# On-Line Core Losses Determination in ACSR Conductors for DLR Applications

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{ac}) of the conductor is a key parameter which allows the ampacity or maximum current capacity of power conductors to be accurately estimated. This can increase their effective capability [16], since Joule and core losses represent the main heat source of the conductor, and can be determined from the alternating current (AC) resistance R

_{ac}[17,18]. Line current, along with ac resistance, are used in some DLR approaches to estimate the effective wind speed [18]. Therefore, DLR enables the development of more accurate power flow analysis approaches if the ac resistance, which changes with temperature, is measured in real time [19].

_{ac}/R

_{dc}ratio [28], R

_{dc}being the direct current (DC) resistance. This is mainly due to the transformer effect [26,27] or current redistribution among the aluminum layers, and only partially due (to a lesser extent) to the eddy currents and hysteresis losses that occur in the steel core [28].

## 2. Theoretical Background

#### 2.1. AC Resistance and Reactance of ACSR Conductors

_{ac}of the conductor above the DC value R

_{dc}at the same temperature. Higher values of the R

_{ac}/R

_{dc}ratio increase the energy losses in the conductor [20].

_{ac}/R

_{dc}. The highest resistance ratio is expected in conductors with a single layer of aluminum, whereas the lowest corresponds to conductors with two layers [3]. The internal inductance of ACSR conductors increases with current up to a maximum value, where the steel core becomes magnetically saturated [36], and then any further increase in current reduces the internal inductance [3] due to the decrease of the magnetic permeability.

_{s}and y

_{p}are, respectively, the skin and proximity effect factors. The effect of temperature is considered as:

_{0}the DC resistance measured at 20 °C and α

_{20}is the temperature coefficient at 20 °C.

_{ac}and reactance X of the conductor expressed in Ω/m as [3,40]:

_{ac}with temperature:

_{ac}

_{,0}is the AC value of the resistance at a given temperature T

_{0}, usually 20 °C, and α

_{ac}is the temperature coefficient of the AC resistance. Note that R

_{ac}

_{,0}is a measured value, which already includes the saturation effect, as proved in [19].

#### 2.2. Power Losses in ACSR Conductors

_{J}), core (P

_{M}) and redistribution (P

_{redi}

_{s}, transformer effect) effects can be expressed as follows:

_{J}+ P

_{M}+ P

_{redis}= I

^{2}R

_{ac}

_{loss}= P

_{J}+ P

_{M}+ P

_{redis}= ΔVIcosφ

#### 2.3. Transient Thermal Balance Equation for DLR Calculation

_{J}+ P

_{M}+ P

_{redi}

_{s}) and solar heat gain must equal the heat loss by convection and radiation [44].

_{J}, P

_{M}, P

_{redis}and P

_{S}being the per unit length heat gain terms of the conductor (Joule, magnetic/core, transformer effect and solar heating terms, respectively, in W/m). P

_{C}and P

_{R}are the per unit length heat loss terms (convective and radiative loss terms, respectively, in W/m), m is the per unit length mass of the conductor in kg/m, c is the specific heat capacity of the conductor expressed in J/(kg °C), T is the average conductor temperature expressed in °C, and t is the time in s. The heat capacity c of the ACSR conductor is calculated as the weighted average of the iron strands in the core and the aluminum strands, and can be expressed as:

_{x}is the specific heat capacity of element x (Al = aluminum or steel), m

_{x}is its mass per unit length, and β is the temperature coefficient of the heat capacity, whose values can be found in [10].

^{2}) as [12]:

_{S}= α

_{s}SD [W/m]

_{s}is the dimensionless solar absorptivity of the conductor surface, where its value is often assumed to be 0.5 [45].

_{R}= πεDσ

_{B}[(T + 273)

^{4}− (T

_{a}+ 273)

^{4}] [W/m]

_{Β}is the Stefan-Boltzmann constant.

