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Evaluation of the loading capacity of a pair of three-phase high voltage cable
systems using the finite-element method
Article in Electric Power Systems Research · July 2011
DOI: 10.1016/j.epsr.2011.03.005
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Electric Power Systems Research 81 (2011) 1550–1555
Contents lists available at ScienceDirect
Electric Power Systems Research
journal homepage: www.elsevier.com/locate/epsr
Evaluation of the loading capacity of a pair of three-phase high voltage cable
systems using the finite-element method
Jovan Nahman a,∗ , Miladin Tanaskovic b
a
b
Faculty of Electrical Engineering, University of Belgrade, Bul. K. Aleksandra 73, 11000 Belgrade, Serbia
Electricity Board Belgrade, Masarikova 1-3, 11000 Belgrade, Serbia
a r t i c l e
i n f o
Article history:
Received 3 February 2010
Received in revised form 31 January 2011
Accepted 6 March 2011
Available online 17 April 2011
Keywords:
High voltage cables
Loading capacity
Cyclic and emergency loads
Finite-element model
a b s t r a c t
A finite-element model is developed for the analysis of the heating of a pair of three-phase 110 kV underground cable systems in normal and emergency conditions. The main contribution of the paper is the
inclusion of the solar emission and radiation in the evaluation of the ampacity of cables and the analysis
of their effects. In addition, alternative approaches in dealing with emergency cases are studied providing
a reference for future applications in practice. The model developed is used for the study of a practical
case being under construction in Belgrade city area.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
The evaluation of the loading capacity of underground HV cable
systems installed in urban area is a complex task as it implies the
consideration of many specific factors including cable trench profile, and the properties of bedding and of various backfill materials
and covers. Thoroughly elaborated analytical approaches addressing the stationary cyclic load diagrams of cables have been provided
in the past [1,2]. Analytical methods have been also developed for
calculating the heating of cable bundles laid out in air in buildings
[3] and buried in ground [4,5], both in steady-state and transient conditions. The boundaries of the drying out area around the
cables have been approximately modeled by a cylindrical isothermal plane at the critical temperature for drying out of the soil,
determined based upon a series of experiments [6]. The aforementioned approaches provide good answers in the most of cases in
practice. However, the idealizations introduced related to various
structural details and thermal characteristics of materials used in
cable trenches reflect upon the results obtained. The method of
finite-elements enables a more detailed modeling of cable trenches.
In [7], this method has been used for the analysis of the temperature
rise of cables in case of step function loads, by assuming a uniform
environment. A more complex model that takes into account the
real shape and structure of cables and trench has been elaborated in
[8]. The present paper uses an enhanced version of the previously
∗ Corresponding author. Tel.: +381 11 337 0104; fax: +381 11 324 8681.
E-mail address: j.nahman@beotel.net (J. Nahman).
0378-7796/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsr.2011.03.005
mentioned model, which now incorporates also the effects of the
heat transfer by radiation and solar emission, to evaluate the loading capacity of a pair of three-phase cable systems sharing the same
trench. These two phenomena partially compensate each other at
specific ambient condition, which explains the good behavior of the
model suggested in [8] when compared with experimental results
[9]. However, the model presented in this paper accounts for all processes affecting the heating of cables and, therefore, can be applied
in all ambient conditions appearing in practice. It is used for analyzing various loading conditions of the considered cable systems
including overloads caused by outages.
2. Finite-element model
2.1. Linear approximation of radiation expression
The heat emitted by the radiation from a not black body per unit
surface equals, with respect to the Stefan–Bolzmann law [10],
R = ε0 (ϑ + 273)
4
(1)
ϑ designates the body temperature, and and ε0 are the radiation
and emissivity constants. The relevant value for the radiation constant is = 5.67 × 10−8 W/(m2 K4 ). For the emissivity constant the
value ε0 = 0.95 is adopted, which is characteristic for a grey body as
is the pavement above the trench. For a modest range of temperatures (ϑ1 , ϑ2 ) Eq. (1) can be approximated by a linear function
R = ε0 g0 ϑ
(2)
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J. Nahman, M. Tanaskovic / Electric Power Systems Research 81 (2011) 1550–1555
with slope g0 satisfying the relationship
ϑ2
ϑ2
4
ϑdϑ = ε0
g0 ε0
ϑ1
(ϑ + 273) dϑ
(3)
ϑ1
that gives
5
g0 =
5
2((ϑ2 + 273) − (ϑ1 + 273) )
5(ϑ22
− ϑ12 )
(4)
If the heat dissipated in the air from the cable trench surface should
be determined, then temperatures ϑ1 and ϑ2 are the extreme seasonal pavement temperatures. Parameter g0 should be separately
determined for each season for which the loading capacity analysis
has to be performed.
