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CHAPTER 6

PREDICTIVE DYNAMIC ARC MODEL

6.1 INTRODUCTION

Laboratory studies show that occurrence of dry-band arcs on

outdoor composite insulators can degrade the polymeric materials surface and

ultimately may lead to insulator failure by flashover. The physical processes

involved are very complex. However it is looked forward to a simple

mathematical model of the arc to help to understand the flashover mechanism.

The LC is a good indication for predicting the flashover. The LC

measurement and instrumentation system in real tower insulator is complex

and expensive. So, alternatively theoretical model has to be used to predict the

LC value. This chapter discusses the concept of AC dynamic arc model and

implements this model to predict pre-flashover LC using calculated pollution

resistance in composite insulator.

A review of recent mathematical models, predicting the leakage

current indicates that they mostly deal with porcelain insulators based on

some assumptions. In order to develop a model which can predict AC

leakage current, the model must take into consideration real polymeric

insulator shape, a dynamic model which accounts for accurate calculation of

surface pollution resistance and rapidly changing arc parameters. This has

motivated the present study. In this study, a dynamic arc model which

considers original shape of a polymeric insulator and adds variable resistance

in series for pollution with the multi arc discharge.

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6.2 CONCEPT OF AC DYNAMIC ARC MODEL

In order to develop a realistic arc model, it is necessary to adopt a

dynamic approach where a model is based on the physical processes

representing the phenomena. The study of arc processes has been

implemented using static models, but most of the parameters involved in the

phenomenon tend to change over time. Dynamic models can be used in order

to understand better the progression of arc parameters as a function of time.

The AC Dynamic Arc model was very well studied and applied in

polluted ceramic insulators, ice covered insulators by Farzaneh (2007). A

mathematical and physical model has been proposed to demonstrate the

mechanism and predict the pollution flashover voltage of composite insulator

by Venkataraman (2006). In order to develop a model which can predict pre

flashover leakage current, the model must take into account all dynamic arc

parameters like voltage, arc length, surface resistance, etc to help better

understanding of the flashover process. Zhang (2000) has clearly explained

arc initiation and propagation under AC voltage in the literature

6.3 ANALYTICAL CALCULATION OF LEAKAGE CURRENT

A predictive dynamic arc model, derived from the physical

considerations and external electric circuit, is that of an arc resistance in series

with a pollution resistance, and applied with AC sinusoidal voltage. The arc is

a multiple series arc and the pollution resistance represents the pollution layer

of the unabridged portion of the insulator. The arc is assumed to move only

along the surface of the insulator and the pollution layer is assumed to be

uniform.

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The root of the arc in composite insulator in the AC pollution

flashover process which was recorded by the high-speed camera is shown in

Figure 6.1(a) and (b). It can be seen from Figure 6.1 that local arcs mainly

start from the electrode of composite insulator because of maximum electrical

field density there, and extend to the nearby sheds. After these local arcs

occur, the polluted composite insulator can be simulated as the following

equivalent model which includes many series of local arcs and pollution

layers, as shown in Figure 6.2.

Figure 6.1 Photograph of Arc Initiation in Polymeric Insulators

Figure 6.2 Eqivalent Circuit of Polluted Polymeric Insulator

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During the arcing period, the leakage current can be calculated using the

model proposed by Obenause(1958), and mathematically developed by Rizk

(1986). This model contains arcs and surface resistances connected in series.

The equations for the sinusoidal supply voltage V (t) and arc voltage are given

by:

pLCarcca RItVVVktV )()()( (6.1)

LCarcarc XIRtV )(

kxxxxkX k )................( 21

where V(t) is applied voltage (v); Va and Vc are the anode and cathode

voltages of local arc, respectively(v); k is the total number of arcs; ILC is the

leakage current ; Rp is the pollution layer resistance. It is assumed that length

of each local arc is x, and total length of the arc is X. The arc resistance per

unit length is obtained dyna given by,

0QQ

arc eR (6.2)

w 0

are constants. The rate of change of the arc resistance with respect to time can

be calculated by using the equation (6.3).

