Electrodynamics, Finite Volume Method

Abstract

Electric breakdown in power transformers is preceded by pre-breakdown events such as streamers. The understanding of these phenomena is important in or-der to optimize liquid insulation systems. Earlier works have or-derived a model that describes streamers in transformer oil and utilized a finite element method to produce numerical solutions. This research investigates the consequences of changing the numerical method to a finite volume-based solver implemented in OpenFOAM. Using a standardized needle-sphere geometry, a number of oil and voltage combinations were simulated, and the results are for the most part similar to those produced by the previous method. In cases with differing results the change is attributed to the more stable numerical performance of the OpenFOAM solver. A proof of concept for the extension of the simulation from a two-dimensional axial symmetry to three dimensions is also presented.

Sammanfattning

Elektriska genomslag i högspänningstransformatorer föregås av bildandet av elekt-riskt ledande kanaler som kallas streamers. En god förståelse av detta fenomen är viktigt vid konstruktionen av oljebaserad elektrisk isolation. Tidigare forskning i ämnet har tagit fram en modell för fortplantningen av streamers. Denna modell har sedan lösts numeriskt av ett beräkningsverktyg baserat på finita elementme-toden. I denna uppsats undersöks konsekvenserna av att byta metod till finita volymsmetoden genom att implementera en lösare i OpenFOAM. En standardi-serad nål-sfär-geometri har ställts upp och ett flertal kombinationer av oljor och spänningar har simulerats. De flesta resultaten visar god överensstämmande med tidigare forskning medan resultat som avviker har tillskrivits de goda numeriska egenskaperna hos OpenFOAM-lösaren. En ny typ av simulering har även genom-förts där simulationen utökas från en tvådimensionell axisymmetrisk geometri til tre dimensioner.

Acknowledgments

First of all I would like to thank my supervisor at ABB, Nils Lavesson. Throughout my thesis-writing period he provided encouragement, sound advice, good company, and lots of good ideas. He also patiently spent hours reviewing and giving feedback on this report and I would have been lost without him.

I would also thank Ola Widlund at ABB for his deep expertise regarding the OpenFOAM solver and providing well-needed numerical wizardry. The simulations would have been of much lower quality were it not for his intricate knowledge.

My many colleagues at ABB have given me support and new insights during discussions and workshops. My gratitude goes to Olof Hjortstam, Christer Törnkvist and many others.

A warm thanks should also be given to my many thesis writer colleagues at the office. Thanks for the fun times and good memories during long hours of research and writing. I am also grateful to the I.T. services department at the ABB office for providing excellent support and keeping the high-performance cluster at its finest.

My supervisor at Linköping University, Weine Olovsson, kindly provided insight and comments on the research for which I am grateful.

I owe my deepest gratitude to my parents Eva and Stefan and my sisters Susanna and Elisabeth. Without their love and support I would not have been where I am today.

Finally, I would like to thank my girlfriend and the woman I love, Anna Jogenfors, for her unconditional love and always being there for me.

Contents

1 Introduction 1

1.1 Challenges of electric insulation . . . 1

1.2 Streamers in transformer oil . . . 2

1.3 Liquid insulation . . . 3

1.4 Research objectives . . . 4

2 Methodology 7 2.1 Deriving the charge continuity equation . . . 7

2.2 Generation and removal of charge carriers . . . 8

2.2.1 Field-dependent molecular ionization . . . 9

2.2.2 Field-dependent ionic dissociation . . . 9

2.2.3 Recombination and attachment . . . 9

2.3 Geometry . . . 10

2.4 Material parameters . . . 11

2.5 Mathematical model . . . 13

2.5.1 Boundary conditions . . . 13

2.5.2 Initial value conditions . . . 14

2.6 Numerical solution . . . 14

2.6.1 Meshing . . . 15

2.6.2 Iterative coupling tolerance . . . 15

2.6.3 Courant number . . . 16 2.6.4 Parallel computing . . . 16 3 Results 17 3.1 Classification of streamers . . . 17 3.2 Aromatic hydrocarbons . . . 18 3.2.1 Applied voltage V0= +80 kV . . . 18 3.2.2 Applied voltage V0= +130 kV . . . 19 3.2.3 Applied voltage V0= +200 kV . . . 19 3.2.4 Applied voltage V0= +300 kV . . . 20 3.3 Naphthenic/paraffinic hydrocarbons . . . 31 3.3.1 Applied voltage V0= +130 kV . . . 31 3.3.2 Applied voltage V0= +200 kV . . . 31 3.3.3 Applied voltage V0= +300 kV . . . 31 xi

xii Contents 3.4 Oil mixture 1 . . . 36 3.4.1 Applied voltage V0= +130 kV . . . 36 3.4.2 Applied voltage V0= +300 kV . . . 36 3.4.3 Applied voltage V0= +400 kV . . . 36 3.5 Oil mixture 2 . . . 36 3.5.1 Applied voltage V0= +80 kV . . . 36 3.5.2 Applied voltage V0= +130 kV . . . 37 3.5.3 Applied voltage V0= +300 kV . . . 37 4 Discussion 49 4.1 Focused streamers . . . 49

4.1.1 Streamer deflection with δr = 1000 nm . . . . 49

4.1.2 Convergence analysis . . . 50

4.2 Bubbly streamers . . . 52

4.3 Streamer classification revisited . . . 54

4.3.1 Charge generation analysis . . . 54

4.3.2 Charge generation in mixed oils . . . 55

4.4 Comparison with previous work . . . 56

4.4.1 Aromatic hydrocarbons . . . 57

4.4.2 Naphthenic/paraffinic hydrocarbons . . . 57

4.4.3 Mixed oils . . . 57

5 Proof of concept: Expanding to the third dimension 59 5.1 Methodology . . . 59

5.2 Results . . . 60

5.3 Discussion . . . 60

6 Conclusions 65 6.1 Future work . . . 65

6.1.1 Lightning impulse voltage . . . 65

Contents

List of Figures

1.1 Examples of hydrocarbons found in crude oil . . . 4 2.1 IEC-standardized testing geometry with a 25 mm gap. Note the

sharp needle electrode (top) and the grounded spherical electrode (bottom). . . 10 2.2 Detail of the initial electric field magnitude V m−1

near the needle electrode. . . 14 3.1 Electric field magnitude plot of the three major streamer types . . 18 3.2 Electric field magnitude V m−1

at t = 25, 50, 75 and 100 ns for aromatic hydrocarbons with δr = 500 nm and V0= +80 kV. . . 21

3.3 Temporal dynamics along the needle-sphere electrode axis [m] for aromatic hydrocarbons with δr = 500 nm and V0= +130 kV. . . . 22

3.4 Electric field magnitude V m−1

at t = 25, 50, 75 and 300 ns for aromatic hydrocarbons with δr = 500 nm and V0= +130 kV. . . . 23

3.5 Temporal dynamics along the needle-sphere electrode axis [m] for aromatic hydrocarbons with δr = 300 nm and V0= +130 kV. . . . 24

3.6 Electric field magnitude V m−1

at t = 25, 50, 75 and 300 ns for aromatic hydrocarbons with δr = 300 nm and V0= +130 kV. . . . 25

