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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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