George E. Vardakis and Michael G. Danikas

Transcription

1 FACTA UNIVERSITATIS (NIŠ) SER.: ELEC. ENERG. vol. 17, December 2004, Simulation of Electrical Tree Propagation in a Solid Insulating Material Containing Spherical Insulating Particle of a Different Permittivity with the Aid of Cellular Automata George E. Vardakis and Michael G. Danikas Abstract: The electrical tree propagation inside a solid insulating material is simulated in the present paper. The effect of a small insulating spherical particle inside the solid insulating material is investigated as far as the electrical field is concerned. Laplace s equation is solved inside the solid insulating material setting the boundary conditions around the spherical particle. An attempt for comparison between the simulation results and experimental data from the technical literature is being made, trying to shed light to the physical mechanisms that are involved in the phenomenon. Keywords: Electrical tree propagation, insulating particle, cellular automata 1 Introduction The insulating capability of various insulating systems is affected by numerous factors. Physical aging, chemical aging and electrical aging contribute to the deterioration of the insulation performance and electrical trees are the visible result of this deterioration [1]. Electrical trees are conducting channels, usually filled with gas. They are strongly connected to the existence of air voids and are also correlated with the space charge formation and partial discharge activity in the vicinity of the electrical tree or near the injected electrode [2, 3]. The words tree and dendrite are used interchangeably in the context of this paper. Electrical trees and their relation to space charges and partial discharge activity have stimulated a lot of research till now. The model proposed by Niemeyer, Manuscript received June 29, 2004 The authors are with Democritus University of Thrace, Department of Electrical and Computer Engineering, Electric Energy Systems Laboratory, Xanthi, Greece ( s:[gvarda, mdanikas]@ee.duth.gr). 377

2 378 G. Vardakis and M. Danikas: Pietronero, and Wiesmann (NPW) [4] recognizes the fractal character of the electrical trees explaining the various tree formations with parameters of the fractal theory. The basic hypothesis of the model is that the growth probability (or the branching) is proportional of the local electric field. Zeller and co-workers in [5]-[8] studied the treeing phenomenon model using some of the NPW model parameters. They actually introduced, among others, the notion of critical breakdown field and of course the well known field limiting space charge (FLSC), admitting the significant role of the space charge cloud around the region of interest. Champion and co-workers [9]-[11] proposed a field driven tree growth model (FDTG) for temporal and spatial analysis of the propagating dendrite extracting useful results for different samples. The aforementioned researchers study and model the partial discharge (PD) behavior inside the tree channels [10, 11]. Dissado et al. [12]-[15] proposed the role of space charge rearrangement in electrical tree propagation and analyzed with chaotic parameters (deterministic chaos, fractal dimensions etc) in combination with the PD activity. Interesting simulation results were presented by Noskov et al. [16]-[18], where the PD activity was combined with treeing phenomena. The simulation method uses parameters such as damage energy, local fluctuation of the electrical field, etc. Sarathi et al. [19]-[22] gave a different point of view in tree growth simulation with laminated dielectrics in the presence of space charges. Furthermore, in [23]-[29] a new simulation method presented, demonstrating bush- like or branch-type dendrites, considering the small local variations of the electric field produced by the dielectric environment of the dendrite. The simulation parameters defining the dendrite behavior are the kind of the electrode arrangement, the presence or absence of space charges, the existence or not of small air voids and the applied voltage. In this paper, the work reported in [23]-[29] is extended by including the role of enclosed insulating particle in tree propagation in a solid insulating material with different permittivity. 2 Cellular Automata and the Related Algorithm The process of the electrical tree propagation is simulated in the present paper with the aid of Cellular Automata (CA) [30, 31] which is a well-known simulation method covering various scientific fields and applications [32, 33]. The treeing phenomena can be employed either as microscopic or as macroscopic phenomena, which is very common for almost every physical dynamic system. The macroscopic observation of the electrical tree patterns can be combined with the study of the microscopic conditions that define the tree propagation.

