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C6-2 10th International Conference on Insulated Power Cables C6-2
Jicable'19 - Paris - Versailles 23-27 June, 2019 1 / 6
Ampacity Calculation of Multiple Independent Cable Systems in Ventilated Tunnels
Damian AEGERTER, Braavos GmbH, Stetten (Switzerland), damian.aegerter@cableizer.com
Stephan MEIER, Emetor AB, Västerås (Sweden), stephan.meier@cableizer.com
ABSTRACT
In 2017, the standard IEC 60287-2-3 for the calculation of
the current rating of cables installed in ventilated tunnels
was published, however, the method is not suited for
applications with multiple independent cable circuits. A new
and extended analytical method was developed to allow for
the calculation of multiple different cable systems or other
heat sources in ventilated tunnels. The numerical method
consists of a thermal network representing axially
connected slices of the tunnel cross sections.
KEYWORDS
Ampacity, Cable Rating, Cables in Tunnel.
INTRODUCTION
In 2017, the International Electrotechnical Commission
(IEC) published the new standard IEC 60287-2-3 for the
calculation of the current rating of cables installed in
ventilated tunnels [1]. The method is not suited for
applications with multiple different cable circuits.
Only one commercially available software tool was found
capable to compute the cable rating for cables in ventilated
tunnels, but with the limitation that only one cable system
can be used. The planning of new energy tunnels often
needs to consider two or more different cable systems.
Sometimes, even 50 Hz three-phase systems and 16.7 Hz
two-phase railway systems are combined in energy
tunnels.
An analytical method capable to calculate multiple different
cable systems of any type or other heat sources such as
gas insulated lines (GIL) or heat and cooling pipes installed
in ventilated tunnels was developed and integrated into an
existing cable rating software [6]. The software also allows
for calculation according to the IEC method which is limited
to four identical cables systems.
CALCULATION OF CABLES IN TUNNELS
IEC 60287-2-3 Method
Capabilities
The IEC 60287-2-3 describes a method for calculating the
continuous current rating for cables of all voltages installed
in ventilated tunnels. The standard was published in 2017.
The main features of the calculation method are based on
the report of a CIGRE working group published in Electra
n°143 and 144 [2].
The air flow in the tunnel removes heat from the cables and
transports it along the tunnel axis, thus gradually increasing
the air temperature. Therefore, calculating the rating for
cables in ventilated tunnels must consider longitudinal
temperature gradients.
The numerical method consists of a thermal network
representing slices of the tunnel cross sections. One slice
is axially connected with the next by the longitudinal heat
transfer of the air flow along the tunnel. For every slice, a
delta-star transformation is applied in order to derive a
thermal network with one thermal resistance each between
the star point and the cable surface, the tunnel wall, and
the ventilating air respectively. This allows the definition of
a fictitious increase of the ambient temperature to account
for the ventilation. The equivalent thermal resistance of the
cable surrounding is used similar to the classical formula in
order to determine the permissible current rating.
The provided iterative method is fast and easy but based
on several simplifications:
• The thermal resistances, computed using temperatures
at the tunnel outlet, are assumed to be constant along
the tunnel route.
• The longitudinal heat transfer within the cables and the
surroundings of the tunnel is assumed to be negligible.
• All cables are assumed to be identical within the tunnel
and it is assumed that the tunnel cross-section does not
change with distance along the tunnel.
• Only steady-state conditions are considered
Limitations
The method is applicable to any type of cable but it has an
important limitation that where multiple circuits are
involved, their characteristics are assumed to be identical.
This means that multiple systems in a tunnel can be
calculated, but all systems are identical and equally loaded.
Extended Method
General Idea
The general idea of the newly developed extended method
is based on the previous work by J. A. Pilgrim et al
published in [3], [4], and [5].
A thermal network was designed such that each cable is
modeled explicitly for slices of 1 m length along the tunnel.
