Download Abstract. This paper deals with the heat generated by ohmic

NUMERICAL SIMULATION OF COUPLED FIELDS
IN LARGE POWER CABLES
Daniela CÂRSTEA1, Ion CÂRSTEA2
Industrial Group CFR, Craiova. Str. Brâncuşi nr. 9, ROMANIA
2
University of Craiova, Craiova. Str. Decebal nr. 5, ROMANIA
Abstract. This paper deals with the heat generated by ohmic losses in different parts of the electromagnetic
devices. The problem is described by a coupled thermal-electric set of equations. The coupling between the two
fields is the thermal effect of the electrical current or a material property as the electrical conductivity. A
computation algorithm is presented for coupled problems in two dimensions.
A numerical algorithm based on the finite element method (FEM) is presented describing the solution of
two-dimensional systems. In our example we consider only steady-state regime although many transient
regimes appear in the behavior of the electromagnetic devices.
A CAD product was developed for coupled problems. The algorithm has been implemented in our CAD
product and several examples from electrical engineering were solved with good results for the engineers.
Keywords: Finite element method; CAD; coupled problems.
1 Introduction
In some electromagnetic devices with strong
current densities, it is important to compute the
currents and the temperature caused by Joule-Lenz
effects. There are a lot of industrial applications in
this area.
Many works in the professional literature present
coupled models for the electromagnetic devices and
this work is toward this direction with emphasis on
the development of efficient algorithms in numerical
computation of the coupled models.
The examples are many. It is the case the
contacts of the HV switcher equipment submitted to
high current densities. The contacts of the circuit
breakers at high temperature generated by high
currents can melt the contacts. Another case is the
dielectric heating in HV cables with polluted or
imperfect
insulation
layers.
An
accurate
computation of the temperature distribution is useful
for designers of these equipments.
In our work we use as target example a highvoltage (HV) and large-power cable in direct
current. The problems that appear at this object are
complex and depend on the cable regime. At a step
voltage the electric field has a variation in time and
the solution at this problem involves an iterative
approach. After the stabilization of the electric
transient regime, the cable is heating. This is a
transient regime and the heating constant is more
large than the electric constants.
The problem is complex because of the
imperfect insulation, that is the solid dielectric has
impurities that destroy the high resistivity of these.
These situations can be solved numerically and
these aspects are presented in this work.
2 A 2D model for coupled electric and
thermal fields
A coupled model for electrical current and its
thermal
effect
involves
development
of
mathematical models for the two distinct physical
fields but the solving of these models must be done
simultaneously. Our models are based on Maxwell
equations and heat conduction [1].
The static field distribution can be modeled by
the following equations:
  E  0;
E  J
with:  - the material resistivity, E - the electric
strength and J – the current density.
A 2D-field model was developed for a resistive
distribution of the electric field. An electric vector
potential P was introduced by the relation [2, 4]:
J   P
Laplace’s equation describes
distribution (for anisotropic materials):
the
field

x
(x
P
x
)

y
( y
P
y
)0
(1)
A finite element model for the above Laplace's
equation can be obtained by Galerkin's procedure.
With this model a resistive distribution of the
electric field is computed and the heat source (JouleLenz’s effect) can be calculated as being (for
isotropic materials):
q  J
2
P 2
P 2
2
2
  ( J x  J y )   [( )  (  ) ] (2)
x
y
Mathematical model for the thermal field is the
conduction equation [1]:

