The Role of AC Resistance of Bare Stranded Conductors for Developing Dynamic Line Rating Approaches

Abstract

Overhead transmission line conductors are usually helically stranded. The current-carrying section is made of aluminum and/or aluminum alloys. Several factors affect their electrical resistance, such as the conductivity of the conductor material, the cross-sectional area, the lay length of the different layers of aluminum, and the presence of a steel core used to increase the mechanical strength of the conductor. The direct current (DC) and alternating current (AC) resistances per unit length of stranded conductors are different due to the effect of the eddy currents. In steel-reinforced conductors, there are other effects, such as the transformer effect due to the magnetization of the steel core, which make the AC resistance dependent on the current. Operating temperature also has an important effect on electrical resistance. Resistive losses are the main source of heating in transmission line conductors, so their temperature rise is highly dominated by such power losses, making it critical to know the value of the AC resistance per unit length when applying dynamic line rating (DLR) methods. They are of great interest especially in congested lines, as by applying DLR approaches it is possible to utilize the full line capacity of the line. This paper highlights the difficulty of accurately calculating the electrical resistance of helically stranded conductors, especially those with a magnetic core, and the importance of accurate measurements for the development of conductor models and DLR approaches.

1. Introduction

Bare aluminum stranded conductors have been used in high-voltage transmission lines for over 100 years. They consist of multiple layers of aluminum or aluminum alloy strands twisted in opposite directions. In the case of steel-reinforced conductors, the core of the conductor, which is used to increase mechanical strength, consists of several galvanized steel wires [1].

As the world increasingly turns to renewable energy and electrification to meet decarbonization targets, ensuring the safe, efficient, and reliable operation of power lines is essential [2,3]. To meet this growing demand, the capacity of existing power lines must be maximized [4], so new approaches are being developed. The replacement of conventional conductors with high-temperature low-sag (HTLS) conductors and dynamic line rating (DLR) strategies based on real-time weather dependent thermal ratings [5,6] are two methods to maximize the capacity of existing overhead transmission lines. DLR solves the conductor heat balance equation based on an on-line measurement of the current flowing through the conductor and its temperature using appropriate sensors, as well as the local weather conditions, including ambient temperature, solar radiation, and wind direction and speed [7,8].

Since ohmic losses are the most significant source of heat in conductors, they have a major impact on the thermal behavior of the conductor. For this reason, it is very important to determine the AC resistance of the conductor with a sufficient degree of accuracy. However, several factors affect the electrical resistance of such conductors, such as the conductivity of the conductor material, the cross-sectional area, the lay length of the different aluminum layers, and the presence of a steel core used to reinforce the mechanical strength of the conductor [1]. Among these, the electrical conductivity of the aluminum strands plays an important role. Since conductor standards typically specify minimum electrical conductivity values, changes in this parameter can cause AC resistance to vary between conductors of the same type. There are other unquantifiable effects such as the contact resistance between adjacent strands, which depends on the condition of the contact surface layer. It is determined by the crystalline structure, the oxidation state, or the presence of grease [9]. All of the above factors will affect the uneven current distribution among the different strands, thus influencing the electrical resistance and the thermal behavior of the conductor. In addition, the DC and AC resistances per unit length of stranded conductors, r_DC [Ω/m] and r_AC [Ω/m], respectively, differ due to the effect of the eddy currents, i.e., the skin and proximity effects, although when the distance between the conductors is much greater than their diameter, the proximity effect becomes much smaller than the skin effect, as commonly occurs in overhead conductors [1]. For steel-reinforced conductors, there are other effects, such as the transformer effect due to the magnetization of the steel core, which make the r_AC dependent on the current. The operating temperature also has an important effect on the electrical resistance [3]. However, the operating temperature of overhead transmission line conductors depends on a thermal balance between heating and cooling effects occurring in the conductor.

This paper deals with the determination of the r_AC of stranded conductors for overhead transmission lines in order to obtain an accurate prediction of ohmic losses in such conductors. These results can be very useful for the development of DLR approaches and models of stranded conductors. For this purpose, three types of conductors with different geometry and composition, and thus different thermal behavior, are analyzed. They are a conductor with all strands made of aluminum alloy, an aluminum conductor with a steel core, and an HTLS conductor. Therefore, the novelties and contributions of this paper are in this area. First, it summarizes the main equations found in the literature to determine the r_DC and r_AC for stranded conductors, highlighting the main difficulties and limitations of such methods. Second, the details of an experimental setup for measuring the AC resistance of stranded conductors based on the temperature rise test are presented. Third, experimental results based on the three stranded conductors of different types are presented, and the experimentally measured DC and AC resistances are compared with the results provided by the different equations. Fourth, the main limitations of the DC and AC resistance equations are summarized in the concluding remarks section, highlighting the need to perform laboratory or on-line tests for the accurate determination of their values in order to apply DLR approaches or develop accurate conductor models.

