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Monte Carlo simulation of corona discharge in SF6
Electric Power Systems Research 80 (2010) 1104–1110
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Monte Carlo simulation of corona discharge in SF6 A. Settaouti ∗ , L. Settaouti Electrotechnic Department, University of Sciences and Technology, P.O. Box 1505, El-M’naouar, Oran, Algeria
a r t i c l e
i n f o
Article history: Received 7 April 2009 Received in revised form 12 January 2010 Accepted 2 March 2010 Available online 7 April 2010 Keywords: SF6 Corona Simulation Monte Carlo Electric field Space charges
a b s t r a c t Sulphur hexafluoride (SF6 ) is one of the most widely used gaseous dielectrics for electric power systems and a number of high-voltage applications. There are many industrial applications where the electric corona discharge is used. In most cases the corona discharge is an inherently dynamic process; all parameters vary in time. Monte Carlo simulation of corona discharges in gas offers several advantages to study fundamental processes. Furthermore, it gives a fair qualitative description of the corona discharge itself as a function of space and time. This paper describes the development of negative coronas in SF6 in a point–plane gap. Detailed structure of avalanches is presented, the total field distribution, propagation of successive avalanches and ion distribution are studied. © 2010 Elsevier B.V. All rights reserved.
1. Introduction With the ever increasing need for electrical energy, high-voltage power transmission lines are constructed. Corona phenomenon is one of the problems associated with high-voltage lines. The corona discharge is important in practical high-voltage insulation systems because it can lead to deterioration of the insulating qualities of the gas as well as to production of toxic or corrosive by-products. Besides generation, transmission and distribution of electrical energy, high voltages are also extensively used for many industrial, scientific and engineering applications. High-voltage equipment is the backbone of modern power systems. In all such applications, the insulation of the high-voltage conductor is of primary importance. Sulphur hexafluoride (SF6 ) is the preferred gaseous dielectric employed by the electric power industry for gas insulated equipment used in the transmission and the distribution of electrical energy. Due to its strong capability of attaching electrons, SF6 gas is characterized by the superior electrical insulation performance. SF6 has excellent dielectric strength and arc-quenching properties. Therefore, SF6 is widely used for high-voltage insulation in electric power equipment such as gas insulated switchgears, gas circuit breaker, gas insulated transmission lines and gas insulated transformers. SF6 has gained wide acceptance as the dielectric for a number of high-voltage applications. Since it has an excellent insulating property, SF6 gas has contributed considerably to advances in miniaturization.
∗ Corresponding author. E-mail address: [email protected] (A. Settaouti). 0378-7796/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2010.03.002
Safe transmission and distribution of electrical power is partly determined by the reliability of the electrical insulation system of the power lines and substations [1–4]. It is well known that the insulation performance of gas is limited, not by its uniform field dielectric strength, but by the effects of local field enhancement and in most typical industrial applications non-uniform field breakdown predominates. By applying a detailed knowledge of the fundamental discharge processes that occur prior to the disruptive discharge, one can predict the behaviour of high-voltage insulator systems in operation under all possible conditions. Both in industrial and in environmental settings, corona-based technologies have been used to treat a variety of liquid and gaseous process effluents. Electrostatic precipitators are employed in electric power plants and many industries such as cement production, chemical processing and domestic air cleaning [5,6]. Owing to its good insulating and heat transfer properties, sulphur hexafluoride (SF6 ) is, besides air, the preferred gaseous dielectric employed by the electric power industry for gas insulated equipment used in the transmission and the distribution of electrical energy. SF6 is an electron-attaching gas widely used as a gaseous insulator in high-voltage systems, these properties allow a reduction in size and enhance the reliability of high-voltage equipment, and so electrical breakdown in SF6 is of interest both to gas discharge physicists and to power engineers. However, despite its advantages, the dielectric strength of SF6 is very sensitive to locally high electric fields. With the increasing importance of SF6 as an insulating medium, it is important to understand the mechanism of corona discharge [4,7,8]. The transient character and the small dimensions make some discharges parameters, like charged particles densities or electric
A. Settaouti, L. Settaouti / Electric Power Systems Research 80 (2010) 1104–1110
field strength, difficult to be accessible to measurements. A full mathematical model of the corona discharge is practically impossible to implement due to the complexity of the problem. Nevertheless, numerical investigations are needed in order to better understand the complex chemical and energetic phenomena triggered during the discharge phase of a corona discharge [9–12]. The purpose of this paper is to analyze the corona discharge in SF6 ; this could be accomplished by a Monte Carlo simulation. The development and quenching of electron avalanches are followed in great detail. The accumulation of positive and negative ions and the development of space charge field are also followed in time sequences. 2. Simulation method In the Monte Carlo simulation the electrons are treated one by one; every electron during its motion performs a succession of free flights punctuated by collisions with particles of the neutral gas. The simulation also included the release of electron ion pairs by photoionization. For any shape or type of ionizing electrode, the corona discharge is characterized by two regions, a thin layer, very close to the active electrode surface called ionization zone, and a drift zone toward a collecting electrode. The complete physical model of the corona discharge is rather complicated, due to the influence of the non-uniform electric fields on the real probability of occurrence of each analyzed collision process. Monte Carlo method takes into account the phenomena of non-equilibrium electron kinetics, incorporating the effects of the distortion due to the space charge on the applied field. We assumed that the applied electric field E is antiparallel to the z-axis. The initial electrons are emitted from the cathode according to a cosine distribution for the entry angles. Whether a collision between an electron and a gas molecule occurs or not is decided by generating a random number R1 uniformly distributed between 0 and 1. A collision is assumed to have occurred at the end of a time step if the condition P ≥ R1 is satisfied, where P is the probability of collision. If the random number is larger than P, no collision has occurred and the particle follows its way as if nothing has happened. The probability of collision over the time step T is