3.7 Electric field magnitude for δr = 1000 nm and V0= +130 kV. . . . 26

3.8 Temporal dynamics along the needle-sphere electrode axis [m] for aromatic hydrocarbons with δr = 500 nm and V0= +200 kV. . . . 27

3.9 Electric field magnitude V m−1

at t = 25, 50, 75 and 100 ns for aromatic hydrocarbons with δr = 500 nm and V0= +200 kV. . . . 28

3.10 Temporal dynamics along the needle-sphere electrode axis [m] for aromatic hydrocarbons with δr = 500 nm and V0= +300 kV. . . . 29

3.11 Electric field magnitude V m−1

at t = 25, 50 and 100 ns for aro-matic hydrocarbons with δr = 500 nm and V0= +300 kV. . . 30

3.12 Electric field magnitude V m−1 _{at t = 25, 50, 75 and 100 ns for} naphthenic/paraffinic hydrocarbons with δr = 500 nm and V0 =

+130 kV. . . 32 xiii

xiv Contents 3.13 Electric field magnitude V m−1

at t = 25, 50, 75 and 100 ns for naphthenic/paraffinic hydrocarbons with δr = 500 nm and V0 =

+200 kV. . . 33 3.14 Temporal dynamics along the needle-sphere electrode axis [m] for

naphthenic/paraffinic hydrocarbons with δr = 500 nm and V0 =

+300 kV. . . 34 3.15 Electric field magnitude V m−1

at t = 25, 75, 100 and 150 ns for naphthenic/paraffinic hydrocarbons with δr = 500 nm and V0 =

+300 kV. . . 35 3.16 Electric field magnitude V m−1

at t = 25, 50, 75 and 100 ns for oil mixture 1 with δr = 500 nm and V0= +130 kV. . . 38

3.17 Temporal dynamics along the needle-sphere electrode axis [m] for oil mixture 1 with δr = 500 nm and V0= +300 kV. . . 39

3.18 Electric field magnitude V m−1

at t = 25, 50, 75 and 100 ns for oil mixture 1 with δr = 500 nm and V0= +300 kV. . . 40

3.19 Temporal dynamics along the needle-sphere electrode axis [m] for oil mixture 1 with δr = 500 nm and V0= +400 kV. . . 41

3.20 Electric field magnitude V m−1

at t = 25, 50 and 300 ns for oil mixture 1 with δr = 500 nm and V0= +400 kV. . . 42

3.21 Electric field magnitude V m−1

at t = 25, 50, 100 and 300 ns for oil mixture 2 with δr = 500 nm and V0= +80 kV. . . 43

3.22 Temporal dynamics along the needle-sphere electrode axis [m] for oil mixture 2 with δr = 500 nm and V0= +130 kV. . . 44

3.23 Electric field magnitude V m−1 _{at t = 25, 50, 100 and 300 ns for} oil mixture 2 with δr = 500 nm and V0= +130 kV. . . 45

3.24 Temporal dynamics along the needle-sphere electrode axis [m] for oil mixture 2 with δr = 500 nm and V0= +300 kV. . . 46

3.25 Electric field magnitude V m−1 _{at t = 25, 100 and 300 ns for oil} mixture 2 with δr = 500 nm and V0= +300 kV. . . 47

4.1 Position and velocity comparison along the electrode axis for aro-matic hydrocarbons with V0= +130 kV. . . 51

4.2 Position and velocity comparison along the electrode axis for aro-matic hydrocarbons with V0= +300 kV. . . 53

5.1 Visualization of the three-dimensional simulation at t = 87 ns. . . . 61 5.2 Visualization of the mesh at t = 87 ns for the three-dimensional

simulation. . . 62 5.3 Temporal dynamics along the needle-sphere electrode axis [m] for

the three-dimensional simulation. δr = 500 nm and V0= +130 kV. 63

List of Tables

2.1 Fundamental physical constants . . . 8 2.2 Material parameters of general transformer oil. . . 11 2.3 Material parameters for aromatic hydrocarbons. . . 11

Contents xv

2.4 Material parameters for naphthenic/paraffinic hydrocarbons. . . . 12

2.5 Material parameters for oil mixture 1 . . . 12

2.6 Material parameters for oil mixture 2 . . . 13

2.7 Parameters to the numerical solver . . . 15

3.1 Simulated oil type and voltage combinations . . . 17

4.1 Coefficients for charge generations for all species in section 2.4 . . 55

4.2 Measured peak Laplacian electric field directly below the needle at t= 0+ _{for different applied voltages. . . . 55}

4.3 Calculated charge generation GF m−3s−1just below the needle tip for each oil type and voltage [kV] at t = 0+ _{. . . . 55}

4.4 Calculated charge generation GFm−3s−1for mixed oils just below the needle tip at t = 0+_{. . . . 56}

4.5 Comparison of peak electric field magnitude 108_{V m}−1_{at 100 ns} and average streamer velocity 103_{m s}−1 _{. . . . 56}

Chapter 1

Introduction

The modern, industrialized world relies on the constant availability of high quality electric power. Basic functions of society from infrastructure and health services to storage and preparation of food, computing and communications have become totally dependent on electricity. Power transmission is a delicate process where supply immediately must match demand; there is no practical way of storing large amounts of electricity for later use. Since power plants commonly are located in remote areas far from the demand, transmission lines have to transmit the generated power over far distances. This will inevitably lead to losses where electric energy is converted to thermal energy in the wires.

These losses are proportional to the line resistance and the electric current squared. It is possible to reduce the electric current, and therefore the losses, by increasing the transmission voltage. This allows the same amount of power to be transmitted over the line while increasing the efficiency. Typically, generators provide electric power with medium voltage, which is then transformed to high voltages before it is sent to the backbone network. Before reaching the consumer (i.e. a populated area or an industry) the power is transformed back into lower

voltages to be handled by simpler, less expensive equipment.

However, high voltages require good insulation to avoid arcing which creates challenges when designing such components. Uncontrolled arcing is a very serious event that can cause structural damage, fire, injuries and death. Insulation can be achieved simply by separating different potentials by large distances, but this leads to the equipment becoming too bulky and expensive. There has therefore been significant research into electrical insulation in order to make insulation systems more compact and cost-effective while maintaining reliability and safety.

1.1

Challenges of electric insulation

The goal of electric insulation is to prevent the flow of electric charges from one point to another, and is an important concept in any device dealing with electricity. Within a transformer, insulators are used to prevent electric faults and protect the surrounding environment from electric shocks. All real insulators are limited

2 Introduction by their breakdown strength, which is the maximum electric field beyond which they become highly conductive. A well-known example of electric breakdown is atmospheric lightning, in which air is stressed by the static buildup between the clouds and the ground. Understanding the mechanisms behind electrical breakdown is of great importance when designing insulation systems.

Electric insulation systems can be categorized into the following groups: Solid insulators such as rubber, plastic, ceramics and pressboard.

Gaseous insulators such as air, sulphur hexafluoride (SF₆) and even vacuum. Liquid insulators such as transformer oil and 3M Fluorinert.

Liquid insulation has the advantage of providing cooling and having self-healing properties. In large high-voltage transformers, insulation is typically achieved by a combination of solid and liquid insulation [5]. Pressboard, a wood-like material made of densely packed paper sheets surround the windings which in turn is submerged into transformer oil. This thesis will focus on liquid insulators due to its wide usage in power transformers.

