Prediction of the transient current capabilities of internally cooled self-contained cable systems

Electric Power Systems Research, 5 (1982) 141 - 149

141

Prediction of the Transient Current Capabilities of Internally Cooled SelfContained Cable Systems B. M. WEEDY

Department of Engineering, University of Southampton, Southampton S09 5NH (Great Britain) (Received November 27, 1981)

SUMMARY

A c o m p u t e r m e t h o d is described for predicting the transient changes in temperature in an internally cooled self-contained cable system with changes in load. It requires small storage and is suited to on-line computation. Being a step-by-step process, changes in physical parameters with temperature'can be readily accommodated. The m e t h o d is applied to a 400 k V lowpressure oil-filled system with a step function o f load applied. It is seen that the conductor temperature rises rapidly for about 2 0 0 0 s when there is then an abrupt reduction in the rate o f rise as the dielectric and soil become influential.

1. INTRODUCTION

Of the various methods for artificially cooling self-contained cables the use of a coolant flow in the c o n d u c t o r duct yields the highest current rating. The reason for this is that the heat from the conductor, the major loss, is removed at source. With low-pressure oil-filled cables the insulating oil may be p u m p e d through the c o n d u c t o r duct. Similarly, in extruded plastic insulation cables a suitable insulating oil could be used as the coolant, if a c o n d u c t o r duct is provided. Owing to the relatively small c o n d u c t o r duct cross-sectional area the constraint on the cooled length or current rating (85 °C at the conductor) is largely determined by the temperature rise of the coolant. The design and field tests of a 400 kV internally cooled oilfilled cable have been reported [1, 2]. In view of the importance of such highp o w e r links and the costs involved it is necessary to predict the cable and coolant tempera0378-7796/82/0000-0000/$02.75

tures for various operational currents, both steady state and transient (for load changes). The present work describes the development of a relatively simple c o m p u t e r method requiring modest storage and suited to microprocessor on-line temperature and load capability prediction. Although the thermal problem is essentially three-dimensional, the method used employs only one
i~lene i

Fig. 1. Cross-section of the cable. © Elsevier Sequoia/Printed in The Netherlands

142

heat

~-I I I I I I

return I I I I i

I I 'l

I I I I I

cooled length

I

I

I

Ls× no. subsections

Fig. 2. Schematic diagram of the cooling system. absorbs heat (i.e. the cable loss) as it flows along the duct and returns to the exchanger via a return pipe. The steady-state analysis of this system is o f primary importance. However, the transient temperature changes occurring with variations in load current are also of interest as the times for which these temperatures exceed the maximum specified for the insulation may be predicted. This will allow judgement to be used in weighing the loss in insulation life against the system advantages of permitting an overload condition. With naturally cooled cables, i.e. normally buried and with no artificial cooling, the rate of change of temperature with current change is relatively slow. With intensely cooled systems the temperature changes are much faster, allowing less scope for overloads. Details of the cable system to be investigated are as follows: C o n d u c t o r duct inner diameter 50.4 mm C o n d u c t o r duct outer diameter 58.0 mm C o n d u c t o r (copper) overall diameter 80.6 mm Dielectric (paper/oil) diameter 125.1 mm Sheath (aluminium) diameter 134.5 mm Outer sheath (H.D.P.E.) diameter 143.0 mm Coolant flow rate 2.33 1/s Coolant (low-viscosity mineral oil) viscosity at 85 ac 0.002 kg/m s at 20 °C 0.0087 kg/m s The physical properties of the soil and constituent materials of the cable are given in Appendix A-1.

small if cross-bonding is used). The heat from the losses flows (a) through the soil to reach the ambient heat sink at the ground surface, and (b) into the coolant. The relative amounts flowing into these t w o heat sinks depend on the coolant temperature. At the entrance where the coolant temperature is low a large proportion of heat will flow into the coolant, b u t along the cables this proportion will decrease as the coolant rises in temperature. Radial flow of heat is assumed through the dielectric, sheath and cover, whose thermal resistances are calculated from the formula g ln(°uterradius) 2u tuner radius where g is the thermal resistivity of the material, i.e. paper/oil, aluminium or H.D.P.E. The effective thermal resistance per unit length presented b y the soil to the inner (hottest cable) is given by gs +

[ln(-d2/~) 2 ~

l n ( 4 / 2 +2a 2 a )1

assuming a horizontal formation of cables. The dielectric loss per unit length is given by

