Relationship between the fractal dimension of

International Review of Electrical Engineering (I.R.E.E.), Vol. xx, n. x

Relationship between the fractal dimension of creeping discharges propagating at solid/gas interfaces and the characteristics parameters of interfaces A. Beroual1, M-L. Coulibaly2, A. Girodet3 and O. Aitken3

Abstract – This paper is aimed at the fractal analysis of surfaces discharges propagating over insulators immersed in gas and the relationship between the physical parameters influencing these discharges and the fractal dimension of these latter. It is shown that: (1) the observed experimentally discharges propagating at the surface of high voltage insulators made of different materials in presence of gas submitted to standard lightning impulse voltage in divergent electric field have a fractal dimension; and (2) there is a relationship between the fractal dimension of these discharges and the physical and geometrical characteristic parameters of solid/gas insulating system that are the thickness (e) and the dielectric constant of insulator (εr) as well as the gas and its pressure. Keywords: Surface discharges, solid/gas insulating system, gas pressure, fractal dimension

I.

Introduction

The solid/gas insulating systems are widely used in high voltage equipment (bushings, insulators, switchgears, gas insulating lines, gas circuit breaker …) where they are exposed to different stresses and particularly to discharges phenomena that can develop within the body of insulator or at the solid/gas interface leading respectively to breakdown or flashover of insulator and hence to the failure of the system. In this work, we are especially interested by the propagation of discharges at the solid/gas interface (i.e., the creeping discharges). The understanding of the mechanisms evolving in the initiation and development of this kind of discharges and the knowledge of their characteristics including their shape and stopping length are of a great interest for the design and the dimensioning of insulation systems used in high voltage components and systems. Different mechanisms have been advanced to explain the mechanisms implicated in the initiation and propagation of creeping discharges [1 – 18]. In previous work, we analyzed the morphology and stopping lengths of creeping discharges propagating over insulators made of polytetrafluoroethylene (PTFE) filled with different kinds of micro- mineral fillers (namely MoS2, Al2CoO4, SiO2, and CaF2) [19], epoxy resin and glass [20] immersed in gas (SF6, N2 and CO2) or gases mixture (SF6-N2 and SF6-CO2), under lightning impulse voltages. These characteristics (morphology and stopping lengths) were investigated versus the dielectric constant and thickness of insulator material, the amplitude and polarity of the voltage, the type of gas (resp. mixture) and its pressure.

Manuscript received January 2007, revised January 2007

We recall here the main experimental results to help the understanding of the goal of this paper. Indeed, we showed that the shape of discharges and their stopping lengths Lf depend significantly on the solid insulator and the type of gas/mixture; the insulator being a disc. For given solid and gas, Lf increases quasi-linearly with the voltage and decreases when the gas pressure increases. For given voltage and pressure, Lf is longer when the point electrode is positive than when it’s negative while the initiation voltage of discharges is higher with a negative point than with a positive one; and Lf is longer with glass than with epoxy [20]. Lf is shorter in SF6 than in CO2 or N2. On the other hand, the increase of SF6 content in SF6-CO2 mixture leads to a significant decrease of Lf. Therefore, the addition of small concentration of SF6 in a given gas mixture improves the dielectric strength of insulating structure. With PTFE, Lf is shorter in SF6 than in CO2. Also, the type of filler greatly influences the characteristics of creeping discharges. In a given gas or mixture, the shortest Lf is obtained with PTFE filled with MoS2 or Al2CoO4 indicating then that insulators with these fillers appear as the best insulators in the point of view of flashover [19]. We also showed that the thickness of insulator is a fundamental parameter that influences significantly the shape and density of branches of discharges as well as their stopping length [19, 20]. When the thickness of insulator is reduced, the density and the length of branches increase indicating the implication of the electric field and the capacitive effects. And for a given thickness of insulator, the density and the length of branches are reduced when the gas/mixture pressure is increased. Copyright © 2007 Praise Worthy Prize S.r.l. - All rights reserved

