Calculation of the Ampacity of High Voltage Cables Laid in Free Air

The paper presents various approaches for determining the current rating of three 110 kV cables in trefoil formation laid in free air including the developed finite element method. All models are possible to consider all circumstances affecting the ampacity of cables including the heat transfer to the surrounding ambient by radiation and convection, as well as the effect of solar heating. It has been shown that all available approaches have provided results that match with a reasonable accuracy. The permissible current rating for the three AC cables in trefoil formation laid in free air are calculated in summer for both, cables protected from solar radiation and directly exposed to solar radiation. Also, the time constants for heating-cooling cycle for the considered bunch of cables are calculated. It is important to note that the ampacity of three cables in trefoil formation, by all methods, determined by omitting the cooling by wind in order to obtain results for the worst conditions.


INTRODUCTION
The investigations concerning the continuous current rating of self-supporting insulated cable bunches have been motivated by a wide application of these cables in distribution networks. The full utilization of the cable material is only possible if the maximum allowable loads are known for the existing ambient conditions. In the past, various valuable analytical approaches have been elaborated addressing the continuous current rating of cables [1], [2]. An analytical method has been developed for calculating the heating of cable bundles laid out in air in buildings [3], in steady state and transient conditions. The methods for the calculation of the continuous current rating have been also proposed in [1], [4]. Analytical approaches concerning the ampacity of overhead lines generally have been presented in [5], [6]. The idealizations introduced in modelling various structural details and materials, inevitable in analytical approaches, reflect to some extent upon the obtained results. A detailed modelling of cable provides the method of finite elements. This method has been used for a transient temperature analysis ____________________ Author's address: Miladin Tanaskovic, freelance consultant for Electrical Power Systems, Zemun Pregrevica 86 e-mail: tanasfam@gmail.com Paper received: 28.09.2017. Paper accepted: 18.12.2017. due to step function loads for cables laid in uniform environment [7]. A finite element mesh generator for thermal analysis of underground cables has been suggested [8], considering only the heat transfer by conduction. Such a model has also been elaborated for the evaluation of the ampacity of cables laid inside retaining ducts [9]. The finite element model for the evaluation of the ampacity of underground cables, elaborated in reference [10], has taken into account the drying out of soil and the heat transfer by convection from the soil surface. An enhanced version of this model, elaborated in reference [11], incorporating the radiation and solar heating effects. The latter model, for determining the current rating of the bunch of self-supporting insulated 10 kV cables, adequately adapted in reference [12], has been applied in this paper for comparison with analytical approaches.

Mathematical model
The Indices c, h, r and s in (2) and (3) refer to conduction, convection, radiation and solar heating, respectively [11]. Index Q refers to the internal heat generation. The Joule losses in phase conductors and metallic sheaths have been taken into account as well as the dielectric losses in the insulating materials. Triangular finite elements have been used in this analysis.
The matrices figuring in (1) -(3) have been obtained by transferring the corresponding relationships for elements written for their nodes in local coordinates [11] to entire network numbering scheme. Expressions (2) and (3) in their general form are relevant only for the boundary elements with edges on the outer surface of cables. For all other nodes only [K c ] and [R Q (t)] on the right hand sides of these expressions are not null matrices.
The coefficient matrices in equations (1) -(3) for a triangular element, in its local coordinates, are as given in [7, 10, 11, and 13]. They are complemented by relationships for heat load vector from specified surface heating, radiation and solar heating (Appendix A).
After discretization for a minor step t, (1) converts into a simple recurrent relationship for determining temperature variation in time [13] )] with  as the integration stability factor. In this analysis =1 and t=2 min have been adopted, assuring a good accuracy for modest calculation time.
The expressions for determination of characteristic parameters figuring in (1) are given in the subsections below.
Here, D s (m) is the overall diameter of the cable bunch, and ϑ s and ϑ a , in o C, are cable surface and ambient temperatures respectively. By inserting ϑ value from Eq.8 into Eq.7 the same value for h can be used when calculating matrix [K h ] and vector [R h ].

