# Design of Power Cable Lines Partially Exposed to Direct Solar Radiation—Special Aspects

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## Abstract

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## 1. Introduction

## 2. Description of the Numerical Model of the Cables

^{2}. The construction of the power cable is shown in Figure 1.

- (1)
- Joule’s heat produced by the current I
_{cc}flowing through the copper conductor—in most of the studied cases, the boundary condition of Joule’s heat flux density q_{Joule}is iteratively introduced to obtain a result for which the max permissible temperature in any area of insulation is reached (70 °C). The current-carrying capacity I_{cc}is included in the following formula:

- q
_{cc}is the thermal power (heat) generated by the current in the power cable given length, W; - A
_{c}is the external/side surface of the copper conductor (dissipating heat to the ambient air), m^{2}; - I
_{cc}is the current-carrying capacity, A; - R
_{AC}is the resistance of the power cable for AC current, Ω; - D
_{c}is the diameter of the copper conductor, m; - L
_{u}is the unit length of the copper conductor, m.

_{cc}separated on the unit length (usually 1 m) of the cable to the side surface of the copper conductor A

_{c}with the unit length. The assumption made in the research is an even distribution of Joule’s heat flux density over the entire lateral surface of the copper conductor.

- (2)
- The heat generated by solar radiation—presented as the authors’ functional determination of the density of heat flux reaching the insulation surface q
_{solar}(x), W/m^{2}. This relationship has been transformed for use in the Cartesian coordinate system. The intensity of solar radiation depends on the height of the sun above the horizon, which is associated with the absorption of solar radiation by the atmosphere.

_{solar}[28]:

- H
_{s}= 1120 W/m^{2}= const; - ks = 1/sin(S
_{h}), where S_{h}is the sun altitude, °,

- f(x, r
_{s}) the function depends on the adopted coordinate system x and the external radius of the cable r_{s}= D_{s}/2, -; - σ
_{abs}is the PVC absorption coefficient, -.

- (1)
- The radiated heat from the system—described by the radiation model P1 and DO (Discrete Ordinates) [29]; implemented in the ANSYS Fluent software.
- (2)
- Heat conduction—mainly in solids, i.e., considered in the PVC insulation and sheath/jacket, but also in the boundary layer. Assuming the isotropicity of the tested materials, the condition regarding the density of conducted energy in the system can be described by the relationship according to Fourier’s law:

_{conduction}= −λ

_{PVC}∇t

- λ
_{PVC}is the thermal conductivity of the PVC insulation and sheath, W/(m·K); - ∇t is the three-dimensional temperature gradient, K/m;

- (3)
- Heat convection—the density of the heat flux transferred by convection is given by the relationship:

- α
_{conv}is the heat transfer coefficient, W/(m^{2}·K); - t
_{PVC}is the power cable external surface temperature, K; - t
_{air}is the air temperature around power cable, K.

## 3. Computer Simulations and Results

#### 3.1. The Effect of the Wind Speed, Wind Direction and Cables Layout

- From North to South;
- From South to North;
- From the top to bottom of the arrangement of the cables.

_{conv}. The convective heat exchange becomes the dominant method of heat exchange between the power cable and the environment.

#### 3.2. Power Cables in the Ground vs. Power Cables in Free Air

_{Joule}determined for the current-carrying capacity of power cables located in the ground was adopted as the reference value and mapped in the above-ground part of the cable line (cables laid in free air). The insulation temperatures of cables laid in the air (without wind—unfavourable conditions) for the boundary condition q

_{Joule}resulting from the conditions in the ground are presented in Figure 9.

_{3m}for the pipe of the length of 3 m, whereas Figure 16b presents the analogical correction factor k

_{6m}for the pipe of the length of 6 m. The values of the correction factors are referred to the case with solar radiation and without a pipe (126 A = 1.00). Multiplying the base value (with solar radiation and without a pipe) of the current-carrying capacity by the correction factor gives the permissible load of the cable (35 mm

^{2}) in the pipe. Such mathematical functions can be helpful for power cable line designers.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Air velocity vectors at a flow: (

**a**) perpendicular to the power cable surface; (

**b**) parallel to the power cable surface [30].

**Figure 4.**Considered horizontal layouts of the power cables: (

**a**) flat formation without spacing; (

**b**) flat formation with spacing 50 mm; (

**c**) trefoil formation without spacing.

**Figure 5.**Effect of the wind direction and speed on the power cables current-carrying capacity, for the layouts of the power cables presented: (

**a**) in Figure 4a (flat formation without spacing, ambient air temp. 30 °C); (

**b**) in Figure 4b (flat formation with spacing 50 mm, ambient air temp. 30 °C); (

**c**) in Figure 4c (trefoil formation without spacing, ambient air temp. 30 °C).

**Figure 7.**Temperature distribution (°C) for wind speed (1 m/s) in the: (

**a**) South–North direction (from the left to the right side); (

**b**) North–South direction (from the right to the left side); (

**c**) Top–Bottom direction [30].

**Figure 8.**Geometry and boundary conditions of the analyzed power cable line; D

_{s}—external diameter of the cable, 1.0 (K·m)/W—the average value of the soil thermal resistivity in many European countries, 2.5 (K·m)/W—unfavourable reference value of the soil thermal resistivity according to [9].

**Figure 9.**Temperature distribution around power cables (load equal to 176.2 A as permissible in the ground of the soil resistivity 1.0 (K·m)/W): (

**a**) with solar radiation (max insulation temp. 102.50 °C); (

**b**) without solar radiation (max insulation temp. 83.13 °C).

**Figure 10.**Temperature distribution around power cables (load equal to 117.5 A as permissible in the ground of the soil resistivity 2.5 (K·m)/W): (

**a**) with solar radiation (max insulation temp. 76.22 °C); (

**b**) without solar radiation (max insulation temp. 53.32 °C).

**Figure 12.**A pole of the overhead line with cables inside the pipe used for the protection of cables against mechanical damage.

**Figure 13.**Air velocity distribution (

**a**) and temperature distribution (

**b**), for the pipe diameter of 100 mm and its length of 3 m.

**Figure 14.**Air velocity distribution (

**a**) and temperature distribution (

**b**), for the pipe diameter of 200 mm and its length of 1 m [5].

**Figure 15.**The current-carrying capacity of the power cable depending on the diameter and length of the casing pipe.

**Figure 16.**The current-carrying capacity correction factors k

_{3m}and k

_{6m}(as a function of the pipe internal diameter D

_{p}) for the cable 35 mm

^{2}installed in the pipe of the length respectively: (

**a**) 3 m; (

**b**) 6 m.

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**MDPI and ACS Style**

Czapp, S.; Szultka, S.; Tomaszewski, A.
Design of Power Cable Lines Partially Exposed to Direct Solar Radiation—Special Aspects. *Energies* **2020**, *13*, 2650.
https://doi.org/10.3390/en13102650

**AMA Style**

Czapp S, Szultka S, Tomaszewski A.
Design of Power Cable Lines Partially Exposed to Direct Solar Radiation—Special Aspects. *Energies*. 2020; 13(10):2650.
https://doi.org/10.3390/en13102650

**Chicago/Turabian Style**

Czapp, Stanislaw, Seweryn Szultka, and Adam Tomaszewski.
2020. "Design of Power Cable Lines Partially Exposed to Direct Solar Radiation—Special Aspects" *Energies* 13, no. 10: 2650.
https://doi.org/10.3390/en13102650