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Streamer inception and propagation models for designing air insulated power devices A. Pedersen1, T. Christen2, A. Blaszczyk2, H. Boehme3 1 ETH Zurich, Switzerland 2 ABB Corporate Research Baden, Switzerland 3 Dresden, Germany (formerly with ABB Corporate Research)
I.
INTRODUCTION
The goal of designing medium voltage (MV) equipment is to predict the withstand voltage Uw in order to pass the required tests. It involves modelling of discharge process including inception and propagation of streamers, which are the basic precursors for dielectric breakdown in inhomogeneous electric fields. Although many publications exists on theory and simulations for simple geometries, based on first principles or kinetic models (see, e.g., [1]-[4]), such approaches are hardly useable for simulation procedures in industrial design for real devices. On the other hand, engineering design criteria are based on a few semi-empirical rules related to streamer inception, streamer propagation, and sometimes leader transition, in air gaps. They are applied as an additional computations based on the numerically calculated electrostatic background field [5]. In Sect. II we present an overview of these rules for the medium voltage range (test voltage levels below 100 kV/peak for AC, and up to 200 kV for standard lightning impulse 1.2/50 μs). For real insulation systems including air and solid dielectrics with complex geometries, the application of the empirical rules is often inappropriate, mainly due to unknown discharge path but also because of a significant influence of surface properties [6] on ionization, attachment, and detachment mechanisms, surface charging etc. Therefore we have started recently research efforts aimed at elaboration of numerical procedures for the computation of streamer behaviour. In Sect. III-V we present our experimental setup, and the first experimental and modelling results concerning discharges near dielectric surfaces and surface flashover [7].
II.
CLASSICAL DIELECTRIC DESIGNINING PROCEDURE
A.
Streamer inception After the appearance of an initial electron in a critical high field volume, an electron avalanche starts to develop. If a sufficient number Nc of electrons are generated, a selfpropagating streamer head forms. The inception criterion is (1) ∫ α eff dx = ln( N c ) , γ where αeff(E) is the field dependent effective ionization coefficient including ionization, electron attachment, and detachment. The integration must be performed along a path γ where αeff > 0, starting from the point with maximum field and ending where αeff =0 (at the inception field). Typical values in normal air are as follows [1,8]: inception field strength 2.6 kV/mm, length of γ up to a few mm, ln(Nc) ≈ 9-18. Based on (1) and the numerical field simulation we calculate iteratively the streamer inception voltage Usi. For illustration, the curve 1 in Fig. 1 shows the relationship between Usi and the electrode distance d of a sphere-plane arrangement. The point P denotes the limit between weakly and strongly inhomogeneous field. For smaller distances d up to the point P, the streamer inception is leading immediately to a discharge. The withstand voltage Uw (defined as flashover Streamer propagation
Streamer inception 2.6 kV/mm
Leader propagation
3 Q
Voltage
Abstract- This paper focuses on simulation based dielectric design rules of air insulated medium voltage power equipment. After a brief overview on the classical dielectric designing procedure, we propose possible new concepts and present first experimental results on streamer inception and propagation affected by dielectric surfaces. For evaluation of the inception voltage we achieved a good experimental agreement for a path between electrodes where the tangential component of the electric (background) field approaches the maximum value. This new calculation method differs from the traditional one based on evaluation along field lines. The streamer propagation models investigated in this paper can be used for a prediction whether streamers can reach the opposite electrode.
P 0.5 kV/mm
0.1-0.2 kV/mm
1
d
2 Distance d
Fig. 1 Sketch of withstand voltage Uw determined by 3 different stages of discharge development. The thick red curve represents Uw for a sphere-plane arrangement. The part beyond Q is valid either for very long gaps (> 1-2 m) or in presence of dielectric surfaces (d is then the discharge path length along this surface rather than the gap width).
voltage Ufl at low probability) is determined by Usi. Thus, Uw can be calculated using (1) and numerical field simulations. B.
