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21 Xynis

This document summarizes research into optimally positioning buried power cables. It presents: 1) A numerical simulation studying different arrangements of parallel underground cables to minimize temperature variations, using COMSOL software. 2) Analytical results showing the power levels cables should carry for an equal temperature rise, with outer cables allowing much higher transmission than central cables. 3) A theoretical analysis of the optimal uniform temperature situation, reducing the problem to an algebraic equation set solvable for the power levels.

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0% found this document useful (0 votes)
61 views5 pages

21 Xynis

This document summarizes research into optimally positioning buried power cables. It presents: 1) A numerical simulation studying different arrangements of parallel underground cables to minimize temperature variations, using COMSOL software. 2) Analytical results showing the power levels cables should carry for an equal temperature rise, with outer cables allowing much higher transmission than central cables. 3) A theoretical analysis of the optimal uniform temperature situation, reducing the problem to an algebraic equation set solvable for the power levels.

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Optimal Position of Buried Power Cables

Article  in  Elektronika ir Elektrotechnika · May 2014


DOI: 10.5755/j01.eee.20.5.7097

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http://dx.doi.org/10.5755/j01.eee ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392–1215, VOL. 20, NO. 5, 2014

Optimal Position of Buried Power Cables


G. De Mey1, P. Xynis2, I. Papagiannopoulos3, V. Chatziathanasiou3, L. Exizidis3, B. Wiecek4
1
Ghent University,
Sint Pietersnieuwstraat 41, 9000 Ghent, Belgium
2
Technological Education Institute of Patras,
Μegalou Alexandrou 1 Koukouli, P.O 263 34 - Patras
3
Aristotle University of Thessaloniki,
54124 Thessaloniki, Greece
4
Technical University of Lodz,
ul. Wolczanska 211-215, 90-924 Lodz, Poland
[email protected]

1
Abstract—The optimum position of parallel underground The reason for this investigation is obvious. If several
cables will be calculated numerically. The criterion is how cables are buried next to each other, it is of practical use to
much joule losses should be dissipated in each cable so that the arrange them in such a way to reduce the extreme
temperature increases of all cables are equal. A simple
temperatures as much as possible.
analytical formula is also given.

Index Terms—Analytical analysis, numerical analysis, power


cables, temperature control.

I. INTRODUCTION
Underground power cables are thermally well insulated.
At a depth of about 1 m the amount of ground surrounding
the cable is quite huge so that it behaves as a high thermal
resistance. As a consequence the temperature of the cable
can be relatively high even for moderate values of the Joule Fig. 1. Cross sectional view of the layout of the underground cables.
heat produced in the cable as compared to the same cable in
air with natural convection cooling. Due to the increasing Related problems exist in other fields such as
demand for electric power, the thermal load of many cables microelectronics. If several heat dissipating components are
is growing continuously because it is not possible in practice placed on a single substrate, a different layout can give rise
to install new cables any time the power demand is to a more uniform temperature distribution and hence to a
increasing. Several papers have treated the thermal problems reduced peak temperature [18]–[22].
of underground cables [1]–[17].
If cables have to be deposited underground, a well has to II. SIMULATION RESULTS
be excavated and several cables are installed parallel next to The problem of underground cables has been simulated
each other. Normally the power in all cables will be numerically using the COMSOL multiphysics software [23].
different. Hence one can ask the question in which order one A steady state analysis has been carried out using the
has to put the cables so that the temperature distribution is geometry shown in Fig. 1.
optimal, i.e. the maximum temperature should be as low as As the cables are very long only a cross section has to be
possible. It can be proved that this problem is equivalent to considered. This gives rise to a two dimensional analysis.
the following one: how to interchange the position of the The dimensions of the box are 5 m × 2 m. The cables are at
cables so that the temperature distribution is as uniform as a depth of d = 1 m. The bottom and the two sides are
possible. Needless to say, a uniform temperature distribution modelled as adiabatic boundary conditions. The ground has
is the optimal situation. a thermal conductivity k = 0.83 W/m.K. The ground level is
Intuitively, it is clear that if all the cables have exactly the cooled convectively. A heat transfer coefficient h = 20
same Joule power, the cable in the middle will have the W/m2K has been used and the ambient air temperature was
highest temperature. The outer cables will be at the lowest
temperatures. If the transmitted power of each cable is set to Tair = 20o C .
different, one should install the cables with the highest Each cable has an external diameter of 5 cm. The metal
power at the ends and the cable with the lowest power in the conducting part with a diameter of 3 cm was assumed to be
middle to approach the optimal situation as good as made of copper with a thermal conductivity k = 400 W/m.K.
possible. The electrical insulation around each cable has a thermal
conductivity k = 0.28 W/m.K. At the copper insulation and
Manuscript received October 10, 2013; accepted December 4, 2013. insulation ground interfaces, the continuity of the
ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392–1215, VOL. 20, NO. 5, 2014

