Department of Physics, Chemistry and Biology

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1 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 SE Linköping, Sweden

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3 Master s Thesis in Applied Physics LiTH-IFM-EX--12/2696--SE Modeling and OpenFOAM simulation of streamers in transformer oil Jonathan Fors Supervisor: Examiner: Nils Lavesson ABB Corporate Research Weine Olovsson ifm, Linköping University Peter Münger ifm, Linköping University Linköping, 21 June, 2012

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5 Avdelning, Institution Division, Department IFM Department of Physics, Chemistry and Biology Linköping University SE Linköping, Sweden Datum Date Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport ISBN ISRN LiTH-IFM-EX--12/2696--SE Serietitel och serienummer Title of series, numbering ISSN URL för elektronisk version 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 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. Nyckelord Keywords Computational Physics, High voltage, OpenFOAM, Streamer, Electric insulation, Electrodynamics, Finite Volume Method

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7 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 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 elektriskt 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 elementmetoden. I denna uppsats undersöks konsekvenserna av att byta metod till finita volymsmetoden genom att implementera en lösare i OpenFOAM. En standardiserad 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 genomförts där simulationen utökas från en tvådimensionell axisymmetrisk geometri til tre dimensioner. v

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9 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. vii

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11 To Anna and my family

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13 Contents 1 Introduction Challenges of electric insulation Streamers in transformer oil Liquid insulation Research objectives Methodology Deriving the charge continuity equation Generation and removal of charge carriers Field-dependent molecular ionization Field-dependent ionic dissociation Recombination and attachment Geometry Material parameters Mathematical model Boundary conditions Initial value conditions Numerical solution Meshing Iterative coupling tolerance Courant number Parallel computing Results Classification of streamers Aromatic hydrocarbons Applied voltage V 0 = +80 kv Applied voltage V 0 = +130 kv Applied voltage V 0 = +200 kv Applied voltage V 0 = +300 kv Naphthenic/paraffinic hydrocarbons Applied voltage V 0 = +130 kv Applied voltage V 0 = +200 kv Applied voltage V 0 = +300 kv xi

14 xii Contents 3.4 Oil mixture Applied voltage V 0 = +130 kv Applied voltage V 0 = +300 kv Applied voltage V 0 = +400 kv Oil mixture Applied voltage V 0 = +80 kv Applied voltage V 0 = +130 kv Applied voltage V 0 = +300 kv Discussion Focused streamers Streamer deflection with δr = 1000 nm Convergence analysis Bubbly streamers Streamer classification revisited Charge generation analysis Charge generation in mixed oils Comparison with previous work Aromatic hydrocarbons Naphthenic/paraffinic hydrocarbons Mixed oils Proof of concept: Expanding to the third dimension Methodology Results Discussion Conclusions Future work Lightning impulse voltage Field-dependent molecular ionization potential

15 Contents List of Figures 1.1 Examples of hydrocarbons found in crude oil IEC-standardized testing geometry with a 25 mm gap. Note the sharp needle electrode (top) and the grounded spherical electrode (bottom) Detail of the initial electric field magnitude [ V m 1] near the needle electrode Electric field magnitude plot of the three major streamer types Electric field magnitude [ V m 1] at t = 25, 50, 75 and 100 ns for aromatic hydrocarbons with δr = 500 nm and V 0 = +80 kv Temporal dynamics along the needle-sphere electrode axis [m] for aromatic hydrocarbons with δr = 500 nm and V 0 = +130 kv Electric field magnitude [ V m 1] at t = 25, 50, 75 and 300 ns for aromatic hydrocarbons with δr = 500 nm and V 0 = +130 kv Temporal dynamics along the needle-sphere electrode axis [m] for aromatic hydrocarbons with δr = 300 nm and V 0 = +130 kv Electric field magnitude [ V m 1] at t = 25, 50, 75 and 300 ns for aromatic hydrocarbons with δr = 300 nm and V 0 = +130 kv Electric field magnitude for δr = 1000 nm and V 0 = +130 kv Temporal dynamics along the needle-sphere electrode axis [m] for aromatic hydrocarbons with δr = 500 nm and V 0 = +200 kv Electric field magnitude [ V m 1] at t = 25, 50, 75 and 100 ns for aromatic hydrocarbons with δr = 500 nm and V 0 = +200 kv Temporal dynamics along the needle-sphere electrode axis [m] for aromatic hydrocarbons with δr = 500 nm and V 0 = +300 kv Electric field magnitude [ V m 1] at t = 25, 50 and 100 ns for aromatic hydrocarbons with δr = 500 nm and V 0 = +300 kv Electric field magnitude [ V m 1] at t = 25, 50, 75 and 100 ns for naphthenic/paraffinic hydrocarbons with δr = 500 nm and V 0 = +130 kv xiii

