Degree Project
Developing Verification Models for Corona Discharge Suppression in High Voltage
Capacitor Banks
Author: Mohammadjavad Javadi Supervisors: Henrik Andersson, Sven Nordebo Examiner: Sven-Erik Sandström Term: Spring 2020 Subject: Electrical engineering Level: Master 30 hp Course code: 5ED36E Department of Physics and Electrical Engineering
Abstract
Due to the universal considerable population and economic growth rate, demands for energy have risen significantly over the past decade. Integration of renewable energies in the power grid has increased as well as requests for reactive power compensation, voltage stability, and mitigation of harmonic filters. Capacitor banks are widely used in the modern electrical transmission system in order to improve power quality and efficiency. In other words, this device aims to contribute in harmonic disturbance elimination, improve the power factor (PF), and provide voltage control and stability which leads into more sustainable energy systems. Utilizing high voltage components, such as shunt capacitors in the power grid can introduce new challenges. One of these challenges is known as corona discharge.
The aim of the presented master thesis is to study and develop corona discharge suppression models on high voltage capacitor banks. The main concerns are, effective factors on corona emergence, corona inception voltage levels, and corona suppression methods. Also, this study evaluates the verification of existing suppression. Two various approaches were applied and compared. The aim of the first approach is to evaluate corona discharge by electric field calculations on three various capacitor banks with different voltage levels. The simulation was implemented based on Maxwell’s equations and finite element method (FEM) by utilizing COMSOL Multiphysics software. The second approach is based on streamer inception and propagation. The calculation on this method is fulfilled with the help of MATLAB software. The results of both approaches were found reasonably compatible. It is discovered that corona discharge can appear at different voltage levels on capacitor banks based on various factors, such as the geometry of the bank. Consequently, the suppression method may vary case by case and different proposals were suggested in order to optimize the corona suppression rings.
Keywords: Corona discharge, electric field, stream inception, negative corona, positive corona, capacitor banks, corona ring, stream propagation.
Sammanfattning
På grund av den allmänna betydande befolknings- och ekonomiska tillväxttakten har kraven på energi ökat markant under det senaste decenniet. Detta innebär att integrationen av förnybara energier i elnätet har eskalerat samt begäran om reaktiv effektkompensering, spänningsstabilitet och mildring av harmoniska filter. kondensatorbatterier används ofta i det moderna elektriska transmissionssystemet för att förbättra strömkvaliteten och effektiviteten. Med andra ord syftar denna enhet till att vara involverad i eliminering av harmonisk störning, förbättra effektfaktorn (PF), tillhandahålla spänningskontroll och stabilitet som leder till mer hållbara energisystem. Att använda högspänningskomponenter, som shuntkondensatorer i elnätet, kan skapa nya utmaningar. En av dessa utmaningar kallas korona-urladdning.
Syftet med den presenterade masteruppsatsen är att studera och utveckla korona- urladdningsmodeller på högspännings-kondensatorbatterier. De viktigaste problemen är effektiva faktorer för korona uppkomst, spänningsnivåer korona och metoder för att underlätta korona. Dessutom utvärderar denna studie verifieringen av befintliga undertryckningsmetoder. Två olika tillvägagångssätt tillämpades och jämfördes. Syftet med det första tillvägagångssättet är att utvärdera korona-urladdning genom elektriska fältberäkningar på tre olika kondensatorbatterier med olika spänningsnivåer. Simuleringen implementerades baserat på Maxwells ekvationer och finita elementmetoden (FEM) genom att använda COMSOL Multiphysics programvara. Det andra tillvägagångssättet är baserat på strömningslinjernas början och utbredning. Beräkningen av denna metod genomförs med hjälp av MATLAB-programvaran. Resultaten från båda metoderna tycktes vara rimligt kompatibla. Det upptäcks att korona-urladdning kan förekomma i olika spänningsnivåer på kondensatorbatterier baserat på olika faktorer, till exempel batteriets geometri. Följaktligen kan undertryckningsmetoden variera från fall till fall och olika förslag föreslogs för att optimera koronaundertryckningsringarna.
Acknowledgments
Special thanks to:
Henrik Andersson, Erik Nylund, Håkan Rörvall, Fredrik Jansson, Göran Eriksson, Peter Holmberg, Emma Petersson, Nils Lavesson, Sven Nordebo, Liliana Arevalo.
Contents
1 Introduction 1 1.1 Overview and background 1 1.2 Aim and objectives 3 1.3 Methodology 3 1.4 Thesis outlines 4 2 Theory 5 2.1 Physics of corona discharge 5 2.2 Electric field theory 9 2.2.1 Electric field calculation 9 2.2.2 Stream inception and propagation 14 2.2.3 Corona discharge threshold 17 2.3 Effective factors on emergence of corona 17 2.3.1 Polarity 18 2.3.2 Pressure 18 2.3.3 Humidity 20 2.3.4 Pollution 21 2.4 Corona discharge suppression methods 22 3 Methods and modeling 26 3.1 Introduction to COMSOL Multiphysics 26 3.1.1 Finite element method (FEM) 27 3.1.2 Boundary conditions 30 3.2 Simulation of capacitor banks 31 3.2.1 DC capacitor bank 31 3.2.2 AC capacitor bank 32 3.3 MATLAB model based on streamer inception and propagation 33 4 Results 35 4.1 Simulation results 35 4.1.1 DC capacitor bank 35 4.1.2 AC capacitor bank 37 4.2 Results of MATLAB model based on streamer inception and propagation 39 5 Discussion and conclusion 40 5.1 Corona supression verification and design analysis 40 5.2 Future work 40 References 41
List of Figures
1.1 Reactive power flow between AC capacitors and load 1 2.1 Avalanche, (a) Individual cloud chamber of an avalanche (b) streamer formation 6 2.2 Discharge transition process 6 2.3 Positive corona discharges under the various impulse voltages 7 2.4 Different discharge modes for positive corona regarding rod-plane model 8 2.5 Type of negative corona discharges 8 2.6 Different discharge modes for negative corona regarding rod-plane model 9 2.7 Rod-plane model 11
2.8 Two cylindrical conductors in parallel (±휌푙) 12 2.9 Two cylindrical conductors in parallel with symmetrical charge lines 13 2.10 Geometry of domain Ω 14 2.11 Streamlines from an electrode towards a grounded plane and equipotential lines 16
2.12 Propagation pattern of the streamers starting from Г0 16 2.13 Paschen’s law 19 2.14 Paschen’s curve 20 2.15 Electric field under three various humidity conditions 21 2.16 Cross section equipotential lines. (a) Block with ring. (b) Block without ring 23 2.17 Corona ring effective parameters 23 2.18 Cross section equipotential lines with tube radius of 0.1m and with tube radius of 0.2m 24 2.19 Cross section equipotential lines with ring diameter of 4m and with ring diameter of 2m 24 2.20 Cross section equipotential lines. (a) Block and ring. (b) Cone and ring 24 3.1 FEM (a) Two-dimensional model. (b) Subdivided two-dimensional region A 28 3.2 Four-element system with 4 known potentials and 1 unknown 30 3.3 Geometry of DC bank 32 3.4 Meshing of DC bank 32 3.5 Geometry of AC bank 33 3.6 Meshing of AC bank 33 3.8 Streamline 34 3.9 Excel table for streamline data 34 3.10 MATLAB model based on streamer inception and propagation 34 4.1 Electric field simulation result for DC capacitor bank (kV/mm) 36 4.2 Electric field simulation result for DC capacitor bank (kV/mm) 35 4.6 Equipotential lines around the top level (kV) 37 4.12 Electric field with (left) and without (right) suppression rings on bank 37 4.13 Equipotential lines with (left) and without (right) suppression rings on bank 38
4.14 Electric field variation along the bank (kV/mm-m) with and without suppression rings 38 4.23 MATLAB model based on streamer inception and propagation 39 4.24 Streamline plot 39
List of Tables
2.1 Gauss’s law and Faraday’s law summary 11
Acronyms
AP Atmospheric pressure
CIV Corona inception voltage
DC Direct current
EF Electric field eqn Equation
Es Electrostatic
FEM Finite element method
GAT Glow to arc transition
HV High voltage
HVDC High voltage direct current kV Kilovolt
LV Low voltage
PF Power factor
Q Reactive power rms Root mean square
SI Stream inception
SP Stream propagation
Chapter 1
Introduction
1.1 Overview and background Capacitors are widely used in AC and DC power systems specifically due to the rising trend of renewable energies utilization. Since loads and transmission devices (transformers and transmission lines) are inductive, additional reactive power (Q) is required to flow in the system due to the lagging power factor. As a result, the capacity of the system decreases, system voltage drops, and losses increase. IEEE standard 1036 [24], has introduced shunt capacitors as an optimal solution for reactive power compensation. Shunt capacitors reduce the power losses and increase the system capacity by preventing the reactive power flow in the system as it is produced locally i.e. near the loads at transmission and distribution substations (Figure 1.1). As it is featured by the blue arrow in the figure, reactive power surges to the load from AC capacitor bank such that Q does not flow through the entire system. Suppose an induction motor which consumes considerable amount of reactive power as the load. The required Q for this motor can be produced locally by shunt capacitors. The moment that induction motor is switched on, reactive power flows instantly from the capacitors. This solution has significant advantages such as improvement in voltage control and PF, system stability increase, and reactive power reduction at generation point. Applying shunt capacitors will result in the voltage rise from the installation point, up to the generation location along with capacity increase of the system. This can be fully utilized during the rapid increase of the load in the system. In addition, studies have shown that the current which flows between the installation point of shunt capacitors and generators, reduces considerably. In other words, power losses reduce substantially in the grid and consequently, the fuel consumption for power generation, maintenance operations and costs decrease [24].