temperature and the heat flux were used as boundary T1 = T2 = T3 = T4 = ... , (2)
conditions.
gives rise to an algebraic set of n-1 equations with n
unknowns: p1, p2, p3,…. However, one can always choose
one; say p1, because if all the power values are multiplied by
the same constant, one still obtains a uniform temperature
distribution. Hence, an algebraic set of n-1 equations and n-
1 unknowns (p2, p3,…) remains. Obviously the problem can
be simplified further on by taking the symmetry into
account. It is clear that p1 = pn, p2 = pn-1,… so that n/2-1
unknowns remain in case n is even. For odd n, the number
of unknowns is reduced to (n-1)/2. Simulations have been
carried out for 3, 6, 9, 12, 15, 21 and 30 cables, the results of
which are displayed in Fig. 2.
All power values have been normalised to the values
obtained in the middle (x = 0). As expected, the outer cables
allow a much higher power transmission. It is surprising to
learn that the power in the outer cables can be several times
the power value in the central cables. A more careful
discussion of these results will be given further on in this
paper.

III. THEORETICAL ANALYSIS


As has been pointed out the optimal situation corresponds
to a uniform temperature of the cables. For the theoretical
Fig. 2. Optimal power distribution in the cables as a function of the analysis the problem will be reversed: given a constant
position of the cables.
temperature which the corresponding power distribution?
In every cable i a certain amount of Joule heat is produced The purpose of the theoretical section is to find a simple
denoted with pi (W/m). As a consequence each cable will analytical formula which can be used on the numerical data
have a temperature rise Ti above the ambient value. Our goal so that a simple design rule could be obtained. As a
now is to find a set of power values pi so that all the cable consequence, the theoretical analysis can be largely
temperatures turn out to be equal or T1 = T2 = …Tn. simplified. First of all, the cables are now replaced by a
For the thermal conduction in the ground the Laplace horizontal flat plate BC at a uniform temperature T0 as
equation has to be solved in order to determine the shown schematically in Fig. 3(a).
temperature distribution. The solution is carried out using
the finite element package COMSOL. With this software
package, as well as with other ones, it is quite easy to find a
temperature distribution provided the heat generation is
known. In this research, we want to solve the inverse
problem: which should be the distribution oft the heat
generation in order to get a uniform temperature distribution
in the cables. The inverse problem will be approached using
the superposition principle.
In order to find the power distribution {pi |, i=1...n} an
arbitrary power p0 is generated in cable 1 and zero power in Fig. 3. Conformal mapping used to determine the optimal power
distribution analytically.
all the other cables. After simulation we obtain the
temperature values T11 in cable 1, T21 in cable 2, T31 in cable Secondly, it is assumed the plate BC is sufficiently deep
3,… The same procedure is then repeated with a constant below the ground level, so that convection can be neglected.
power p0 in cable 2 and zero power in the all the remaining Moreover, we are mainly interested in the temperature field
cables. The corresponding temperature values is then T12 in near the plate BC. The plate BC with a width 2a is now
cable 1, T22 in cable 2, T32 in cable 3,…. This procedure is placed in an infinite medium with thermal conductivity k.
repeated for all n cables. If a power distribution p1 ,p2, p3,… Due to symmetry it is sufficient to consider only the upper
is applied, the temperature of cable i is found by half part y > 0 of Fig. 3(a). This problem has been described
superposition in several textbooks mainly devoted to potential theory [24],
[25]. The most obvious way to solve the problem of Fig.
p p2 p3
Ti = Ti1 1 + Ti 2 + Ti 3 + ... . (1) 3(a) is to use a conformal mapping from the z = x + jy to the
p0 p0 p0 w = u + jv plane
⎛z⎞
Requiring that all cable temperatures must be equal z = a sin( w) or w = arcsin ⎜ ⎟ . (3)
⎝a⎠
ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392–1215, VOL. 20, NO. 5, 2014