16 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 V 0 = +200 kv Temporal dynamics along the needle-sphere electrode axis [m] for naphthenic/paraffinic hydrocarbons with δr = 500 nm and V 0 = +300 kv Electric field magnitude [ V m 1] at t = 25, 75, 100 and 150 ns for naphthenic/paraffinic hydrocarbons with δr = 500 nm and V 0 = +300 kv Electric field magnitude [ V m 1] at t = 25, 50, 75 and 100 ns for oil mixture 1 with δr = 500 nm and V 0 = +130 kv Temporal dynamics along the needle-sphere electrode axis [m] for oil mixture 1 with δr = 500 nm and V 0 = +300 kv Electric field magnitude [ V m 1] at t = 25, 50, 75 and 100 ns for oil mixture 1 with δr = 500 nm and V 0 = +300 kv Temporal dynamics along the needle-sphere electrode axis [m] for oil mixture 1 with δr = 500 nm and V 0 = +400 kv Electric field magnitude [ V m 1] at t = 25, 50 and 300 ns for oil mixture 1 with δr = 500 nm and V 0 = +400 kv Electric field magnitude [ V m 1] at t = 25, 50, 100 and 300 ns for oil mixture 2 with δr = 500 nm and V 0 = +80 kv Temporal dynamics along the needle-sphere electrode axis [m] for oil mixture 2 with δr = 500 nm and V 0 = +130 kv Electric field magnitude [ V m 1] at t = 25, 50, 100 and 300 ns for oil mixture 2 with δr = 500 nm and V 0 = +130 kv Temporal dynamics along the needle-sphere electrode axis [m] for oil mixture 2 with δr = 500 nm and V 0 = +300 kv Electric field magnitude [ V m 1] at t = 25, 100 and 300 ns for oil mixture 2 with δr = 500 nm and V 0 = +300 kv Position and velocity comparison along the electrode axis for aromatic hydrocarbons with V 0 = +130 kv Position and velocity comparison along the electrode axis for aromatic hydrocarbons with V 0 = +300 kv Visualization of the three-dimensional simulation at t = 87 ns Visualization of the mesh at t = 87 ns for the three-dimensional simulation Temporal dynamics along the needle-sphere electrode axis [m] for the three-dimensional simulation. δr = 500 nm and V 0 = +130 kv. 63 List of Tables 2.1 Fundamental physical constants Material parameters of general transformer oil Material parameters for aromatic hydrocarbons

17 Contents xv 2.4 Material parameters for naphthenic/paraffinic hydrocarbons Material parameters for oil mixture Material parameters for oil mixture Parameters to the numerical solver Simulated oil type and voltage combinations Coefficients for charge generations for all species in section Measured peak Laplacian electric field directly below the needle at t = 0 + for different applied voltages Calculated charge generation G F [ m 3 s 1] just below the needle tip for each oil type and voltage [kv] at t = Calculated charge generation G F [ m 3 s 1] for mixed oils just below the needle tip at t = Comparison of peak electric field magnitude [ 10 8 V m 1] at 100 ns and average streamer velocity [ 10 3 m s 1]

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19 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 1

20 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 (SF 6 ) 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.

21 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 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 cyclohexane (fig. 1.1b). Paraffinic hydrocarbons consist of straight alkane chains with the chemical formula C n H 2n+2. Examples include hexane, octane (fig. 1.1c) and heptane. When designing transformer oil there are several factors to take into consideration. 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].

22 4 Introduction (a) Toluene ( C 6 H 5 CH 3 ), an aromatic hydrocarbon ( ) (b) Cyclohexane C 6 H 12, a naphthenic hydrocarbon (c) Octane ( C 8 H 18 ), 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]. Advancements 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

23 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.

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25 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 thermodynamic 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 n e, n p and n n, 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

26 8 Methodology Constant Symbol Numerical value Unit Speed of light in vacuum c m s 1 Permeability of vacuum µ 0 4π 10 7 H m 1 Permittivity of vacuum ɛ 0 = ( µ 0 c0) F m 1 Elementary charge q C Electron mass m e kg Planck constant h J s Table 2.1. Fundamental physical constants is ρ i t + j i = q (G i R i ) (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: v i = µ i E (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 j i = n i qv i = n i qµ i E (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: n i t + (µ in i E) = G i R i (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 to be true at all times. G p R p = G n R n + G e R e (2.6)

27 2.2 Generation and removal of charge carriers 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 G F (E) = qn 0aE h exp ( π2 m a 2 ) qh 2 E (2.7) where is the molecular ionization potential, E the local electric field magnitude, a the molecular separation, n 0 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 G i it must be noted that it only applies to positive ions and electrons Field-dependent ionic dissociation G p = G F (2.8) G n = 0 (2.9) G e = G F (2.10) 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 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 R pn and with electrons with a rate of R pe. Negative ions are removed by recombination with positive ions with a rate of R pn and are generated by neutral molecules attaching to electrons with a rate of τ 1 a

28 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 = np nn Rpn + np ne Rpe ne Rn = np nn Rpn τa ne Re = + np ne Rpe τa 2.3 (2.11) (2.12) (2.13) Geometry The simulations are performed over a geometry as defined by the IEC 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

