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C I R E D C I R E D C I R E D C I R E D 20th International Conference on Electricity Distribution Prague, 8-11 June 2009
Paper 0209
CIRED2009 Session 1 Paper No 0209
THERMAL RESPONSE OF A THREE CORE BELTED PILC CABLE UNDER
VARYING LOAD CONDITIONS
Peter A WALLACE Donald M HEPBURN Chengke ZHOU
Glasgow Caledonian University – UK Glasgow Caledonian University – UK Glasgow Caledonian University – UK
p.wallace@gcal.ac.uk d.m.hepburn@gcal.ac.uk c.zhou@gcal.ac.uk
Mohamed ALSHARIF
Glasgow Caledonian University – UK
mohamed.alsharif@gcal.ac.uk
ABSTRACT
The most important indicator of the health of electrical
plant items is the condition of their insulation. In the case of
underground cables an important issue is the operating
temperature and indeed the thermal history of the cable.
There are several factors which will determine the thermal
behaviour of a given cable installation. These include the
assumed ampacity, the cable construction and
circumstances of installation, the thermal properties of the
surrounding soil and the ambient temperature. The work
presented in this paper involves the use of COMSOL
multiphysics finite element software to develop an
integrated electrical, thermal and mechanical model of
buried single or multiphase cables that simulates the
behaviour occasioned by a varying load. The principal heat
source in the problem is the Joule heat dissipated in the
conductor(s). The transfer of this heat to the surroundings
is governed by the geometry and material properties of the
conductor, insulation, screening, sheathing and trench fill
material as well as the ambient conditions. The thermal and
electrical systems are coupled via the temperature
dependence of the resistivities of the conductor and sheath
materials. The input data to the model is provided in the
form of time series describing the variations in load over a
24-hour cycle and the output takes the form of the thermal
response of the cable for given installation and ambient
conditions.
INTRODUCTION
From on-line investigation of partial discharge (PD) activity
in distribution cable networks previously reported [1, 2], it
was found that the PD activity level in any given cable
changed over the course of the day. The authors attempted
to relate the level of current load in the cable to the PD
activity. Figure 1 shows a comparison of the total PD
energy detected in a particular cable over a 24-hour period
and the load current, which was being transported through
the cable in this period. As can be seen from Figure 1, there
is no direct relationship between the factors. It was
considered that an inverse relationship might exist, i.e. as
load increased PD activity decreased. Data analysis also
pointed to the possibility of a time shift affecting
correlation.
Figure1: PD activity and load current in cable over a 24-
hour period (normalised values).
After considering the factors that would influence the PD
activity in cables the authors started a programme of
research to investigate, using finite element analysis
techniques, the changes in cable properties that would result
from the changes in load current. This paper presents results
of initial simulations to simulate changes in the temperature
of the various components of the cable under the changes in
load from the cable shown in Figure 1. Simulation of
electrical stress in a gas filled void introduced into cable
insulation is also undertaken.
COMSOL SIMULATIONS
Overview
The model system considered in this work is a three-core
paper insulated lead covered (PILC) cable of belted
construction. The cable dimensions and characteristics are
taken from data sheets of old cables. The conductors are
made of copper and each is 70 mm2, the overall diameter of
the cable is 45 mm.
Finite element simulations of its electrostatic and thermal
CIRED2009 Session 1 Paper No 0209
characteristics have been performed using COMSOL
MultiphysicsTM
[3]. The salient advantage of this software
package is its ability to treat simultaneously several coupled
physics phemomena, e.g. the coupled electrical, thermal and
mechanical dynamics of a power cable under load.
While the inherent multiphysics capabilities of the platform
will be utilised in future work the results presented here for
the electrostatic and thermal domains are in fact the product
of separate calculations. The reason for this is that the
timescales of interest are very different. The thermal
simulation needs to take into account the diurnal load
variation and, indeed, the response times of a buried cable
may extend to hundreds of hours. The electrostatic
calculation, in contrast, is concerned with changes that
occur over a single a.c. cycle of 20 ms.
