# Numerical Simulation of the Dynamic Responses of Electric Power Fitting of Transmission Lines after Ice Shedding

Authors: Zhongfei Ye; Chuan Wu; Xiaohui Liu; Fei Huang
Journal: Shock and Vibration
 (2022)

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## Abstract

As the key connecting components between tower and conductors, electric power fittings play an important role in the provision of the safe operation of transmission lines. Taking 500 kV transmission lines as the research object, electric power fittings failure in heavy icing areas is studied. The following three questions are considered: first, the influence of different ice-shedding methods on the tension of quad bundle conductors; second, the influence of the spans number, span length, height difference, and ice thickness on the conductor tension under the most dangerous ice-shedding mode; lastly, the mechanical characteristics of tension string and suspension string under ice-shedding condition in the stable wind. The results show that the von Mises stress of the parts that connect with the iced conductor in the tension string model is greater than those of the parts connected with the ice-shedding conductor. What is more, the shackle is the most vulnerable part of the whole model, which is prone to wear and damage, and the failure is most likely to occur at the upper shackle. The middle part in the rectangular hanging plate which is connected with the iced conductor is prone to damage, for its maximum stress has already exceeded the yield stress, with a danger degree inferior only to the shackle. In the suspension string model, the von Mises stress of each part is relatively small and does not reach the yield stress. Similarly, the shackle is most likely to be damaged in the overall model, while other parts maintain larger safety margins. The structure of the parts with smaller stress can be optimized according to the simulation results where design margins may be reduced and additional cost benefits realized.


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## Body

## 1. Introduction

In an environment with high humidity and near-zero temperature, the ice shedding and icing of the conductor may cause the accidents such as conductors fracture and electric power fittings damage. These accidents will directly affect the power system and the national economy. Therefore, some scholars have carried out experimental research on the ice shedding of conductors. Kollar et al. [1] and Jamaleddine et al. [2] carried out the ice-shedding experiment of transmission lines model and measured the displacement and other parameters after ice shedding. Morgan et al. [3] carried out an ice-shedding experiment on five-span transmission lines, and simulated the process of icing and ice shedding on transmission lines by hanging and releasing weight in the middle part of the span. Chen et al. [4] conducted ice-shedding experiments on single-span and double-span transmission lines. They analyzed the displacement and dynamic tension patterns of the transmission lines under various ice-shedding modes.With the development of computer hardware and finite element software, the numerical simulation method has been widely used in the problem of ice shedding on transmission lines.Kollar et al. [5] created a numerical model of four-bundle transmission lines and studied the dynamic response of each subconductor after ice shedding. McClure et al. [6] and Mirshafiei et al. [7] simulated the process of conductor fracture of single-span transmission line and analyzed the influence of conductor fracture on the dynamic response of ice shedding. Meng et al. [8] set different span lengths combination conditions and analyzed the influence of relevant parameters on dynamic characteristics of conductor ice shedding under different conditions. Ji et al. [9] proposed the ice-shedding failure model of transmission lines to predict the transient response of the transmission line system under impact load. Juraj et al. [10] and Liu et al. [11] analyzed the ice shedding of different conductors and studied the influence of conductor types and icing positions on overhead transmission lines. Shen et al. [12] analyzed the dynamic response of the tower line coupling system under ice-shedding condition. Yan et al. [13] studied the influence of different structural parameters on the jumping of transmission lines and obtained the simplified theoretical formula of ice-shedding jump height. Fan et al. [14] studied the icing patterns of transmission lines with different diameters under four natural ice types.As key connecting components between conductors and towers, electric power fittings play an important role in the provision of the safe operation of overhead transmission lines. The icing of the conductor will increase its tension. But the tension may vary greatly due to ice shedding, and then the fittings will be damaged. Figure1 shows the case of atmospheric icing of the conductor, and Figure 2 shows the case of the yoke plate being damaged.Figure 1
Atmospheric icing of conductors.Figure 2
Yoke plate damaged.Xie et al. [15–17] studied the fatigue failure mechanism of composite insulators and ball-eye. At the same time, they carried out fatigue experiments on the ball-eye and analyzed the influence of different heat treatment processes on the mechanical performance of the structure. Yang [18] carried out experimental research and numerical simulation to study and predict the wear problem of shackles. Prenleloup et al. [19] and Kumosa et al. [20] analyzed the damage initiation process of the material by using experimental and numerical research on the bending joint of the insulator and properly optimized the stress corrosion resistance of the material. Han et al. [21] studied the icing characteristics of the ice-free postinsulator and derived the expressions of local icing amount and icing thickness on the insulator surface. Wang et al. [22] carried out contact analysis on the crimping position of tension clamp and analyzed the stress and deformation in the process of crimping.Based on the above analysis, most studies focus on the dynamic response of conductors following ice shedding. However, the fitting failures should also be analyzed. The existing research on fitting failures is all based on independent parts, ignoring the relative motion between fittings. Therefore, it is necessary to analyze the influence of stress distribution of the whole electric power fittings under the asynchronous ice shedding on the reliability of the transmission line.


## 2. Validation of the Ice-Shedding Model

### 2.1. Finite Element Model of Ice Shedding

On the conductor, affected by the external environment, the ice is found to be unevenly distributed along the horizontal axis of the conductor. It is not realistic to achieve a completely real simulation according to the actual situation. It is assumed that the outer surface of the conductor is uniformly covered with ice in the design of the overhead transmission line.For an iced conductor, static ice loads can be simulated by increasing the density of conductors [13]. The equivalent density of the iced conductor could be defined by(1)ρ′=W1+W2A,where W1 is the mass per unit length of the conductor (kg/m); W2 is the mass per unit length of the ice on the conductor (kg/m); A is the cross-sectional area of the conductor.For an ice-shedding conductor, the ice load attached to the conductors can be regarded as a body force, and the conductor’s ice shedding can be simulated by changing the inertial acceleration [13]. The equivalent density ρ″ of a conductor after ice shedding could be described as follows:(2)ρ″=W1+W21−βA,where β is the ice-shedding rate of the conductor. The equivalent density of conductor after icing and ice shedding could be described as follows:(3)ρ′=W1+W2A=W1+W21−βA+βW2A=ρ″+ρ¯.where ρ¯ is the equivalent density caused by the shedding part. The gravity per unit volume of the conductor before ice shedding could be divided into two parts(4)ρ′g=ρ″g+ρ¯g.If the equivalent density of conductor after ice shedding isρ″, the gravity of the conductor must remain unchanged before ice shedding, and the following formula could be (5)ρ″g′=ρ″g+ρ¯g.Therefore, the equivalent inertial acceleration of the conductor is(6)g′=1+ρ¯ρ″,g=W1+W2W1+W21−β.The relationship between icing thickness and icing weight is given in the literature [23]:(7)δ=d022+W2πρi×106−d02.where δ is the icing thickness of the conductor (mm); d0 is the initial outer diameter of the conductor (mm); ρi is the density of ice (g/cm3); W2 is the mass per unit length of the icing conductor (kg/m).


### 2.2. Validation of Finite Element Model

To verify the rationality of the simulated ice-shedding method, the conductor tension is compared with [4]. For the single conductor used in the experiment in [4], the parameters of the conductor are shown in Table 1, and the original experimental conditions are shown in Table 2.Table 1
Conductor parameters.
Conductor typeOuter diameter (mm)Cross-sectional area (mm2)Mass per unit length (kg/m)Rated tensile strength (kN)Equivalent elastic modulus (kN/mm2)LGJ-300/4023.94338.991.13387.673Table 2
Parameters of iced transmission line.
Span length (m)Conductor initial tension (kN)Ice thickness (mm)Ice-shedding rate (%)23525.315100According to the experimental model, the finite element model of a single-span conductor is established by ABAQUS software. Both ends of the single-span conductor are fixed in the finite element model. The conductor tension obtained in this paper is compared to the experimental data and other numerical simulations, as shown in Figure3. It can be seen that the tension-time curves in this paper are close to the experiment and simulation results in the literature [4].Figure 3
Comparison of conductor tension between this paper and the literature.Therefore, it can be found that under the same conductor parameters and simulation conditions, the tension simulated in this paper has little error with that in literature and experiment. For the nonlinear vibration with large displacement, the simulation results in this paper are very satisfactory. So it can be considered that the method and model in this paper are accurate.


