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Journal of Electrostatics 69 (2011) 15e22

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Journal of Electrostatics

journal homepage: www.elsevier .com/locate/elstat

Finite element analysis of electric field around an ice-covered semi-conductingglazed insulator using a form of Kelvin transformation

Vinay JaiswalNSERC / Hydro-Quebec / UQAC Industrial Chair on Atmospheric Icing of Power Network Equipment (CIGELE) and Canada Research Chair on Engineering of Power NetworkAtmospheric Icing, Québec, Canada G7H 2B1

a r t i c l e i n f o

Article history:Received 16 May 2010Received in revised form26 July 2010Accepted 22 October 2010Available online 18 November 2010

Keywords:Electric fieldFEMInsulatorKelvin transformationLightning impulseSwitching impulse

E-mail addresses: jaiswal_vinay@yahoo.com, vinay

0304-3886/$ e see front matter � 2010 Elsevier B.V.doi:10.1016/j.elstat.2010.10.003

a b s t r a c t

This paper describes electric field distributions around a semi-conducting glazed standard post insulatorfor power frequency, lightning impulse(1.2/50 ms) and switching impulse(250/2500 ms) voltages undericing conditions, which were computed numerically using finite element method. Thin glaze coating onthe insulator requires a very large number of elements for finite element analysis because of an openboundary around an ice-covered insulator. To reduce number of elements and hence computation time,the region between domain of interest and infinity was modeled by a form of Kelvin transformation.Simulation results are compared with laboratory experiments and are found to be in agreement.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Insulators function in an outdoor environment in many cases. Insuch applications in cold climate regions, the high voltage insulatorsurfaces are exposed to atmospheric pollution and also accretion ofsnow or ice of different natures. Flashover phenomena on ice-covered insulators have been reported from a large number of coldclimate countries [1e9]. The performance of a high voltage insu-lator under an ice-free condition is quite different from those underice-covered conditions. This is because, for a wet ice-coveredinsulator, the field distribution is capacitive-resistive dependingupon the severity of the ice, while for a clean ice-free insulator thefield is purely capacitive. Therefore, it is very important to know thechanges in the field distribution around an insulator caused bydifferent severities of the ice accretion. The main objective of thepresent work is to improve the flashover performance of a semi-conducting glazed post station insulator typically used in 735-kVHydro-Quebec substations in case of heavy ice accretion.

Studies of flashover characteristics of insulators are necessary todetermine suitable insulator profiles for a particular application.The flashover voltage depends on the electric field distribution,which is mainly distorted by the presence of a water film and airgaps along the ice-covered insulators.

_kjaiswal@hotmail.com.

All rights reserved.

Theoretical models that compute the flashover voltage of ice-covered insulators are useful for two main purposes: (i) to betterunderstand the physical processes responsible for flashover, and (ii)to predict the flashover voltages over a wide range of field condi-tions, if the model can be shown to be well correlated with field orlaboratory experiments. Such a model could help reduce thenumber of laboratory experiments needed for the better design ofinsulators.

An ice-covered insulator is an unbounded problem, and henceto study the effects of semi-conducting glaze coating on the elec-trical performance by computing potential and electric fielddistributions around an insulator, an artificial boundary, far awayfrom the device where electric fields are effectively zero, has to bedefined when the mesh is generated. Considerable memory andcomputation time are required for high voltage applications.

In Kelvin transformation, the region between the domain ofinterest and infinity is modeled simply by adding a circularboundary, a second mesh of the same size, and boundaryconstraints to force equivalent boundary potentials to be identical.Infinity lies at the centre of the second mesh. In effect, the secondmesh represents the conformal transformation of all the spaceexterior to the first mesh into a circle [10]. Using this idea, no deepknowledge of analysis is needed; very little extra computer time isnecessary and, frequently, fewer elements are required than formost other methods of calculating the potential and electric field

Table 1Simulation parameters.

Porcelain Semi-conductingGlaze

Ice layer Water film

Permittivity (3r) 6.0 16 75.0 for 60 Hz(variable)

81.0 (variable)

Conductivity (mS/cm) 0 0.22 0 300Thickness (mm) e 0.5 15 1.0

V. Jaiswal / Journal of Electrostatics 69 (2011) 15e2216

distributions around ice-covered insulators for high voltageapplications.

