In recent years, there has been the problem of imbalance between power supply and demand (EPRI 2009; Li et al., 2021a; Li et al., 2021b; Li et al., 2022). Therefore, it is necessary to address ultra-long distance and cross-regional energy transmission. Of course, the safe operation of UHV transmission lines will be affected by the working environment. Due to the sudden change and complexity of the external environment, the normal operation of transmission lines faces serious challenges (Jafari et al., 2020). Aerodynamic instability of iced conductors with asymmetric cross-section area would cause iced conductors to gallop. This phenomenon is typical self-excited fluid–solid coupling vibration, which may cause breakage and short-circuit of transmission lines, or even tower toppling (Nigol et al., 1977; Cai et al., 2019a; Cai et al., 2019b). Research on the galloping phenomenon of UHV transmission lines is very limited, which cannot yet meet the actual engineering needs. In recent years, domestic scholars have begun to study the galloping mechanism of UHV transmission lines and its prevention and control technology, but much research work is still in the exploratory stage.

Numerical models for sub-span oscillation analysis of subconductors are usually dependent on quasi-stationary theory (QST). Since it is very difficult to design a corresponding small-scale model of transmission lines in a long-span length wind tunnel, numerical simulation is an effective method for studying wake-induced oscillation phenomena. Most researchers focused on the wake-induced oscillation responses of twin bundle conductors. Rawlins (1976) and Rawlins (1977) expressed the dynamic properties of twin conductors in the normal propagation mode using the transfer matrix method. Tsui and Tsui (1980) used the 2D and 3D finite element method (FEM) to study sub-span oscillation. Williams and Suaris (2006) presented an interference model to discuss the effects of space on aerodynamic response, and they highlighted that there were three dominant regions: the proximity interference region, induced sub-span oscillation region, and wake interference region. With the wide use of quad-bundle conductors, the sub-span oscillation of quad bundle conductors attracts the interest of researchers. Diana et al. (2014a) and Diana et al. (2014b) proposed a numerical approach to reproduce sub-span oscillation and investigated the quad spacer damper for controlling sub-span oscillations using aerodynamic coefficients obtained in wind tunnel tests. Although there are some studies on the oscillation of bundle conductors, the effects of complicated meteorological conditions of sub-span oscillation behaviors of bundle conductor transmission lines, especially eight-bundle conductors, require further investigation.

In recent years, lots of experts have studied the problem of galloping of transmission lines by numerical methods. Cai et al. (2015) studied the variation of the aerodynamic coefficient varying with the angle of wind attack by using the finite element method (FEM). Zhou et al. (2018) carried out a wind tunnel simulation to simulate the galloping of iced eight-bundle conductors and investigated different galloping behaviors of the parameters. Cai et al. (2019b) used the nonlinear FEM to analyze the galloping morphology of a sector-shaped eight-bundle conductor with different wind speeds, span lengths, and initial angle of wind attack. Talib et al. (2019) proposed a new dynamic model for the simulation of transmission line galloping. Liu et al. (2019), Liu et al. (2020a), Liu et al. (2020b), Liu et al. (2020c), Liu et al. (2021a), Liu et al. (2021b), Liu et al. (2021c), Liu et al. (2021d) and Liu et al. (2021e) and Min et al. (2021) obtained the aerodynamic coefficients of the conductor by applying the wind tunnel test and examined the stability and galloping characteristics of the iced conductor. The galloping behaviors of D-shape six bundles in the random wind test line are numerically simulated. Due to the continued existence of the galloping phenomenon and great harm, Oh and Sohn (2020) analyzed conductor galloping through the study of transmission line stability. Cai et al. (2020a) analyzed sector-shape eight-bundle conductor galloping through the FEM; furthermore, Cai et al. (2020b) analyzed galloping behaviors results of the test transmission tower-line.

Recently, there is still a lack of the influence of sub-span oscillation of iced eight-bundle conductors during galloping, especially the parameter analysis of UHV transmission lines under different turbulence intensity. In this article, the conductor galloping process of iced eight-bundle conductors is simulated by the numerical simulation method, the influence of wind speed and span length on sub-span oscillation during the conductor galloping process is analyzed, and the effects of wind speed, the angle of wind attack and the wind turbulence intensity are studied. Therefore, the follow-up research on galloping and anti-galloping characteristics of the frozen eight-bundle conductor has significant reference meaning.

Wind-driven wet snow may pack onto the windward sides of conductors, forming a hard, tenacious deposit with a sharp leading edge. The resulting ice shape may permit galloping. Combined with actual observation, the crescent shape can be generalized with respect to the great variety of natural heavy ice shapes (Hu et al., 2012; Yan et al., 2016). The aerodynamic forces of bundle conductors are the foundations of the analysis of the galloping of transmission lines (Liu et al., 2019). Here, the aerodynamic coefficients of crescent-shaped iced eight-bundle conductors are experimentally measured by wind tunnel tests.

