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.