A New Method for Solving Ionized Fields
Associated with HVDC Transmission Lines
Xin Li
M.R.Raghuveer I.M.R.Ciric
Dept. of Electrical & Computer Engineering, University of Manitoba
Winnipeg, Manitoba, R3T 2N2
CANADA
Abstract
In this paper, a new method for solving the ionized fields
associated with HVDC transmission lines is presented.
The boundary value problem is transformed into an optimization problem by using the finite element method.
Numerical tests on a coaxial cylindrical geometry show
that the new method is very efficient. This method has
also been applied to a practical line-plane geometry.
Introduction
The ionized field associated with unipolar HVDC transmission lines is mainly characterized by two quantities:
the electric potential u and the space charge density e ,
which are governed by Poisson's equation and the current continuity equation subject to appropriate boundary conditions. This problem is nonlinear and extremely
difficult to solve.
Townsend [I] was the first to carry out ionized field analysis analytically on a coaxial cylindrical geometry. Later,
Deutsch [2] and Popkov [3] obtained approximate analytical solutions to the ionized field of a line-plane geometry. In the case of a line-plane geometry or other
practical transmission line systems, a rigorous solution
can only be obtained by employing numerical, iterative
methods.
Sarma et. al [4--6] was the first to develop an iterative
method based on Deutsch's assumption, in which the
problem was reduced to one dimension. This method has
been applied to several practical cases, but it is basically valid only for a unipolar de system in the absence
of wind.
In 1979, Janischewskyj and Gela [7] presented a method which does not employ Deutsch's assumption. Their
basic idea is: (1) to assume an initial distribution for
the charge density (J , (2) obtain a solution Ua of potential by solving Poisson's equation, (3) obtain a solution
Ub by solving the current continuity equation, and (4)
update the charge density by using the formula
fluw = eo1d + b(ua - uh) . Steps (2)-( 4) are repeated until
convergence is obtained. Unfortunately, no rigorous
theoretical guidelines have been proposed for the
choice of the error function O(u
0
-
ub) .
In the last two decades, new techniques have been introduced in which u is solved from Poisson's equation
by using the charge simulation method (9, 11] or the
boundary element method [ 10, 12], and (!,,,.,, is directly solved from the current continuity equation by using
the weighted residual method [11,12], the upwind finite
element method [9] or by reducing it to a set of onedimensional differential equations along flux lines [I OJ.
Although different methods are used for solving the two
differential equations, a common iterative philosophy is
employed, i.e, ( 1) to assume an initial approximation to
(! , (2) solve Poisson's equation for u, and (3) solve
the current continuity equation for
e
in order to update
it. Steps (2) and (3) are repeated until convergence is
obtained.
The iterative methods hitherto employed do not use standard iterative techniques. In this paper, a new method is
presented based on a completely different approach.
Principle of New Method
The basic idea of the new method is to transform the
boundary value problem of an ionized field into an optimization problem; the field distribution is then obtained
by minimizing an objective function. In this method, the
finite element method is utilized for the discretization of
the boundary value problem.
Let us define two vectors
where u; and (J; ( i=l ,2, ... , n) denote the ith nodal
values of u and (! respectively, and n is the number
of the nodes contained in the finite element mesh.
Poisson's equation, within the domain, can be transformed equivalently to the following linear equation system
.li(q, u)
=
0
l = l,., µ = 1, 2, ... , n;
where I
= lw µ = I, 2, ... , n; represent the interior nodes
of the finite element mesh.
The current continuity equation, in which the effect of
wind may be considered, can be transformed to the following nonlinear equation system
t/J1(Q, u)
=0
I= 1,2, ... ,n
Thus the ionized field can be solved by finding
q' and u' which satisfy
The domain is subdivided into a mesh with 1680 triangular elements and 880 nodes. The minimization of the
formulated objective function is carried out by using
Newton's method [ 14, 15] which has the fastest rate of
convergence.
The numerical solutions for
(!
and
u ,
shown in
Figs.2 and 3 respectively, correspond to the case when
the potentials of the electrodes and the field strength at
the inner conductor surface are taken as the boundary
conditions.