Figure 1.1 Reactive power flow between AC capacitors and load
DC capacitors are used extensively in HVDC systems as well. Long distance power transmissions as well as renewable power plants such as solar and wind are connected via HVDC systems to the power grid. One of the main challenges on these systems is voltage stability. Due to the voltage instability in long-distance transmission systems, distribution
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systems and equipment may damage or fail. DC capacitor banks with high current capability that can withstand significant voltage fluctuations is one of the effective solutions. These capacitors should have high reliability and durability as they have direct impact on the sensitive infrastructures. These banks are normally installed in converter stations at the end of each HVDC link. Utilizing capacitors in high voltage levels regardless of their type (AC or DC) can introduce challenges to the system. One of these challenges is corona discharge.
Corona discharge is a common and undesirable effect in high voltage power transmission systems. This phenomenon depends on the electric field strength and distribution. It is defined as the ionization of the medium (gas) that surrounds the charged surface and allows a current flow into the subjected medium. Corona can be followed by complete breakdown which is created by ionized atoms and molecules within the form of a conductive path where charges can pass through. As a result of corona discharge, insulators will be damaged, the equipment will fail, power losses will rise, and radio interferences will grow. One of the solutions is to increase the apparent radius of the active area and reduce the electric potential gradient. This can be fulfilled by using corona suppression rings that cover critical areas where the electric field is severely high. To be more explicit, these rings partially cover each level of high voltage capacitor banks and decrease the electric field gradient on the associated active surfaces and structural elements.
Having a closer look at this phenomenon, it is defined as discharges within the uniform and non-uniform electric field gaps that can be observed as glowing areas and heard as audible noises. It can be recognized as the primary long step before the flashover stage in high voltage equipment and transmission lines. Also, it is one of the reasons for gradual degradation of insulators due to the surface bombardment with discharged ions. Corona is classified within two categories, Positive corona (Anode) and Negative corona (Cathode). The visual emergence between positive and negative corona depends on the polarity of the voltage. Although positive corona appears in the form of purplish discharges, negative corona emerges as reddish discharges [1].
There have been numerous studies regarding the corona rings on the insulators for transmission lines and transformers [29]-[32]. However, there is no specific study regarding the high voltage capacitor banks which makes this study distinctive from the former ones. The optimization of corona ring on various high voltage transmission lines was studied by N. Mohan [2] and it is suggested that three factors can be modified in order to enhance the performance of the ring, tube thickness, distance from high voltage end, and ring radius. H. Terrab and A. Kara [3] proposed changes on the tube thickness and ring dimensions regarding 230 kV insulators. Barros et al. [4] investigated the same subject on the polymeric insulators and suggested that by changing all dimensional parameters simultaneously, the ring can be optimized. In addition, vast number of studies have been implemented regarding the physical theory behind the corona appearance and its behavior in gas which will be discussed further in chapter 2. It can be concluded from the previous investigations that modifying the ring dimensions can optimize and improve its functionality. However, this study proposes structural alterations and different approaches regarding the suppression methods.
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1.2 Aim and objectives The main purpose of the presented master thesis is to investigate and develop corona discharge suppression models on high voltage capacitor banks. More precisely, this study covers the following questions:
1. How does the electric field behave in various high voltage capacitor banks? 2. What factors are effective on emergence of the corona discharge? 3. What is the threshold voltage level for the corona inception in capacitor banks? 4. How effective is the existing suppression method? 5. What are the critical areas of the present corona suppression ring design? 6. What are the key parameters that can be modified in order to enhance the current design?
Also, creating 3D-models with respect to the corona discharge by using COMSOL Multiphysics software and suggesting possible structural alterations based on the simulations is one of the important aims. Moreover, the results of this project can be beneficial mainly for two sectors, electric power industry and environment. Performing a reliable assessment regarding the corona discharge is essential in order to be utilized during the design stages of capacitor banks. In addition, optimizing the corona suppression ring design may result in material consumption reduction as well as cost and carbon emission decrease which is considered as one step towards sustainability.
1.3 Methodology Due to the complex nature of corona discharge two various approaches were applied in this study, finite element method (FEM) by utilizing COMSOL Multiphysics software and streamer inception and propagation. The aim of the first approach is to evaluate corona discharge by utilizing electric field calculations on three various capacitor banks with different voltage levels. The simulation was implemented based on the Maxwell’s equations and finite element method. FEM is a computational method which divides an object into the small size elements. Then it assigns a set of characteristics, such as equations which describe the physical properties, boundary conditions, and applied forces to each element. Finally, these equations are solved as the form of simultaneous equations to forecast the object’s behavior. This method consists of boundary element method (BEM). BEM is used in order to define the mesh on the boundaries of the modeled geometry and solve partial differential equations. It is considered as complementary method for FEM which means a volumetric finite element mesh can be integrated with BEM-based physics. One of the significant advantages of the finite element method is the adjustability of the mesh. In other words, on the critical areas where the electric field behavior should be simulated with higher accuracy the mesh can be defined relatively smaller.
The second approach in this study is based on streamer inception and propagation. This method is based on the extracted data from the streamline which connects two surfaces with different potentials, for instance between a charged electrode and grounded plane. The streamline data is the function of two parameters, electric field and distance. These elements are used in calculations in order to identify the corona inception voltage as well as corona discharge appearance. Calculations in this method are fulfilled with the help of MATLAB software.
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Simulations are implemented for three different capacitor banks with various voltage levels. Capacitor banks are simulated once without the rings to identify the critical electric field areas and once with the rings to evaluate their effects on the model. The streamer inception and propagation calculations are implemented for extracted data from the critical and non-critical points.
1.4 Thesis outline This thesis is divided into five chapters. The content of each section is explained briefly as follows.
▪ Chapter 2: Theory This section presents different types of corona discharges and the physical aspects behind it i.e. the behavior of gas atoms and molecules during their exposure to the electric field. The theory behind the electric field calculations, stream inception and propagation, and inception voltage calculations are discussed in depth. Moreover, effective factors that may have influences on the corona inception and the application of the current suppression method are studied.
▪ Chapter 3: Methods and modeling This section contains the methods that are employed as well as simulation of three different capacitor banks with various voltage levels with the help of COMSOL Multiphysics software. The process of simulation and factors that have been considered are explained in more detail. A MATLAB program is introduced in order to calculate corona discharge by utilizing streamer inception and propagation method.
▪ Chapter 4: Results This section demonstrates the results of the simulations and calculations from the previous chapter.
▪ Chapter 5: Discussion and conclusion This section consists of analyses and discussions based on the simulation results. The critical areas of the current corona suppression method are identified. A comparison is carried out between current and proposed designs in order to illustrate the feasibility of modifications. Also, recommendations for future work is expressed in this part.
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Chapter 2
Theory
2.1 Physics of corona discharge Electrical discharges depend on 5 groups of participants, photons, free electrons, positive ions, negative ions, and excited atoms and molecules. Therefore, identifying the material property of the medium (in this case air) and recognizing the behavior of molecules is essential. In the absence of electric field, gas molecules behave as insulators. However, when these molecules are exposed to a sufficiently high field, they become conductive due to the formation of low- impedance path as a result of free charge generation and ionization [16]. Air mainly consists of 78% nitrogen, 20% oxygen and 1% water vapor. Nitrogen is free-electron gas, while oxygen and water vapor are electronegative gases. Electronegative gases are defined as gases with high attraction to electrons while free-electron gases do not have any affinity for electrons [36]. The process of ionization and subsequently corona discharge highly depends on free electrons. A free electron can be created as a result of photon impact, electron impact, and detachment procedure. However, it can be lost during the attachment to other atoms and produce a negative ion as well as recombination with positive ions. The number of free electrons under the electric field can be calculated by the following equation [20].
푛푒(푡) = 푛0exp[(푣푖 − 푣푎)푡] (2.1) where t is time, 푣푖 is ionization frequency, 푣푎 is attachment frequency, and 푛0 is the gas constant. It can be concluded that once 푣푖 exceeds 푣푎, number of electrons increase exponentially with the time. This phenomenon is known as avalanche and at atmospheric pressure (AP), approximately 3 kV/mm electric field strength is needed for air to create an avalanche. An avalanche is described as the electrons that become free during the impact of the other electrons. Note that while the free electrons travel alongside the electric field direction, their movement is not linear due to the constant collisions with molecules and atoms. This motion pattern implies the high occurrence probability of electron avalanches from a free charge [16]. Figure 2.1 (a) shows an individual cloud chamber of an avalanche. While positive ions are moving towards the cathode, another collision occurs between them and the electrons which are moving towards the anode. This can cause the emission of photons and inception of secondary avalanches, Figure 2.1 (b). By accumulating these avalanches, the primary one becomes sufficiently strong to generate streamers as it is shown in Figure 2.1 (c) [19], [20], [8].
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Figure 2.1 Avalanche (a) Individual cloud chamber of an avalanche. (b) Streamer formation
In general, the discharge transition process from an avalanche initiation to a complete breakdown can be divided into 3 stages based on the point-plane model as it is demonstrated in Figure 2.2. After the avalanche formation and enrichment of 1011푚푚−3 electron density at atmospheric pressure in the air, the first stage occurs (A) which is known as avalanche to streamer transition. Depending on the voltage levels and distances, a streamer can appear into two forms, corona streamer and breakdown streamer. If the voltage is not sufficiently high, the streamer emerges as the form of steady glow around the surface which is known as corona. Note that in every step that an electron changes the energy level, it absorbs or releases photon as the form of light. Therefore, the emitted light during corona discharges is due to the electron recombination with ions [20], [28]. The radiated photons can be absorbed by more atoms and release more electrons which is known as secondary avalanche [5]. On the other hand, breakdown streamer appears for shorter gap distances and may lead to a complete breakdown. The second stage (B) is known as streamer to leader transition. When the stem temperature of the streamer reaches to 1727°C, a leader discharge forms. In other words, the ionization rate increases and consequently the conductivity becomes high enough that allows the streamer to propagate completely independent from the external electric field. Finally, the last stage (C) is the complete breakdown which occurs with the shape of flashover (arc). It is necessary to mention that corona and streamers are known as cold plasmas due to their non- thermal reaction while arcs and leader discharges are considered as thermal plasmas (above 1727°C) [20], [18], [17].