A. Beroual, M-L. Coulibaly, A. Girodet, O. Aitken

The polarity of voltage also influences the morphology of discharges. Under positive lightning impulse voltage, the discharges consist of radial ramified filamentary structure while under positive lightning impulse voltage they don’t present always a radial aspect [19, 20]. These filamentary structures remind Lichtenberg figures [21]. Different investigators developed soft wares that allowed us to get such shapes by numerical simulation using various criteria based on the electric field and/or voltage, and using stochastic laws [22 - 25]. Since these figures do not cover the whole space (3D) or surface (2D) but a fraction, one speaks about the fractal geometry. However, if the shapes of discharges so generated by numerical simulation resemble to those observed experimentally, they are far of the experimental ones which depend on physicochemical characteristics of the solid/gas insulating system and experimental conditions. This paper is aimed at: (1) the fractal analysis of observed experimentally creeping discharges propagating over insulators made of polytetrafluoroethylene (PTFE), glass and epoxy resin immersed in SF6 at different pressures, under positive lightning impulse voltage in a point-plane electrode arrangement; and (2) the search of possible relationship between the fractal dimension of such discharges and the physical parameters of solid/gas insulating system, i.e., the thickness (e) and the dielectric constant of insulator (εr) as well as the gas and its pressure. Note that similar study has been already achieved by our group for solid/liquid interface [26].

depending on the investigated parameter, we vary the thickness of insulators or gas pressure. For example, the voltage required to have a stopping length of 44 mm for the tested insulators in SF6 at 0.10 MPa for different thicknesses are summarized in Table I. We observe that for a given insulator, the voltage required increases with the thickness of insulator. Similarly, for a given thickness of insulator, the voltage enabling to reach S increases when the gas pressure is increased (Table II). We note that the voltage is increased when the dielectric constant is reduced. One can do the same for a given insulator of a given thickness in different gases at the same pressure. Table III gives the voltage required to have a discharge of length S = 40 mm on insulator of glass of 2 mm thickness in the three considered gases at 0.20 MPa. It appears that the voltage necessary to get the same S is higher with SF6 than with CO2 and N2. This is due to its dielectric strength which is higher than these two latter gases which have very close dielectric strength. PC

HV Impulse Marx Generator

CCD Camera Resistor

Oil Gas Point electrode Insulator

II.

Experiment

The experimental arrangement is the same as that we used in previous work [19, 20]. Fig. 1 gives a schematic diagram of the set-up the details of which can be found in [20]. The solid insulating samples are discs of 100 mm diameter and 2, 4 and 6 mm thickness. These are made of glass, epoxy resin and polytetrafluoroethylene (PTFE), the dielectric constants ε of which are 2.1, 3.4 and 5 respectively; note that PTFE is filled with SiO2 micrometric charges the dimensions of which are smaller than 10 µm. The test cell is filled with hexafluoride gas (SF6) at different pressures. Figs 2 and 3 give some examples of discharges and their evolution versus the thickness of insulator and gas pressure respectively.

III. Fractal Analysis of Creeping Discharges To analyze and compare the fractal dimension D of creeping discharges developed over different solid/gas interfaces and the influence of parameters that can influence D, we consider discharges of the same stopping length S. For this, we proceed as follows: for given insulator, gas and pressure, we progressively increase the crest value of voltage up to reach the chosen S. And Copyright © 2007 Praise Worthy Prize S.r.l. - All rights reserved

Plane electrode Current probe

Oscilloscope

Fig. 1. Experimental set-up +34 kV – SF6

0.1 MPa

+38 kV – SF6

+35 kV – SF6

e = 2 mm

0.1 MPa

e = 4 mm

0.1 MPa

e = 6 mm

Fig. 2. Influence of insulator thickness on discharges propagating on PTFE surface samples in SF6 subjected to positive lightning impulse voltage.

e = 2 mm

+40 kV – SF6

+37 kV – SF6

+35 kV – SF6

0,15 MPa

e = 2 mm

0,3 MPa

e = 2 mm

0,5 MPa

Fig. 3. Influence of gas pressure on discharges propagating on PTFE surface samples of 2 mm thickness in SF6 subjected to positive lightning impulse voltage.