Linear approximation of the heat transfer by radiation
The heat flux emitted by the radiation from a body at temperature ϑ equals (9) with σ=5.67·10 -8 W/(m 2 K 4 ) designating the radiation constant and ε 0 =0.95 being the emissivity constant for the cable outer sheath [2,5]. For a modest range of temperatures (ϑ 1 , ϑ 2 ) expression (9) can be approximated by a linear function satisfying the relationship  (11) which is an alternative, enhanced version of the approximation suggested in [11]. If the heat dissipated in the air from the surface of the cable should be determined, then for  1 and  2 the minimum and maximum temperatures for the outer sheathshould be inserted. The peak temperatures of the outer sheaths of cables have been assessed to be within the interval (40 0 C, 65 0 C). By inserting these extreme temperature values in (11) we obtain (12) The ambient (air) radiation, being thermal load for cables, can also be approximated by a linear function. The slope of this function can be determined using (11) TEHNIKA -ELEKTROTEHNIKA 67 (2018) 1 91 with ϑ 1 , and ϑ 2 being extreme air temperatures during the period under consideration. For winter air temperatures, ranging from -20 0 C to 10 0 C, in extreme cases, the slope of the approximation function equals (13) For summer air temperatures, ranging from 10 0 C to 40 0 C, Eq.11 gives (14) 2

.3 Solar heating
The maximum heat emitted from the sun to the earth surface is E=0.65kW/m 2 in winter, and E=1.353kW/m 2 in summer. For Belgrade the maximum heat emitted from the sun to the earth surface is E=0.532kW/m 2 in winter, and E=1.096kW/m 2 in summer. The exposed bodies partially absorb this heat. The body absorptivity constant α s determines the grade of the absorption. An appropriate value for gray coloured cable surface [1,2] is adopted as  s = 0.4.

Basic ampacity expression
The permissible current rating of an AC cable laid in free air [1], protected from solar radiation and directly exposed to solar radiation, respectively, is given by: where ϑ c is the maximum operating temperature of the conductor, ϑ a is the ambient temperature, R is the alternating current resistance per unit length of the conductor at its maximum operating temperature. Parameter n is the number of conductors in the bunch of cables, T 1 is the thermal resistance per unit length between one conductor and the sheath, T 2 is the thermal resistance per unit length of the bedding between the sheath and the armor, and T 3 is the thermal resistance per unit length of the external insulation of the cable.
Symbol T 4 designates the thermal resistance per unit length between the cable surface and the surrounding medium for cables protected from solar radiation, T 4r designates the thermal resistance per unit length between the cable surface and the surrounding medium adjusted to take into account solar radiation, λ 1 and λ 2 are ratios of the total losses in metallic sheaths and armour to the total conductor losses, P d are dielectric losses per unit length per phase, and P s is the solar heating per unit length.
The permissible current rating of an AC cable laid in free air [5], in the general form, is given by: All methods of analytical calculation for load capacity relate to cables installed in the horizontal plane. Whether the mounting is vertical or horizontal has, in principle, no influence on the heat dissipated by radiation [2].
The heat dissipated by convection from a vertical cylinder of length l and diameter d s with all other condition equal, more favourable than that of the cylinder in horizontal position, provided that [2] 77 Since this relationship is always satisfied in respect of cables installed in a vertical plane it follows that they can normally withstand heavier loads then when they are installed horizontally. Thus the same load capacity can be applied to both conditions [2].
In this paper, we shall particularly consider the outer thermal resistances T 4 , T 4r and P s as the available analytical approaches differ in calculation of this parameter.
The expressions for the calculation of other variables are the same for all analytical methods under consideration, and are as given in [1.

The IEC Method
The thermal resistance per unit length between the cable surface and the surrounding medium for cables protected from solar radiation is expressed in the form where D s is the overall diameter of the cable bunch, h 1 is the heat dissipation coefficient and Δϑ s is excess of cable surface temperature above ambient temperature.
In case under consideration overall diameter of the cable bunch D s =2.312d s , where d s is the overall diameter for one cable.
The heat dissipation coefficient is calculated as The excess of cable surface temperature above ambient temperature is obtainable by a simple iterative method.
where  d is temperature rise above ambient temperature due to the dielectric losses and index n designating the number of iterations.
Where cables are directly exposed to solar radiation, thermal resistance T 4r is calculate from equation by the previous method except that in the iterative method ( s ) 1/4 is calculated using the following formula: where P s is the solar heating per unit length, α s is the absorption coefficient of solar radiation for the cable surface and E is maximum heat emitted from the sun to the earth surface.