## 2.1. Finite Element Model of Ice Shedding

On the conductor, affected by the external environment, the ice is found to be unevenly distributed along the horizontal axis of the conductor. It is not realistic to achieve a completely real simulation according to the actual situation. It is assumed that the outer surface of the conductor is uniformly covered with ice in the design of the overhead transmission line.For an iced conductor, static ice loads can be simulated by increasing the density of conductors [13]. The equivalent density of the iced conductor could be defined by(1)ρ′=W1+W2A,where W1 is the mass per unit length of the conductor (kg/m); W2 is the mass per unit length of the ice on the conductor (kg/m); A is the cross-sectional area of the conductor.For an ice-shedding conductor, the ice load attached to the conductors can be regarded as a body force, and the conductor’s ice shedding can be simulated by changing the inertial acceleration [13]. The equivalent density ρ″ of a conductor after ice shedding could be described as follows:(2)ρ″=W1+W21−βA,where β is the ice-shedding rate of the conductor. The equivalent density of conductor after icing and ice shedding could be described as follows:(3)ρ′=W1+W2A=W1+W21−βA+βW2A=ρ″+ρ¯.where ρ¯ is the equivalent density caused by the shedding part. The gravity per unit volume of the conductor before ice shedding could be divided into two parts(4)ρ′g=ρ″g+ρ¯g.If the equivalent density of conductor after ice shedding isρ″, the gravity of the conductor must remain unchanged before ice shedding, and the following formula could be (5)ρ″g′=ρ″g+ρ¯g.Therefore, the equivalent inertial acceleration of the conductor is(6)g′=1+ρ¯ρ″,g=W1+W2W1+W21−β.The relationship between icing thickness and icing weight is given in the literature [23]:(7)δ=d022+W2πρi×106−d02.where δ is the icing thickness of the conductor (mm); d0 is the initial outer diameter of the conductor (mm); ρi is the density of ice (g/cm3); W2 is the mass per unit length of the icing conductor (kg/m).


## 2.2. Validation of Finite Element Model

To verify the rationality of the simulated ice-shedding method, the conductor tension is compared with [4]. For the single conductor used in the experiment in [4], the parameters of the conductor are shown in Table 1, and the original experimental conditions are shown in Table 2.Table 1
Conductor parameters.
Conductor typeOuter diameter (mm)Cross-sectional area (mm2)Mass per unit length (kg/m)Rated tensile strength (kN)Equivalent elastic modulus (kN/mm2)LGJ-300/4023.94338.991.13387.673Table 2
Parameters of iced transmission line.
Span length (m)Conductor initial tension (kN)Ice thickness (mm)Ice-shedding rate (%)23525.315100According to the experimental model, the finite element model of a single-span conductor is established by ABAQUS software. Both ends of the single-span conductor are fixed in the finite element model. The conductor tension obtained in this paper is compared to the experimental data and other numerical simulations, as shown in Figure3. It can be seen that the tension-time curves in this paper are close to the experiment and simulation results in the literature [4].Figure 3
Comparison of conductor tension between this paper and the literature.Therefore, it can be found that under the same conductor parameters and simulation conditions, the tension simulated in this paper has little error with that in literature and experiment. For the nonlinear vibration with large displacement, the simulation results in this paper are very satisfactory. So it can be considered that the method and model in this paper are accurate.


## 3. Finite Element Model of Quad Bundle Conductors

In the finite element model of quad bundle conductors, the conductor is simulated by a three-dimensional truss element, the element type is T3D2. The suspension string and spacer are simulated by a three-dimensional beam element, and the element type is B31.


### 3.1. Transmission Line Parameters

This research is based on the 500 kV transmission line from Nanyang middle to Nanyang south, Henan Province, China. The range of height difference ratio and span length are shown in Table3 and 4 in the transmission line, and the conductor material parameters are shown in Table 5. According to the engineering design, the initial tension of the conductor is taken as 25% of the rated tensile strength [23].Table 3
The range of height difference ratio in transmission lines.
Height difference ratio0(0∼0.1)(0.1∼0.2)Number of cases251114Table 4
The range of span length.
Span length100–200 m200–300 m300–400 m400–500 m500–600 mNumber of cases110594624Table 5
Conductor material parameters.
Conductor typeOuter diameter (mm)Calculated cross section (mm2)Mass per unit length (kg/km)Rated tensile strength (kN)Modulus of elasticity (GPa)Poisson’s ratioJL/G1A-630/45–45/733.8674.02079.2150.4563.00.3


### 3.2. Parameters of the Iced Conductors

According to the statistical results of transmission line parameters in Table3 and 4, span number, span length, icing thickness, and height difference ratio in the transmission line are selected, respectively. To study the effect of different parameters of iced transmission line on the dynamic tension of conductor due to ice shedding, a standard finite element model is established to simulate ice shedding as shown in Figure 4. The standard finite element model is a single-span transmission line. The parameters of the iced transmission lines with single-span are shown in Table 6. In the following sections, based on the standard finite element model, the influence of different parameters on the dynamic tension of the conductor will be studied.Figure 4
Single-span transmission line.Table 6
Standard working conditions of conductor icing.
Conductor typeNumber of transmission linesSpan length (m)Ice thickness (mm)Wind speed (m/s)Height difference ratioJL/G1A-630/45–45/7130020150According to the relevant literature, the dynamic responses under different ice-shedding rates of 50%, 60%, 70%, 80%, 90% and 100% were simulated, respectively. The results show that the jump height, and tension variations of the conductor increase with the increase of ice-shedding rate obviously when only the ice-shedding rate is changed. Therefore, tension variations of the conductor under the ice-shedding rate of 100% and stresses of electric power fittings are studied in this paper.


### 3.3. Selection of Ice-Shedding Mode

As the ice-shedding process of the conductor is complex and random, the dynamic tension of the conductor is studied under synchronous and asynchronous ice shedding. The dynamic response of the electric power fittings is simulated under the action of obtained dynamic tension. The four modes of asynchronous ice shedding are ① A1; ② A1, A2; ③ A1, A3; ④ A1, A2, A3. The dynamic response of conductors following ice shedding with a rate of 100%, respectively, from subconductor 1, subconductors 1 and 2, subconductors 1 and 3, and subconductors 1–3 are investigated. The finite element model of quad bundle conductors with single-span is shown in Figure5.Figure 5
Finite element model of quad bundle conductors with single-span.When using symmetrical ice shedding, the dynamic response of the conductor following ice shedding with a rate of 100% from subconductors 1–4 is studied. After the calculation, the tension-time curves at the hanging points at both ends of the conductor are shown in Figure6.Figure 6
Tension-time curve of conductor hanging point (synchronous ice shedding).Figure6 shows that the tension of each iced subconductor under the action of gravity is 70.20 kN. The tension of each subconductor is the same under ice shedding synchronously. The maximum dynamic tension of each subconductor is close to 61.59 kN. The quad bundle conductors can be considered as a whole under ice shedding synchronously.After calculation, the hanging point tension at both ends of the conductor is extracted. Figure7 shows the tension of each subconductor under asynchronous ice shedding. The maximum dynamic tension of each subconductor is shown in Table 7 under asynchronous ice shedding.Figure 7
Tension-time curve of conductor hanging point: (a) working condition, (b) working condition, (c) working condition, (d) working condition (asynchronous ice shedding).
(a)(b)(c)(d)Table 7
Maximum dynamic tension of each subconductor.
Working condition① (kN)② (kN)③ (kN)④ (kN)A167.8167.8466.1163.74A274.5467.9077.5868.93A369.2568.5864.3663.22A472.9868.6871.8269.97By analyzing the results, it can be found that the dynamic tension of each subconductor is less than the static tension under synchronous ice shedding. In the process of asynchronous ice shedding, the dynamic tension of each subconductor in condition ② and condition ④ is also less than the static tension. In condition ① and condition ③, the maximum dynamic tension of A2 and A4 conductors is greater than the static tension, and the maximum tension of the iced conductor in condition ③ is greater than that in condition ①. The reason is that once the ice on one side of the four-bundle conductor model is shedding, and the other side is iced, the conductor has a larger torsion under asymmetric load. Therefore, the most dangerous way of ice shedding is one side of the conductor is completely ice shedding, and the other side of the conductor is still covered with ice.In fact, for the iced quad bundle conductors, the whole bundle conductors will be subject to impact loading when the icing on the partial subconductors fall off, the icing of other subconductors may fall under the impact load, but it is also possible that the impact load is less than the bonding force between ice and conductor, and the icing of other subconductors remains. To study the stress characteristics of electric power fittings under the maximum dynamic tension of conductors, this paper assumes that the icing on one side of quad bundle conductors falls off and the icing on the other side does not fall off again, and this way of ice shedding is the main object of study.


## 3.1. Transmission Line Parameters

This research is based on the 500 kV transmission line from Nanyang middle to Nanyang south, Henan Province, China. The range of height difference ratio and span length are shown in Table3 and 4 in the transmission line, and the conductor material parameters are shown in Table 5. According to the engineering design, the initial tension of the conductor is taken as 25% of the rated tensile strength [23].Table 3
The range of height difference ratio in transmission lines.
Height difference ratio0(0∼0.1)(0.1∼0.2)Number of cases251114Table 4
The range of span length.
Span length100–200 m200–300 m300–400 m400–500 m500–600 mNumber of cases110594624Table 5
Conductor material parameters.
Conductor typeOuter diameter (mm)Calculated cross section (mm2)Mass per unit length (kg/km)Rated tensile strength (kN)Modulus of elasticity (GPa)Poisson’s ratioJL/G1A-630/45–45/733.8674.02079.2150.4563.00.3


## 3.2. Parameters of the Iced Conductors

According to the statistical results of transmission line parameters in Table3 and 4, span number, span length, icing thickness, and height difference ratio in the transmission line are selected, respectively. To study the effect of different parameters of iced transmission line on the dynamic tension of conductor due to ice shedding, a standard finite element model is established to simulate ice shedding as shown in Figure 4. The standard finite element model is a single-span transmission line. The parameters of the iced transmission lines with single-span are shown in Table 6. In the following sections, based on the standard finite element model, the influence of different parameters on the dynamic tension of the conductor will be studied.Figure 4
Single-span transmission line.Table 6
Standard working conditions of conductor icing.
Conductor typeNumber of transmission linesSpan length (m)Ice thickness (mm)Wind speed (m/s)Height difference ratioJL/G1A-630/45–45/7130020150According to the relevant literature, the dynamic responses under different ice-shedding rates of 50%, 60%, 70%, 80%, 90% and 100% were simulated, respectively. The results show that the jump height, and tension variations of the conductor increase with the increase of ice-shedding rate obviously when only the ice-shedding rate is changed. Therefore, tension variations of the conductor under the ice-shedding rate of 100% and stresses of electric power fittings are studied in this paper.