29 2.4 Material parameters 11 Parameter Symbol Value Dimension References Positive ion mobility µ p m 2 V 1 s 1 [1, 5] Negative ion mobility µ n m 2 V 1 s 1 [1, 5] Electron mobility µ n m 2 V 1 s 1 [2, 12, 20] Relative permittivity ɛ r [5, 15] Ion-ion recombination rate R pn m 3 s 1 [9] Ion-electron recombination rate R pe m 3 s 1 [9] Electron attachment time τ a s [2] Table 2.2. Material parameters of general transformer oil. Parameter Symbol Value Unit References Number density n m 3 [9] Ionization potential J 6.20 ev [6] Molecular separation a m [9, 19] Effective electron mass m 0.1m e = 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 needlesphere 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 potentials [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.

30 12 Methodology Parameter Symbol Value Unit References Number density n m 3 [9] Ionization potential J 9.86 ev [6] Molecular separation a m [9, 19] Effective electron mass m 0.1m e = kg [9] Table 2.4. Material parameters for naphthenic/paraffinic hydrocarbons. Parameter Symbol Species 1 Species 2 Unit Number density n m 3 Ionization potential J ev Molecular separation a m Effective electron mass m 0.1m e = 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.

31 2.5 Mathematical model 13 Parameter Symbol Species 1 Species 2 Unit Number density n m 3 Ionization potential J ev Molecular separation a m Effective electron mass m 0.1m e = 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. (ɛ r ɛ 0 V ) = q (n p n n n e ) (2.14) n p t + (µ pn p E) = G F (E) n p n n R pn n p n e R pe (2.15) n n (µ n n n E) = n p n n R pn + n e t τ a (2.16) n e t (µ en e E) = G F (E) n e n p n e R pe τ a (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: Boundary conditions The following boundary conditions apply: G F = G F1 + G F2 (2.18) 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: b Electric potential, eq. (2.3) n n e = n n p = n n n = 0 (2.19) 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

32 14 Methodology 8 4 x x 10 (a) Electric field distribution as a function of the (b) Electric field distribution along the electrode spatial coordinates r [m] and z [m]. 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 = (2.20) 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: ( 0 r V ) = 0 (2.21) E = V (2.22) where E is found by The resulting field distribution for an applied voltage of V0 = 130 kv can be seen in fig 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.

33 2.6 Numerical solution 15 Parameter Symbol Default value Smallest mesh size δr 500 nm Maximum Courant number C max 2.0 Coupling Tolerance couplingtolerance 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 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 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

34 16 Methodology performed without under-relaxation, which might increase the overall accuracy. Enabling this setting results in a marked increase in simulation time 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 C max, 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 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.

35 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 Classification of streamers It will be seen that the resulting streamers can be categorized into the three general groups listed in fig 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. V 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]. 17

36 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 parameters from table Applied voltage V0 = +80 kv The first case to be investigated is the applied voltage of 80 kv. The Laplacian field just below the needle electrode is 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.

37 3.2 Aromatic hydrocarbons Applied voltage V 0 = +130 kv The Laplacian field just below the needle electrode at this voltage is kv. In this case, several meshes have been investigated in order to perform the convergence analysis in section 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 n p n n n e along 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 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 n p n n n e along 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 m s 1 which is slightly higher than in section a 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 a, showing that this solution is mesh-limited. A more detailed investigation into the numerical convergence is performed in section Applied voltage V 0 = +200 kv The Laplacian field just below the needle electrode at this voltage is kv. With the increased voltage the electric field becomes very high, causing more

38 20 Results stress on the oil. Figure 3.8 shows the electric field magnitude and charge density n p n n n e along 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 Applied voltage V 0 = +300 kv Figure 3.10 shows the electric field magnitude and charge density n p n n n e along 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 kv near the needle tip. With this high voltage, a bubbly streamer is formed.

39 Figure 3.2. Electric field magnitude [ V m 1] at t = 25, 50, 75 and 100 ns for aromatic hydrocarbons with δr = 500 nm and V 0 = +80 kv. 3.2 Aromatic hydrocarbons 21

40 22 Results 4 x 108 t= t=25ns t=50ns t=75ns t=100ns t=150ns t=200ns x 10 3 (a) Electric field magnitude [ V m 1]. Note that the field at t = 0 + corresponds to the Laplacian field t=0 + t=25ns t=50ns t=75ns t=100ns t=150ns t=200ns x 10 3 (b) Space charge density [ C m 3] Figure 3.3. Temporal dynamics along the needle-sphere electrode axis [m] for aromatic hydrocarbons with δr = 500 nm and V 0 = +130 kv.