A linear function can also approximate the ambient (air) radiation, acting as a thermal load for cables. The slope of this function
is determined by Eq. (4) with ϑ1 and ϑ2 being extreme air temperatures in the period under consideration.
2.2. Solar emission
in winter and E = 1.35 kW/m2 in summer, during characteristic periods of
day. The exposed bodies absorb a portion of this heat, proportionally to the body apsorptivity constant. For grey colored pavement
surface this constant equals ˛0 = 0.80.
Based upon a wide set of experimental results, the following
empirical expression is constructed for determining the temperature of the boundary isotherm separating the drying out and
non-drying out areas [6]
100
3
(5)
with ϑS and m being the soil temperature and cable load factor,
respectively. In Eq. (5), the temperatures are in ◦ C.
2.4. Transient temperature equation
The transient temperature equation for the cables’ subspace partitioned into a network of finite elements with N nodes has the
following general form
d[ϑ(t)]
+ [K][ϑ(t)] = [R(t)]
dt
(6)
The mathematical model developed in [8] is enhanced by including the radiation and solar emission effects, for completeness. In Eq.
(6), [ϑ(t)] is N dimensional column vector of temperatures of nodes,
[C] is N by N heat capacitance matrix, [K] is N by N heat conductance, convection and radiation matrix, and [R(t)] is N dimensional
heat load vector stemming from internal heat generation and surface convection, ambient radiation, and solar emission. All these
matrices and vectors are formed from the corresponding data for
the finite elements covering the space domain under consideration.
The parameter matrices in Eq. (6) are
[K] = [Kc ] + [Kh ] + [Kr ]
(7)
[R(t)] = [RQ (t)] + [Rh (t)] + [Rr (t)] + [Rs ]
(8)
Indices c, h, r and s in Eqs. (7) and (8) denote conduction, convection,
radiation and solar emission, respectively. Index Q refers to the element internal heat generation. The Joule losses in phase conductors
and metallic sheaths including the change of associated resistances
with temperature are taken into account as well as the dielectric losses in the insulating materials. Triangular finite elements
(9)
where
[KE ] = [K] +
[C]
t
[RE (tn+1 )] = −(1 − )[K] +
2.3. Effect of draining out of soil
[C]
are used in this analysis. The matrices figuring in Eqs. (6)–(8) are
obtained by transferring the corresponding relationships for elements written for their nodes in local coordinates (Appendix A) to
entire network numbering scheme. Eqs. (7) and (8) in their general
form relate only to the boundary elements with edges lying on the
pavement surface. For all other nodes, only [Kc ] and [RQ (t)] are not
null matrices. For nodes remote from cables, temperatures are fixed
at the presumed ambient temperature. It is clear that for equally
loaded pair of cable systems the analysis of the temperature field
could be performed for a half of the space under consideration separated from the other half by the plane of symmetry between the
systems. Then, the boundary conditions for the edges of elements
lying in this plane would be no heat transfer through the plane.
However, as this paper investigates emergencies with significantly
different current flows through the considered two cable systems,
such a simplification of the approach was not possible.
By discretization for a minor step t Eq. (6) converts into a simple recurrent relationship for determining temperature variation
in time [8,11]
[KE ][ϑ(tn+1 )] = [RE (tn+1 )]
The solar emission is, in authors’ country, E = 0.65 kW/m2
ϑB = ϑS + 15 + (1 − m)
1551
(10)
1
[C] · [ϑ(tn )]
t
+ (1 − )[R(tn )] + [R(tn+1 )]
(11)
with being the integration stability factor. In this analysis = 1/2
and t = 5 min has been adopted as in [8], providing a good accuracy
with the modest computer time consumption.