2
.fi(q,u)=O
(!/t: 0 (MV /m )
l=lw µ= 1,2, ... ,n,
2001.-~~~~~~~~~~---,
tp 1(q,u)=0
/=1,2, ... ,n
subject to specified boundary conditions
analytical solution
150
This problem is equivalent to the following unconstrained optimization problem:
+ numerical solution
100
Minimize
n;
n
F(q, u)
1/Jf(q, u) + L~(q, u) + Fb(q, u)
=L
1-1
µ-I
50
0-h,-,--,--,,,.,-,-,-,.-,--,-,-,-,-,-,-,-.,-"
I
where F(q, u) is called the objective function. The term
Fb(q, u) corresponds to the boundary conditions.
3
5
7
9
II 13 15 17 19
T(mm)
and any form of boundary condition.
Fig.2. Analytical and numerical solutions
for(! along a radius. (Field strength
at inner conductor is specified).
The problem can now be solved by using standard iterative methods such as Newton's method.
u (kV)
Fb(q, u) can be formulated for any shape of boundary
Numerical Tests
The new method is applied to a coaxial cylindrical model shown in Fig. I, for which the analytical solution is
available [7].
0
V0
(! 0
-
20
numerical solution
15
01+.-.,......,.-,-..,.....,""'"T"-.--r--r-r-..,.....,-,-..,-.,......,.-,-..,......
l
= 17.5 kV
= 1.03994 mC/m3
u=O
Fig. I. Coaxial cylindrical geometry,
V0 - onset voltage,
(!
space charge density at T =
0
analytical solution
5
R =20 mm
T0 =I mm
V = 25kV
u= V
25
IO
radius R
radius To
30-,....-~~~~~~~~~~--,
3
5
7
9 11 13 15 17 19
T(mm)
Fig.3. Analytical and numerical solutions
for u along a radius. (Field strength
at inner conductor is specified).
The errors in the numerical solutions for
T0 •
and u are
shown in Fig. 4. The performance of the iterative algorithm is shown in Table I .
(!
op(%)
When the space charge density at the inner conductor
surface is specified as a boundary condition instead of
the field strength, the performance of Newton's iterative method is similar. The errors in the numerical solutions for Q and u , in this case, are shown in Fig.5.
3
0,.(%) 2.5
2
1.5
The above results show that the proposed algorithm is
very efficient. It takes only 4 iterations and I '25" of
CPU time for convergence. The calculated results
show a good agreement with the exact values in a coaxial cylindrical geometry.
0,.(%)
0.5
- --
0 I
3
5
7
9
11 13 15 17 19
r(mm)
Fig.4. Percentage errors in numerical solutions
for (J and u . (Field strength at inner
conductor is specified).
Table I. Performance of the i1erative algorithm.
Initial value q(01 = [0.5, .. ., 0.5Y(Jo/fo
In addition, the proposed method is applied to a lineplane geometry (Fig.6) with absence of wind. The potentials at the electrodes and the space charge density at
the line conductor surface are taken as the boundary conditions. A rectangle with height= 2H (m) and width= 50
(m) is taken as the solution domain. The boundary conditions at locations which do not coincide with the conductor surface and ground plane are determined from the associated space charge free field [9]. The domain is divided into a mesh with 2560 triangular elements and
1320 nodes.
k(No. of iterations)
F( q(k>, u(k')
0
6.70 x 102
r0 =
I
7.31 x 10°
1.67 x 10- 2
V = 275 kV
(Jo= 88.54 µC/m 3
1.91 x 10-
H= 6.144 m
2
4
3
4
1.84 x 10-4
5
1.84 x 10-4
x (m)
Fig.6. Line-plane geometry,
SPARC 10 workstation: CPU time : l '25"
op(%)
3 mm
Qo - space charge density at r = ro.
Eg(kV/m)
3
40~~~~~~~~~~~~~~~~~
ionized field
space charge free field
0.(%)2.5
30
2
1.5
20
o"(o/o)
o.(%)
~
0.5
0
I
3
5
7
9
11 13 15 17 19
r(mm)
Fig.5. Percentage errors in numerical solutions
for (} and u . (Space charge density at inner
conductor is specified).