41 Figure 3.4. Electric field magnitude [ V m 1] at t = 25, 50, 75 and 300 ns for aromatic hydrocarbons with δr = 500 nm and V 0 = +130 kv. 3.2 Aromatic hydrocarbons 23

42 24 Results 4 x 108 t= t=25ns t=50ns t=75ns t=100ns t=150ns t=200ns x 10 3 (a) Electric field magnitude [ V m 1]. Note that the field at t = 0 + corresponds to the Laplacian field t=0 + t=25ns t=50ns t=75ns t=100ns t=150ns t=200ns x 10 3 (b) Space charge density [ C m 3] Figure 3.5. Temporal dynamics along the needle-sphere electrode axis [m] for aromatic hydrocarbons with δr = 300 nm and V 0 = +130 kv.

43 Figure 3.6. Electric field magnitude [ V m 1] at t = 25, 50, 75 and 300 ns for aromatic hydrocarbons with δr = 300 nm and V 0 = +130 kv. 3.2 Aromatic hydrocarbons 25

44 Figure 3.7. Electric field magnitude [ V m 1] at t = 25, 50, 75 and 300 ns for aromatic hydrocarbons with δr = 1000 nm and V 0 = +130 kv. Note that the streamer moves away from the needle-electrode axis, indicating failed numerical convergence. 26 Results

45 3.2 Aromatic hydrocarbons 27 6 x 108 t=0 + 5 t=5ns t=20ns t=60ns t=100ns x 10 3 (a) Electric field magnitude [ V m 1]. Note that the field at t = 0 + corresponds to the Laplacian field t=0 + t=5ns t=20ns t=60ns t=100ns x 10 3 (b) Space charge density [ C m 3] Figure 3.8. Temporal dynamics along the needle-sphere electrode axis [m] for aromatic hydrocarbons with δr = 500 nm and V 0 = +200 kv.

46 Figure 3.9. Electric field magnitude [ V m 1] at t = 25, 50, 75 and 100 ns for aromatic hydrocarbons with δr = 500 nm and V 0 = +200 kv. 28 Results

47 3.2 Aromatic hydrocarbons 29 9 x 108 t= t=5ns t=20ns t=60ns t=100ns x 10 3 (a) Electric field magnitude [ V m 1]. Note that the field at t = 0 + corresponds to the Laplacian field t=0 + t=5ns t=20ns t=60ns t=100ns x 10 3 (b) Space charge density [ C m 3] Figure Temporal dynamics along the needle-sphere electrode axis [m] for aromatic hydrocarbons with δr = 500 nm and V 0 = +300 kv.

48 30 Figure Electric field magnitude V m 1 at t = 25, 50 and 100 ns for aromatic hydrocarbons with δr = 500 nm and V0 = +300 kv. Results

49 3.3 Naphthenic/paraffinic hydrocarbons Naphthenic/paraffinic hydrocarbons This section concerns the simulation results for naphthenic/paraffinic hydrocarbons with a high ionization potential Applied voltage V 0 = +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 Applied voltage V 0 = +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 Applied voltage V 0 = +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 V 0 = +300 kv.

50 Figure Electric field magnitude [ V m 1] at t = 25, 50, 75 and 100 ns for naphthenic/paraffinic hydrocarbons with δr = 500 nm and V 0 = +130 kv. 32 Results

51 Figure Electric field magnitude [ V m 1] at t = 25, 50, 75 and 100 ns for naphthenic/paraffinic hydrocarbons with δr = 500 nm and V 0 = +200 kv. 3.3 Naphthenic/paraffinic hydrocarbons 33

52 34 Results 9 x 108 t= t=50ns t=75ns t=100ns t=125ns t=150ns x 10 3 (a) Electric field magnitude [ V m 1]. Note that the field at t = 0 + corresponds to the Laplacian field t=0 + t=50ns t=75ns t=100ns t=125ns t=150ns x 10 3 (b) Space charge density [ C m 3] Figure Temporal dynamics along the needle-sphere electrode axis [m] for naphthenic/paraffinic hydrocarbons with δr = 500 nm and V 0 = +300 kv.

53 Figure Electric field magnitude [ V m 1] at t = 25, 75, 100 and 150 ns for naphthenic/paraffinic hydrocarbons with δr = 500 nm and V 0 = +300 kv. 3.3 Naphthenic/paraffinic hydrocarbons 35

54 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 naphthenic/paraffinic molecules with a small addition (1%) of molecules with a mediumhigh ionization potential Applied voltage V 0 = +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 Applied voltage V 0 = +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 Applied voltage V 0 = +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 naphthenic/paraffinic molecules with a small addition (1%) of aromatic molecules Applied voltage V 0 = +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.

55 3.5 Oil mixture Applied voltage V 0 = +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 Applied voltage V 0 = +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.

56 Figure Electric field magnitude [ V m 1] at t = 25, 50, 75 and 100 ns for oil mixture 1 with δr = 500 nm and V 0 = +130 kv. 38 Results

57 3.5 Oil mixture x 108 t= t=10ns t=20ns t=30ns t=40ns t=50ns t=60ns t=70ns t=80ns t=90ns t=100ns x 10 3 (a) Electric field magnitude [ V m 1]. Note that the field at t = 0 + corresponds to the Laplacian field t=0 + t=10ns t=20ns t=30ns t=40ns t=50ns t=60ns t=70ns t=80ns t=90ns t=100ns x 10 3 (b) Space charge density [ C m 3] Figure Temporal dynamics along the needle-sphere electrode axis [m] for oil mixture 1 with δr = 500 nm and V 0 = +300 kv.