## 3.3. Selection of Ice-Shedding Mode

As the ice-shedding process of the conductor is complex and random, the dynamic tension of the conductor is studied under synchronous and asynchronous ice shedding. The dynamic response of the electric power fittings is simulated under the action of obtained dynamic tension. The four modes of asynchronous ice shedding are ① A1; ② A1, A2; ③ A1, A3; ④ A1, A2, A3. The dynamic response of conductors following ice shedding with a rate of 100%, respectively, from subconductor 1, subconductors 1 and 2, subconductors 1 and 3, and subconductors 1–3 are investigated. The finite element model of quad bundle conductors with single-span is shown in Figure5.Figure 5
Finite element model of quad bundle conductors with single-span.When using symmetrical ice shedding, the dynamic response of the conductor following ice shedding with a rate of 100% from subconductors 1–4 is studied. After the calculation, the tension-time curves at the hanging points at both ends of the conductor are shown in Figure6.Figure 6
Tension-time curve of conductor hanging point (synchronous ice shedding).Figure6 shows that the tension of each iced subconductor under the action of gravity is 70.20 kN. The tension of each subconductor is the same under ice shedding synchronously. The maximum dynamic tension of each subconductor is close to 61.59 kN. The quad bundle conductors can be considered as a whole under ice shedding synchronously.After calculation, the hanging point tension at both ends of the conductor is extracted. Figure7 shows the tension of each subconductor under asynchronous ice shedding. The maximum dynamic tension of each subconductor is shown in Table 7 under asynchronous ice shedding.Figure 7
Tension-time curve of conductor hanging point: (a) working condition, (b) working condition, (c) working condition, (d) working condition (asynchronous ice shedding).
(a)(b)(c)(d)Table 7
Maximum dynamic tension of each subconductor.
Working condition① (kN)② (kN)③ (kN)④ (kN)A167.8167.8466.1163.74A274.5467.9077.5868.93A369.2568.5864.3663.22A472.9868.6871.8269.97By analyzing the results, it can be found that the dynamic tension of each subconductor is less than the static tension under synchronous ice shedding. In the process of asynchronous ice shedding, the dynamic tension of each subconductor in condition ② and condition ④ is also less than the static tension. In condition ① and condition ③, the maximum dynamic tension of A2 and A4 conductors is greater than the static tension, and the maximum tension of the iced conductor in condition ③ is greater than that in condition ①. The reason is that once the ice on one side of the four-bundle conductor model is shedding, and the other side is iced, the conductor has a larger torsion under asymmetric load. Therefore, the most dangerous way of ice shedding is one side of the conductor is completely ice shedding, and the other side of the conductor is still covered with ice.In fact, for the iced quad bundle conductors, the whole bundle conductors will be subject to impact loading when the icing on the partial subconductors fall off, the icing of other subconductors may fall under the impact load, but it is also possible that the impact load is less than the bonding force between ice and conductor, and the icing of other subconductors remains. To study the stress characteristics of electric power fittings under the maximum dynamic tension of conductors, this paper assumes that the icing on one side of quad bundle conductors falls off and the icing on the other side does not fall off again, and this way of ice shedding is the main object of study.


## 4. Parametric Analysis

### 4.1. Influence of Ice Thickness on Conductor Tension

In literature [23], the degree of icing is divided according to the ice thickness of the conductor. The conductor icing radius less than 20 mm was defined as a mild icing area, while if it is larger than 20 mm was named a heavy icing area. To find out the stress characteristics of the conductors and fittings in an extreme environment, the icing thickness in the heavy ice area is selected as the research object. According to the working conditions in Table 7, the icing thickness of each subconductor is set as 20 mm, 25 mm, 30 mm, 35 mm and 40 mm, respectively, with other parameters remaining unchanged. By using the most dangerous way of conductor ice shedding, the above five working conditions of icing thickness are simulated.The tensions of the hanging points at the left and right ends of the conductors are extracted by calculation. Figure8 shows the tension time-history curve of each subconductor under different icing thicknesses, A1, A2, A3, and A4 in the figure are the same as Figure 5. It can be seen that the tension variation trend of the same subconductor is roughly the same. In the first 30 seconds after ice shedding, the dynamic tension of each subconductor hanging point fluctuates greatly and shows a sharp attenuation trend. In the interval of 30 seconds to 60 seconds, it is much slower. After the 60s, each conductor is basically in a stable state, and the jump and dynamic tension fluctuation of each subconductor can be ignored. Therefore, the data of the first 30 seconds after conductor ice shedding are used as input conditions for tension string and suspension string models. Besides, the ice thickness has a significant effect on the dynamic tension of the conductor at the hanging point. With the increase of icing thickness, the maximum dynamic tension and average tension of the conductor hanging point increase linearly, as shown in Figure 9.Figure 8
Tension-time curves of subconductor with different icing thickness: (a) A1 conductor, (b) A2 conductor, (c) A3 conductor, (d) A4 conductor.
(a)(b)(c)(d)Figure 9
Relationship between icing thickness and conductor hanging point tension: (a) Average tension, (b) Maximum tension.
(a)(b)


### 4.2. Influence of Span Length on Conductor Tension

Based on the working conditions in Table8, the ice-shedding working conditions of 300 m, 400 m, 500 m, 600 m, 700 m, and 800 m span length are simulated, respectively. The most dangerous ice-shedding mode could be adopted, and the tensions of the hanging points at the left and right ends of the conductors are extracted by calculation. Figure 10 shows the tension time-history curve of each subconductor under different span lengths.Table 8
Material parameters of tension string parts.
Number and name(1) shackle 1(11) socket-clevis eye(9) ball-eye(10) tensioninsulator stringBolt(2) shackle 2(12) trapezoidal yoke plate(3) DB adjusting plate 1(13) rectangular hanging plate(4) parallel hanging plate 1(14) triangle hanging plate(5) PT adjusting plate 1(15) upper shackle 3(6) PT adjusting plate 2(16) lower shackle 3(7) towing plate(17) DB adjusting plate 2(8) parallel hanging plate 2(18) shackle 4MaterialSteel 3540CRQ235 BSteel 45Elastic modulus (MPa)2.12 × 1052.11 × 1052.10 × 1052.09 × 105Poisson’s ratio0.2910.2770.2740.269Density (kg/m3)7.87 × 1037.82 × 1037.83 × 1037.89 × 103Yield stress (MPa)3.15 × 1027.85 × 1022.35 × 1023.55 × 102Figure 10
Tension-time curves of subconductor with different span length: (a) A1 conductor (b) A2 conductor (c) A3 conductor (d) A4 conductor.
(a)(b)(c)(d)The observation results show that with the increase in span length, the time-history curves of the conductor’s hanging point tension with different span lengths are quite different. The reason is that when the span length is small, the vibration frequency of the conductor is high. What’s more, the sag of the conductor will increase as the span length increases and the vibration frequency of the conductor decreases significantly.It can be seen from Figure11 that with the increase of span length, the conductor hanging point on static, maximum, and average tension of icing are increasing, and the static tension of icing increases parabola with the increase of span length. The A1 and A3 conductors are ice-shedding subconductors. During ice shedding, the conductor hanging point of the maximum dynamic tension and average tension is less than those of A2 and A4. Moreover, the maximum dynamic tension and average tension of the A2 conductor are significantly greater than that of the other three subconductors. This will lead to a large torsion of the four-bundle conductor in the process of ice shedding, and then the tension string model will be subjected to a large torque.Figure 11
Relationship between span length and conductor hanging point tension: (a) Static tension, (b) Maximum tension, (c) Average tension.
(a)(b)(c)