_{max}is calculated at the maximum allowable temperature of the conductor, assuming that the temperature of the conductor is in thermal equilibrium, resulting in [10,46]:

## 3. Experimental Setup

#### 3.1. The Analyzed Single-, Two- and Three-Layer ACSR Conductors

#### 3.2. The High-Current Transformer Used to Test the Conductors

#### 3.3. Measuring Devices

## 4. Experimental Results

#### 4.1. Results Obtained with a Single-Layer ACSR Conductor

#### 4.2. Results Obtained with a Two-Layer ACSR Conductor

_{ac}value below 5%.

#### 4.3. Results Obtained with a Three-Layer ACSR Conductor

#### 4.4. Results Summary

_{ac}

_{,0}and α parameters obtained from a linear fit of the experimental data shown in Figure 4, Figure 5 and Figure 6 according to Equation (6), where R

^{2}is the coefficient of determination of the linear regression.

_{ac}

_{,0}measured at 20 °C for the three analyzed current levels. This was due to the effect of the axial component of the magnetic flux. However, for two- and three-layer conductors, R

_{ac}

_{,0}was almost independent of the current level. Results presented in Table 2 also prove that the temperature coefficient of the resistance was almost independent of the current level and the topology of the conductor.

- The AC resistance of two- and three-layer ACSR conductors was nearly independent of the current level, but this simplification cannot be applied to single-layer ACSR conductors. Therefore, for two- and three-layer ACSR conductors, it can be assumed that R
_{ac}= R_{ac}(T), so that the heat gain due to the conductor losses P_{loss}only depends on the conductor temperature, but not on the current level, i.e., P_{loss}= P_{loss}(T). In contrast, for single-layer conductors, R_{ac}depends on both conductor temperature and current level, i.e., R_{ac}= R_{ac}(T,I), and hence P_{loss}= P_{loss}(T,I). - In DLR applications, the conductor surface temperature is often measured, although it differs from the temperature of the internal strands. In strong wind conditions, the temperature difference between the surface of the conductor and the internal parts is typically greater. Therefore, in this study, for a given conductor surface temperature, the apparent AC resistance R
_{ac}measured in strong winds was larger than when measured without wind due to the increased radial temperature gradient under strong wind conditions. However, this difference was always below 5%, so it would not have a significant effect on the calculation of the DLR rating.

- Approach 1, which is valid for ACSR conductors with any number of layers. The current, conductor temperature, voltage drop and the phase shift between the voltage drop and the current must be measured, so that, by applying (4), the actual value of the AC resistance can be determined.
- Approach 2: Two- and three-layer ACSR conductors. For these conductors, the AC resistance R
_{ac}and thus, the heat gain due to conductor losses P_{loss}, are almost independent of current level. Therefore, if the parameters R_{ac}_{,0}and α_{ac}are known, it is possible to measure only the current and the temperature of the conductor, thus avoiding the need to measure the voltage drop and the phase shift between the voltage drop and the current. This is advantageous because the voltage drop measurement has some drawbacks related to the addition of wires placed on the surface of the high-voltage ACSR conductors, with the consequent problems related to outdoor environments. Since R_{ac}cannot be measured without measuring the voltage drop, if R_{ac}_{,0}and α_{ac}are known, R_{ac}can be obtained by applying R_{ac,T}= R_{ac}_{,0}[1 + α_{ac}(T − T_{0})]. According to this equation, the temperature of the conductor, the parameters R_{ac}_{,0}and α_{ac}can be measured in the laboratory for a sample of the conductor, in a similar way as has been done in this paper. - Approach 2: Single-layer conductor. In single-layer conductors, both the AC resistance R
_{ac}and the heat gain due to conductor losses P_{loss}, depend on the current level and the temperature of the conductor. In this case it is also possible to avoid measuring the voltage drop. According to the values presented in Table 2, α_{ac}can be considered as a constant value, so the current level determines R_{ac}_{,0}. Then, R_{ac}can be obtained by applying R_{ac}_{,T}= R_{ac}_{,0}[1 + α_{ac}(T − T_{0})]. Once the values of the parameters R_{ac}_{,0}and α_{ac}summarized in Table 2 are known, they can be interpolated for any current level.