Electrostatic Model
COMSOL’s electrostatic application mode solves Poisson’s
equation ,
( ) (1) 0 ρ=∇εε⋅∇− Vr
and obtains the electric field E from the gradient of the
potential
(2) V−∇=E
The PDE (1) is solved subject to the following boundary
conditions :
Sheath : V = 0
Conductors : V(t) = V0cos(ωt + 2nπ/3) n = 0,1,2
on the 2 dimensional domain represented by the cable cross-
section (Figure 2). V0 is set at 11 kV, εr is taken as 6 for the
insulation and filler and space charge effects are presently
neglected.
Figure 2 shows the electric field distribution within the
cable at the point in the a.c. cycle where the potential of the
right hand conductor is at its maximum value. The electric
field distribution is complicated and continuously varying.
In most locations, at a given point in time, within the cable
the electric field is more or less normal to the direction in
which the paper insulation is laid, which is to say the
direction in which the strength of the insulation is greatest.
However there are locations where the electric field has a
significant component tangential to the paper direction.
Figure 2 shows the field distribution in an ideal, flawless,
cable. The presence of a void within the insulation will
result in a local increase in the electric stress. To
demonstrate this effect a 0.5 mm diameter void was
introduced into the insulation belt of the right hand
conductor. The void is placed on the horizontal symmetry
axis of the cable, within the insulation belt, just to the right
of the cable centre.
Figure 2: The electric field distribution within the cable at
the point in the a.c. cycle where the potential of the right
hand conductor is at its maximum value.
Figure 3 shows the horizontal component of the electric
field, Ex, plotted along the horizontal symmetry axis. The
curve represents Ex at the point during the a.c. cycle when
the right hand conductor is at maximum (negative) potential.
This axis runs between the two left hand conductors,
through the void and then through the right hand conductor.
The sharp increase in electric stress occasioned by the
presence of the void is clearly seen in the figure (Fig. 3) at
x = 0.03 m.
Figure 3: The horizontal component of the electric field
plotted along the horizontal symmetry axis of the cable at
the point when the conductor located at x = 0.01 m is at
maximum (negative) potential. Note the presence of the void
at a position of approximately x = 0.003 m
Thermal Transport Model
The COMSOL Conduction application mode solves the
following PDE :
( ) (3) QTkt
TC p =∇⋅∇+
∂
∂ρ
CIRED2009 Session 1 Paper No 0209
where ρ is the density, Cp is the specific heat capacity, k the
thermal conductivity and Q is the heat source term. The
cable is modelled as being buried in a 0.6 m square, sand
filled, trench. The cable centre is at a depth of 0.3 m and the
surface of the trench is flush with the surrounding soil
surface. Equation (3) is solved on a 2 dimensional domain
which comprises the cable cross-section of Figure 2 set in a
6 m diameter semicircle of surrounding soil and trench-fill.
The boundary of the domain of solution comprises (a) the
soil surface, (b) the 3 m distant semicircular boundary
within the native soil. Both parts of the boundary are set to
288 K (15 °C) following [4]. The thermal conductivities of
the soil and sand are set to 0.833 Wm-1K
-1 and 0.2 Wm
-1K
-1
respectively [4]. The thermal conductivity of the cable
insulation and filler is assumed to be 0.16 Wm-1K
-1.
The two heat source terms considered are the Ohmic loss
due to the current flowing in the conductor, Qc, and the loss
in the sheath due to the induction of eddy currents, Qs,. In
both cases the heat source terms are calculated by dividing
the total loss in the conductor (sheath) by the cross-sectional
area of the conductor (sheath). Hence the term Qc is given
by
(4) 2
cc
A
RIQ =
where R is the a.c. resistance of the conductor per unit
length
( ) (5) 1 ps yyRR ++′=
and R′(T) is the d.c. resistance per unit length. ys and yp are
the skin and proximity effects respectively and are
calculated according to [5]. Qs the sheath loss is defined in
terms of Qc.
(6) s
cs
A
QQ λ=
where the loss factor λ is calculated in terms of R, Rs (the
sheath resistance/m) and the cable geometry according to
the prescription of [5].