Figure 2.2 Discharge transition process
Corona discharges are categorized in to two groups, anode or positive corona and cathode or negative corona. In order to comprehend these phenomena, it is necessary to study the physics underlying these discharges. The most suitable model that does not involve with negative and positive charge interactions simultaneously, is the spherical capped rod and a plane as it is illustrated in Figure 2.3. Variations in the radius of the rod tip create different levels of electric fields and implements non-uniformity. Due to the close boundaries between corona and
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breakdown (arc), it is essential to recognize the characteristics of the impulse corona (which occurs under a pulsed voltage in short period) and static field corona (which occurs under stable DC voltages).
Positive or anode corona is recognized with the help of Lichtenberg figures. By applying positive impulse voltages on the rod, the first ionization occurs which is known as burst pulse corona as it is presented in Figure 2.3 (a). By increasing the magnitude of the voltage, streamers are created and expanded which is called streamer corona, Figure 2.3 (b). Glow corona appears before the complete breakdown, Figure 2.3 (c); and if the voltage level becomes sufficiently high, a complete flash over (arc) occurs, Figure 2.3 (d). It is essential to mention that applying voltage (DC or AC) for a long period of time can cause field distortion due to the enough time gap for ionized atoms to drift and accumulate in the air [1], [6].
Figure 2.3 Positive corona discharges under the various impulse voltages
Figure 2.4 illustrates different discharge scenarios including breakdown characteristic. The study contains a plane-rod model with 1 cm tip radius. The gap between the rod and the plane is filled with the atmospheric air. By decreasing the gap distance to less than 2 cm and rising the voltage gradually, it can be observed that breakdown occurs without any detectible ionizations. As the distance increases, the electric field evolves towards non-uniformity and by slight voltage rise, the first discharge reveals. These discharges have the same pattern as the impulse voltage experiment. Thus, they are called streamers as well. By increasing the voltage, streamers appear more frequent and this situation continues until the transition to self- dependent streamer stage occurs [1].
Self-sustained streamers appear when the number of positive ions created by the primary avalanche becomes equal to the ones from secondary avalanche [5]. On this stage a sustain glow reveals near the rod. By increasing the voltage, the glowing area becomes more intense and it expands. Rising the voltage levels further, streamers become more powerful until flashover occurs. As Figure 2.4 demonstrates, for small distances and lower voltages a uniform streamer appears. This streamer reaches into the weaker parts of the field, crosses the gap and starts flashover. The striped area between curves 1 and 4 shows that for gap distances below 10 cm the situation from No ionization turns to Spark directly without getting into the glowing area (curve 1). By rising the gap distance to 10 centimeter and more, streamers cannot reach the negative plane as presented by curve 2. Curve 3 demonstrates the stage which streamers become independent and transit to steady corona. Ultimately by exceeding the Glow area a complete breakdown occurs (curve 4) [1].
Negative or cathode corona has the same rod-plane model and the rod is loaded with negative charges. Also, the radius of the spherical capped electrode is reduced to 0.75 mm. As it is presented in Figure 2.5, by applying voltages above the onset, current flows with form of
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Figure 2.4 Different discharge modes for positive corona regarding rod-plane model (see Fig 5.33 in [1]) regular pulses known as Trichel pulse corona (a), followed by pulseless corona (b) and arc (c). Negative corona normally appears with the form of a fast propagating glow or concentrated tiny points known as tufts or beads [6]. Increasing the voltage leads to pulse frequency rise. Variables such as, voltage, radius of the cathode, gap distance, and pressure are effective factors on emergence of negative corona. Figure 2.6 presents various discharge modes for the negative corona. Curve 1 shows the non-ionization zone where the gas behavior is independent from the gap distance. Increasing the voltage does not make any alterations in the discharge mode for a certain voltage range (curve 2). Finally, under the higher voltages a steady corona appears (curve 3) and the transition region is shown between the curves 2 and 3. Eventually by raising the voltage the corona discharge is followed by spark [1], [15].
Figure 2.5 Type of negative corona discharges
In general, negative corona propagation relies on the gas molecule ionizations while positive corona requires mostly photoionization in order to propagate. The breakdown and corona inception voltages are lower for positive corona due to the considerable number of ions produced near the anode in comparison with cathode [10], [1]. The area between opposite electrodes is divided in to two regions, ionization and drifting areas. Ionization region does not exceed more than few millimeters from the high-voltage conductive surface. For instance, in negative corona electrons may transfer fractions of the current beyond the plasma region however, these electrons are lost during the attachment reactions. Drifting region is between the ionization region and collecting electrode. This area has low electric field values. Apart from streamer corona and arc, the cold plasma area does not exceed more than ionization region due to the field inability for maintaining the ionization process [6], [9].
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Figure 2.6 Different discharge modes for negative corona regarding rod-plane model (see Fig 5.35 in [1])
2.2 Electric field theory In order to evaluate the functionality of the corona suppression rings and existing design, it is essential to calculate the electric field. There are various methods that can be used for calculating electric field distribution such as analytical, experimental, and numerical. The analytical method is based on the Gaussian equation and it is suitable for modeling simple geometries. For instance, electrical field distribution around a negative or positive charge. This technique cannot be used for complex systems due to the limits of defining geometry and boundary. Experimental method can be used in laboratories by using detectors within the electric field. The laboratory experiments are costly and time consuming. Numerical method can be used for complicated geometries and non-uniformed fields. It can be categorized into three primary methods such as, finite difference method (FDM), charge simulation method (CSM), and finite element method (FEM) which the latter will be discussed further in chapter 3 [2].
2.2.1 Electric field calculation This section is reflecting a summary of chapter 4 in [1]. The purpose of the electric field calculation is to determine the critical regions where streamers initiate, and to evaluate their propagation distances. Electric field calculation is based on electrostatic which is considered as a subfield of electromagnetics. However, there is a fundamental difference between them which is the source of the field (immobile static charges). Assume a space charge with density of 휌 in free space, the volume flux density of outward field vectors can be defined as the following equation.
훻. 퐸 = 휌⁄휖0 (2.2) where 휖0 is electric permittivity of free space and it is a universal constant [21]. The resistance ability of materials against electric field is called electric permittivity. In order to take this factor into consideration, relative permittivity of material is introduced. Relative permittivity is defined as 휖푟 = 휖⁄휖0 where 휖 is the material permittivity and 휖0 is the vacuum permittivity. Higher values of relative permittivity present higher resistance of the material against the electric field. For instance, silicon relative permittivity is 11.68 while this number for air is 1
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[22]. It can be concluded from eqn (2.2) that in electrostatics, the behavior of space charge density is similar to a volume source. Although this relationship is insufficient, it can be completed by adding irrotationality condition (curl free) to the electric field from Maxwell’s equation (훻 × 퐸 = 0). The existence of a scalar potential for each irrotational field can be proved which leads to the electric potential equation as (2.3).
휕 휕 휕 퐸 = − [ 푖 + 푗 + 푘] 푉 (2.3) 휕푥 휕푦 휕푧 The following relation is applicable for any adequately smooth field and confirms the irrotationality of the field [21].
훻 × 훻푉 = 0 (2.4) The electrostatic behavior of dielectric material is different from free space which leads to introduction of polarization vector field and polarization charge density. An ideal dielectric material is assumed to have bound charges rather than free charges. Bound charges can be demonstrated by exposing the dielectric material into the external electric field which creates induced electric dipoles. In other words, the induced electric field causes the formation of negative and positive charge pairs which are aligned alongside the field direction. This phenomenon produces an electric field inside the material which resists against external field and it is known as polarization vector field (P) and the compactness of the charges is called polarization charge density (휌푝) which can be shown as:
휌푝 = 훻. 푃 (2.5) The polarization that modifies the material’s internal electric field is based on:
휌 = 훻. (휖0퐸 + 푃) (2.6) According to eqn (2.6) a new parameter can be introduced by the name of electric displacement field (D) which is defined as:
퐷 = 휖0퐸 + 푃 (2.7) By using this definition and inserting it to the Gauss’s law (eqn 2.3) along with fulfillment of the field irrotationality, the equations of electrostatics can be expressed as a single equation covering all requirements.
휌 = −훻. (휖0훻푉 − 푃) (2.8) Faraday’s law and Gauss’s law specify circumstances on divergence and curl for the electric field. The former implies a situation on the normal field character and the later implies a condition on the tangential character of the field. The definition and interpretation for both laws can be summarized in tables 2.1 [21]. Apart from the electric field strength, its distribution is an essential factor for field stress identification. In order to understand this factor, let us assume a rod-plane model (Figure 2.7) where the distance between the electrode and plane is filled with a homogenous material and the breakdown strength of this material is
퐸푏(constant). The essential separation distance between the plane and rod can be calculated as 푑 = 푉/퐸푏 where 푉 is the applied voltage on the rod. Suppose the gap between the rod and plane remains constant and it is filled with atmospheric air at atmospheric pressure with steady air density. As it is presented by dashed lines, the diameter of the electrode (D) is changed
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over a broad range. One can conclude that for any diameter two field quantities can be defined as mean value of the field strength (퐸푚푒푎푛 = 푣/푑) and maximum field strength at the tip of the electrode (퐸푚푎푥). With these two factors field efficiency factor (휂) can be defined as eqn (2.9)
Table 2.1 Gauss’s law and Faraday’s law summary Name Equation Meaning Start and end point of each field line Gauss’s law 훻. 퐷 = 휌 is on charges. 훻 × 퐸 = 0 Faraday’s law The electric field is irrotational. −훻푉 = 퐸
The total flux passing through a Gauss’s law ∮퐷. 푛푑푠 = 푄 closed surface is equal to its enclosed 푆 charge.
Faraday’s law ∮퐸푑푙 = 0 The electric field is conservative. 푙
퐸 푉 휂 = 푚푒푎푛 = (2.9) 퐸푚푎푥 푑퐸푚푎푥 where 휂 is equal to 1 for uniform field distribution and 0 for non-uniform distribution such as, a rod with zero edge radius [1]. Due to the realignment of dipoles inside the dielectric under the electric field influence, they compensate each other except on the surface [13]. Therefore, in more complex geometries 퐸푚푎푥 reveals at any point of the surface and not essentially corresponding to the points creating shortest gap distance (d) [1].