2. Simulation parameters

Fig. 1 shows an ice-covered station post insulator, simulated inthis work. For simulation, it is assumed that the ice is accumulateduniformly around the insulator to determine the performance invarious conditions.

The real relative permittivity of the ice, 3r, depends on thefrequency of the applied voltage, according to the following expres-sion [11],

3r ¼ 3N þ 3s � 3N1þ u2s2

where,3r is the real relative permittivity of the ice,3N is the real relative permittivity of the ice at very high

frequency,3s is the static permittivity of the ice, u is the angular frequency,

ands is the relaxation constant.The relaxation constant, s, is given by the following expression,

s ¼ Csexp�EsKT

�

where,Cs is a constant of value 7.7 � 10�16 s,Es is the activation energy of value 9.29 � 10�20 JT is the temperature (�K), and.K is the Boltzmann constant.Table 1 shows the set of parameters used in the simulationwork.

The conductivities of ice and porcelain are neglected assuming thatall the leakage current flows in the semi-conducting glaze andwater film.

3. An ice-covered insulator (an open boundary problem)

Since this consists of an open boundary problem, two approxi-mations were used to compute the potential and electric fielddistributions along an ice-covered insulator.

Fig. 1. An ice-covered insulator (laboratory model).

3.1. An artificial boundary

An artificial boundary is defined three to four times the deviceradius away from the device. At this boundary, it is assumed thatthe electric fields fall off to zero. The position of the artificialboundary is important because the entire interior region up to theboundary must be discretized, which increases the size of theproblem without adding any useful information to the desiredsolution [12]. Fig. 2 shows an insulator model with an artificialboundary.

3.2. An open boundary simulated by Kelvin transformation

The Kelvin transformation is a process by which the exterior ofa circle is transformed into its interior [13]. Hence, the entire x-yplane can be transposed onto the two faces of a circular disk: thecircular interior region onto one face, the exterior of this circle ontothe other.

The Kelvin transformation only transforms space and, strictlyspeaking, is not applicable to three-dimensional or rotationallysymmetric geometries. It does not of itself in any way transform thematerial properties. It is for this reason that an extra step in thederivation is necessary. As a starting point, it is useful to considerthe problem in network terms. Imagine a three-dimensional modelbased on a spherical geometry, with a defined center. The mainregion of interest lies immediately around the center and extendsto a radius “a”. Beyond that radius there exists only non-electric,non-conducting material.

The network model is built-up as a series of spherical shells. Theouter region network could be rebuilt as a network within a sphereof radius “a”. The successive radii within the new network are to bethe transformed equivalent radii of the original outer network.

To complete the model, the final step consists in modifying theimpedances of the network so that the impedance, looking into thenew model from any pair of points, is identical to the impedance

Artificial boundary

Ice-covered insulatormodel

0.15m

1.35 m

Fig. 2. An ice-covered insulator with an artificial boundary.

Table 2Relative permittivities of a series of spherical shells for an ice-covered insulator.

Shell Number Relative Permittivity

Exterior 1 1.28125Exterior 2 2.17014Exterior 3 4.51388Exterior 4 15.6250Exterior 5 500012.5

V. Jaiswal / Journal of Electrostatics 69 (2011) 15e22 17

seen looking out from those same two points, in the original model.This change in impedance is achieved by applying a transformationto the material properties. This was first done by Ciric and Wong[14,15]. Their approach was based on algebraic transformations,and they were able to effect great economy of effort in achievingsolutions for a range of basic geometries. Later Freeman and Low-ther [10] developed the technique to use an all-enveloping spherethat can be applied to the solution of a general range of fieldproblems using finite elements.

The transformation of the material properties is given by Ciricand Wong [14,15] as:where,

30 ¼ 3� rr0�

r0 ¼ a2

r

30 is the permittivity at the transformed inner radius r0,3 is the permittivity at a radius, r.Fig. 3 shows an insulator model for the computation of electric

field and potential distributions by simulating an open boundary bya form of Kelvin transformation.

The relative permittivities calculated of a series of sphericalshells for an ice-covered insulator are shown in Table 2.