x
(kx
T
x
)+

y
(ky
T
y
)+ q =  c
T
t
(3)
with: T (x, y, t) - temperature in the point with coordinates (x, y) at the time t; kx , ky – thermal
conductivities;  -specific mass; c – specific
heating; q – heating source.
It is obviously that there is a natural coupling
between electrical and thermal fields. Thus, the
resistivity in equation (1) is a function of T, and the
heating source q in (2) depend on J. Numerical
models for the two field problems can be obtained
by the finite element method. An iterative procedure
was used for the temperature distribution.
The algorithm in pseudo-code has the following
structure:
1. Choose the initial value of the temperature
2. Repeat
{Computations for electrical field}
 Compute the resistivity ρ
 Solve the numerical model for electric potential
{Computations for thermal field}
 Compute the heating source q by (2)
 Solve the numerical model for the temperature
Until the convergence_test is TRUE
The convergence test is the final time of the
physical process but in the internal loop of the cycle
repeat-until we have an iterative process because the
electric conductivity depends highly on temperature
T and electric field E. In a first approximation we
neglect the dependency of the resistivity by the
electric field strength.
In insulation the resistivity decreases with the
temperature so that we can use the relation:
 (T )   (T0 )[1   (T  T0 )]
ρ (T0) is the resistance at ambient temperature T0
(usually +20 0C) and ρ (T) is the resistivity at the
instantaneous temperature T.
3. The case of a large-power cable
Our example is a high-voltage direct-current
(HVDC) cable with two insulation layers. The
leakage current in dielectric is caused by the finite
resistivity of the dielectric insulation.
At the application of a high voltage the field has
a capacitive distribution initially. This distribution is
for a short time so that it is not interset for the
temperature distribution. Finally the field has a
resistive distribution. Between these limits there is
an intermediate field that can be computed by an
iterative procedure. We limit our discussion at the
final distribution of the electric field (resistive case).
Generally speaking there is no perfect dielectric
insulation so that a leakage current exists. Ohmic
losses cause the dielectric heating. A parallel-plane
model can be used to compute the electric and
thermal fields.
Fig.1 –Analysis domain and mesh
The physical properties of this example (relative
permitivity, electric resistivity, and thermal
conductivity) are given below for a cable with the
following geometrical dimensions: r1=15 [mm];
r2=30 [mm]
Electrical properties are:
 =1.1014 [. m]; U=10 000 [V]
Thermal properties are:
 kx=ky=0.15 [W/m. K]
 convective coefficient h=12 [W/m.K]; medium
temperature: 40 0C.
 c=1800 [J/kg.K ];γ=1300 [kg/m3]
The time domain for the transient regime is
1800[s]. The analysis domain is the insulation space.
The symmetry of the problem can reduce the analysis
domain to a quarter (Figure 1).
In Figure 2 the map of the density current and
with Pcond - the ohmic losses per cable meter in the
inner conductor as Joule-Lenz’s effect.
Thus, Neumann’s condition is:
T
p
n C1
with C1 – the interface of the cable conductor and
insulation.
At the interface insulation-environment we
consider a convective condition by the form:
T
 h(T  T )
n C 2
with h – the convective coefficient, T∞ - the ambient
temperature and C2 – the boundary of the cable and
the external medium.
In Figure 3 the map and the vectors of the heat
transfer are plotted at the final time of the time
domain.
Fig.2 – Map and vectors of the current
density
vectors of the current
densities are plotted. The
highest current density is near the conductor surface.
Initially, the field strength has the highest
strength near the conductor surface but this
diustribution can be changed by the thermal field. It
is possible to have the highest field strength at the
boundary of insulation with external medium.
The second stage of the numerical algorithm is
the simulation of the thermal field. The heat source
is the thermal effect of the current in the dielectric
insulation and the load current of the cable. It is
obviously that the ohmic losses in the cable
conductor is the most important heat source. In our
model we consider that there is a constant heat flux
on the interface conductor-insulation. The source of
this flux is the Joule-Lenz’s effect of the load in the
cable. The mathematical model for the heat transfer
is the conduction equation (3). The boundary
conditions are: Neuman’s condition at the interface
conductor-insulation, and convective condition at
the boundary insulation-environment.
The Fig.4
Neumann's
condition
can and
be computed
–Analysis
domain
mesh by the
conductor losses in the case the cable was loaded
before switching of the step voltage, that is the
current in the cable has been raised long before and
the temperature distribution in the cable is stable. In
this case the value of the heat flux is computed with
the relation:
p
Pcond
2r0
Fig.3 –Map and vectors of the heat flux
The variation of the temperatures versus time at
the interfaces of the insulation with the conductor
and environment are plotted in Figure 4. The highest
temperature is near the conductor and this fact leads
to important changes in the electric field distribution
in the insulation.
The solution plotted in the Figure 3 is obtained in
hypothesis that the resistivity of the insulation is
constant on the time domain and space. Practically,
the heat sources depends on the temperature because
the electrical properties (resistivity of insulation and
conductor) vary with the temperature and space
variables. The highest temperature is near the
conductor and the resistivity of the insulation
decreases in this zone much more than at zone near
the environment.The field distribution is a
hyperbolic function if there is no temperature drop
in insulation. With no temperature drop in
insulation, the maximum value of the electric field
T (K)
80
70
60
50
40
30
20
10
0
0
300
600
900
1200
1500
1800
Time (s)
Fig. 4 - Temperatures vs. time at conductor (green) and environment (red) surfaces
strength is near the conductor.For large loads that
lead to high temperatures in conductor, the field
near the environment may become higher than the
highest field strength near the conductor. An
analytical relation is proposed in the work [5].
In figure 4 the variations in time of the
temperature are plotted at two points of interest : the
conductor surface and environment surface.
4 Conclusions
In our work we presented some examples that
illustrate the real coupling between the electric and
thermal fields. The mathematical models are solved
using a finite element formulation. A CAD product
for 2D or axi-symmetric problems was developed
by the authors [1,4]. The validation of the
simulation for our examples was achieved by
comparing the results obtained from our program
with results from open literature or obtained with
similar software as program Quickfield [6].
In the models used in the work we have both
explicit and implicit couplage between the two
phenomena. The couplage is achieved both by the
right term of the conduction equation (heat source)
and electric conductivity of the material (in Laplace
equation of the electric field). At each time step, the
physical properties are computed using the
temperature distribution computed by a distributedparameter model. In another variant of our software
product we use a lumped-parameter model for
temperature distribution in the case of HVDC cables.
References:
[1]. Cârstea, D. Numerical simulation of magnetothermal processes using automatic computation
systems. Ph.D. Thesis, 1999. Romania.
[2]. Bastos, J.P.A., Sadowski, N., Carlson, R. “A
modelling approach of a coupled problem
between electrical current and its thermal
effects”. In: IEEE Transactions on Magnetics,
vol.26, No.2, March 1990.
[3]. Cârstea, D., Cârstea. I. “A CAD product for
inverse thermal problems”. AMSE Press,
Modelling, Measurement & Control, B, 2002Vol.71, N0 3,4. ISSN 0761-2516, pg. 61-70.
[4]. Cârstea, D., Cârstea. I., CAD in electrical
engineering. The finite element method.
Publisher: Sitech, Craiova, 2000. Romania.
[5]. Jeroense, M.J.P., Morshuis, P.H.F. Electric
Fields in HVDC Paper-Insulated Cables. In:
IEEE Transactions on Dielectrics and Electrical
Insulation. Vol.5, No.2, April 1998.
[6]. Quickfield package. Tera Analysis Co.