International Review of Electrical Engineering, Vol. xx, n. x


TABLE I VOLTAGE REQUIRED TO GET DISCHARGES OF S = 44 mm LENGTH ON DIFFERENT TYPE OF INSULATOR MATERIALS FOR THREE THICKNESSES IN SF6 AT 0.10 MPa SUBJECTED TO POSITIVE LIGHTNING IMPULSE VOLTAGE. Insulator thickness e

2 mm

4 mm Voltage (kV) 33,6 35,5 32.0 34.0 24.0 27.0

PTFE (εr = 2,1) Epoxy resin (εr = 3,4) Glass (εr = 5)

6 mm 38.0 35,5 31.0

TABLE II VOLTAGE REQUIRED TO GET DISCHARGES OF S = 44 mm LENGTH ON DIFFERENT TYPE OF INSULATOR MATERIALS OF 2 mm THICKNESS IN SF6 AT DIFFERENT GAS PRESSURES SUBJECTED TO POSITIVE LIGHTNING IMPULSE VOLTAGE. Gas pressure (MPa) Insulators PTFE (εr = 2,1) Epoxy resin (εr = 3,4) Glass (εr = 5)

0.15 35.0 33.4 25.0

0.30 Voltage (kV) 36.8 35.6 28.0

The principle of this software consists in generating square boxes of slide l the magnitude of which are changed at each step up to cover totally the considered figure. The total number of boxes N(l) containing any branch of discharge for each value of l enables to determine the fractal dimension D according to equation

N (l ) ~ l − D

Indeed, by plotting N versus l in log-log representation, we deduce D which is the slope of linear part of this characteristic knowing that

D = − lim l →0

0.50 41 37 30.2

IV.

10

Nombre de boites N

Voltage (kV)

Gas CO2 23

N2 22.5

log N ( l ) log l

(2)

Fig. 5 shows an example of the evolution of the total number of boxes N versus the side length l of boxes for discharges propagating over epoxy resin samples of different thicknesses in SF6 at 0.10 MPa.

TABLE III VOLTAGE REQUIRED TO GET DISCHARGES OF S = 44 mm LENGTH ON GLASS INSULATOR OF 2 mm THICKNESS IN DIFFERENT GASES AT 0.20 MPa SUBJECTED TO POSITIVE LIGHTNING IMPULSE VOLTAGE. SF6 28

(1)

10

e = 2 mm e = 4 mm e = 6 mm

3

2

Computation of Fractal Dimension

There are many methods to estimate the fractal dimension D of a self-similar structure [27]. However, we showed in previous work that the most adapted method for surface discharges is the box counting method [26]. This method consists in converting the original figures of discharges into binary figures that are then processed with software we developed in our lab [27] (Fig. 4).

10

1

10

1

10

2

Taille des boites l (Pixel)

Fig. 5. Total number of boxes N vs. their side length l for surface discharges on epoxy resin insulators for 2, 4 and 6 mm thicknesses in SF6 gas at 0.10 MPa.

IV.1. Influence of Insulator Thickness on Fractal (a)

(b)

By plotting the fractal dimension D versus the thickness e for the three investigated insulator materials at a given gas pressure (Fig. 6), we observe that D decreases linearly when the insulator thickness increases. And for a given insulator thickness, D increases also linearly with the dielectric constant of insulator (Fig. 7). 1.9

(d)

1.8 Fractal dimension D

(c)

1.7 1.6 1.5 1.4 1.3

PTFE

1.2

Epoxy resin

1.1

Glass

1

Fig. 4. Different steps of box counting processing: (a) original figure, (b) the corresponding binary figure, (c) and (d) the binary figure covered by squares of side l= 45 pixels and l= 15 pixels, respectively. The treated figure is that of surface discharge propagating on PTFE insulator of 2 mm thickness. Copyright © 2007 Praise Worthy Prize S.r.l. - All rights reserved

0

1

2

3

4

5

6

7

Thickness e (mm)

Fig. 6. Fractal dimension D versus the insulator thickness e for a gas pressure P = 0.10 MPa.