Vi2 o~Ce tan 6 where Vs is the voltage across annulus j, co = 2u X frequency, and tan 5 is the loss factor (taken as 0.003). The steady-state distribution of temperature may be found by numerical iteration or by an analytical process. The latter will be described and is based on the heat balance at a distance x from the coolant entry into the cable duct. Consider Fig. 3 in which, for simplicity, the whole dielectric loss is injected J J- J i q dx qddx

~

/

/

ambient

R ~/dx (,including sheath

~ Rd/dx &cover)

q(dx 2. STEADY-STATE ANALYSIS

coolant flow

~dx~x~ (e+do)

Heat is developed via the losses in the conductor, dielectric and sheath (this latter loss is

o

Fig. 3. Thermal network for steady-state heat flow.

143

at the midpoint of the thermal resistance representing the dielectric. The refinement of the injection of this loss (as in the transient calculation) presents no problems but makes the analysis more cumbersome. Also, because the heat transfer resistance is very small (see § 3.4) compared with the dielectric and soil thermal resistances, it will be neglected, thus making the oil and conductor temperatures equal. Let 0 be the temperature rise above ambient o f the oil and conductor at distance x from entry. Let R~ be the effective thermal resistance of the soil per unit length presented to the cable under study. R d is the thermal resistance of the dielectric per unit length. If the heat flow into the coolant is Y, then the following equations apply over length dx:

ture of 20 °C above an ambient of 10 °C. All quantities are for 1 cm length of cable. Rs = 133.6 °C/W,

Rd = 38.4 °C/W

qc = 32002 X 0.0124 × 10 -5 = 1.27 W qd

=

0.15

qs = 0

W,

Oo = 20 °C A

= 22.9,

Bcp Wo

B=172.0

= 172 × 2330 × 0.89 X 2.0 =

7.13 X

l0 s

Substituting in (1), 0 at end of cable, i.e. at x = 10 s cm, = 49 °C. Similarly, for a current of 4200 A, 0 (x = 105 cm) = 7 0 ° C . Hence, the conductor temperature = 80 °C.

d0 Cp Wo

- Y

dx

0 = qsRs

3. ANALYSIS OF T R A N S I E N T P E R F O R M A N C E +

1

qd (~Rd + Rs) +

3.1. I n t r o d u c t i o n

+ (qc - - Y ) ( R s + R d )

Hence dO cpWo

dx

-

Y

=

qc m

1

0 -- [qsRs + qd(~Rd + Rs) ] R s + R d

qcB--O + A B dO

dx

Bq~ + A - - 0

BcpWo

the solution of which, using the constraint t h a t at x = 0 the coolant temperature above ambient is 0 = 0o, is Bq¢ + A - - 0 = (Bqc + A - - 0 o ) exp

(1) where 1 A = qsRs + qd (~Rd + Rs)

B = R~ + Rd

The explicit finite difference m e t h o d is used to solve the temperature in onedimensional heat flow. The m e t h o d [4] is facilitated by the use o f lumped-constant thermal networks comprised of the thermal resistances (°C/W) and capacities (J/°C) o f the annular volumes into which the cable is divided. A disadvantage of the explicit method is the limitation on the time-step magnitude for reasons of stability of solution and more sophisticated methods have been applied to similar problems [5, 6]. However, it will be seen t h a t the time-step limit due to the coolant flow is lower than that due to the cable and there is little purpose in using a more complex method in this instance. There are three interconnected regimes to be considered in calculating the transient temperatures. These are: (a) the cable itself in which heat flow is assumed to be radial -- the sheath and covering are assumed isothermal; (b) the surrounding soil -- the soil surface is considered isothermal and at ambient temperature; (c) the coolant which absorbs heat due to both its flow and thermal capacity. 3.2. T h e cable

Consider a current of 3200 A with the oil flow rate specified and an oil inlet tempera-

The dielectric is divided into a number of annular cylinders (preferably more than six

144 qc

qs

qd

to coolant

???