Using multi-level iterations and weighting of the data from
previous iterations, the tunnel air temperature and the
temperature of the inner tunnel wall can be calculated for
each slice. The axial thermal resistance due to the
movement of air through the tunnel is used to consider the
longitudinal temperature gradients.
Tunnel Equations
First, the conditions at the beginning of the tunnel are set
as follows:
The tunnel air temperature of previous slice
at(z-1)
and the
tunnel air temperature of the current slice
at(z)
are initially
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set to be equal to the inlet air temperature
at(0)
. The
temperatures of the inner tunnel wall
t(z-1)
and
t(z)
are
likewise initially set to the inlet air temperature
at(0)
. All
temperatures are in degrees Celsius.
And the initial temperature of the outer tunnel wall is
calculated
()=
() [1]
a
Ambient temperature [°C]
Ttw
Thermal resistance of the tunnel wall [K·m/W]
Te
External thermal resistance of tunnel including the thermal
resistance of the tunnel wall [K·m/W]
Now the conditions along the z-axis are being calculated
Prandtl number for air
()= 0.715 0.00025 () [2]
Kinematic viscosity for air
()= 9.5 10()+ 1.32 10 [3]
Thermal conductivity of air [W/(m·K)]
()= 7.2 10()+ 0.0242 [4]
Reynolds number for air to tunnel wall
()=/() [5]
Dit
Inner tunnel diameter [m]
Vair
Air velocity [m/s]
Note: The air velocity could also depend on the location
z
along the tunnel.
Volumetric heat capacity of air [Ws/(m3·K)]
()=()()/() [6]
Convection thermal resistance air-tunnel [K·m /W]
()=
().()..(). [7]
Heat capacity of the air flow [W/K]
()=() [8]
At
Cross-sectional area of the tunnel [m2]
Axial thermal resistance due to the movement of air
through the tunnel [K·m/W]
()= 1/() [9]
Then, the following properties are calculated for each cable
system:
Thermal resistance radiation surface-tunnel [K·m/W]
()=
()()
()()
[10]
Do
Outer diameter of the cable [m]
Stefan Boltzmann constant, 5.67036713·10-8
Kr
Radiation shape factor, refer to [1]
Kt
Effective emissivity of the cable surface
e
External temperature of cable [°C]
Reynolds number for air to cable is calculated
()=/() [11]
Thermal resistance convection surface-air [K·m/W ], for
ReairC
2000:
=
()().
.() [12a]
hbs
Heat dissipation coefficient for black surfaces in free air
and for ReairC > 2000:
=
()(). [12b]
Kcv
Convection factor, refer to [1]
External temperature of cable
()=()()()()()()()
()()
[13]
Wtot
Total losses per cable [W/m]
The following summation parameters over all
n
cables are
introduced in order to calculate the temperature of the inner
tunnel wall and the tunnel air temperature:
()=
()
[14]
()=()
()
[15]
()=
()
[16]
()=()
()
[17]
Now the temperature of the inner tunnel wall can be
calculated using the tunnel air temperature of the previous
slice
at(z-1)
:
()=
()
()()()
()
()()
()
()
()
()
()
()()
()
[18]
With the new temperature of the inner tunnel wall, the new
tunnel air temperature is calculated:
()=()1 + ()
+()()()
()()
[19]
And the temperature of the outer tunnel wall
to(z)
is
calculated using equation [1].
Cable System Equations
For each iteration, all current and temperature relevant
cable characteristics are updated for each system based
on the new tunnel temperature. This includes recalculating
the loss factors for sheath/screen and armour, the
temperatures and resistances of conductor, screen and
armour, and the ohmic losses using the set of equations
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from IEC 60287. Finally, the external temperature of the
cable is updated.
As the kinematic viscosity for air depends on the tunnel air
temperature, the Reynolds number for air to cable
Reair
is
recalculated using equation [11]. And the thermal
resistances for radiation surface-tunnel
Tst
and for
convection surface-air
Tsa
are recalculated using equations
[10] and [12].
And finally, the new permissible current rating is calculated
with respect to the difference of conductor temperature and
external temperature of the cable.