The paper is organized as follows. Section 2 summarizes the methods for calculating the DC resistance of stranded conductors, while Section 3 summarizes the methods for calculating the AC resistance of stranded conductors. Section 4 describes the conductors analyzed and the experimental setup. Section 5 presents the experimental results in detail. Section 6 presents the concluding remarks, and finally, Section 7 develops the conclusions.

2. Conductor Resistance under DC Supply

According to the international standard IEC 60287-1-1-2023 [10], the effect of temperature on the DC resistance of a conductor per unit length can be described by the following expression:

$$r_{DC}(T)=r_{DC,20°\mathrm{C}}[1+{\alpha }_{20°\mathrm{C}}\left(T-20\right)]\hspace{1em}\left[\Omega /\mathrm{m}\right]$$

(1)

where r_DC(T) is the DC resistance of the conductor per unit length [Ω/m] at a given temperature T [°C], r_DC,20°C is the DC resistance of the conductor per unit length [Ω/m] at 20 °C, and α_20°C [1/°C] is the linear temperature coefficient of the resistance at the reference temperature of 20 °C. Note that the temperature of the conductor in the radial direction is not uniform, especially for stranded conductors with large cross-sections, so the temperature difference between the surface and the center of the conductor also affects the resistance of the conductor.

According to the Aluminum Conductor Electrical Handbook [11] and based on values from ASTM standards, the temperature coefficient of Al at 20 °C depends on its conductivity, so that α_{20°C,Al52.5%IACS} = 0.00347 [°C⁻¹], and α_{20°C,Al61.2%IACS} = 0.00404 [°C⁻¹], while for steel it is suggested α_{20°C,steel,9%IACS} = 0.00320 [°C⁻¹] and for copper α_{20°C,copper,97%IACS} = 0.00381 [°C⁻¹]. These values indicate that for each 10 °C increase in temperature, the DC resistance increases by 3.47% (Al 52.5% IACS) to 4.04% (Al 61.2% IACS). Any increase in resistance is accompanied by an increase in ohmic losses in the conductor. Note that IACS refers to the International Annealed Copper Standard (IACS, https://nvlpubs.nist.gov/nistpubs/Legacy/hb/nbshandbook100.pdf, accessed on 17 September 2024), which provides a comparative reference for conductivity, with 100% IACS corresponding to an electrical conductivity of 58·10⁶ Ω⁻¹·m⁻¹ at 20 °C.

2.1. Solid Conductors of Circular Cross-Section

Assuming an infinitely long solid conductor with a circular cross-section, r_DC,20°C can be calculated as follows:

$$r_{DC,20°\mathrm{C}}={\displaystyle \frac{4}{\pi D_c^2}} {\rho }_{20°\mathrm{C}}\hspace{1em}\left[\Omega /\mathrm{m}\right]$$

(2)

where D_c [m] is the diameter of the conductor and ρ_20°C [Ωm] is the electrical resistivity of the conductor material at 20 °C.

2.2. Multi-Stranded Monometallic and Non-Magnetic Conductors of Circular Cross-Section

When considering stranded conductors, it is important to note that stranding increases the length of the various layers of aluminum strands. For stranded monometallic conductors with n layers, r_DC,20°C can be calculated according to the formulation proposed by Cigré TB 345 as [1] the following:

$${\displaystyle \frac{1}{r_{DC,20°\mathrm{C}}}}={\displaystyle \frac{\pi d^2}{4{\rho }_{20°\mathrm{C}}}} \left[1+{\displaystyle \sum _{i=1}^n\frac{6i}{\sqrt{1+{(\pi D_i/{\lambda }_i)}^2}}}\right]\hspace{1em}\left[\mathrm{m}/\Omega \right]$$

(3)

where d [m] is the diameter of each strand, i = 1,2, … n is the number of layers, and D_i [m] and λ_i [m] are the mean diameter and the lay length of the i-th layer, respectively.

Figure 1 shows a stranded conductor, indicating the lay length L_l. It is defined as the axial distance required for the strand to complete one helical revolution. In addition, the lay ratio is defined as the ratio of the lay length to the mean diameter D_c,mean of the helix [12].