3 Simulation of Electrical Tree Propagation in a Solid Insulating Material The physical system in the present paper consists of a needle-plane electrode arrangement, the solid insulating material which is considered to be polyethylene with mean value for relative permittivity ε and the spherical insulating particle with relative permittivity ε 2 8 (glass) placed in the middle almost of polyethylene. The whole system is divided into cells with dimensions of every cell 10µm 10µm, so the sample has dimensions 5mm 5mm (Fig.1). The correct calculation of the electric field in the main volume of the solid insulating material is calculated with the Partial Differential Equation Toolbox (PDE) of MATLAB, which gives all the necessary tools for the solution of such electrostatics problems. Fig. 1. Needle-plane electrode arrangement with a spherical insulating particle almost in the middle of the solid insulating material. Laplace s equation is solved and a matrix of potential values is gained as the solution of the equation. In the present paper, the space charges are not taken into consideration so the electric field is defined by the potential at the electrodes, the presence of the insulating particle and finally the local slight variations of the dielectric environment. The algorithm corresponds to each cell belonging to the solid insulating material two values. One value is the potential value as is calculated from the solution of Laplace s equation, and another is a value g between that is produced in a random way, inserting the notion of inhomogeneity of the material. The g-factor is called dielectric inhomogeneity factor (d.i.f.) and is an indicative dimensionless magnitude giving a necessary variation of the electrical field values E at every step of the dendrite propagation. The field E for very small distances is approximated [2, 3, 23]-[29, 34] with the

4 380 G. Vardakis and M. Danikas: Eq. (1) E g V l where g is dielectric inhomogeneity factor of a cell produced in a random way, V is potential difference between the two neighboring cells, l is distance between the centers of the cells. The electrical field is calculated in every step of tree propagation. The algorithm checks the neighbor cells around the tip of the needle during the inception stage or around every tree cell during the propagation stage. The whole system consists of four kinds of cells. The electrode cells. These cells have infinite permittivity and potential equal to the applied potential. There are two subcategories. 1. The upper electrode cells, which are under high voltage. 2. The lower electrode cells with zero potential upon them. The insulating sample cells. They consist the main volume of the arrangement. The electric field is calculated in their interior using boundary conditions applied to the other three categories cells. The insulating particle cells with different permittivity than the surrounding insulating material. The tree cells. If the electrical field, as is calculated with the aid of Eq. (1), is greater than a predefined critical value then the dendrite moves towards that direction. This means that a tree cell is formed in the same place where an insulating sample cell was previously. The new dendrite cell will contribute to the next application of the boundary conditions. The critical value in the present paper is taken to be E c =50 kv/mm which is the dielectric strength for polyethylene. (1) 3 Definition of Boundary Conditions The definition of the appropriate boundary conditions is crucial for the correct calculation of the electric field. Three simulations are presented in this paper. For the first case, the value of +80 kv is applied at the upper needle electrode. Fig. 2 shows the equipotential lines of the electrode arrangement with applied voltage of +80 kv. The voltage of 0 kv is applied at the opposite plane electrode and the boundary values around the dielectric sphere are calculated with the aid of the fol-

5 Simulation of Electrical Tree Propagation in a Solid Insulating Material lowing equation [35, 36], Φ in 3 E 0 r ε cos θ (2) 2 2 ε 1 where Φ in is the potential inside the spherical dielectric, ε 1 is the relative permittivity of polyethylene, ε 2 is the relative permittivity of the spherical dielectric, E 0 is the value of the external electric field considering it as homogeneous, without the influence of the insulating particle. The radius of the insulating spherical particle is r and θ is the angle between the vector E 0 of the external electric field and the radius vector r. The external field is calculated dividing the applied voltage with the electrode separation considering that the external field in the region around the particle is homogeneous. In the present case 80/3=26.66 kv/mm (3mm being the distance between the tip of the needle and the plane electrode) which is confirmed also by Fig. 3 where the same value for the external field is calculated again (number of equipotential lines divided by the relative distance). Fig. 2. Equipotential lines at the needleplane electrode arrangement. Applied voltage +80 kv. The insulating particle with the different permittivity inside the solid insulating material is shown in the middle of the material. The potential distribution is demonstrated without the influence of the insulating particle. Fig. 3. Magnification of the equipotential lines around the insulating particle. Applied voltage of +80 kv at the upper needle electrode, without the influence of the particle. Every line corresponds to 100V (0.1 kv). The first arrow shows the vector E 0 as it is approximated around the particle and the second arrow shows the radius vector r. For r=0.025 mm, ε 2 =8, ε 1 =2.3 and E 0 =26.66 kv/mm we calculate the potential values at the circumference of the insulating spherical particle with the aid of Eq. (2). The angle θ takes its extreme values for θ 0 and θ π rad resulting cosθ 1 and cos θ 1 respectively. The extreme potential values at the circumference of the particle are in the range of 365 V, which are quite small potential values in

