# Optimization of Thermal Backfill Configurations for Desired High-Voltage Power Cables Ampacity

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}have been compared.

## 1. Introduction

## 2. Theoretical Background

_{AMP}is the ampacity of a cable in ground, A;

_{d}is the dielectric losses per phase, W/m;

_{1}is thermal resistance (per core) between the conductor and sheath/insulation, (K·m)/W;

_{2}is the thermal resistance between the sheath/insulation and armor, (K·m)/W;

_{3}is the thermal resistance of external serving of the cable, (K·m)/W;

_{4}is the external thermal resistance of the surrounding medium, (soil/backfill), (K·m)/W;

_{co}is the number of conductors in a power cable;

_{co}is the AC resistance of a conductor (core) at its permissible temperature, Ω/m;

_{co}is the conductivity of a conductor (core) at its permissible temperature, m/(Ω/mm

^{2});

_{co}is the cross-sectional area of a conductor (core), mm

^{2};

_{1}is the ratio of the total losses in metallic sheaths (if any) to the total conductor losses;

_{2}is the ratio of the total losses in metallic armor (if any) to the total conductor losses.

_{4}of the medium surrounding the power cables. Hence, the increase in the ampacity can be done by creating a favorable heat transfer from the surface of cables to the ground. The favorable heat transfer from cables is usually ensured by replacing the native soil with the thermal backfill of relatively low thermal resistivity—less than 1.0 (K·m)/W. Figure 1 shows a typical arrangement of the cable line in a flat formation when thermal backfill of width w and height h is used. The presented arrangement reflects real cables placement—it includes a backfill, pavement slabs (grey rectangular element) for the protection against mechanical damage, and a warning tape (red line). The modeled geometry for the trefoil formation is presented in Figure 2.

_{4}of the uniform single cable surrounding can be written according to the Kennelly’s theorem [29] as:

_{ext}is the external diameter of the cable, mm;

_{e}is the thermal resistivity of the native soil, (K·m)/W.

_{ext}(more than 10) the aforementioned relation (2) can be simplified to the following expression:

_{c}and thermal resistivity of the surrounding native soil is equal to ρ

_{e}, based on the provisions of the standard IEC 60287-2-1 [30], the correction factor T

_{4}

^{corr}to the thermal resistance should be calculated as follows:

_{G}is the distance from the ground surface to the center of the backfill, mm;

_{b}is the equivalent radius of the backfill, mm;

_{e}is the thermal resistivity of the native soil, (K·m)/W;

_{c}is the thermal resistivity of the backfill, (K·m)/W.

_{b}:

_{b}with (5) gives inaccurate values—in such cases data from tables included in [19] (p. 233) can be used.

_{4}* is expressed as follows:

_{4}* depends on the dimensions of the backfill. In further analysis, when the optimal dimension of the backfill is searched for, the following constraints are introduced:

## 3. A Proposal of the Cable System Optimization

_{K}is the cable cost;

_{E}is the cost of the performed excavation;

_{ZZ}is the cost of backfilling with the thermal backfill, sand and native soil as well as compaction;

_{B}is the backfill material cost;

_{ES}other costs;

_{max}is searched for, as follows:

_{G}included in (13). In consequence, one can obtain maximal ampacity of power cables with relation to the optimal geometry of the backfill.

- The ambient temperature of the ground is equal to 20 °C;
- The max permissible temperature for the cable is equal to 90 °C;
- The thermal resistivity of the backfill (cement–sand mixture) is assumed within the range ρ
_{c}= 0.5–0.9 (K·m)/W; - The angle of the excavation trench is constant (α = 15°);
- The distance L from the ground surface to the cable/cables center for particular formation (flat or trefoil) is constant (for the flat formation: L = 1200 mm + D
_{ext}/2; for the trefoil formation: L = 1200 mm + D_{ext}/2 + D_{ext}/√3); - The maximum accepted price of the cable line is equal to 720 EUR/m.