4. Computation of the transient electric field strengths

The general solution for the low frequency electromagneticfields in insulating systems has been obtained by solving Maxwell’sequations. To apply the finite element method, a functional basedon the system energy is defined that can be applied to the statedproblem. Minimization of the system energy leads to a system ofalgebraic equations, whose solution, under the application of cor-responding boundary conditions (Dirichlet, Neuman or mixed),

Exterior 1

Exterior 4

Interior Region

Exterior 2

Exterior 3

Exterior 5

Fig. 3. An insulator model for computation of potential and electric field distributionsusing a form of Kelvin transformation.

supplies the required node potentials, and then in turn, electricfield strengths can be computed.

Electric field strengths as a function of time under the transientconditions were computed at the four critical points near the highvoltage electrode in the radial outward direction from the axis of aninsulator. These points were considered because the electric fieldstrength was very high near the high voltage electrode. The coor-dinates of the four critical points are (0.15, 1.35), (0.16, 1.35), (0.17,1.35), and (0.20, 1.35) meters, respectively.

5. Flashover performance based on potential distribution

Computations were also carried out for the sinusoidal voltagesequivalent (same rise times) to the lightning and switching impulsevoltages. Equivalent sinusoidal voltages were determined for light-ning and switching impulse voltages [16]. This faster computingprocess avoids long transient solutions and yields results that arealmost as useful. Flashover performance can be studied by com-puting the electric field strengths around an insulator.

When the electric field strength, at any point, exceeds thecorona onset field, discharges occur at that point and may lead toflashover. Computation of the electric field strength can introducea large error because of the numerical differentiation of thepotential at any point. Hence, potential distributions along theinsulators were also computed in this work. By visible inspectionof the potential distributions, flashover performance has beenstudied.

5.1. Computation of potential distribution

The potential distributions were computed on a vertical planecutting the insulator into four symmetrical parts.

5.2. The clean and wet ice-covered insulators

Computations were carried out for power frequency andequivalent sinusoidal voltages of switching impulse (SI) and light-ning impulse (LI) voltages [16]. The potential distributions werecomputed along the insulator surface from the high voltage elec-trode to the ground electrode for uniform ice accretions.

For a switching impulse.

tr ¼ 250ms

Feq ¼ 1=ptr ¼ 1:3KHz

For a lightning impulse.

tr ¼ 1ms

Feq ¼ 1=ptr ¼ 320KHz

Where tr is the rise time of an impulse.

V. Jaiswal / Journal of Electrostatics 69 (2011) 15e2218

6. Experimental validation

To verify the simulation results, flashover tests were carried outin a 6.2 � 5.7 � 3.8-m climate room on a semi-conducting glazedinsulator covered with a 1.5-cm layer of glaze ice, in the presence ofa water film and air gap. During the accumulation period, theinsulator was energized by a high voltage AC system consisting ofa 240 kVA, 120 kV, transformer and its 240 kVA regulator. Theoverall short-circuit current of the high voltage system was about28 A at the maximum operating voltage of 120 kVrms. Theconductivity of the water used to form the glaze ice was set at80 mS/cm. Dripping water conductivity was measured after theflashover test and it was found to be 300 mS/cm. This increase inconductivity is due to the fact that the salt migrates to ice surfaceduring the ice accumulation process [17]. The length of the testedinsulator was 80 cm and the flashover tests were performed forpositive and negative lightning and switching impulse voltages.

6.1. Experimental setup

Figs. 4 and 5 show the experimental setups used during the icedeposit process and flashover performance test, respectively.Impulse voltages were generated using an 800 kV, 40 kJ, Marxgenerator and shaped to a standard lightning impulse (1.2/50ms)and a switching impulse voltage (250/2500ms) using externalresistors. The test sample was kept in a climate room and thevoltage was applied through a bushing.

6.2. Ice deposit method

Natural conditions were simulated as closely as possible bydepositing artificial ice on energized insulators in the climate room.The testing conditions were adjusted to form a wet-grown ice onenergized insulators. This type of ice is associated with the highestprobability offlashover [18]. In the climate room,parameters such asair temperature and wind velocity, as well as both the vertical andhorizontal components of precipitation intensities, were controlledand kept constant following the instructions provided in [19].