6 382 G. Vardakis and M. Danikas: the middle of the solid insulating material comparing with the potential values in the same region without the influence of the insulating particle. The electric field inside the sphere E in is smaller than outside the sphere E 0. This is the direct influence of the polarized bound charges that appear at both sides of the spherical particle depending on their polarity. This results to an internal field parallel to the external E 0 but with opposite direction to that. So the total field E in is given by the following equations Ein E 0 P 2ε 1 (3) and 3 E in ε E 2 0 (4) 2 ε 1 where P is the polarization vector. Substituting the values ε 2 8, ε and E 0 =26.6 kv/mm in Eq. (4), the calculated electric field inside the sphere is E in =14.62 kv/mm which is much smaller than the dielectric strength of the glass which is E c =30 kv/mm [37]. Following the same methodology, the voltages of +45 kv and +30 kv are applied at the upper electrode in two more cases. The critical value for the dielectric strength is again 50 kv/mm (which is the dielectric strength for PE). In the case of +30 kv, the insulating particle is placed closer to the injecting electrode. The various values, which are used or are calculated in these three simulations, are summarized in the Table 1 below: Table 1. Values for the various physical quantities used in the three simulations Applied Electrode Mean Φ in (Eq. (2)) Range Electric Dielectric DC separation Electric of potential vales field inside strength voltage (mm) field E 0 at the the sphere of the (kv) (kv/mm) circumference of E in material the particle (V) (kv/mm) E c (kv/mm) Simulation results The electrical tree propagation is simulated in three cases as is shown in Figs It is assumed that at the opposite plane electrode 0 kv is applied (Dirichlet boundary conditions). The interfaces between the insulating material and the surrounding

7 Simulation of Electrical Tree Propagation in a Solid Insulating Material air has no surface charge, so Neumann boundary conditions are applied to these interfaces where the surface charge is taken equal to zero. The low potential values at the circumference of the insulating particle do not affect tree propagation in the case of +80 kv. The dendrite clearly belongs to the bush-like type. After several steps, the electrical tree would form a dense and wide dendritic formation which eventually will result to a complete electrical breakdown. In the case of +45 kv, two main branches show almost the same propagation activity, which is the result of almost the same electric field towards these two directions. The electric field at lower potential values is slightly affected by the presence of the insulating particle creating more branches at the right main branch as it is clearly shown in Fig. 5. The left main branch of the structure shows a thin channel formulation representing slightly lower activity in contradiction to the right main branch. The comparison between the left and the right main branches clearly demonstrates the significant role of the small insulating particle in the structure. Finally, the case of +30 kv applied voltage is studied with the change in the particle s position (Fig. 6). The particle closer to the electrode, affects the electrical tree right main branch creating complicated tree structures with many secondary branches lying upon the main branch. Moreover, after several steps, a small bushlike region is formed at the tip of the right main branch of the dendrite as it is reaching the insulating particle. In contradiction to this behavior, the left main branch has small branching activity because the electric field is low towards that direction and after few steps stops. 5 Discussion The chemical bond breaking inside the solid insulating material is the mechanism that is represented by the formation of one tree cell. The tree channel consists of totally conducting material (gas) so the potential value applied at the needle electrode is considered to be the same at every tree cell in contact with the electrode cell. As it is shown in Figs. 4-6, no tree inception is noticed from the insulating particle. The physical meaning of this behavior is that the bound charges do not have sufficient energy to break their chemical bonds. The case of electrical trees emanating from insulating particles should not be excluded because from Eq. (2) as is stated in the previous paragraph, it is clear that the position of the particle and the potential values at the circumference of the particle are crucial factors for its electrical behavior. If insulating particles were placed at regions with high electric field values (i.e. very close to the high voltage electrode), the potential values would significantly increase causing field enhancement towards the direction of the

8 384 G. Vardakis and M. Danikas: Fig. 4. Bush-type dendrite emanating from the needle electrode. Applied voltage +80 kv at the upper electrode. The boundary conditions around the insulating particle don t create the necessary physical parameters for tree inception and propagation from it. The presence of the insulating particle modifies the electric field around it. Fig. 5. Applied voltage +45 kv at the upper electrode. The electrical tree emanating from the needle electrode shows two branches moving diagonally towards the opposite plane electrode. The right main branch of the dendrite shows slightly more activity than the left main branch. The presence of the insulating particle inside the dielectric slightly enhances the electrical field towards that direction.

