Sag and Tension Calculations for High-Voltage Overhead Line Conductors

Abstract

Overhead lines are used to transmit electricity from where it is generated to the receiving stations. The correct design of an overhead line affects public safety, because it should ensure the required clearances between conductors and the ground and objects located in the space under the overhead line. The temperature of conductors in overhead lines depends on the load current and weather conditions, and affects the sag and tension of the conductors. Calculations of sags and tensions of overhead conductors can be performed using simplified calculation methods that do not consider insulator sets. In some situations, this approach may cause calculation errors. This article discusses algorithms for calculating overhead conductor tensions and sags in the tensioning sections of high-voltage overhead lines, accounting for and excluding insulator sets. The analysis is carried out for different lengths of tensioning sections and various thermal and mechanical states of the conductors.

1. Introduction

An overhead power line can be defined as an overhead device intended for the transmission and distribution of electrical energy. The components of a high- and extra-high-voltage overhead line are working and lightning conductors, insulator sets, supporting structures, as well as line equipment (i.e., connectors, clamps, and insulator equipment). Overhead line wires are usually made as multi-wire lines arranged alternately in left-handed and right-handed layers. Most overhead lines around the world use aluminum–steel (aluminum conductor steel reinforced (ACSR)) conductors [1,2]. During their installation in the overhead line, conductors are suspended with a tension of no more than 10–30% of the rated tensile strength (RTS) in conditions where the conductor temperature is in the range of 5–30 °C. During the continued operation of the overhead line, which is expected to last several dozen years, the working conductors reach high temperatures, especially during peak summer loads. All working and lightning conductors are also periodically subject to high mechanical loads in winter conditions during ice loading or ice loading combined with wind loading. The conductors are also exposed to troublesome aeolian vibrations and high offsets caused by the wind. During all the above-mentioned states, the conditions for the reliable and environmentally safe operation of the line must be maintained throughout the entire long-term operation period [3]. Due to the need to ensure high power supply reliability, overhead lines are considered critical infrastructure that should be protected against terrorist activities and attacks by unmanned aerial vehicles [4]. Power systems are exposed to cyber attacks, and protection against these attacks is currently a very important issue for transmission network operators. Methods of protecting power systems against the injection of false data regarding the estimation of the operating status of these systems are described in [5,6].

The key condition for the operation of overhead network infrastructure is to ensure public safety by maintaining the required vertical distances between the phase conductors of the line and objects located in the space under the overhead line. Despite the influence of weather and line load, the conductors should be located at a safe distance from buildings, objects, and people or vehicles near the line. An important issue is also to guarantee the required distances between circuits with different rated voltage levels running on common supporting structures [3,7,8,9]. Ensuring the safe operation of an overhead line in various combinations of weather conditions requires carrying out many variants of calculations of the mechanical components of the line, both at the design stage and later, e.g., during modernization aimed at increasing the current carrying capacity of a modernized line. The conductor’s behavior in the future is determined through calculations, commonly called sag–tension calculations. Sag–tension calculation predicts conductor behavior based on the suggested tension limits under various load conditions. This tension limit determines the percentage value of rated tensile strength which cannot be exceeded at the time of installation or during operation of the line.

The “ruling span method” is a commonly used method for determining stresses and sags in overhead lines around the world. One of the limitations of this method is the significant calculation errors in the tensioning sections with uneven lengths of individual spans at high operating temperatures of the conductor [10,11]. According to [12], the mechanical analysis carried out for tensioning sections can be performed, taking into account or omitting insulator sets. Omitting insulator sets from the computational model may in some cases lead to significant computational errors. The assessment of the excessive proximity of overhead line conductors due to the flow of fault currents can be performed using the finite element method [13]. Issues regarding measurement methods for determining stresses and sags are discussed in the paper [14]. According to [15], in bimaterial conductors, there is a phenomenon of the distribution of the tension forces between the aluminum braid and the conductor core. For conductors with nonlinear tensile characteristics, braid strain relief occurs at lower temperatures than for ACSR conductors, which results in a much lower increase in conductor sag as a function of temperature.

In the calculations presented in this paper, the linear elastic model (LE) used to analyze ACSR conductors is employed to model the overhead conductors [3]. In the LE model, the conductor is modeled as a spring, with one value of the thermal coefficient α and one value of the elastic modulus E. The main purpose of this paper is to determine the influence of insulator sets on the sag and stress of the conductor at the various operating temperatures of the overhead line. The main achievement presented in this paper is the development and verification of a mathematical model used for the analysis of single-span tensioning sections, taking into account insulator sets. Additionally, a computational tool was developed and tested for the mechanical calculations of multi-span tensioning sections, taking into account insulator sets. This paper presents the calculation results for single-span tensioning sections and multi-span tensioning sections. In each case, the calculations are made assuming that insulator sets are included or omitted. The analysis assumes AFL-6 240 mm² and AFL-8 525 mm² conductors, different span lengths, and different values of initial conductor tension.