Service voltagewas applied to the test insulatorduring thewholeicing period. This is necessary for a more realistic simulation of thenatural icing of insulator equipment in the field [19]. In fact, the

Fig. 4. The ice dep

electric field affects the ice as it accretes [20], particularly theprogression of icicles and, consequently, the formation of air gaps.The location, number, and length of these air gaps are major factorsin electric field distortion [21e23], and consequently, flashovervoltage level.

6.3. Ice test preparation prior to flashover

A preparation period is needed between the end of the iceaccretions at subzero air temperature and the moment when thetest voltage is applied for the flashover voltage evaluation. Theprocedure may be adjusted according to two approaches, that is,performing the test under the “icing regime” or under the “meltingregime” [19].

In this work, testing was done under the icing regime. The testinsulator was exposed to conditions as close as possible to those inthe field (e.g. freezing rain conditions).

In icing regime testing the flashover performance test is carriedout shortly after the ice accretion is completed and a water film isstill present on the ice surface [20]. In such a case, the preparationperiod is short, typically about 2e3 min. This leaves enough timefor taking pictures, adjusting the test setup, installing/removing thecollector receiving dripping water from the test insulator, andmeasuring ice thickness. Immediately after this period, the flash-over test voltage is applied.

Table 3 shows the test conditions used for the experiments.

6.4. A.Ice test flashover performance evaluation

High leakage currents from an appropriate power supply,including both partial and flashover arcs, generally change thelength and shape of air gaps i.e. ice-free areas formed along an ice-covered insulator, modify the ice surface conductivity, and willsometimes totally shed the ice deposit. These changes will affectthe flashover voltage of test insulators [19]. Therefore, the flashovertest for one condition was achieved for any ice accretion sequence.

7. Results and discussion

The potential and electric field distributions were computedaround insulators by a program, based on the finite element

osit process.

Fig. 5. Impulse flashover performance test.

4.0E+05

5.0E+05

6.0E+05

7.0E+05

8.0E+05

9.0E+05

1.0E+06

lect

ric fi

eld

|E| (

V/m

) Clean insulator

Ice-covered insulator (inpresence of a water film andan air-gap)

V. Jaiswal / Journal of Electrostatics 69 (2011) 15e22 19

method. For each simulation, the peak voltagewas 100 kV. Using anartificial boundary, time taken for the computation of potentialdistribution on a Pentium 4, 1.8 GHz computer - for polynomialorder 4 and Conjugate gradient tolerance 0.0001% - was 68 h45 min. Time taken for the computation of potential distributionusing a form of Kelvin transformation was 25 h 57 min only.Potential and electric field distributions along the semi-conductingglazed insulator, computed using both the approximations, arecompared, and are found to be in agreement.

Figs. 6 and 7 show the transient electric field strengths at thefirst critical point (0.15, 1.35) excited by the lightning and switchingimpulse voltages, respectively, for a clean and a wet ice-coveredsemi-conducting glazed insulator.

Fig. 8 shows the comparison of the maximum electric fieldstrengths at the four critical points of a clean semi-conductingglazed insulator. It is observed that the maximum electric fieldstrength is higher for a lightning impulse voltage than that fora switching impulse voltage. Based on the simulation results, it isconcluded that a lightning impulse is the limiting factor in thedesign of a clean semi-conducting glazed insulator. It is alsoobserved from Figs. 6 and 7 that the time instants at the maximumelectric field strengths for the lightning impulse and switchingimpulse voltages are 2.0 ms (peak voltage 1.2 ms) and 200.0 ms (peakvoltage 250.0 ms), respectively. It is concluded that the electric fieldstrength reaches maximum after the peak voltage under a lightningimpulse voltage, and before the peak voltage under a switchingimpulse voltage, for the clean semi-conducting glazed insulator.This is due to the fact that a continuous leakage current flows in thesemi-conducting glaze of the insulator, which has a major effect onthe equivalent impedance of the system under ice-free conditions.

Fig. 9 shows the comparison of the maximum electric fieldstrengths at the four critical points near the high voltage electrode

Table 3Experimental test conditions.