1.2

Streamers in transformer oil

In the case of transformer oil, electrical breakdown is preceded by a number of pre-breakdown events. Given enough electrical stress, the oil will start to ionize and release free charge carriers. The path along which the ionization propagates is called a streamer, and these charge carriers will cause its conductivity to be higher than the surrounding oil. If the streamer completely bridges the insulation it will cause an electric breakdown. The now-decreased resistance along the streamer causes a current to flow, which in turn increases the ionization. This cycle of positive feedback quickly causes a short-circuit to be formed which results in arcing and large energy dissipation.

When designing liquid insulation that is intended to prevent streamer formation, two factors are important: initialization (breakdown) voltage and streamer velocity. The breakdown voltage shows at which voltage a streamer is formed. Naturally, it is desirable to make this value as large as possible in order to delay the formation of a streamer. The streamer velocity measures the speed at which the streamer propagates and is especially important at voltages significantly higher than the breakdown voltage. A low streamer velocity could allow a voltage spike to subside before the streamer completely bridges the insulation, avoiding a full breakdown. Streamers can be divided into two groups: positive and negative [2]. Positive streamers are formed around positive electrodes and negative streamers come from negative electrodes. The underlying difference between these two types is because of the differing mobility between the negative and positive charge carriers. Electrons are highly mobile in comparison to positive charge carriers such as positive ions. Positive electrodes rapidly attract electrons while slowly pushing away positive ions. This results in a focused streamer forming near sharp edges because of the electrons causing an enhancement to the electric field.

1.3 Liquid insulation 3 Negative streamers require a higher initialization voltage because of the lower level of field enhancement. This is due to the fact that the mobile electrons quickly move away from the negative electrode in all directions, leaving behind the positive ions. By distributing the electrons over a larger region, the field enhancement will become weaker. The lower initialization voltage of positive streamers has been verified experimentally [7]. In altering-current applications, only positive streamers are of interest because of the lower initialization voltage. This thesis will only focus on positive streamer propagation because of this reason.

1.3

Liquid insulation

The majority of power transformers make use of crude-derived oils for insulation. It is possible to use synthetic oils for this purpose, but this thesis will only discuss mineral-based oils due to their widespread [9] use. Any given mineral oil has a complicated composition and contains well over 100 chemical compounds [15]. The bulk of the oil, however, is made up of hydrocarbons that are usually categorized as in fig. 1.1.

Aromatic hydrocarbons contain benzene rings that have alternating single and double bonds. Examples include benzene and toluene (fig. 1.1a).

Naphthenic hydrocarbons are cyclic alkanes such as cyclopentane and cyclohex-ane (fig. 1.1b).

Paraffinic hydrocarbons consist of straight alkane chains with the chemical for-mula CnH2n+2. Examples include hexane, octane (fig. 1.1c) and heptane.

When designing transformer oil there are several factors to take into considera-tion. Low viscosity simplifies handling by lowering the pour point and allows the oil to fill voids inside the transformer and avoid air pockets. This also allows good impregnation of cellulose materials commonly used in insulation in conjunction with the oil. Chemical stability is desired in order to reduce the maintenance costs that arise from frequent oil replacements. Environmental concerns and toxicity are also of great concern, especially due to the historic use of polychlorinated biphenyls (PCB:s) in transformer oil. These additives were found to be highly toxic and have

been banned in most countries.

An oil with a majority of naphthenic or paraffinic content is called ”naphthenic” or ”paraffinic”, respectively [15]. The terms ”weakly aromatic” and ”highly aromatic” describe oils that contain less than 5% and more than 10% of aromatic molecules, respectively [15]. Transformer oil is mostly of the naphthenic kind due to the tendency of paraffinic-based oils to form wax at low temperatures [15] and the lower viscosity of the naphthenes [9].

4 Introduction

(a) Toluene C₆H₅CH₃
, an aromatic hydrocarbon

(b) Cyclohexane C₆H₁₂
, a naphthenic hy-drocarbon

(c) Octane C₈H₁₈
, a paraffinic hydrocarbon

Figure 1.1. Examples of hydrocarbons found in crude oil [3, 4].

1.4

Research objectives

Previous published research on the subject of streamers and electrical breakdown in liquid insulation systems have mostly been empirical [2, 5, 7, 14, 16]. Advance-ments in the field of measurement technology have in the last decades made new investigations into these phenomena possible.

ABB Corporate Research in Västerås, Sweden is the largest central research and development center within the ABB Group. Research conducted within this center includes investigations into power transmission and high-voltage insulation systems. As a part of this ongoing research, ABB has sponsored two Ph.D. theses [9, 19] in collaboration Massachusetts Institute of Technology in Cambridge, MA, USA. These theses focused on theoretical and computational aspects of streamer physics and derived a mathematical model for charge carrier transport. This model was

1.4 Research objectives 5 solved using Comsol, a multiphysics computational tool based on the finite element method. The current ABB-sponsored research at MIT has so far produced results such as in Jadidian et al. [10], where further improvements to the model have been presented.

More recently, ABB Corporate Research has implemented the model as a custom solver in OpenFOAM [11], a C++ toolbox that uses the finite volume method. This method might be better suited for transport equations than the finite element method used in [9, 19]. It is the goal of this thesis to investigate and analyze the results from the OpenFOAM solver and to compare the results to previous work. The model used in this thesis is analogous to those used in O’Sullivan [19] and Hwang [9], however a minor change has been made in neglecting thermal effects. This is due to the decoupling of the thermal diffusion equation from the charge carrier transport equations. Thus, the thermal effects have had no impact on the streamer propagation itself and can be disregarded without influencing the results. The derivation of the model and a description of the numerical solver are presented in chapter 2. Chapter 3 presents the results of the simulations, which are then discussed in chapter 4. A proof of concept using three-dimensional simulations has been performed and is presented in chapter 5. The research is concluded in chapter 6 together with several ideas for future research.

Chapter 2

Methodology

Dielectrics contain very few free charge carriers under normal conditions, which makes them good electrical insulators. When large electrical fields are applied, however, charge generation can occur which increases the number of free carriers. Due to the same electric field the free charge carriers are forced to move through the medium. This causes field enhancement due to the displaced free charge carriers, enhancing the electric field around the electrode.

A streamer is formed when the field enhancement is large enough to repeat the above process in the neighboring area, creating a propagating wave of charge generation and electric fields [7]. Due to the large amount of free charge carriers contained, the conductivity of the streamer is several orders of magnitude greater than that of the medium. If the field becomes weak enough, the streamer comes to a stop and leaves a trail of micro-bubbles in its wake. However, a streamer that travels all the way to the other electrode acts as a short-circuit because of the large number of charge carriers in the streamer tail. The understanding of charge generation, field enhancement and free charge movement is therefore essential for modeling streamers.

This chapter investigates charge generation mechanisms and applies this to a model for charge transport. The numerical method with which the model is solved is also discussed.

2.1

Deriving the charge continuity equation

For the duration of this thesis all quantities are assumed to be in local thermody-namic equilibrium and therefore have definite values. The effect of phase transitions that normally occur within the streamer are neglected. The charges in transformer oil are assumed to consist of free electrons and positive and negative ions with number densities of ne, np and nn, respectively. For the duration of this thesis,

the subscript i represents either a positive ion p, negative ion n or a free electron e. All ions are also assumed to be generic and have a charge of ±q.

