It has become rather confusing to get a whole picture of current carrying capacities of covered overhead lines. Current ratings of similar conductor constructions vary depending where they are calculated. Especially, there have been some major dif- ferences between the calculations made in Scandinavia and UK. The Scandinavian way follows current European thinking and will most likely form the basis for new legislation. The current rating calculation presented here now is based mainly on the method devised by Cigré Working Group 22.12 and published in Électra no. 144 [2]. However, as this method was developed for bare overhead conductors, some modi- fications have been made to take account of the sheath. The author is grateful to Mr Jaako Pitkänen of Pirelli Cables and Systems Oy, Pikala, Finland for providing the following detailed calculation.
It is obvious that different calculation methods will give slightly different cur- rent ratings for the same kinds of conductors, but this difference is not always fully due to the differences between the calculation methods. There are also different environmental parameters considered in the calculations.
8.7.4.1 Principles of calculation method used
As in section 8.7.3, the current carrying capacity calculation method is based on a simple heat balance equation
heat gain= heat loss which is formulated as:
PJ + PS = PR+ PC (8.8)
where PJ is joule losses generated in the conductor, Wm−1, PS is solar heating, Wm−1, PRis radiative cooling, Wm−1and PCis convective cooling, Wm−1.
Joule heating can be calculated by means of the AC resistance of the conductor:
PJ = I2RAC (8.9)
Thus the current can be obtained from equations (8.8) and (8.9): I = PR+ PC− PS RAC (8.10) The current obtained from equation (8.10) can be counted as the current carrying capacity of a covered conductor if the AC resistance, radiative cooling etc. are calculated at the maximum allowable temperature of the conductor.
However, as has already been shown, the calculation of current carrying capacity is an iterative process, because radiative and convective cooling are functions of the surface temperature of the sheath that is itself dependent on the load current. Since the surface temperature is actually a function of joule heating the cooling expressions might be written as:
PR = PR(PJ)= PR(I2RAC)
PC = PC(PJ)= PC(I2RAC)
(8.11) When calculating current carrying capacity the iteration steps are as follows: 1 Estimate an approximate value for current I0.
2 Calculate cooling powers PR(I02RAC)and PC(I02RAC). 3 Calculate current I1from equation (8.10).
4 Calculate new values of cooling powers PR(I12RAC)and PC(I12RAC)using current
I1in equation (8.11).
5 Calculate new value of current from equation (8.10).
6 Repeat steps 2 to 5 n times until the differential|In− In−1| is small enough (for example smaller than 0.1 A).
The maximum current carrying capacity of the covered conductor is then finally obtained. This is exactly the same process as in the UK method.
8.7.4.2 Calculation of AC resistance RAC[8]
AC resistance of a covered conductor can be calculated according to the standard IEC 287-1-1:1994 using sub-clauses 2.1, 2.1.1 and 2.1.2 [8]. However, the proximity effect factor described in sub-clause 2.1 can be neglected in the case of widely spaced covered overhead conductors.
8.7.4.3 Calculation of radiative cooling PR
From the Cigré method [2] the formula for radiative cooling is:
PR = πDεσB[(θS+ 273)4− (θa+ 273)4] (8.12) where
D= outer diameter of the covered conductor, m ε= emissivity coefficient
Conductor characteristics and selection 133 θS = surface temperature of sheath, ◦C
θa= ambient temperature, ◦C
The main problem when using formula (8.12) is that the surface temperature of the conductor is not known. In some cases it may be assumed that the surface temperature of the sheath is a few◦C below the conductor temperature. However, an exact approach is given next.
8.7.4.4 Calculation of surface temperature of the sheath
By using the analogy between electrical circuit theory and heat transfer phenomena, the surface temperature can be calculated as follows:
θS = θC− T3PJ (8.13)
where T3is the thermal resistance of sheath, KmW−1.
From IEC 60287 [9] we can get a formula for the thermal resistance of the sheath:
T3= ρT 2π ln 1+2t1 dc (8.14) where ρT is the thermal resistivity of sheath, KmW−1, t1is the thickness of sheath, mm and dcis the diameter of AlMgSi conductor, mm.
8.7.4.5 Radiative cooling as a function of joule heating
As mentioned earlier, the radiative cooling is a function of the joule heating, PJ, and it can be obtained by substituting equation (8.13) into equation (8.12):
PR(PJ)= πDεσB[((θC− PJT3)+ 273)4− (θa+ 273)4] (8.15) 8.7.4.6 Calculation of convective cooling PC
From the Cigré calculation method [2] we get a formula for convective cooling
PC = πλf(θS− θa)Nu (8.16)
where λf is the thermal conductivity of air, Wm−1K−1, θSis the surface temperature of the sheath,◦C, θais the ambient temperature,◦C and Nu is the Nusselt number.
An empirical value for the thermal conductivity of air is [2]:
λf = 2.42 · 10−2+ 7.2 · 10−5· 0.5(θS− θa) (8.17) Convective cooling is often divided into natural and forced cooling. It can safely be assumed that cooling is forced by the wind in this case. It is also more convenient to assume that the wind direction is normal to the span (angle of wind attack is 90◦) and that the wind speed is more than 0.5 m/s.
The Nusselt number for forced cooling that is valid in these circumstances can be calculated from equation (8.18). If it is necessary to calculate the Nusselt number in other conditions than those mentioned earlier the relevant formulae can be found [2]:
where Re is the Reynolds number and B1and n are coefficients depending on the Reynolds number and roughness of the sheath’s outer surface.
The Reynolds number can be calculated from equation (8.19) [2]:
Re= ρrvD
ν (8.19)
where ρr is the relative air density, v is the wind velocity, ms−1, D is the outer diameter of the covered conductor, m and ν is the kinematic viscosity of air, m2s−1. Empirical equations for relative density of air and for kinematic viscosity of air are [2]:
ρr = e(−1.16·10 −4y)
(8.20)
ν= 1.32 · 10−5+ 9.5 · 10−8· 0.5(θS+ θa) (8.21) where y is the height above sea level (m).
Values for coefficients in equation (8.18) are shown in Table 8.7. 8.7.4.7 Convective cooling as a function of joule heating
Convective cooling is a function of the surface temperature of the sheath in the same way as radiative cooling. This can be seen especially from equations (8.9), (8.10) and (8.14). Because the surface temperature is a function of joule heating, the convective cooling might be written in terms of joule heating.
8.7.4.8 Calculation of solar heating PS [2]
The formula for solar heating using global solar radiation can be obtained from equation (8.22):
PS = αsSD (8.22)
where S is the global solar radiation, Wm−2, D is the outer diameter of covered conductor, m and αsis the absorptivity of sheath surface.
Table 8.7 Constants B1 and n as a function of
Reynolds number [2]
Reynolds number B1 n
from to
102 2.65· 103 0.641 0.471 2.65· 103 5· 104 0.178 0.633
Values given in the table are valid for covered conductors with surface roughness of less than 0.05. That is why the given values are suitable for the UK smooth-surfaced covered conductors.
Conductor characteristics and selection 135