=
()() [20]
c
Conductor temperature [°C]
∆θd
Temperature rise by dielectric losses [K]
λ
1 Loss factor for sheath and screen
λ
2 Loss factor for armour
T1 Thermal resistance between one conductor and sheath
[K·m/W ]
T2 Thermal resistance between sheath and armour [K·m/W]
T3 Thermal resistance of jacket [K·m/W]
nc Number of conductors in cable
Solver Convergence Strategy
A certain air temperature at the end of the tunnel
at
results
in a corresponding permissible current rating
Ic
[A]. This
current should result in the same air temperature once the
solution has converged. For every iteration step
i
, the initial
tunnel air temperature
at(i-1)
(input to the current rating
calculation) and the new tunnel air temperature
at(i)
(resulting from the new current rating) are stored. The
correct tunnel air temperature will be located somewhere in
between these two temperatures. In order to make the
solver converge better, the data from each iteration step is
being weighted.
The weight of iteration step over all iteration steps are
calculated
=
[21]
Using the weight values, the temperature values are
calculated over all iteration steps
=
+
[22]
In order to improve the convergence, the temperature
increase over one iteration may be limited. Using the total
temperature value and the total weight, the temperature
increase over one iteration was limited to ± 10°C and a new
tunnel air temperature
at
is calculated as input to the next
iteration step.
Before the next iteration step, the values
Prair, air, kair, ReairT,
Cvair, Tat, Cav, Tf
need to be recalculated based on the new
tunnel air temperature. Also the values
Tst, Tsa, e,
st,
sa,
st, sa
are recalculated for each system.
Finally, the temperature of the inner tunnel wall is
recalculated as follows:
()=()()
()()
()
()()
[23]
IEC EXAMPLE CALCULATION
Description and Modeling
In the IEC standard 60287-2-3 Annex A, a calculation
example is given for three single-core cables without
armour and spaced vertically with a spacing between the
cables being three times their diameter within a circular
ventilated tunnel.
All installation data is listed in Table 1. Neither voltage level
nor the conductor cross-section is defined. Instead, the
cable outer diameter, the alternating current conductor
resistance at 90°C, the dielectric losses, the screen loss
factor and the thermal resistances are given. Furthermore,
the parameters of tunnel and surroundings are given and
the coefficient
Kcv
used to calculate the convection from
cable surface to air as well as the coefficient
Kr
and the
effective emissivity of the cable surface used to calculate
the radiation from cable surface to tunnel wall are given.
Table 1 Installation data [1]
The described extended method was integrated in [6] and
modeling of the cables and arrangement as well as all
testing took place using this software.
Fig. 1 depicts the layout of the IEC example calculation as
a two-dimensional preview. The tunnel wall is considered
to be of identical thermal resistance as the surrounding soil.
In order to emphasize this, the thickness of the tunnel wall
was set to zero for the preview and for the simulation.
Fig. 1 Arrangement of IEC calculation example
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Fig. 2 shows a preview of the modeled cable. A conductor
cross section of 2500 mm2 and a voltage of 220 kV was
assumed. However, for the calculation example, the given
parameters from Table 1 were hard-coded as constants.
Fig. 2 Cable cross section
Calculation Results
Using IEC Method
The IEC standard gives a resulting current rating of 2755 A
after three iterations for the 1 km long tunnel. The software
calculated a current rating of 2754.6 A after seven
iterations using the fixed parameters as given in Table 2.
Furthermore, the IEC standard gives a current rating of
1999 A for a 10 km long tunnel. The software calculated
1997.3 A. The difference to the value in the IEC standard
may be due to rounding of values during intermediate steps
as verification by hand calculation resulted in 1997.8 A.