Calculation of resistances T 4 , T 4r and solar heating P s based on non-dimensional numbers
The thermal resistance per unit length between the surface of one circular cylinder protected from solar radiation and the surrounding medium T 4 is expressed in the form [5] ) ( where d s is the overall diameter for one circular cylinder, and k tr , and k tc are the coefficients for heat transfers by radiation and convection, respectively. According to the Stefan-Boltzmann law, the heat transfer coefficient for radiation is: The Nusselt number for natural convection for three cables in trefoil formation in horizontal position is also calculated by equation (33) where the Grashof number for one circular cylinder is substituted by Grashof number for three cables in trefoil formation.
The Grashof number for three cables in the trefoil formation is calculated by equation (34) where d s are substituted by d ek . In the case of three cables in trefoil formation d ek is equal to 2.155d s [5].According to the previous Nusselt number for natural convection for three cables in trefoil formation is calculated by equation (33) where d s is substituted by d ek . Finally, the thermal resistance per unit length between the cable surface and the surrounding medium for cables protected from solar radiation T 4 is expressed in the form [5] ) ( where f r is the factor for the interchange of radiation energy between the surfaces of the conductors of the cable bunch. The factor f r for two long parallel circular cylinders is, according to [5], (39) where s is the spacing between surface of cylinders. In the case of three cables in trefoil formation, this factor is equal to 0.636.
The thermal resistance between the cable surface and the surrounding medium adjusted to take into account solar radiation is expressed in the form The factors that define the incident surface of radiation are specifically determined for each type of incident solar radiation.
The total solar heat input per unit length may be expressed as the sum from beam radiation, diffuse radiation, long-wave radiation and reflected radiation. In case of consideration we assumed that diffuse radiation, long-wave radiation and reflected radiation are neglected.
The heat input from the beam radiation for one circular cylinder is defined by where  s is the absorptive surface to short-wave radiation, I B is the direct-beam short-wave radiation flux at high H a above sea level, f SB is the coefficient of exposed projected area normal to the direct beam defined for each conductor, and  is the angle between the solar beam and the axis of the three cables in the trefoil formation. Average high above sea level in Belgrade is 150 m.
The heat input from the beam radiation for three cables in the trefoil formation is calculated by equation (41)  where t s is local apparent time. The local apparent time is given by where t is official time for the first standard meridian, t φ is time correction due to geographical length, and t et is uniformity of time.
The correction of time due to the geographical length is four minutes per degree and is positive for the local meridian east of the standard meridian. The geographical length for Belgrade is 20 28' east.
The uniformity of time is given by The angle between the solar beam and the axis of the three cables in the trefoil formation is calculate from equation where  is inclination of cable bunch to horizontal,  s is azimuth of the sun, and  L is azimuth of the cable bunch. The azimuth of the cable bunch is measured in the direction of the cursor on the clock from the south.
Azimuth of the sun is given by

Cable data
The approaches presented in the preceding sections have been applied to determine the maximum allowable continuous current of the three phases 110 kV XHE 49 -A 3 x 1000 mm 2 cable bunch in trefoil formation, presented in Figure 1. Table 1 gives the main data on relevant materials of the cable under consideration.  Maximum permissible conductor temperature for normal conditions is ϑ m =90 o C. Aluminium conductor and copper screen Joule losses at ϑ m are 0.0408 x I 2 x 10 -2 and 0.0271 x I 2 x 10 -2 , while dielectric losses at rated voltage are 1.193, respectively, with all losses expressed in W/m. I is the rms of the continuous load current at which the conductors have reached the maximum allowable temperature.
TEHNIKA -ELEKTROTEHNIKA 67 (2018) 1 95 Figure 2 displays the finite element mesh for three cables in trefoil formation constructed for the analysis of temperature distribution.

The finite element mesh
Total number of nodes for the network of finiteelements mesh is 98. The number of triangular finiteelement of: aluminium conductor is 24, insulation is 48, cooper screen is 48, and outer sheath is 52.