Chamber dimensions 6.2 � 5.7 � 3.8 mAccumulation voltage 64 kV, 60 Hz, ACTest voltages Lightning and Switching Impulse,

up-and-down method (V50%)Accumulation temperature �12 �CIce thickness 15 mmType of ice GlazeConductivity 80 mS/cm at 20 �C

of a wet ice-covered semi-conducting glazed insulator. It isobserved that the maximum electric field strength is higher fora switching impulse than it is for a lightning impulse voltage. Basedon the simulation results, it is concluded that a switching impulse isthe limiting factor in the design of a semi-conducting glazedinsulator under icing conditions. It is also seen from Figs. 6 and 7that the time instants at the maximum electric field strengths forthe lightning impulse and the switching impulse voltages are21.6 ms (peak voltage 1.2 ms) and 550.0 ms (peak voltage 250.0 ms),respectively. It is concluded that the electric field strength reachesmaximum after the peak voltage for a wet ice-covered semi-con-ducting glazed insulator under a lightning impulse and a switchingimpulse voltage. This is due to a change in the equivalent imped-ance of the system due to high value of conduction current in thewater film under icing conditions.

It is observed from Figs. 8 and 9 that the wet ice accumulated onthe insulator has a little effect on the electric field distribution

0.0E+00

1.0E+05

2.0E+05

3.0E+05

0 0.02 0.04 0.06Time instants (ms)

E

Fig. 6. Electric field strengths at the first critical point for a clean and a wet ice-coveredsemi-conducting glazed insulator for a lightning impulse voltage.

0.00E+00

2.50E+05

5.00E+05

7.50E+05

1.00E+06

1.25E+06

1.50E+06

1.75E+06

2.00E+06

2.25E+06

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Time instants (ms)

Ele

ctric

fie

ld

|E

| (V

/m

)

Clean insulator

Ice-covered insulator (in presence of awater film and an air-gap)

Fig. 7. Electric field strengths at the first critical point for a clean and a wet ice-coveredsemi-conducting glazed insulator for a switching impulse voltage.

0

250

500

750

1000

1250

1500

1750

2000

2250

0.15 0.16 0.17 0.2

ma

x |E

| (K

V/m

)

Radial distance (m)

Lightning impulse

Switching impulse

Fig. 9. The maximum electric field strengths for the lightning and switching impulsevoltages near the high voltage electrode of an ice-covered semi-conducting glazedinsulator in the presence of a water film and an air gap.

V. Jaiswal / Journal of Electrostatics 69 (2011) 15e2220

around the insulator for a lightning impulse voltage, but it hasa major effect for a switching impulse voltage. This is because of therise time of a lightning impulse is much lower than that ofa switching impulse voltage.

Fig. 10 shows the potential distributions along a semi-con-ducting glazed insulator covered with wet glaze ice in the presenceof an air gap 7.5 cm in length at the tip of the sheds, at a radialdistance of 15.0 cm from the center of the insulator and down thelength of the insulator. The insulator is coated with a semi-con-ducting glaze of thickness 0.5 mm with a conductivity of2.2 � 10�5 S/m for an equivalent lightning impulse, an equivalentswitching impulse and power frequency voltages. It is observedthat the potential distribution is more linear for a lightning impulsevoltage than that for a switching impulse voltage. It is concluded

0

25

50

75

100

125

150

175

200

225

250

0.15 0.16 0.17 0.2

max |E

| (K

V/m

)

Radial distance (m)

Lightning impulseSwitching impulse

Fig. 8. The maximum electric field strengths for the lightning and switching impulsevoltages near the high voltage electrode of a clean semi-conducting glazed insulator.

that a switching impulse is the limiting factor in the design of a wetice-covered semi-conducting glazed insulator.

Fig.11 shows the potential distributions along a clean insulator atthe tip of the sheds at a radial distance of 15.0 cm from the center ofthe insulator and down the length of the insulator. The insulator iscoated with a semi-conducting glaze of thickness 0.5 mm witha conductivity of 2.2 � 10�5 S/m, for 320 kHz and 1.3 kHz sinusoidalvoltages, equivalent to lightning and switching impulse voltages,respectively. It is observed that the potential distribution is morelinear for a switching impulse voltage than that for a lightningimpulse voltage. It is concluded that a lightning impulse is thelimiting factor in the design of a clean semi-conducting glazedinsulator.