LT
θt
θat
θe
Ic
Wsys
Ncalc
500
31.3
30.1
47.1
2933.6
357.6
6
1000
37.9
37.3
52.0
2754.6
316.7
7
5000
56.6
58.0
66.2
2162.6
199.8
7
10000
60.9
62.8
69.6
1997.3
172.2
8
Table 2. Results for IEC example, IEC method
LT
Length of the tunnel [m]
Wsys
Total losses of system [W/m]
Ncalc
Number of iterations
The number of iterations is quite stable but higher for longer
tunnels.
The temperatures along the tunnel are given in Fig. 3. Note
that the temperature of the outer tunnel wall
to
is identical
to the temperature of the inner tunnel wall
t
because the
thickness of the tunnel wall is zero.
Fig. 3 Temperature distribution 10 km, IEC method
Initially, the temperature of the tunnel wall is higher than the
air temperature. After about 2 km, the air temperature
becomes higher than the temperature of the inner tunnel
wall.
Using Extended Method
Using the newly developed extended method for the same
cases results in higher current ratings as shown in Table 3.
It is interesting to see that the number of iterations does
now correlate inversely with the tunnel length.
LT
θt
θat
θe
Ic
Wsys
Ncalc
500
31.1
29.9
46.9
2939.6
359.0
37
1000
37.6
37.0
51.8
2764.3
318.8
23
5000
56.1
57.5
65.9
2178.3
202.5
12
10000
60.7
62.6
69.4
2004.5
173.3
12
Table 3. Results for IEC example, extended method
The distribution of the temperatures for the 1 km long tunnel
using the extended method with constant parameters is
given in Fig. 4. It correlates well with the Figure A.1 in [1].
Fig. 4 Temperature distribution 1 km, extended method
Changing the inlet air temperature on the 1 km long tunnel
with an air velocity of 2 m/s gives the results in Table 4.
θat0
θt
θat
θe
Ic
Wsys
Ncalc
0.0
25.3
23.5
42.5
3087.1
394.7
22
10.0
31.6
30.3
47.2
2927.7
356.2
23
20.0
37.6
37.0
51.8
2764.3
318.8
23
30.0
43.4
43.4
56.2
2595.4
282.5
34
Table 4. Results at different inlet air temperatures
As an example, the distribution of the temperatures at 0°C
inlet air temperatures is shown in Fig. 5.
Fig. 5 Temperature distribution, 0°C inlet temperature
Changing the air velocity on the 1 km long tunnel with an
inlet air temperature of 20°C gives the results in Table 5.
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Vair
θt
θat
θe
Ic
Wsys
Ncalc
0.5
51.2
51.4
65.9
2175.9
202.1
36
1.0
44.8
44.3
59.6
2456.1
254.2
25
2.0
37.6
37.0
51.8
2764.3
318.8
23
4.0
31.2
30.8
43.7
3049.0
385.3
22
6.0
28.3
28.0
39.4
3191.4
421.0
20
Table 5. Results at different inlet air temperatures
The distribution of the temperatures at 0.5 m/s and 6.0 m/s
air velocity is shown in Fig. 6 and Fig. 7.
Fig. 6 Temperature distribution, at 0.5 m/s
Fig. 7 Temperature distribution, at 6.0 m/s
At low air velocity, the heat removal is also low and the
temperature increases quickly. At high air velocity, the heat
removal is high and the curves become more flat.
Using Extended Method with Calculated Cable
Characteristics
The IEC method assumes identical cable characteristics.
However, even three identical cables aligned vertically or
horizontally do have a different value for the radiation
shape factor Kr due to a different radiation view factor
coefficient Fm. This results in different radiation thermal
resistance surface-tunnel Tst and thus in different
temperatures for the center cable and the two outer cables.
In case of the calculation example from IEC standard, the
values for Kr using the formulas for Fm in Annex C of IEC
60287-2-3 and for Tst are shown in Table 6:
Cable
Fm
Kr
Tst
Center
0.903
0.107
0.367
Outer
0.952
0.054
0.349
Table 6. Calculated cable characteristics
The results for the different tunnel lengths at 20°C inlet air
temperature and 2 m/s air velocity are given in Table 7.