9 Simulation of Electrical Tree Propagation in a Solid Insulating Material Fig. 6. Applied voltage +30 kv at the upper electrode. The insulating particle is placed close to upper electrode, so the boundary conditions around it, create the circumstances for tree inception and propagation towards specifically the direction of the particle. The dendrite in the initial stage of propagation creates few branches but reaching the insulating particle bush-like regions appear at the tip of the dendrite. external electric field. The carriers at the surface of the insulating particle (interface between two insulating materials) would have now more energy to populate traps in the middle of the forbidden zone or, even for weaker dielectric materials to move into the conduction band. Moreover according to Eq. (4) the electric field inside the insulating particle would increase, so the probability for deterioration activity due to ionization effects in the interior of the insulating particle would increase too. Additionally, ionization incidents at the interface between the solid insulating material and the insulating particle are possible under certain electric field conditions. In [38], it was assumed that space charges in the form of ionized gas molecules might be injected in the interior of any insulating material through a network of minute channels connecting voids and microvoids. Although theoretically speaking, at high applied electric fields it is possible for an insulating particle to become the source of a dendrite, nowhere in the scientific literature is mentioned this case. Such a behavior is difficult to be observed experimentally because electrical trees would emanate both from the edges of the insulating particle (due to the high electric fields) and from the high voltage electrode (as the region where free charges are injected into the insulating material). This behavior leads to complete electrical breakdown of the insulating material because the electrical field would be many times stronger than the dielectric strength of either the insulating material or the insulating particle. In the scientific literature,

10 386 G. Vardakis and M. Danikas: it is considered that electrical trees are formed from air voids, metallic impurities, moisture, byproducts etc. [1, 39]-[51], but not from insulating particles. Concluding, we may say that the existence of insulating particles such as those investigated in the present paper and with the electric fields commented upon, does not cause any treeing structures but strongly modifies the electrical field affecting thus at the same time the electrical tree emanating from the high voltage electrode. In the present paper, the case of the enclosed insulating particle and the related electrical phenomena are studied with the hypothesis that the particle is in perfect contact with its surrounding bulk. Imperfect contacts may create local field enhancements and so the electric field distribution is quite complex to be analyzed. In the present paper a perfect contact between the insulating particle and polyethylene was assumed. 6 Conclusions Treeing phenomena in the case of a small insulating spherical particle inside a solid insulating material are simulated in the present paper. Using basic concepts and equations of the Electromagnetic Field Theory, the appropriate boundary conditions were applied at the insulating particle boundaries producing interesting results in the simulation of dendrite propagation. The existence of the insulating particle inside the dielectric material may be a a significant factor for the electrical tree propagation but not for its initiation. Acknowledgments One of the authors (G.E.V) is indebted to the Greek State Scholarships Foundation (I.K.Y., Athens, Greece) for financial assistance. Moreover the authors would like to thank Associate Professor G. Anagnostopulos for this help. References [1] L. Dissado and J. Fothergill, Electrical degradation and breakdown in polymers. Peter Peregrinus Ltd, [2] M. Danikas, I. Karafyllidis, A. Thanailakis, and A. Bruning, Simulation of electrical tree growth in solid dielectrics containing voids of arbitrary shape, Modelling Simul. Mater. Sci. Eng., vol. 4, pp , [3] I. Karafyllidis, M. Danikas, A. Thanailakis, and A. Bruning, Simulation of electrical tree growth in solid insulating materials, Archiv. f. Elektr., vol. 81, pp , 1998.