Neglecting all diffusion currents, the continuity equation for each charge carrier 7

8 Methodology

Constant Symbol Numerical value Unit

Speed of light in vacuum c0 299 792 458 m s−1

Permeability of vacuum µ0 4π × 10−7 H m−1 Permittivity of vacuum 0= µ0c20
−1 _{8.854 187 817 × 10}−12 _{F m}−1 Elementary charge q 1.602 177 33 × 10−19 C Electron mass me 9.109 389 7 × 10−31 kg Planck constant h 6.626 075 5 × 10−34 J s

Table 2.1. Fundamental physical constants

is

∂ρi

∂t + ∇ · ji = q (Gi− Ri) (2.1)

where ρ is the charge density, j the current density, G the rate with which charges are generated and R the rate with which charges are removed. See table 2.1 for the fundamental physical constants used in this thesis. Now, all charge carriers are assumed to have a linear mobility for the drift v:

vi= µiE (2.2)

The electric field is the negative gradient of the electric potential

E= −∇V (2.3)

The charge density for a current of charge carriers of type i can then be written as

ji= niqvi= niqµiE (2.4)

The following expression is the result of combining eq. (2.1) and eq. (2.4) and dividing both sides by the elementary charge:

∂ni

∂t + ∇ · (µiniE) = Gi− Ri (2.5)

which is the equation that is used in the following sections.

2.2

Generation and removal of charge carriers

A number of different charge sources G and sinks R are discussed in this section. First, it can be noted that conservation of charge requires the relation

Gp− Rp= Gn− Rn+ Ge− Re (2.6)

2.2 Generation and removal of charge carriers 9

2.2.1

Field-dependent molecular ionization

O’Sullivan [19] studied field-dependent molecular ionization in which electrons are extracted from neutral molecules by large electric fields. This theory is based on the research by Zener [22], where a model for electric breakdown in solid dielectrics was derived. The charge generation rate C m−3_s−1_{is given by}

GF(E) = qn0aE h exp  −π 2_m∗_a∆2 qh2_E  (2.7) where ∆ is the molecular ionization potential, E the local electric field magnitude, athe molecular separation, n0 the number density of ionizable molecules and m∗

the effective electron mass.

However, this model assumes the material to be a solid and have a periodic structure, both of which a heterogeneous liquid such as transformer oil fails to be. It might not even be possible to find single values for parameters such as the number density or molecular separation since the different oil components have different values. While problematic, this model is to the author’s best knowledge the only one that can be applied to streamer modeling at the time for writing. Field ionization is therefore used in this thesis as the primary charge generation mechanism.

In order to apply eq. (2.7) to the generation terms Gi it must be noted that it

only applies to positive ions and electrons.

Gp= GF (2.8)

Gn= 0 (2.9)

Ge= GF (2.10)

2.2.2

Field-dependent ionic dissociation

In the theory of ionic dissociation [18], neutral ion pairs are dissociated by strong electric fields into positive and negative free ions. Oil conductivity is hypothesized to increase with larger fields due to the dissociated ions being charge carriers. However, the low mobility of the ions in relation to that of the electrons (table 2.2) makes this mechanism unlikely to have a significant influence on the formation of streamers [19]. This mechanism is not discussed in further detail in this thesis.

2.2.3

Recombination and attachment

To find the sink terms R, Langevin recombination and electron-molecule attachment are used.

Positive ions are removed by recombination with negative ions with a rate of Rpnand with electrons with a rate of Rpe.

Negative ions are removed by recombination with positive ions with a rate of Rpnand are generated by neutral molecules attaching to electrons with a

rate of τ−1

10 Methodology

(a) Experimental setup of needle-sphere geometry at ABB Corporate Research, Västerås. Courtsey of Rongsheng Liu.

(b) 3D computer model (Outer walls not shown)

Figure 2.1. IEC-standardized testing geometry with a 25 mm gap. Note the sharp

needle electrode (top) and the grounded spherical electrode (bottom).

Electrons are removed by recombination with positive electrons with a rate of Rpe and by attachment to neutral molecules with a rate τa−1.

The sink terms then become

Rp= npnnRpn+ npneRpe (2.11) Rn= npnnRpn− ne τa (2.12) Re= ne τa + n pneRpe (2.13)

2.3

Geometry

The simulations are performed over a geometry as defined by the IEC 60897 standardized needle-sphere model [17]. An experimental realization of this geometry can be seen in fig. 2.1a. In order to create the extreme electric fields required

2.4 Material parameters 11

Parameter Symbol Value Dimension References

Positive ion mobility µp 1 × 10−9 m2V−1s−1 [1, 5]

Negative ion mobility µn 1 × 10−9 m2V−1s−1 [1, 5]

Electron mobility µn 1 × 10−4 m2V−1s−1 [2, 12, 20]

Relative permittivity r 2.2 1 [5, 15]

Ion-ion recombination rate Rpn 1.64 × 1017 m3s−1 [9]

Ion-electron recombination rate Rpe 1.64 × 1017 m3s−1 [9]

Electron attachment time τa 2 × 10−7 s [2]

Table 2.2. Material parameters of general transformer oil.

Parameter Symbol Value Unit References

Number density n0 1023 m−3 [9]

Ionization potential ∆ 9.92 × 10_6.20−19 _eVJ [6]

Molecular separation a 3.0 × 10−10 m [9, 19]

Effective electron mass m∗ 0.1me= 9.11 × 10−32 kg [9]

Table 2.3. Material parameters for aromatic hydrocarbons.

for streamer initiation, a sharp electrode tip is used to increase the breakdown probability. Naturally, when designing real high-voltage systems sharp edges are highly undesirable and are avoided precisely because of high fields. The needle-sphere model does therefore not correspond to a real situation, however it is very useful for testing the insulating strength of the oil.

This geometry has a rotational symmetry, allowing the use of a cylindrical coordinate system with its origin on the needle head. In this thesis, the tip radius is 40 µm and the grounded spherical electrode of radius 6.35 mm has a gap distance of 25 mm to the needle. The electrode axis is defined as the symmetry axis. z is positive below the needle head, r is the distance from the electrode axis while ϕ follows the conventional definition.

2.4

Material parameters

As seen in section 1.3, transformer oil contains a mixture of mostly naphthenic and paraffinic molecules with the addition of a small amount of aromatic hydrocarbons. Since the naphthenic and paraffinic molecules have similarly high ionization poten-tials [9] they are not distinguished from each other in the simulations. The material parameters for naphthenic/paraffinic molecules are found in table 2.4. Aromatic molecules have a lower ionization potential as can be seen in table 2.3. Table 2.2 contains the material parameters that apply to both naphthenic/paraffinic and aromatic molecules.

12 Methodology

Parameter Symbol Value Unit References

Number density n0 1025 m−3 [9]

Ionization potential ∆ 1.58 × 10_9.86−18 _eVJ [6]

Molecular separation a 3.0 × 10−10 m [9, 19]

Effective electron mass m∗ 0.1me= 9.11 × 10−32 kg [9]

Table 2.4. Material parameters for naphthenic/paraffinic hydrocarbons.