2.3. Multi-Stranded Bimetallic Conductors of Circular Cross-Section

For stranded bimetallic conductors with n_Al aluminum layers and n_steel layers, as is the case for aluminum conductor steel-reinforced (ACSR) conductors, r_D,20°C can be calculated as Cigré 345 [1] by the following formula:

$${\displaystyle \frac{1}{r_{DC,20°\mathrm{C}}}}={\displaystyle \frac{\pi d_{steel}^2}{4{\rho }_{steel,20°\mathrm{C}}}} \left[1+{\displaystyle \sum _{j=1}^{n_{steel}}\frac{6j}{\sqrt{1+{(\pi D_{steel,j}/{\lambda }_{steel,j})}^2}}}\right]+{\displaystyle \frac{\pi d_{Al}^2}{4{\rho }_{Al,20°\mathrm{C}}}} \left[1+{\displaystyle \sum _{i=n_{steel}+1}^{n_{steel}+n_{Al}}\frac{6i}{\sqrt{1+{(\pi D_{Al,i}/{\lambda }_{Al,i})}^2}}}\right]\hspace{1em}\left[\mathrm{m}/\mathsf{\Omega }\right]$$

(4)

where d_Al [m] and d_steel [m] are the diameters of the aluminum and steel strands, respectively; i = 1, 2, … n_Al is the aluminum layer number; j = 1, 2, … n_steel is the steel layer number, D_Al,_i [m] and D_steel,_j [m] are the mean diameters of the i-th aluminum and j-th steel layers; and λ_Al,_i [m] and λ_steel,_j [m] are the lay lengths of the i-th aluminum and j-th steel layers. Note that the lay length is the axial distance required for the wire to complete one full revolution around the conductor diameter.

However, according to Cigré Working Group 22.12 [13], for stranded bimetallic conductors with n_Al aluminum layers and n_steel layers, r_DC,20°C should be calculated as follows:

$${\displaystyle \frac{1}{r_{DC,20°\mathrm{C}}}}={\displaystyle \frac{\pi d_{steel}^2}{4{\rho }_{steel,20°\mathrm{C}}}} \left[1+{\displaystyle \sum _{j=1}^{n_{steel}}\frac{6j}{\sqrt{1+{(\pi D_{steel,j}/{\lambda }_{steel,j})}^2}}}\right]+{\displaystyle \sum _{i=1}^{n_{Al}}\frac{\pi d_{Al,i}^2{n}_{Al,strands,i}}{4{\rho }_{Al,20°\mathrm{C}}\sqrt{1+{(\pi D_{Al,i}/{\lambda }_{Al,i})}^2}}}\hspace{1em}\left[\mathrm{m}/\mathsf{\Omega }\right]$$

(5)

where n_Al,strands,i [-] is the number of strands in the i-th aluminum layer.

From a simple observation of Equations (3)–(5) it can be deduced that in order to determine r_DC,20°C, many parameters of the conductor are required, which are not always available; they are difficult to measure or at least are not known with accuracy. The results are also highly dependent on the exact value of the electrical resistivity of the specific composition of the aluminum strands, which is not exactly known a priori, because it requires a specific measurement.

2.4. Lay Ratios According to the EN 50182 Standard

According to Equations (3)–(5), the lay ratios of the different constitutive layers of the conductor play a key role in determining the DC resistance r_DC. The lay ratios for aluminum stranded conductors and ACR conductors are specified in the EN 50182 standard [14].

Table 1. Lay ratios of the analyzed cables according to the EN 50182 standard [14].

Conductor Type	Wires in Steel Layers	Strands in Al Layers	Lay Ratio Steel Layers	Lay Ratio Al Layers
AAAC 4 layers	-	61 (1/6/12/18/24)	-	15/13.5/12.5/11
ACSR 3 layers	7 (1/6)	54 (12/18/24)	19	15/13/11.5
ACSR 2 layers	7 (1/6)	26 (10/16)	19	14/11.5
ACSR 1 layer	7 (1/6)	10	19	14
ACCC/TW	-	36 (8/12/16)	-	15/13.5/11.5

summarizes the lay ratios of the different layers of the conductors analyzed in this paper. Note that each conductor topology has its own lay ratio distribution for the different layers.

3. Conductor Resistance under AC Supply

At the same temperature, the AC resistance r_AC tends to be higher than the DC resistance r_DC due to the effects described in the following subsections.

3.1. Monometallic and Non-Magnetic Stranded Conductor under Power Frequency Supply

Under power frequency supply, the current density within a stranded conductor is not uniform due to skin and proximity effects as well as variable contact resistance between strands [1]. As a result, at the same temperature, the resistance of a monometallic and non-magnetic conductor is greater under alternating current than under direct current. The magnitude of the skin effect depends on the supply frequency, the conductivity of the material, and the geometry of the conductor. Therefore, the radial current density distribution in the aluminum section and the power losses in the magnetic core are frequency dependent. The effective transverse resistivity also has a profound effect on the uneven current distribution of stranded conductors. It is greatly affected by the inter-strand contact resistance, the contact resistance at the crossings of the strands, and the characteristics of the contact surface layer, such as the degree of oxidation, the presence of grease, or the crystalline structure [9]. Therefore, the condition and the size of the contact area between the strands play an important role, along with the compression tension to which the conductor is subjected [15] and the angle of intersection between the strands [9].