Results of calculations for various approaches
The maximum allowable continues current for the three cables in trefoil formation has been calculated by applying the finite element method and the numerical approaches presented before. Table 2 gives the maximum allowable load currents for the cable bunch determined by applying different calculation approaches discussed in this paper. The methods under consideration have been applied to determine the ampacity of the bunch of cables in the cases that the ambient temperatures are 25 o C and 40 o C for cables protected from solar radiation and directly exposed to solar radiation and compare with catalogue data [15]. The finite element method and analytical approach based on non-dimensional numbers [5] respect heat exchange between the cable surface and the surrounding medium by convection and radiation as opposed to the IEC method that only regards radiation. Therefore, in both cases for cables protected from solar radiation and cables directly exposed to solar radiation, the IEC method gives lower values of permitted current loads from the finite element method and analytical approach based on non-dimensional numbers.
As can be seen, ampacity calculated by finite element method and analytical approach based on nondimensional numbers for cables protected from solar radiation, gives value 1 A higher and 11.5 A lower to the given catalog value for ambient temperature  a =25C, respectively. The comparative calculations have been also performed for the ambient temperature  a =40C, and finite element method has again gives the results higher than the analytical approaches.
Higher values for ampacity calculated by finite element method for cables protected from solar radiation is due to the fact that the actual temperature is taken into account for each point on the outer surface of the cable bunch, while the analytical approach based on non-dimensional numbers for all points on the outer surface of the cable bunch takes into account only one temperature (in our case  s = 65C).
The influence of solar radiation in summer on the reduction of the maximum allowable continues current for the three cables in trefoil formation is in the finite element method about 145 A, in the IEC method about 190 A, and in the case of analytical approach based on non-dimensional numbers about 95 A. The effect of the TEHNIKA -ELEKTROTEHNIKA 67 (2018) 1 incident angle between the solar beam and the axis of the three cables in the trefoil formation to reduce the maximum allowable continues current is not respected in the IEC method, and therefore the reduction of the allowable continues current due to solar radiation is the greatest in this method. The incident angle of the solar beam is calculated in the case of the analytical approach based on non-dimensional numbers in relation to the three cables in the trefoil formation, while in the FEM, the incident angle of the solar beam is calculated for each individual triangular finite element.

TIME CHANGES OF CONDUCTOR TEMPERATURE DURING HEATING-COOLING CYCLE
It is of particular importance to determine the time constants for heating and cooling cycles for the three HV cables in trefoil formation laid in free air. Specifically, if the time constants of heating and cooling cycles are greater than 1 hour, then the thermal capacity of the cable bunch can be used in the control of the load flow in real-time.

CONCLUSION
The paper presents various methods for the calculation of the continuous current rating of a three AC cables in trefoil formation laid in free air. A finite element model conceived for this analysis has been presented as well as two different analytical approaches used in practice. The results obtained by all considered methods have been compared to the catalogue data. It has been shown that MKE and analytical approach based on non-dimensional numbers give ratings that very slightly differ from catalogue data. The best agreement to the catalogue data has been achieved by the finite element method. Particular consideration is given to the influence of solar radiation on the permissible current rating for all shown methods. The developed FE model has been also used for the calculation of the time constants for heating-cooling cycle for the considered bunch of cables.

APPENDIX A. PARAMETERS OF TRIANGULAR FINITE ELEMENTS
Triangular finite-elements that are adopted for modelling the interaction of cables and pipelines are shown in Fig.5.

(A10)
In above given expressions ρ 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 conduction and convection factors, x k and y k connecting nodes 1 and 2, Q(t) is the heat generated inside the element, ϑ e (t) is the air temperature, E is emitted heat from the sun to the earth surface during an average winter or summer day, and α s is the absorption coefficient of the gray coloured cable surface.
As can be seen from the expressions above, all terms in (A.1), (A.2) and (A.5) to (A.10) are proportional to d and these relationships hold for all values of this parameter. The adopted value in this analysis was d=1 m. x k and y k , k=1,2,3, are plane rectangular coordinates of triangular element nodes. l 12 is the length of triangle edge connecting points 1 and 2. In deriving the expressions listed above, it is assumed that the triangular element exchanges the heat by convection through its edge connecting nodes 1 and 2.