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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.