There is an exact solution for the AC resistance of an infinitely long solid non-magnetic conductor of circular cross-section and also for an infinitely long tubular round conductor [16], so these solution take into account the skin effect. There is also an approximate analytical solution for two infinitely long parallel conductors of circular cross-section forming a single-phase system [16,17], so this solution takes into account both skin and proximity effects. However, there are no exact methods for stranded conductors and for conductors with a magnetic core, so that the methods found in the technical literature [15] are based on many assumptions and the calculation process is complex.

For stranded aluminum conductors with or without a magnetic core, in the 20–50 mm diameter range, the skin effect increases the resistance by approximately 1% and 10%, respectively [1].

In the case of having tabulated values of r_AC at two temperatures T₂ > T₁, the IEEE Std-738 for calculating the current–temperature of bare overhead conductors [18] suggests determining the conductor resistance at a given temperature T_avg, where T₁ ≤ T_avg ≤ T₂, by applying the following linear fit:

$$r_{AC}(T_{avg})={\displaystyle \frac{r_{AC}(T_2)-r_{AC}(T_1)}{T_2-T_1}}·(T_{avg}-T_1)+r_{AC}(T_1)\hspace{1em}\left[\mathsf{\Omega }/\mathrm{m}\right]$$

(6)

Note that when using Equation (1), the tabulated values of r_AC should include the skin effect, as well as the effects of the lay ratios and the magnetic core. However, some conductor manufacturers only provide the value of r_DC at 20 °C. It should be noted that the conductor temperature is not homogeneous within the cross-section, as it varies radially, with a maximum at the center and a minimum at the surface, so this effect must be considered as it affects the value of r_AC.

The IEC 60287-1-1 standard [10], proposes a method for calculating the r_AC/r_DC ratio for non-magnetic conductors taking into account the skin and proximity factors as follows:

$$r_{AC}/r_{DC}=1+y_s+y_p [-]$$

(7)

The skin effect factor y_s is calculated as follows

$$y_s={\displaystyle \frac{x_s^4}{192+0.8x_s^4}}$$

(8)

where the following applies:

$$x_s^4={\left({\displaystyle \frac{8\pi f{K}_s}{{10}^7{r}_{DC,T}}}\right)}^2$$

(9)

where K_s = 1 for stranded Al conductors and r_DC,T is the DC resistance per unit length at the operating temperature T. Note that Equation (10) is accurate when x_s ≤ 2.8.

The proximity effect factor y_p for parallel conductors can be calculated as follows:

$$y_p={\displaystyle \frac{2.9x_p^4}{192+0.8x_p^4}}{(d_c/s)}^2$$

(10)

where the following applies:

$$x_p^4={\left({\displaystyle \frac{8\pi f{K}_p}{{10}^7{r}_{DC,T}}}\right)}^2$$

(11)

where d_c [mm] is the conductor diameter, s [mm] is the distance between the axes of the conductors, and K_p = 1 for stranded conductors. Note that the contribution of the proximity factor y_p is typically very small for transmission line conductors, since s >> d_c [1].

3.2. Bimetallic Stranded Conductor with Magnetic Core under Power Frequency Supply

For magnetic core conductors, r_AC depends not only on temperature and power frequency, but also on the magnitude of the electric current [18]. Due to the much lower electrical conductivity of the steel core, most of the AC current flow in ACSR conductors is through the aluminum strands [19]. The aluminum strands are wound helically around the magnetic core, creating an axial AC magnetic flux within the steel core, similar to a solenoid, which increases with the magnitude of the electric current [18]. The adjacent layers of helically stranded conductors are stranded in opposite directions so that their respective currents produce magnetic fields whose axial components are in opposite directions and tend to cancel each other out. Under power frequency supply, the axial component of the total magnetic field induces eddy currents and hysteresis losses in the magnetic core, while modifying the current distribution of the different aluminum layers. The magnetic coupling of the current through the steel core is known as the transformer effect [1], and its intensity is determined by the magnitude of the axial magnetic flux. The amplitude of the axial magnetic flux depends on the magnitude of the electric current, the supply frequency, the magnetic permeability of the magnetic core, and the lay length [20]. The magnetic permeability of the steel core depends on its temperature and tensile stress [21], which affect the AC resistance of steel core stranded conductors [1].

The cancellation of the axial component of the magnetic field generated by the different aluminum layers is not possible in a single-layer aluminum conductor, and it is maximum in two-layer aluminum conductors. Therefore, this cancellation is generally more important in conductors with an even number of layers than in conductors with an odd number of layers [22,23,24]. As a result, the core losses and the uneven distribution of current density among the different aluminum layers are more pronounced in ACSR conductors with an odd number of aluminum layers than in conductors with an even number of aluminum layers [1,25]. Therefore, the effect of the core on r_AC depends on the topology of the stranded conductor. It is very small for two-layer ACSR conductors, maximum for single-layer conductors, where r_AC increases up to 20%, and moderate for three-layer ACSR conductors, where r_AC increases up to 5% [18]. However, these values depend on the current density through the aluminum strands, so if tabulated values of r_AC are used, they must be transformed using correction curves to obtain a multiplier factor that depends on the current density [18]. However, this method is impractical because it requires detailed conductor information, which is not always available.

