CURRENT RATING OF CABLES INSTALLED IN DEEP OR VENTILATED TUNNELS
Eric DORISON - EDF R&D, Moret-sur-Loing (France) eric.dorison@edf.fr
George ANDERS - Technical University of Lodz, george.anders@attglobal.net
ABSTRACT
The series of IEC 60287 standards provides methods for
calculating the permissible current rating of cables.
This paper deals with the present work carried out within
the IEC TC 20 WG 19, intending to extend the scope of
these standards, to groups of cables installed in deep or
ventilated tunnels.
Generally, the thermal resistance T4 of the surroundings
of a cable is defined by:
T4 
The IEC 60287-1-1 [1] and 2-1 [2] standards address the
thermal rating of a cable circuit installed in still air with
given temperature with the IEC 60287-2-2 [3] providing
an extension to some homogeneous groups of cables.
The same calculations can be used to rate cables in
tunnels, however, the external thermal resistance of the
tunnel itself requires special considerations..
The paper deals with the work carried out by WG19 of
the Technical Committee 20 of the IEC intended to extend
the scope of these standards to cables installed in
ventilated deep tunnels.
The IEC standards combine the effect of heat transfer by
radiation and convection into one coefficient. In order to
properly model the effect of air movement inside the
tunnel, the convective and radiative heat transfers must
be treated separately. Hence, first, the IEC method for
rating cables installed in still air is reviewed and
considerations are given to the modeling of heat transfer
by radiation. In particular, the radiative heat transfer for a
group of cables is addressed and the extension of the IEC
method to the groups of cables with different designs is
proposed considering the effect of dielectric losses.
Next, a rating method for cables in ventilated tunnels is
presented, based on an analytical approach, originally
developed by CIGRE and published in Electra [4,5].
Finally, for deep tunnels, a fictitious equivalent depth is
introduced, to optimize cable rating, taking into account
the soil thermal inertia, without performing transient
analysis.
BASIC MODEL FOR CABLES IN AIR
According to the IEC 60287 standards, the rating of air
installed cables, , is based on a relationship that links the
total heat loss of a cable Wt with the temperature rise of
its surface s above the ambient a by:
Wt   .De .h. s   a q
[1]
where De is the cable diameter, h is a heat dissipation
coefficient depending on the installation and q is a
constant, set equal to 1,25.
[2]
So that the thermal resistance T4 for a cable in air is:
T4 
INTRODUCTION
Wt
s  a
1
 .De .h. s0,25
 s   s   a
[3]
The heat dissipation coefficient h is given as a function of
the cable diameter:
Z
h
Deg
E
[4]
where Z, E, g are constants, whose values depend on the
type of installation.
As T4 is a function of s, an iterative process has to be
conducted, taking into account the temperature drop
between the cable conductor(s) and its surface:
 c  s  n.Wd .Td  Tint   Wt .Tint
[5]
c is the conductor(s) temperature, n is the number of
conductors, Wd represents the dielectric losses per
conductor and Td and Tint are the equivalent thermal
resistances used in expressing the transfer of dielectric
losses and Joule losses within the cable, respectively:
T1
 T2  T3
2.n
1
T


. 1  1  1 .T2  1  1  2 .T3 
1  1  2  n

Td 
Tint
[6]
The permissible current I is obtained from Wt as follows:
Wt  n.Wc .1  1   2   Wd 
[7]
Wc  R.I ²
[8]
R being the conductor resistance at the temperature c.
For installations involving several circuits, the value of the
heat dissipation coefficient is defined in IEC 60287-2-2
from the value for a single circuit given in IEC 60287-2-1.
This approach is based on modelling of the heat transfers
by radiation and convection, using Ohm’s thermal law,
linking temperature drop and heat rate through a thermal
resistance with the following assumptions :

The construction of all cables in the tunnel is the
same. This means that the surface temperatures of
the cables are similar; and, therefore, the radiative
heat transfers between the cables may be ignored.
Cable losses
Radiation cable / tunnel walls
n.Wd
n.Wc
n.1.Wc
n.2.Wc
s
c
T1/2n
T1/2n
Wrad
T3
T2
t
Te
Tst
Wk
Ww
Tsa
Tat
Convection cable /air
Convection air / walls
Wconv
a
Figure 1 : Heat transfer modeling – cables in still air
Tst 
1
 .De .hrad
Tij 
F12 



hrad  K r . s . .  s  2732   w  2732 . s  273   w  273

  c
0, 5
Fsw
1   s .Fsw   s


hij  Fij . s . .  si  2732   sj  273 2 .  si  273   sj  273
1
 .Dei .hij
1 
.  c 2  r  12
2 
 