Parameter Symbol Species 1 Species 2 Unit

Number density n0 1023 1025 m−3

Ionization potential ∆ 1.36 × 10_8.50−18 1.58 × 10_9.86−18 _eVJ

Molecular separation a 3.0 × 10−10 3.0 × 10−10 m

Effective electron mass m∗ 0.1me= 9.11 × 10−32 9.11 × 10−32 kg

Table 2.5. Material parameters for oil mixture 1 from Hwang [9].

Due to their low ionization potential, aromatic molecules ionize more easily than their naphthenic/paraffinic counterparts. However, if more ionization energy is available (which is the case when simulating high voltages) the naphthenic/paraffinic molecules can release more free charges and cause more dangerous streamers. Therefore, following the organization of results in Hwang [9], the following oils are simulated:

• Aromatic hydrocarbons with low number density.

• Naphthenic/paraffinic hydrocarbons with high number density. • Oil mixtures containing several molecular species.

The mixtures consist of naphthenic/paraffinic molecules with a small addition of molecules with a lower potential. Two mixtures are simulated, and they are identical except for the ionization potential of this additive. The material data for the mixtures are found in tables 2.5 and 2.6. Note that mixture 2 is precisely the naphthenic/paraffinic and aromatic hydrocarbons from tables 2.3 and 2.4 combined. This is the same configuration as found in Hwang [9], where oil mixture 1 is named “Case 1” and oil mixture 2 is referred to as “Case 2”.

2.5 Mathematical model 13

Parameter Symbol Species 1 Species 2 Unit

Number density n0 1023 1025 m−3

Ionization potential ∆ 9.92 × 10_6.20−19 1.58 × 10_9.86−18 _eVJ

Molecular separation a 3.0 × 10−10 3.0 × 10−10 m

Effective electron mass m∗ 0.1me= 9.11 × 10−32 9.11 × 10−32 kg

Table 2.6. Material parameters for oil mixture 2 from Hwang [9].

2.5

Mathematical model

The charge continuities in eq. (2.1) are now explicitly stated together with Gauss’ law on the differential form to achieve the complete description.

−∇ ·(r0∇V) = q (np− nn− ne) (2.14) ∂np ∂t + ∇ · (µpnpE) = GF(E) − npnnRpn− npneRpe (2.15) ∂nn ∂t − ∇ ·(µnnnE) = −npnnRpn+ ne τa (2.16) ∂ne ∂t − ∇ ·(µeneE) = GF(E) − ne τa − npneRpe (2.17)

This model is similar to the models derived in O’Sullivan [19] and Hwang [9] with the main difference being the removal of the thermal equation (see section 1.4).

In the case of mixed oils, the charge generation parameter is modified to be the sum of the individual charge generation terms:

GF = GF1+ GF2 (2.18)

2.5.1

Boundary conditions

The following boundary conditions apply:

2.5.1.a Charge carrier continuity, eqs. (2.15) to (2.17)

Both the walls and electrodes are set to be penetrable by the charge carriers. This is formulated as a Neumann boundary condition by setting the normal gradient of the corresponding number densities to zero:

n · ∇ne= n · ∇np= n · ∇nn= 0 (2.19)

2.5.1.b Electric potential, eq. (2.3)

The sphere electrode is a conductor and has a potential fixed at (V = 0). The needle electrodes is a conductor and goes from an electric potential of V = 0 to

14 Methodology

(a) Electric field distribution as a function of the spatial coordinates r [m] and z [m].

0 0.2 0.4 0.6 0.8 1 x 10−3 0 0.5 1 1.5 2 2.5 3 3.5 4x 10 8

(b) Electric field distribution along the electrode axis z [m].

Figure 2.2. Detail of the initial electric field magnitude V m−1 near the needle

electrode. The applied voltage V0 is +130 kV. The sharp needle electrode creates a very

large electric field in the surrounding area that quickly goes to zero at further distances.

V = V0 at t = 0 as a step Heaviside function. Insulating boundaries (outer walls)

are set as a Neumann boundary condition to have no normal component of the electric potential gradient:

n · ∇V = 0 (2.20)

2.5.2

Initial value conditions

At t = 0+_{, the electric electric field for all points in the geometry is given by the}

solution to Laplace’s equation:

∇ ·(0r∇V) = 0 (2.21)

where E is found by

E= −∇V (2.22)

The resulting field distribution for an applied voltage of V0= 130 kV can be seen

in fig. 2.2.

2.6

Numerical solution

The intricate coupling of the system of equations in section 2.5 together with the complicated geometry makes anything but a numerical solution for the streamer model infeasible. This thesis uses the solver implemented in OpenFOAM 1.6-ext by ABB Corporate Research that was introduced in section 1.4.

2.6 Numerical solution 15

Parameter Symbol Default value

Smallest mesh size δr 500 nm

Maximum Courant number Cmax 2.0

Coupling Tolerance couplingTolerance 5 × 10−3

Maximum number of iterations nMaxIter 20

Extra non-relaxed iteration extraNoRelax false

Number of parallel cores used nCores

-Table 2.7. Parameters to the numerical solver

The model is computationally expensive to solve even numerically. To reduce computational complexity, the model is simplified to a two-dimensional geometry with axial symmetry. The solver takes several arguments that controls its numerical behavior. The most important parameters are listed together with default values (if applicable) in table 2.7. These parameters are discussed in the following sections.

2.6.1

Meshing

Since the finite volume method uses space discretization (meshing), great care must be taken to generate a good mesh. If the mesh is too fine-grained, the computation will be unwieldy but a mesh that is too coarse causes problems with convergence and numerical stability. The bulk of the oil is meshed as a square mesh, thereby minimizing the number of mesh interfaces that the streamer must pass through at an acute angle. The geometry is very large compared to the size of the area of interest which makes a mesh with varying resolution suitable. The region around the needle is the densest, and the mesh size in this region is defined as δr. This high-resolution region is made large enough for each simulation to contain the streamer for all time steps.

Since computational complexity is strongly dependent of the mesh resolution, care must be taken to avoid excessive simulation times from a mesh resolution that is too high.

2.6.2

Iterative coupling tolerance

The numerical solver uses an iterative, under-relaxed algorithm to find the solution to the coupled equations in each time step. In every iteration the residual is checked against a threshold level, couplingTolerance. Varying this threshold is a powerful method to change the numerical accuracy, but a tolerance that is too small leads to excessive simulation times. If the solver does not reach the requested coupling tolerance within nMaxIter iterations, the simulation is halted and returns an error. Therefore, a lowering of couplingTolerance should be accompanied by an increase of nMaxIter.

It is possible to perform a final iteration after couplingTolerance has been reached by setting the variable extraNoRelax to true. The final iteration is then

16 Methodology performed without under-relaxation, which might increase the overall accuracy. Enabling this setting results in a marked increase in simulation time.

2.6.3

Courant number

The Courant number is an important parameter when numerically solving partial differential equations using discretization. On a conceptual level, the Courant number C can be understood as

C ∝ uδt

δr (2.23)

for each cell, where u is the solution (streamer) velocity, δt is the time step and δr is the cell size. For static meshes and a given maximum Courant number Cmax,

the solver then chooses the largest possible time step according to eq. (2.23). The simulation time increases inversely proportional to the Courant number and value that is too large can lead to numerical problems.