According to [20], the r_AC/r_DC resistance ratio of three-layer ACSR conductors can be found by correcting r_DC for temperature and skin effect at power frequency, but for conductors with an odd number of aluminum layers, a current-dependent resistance multiplier r_AC,total/r_AC,skin,T for core loss must also be applied, where r_AC,total is the total AC resistance and r_AC,skin,T is the AC resistance considering only temperature and skin effect. However, the resistance multiplier factor r_AC,total/r_AC,skin,T obtained from the experimental data of different three-layer conductors at different current densities shows a wide dispersion of results.

In many papers, the conductor is considered as a homogeneous heat source, and the ohmic losses are calculated based on its DC resistance, as it can be found in manufacturers’ datasheets and relevant standards. However, skin and proximity effects, the transformer effect, and the associated uneven current distribution between aluminum layers or radial temperature gradients cause temperature differences of about 10% between calculated and measured temperature values [1].

Cigré [13] provides current-dependent correction factors to determine r_AC from r_DC for ACSR conductors that account for the transformer effect, but they are specific to each conductor configuration. These correction factors are summarized in Equation (12):

$$\left\{\begin{array}{ll}3\text{-}\mathrm{layer} \mathrm{ACSR} \mathrm{conductors}:& r_{AC}/r_{DC}=1.0123+2.36·{10}^{-5}{I}_{AC}\\ 1\text{-}\mathrm{layer} \mathrm{or} 2\text{-}\mathrm{layer} \mathrm{ACSR} \mathrm{conductors}:& r_{AC}/r_{DC}=1.0045+9·{10}^{-8}{I}_{AC}\end{array}\right.$$

(12)

Note that the formulas in Equation (12) were derived for a three-layer Zebra ACSR conductor with a 428-Al/S1A-54/7 structure and for single-layer or two-layer ACSR conductors with a cross-section greater than 175 mm². The current-dependent term in Equation (12) is approximate because it depends on factors such as the lay length, which varies from conductor to conductor.

4. The Conductors Analyzed and the Experimental Setup

This section describes the conductors analyzed, which include three types of transmission line conductors with very different characteristics. These conductor types were selected because they are among the most commonly used conductors in transmission lines and exhibit different electrical resistance behaviors.

4.1. All Aluminum Alloy Conductors (AAAC)

AAAC conductors are composed of high-tensile-strength aluminum alloy strands. This type of conductor does not require a steel core, so it has high corrosion resistance and good sag-tension characteristics.

This paper analyzes the Aster 570 AAAC conductor, which consists of 61 aluminum strands distributed as shown in Figure 2.

4.2. Aluminum Conductor Steel-Reinforced (ACSR) Conductors

ACSR conductors have been used extensively in transmission lines for the past 100 years [26,27]. ACSR conductors consist of a core of steel strands surrounded by aluminum strands. All strands are wound helically, while the strands of adjacent layers are wound in opposite directions [20]. Due to aluminum’s much higher electrical conductivity, the aluminum strands carry most of the current [28], while the steel strands provide the mechanical strength [22]. However, eddy current and hysteresis losses occur in the core due to the 50 Hz magnetic flux [23]. The combined effect of the uneven current distribution in the aluminum layers, the magnetism in the core, and the transformer effect tends to increase the r_AC [22].

Figure 3 shows the configuration of the studied ACSR conductors.

By carefully removing the outer layer of the aluminum strands, the two-layer ACSR conductor was converted to a single-layer ACSR conductor.

4.3. High-Temperature Low-Sag (HTLS) Conductors

Due to their much higher ampacity, HTLS conductors are used to replace conventional conductors in congested areas. The increased ampacity at nearly the same cross-sectional area is due to higher temperature operation and low conductor sag due to the exceptional properties of the core [29]. An HTLS conductor with a non-conductive hybrid core and aluminum trapezoidal wires (TW) helically wound around the core is analyzed in this section. It is known as an aluminum conductor composite core (ACCC/TW). This conductor can operate up to 180 °C [30] using 1350-O annealed aluminum wires [31].

Figure 4 shows the cross-section of the Dhaka ACCC/TW conductor studied in this section.

4.4. Experimental Setup for Heating the Conductors

To obtain the experimental relationship between the AC resistance per unit length r_AC and the temperature T, a 50 Hz AC current was injected into the studied conductors to change their temperature T. The conductors were connected to a controlled high-current transformer (150 kVA, 0–400 Vrms input voltage, 0–10 Vrms output voltage). The current flowing through the conductors was measured with a Rogowski current sensor (CWT60LF, ±1%, PEM, Nottingham, UK). Low-inertia T-type thermocouples were used in conjunction with a thermocouple data acquisition module (TC08, ±1 °C, Omega, Northbank, UK) to acquire the temperatures at different points on the conductors, which were averaged.