Kr 
2
 r  12

0, 5
 r 1 
1  r  1  
 r  1. cos 1 

  r  1. cos 
 c 
 c 
r
De 2
De1
c  2.
S
De1
Table 1 : Radiative thermal resistances
The temperature of the tunnel walls t is close to the
temperature of the air a in the tunnel.
 The air in the tunnel is still, so that the convective
transfers can be described through the well known and
experimentally checked correlations.
In reality, the heat transfer from the cable surface is
composed of several components as illustrated in Fig. 1.
Referring to this figure, Tst represents the radiative heat
transfer between the cable surface and the tunnel walls,
Tsa and Tat represent the convective heat transfer from the
cable surface to the air in the tunnel and from the air to
the tunnel wall, respectively.

The thermal resistance T4 is a combination of the thermal
resistances expressing heat transfers by radiation and
convection from the cable surface. The thermal resistance
representing heat transfer between the air and the tunnel
wall is assumed to be negligible.
Aw represent the cable and wall surface areas, s and w
are the reflectivity of the cable outside surface; and the
wall inner surface, respectively.
When several cables are present, the mutual radiant area
between them must be subtracted from the area radiating
to the tunnel inner surface. The effective radiating area to
the tunnel walls is obtained as follows, using Hotel’s
method of stretched bands [6].
Elastic bands are imagined stretched around two arbitrary
concave surfaces (see Fig. 2).
C
2
C'
K r  1   s /  s  1 / Fsw  As w / Aw w 1
[9]
where w is the emissivity of the wall inner surface; As and
Length of the internal band :
Iint  AD  BC' C
where C' is the point of
intersection of BC with
surface CD.
RADIATIVE HEAT TRANSFERS
The expression for the thermal resistance Tst is given in
Table 1, where  is the Stefan-Boltzman constant; s is
the cable surface emissivity, Kr is the radiation shape
factor, which depends on the view factor Fsw defined as a
fraction of the radiation that leaves the considered cable
and is intercepted by the tunnel walls.
D
A
1
Length of the external band:
B
Iext  AC  BD .
Figure 2 : Calculation of the mutual radiation area [6]
The mutual radiation area Am per unit axial length is equal
to half the difference in lengths of internal and external
bands, that is,
Am 
Iint  Iext AD  BC' C AC  BD


2
2
2
[10]
Application of equation [10] can become quite
cumbersome. In practice, as the tunnel size is large
compared to the cable surface, and as s and Fsw are
close to one, the following expression given in [7] may be
used:
K r  1   s .1  Fsw 
[11]
From the relationship [2] and the definition of T4 :
T4 n h0  s00, 25
 .
T40 hn  n0, 25
 s 0 Wt 0 T40 1  k0


 k0
.
T4 n
 sn Wtn T4 n
T40
It is found that T4n/T40 is a solution of
Fsw and corresponding Kr (for s =0.9) are given in Table 2
for installations described in the IEC 60287-2-1 standard.
Installation
Single cable
Fsw
1,000
Kr
1,000
2 cables touching
0,818
0,833
2 cables - spacing 2.De
0,919
0,927
3 cables touching – middle cable
0,636
0,660
3 cables spacing 2.De – middle cable
0,838
0,852
Trefoil
0,652
0,676
T4n h0

T40 hn
The derating factor is derived from the relationship given
below, which reduces to the IEC formula if the dielectric
losses are small compared to the total losses.
I tn2
I t20
Table 2 : View factor and radiation shape factor
Table 1 gives the formula for the thermal resistance Tij
expressing the radiative transfers between two cables
[8,9], which is of interest when the rating cables with
different designs or in ventilated tunnels are sought.
0,25