10
0+-~--,r--..,...--r-~~~~~~~~~..,...-~
-15
-IO
-5
0
5
IO
15
x (m)
Fig.7. Electric field profile at ground level.
The calculated field profile at ground level is shown in
Fig.7. The performance of the iterative algorithm is
shown in Table 2. In this case also the iteration converges fast and stably.
[2] W.Deutsch, Ann. Physik, Vol.5, pp. 589-613, 1933.
[3) V.I.Popkov, Elektrichestvo, No.I, pp.33-48, 1949.
Table 2. Performance of the iterative algorithm.
Initial value q<0> = (0.4, ..., 0.4Y(>0 /Eo
k(No. of iterations)
F( q(*>, uCk')
x
102
0
4.51
2
f.33 X J0 1
4
x 10- 1
2.37 x 10- 2
2.19 x 10-)
1.39 x 10-4
1.27 x 10-4
5.57
6
8
10
11
SPARC 5 workstation: CPU time: 4'43"
Conclusions
A new method is proposed for the solution of ionized
fields by transforming the boundary value problem to an
equivalent optimization problem. Several standard iterative methods (such as Newton's method used in this paper) developed on firm mathematical bases can be
applied to solve the transformed problem.
Numerical tests show that the proposed method is very
effective in terms of speed and accuracy. The calculated
results show a good agreement with the exact solutions
in the case of the coaxial cylindrical geometry. The
method has also been successfully applied to a lineplane geometry.
This method is generally applicable for solving ionized
fields. The form or structure of the objective function
does not depend on the domain geometry. The boundary
conditions can be easily treated. The effect of wind can
be easily included into the objective function.
In addition, this method can also be extended to the bipolar cases. This problem is under formulation and the results will be published in the near future.
[4) M.P.Sarama and W.Janischewskyj, IEEE Trans.,
Vol.Pas-88, pp.718-731, 1969.
[5] M.P.Sarma and W.Janischewskyi, IEEE Trans., Vol.
PAS-88, pp.1476--1491, 1969.
[6] M.P.Sarma, IEEE Trans. on Elec. Insul., Vol. EI-17,
No.2, pp.125-130, 1982.
[7] W.Janischewskyj and Gela, IEEE Trans., Vol.
PAS-98, pp. I000-1011, 1979.
(8) M.Abdel-salam, M.Farghally and S.Abdel-Sattar,
IEEE, Trans. on Electrical Insulation, Vol.El-18,
No.2, 1983,pp.110-119.
(9) T.Takuma, T.Ikeda and T.Kawamoto, IEEE Trans.,
Vol.PAS-I 00, pp.4802-4810, 1981.
[IO] E.Kuffel, A.Dzierzynski and J.Poltz, "Final Report
on: Development of Numerical Method for
Analysis of Bipolar Corona on HVdc System",
The University of Manitoba, January 28, 1987.
[ 11] B.L.Qin, J .N .Sheng, Z. Yan and Gel a, IEEE Trans.
on Power Delivery, Vol.2, No. I, pp.368-376, 1988. [ 12] Ming Yu, Ph.d Dissertation, The University of
Manitoba, 1993.
[ 13] O.C.Zienkiewicz and K.Morgan, "Finite Elements
And Approximation", John Wiley & Sons, Inc.,
New York, 1983.
[ 14) S.L.S.Jacoby, '1terative Methods for Nonlinear
Optimization Problems", Prentice-Hall, Inc.,
Englewood Cliffs, New Jersey, 1972.
[ 15] J.Kowallk and M.R.Osbome, "Methods for
Unconstrained Optimization Problems", American
Elsevier Publishing Company, Inc., New York,
1986.
Acknowledgement
References
[I] J.S.Townsend, Phil. Mag., Vol. 28, p.83, 1914.
Financial support from NSERC, Canada and Manitoba
Hydro through the Manitoba Hydro Research and Development Committee is gratefully acknowledged.