The DC resistance r_DC was measured using a μΩ meter (Centurion II, 10–200 A, ±0.1 μΩ, Raytech, Wohlen, Switzerland), while the AC resistance r_AC = (ΔV/I)·cosφ was measured by measuring the voltage drop ΔV along the length of one meter of conductor, the current I flowing through the conductor, and the phase shift between ΔV and I. A data acquisition card (USB-6343, 16 bits, 500 kS/s, National Instruments, Austin, TX, USA) was used to acquire the ΔV and I values. This setup allows an accuracy in measuring r_AC better than 1 μΩ.

Figure 5 shows a schematic of the indoor experimental setup consisting of a conductor loop, a high-current transformer, and all associated instrumentation. Note that the length of the conductors analyzed is approximately 8 m.

5. Experimental Results

This section summarizes the experimental results with the different conductors analyzed and explains the practical difficulties in calculating the DC and AC resistances.

5.1. Aster 570 AAAC Conductor

Figure 6 shows the experimental results of the r_AC versus the temperature for the Aster 570 AAAC conductor. To obtain these results, the conductor was heated at different current levels under power frequency supply using the high-current transformer.

Table 2. Main parameters of the Aster 570 AAAC conductor.

Conductor Parameters	Values
Number of Al alloy strands	61 (1/6/12/18/24)
Lay ratios [14]	15/13.5/12.5/11
Diameter of the strands [mm]	3.45
Measured conductor diameter [mm]	31.15
Aluminum alloy cross-section [mm2]	570.22
Minimum electrical resistivity of Al alloy [11]	52.5% IACS
Measured temperature coefficient of resistance, α [°C−1]	0.00330
Measured rDC at 20 °C [μΩ/m]	56.77
Calculated rDC at 20 °C [μΩ/m] from (3)	59.34
Calculated rAC/rDC ratio from (7)	1.0246
Calculated rAC at 20 °C [μΩ/m] from (3) + (7)	60.80
Measured AC resistance per unit length (20 °C) [μΩ/m]	57.69
Difference between measured and calculated values of rAC [%]	−5.39%

shows the main parameters of the Aster 570 AAAC conductor. It combines datasheet values with calculated values from Equations (3) and (7), and experimental measurements performed in this work, some of which are extracted from the results shown in Figure 6.

Note that the values shown in Table 2 may vary between different conductors of the same type. According to [11], the minimum electrical resistivity of the Al alloy must be higher than 52.5% IACS. For example, if this parameter is changed from 52.5% IACS to 57.5% IACS, the calculated value of r_DC by applying Equation (3) changes from 59.34 to 54.18 μΩ/m. This is a difference of about 10%, so this parameter has an important influence on the final result. In addition, the outer diameter is never completely uniform and depends on the mechanical stress, which affects the pitch ratio. The pitch ratios shown in Table 2 are taken from the EN 50182 standard [14], so their values are approximate in the sense that they can vary according to the above constraints, while affecting the calculated value of r_DC derived from Equations (3)–(5).

5.2. Three-Layer, Two-Layer, and Single-Layer ACSR Conductors

Figure 7 shows the experimental results of r_AC versus temperature for the three ACSR conductors studied. These results were obtained by heating the conductors at different current levels using the experimental setup shown in Figure 5.

The results shown in Figure 3 show that, as expected, r_AC depends on the current level, but its behavior is very different in three-layer, two-layer, and single-layer conductors. While the dependence of r_AC on the current level is very pronounced in single-layer conductors, this dependence is almost absent in two-layer conductors and very weak in three-layer conductors. This behavior is due to the total cancellation, partial cancellation, or absence of cancellation of the axial component of the magnetic field generated by the different aluminum layers, depending on the number of aluminum layers of the conductor, as explained in Section 3.2.

Table 3. Main parameters of the analyzed ACSR conductors.