### 4.3. Influence of Height Difference Ratio on Conductor Tension

To study the influence of height difference ratio on the dynamic response of transmission line, the no height difference model was used to compare the five working conditions with the height difference ratio of 0.2, 0.4, 0.6, 0.8, and 1.0 were simulation.The tensions of the hanging points at the left and right ends of the conductors are extracted by calculation. Figure12 shows the tension time-history curves of the subconductor with different height difference ratios. Figure 13 shows the relationship between the height difference ratio and the average tension and maximum tension of the hanging point conductor. It can be found that the height difference ratio has a different influence on the tension of each subconductor. When the height difference ratio is in the range of 0–0.2, with the increase of the ratio of height difference, the average and maximum dynamic tensions of each subconductor have no obvious change. When the height difference ratio is in the range of 0.2–1.0, the hanging point tension of conductors A1 and A3 increases continuously with the increase of height difference ratio, and the average and maximum tension tend to increase linearly. However, the average tension fluctuation of conductors A2 and A4 is much smaller and tends to be stable, while the maximum dynamic tension only has a slight upward trend.Figure 12
Tension-time curves of subconductor with different height difference ratios: (a) A1 conductor, (b) A2 conductor, (c) A3 conductor, (d) A4 conductor.
(a)(b)(c)(d)Figure 13
Relationship between height difference ratio and conductor hanging point tension: (a) Average tension, (b) Maximum tension.
(a)(b)Based on the above analysis, it can be concluded that in the small range of height difference ratio (0–0.2), the influence of height difference ratio on conductor tension is too small to put much attention. When the height difference ratio is large (0.2–1.0), the influence of the height difference ratio on the ice-shedding subconductor is larger, which is approximately linear correlation. Compared with the ice-shedding subconductor, the influence of the height difference ratio on the tension of the iced subconductor is relatively smaller.Based on the above analysis, it is suggested that the height difference ratio should be controlled within the range of 0–0.2 in line design, and the design of large height difference lines should be minimized in heavy ice areas.


### 4.4. Influence of Span Number on Conductor Tension

To study the influence of span number on conductor tension, the finite element models of quad bundle conductors with single-span, three-span, and five-span are constructed, as shown in Figure14. Each model is the middle-span ice shedding.Figure 14
Finite element model of quad bundle conductors with multispan: (a) Single-span, (b) Three-span, (c) Five-span.
(a)(b)(c)The tensions of the hanging points at the left and right ends of the conductors are extracted by calculation. Figure15 shows the relationship between the number of spans and hanging point tension of middle-span conductors. It can be seen that for ice-shedding subconductors A1 and A3, the conductor hanging point tension of single span is less than that of three-span and five-span models. As the number of spans increases, the conductor hanging point tension of the middle span increases continuously. Figure 16 is the relationship between span number and average tension of conductor hanging point. For the five-span quad bundle conductors model, the average tension and maximum tension of the conductor hanging point are significantly greater than that of the single-span and three-span models. For iced subconductors A2 and A4, the hanging point tension of the single-span model varies greatly, but the average tension of the hanging point is smaller than that of the three-span and five-span finite element models.Figure 15
Tension-time curves of subconductors with span number: (a) A1 conductor, (b) A2 conductor, (c). A3 conductor, (d) A4 conductor.
(a)(b)(c)(d)Figure 16
Relationship between span number and average tension of conductor hanging point.On the whole, the number of spans has a different influence on the ice-shedding and icing subconductor tension, but the average tension of each subconductor in the five-span model is greater than that in the single-span and three-span models. So the following research focuses on the five-span quad bundle conductors model.


## 4.1. Influence of Ice Thickness on Conductor Tension

In literature [23], the degree of icing is divided according to the ice thickness of the conductor. The conductor icing radius less than 20 mm was defined as a mild icing area, while if it is larger than 20 mm was named a heavy icing area. To find out the stress characteristics of the conductors and fittings in an extreme environment, the icing thickness in the heavy ice area is selected as the research object. According to the working conditions in Table 7, the icing thickness of each subconductor is set as 20 mm, 25 mm, 30 mm, 35 mm and 40 mm, respectively, with other parameters remaining unchanged. By using the most dangerous way of conductor ice shedding, the above five working conditions of icing thickness are simulated.The tensions of the hanging points at the left and right ends of the conductors are extracted by calculation. Figure8 shows the tension time-history curve of each subconductor under different icing thicknesses, A1, A2, A3, and A4 in the figure are the same as Figure 5. It can be seen that the tension variation trend of the same subconductor is roughly the same. In the first 30 seconds after ice shedding, the dynamic tension of each subconductor hanging point fluctuates greatly and shows a sharp attenuation trend. In the interval of 30 seconds to 60 seconds, it is much slower. After the 60s, each conductor is basically in a stable state, and the jump and dynamic tension fluctuation of each subconductor can be ignored. Therefore, the data of the first 30 seconds after conductor ice shedding are used as input conditions for tension string and suspension string models. Besides, the ice thickness has a significant effect on the dynamic tension of the conductor at the hanging point. With the increase of icing thickness, the maximum dynamic tension and average tension of the conductor hanging point increase linearly, as shown in Figure 9.Figure 8
Tension-time curves of subconductor with different icing thickness: (a) A1 conductor, (b) A2 conductor, (c) A3 conductor, (d) A4 conductor.
(a)(b)(c)(d)Figure 9
Relationship between icing thickness and conductor hanging point tension: (a) Average tension, (b) Maximum tension.
(a)(b)


## 4.2. Influence of Span Length on Conductor Tension

Based on the working conditions in Table8, the ice-shedding working conditions of 300 m, 400 m, 500 m, 600 m, 700 m, and 800 m span length are simulated, respectively. The most dangerous ice-shedding mode could be adopted, and the tensions of the hanging points at the left and right ends of the conductors are extracted by calculation. Figure 10 shows the tension time-history curve of each subconductor under different span lengths.Table 8
Material parameters of tension string parts.
Number and name(1) shackle 1(11) socket-clevis eye(9) ball-eye(10) tensioninsulator stringBolt(2) shackle 2(12) trapezoidal yoke plate(3) DB adjusting plate 1(13) rectangular hanging plate(4) parallel hanging plate 1(14) triangle hanging plate(5) PT adjusting plate 1(15) upper shackle 3(6) PT adjusting plate 2(16) lower shackle 3(7) towing plate(17) DB adjusting plate 2(8) parallel hanging plate 2(18) shackle 4MaterialSteel 3540CRQ235 BSteel 45Elastic modulus (MPa)2.12 × 1052.11 × 1052.10 × 1052.09 × 105Poisson’s ratio0.2910.2770.2740.269Density (kg/m3)7.87 × 1037.82 × 1037.83 × 1037.89 × 103Yield stress (MPa)3.15 × 1027.85 × 1022.35 × 1023.55 × 102Figure 10
Tension-time curves of subconductor with different span length: (a) A1 conductor (b) A2 conductor (c) A3 conductor (d) A4 conductor.
(a)(b)(c)(d)The observation results show that with the increase in span length, the time-history curves of the conductor’s hanging point tension with different span lengths are quite different. The reason is that when the span length is small, the vibration frequency of the conductor is high. What’s more, the sag of the conductor will increase as the span length increases and the vibration frequency of the conductor decreases significantly.It can be seen from Figure11 that with the increase of span length, the conductor hanging point on static, maximum, and average tension of icing are increasing, and the static tension of icing increases parabola with the increase of span length. The A1 and A3 conductors are ice-shedding subconductors. During ice shedding, the conductor hanging point of the maximum dynamic tension and average tension is less than those of A2 and A4. Moreover, the maximum dynamic tension and average tension of the A2 conductor are significantly greater than that of the other three subconductors. This will lead to a large torsion of the four-bundle conductor in the process of ice shedding, and then the tension string model will be subjected to a large torque.Figure 11
Relationship between span length and conductor hanging point tension: (a) Static tension, (b) Maximum tension, (c) Average tension.
(a)(b)(c)


## 4.3. Influence of Height Difference Ratio on Conductor Tension

To study the influence of height difference ratio on the dynamic response of transmission line, the no height difference model was used to compare the five working conditions with the height difference ratio of 0.2, 0.4, 0.6, 0.8, and 1.0 were simulation.The tensions of the hanging points at the left and right ends of the conductors are extracted by calculation. Figure12 shows the tension time-history curves of the subconductor with different height difference ratios. Figure 13 shows the relationship between the height difference ratio and the average tension and maximum tension of the hanging point conductor. It can be found that the height difference ratio has a different influence on the tension of each subconductor. When the height difference ratio is in the range of 0–0.2, with the increase of the ratio of height difference, the average and maximum dynamic tensions of each subconductor have no obvious change. When the height difference ratio is in the range of 0.2–1.0, the hanging point tension of conductors A1 and A3 increases continuously with the increase of height difference ratio, and the average and maximum tension tend to increase linearly. However, the average tension fluctuation of conductors A2 and A4 is much smaller and tends to be stable, while the maximum dynamic tension only has a slight upward trend.Figure 12
Tension-time curves of subconductor with different height difference ratios: (a) A1 conductor, (b) A2 conductor, (c) A3 conductor, (d) A4 conductor.
(a)(b)(c)(d)Figure 13
Relationship between height difference ratio and conductor hanging point tension: (a) Average tension, (b) Maximum tension.
(a)(b)Based on the above analysis, it can be concluded that in the small range of height difference ratio (0–0.2), the influence of height difference ratio on conductor tension is too small to put much attention. When the height difference ratio is large (0.2–1.0), the influence of the height difference ratio on the ice-shedding subconductor is larger, which is approximately linear correlation. Compared with the ice-shedding subconductor, the influence of the height difference ratio on the tension of the iced subconductor is relatively smaller.Based on the above analysis, it is suggested that the height difference ratio should be controlled within the range of 0–0.2 in line design, and the design of large height difference lines should be minimized in heavy ice areas.