58 Figure Electric field magnitude [ V m 1] at t = 25, 50, 75 and 100 ns for oil mixture 1 with δr = 500 nm and V 0 = +300 kv. 40 Results

59 3.5 Oil mixture x 108 t= t=5ns t=25ns t=100ns x 10 3 (a) Electric field magnitude [ V m 1]. Note that the field at t = 0 + corresponds to the Laplacian field t=0 + t=5ns t=25ns t=100ns x 10 3 (b) Space charge density [ C m 3] Figure Temporal dynamics along the needle-sphere electrode axis [m] for oil mixture 1 with δr = 500 nm and V 0 = +400 kv.

60 Figure Electric field magnitude [ V m 1] at t = 25, 50 and 300 ns for oil mixture 1 with δr = 500 nm and V 0 = +400 kv. 42 Results

61 Figure Electric field magnitude [ V m 1] at t = 25, 50, 100 and 300 ns for oil mixture 2 with δr = 500 nm and V 0 = +80 kv. 3.5 Oil mixture 2 43

62 44 Results 4 x 108 t= t=25ns t=50ns t=75ns t=100ns t=125ns t=150ns t=175ns t=200ns x 10 3 (a) Electric field magnitude [ V m 1]. Note that the field at t = 0 + corresponds to the Laplacian field t=0 + t=25ns t=50ns t=75ns t=100ns t=125ns t=150ns t=175ns t=200ns x 10 3 (b) Space charge density [ C m 3] Figure Temporal dynamics along the needle-sphere electrode axis [m] for oil mixture 2 with δr = 500 nm and V 0 = +130 kv.

63 Figure Electric field magnitude [ V m 1] at t = 25, 50, 100 and 300 ns for oil mixture 2 with δr = 500 nm and V 0 = +130 kv. 3.5 Oil mixture 2 45

64 46 Results 9 x 108 t= t=5ns t=10ns t=15ns t=20ns t=30ns t=50ns t=75ns t=100ns t=150ns t=200ns t=300ns x 10 3 (a) Electric field magnitude [ V m 1]. Note that the field at t = 0 + corresponds to the Laplacian field t=0 + t=5ns t=10ns t=15ns t=20ns t=30ns t=50ns t=75ns t=100ns t=150ns t=200ns t=300ns x 10 3 (b) Space charge density [ C 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 Temporal dynamics along the needle-sphere electrode axis [m] for oil mixture 2 with δr = 500 nm and V 0 = +300 kv.

65 3.5 Oil mixture 2 Figure 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

66

67 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, V 0 = +130 kv. Section 3.3.3: Naphthenic hydrocarbons, V 0 = +300 kv. Section 3.4.2: Oil mixture 1, V 0 = +300 kv. Section 3.5.2: Oil mixture 2, V 0 = +130 kv. Only the case with aromatic hydrocarbons and V 0 = +130 kv is discussed and the convergence analysis will then be assumed to be valid for all focused streamer results Streamer deflection with δr = 1000 nm The results that are achieved with a mesh of δr = 1000 nm can be found in section 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

68 50 Discussion Extra non-relaxed iteration. Lowering C max from 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 G F 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 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 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 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 C max from 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 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 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.

69 4.1 Focused streamers 51 1 x δ r = 1000nm δ r = 500nm δ r = 500nm, coupling δ r = 500nm, extranorelax δ r = 500nm, C max =0.5 δ r = 300nm x 10 7 (a) Position δ r = 1000nm δ r = 500nm δ r = 500nm, coupling δ r = 500nm, extranorelax δ r = 500nm, C max =0.5 δ r = 300nm x 10 7 (b) Velocity Figure 4.1. Position and velocity comparison along the electrode axis for aromatic hydrocarbons with V 0 = +130 kv.