2. Methodology

2.1. Mechanical Calculations in Overhead Lines—Simplified Methods

The current value of the conductor tension and sag depends on the changes in the length of the conductor suspended in the span due to various line operating conditions. For the mechanical calculations of conductors in overhead power lines, an equation describing the sag curve of a perfectly flexible cable suspended rigidly at both ends is used [12,16]. One of the basic quantities employed in the mechanical analysis of overhead lines is the conductor sag, which expresses the greatest vertical distance between the conductor and the straight line connecting its suspension points. In the mechanical calculations of the overhead conductors, the chain curve method or parabola method is utilized. The sag calculated using the chain curve equation has lower values than the sag calculated according to the parabola equation. In the case of spans with a significant span value, as well as spans with a large slope, the use of the parabolic approximation leads to calculation errors that are too large, and the recommended approach is to use the chain curve method. Changes in the tension and sag of the conductor in the current operating conditions of the line occur as a result of changes in the length of the conductor. In the case of classic aluminum–steel conductors, changes in the conductor length occur as a result of linear elastic and thermal elongations. The calculation method combines relationships that allow for the determination of the slack of the conductor and the linear elongation of the conductor; this method allows us to find the conductor state change equation, as illustrated in Equation (1). The conductor state change equation combines any two operating states of conductor “1” and “2” determined by conductor temperature, weight per unit length, and stress.

Using the conductor state change equation, shown in Equation (1), knowing the conditions describing the operating condition of the conductor in the initial state “1”, it is possible to determine the stress σ₂ in state “2” after changing the operating conditions of the conductor [16,17]:

$$\begin{array}{c}{\cos }^3\psi \frac{a^{2}E{w}_2^2}{24A_{\mathrm{c}}^2}-{\cos }^3\psi \frac{a^{2}E{w}_1^2}{24A_{\mathrm{c}}^2{\sigma }_1^2}{\sigma }_2^2=\cos \psi ·\alpha \left(T_2-T_1\right)E{\sigma }_2^2+\\ +\cos \psi ·{\epsilon }^{plast}E{\sigma }_2^2+{\sigma }_2^3-{{\sigma }_1\sigma }_2^2,\end{array}$$

(1)

where ψ signifies the angle of tilt of the span, a is the span length, E is the modulus of the elasticity of the conductor, α is the thermal elongation coefficient, A_c is the cross-section of the conductor, w₁ and w₂ are the weight per unit length of the conductor in states “1” and “2”, respectively, σ₁ and T₁ are the stress and temperature of the conductor, respectively, in the initial state “1”, σ₂ and T₂ are the stress and temperature, respectively, in state “2”, after changing the operating conditions of the conductor, and ε^plast is the plastic elongation of the conductor.

After rearranging Equation (1), the conductor state change equation is obtained in the final form:

$${\sigma }_2^2\left({\sigma }_2+A\right)=B.$$

(2)

The dependencies for calculating coefficients A and B take the following form:

$$A={\mathrm{c}\mathrm{o}\mathrm{s}}^3\psi \frac{a^{2}E{w}_1^2}{24A_{\mathrm{c}}^2{\sigma }_1^2}+\mathrm{c}\mathrm{o}\mathrm{s}\psi ·\alpha \left(T_2-T_1\right)E+\mathrm{c}\mathrm{o}\mathrm{s}\psi ·{\epsilon }^{\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}}E-{\sigma }_1,$$

(3)

$$B={\mathrm{c}\mathrm{o}\mathrm{s}}^3\psi \frac{a^{2}E{w}_2^2}{24A_{\mathrm{c}}^2},$$

(4)

Knowing the material data of the conductor, as well as the initial (“1”) and final (“2”) operating conditions of the conductor, Equation (1) is solved to obtain the value of the stress in the final state. The conductor state change equation described above can be used for the analysis of single spans of overhead lines or for a simplified mechanical analysis of tensioning sections. In the case of small differences in the values of the vertical coordinates of the pylon positions, Equations (3) and (4) can be used to calculate the conductor sag, replacing the span a with the span of the equivalent length a_r given by the formula:

$$a_r=\sqrt{\frac{{\displaystyle \sum _i{a}_i^3}}{{\displaystyle \sum _i{a}_i}}}$$

(5)

where a_i is the span length in the tensioning section.

Solving the conductor state change equation, assuming a = a_r under the known operating conditions of the conductor in state “1”, allows for the calculation of the stress σ₂, i.e., the stress in the conductor after a change in operating conditions. With this approach, the calculated value of stress σ₂ is the same for all spans in the tensioning section, while the value of sag in the i-th span in state “2” is determined by the relation:

$$f_i=\frac{a_i^2{w}_2}{8A_c{\sigma }_2}$$

(6)

The calculation method described above is called the “ruling span method” and allows for the determination of stress and sags in individual spans of the tensioning section, assuming the similar suspension heights of the conductors and their direct attachment to the supporting structure, i.e., ignoring the influence of insulator sets on the shape of the conductor suspension curve [10,11,16].