## 4. Discussion

_{c}. However, when the difference between thermal resistivities of backfill ρ

_{c}and native soil ρ

_{e}increases, one can observe the intersection of planes for particular cable cross-sections—compare results for the flat formation (Figure 3a vs. Figure 3c and vs. Figure 3d) and for the trefoil formation, compare Figure 4a vs. Figure 4c and vs. Figure 4d. The planes’ intersection points indicate that a given ampacity can be obtained by various cross-sectional areas of cables. The intersection points also indicate the optimal transition to the cable of the higher cross-sectional area.

_{e}= 2.5 (K·m)/W and thermal resistivity of the backfill ρ

_{c}= 0.6 (K·m)/W (Figure 3d), the ampacity of the 630 mm

^{2}cable starts from 700 A at the cost below EUR 400 (green plane at the left down part of the Figure 3d). Investing in cable backfill for the 630 mm

^{2}cable increases its ampacity but in consequence, also the cost. However, it is visible that within the cost range of EUR 475–525, the green plane (630 mm

^{2}cable) is located higher than the orange plane (800 mm

^{2}cable). It means that the use of the 630 mm

^{2}cable instead of 800 mm

^{2}is more profitable because the ampacity of the 630 mm

^{2}cable is higher than that of the 800 mm

^{2}cable at the same cost. For the cost EUR 475, the ampacity of the 630 mm

^{2}cable is equal to I

_{630}= 884 A whereas the 800 mm

^{2}cable has I

_{800}= 771 A, which gives around a 15% difference in ampacities (the backfill of the cable line with cables 630 mm

^{2}has a greater volume). The ampacity of two cables equalizes at point EUR 525, 970 A. This point indicates that it is better to use the 800 mm

^{2}cable, instead of the 630 mm

^{2}cable, if the desired ampacity is higher than 970 A.

^{2}cable to the 1000 mm

^{2}cable. If the desired ampacity is higher than 1040 A, the use of the 1000 mm

^{2}cable is the most profitable solution. After analysis of Figure 3d, for the native soil ρ

_{e}= 2.5 (K·m)/W and thermal resistivity of the backfill ρ

_{c}= 0.6 (K·m)/W the conclusions are as follows:

- The 630 mm
^{2}cable is profitable for ampacities below 970 A; - The 800 mm
^{2}cable is profitable for ampacities between 970 and 1040 A; - The 1000 mm
^{2}cable is profitable for ampacities higher than 1040.

^{2}cable or the 800 mm

^{2}cable, whereas the ampacity 1040 A can be achieved by either the 800 mm

^{2}cable or the 1000 mm

^{2}cable.

^{2}cable or 800 mm

^{2}with relevant backfill. It gives a cost of EUR 525. If the 1000 mm

^{2}cable is used (for the ampacity 970 A), the unit cost of the cable line is equal to EUR 566, which is 8% more. Generally, the plane in Figure 3 located at the highest level gives the most profitable solution.

_{e}= 2.5 (K·m)/W and thermal resistivity of the backfill ρ

_{c}= 0.6 (K·m)/W—Figure 4d) gives the following conclusions:

- The 630 mm
^{2}cable is profitable for ampacities below 910 A; - The 800 mm
^{2}cable is profitable for ampacities between 910 and 970 A; - The 1000 mm
^{2}cable is profitable for ampacities higher than 970.

^{2}cable or the 800 mm

^{2}cable, whereas the ampacity 970 A can be achieved by either the 800 mm

^{2}cable or the 1000 mm

^{2}cable.

^{2}cables laid in the trefoil formation where the native soil is ρ

_{e}= 2.5 (K·m)/W and thermal resistivity of the backfill is ρ

_{c}= 0.6 (K·m)/W. If the backfill area is the smallest, the ampacity is equal to 706 A (point 1 in Figure 5). For the largest possible backfill area in the considered case (point 3 in Figure 5), the ampacity is as high as 926 A. The green dashed trace indicates the variation of the ampacity as a function of cost P and the direction of the extension of the backfill shape. The shape between point 1 and point 2 has a constant width—only its height varies. It is an assumption made by the authors, based on practical applications. If the maximal height is achieved (point 2), the backfill width is increased (along the green trace between point 2 and point 3).