Figure 2.7 Rod-plane model (see Fig 4.1 in [1])
In order to grasp the necessity of the field efficiency factor, eqn (2.9) can be expressed as:
푉푏 = 퐸푚푎푥. 푑휂 = 퐸푏푑휂 (2.10) where 푉푏 is breakdown voltage and 퐸푏 is field breakdown of the material (in this case air) and it is equal to 퐸푚푎푥. As 0 ≤ 휂 ≤ 1 for any electric field distribution, one can conclude that non- uniformities in the field, decrease the flashover voltages. Since breakdown stresses and corona discharges rely on the field distribution, models with demonstration of highly stressed regions
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are adequate. Thus, identifying the breakdown voltages with the aid of field efficiency factor is essential [1]. It is necessary to underline that due to the interconnected relationship between the field strength and field distribution, solving problems with complex geometries is extremely sophisticated. Therefore, powerful methods are needed to be employed, such as finite element method (FEM) which will be discussed in chapter 3. Simple electric field calculations can be done for various models for instance, cylinder and sphere. Due to the high possibility of E-field stresses on smaller parts of the geometry which are loaded with high potentials, connecting cables in capacitor banks are more likely to show corona. Two- cylindrical-conductor model is selected for E-field calculation and interaction investigation as it shows the interaction between two cables in the bank [1].
E. Kuffel et al. [1] stated that the corona inception voltage can be escalated considerably among the bundle of conductors in comparison with a single cylindrical conductor. This implies that there is an interaction between the conductors which can result in the maximum field reduction. In order to prove this statement, consider two cylindrical conductors which are loaded with opposite charges. Line charges can be shown as:
±휌l = ±푄/푙 (2.11) where l is the length of the conductor and Q is the number of charges. It should be emphasized that the two conductors are in parallel with each other. Figure 2.8 shows two-line charges
(±휌l) with the distance of b. The potential (휙푝) can be calculated at any point (P) with in the plane. Let us define the field intensity of a single line charge as:
퐸(푟) = 휌/2휋휀푟 (2.12) where r is distance from the charge and with the help of superposition principle, the potentials can be calculated as:
휌 푟′′ 휙 = l ln( ) + 푘 (2.13) 푝 2휋휀 푟′ where푟′′ and 푟′ are distances presented in Figure 2.8 and 푘 is constant value corresponding to the boundary conditions [1].
Figure 2.8 Two cylindrical conductors in parallel (±휌푙) (see Fig 4.9 in [1])
Suppose the lines are equally loaded with opposite charges and 푟′′ = 푟′. As a result, potential is zero at the plane (휙푝 = 0) which means 푘 is zero. It is concluded that by maintaining the ′′ ′ ′′ ′ ratio of 푟 /푟 constant, 휙푝 remains constant. However, any constant ratios of the 푟 /푟 create cylindrical surfaces. These cylindrical surfaces are assumed to be conductors with different diameters [1]. Figure 2.9 demonstrates two conductors with equal diameters which are charged equally with opposite polarities. Distance c is the eccentric position between the centers M
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′′ ′ ′′ ′ and line charges (±휌l) which can be found for constant values of 푟 1/푟 1 and 푟 2/푟 2 for the points 푃1 or 푃2 (located at A, B or C, D). By considering the radius of both cylinders as r, distance c can be shown as:
푏 2 푏 푎 푎 2 푐 = √( ) + 푟2 − = − √( ) − 푟2푎푛푑푎 = √(푏)2 + (2푟)2 (2.14) 2 2 2 2
Figure 2.9 Two cylindrical conductors in parallel with symmetrical charge lines (see Fig 4.10 in [1])
For radii (푟) which are significantly smaller than 푎/2, distance c approaches to zero (a=b). It can be interpreted that for larger gaps the line charges locate at the center of the conductors i.e. in comparison with a single conductor, fields will not be distributed substantially around the conductor surface. For the conductors with small diameters, the field distribution can be calculated along the highest dense flux line (between B and C). Equation (2.15) expresses the potential calculation along this line initiating at 퐵(푦 = 0) based on eqn (2.13) [1].
푏+푆 ′′ ( )−푦 푟 2 휙(푦) = 퐴 ln ( ) = 퐴 ln [ 푏−푆 ] (2.15) 푟′ ( )+푦 2 where S is the gap distance and A is a constant value derived from the boundary conditions. Suppose the voltage difference between the two conductors is 푉. As a result, 휙(푦) for 푦 = 0 is 푉/2 and 퐴 is equal to:
푉/2 퐴 = (2.16) 푏 + 푆 ln ( ) 푏 − 푆
Based on the Gauss’s law (퐸 = −∇푉) the field strength 퐸(푦) can be written as:
휕휙(푦) 1 1 푉 푏 퐸(푦) = − = 퐴 [ + ] = . (2.17) 휕푦 푏 + 푆 푏 − 푆 2 푏 2 푆 2 푏 + 푆 ( ) − 푦 ( ) + 푦 [( ) + (푦 − ) ] ln ( ) 2 2 2 2 푏 − 푆
Since the field distribution is symmetrical from the surface of the conductors (푦 = 푆/2), for simplification purposes the distance 푏 = 푓(푎, 푟) can be shown by the gap distance as:
푆 2 푆 √( ) + ( ) 푉 2푟 푟 퐸(푦) = (2.18) 푆 푦 푦2 푆 푆 2 푆 [1 + − ] ln (1 + + √( ) + ( )) 푟 푟푆 2푟 2푟 푟
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In order to explain the field distribution in a comprehensible way, it is essential to identify the relationship between maximum field stress (퐸푚푎푥) and equation (2.18) for the surface of conductor (푦 = 0):
퐸(푦) 1 푟 = = (2.19) 퐸 푦 푦2 푦 푚푎푥 1 + − 푟 + 푦 (1 − ) 푟 푟푆 푆 For a single charged conductor, the mentioned ratio would be equal to 푟⁄(푟 + 푦). Thus, for any values that 푦⁄푆 is significantly smaller than 1, the corresponding conductor reduces the field effect. By reaching to the maximum E value at the point 푦 = 푆/2, the ratio between
퐸푚푎푥and 퐸푚푖푛 becomes: 퐸 1 푚푖푛 = (2.20) 푆 퐸푚푎푥 1 + ( ) 4푟
2.2.2 Stream inception and propagation The aim of the streamer propagation model is to predict the strength and to determine direction of the streamers. In other words, this model forecasts how far a streamer can propagate and towards which direction. Obviously, if streamers succeed to reach the opposite electrode, a flashover takes place. As presented in Figure 2.10, Ω is the geometry domain, ∂Ω0 is the positive electrode Dirichlet boundary condition, and ∂Ω is the Dirichlet boundary condition for the domain Ω. Suppose that the potential of the electrode is positive (푉0) and the potential of counter electrode is small (let’s say grounded). Therefore, electric field lines (퐸 = −훻푉) at ∂Ω0 flow from the high potential surface towards the lower one [25].
Figure 2.10 Geometry of domain Ω
The critical size of electron avalanche is called streamer inception criterion which can be expressed as:
∫ 훼 (퐸)푑푙 > 퐶 (2.21) 푙 푒푓푓 푐푟푖푡푖푐푎푙
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where l is the streamline and 훼푒푓푓 is the production rate of free charges (electrons) in the gas which it is called effective ionization. Effective ionization can be calculated as the subtraction of ionization rate (which is highly dependent on the electric field) and recombination rate.
Therefore, 훼푒푓푓 is a function of E-field and the shape of this function may differ for various gases. There are two conditions that must be fulfilled in order to apply equation (2.21). First, the integral should be along the field line (l) and second, 훼푒푓푓 should be positive [19]. The main goal of these calculations is to find critical regions on Ω where the streamer inception region (see Fig 2.10) is determined and eqn (2.21) is satisfied. Consequently, the critical electrode region can be detected as:
휕Г0 = Г ∩ ∂Ω0 (2.22) where휕Г0 is the starting point of the streamlines fed from the electrode, Г is the inception region, and ∂Ω0 is the electrode boundary. With the intention of achieving this aim, it is essential to define a new scalar field variable Ø(푥⃗) which can satisfy the partial differential equation (2.23) [25], [7].
−푣⃗. ∇⃗⃗⃗Ø = 훼(퐸)Ө(훼) (2.23) where Ө is Heaviside theta version function defined as:
1 Ө(훼) = {0, , 1} for{훼 < 0, 훼 = 0, 훼 > 0} (2,24) 2
And 푣⃗ is the normalized vector field along the field lines as:
푣⃗ = 퐸⃗⃗/퐸 (2.25)
Equation (2.23) can be solved as the integral of the 훼 along the field lines. Obviously, Ө function is defined to ensure that Ø is equal to zero when 훼 ≤ 0 and the integration is applicable for positive amounts of α. As a result, the inception region Г appears under the
Ø(푥⃗) > 퐶푐푟푖푡푖푐푎푙 condition [25]. Assuming a simple stream propagation which has a constant internal field (퐸푠) along its route, the potential loss along the path can be expressed as:
푉푠(푠) = 푉푠10 + 퐸푠푙 (2.26) where l is the streamer length and 푉푠10 is constant and can be defined as the required voltage by the stream head. This constant is used for the model reproduction improvement purposes. Suppose that a streamer propagates along the E-field lines, propagation path can be determined by solving differential equation (2.27) which indicates the location of the streamer head [25].
푑푥⃗⁄푑푡 = 푣⃗(푥⃗)ℎ(∆푉, 푡) (2.27) where 푥⃗(푡) has the initial condition of 푥⃗(t = 0) ∈ 휕Г0 (see Figure 2.10), and ∆푉 is the voltage drop between the starting point of the streamer (on the electrode surface) and the head position of vector 푥⃗ presented in eqn (2.28). It is noteworthy to mention that t is equal to the length of streamer (l) and since the vector field is normalized, the absolute value of 푣⃗ is equal to one [25], [7].