2.2. Mechanical Calculations in Overhead Lines—Exact Methods Analysis of a Single-Span Tensioning Section, including Insulator Sets

For most high- and extra-high-voltage overhead lines, a simplified calculation method allows for the tension and sag results to be achieved with an acceptable uncertainty. However, in tensioning sections running in terrain with a large tilt, as well as in spans with short lengths, the simplified calculation approach is insufficient. Conductors in the tensioning sections of the line are suspended on supporting structures via insulator sets. The impact of insulator sets on the mechanics of the conductor is particularly visible in spans with short lengths, where the weight and dimensions of the insulator sets are quite large in relation to the weight and length of the conductor (Figure 1). Short spans usually occur in or near high-voltage power substations [12].

In practice, overhead lines consist of tensioning sections that are limited by tension supports used to transfer the conductor tension force. Tensioning sections consist of several or even a dozen or so spans with specific span lengths, separated by suspension supports used only to support the conductor [12,16]. Calculations of the tensions and sags of conductors in overhead lines are often carried out without the influence of insulator sets. The latter are usually modeled using a flexible wire constituting a fragment of the conductor. In reality, however, the working conductors are connected to the supporting structures of the tension supports through rigid insulator sets (Figure 1).

During the operation of the overhead line, depending on the current tensile force in the conductor, the angle of the tension insulator sets changes (for example, from α₁ to α₂). These changes cause changes in the position of the conductor suspension points, which result in a small change in the span length Δa₀/2. This situation is shown in Figure 1 and Figure 2. The above factors, as well as the lengths (L_ins) and weights (w_ins) of the insulator sets, must be considered for inaccurate calculations of the conductor sags. In the case of single-span tensioning sections, the shape of the conductor sag depends mainly on the current position of the insulator sets. In simplified calculations, omitting the effect of the tension insulators, the final sag of the conductor is determined using the relationship (6). In order to determine sags and stresses in single-span tensioning sections, taking into account the insulator sets, the calculation model described by Equations (1)–(6) should be modified, accounting for the correction of the span caused by the offset of the position of the conductors’ suspension points.

Considering a rigid insulator set leads to a shortening of the actual span length. Therefore, in the exact model, the actual span a₀ should be used to calculate the conductor sag (Figure 1 and Figure 2). Due to the large geometric dimensions and large masses of insulator sets, the calculations of sags in short spans carried out using the simplified method may result in calculation errors that are too large. The inaccurate calculation of conductor sags may result in exceeding the required vertical distances of high voltage conductors from the ground and from objects located in the space under the overhead line, which may result in a loss of safety for people in the area. To analyze the sag of conductors operating in a single-span tensioning section, a computational model was developed that considered the length and mass of the insulator sets. The modified form of the coefficients of the conductor state changing equations, as in Equations (1)–(4), allowing for the analysis of the mechanics of conductors in single-span tensioning sections and considering the length and mass of the insulator sets, takes the form of Equations (7) and (8). In turn, the sag of the conductor in the span ending with insulator sets on both sides can be determined from Equation (9):

$$A=\frac{a_0^{2}E{w}_1^2}{24A_{\mathrm{c}}^2{\sigma }_1^2}+\alpha \left(T_2-T_1\right)E-{\sigma }_1+\frac{L_{\mathrm{i}\mathrm{n}\mathrm{s}}E}{4A_{\mathrm{c}}^2{\sigma }_1^2{a}_0}{\left(G_{\mathrm{i}\mathrm{n}\mathrm{s}1}+w_1{a}_0\right)}^2$$

(7)

$$B=\frac{a_0^{2}E{w}_2^2}{24A_{\mathrm{c}}^2}+\frac{L_{\mathrm{i}\mathrm{n}\mathrm{s}}E}{4A_{\mathrm{c}}^2{a}_0}{\left(G_{\mathrm{i}\mathrm{n}\mathrm{s}2}+w_2{a}_0\right)}^2$$

(8)

$$f_2=\frac{L_{\mathrm{i}\mathrm{n}\mathrm{s}}\left(G_{\mathrm{i}\mathrm{n}\mathrm{s}2}+w_2{a}_0\right)}{2{\sigma }_2{A}_c}+\frac{a_0^2{w}_2}{8{\sigma }_2{A}_c}$$

(9)

where L_ins is the length of the insulator set, and G_ins is the weight of the insulator set.