2.6.4

Parallel computing

OpenFOAM performs very well with parallel computation. Combined with the large computational complexity of the model it is possible to gain a remarkable speedup with parallelized computation. The high-performance cluster at ABB Corporate Research in Västerås is supplied by Gridcore AB. The cluster uses quad-core Intel Xeon processors with a clock frequency of 2.67 GHz. The solver parameter nCores sets the number of cores that the solver uses. Most simulations in this thesis use 16 cores, whereas the three-dimensional simulations in chapter 5 use 64.

Chapter 3

Results

The numerical solutions to the mathematical model in section 2.5 are presented in this chapter. A large number of simulations have been performed, and the simulation parameters are the same as in Hwang [9]. The results are grouped by oil type (see section 2.4), voltage and, if applicable, mesh size. Table 3.1 lists the investigated voltage and oil combinations. Note that the mixed oil cases with voltages other than 300 kV were not investigated by Hwang. The significance of the findings in this chapter are discussed in chapter 4.

3.1

Classification of streamers

It will be seen that the resulting streamers can be categorized into the three general groups listed in fig. 3.1. If the electric field is low, field ionization will be very weak due to its exponential relation to the field. This might lead to a lack of streamer formation as in fig. 3.1a, which is called glow. Focused streamers are streamers that focus to become narrow and have continued propagation, as seen in fig. 3.1b Finally, bubbly streamers are streamers that grow much wider than the needle and have an almost spherical appearance, see fig. 3.1c.

V0 +80 kV +130 kV +200 kV +300 kV +400 kV

Aromatic • • • •

Naphthenic/paraffinic • • •

Oil mixture 1 ◦ • ◦

Oil mixture 2 ◦ ◦ •

Table 3.1. Simulated oil type and voltage combinations. Entries marked with • have

been simulated. Entries marked with ◦ have been simulated here but not in Hwang [9].

18 Results

(a) Glow (b) Focused (c) Bubbly

Figure 3.1. Electric field magnitude plot of the three major streamer types

3.2

Aromatic hydrocarbons

This section concerns the results for aromatic hydrocarbons with material parame-ters from table 2.3.

3.2.1

Applied voltage V

= +80 kV

The first case to be investigated is the applied voltage of 80 kV. The Laplacian field just below the needle electrode is 2.2 × 108_{V m}−1_{, which is enough to cause}

some field ionization. As can be seen in fig. 3.2 it is however not enough to initiate a propagating streamer. This plot shows the electric field magnitude as a function of the spatial coordinates. Due to the lack of streamer formation this result is classified as glow.

3.2 Aromatic hydrocarbons 19

3.2.2

Applied voltage V

= +130 kV

The Laplacian field just below the needle electrode at this voltage is 4 × 108_{kV. In}

this case, several meshes have been investigated in order to perform the convergence analysis in section 4.1.

3.2.2.a Mesh with δr= 500 nm

The solution for a mesh with a resolution of 500 nm is presented in this section. Figure 3.3 shows the movement of the streamer tip by measuring the electric field magnitude and charge density np− nn− nealong the electrode axis. A propagating

streamer that starts at the needle electrode and moves towards the grounded sphere can clearly be seen. The extreme electric field around the needle leads to field ionization, causing neutral molecules to be torn apart into free electrons and positive ions. The highly mobile electrons will quickly drift towards the positive needle while ions move much slower. This results in a displacement between the negative electrons and positive ions which in turn causes an electric field of its own. The result is a field that is weak at the needle electrode and with a sharp peak at a point z > 0. As seen in fig. 3.3, this field then ionizes new molecules and repeats this process further out along the electrode axis.

Figure 3.4 shows the electric field magnitude as a function of the spatial coordinates. It can be seen that the streamer becomes focused at 75 ns and retains this width for the duration of the simulation. Thus, this result is of the focused type and the average velocity is approximately 3.1 km s−1_.

3.2.2.b Mesh with δr= 300 nm

In order to investigate the numerical convergence, an even higher mesh resolution was investigated. This section concerns the numerical solution using a mesh with δr = 300 nm. Figure 3.5 shows the electric field magnitude and charge density np− nn− nealong the electrode axis. Figure 3.6 shows the electric field magnitude

as a function of the spatial coordinates. The resulting streamer is focused and has an average velocity of 3.3 × 103_{m s}−1_{which is slightly higher than in section 3.2.2.a.}

3.2.2.c Mesh with δr= 1000 nm

The numerical solution over a mesh with δr = 1000 nm is presented in this section. Figure 3.7 shows the electric field magnitude as a function of the spatial coordinates. It can clearly be seen that the streamer is deflected from the electrode axis at a 45° angle. This result differs strongly from the finer mesh in section 3.2.2.a, showing that this solution is mesh-limited. A more detailed investigation into the numerical convergence is performed in section 4.1.1.

3.2.3

Applied voltage V

= +200 kV

The Laplacian field just below the needle electrode at this voltage is 6 × 108_kV.

20 Results stress on the oil. Figure 3.8 shows the electric field magnitude and charge density np− nn− nealong the electrode axis. Figure 3.9 shows the electric field magnitude

as a function of the spatial coordinates. It can be seen that the streamer type has changed to become bubbly.

3.2.4

Applied voltage V

= +300 kV

Figure 3.10 shows the electric field magnitude and charge density np− nn− nealong

the electrode axis. Figure 3.11 shows the electric field magnitude as a function of the spatial coordinates. An applied voltage of +300 kV clearly increases the electric stress on the oil, with the Laplacian field exceeding 8 × 108_{kV near the needle tip.}

3.2 Aromatic h ydro carb ons 21

22 Results 0 0.2 0.4 0.6 0.8 1 x 10−3 0 0.5 1 1.5 2 2.5 3 3.5 4x 10 8 t=0+ t=25ns t=50ns t=75ns t=100ns t=150ns t=200ns

(a) Electric field magnitudeV m−1. Note that the field at t = 0+ corresponds to the Laplacian field.

0 0.2 0.4 0.6 0.8 1 x 10−3 −500 0 500 1000 1500 2000 2500 t=0+ t=25ns t=50ns t=75ns t=100ns t=150ns t=200ns

(b) Space charge densityC m−3

Figure 3.3. Temporal dynamics along the needle-sphere electrode axis [m] for aromatic

3.2 Aromatic h ydro carb ons 23

24 Results 0 0.2 0.4 0.6 0.8 1 x 10−3 0 0.5 1 1.5 2 2.5 3 3.5 4x 10 8 t=0+ t=25ns t=50ns t=75ns t=100ns t=150ns t=200ns

(a) Electric field magnitudeV m−1. Note that the field at t = 0+ corresponds to the Laplacian field.

0 0.2 0.4 0.6 0.8 1 x 10−3 −500 0 500 1000 1500 2000 2500 3000 3500 t=0+ t=25ns t=50ns t=75ns t=100ns t=150ns t=200ns

(b) Space charge densityC m−3

Figure 3.5. Temporal dynamics along the needle-sphere electrode axis [m] for aromatic

3.2 Aromatic h ydro carb ons 25

26

Results

Figure 3.7. Electric field magnitudeV m−1at t = 25, 50, 75 and 300 ns for aromatic hydrocarbons with δr = 1000 nm and V0= +130 kV.