∆푉=푉0 − 푉(푥⃗) (2.28)
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The functionality of ℎ(∆푉, 푡) in eqn (2.27) is to ensure that the streamer stops when the regional potential difference is inadequate for further propagation. Meaning that if the inception and propagation criteria is satisfied, h is equal to 1 otherwise it is 0. The simplest model can be defined as the 푉푠10(constant) is equal to 0 and h is chosen to be the Heaviside theta function as h=Ө(∆푉 − 퐸푠푙). Since on the surface of the electrode the voltage difference and distance are equal to zero (∆푉 = 푠 = 0), h becomes 0 which means that a streamer does not start. Therefore, h should be modified to eqn (2.29) by adding the term Ө(Ø − 퐶푐푟푖푡푖푐푎푙) which is positive at Г0(surface of the electrode) and ensures that 푣⃗ ≠ 0 at starting point (푥⃗(0) ∈ 휕Г0). Now the streamer can start propagation from all the points of the surface if it meets the mentioned criterions. Also, the most common streamer propagation condition for outside the Г is ∆푉 > 퐸푠푙 [25], [7].
h =Ө(Ө(Ø − 퐶푐푟푖푡푖푐푎푙) + ∆푉 − 퐸푠푙) (2.29)
Figure 2.11 presents streamlines from an electrode towards a grounded plane and equipotential lines. It can be observed that the equipotential lines are curvature and closer to each other near the electrode tip and parallel in a distance. This means that the intensity of voltage and consequently electric field is higher near the electrode and it fades away by distance. Also,
Figure 2.12 demonstrates the propagation pattern of the streamer lines starting from Г0.
Figure 2.11 Streamlines from an electrode towards a grounded plane and equipotential lines
Figure 2.12 Propagation pattern of the streamers starting from Г0
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2.2.3 Corona discharge threshold In order to evaluate the existence of corona discharge, the first step after calculations of E- field is to detect the areas with highest intensity. These regions should be compared with the limits and if they are high enough to exceed the threshold, then it means that corona discharge occurs. Now the question is, what is the criterion for corona discharge appearance? A short answer to this question can be; adequate effective ionization amount (훼푒푓푓)[10].
As it is mentioned previously, the necessary condition for a discharge formation is positive effective ionization (훼푒푓푓 > 0). However, this condition is insufficient, and an electron avalanche must be integrated with it. Electron avalanche is described as free electron mobility accelerations under the effect of the electric field which can cause further ionization at an increasing rate. This will result into the self-sustained discharge formation i.e. independent from additional charge injection. Assume that the formation of such an avalanche is in the direction of electric field lines, the requirement for self-sustained avalanche can be expressed as eqn (2.21). The critical number (퐶푐푟푖푡푖푐푎푙) may vary between 15 to 20 which 18.4 is considered as the corona inception criterion. In other words, if the result of the eqn (2.21) reveals 18.4 and higher, stationary streamer (corona discharge) emergence is ensured. Furthermore, by achieving eqn (2.21) and maintaining effective ionization factor positive along the field line (starting from the surface of conductive where E-field is maximum until it reaches to the earth), an instant breakdown occurs over the gap. It is essential to mention that in this scenario the leader propagation effect is neglected. Leader propagation occurs when the plasma channel conductivity is high enough, and the field streamer head growth allows the streamer to propagate further almost independent from the extremal E-field [10].
Calculation of the integral along the streamline depends on the effective ionization (훼푒푓푓) and it may vary in different gases. For air this factor can be evaluated as:
퐸 2 훼푒푓푓(퐸) = 푝[푘( ⁄푝 − 훬) − 퐴] (2.30) where 푘=1.6 (mm bar/푘푉2), 훬=2.2 (kV/mm bar), A=0.3 (I/mm bar), and 푝 is pressure in bar. One can calculate that in standard atmospheric pressure (1 bar) and for electric field values above 2.6 kV/mm, 훼푒푓푓(퐸) is positive which means the field strength should be higher than this number in order to create net charge carriers. As a result, the critical E-field for corona emergence in the air is 2.6 kV/mm without considering the environmental factors, such as humidity and pollution that will be discussed in section 2.3 [10].
2.3 Effective factors on emergence of corona Apart from the type of discharge (breakdown or steady corona), factors such as voltage polarity, humidity, gas pressure, pollution, geometry, and gas nature are important and influential. The first four elements will be discussed further in more detail and the geometry factor is explained in section 2.4. However, since the focus of this study is on the high voltage capacitor banks, the only surrounding gas is air and investigations regarding the nature of other gases is beyond the scope of this study. In addition, an experimental study was conducted by D.Ariza et al. [27] regarding the wind effects on DC positive corona. The study was implemented for eight voltage levels between 5 and 12 kV and air stream with three various velocities. The results have shown that wind speed does not have any effect on corona onset electric field although, the corona current increases with the wind velocity [27]. Another study
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regarding glow to arc transition (GAT) implemented by C. Hsu and C. Wu [18]. It is shown that by increasing the gas flow rate from 0 to 50 m/s, the transition voltage rises from 10% to 20%. On the other hand, an erratic behavior was observed during an experimental study regarding HVDC systems conducted by J.Kuffel et al. [34]. In other words, it is stated that wind can either ignite or extinguish corona discharges.
2.3.1 Polarity An experimental study was conducted by J.Riba et al. [9] regarding visualization of corona on AC and DC (positive and negative) voltages. The results have shown that positive DC corona appears in lower voltages in comparison with AC. This behavior can be explained due to the DC steady electric field. During the exposure of atoms and molecules to the DC sustained field, ionization occurs with lower voltages. On the other hand, because of the alternating nature of AC voltages (50 Hz) ionization process is slower as the electrons are getting repelled and attracted constantly during the wave transition from positive half-cycle to negative cycle. Thus, more energy and higher voltages are needed to initiate corona [9]. By applying the pressure factor in this relation, one can conclude that complete breakdown voltages are significantly higher under the negative polarity except in low pressure circumstances. Consequently, under the alternating voltages the complete breakdown occurs consistently in the positive cycle of the wave [1].
Research has shown that for small gap distances, steady corona does not appear and the breakdown behavior is similar for both positive and negative polarities. By increasing the gap distances within the pressure range of 0 to 7 bars, a strong steady corona appears for the positive voltages. For the further pressures the breakdown voltage falls unexpectedly and rises again. Negative polarity has shown that steady corona and breakdown appear in much higher voltages and pressures [1].
2.3.2 Pressure With the help of cloud tracking pictures during the breakdown, one can observe that breakdown appears through extremely narrow filaments and channels in higher pressures. A study conducted by Leob and Meek [11] regarding the relationship between pressure and breakdown voltages. The first scenario is to have two planes in parallel with the gap distance of one centimeter and expose the cathodic plane to the ultraviolet light in the way that one electron moves 1 mm2 from cathode region. Therefore, the potential on the plates must be 31,600 volts which is considered as sparking potential and its ratio at atmospheric pressure is 41.6 volts per centimeter for each mmHg [11].
Having a closer look on electrons’ behavior during avalanche, the initial average energy for each electron can be calculated from eqn (2.31) which is 3.6 volts per electron. The moving electron in the field direction creates more electrons at the ratio of α per centimeter where α is the number of ions per centimeter. As the avalanche grows, the movements of electrons become more random and the average diffusion radius can be calculated by eqn (2.32) where 푡 is inserted from eqn (2.33), 퐷 is diffusion coefficient that can be projected from 푣 which is the velocity of diffusion, and 푥 is the desired distance. Equation (2.34) presents the relationship between the gap distance and number of electrons where 훿 is the gap distance. For instance, the number of electrons at 1 cm distance is 푒17or 2.4×107by considering α as 17 in these situations. It is necessary to mention that an avalanche cannot create conducting
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filaments within the gap therefore the one that can succeed to pass the gap does not create breakdown [11].
퐸 = 1/2푚푐2 (2.31)
1 푟 = (2퐷푡)2 (2.32) 푥 푡 = ⁄푣 (2.33) 퐴 = 푒훼훿 (2.34) Corona discharge will be followed by an extremely high current caused from electrons’ recombination with the speed of 109 up to 1010cm/sec known as breakdown. It is observed that for longer distances due to the considerable avalanche feed to streamers and noticeable strength of them an arc appears with the crooked or forked forms. The effect of this phenomenon becomes more powerful when the gap is longer (within the limits) and pressure is lower. Obviously, under the overvoltage situations the whole process accelerates and the chance of every avalanche to create a breakdown increases [11].
Paschen [12] demonstrated that there is a relation between the breakdown voltages (푣), gap distances (cm) and pressures (mmHg) for various gases which is illustrated in Figure 2.14. By maintaining the pressure and increasing the gap distance, the breakdown voltage drops initially. Then by maintaining the distance and rising the pressure, the break down voltage increases gradually. This means that for higher altitudes because of the low gas pressures, it is more likely that discharges reveal as the form of corona followed by spark [12]. This behavior of various gases can be explained by a simple model. Suppose an electric field between two planes inside a sealed chamber, as presented in Figure 2.13. Under the electric field effects, electron detachment occurs near the grounded plane. The separated electron accelerates and impacts with another electron on its way towards the negative plane which creates two free electrons as well as two ionized atoms (a). If the gap distance between the planes is not large enough, it is less likely to have the impact (b). However, less gap distances means lower breakdown voltages as it is shown by the initial drop in Paschen’s curve. Also, by increasing the pressure the free electrons do not accelerate enough to release other electrons after the impact (c). This effect agrees with the rising trend of the Paschen’s curve. Although Paschen’s law is stated for complete breakdown, it is applicable for corona inception voltages since corona discharge is considered as early stage of the breakdown occurrence.
Figure 2.13 Paschen’s law. (a) Electron impact in normal condition. (b) Electron impact in short gap distance. (c) Electron impact in high pressure
Streamer establishment requires specific length of electron avalanches for each individual amount of field strength divided by pressure ratio (푋⁄푝). In the case that avalanche length is equal or less than the gap distances, an arc appears. The reason that this rule is applicable for less lengths is that a mid-gap streamer interferes and helps to shape a spark. Under the conditions that the gap is longer than δ, streamers cannot shape and spark does not appear
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because of the electron density loss during the long path lengths. As a result, a spark field strength (푋푠) relies on sufficient charge density created by an avalanche within the gap distance or less. The length of avalanche should have two characteristics, adequate charge density and photoelectron production to guarantee the streamer propagation (SP) [12].