In Eq. (6), we assumed that the heat capacities of materials under
consideration do not depend on the temperature. In the processes
of heating of underground cables, the temperatures of materials
rang from the ambient temperature to maximum 90 ◦ C, which is a
quite narrow interval. Therefore, such an assumption is generally
taken as justified [2,5–7]. Otherwise, the variation of heat capacities with temperature could be easily modeled in the stepwise
calculation process described in this paper.
2.5. Loading capacity evaluation
The practical task is to determine the maximum loading capacity of a pair of three-phase cable systems supplying the same
consumption area by taking into account as detailed as possible
the cable trench shape and impacts of various materials including
non-draining mixture, backfill and covers. Possible emergencies of
various durations are also considered. These situations can arise
after the outage of one of the cable systems causing an extra load
for the sound system. The maximum allowable conductor temperature during the emergencies must not exceed this maximum
temperature for normal daily cycling loads, for security reasons.
The required input data for the calculation process are the per
unit cyclic load diagram for cables in normal operation, and the
duration, the magnitude and the shape of the superimposed emergency load.
The calculation procedure implies the following steps:
1. Adopt the magnitudes of the considered cyclic diagram that
should serve as an initial guess at the final solution.
2. For a set of succeeding daily diagrams, using Eq. (12), calculate
the temperatures of network nodes. Proceed with this calculation until the stationary temperature pattern is reached. If the
draining out condition is met during the calculation flow, change
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J. Nahman, M. Tanaskovic / Electric Power Systems Research 81 (2011) 1550–1555
0.1
1552
0.1
1
3
0.75
0.15
0.16 0.34
1.65
2
4
5
0.37
0.4
1.0
0.37
0.4
0.6
0.17
0.17
1.74
Fig. 1. Cross section of the trench for two three-phase systems of 110 kV cables: 1 – asphalt cover, 2 – concrete covers, 3 – backfilling material, 4 – special mixture bedding,
5 – cables (all measures are in m).
adequately the thermal conductivity of the trench backfilling
material in the succeeding calculation steps.
3. The emergency load superimposes upon the cycling load of the
sound cable system at the instant of the highest conductors’ temperature. The calculation of temperatures continues for a period
lasting approximately twice the period with emergency load.
4. If the temperature of cable conductors has exceeded the maximum allowable value at any time instant, return to step No.
1 with decreased magnitudes of the cycling daily diagram.
However, if this temperature is lower than the allowable temperature, return to step No. 1 with increased magnitudes of the
daily load diagram.
5. If the maximum temperatures of cable conductors match with
the maximum permissible temperatures, the calculation process
terminates. The cycling load diagram that satisfies the aforementioned criterion is the required solution.
3. Application
3.1. Application example description and data
The finite-element model presented in the previous section has
been applied to determine the maximum loading capacity of a pair
of 110 kV XHE – a three-phase cable systems to be installed in
Belgrade area. Fig. 1 shows the cross section of the trench used
for laying the cables. The cable bedding is made of a special nondraining out mixture. The winter is the critical period of the year
for load capacity rating as a considerable portion of households use
electricity for heating. Cable systems have two separate and a common concrete covers as mechanical protection. Fig. 2 presents the
per unit form of the typical winter weekday load diagram.
Table 1 quotes the data for relevant materials. For soil, values
for conduction coefficient are given for non-draining and draining
out cases.
In authors’ country the trench surface temperature for winter
lies between 0 ◦ C and 6 ◦ C. By inserting these extreme temperature values in Eq. (4) we obtain for the slope of the linear function
Fig. 2. Weekday per unit load diagram during winter.
approximating the heat discharging from the cable trench cover
out into air
g0c = 21, 029, 544.05 (K3 )
(12)
The average values of air temperatures during a winter day rang
from −20 ◦ C to 5 ◦ C. For these extreme temperatures the slope for
the expression modeling the heat radiated to the trench cover by
the air equals
g0a = 18, 798, 148.5 (K3 )
(13)
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J. Nahman, M. Tanaskovic / Electric Power Systems Research 81 (2011) 1550–1555
1553
Table 2
Maximum allowable peaks of cycling loads.