3.2 Aromatic hydrocarbons 27 0 0.2 0.4 0.6 0.8 1 x 10−3 0 1 2 3 4 5 6x 10 8 t=0+ t=5ns t=20ns t=60ns t=100ns

(a) Electric field magnitudeV m−1. Note that the field at t = 0+ corresponds to the Laplacian field.

0 0.2 0.4 0.6 0.8 1 x 10−3 −50 0 50 100 150 200 250 t=0+ t=5ns t=20ns t=60ns t=100ns

(b) Space charge densityC m−3

Figure 3.8. Temporal dynamics along the needle-sphere electrode axis [m] for aromatic

28

Results

3.2 Aromatic hydrocarbons 29 0 0.2 0.4 0.6 0.8 1 x 10−3 0 1 2 3 4 5 6 7 8 9x 10 8 t=0+ t=5ns t=20ns t=60ns t=100ns

(a) Electric field magnitudeV m−1. Note that the field at t = 0+ corresponds to the Laplacian field.

0 0.2 0.4 0.6 0.8 1 x 10−3 −20 0 20 40 60 80 100 120 140 t=0+ t=5ns t=20ns t=60ns t=100ns

(b) Space charge densityC m−3

Figure 3.10. Temporal dynamics along the needle-sphere electrode axis [m] for aromatic

30

Results

3.3 Naphthenic/paraffinic hydrocarbons 31

3.3

Naphthenic/paraffinic hydrocarbons

This section concerns the simulation results for naphthenic/paraffinic hydrocarbons with a high ionization potential.

3.3.1

Applied voltage V

= +130 kV

The numerical solution to the mathematical model over a mesh with δr = 500 nm is presented in this section. Figure 3.12 shows the electric field magnitude as a function of the spatial coordinates. Due to the higher ionization potential of the naphthenic/paraffinic hydrocarbons in comparison to the aromatic hydrocarbons a voltage of +130 kV is not enough to initiate a streamer. This result is therefore of the glow type.

3.3.2

Applied voltage V

= +200 kV

The numerical solution to the mathematical model over a mesh with δr = 500 nm is presented in this section. Figure 3.13 shows the electric field magnitude as a function of the spatial coordinates. Even though the voltage has been increased to +200 kV it is not enough to initiate streamer propagation, resulting in glow.

3.3.3

Applied voltage V

= +300 kV

The numerical solution to the mathematical model over a mesh with δr = 500 nm is presented in this section. Figure 3.14 shows the electric field magnitude and charge density along the electrode axis. Figure 3.15 shows the electric field magnitude as a function of the spatial coordinates. This streamer is focused and has an average velocity of approximately 4.3 km s−1_{. It can be seen that the higher ionization}

potential of the naphthenic/paraffinic hydrocarbons delayed the inception of a streamer to a voltage of V0= +300 kV.

32

Results

Figure 3.12. Electric field magnitudeV m−1at t = 25, 50, 75 and 100 ns for naphthenic/paraffinic hydrocarbons with δr = 500 nm and V0= +130 kV.

3.3 Naph thenic/paraffinic h ydro carb ons 33

Figure 3.13. Electric field magnitudeV m−1at t = 25, 50, 75 and 100 ns for naphthenic/paraffinic hydrocarbons with δr = 500 nm and V0= +200 kV.

34 Results 0 0.2 0.4 0.6 0.8 1 x 10−3 0 1 2 3 4 5 6 7 8 9x 10 8 t=0+ t=50ns t=75ns t=100ns t=125ns t=150ns

(a) Electric field magnitudeV m−1. Note that the field at t = 0+ corresponds to the Laplacian field.

0 0.2 0.4 0.6 0.8 1 x 10−3 −1000 0 1000 2000 3000 4000 5000 6000 t=0+ t=50ns t=75ns t=100ns t=125ns t=150ns

(b) Space charge densityC m−3

Figure 3.14. Temporal dynamics along the needle-sphere electrode axis [m] for

3.3 Naph thenic/paraffinic h ydro carb ons 35

Figure 3.15. Electric field magnitudeV m−1at t = 25, 75, 100 and 150 ns for naphthenic/paraffinic hydrocarbons with δr = 500 nm and V0= +300 kV.

36 Results

3.4

Oil mixture 1

This section concerns the results for a mixed oil with parameters according to table 2.5. It can be seen that this oil consists of a large number of naph-thenic/paraffinic molecules with a small addition (1%) of molecules with a medium-high ionization potential.

3.4.1

Applied voltage V

= +130 kV

The numerical solution to the mathematical model over a mesh with δr = 500 nm for oil mixture 1 is presented in this section. Figure 3.16 shows the electric field magnitude as a function of the spatial coordinates. It can be seen that no streamer is formed; only glow is apparent.

3.4.2

Applied voltage V

= +300 kV

The numerical solution to the mathematical model over a mesh with δr = 500 nm for oil mixture 1 is presented in this section.

Figure 3.17 shows the electric field magnitude and charge density along the electrode axis. Figure 3.18 shows the electric field magnitude as a function of the spatial coordinates. It is clear that this result is of the focused streamer type and the average velocity is approximately 7.5 km s−1_.

3.4.3

Applied voltage V

= +400 kV

The numerical solution to the mathematical model over a mesh with δr = 500 nm for oil mixture 1 is presented in this section.

Figure 3.19 shows the electric field magnitude and charge density along the electrode axis. Figure 3.20 shows the electric field magnitude as a function of the spatial coordinates. This result clearly shows a bubbly streamer.

3.5

Oil mixture 2

This section concerns the results for a mixed oil with parameters according to table 2.6. It can be seen that this oil consists of a large number of naph-thenic/paraffinic molecules with a small addition (1%) of aromatic molecules.

3.5.1

Applied voltage V

= +80 kV

The numerical solution to the mathematical model over a mesh with δr = 500 nm for oil mixture 2 is presented in this section.

Figure 3.21 shows the electric field magnitude as a function of the spatial coordinates.

3.5 Oil mixture 2 37

3.5.2

Applied voltage V

= +130 kV

The numerical solution to the mathematical model over a mesh with δr = 500 nm for oil mixture 2 is presented in this section.

Figure 3.22 shows the electric field magnitude and charge density along the electrode axis. Figure 3.23 shows the electric field magnitude as a function of the spatial coordinates. The resulting streamer is of the focused type and its average velocity is approximately 3.4 km s−1_.

3.5.3

Applied voltage V

= +300 kV

The numerical solution to the mathematical model over a mesh with δr = 500 nm for oil mixture 2 is presented in this section.

Figure 3.24 shows the electric field magnitude and charge density along the electrode axis. Figure 3.25 shows the electric field magnitude as a function of the spatial coordinates. It can be seen that a bubbly streamer is formed.

38

Results

3.5 Oil mixture 2 39 0 0.2 0.4 0.6 0.8 1 x 10−3 0 1 2 3 4 5 6 7 8 9x 10 8 t=0+ t=10ns t=20ns t=30ns t=40ns t=50ns t=60ns t=70ns t=80ns t=90ns t=100ns

(a) Electric field magnitudeV m−1. Note that the field at t = 0+ corresponds to the Laplacian field.