Similarly, by plotting the fractal dimension D versus the gas pressure P for insulators of 2 mm thickness, we observe that D decreases linearly when the gas pressure increases (Fig. 8). And for a given P, D increases also linearly when the dielectric constant ε of insulator increases (Fig. 9). Therefore, the fractal dimension of surface discharges also depends on the gas pressure. One can also deduce that the fractal dimensions will be lower with SF6 than with CO2 and N2. The higher the dielectric strength of gas, the lower the fractal dimension is. 1.9 1.8 Fractal dimension D

Thus, the ramification degree of discharges is well correlated with the characteristics of insulator (dielectric constant ε and thickness e). The fact that the density of discharge branches expressed by the fractal dimension depends on the thickness and the type of insulator (dielectric constant) evidences the implication of capacitive effects on the propagation phenomena of creeping discharges. Also, the fact that the charge stored at the insulator surface increases when increasing the dielectric constant and/or decreasing the thickness of insulator [19] leads us to conclude that the fractal dimension increases with the ability of insulator to store electrical charge: when the charge increases, D increases. 1.9 1.8 Fractal dimension D

1.7 1.6 1.5

1.3

e=2mm

1.2

e=4mm

1.1

e=6mm 0

1

2

3

4

5

6

Epoxy resin 1.2 Glass

0.2

0.3

0.4

0.5

0.6

1.9 1.8 1.7 1.6 1.5 1.4

P=0,15 MPa

1.3

P=0,5 MPa

1.2 P=0,3 MPa 1.1 1 0

1

2

3

4

5

6

Dielectric constant ε

Fig. 10. Fractal dimension D versus the dielectric constant ε for three gas pressure; the insulator thickness being e = 2 mm.

V.

P = 0.15 MPa P = 0.3 MPa P = 0.5 MPa

3

0.1

Fig. 9. Fractal dimension D versus gas pressure for the three investigated insulator materials; the insulator thickness being e = 2 mm.

Fractal dimension D

In this test, the thickness of insulator is maintained constant and for a given gas pressure, we increase the voltage up to reach a given stopping length S we chose. We proceed in this way for different pressures. Fig. 8 gives an example of the evolution of the total number of boxes N versus the side length l of boxes for discharges of length S = 46 mm, propagating over glass samples of 2 mm thickness in SF6 at different pressures namely 0.15, 0.3 and 0.5 MPa.

Nombre de boites N

PTFE 1.3

Gas pressure P (MPa)

IV.2. Influence of Gas Pressure on Fractal Dimension

Conclusion

This work evidences the existence of a correlation between the fractal dimension D of creeping discharges propagating over insulators made of different materials, and the physical parameters of solid/gas interface namely: the thickness (e) and the dielectric constant of insulator (ε), the gas and its pressure. We especially showed that D decreases when the thickness (e) of insulator increases and/or the gas pressure P is increased; and it increases with the dielectric constant of insulator (ε).

2

1

10

1.4

0

Fig. 7. Fractal dimension D versus the dielectric constant εr of insulator materials for a gas pressure P = 0.10 MPa.

10

1.5

1

Dielectric constant ε

10

1.6

1.1

1.4

1

10

1.7

1

10

2

Taille des boites l (Pixel)

Fig. 8. Total number of boxes N vs. their side length l for surface discharges on glass insulators of 2 mm thickness for three SF6 gas pressures.

