Download Communication BVI-3 Une nouvelle methode pour !`analyse par

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SESSION B. VI : CAO ET MODELISATION
Communication BVI-3
Une nouvelle methode pour !'analyse par elements finis des champs ionises dus
courant continu unipolaire.
A new method for the finite-element analysis
a l'effet
Corona en
of Wlipolar DC Corona ionized fields.
ABOELSAAD M.M., CIRIC I.R., RAGHUVEER \1.R.
UNIVERSITY OF MANITOBA
Department of Electrical Engineering
WINNIPEG, R3T 2N2
CANADA
ABSTRACT
Solution of the DC corona ionized field problem is, in
general, extremely difficult because of the nonlinearity of the differential equations describing it.
Furthermore, except for the potential, the boundary
conditions at the coronating electrodes are not known.
Various simplifying assumptions have been employed to
determine corona power loss and field quantities for
practical transmission line configurations.
In this
paper, a new method based on a finite-element iterative algorithm, without invoking simplifying assump-
tions used so far, is proposed.
It consists of a
simple and realistic modelling of the initial space
charge distribution, and of a new, efficient updating
criterion.
The method is applied to the unipolar DC
field in a cylinder-to-plane configuration and the
numerical results agree well with experimental data.
Comparison with previous methods shows that the proposed method is highly efficient and more accurate,
yielding also the correct distribution of the electric
field intensity over the coronating conductor surface.
610
INTRODUCTION
tions (1) and (4),
The corona performance of high voltage direct current
(IWDC) transmission lines ls one of the critical criteria in their design due to associated, undesirable
effects, such as electromagnetic interference and
audible noise, and to supplementary power losses. The
presence of ionic charges in the interelectrode region
modifies substantially the electrostatic field and,
consequently, has an important environmental impact
[l,
2J.
First attempts to solve the nonlinear partial differential equations describing the corona ionized fields
were based on the Deutsch assumption [3], which states
that the presence of the space charge affects only the
magnitude and not the direction of the electric field.
As expected, the corresponding methods of analysis
proposed {4, SJ yield ,acceptable results only at low
corona levels.
Recenifly, the Deutsch assumption has
been waived in the analysis of corona ionized fields
from HVDC transmission lines by applying iteratively
the flnl te element method (FEM). Most of the techniques proposed [6, 7J are constructed on the basis of
Kaptzov' s assumption [10) that the magnitude of the
electric field intensity at the coronating conductor
surface remains constant at its onset value regardless
of the level of the applied voltage. The validity of
these assumptions, as applied to the analysis of HVDC
transmission lines, has been questioned in literature
[11-14].
lt was claimed that they could have a
significant effect on the magnitude and spatial
distribution of the ionic space charge, electric field
and current density in. the close vicinity of these
lines [11-13). In fact, the electric field intensity
at the coronating conductor surfaces does depend on
the applied voltage [15, 16) and the field quantities
in the interelectrode region, at a given voltage, are
significantly influencf!'d by the field distribution
around the conductors in corona. On the other hand,
the experience of the authors of this paper shows that
finite-element techniques based on an imposed, constant conductor surface potential gradient yield results of a limited accuracy, which require, in general, a huge amount of computation.
The method proposed here employs a novel finiteelement based iterative algorithm, which uses isoparsmetric finite elements and a new updating criterion,
in order to achieve accurate values of the nodal potentials "with a substantially reduced computational
effort.
Following an iterative procedure, with an
approximate initial volume charge distribution corresponding to the corona onset electric field intensity,
it finally yields all the field quantities, including
the actual potential gradient variation around the
coronating conductor surface.
The method is illustrated for the case of the unipolar DC corona field of
a circular cylindrical conductor parallel to the
ground plane.
FEM FIELD ANALYSIS
The general equations governing the steady-state unipolar DC corona field represent the Gauss's law, the
relationship between the current density and field
intensity and the current continuity equation, respectively:
(1)
J • koE,
v.J - o,
(2)
(3)
where the electric field intensity can be derived from
a scalar potential ~.
E • -Vi'/>.
(4)
.:
is the permittivity of free space and k the ionic
0
mobility, which is assumed to be constant. From equa-
-.
0
(5)
~o
and from equations (2) - (4),
11-( pVc"l) • 0,
(6)
with the volume charge density p being an unknown
function of point. Equations (5) and (6) are equivalent to equations (l)-(4) and their simultaneous solution for i'fl and n provides the solution to the corona
problem.
The following boundary conditions are used:
- the potential on the coronating conductor is given,
(t) •
V;
(7)
- the potential on the ground plane
i'fl •
O;
(8)
since for transmission line configurations the
domain is unbounded, an artificial boundary is placed
sufficiently far sway from conductors, with the
potential on that boundary assumed to be approximately
equal to the corresponding electrostatic value,
a;•
""es·
(9)
The boundary conditions {7)-(9) are not sufficient for
obtaining a unique solution to the problem. As shown
below, in the proposed method only an initial distribution of charge within the interelectrode region,
which is sufficiently close to that corresponding to
the corona onset value of the conductor potential
gradient, is needed when starting the iterative procedure, without being necessary to specify any other
boundary condition.
