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9th International Conference on Insulated Power Cables
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Calculation of the current rating for complex cable arrangement
in a deep tunnel
George J. ANDERS; Lodz University of Technology, Lodz, Poland, george.anders@bell.net,
Boguslaw BOCHENSKI, Kinectrics Inc., Toronto, Canada, Boguslaw.bochenski@kinecrics.com
Gunnar HENNING, ABB HV Cables, Karlskrona, Sweden, gunnar.henning@se.abb.com
ABSTRACT
This paper addresses several issues related to the
calculation of rating of cables located in deep tunnels. It
shows the results for ventilated and unventilated tunnel 3 m
in diameter 30 m deep. The paper also introduces of a
simplified model for the temperature calculations in a
ventilated tunnel.
KEYWORDS
power cables, rating calculations, tunnel installations
tunnel. The paper discusses also how such installations
can be rated and shows the results for ventilated and
unventilated tunnel 3 m in diameter 30 m deep.
SIMPLIFIED MODEL FOR RATING OF
CABLES IN A VENTILATED TUNNEL
Rating of cables installed in a deep ventilated tunnel was
discussed in detail in [1]. A comprehensive model shown
there is fairly complex. Therefore, the authors of this paper
decided to introduce a simplified set of equations that can
be used in a first approximation of current rating
calculations for such installations.
INTRODUCTION
List of symbols
Due to congested infrastructure in urban centres, increasing
number of cable circuits is installed in deep tunnels. An
analytical model for rating of cables in such installations has
been described in the Jicable’11 paper by Dorison and
Anders [1]. They observed that due to the soil thermal
inertia, a long duration is necessary to reach the steadystate value; thus, instead of using the standard IEC formula
for the cable external thermal resistance, 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. They defined a fictitious equivalent depth 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.
D
H
L
z0
m
m
m
m
Tunnel diameter
Tunnel depth to centre
Tunnel length
Characteristic length of the tunnel
v
T0
m/s
K×m/W
air velocity in the tunnel
Thermal resistance of the tunnel
T0e
K×m/W
Equivalent thermal resistance of the
Wc
W/m
tunnel
Cable losses
Wa ( z )
W
Heat transported by air flow
Wo ( z )
W/m
Heat transferred through tunnel wall
θ ( z)
°C
θi
θ0
∆θ
°C
Temperature of air in the tunnel at
distance z.
Temperature of air at the inlet
°C
Temperature of the ground level
K
c pa
Ws/K⋅m
Temperature drop in the transition
layer at the tunnel wall
Specific heat of air
h
W/K⋅m
The analytical approach discussed in the above mentioned
paper is applicable to simple cable geometry. However, in a
recent project involving installation of four 230 kV cable
circuits located in a new concrete tunnel, due to personnel
safety of people entering the tunnel for maintenance and
inspection, two of the four circuits will be installed inside
ducts embedded in concrete on the tunnel side wall (see
Fig. 1 below) or in trough at the bottom floor of the tunnel.
The calculation of the current rating of such installations is
not easily amenable to analytical approaches. One of the
contributions of the paper is an introduction of a simplified
model for the temperature calculations in a ventilated
3
2
Heat transfer coefficient at the
tunnel wall.
Heat balance equation
Figure 1 shows heat balance at distance z in the tunnel.
The heat dissipated from the cable is transported by the air
flow along the tunnel and conducted through the tunnel wall.
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9th International Conference on Insulated Power Cables
dθ
=
Wc ⋅ T0 e + θ 0 − θ
dz
π⋅
(7)
2
D
⋅ v ⋅ c pa ⋅ T0 e
4
Introduction of the characteristic tunnel length zo gives:
dθ
dz
=
Wc ⋅ T0 e + θ 0 − θ z0
(8)
Integrating along the tunnel, we obtain:
z
 z
θ
ln (Wc ⋅ T0 e + θ 0 − θ ) θ =  − 
i
 z0  0
Fig. 1 Illustration of the heat balance equation
The temperature along the tunnel length is thus given by:
The equation for heat balance in the tunnel is:
Wc ⋅ dz =
θ − ∆θ − θ 0
T0
(1)
The temperature θ(z) of the air in the tunnel is a function of
the distance z along the tunnel length. In the equations this
dependency is omitted for clarity of the presentation. The
heat transfer coefficient h at the tunnel wall gives rise to a
temperature drop ∆θ over a transition layer.
The
temperature drop may be eliminated through introduction of
an equivalent thermal resistance Toe as shown below.