Copyright © 2007 Praise Worthy Prize S.r.l. - All rights reserved

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Authors’ information 1

Ecole Centrale de Lyon, University of Lyon, Ampere CNRS UMR 5005, 36 avenue Guy de Collongue, 69134 Ecully, France 2 ALSTOM Grid Power Transformers, TICC – 129, Avenue de Paris, 91300 Massy, France 3 ALSTOM Grid – 130, Rue Léon Blum, BP 1321, 69611 Villeurbanne Cedex, France Abderrahmane Beroual (M’92, SM’09, F’11) received the Dipl. Ing. and Ph.D. degrees in electrical engineering, respectively in 1976 from the Ecole Nationale Polytechnique of Algiers, Algeria and in 1979 from the Technical University of Wroclaw, Poland. In 1987, he obtained the Doctorat d’Etat ès Sciences Physiques degree from the Institut National Polytechnique of Grenoble, France. He is currently Professor at the Ecole Centrale de Lyon, University of Lyon, France. Presently, he is the head of Materials Group at AMPERE Lab - CNRS as well as Responsible of the Master Research Program in electrical engineering. From 1994 to 1998, he chaired the International Study Group on Streamer Propagation in Liquids of the IEEE – DEIS. He is/was member of many Advisory Committees of International Conferences (ICDL, ISH, APTADM, Gas Discharge, ICHVE, CMD, INSUCON, NIWE, iREED, Technical Committee of the IEEE CEIDP, UF10 Technical Commission – MT30 of IEC) and associate editor for IEEE TDEI. He was/is responsible of various research projects. His main research interests include high voltage engineering, dielectric materials, long air gaps discharge and lightning, outdoor insulation, modelling of discharges and composite materials. He supervised more than 40 Ph.D. Students. He is author/co-author of more than 350 technical papers including more than 140 refereed journal papers, 4 patents, 1 book and 6 book chapters Mamadou Lamine Coulibaly received the Engineer degree in Electromechanical from National Institute of Hydrocarbons and Chemistry (INHC), Boumerdès, Algeria in 2003; Master Sc. (DEA) in Mechanical Engineering and Master Sc. in Electrical Engineering in 2004 and 2005, respectively from Université de Technologie de Compiègne (UTC) and Ecole Centrale de Lyon, France; and the PhD degree in Electrical Engineering (Dielectric Materials and High Voltage) from the Ecole Centrale de Lyon, France in 2009. His PhD thesis reported on creeping discharges phenomena and alternative gases for GIS/GCB applications he achieved in collaboration with Areva T&D - Villeurbanne Research Centre, France. After receiving the Ph.D degree, he joined the Technology, Innovation and Competence Development Centre (TICC) of Areva T&D (presently Alstom Grid) at Massy, France and extended the physicochemical laboratory (dedicated to insulating liquids) to insulating solids for transformers. He is currently with Alstom Grid - TICC Massy on R&D projects linked to insulating materials involving solids and mixed International Review of Electrical Engineering, Vol. xx, n. x


solids/liquids for power transformers. He is member of CIGRE and IEC Working Groups. Alain Girodet was graduated from the Ecole Nationale des Arts et Métiers, France, in 1981. He joined ALSTOM in 1982 as an engineer in the Air-blast Circuit Breaker Research Group. In 1987 he was in charge of the Material and Technology Laboratory of HV Technical Department. In 1992 he took the head of the GIS Research Group. Since 2002 he is in charge of High Voltage Technology and Material Research Group. Oana Aitken was born in Romania. She received the Electrical Engineer degree from Politehnica University of Bucharest, Romania in 1995 and the Ph.D degree in Electrical Engineering from University Joseph Fourrier of Grenoble (France) and Politehnica University of Bucharest (Romania) in 2001. Since 2002, she is working as research engineer on the materials filed in Switchgear Research Center of Alstom Grid (Villeurbanne, France) ; she is an expert in Materials Field for High Voltage Switchgears.