2.3. Mechanical Calculations in Overhead Lines—Exact Methods Analysis of a Multi-Span Tensioning Section, including Insulator Sets

Due to the phenomenon of tension equalization in the tensioning section, which occurs as a result of the offset of insulator sets on the tension and suspension supports, the analysis of the mechanics of conductors in multi-span tensioning sections cannot be carried out, assuming the division of the tensioning section into single spans. To determine the tension in a multi-span tensioning section, a mathematical model should be built using information about the material parameters of the conductor and the dimensions and weight of the insulator sets, as well as the coordinates of the suspension points of the working conductors and the length of individual spans. A detailed analysis of the mechanics of the conductors in the tensioning section also allows for a consideration of different wind and ice loads in individual spans. In the tensioning section, a change in the operating state of the conductor due to a change in temperature, or temperature and mechanical load, results in the equalization of the tension through small offsets of the insulator sets and a change in the tilt angle of the insulator sets. In practice, the offsets of the insulator sets are small and amount to a few centimeters, but they result in a significant equalization of tension in the entire tensioning section, which is shown in Figure 3.

The change in the span Δa_i and the change in the position of the conductor suspension point Δh_i can be expressed using equations describing the equilibrium state of insulator sets [12], i.e.,:

Δa_i = (δ_i+1 – δ_1,i+1) − (δ₁ – δ_1,i)

(10)

Δh_i = (ε_i+1 – ε_1,i+1) − (ε₁ – ε_1,i)

(11)

where Δa_i denotes the change in span length, Δh_i is the change in span tilt, δ_i and δ_i+1 are the horizontal offset of the insulator set on pylon i and i + 1, respectively, and ε_i and ε_i+1 are the vertical deflection of the insulator sets on pylon i and i + 1, respectively, as depicted in Figure 3.

Values marked with the index “1” refer to offsets in the initial state, and it is assumed that they are zero for the suspension insulator sets. Due to the occurrence of different tension forces in individual spans of the tensioning sections, after changing the operating conditions of the conductor, the suspension insulator sets deflect, thus ensuring a state of balance in the forces acting on the insulator sets. An accurate computational model requires complex transformations of the general form of the conductor equation of states, considering the coordinates of the pylon position, the height of the crossbar, and the parameters of the tension insulator sets and the suspension insulator set [12]. The structure of the algorithm used for mechanical calculations in the tensioning sections in the exact method is shown in Figure 4.

At the beginning of the calculations, the conductor type must be selected, providing information about the tensioning section, including the following parameters:

• value of the tension force in the initial state (“1”),
• conductor temperature values in the initial and final operating state,
• numbering of individual poles,
• coordinates of the pole position,
• height of the crossbar,
• length and mass of insulator sets,
• value of additional ice load and wind pressure in the final state (“2”).

The next step is to determine the basic parameters of the tensioning section, such as the lengths and slopes of individual spans. Then, for a given value of tension and temperature in the initial state, the offsets of the tension insulator sets in the initial state are determined. In this state, zero deflection of the suspension insulator sets is assumed, which is consistent with the conditions occurring in the initial state. In the next stage, a system of non-linear equations is created; the number of the equations coincides with the number of spans in the considered tensioning section. Then, the coefficients occurring at the unknowns, which are the values of the conductor stress in the final state, are determined. These coefficients depend on the slack of the conductor in the initial state, the material parameters of the conductor, the temperature difference in the initial and final state, and the span and tilt of individual spans, as well as the initial tension in the initial state. In the further steps of the computational procedure, a starting solution vector is adopted and the iterative solution of the system of equations begins using Newton’s method. As a result of these calculations, the stress values in each span of the tensioning section, the horizontal and vertical offsets of the tension, and the suspension insulator sets are attained. The obtained values can be used to determine a number of subsequent parameters describing the condition of the tensioning section, such as the balance of forces of the insulator set, the value of the vertical force acting on the insulator set, or the length of the conductor in each span. An analysis of a multi-span tensioning section, including the insulator sets in this article, was performed using ZikOS software [18].

3. Results

3.1. Mechanical Calculations of Conductors in Single-Span Tensioning Sections of Short Span (30 m and 50 m)

This paper presents the results of mechanical calculations performed for the 110 kV and 220 kV tensioning sections consisting of five spans with different lengths, and for the so-called “short span”. The latter can be defined as a span with a short length, in which the conductors have been suspended with a small value of initial tension. In practice, short spans are most often found at the beginning and end of the line, and connect the station gate with the first and last poles of the line. The values of stresses and sags of typical conductors used in high-voltage and extra-high-voltage lines were analyzed, i.e., for AFL-6 240 mm² and AFL-8 525 mm² conductors. In the case of analyses for the 110 kV line, an AFL-6 240 mm² conductor and insulator sets with a length of L_ins = 2 m and a mass of m_ins = 42 kg were assumed. The spans of the 220 kV line were modeled with the assumption of a conductor with a cross-section of AFL-8 525 mm² and insulator sets with a length of L_ins = 4.5 m and a mass of m_ins = 179 kg.