Figure 2.14 Paschen’s curve [37]
2.3.3 Humidity Humidity is one of the factors that has a significant impact on corona emergence, and it can be defined as any incident that rises the number of drifting water molecules in the air such as, fog. An experimental study was conducted by Zhang et al. [26] for different DC voltage levels in various humidity conditions. Measurements and calculations on this study was based on the released number of photons per second which shows the severity of corona discharges. It is observed that by maintaining the voltage and rising the humidity levels from 45% to 85%, number of released photons is increased by 40% which means the intensity of corona discharge is risen [26]. In other words, emitted photons ignite more avalanches which means more ionized atoms and molecules are generated. Calculations depict that escalation of emitted photons from 500/m to 1000/m may result in corona inception voltage reduction from 12.59 kV to 12.48kV [5]. Obviously, this depletion is based on the electric field increase. However, for voltages below 70 kV the humidity impact is not significant as it is illustrated in Figure 2.15 [26]. The results of this study have shown that when corona initiates near the conductor surface, floating water particles in the air (larger than 2 × 10−7푚) will be charged. Therefore, they can be treated as homogenous dielectric sphere which is charged mostly by the field. The water particles’ charge saturation (q) can be written as: 휀 휀 푞 = 12휋푟2퐸 0 푟 (2.35) 휀푟 + 2 Also, the Coulomb force applied on the water particle can be calculated as:
2 2 휀0휀푟 퐹푞 = 퐸푞 = 12휋푟 퐸 (2.36) 휀푟 + 2 where r is the radius of the water particle, 휀푟 is relative permittivity of the water particle, 휀0 is relative permittivity of free space, and E represents the exterior electric field [5].
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Figure 2.15 Electric field under three various humidity conditions (see Fig 6 in [26])
Three forces are interacting on the suspended water particle, Coulomb force 퐹푞, Stokes’ drag force, and gravitational force. The gravitational force is negligible and Stokes’ drag force can be expressed as 퐹푠 = 6휋푟휂푣 where r is the particle radius, v is particle velocity, and 휂 is the air viscosity. When the mentioned forces are balanced, the particle is in dynamic equilibrium and the mobility (k) of the charged water particle can be expressed as:
푣 2퐸푟휀 휀 푘 = = 0 푟 (2.37) 퐸 휂(휀푟 + 2) It can be observed that there is a linear relationship between the water particles’ mobility, the radius, and electric field strength. One can conclude from the calculations and experimental studies that when the mobility of the water particle reduces, number of floating ions in the air rises [26]. Since the drift velocity of ions in humid air is lower than the dry air, less charges flow to the grounded plane [33]. The effect of humidity can also be expressed as the form of the effective ionization integral along the length of ionization zone and it shows the intensity of the area:
푧푖 휉 = ∫ [훼(푧) − 휂(푧)]푑푧(2.38) 푧푚 where 훼 − 휂 is the effective ionization coefficient, 푧푖 is the area with 훼 − 휂 = 0, and 푧푚 is the point where the net charge density is maximum (streamer head). At sustained air pressure, by humidity rise, the values of effective ionization (훼 − 휂) increases as well as the critical avalanche size. As a result, corona emergence occurs on lower voltages [5].
2.3.4 Pollution Pollution can be generated from various sources, such as sea salt which is carried by wind, industrial air pollutants, road salts, and bird excrement. Pollutants can be divided into two categories, covering contaminants and floating contaminants. Covered areas with a layer of pollution are neutral as long as the surface is dry; however, a light rain, fog, or mist can turn the pollution layer to a conductive which increases the electric field non-uniformity. Also, air pollution has a considerable impact on discharges as the suspended particles can be charged and create a conductive path which current can leak into the air. The measurement of pollution severity is essential as this information can be utilized for separating different sever zones
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based on characteristic of contamination. Pollution severity can be classified into four sections, light, medium, heavy, and very heavy. Light zone is the region with low density of industries and houses where it is exposed to the frequent winds and rainfall. Medium zone is referred to the area with average and high density of industries and houses which is subjected to the frequent rainfall and winds. Heavy zone is large cities with high density of industries and areas near the sea with exposure to the strong winds from the sea. Finally, very heavy zone is the subjected location to the conductive dusts and industrial smoke. In addition, coastal areas as well as desert areas are categorized in this group [1].
2.4 Corona discharge suppression methods As shown by Cohen [23], electric field on the sharp edges is reduced when the radius of the edge decreases. In other words, the aggregation of charges on the sharp edges is extremely high in comparison with round surfaces. Therefore, it is more likely that corona may appear on sharp edges and corners. One of the methods to suppress corona on high voltage devices is to make the corners and edges smooth and round. One can show that by increasing the size of the conductors the electric field declines as charges are distributed on the larger area. Suppose two conductive cylinders with the same length and material. The radius of the first cylinder (A) is chosen to be twice as the radius of second cylinder (B). By applying the same voltage on them and maintaining the same condition, cylinder A shows half of the electric field values in comparison with B. This means that charges are distributed over less area which rises the intensity of E-field. It can be concluded that the field’s drop rate along the streamlines in A is less than B i.e. the energy can be maintained longer in A.
Although the mentioned methods are effective, there are some constraints that can limit the utilization of these techniques in the high voltage component design. For instance, the radius of the conductors (specifically cables) depend on various factors that cannot be increased in order to just control the corona effect. Therefore, the last method is more feasible than the others which is the corona ring’s utilization. This method does not interfere with any electrical element design and reduces corona efficiently. Since the conductor curvature shape is important and sharp edges should be avoided, two alternatives can be suggested, ring or sphere. Choosing between mentioned options depends on various factors which will be discussed further in chapter 3. Let us assume a simple block which is surrounded by a ring (as shown in Figure 2.17) with the same voltage applied on both. On the other hand, the same condition for a single block without the ring. It can be seen clearly that the surface of the block surrounded by a ring has less electric field rather than a simple block. One can conclude that the ring field is interacting with the block. As a result, the E-field intensity is reduced since the maximum values of it are shifted away.
As it is expressed by E. Kuffel et al. [1] a direct relationship between the concentration of the equipotential lines, and the strength of the electric field exists. Thus, the aforementioned statement can be elucidated by detecting the equipotential lines as presented in Figure 2.16. It can be observed that these lines are closer to each other for a single block in (b) and they are more evenly distributed between the ring and block in (a).
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Figure 2.16 Cross section equipotential lines. (a) Block with ring. (b) Block without ring
Corona ring is a metallic hollow tube which is normally made by aluminum. It is assembled on high voltage components in order to distribute the electric field gradient in the way that the intensity of the area decreases, and maximum E-field location alters. As a result, the air does not get ionized, and corona does not emerge. These rings have been used in high voltage industry for several decades and conventionally it has been recommended to be installed on AC capacitor banks with 180 kV and above. However, this voltage level may vary in different cases which will be discussed further in chapter 3. Suppression ring has three main parameters that can affect the position and magnitude of the maximum electric field. These factors are tube radius (R), diameter of the ring (D), and geometry. The effects of these parameters can be presented by a simple model which consists of a block and a surrounding ring as illustrated in Figure 2.17.
Figure 2.17 Corona ring effective parameters
The model consists of a block with 1 m width, depth and height. The ring diameter (D) is 3 m, the tube radius (R) is 0.1 m, and 1 kV is applied on the block and the ring. The first scenario is to evaluate the E-field behavior by changing the tube radius (R). By increasing the tube radius, the electric field reduces on the block. It can be explained that the ring is covering more areas and the electric field has more even distribution in this situation. Figure 2.18 shows the equipotential lines for both cases. One can observe that as the tube radius increases, the lines become less concentrated and further from each other.
For the second factor, the same model with the same condition is maintained except the ring diameter which is changed from 4 m in to 2 m. Although reducing the ring diameter decreases the E-field intensity at the middle part of the block, it rises on the top and bottom sides. When the ring approaches the block, charges are pushed and concentrated on the area that ring has less effect. This interpretation agrees with the pattern of the equipotential lines in Figure 2.19. As the ring diameter increases, the equipotential lines between the ring and block become
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denser. Also, as the ring approaches to the block, lines are more concentrated at the top and bottom sides of the block.
Geometry of the object has significant impact on the electric field distribution. Figure 2.20 compares the cross section equipotential lines for the same block-ring model in (a) and a cone- ring model in (b). As it is illustrated by black arrows, the equipotential lines are more intense at the tips of the cone and the line between the ring and cone is elongated in comparison with the block (a).
Figure 2.18 Cross section equipotential lines. (a) With tube radius of 0.1 m. (b) With tube radius of 0.2 m
Figure 2.19 Cross section equipotential lines. (a) With ring diameter of 4 m. (b) With ring diameter of 2 m
Figure 2.20 Cross section equipotential lines. (a) Block and ring. (b) Cone and ring
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In order to discuss in more detail regarding the geometry effects on corona discharge and breakdown emergence, a sphere-plane model is a suitable example. Suppose a sphere with the diameter of 퐷푠푝ℎ푒푟푒 is attached to a tube with the diameter of 퐷푡푢푏푒. The tube is located vertically while the sphere is attached to the bottom end of it. A plane is located beneath the sphere where there is a gap between the lowest point of the sphere and the plane. The diameter of the sphere changes within the range of 0.25 m to 2 m. the gap length varies between 0.05m to 30m. Applied voltage on the sphere is 1000 kV and the plane is grounded. It is detected that by reaching to the ratio of 4.36 between the 퐷푠푝ℎ푒푟푒 and 퐷푡푢푏푒, the maximum electric field gradient appears at the bottom of the sphere [14]. C. Menemenlis et al. [35] proposed that the corona inception surface gradient (퐸푏) can be calculated as: 1 + 푑/푅 퐸 = (2.39) 푏 0.42 + 0.30(푑/푅) where 퐸푏 (MV/m) is a function of the geometry, d is the gap length, and R is the radius of the sphere. By having 퐸푏, The breakdown voltage can be calculated from the equation (2.40) [14].