Equivalent thermal resistance
The heat transfer through the transition layer of a tunnel
length of 1 meter is:
h ⋅ ∆θ ⋅ π ⋅ D =
θ − ∆θ − θ 0
(2)
T0
z
zo




(10)
where T0 e is given in (5) and:
z0 = π ⋅
D2
⋅ v ⋅ c pa ⋅ T0e
4
(11)
Comparison with the IEC Standard
In 2014 IEC has published a new document 60287-2-3 [2]
giving equations for the rating of cables installed in a
ventilated tunnel. It is interesting to observe that equation
(10) is almost identical to the one in the standard. A
numerical example illustrates this point.
Let’s consider three identical cables installed in a tunnel.
Cable losses are 36.7 W/m×phase; that is,in total 330 W/m.
The remaining parameters are as follows:
H = 20 m
θ − θ0
1 + h ⋅ π ⋅ D ⋅ T0
∆θ =
(3)
An equivalent thermal resistance is defined from the
equation:
θ − θ0
− θ0
1 + h ⋅ π ⋅ D ⋅ T0
T0
T0 e =
=
θ − θ0
(4)
T0 e
1 + h ⋅ π ⋅ D ⋅ T0
h ⋅π ⋅ D
(5)
It is clear from the last equation that the heat transfer
coefficient at the tunnel wall has small influence on the heat
transfer to the ambient. The heat transfer coefficient is at
2
least h = 2 W ⋅s/K⋅m .
Solution of the heat balance equation
The heat balance equation is modified as follows:
θ − θ0
T0 e
⋅ dz + dθ ⋅ c pa ⋅ v ⋅ π ⋅
Separation of variables gives:
L = 800 m
v = 2 m/s
W c = 330 W/m
θI = 20 °C
Thus:
Wc ⋅ dz =
−
D=4m
From (2), we obtain:
θ−



θ ( z ) = θ i + (Wc ⋅ T0e + θ 0 − θ i ) ⋅ 1 − e
2
D
⋅ dz + dθ ⋅ c pa ⋅ v ⋅ π ⋅
4
(9)
D2
4
(6)
θ0 = 20 °C
h = 1 W/K⋅m
2
In the simplified method temperature of air at the tunnel
outlet is calculated. The cable conductor temperature is
calculated with use of thermal resistances and the heat
transfer coefficient between cable surface and air. The
2
value of the coefficient is h = 10 W/K⋅m .
The air temperature at the tunnel outlet computed using the
IEC approach is θIEC(L) = 28.6 °C and the simplified method
gives θS(L) = 28.9 °C. As mentioned earlier in the text the
agreement should be good between the methods.
Of course, the problem lies in the details. One should
remember that in (5) we did not say how the heat transfer
coefficient is computed, nor how the value of T0 is obtained.
These can be found in [1] and [2]. However, an important
observation is made below equation (5); namely, that the
value of this coefficient dos not play an important role in the
calculations.
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COMPLEX TUNNEL EXAMPLE
This section addresses ampacity issues for 230 kV cables
located in a tunnel. A new concrete tunnel in downtown
Toronto approx. 3m in diameter, 30m below grade, 650 m
long, and connect to the existing tunnel. There will be four
circuits inside the new tunnel’.
Due to personnel safety of people entering the ‘new tunnel’
for maintenance and inspection, two of the above four
circuits will be installed inside ducts (PVC, FGBR or HDPE)
embedded in concrete on the tunnel side wall and in trough
at the bottom floor of the tunnel.
The following sections review the input parameters and
discuss the assumptions made during the calculations. The
results are presented next followed by a discussion.
Input data
Given values
Cable: XLPE 230 kV, 2000 kcmil.
Cable maximum operating conditions:
o
maximum voltage of 127 kV,
o
maximum continuous load 1200 A in
summer or 1300 A in winter,
o
emergency 2-hour limited time rating of
1700 A,
o
maximum
allowable
continuous
conductor temperature of 90°C,
o
maximum allowable emergency (2-hr)
conductor temperature of 105°C,
Daily load factor of 0.85,
Maximum soil temperature at tunnel depth of
15°C,
Assumed thermal resistivity of soil of 1.0 (K*m/W).
Cable installations presented in Fig. 2.
Bonding – single point bonded.
•
•
•
•
•
•
•
(a)
(b)
Derived values – Losses
Fig. 2 Cable installations: (a) Option A, (b) Option 1
Conductor
29.5 W/m @90°C, 1200 A,
Load Factor=0.85
Conductor:
34.6 W/m @90°C, 1300 A,
Load Factor =0.85
Sheath:
3.7 W/m @90°C, 1200 A
Sheath:
4.3 W/m @90°C, 1300 A
Dielectric:
1.3 W/m
Two different tunnel constructions were investigated as
shown in Fig. 2.