T

conductor

sheath

_i'°'°''

Fig. 4. T h e r m a l n e t w o r k for radial h e a t flow in cable.

and in this case ten) o f length dependent on the axial spacing between sections. Each cylinder is represented by its radial thermal resistance and heat storage capacity, as are also the sheath, covering and conductor. The thermal capacity of each nodal section is given by

Cj = Tr(r2+s

-

-

r2)pcp J/°C per unit length

where r~÷l is the outer radius o f the j t h annulus. Thermal resistances of the annular cylinders are derived as explained in § 2. The reference ring or node for each annulus is at its radial midpoint and the thermal conductance between adjacent nodes is the sum of the two half values of the conductances of the two annuli containing the nodes. The equivalent network is shown in Fig. 4. The longitudinal thermal resistances in the dielectric between axial sections are very large and zero heat flow is assumed in this direction. The equation to be solved for each node j over each time step A t is as follows:

O~ =Aj_IOj_ 1 +Bj+IOj+I +DiOj +Gj

(2)

The coefficients are defined in Appendix A-2.

3.3. The soil In any section, as shown in Fig. 5, the heat flow from the cables to the surface of the ground (assumed isothermal) is twoKlimen-

1'\ -~, /-3' images

r 1,1

r13

Fig. 5. D i s p o s i t i o n o f cables and images: rl, 1 310 m m , d = 145 ram.

= rl, 3 =

sional. The m e t h o d outlined below avoids the solution of the full finite difference (or element) thermal field. The temperature rise on the outer surfaces of the cables at any instant may be obtained by the m e t h o d of images (see Fig. 5) and for the centre cable (the hottest) is given by [7]

O ( t ) - q(t)gs 4~ ~ ~

[I

_Ei --Ei-

d') I l')

16c~t 1,i

+Ei

+Ei-

-----

at

+

2,i

i=1,i~:2

(3) where -- Ei(-- x) is the exponential integral, c o m p u t e r subroutines for the evaluation of which are available. Only the centre cable will be considered as this represents the hottest cable owing to mutual heating from the two outer cables. A problem associated with this equation is the value of q(t), the heat flowing into the soil at the instant in time t. In the present m e t h o d q is evaluated as follows. Keeping the cable surface temperature constant at its previous or known value 0 (10), the temperature rises in the cable are computed for several time steps, in the present case 50 steps each of 20 s. The heat flow from node 9 to node 10 (Fig. 4) is evaluated using the latest value of 0(9), i.e. q(t) = [0(9) -0(10)]K(9). The rise in 0(10), the cable surface temperature, is then c o m p u t e d using eqn. (3). Holding 0(10) at its new value, a further 50 time steps are evaluated in the cable, the cable surface temperature recomputed, and so on. The process is illustrated in Fig. 6, where the transient is assumed to commence from a steady-state condition relevant to a previous load, i.e. an initial cable surface steady-state rise 0sl above ambient. The transient rise in the cable itself (with 0sl held constant) resulting from the new load yields, after say 50 steps, a value of heat flow to the soil of q(ta). The t e m p e r a t u r e - t i m e curve for this flow from eqn. (3) is shown in Fig. 6. It is

145

(4] ~

[_

1"

_1_ I_ 50~

T

=l

q(2)

time

Fig. 6. Rise in cable o u t e r s u r f a c e t e m p e r a t u r e

from

an initial steady-state temperature 0sl. (Not to scale.) necessary to establish the origin of this curve and this is achieved b y generating a range of values from eqn. (3}. Time tl is then obtained simply in the c o m p u t e r program by sampling the (0s, t) values. Point 2 is the temperature rise after a time interval o f 50A t, this being the new cable surface temperature. The cable surface is n o w held at the new temperature 0s and further temperature rises calculated in the cable itself, giving a new 0 (9) and in turn q(t2). The origin of q(t2) is established and the new surface rise along curve q ( t 2 ) obtained, and so on. It should be noted that a value of 50 time steps is used for the cable for one soil up-date because the thermal time constant in the soil is much longer than that in the cable. However, it is at the discretion o f the analyst to vary this quantity to suit the circumstances. If required the soil transient can be recalculated on each time step of the cable transient with an increase in the c o m p u t e r time required. Also, because q ( t ) varies over the time interval o f the computation, the average value of the old and new temperatures could be used. However, comparison of results indicates that with the short time involved this complication is unwarranted. Should the analyst have quantitative knowledge o f the changes in the soil parameters with temperature (due to change in water content, etc.), such changes may be readily incorporated at each soil up-date. 3.4. T h e c o o l a n t