P = 1 − exp Tm
−T Tm
1 = N.QT (ε).v(ε)
P2,j =
(3)
where Qel is the elastic cross-section; Qatt is the attachment crosssection; Qex the total electronic excitation cross-section; Qv the
Qj
(4)
QT
where P2,j are the fractional probability of a attachment, ionization, excitation, elastic and vibration collision for an electron. At a given electron energy ε, the sum of the fractional probabilities is equal to unity and the interval [0,1] is divided into segments of lengths corresponding to these fractional probabilities. The nature of the collision is determined in the following way: P2,j is the probability that collision process j (the elastic, attachment, vibration, excitation and the ionization collisions) takes place, j = 1,2,3,. . .,n:
P2,j = 1,
P2,1 ≤ P2,2 ≤ P2,j ≤ P2,n ,
P2,1 + P2,2 + P2,j−1 < R2 ≤ P2,1 + P2,2 + . . . + P2,j
(5)
This leads to determine the jth type of collision. After the collision, the particle follows its way during the next time step and procedure is repeated [11]. The temporal development of electron and ion populations in a point–plane gap is calculated in the simulation by simultaneously following the trajectories of electrons. The development of the electron avalanche is assumed to occur along the axis of the gap. The model is one-dimensional in position space and three dimensional in velocity space. In the simulation of a corona discharge, the time step and the cell size should receive much attention. This study is performed in a point–plane gap, note that it is necessary to follow the region where the electric field changes most rapidly, with a fine mesh. For the purpose of simulation, the gap between the point electrode and the plane is divided into two regions, regions 1 close to the point electrode where the electric field is high and most electrons and ions are located, and region 2 the rest of the gap. In region 1, electron motion is simulated by dividing the region into a number of cells; a small cell size improves the accuracy. Also the accumulation of space charge causes the field in region 1 to change abruptly over space, and therefore a smaller cell size z1 is used. In contrast, in region 2, a larger cell size z2 is adequate. The length of region 1 may vary with various voltages and gap separations. z1 =
in which v(ε) is the velocity of an electron, QT the total collision cross-section, N the gas number density, ε the electron energy and Tm is the electron mean collision time. When a collision occurs between an electron and a gas molecule, the type of collision is determined by generating a random number R2 , this too being uniformly weighted between 0 and 1. The segment into which R2 falls determines the type of collision that has occurred. Where no new particles are produced the electron loses the energy corresponding to the type of collision. However, in the case of an ionizing collision, after subtracting the threshold energy, the remaining energy is evenly ascribed to the two electrons. To determine the nature of the collision, the total collision crosssection QT is split up into its component collision cross-sections for all possible collision processes of the electron, and the fractional probability of a certain kind of collision is computed. The total collision cross-section is defined as QT = Qel + Qatt + Qv + Qex + Qion
vibration cross-section; Qion is the ionization cross-section of the electron with energy ε. To determine which type of collision takes place, the fractional probabilities of all the collisions are computed:
(1) (2)
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d1 , M1
z2 =
d − d1 M2
(6)
where M1 and M2 are the cell numbers in region 1 and region 2, respectively, d is the gap length and d1 is the length of region 1. In a simulation of corona discharge based on the Monte Carlo method, the electrons and ions, if exceeding a certain number, distort the electric field and all the electrons are followed over the same time interval. When all electrons have been followed, we move to the next time step to follow all the electrons again, i.e., the electrons are stored after being followed during the previous time step, and new electrons are created by ionization collisions during the previous time step [12]. The time interval should be small, and less than the mean flight time. At the end of each time step, the space charge field is calculated from the Poisson’s equation as a function of charge distribution and is stored for use over the next time step, the total field is the sum of the Laplacian field and the space charge field. The total number of electrons in the gap increases over many orders of magnitude, hence scaling is necessary to limit the number of simulation particles. To circumvent this limitation, a renormalization and weighting procedure has been developed which maps the electron assembly into another consisting of fewer test particles. When total number of simulation particles (electrons) exceeds the maximum allowable number of simulation particles which depends on the
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computational resources utilized for the simulation, the latter being specified with the program input data, a statistical subroutine is introduced to choose an approximately equivalent group containing fewer simulation particles which replace the existing group of simulation particles. Each new particle represents several times as many actual electrons as the old particles. The subroutine contains a weighting of velocity distribution of the old group, so that the new group is equivalent in phase space to the old group. It should be noted that the principal advantage of the Monte Carlo method lies in the fact that swarm parameters are not required for the simulation. 3. Results and discussion Corona discharges, in general, are generated in electrode systems characterized by high non-uniformity of electric field. In many cases the numerical simulation of the corona discharge is necessary for predicting the performance of a device. We use the Monte Carlo method for the simulation of the corona discharge; it lets us study all temporal and space aspects of the electron avalanche development. Results are obtained for a negative corona discharge in SF6 in a point–plane geometry. The calculations are performed at a gas number density N = 2.12 × 1024 m−3 at 3 kV. The cross-section set of SF6 employed is that referred in [13,14]. The axial electric field distribution at any point along the axis of the gap is approximated as [15,16]: E(z) =