0 0.2 0.4 0.6 0.8 1 x 10−3 −500 0 500 1000 1500 2000 2500 3000 3500 t=0+ t=10ns t=20ns t=30ns t=40ns t=50ns t=60ns t=70ns t=80ns t=90ns t=100ns

(b) Space charge densityC m−3

Figure 3.17. Temporal dynamics along the needle-sphere electrode axis [m] for oil

40

Results

3.5 Oil mixture 2 41 0 0.2 0.4 0.6 0.8 1 x 10−3 0 2 4 6 8 10 12x 10 8 t=0+ t=5ns t=25ns t=100ns

(a) Electric field magnitudeV m−1. Note that the field at t = 0+ corresponds to the Laplacian field.

0 0.2 0.4 0.6 0.8 1 x 10−3 −50 0 50 100 150 200 250 300 350 400 t=0+ t=5ns t=25ns t=100ns

(b) Space charge densityC m−3

Figure 3.19. Temporal dynamics along the needle-sphere electrode axis [m] for oil

42

Results

3.5

Oil

mixture

2

43

44 Results 0 0.2 0.4 0.6 0.8 1 x 10−3 0 0.5 1 1.5 2 2.5 3 3.5 4x 10 8 t=0+ t=25ns t=50ns t=75ns t=100ns t=125ns t=150ns t=175ns t=200ns

(a) Electric field magnitudeV m−1. Note that the field at t = 0+ corresponds to the Laplacian field.

0 0.2 0.4 0.6 0.8 1 x 10−3 −500 0 500 1000 1500 2000 2500 t=0+ t=25ns t=50ns t=75ns t=100ns t=125ns t=150ns t=175ns t=200ns

(b) Space charge densityC m−3

Figure 3.22. Temporal dynamics along the needle-sphere electrode axis [m] for oil

3.5

Oil

mixture

2

45

46 Results 0 0.2 0.4 0.6 0.8 1 x 10−3 0 1 2 3 4 5 6 7 8 9x 10 8 t=0+ t=5ns t=10ns t=15ns t=20ns t=30ns t=50ns t=75ns t=100ns t=150ns t=200ns t=300ns

(a) Electric field magnitudeV m−1. Note that the field at t = 0+ corresponds to the Laplacian field.

0 0.2 0.4 0.6 0.8 1 x 10−3 −4000 −2000 0 2000 4000 6000 8000 10000 12000 14000 t=0+ t=5ns t=10ns t=15ns t=20ns t=30ns t=50ns t=75ns t=100ns t=150ns t=200ns t=300ns

(b) Space charge densityC m−3. This simulation shows bad and noisy results for the charge density. The reason for this is unknown, but the rest of the simulation appears unaffected by this problem.

Figure 3.24. Temporal dynamics along the needle-sphere electrode axis [m] for oil

3.5

Oil

mixture

2

47

Chapter 4

Discussion

This chapter will discuss and analyze the results presented in chapter 3. Convergence analysis is very important when performing numerical calculations with space discretization. A detailed discussion of the convergence of all streamer types is performed in sections 4.1 to 4.2. The categorization of streamers into the three major types is revisited, and the results will then be compared with Hwang [9].

4.1

Focused streamers

This section will analyze the impact of numerical parameters on focused streamer results. As discussed in section 3.1, focused streamers are narrow and have continued propagation. The following oil and voltage combinations exhibited focused streamers with at least one mesh:

• Section 3.2.2: Aromatic hydrocarbons, V0= +130 kV.

• Section 3.3.3: Naphthenic hydrocarbons, V0= +300 kV.

• Section 3.4.2: Oil mixture 1, V0= +300 kV.

• Section 3.5.2: Oil mixture 2, V0= +130 kV.

Only the case with aromatic hydrocarbons and V0 = +130 kV is discussed and

the convergence analysis will then be assumed to be valid for all focused streamer results.

4.1.1

Streamer deflection with δr = 1000 nm

The results that are achieved with a mesh of δr = 1000 nm can be found in section 3.2.2.c. This result differs distinctly from those with higher resolutions, which suggests a mesh-based convergence issue. Several attempts have been made to try to work around the convergence issues. In turn, these are:

• Lowering couplingTolerance by an order of magnitude. 49

50 Discussion • Extra non-relaxed iteration.

• Lowering Cmaxfrom 2.0 to 0.5.

All of these modified simulations take much longer to run due to the added complexity for the solver. These modifications barely made a difference for the results, and the overall problem remained. This can be seen by close inspection of the charge generation field GF at the streamer head. Focused streamers have

very thin streamer heads, and confining these to a resolution of 1000 nm loses too much information. The longer the simulation is run, the narrower the streamer head becomes until the charge generation is too small to continue the streamer propagation. The field is less weakened along the diagonal axis of the mesh blocks due to the length of that axis. What follows is a weaker field along the electrode axis and a slightly stronger field at a 45◦ _{angle. This results in a streamer that}

deflects from the electrode axis, moving out into the oil as seen in fig. 3.7. Since the simulations are axisymmetric, a streamer that deflects from the electrode axis as in fig. 3.7 is unrealistic. Such behavior would mean that the focused streamer turns into an inverted-funnel-like shape, which to the author’s best knowledge has not been seen in experiment. It can therefore be concluded that a mesh with δr = 1000 nm is too coarse to work with focused streamers.

4.1.2

Convergence analysis

Having established that δr = 1000 nm is too coarse to work with focused streamers, finer meshes will be used. The results in section 3.2.2.a show no streamer deflection, meaning that a doubling of mesh resolution had a dramatic impact. The question now is if 500 nm is fine enough and if the results are valid. As with δr = 1000 nm, the following modifications were attempted

• Lowering couplingTolerance by an order of magnitude. • Extra non-relaxed iteration.

• Lowering Cmaxfrom 2.0 to 0.5.

The results are very similar to the unmodified case and the corresponding plots have therefore been omitted for brevity. The results from section 3.2.2.b are from a mesh with even finer resolution, δr = 300 nm.

A comparison of the streamer positions and velocities from the above simulations can be found in fig. 4.1. This plot shows the impact of the mesh resolution and numerical settings on the streamer propagation. As already noted above, δr = 1000 nm deviates significantly from the other simulations. It can also be seen that there is a small yet noticeable difference between δr = 300 nm and δr= 500 nm, indicating that mesh-based convergence hasn’t been fully reached for δr = 500 nm. However, the running time of δr = 300 nm is very long due to the very fine mesh. In this case, a single run with this resolution took over 2600 CPU-hours, making it infeasible for a large number of simulations. Therefore, for the remaining simulations, δr = 500 nm must be seen as ”good enough” for focused streamers. This resolution is used in all focused streamer cases.

4.1 Focused streamers 51 0 0.5 1 1.5 2 2.5 3 x 10−7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1x 10 −3 δ _{r = 1000nm} δ r = 500nm δ r = 500nm, coupling δ r = 500nm, extraNoRelax δ r = 500nm, C max=0.5 δ r = 300nm (a) Position 0 0.5 1 1.5 2 2.5 3 x 10−7 0 1000 2000 3000 4000 5000 6000 7000 δ _{r = 1000nm} δ r = 500nm δ r = 500nm, coupling δ r = 500nm, extraNoRelax δ r = 500nm, C max=0.5 δ r = 300nm (b) Velocity

Figure 4.1. Position and velocity comparison along the electrode axis for aromatic