3.1. The Effect of the Wind Speed, Wind Direction and Cables Layout
In practice, the wind speed can take different directions. Usually, it will be possible to distinguish the perpendicular and parallel components in relation to the surface of the power cable. Wind directions affect the intensity of heat exchange between the cable and the environment.
a presents vectors of air velocity flowing perpendicular to the axis of a single power cable. At the inlet on the left plane of the computational domain, speed equal to 1 m/s perpendicular to the axis of the power cable was set in the model.
b shows the distribution of air vectors with the flow parallel to the surface of the power cable. Similar to the considerations of a cable laid perpendicular to the air flow, the set airspeed is equal to 1 m/s. The air velocity near the insulation surface (blue velocity vectors in Figure 3
b) is significantly lower than the velocity at some distance from this surface. This is due to the occurrence of a boundary layer formed in the vicinity of insulation and the viscosity of air flowing around the cable shape. The heat exchange, in this case, occurs mainly by conduction in the hydraulic boundary layer.
It can be clearly seen that the air movement in close proximity to the insulation wall in the case of parallel flow (Figure 3
b) is less turbulent compared to the flow perpendicular to the axis of the power cable (Figure 3
a), where the laminar boundary layer is definitely thinner and there are more vortex structures in the flow.
As a result, it turns out that even with a very small path traversed along the power cable and with a similar insulation temperature (70 °C), the air in the flow parallel to the power cable axis is able to receive twice less heat than the air with a direction perpendicular to the power cable axis, at a constant direction and speed of air, and a constant air temperature [30
The current-carrying capacity of the three-phase power cable system (cable type from Figure 1
) has been analyzed for the three cables layouts presented in Figure 4
. For every layout, the wind direction has been taken into account according to Figure 2
. The effects of the following wind directions have been investigated:
Results of this analysis are presented in Figure 5
. A higher wind speed leads to the increased current-carrying capacity of the cable system. However, the highest increment of this capacity is for changing the wind speed from 0.5 m/s to 1 m/s or from 1 m/s to 2 m/s. For relatively high wind speed, when the flow is already turbulent, the increase of 1 m/s is not very effective. It can also be concluded that the most intense heat exchange occurs in the case of the air flow in the top-bottom direction, compared to the direction South–North or North–South. For the top–bottom flow, comparatively higher values of the current-carrying capacity of power cables were obtained. According to this, placing a cable line vertically instead of horizontally could be a reasonable option.
If one compares the trefoil formation with the flat formation without spacing, the first of these two mentioned gives a higher current-carrying capacity of the cable line, when the wind speed is relatively low (compare Figure 5
a vs. Figure 5
c for the wind speed up to 2 m/s).
The best of the power cables layouts, among the analyzed ones, is the flat formation with cables spacing of 50 mm (Figure 5
b). For this type of arrangement, the current-carrying capacity of the cable line is the highest (among the considered).
As the results in Figure 5
show, the speed of air inflow has a significant impact on the current-carrying capacity of power cables. This effect is particularly noticeable for the speed of 0.5–3.0 m/s. The justification for this phenomenon is the fact that the increase in the air flow velocity causes a proportional increase in the Reynolds number. According to the relationships attributed to the forced convective heat exchange, an increase in the Reynolds number results in an increase in the Nusselt number. Increasing the Nusselt number results in increasing the heat transfer coefficient αconv
. The convective heat exchange becomes the dominant method of heat exchange between the power cable and the environment.
presents the distribution of air velocity vectors around the cables. In Figure 6
a, air velocity vectors around each power cable show vortex structures (density of velocity vectors) that intensify convective heat exchange between the cable and air. The air velocity increases to 1.39 m/s, where the set starting value is 1 m/s in the top-bottom flow. The airflow around the power cables is based on the principle that the first power cable is cooled by air like a single cable, while the next cable/cables are located in the flow structures generated by the preceding cables, as shown in Figure 6
b,c. More turbulent air flow (higher air speed) leads to the more efficient convective heat transfer. Figure 6
b shows the distribution of air velocity vectors for the South–North inflow (from the left to the right in the picture), where it can be seen that near the first cable (left, L1) a higher air velocity is recorded compared to the velocity distribution around the L2 and L3 cables, and therefore, locally, the heat transfer around the first (left) cable is more intense. In the trefoil formation (Figure 6
c), two cables are directly exposed to the wind.
a shows the temperature distribution in the “flat formation without spacing” case. The first power cable (left, L1) receives the highest solar heat (from the left as in Figure 2
) compared to others, but it is not the hottest one. The other cables (L2 and L3) are screened by the cable L1 from the cooling effect of the wind in the direction South–North, reaching a higher temperature. In consequence, the temperature of the cables L2 and L3 determines the current-carrying capacity of the cable system. If the wind direction is changed to North–South (from the right to the left side in Figure 7
b), the current-carrying capacity of the cable line is determined by the first cable (left, L1), irradiated, and the worst (last) cooled by the air stream.