Streamer propagation A streamer head will propagate towards the opposite electrode. It will reach this electrode only if the applied voltage is large enough to maintain the propagation process. Locally, this means that the field in front of the streamer head must satisfy a criterion analogous to the inception criterion; this requires a sufficiently high voltage drop between streamer head and counter electrode. The lowest voltage value that enables streamer propagation in an inhomogeneous field can be approximately expressed (in kV, valid for distances larger than 40-50 mm) as: U w = U 0 + d ⋅ Est , (2) where d is the distance between electrodes in mm, while Est ≈ 0.5 kV/mm is the internal field strength along the (positive) streamer behind its head, and has the same value as the required external field for stable streamer propagation. The voltage U0 ≈ 20-30 kV is equivalent to the needed potential of the streamer head in order to generate a breakdown [10]. The value of Est may vary in the range of +/-10-20%, depending on various conditions like humidity (dry air: 0.4 kV/mm), voltage shape (ac: 0.45 kV/mm; impulse: 0.54 kV/mm) etc. The negative streamers require considerably higher internal field strength for stable propagation (up to Est ≈ 1.2kV/mm). Therefore, discharges from negative electrodes are less critical. Equation (2) is shown in Fig 1 by the dashed curve 2 and provides Uw based on streamer propagation for a point-plane, or equivalently needle-plane, arrangement beyond point P where the field is strongly inhomogeneous.
To determine Uw of the test arrangement the streamer inception and streamer propagation as well as the transition to leader needs to be predicted. In this section we focus on modelling of streamer inception and propagation. Using (1) and (2) for real geometries requires an appropriate path for both, avalanche formation and streamer propagation. Fig. 3 shows possible paths L1 to L4. The paths L1 to L3 are used for calculation of the streamer inception voltage, while path L4 shows a proposed method of calculation of streamer propagation.
C.
A.
Leader transition and propagation The transition to a leader for medium voltage is typically restricted to discharges propagating along dielectric surfaces. The internal field strength along a leader is 0.1-0.2 kV/mm. Hence the leader can propagate over longer distances than a streamer. The leader transition depends on its capacitive coupling with the surrounding electrodes and can be predicted for simple geometries from empirical rules [9]. But they are often inappropriate for real geometries. In Fig 1 the dotted curve 3 represents the withstand voltage determined by leader propagation. It illustrates the significant reduction of Uw beyond the leader transition point Q. III.
TEST ARRANGEMENT
The test object is an insulator with two cylindrical aluminium electrodes moulded in epoxy (Bakelite® EPR 845+EPH 845 + EPC 845). We use the vacuum cast moulding technology to avoid voids inside epoxy, and punctures of insulator during the tests. The length of the test object is 30 cm and the diameter is 3.5 cm. Fig 2 shows the main dimensions of the test arrangement. Two cylindrical ring electrodes, are
d
Fig. 2. Cross section of the test arrangement with cylindrical inner and outer electrodes. The main dimensions are shown in the 2D sketch in mm.
symmetrically mounted outside of the insulator. The distance d between the rings is the length of the air gap. IV.
MODELING
Streamer inception Typically, field lines, like L1, starting from a point of maximum field strength high voltage electrode are used for calculation of streamer inception voltage. At insulator surfaces, the tangential field along a path L2 is used. In addition, a proposed path L3 is determined as follows: It starts from the point with maximum electric field, Emax2, on the insulator surface. The algorithm is based on searching for the local maximum of electric field along each equipotential line between the electrodes. It is equivalent to following the local maximum of tangential field strength in both directions. When integrating (1) along L3 the total length of the path is considered, including all high field locations at the surface of insulator and the shielding rings. Fig. 4. shows Usi calculated for the integration paths L1 L3. For L1 one finds Usi =20 kV and 175 kV at 5 mm and 200 mm ring electrode distance, respectively. For L2 Usi = 40 kV at 5 mm, increases up to 67 kV at 40 mm, and stays constant for longer distances. Finally, for L3 Usi follows initially Usi for L1, while after 20 mm it deviates from L1 and becomes constant at the level of 60 kV for distances larger than 95 mm.