푉푎푝푝푙푖푒푑 푉푏 = 퐸푏(2.40) 퐸푚푎푥 By calculating breakdown voltages for various sphere diameters, one can conclude that when the diameter increases, the breakdown voltage rises and vice versa [14]. Consequently, the corona inception voltage has the same pattern and as it was expected, the results agree with the proposed calculations by E. Kuffel et al. [1].
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Chapter 3
Methods and modeling
3.1 Introduction to COMSOL Multiphysics COMSOL Multiphysics is a powerful simulation tool which allows scientists and engineers to model and solve wide range of problems. Within this software the integration of the desktop environment and model builder has developed in the way that having a full view of the model with access to all functions is viable. It provides a wide range of options to apply several types of physical conditions on the model simultaneously. By applying physical quantities, such as loads, material properties, fluxes, and sources on the model, one can easily solve problems without any requirements of defining the underlying equations. However, it is always possible to apply numbers, variables, and expressions directly to the boundaries, domains, points, and edges in the model. Due to the complexity of the geometries, the numerical method that COMSOL has employed is finite element method (FEM). This software provides several studies, such as stationary and time dependent that can be utilized during the model analysis. The program is structured in the way that sequences can be followed easily and required alterations can be executed by changing the corresponding nodes in the model tree. Sequences in COMSOL are defined as the following.
▪ Global definitions • Parameters • Model inputs • Materials ▪ Component • Definitions • Geometry • Materials • Electrostatics (es) • Mesh ▪ Study • Stationary • Solver configurations ▪ Results • Datasets • Derived values
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• Tables • Exports • Reports
By running the analysis alongside with the adaptive mesh, errors are identified and controlled by software. It is necessary to mention that all steps such as, meshing, boundary conditions, and post-processing are recorded which provides considerable freedom for the user to follow and control the process of calculations. Since the accuracy of the mesh has significant impact on final results and computation processes, different meshing alternatives are designed in COMSOL Multiphysics such as, hexahedral, tetrahedral, triangular, and quadrilateral. In addition, post-processing stage consists of a wide range of visualization tools that can present the results in comprehensible way. This software has numerous advantages, such as reliability, accuracy, user friendly, and high-speed calculation which makes it suitable for simulating physical phenomena.
3.1.1 Finite element method (FEM) This section is based on chapter 4, section 4.4 in [1]. Although there are various mathematical approaches to model the finite element method (FEM), the most common process is based on the physical property of different fields. A brief description for this method is based on concerns regarding diminishing the energy in the entire area of interest as much as possible when the field belongs to Laplacian or Poissonian type (magnetic or electric). In other words, this method is focused on the energy derivatives with respect to the corresponding potential distribution. Assume an equilibrium electric field penetrating a dielectric substance which has the permittivity independence from the field strength or reliance on it (anisotropic or isotropic) and its conductivity is neglected. Since space charges are neglected, the field is generated between metal boundaries where the dielectric material is located. In Cartesian coordinate system, the stored electric energy (W) inside the volume R (where it is the region under the study) is:
1 휕∅ 2 휕∅ 2 휕∅ 2 푊 = ∭ [ {휀푥 ( ) + 휀푦 ( ) + 휀푧 ( ) }] 푑푥푑푦푑푧 (3.1) 푉 2 휕푥 휕푦 휕푧 where ∅ is potential, 휀푥, 휀푦, and 휀푧 are defined as permittivity of anisotropic materials and for isotropic material this element is equal to 휀 in x, y, and z directions (휀푥 = 휀푦 = 휀푧 = 휀). It should be emphasized that 휀 may vary on boundaries for different isotropic substances. One can easily identify that applied expressions (휀∇2∅/2) inside the eqn (3.1) are energy densities per unit in volume R. Suppose the potential distribution variation rate is zero in the Z direction which means the gradient in that direction is zero. Therefore, the volume R alters to a two- dimensional model A. Figure 3.1 (a) demonstrates the mentioned area (A) which two potentials
∅푎 and ∅푏 are given as boundary conditions and it is divided in to two parts, Ⅰ and Ⅱ where they are designated by dashed line. By considering the absence of free charges on the dashed line interface, the stored electric energy is given by:
1 휕∅ 2 휕∅ 2 푊 = 푧 ∬ [ {휀푥 ( ) + 휀푦 ( ) }] 푑푥푑푦(3.2) 퐴 2 휕푥 휕푦 where W/z is the energy density and z is a constant value. For further calculations triangular shapes are chosen in order to subdivide the two-dimensional model into the smaller elements
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(Figure 3.1, b). Choosing triangle provides a suitable freedom to cover any type of curvature boundary [1].
Figure 3.1 FEM. (a) Two-dimensional model. (b) Subdivided two-dimensional region A (see Fig 4.28 and 4.29 in [1])
Suppose one of these triangles which is shown as e and consists of m, i, and j nodes. Potential distribution in this element is based on the polynomials that can be expressed as:
∅(푥, 푦) = 훼1 + 훼2푥 + 훼3푦 (3.3) This equation shows that potentials are distributed linearly inside each element and the field severity can be calculated by taking the derivative of ∅(푥, 푦) alongside x and y axes and since this equation is first order, its derivative is constant. Also, coefficient factors (훼) for element e can be calculated by following equations:
∅푖 = 훼1 + 훼2푥푖 + 훼3푦푖 ∅푗 = 훼1 + 훼2푥푗 + 훼3푦푗 (3.4)
∅푚 = 훼1 + 훼2푥푚 + 훼3푦푚
By applying Cramer’s rule coefficient factors (훼) can be shown as:
1 훼1 = (푎푖∅푖 + 푎푗∅푗 + 푎푚∅푚) 2∆e 1 훼2 = (푏푖∅푖 + 푏푗∅푗 + 푏푚∅푚)(3.5) 2∆e 1 훼3 = (푐푖∅푖 + 푐푗∅푗 + 푐푚∅푚) 2∆e where a, b, and c are defined as:
푎푖 = 푥푗푦푚 −푥푚푦푗 푏푖 = 푦푗 −푦푚 푐푖 = 푥푚 −푥푗
푎푗 = 푥푚푦푖 −푥푖푦푚 푏푗 = 푦푚 −푦푖 푐푗 = 푥푖 −푥푚 (3.6)
푎푚 = 푥푖푦푗 −푥푗푦푖 푏푚 = 푦푖 −푦푗 푐푚 = 푥푗 −푥푖 and
2∆푒= 푎푖 + 푎푗 + 푎푚 (3.7)
= 푏푖푐푗 − 푏푗푐푖
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It can be observed that ∆푒 is the description of the triangular element e and distributed potentials within the element which are related to the adjacent nodes. By inserting eqn (3.5) in to eqn (3.3), potentials can be shown as:
1 ∅푒(푥, 푦) = ([푎푖 + 푏푖 + 푐푖]∅푖 + [푎푗 + 푏푗 + 푐푗]∅푗 + [푎푚 + 푏푚 + 푐푚]∅푚)(3.8) 2∆e
Let us define a function which relies on the shape of the used finite element and it is called shape functions (N). Therefore, eqn (3.8) can be expressed as:
∅푖 ∅푒 = [푁푖, 푁푗, 푁푚]{ ∅푗 }(3.9) ∅푚
The partial derivatives in eqn (3.2) can be considered as:
휕∅ 휕∅ = 훼 = 푓(∅ , ∅ , ∅ )푎푛푑 = 훼 = 푓(∅ , ∅ , ∅ )(3.10) 휕푥 2 푖 푗 푚 휕푦 3 푖 푗 푚
Consequently, the stored energy per unit can be written from eqn (3.2) in the z-direction:
푊 1 휕∅ 2 휕∅ 2 푒 = ∆ {휀 ( ) + 휀 ( ) } = 푋(3.11) 푧 2 푒 푥 휕푥 푦 휕푦 푒
This equation only relies on the node potentials for each element and ∬ 푑푥푑푦creates the area of element e (∆푒). Also, for simplicity purposes, the material is assumed to be isotropic i.e. 휀푥 = 휀푦 = 휀푒 and eqn (3.11) is named X. As it is mentioned in the beginning of this section, the aim of this method is to minimize the energy within the system. Therefore, the minimization equation can be written as: 휕푋 = 0(3.12) 휕{∅} where potential vector is presented by {∅} for all nodes inside the study area. Furthermore, by considering equations (3.10), (3.11), and (3.6), the minimization equations can be shown as the form of the following symmetric matrix [1].
2 2 (푏푖 + 푐푖 ) (푏푖푏푗 + 푐푖푐푗) (푏푖푏푚 + 푐푖푐푚) ∅ 휕푋 휀 푖 = 푒 [ (푏 푏 + 푐 푐 ) (푏 2 + 푐 2) (푏 푏 + 푐 푐 )] { ∅ }(3.13) 휕{∅} 4∆ 푖 푗 푖 푗 푗 푗 푗 푚 푗 푚 푗 푒 2 2 ∅ (푏푖푏푚 + 푐푖푐푚) (푏푗푏푚 + 푐푗푐푚) (푏푚 + 푐푚 ) 푚
= [ℎ]푒{∅푒} where [ℎ]푒 is known as stiffness matrix for each single element. Stiffness matrix consists of functional sensitivity regarding the potentials. It has the material property (휀푒) as well as geometric quantities (eqn 3.6). After defining the necessary equations, it is possible now to calculate the unknown potentials by utilizing algebraic equations. For example, consider an area with 4 elements and 5 nodes as it is illustrated in Figure 3.2 and unknown potential is node 5. Implementation of eqn (3.12) to these group of elements can be fulfilled by:
휕푋 = 0(3.14) 휕∅5
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where X is the energy function of the four-element system defined by eqn (3.11). By computing the stiffness matrix and insert it to eqn (3.14) and knowing ∅1 to ∅4 potentials, the potential at point 5 can be calculated with the following equation [1].