The third c o m p o n e n t of the transient process comprises the effect of the coolant flow. The flow o f heat into the coolant is impeded b y a thermal resistance quantified b y the heat transfer coefficient h, i.e. t h e heat transferred per m 2 of duct surface per °C. As the coolant varies in temperature with both time and axial length along the cable

route it is necessary to subdivide the system into a number o f axial lengths, e.g. 20. The resulting thermal network is shown in Fig. 7, assuming the coolant flows to be in the same direction in each c o n d u c t o r duct and the return flow to be in a separate pipe. It is seen that the cable c o n d u c t o r is connected to the coolant nodes at the midpoints of the axial subdivisions b y the heat transfer conductances H, where H is evaluated b y the wellk n o w n formula [8] for turbulent flow,

where the parameters relate to the coolant and H = •dhL s W/°C, i.e.,

h × 0.0504 _ 0.023 ( 2330 x 10-6 0.133

~ Ir(0"0252) 2

X

X 0"0504 ° 'X 890)°'8 4 0 .\( 2 000 00 . X 01 -0"002 ~2 ) if quantities are in metre and kg units and pertinent to 85 °C, h = 810 W/m 2 °C and H = 6412 W/°C per 50 m length of cable. The corresponding thermal conductance of the dielectric is 130.2 W/°C, and hence the heat transfer conductance is o f small influence. In each iteration the value for coolant viscosity was based on the average temperature, although because of the small influence o f H this is not critical. The nodes representing the mean temperatures in the c o n d u c t o r sections are connected b y the axial thermal conductances of the cond u c t o r subsection lengths. soil

|

d

;:

J

I

uv[

I

I

1I

I

! en ]H -

~

eo ~H

t radial

heo'f'ow'

lU

longitudinal

.__-c~%ct v

~4~÷1 ~

Co: ~o2&,:m., __-__Z T mT

T

I IH heat transfer -) coolantflow

_k T

Fig. 7. Complete thermal network including coolant temperatures.

146

The value o f t h e oil t e m p e r a t u r e rise at the end o f a cooled subsection, OomZ+z, is shown in Appendix A-3 to be

HAt 1 0ore +i

(2Woc~o

--

H)At

On +

--

OOm +

Co

2Co

[ +

[ .... s. . . . . . . . . . . .

(2W°%°+H)Atlo I

--

2Co

+1 (5)

OIL

om

]--~.ENG~OO~FFID,~NTB' .........

........

I ................

f .................... _o .... ! _

H < 2Wocpo A

t<

time criterion

CALCULATE

SSIGN TEMP'S OF NODES FROM CALCULATION OF SSTOIL

i ....

where Wo = volumetric flow rate X density = 2330 X 10 -6 X 890 kg/s, %o = 2000 J / k g °C, and H < 8295 W/°C (50 m length). F o r a subsection length L s of 50 m, H = 6864 W/°C. Also, f or L~ = 50 m, A t = 22.5 s. The m i n i m u m time step permissible in the cable n e t w o r k is 33 s (see §A-2) and hence the At to be used is t ha t for the coolant, i.e. a value less th an 22.5 s.

SSTOIL

INITIAL TEMP'S DUE TO ELEVATED TEMP OF COOLANT

~" i

2Woc,o + H

I

CALL SUBROUTINE SSTOI L

space criterion

2Co

CALCULATE am~ c',

t

RESISTANCE OF THE SOIL

There are limits for the stability of the computational process (coefficients of temperatures in eqn. (5) must be positive) as follows:

OIL

q t

IDALCOLATEC...... OSSES [ CA L CEULA] NEoLDOESS. . . . . . . . .

"

]

hTU]

NEWLY CALCULATED TEMP'S NOW I BECOME LATEST TEMP'S OF NODES I

IT=IT+I

+

. . . . . . .

IRESNO = IT + 50 Y M

..... l

. . . . .