Response to a static load
When this system is subject to a load step of 100 amperes
the response is as shown in Figure 4. The upper curve
represents the temperature at a point within the conductor
while the lower (cooler) curve represents the temperature of
the surface of the sheath. The response curves exhibit a fast
and a slow component. The fast component may be
interpreted as the relatively rapid heating of the cable itself
while the slower component describes the effect of the
gradual heating of the trench fill material surrounding the
cable. Even after two weeks the system has not quite
reached thermal equilibrium, although a steady temperature
difference of approximately 10 K has bee n established
between the conductor cores and the sheath.
Figure 4: Thermal response of the installed cable to a load
step input impressed at time t = 0 assuming an ambient
temperature of 288 K. The upper curve relates to the
temperature of a conductor core, the lower curve refers to
the temperture of the surface of the sheath.
Response to diurnal variation in load
The simulation was re-run using as input data a time series
representing a real load measured over a one week period
[1]. The diurnal variation in load produces a corresponding
variation in cable operating temperature of 15-18 K. As
before the heating load was impressed into a pre-existing
state of thermal equilibrium where the ambient temperature
was a uniform 288 K. By the end of the week the cable
operating temperature had climbed to a maximum value of
326 K (53 °C). In contrast to the static load case the
temperature difference between cable centre and sheath
varies between 2 and 10 K depending on the load
conditions.
Figure 5 shows the temperature variation occurring across
the section of the cable at the point of maximum
temperature on day 7 of the simulation. The temperature
across the conductor faces is uniform, consistent with the
high thermal conductivity of copper and the assumption of a
uniform heating term. There is, however, a noticeable trefoil
pattern in the temperature distribution within the insulation,
see Figure 6, indicating the triple sources of the three cable
cores and the effect of the thermal conductivity of the
insulation material.
CONCLUSIONS AND FURTHER WORK
The initial research has demonstrated the ability of the
software to simulate both electrical stress levels in a gas
filled void in a cable and thermal effects of current flow in
the cable.
The results indicate that for a buried cable operating unde
realistic load conditions there is a significant diurnal
CIRED2009 Session 1 Paper No 0209
Figure 5: Upper portion of figure describes the thermal
response of the cable to a diurnal variation in load over a
period of seven days. The two curves correspond to a
conductor core and to the sheath. Lower portion of figure
displays the input load data.
Figure 6: Cross section of cable showing the variation in
temperature extant over the cable section and also the
heating of the surrounding trench fill. The scale to the right
hand side shows greyscale indication of temperature (K).
variation in the temperature of the cable superimposed on
the quasi-static temperature rise experienced by the material
immediately surrounding the cable. Furthermore there is a
significant, time varying, temperature gradient extant within
the cable.
Further work will seek to determine the mechanical strain
occurring within the cable as a result of the stress induced
by differential thermal expansion of the cable components.
By looking at the variation in expansion of the materials
under the thermal stresses produced it is expected that
changes in internal pressure can be determined. As PD
activity is controlled, to an extent, by the pressure in a gas
filled void, any variation in pressure as a result of thermal
changes in cable materials will affect PD activity.
Development of a suite of models to simulate alteration of
PD activity resulting from changes in properties of cable
components under in-service stress will allow more accurate
determination of remnant life of utilities’ assets.
Confirmation of the predicted changes from the simulations
will be sought from data gathered on-line from in-service
cables and subsequent repair and replacement programmes.
REFERENCES
[1] D M Hepburn, C Zhou, X Song, G Zhang and M
Michel, 2008, “Analysis of On-line Power Cable
Signals”, Int. Conf. Condition Monitoring and
Diagnosis, Beijing, China, April 21-24 2008,
p.1175-1178
[2] C Zhou, D M Hepburn, M Michel, X Song and G
Zhang, 2008, “Partial Discharge Monitoring in
medium Voltage Cables”, Int. Conf. Condition
Monitoring and Diagnosis, Beijing, China, April
21-24 2008, p.1021-1024
[3] www.comsol.com
[4] IEC 60287-3-1:1995, Electric cables – calculation
of current rating - Part 3: sections on operating
conditons
[5] BS IEC 60287-1-1:2006, Electric cables –
calculation of current rating - Part 1-1: Current
rating equations (100% loss factor) and
calculation of losses – General.