1  k

0 k 
.
0
 T4 n



T
 40

Wtn n.Wd

Wtn  n.Wd
Wt 0 Wt 0
W
 tn

.
n
W
.
Wt 0  n.Wd
Wt 0
d
1
Wt 0
Cables with different designs
GROUPS OF CABLES
If there are cables with different designs, the radiative
heat transfers between the cables have to be taken into
account [10]. This is achieved by computing first the
thermal resistances involving the view-factor between
cables.
Cables with the same design
Basic equations are as follows:
The IEC standard 60287-2-2 extends the calculation
method in the IEC 60287-2-1 to some homogeneous
groups of cables. The formulae do not consider the effect
of the dielectric losses. The following developments
extend the standard method by considering these losses.
Let
 c*
and
 c*
 ci*   si  Wi .Tint_ i

 si   a  Wi 
are defined as follows :
   c  n.Wd .Td  Tint 
*
c
     a
*
c
*
c
Using the relationships [5] and [2],  c* is expressed as:
 c*  Wt .Tint  T4 
For given air temperature a and the maximum
permissible conductor temperature c , as Td and Tint
depend only on the parameters internal to the cable,  c*
is the same whether the cable is alone or is a part of a
group. Consequently, defining Wt0 and Wtn as the losses
for a cable alone or a part of a group, respectively, we
have:
Wt 0 Tint  T4 n

Wtn Tint  T40
[12]
Iintroducing the ratio k0 of the cable surface temperature
rise to the conductor temperature rise due to Joule losses
in the metal components only, i.e., neglecting dielectric
losses, we can write equation [12] as:.
Wt 0
T
 1  k 0   k 0 . 4 n
Wtn
T40
k0 
T40
Tint  T40
 si   sj


W ij .Tij

Wij .Text _ i

j i


[13]
where Text_I is the thermal resistance between the cable
surface and the tunnel walls, representing a combination
of the radiative and the convective thermal resistances,
respectively Tst andTas with the former one defined in
Table 1.
Tas is determined as:
Tas 
1
 .De .hconv
[14]
hconv  h. s   a 0, 25  hrad _ 0
where h is the global heat dissipation coefficient given in
IEC 60287 and hrad_0 is the radiative heat transfer
coefficient for an isolated cable.
Substituting Wi and Wij in the second equation in [13],
from the first and third one respectively, leads to a set of
equations linking the cables surface temperatures:

1
 1


T
T
int_ i
 ext _ i
 si .

j i

1

Tij 

 sj
T
j i
ij

a
Text _ i

 ci*
Tint_ i
Once the surface temperatures are determined, the
permissible heat rate of the cable i is derived using the
first equation in [13], and, thus, the ampacity is obtained
from relationships [7] and [8].
240 mm² Cu
Cables 3-core SA type
63 kV
Outer diameter (mm)
94.3
Tint (K.m/W)
0.189
Each cable alone
Text (K.m/W)
0.293
Surface temperature (°C)
66.5
IEC Ampacity (A)
637.6
S = 1.75.(De1+De2)/2
Text (K.m/W)
0.293
Surface temperature (°C)
68.1
Ampacity (A)
633.9
S = 100000 m
Text (K.m/W)
0.286
Surface temperature (°C)
66.2
Ampacity (A)
638.3
As the radiative thermal resistances between the cables
depend on the cables surface temperatures, there is a
need for an iterative process.
Following table gives sample results for 2 different 3-core
cables, installed horizontally with a spacing equal to 1,75
times the mean diameter.
The results in Table 2 confirm the assumptions in the IEC
60287-2-2 that, for the clearances between cables, as
given in Table 1 of this standard, the thermal proximity
effects are negligible.
This approach is quite correct if the clearance between
the cables is large enough so that the proximity does not
lead to significant disturbance in the convection transfer.
If not, it is suggested to determine the convective thermal
resistance using for the global dissipation coefficient h in
[14], the IEC value for a cable clipped to a wall.
50 mm² Al
20 kV
53.6
0.240
0.475
69.9
186.2
0.492
68.1
182.2
0.460
69.4
186.4
Table 2. Sample results of two different cable ratings
installed in a tunnel.
STILL AIR MODELLING
 s   e  Ts . N .W k
STAR – DELTA TRANSFORMATION
 e   t  Tt . N .W k  W a 
 t   0  Te . N .W k  W a 
Te
Ts
Tt
e
(N)
 a   e  Ta .W a
W a  C av .
t
s
T3
T3
N.Wk
Ta
 a
z
Heat removed
by the air
Wa
a
Figure 3 : Model for heat transfer in a ventilated tunnel
VENTILATED TUNNELS
In the case of ventilated tunnels, the same models as for
still air apply, but the convective heat transfer coefficient is
computed taking into account the effect of the air
movement. In addition, the heat removed by the flowing
air has to be taken into account in the energy balance
equations.
Figure 3 presents a heat transfer model and gives the
corresponding set of equations, obtained from the initial
diagram, using a star-delta transformation. The tunnel
axis is assumed to be perpendicular to the page surface
(z coordinate).
Convective transfer between the cables and
air
The following formula for the thermal resistance Tsa
applies if the Reynolds number Re is larger than 2000.
Tsa 
1
 .k t .K p . Re 0.65
with
Re 
V .De

