Flashover Dynamic Model of Polluted HV Insulators using FEM & Lagrange

IEEE Transactions on Dielectrics and Electrical Insulation Vol. 11, No. 4; August 2004 577
A New Flashover Dynamic Model of Polluted HV
Insulators
Zafer Aydogmus
Fırat University, Technical Education Faculty
Electrical Education Department, Elazıg, 23119 Turkey
ˇ
and Mehmet Cebeci
Fırat University, Engineering Faculty
Electrical-Electronics Department, Elazıg, 23119 Turkey
ˇ
ABSTRACT
In this work, a model based on field criterion has been developed to represent the
flashover phenomenon, which occurs due to surface pollution on high voltage insu-
lators, under ac voltage. The values of potential and electric field on an insulator
()
surface have been determined using the finite element method FEM . The open
model of the insulator has been used for calculating the resistance in series with
the arc, in addition to the values of the leakage current and the arc gradient. As a
new approach, this dynamic model uses Lagrange multipliers for the solution of the
pollution flashover problem. Both the impedance and the electric field criterion
have been used for the propagation of arc on the surface. A computer program
()
called NFDM new flashover dynamic model has been developed to achieve this.
The results obtained from the program have been compared with theoretical and
experimental results of other researchers.
Index Terms — Dynamic model, pollution, flashover, FEM.
1 NOMENCLATURE
Symbol Variable Unit
EArc gradient kVrcm
arc
EPollution gradient kVrcm
poll
⌬EDifference between arc gradient and
pollution gradient on discharge root kVrcm
RPollution resistance in series with the arc ⍀
iLeakage current A
xLength of the arc cm
a
VVoltage drop along the arc V
arc
VVoltage drop between anode and cathode V
ca
VVoltage drop along the arcless zone V
r
EField strength along the x direction kVrcm
x
EField strength along the y direction kVrcm
y
2 INTRODUCTION
NE of the faults that exist in high voltage
OŽ.
HV insulators is the pollution flashover. Flashover
of a polluted insulator can occur when the surface is wet
due to fog, dew, or rain. Development of partial dis-
charges on the insulator surface and propagation of these
discharges over a period of time cause the leakage current
Manuscript recei
®ed on 5 October 2001, in final form 6 February 2004.
to increase on the insulator surface that may result in a
flashover. This occurs in three stages. These are the for-
mation of electrolytic conductive film layer, the formation
of dry bands and the starting of pre-discharges and propa-
gation of pre-discharges. The first two stages can occur
frequently, however, the last stage does not occur as often
as the others.
Most commonly seen pollution related problems to
Ž.
flashover exist in coastal areas sea salt , industrial areas
Ž.Ž.w
chemical pollution , and other areas desert sand, etc. 1,
x
2 . In the case of the cap-and-pin porcelain insulator, usu-
ally dry bands initially occur near the cap andror the pin
of the HV insulator, and when the voltage drop on a dry
band exceeds the air withstand-voltage sparks occur. If this
dry band is bridged by a discharge, voltage drop on an-
other dry band increases and therefore a discharge chain
wx
starts 3᎐5.
Analytical and experimental methods have been devel-
oped for calculating the flashover characteristics of pol-
luted insulators, and several mathematical approaches
wx
have been analyzed 6 . However difficulties still exist and
further studies are needed.
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Aydogmus and Cebeci: A New Flasho©er Dynamic Model of Polluted HV Insulators578
To explain the flashover problems, the geometry of an
insulator is modeled in two ways: the flat plate model and
the open model. The flat plate model involves converting
the complex insulator surface into equivalent rectangular
wx
shapes 7-13 . This model is advantageous for both theo-
retical modeling and experimental work due to its simple
geometry. An equivalent electrical circuit based on the flat
plate model to predict the critical voltage, current and arc
wx
length for polluted insulators has been developed 9, 10 .
In 2-D, the flat plate model represents the insulator as a
basic rectangle with one edge equal to the leakage dis-
tance and the other edge obtained from the ratio of the
leakage distance and the form factor. This model however
cannot show the effects of the complex insulator geometry
and of the arc state on the flashover voltage. Thus, using
the open model, which displays the open shape of the in-
sulator surface and corresponds completely to the actual
wx
insulator geometry is a more suitable approach 4, 5, 14 .