The variant, using a simplified calculation method, uses the mathematical relationships described in Section 2.1, i.e., Equations (1)–(6). Precise calculations that consider the length and mass of insulator sets were carried out based on the formulas described in Section 2.2, i.e., Equations (7)–(9). In both of the above cases, Newton’s iterative method is used to solve the equations describing the tensioning section.

Table 1. Calculated values of stresses and sags of the AFL-6 240 mm² conductor in selected operating conditions of the conductor, considering and omitting insulator sets in a span of 30 m with an initial stress of 8.46 MPa at a temperature of +10 °C.

Span Length, m	T2, °C	Simplified Computational Model (Ruling Span Method)		Accurate Computational Model
		σ, MPa	f, m	σ, MPa	f, m	α, °
30	−5 + Ik	18.53	0.40	13.39	0.66	6.9
	−25	19.58	0.20	10.21	0.52	6.7
	10	8.46	0.46	8.46	0.63	8.1
	40	6.22	0.62	7.50	0.71	9.1
	60	5.42	0.72	7.01	0.76	9.7
	80	4.86	0.80	6.61	0.80	10.3

,

Table 2. Calculated values of stresses and sags of the AFL-6 240 mm² conductor in selected operating conditions of the conductor, considering and omitting insulator sets in a span of 50 m with an initial stress of 8.46 MPa at a temperature of +10 °C.

Span Length, m	T2, °C	Simplified Computational Model (Ruling Span Method)		Accurate Computational Model
		σ, MPa	f, m	σ, MPa	f, m	α, °
50	−5 + Ik	17.05	1.21	14.96	1.47	8.7
	−25	10.59	1.02	9.58	1.28	9.2
	10	8.46	1.27	8.46	1.44	10.4
	40	7.37	1.46	7.76	1.57	11.4
	60	6.84	1.57	7.38	1.66	11.9
	80	6.41	1.68	7.04	1.73	12.5

,

Table 3. Calculated values of stresses and sags of the AFL-6 240 mm² conductor in selected operating conditions of the conductor, considering and omitting insulator sets in a span of 30 m with an initial stress of 14.45 MPa at a temperature of +10 °C.

Span Length, m	T2, °C	Simplified Computational Model (Ruling Span Method)		Accurate Computational Model
		σ, MPa	f, m	σ, MPa	f, m	α, °
30	−5 + Ik	31,71	0.23	24.39	0.36	3.8
	−25	49.01	0.08	26.82	0.20	2.5
	10	14.45	0.27	14.45	0.37	4.7
	40	8.04	0.48	10.94	0.49	6.2
	60	6.54	0.59	9.61	0.55	7.1
	80	5.63	0.69	8.66	0.61	7.9

and

Table 4. Calculated values of stresses and sags of the AFL-6 240 mm² conductor in selected operating conditions of the conductor, considering and omitting insulator sets in a span of 50 m with an initial stress of 14.45 MPa at a temperature of +10 °C.

Span Length, m	T2, °C	Simplified Computational Model (Ruling Span Method)		Accurate Computational Model
		σ, MPa	f, m	σ, MPa	f, m	α, °
50	−5 + Ik	29.69	0.69	26.35	0.84	4.9
	−25	29.66	0.36	21.74	0.56	4.1
	10	14.45	0.75	14.45	0.84	6.1
	40	10.6	1.02	11.7	1.04	7.5
	60	9.21	1.17	10.53	1.16	8.4
	80	8.25	1.31	9.64	1.27	9.1

contain the results of the calculations of the sags and stresses of the AFL-6 240 mm² conductor in selected operating states for two different span lengths (30 m and 50 m) and for two different values of initial tension (8.46 MPa and 14.45 MPa).

Table 5. Calculated values of stresses and sags of the AFL-8 525 mm² conductor in selected operating conditions of the conductor, considering and omitting insulator sets in a span of 30 m with an initial stress of 9.98 MPa at a temperature of +10 °C.

Span Length, m	T2, °C	Simplified Computational Model (Ruling Span Method)		Accurate Computational Model
		σ, MPa	f, m	σ, MPa	f, m	α, °
30	−5 + Ik	19.99	0.29	11.45	1.06	10.2
	−25	32.15	0.12	10.46	0.97	10.1
	10	9.98	0.37	9.98	1.01	10.6
	40	6.57	0.56	9.63	1.05	11.0
	60	5.56	0.67	9.41	1.07	11.2
	80	4.9	0.76	9.21	1.10	11.5

,

Table 6. Calculated values of stresses and sags of the AFL-8 525 mm² conductor in selected operating conditions of the conductor, considering and omitting insulator sets in a span of 50 m with an initial stress of 9.98 MPa at a temperature of +10 °C.