2V0 1 · ln(4d/r1 ) 2z + r1 − z 2 /d
(7)
where r1 , d, z and V0 are, respectively, the tip radius, gap length, distance from the point electrode and applied voltage. For the purpose of simulation, the gap between the point and the plane is divided into two regions, the gap parameters are d = 5 × 10−3 m, r1 = 5 × 10−4 m, d1 = 1 × 10−3 m, M1 = 50 and M2 = 100. According to the calculation of mean flight times, the time step 0.8 × 10−12 s is chosen in this study. The simulation is initiated by introducing a number of electrons released from the cathode with small energy (0.1 eV) at t = 0. As the electrons move towards the anode, they ionize the neutral gas, creating an avalanche of electrons. Radiation is also generated, resulting in the photoionization of the gas [17,18], and the photoelectrons create new avalanches. The photoionization position is determined by comparing the probability of the photoionization point, which is proportional to the solid angle extended from the excitation point to the plane contained photoionization point, with computer generated random numbers at the end of each step, when one ion–electron pair is produced due to photoionization. Then, an ion electron pair is stored, with 0.1 eV energy for the electron generated by photoionization. It is well known that the insulation performance of SF6 is limited, not by its uniform field dielectric strength, but by the effects of local field enhancement and in most typical industrial applications non-uniform field breakdown predominates, the electric field plays a key role in corona discharge development. The dynamics of corona discharge are determined strongly by the local electric field, it is important to analyze the time evolution of the electric field. In order to understand the physical process in corona discharge development, the evolution of several electric field profiles in the gap is shown in Figs. 1 and 2. While the density profile of electrons and positive ions are shown in Figs. 3–6, respectively. From Fig. 1, it is clear that up to 1.5 ns, there is a negligible distortion of the electric field from the Laplacian field. Cloud of ions almost stationary in the time scale of motion of the electron avalanche, remains behind. Later, the space charge enhances the field at both sides of the electron avalanche and weakens the field in between. The negative sign means the space charge field exceeds the applied field at the same position, but with a reversed direction. This is due to the
Fig. 1. Total field distribution in the gap at various points in time.
fact that ionization processes in the high-field region cause more positive ions in a small region; the space charge field due to ions is very high, even higher than the applied field. With the shifting of the high density region towards the cathode (Figs. 3–6), a denser space charge develops closer to the cathode that further leads to an enhancement of the electric field on the cathode side and reduction within it. The initiation and development of successive avalanches of electrons are traced as a function of time (Figs. 3–5). When the initial electrons are released from the cathode, the primary avalanche drifts toward the anode and multiplies fast to 0.9 ns. At the time 1.1 ns, a second avalanche is initiated by photoionization and grows faster than the primary avalanche because the former (secondary
Fig. 2. Total field distribution in the gap at various points in time.
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Fig. 5. Temporal and spatial variations of electron density distribution.
avalanche) is in the high-field region close to cathode (Fig. 3). It should be noted that photoionization may occur in a low-field region, but the avalanche will extinguish soon there after because the SF6 is an electron-attaching gas. Only those avalanches started in a high-field region will develop. The electric field has a steep gradient near the ionization front where the ionization is proceeding. While the primary avalanche builds up toward anode, excitation of atoms has been taking place at the same time ionization events have been occurring. Thus, before the primary avalanche has reached its full size, photons will be emitted from these excited states as they return to the ground state. These photons will be emitted in all directions and will be absorbed at various distances from their origin. Many processes can take place when a photon is
absorbed and many processes combined may lead to photoionization of the SF6 . With the availability of photoelectrons in the SF6 , successor avalanches of the second generation start at various distances from the primary avalanche. Like with the growth of the primary avalanche, all successor avalanches of the second generation will emit photons once they have been formed. These photons create new photoelectrons which will start a third generation of avalanches and so on. From the time 1.3 ns, the second avalanche exceeds the primary avalanche (Fig. 4). At t = 1.7 ns, a third avalanche is initiated (Fig. 5) in the high-field region. The initiation of the electron avalanche near the cathode is due to generation of photoionized ions–electrons. The charged particles density profiles of Figs. 5 and 6 extend to the plane electrode, and their peak values decrease gradually, this effect is due to the decreases in the electric field. A process will still be needed to provide an electron to initiate each new corona pulse. In the meantime, successive avalanches are produced by photoionization. Without the photoionization process, even though an electron may multiply into an avalanche through ionization processes, all electrons will eventually travel to the anode and the discharge will be extinguished. The second avalanche decays as time increases, the third avalanche grows and starts to decrease
Fig. 4. Electron density distributions at various times.