The algorithm will modify this
initial distribution st each iteration, yielding
finally the solution to the problem.
The coupled partial differential equations (5) and (6)
are solved iteratively by implementing a standard
two-dimensional FEM technique [17, 18], with isoparametric quadratic elements.
A semi-automatic mesh generation routine was developed
and used for the problem region. Initially, a number
of equidistant nodes is chosen on the coronating conductor contour. Starting from one conductor node, the
subsequent nodes are placed approximately on the same
electrostatic field line. The second node is removed
from the first one by s distance equal to rAc' where r
is the conductor radius and Ac is the angle subtended
at the conductor centre by a chord joining two
consecutive conductor nodes.
Any subsequent node is
at a distance equal to 1r 1 Ac from the previous one,
where r 1 is the radial distance from the latter node
to the conductor centre and 1 is a parameter that
controls the mesh growth pattern. A value of .t•l. 5
was found to result, generally, in good element shapes
throughout the whole mesh, for problems relative to
transmission line configurations.
The initial distribution of charge density n at the
finite-element nodes is obtained by exploiting the way
in which the mesh is constructed. As mentioned above,
the mesh is generated by tracing, approximately, the
electrostatic field lines starting from the conductor
surface towards the outer boundary.
The length of
each field line is considered to be equal to the outer
radius of a coaxial cylinder system whose inner radius
is the conductor radius. With the potential gradient
on the inner conductor taken equal to the approximate
corona onset value given by the empirical Peek's
formula [19), the known analytical solution for the
coaxial cylinder geometry (20] is then applied to
obtain the initial space charge distribution along
that line. This calculation is repeated for all the
field lines in order to obtain the charge distribution
6-11
at all the nodes.
Such a procedure for detennining
the initial charge distribution is much simpler and
easier to implement than previous techniques [21, 9].
The proposed algorithm is based on the simultaneous
solution of equations (5) and (6), iteratively. Each
equation is solved separately by applying the FEM,
with the boundary conditions mentioned before.
The
algorithm begins by using the initial distribution of
space charge density P•
Since this distributi~n, is
not the one corresponding to the actual conf igur!\\.Jon,
the solution ~ from equation (5) and ~2 from equation
(6) will differ.
The charge distribution is then
updated at each node according to the following criterion:
Pnew • f(E le' E 2c•Pc•Pold}
+ g (11\1• 1112• v ,pold)
-float" such that all the field quantities will satisfy the problem equations with a desired accuracy at
all the nodes.
This "relaxation· procedure yielda,
finally, the actual variation of the electric field
intensity over the coronating conductor surface.
As
shown in the following section, the theoretical
results agree well with existing experimental dat~.
(10)
where pnew is the updated volume charge density at the
node, pold is the charge density at the same node used
in the previous iteration, E1c and E2 c are the electric field intensities determined from equations (5)
and (6), respectively, at the node on the conductor
surface lying on the same field line as the node considered, and Pc is the volume charge density at this
node on the conductor surface.
For the unipolar DC
field case, the functions f and g are chosen to be
4>= 0
GROUND PLANE
(11)
Fig. 1
Single conductor above ground
and
(12)
in which V is the applied voltage, ~av is the arithmetic average of ~ and 4>2, Eav is the arithmetic
c
average of E1 c and E2c, and a, b, c, d are constants
determined so that the convergence of the iterative
process is ss rapid as possible. The updating criterion (10)-(12) reflects the fact that the space charge
distribution is affected not by the conductor surface
potential gradient alone, or by the local potential
alone, but it is actually affected by their combined
effect. This is an improvellll!nt over previous updating
criteria [7, 9), where only the effect of the corona
onset potential gradient alone, or that of the local
potential alone, were taken into account.
Following the first iteration, which uses the initial
distribution, p, the FEM solution of equations (5) and
(6) is obtained again by using the new distribution
pnew. The iterative process is performed by continuously inserting into the updating criterion (10)-(12)
the most recent values of all the corresponding quantities.
In order to ensure an accurate solution for
all the field quantities, this process is continued
until the following conditions are satisfied si111Ultaneously at all the nodes except those lying on the
artificial boundary:
1¢1. I pnew
il?av1 < 'tl<l>av•
(13)
_ poldl < "£2Pnew,
(14)
1E1c - Eave
I
< 't3Eavc'
TEST EXAMPLE AND DISCUSSIONS
The proposed method was applied to the solution of the
unipolar DC ionized field problem in a cylinder-toplane configuration.
Figure 1 presents the system
geometry, with r-0.0025m and H•2m, hence a ratio
H/r•800, which is typical for a practical DC transmission line. The applied voltage is V•200kV, Due to
the symmetry of the configuration chosen for illustration, the normal derivative of the potential over
the conductor vertical plane is
M. O.
an
The finite-element region and the generated mesh are
shown in Figure 2, where the number of elements is 333
and the total number of nodes is 722.