## 5. Conclusions

_{ac}

_{,0}and α

_{ac}.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Cross section of the analyzed ACSR conductors. (

**a**) Single-layer 7/10 conductor. (

**b**) Two-layers 7/26 conductor. (

**c**) Three-layer 7/54 conductor.

**Figure 4.**Single-layer ACSR conductor. R

_{ac}versus temperature measured for heating-cooling cycles at 220 A, 145 A and 75 A.

**Figure 5.**Two-layer ACSR conductor. R

_{ac}versus temperature measured for heating-cooling cycles at 430 A, 280 A and 130 A.

**Figure 6.**Three-layer ACSR conductor. R

_{ac}versus temperature measured for heating-cooling cycles at 1080 A, 650 A and 310 A.

**Figure 7.**Proposed strategies to measure the AC resistance of the conductor as a function of the temperature and current level.

**Table 1.**Main parameters of the three-layer 550-AL1/71-ST1A ACSR conductor from HAASE Gesellschaft and the two-layer 135-AL1/22-ST1A ACSR conductor from EMTA Kablo.

Symbol | Description | Three-Layer | Two-Layer | Unit |
---|---|---|---|---|

${A}_{Al}$ | Area of aluminum | 549.7 | 134.9 | mm^{2} |

${A}_{steel}$ | Area of steel | 71.3 | 22 | mm^{2} |

${N}_{Al}$ | Number of aluminum wires | 54 (12/18/24) | 26 (10/16) | - |

${N}_{Steel}$ | Number of steel wires | 7 | 7 | - |

${D}_{Al}$ | Aluminum wire diameter | 3.6 | 2.57 | mm |

${D}_{steel}$ | Steel wire diameter | 3.6 | 2.0 | mm |

D | Conductor diameter | 32.4 | 16.3 | mm |

${m}_{AL}$ | Mass per unit length of aluminum | 1.5183 | - | kg/m |

${m}_{steel}$ | Mass per unit length of steel | 0.5583 | - | kg/m |

${R}_{20\xb0\mathrm{C}}$ | DC resistance of the conductor | 0.0526 | 0.2038 | Ω/km |

${I}_{max}$ | Current carrying capacity | 1020 | 430 | A |

Cable Type | Current | R_{ac}_{,0} | α_{ac} | R^{2} |
---|---|---|---|---|

Single-layer | 220 A | 602.4 μΩ | 0.0046 °C^{−1} | 0.9997 |

145 A | 535.2 μΩ | 0.0048 °C^{−1} | 0.9991 | |

75 A | 498.5 μΩ | 0.0049 °C^{−1} | 0.9827 | |

Two-layer | 430 A | 200.8 μΩ | 0.0044 °C^{−1} | 0.9999 |

280 A | 200.2 μΩ | 0.0046 °C^{−1} | 0.9996 | |

130 A | 201.9 μΩ | 0.0044 °C^{−1} | 0.9747 | |

Three-layer | 1080 A | 52.3 μΩ | 0.0046 °C^{−1} | 0.9987 |

650 A | 51.0 μΩ | 0.0049 °C^{−1} | 0.9990 | |

310 A | 51.4 μΩ | 0.0047 °C^{−1} | 0.9843 |

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**MDPI and ACS Style**

Riba, J.-R.; Liu, Y.; Moreno-Eguilaz, M.; Sanllehí, J. On-Line Core Losses Determination in ACSR Conductors for DLR Applications. *Materials* **2022**, *15*, 6143.
https://doi.org/10.3390/ma15176143

**AMA Style**

Riba J-R, Liu Y, Moreno-Eguilaz M, Sanllehí J. On-Line Core Losses Determination in ACSR Conductors for DLR Applications. *Materials*. 2022; 15(17):6143.
https://doi.org/10.3390/ma15176143

**Chicago/Turabian Style**

Riba, Jordi-Roger, Yuming Liu, Manuel Moreno-Eguilaz, and Josep Sanllehí. 2022. "On-Line Core Losses Determination in ACSR Conductors for DLR Applications" *Materials* 15, no. 17: 6143.
https://doi.org/10.3390/ma15176143