Project requirements
The main goal of the calculations was to assess whether
the proposed cable arrangements satisfy the ampacity
requirements specified above. The following calculations
were specified by the client:
•
Assumptions
The following assumptions were made either due to lack of
data or for modeling purposes:
•
•
•
•
The isothermal boundary conditions were
assumed on all sides of the examined installation.
The tunnel is long enough for two-dimensional
models to be applicable.
The tunnel is not ventilated during normal
operation.
Ampacity of the ventilated tunnel will also be
computed.
•
•
Calculate the current rating for the following preferred
cables – 230 kV, 2000 kcmil copper conductor, XLPE
insulated, HDPE served - in ducts embedded in
concrete for the configuration shown in Fig.2a. Suggest
suitable spacing between the ducts, for different type of
ducts PVC, FGBR and HDPE.
If the preferred cables conductor (2000 kcmil copper) is
not able to carry required current of 1200/1300 A @ 90°
C for both summer/winter continuous when installed in
ducts embedded in concrete wall of the THES tunnel,
suggest a cable conductor size that is adequate and
available to meet all the ratings for conditions specified
above. Also, suggest the required spacing between
phases for ducts to be placed inside the tunnel wall.
Create a thermal model for the whole tunnel system with
two cable circuits on the rack/brackets in air and two
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•
•
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9th International Conference on Insulated Power Cables
circuits in ducts embedded in concrete. Calculate the
ambient air temperature inside the tunnel for a
conductor temperature of 90°C and 105°C operation
conditions and specified currents.
Perform similar calculations as above for the trough
section of the tunnel and suggest dimension of the
troughs to be built at the floor of the tunnel.
Perform calculations assuming forced longitudinal air
flow at the speed of 1200 ft/min (6.096 m/s).
solution which will result in the installation being capable to
carry the required load.
Calculation method
For ampacity calculations commercial software called
COMSOL was used which allows analysis of complex
installations using the finite element method (FEM).
The FEM was described in many textbooks; thus, the details
will not be provided here. The models of the two tunnels are
shown in Fig. 3. An application of the program involves
preparation of:
a)
b)
c)
d)
graphics,
definition of materials,
domain settings and boundary conditions, and
creation of the mesh.
The mesh is presented in Fig. 3. Materials were defined
using either provided physical properties or from tables.
Fig. 4 Temperature distribution and air velocity field in
rated conditions for Option A
Continuous winter ratings for the proposed
installations
The results for the winter steady state conditions are shown
in Table 1.
Table 1 Winter continuous ratings for both tunnels in
original configuration
Installation
Ampacity (A)
Option A
860
Option 1
866
Fig. 3 Mesh and contours of the cable installation
models; the coordinates are expressed in meters [m]:
Option A left and Option 1 right
The heat transfer employs three physical phenomena:
conduction (mainly in solids), convection (in air) and
radiation from surfaces. Convection, however; requires
some minimal volume of air to occur and is very limited in
ducts where most of the heat is transferred by radiation.
Results
In order to determine whether the cables in the installations
presented in Fig. 2 are capable to carry the load specified in
the requirements, a number of computations (according to
the specifications presented above) had to be performed.
However, the order of the tasks was modified to avoid
repeating unnecessary calculations. The results of each of
them are presented below.
Figs. 4, 5 and 6 present the results of the FEA analysis in a
form of the temperature distribution and air flow maps for
Option A and Option 1. The picture is similar for all further
cases and the only difference is in the current which
generates the losses.
Fig. 5 Temperature distribution and air velocity field
in rated conditions for Option 1
Result of the calculations show that the continuous rating of
the proposed installations is lower than the requirements.
The goal of the remaining studies was to recommend a
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Table 3 The effect of the trough size on the cable
rating – Winter continuous rating
Trough size (H x W)
Rating (A)
0.35m x 0.5m
866
0.35m x 0.75m (wide trough)
868
1m x 0.5m (tall trough)
895
1m x 0.75m (wide and tall trough)
900
Two additional conductor sizes
Fig. 6 Temperature distribution in tunnels and in soil
above the tunnels for Option A (left) and Option 1 (right)
Three different types of ducts PVC, FRE and
HDPE and different spacing between the ducts
Study of the impact of the type of cable ducts and different
spacing between ducts is presented in Table 2. The cables
are planned to be laid in the 8” ducts; therefore, three ducts
of the same size but different materials were tested.