## 4.4. Influence of Span Number on Conductor Tension

To study the influence of span number on conductor tension, the finite element models of quad bundle conductors with single-span, three-span, and five-span are constructed, as shown in Figure14. Each model is the middle-span ice shedding.Figure 14
Finite element model of quad bundle conductors with multispan: (a) Single-span, (b) Three-span, (c) Five-span.
(a)(b)(c)The tensions of the hanging points at the left and right ends of the conductors are extracted by calculation. Figure15 shows the relationship between the number of spans and hanging point tension of middle-span conductors. It can be seen that for ice-shedding subconductors A1 and A3, the conductor hanging point tension of single span is less than that of three-span and five-span models. As the number of spans increases, the conductor hanging point tension of the middle span increases continuously. Figure 16 is the relationship between span number and average tension of conductor hanging point. For the five-span quad bundle conductors model, the average tension and maximum tension of the conductor hanging point are significantly greater than that of the single-span and three-span models. For iced subconductors A2 and A4, the hanging point tension of the single-span model varies greatly, but the average tension of the hanging point is smaller than that of the three-span and five-span finite element models.Figure 15
Tension-time curves of subconductors with span number: (a) A1 conductor, (b) A2 conductor, (c). A3 conductor, (d) A4 conductor.
(a)(b)(c)(d)Figure 16
Relationship between span number and average tension of conductor hanging point.On the whole, the number of spans has a different influence on the ice-shedding and icing subconductor tension, but the average tension of each subconductor in the five-span model is greater than that in the single-span and three-span models. So the following research focuses on the five-span quad bundle conductors model.


## 5. Overall Electric Power Fitting Model

### 5.1. Establishment of Solid Model

According to the two-dimensional chart of electric power fitting, the three-dimensional solid models of tension string and suspension string are set, as shown in Figure17 and 18. There are 84 parts in the tension string model and 27 parts in the suspension string model.Figure 17
Integral model of tension string.Figure 18
Integral model of suspension string.


### 5.2. Material Parameters

The material parameters of each part of the tension string are shown in Table8, and the material parameters of each part of the suspension string are shown in Table 9. The plastic model of the steel is bilinear constitutive relation, and the tangent modulus after yielding is 1% of the elastic modulus before yielding.Table 9
Material parameters of suspension string parts.
Number and name(1) trunnion hanging plate(4) ball-eye(5) suspensioninsulator(9) suspensioninsulator stringBolt(2) upper shackle(3) lower shackle(6) socket-clevis eye(7) plate(8) U-plateMaterialSteel 3540CRQ235 BZL104Steel 45Elastic modulus (MPa)2.12 × 1052.11 × 1052.10 × 1056.9 × 1042.09 × 105Poisson’s ratio0.2910.2770.2740.340.269Density (kg/m3)7.87 × 1037.82 × 1037.83 × 1032.7 × 1037.89 × 103Yield stress (MPa)3.15 × 1027.85 × 1022.35 × 1021.95 × 1023.55 × 102


### 5.3. Interaction between Fittings

In the whole model of tension string and suspension string, the “Coupling” constraints between two parts are set by two reference points, such as the bolt and DB adjusting plate. For the connection between the three parts, the connection between the bolt and the part is the “Tie” constraint, such as bolt and parallel hanging plate, while the relationship between a bolt and another part is the “Coupling” constraint. The constraint relationship is shown in Figure19. According to the actual force situation of the parts, the line feature “Wire” is set by two adjacent reference points, and the connection property is determined. There are many spherical hinge constraints in the models of tension string and suspension string, and the parts can rotate in any direction at the spherical hinge constraints. Therefore, the connection section type is defined as “Join” and “Flexion-Torsion”. Only one direction of rotation occurs at the connection position of the bolt and other parts are the connection section type defined as “hinge.” The type of connection section between parts is shown in Figure 20.Figure 19
Constraint relationship: (a) “Coupling” constraint, (b) “Tie” constraint.
(a)(b)Figure 20
Connection section type: (a) “Hinge” (b) “Join” and “Flexion-Torsion.”
(a)(b)


### 5.4. Meshing

For the whole model of tension string and suspension string, each part is simulated by solid element. After reasonably splitting the solid parts, the mesh attributes of the parts are assigned as hexahedral elements, and the element type is C3D8R. Considering that the greater the mesh density of parts, the higher the calculation cost of the model, and the whole model of tension string and suspension string is very complex, the mesh size of each part in the tension string model is set to 8 mm, and the mesh size of each part in the suspension string model is set to 6 mm. The grid model is shown in Figure21. After meshing, the total number of model units of tension string and suspension string is 225104 and 181706, respectively.Figure 21
Meshed fittings in finite element model: (a) Tension string, (b) Suspension string.
(a)(b)


### 5.5. Boundary Conditions and Loads

Based on the results of parametric analysis, when the height difference ratio is 0–0.2, the influence on the conductor tension can be ignored. Combined with the height difference ratio and span length range of the engineering line, the typical working condition is set as the input condition of the electric power fitting research, as shown in Table10.Table 10
Load input conditions of electric power fittings.
Number of spansSpan length (m)Ice thickness (mm)Wind speed (m/s)Height difference ratio540020150After calculating the quad bundle conductors model, the conductor hanging point tension, stress, and coordinates of adjacent elements could be extracted respectively. The hanging point tension of tension string conductors is shown in Figure22 The conductor tension applied to the suspension string could be obtained after spatial coordinate transformation and load part synthesis, as shown in Figure 23.Figure 22
Conductor tension applied on tension string: (a)Fx, (b) Fy, (c) Fz.
(a)(b)(c)Figure 23
Conductor tension applied on suspension string: (a)Fx, (b) Fy, (c) Fz.
(a)(b)(c)In the models of tension string and suspension string, the top hanging point fittings cannot translate and rotate in theX, Y, and Z directions, so the degrees of freedom in six directions are constrained, as shown in Figure 24.Figure 24
Boundary conditions and loads: (a) Tension string; (b) Suspension string.
(a)(b)


## 5.1. Establishment of Solid Model

According to the two-dimensional chart of electric power fitting, the three-dimensional solid models of tension string and suspension string are set, as shown in Figure17 and 18. There are 84 parts in the tension string model and 27 parts in the suspension string model.Figure 17
Integral model of tension string.Figure 18
Integral model of suspension string.


## 5.2. Material Parameters

The material parameters of each part of the tension string are shown in Table8, and the material parameters of each part of the suspension string are shown in Table 9. The plastic model of the steel is bilinear constitutive relation, and the tangent modulus after yielding is 1% of the elastic modulus before yielding.Table 9
Material parameters of suspension string parts.
Number and name(1) trunnion hanging plate(4) ball-eye(5) suspensioninsulator(9) suspensioninsulator stringBolt(2) upper shackle(3) lower shackle(6) socket-clevis eye(7) plate(8) U-plateMaterialSteel 3540CRQ235 BZL104Steel 45Elastic modulus (MPa)2.12 × 1052.11 × 1052.10 × 1056.9 × 1042.09 × 105Poisson’s ratio0.2910.2770.2740.340.269Density (kg/m3)7.87 × 1037.82 × 1037.83 × 1032.7 × 1037.89 × 103Yield stress (MPa)3.15 × 1027.85 × 1022.35 × 1021.95 × 1023.55 × 102


## 5.3. Interaction between Fittings

In the whole model of tension string and suspension string, the “Coupling” constraints between two parts are set by two reference points, such as the bolt and DB adjusting plate. For the connection between the three parts, the connection between the bolt and the part is the “Tie” constraint, such as bolt and parallel hanging plate, while the relationship between a bolt and another part is the “Coupling” constraint. The constraint relationship is shown in Figure19. According to the actual force situation of the parts, the line feature “Wire” is set by two adjacent reference points, and the connection property is determined. There are many spherical hinge constraints in the models of tension string and suspension string, and the parts can rotate in any direction at the spherical hinge constraints. Therefore, the connection section type is defined as “Join” and “Flexion-Torsion”. Only one direction of rotation occurs at the connection position of the bolt and other parts are the connection section type defined as “hinge.” The type of connection section between parts is shown in Figure 20.Figure 19
Constraint relationship: (a) “Coupling” constraint, (b) “Tie” constraint.
(a)(b)Figure 20
Connection section type: (a) “Hinge” (b) “Join” and “Flexion-Torsion.”
(a)(b)