For short ring distances inception occurs at the high voltage ring edge with the maximum field strength. Thus, L1 and L3 give the lowest Usi–values. The surface of the insulator is shielded by the rings, which results in a low field there and larger Usi for L2. When the distance between rings increases a cross-over (for curves L1 and L2 in Fig. 4) occurs at about 20 mm. Then the electric field on the surface is high enough that the electrodeless inception may start. The difference of 10 kV between L2 and L3, is because the L3 follows the line of maximum field, which results in a lower Usi. Note, there is a step of L3 at 95 mm, which is related to additional integration path at the rings. This part of the integration path is not contributing significantly to the resulting inception voltage (2-3 kV) and it disappears for distances longer than 95 mm since the maximum field at the rings is lower than 2.6 kV/mm. B.
Streamer propagation Most of publications on streamer propagation report on symmetric geometries where the streamer propagates along an electric field line of the background field with velocity G G v = AE . The scalar A must then be determined as a function of various quantities like local electric background field, streamer radius etc. A becomes zero if the field drops below Est. In typical real geometries, however, one often observes that a streamer takes a path to the opposite electrode that deviates from field lines. One reason for this behaviour could
180 160
Inception voltage Usi [kV]
Fig. 3 Equipotential lines (coloured curves) and field strength (surface plot; between 3.6 kV/mm (red) and 0.5 kV/mm (dark blue)), calculated for voltage 60 kV and the distance d=50 mm. The paths L1-L3 have been calculated for evaluation of the streamer inception voltage as described in section IV A. The streamer propagation path L4 is simulated with the model described in section IV B. (B 0.5 kV/mm, otherwise A=0).
be the self-consistent field of the streamer, i.e., the attractive interaction with its image charges on electrodes. Another G G G mechanism could occur if the vectors ∇( E 2 ) and E are not collinear. Then, there is a lateral preference in electron avalanching in front of the streamer head towards the direction of increasing electric field. Despite its weakness the effect could become relevant near the plane defined by E = Est, because there A vanishes. An streamer propagation model G G G G could be v = AE + B∇( E 2 ) . A and B must be determined by basic theory or experiments. With such a model, the streamer path can be determined (see Fig. 3). The path follows first the field line as long as the field is above 0.5 kV/mm. If it reaches the boundary E = Est (boundary of coloured region in Fig. 3), G G G the behaviour depends on whether ∇( E 2 ) × E = 0 or not. If it is zero, the streamer dies. If not, there is a deflection of the streamer in direction of the higher field into the streamer stability region. The combined action of the two terms A and B, will force the streamer along the boundary of this region. G G G The streamer either stops because ∇( E 2 ) ∝ E , or it reaches the electrode triggering breakdown (L4 in Fig. 3). If inception occurs at a dielectric surface (electrode free), double headed streamers occur. Although surface effects are unclear, it seems obvious that if inception occurs it will start in the red region at the insulator surface in (around location of Emax2). The most simple model assumption for propagation along the surface is streamer propagation according to the tangential field, as long as the electric field has a field component pointing inwards (positive streamer head) or outwards (negative streamer head). If the normal field direction changes, the streamer head will follow the field line away from the surface. Still, when the streamers reach the regions where A becomes small, they will propagate according to the same principles as described for L4. So it may reach the electrodes (similar to path L3). Of course the different stability field values for the different polarities must be taken into account.
140
L1
120 100
L2
80 60 40
L3 Measured Usi
20 0 0
20
40
60
80
100
120
140
160
180
200
Distance d [mm]
Fig. 4. Calculated inception voltage for the paths L1-L3. The calculation has been performed according to (1) using the streamer constant of 18.0 and ionization coefficients αeff(E) according to [8]. The circles represent the mean value of inception voltage obtained experimentally; the vertical bars show the standard deviation.
C.
V.
EXPERIMENTAL RESULTS
The inception and flashover voltage is measured for 50 * Hz ac voltage. The voltage amplitude was slowly increased until a partial discharge was observed with an uv-camera. The results shown Fig. 4 indicate an agreement between the experimental and the computed inception voltages (path L3). At short ring distances, the measured inception voltage is well described by the paths L1 and L3, and we conclude that the streamer starts from the electrode. For longer distances the experimental results are lower than the inception voltage calculated using tangential path L2. The path L3 describes the observations much better. The ac-flashover test results shown in Fig. 5 confirm the expectations indicated in Fig. 1. In the region of weakly inhomogeneous field (distance d < 100 mm) the flashover due to streamer inception is expected. It is true for clean surfaces. At a distance of d = 100 mm immediate streamer-leadertransitions were observed. This coincides with the expected border between weakly and strongly inhomogeneous field, which is characterized by the crossing of calculated streamer inception (1) (Usi curve) and the withstand voltage (2) (streamer gradient curve). At higher distances (e.g. d = 220 mm) we have observed streamer inceptions according to calculated value of 60 kV. After that, at higher voltages above 70 kV, leader inceptions, and consequently leader propagations determined the flashover that occurred at 90 kV. This corresponds to the leader gradient of approximately 0.18 kV/mm.