Fig. 6. Temporal and spatial variations of positive ion density distribution.
Fig. 3. Electron density distributions at various times.
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Fig. 9. Temporal and spatial variations of electron density distribution.
Fig. 7. Total field distribution in the gap at various points in time.
(Figs. 5 and 6) later because the total field in its region decreases. The second and third avalanches continue to decline, and then appears a fourth avalanche close to the cathode which increases, and then decreases. The electric field plays a key role in corona discharge development; the evolution of several electric field profiles in the gap is shown in Figs. 1 and 2 and Figs. 7 and 8. The field distortion increases with increasing time. For the negative corona, the positive charge near the electrode increases the positive slope of the electric field, enhancing the field at the cathode surface. The enhancement of the electric field adds to an already high Laplacian field towards the point electrode, results in enhanced ionization in this region, causing a further increase of electrons and positive ion densities towards the cathode side of space charge. Due to reduction of the elec-
Fig. 8. Total field distribution in the gap at various points in time.
tric field within the space charge, the ionization reduces while the attachment starts playing its role. Elastic collisions of the electron with background gas molecules slow down the electron’s motion toward the anode and randomize the directed kinetic energy along the electric field in all three directions. Figs. 3–6 and Figs. 9–11 show the development of successive avalanches, the electron and ion densities increase exponentially at a constant rate (Figs. 3–6). We also observe that the location of the respective peaks of the electron and positive ion densities remain nearly stationary. The density profiles of Figs. 9–11 extend to the plane electrode, and their peak values decrease gradually, this effect is due to the decreases in the electric field. The combined effect of enhanced ionization in the region between cathode and the space charge, with reduced ionization along space charge, increased attachment in space charge region, leads to the shifting of the electron and ion density peaks, hence the space charge shifts towards the cathode. When the electrons are in the low-field region and finally attach to molecules and decrease with time, there is only a narrow high-field region close to cathode usually called the cathode fall region. A process will still be needed to provide an electron to initiate each new corona pulse. The peak of the positive ion distribution moves toward the cathode because at later stages ionization can only occur in the high-field region. The negative ion distributions (Fig. 11) are similar
Fig. 10. Temporal and spatial variations of positive ion density distribution.
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depends on the variations of the electric field distributions. The simulation provides a detailed structure of avalanches, and the propagation of successive avalanches.
4. Conclusion
Fig. 11. Temporal and spatial variations of negative ion density distribution.
In this paper, negative corona discharge in SF6 gas has been simulated by using Monte Carlo method. The principal advantage of the Monte Carlo method lies in the fact that swarm parameters are not required for the simulation. From the results, it is found that the space charges and the photoionization play an important role in the progress of the corona discharge. The results also show that space charge effects by positive ions swarms intensify the electric field between cathode and positive ion so that the discharge in this region becomes more stable. At the same time they weaken the electric field between positive ion swarm and anode so that the corona expansion toward anode is restrained as time goes by. The simulation provides a detailed structure of avalanches, and propagation of successive avalanches can be discerned.
References
Fig. 12. Net charge density distributions at various times.