(15)
in which 'tl, "12 and 1'3 are Slllllll deviations specified
in terms of the desired accuracy.
It should be noted that the updating formulas (10)(12) do not contain an imposed value for the electric
field intensity on the corona.ting conductor surface,
therefore the presented algorithm does not rely on
Kaptzov's assumption. Only the corona onset value of
the conductor potential gradient is necessary for
determining the initial space charge distribution,
which is used for the first iteration.
In all the
subsequent iterations the field intensity on the surface of the coronating conductors is let free to
(16)
Fig. 2
Finite-element mesh
For calculating the initial volume charge distribution
a value of 50.B4kV/cm was used for the corona onset
potential gradient on the cylinder surface {19]. To
achieve a good rate of convergence for the iterative
algorithm, the numerical values for the constants in
formulas (10)-(12) were chosen as follows:
a•l,
b•O.fJ, c•0.01, d•0.21.
Highly accurate results were
obtained by imposing the following relative errors in
equations (13)-(15): , 1 •0.025, •2•0.01, , 3 •0.015.
0.4
.....,
E
'
(.)
::l 0.2
cf'
0.00
2
flt')
E
Xg/H
a.
~barge
'
u
density
::l 2.5
Q...u
120.0
0.00
30
60
90
120
150
180
150
180
90
a. Charge density
E
'.x:
>
w0t40.0
e
Corona onset value [191
5o.
u
0.00
2
3
X /H
9
'>.x:
u
45.0
w
b. Electric field intensity
40.00
30
60
6.0
90
eo
120
b. Electric field intensity
4.0
(\I
3.0
E
......
<t
3-
..,
OI
(\I
2.0
E
......
2.0
<t
E
u
..,
0.00
1.0
2
Xg/H
0.00
c. Current density
Fig. 3
Distributions over the ground plane for
V•200kV:
Computed results by presented method
--- Computed results with Kaptzov' s as sumpt1on
' ' ' Experimental values {22}
30
60
90
eo
120
150
180
c. Current density
Fig. 4
Distributions around the conductor periphery
for V•200kV
Only eighteen iterations were necessary to achieve
this level of accuracy, which is far better than for
previous finite-element based algorithms.
1ndeed, in
reference {9 J it was reported that seventy iterations
were needed to obtain a 5% accuracy in the potential
values, and in reference [8J the accuracy of the electric field at the conductor surface was reported to be
about 7%.
The variation of charge density, electric field intensity and current density over the ground plane and
around the conductor periphery is presented in Figures
3 and 4, respectively. In Figures 3b and 3c computed
results by the proposed method are compared with those
obtained by using Kaptzov' s assumption, for the same
mesh and the same number of iterations, and with
experimental data [22 J. The results obtained by the
method presented in this paper are in better agreement
with experimental values.
Furthemore, as shown in
Figure 4b, the potential gradient is not constant over
the coronating conductor periphery, presenting a variation of about 4%, with the minimum value being at the
bottom of the conductor. On the other hand, Figures
4b and 5 show that the average value of the electric
field intensity at the surface of the conductor in
corona drops by about 9% from its onset value, which
is in agreement with available experimental observations [15, 16].
52.0
Corona onset value C191
50.0
E
(.)
'~
-,."
~
Q)
c:
~
-0
15
2.5
0.01---+1--+----+---+--t----1
I
Q...
I
:ic:
7
10
13
16
Number of iterations
19
Q...
-5.0
Fig. 7
Maximum percentage deviation in nodal charge
densities versus the number of iterations
CONCLUSIONS
A new method for the finite-element analysis of uni
polar DC corona ionized fields has been developed.
The method does not rely on simplifying assumptions
used in methods proposed so far. It employs a simple
procedure for determining the initial charge distribution within the interelectrode region and an iterative
algorithm based on a new updating criterion.
The
"'relaxation" of the potential gradient on the conduc
tor surface from its corona onset value yields a relatively rapid convergence of the iterative process and
also a distribution of the coronating conductor potential gradient in good agreement with experimental
data. Comparison with previous techniques shows that
the method presented in this paper is more accurate
and computationally much more efficient. The authors
intend to extend this general uiethod to the more practical case of bundled conductors.
ACKNOWLEDGEMENTS
uP46.o
This work was supported by a grant from Natural
Sciences and Engineering Research Council (NSERC} of
Canada.
Financial support of NSERC is gratefully
acknowledged.
4
7
IO
13
16
19
Number of iterations
REFERENCES
[l]
Fig. 5
Average electric field around the conductor
surface in terms of the number of iterations
The convergence of the proposed iterative algorithm is
illustrated in Figures 5, 6 and 7.
~ 7.5
~
>
0
'9 5.0
............
J 2.5
I
i9
5.0
~
~
0.01
Fig. 6
4
10
13
Number of iterations
7
16
19
Maximum percentage deviation in nodal potentials versus the number of iterations
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(19]
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