V is the bulk air velocity,  is the kinematic viscosity and kt
is the thermal conductivity of the air. Kp is an
experimentally determined constant with the values given
in Table 3 taken from reference [11].
Cable Arrangement
Single Cable
3 cables spaced horizontally (spacing>2Diameter)
3 cables touching horizontally
3 cables spaced vertically (spacing>2Diameter)
3 cables touching vertically (spacing<=2Diameter)
3 cables touching in trefoil
Kp
0.130
0.115
0.086
0.115
0.086
0.070
Table 3 : Experimental constant for cables / air convection
Convective transfer between air and walls.
The Dittus-Boelter correlation, in agreement with the
Kitagawa works, is used in the formula of the thermal
resistance Tat.
This applies if the air flow is turbulent, i.e. if the Reynolds
number is larger than 2500. If the Reynolds number is
smaller than 2500, this thermal resistance may be
considered negligible .
Tat 
with
1
 .k t .0,023. Re 0,8 . Pr 0,4

Pr  C pair .
Re 
kt

 T1

 max   0   0   Wd .  n.T2  T3  T4t  
2

I 
 R.T1  n.1  1 .T2  n.1  1   2 
. T3  T4t  





2
with:
 0   a 0    0 .
Tt  Te
L
.e 
Ta  Tt  Te


Tt  Te
L
T4t  N .Ts  Tt  Te .1 
.e 
 Ta  Tt  Te

V . Dt
1



Cpair is the specific heat of the air per unit volume.
This formula does not take into account the wall
roughness.
Conductive heat transfer in the tunnel
surroundings
The thermal resistance Te is given for circular shallow
tunnels as:
Te 


.Ln u  u ²  1
2

u
2.L
Dt
[15]
where  is the soil thermal resistivity, L is the depth of
tunnel axis and Dt is the tunnel diameter.
.For a square tunnel, the expression derived by
Goldenberg for a buried square trough and reported by
Symm (1969) can be used. This expression is
Te 

2
L

ln  3.388 
a

  Ta  Tt  Te .C av
Temperature distribution
The air temperature a(L) at tunnel outlet is estimated as :

 a L    a 0    0  Tt  Te .N .Wk   a 0 .1  e


L




The cable surface temperature and the tunnel wall
temperature at the tunnel outlet are derived from the air
temperature as :
 s L    a L   Ta .Wa L   Ts .N .Wk
 t L    a L   Ta .Wa L   Tt .N .Wk  Wa L 
where a is the height and width of a square cable tunnel,
where Wa(L) is the heat removed by the air at tunnel
outlet, given by :
Heat removed by the air
The heat removed by the air is linked to the air
temperature variations according to:
Wa  C av .
 a
z
W a L  
Tt
C av  C pair .V . At
where At is the tunnel cross-section.
Permissible current
CIGRE works show that, for typical installations, air
properties may be considered constant along the tunnel
route and computed using air temperature at the tunnel
outlet. With this assumption, solving the set of heat
balance equations is straightforward and the temperatures
of the cables’ surface, air and tunnel wall are easily
derived as a function of the cables losses.
As regards the heat generated by the cables, it is
assumed to be constant along the cable route, computed
for the core maximum permissible temperature, leading to
an estimate of the current carrying capability on the safe
side.
The permissible current rating is obtained from the
following formula which is similar to the classical formula
for cable rating:
 Te .N .Wk   a L    0 
Ta  Tt  Te
DEEP TUNNELS
The losses of the cables installed in a tunnel are
dissipated in its surroundings, leading to a temperature
rise of the tunnel wall.
Due to the soil thermal inertia, a long duration is
necessary to reach the steady-state value. Considering a
circular tunnel (2 m in diameter) and assuming a constant
heat rate, Figure 4 gives the tunnel wall temperature rise
evolution with time, reported to the steady-state value of
the current, for 3 laying depths (10, 20, 40 m).
The temperature rises quickly at first but then the growth
is very slow. Applying standard steady state calculation
algorithm would yield ampacities that are too small.
A more appropriate approach would be to use the
transient analysis algorithm and iteratively find out what
value of the current would give desired temperature at the
end of the study period.
Tunnel w all tem perature rise ratio vs tim e
Ratio Equivalent / Actual depth vs Duration
100
0.8
80
0.7
Ratio
Temperature rise ratio (%)
1
0.9
90
70
10 m
60
0.6
0.5
0.4
20 m
0.3
40 m
0.2
'10 m'
'20 m'
50
'40 m'
0.1
0
40
0
0
10
20
30
Figure 4 :Tunnel wall temperature rise
The formula for the steady-state external thermal
resistance is given above as relationship [9].
The transient thermal resistance of the tunnel
surroundings was computed with a standard equation
using the exponential integrals given in [12].