In addition to these two models, the circular strip model
wx
has also been used 15 .
Ž.
Furthermore, two mathematical approaches models
namely, the static model and the dynamic model have been
used. Of these two, the recently developed dynamic mod-
els allow the prediction of discharge activity leading to the
flashover of polluted insulators. The dynamic models ac-
count for the instantaneous changes of the arc parameters
wx
10᎐13, 17᎐18 , Static models that take the arc into ac-
count statically are therefore not as efficient.
Several criteria are used to control the arc propagation
on the insulator surface; one of them is to analyze the arc
dynamics till flashover is established by the equivalent
wx
impedance 9, 10 . The difference between the arc gradi-
ent and the pollution gradient on the discharge root, ⌬E
Ž.
sE -E , is the most important factor to control the
poll arc
Ž.wx
flashover electric field criterion 3, 4, 11᎐13 . The oth-
Ž. wx
ers include currentrarc length dirdx )0 criterion 15 ,
wx
the geometric shape criterion 16 , and the energy crite-
wx
rion 18 . Furthermore, the pollution resistance in series
Ž.
with the arc that varies with time dRrdt should also be
wx
considered 4᎐6, 19 .
The initial studies assumed the arc propagation to start
from the pin and elongate toward the cap. However, ac-
wx
cording to 3᎐5 the pre-discharges not only elongate by
starting from one point, but can also occur on several
points at the same instant depending on the state of the
wx
dry bands 3᎐5 . This case has encouraged our investiga-
tion to develop this presented dynamic model, which con-
siders the actual insulator geometry and the arc formation
and propagation on different zones of the leakage dis-
tance.
3 FLASHOVER EQUATION
In most analytical studies, flashover problem is repre-
sented with the flat plate model covered by uniform pollu-
Figure 1. Partial arc discharges on the insulator surface.
tion. The model shown in Figure 1 is an ideal model which
represents the ignited arc on a plane surface.
Researchers have used this model using different ap-
proaches and obtained the same results. The following as-
sumptions are considered for this model:
䢇
Length of the arc is equal to that of the dry bands
Ž.
xsxqxqxq... .
aa 1a2a3
䢇
Pollution resistance, R is only a function of the arc-
less part of the surface.
䢇
Temperature and humidity effects are neglected.
䢇
Irregularity of the current density distribution at the
arc ends is ignored.
Ž.
Equation 1 can be written for Vs according to the
model shown in Figure 1.
VsVqVqV1
Ž.
sarcrca
where, V,V,Vare the voltage drop along the arc, the
arc r ca
voltage drop along the arcless zone and the voltage drop
between the anode and the cathode respectively. The volt-
age drop along the arc can be calculated by
VsxEi 2
Ž. Ž.
arc a
Ž.
where xis the length of the arc and E i is the arc volt-
a
age gradient. Researchers have considered the expression,
Ž.
y n
Ei sAi for the arc voltage gradient. Values of the
coefficients A and n change due to the environment con-
ditions where the arc is ignited. When the arc is ignited in
wx
air, A s63 and n s0.76 are used 3, 11᎐14 .
Dry bands can cause the arc to occur on different zones
of the leakage distance at the same time. Thus, the total
voltage drop along the arc becomes
VsVqVqVq... 3
Ž.
arc arc1 arc2 arc3
If R is the total pollution resistance in series with the
arc then
RsRx sRx qRx qRx q... 4
Ž. Ž . Ž . Ž . Ž.
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and the voltage drop on the pollution resistance along the
arcless zone is
VsiR 5
Ž.
r
Thus, the flashover equation as given below is derived from
Ž.Ž. Ž.
equations 1 , 2 and 5
VsxA.i
y nqiRqV6
Ž.
sa ca
where Vis the total voltage drop on anode and cathode
ca
Ž.
VsVqV.
ca c a
The pollution resistance Ris not only dependent on
the arcless zone length xbut also on the heat energy w
P
ŽŽ..