Table 1
Thermal parameters of materials.
Material
Conduction
coefficient k
(W/(K m))
Density (kg/dm3 )
Specific heat c
((J/(K kg)) × 10−3 )
Aluminum conductor
Insulation
Copper screen
Outer sheath
Mixture bedding
Concrete
Asphalt
Surrounding soil
220
0.286
385
0.286
1.219
1.111
0.075
0.83/2.5
2.70
0.93
8.92
0.92
1.95
2.20
2.10
1.49
0.919
3.978
0.393
2.610
1.026
0.832
1.657
1.054
CASE 1
CASE 2
CASE 3
Emergency situation
duration (h)
Load peak for
one system (A)
Total load peak for
two systems (A)
0
2
4
8
No limits
863
721
662
652
890
1726
1442
1324
1304
890
3.2. Maximum allowable loads
The maximum allowable peaks of the cyclic load for the pair of
systems of cables were determined for the following cases:
Convection coefficient at the ground surface is h = 5 W/(K m2 ).
Maximum permissible conductor temperature equals ϑm ≈ 90 ◦ C
Conductor and copper screen Joule losses at ϑm are 0.0408
I(t)2 and 0.027 I(t)2 , respectively, whereas dielectric losses
at rated voltage equal 119.3, with all losses expressed in
W/m3 . I(t) is the rms of the cable load current at instant
t when the maximum conductor temperature ϑm is attained.
The variation of the specific conductor conductivity with temperature is accounted for during the entire calculation flow.
The heat transfer by convection and radiation are determined
by referring to the variation of the air temperatures during
day.
Fig. 3 displays the global finite-element network constructed
for the analysis of the temperature distribution indicating the
plane coordinates of the vertices of triangles: network nodes.
The model of a system of cables is detailed presented in
Fig. 4.
CASE 1: Both cable systems are equally loaded with no concern
about emergency conditions.
CASE 2: Both cable systems are equally loaded. In the case of the
outage of a system, the sound system overtakes the load of the
faulted system for a short period needed for adequate remedy
actions in the distribution system.
CASE 3: Only one cable system is loaded while the other one serves
as a cold reserve.
Table 2 gives the maximum allowable peaks of cycling loads for
cables in the cases under consideration obtained by applying the
finite-element model developed in this paper.
It is noteworthy to stress that in CASE 1, if the solar emission
were not considered, the maximum calculated allowable load current would be 890 A, which is 27 A higher than the value obtained
with the complete model. If the radiation were not considered, the
Fig. 3. Finite-element network for the cables’ trench and surrounding environment (all measures are in m).
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J. Nahman, M. Tanaskovic / Electric Power Systems Research 81 (2011) 1550–1555
Fig. 4. Finite-element network modeling a cable system (all measures are in m).
maximum allowable load current would be 855 A. This is slightly
lower than the correct value.
For illustration, Figs. 5 and 6 display the load diagram and the
variations of the hottest spot conductor temperatures of the sound
cable system for CASE 2 for the overload lasting 8 h.
As can be seen from Table 2, the allowable load peaks for emergencies are remarkably lower than this peak determined when
no emergency is considered. That means that a serious analysis
of such situations must be conducted during the design phase
1350
1300
1250
1200
1150
1100
Load current (A)
1050
1000
950
900
850
800
750
700
650
600
550
500
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Time (hours)
Fig. 5. Load diagram of sound cables during the emergency situation lasting 8 h.
Fig. 6. Cable conductor hottest spot temperature variation for overloads lasting 8 h.
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J. Nahman, M. Tanaskovic / Electric Power Systems Research 81 (2011) 1550–1555
to determine an optimal solution regarding both the capital and
undelivered energy costs and risks. Moreover, the results in the
last column of Table 2 clearly show that the load supply security concept modeled by CASE 3, which is sometimes advocated,
could not be considered as economically justified. For the same
capital costs, the maximum load that can be delivered to the
customers in CASE 3 is much smaller than in all other studied
cases.