## 5.4. Meshing

For the whole model of tension string and suspension string, each part is simulated by solid element. After reasonably splitting the solid parts, the mesh attributes of the parts are assigned as hexahedral elements, and the element type is C3D8R. Considering that the greater the mesh density of parts, the higher the calculation cost of the model, and the whole model of tension string and suspension string is very complex, the mesh size of each part in the tension string model is set to 8 mm, and the mesh size of each part in the suspension string model is set to 6 mm. The grid model is shown in Figure21. After meshing, the total number of model units of tension string and suspension string is 225104 and 181706, respectively.Figure 21
Meshed fittings in finite element model: (a) Tension string, (b) Suspension string.
(a)(b)


## 5.5. Boundary Conditions and Loads

Based on the results of parametric analysis, when the height difference ratio is 0–0.2, the influence on the conductor tension can be ignored. Combined with the height difference ratio and span length range of the engineering line, the typical working condition is set as the input condition of the electric power fitting research, as shown in Table10.Table 10
Load input conditions of electric power fittings.
Number of spansSpan length (m)Ice thickness (mm)Wind speed (m/s)Height difference ratio540020150After calculating the quad bundle conductors model, the conductor hanging point tension, stress, and coordinates of adjacent elements could be extracted respectively. The hanging point tension of tension string conductors is shown in Figure22 The conductor tension applied to the suspension string could be obtained after spatial coordinate transformation and load part synthesis, as shown in Figure 23.Figure 22
Conductor tension applied on tension string: (a)Fx, (b) Fy, (c) Fz.
(a)(b)(c)Figure 23
Conductor tension applied on suspension string: (a)Fx, (b) Fy, (c) Fz.
(a)(b)(c)In the models of tension string and suspension string, the top hanging point fittings cannot translate and rotate in theX, Y, and Z directions, so the degrees of freedom in six directions are constrained, as shown in Figure 24.Figure 24
Boundary conditions and loads: (a) Tension string; (b) Suspension string.
(a)(b)


## 6. Analysis of Calculation Results

### 6.1. Overall Displacement Analysis

The displacement cloud picture of the tension string and suspension string model is shown in Figure25 and 26. The black tension string in the figure is the initial shape of the whole model before being stressed. It can be found that under the action of steady wind and ice-shedding load, the tension string has displacement in X, Y, and Z directions and rotation around the Y and Z axes. In the X–Y plane, the displacement range of the tension string in the Y direction is –10.57 mm–738.70 mm. In the X–Z plane, the displacement range of the tension string in the Z direction is –1.28 mm–389.10 mm. The suspension string has displacement in X, Y, and Z directions and rotation around the X and Z axes. In the X–Y plane, the displacement range of the overhanging string in the X direction is –0.75 mm–80.73 mm. In the Y–Z plane, the displacement range of the overhanging string in the Z direction is –106.30 mm–2.30 mm. Due to the large tension of the conductor, the tension string, and the suspension string are mainly dominated by the overall rotation, and the rotation between adjacent components is not obvious. After defining corresponding constraints, each part can rotate correctly, which is the same as the actual engineering situation. Therefore, it can be considered that the interaction and boundary conditions between the parts of the two models are reasonable.Figure 25
Displacement cloud picture of tension string (mm): (a) In X–Y plane, (b) In X–Z plane.
(a)(b)Figure 26
Displacement cloud picture of suspension string (mm): (a) In X–Y plane, (b) In Y–Z plane.
(a)(b)


### 6.2. Overall Stress Analysis

Figure27 shows the ratio of maximum stress to yield stress of each part of the tension string and suspension string. The stress distribution of tension string is shown in Figure 28, it can be found that all shackles have exceeded the yield stress of materials. The maximum stress appears on shackle 1, and the maximum stress is 325.5 MPa, the ratio of maximum stress to yield stress is 1.03. What is more, the stress concentration occurs in the rectangular hanging plate, and the stress at the abrupt change of section reaches 317.1 MPa, the ratio of maximum stress to yield stress is 1.01. The maximum stress of the Trapezoidal yoke plate, insulator string, and triangular hanging plate accounts for 0.85, 0.83, and 0.80 of the yield stress, respectively, which still has a certain degree of safety redundancy. The ratio of stress to yield stress of other parts is relatively small, which is much safer.Figure 27
The ratio of maximum stress to yield stress of parts: (a) Tension string; (b) Suspension string.
(a)(b)Figure 28
Stress cloud picture of tension string.The stress distribution of the suspension string is shown in Figure29. Different from the tension string model, the stress of each electric power fitting in the suspension string model is smaller, and it does not reach the yield stress of the material. The maximum stress of the whole model occurs at the hanging point of the upper shackle, with the maximum stress 239.3 MPa accounting for 0.76 of the yield stress. There is a stress concentration phenomenon between the hull and the hole, and the maximum stress is 144.2 MPa, accounting for 0.74 of the yield stress. The maximum stress of the suspension insulator is 170.4 MPa, accounting for 0.73 of yield stress. The other parts have lower maximum stress and higher safety.Figure 29
Stress cloud picture of suspension string.


### 6.3. Analysis of Key Fittings for Tension String

As the maximum stress in the contact area of each shackle exceeds the yield stress, as shown in Figure30, it can be concluded that the shackle is the key part of the whole model. The maximum stress position of each shackle in the time-history curve is shown in Figure 31. Under the ice-shedding load, the stress of shackle 1 is the largest, followed by shackle 3, shackle 4, and shackle 2. It can be seen from Figure 32 that the average stress of shackle 3 is significantly greater than other shackles. Therefore, it can be concluded that among all the shackles, shackle 1 and shackle 3 are much more dangerous.Figure 30
Stress cloud picture at maximum stress time for each shackle. (a) Shackle 1/part-(1), (b) Shackle 2/part-(2), (c) Upper shackle 3/part-(15), (d) Lower shackle 3/part-(16), (e) Shackle 4/part-(18).
(a)(b)(c)(d)(e)Figure 31
Stress-time curve of each shackle.Figure 32
Average stress of each shackle.In literature [18], the wear experiments of two shackles were carried out, the results show that the wear degree of the upper shackle is greater than that of the lower shackle. In the tension string model, shackle 1 and shackle 2 are upper and lower connection relations, and upper shackle 3 and lower shackle 3 are also upper and lower connection relations. It can be found from Figure 32 that when a couple of shackles are in the upper and lower connection relationship, the average stress of the upper shackle is greater than that of the lower one, which is the same as the experimental results in the literature.Figure33 is the failure diagram of the shackle on the engineering site. The failure location of the part is the same as the simulation result in this paper, and the upper shackle breaks before the lower shackle, which indicates the rationality of the simulation method in this paper.Figure 33
The fracture mode of the shackle.Figure34 is the cloud picture of the maximum stress of the rectangular hanging plate, it can be seen that the maximum stress of the two rectangular hanging plates occurs at the mutation of the middle section. The reason is that under the action of stable wind and ice-shedding load, the central position of the rectangular hanging plate bears the bending moment in the Y and Z directions, resulting in large stress. The maximum stress of the rectangular hanging plate connected with the iced conductor is 317.1 MPa, accounting for 1.01 of the yield stress. The maximum stress of the rectangular hanging plate connected with the ice-shedding conductor is 256.6 MPa, accounting for 0.81 of the yield stress. Therefore, the rectangular hanging plate connected with the iced conductor is easier to fracture than the rectangular hanging plate connected with the iced conductor. To avoid the fracture of the rectangular hanging plate, the middle position can be properly thickened according to the local meteorological conditions.Figure 34
Stress cloud picture of rectangular hanging plate: (a) connected with the iced conductor; (b) connected with the ice-shedding conductor.
(a)(b)Figure35 shows the stress cloud picture of the trapezoidal yoke plate. Points 1 and 2 are the maximum stress positions for connecting ice-shedding conductors, and points 3 and 4 are the maximum stress points for connecting iced conductors. Point 4 is the maximum stress position of the trapezoidal yoke plate, and the maximum stress is 269 MPa, accounting for 0.85 of the yield stress. The stress time-history curve of the trapezoidal yoke plate at each hanging point is shown in Figure 36. For the trapezoidal yoke plate, the stress on the side connected with the iced conductor is greater than that of the side connected with the iced conductor, and the stress of the lower hanging point is greater than that of the upper hanging point. On the whole, the trapezoidal yoke plate still has a certain degree of safety redundancy. If the ice thickness increases, the hanging point on the side of the iced conductor may yield.Figure 35
Stress cloud picture of trapezoidal yoke plate.Figure 36
Stress-time curve of each point.Figure37 shows the stress distribution of insulator strings. The maximum stress position of two insulator strings is at the corner of the bottom straight bar, the maximum stress of the insulator string connected with the ice-shedding conductor is 181.7 MPa, accounting for 0.77 of the yield stress. The maximum stress of the insulator string connected with the iced conductor is 195.5 MPa, accounting for 0.83 of the yield stress. Figure 38 shows the actual damage to the insulator string, the damage position is the same as the maximum stress position simulated in this paper. The reason is that this position is connected with the socket-clevis eye, which can rotate in any direction. Under the action of external load, the position is vulnerable to alternating stress in two directions, which leads to fatigue failure of the structure. Therefore, in the processing of the following structural design, the insulator string should be properly optimized to avoid failure.Figure 37
Stress cloud picture of insulator string: (a) connected with the ice-shedding conductor; (b) connected with the iced conductor.
(a)(b)Figure 38
Failure mode of ball-joint at the lower end of insulator string.The stress results of the triangular hanging plate are shown in Figure39 and 40. Points 1, 2, and 3 are the maximum stress positions of each hanging point, respectively. The maximum stress of two triangular hanging plates is at the lower right hanging point, and the stress of the lower hanging point is significantly greater than that of the upper hanging point. The maximum stress of the triangle hanging plate connected with the iced conductor is 251 MPa, accounting for 0.8 of the yield stress, and the average stress is 206 MPa. The maximum stress of the triangle hanging plate connected with the ice-shedding conductor is 243 MPa, accounting for 0.77 of the yield stress, and the average stress is 199 MPa. On the whole, the two triangle hanging plates are relatively safe. If they are in a more extreme environment, the lower hanging point of the triangle hanging plate connected with the iced conductor may be destroyed first.Figure 39
Triangle hanging plate connected with the iced conductor: (a) stress cloud picture; (b) stress-time curve of each hanging point.
(a)(b)Figure 40
Triangle hanging plate connected with the ice-shedding conductor: (a) stress cloud picture; (b) stress-time curve of each hanging point.
(a)(b)The stress cloud picture in Figure41 includes (4) parallel hanging plate 1, (8) parallel hanging plate 2, (5) PT adjusting plate 1, (6) PT adjusting plate 2, (7) traction plate, (9) ball-eye, (11) socket-clevis eye, (3) DB adjusting plate 1, and (17) DB adjusting plate 2. It is worth noticing that the ball-eye is prone to stress concentration at the corner of the straight rod. Under the action of external load, the high-frequency rotation of each electric power fittings will occur, resulting in repeated bending moment action at the cross section of the ball-eye, which may lead to fatigue fracture. Therefore, in the engineering design, the ball-eye should be properly optimized. The ratio of maximum stress to yield stress of other parts is relatively small, which has much more safety redundancy. Therefore, in engineering design, the material strength could be reasonably selected according to the actual situation.Figure 41
Stress cloud picture of some fittings of tension string.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