RsRx,w. The heat energy occurs on resistance R,
p
due to the current flow on the polluted layer of the insula-
tor surface, which causes variations to occur on the thin
wet film layer and can be expressed as
t
2
wsR.i.dt 7
Ž.
H
0
4 DEVELOPED DYNAMIC MODEL
Ž
In this study the cap-and-pin insulator BSFT-9336,
.
Bullers and its open model are used to display the poten-
Ž
tial and electric field calculations see Figures 2a and 2b
.
for geometry and open model of the insulator .
An open model of an insulator can be obtained from
the following
yx s
␲
Dx 8
Ž. Ž. Ž.
where xis the distance from the cap to the point which is
Ž.
on the leakage distance, Dx is the diameter which has
been determined from the distance of a node to the insu-
Ž.
lator axis as in Figure 2a, and yx is the vertical distance
on the open model.
In this dynamic model, the insulator has been consid-
Ž.
ered as a two-dimensional 2D surface for the values of
potential and field strength along the leakage distance
Ž.
calculations Figure 3a . However, the pollution resistance
in series with the arc has been calculated from the instan-
taneous potential values on the open model of the insula-
tor. The potential and the electric field values have been
Ž.
computed by the finite element model FEM .
A computer program is developed to achieve the open
model for resistance calculations. The program uses the
x-y coordinates of the nodes on the leakage distance by
Ž
rotating each node for a desired step angle 3⬚,5⬚,10⬚,15⬚
.
etc. until 360⬚and then unifying all those newly obtained
Ž. Ž
nodes to form a three-dimensional 3D surface Figure
.Ž.
3b . This 3D surface is then flattened using equation 8
to obtain the coordinates of the nodes of the open model
Ž.
in 2D. Thus, when the surface of the insulator in 3D as
shown in Figure 3b has been opened and flattened, the
Ž.
open model in 2D is achieved as shown in Figure 3c.
The conductivity of the pollution is assumed to be uni-
form. However, the conductivity can be assumed to vary
linearly or exponentially over the whole area.
The following steps are applied in this model:
Step 1: Calculation of the potential distribution in the insu-
()
lator and selected external region Figure 3a is considered as
a Laplace problem.
The functional of the Laplace equation for the problem
region is given as follows.
2
2
⭸ V ⭸ V
Ws
␳ qdx.dy 9
Ž.
H
es
ž/
ž/
⭸ x ⭸ y
S
where ␳ s ␳ rh, is the surface conductivity and hthe
s
polluted layer thickness. The potential distribution for the
investigated region is obtained with the minimization of
energy function.
Step 2: The field strength ©alues along the leakage distance
on the insulator surface are determined.
These values are calculated from the nodal potentials
obtained by the FEM. xand ycomponents of the field
Figure 2. BSFT-9336 insulator, a, Geometry; b, Open model of the insulator.
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Aydogmus and Cebeci: A New Flasho©er Dynamic Model of Polluted HV Insulators580
Figure 3. a, Model of the BSFT-9336 insulator meshed with triangular elements for FEM calculations; b, A part of the insulator surface
Ž
meshed with the triangular elements for transformation to the open model; c, The meshed open model for the FEM calculations for the
.
axi-symmetry .
are given as follows:
⭸ V
Esy 10
Ž.
x
⭸ x
⭸
V
Esy 11
Ž.
y
⭸ y
From above equations the magnitude of Eis
™
22
<<
EsEqE12
Ž.
'
xy
For calculating the differentials, corner potentials be-
longing to the triangular element, the area of the triangle,
and the interpolation functions are used.
()
Step 3: The resistance R x in series with the arc is calcu-
p
lated using the FEM on the open model.
The xcoordinates are the leakage distances calculated
by using 3D coordinates, and ycoordinates are circumfer-
ential distances, where their radii are the length between
the nodes and the insulator axis. To calculate the varia-
Ž.
tion of Rx using the potentials on the open model of
p
BSFT-9336 insulator shown in Figure 3c, the solution re-
gion has been meshed by triangular elements and then,
the initial conditions to the cap and pin parts of the insu-
lator are applied.