Note that the external cable temperature is given for both
the center cable and the two outer cables.
LT
θt
θat
θe
Ic
Wsys
Ncalc
0.5
31.1
29.8
46.8/46.6
2963.4
356.7
57
1.0
37.5
36.8
51.6/51.4
2791.1
316.7
23
5.0
55.9
57.2
65.7/65.5
2217.9
201.2
12
10.0
60.5
62.3
69.2/69.0
2048.3
172.1
14
Table 7. Results with calculated characteristics
Comparing Table 7 with Table 3 shows higher ratings. The
increase is 0.8%, 1.0%, 1.8%, 2.2% with higher difference
for longer tunnels.
MULTI-SYSTEM CALCULATION
Description and Modeling
Fig. 8 depicts the layout of four systems in a round tunnel.
The tunnel wall thickness is set to 20 cm with a thermal
resistivity of 0.7 K.m/W. The cables in the two systems on
the left (A above, C below) are identical to the previous
calculations whereas for the cables in the two systems on
the right (B above, D below) the conductor cross-section
was reduced to 1600mm2 and the conductor diameter
accordingly.
Fig. 8 Arrangement of multi-system calculation
Calculation Results
The current rating was calculated for a maximal permissible
conductor temperature of 90°C for all four systems in a
1 km long tunnel with an air velocity of 2 m/s and an inlet
air temperature of 20°C. Results in Table 8 show ambient
and tunnel parameters and in Table 9 system parameters.
This arrangement needed 59 iterations to converge.
θat0
θt
θat
Tat
Ttw
Te
20.0
60.5
58.7
0.022
0.014
0.253
Table 8. Ambient and tunnel parameters
Sys
Tst
Tsa
θe
θc
Ic
Wsys
A
0.370/0.351
0.206
67.5/67.3
90.0/89.9
2132.2
186.2
B
0.404/0.385
0.219
67.0/66.9
90.0/89.9
1725.9
165.6
C
0.485
0.339
72.0
90.0
1781.1
191.9
D
0.534
0.360
71.0
90.0
1518.2
164.2
Table 9. System parameters
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When the the current ratings are set to the calculated
values only 19 iterations were needed until the solver
converged with similar results.
SUMMARY AND OUTLOOK
An extended analytical method was developed to allow for
the calculation of multiple different cable systems or other
heat sources in ventilated tunnels. This overcomes the
important limitation of the IEC method that all systems have
to be identical and equally loaded.
The software was recently extended to calculate the rating
and temperatures of GIL in ventilated tunnels.
REFERENCES
[1] IEC 60287-2-3, 2017, “Electric cables – Calculation of
the current rating – Part 2-3: Thermal resistance –
Cables installed in ventilated tunnels”, International
Electrotechnical Commission, Geneva, Switzerland
[2] CIGRE Electra No. 143 and No. 144, 1992,
“Calculation of temperatures in ventilated cable
tunnels”, WG 21.08, International Council on Large
Electric Systems
[3] J.A. Pilgrim, D.J. Swaffield, P.L. Lewin, S.T. Larsen, D.
Payne, 2009, “Towards a more flexible rating method
for cables in tunnels”, 11th International Electrical
Insulation Conference, Vol. 25, Issue 1, Birmingham
[4] J.A. Pilgrim, D.J. Swaffield, P.L. Lewin, S.T. Larsen, D.
Payne, F. Waite, 2010, “Thermal rating implications of
the co-location of HV cable circuits in tunnels”, IEEE
International Symposium on Electrical Insulation, San
Diego
[5] J.A. Pilgrim, D.J. Swaffield, P.L. Lewin, S.T. Larsen, F.
Waite, D. Payne, 2010, “Rating independent cable
circuits in forced-ventilated cable tunnels”, IEEE
Transactions on Power Delivery, Vol. 25, No. 4, 2046-
2053.
[6] Rating software https://www.cableizer.com
GLOSSARY
IEC: International Electrotechnical Commission
GIL: Gas Insulated Lines
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