11 Simulation of Electrical Tree Propagation in a Solid Insulating Material [4] L. Niemeyer, L. Pietronero, and H. Wiesmann, Fractal dimension of dielectric breakdown, Phys. Rev. Lett., vol. 52, pp , [5] H. Wiesmann and H. Zeller, A fractal model of dielectric breakdown and prebreakdown in solid dielectrics, Journal of Applied Physics, vol. 60, no. 5, pp , [6] T. Hibma and H. Zeller, Direct measurement of space charge injection from a needle electrode into dielectrics, Journal of Applied Physics, vol. 59, no. 5, pp , [7] H. Zeller and W. Schneider, Electrofracture mechanics of dielectric aging, Journal of Applied Physics, vol. 56, no. 2, pp , [8] H. Zeller, Breakdown and prebreakdown phenomena in solid dielectrics, IEEE Trans. On Electrical Insulation, vol. 22, no. 2, pp , [9] J. C. A. S. Dodd and G. Stevens, Analysis and modeling of electrical tree growth in synthetic resins over a wide range of stressing voltages, J. Phys. D: Appl. Phys., vol. 27, pp , [10] J. Champion and S. Dodd, An approach to the modelling of partial discharges in electrical trees, J. Phys. D: Appl. Phys., vol. 31, pp , [11], Simulation of partial discharges in conducting and non-conducting electrical tree structure, J. Phys. D: Appl. Phys., vol. 34, pp , [12] L. Dissado, S. Dodd, J. Champion, P. Williams, and J. Alison, Propagation of electrical tree structures in solid polymeric insulation, IEEE trans. on Diel and Electr. Insul., vol. 4, no. 3, pp , [13] L. Dissado and P. Sweeny, Physical model for breakdown structures in solid dielectrics, Phys. Review B, vol. 48, no. 22, pp , [14] J. Fothergill, L. Dissado, and P. Sweeny, A discharge avalanche theory for the propagation of electrical trees, IEEE Trans. on Diel. and Electr. Insulation, vol. 1, no. 3, pp , [15] L. Dissado, G. Mazzanti, and G. Montanari, The role of trapped space charges in the electrical aging of insulating materials, IEEE Trans. on Diel. and Electr. Insulation, vol. 4, no. 5, pp , [16] M. Noskov, A. Malinovski, M. Sack, and A. Schwab, Self-consistent modeling of electrical tree propagation and pd activity, IEEE Trans. On Diel. and Electr. Insulation, vol. 7, no. 6, pp , [17] M. Noskov, M. Sack, A. Malinovski, and A. Schwab, Measurement and simulation of electrical tree growth and partial discharge activity in epoxy resin, J. Phys. D: Appl. Phys., vol. 34, pp , [18] M. Noskov, A. Malinovski, M.Sack, and Schwab, Simulation of partial discharge development in long narrow channel, in OPTIM 2002, Brasov, Romania, 2002, pp [19] R. Sarathi, Probabilistic aspects of failure of hv xlpe cables in service, in 3rd Workshop & Conference on EHV Technology, Bangalore, India, 1995, pp

12 388 G. Vardakis and M. Danikas: [20] R. Sarathi and S. Rao, Effect of space charges on the dynamical aspects of electrical trees, in Fourth Workshop & Conference on EHV Technology, India, 1998, pp [21] R. Sarathi, Stochastic modeling and characterization of electrical tree, Solid State Communications, pp , [22] R. Sarathi and A. Saradhi, Modeling and characterization of electrical trees in a laminated dielectric structure, in Conference on Electrical Insulation and Dielectric Phenomena, India, [23] G. Vardakis and M. Danikas, Simulation of tree propagation in polyethylene containing air voids at various positions using cellular automata, in OPTIM 8th International Conference on Optimization of Electrical and Electronic Equipments, Brasov, Romania, 2002, pp [24] G. Vardakis, M. Danikas, and I. Karafyllidis, Simulation of space-charge effects in electrical tree propagation using cellular automata, Materials Letters, vol. 56, no. 4, pp , [25] M. Danikas and G. Vardakis, Propagation of electrical trees in solid dielectrics containing conducting particles by using cellular automata, in XVIth Conference on Electromagnetic Fields and Materials, Bratislava, Slovakia, [26] G. Vardakis and M. Danikas, Simulation of tree propagation (by using cellular automata) in polyethylene including insulating particles: the effect of space charges, in Medpower, Athens, Greece, [27], Simulation of tree propagation (by using cellular automata) in polyethylene in plane-plane electrode arrangement, in CIRED 17th International Conference on Electricity Distribution, Barcelona, Spain, [28], Simulation of tree propagation in polyethylene including air void by using cellular automata: The effect of space charges, Arch. f. Elektr., vol. 84, no. 4, pp , [29], Simulation of tree propagation in polyethylene in plane-plane electrode arrangement using cellular automata: the effect of homocharges and heterocharges, in 38th International University Power Engineering Conference UPEC, Thessaloniki, Greece, 2003, pp [30] J. V. Neuman, Theory of self-reproducing automata. Urbana: University of Illinois, [31] B. Chopard and M. Droz, Cellular automata modeling of physical systems. Cambridge University Press, [32] M. Marin, V. Rauch, A. Rojas-Molina, C. Lopez-Cajun, A. Herrera, and M. Castano, Cellular automata simulation of dispersion of pollutants, Computational Materials Science, vol. 18, no. 2, pp , [33] P. Matic and A. Geltmacher, A cellular automaton-based technique for modeling mesoscale damage evolution, Computational Materials Science, vol. 20, no. 1, pp , 2001.