to those of positive ions with a slightly smaller peak. The net charge also depends on electron density, which changes rapidly with time, the net charge distributions do not stabilize. A large positive peak always seems to be followed by a negative peak (Fig. 12). The electron and ion densities inside the avalanche body are on the order of 1012 cm−3 , which agrees with the results reported in [19,20]. The propagation velocity of the ionization front is in the range of 3 × 107 to 4 × 107 cm/s. We find that the calculated values for the frontal velocities agree well with the experimental results [21–23] and with those reported in [20,24–27]. Measurements [21,22] of the average propagation velocity of the ionization front extend from 2 × 107 to 5 × 107 cm/s, the propagation average velocity [24,25] of the ionization wave is approximately of 5 × 107 cm/s, which agrees with our results. From the results of numerical analyzes by Monte Carlo technique, it is certain that SF6 as an electronegative gas can extinguish corona discharge rapidly. In the case of negative corona discharge, it is confirmed that the effect of the space charge has been stronger as time goes by and the characteristics of the corona discharge
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Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Monte Carlo simulation of corona discharge in SF6 A. Settaouti ∗ , L. Settaouti Electrotechnic Department, University of Sciences and Technology, P.O. Box 1505, El-M’naouar, Oran, Algeria
a r t i c l e
i n f o
Article history: Received 7 April 2009 Received in revised form 12 January 2010 Accepted 2 March 2010 Available online 7 April 2010 Keywords: SF6 Corona Simulation Monte Carlo Electric field Space charges
a b s t r a c t Sulphur hexafluoride (SF6 ) is one of the most widely used gaseous dielectrics for electric power systems and a number of high-voltage applications. There are many industrial applications where the electric corona discharge is used. In most cases the corona discharge is an inherently dynamic process; all parameters vary in time. Monte Carlo simulation of corona discharges in gas offers several advantages to study fundamental processes. Furthermore, it gives a fair qualitative description of the corona discharge itself as a function of space and time. This paper describes the development of negative coronas in SF6 in a point–plane gap. Detailed structure of avalanches is presented, the total field distribution, propagation of successive avalanches and ion distribution are studied. © 2010 Elsevier B.V. All rights reserved.
1. Introduction With the ever increasing need for electrical energy, high-voltage power transmission lines are constructed. Corona phenomenon is one of the problems associated with high-voltage lines. The corona discharge is important in practical high-voltage insulation systems because it can lead to deterioration of the insulating qualities of the gas as well as to production of toxic or corrosive by-products. Besides generation, transmission and distribution of electrical energy, high voltages are also extensively used for many industrial, scientific and engineering applications. High-voltage equipment is the backbone of modern power systems. In all such applications, the insulation of the high-voltage conductor is of primary importance. Sulphur hexafluoride (SF6 ) is the preferred gaseous dielectric employed by the electric power industry for gas insulated equipment used in the transmission and the distribution of electrical energy. Due to its strong capability of attaching electrons, SF6 gas is characterized by the superior electrical insulation performance. SF6 has excellent dielectric strength and arc-quenching properties. Therefore, SF6 is widely used for high-voltage insulation in electric power equipment such as gas insulated switchgears, gas circuit breaker, gas insulated transmission lines and gas insulated transformers. SF6 has gained wide acceptance as the dielectric for a number of high-voltage applications. Since it has an excellent insulating property, SF6 gas has contributed considerably to advances in miniaturization.
∗ Corresponding author. E-mail address: [email protected] (A. Settaouti). 0378-7796/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2010.03.002
Safe transmission and distribution of electrical power is partly determined by the reliability of the electrical insulation system of the power lines and substations [1–4]. It is well known that the insulation performance of gas is limited, not by its uniform field dielectric strength, but by the effects of local field enhancement and in most typical industrial applications non-uniform field breakdown predominates. By applying a detailed knowledge of the fundamental discharge processes that occur prior to the disruptive discharge, one can predict the behaviour of high-voltage insulator systems in operation under all possible conditions. Both in industrial and in environmental settings, corona-based technologies have been used to treat a variety of liquid and gaseous process effluents. Electrostatic precipitators are employed in electric power plants and many industries such as cement production, chemical processing and domestic air cleaning [5,6]. Owing to its good insulating and heat transfer properties, sulphur hexafluoride (SF6 ) is, besides air, the preferred gaseous dielectric employed by the electric power industry for gas insulated equipment used in the transmission and the distribution of electrical energy. SF6 is an electron-attaching gas widely used as a gaseous insulator in high-voltage systems, these properties allow a reduction in size and enhance the reliability of high-voltage equipment, and so electrical breakdown in SF6 is of interest both to gas discharge physicists and to power engineers. However, despite its advantages, the dielectric strength of SF6 is very sensitive to locally high electric fields. With the increasing importance of SF6 as an insulating medium, it is important to understand the mechanism of corona discharge [4,7,8]. The transient character and the small dimensions make some discharges parameters, like charged particles densities or electric
A. Settaouti, L. Settaouti / Electric Power Systems Research 80 (2010) 1104–1110