Table 2 The effect of the duct type on the cable rating
The results of the analysis are presented in Table 4.
Table 4 The effect of the conductor size on the cable
rating – Winter continuous rating
Cable
Option A (A)
Option 1 (A)
1000 mm2
860
866
2000 mm2
1229
1234
2500 mm2
1384
1388
Ampacity (A)
The required value of ampacity for winter conditions is 1300
A and only the 2500 mm2 cable is capable to carry this load
continuously without exceeding the temperature limit.
Duct
Distance 10.5°
Distance 9.5°
Distance 11.5°
PVC
858
857
858
Continuous and 2-h winter ratings
HDPE
857
856
858
FRE
860
859
859
The previous sections proved that the only solution to
significantly increase the ampacity is by increasing the size
of the cable; therefore, the final calculations were performed
for these conditions.
2
The ducts were moved within the same boundaries as the
initial proposed installation. The distance is expressed in
degrees due to the way the ducts were drawn – one duct is
copied with rotation referenced to the geometrical centre of
the tunnel (coordinates 0,0). This approach allows keeping
a constant distance from the outer surface of the tunnel and
from its central point. There is limited space for ducts
between small PVC ducts at the bottom and the top
concrete surface on top; hence, limited range within which
the distance between ducts can be changed. The results
show an insignificant change of ampacity within the
aforementioned boundaries.
Table 5 Ampacity (A) of the 2500 mm cables without
forced ventilation
Winter
Summer
Continuous
2-hr
rating
Continuous
2-hr
rating
Option
A
1384
2640
1338
2610
Option
1
1388
2650
1343
2625
Continuous and 2-hr ratings with forced air flow
Different troughs dimensions
Three different through dimensions were investigated for
Option 1 as shown in Table 3.
In order to increase the ampacity, forced air flow at the rate
of 1200 ft/min is to be used. The calculations were
performed for the initial cable size of 2000 kcmil and the
results are presented in Table 6.
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Table 6 Ampacity of the 2000 kcmil (1000 mm ) cable in the tunnels with forced ventilation
Ampacity
Option A
Ampacity
Option 1
Air temperature
(inlet) (°C)
Continuous
(A)
2-hr rating
(A)
Air temperature
(inlet) (°C)
Continuous
(A)
2-hr rating
(A)
20
1636
1977
20
1346
1743
25
1570
1924
25
1296
1705
30
1502
1871
30
1244
1668
35
1431
1817
35
1191
1630
Introducing the forced air flow significantly increases the
ampacity. In fact, the ampacity of the cables in tunnels
(trefoils) is much higher than presented in the table while
the limiting cables are these in the ducts/troughs. In
addition, Option A has higher ampacity than Option 1
because of better cooling conditions – the cables are
separated and the mutual heating is decreased and at the
same time the cooling surface is larger.
c.
Situation improves dramatically when the forced
ventilation is introduced in the tunnel. With air
velocity of 1200 ft/min, the 2000 kcmil cable will
satisfy both the winter and summer continuous and
emergency ratings when the inlet air temperature
does not exceed 25°C. This is a reasonable
assumption since the tunnel is located at the 30 m
depth. At such depths, the soil temperature is
expected to be constant during the year and likely
will not exceed 10°C.
Even with hot air
temperature at the earth surface, the tunnel inlet
temperature should be below 20°C.
d.
Because of cable separation and better cooling
conditions, Option A shows better thermal
performance.
CONCLUSIONS
The following conclusions can be drawn from the ampacity
studies performed for the two tunnel cross sections shown
in Fig. 2.
a.
When the tunnels are not ventilated, the existing
2000 kcmil cable construction will not meet
ampacity requirements.
b.
The sensitivity studies have shown that only the
2
2500 mm cable will have sufficient rating to meet
the steady state ampacity requirements in the
unventilated tunnel. The requirements are also met
2
when the 1000 mm cables are on racks and the
2
2500 mm circuits are placed in ducts/troughs.
Acknowledgements
The authors would like to thank Mr. Badrinath Comar and
Mr. Arnold Brakel from Hydro One Networks Inc. in Toronto,
Canada for their support, encouragement and active
involvement in performing these studies.
REFERENCES
[1]
E. Dorison, G.J. Anders Current rating of cables installed in
deep or ventilated tunnels. Jicable 2011 rapport C.8.3.
[2]
IEC Standard 60287 – ELECTRIC CABLES – CALCULATION
OF THE CURRENT RATING – Part 2-3: Cables installed in
ventilated tunnels.
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