4.

 D2
. Ei  t
 16.t.




  Ei   L ² 

 t. 

where  is the soil thermal diffusivity;  is the soil thermal
resistivity, Dt is the tunnel diameter and L its burial depth,
to tunnel centerline.
One can define a fictitious equivalent depth [13] of the
cable circuit that with the application of the steady state
algorithm would give the same value of the current as the
one obtained from the transient analysis (see Figure 5).
LT 
20
30
40
50
60
Duration (years)
Tim e (years)
T4 t  
10
40
1 
D
D² 
 L ²  

. cosh  . Ei 
 
  Ei  
2
 t.  
 16.t. 
2 
CONCLUSION
As the number of cable tunnels has been increasing over
the last decade, IEC TC20 WG 19 started a new work
item aiming at extending the scope of the present cable
rating methods, within IEC 60287 standard series.
This paper summarizes some of these works.
As distribution and transmission systems may be installed
in a tunnel, to take full benefit of the costly equipment, the
rating of groups of cables, with different designs, is
addressed.
The main issues addressed in the standard to be issued
shortly are presented, dealing with the rating of cables in
ventilated tunnels .
Finally, as it was recognized that, for deep tunnels,
applying standard steady state calculation algorithm
would yield ampacities that are too small, a fictitious
equivalent depth is introduced, to optimize cable rating,
taking into account the soil thermal inertia, without a
transient analysis
Figure 5 : Equivalent depth for deep tunnels
REFERENCES
[1] IEC 60287-1-1, " Electrical cables – calculation of the
current rating – Part 1-1 Current rating equations
(100% load factor and calculation of loses – General”
[2] IEC 60287-2-1, " Electrical cables – calculation of the
current rating – Part 2-1 Thermal resistance –
Calculation of thermal resistance”
[3] IEC 60287-2-2, " Electrical cables – calculation of the
current rating – Part 2-2 Thermal resistance – A
method for calculating reduction factors for groups of
cables in free air, protected from solar radiation”
[4]. CIGRE, (1992a) "Calculation of Temperatures in
Ventilated Cable Tunnels - Part 1", Electra, No.143
[5] CIGRE, (1992b) "Calculation of Temperatures in
Ventilated Cable Tunnels - Part 2", Electra, No.144
[6] G.J. Anders, “Rating of Electric Power Cables –
Ampacity Calculations for Transmission, Distribution
and Industrial Applications" (1997) IEEE Press, New
nd
York. 2 printing jointly IEEE Press and McGraw-Hill,
(1998) New York.
[7] P.Slaninka, 1969, " External thermal resistance of airinstalled power cables", Proc. IEE Vol. 116 n°9,
September 1969
[8] F.P. Incropera – D.P. De Witt, WWWW, McGraw-Hill
Handbook of heat transfer [Third Edition]
[9] W.M. Rohsenow – J.P. Hartnett – Y.I. Cho –
“Handbook of heat transfer” – third edition – McGrawHill Handbooks – page 7.83
[10] J.A. Pilgrim, D.J. Swaffield, P.L. Lewin et alii, “Rating
independent cable circuits in ventilated tunnels”, IEE
Trans. On power delivery, Vol. 25 n°4, October 2010
[11] Weedy – El Zayyat “Heat transfer from cables in
tunnels and shafts”, IEEE Con Paper C72 506-4
[12] IEC 60853-2, " Calculation of the cyclic and
emergency current rating of cables – Part 2: Cyclic
rating of cables greater than 18/30 (36) kV and
emergency ratings for cables of all voltages”
[13] E .Dorison, G.J. Anders, F. Lesur, “Ampacity
calculations for deeply installed cables”, IEEE Trans.
on Power Delivery, TPWD, Vol. 25, No. 2, April 2010