## 5. Conclusions

_{e}= 2.5 (K·m)/W and the backfill has ρ

_{c}= 0.6 (K·m)/W, the cost-effective transition point of cable cross-section is observed, meaning that the given cost enables to achieve various ampacities of the cable line depending on which cross-section of the cable and shape of the backfill area are applied. For example, if the assumed cost is EUR 475, the ampacity of the 630 mm

^{2}cable (with large backfill area) can be higher by 15% than the ampacity of the 800 mm

^{2}cable. Thus, in some cases, it is more beneficial to invest in the thermal backfill rather than in the high cross-sectional area of the cables.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Cables in flat formation with spacing, placed in a backfill; L—distance from the ground surface to the cable center, mm; h—height of the backfill, mm; w—width of the backfill, mm; D

_{ext}—axial spacing of the adjacent cables, mm; α—angle of the excavation trench, (°).

**Figure 2.**Cables in trefoil formation, placed in a backfill (for symbols see Figure 1).

**Figure 3.**Ampacity I of the cable line with cables (630 mm

^{2}, 800 mm

^{2}and 1000 mm

^{2}) laid in the flat formation, as a function of costs P and thermal resistivity of the backfill ρ

_{c}; thermal resistivity of the native soil: (

**a**) ρ

_{e}= 1.0 (K·m)/W; (

**b**) ρ

_{e}= 1.5 (K·m)/W; (

**c**) ρ

_{e}= 2.0 (K·m)/W; (

**d**) ρ

_{e}= 2.5 (K·m)/W. Characteristic example points in (

**d**): (EUR 525, 970 A) and (EUR 570, 1040 A)—the cost efficient transition to the cable of higher cross-section (for ρ

_{c}= 0.6 (K·m)/W).

**Figure 4.**Ampacity I of the cable line with cables (630 mm

^{2}, 800 mm

^{2}and 1000 mm

^{2}) laid in the trefoil formation, as a function of costs P and thermal resistivity of the backfill ρ

_{c}; thermal resistivity of the native soil: (

**a**) ρ

_{e}= 1.0 (K·m)/W; (

**b**) ρ

_{e}= 1.5 (K·m)/W; (

**c**) ρ

_{e}= 2.0 (K·m)/W; (

**d**) ρ

_{e}= 2.5 (K·m)/W. Characteristic example points in (

**d**): (EUR 530, 910 A) and (EUR 570, 970 A)—the cost efficient transition to the cable of higher cross-section (for ρ

_{c}= 0.6 (K·m)/W).

**Figure 5.**Ampacity I of the power cable line as a function of the cost P of the line performance and the backfill geometry; 630 mm

^{2}cables laid in the trefoil formation. Characteristic points: (1) I = 706 A (the smallest backfill area); (2) I = 851 A; (3) I = 926 A (the largest backfill area). Thermal resistivities: ρ

_{e}= 2.5 (K·m)/W, ρ

_{c}= 0.6 (K·m)/W.

Parameter | Coefficient | Cost (EUR/m) |
---|---|---|

Manual ground preparation | c_{1} | 13.65 |

Mechanical ground preparation | c_{2} | 10.06 |

Warning tape and concrete slab | c_{3} | 12.33 |

Trench burial | c_{4} | 4.38 |

Native soil disposal, backfilling and compaction | c_{5} | 74.62 |

Laying cables | c_{6} | 2.30 |

630 mm^{2} cable | c_{n},n= 630, 800, 1000 | 75.17 |

800 mm^{2} cable | 106.04 | |

1000 mm^{2} cable | 122.49 |

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**MDPI and ACS Style**

Czapp, S.; Ratkowski, F. Optimization of Thermal Backfill Configurations for Desired High-Voltage Power Cables Ampacity. *Energies* **2021**, *14*, 1452.
https://doi.org/10.3390/en14051452

**AMA Style**

Czapp S, Ratkowski F. Optimization of Thermal Backfill Configurations for Desired High-Voltage Power Cables Ampacity. *Energies*. 2021; 14(5):1452.
https://doi.org/10.3390/en14051452

**Chicago/Turabian Style**

Czapp, Stanislaw, and Filip Ratkowski. 2021. "Optimization of Thermal Backfill Configurations for Desired High-Voltage Power Cables Ampacity" *Energies* 14, no. 5: 1452.
https://doi.org/10.3390/en14051452