0

2

1N

IRR

dt

dR arcarcarcarc (6.3)

0

0

N

Q

w and No arc heat conduction loss

constant

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6.3.1 The Pollution Layer Resistance

The pollution layer resistance is calculated from the shape factor

cl

s

plr

dlR

0)(2

1 (6.4)

where dl is a small increment of composite insulator creepage distance, r(l) is

the radius of the insulator at a distance of l from the upper electrode. lc is

length of pollution layer s is the surface conductivity of composite

).

When the length of local arcs x is not more than the distance

p of the local pollution layer shown in

Figure 6.3 can be divided into two parts namely, local pollution layer

resistance of sheath which is not covered by the arcs and local pollution layer

resistance of shed between nearly local arcs, which can be expressed as

follows:

s

shedsheathshedsheathp

ffrrr (6.5)

2/

2/2

2D

dss

pr

dr

d

xhr (6.6)

d

xh

d

Dr

s

p ln1

(6.7)

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Figure 6.3 Pollution Layer Resistances when the Local Arc is short

Let fsheath ,fshed be the profile factor of sheath and shed respectively, D,d be the

diameter of shed and sheath respectively(cm), and h be the distance between

two sheds(cm).

Based on the equation (6.4), the pollution layer resistance Rp of the

composite insulator

pnpppp rshedrshedrshedrshedR .........321 (6.8)

3,2,1

3,2,1

ln1

SSS

DDD s

pd

xh

d

DR (6.9)

where S1,S2,S3 are the number of sheds with diameter D1,D2,D3 respectively in

composite insulator.

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When local arc elongates to the surface of sheds and the length of

the arc x is more than the distance between sheds (x>h), resistance rp of the

local pollution layer shown in Figure 6.4 can also be divided into two parts,

local pollution layer resistance of the top and bottom surfaces not covered by

local arcs on the shed with diameter D, assume that the shape factor for top

and bottom surface of the insulator shown in Figure 6.5, is the same.

Figure 6.4 Pollution Layer Resistances when the Local Arc is Long

Figure 6.5 Model of Pollution Layer in Shed and Sheath

(D-d-x+h)/2

(x-h)/2 2r0

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s

bottomtop

p

ffDr )( (6.10)

hxdD

s

pls

dlr

0)(

2 (6.11)

00 2

)(ln

2

)(

r

dD

rdD

hxdDrp (6.12)

Where r0 is the radius of the local arc root,(cm), which can be expressed as

(jolly 1971)

c

Ir arc

0 (6.13)

where c is the influence exponent (0.875).

Based on equation (6.4), the pollution layer resistance Rp of the composite

insulator can be expressed as

3,2,1

3,2,100 2

)(ln

)2(

)(SSS

DDD s

pr

dD

rdD

hxdDR (6.15)

Finally substituting all the parameter values in the equation (6.1), the leakage

current value can be expressed as,

parc

caLC

RXR

VVktVI

)()( (6.16)

6.3.2 Arc Reignition and Propagation

The arc re-ignition voltage can be obtained by an empirical formula

given in Changiz (2004).

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4.023 pig RV (6.17)

The arc will propagate only if the electrical field is lower in the arc than in the

pollution layer (Earc<Ep) . This is the criterion for propagation.

n

LCarc AIE (6.18)

where ILC is the leakage current, and A and n are arc constants. These

constants may vary in accordance with material of the arc medium and

ambient conditions.

A literature survey shows that the values of A and n, utilized by

different investigators, vary over a wide range for different types of arcs.

These values depend not only on the arc medium but also on the electrolyte

used to form the pollution layer. In this work Farzaneh (2007), values are

used.

33.067.0

pp RAE (6.19)

If the condition for propagation is satisfied, then the velocity of propagation is

given by the model which determines arc velocity as a proportional function

of the electric field within the arc reads as follows

arcEdt

dXtv )( (6.20)

where is arc mobility (5 to 50 cm2/Vs), and Earc represents the arc voltage

gradient.