field strength, difficult to be accessible to measurements. A full mathematical model of the corona discharge is practically impossible to implement due to the complexity of the problem. Nevertheless, numerical investigations are needed in order to better understand the complex chemical and energetic phenomena triggered during the discharge phase of a corona discharge [9–12]. The purpose of this paper is to analyze the corona discharge in SF6 ; this could be accomplished by a Monte Carlo simulation. The development and quenching of electron avalanches are followed in great detail. The accumulation of positive and negative ions and the development of space charge field are also followed in time sequences. 2. Simulation method In the Monte Carlo simulation the electrons are treated one by one; every electron during its motion performs a succession of free flights punctuated by collisions with particles of the neutral gas. The simulation also included the release of electron ion pairs by photoionization. For any shape or type of ionizing electrode, the corona discharge is characterized by two regions, a thin layer, very close to the active electrode surface called ionization zone, and a drift zone toward a collecting electrode. The complete physical model of the corona discharge is rather complicated, due to the influence of the non-uniform electric fields on the real probability of occurrence of each analyzed collision process. Monte Carlo method takes into account the phenomena of non-equilibrium electron kinetics, incorporating the effects of the distortion due to the space charge on the applied field. We assumed that the applied electric field E is antiparallel to the z-axis. The initial electrons are emitted from the cathode according to a cosine distribution for the entry angles. Whether a collision between an electron and a gas molecule occurs or not is decided by generating a random number R1 uniformly distributed between 0 and 1. A collision is assumed to have occurred at the end of a time step if the condition P ≥ R1 is satisfied, where P is the probability of collision. If the random number is larger than P, no collision has occurred and the particle follows its way as if nothing has happened. The probability of collision over the time step T is
P = 1 − exp Tm
−T Tm
1 = N.QT (ε).v(ε)
P2,j =
(3)
where Qel is the elastic cross-section; Qatt is the attachment crosssection; Qex the total electronic excitation cross-section; Qv the
Qj
(4)
QT
where P2,j are the fractional probability of a attachment, ionization, excitation, elastic and vibration collision for an electron. At a given electron energy ε, the sum of the fractional probabilities is equal to unity and the interval [0,1] is divided into segments of lengths corresponding to these fractional probabilities. The nature of the collision is determined in the following way: P2,j is the probability that collision process j (the elastic, attachment, vibration, excitation and the ionization collisions) takes place, j = 1,2,3,. . .,n:
P2,j = 1,
P2,1 ≤ P2,2 ≤ P2,j ≤ P2,n ,
P2,1 + P2,2 + P2,j−1 < R2 ≤ P2,1 + P2,2 + . . . + P2,j
(5)
This leads to determine the jth type of collision. After the collision, the particle follows its way during the next time step and procedure is repeated [11]. The temporal development of electron and ion populations in a point–plane gap is calculated in the simulation by simultaneously following the trajectories of electrons. The development of the electron avalanche is assumed to occur along the axis of the gap. The model is one-dimensional in position space and three dimensional in velocity space. In the simulation of a corona discharge, the time step and the cell size should receive much attention. This study is performed in a point–plane gap, note that it is necessary to follow the region where the electric field changes most rapidly, with a fine mesh. For the purpose of simulation, the gap between the point electrode and the plane is divided into two regions, regions 1 close to the point electrode where the electric field is high and most electrons and ions are located, and region 2 the rest of the gap. In region 1, electron motion is simulated by dividing the region into a number of cells; a small cell size improves the accuracy. Also the accumulation of space charge causes the field in region 1 to change abruptly over space, and therefore a smaller cell size z1 is used. In contrast, in region 2, a larger cell size z2 is adequate. The length of region 1 may vary with various voltages and gap separations. z1 =
in which v(ε) is the velocity of an electron, QT the total collision cross-section, N the gas number density, ε the electron energy and Tm is the electron mean collision time. When a collision occurs between an electron and a gas molecule, the type of collision is determined by generating a random number R2 , this too being uniformly weighted between 0 and 1. The segment into which R2 falls determines the type of collision that has occurred. Where no new particles are produced the electron loses the energy corresponding to the type of collision. However, in the case of an ionizing collision, after subtracting the threshold energy, the remaining energy is evenly ascribed to the two electrons. To determine the nature of the collision, the total collision crosssection QT is split up into its component collision cross-sections for all possible collision processes of the electron, and the fractional probability of a certain kind of collision is computed. The total collision cross-section is defined as QT = Qel + Qatt + Qv + Qex + Qion
vibration cross-section; Qion is the ionization cross-section of the electron with energy ε. To determine which type of collision takes place, the fractional probabilities of all the collisions are computed:
(1) (2)
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d1 , M1
z2 =
d − d1 M2
(6)
where M1 and M2 are the cell numbers in region 1 and region 2, respectively, d is the gap length and d1 is the length of region 1. In a simulation of corona discharge based on the Monte Carlo method, the electrons and ions, if exceeding a certain number, distort the electric field and all the electrons are followed over the same time interval. When all electrons have been followed, we move to the next time step to follow all the electrons again, i.e., the electrons are stored after being followed during the previous time step, and new electrons are created by ionization collisions during the previous time step [12]. The time interval should be small, and less than the mean flight time. At the end of each time step, the space charge field is calculated from the Poisson’s equation as a function of charge distribution and is stored for use over the next time step, the total field is the sum of the Laplacian field and the space charge field. The total number of electrons in the gap increases over many orders of magnitude, hence scaling is necessary to limit the number of simulation particles. To circumvent this limitation, a renormalization and weighting procedure has been developed which maps the electron assembly into another consisting of fewer test particles. When total number of simulation particles (electrons) exceeds the maximum allowable number of simulation particles which depends on the