Step 4: The pollution layer near the grounded electrode is
meshed densely to calculate the current flowing from the sur-
face.
The current flow, Iin this region can be expressed as
T
mV
AV k
is;ks1,2,3. ..., m13
Ž.Ž.
Ý
T
r
k
ks1
where kis the triangle number adjacent to the part of the
pollution layer next to the grounded electrode. ris the
k
resistance of each triangular element, and Vis the av-
AV k
erage voltage of the corner nodes of each triangular ele-
wx
ment 3 .
As stated earlier the arc voltage gradient, Eis
arc
EsA.i
y n14
Ž.
arc T
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Step 5: The condition E )E , is used for arc propaga-
poll arc
tion criterion.
If the potential gradient of the pollution layer is greater
than the arc gradient, ionization in the arc strip will occur
due to an increase in the current. In this case, the arc will
propagate because of the leading potential gradient. If
EFE, then the arc will extinguish. If the arc length
poll arc
is smaller than the leakage distance and the arc criterion
Ž.
is met; the dynamic change in Rx is recalculated on
p
the open model.
Therefore, the arc gradient between two neighbor nodes
on the leakage distance is calculated and checked to find
if an arc would occur. This process is repeated for all the
nodes on the leakage distance. Thus, the propagation of
arc can be followed step by step easily.
Step 6: If an arc occurs between any two nodes on the leak-
age distance, the potentials of these nodes should be taken
equal.
This case will be used as an initial condition for the
calculations in the next step. Thus, according to the arc
criterion, if an arc occurs between two points, the condi-
tion VsVshould be incorporated in the equation sys-
12
tem. This can be made easily by using Lagrange Multipli-
wx
ers 21 .
Step 7: The formation of the dry band between two nodes
where the arc has occurred is considered in the calculation of
the resistance in series with the arc.
If an arc occurs on two or more nodes, the calculations
are repeated from step 2 through 7 until the arc is ob-
tained on all nodes.
A flow chart of this model is shown in Figure 4.
5 APPLICATION
In this study, the BSFT-9336 type chain insulator unit is
investigated, and the specifications of this unit are given
in Table 1.
Potential distributions are calculated for different val-
ues of conductivity and the normalized equipotential con-
Ž.
tours are plotted 10%, 20%, ..., 90% . The potential dis-
tributions for conductivities
␴ s2,5
␮ S and ␴ s10
␮ S
of pollution layer are shown in Figures 5a and 5b, respec-
tively.
For a certain value of conductivity, the field strength
increases on nodes of the leakage distance due to the in-
crease in voltage. This is shown in Figure 6.
Figure 4. Flow chart of the dynamic arc model.
Under service voltage, variation of the field strength for
values of different conductivity can be seen from Figure 7.
The calculated flashover voltages for the chain insulator
with 9 units are given in Figure 8 and compared with the
wx
literature data 3 .
Potential variations related to pre-discharges which oc-
cur on the nodes of the different regions along the leak-
age distance are shown in Figure 9a for
␴ s15 ␮ S. When
an arc occurs on the nodes of the leakage distance, the
potentials of these nodes would be equal.
In Figure 9a, the curve for N s0 is the potential varia-
tion for the clean insulator. As no pre-discharge occurs
for 10 kV potential, there are no equipotential nodes and
N is zero. In polluted conditions, the potential has started
from 4 kV and increased in 0.5 kV intervals, and then at
7.5 kV arcs were observed at N s13 nodes on 4 different
regions of leakage distance. The location of these arcs is
shown in Figure 9b. For 12.5 kV potential, the arc oc-
curred at N s26 points on 7 different regions of leakage
distance. For 15.5 kV potential, the arc occurred at N s
39 points on 8 different regions.
Table 1. Some specifications for the BSFT-9336 insulator.
The values of used in this work
Number of Number of Number of
Shield Leakage Form nodes on 2-D nodes on open nodes on the
diameter Height distance factor geometry for model for the leakage
Ž. Ž. Ž.
Type mm H mm L mm F the FEM FEM distance
Fog 288 140 405 1.09 352 392 56
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