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6.4 COMPUTER SIMULATION

The above equations form a system of coupled differential-

algebraic equations. The system is solved on a Core 2 PC using MATLAB

software. A flowchart of the program is shown in Figure 6.6. The parameters

and constant values used for the simulation are shown in the Table 6.1

Table 6.1 Constant of the Model and Parameters of the Composite

Insulators

Parameter and constant symbol Values

Creepage distance L 323mm

Shed diameter D 90mm

Sheath diameter d 30 mm

Distance between two shed h 48mm

Anode voltage drop Va 200 V

Cathode voltage drop Vc 700V

Arc energy content constant Qo 0.16W/cm

Arc heat conduction loss

constant

No 1000 W/cm.s

Influence exponent c 0.875

Arc constant A 980

Arc parameter n 0.41

Arc mobility 25 cm2/Vs

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Figure 6.6 Flowchart for Continuous Calculation of Pre-Flashover

Leakage Current

The initial values of the arc resistance, length and number of arc,

are respectively /cm a 15% of total length respectively and 1,

for solving the coupled differential equitation by Mfile programme in

MATLAB.

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6.5 RESULTS AND DISCUSSION

The measured and calculated LC results have been shown in

Figures 6.7(i) and (ii) respectively at different conductivities. There is no

arcing, taking place at 0.06 and 0.08 conductivity as shown in Figure 6.7a,b.

At conductivity 0.12 the arc takes place,

(a) C

(b) C

(c) C S

(d) C

(i) Measured from testing (ii) calculated from model

Figure 6.7 Comparison of Calculated and Measured Pre-Flashover

Leakage Current

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which is less than the distance between two sheds. At conductivity 25 severe

arc took place, which is greater than the distance between two sheds. This is

the current that normally flows over the surface of the insulator before

flashover occurs. This current is called as the pre-flashover LC current. The

simulated and experimentally measured LC is almost similar except distortion

in measured LC and low magnitude value at initial cycle in simulated LC

current.

6.5.1 Validation of Model

In order to validate the model, the results of the simulation are

compared with similar results obtained from the experimental work described

in chapter2. The experiment was conducted to measure leakage current in the

composite insulator energized at constant voltage at different pollution levels.

There is a relation between the conductivity s , and pollution level

ESDD(mg/cm2), given by Looms (1999)

ESDDs 100 (6.21)

The voltage applied to the composite insulator is 11kV. Figure

6.6&6.7 shows the analytically calculated and experimentally measured

leakage current for no arc condition and short duration arc respectively

s s

11kv rms). Figure 6.7(c) shows the analytically calculated and experimentally

measured leakage current for long s d

supply voltage 11kV rms). A comparison of pre flashover LC for measured

and calculated leakage currents is provided in Figure 6.7(d). The LC is

experimentally measured at a contamination level of about 0.25 ESDD. The

RMS and Peak currents obtained from both analytical model and

experimentally are compared and shown in Figures 6.8 and 6.9 respectively.

The RMS and Peak current values calculated from the model are not exactly

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coinciding with the experimentally measured one. This is happening because

of the various parameter values and assumptions in the dynamic are model.

(a) C

(b) C

(c) C

(d) C

Figure 6.8 Comparison of IRMS(LC) at Different Conductivity

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(a) C

(b) C

(c) C

(d) C

Figure 6.9 Comparison of Ipeak(LC) at Different Conductivity

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6.6 CONCLUSION

In this chapter, the dynamic arc model is proposed which simulates

LC on polluted polymeric high voltage 11 kV insulators under various

polluted conditions and the pre-flashover leakage current is predicated. The

model incorporates multi discharge arc, rate of change of arc length and arc

resistance. The leakage current was simulated with different conductivity and

the simulated results are verified by experimental results. The model could be

helpful in the simulation of polymeric insulators designed for ultra high

voltage outdoor applications or fabricated with different polymeric materials.

Even though the model presented above still needs modifications, the degree

of agreement between current magnitude for the analytical and experimental

cases is satisfactory. Further this model could be extended to predict the

flashover voltage of polymeric insulators.