Figs. 10 and 11 show that the potential distributions for a light-ning impulse voltage both along the wet ice-covered semi-con-ducting glazed insulator and along the clean semi-conductingglazed insulator, are almost linear. It is concluded that wet iceaccumulated on the insulator has very little effect on the potential

0

20000

40000

60000

80000

100000

120000

0 0.5 1 1.5 2Distance (m)

)V

(l

ai

tn

et

oP

Switching impulse(semiconducting glazeconductivity 0.000022 S/m)

Lightning impulse

Power frequency

Fig. 10. Potential distributions along a wet ice-covered insulator coated with a semi-conducting glaze for equivalent switching impulse, equivalent lightning impulse andpower frequency voltages.

0

20000

40000

60000

80000

100000

120000

0 0.5 1 1.5 2Distance (m)

Po

ten

tial (V

)

Lightning impulse(conductivity0.000022 S/m)Switching impulse

Fig. 11. Potential distributions along a clean standard insulator coated with a semi-conducting glaze for equivalent lightning and equivalent switching impulse voltages.

V. Jaiswal / Journal of Electrostatics 69 (2011) 15e22 21

distribution along the insulator for a lightning impulse voltage. Itcan also be observed from these figures that while the potentialdistribution for a switching impulse voltage is not linear along theinsulator when it is covered with wet ice, it is linear when theinsulator is clean. It is concluded that the wet ice accumulated onthe insulator has a major effect on the potential distribution, andhence on its flashover voltage for a switching impulse voltage. Thisis because the conduction current is negligible in the ice layer andthe displacement current is much higher for an equivalent light-ning impulse than that for an equivalent switching impulse voltage.

Fig. 12 shows the 50% flashover voltages measured in icing andclean conditions under the test conditions mentioned in theprevious section. It is observed that the switching impulse is thelimiting factor in the design of the semi-conducting glazed insulatorunder icing conditions, as shown by the simulation results. This iscontrary to clean conditions, where it is observed that the lightningimpulse is the limiting factor for the design of semi-conductingglazed insulators, as demonstrated by the simulation results.

The laboratory experiments show that flashover occurs after thepeak voltage for the clean as well as the wet ice-covered semi-

0

100

200

300

400

500

600

clean ice

V(

eg

at

lo

vr

ev

oh

sa

lF

%0

5V

k,

)

Lightning impulse

Switching impulse

Fig. 12. Impulse flashover performance of a semi-conducting glazed standard postinsulator (experimental results).

conducting glazed insulators under a lightning impulse voltage. Onthe other hand, flashover occurs before the peak voltage for theclean semi-conducting glazed insulator and after the peak voltagefor the wet ice-covered semi-conducting glazed insulator undera switching impulse voltage.

Fig. 12 also shows that the ice, accumulated on the insulator, haslittle effect on the flashover voltage of the semi-conducting glazedinsulator for a lightning impulse voltage, but it significantly reducesthe flashover voltage for a switching impulse voltage. Theseexperimental results are in agreement with the simulation results.

8. Conclusion

The following conclusions have been drawn from this work:

1. In the case of an open boundary, simulated by a form of Kelvintransformation, computation time for the potential and electricfield distributions along a wet ice-covered semi-conductingglazed station post insulator in the presence of an air gap ismuch less than that of an artificial boundary method.

2. From the simulation results, confirmed by the experimentalresults, it has been found that a switching impulse is thelimiting factor in the design of a semi-conducting glazedinsulator under icing conditions.

3. This is opposite to clean conditions where it has been foundthat a lightning impulse is the limiting factor in the design ofthe semi-conducting glazed insulator under clean conditions.

4. It has been found that flashover occurs after the peak voltagefor a clean as well as a wet ice-covered semi-conducting glazedinsulator under the lightning impulse voltage.

5. Flashover occurs before the peak voltage for the clean semi-conducting glazed insulator, and after the peak voltage for thewet ice-covered semi-conducting glazed insulator under theswitching impulse voltage.

6. From the simulation results, confirmed by the experimentalresults, it has been found that the accumulation of ice on theinsulator does not have a major effect on the electric fielddistribution, and hence on its flashover voltage for a lightningimpulse voltage, but ice has a major effect on the electric fielddistribution, and hence significantly reduces the flashovervoltage of the semi-conducting glazed insulator for a switchingimpulse voltage.

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