13 Simulation of Electrical Tree Propagation in a Solid Insulating Material [34] F. Kreuger, Industrial High Voltage. Eds. Delft University Press, [35] W. Roald, Electromagnetic Fields. John Wiley & Sons, [36] J. Jackson, Classical Electrodynamics. John Wiley & Sons, [37] D. Halliday and R. Resnick, Physics II. John Wiley and Sons, [38] G. Bahder, T. Garrity, M. Sosnowski, R. Eaton, and C. Gatz, Physical model of electrical aging and breakdown of extruded polymeric insulated power cables, IEEE Trans on Power Apparatus and Systems, pp , [39] Y. Cho, M. Shim, and S. Kim, Electrical tree initiation mechanisms of artificial defects filled xlpe, Material Chemistry and Physics, vol. 56, pp , [40] M. Lakner, J. Rhyner, and D. Sologuren, Effect of small metal particles on ageing of epoxy insulation, IEEE, pp , [41] A. Pillai and R. Hackam, Modification of electric field at the solid-insulator vacuum interface arising from surface charges on the solid insulator, J. Appl. Phys., vol. 3, pp , [42] K. Honda, A. Ohsawa, and N. Toyokura, Breakdown in silicon oxides-correlation with fe precipitates, Appl. Phys. Lett., vol. 46, no. 6, pp , [43], Breakdown in silicon oxides-correlation with cu precipitates, Appl. Phys. Lett., vol. 45, no. 3, pp , [44] S. Rasikawan and N. Shimizu, Effect of additives on treeing initiation as a function of oxygen concentration in polyethylene, IEEE Trans. on Electr. Insulation, vol. 27, no. 6, pp , [45] J. Svahn and S. M. Gubanski, Influence of metal inclusion on ac breakdown strength in epdm, in IEEE Int. Conf. on Conduction and Breakdown in Solid Dielectrics, Sweden, 1998, pp [46] M. Khalil, A. Cherifi, A. Toureille, and J. Reboul, Influence of batio3 additive and electrode material on space charge formation in polyethylene evidence from thermal step space-charge measurements, IEEE Trans. Diel. and on Electr. Insulation, vol. 3, no. 6, pp , [47] C. Laurent, E. Kay, and N. Souag, Dielectric breakdown of polymer films containing metal clusters, Journal of Appl. Phys., vol. 64, no. 1, pp , [48] D. Auckland, A. Taha, and B. Varlow, Mechanical interaction of electrical trees and barriers in insulating resins, IEE Proc.-Sci. Meas. Technol., vol. 143, no. 5, pp , [49] D. Auckland, A. Rashid, and B. Varlow, Effect of barriers on tree growth in solid insulation, IEE Proc.-Sci. Meas. Technol., vol. 142, no. 4, pp , [50] L. Dascalescu, S. Adrian, and R. Tobazeon, Cylindrical conductive particles in the proximity of an electrode affected by a high-intensity electric field, Journal of Electrostatics, vol. 37, pp , [51] P. Olivo, B. Ricco, T. Nguyen, T. Kuan, and S. Jeng, Evidence of the role of defects near the injecting interface in determining sio2 breakdown, Appl. Phys. Lett., vol. 51, no. 26, pp , 1987.