Description	Three-Layer	Two-Layer	Single-Layer
Designation	550-AL1/71-ST1A	135-AL1/22-ST1A	-
Number of Al strands	54 (12/18/24)	26 (10/16)	10
Number of steel wires	7 (1/6)	7 (1/6)	7 (1/6)
Area of aluminum [mm2]	549.7	134.9	51.87
Area of steel [mm2]	71.3	22	22
Steel layer lay ratio [14]	19	19	19
Al layers lay ratios [14]	15/13/11.5	14/11.5	14
Diameter of Al strands [mm]	3.6	2.57	2.57
Diameter of steel wires [mm]	3.6	2.0	2.0
Measured outer diameter [mm]	32.4	16.3	11.2
Area of aluminum [mm2]	549.7	134.9	51.87
Area of steel [mm2]	71.3	22.0	22.0
Minim. conductivity of Al [32]	61% IACS	61% IACS	61% IACS
Min. conductivity of steel [33]	8% IACS	8% IACS	8% IACS
Measured temperature coefficient of resistance, α [°C−1]	0.00434 (1080 A) 0.00459 (650 A) 0.00425 (310 A)	0.004435	0.00462 (220 A) 0.00482 (145 A) 0.00487 (75 A)
Measured rDC at 20 °C [μΩ/m]	51.75	200.80	489.70
Calculated rDC at 20 °C [μΩ/m] from (4)	51.04	177.60	410.19
Calculated rDC at 20 °C [μΩ/m] from (5)	52.00	211.27	527.65
Calculated rAC at 20 °C [μΩ/m] from (4) + (12)	52.97 (1080 A) 52.45 (650 A) 52.04 (310 A)	178.40 (all currents)	412.04 (all currents)
Calculated rAC at 20 °C [μΩ/m] from (5) + (12)	53.96 (1080 A) 53.48 (650 A) 53.02 (310 A)	212.22 (all currents)	530.02 (all currents)
Measured rAC at 20 °C [μΩ/m]	53.93 (1080 A) 53.10 (650 A) 52.90 (310 A)	201.52 (all currents)	602.38 (220 A) 535.12 (145 A) 498.78 (75 A)
Difference between measured and calculated values of rAC [%] from (4) + (12)	+1.78% (1080 A) +1.22% (650 A) +1.63% (310 A)	+11.47%	+31.60% (220 A) +21.31% (145 A) +17.39% (75 A)
Difference between rAC,measured and rAC,calculated [%] from (5) + (12)	−0.06% (1080 A) −0.72% (650 A) −0.25% (310 A)	−5.31%	+12.01% (220 A) +0.95% (145 A) −6.26% (75 A)

summarizes the main parameters of the analyzed ACSR conductors. It combines the datasheet values with the calculated values from Equations (4), (5) and (12), and the experimental measurements performed in this work, some of which are extracted from the results shown in Figure 7. The results presented in Table 3 show that the smallest differences between the measured and calculated values of the AC resistance are found in three-layer ACSR conductors, while for two-layer and single-layer conductors the differences are much larger, especially when using Equation (4), so it seems that Equation (5) gives more accurate results than Equation (4). Note that these results are subject to an appropriate choice of the electrical conductivity of the Al strands and, to a lesser extent, of the steel wires, as this is a critical parameter. In addition, Equation (12) is not very suitable for single-layer conductors because the dependence on current is very weak (factor 9·10⁻⁸), while experimental tests show that the current level has a large effect on the r_AC/r_DC ratio, as shown in Figure 7c.

Once again, the electrical conductivity of the aluminum strands plays a critical role since the percentage change in this parameter is directly and proportionally reflected in the r_DC value given by Equations (4) and (5).

5.3. HTLS (Dhaka ACCC/TW) Conductor

Figure 8 shows the experimental results of the r_AC versus the temperature of the Dhaka ACCC/TW conductor. These results were obtained by heating the conductor at different current levels under power frequency supply up to nearly 190 °C with a high-current transformer.

Note that the temperature coefficient of resistance α plays a key role when analyzing a wide temperature range. For example, assuming r_AC = 39.445·(1 + 0.00436·ΔT) or r_AC = 39.445·(1 + 0.00390·ΔT) with ΔT = 180 °C, in the first case it results in r_AC = 70.401, while in the second case it results in r_AC = 67.135, so the difference is Δr_AC ≈ 5%.

Table 4. Main parameters of the HTLS (Dhaka ACCC/TW) conductor.

Conductor Parameters	Value
Number of trapezoidal Al strands	36 (8/12/16)
Measured conductor diameter [mm]	32.87
Core diameter [mm]	9.53
Layer thickness [mm]	3.89
Aluminum area [mm2]	723.975
Equivalent diameter of strands [mm]	5.06
Lay ratios EN 50182 [14]	15/13.5/11.5
Minimum conductivity of Al 1350-0 [31]	61.8% IACS
Measured temperature coefficient of resistance, α [°C−1]	0.00436
Measured rDC at 20 °C [μΩ/m]	39.00
Calculated rDC at 20 °C [μΩ/m] from (3)	38.55
Calculated rAC/rDC ratio from (7)	1.0539
Calculated rAC at 20 °C [μΩ/m] from (3) + (7)	40.63
Measured AC resistance per unit length (20 °C) [μΩ/m]	39.45
Difference between measured and calculated values of rAC [%]	−2.90%

summarizes the main parameters of the analyzed Dhaka ACCC/TW conductor. It combines datasheet values with calculated values from Equations (3) and (7), and experimental measurements performed in this work, some of which are extracted from the results shown in Figure 8.