퐻15∅1 + 퐻25∅2 + 퐻35∅3 + 퐻45∅4 + 퐻55∅5 = 0(3.15)
Figure 3.2 Four-element system with 4 known potentials and 1 unknown (see Fig 4.30 in [1])
Finite element method is a powerful tool for calculations and has considerable advantages such as, it can be applied for non-homogenous and anisotropic systems, it gives a significant freedom regarding the element size and shape to cover various boundaries in the way that grid can be adjusted to the gradient of the potentials, and the accuracy can be enhanced by utilizing higher order elements without any boundary complications [1].
3.1.2 Boundary conditions Boundary conditions are applied on the geometric entities in order to segregate the interior region of the geometry from unspecified outside area as well as interfaces within the same geometry. In other words, boundary conditions are applicable on the 3D solid geometry exterior surfaces as well as interior regions embedded inside the object. Applying these conditions on the hollow 3D objects is limited to the edges of the surface. In COMSOL all physical interfaces consist of default model equations, are inseparable from default boundary conditions. In general, boundary conditions are categorized into Neumann boundary condition (Flux conditions) and Dirichlet boundary condition (Constraints). The former specifies the interaction between the model and the surrounding environment at the boundary. For example, applied forces, currents, and fluxes on the surface. The latter defines the consequences of these interactions, such as deformation of the boundary which is caused by forces. These two conditions are interconnected to each other. COMSOL is able to distinguish between the exterior and interior boundaries. The interior boundary is an interface that divides two domains in a single geometry while, an exterior boundary is defined as an outer region of the geometry. Since the focus of this study is on the corona discharge on capacitor banks, the boundary condition is based on the electrostatic study. As it is explained in chapter 2, the geometry surface has the highest voltage values which means corona discharge initiates on the surface and while it propagates, the field strength reduces. Therefore, the boundary condition is applied on the surface of the capacitor bank. Boundary condition consists of material type and property, relative permittivity, potential (applied voltage), pressure, and temperature.
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3.2 Simulation of capacitor banks Three various capacitor banks with different voltage levels were selected to be modeled and investigated. The main purpose of the simulation is to calculate the electric field and evaluate the critical areas on the surface of the banks as it has the maximum electric field. The field values which are equal or higher than 2.6 kV/mm are considered as corona discharge areas as it is discussed in section 2.2.3. There are some common factors that are considered in each simulation regardless of the type or shape of the bank. These factors are:
▪ The chosen study is electrostatic (es) and stationary. ▪ The ground is modeled with a large plane under the lowest level insulators. ▪ The capacitor shell is just a stainless-steel box which is insulated internally from the folded aluminum planes inside the unit. ▪ The rack is made of metal and has the same potential as the middle connecting cable between units as it is shown in Figure 3.3 with the black arrow. ▪ The capacitor shell has the same potential as the rack. ▪ Insulators are considered as silicon with relative permittivity of 11.7. ▪ In case that corona ring is designed for the level, it has the same potential as the rack. ▪ For more proper meshing the diameter of the connecting cables is considered as 18.88 mm which is two times larger than the actual size. ▪ The results of the electric field calculation for cables should be multiplied by the factor of two, as this is discussed in section 2.4. ▪ The default temperature and pressure are considered 20°C and 1 atm respectively for all models. ▪ Bushings and insulators are simplified with cylinders.
3.2.1 DC capacitor bank Figure 3.3 demonstrates a DC capacitor bank consisting of 8 levels with 4 units on each individual level. All levels are segregated with proper insulators as well as HV and LV terminals. The bank height, width and length are 5.9m, 2.3m, and 2m respectively. The bank consists of 32 capacitor units. These capacitors create two parallel circuits consisting of 16 units in series. The distance between the lowest level and ground is 0.8m. An air domain is defined around the bank with dimensions of 33×32×25m without any free charges. Since the accuracy of the results highly depends on the mesh quality, free tetrahedral mesh is chosen as shown in Figure 3.4. cables and terminals are meshed with the size of less than 1 mm. As a result, more than 3 million elements were created and covered the geometry.
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Figure 3.3 Geometry of DC bank
Figure 3.4 Meshing of DC bank
3.2.2 AC capacitor bank Figure 3.5 (a) depicts AC capacitor bank consists of 5 levels and 12 units on each level. This bank is not symmetrical since the distribution of the units on each side is different. It is designed by 60 units within two parallel circuits. One circuit includes 20 units in series (Figure 3.5 b) and the other one is made of 20 groups of two by two parallel units. These groups are all in series, Figure 3.5(c). All levels are separated with proper insulators as well as HV busbar. The size of the bank is 8.2×2.2×3.3m. Connecting cables and terminals are shielded with corona rings on the top 4 levels. Since the bank is 3-phase, the voltage should be
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converted from phase-to-phase into phase-to-ground. The distance between the ground and the lowest level is 2.6m. The air domain dimensions are 33×32×25m and it does not include any free charges. Cables, terminals, HV bar, and corona rings are meshed with elements less than 7 mm as shown in figure 3.6. Consequently, the entire geometry is covered by more than 4.4 million elements. In order to simplify calculations, the hollow frame is filled with a block and due to the close location of the units, they are unified. Since capacitor shells have the same potential, unifying them does not interfere with results accuracy.
Figure 3.5 Geometry of AC bank
Figure 3.6 Meshing of AC bank
3.3 MATLAB model based on streamer inception and propagation Based on the previous discussion in section 2.2.2, an integral of the ionization efficiency along a streamline determines the corona discharge appearance. Streamline data can be extracted by creating streamline sub-node in COMSOL. Since the electric field has the maximum value on the surface and this value drops considerably by distance, data extraction is sufficient for few centimeters out of the surface as shown in Figure 3.8. Based on the equations (2.21) and (2.30), a code is written in MATLAB in order to calculate and evaluate the corona inception. Also, for convenience it is linked with an Excel table (Figure 3.9), so that the extracted data from the streamlines can be inserted and read directly by the code as shown in Figure 3.10.
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Figure 3.8 Streamline
Figure 3.9 Excel table for streamline data
Figure 3.10 MATLAB model based on streamer inception and propagation
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Chapter 4
Results
4.1 Simulation results Results are presented within the following scenarios:
▪ Model without corona rings in order to identify the critical areas ▪ Original design model with corona suppression rings in order to evaluate the current design efficiency
4.1.1 DC capacitor bank Figure 4.1 illustrates the electric field strength (kV/mm) and distribution on the surface of the DC capacitor bank. High field values were observed on the top cables, Figure 4.2.
Figure 4.2 Electric field simulation result for DC capacitor bank (kV/mm)
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Figure 4.1 Electric field simulation result for DC capacitor bank (kV/mm) Note that the electric field calculations for the cables should be multiplied by two as explained in section 3.2 and this is considered in the presented results. By creating a Cut Line 3D from the ground to the first terminal, the electric field and voltage variations along the capacitor bank can be evaluated and results have shown electric potential increase as the distance from the ground increases. Consequently, electric field rises since it behaves as the function of the voltage. It is essential to analyze the equipotential lines in order to assess the distribution of the electric field along the surface. By creating a cut plane from the top rack, equipotential lines can be plotted as shown in Figure 4.6. The intensity of the lines shows the critical areas and as it is expected these critical areas are around the cables and terminals rather than capacitor shells and the holding frame.
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Figure 4.6 Equipotential lines around the top level (kV)
4.1.2 AC capacitor bank The results are presented with and without corona shields. The model without corona ring has several critical areas which are identified with high electric field values. Top level cables are overstressed due to the high potentials on them. The original model with suppression rings has no critical areas and electric field intensity is eliminated with the rings. Figure 4.12 illustrates the difference between two models, and it can be observed that the existence of corona rings is essential in order to decrease field severity and terminate discharges. This effect can be identified with the help of equipotential lines in Figure 4.13. Substantial decrease can be identified on the cables and terminals as a result of the suppression rings.
Figure 4.12 Electric field with (left) and without (right) suppression rings on bank (kV/mm)
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Figure 4.13 Equipotential lines (left) and without (right) suppression rings on bank (kV)
Electric field variation along the entire capacitor bank shows the same behavior. Figure 4.14 illustrates the field variation and the plotted red line is representing corona inception limit. One can observe that without the rings, electric field violates the threshold value on the top level as shown with the black arrow in (a). On the other hand, by utilizing the rings, the field sustains below the limit (b).
Figure 4.14 (a) Electric field variation along the bank (kV/mm-m) without suppression rings
Figure 4.14 (b) Electric field variation along the bank (kV/mm-m) with suppression rings
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4.2 Results of MATLAB model based on streamer inception and propagation After the MATLAB calculations, the results will be shown with the form of “Corona appears” or “Corona DOES NOT appear”, as presented in Figure 4.23. Results have indicated complete compatibility between this approach and simulation results. By plotting the streamline data in Figure 4.24, it is observed that the field strength declines significantly by distance as it is discussed in chapter 2. Therefore, a few centimeters further from the surface, the field becomes extremely low until it reaches to the ground.
Figure 4.23 MATLAB model based on streamer inception and propagation
Figure 4.24 Streamline plot
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Chapter 5
Discussion and conclusion
5.1 Corona suppression verification and design analysis The simulation results illustrate different values. These differences can be explained according to the various factors that can affect electric field strength and distribution, such as voltage levels and geometry. However, based on the investigated capacitor banks, the presence of the corona suppression rings is vital and without these rings, corona discharges are expected. All the models are simulated with two scenarios, with and without the original rings. The intention of this approach is to identify critical areas on the capacitor banks without the suppression rings and compare the results with the same model including the rings. Presented results verify the effects of the geometry and voltage levels on the corona discharge appearance. It is essential to mention that under the certain circumstances when corona discharge is distributed over the entire level, utilizing full rings is the best solution. It can be concluded that the original design should be used when a wide area is required to be covered.
5.2 Future work It is necessary to mention that the environmental impact assessment is beyond the scope of this paper and can be considered as future work. Also, small scale experimental studies can identify differences between the simulation and actual results. It is highly recommended that the ambient factors on corona discharge emergence (section 2.3) can be inserted into the simulations and investigated in a separate study as well as a comparison between their effectiveness degree.
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