70 52 Discussion 4.2 Bubbly streamers This section will analyze the impact of numerical parameters on bubbly streamer results. As discussed in section 3.1, streamers are called bubbly if they grow wider than the needle electrode. The following oil and voltage combinations exhibited bubbly streamers: Section 3.2.3: Aromatic hydrocarbons, V 0 = +200 kv. Section 3.2.4: Aromatic hydrocarbons, V 0 = +300 kv. Section 3.4.3: Oil mixture 1, V 0 = +400 kv. Section 3.5.3: Oil mixture 2, V 0 = +300 kv. Only the case with aromatic hydrocarbons and V 0 = +300 kv is discussed and the convergence analysis will then be assumed to be valid for all bubbly streamer results. Interestingly, all bubbly streamer cases tend to generally similar states as t with a bubble that comes to a stop with a weakened field at the electrode axis. This is a problem, because as discussed in section 4.1.1, a streamer that deviates from the electrode axis is unrealistic. The deflection causes the streamer to come to a stop shortly after the field along the electrode axis is weakened. Bubbly streamers are caused by high voltages, and this voltage forces the charge carriers to be dispersed over a larger region than for focused streamers. This is probably the mechanism that leads to the slowing down within this model. Figure 4.2 shows a plot with the streamer positions and velocities for different meshes in the case of aromatic hydrocarbons and V 0 = +300 kv. It can be seen that the results for δr = 500 nm and δr = 1000 nm are identical, showing that mesh-based convergence has been reached at δr = 1000 nm. In difference to the focused streamers, the bubbly cases have wide wavefronts which allows them to be described with larger meshes. However, even with mesh-based convergence, the weakening of the field at the electrode axis inevitably leads to the questioning of the validity of the results. The following modifications were attempted to rectify the deflection: Lowering couplingtolerance by an order of magnitude. Extra non-relaxed iteration. Lowering C max from 2.0 to 0.5. The result plots for these modifications have been omitted for brevity, but none of them had any noticeable impact. Consequently, the results for bubbly streamers have converged to the correct solution given by the mathematical model in section 2.5. This does not mean it is the correct solution in the physical sense; the problematic results might be a modeling problem. Indeed, the same results for bubbly streamers is found in Hwang [9], where a similar model is used. The deflections therefore require the questioning of any results from these bubbly streamer simulations. Jadidian et al. [10] suggests that models using constant ionization

71 4.2 Bubbly streamers x δ r = 2000nm δ r = 1000nm δ r = 500nm x 10 7 (a) Position δ r = 2000nm δ r = 1000nm δ r = 500nm x 10 7 (b) Velocity Figure 4.2. Position and velocity comparison along the electrode axis for aromatic hydrocarbons with V 0 = +300 kv.

72 54 Discussion potentials are not able to describe positive streamers with applied voltages above +200 kv. A number of model extensions, including an electric field-dependent ionization potential is discussed briefly in section Streamer classification revisited This section will revisit the discussion in section 3.1 in light of the results in chapter 3. It can be seen that the streamer type is strongly dependent on the applied voltage V 0. At low voltages, the lack of field ionization results in a largely unchanging glow state. When the applied voltage increases, the result is a focused streamer, followed by bubbly streamer at large overvoltages. This trend applies to all oil configurations, the only difference is the voltages at which the transitions occur. Aromatic hydrocarbons glow at +80 kv, give a focused streamer at +130 kv and bubble up at +200 kv and above. Naphthenic/paraffinic hydrocarbons glow at +130 kv, have a focused streamer at +300 kv and result in a bubble at +400 kv. The increase is expected, since the naphthenic/paraffinic hydrocarbons have a higher ionization potential Charge generation analysis Much can be learned by rewriting the charge generation term in eq. (2.7): ( G F (E) = γe exp E ) ref (4.1) E where γ := qn 0a (4.2) h E ref := π2 m a 2 qh 2 (4.3) The negative exponential form of G F gives rise to a switch-like behavior determined by the ratio E ref /E. If E E ref, the argument to the exponential function tends to and thus G F 0. However, if E E ref, then G F γe, causing charge generation proportional to γe. Hence, the E ref term corresponds to the when while the γ term relates to the how much when talking about charge generation. A valid question is if it is possible to more closely predict the resulting streamer behavior from just the initial state. Table 4.1 shows the γ and E ref terms for all oil species. Using these values, the charge generation at t = 0 + can be computed and compared with the resulting streamer types. The measured electric fields just below the needle electrode can be found in table 4.2. These values are taken from the electric field plots in chapter 3. Note that these fields only depend on the voltage and not on oil type. It is now possible to compute the charge generation terms G F as has been done in table 4.3. It can be seen that charge generation values lower than approximately

73 4.3 Streamer classification revisited 55 Parameter γ E ref Aromatic hydrocarbons Naphthenic/paraffinic hydrocarbons Mixture 1, species Mixture 1, species Mixture 2, species Mixture 2, species Unit C J 1 s 1 m 2 V m 1 Table 4.1. Coefficients for charge generations for all species in section 2.4 V kv +130 kv +200 kv +300 kv +400 kv E max [ 10 8 V m 1] Table 4.2. Measured peak Laplacian electric field directly below the needle at t = 0 + for different applied voltages C m 3 s 1 cause glow, values roughly between and C m 3 s 1 cause focused streamers while even higher charge generation leads to bubbly streamers. Note that the charge generation for the naphthenic/paraffinic case at V 0 = +200 kv is relatively high for a glowing streamer. This is an important correlation, as it might be possible to predict the behavior from relatively simple calculations. This analysis is a new contribution to streamer research, however more research is needed to verify the correlation between charge generation and streamer type Charge generation in mixed oils The mixed oils contain several species with charge generation terms defined in eq. (2.18). In mixture 1, a low voltage (section 3.4.1) is not enough to ionize the required number of molecules to initiate a streamer. When increasing the voltage (section 3.4.2), the ionization potentials of the two species are close enough to generate charges simultaneously. In table 4.4 this can be seen through the small V kv +130 kv +200 kv +300 kv +400 kv Aromatic Naphthenic/paraffinic Oil mixture Oil mixture [ Table 4.3. Calculated charge generation G F m 3 s 1] just below the needle tip for each oil type and voltage [kv] at t = 0 +. symbolizes glow, focused streamers and bubbly streamers. All values are rounded 10-logarithms.