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computational resources utilized for the simulation, the latter being specified with the program input data, a statistical subroutine is introduced to choose an approximately equivalent group containing fewer simulation particles which replace the existing group of simulation particles. Each new particle represents several times as many actual electrons as the old particles. The subroutine contains a weighting of velocity distribution of the old group, so that the new group is equivalent in phase space to the old group. It should be noted that the principal advantage of the Monte Carlo method lies in the fact that swarm parameters are not required for the simulation. 3. Results and discussion Corona discharges, in general, are generated in electrode systems characterized by high non-uniformity of electric field. In many cases the numerical simulation of the corona discharge is necessary for predicting the performance of a device. We use the Monte Carlo method for the simulation of the corona discharge; it lets us study all temporal and space aspects of the electron avalanche development. Results are obtained for a negative corona discharge in SF6 in a point–plane geometry. The calculations are performed at a gas number density N = 2.12 × 1024 m−3 at 3 kV. The cross-section set of SF6 employed is that referred in [13,14]. The axial electric field distribution at any point along the axis of the gap is approximated as [15,16]: E(z) =
2V0 1 · ln(4d/r1 ) 2z + r1 − z 2 /d
(7)
where r1 , d, z and V0 are, respectively, the tip radius, gap length, distance from the point electrode and applied voltage. For the purpose of simulation, the gap between the point and the plane is divided into two regions, the gap parameters are d = 5 × 10−3 m, r1 = 5 × 10−4 m, d1 = 1 × 10−3 m, M1 = 50 and M2 = 100. According to the calculation of mean flight times, the time step 0.8 × 10−12 s is chosen in this study. The simulation is initiated by introducing a number of electrons released from the cathode with small energy (0.1 eV) at t = 0. As the electrons move towards the anode, they ionize the neutral gas, creating an avalanche of electrons. Radiation is also generated, resulting in the photoionization of the gas [17,18], and the photoelectrons create new avalanches. The photoionization position is determined by comparing the probability of the photoionization point, which is proportional to the solid angle extended from the excitation point to the plane contained photoionization point, with computer generated random numbers at the end of each step, when one ion–electron pair is produced due to photoionization. Then, an ion electron pair is stored, with 0.1 eV energy for the electron generated by photoionization. It is well known that the insulation performance of SF6 is limited, not by its uniform field dielectric strength, but by the effects of local field enhancement and in most typical industrial applications non-uniform field breakdown predominates, the electric field plays a key role in corona discharge development. The dynamics of corona discharge are determined strongly by the local electric field, it is important to analyze the time evolution of the electric field. In order to understand the physical process in corona discharge development, the evolution of several electric field profiles in the gap is shown in Figs. 1 and 2. While the density profile of electrons and positive ions are shown in Figs. 3–6, respectively. From Fig. 1, it is clear that up to 1.5 ns, there is a negligible distortion of the electric field from the Laplacian field. Cloud of ions almost stationary in the time scale of motion of the electron avalanche, remains behind. Later, the space charge enhances the field at both sides of the electron avalanche and weakens the field in between. The negative sign means the space charge field exceeds the applied field at the same position, but with a reversed direction. This is due to the
Fig. 1. Total field distribution in the gap at various points in time.
fact that ionization processes in the high-field region cause more positive ions in a small region; the space charge field due to ions is very high, even higher than the applied field. With the shifting of the high density region towards the cathode (Figs. 3–6), a denser space charge develops closer to the cathode that further leads to an enhancement of the electric field on the cathode side and reduction within it. The initiation and development of successive avalanches of electrons are traced as a function of time (Figs. 3–5). When the initial electrons are released from the cathode, the primary avalanche drifts toward the anode and multiplies fast to 0.9 ns. At the time 1.1 ns, a second avalanche is initiated by photoionization and grows faster than the primary avalanche because the former (secondary
Fig. 2. Total field distribution in the gap at various points in time.
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Fig. 5. Temporal and spatial variations of electron density distribution.
avalanche) is in the high-field region close to cathode (Fig. 3). It should be noted that photoionization may occur in a low-field region, but the avalanche will extinguish soon there after because the SF6 is an electron-attaching gas. Only those avalanches started in a high-field region will develop. The electric field has a steep gradient near the ionization front where the ionization is proceeding. While the primary avalanche builds up toward anode, excitation of atoms has been taking place at the same time ionization events have been occurring. Thus, before the primary avalanche has reached its full size, photons will be emitted from these excited states as they return to the ground state. These photons will be emitted in all directions and will be absorbed at various distances from their origin. Many processes can take place when a photon is
absorbed and many processes combined may lead to photoionization of the SF6 . With the availability of photoelectrons in the SF6 , successor avalanches of the second generation start at various distances from the primary avalanche. Like with the growth of the primary avalanche, all successor avalanches of the second generation will emit photons once they have been formed. These photons create new photoelectrons which will start a third generation of avalanches and so on. From the time 1.3 ns, the second avalanche exceeds the primary avalanche (Fig. 4). At t = 1.7 ns, a third avalanche is initiated (Fig. 5) in the high-field region. The initiation of the electron avalanche near the cathode is due to generation of photoionized ions–electrons. The charged particles density profiles of Figs. 5 and 6 extend to the plane electrode, and their peak values decrease gradually, this effect is due to the decreases in the electric field. A process will still be needed to provide an electron to initiate each new corona pulse. In the meantime, successive avalanches are produced by photoionization. Without the photoionization process, even though an electron may multiply into an avalanche through ionization processes, all electrons will eventually travel to the anode and the discharge will be extinguished. The second avalanche decays as time increases, the third avalanche grows and starts to decrease
Fig. 4. Electron density distributions at various times.