100
U fl flashover due to streamer inception
70
Uli U si
60 U si calculated calculated 50
flashover due to streamer inception and immediate transition to leader
40
flashover due to streamer and leader propagation
30 20 10 0 0
50
100
150
200
250
Distance d [mm]
Fig. 5. Ac-peak voltages measured for the test arrangement: Usi: Streamer inception voltage, Uli: Leader inception voltage, Ufl: Flashover voltage
properties and its path are essential for reliable evaluation of the withstand voltage. This topic should a subject of future investigations. REFERENCES [1] [2]
[3] [4] [5] [6] [7]
CONCLUSIONS
The experimental results indicate that the withstand voltage in arrangements with insulating materials can be computationally predicted based on typical streamer/leader inception and propagations models. The crucial point in such a prediction is calculation of the inception and propagation paths. For the evaluation of the inception voltage we achieved the best experimental agreement for path L3 between electrodes where the tangential component of the electric field approaches the maximum value. This is a different approach than the traditional one based on field lines and can be computed based on the background field. The streamer propagation models investigated in this paper can be used for a prediction whether streamers can reach the opposite electrode. However, the experimental results show that in case of propagation along dielectric surfaces the prediction of streamer/leader transition as well as the leader
0.18 kV/mm; leader gradient
2.6 kV/mm 80
[8] VI.
0.45 kV/mm; streamer gradient.
90
Widthstand voltage Uw [kV]
Surface Streamer-Leader Transition According to literature, the presence of a dielectric surface can influence the streamer and leader mechanism in manifold ways. The leader transition can be triggered already in the propagation phase of a surface streamer. However, its quantitative assessment has been beyond the scope of this paper.
[9] [10]
Gallimberti, “The mechanism of the long spark formation,” Colloque de physique, vol. 40, 1979, pp. 193-250. Y.V. Serdyuk, A. Larsson, S.M. Gubanski, and M. Akyuz, “The propagation of positive streamers in a weak and uniform background electric field,” Journal of Physics D: Applied Physics, vol. 34, 2001, pp. 614-623. A. Luque, V. Ratushnaya, and U. Ebert, “Positive and negative streamers in ambient air: modeling evolution and velocities” J. Phys. D: Appl. Phys. 41, 234005 (2008). M. Akyuz, A. Larsson, V. Cooray, and G. Strandberg, “3D simulations of streamer branching in air,” Journal of Electrostatics, vol. 59, 2003, pp. 115-141 A. Blaszczyk, H. Boehme, A. Pedersen, M. Piemontesi, “Simulation based sparkover prediction in the medium voltage range,” ISH Bangalore 2001. L. Pritchard and N. Allen, “Streamer propagation along profiled insulator surfaces,” Dielectrics and Electrical Insulation, IEEE Transactions on, vol. 9, 2002, pp. 371-380. H. Miller, “Surface flashover of insulators,” Electrical Insulation, IEEE Transactions on, vol. 24, 1989, pp. 765-786. K. Petcharaks: “Applicability of the Streamer Breakdown Criterion to Inhomogenous Gas Gaps”, Ph. D. Thesis No. 11192, ETH Zurich, 1995. A. Kuechler, Hochspannungstechnik: Grundlagen – Technologie – Andwendungen, VDI-Verlag 1996. ISBN 3-18-401530-0 E. Nasser; Der räumliche und zeitliche Entladungsaufbau im ungleichförmigen Feld bei positiver Spitze in atmosphärischer Luft; Arch.f. Elektrotechn, XLIV (1959), 157