74 56 Discussion V kv +130 kv +300 kv +400 kv Oil mixture 1, species Oil mixture 1, species Oil mixture 2, species Oil mixture 2, species Oil mixture 1, cumulative Oil mixture 2, cumulative [ Table 4.4. Calculated charge generation G F m 3 s 1] for mixed oils just below the needle tip at t = 0 +. The cumulative charge generation for each mixture is calculated according to eq. (2.18). All values are rounded 10-logarithms. Oil type V 0 E 1 max v 1 avg E 2 max v 2 avg Aromatic 130 kv Aromatic 200 kv Aromatic 300 kv Naphthenic/paraffinic 130 kv Naphthenic/paraffinic 300 kv Oil mixture kv Oil mixture kv Table 4.5. Comparison of peak electric field magnitude [ 10 8 V m 1] at 100 ns and average streamer velocity [ 10 3 m s 1]. Superscript 1 refers to the results in this thesis and superscript 2 to the results in Hwang [9]. difference between species 1 and 2 for oil mixture 1. This is also true for very high voltages (section 3.4.3), where the difference only corresponds to an order of magnitude. Thus, the two species in mixture 1 are ionized at roughly the same time and no species dominate the other in terms of charge generation. In oil mixture 2, species 1 consists of a low ionization potential molecule with a large potential difference to species 2. Species 1 will therefore dominate the charge generation (table 4.4) even though it has a much lower number density than species 2. In the focused streamer case (section 3.5.2), the charge generation difference is as large as four orders of magnitude. Here, species 1 takes care of the large electric field before it can ionize the higher potential molecules of species Comparison with previous work As stated in section 1.4, the main focus of this thesis is to reproduce and/or improve upon the results in Hwang [9] using the OpenFOAM-based solver developed at ABB. This section will make detailed comparisons of all relevant cases. A comparison of peak electric fields and velocities is presented in table 4.5.

75 4.4 Comparison with previous work Aromatic hydrocarbons The results are similar for all aromatic hydrocarbon simulations. All streamers are smooth and exhibit no disturbances that could influence the results. Indeed, neither table 4.5 nor the result plots in section 3.2 show anything but minuscule differences. It is therefore safe to assume that the same results have been achieved as in Hwang [9] for this configuration Naphthenic/paraffinic hydrocarbons In the case of naphthenic/paraffinic hydrocarbons, the situation is slightly different. The +130 kv simulations are identical, but that is mostly due to the very simple outcome: glow. The results for +300 kv are more interesting, and a large difference can be seen to Hwang [9]. The electric field magnitude plot in fig shows a smooth, focused streamer without any apparent instability ripples. In contrast, the corresponding plot in Hwang exhibits several branches that deflect from the electrode axis with large field peaks. It is difficult to even speak of a streamer head and much less its velocity when there appears to be an instability in the solution. For this oil, the results in Hwang might not be fully converged while the results in this thesis appear to be Mixed oils One of the major results in Hwang [9] was the large difference between the streamers in the two oil mixture cases at +300 kv. It was seen that the streamer in mixture 1 was much faster than the one in mixture 2 for the same voltage. This is interesting and counter-intuitive due to the lower ionization potential of the low number density additive in mixture 2. However, an analysis of the corresponding electric field magnitude plot in Hwang reveals numerical challenges for mixture 1. At t = 50 ns, small fringes begin to appear on the side of the streamer that start to grow as time goes on. These fringes are similar to the deflections discussed in section and are unrealistic from a physical perspective. The corresponding results in section of this thesis are very smooth and of the focused type. The relevant line in table 4.5 shows a moderate difference between these results and Hwang. This is probably due to the lack of a focused streamer propagation in the latter case. As in the above discussion of naphthenic/paraffinic hydrocarbons, it can be concluded that the results in this thesis have achieved better convergence than those in Hwang. For oil mixture 2 at +300 kv, the results in both this thesis and Hwang are of the bubbly type and very similar. As discussed in section 4.2, one must question any conclusions drawn from bubbly streamers computed with this model including streamer velocity and shape. Hwang [9] referred to this streamer as slow and inferred that the streamer velocity voltage is lower than in mixture 1. This result was then explained to be in agreement with the experimental studies of Hebner et al. [7] and Lesaint and Jung [13]. It is indeed possible that the streamer velocity is lower for bubbly streamers, however the discussion above indicates model limitations.