When comparing the results of temperature distributions in Figure 7
a,b, it is important to note that the current-carrying capacities differ by as much as 27%. For the example from Figure 7
a, the current-carrying capacity (maximum permissible symmetrical load of the all cables) is equal to 78.8 A, whereas for the example in Figure 7
b, the capacity is equal to 61.9 A.
c shows an example analysis of the temperature distribution for the trefoil formation. For this type of arrangement, it is seen that the heat supplied from solar radiation to the power cable L3 (the bottom cable on the right) is significantly limited due to the shadow zone of the other cables. In case of the wind direction from top to bottom, each cable is cooled by the wind. For wind direction from South to North (from left to right), two cables at the left side of the picture are subject to wind cooling (see air velocity vectors in Figure 6
3.2. Power Cables in the Ground vs. Power Cables in Free Air
The current-carrying capacity of the power cable lines is usually calculated for cables buried in the ground. The average value of the soil thermal resistivity in many European countries (e.g., Italy, Norway, Poland) is 1.0 (K·m)/W [31
]. With this in mind, the value of the current-carrying capacity for a cable line, for example, arranged in accordance with Figure 8
(if only the conditions in the ground are taken as the criterion) is 176.2 A. The results are consistent with the data contained in the standard [9
In the further analysis, the heat flux density qJoule
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 qJoule
resulting from the conditions in the ground are presented in Figure 9
Based on the results obtained in Figure 9
, it can be concluded that there is a real risk of exceeding the long-term permissible temperature (70 °C) in a section of the cable line arranged in the air, both in the presence and absence of solar radiation. As a result, the reliability of the power grid is below expectations [32
Power cables in the air cannot be loaded as much as in the ground characterized by the soil resistivity equal to 1.0 (K·m)/W. In the air, their maximum permissible load is only 96 A (with solar radiation) and 152 A (without solar radiation). Even worse, in the presence of solar radiation, the insulation of the cable may be overheated (76.22 °C—Figure 10
a) for the load equal to 117.5 A, which is the current-carrying capacity of cables in the ground for the soil resistivity is equal to 2.5 (K·m)/W (relatively high soil thermal resistivity—bad heat dissipation).
In order to improve heat exchange conditions for cables laid in the air and reduce the adverse factor (solar radiation) affecting the current-carrying capacity of power cables, a method of increasing the permissible load using a passive cooling system has been proposed. The idea of the solution is based on the use of a casing/shielding pipe made of PVC, in which the power cable is laid coaxially (Figure 11
). Such a pipe is usually installed for the protection of the cable/cables against mechanical damage, but it is closed at the top and its dimensions are not optimized from the point of view of thermal effects (Figure 12
). If the pipe is opened at both sides and optimized, it creates a pipe-channel and can be a solution for improving the current-carrying capacity in the presence of solar radiation. The heat exchange occurring in the power cable system with the casing pipe is mainly based on the principle of free convection, in which air flow is the result of buoyancy force resulting from the difference in air density arising due to the change in air temperature in the pipe.
To illustrate physical phenomena in the arrangement “the cable in a pipe” (during solar radiation as in Figure 11
), numerical simulations of natural convection around the power cable have been performed. Figure 13
presents results of these simulations for the case in which the pipe has a length of 3 m and the internal diameter is 100 mm, whereas Figure 14
presents the case with the pipe length equal to 1 m and its diameter equal to 200 mm.
presents the results of the numerically-based calculations of the current-carrying capacity of a single cable located in the pipe. The calculations have been performed for two lengths of the pipe (3 m and 6 m), and for pipe internal diameters 40 mm, 100 mm, 150 mm, 200 mm and 300 mm. In Figure 15
, the reference current-carrying capacity, without the pipe and with solar radiation is marked as well (126 A). The key factor is the diameter of the casing pipe, because the smaller the diameter, the weaker the air movement and velocity in the pipe, so the heat exchange is difficult. A positive effect of the pipe is obtained for the pipe diameters 100 mm, 150 mm, 200 mm and 300 mm. The diameter 40 mm is too low for this cable—the current-carrying capacity is lower than for the reference case 126 A. The length of the cable is important as well. For the pipe (cable) length of 3 m and the diameter 100 mm, the current-carrying capacity is practically the same as for the length of 6 m and the diameter 200 mm.
From the considered cases, the highest current-carrying capacity (145 A) has been obtained, when a pipe of the length of 3 m and the diameter 300 mm is applied. This is around 15% more than in the case without a pipe and in the presence of solar radiation (126 A). It should be noted that the result for the pipe diameter 300 mm (length 3 m) is very similar to the case with the pipe diameter equal to 200 mm (length 3 m)—in practice, it is not necessary to increase the diameter above 200 mm. Thus, it is clearly visible that the parameters of the cable-in-pipe arrangement have to be optimized from a thermal effects point of view. Especially, since the length of the cable section in the air can be different in each case. This length influences the pipe diameter.
The authors have estimated mathematical functions, which enable the calculation of the current-carrying capacity correction factors for the case where the analyzed cable is placed in the casing pipe. Figure 16
a presents the correction factor k3m
for the pipe of the length of 3 m, whereas Figure 16
b presents the analogical correction factor k6m
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 mm2
) in the pipe. Such mathematical functions can be helpful for power cable line designers.