# A Multiphysics Analysis of Coupled Electromagnetic-Thermal Phenomena in Cable Lines

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. General Remarks

- description of a laboratory stand for examination of current distribution in multi-strand lines,
- development of a numerical model taking into account coupled electromagnetic-thermal phenomena,
- comparison of computation results from ICEPACK and MAXWELL-MECHANICAL codes for chosen geometries,
- variations of temperature and RMS current values during successive iterations using the coupled electromagnetic-thermal model.

#### 1.2. Literature Review Concerning Ampacity Computation

## 2. Materials and Methods

^{2}and 240 mm

^{2}). The strand was made of aluminum, insulation—of cross-linked polyethylene (XLPE) and the coating—of polyvinyl chloride, cf. Figure 4.

_{n}= 1000 A, minimal fundamental accuracy 1%, according to the producer).

## 3. Numerical Modeling

- modeling was carried out for 2D geometry. The full 3D modeling is possible in ANSYS, yet it is very time consuming; some tests carried out by the first author have shown that it may take several hours even for relatively simple geometries using a state-of-the-art desktop computer;
- individual strands twisted together to form a cable line (cf. Figure 2) are treated as a whole. This simplification may be called a geometric homogenization on the local scale. The proximity effects are accounted only between “clusters” of strands from the macroscopic standpoint. It can be remarked that this approach is a typical one; yet a recent publication on the wiring system for electrical vehicles focused on a more detailed scale [53];
- the cables are cooled by natural convection. It is assumed that their distance from the floor is sufficient to neglect the floor heating due to radiation emitted from the cables.

^{−1}, respectively.

#### Determination of Free Convection Coefficient and Modeling

- the working strand—79.0 °C
- insulation—69.8 °C
- outer coating (measurement point below the strand)—62.0 °C
- outer coating (measurement point above the strand)—67.1 °C

^{2}. Because the strand temperature varied from 62.0 to 67.1 °C along the perimeter, the average value 64.55 °C was assumed for further calculations. The computed value of natural convection coefficient was equal to h = 10.5 W/(m

^{2}K).

- −
- unit loss per 1 cm length was calculated in Maxwell using the 3D model, it was equal to 0.557 W;
- −
- the values of thermal conductivity coefficients were taken from relevant materials science publications. They were as follows: PVC = 0.17 W/(m K), XPLE = 0.52 W/(m K) and ALU = 237 W/(m K). The radiation coefficient was 0.94. The ambient temperature was assumed as 26.1 °C.

^{2}K).

_{amb}= 25 °C was used as the starting value for computation of aluminum conductivity in the MAXWELL module. In accordance with the assumed algorithm the computed current values after the first iteration were imported to the thermal module and subsequently the temperature attained by individual strands was computed. The highest temperature values (90.3 °C) were obtained for the strands at the edges (I

_{RMS}= 722 A), the lowest ones (52.1 °C)—for the strands in the middle (I

_{RMS}= 329 A). The temperature distribution after the first and the last iteration is shown in Figure 18.

## 4. Conclusions

- further validation of the barycenter criterion for optimal ampacity of multi-strand cable lines, outlined in Reference [7], considering the effect of harmonics and the presence of conducting shelves for placement of cables; a comparison of results obtained with the aforementioned criterion and with the VIS algorithm shall be carried out; and
- FEM-based studies of more complicated cases (distorted current waveforms, chosen three phase configurations, consideration of conducting power cable tray, forced air flow etc.).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Field interactions in a general electromagnetic problem. Source: own work, based on Driesen’s Ph.D. Thesis [20].

**Figure 3.**The flowchart of coupled-field computations. Source: own work, based on Driesen’s Ph.D. Thesis [20].

**Figure 6.**An exemplary model of a six stranded YAKXS 1 × 240 cable (

**a**) a single strand (

**b**) the whole setup.

**Figure 10.**Measured heating curves for chosen locations indicated in Figure 9.

**Figure 12.**Temperature distribution for a single strand: (

**a**) from ANSYS-MECHANICAL computations for the strand itself, (

**b**) ICEPACK, local temperature distribution of surrounding free air is also shown.

**Figure 13.**Temperature distribution for the single strand from the coupled electromagnetic-thermal computations using ANSYS-MECHANICAL module.

**Figure 14.**Computed temperature profiles for the congested trefoil configuration (

**a**) with ICEPACK and (

**b**) with ANSYS-MECHANICAL coupled module.

**Figure 15.**Computed temperature profiles for the distant trefoil configuration (

**a**) with ICEPACK (

**b**) with ANSYS-MECHANICAL coupled module.

**Figure 16.**Computed distributions of (

**a**) power density (

**b**) temperature for the configuration ABC/ABC.

**Figure 19.**An exemplary three-phase system. Values in the middle of circles indicate the computed RMS current values, obtained after the final iteration.

**Figure 20.**Computed temperature distribution (

**a**) after the first, (

**b**) after the last iteration, (

**c**) for the excitation that includes odd harmonics.

YAKXS 1 × 240 | YAKXS 1 × 70 | |
---|---|---|

Working aluminum strand—RMC | 240 mm^{2} | 70 mm^{2} |

Insulation—cross-linked polyethylene | 1.7 mm | 1.1 mm |

Coating—polyvinyl chloride | 1.7 mm | 1.4 mm |

Outer diameter | 24.8 mm | 14.7 mm |

Maximum resistance at 20 °C | 0.125 Ω/km | 0.443 Ω/km |

Maximum working temperature | 90 °C | 90 °C |

A1B1C1A2B2C2 | A1B1C1/C2B2A2 | A1B1C1/A2B2C2 | ||||
---|---|---|---|---|---|---|

P (W) | Tmax (°C) | P (W) | Tmax (°C) | P (W) | Tmax (°C) | |

A1 | 22.91 | 24.73 | 24.48 | |||

B1 | 24.61 | 25.19 | 90.85 | 26.87 | 93.3 | |

C1 | 24.27 | 74.1 | 24.40 | 24.52 | ||

A2 | 22.91 | - | 24.37 | 24.53 | ||

B2 | 24.61 | - | 25.21 | 90.85 | 26.89 | 93.3 |

C2 | 24.26 | 74.1 | 24.40 | 24.49 | ||

ΣP [W] | - | 143.57 | - | 148.3 | - | 151.78 |

Strand No. | RMS Current after the First Iteration (A) | Final RMS Current (A) |
---|---|---|

1 | 722 | 687 |

2 | 411 | 437 |

3 | 329 | 348 |

4 | 329 | 348 |

5 | 411 | 437 |

6 | 722 | 687 |

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**MDPI and ACS Style**

Cywiński, A.; Chwastek, K.
A Multiphysics Analysis of Coupled Electromagnetic-Thermal Phenomena in Cable Lines. *Energies* **2021**, *14*, 2008.
https://doi.org/10.3390/en14072008

**AMA Style**

Cywiński A, Chwastek K.
A Multiphysics Analysis of Coupled Electromagnetic-Thermal Phenomena in Cable Lines. *Energies*. 2021; 14(7):2008.
https://doi.org/10.3390/en14072008

**Chicago/Turabian Style**

Cywiński, Artur, and Krzysztof Chwastek.
2021. "A Multiphysics Analysis of Coupled Electromagnetic-Thermal Phenomena in Cable Lines" *Energies* 14, no. 7: 2008.
https://doi.org/10.3390/en14072008