J

]

4. C O M P L E T E P R O C E S S

All the quantities involved in the c o m p l e t e transient calculation are shown in Fig. 7. Cable thermal quantities are initially calculated per metr e and later modified for the subsection length Ls. The heat q injected into the co o lan t nodes o f the cable networks at each time step is obtained from a simple heat balance which gives the following equation:

q = qcL +

--

[(

H O, --

20. - - 0 . - z -- 0 RL

0o +0om., )+ 2 +I

"

1

J

YES TEM, iRESNO • ]POLL

&

YES M.M+1

t

l i CALCULATE TIME ~CALCULATE ',qE~T INTO bOil

(6)

Th e initial t e m p e r a t u r e s are entered into a two-dimensional array T(j, N) where j refers to the cable ladder n e t w o r k nodes and N the axial subsection node num ber . These temperatures may be due to t he previous steady-state load or, if initially at no load, the t e m p e r a t u r e distribution d u e to t he dielectric loss alone or c o o lan t t e m p e r a t u r e . Equations (2), (5) and (6) are th en solved for each time step and

_

_

i

CALC EFFECTIVE TEMP OF CABLE SURFACE USING EXPONENTIAl INTEGRAL FORMULA

<

~

NH

Fig. 8. F l o w c h a r t o f t h e p r o g r a m . IT = n u m b e r o f i t e r a t i o n s since l a s t soil u p - d a t e ; I R E S N O = t o t a l n u m b e r o f i t e r a t i o n s since s t a r t o f t r a n s i e n t ; N I T T = total n u m b e r of iterations.

147

then after say 50 steps the soil influence is accounted for. Although the soil influence is small at the initial low coolant temperatures, progressively higher coolant temperatures result in significant heat flows to the soil. Temperature rises resulting from a loss of coolant flow through the ducts are readily obtained b y making the heat transfer conductance to the coolant zero in value. The operations are summarized in the flow chart in Fig. 8.

100'

80

6O ¢ .D

o40

5. RESULTS

~J

The m e t h o d has been applied to a typical 400 kV cable system and a few illustrative results are presented. Variations in coolant viscosity with temperature are incorporated as the program progresses along with the increase in c o n d u c t o r AC resistance with temperature. As specific heat and density varied b y less than 5% over the temperature range they are assumed constant. A cooled length o f 1 km is considered and t w e n t y longitudinal subsections each of 50 m are used. Crossbonding is assumed and the sheath loss is assumed to be negligible. In Fig. 9 are shown the temperature rises of the conductor, coolant, o u t e r ring of cable dielectric and cable o u t e r surface for the application of a step-function of load current to a previously unenergized cable. Originally the steady-state temperatures due to the dielectric loss alone were established before the load was applied. However, the difference between the results o f this and the application o f the copper and dielectric losses simultaneously was very small after the first few minutes and the extra complication o f the separate dielectric loss injection not warranted. The oil entered the cable at a temperature above ambient (10 °C and 20 °(3 were assumed) and the resultant temperature rises above ambient in the unenergized cables were c o m p u t e d prior to the main calculation for the energized system. In Fig. 9 the temperature rises above ambient are shown for the entrance subsection of the cooled length for inlet temperatures of 20 °C and 10 °C above ambient. The increase in steady-state c o n d u c t o r and coolant temperatures along the cable length is approximately linear. Owing to good heat transfer there is only a small temperature difference

~J O

~20

10 4

2 ,, 10 4

3,10 4

time ( s )

Fig. 9. Transient temperature rises in the entrance section, current and dielectric losses being applied as a step function. Curves a, b, c and d: conductor, coolant, outer ring o f dielectric and cable surface, respectively, for a current o f 7.7 kA; coolant inlet .temperature = 10 °C above ambient. Curves e and f: c o n d u c t o r and cable outer surface for 3.2 kA; coolant inlet temperature = 20 °C above ambient.

between the c o n d u c t o r and coolant. The initial rise of temperature o f the c o n d u c t o r is relatively fast and at about 2000 s the rate of change decreases quickly and a much slower temperature rise occurs as the effects of the cable dielectric and the soil take over the thermal process. That the initial rise is largely a function o f the c o n d u c t o r and coolant is borne o u t b y the curves for the o u t e r dielectric and cable surface, where little rise is seen over the first 0.5 h. With an oil inlet and ambient temperatures o f 20 °C and 10 °C respectively and a cooled length o f 1 km, a current rating (steady state) o f 4500 A is achievable. The limited available field test data relate to a system using external water pipe cooling in addition to internal off cooling. However, the initial rate of rise o f temperature for 3200 A is approximately the same as predicted in the present work. This is a reasonable comparison as over the initial heating period the influence o f the soil (and of associated water pipes adjacent to the cables) will be small.

148 5. CONCLUSIONS

U V

A fast and economic program has been developed which is suitable for the prediction of temperatures, and hence currents, in internally cooled self-contained cable systems. A sudden change in cable load is seen to produce an initial fast change in temperature in the c o n d u c t o r and coolant, followed abruptly b y a much slower rise. Over this initial period, little change in the outer dielectric and soil temperature occurs. The fast initial transient indicates little scope for short-term overload capability.