### 6.4. Analysis of Key Fittings for Suspension String

According to the analysis results of the whole model of the suspension string, the shackle is the maximum stress part, which indicates that it is the most dangerous part. Figures42 and 43 are the stress cloud picture and stress time-history curve of shackle, respectively. It can be seen that the maximum stress of the shackle also appears in the contact area, and the average stress of the upper shackle is significantly greater than that of the lower one, which is consistent with the analysis results of the shackle in the tension string model.Figure 42
Stress cloud picture of shackle.Figure 43
Stress-time curve of shackle.Figure44 shows the stress cloud picture of the suspension clamp, the stress of the parts is relatively small and mainly distributed near the hull and the hole. The stress concentration occurs at the abrupt change of the cross section of the hull and the hole. The maximum stress is 144.2 MPa, accounting for 0.74 of the yield stress, and the structure has a certain degree of safety redundancy.Figure 44
Stress cloud picture of suspension clamp.Figure45 shows the stress cloud picture of the suspension insulator string, the maximum stress is 170.4 MPa, accounting for 0.73 of yield stress. The maximum stress position of the part is at the joint with the ball-eye. Under the action of alternating load, the location may be damaged. Therefore, local enhancement can be considered for this position.Figure 45
Stress cloud picture of suspension insulator.The stress cloud picture in Figure46 includes (1) trunnion hanging plate, (3) ball-eye, (5) socket-clevis eye, (6) yoke plate, and (7) U-plate. The maximum stress of the ball-eye is at the corner of the straight bar, and the maximum stress is 158.3 MPa, accounting for 0.2 of the yield stress. The maximum stress of the yoke plate appears near the hole, and the maximum stress is 74.26 MPa, accounting for 0.24 of the yield stress. The position of maximum stress of the U-plate is near the groove, and the maximum stress is 102.8 MPa, accounting for 0.33 of the yield stress. The maximum stress of the trunnion hanging plate and the socket-clevis eye is at the hanging point of bolt connection, and the maximum stress is 66.4 MPa and 144.3 MPa, accounting for 0.21 and 0.46 of the yield stress, respectively. The ratio of the maximum stress to the yield stress of the five parts is less than 0.5, which has much higher safety redundancy during conductor ice shedding.Figure 46
Stress cloud picture of some fittings in suspension string.
(a)(b)(c)(d)(e)


## 6.1. Overall Displacement Analysis

The displacement cloud picture of the tension string and suspension string model is shown in Figure25 and 26. The black tension string in the figure is the initial shape of the whole model before being stressed. It can be found that under the action of steady wind and ice-shedding load, the tension string has displacement in X, Y, and Z directions and rotation around the Y and Z axes. In the X–Y plane, the displacement range of the tension string in the Y direction is –10.57 mm–738.70 mm. In the X–Z plane, the displacement range of the tension string in the Z direction is –1.28 mm–389.10 mm. The suspension string has displacement in X, Y, and Z directions and rotation around the X and Z axes. In the X–Y plane, the displacement range of the overhanging string in the X direction is –0.75 mm–80.73 mm. In the Y–Z plane, the displacement range of the overhanging string in the Z direction is –106.30 mm–2.30 mm. Due to the large tension of the conductor, the tension string, and the suspension string are mainly dominated by the overall rotation, and the rotation between adjacent components is not obvious. After defining corresponding constraints, each part can rotate correctly, which is the same as the actual engineering situation. Therefore, it can be considered that the interaction and boundary conditions between the parts of the two models are reasonable.Figure 25
Displacement cloud picture of tension string (mm): (a) In X–Y plane, (b) In X–Z plane.
(a)(b)Figure 26
Displacement cloud picture of suspension string (mm): (a) In X–Y plane, (b) In Y–Z plane.
(a)(b)


## 6.2. Overall Stress Analysis

Figure27 shows the ratio of maximum stress to yield stress of each part of the tension string and suspension string. The stress distribution of tension string is shown in Figure 28, it can be found that all shackles have exceeded the yield stress of materials. The maximum stress appears on shackle 1, and the maximum stress is 325.5 MPa, the ratio of maximum stress to yield stress is 1.03. What is more, the stress concentration occurs in the rectangular hanging plate, and the stress at the abrupt change of section reaches 317.1 MPa, the ratio of maximum stress to yield stress is 1.01. The maximum stress of the Trapezoidal yoke plate, insulator string, and triangular hanging plate accounts for 0.85, 0.83, and 0.80 of the yield stress, respectively, which still has a certain degree of safety redundancy. The ratio of stress to yield stress of other parts is relatively small, which is much safer.Figure 27
The ratio of maximum stress to yield stress of parts: (a) Tension string; (b) Suspension string.
(a)(b)Figure 28
Stress cloud picture of tension string.The stress distribution of the suspension string is shown in Figure29. Different from the tension string model, the stress of each electric power fitting in the suspension string model is smaller, and it does not reach the yield stress of the material. The maximum stress of the whole model occurs at the hanging point of the upper shackle, with the maximum stress 239.3 MPa accounting for 0.76 of the yield stress. There is a stress concentration phenomenon between the hull and the hole, and the maximum stress is 144.2 MPa, accounting for 0.74 of the yield stress. The maximum stress of the suspension insulator is 170.4 MPa, accounting for 0.73 of yield stress. The other parts have lower maximum stress and higher safety.Figure 29
Stress cloud picture of suspension string.