⎡ ⎤
1
1
⎢ ⎥
[RQ ] = Q (t)dS ⎣ 1 ⎦
3
1
(A7)
1
1
⎢ ⎥
[Rh (t)] = ϑe (t)dhl12 ⎣ 1 ⎦
2
0
(A8)
⎡ ⎤
⎡
4. Conclusions
The finite-element approach described in the paper makes it
possible to evaluate with a reasonable accuracy the loading capacity of cables sharing the same trench by detailed modeling of the
trench profile and bedding properties including the effects of the
drying out of soil. As such, it can be used for optimal construction of cable trenches by properly accounting for both the capital
and the outage costs. The calculations performed have shown that
both the radiation and solar emission affect the heating of cables.
The effect of solar emission would be particularly pronounced if
the summer period was critical for the rating of cables and the
peak load was attained at the time of the day with maximum solar
activity.
The analysis conducted in the paper has shown that the emergencies considerably affect the allowable loading of cables and
therefore deserve a serious consideration. It was also demonstrated
that the security concept to use a cable system as the cold reserve
could not be generally justified from the economical standpoint as
it can lead to an inefficient exploitation of the installed equipment.
The recommendable solution is to load equally the cables during
normal operation.
Appendix A.
A.1. Parameters of triangular elements
The coefficient matrices in Eqs. (5)–(7) for a triangular element,
in its local coordinates, are [8,10]:
⎡
2 1
1
1
⎢
dcS ⎣ 1 2
12
1 1
[C] =
⎤
⎥
1⎦
(A1)
2
[Kc ] = kdS[B]T [B]
[B] =
1
2S
(A2)
b1
b2
b3
c1
c2
c3
(A3)
b1 = y2 − y3 ,
b2 = y3 − y1 ,
c1 = x2 − x3 ,
c2 = x3 − x1 ,
[Kh ] =
⎡
2 1
1
⎢
dhl12 ⎣ 1 2
6
0 0
⎡
[Kr ] =
0
2
1
⎢
dg0c l12 ⎣ 1
6
0
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⎤
⎥
0⎦
0
b3 = y1 − y2
c3 = x1 − x2
(A4)
(A5)
⎤
1
0
2
0⎦
0
0
⎥
(A6)
1555
[Rr (t)] =
1
⎤
1
⎢ ⎥
ϑe (t)g0a ε0 dl12 ⎣ 1 ⎦
2
Y0
(A9)
⎡ ⎤
1
⎢ ⎥
[Rs ] = ˛0 Edl12 ⎣ 1 ⎦
(A10)
0
Here and c are material density and specific heat capacity, S and
d are triangle area and axial length of cable section, k and h are
convection and conduction factors, xk and yk , k = 1, 2, 3, are plane
rectangular coordinates of element nodes, l12 is the length of triangle edge connecting nodes 1 and 2, Q(t) is the heat generated inside
the element, ϑe (t) is the air temperature, E and ˛0 are the specific
solar heat emission and the absorption coefficient of the pavement
surface. It is implied that the edge connecting nodes 1 and 2 lies on
the pavement surface. As all terms of Eq. (6) are proportional to d,
this relationship is valid for all d. Hence, d = 1 m is adopted.
References
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[5] International Standard IEC 853-2, 1989.
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[7] N.N. Flatabo, Transient heat conduction problems in power cables solved by
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cables using the finite element method, EPSR 61 (2002) 109–117.
[9] M. Sredojevic, R. Naumov, D. Popovic, M. Simic, Long-term investigation of
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[10] A. Chapman, Heat Transfer, Macmillan Publ, Nerw York, 1974.
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Sons, New York, 1982.
Jovan Nahman was born in Belgrade, Serbia. He received his Dipl.Eng. grade in
electrical power engineering from the Faculty of Electrical Engineering, University of
Belgrade, in 1960, and Tech.D. from the same University in 1969. From 1960–2001 he
was with the Faculty of Electrical Engineering, Belgrade, as a professor at the Power
System Department. Currently he is active as a freelance consultant for electric
power systems.
Miladin Tanaskovic was born on December 24, 1956 in Belgrade, Serbia. He
attended Faculty of Electrical Engineering, University of Belgrade, where he graduated in 1981. He got master’s degree in 1993 and earned his doctorate in 2003, at
the same University. Since 1988, he is with the Electricity Board Belgrade. Currently
he holds the post of the general manager’s adviser for research and development
activities.