Span Length, m	T2, °C	Simplified Computational Model (Ruling Span Method)		Accurate Computational Model
		σ, MPa	f, m	σ, MPa	f, m	α, °
50	−5 + Ik	17.38	0.94	12.38	1.78	11.6
	−25	14.72	0.70	10.57	1.58	11.7
	10	9.98	1.03	9.98	1.67	12.5
	40	8.17	1.26	9.54	1.75	13.0
	60	7.39	1.39	9.28	1.8	13.4
	80	6.79	1.51	9.01	1.85	13.8

,

Table 7. Calculated values of stresses and sags of the AFL-8 525 mm² conductor in selected operating conditions of the conductor, considering and omitting insulator sets in a span of 30 m with an initial stress of 15.68 MPa at a temperature of +10 °C.

Span Length, m	T2, °C	Simplified Computational Model (Ruling Span Method)		Accurate Computational Model
		σ, MPa	f, m	σ, MPa	f, m	α, °
30	−5 + Ik	31.88	0.18	18.44	0.66	6.4
	−25	53.39	0.07	17.61	0.57	6.0
	10	15.36	0.24	15.68	0.64	6.7
	40	7.98	0.46	14.44	0.70	7.3
	60	6.36	0.58	13.75	0.73	7.7
	80	5.43	0.68	13.16	0.77	8.1

and

Table 8. Calculated values of stresses and sags of the AFL-8 525 mm² conductor in selected operating conditions of the conductor, considering and omitting insulator sets in a span of 50 m with an initial stress of 15.68 MPa at a temperature of +10 °C.

Span Length, m	T2, °C	Simplified Computational Model (Ruling Span Method)		Accurate Computational Model
		σ, MPa	f, m	σ, MPa	f, m	α, °
50	−5 + Ik	26.80	0.85	20.47	1.1	7.2
	−25	37.21	0.28	18.31	0.91	6.8
	10	15.68	0.66	15.68	1.06	7.9
	40	10.77	0.95	14.17	1.18	8.8
	60	9.18	1.12	13.37	1.25	9.3
	80	8.12	1.27	12.61	1.32	9.9

contain the results of the calculations of the sags and stresses of the AFL-8 525 mm² conductor in selected operating states for two different span lengths (30 m and 50 m) and for two dissimilar values of initial tension (9.98 MPa and 15.68 MPa).

In the above cases, comparative calculations were made considering and omitting the insulator sets. The analyses were performed for several selected values of the conductor operating temperature and for the condition of –5 °C with ice load (I_k), determined in accordance with ref. [19]. For the AFL-6 240 mm² conductor, the calculated value of the ice load is 8.72 N/m, while the assumed ice load value is 11.41 N/m for the AFL-8 525 mm² conductor.

Figure 5, Figure 6, Figure 7 and Figure 8 show the conductor sag as a function of the conductor operating temperature in the range of −25 °C to +80 °C for the AFL-6 240 mm² and AFL-8 525 mm² conductors for two short spans. An analysis of the sag as a function of temperature was performed for two values of conductor initial tension at an initial temperature of +10 °C, considering and disregarding the influence of the length and mass of the insulator sets on the final conductor sag.

3.2. Mechanical Calculations of Overhead Conductors in the Multi-Tensioning Section of High-Voltage Lines with Various Span Lengths

The computational algorithm, previously described in Section 2.3, was used to analyze the operating states of overhead conductors suspended in the multi-span 110 kV and 220 kV tensioning section. Parameters of the tensioning sections used in the analysis are presented in

Table 9. Parameters of the tensioning sections used in the analysis.

	110 kV Tensioning Section	220 kV Tensioning Section
Conductor type	AFL-6 240 mm2	AFL-5 525 mm2
Initial temperature	+10 °C	+10 °C
Initial value of tensile force	15.8 kN	37.6 kN
Initial stress of the conductor	57.64 MPa	64.10 MPa
Conductor weight per unit length	9.52 N/m	19.32 N/m
Mass of the tension insulator set	42 kg	179 kg
Length of the tension insulator set	2.0 m	4.5 m
Mass of the suspension insulator set	42 kg	181 kg
Length of the suspension insulator set	1.8 m	3.1 m
Span length	390 m; 280 m; 210 m; 150 m; 300 m	450 m; 400 m; 420 m; 390 m; 470 m

. Figure 9 shows the analyzed multi-span 110 kV tensioning section.

The analysis assumed an installation (initial) temperature of +10 °C, the same height for the supporting structures, and different span lengths. In the case of the 110 kV tensioning section (

Table 10. Results of stresses and sags of the AFL-6 240 mm² conductor at –25 °C in the five-span tensioning section.

Span	Span Length, m	Simplified Computational Model (Ruling Span Method)		Accurate Computational Model
		σ, MPa	f, m	σ, MPa	f, m
1–2	390	71.1	9.22	69.74	10.39
2–3	280		4.75	70.48	6.64
3–4	210		2.67	70.96	4.51
4–5	150		1.36	71.13	3.19
5–6	300		5.45	71.10	6.42

and

Table 11. Results of stresses and sags of the AFL-6 240 mm² conductor at +80 °C in the five-span tensioning section.