The half upper plane y > 0 is then mapped into a strip in will be valid for all values of n used in the simulations.
the w plane borned by −π / 2 < u < +π / 2 and v > 0 (Fig.
3(b)). In the w-plane the temperature distribution T depends
only on v

T (v) = T0 − α v, (4)

which corresponds to a uniform heat flux p(u) = k α in the


plane. The complex temperature distribution in the w-plane
is then

T (v) = T0 + jα w, (5)

which can be easily found because (4) is nothing else than


the real part of (5). In the z-plane the complex temperature
distribution is then given by

⎛z⎞
T ( z ) = T0 + jα arcsin ⎜ ⎟ . (6)
⎝a⎠ Fig. 4. Optimal power distribution (pi) versus cable position (x).1/pi2 is
plotted versus x2 in order to obtain a linear relationship.
The complex temperature gradient is found by taking the
derivative of (6) In Fig. 4 1/pi2, as obtained from the simulations, has been
plotted as a function of xi2. It is remarkable that a linear
dT ( z ) d z a relation appears which is in full agreement with the
= jα arcsin = jα . (7) theoretical analysis. Moreover, all lines intersect the
dz dz a a − z2
2
horizontal axis in points with abscissae given by an2, in full
agreement with (11).
Putting z = x and bearing in mind that the imaginary part
Taking into account that the theoretical analysis, outlined
of (7) is the heat flux in the y direction, one gets the power
in the previous section, assumed an infinitely thin flat plate
density distribution along the plate
as the heat source at a uniform temperature whereas the
simulation used cables for the heat generation, the
1
p( x) ∝ . (8) agreement between both can be described as surprisingly
a − x2
2
good.
In the theoretical section, convection on the ground level
The proportionality constant is not important because the was not taken into account. The ground layer was assumed
final solution can only be determined with respect to a to be infinite. It proves that the power distribution is entirely
constant. determined by the thermal conduction in the near
neighbourhood of the underground cables.
IV. DISCUSSION The same set of simulations has also been carried out for
In this section it will be investigated how well the different cable diameters giving the exact same results.
theoretical formulae (8) can be fitted to the simulation
results shown in Fig. 2. To each cable a coordinate xi can be V. CONCLUSION
assigned, given by The optimal power distribution for underground cables
has been determined using numerical simulations. The
n +1 criterion was that all cables should have the same
xi = [i − ] × 5 cm, (9)
2 temperature increase. The same problem was also solved
using an analytical calculation and it was found that the
where n = 3,6,9,...30 , which corresponds to the centre of numerical results agree very well with the theoretical
each cable. The half width of the cable set is then an = (n/2) analysis. As a consequence, one has now a simple formula
× 5 cm. According to the theoretical results (8), one has to to find the optimal power distribution for a set of
to verify whether underground cables.
1
pi ∝ , (10) ACKNOWLEDGMENT
an2 − xi2 P. Xynis and I. Papagiannopoulos want to express their
gratitude to the EU for their Erasmus fellowships.
turns out to be valid or not. It is more convenient to verify
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