## 6.3. Analysis of Key Fittings for Tension String

As the maximum stress in the contact area of each shackle exceeds the yield stress, as shown in Figure30, it can be concluded that the shackle is the key part of the whole model. The maximum stress position of each shackle in the time-history curve is shown in Figure 31. Under the ice-shedding load, the stress of shackle 1 is the largest, followed by shackle 3, shackle 4, and shackle 2. It can be seen from Figure 32 that the average stress of shackle 3 is significantly greater than other shackles. Therefore, it can be concluded that among all the shackles, shackle 1 and shackle 3 are much more dangerous.Figure 30
Stress cloud picture at maximum stress time for each shackle. (a) Shackle 1/part-(1), (b) Shackle 2/part-(2), (c) Upper shackle 3/part-(15), (d) Lower shackle 3/part-(16), (e) Shackle 4/part-(18).
(a)(b)(c)(d)(e)Figure 31
Stress-time curve of each shackle.Figure 32
Average stress of each shackle.In literature [18], the wear experiments of two shackles were carried out, the results show that the wear degree of the upper shackle is greater than that of the lower shackle. In the tension string model, shackle 1 and shackle 2 are upper and lower connection relations, and upper shackle 3 and lower shackle 3 are also upper and lower connection relations. It can be found from Figure 32 that when a couple of shackles are in the upper and lower connection relationship, the average stress of the upper shackle is greater than that of the lower one, which is the same as the experimental results in the literature.Figure33 is the failure diagram of the shackle on the engineering site. The failure location of the part is the same as the simulation result in this paper, and the upper shackle breaks before the lower shackle, which indicates the rationality of the simulation method in this paper.Figure 33
The fracture mode of the shackle.Figure34 is the cloud picture of the maximum stress of the rectangular hanging plate, it can be seen that the maximum stress of the two rectangular hanging plates occurs at the mutation of the middle section. The reason is that under the action of stable wind and ice-shedding load, the central position of the rectangular hanging plate bears the bending moment in the Y and Z directions, resulting in large stress. The maximum stress of the rectangular hanging plate connected with the iced conductor is 317.1 MPa, accounting for 1.01 of the yield stress. The maximum stress of the rectangular hanging plate connected with the ice-shedding conductor is 256.6 MPa, accounting for 0.81 of the yield stress. Therefore, the rectangular hanging plate connected with the iced conductor is easier to fracture than the rectangular hanging plate connected with the iced conductor. To avoid the fracture of the rectangular hanging plate, the middle position can be properly thickened according to the local meteorological conditions.Figure 34
Stress cloud picture of rectangular hanging plate: (a) connected with the iced conductor; (b) connected with the ice-shedding conductor.
(a)(b)Figure35 shows the stress cloud picture of the trapezoidal yoke plate. Points 1 and 2 are the maximum stress positions for connecting ice-shedding conductors, and points 3 and 4 are the maximum stress points for connecting iced conductors. Point 4 is the maximum stress position of the trapezoidal yoke plate, and the maximum stress is 269 MPa, accounting for 0.85 of the yield stress. The stress time-history curve of the trapezoidal yoke plate at each hanging point is shown in Figure 36. For the trapezoidal yoke plate, the stress on the side connected with the iced conductor is greater than that of the side connected with the iced conductor, and the stress of the lower hanging point is greater than that of the upper hanging point. On the whole, the trapezoidal yoke plate still has a certain degree of safety redundancy. If the ice thickness increases, the hanging point on the side of the iced conductor may yield.Figure 35
Stress cloud picture of trapezoidal yoke plate.Figure 36
Stress-time curve of each point.Figure37 shows the stress distribution of insulator strings. The maximum stress position of two insulator strings is at the corner of the bottom straight bar, the maximum stress of the insulator string connected with the ice-shedding conductor is 181.7 MPa, accounting for 0.77 of the yield stress. The maximum stress of the insulator string connected with the iced conductor is 195.5 MPa, accounting for 0.83 of the yield stress. Figure 38 shows the actual damage to the insulator string, the damage position is the same as the maximum stress position simulated in this paper. The reason is that this position is connected with the socket-clevis eye, which can rotate in any direction. Under the action of external load, the position is vulnerable to alternating stress in two directions, which leads to fatigue failure of the structure. Therefore, in the processing of the following structural design, the insulator string should be properly optimized to avoid failure.Figure 37
Stress cloud picture of insulator string: (a) connected with the ice-shedding conductor; (b) connected with the iced conductor.
(a)(b)Figure 38
Failure mode of ball-joint at the lower end of insulator string.The stress results of the triangular hanging plate are shown in Figure39 and 40. Points 1, 2, and 3 are the maximum stress positions of each hanging point, respectively. The maximum stress of two triangular hanging plates is at the lower right hanging point, and the stress of the lower hanging point is significantly greater than that of the upper hanging point. The maximum stress of the triangle hanging plate connected with the iced conductor is 251 MPa, accounting for 0.8 of the yield stress, and the average stress is 206 MPa. The maximum stress of the triangle hanging plate connected with the ice-shedding conductor is 243 MPa, accounting for 0.77 of the yield stress, and the average stress is 199 MPa. On the whole, the two triangle hanging plates are relatively safe. If they are in a more extreme environment, the lower hanging point of the triangle hanging plate connected with the iced conductor may be destroyed first.Figure 39
Triangle hanging plate connected with the iced conductor: (a) stress cloud picture; (b) stress-time curve of each hanging point.
(a)(b)Figure 40
Triangle hanging plate connected with the ice-shedding conductor: (a) stress cloud picture; (b) stress-time curve of each hanging point.
(a)(b)The stress cloud picture in Figure41 includes (4) parallel hanging plate 1, (8) parallel hanging plate 2, (5) PT adjusting plate 1, (6) PT adjusting plate 2, (7) traction plate, (9) ball-eye, (11) socket-clevis eye, (3) DB adjusting plate 1, and (17) DB adjusting plate 2. It is worth noticing that the ball-eye is prone to stress concentration at the corner of the straight rod. Under the action of external load, the high-frequency rotation of each electric power fittings will occur, resulting in repeated bending moment action at the cross section of the ball-eye, which may lead to fatigue fracture. Therefore, in the engineering design, the ball-eye should be properly optimized. The ratio of maximum stress to yield stress of other parts is relatively small, which has much more safety redundancy. Therefore, in engineering design, the material strength could be reasonably selected according to the actual situation.Figure 41
Stress cloud picture of some fittings of tension string.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 6.4. Analysis of Key Fittings for Suspension String

According to the analysis results of the whole model of the suspension string, the shackle is the maximum stress part, which indicates that it is the most dangerous part. Figures42 and 43 are the stress cloud picture and stress time-history curve of shackle, respectively. It can be seen that the maximum stress of the shackle also appears in the contact area, and the average stress of the upper shackle is significantly greater than that of the lower one, which is consistent with the analysis results of the shackle in the tension string model.Figure 42
Stress cloud picture of shackle.Figure 43
Stress-time curve of shackle.Figure44 shows the stress cloud picture of the suspension clamp, the stress of the parts is relatively small and mainly distributed near the hull and the hole. The stress concentration occurs at the abrupt change of the cross section of the hull and the hole. The maximum stress is 144.2 MPa, accounting for 0.74 of the yield stress, and the structure has a certain degree of safety redundancy.Figure 44
Stress cloud picture of suspension clamp.Figure45 shows the stress cloud picture of the suspension insulator string, the maximum stress is 170.4 MPa, accounting for 0.73 of yield stress. The maximum stress position of the part is at the joint with the ball-eye. Under the action of alternating load, the location may be damaged. Therefore, local enhancement can be considered for this position.Figure 45
Stress cloud picture of suspension insulator.The stress cloud picture in Figure46 includes (1) trunnion hanging plate, (3) ball-eye, (5) socket-clevis eye, (6) yoke plate, and (7) U-plate. The maximum stress of the ball-eye is at the corner of the straight bar, and the maximum stress is 158.3 MPa, accounting for 0.2 of the yield stress. The maximum stress of the yoke plate appears near the hole, and the maximum stress is 74.26 MPa, accounting for 0.24 of the yield stress. The position of maximum stress of the U-plate is near the groove, and the maximum stress is 102.8 MPa, accounting for 0.33 of the yield stress. The maximum stress of the trunnion hanging plate and the socket-clevis eye is at the hanging point of bolt connection, and the maximum stress is 66.4 MPa and 144.3 MPa, accounting for 0.21 and 0.46 of the yield stress, respectively. The ratio of the maximum stress to the yield stress of the five parts is less than 0.5, which has much higher safety redundancy during conductor ice shedding.Figure 46
Stress cloud picture of some fittings in suspension string.
(a)(b)(c)(d)(e)


## 7. Conclusion

The finite element model of 500 kV transmission lines is established, and the influence of icing thickness, span length, height difference ratio, and span number on conductor tension is studied under the most dangerous ice-shedding mode. Based on the numerically simulated results, the most dangerous tension of the conductor under different conditions is determined. The displacement and stress of tension string and suspension string are numerically simulated after ice shedding from conductors, and the vulnerable electric power fittings are discussed. The numerical simulation results show that(1)
Under the most dangerous ice-shedding mode, there is a linear correlation between ice thickness and conductor hanging point tension. The influence of span number on the tension of the conductor is significantly different, and the average tension of each subconductor in the five-span model is greater than that in the single-span and three-span models. The static tension, maximum tension, and average tension of the conductor increase with the increase of span length. When the ratio of height difference to span length is in the range of 0–0.2, it has little influence on the conductor tension. When the ratio of height difference to span length is in the range of 0.2–1.0, the conductor tension increases rapidly, and the influence of the height difference ratio on the tension of ice-shedding conductor is greater than that of the iced conductor. Therefore, it is suggested that the height difference ratio should be controlled within the range of 0–0.2 in line design, and the design of large height difference lines should be minimized in heavy ice areas.(2)
In the tension string and suspension string models, the shackle is the most vulnerable component, and the position where the shackle connects with other components is more prone to damage. Therefore, this area should be optimized in future structural design to increase its ability to resist heavy ice. In the harsh environment, strengthen the detection of shackle to ensure the safe operation of the whole line.(3)
In the engineering design, the weak areas of each component can be analyzed with the stress cloud diagram, the areas with higher stress can be strengthened, and the areas with lower stress can be optimized. For example, the area with less stress in the middle of the trapezoidal yoke plate and triangle hanging plate can be increased with weightless holes, which can reduce the self-weight of the structure and also improve the safety performance of the members and save the engineering cost.



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*Source: 3675271-2022-06-30.xml*