76 58 Discussion Further research could shed light on the transition between focused and bubbly streamers. The results in section show that mixture 2 forms a focused streamer at voltages as low as +130 kv. We therefore draw the conclusion that mixture 2 has a lower breakdown voltage than mixture 1, which at this voltage only exhibits glow. The lowering of the breakdown voltage by a small addition of low ionization potential polyaromatic compounds such as N,N-dimethylaniline (DMA) has been reported in an empirical study by Beroual et al. [2]. It would have been valuable to compare the mixed oil results at +130 kv to Hwang [9]; those configurations were not attempted there.

77 Chapter 5 Proof of concept: Expanding to the third dimension Streamers found in empirical studies often exhibit a semi-chaotic structure with branching, filaments etc. with a tree-like appearance [2]. The results in this thesis and Hwang [9], Jadidian et al. [10], O Sullivan [19] are very symmetric and smooth except for results with numerical problems. It would be interesting to reproduce the chaotic structure found in experiment using this model, however this is not possible when only performing axisymmetric simulations. Therefore it might be fruitful to extend the simulations to be fully three-dimensional. 5.1 Methodology Three-dimensional simulations have a considerable impact on the model complexity in comparison to the previous axisymmetric results. Therefore, care must be taken to create high-resolution areas only in the region of interest. By creating a new mesh when the edge of the previous one is reached it is possible to reduce simulation times enough to make focused simulations realistic. Bubbly streamers remain difficult to simulate in three dimensions due to the large area occupied by the streamer head. The configuration used for this three-dimensional simulation is the same as in section 3.2.2, i.e. aromatic hydrocarbons with an applied voltage V 0 = +130 kv. The solution must be interpolated between meshes, and this functionality is provided by the OpenFOAM tool mapfields. Determining the correct times to switch meshes is a task that needs to be performed manually, which makes the process rather tedious. Just as with the mesh in the two-dimensional case, the 3D mesh primarily consists of hexahedrons (blocks). Using the OpenFOAM tool blockmesh, a lowresolution block mesh is generated over the whole geometry. The primary mesh generator, snappyhexmesh then refines the blocks until the desired resolution is reached at each point. This second step repeatedly splits the blocks along each 59

78 60 Proof of concept: Expanding to the third dimension dimension, dividing them into eight hexahedrons each. When the desired resolution or the maximum allowed number of mesh blocks is reached the splitting process is stopped. For three dimensions a doubling of the resolution results in approximately an eightfold increase in the number of mesh blocks. Just as with focused streamers in two dimensions, a resolution of 500 nm is used. Due to this chapter only being a proof of concept, no convergence analysis is performed. The maximum number of mesh blocks in each mesh step is set to be This number is much higher than in the two-dimensional case, and therefore ncores is set to be as high as 64 nodes instead of the usual 16. Still, each mesh step takes between 8 and 16 hours to finish, resulting in a total simulation time of close to 4000 CPU hours. 5.2 Results The model was simulated until t = 120 ns using five different meshes. Figure 5.1 shows a three-dimensional visualization of the results. Interpolation was performed at 1.5, 17, 63 and 87 ns. Mesh step four at 87 ns is shown in fig It can be seen that the streamer has reached the end of this mesh and the solution was therefore interpolated to the next mesh at this point. Figure 5.3 shows the electric field magnitude and charge density n p n n n e along the electrode axis. The streamer is focused and has an average velocity of approximately 3.3 km s Discussion The results in section 5.2 show that it is possible to perform three-dimensional simulations with similar results to 2D. A focused streamer is formed and propagates for 100 ns without deflection. After this time the streamer steers slightly off-axis. The reason for this is unknown but we believe it is caused by numerical noise. The average velocity and streamer position at t = 100 ns is very similar to those in section b. In the future, dynamic meshing could be used to let the high-density mesh move with the streamer head. This would reduce the manual work needed between mesh steps. It would be interesting to simulate all cases in chapter 3 using three dimensions. Bubbly streamers are of special interest as it would be interesting to see the deflection in three dimensions. It could also be possible to get results that show branching and filaments by introducing imbalances. For instance, the needle electrode could be made asymmetric which would severely affect the initial Laplacian field. It can be concluded that it is possible to simulate streamers in three dimensions with satisfactory results. Although the computational load currently limits us to focused streamers it should be possible to extend to other types in the future.

79 Figure 5.1. Visualization of the three-dimensional simulation at t = 87 ns. The needle electrode is in the upper right corner. A surface plot of the electric field can be seen together with isosurface plot of the charge density n p n e n n at C m 3. Both the surface and isosurfaces are semitransparent, and the needle tip has a radius of 40 µm. 5.3 Discussion 61

80 Figure 5.2. Visualization of the fourth mesh step at t = 87 ns for the three-dimensional simulation. This plot shows a plane cut along the electrode axis with the mesh borders highlighted. Due to limitations in the visualization software the mesh blocks are drawn incorrectly. The blocks are supposed to be square, however they are drawn with lines along the diagonal. The semitransparent isosurface shows the charge density n p n e n n at 160 C m 3. The needle tip has a radius of 40 µm. Note the low resolution of the mesh outside of the streamer head. 62 Proof of concept: Expanding to the third dimension