Fig. 6. Temporal and spatial variations of positive ion density distribution.
Fig. 3. Electron density distributions at various times.
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Fig. 9. Temporal and spatial variations of electron density distribution.
Fig. 7. Total field distribution in the gap at various points in time.
(Figs. 5 and 6) later because the total field in its region decreases. The second and third avalanches continue to decline, and then appears a fourth avalanche close to the cathode which increases, and then decreases. The electric field plays a key role in corona discharge development; the evolution of several electric field profiles in the gap is shown in Figs. 1 and 2 and Figs. 7 and 8. The field distortion increases with increasing time. For the negative corona, the positive charge near the electrode increases the positive slope of the electric field, enhancing the field at the cathode surface. The enhancement of the electric field adds to an already high Laplacian field towards the point electrode, results in enhanced ionization in this region, causing a further increase of electrons and positive ion densities towards the cathode side of space charge. Due to reduction of the elec-
Fig. 8. Total field distribution in the gap at various points in time.
tric field within the space charge, the ionization reduces while the attachment starts playing its role. Elastic collisions of the electron with background gas molecules slow down the electron’s motion toward the anode and randomize the directed kinetic energy along the electric field in all three directions. Figs. 3–6 and Figs. 9–11 show the development of successive avalanches, the electron and ion densities increase exponentially at a constant rate (Figs. 3–6). We also observe that the location of the respective peaks of the electron and positive ion densities remain nearly stationary. The density profiles of Figs. 9–11 extend to the plane electrode, and their peak values decrease gradually, this effect is due to the decreases in the electric field. The combined effect of enhanced ionization in the region between cathode and the space charge, with reduced ionization along space charge, increased attachment in space charge region, leads to the shifting of the electron and ion density peaks, hence the space charge shifts towards the cathode. When the electrons are in the low-field region and finally attach to molecules and decrease with time, there is only a narrow high-field region close to cathode usually called the cathode fall region. A process will still be needed to provide an electron to initiate each new corona pulse. The peak of the positive ion distribution moves toward the cathode because at later stages ionization can only occur in the high-field region. The negative ion distributions (Fig. 11) are similar
Fig. 10. Temporal and spatial variations of positive ion density distribution.
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depends on the variations of the electric field distributions. The simulation provides a detailed structure of avalanches, and the propagation of successive avalanches.
4. Conclusion
Fig. 11. Temporal and spatial variations of negative ion density distribution.
In this paper, negative corona discharge in SF6 gas has been simulated by using Monte Carlo method. The principal advantage of the Monte Carlo method lies in the fact that swarm parameters are not required for the simulation. From the results, it is found that the space charges and the photoionization play an important role in the progress of the corona discharge. The results also show that space charge effects by positive ions swarms intensify the electric field between cathode and positive ion so that the discharge in this region becomes more stable. At the same time they weaken the electric field between positive ion swarm and anode so that the corona expansion toward anode is restrained as time goes by. The simulation provides a detailed structure of avalanches, and propagation of successive avalanches can be discerned.
References
Fig. 12. Net charge density distributions at various times.
to those of positive ions with a slightly smaller peak. The net charge also depends on electron density, which changes rapidly with time, the net charge distributions do not stabilize. A large positive peak always seems to be followed by a negative peak (Fig. 12). The electron and ion densities inside the avalanche body are on the order of 1012 cm−3 , which agrees with the results reported in [19,20]. The propagation velocity of the ionization front is in the range of 3 × 107 to 4 × 107 cm/s. We find that the calculated values for the frontal velocities agree well with the experimental results [21–23] and with those reported in [20,24–27]. Measurements [21,22] of the average propagation velocity of the ionization front extend from 2 × 107 to 5 × 107 cm/s, the propagation average velocity [24,25] of the ionization wave is approximately of 5 × 107 cm/s, which agrees with our results. From the results of numerical analyzes by Monte Carlo technique, it is certain that SF6 as an electronegative gas can extinguish corona discharge rapidly. In the case of negative corona discharge, it is confirmed that the effect of the space charge has been stronger as time goes by and the characteristics of the corona discharge
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