Span	Span Length, m	Simplified Computational Model (Ruling Span Method)		Accurate Computational Model
		σ, MPa	f, m	σ, MPa	f, m
1–2	390	42.65	15.37	43.74	16.02
2–3	280		7.92	42.34	9.85
3–4	210		4.45	41.46	6.44
4–5	150		2.27	41.20	4.16
5–6	300		9.09	41.42	10.36

), an initial stress of 57.46 MPa was assumed.

Table 12. Values of horizontal offsets of the insulator sets and the balances of the forces acting on the insulator sets at the analyzed operating temperatures of the AFL-6 240 mm² conductor in the 110 kV tensioning section.

Pole Number	T1 = T2 = +10 °C		T2 = −25 °C		T2 = +80 °C
	δ, m	Force Balance of Insulator Set, N	δ, m	Force Balance of Insulator Set, N	δ, m	Force Balance of Insulator Set, N
1	2.009	15.9 × 103	2.009	19.3 × 103	1.991	12.1 × 103
2	0	0	0.112	205	−0.207	−386
3	0	0	0.093	131	−0.172	−243
4	0	0	0.044	47	−0.068	−73
5	0	0	−0.008	−10	0.049	61
6	−2.009	−15.9 × 103	−2.013	−19.6 × 103	−1.999	−11.4 × 103

show the values of the horizontal offset of insulator sets (δ) and the balance of forces acting on the insulator sets in the 110 kV tensioning section with the AFL-6 240 mm² conductor. In the case of the 220 kV tensioning section (

Table 13. Results of stresses and sags of the AFL-8 525 mm² conductor at −25 °C in the five-span tensioning section.

Span	Span Length, m	Simplified Computational Model (Ruling Span Method)		Accurate Computational Model
		σ, MPa	f, m	σ, MPa	f, m
1–2	450	74.64	11.16	74.46	12.84
2–3	400		8.82	74.59	11.97
3–4	420		9.72	74.58	12.88
4–5	390		8.39	74.53	11.54
5–6	470		12.18	74.30	13.89

and

Table 14. Results of stresses and sags of the AFL-8 525 mm² conductor at +80 °C in the five-span tensioning section.

Span	Span Length, m	Simplified Computational Model (Ruling Span Method)		Accurate Computational Model
		σ, MPa	f, m	σ, MPa	f, m
1–2	450	50.99	16.34	51.20	17.98
2–3	400		12.91	50.93	16.11
3–4	420		14.24	50.93	17.44
4–5	390		12.27	51.00	15.45
5–6	470		17.83	51.46	19.38

), an initial stress of 64.10 MPa was assumed.

Table 15. Values of horizontal offsets of the insulator sets and the balances of the forces acting on the insulator sets at the analyzed operating temperatures of the AFL-6 240 mm² conductor in the 220 kV tensioning section.

Pole Number	T1 = T2 = +10 °C		T2 = −25 °C		T2 = +80 °C
	δ, m	Force Balance of Insulator Set, N	δ, m	Force Balance of Insulator Set, N	δ, m	Force Balance of Insulator Set, N
1	4.454	37.6 × 103	4.465	43.7 × 103	4.430	30.0 × 103
2	0	0	0.026	75	−0.055	−158
3	0	0	−0.001	−4	0.001	2
4	0	0	−0.011	−30	0.014	40
5	0	0	−0.047	−137	0.092	267
6	−4.451	−37.6 × 103	−4.463	−43.6 × 103	−4.426	−30.2 × 103

show the values of the horizontal offset of insulator sets (δ) and the balance of forces acting on the insulator sets in the 220 kV tensioning section with the AFL-8 525 mm² conductor, at three selected conductor operating temperatures. The starting point for the calculations was the installation (initial) condition of the conductor at a temperature of +10 °C. The horizontal offsets of the insulator sets δ are illustrated in Figure 3. In each case, the calculations for sags and stresses were carried out using the ruling span method and the exact method, accounting for the insulator sets for two conductor operating temperatures T₂ = −25 °C and T₂ = +80 °C (Table 12 and Table 15).