Wo

velocity of coolant, m/s line-to-line voltage mass flow rate of coolant, kg/s

Greek symbols a = 0.0047 × 10 -4 m 2 s - 1 , thermal diffusivity o f soil 0 temperature rise above ambient, °C p density

APPENDIX

A-1. Physical properties ACKNOWLEDGEMENTS

The author thanks S o u t h a m p t o n for the facilities.

the use

University of of computing

NOMENCLATURE

a

axial spacing of cables in horizontal formation, 310 mm A, B, C coefficients in heat flow equations cp specific heat, J/°C kg C thermal capacitance per unit length, J/°C Ce electrical capacitance per unit length, F Co thermal capacitance of oil per subsection outer diameter of cables, m d thermal resistivity of the soft, taken gs as 1.2 °C m/W heat transfer coefficient, W/°C m 2 h H heat transfer conductance from duct wall to coolant per subsection W/°C k thermal conductivity, W/°C m K thermal conductance per unit length, W/°C l depth of cables, 971 mm Ls length of subsection of system, m M counter for soil temperature up
Oil Conductor Dielectric Sheath Outer sheath Soil

Specific heat (kJ/°C kg)

Thermal resistivity (°C m/W)

2.0 0.39 1.6 0.127 1.5 1.02

7.5 0.0026 5.5 0.029 5.7 1.2

Densit~ (kg/m)

890 8930 1050 2700 920 1750

A-2. Derivation o f nodal equation for cable ladder network Consider the network shown in Fig. 4. The heat balance at node j after a time step & t is as follows: Kj-l,i(01-1 -- 0i) + Ki+ 1,j(0/+ 1 - - Oj) + q1

= Q(O] -- Os)l~t where Kj-I,j is the thermal conductance between nodes j -- 1 and j. From which, O] = Ai_ 10i_ 1 + Bj+ 16j+ 1 + Di0i + G~ where A i - 1 _ K~_I,jAt

Cj gj+ l, lA t ej+ l

-

Cj

D 1 = I - - A j _ 1 --Bj+ 1

6i-

qjA t Ci

For stability Dj must be positive and hence

149

At

,.,) Thus, f or any n o d e j , At<

q ZKs connected to j

This criterion must be checked at each node and the lowest value of A t used. A subroutine for the calculation of these coefficients and checking stability is incorporated in the program.

A-3. Equations for the coolant In Fig. 7 the coolant nodes are shown. In its passage across one subsection, Ls, the coolant gains heat from the duct and increases in temperature from 0,, to 0o~ +1 owing to convective heat transfer and to 01o~+ 1 owing to its thermal capacity Co over the length Ls. The heat balance equation is

A slight approximation arises in taking the capacity rise as the increase in 0ore +z and not the average value for the subsection. However, the error is small and much complication is avoided. REFERENCES 1

2 3

4 5

6

H[ O, -- O°m +0°m+112 7

8

from which 0~,~ +1 is obtained (see eqn. (5)).

W. Brotherton, H. N. Cox, R. F. Frost and J. Selves, Field trails of 400 k V internally oilcooled cables,Proc. Inst. Electr. Eng., 124 (1977) 326 - 344. C . A . Arkell, R. B. Hutson and J. A. Nicholson, Development o f internally cooled cable systems, Proc. Inst. Electr. Eng., 124 (1977) 317 - 326. C . R . Gane, J. A. Hitchcock and D. R. Soulsby, Digital computation methods for the determination of ehv cable transient ratings, Proc. CIGRE, P a p e r 21-07, Paris, 1980. B. M. Weedy, Underground Transmission o f Electric Power, Wiley, New York, 1980. B . M . Weedy and J. E. J. Cottrill, Prediction of thermal instability in ehv cable joints, Proceedings o f IEEE Conference on Underground Transmission and Distribution, Atlantic City, 1976. B. M. Weedy, Effects of environment on the transient thermal performance of underground cables, Proc. Inst. Electr. Eng., 119 (1972) 225 230. A. Morello, Transient temperature variations in power cables, Eiectrotechnica, 45 (1958) 213 218. W. H. McAdams, Heat Transmission, McGrawHill, New York, 3rd edn., 1954.