Note that Equation (3) assumes strands of circular cross-section. However, the Dhaka ACCC/TW conductor is composed of trapezoidal strands, so Equation (3) cannot be directly applied. The calculated value of r_DC was obtained by dividing the total aluminum area (723.975 mm²) by the 36 aluminum strands (20.11 mm²/strand), so assuming that they are all identical and of round cross-section, the diameter of the equivalent strands of round cross-section is 5.06 mm. Note also that any deviation of the Al strand conductivity from the tabulated value is directly reflected in the r_DC value given by Equation (3), and by the same percentage change.

6. Concluding Remarks

The previous sections have shown that there are many factors that affect the calculation of the r_DC resistance, such as the following:

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The exact value of the electrical conductivity of the conductor materials, which has a large influence on the r_DC value.

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The temperature coefficient of resistance, which is an intrinsic property of the conductor material that depends on the specific alloy composition and can vary between similar conductors. Its effect is more pronounced in HTLS conductors, due to the wide range of operating temperatures.

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Knowledge of the lay ratio values.

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Deviation of core and outer diameters from datasheet values.

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Uneven temperature distribution along the conductor.

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Differences in r_DC and r_AC values obtained with different formulas suggested by regulatory and internationally recognized organizations.

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The inability to quantify the effect of the contact resistance between the strands, which affects the effective transverse resistivity and the uneven current distribution of the stranded conductors.

In addition, the calculation of the r_AC is affected by other factors:

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The inability to obtain explicit and exact equations to determine the effect of eddy currents in stranded conductors and, even worse, in steel core conductors.

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The transformer effect in steel core conductors, which makes the r_AC value dependent on the current level due to the non-linear magnetization of the steel core and the dependence of the magnetic permeability on the current level, temperature, and tensile stress.

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The methods found in the technical literature to determine the r_AC/r_DC ratio usually make important assumptions, such as a homogeneous current distribution among the different layers of the aluminum strands, or a homogeneous distribution of the magnetic field in the steel core, which are typically not met by real conductors.

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There are many stranded conductor topologies and materials used for overhead transmission lines. Such conductors present a wide range of operating currents and temperatures, coupled with the impossibility of obtaining highly accurate explicit formulas to determine the r_AC resistance.

As a result, the AC resistance of multi-stranded conductors, with or without a magnetic core, is therefore a complex multiphysics problem that depends on many parameters that are difficult to know and control, and some of which are likely to evolve with the aging of the conductor. As a result, it is extremely complex to develop accurate analytical formulations to determine the r_AC/r_DC ratio of stranded conductors, so it is important to perform laboratory or on-line measurements of r_AC to develop accurate and reliable DLR approaches and conductor models.

7. Conclusions

This paper has analyzed the DC and AC resistance of stranded conductors for overhead transmission lines, since their temperature is highly dominated by resistive losses, as they are the main heat source. Although stranded conductors seem to be simple elements, the reality is quite different, since the resistance depends on several parameters, some of which are not quantifiable or can vary between different conductors. The electrical conductivity of the aluminum strands has a profound effect on the resistance of the conductor, but other factors also play a role, such as the lay length of the various aluminum layers, the effective cross-sectional area of the conductor, or the contact resistance between adjacent strands, which depends on the condition of the surface layer between the strands. It is reasonable to assume that some of these parameters may evolve as the conductor ages. In addition, eddy current effects in stranded conductors do not have explicit exact equations. In the case of ACSR conductors, the magnetization of the core wires also plays an important role. The combined action of the above effects results in the impossibility of obtaining exact equations to predict the AC resistance of stranded conductors with or without a magnetic core.

A comparison between experimental results and those obtained from explicit equations has been presented for three types of stranded conductors, i.e., AAAC, ACSR, and ACCC/TW-HTLS conductors. It has been shown that the largest differences between measured and calculated values are found in the case of ACSR conductors, especially for single-layer and two-layer configurations. This is because ACSR conductors have a magnetic core, so the current distribution between the strands is more difficult to predict and formulate due to the transformer effect, the intensity of which depends on effects such as the current dependence of the magnetic permeability and the non-linear magnetization of the magnetic core. Therefore, in ACSR conductors, the AC resistance depends on the current level, and this dependence is characteristic of each specific conductor, so there are no explicit exact equations to quantify this effect.

The results presented in this paper have shown the difficulty of accurately calculating the electrical resistance of helically stranded conductors, especially those with a magnetic core, and the importance of performing accurate laboratory or on-line measurements of the AC resistance. These results enable the development of accurate line models, which are key elements in the development of reliable DLR approaches. DLR provides valuable tools for grid operators to provide information about current line capacity, make informed decisions about the amount of power a particular line can safely carry, manage power flows more efficiently, manage alerts when line ratings are close to their safe limits, and optimize the use of transmission lines during periods of congestion or high demand.