4. Discussion

The results of the comparative analysis of the calculations of sags and stresses in single-span tensioning sections with aluminum–steel conductors commonly used in high-voltage and extra-high-voltage overhead lines are presented in Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8 and Figure 5, Figure 6, Figure 7 and Figure 8. The calculations in each of the above cases were made in a variant that considered and excluded the insulator sets for various operating conditions of the conductor. The calculations show that in short spans (i.e., 30 m and 50 m), accounting for the insulator sets leads to higher sag values than in the case of calculations performed using the ruling span method. This phenomenon is particularly visible in the range of low conductor operating temperatures and when low values of initial stress are applied. For the AFL-6 240 mm² conductor, the largest difference in the conductor sag value occurred at −25 °C and amounted to 0.32 m in a span of 30 m, assuming an initial stress of 8.46 MPa. The sag value for this case determined using the simplified model is 0.20 m, so it is more than twice as small as the value obtained by the exact method (0.52 m). The comparison of the sags of the AFL-8 525 mm² conductor indicates the largest sag difference of 0.88 m at a temperature of −25 °C in a span of 50 m, with a conductor initial stress of 9.98 MPa. As the operating temperature of the conductor increases, the differences in the sag values calculated using the exact model and the simplified model decrease. At the maximum operating temperature of ACSR conductors, i.e., at +80 °C, in the case of the AFL-6 240 mm² conductor, the obtained differences in the conductor sag are negligible, while for the AFL-8 525 mm² conductor, the differences in the conductor sag values are 0.33–0.34 m using an assembly stress of 9.98 MPa, regardless of the span length. The calculations showed that the differences in the values of sags and stresses modeled using the exact method and the simplified method strongly depend on the assumed initial stress and the span length. Differences in the values of sags are significant in the spans in which low values of initial stress were applied, and these differences decrease as the span increases.

According to the results in Table 10 and Table 11, slight differences in stress occurring in subsequent spans can be noticed. In both the considered operating states, the largest difference in sag occurs in the middle spans of the analyzed tensioning section and is approximately equal to the length of the insulator set (i.e., 1.8 m). Much smaller differences in the conductor sag occur in the first and last span of the analyzed tensioning section, which result from the presence of a tension insulator set installed on the tension support, limiting the tensioning section. The stress values presented in Table 10 and Table 11 are achieved as a result of the offset of the insulator sets (δ), the values of which are listed in Table 12. For example, on pylon no. 2 at a conductor temperature of +80 °C, the suspension insulator set deflects by 0.207 m towards support no. 1. The balance of forces acting on the insulators set on pole no. 2 is, in this case, −386 N; this is the largest difference in forces acting on the suspension insulator set in the analyzed tensioning section. A similar situation occurs in the tensioning section with the AFL-8 525 mm² conductor, where the offset of the insulator sets ensures the equalization of the tension in the tensioning section. In the case under consideration, the highest value of the offset of the insulator set occurs on the fifth pylon at a temperature of +80 °C and amounts to 0.092 m (Table 15).

The obtained results of the comparative analysis of sags, in some cases, exceed the design reserves of sags adopted at the stage of creating the line design documentation (0.5 m). Therefore, the preferred solution is to perform accurate calculations of sags and stresses for the specific analyzed span. An increase in sags can also be noticed for operating temperatures lower than 40 °C. Therefore, accounting for insulator sets in the above operating temperature range in the calculations may be important, especially in short spans.

The comparative analysis carried out showed that the omission of insulator sets in the calculation of tensions and sags in some cases may result in incorrect calculation of the required vertical distances between phase conductors and objects located in the space under the overhead line. If the required vertical distances are exceeded, the overhead conductors may come too close to objects under the overhead line and electric shock may occur to people in this space. This situation may occur during operational work in the immediate vicinity of high-voltage power stations.

5. Conclusions

The correct determination of sags and stresses for overhead line conductors is extremely important from the point of view of maintaining the required vertical distances between phase conductors and objects located under the overhead line, and due to the need to determine the values of forces acting on the supporting structure. Overhead lines are made of tensioning sections limited by tension support. Therefore, the shape of the conductor sag curve in a given span also depends on the condition of the conductor in the adjacent spans and the offset of insulator sets.

The values of conductor sags and stresses can be determined using simplified or exact calculation methods. The algorithm used in the exact method is based on a modification of the conductor state change equation, considering the length and mass of the insulator sets and the coordinates of the position of the support, as well as the coordinates of the conductor suspension points. The use of accurate computational models that allow for the determination of the offset of insulator sets requires the implementation of numerical methods, e.g., Newton’s iterative method, which, due to the size of the computational task, can be accomplished using computer computational tools.

The influence of the parameters of the insulator sets on the conductor mechanics is particularly visible in spans with short lengths, in which the working conductors are suspended with a low value of initial stress. In short spans, both the length and mass of the insulator sets have relatively high values compared to the span length and length of the conductor. Hence, the use of computational models that account for the parameters of the insulator sets is recommended in this case. The phenomenon described above occurs mainly in high- and extra-high-voltage lines, where, due to the required insulation distances and the high values of stresses occurring in the conductors, the insulator sets reach significant geometric dimensions. The computational method described in this paper allows for the analysis of conductors modeled using linear tensile characteristics and taking into account rigid strings of insulator sets. In the perspective of further work, the development of a model for the mechanical analysis of conductors with non-linear stretching characteristics, i.e., ACSS, ACCC or ACCR conductors, is planned.