Eight-bundle iced transmission lines are major research objects. The sub-spans of each span are the same. The conductor model is 8×LGJ-400/50, and the diameter of the sub-conductor is 30 mm. The model of the sub-spacer is FJZ-400, each with a mass of 17.5 kg. Its parameters are given in Table 1, and the model diagrams of 200- and 400-m lines are shown in Figures 1, 2.

The physical parameters of conductors and ice are listed in Table 2. The cross section of the iced conductor is simplified as a circular section when the galloping of the iced conductor is simulated by ABAQUS software. It is noted that the axial rigidity, torsional rigidity, mass per unit length, and moment of inertia of the equivalent cable and those of the original cable should be equal, which can be expressed as E ′ π d ′ 2 / 4 = E A ; G ′ π d ′ 2 / 32 = G I ρ ′ π d ′ 2 / 4 = μ ; ρ ′ π d ′ 4 / 32 = J , (1) where E′, G′, ρ′, and d’ are, respectively, the elastic modulus, shear modulus, density, and diameter of the equivalent cable, which can be obtained by solving (1) whose right hand sides are the corresponding quantities of the original iced conductor. The physical parameters of the conductors and ice are listed in Table 2.

The influence of the initial axial tension in the main cable and side cable on the element stiffness matrix cannot be ignored. In addition, the cable sags due to its own weight, resulting in a certain decrease or loss of its elastic modulus. In order to consider the influence of cable sag, the concept of equivalent elastic modulus is used to modify the elastic modulus of the cable. The equivalent modulus of elasticity generally adopts the E _Ernst formula: E Ernst = E 1 + ( q l ) 2 12 T 3 A E , (2) where E _Ernst is the equivalent elastic modulus of the material; E is the elastic modulus of the material; q is the weight of the unit length of the cable; l is the projection length of the cable element in the horizontal direction; A is the cross-sectional area of the cable; and T is cable tension.

The numerical simulations were carried out on a personal computer Dell Studio Desktop D540, and each process to arrive at a steady result took about 5 h. To speed up the efficiency, several simulations were submitted at the same time. The dynamic implicit analysis is used in the numerical method. The dynamic responses of the transmission line with different damping ratios in different directions are analyzed by ABAQUS with the user-defined cable element. The damping ratios ξ _z1 , ξ _y1 , and ξ _θ1 , in the horizontal, vertical, and torsional directions are set to be 0, 0.5, and 2%, respectively, determined and verified by the reference (Zhou et al., 2016). The time step is set to be 0.01; we have already calculated that the step is set to be 0.005, and the error of the vertical amplitude is 2.74%. Considering the efficiency of the numerical simulation, the time step is set to be 0.01.

In the research, the aerodynamic parameters of the iced UHV transmission line are analyzed using the results of wind tunnel experiments (Yan et al., 2016). The aerodynamic forces are applied to the conductor to numerically simulate the galloping of the conductor. In the stiffness and massless user-defined element, the torsion angle, velocity, and displacement under the condition of applying aerodynamic force could help obtain the node of the cable element used in ABAQUS software (Zhou et al., 2016). The arrangement of the spacers is shown in Table 3.

The Rayleigh attenuation model is general for cable attenuation, shown in Eq. 3: C = α G + β K . (3)

C, M, and K are damping matrices, quality matrices, and stiffness matrices, respectively. α and β is the Rayleigh damping coefficient, which are determined by the natural frequency and damping ratio. In order to improve the accuracy of the galloping characteristics, according to the previous literature (Hu et al., 2012, Cai et al., 2015) the calculation repetition time step is set to 0.01.

When the transmission conductor galloping occurs, the line is often accompanied by the vibration between the sub-spans in addition to the whole-span galloping. The natural frequency and mode affect the characteristics of conductor galloping (Zhang et al., 2000; Liu et al., 2021e). The natural frequency and whole-span mode are obtained, shown in Table 4. In the case of a transmission line with a distance of 200 m, the natural frequency of the in-plane single and half-wave 0.26 Hz is less than the natural frequency of the in-plane double half-wave 0.52 Hz, and the frequency of the in-plane single-half-wave is almost the natural frequency of the double half-wave; the natural frequency of the out-of-plane double–half-wave 0.52 Hz is less than the natural frequency of the out-of-plane four-half wave, and the natural frequency of the out-of-plane double half-wave is almost half of the natural frequency of the out-of-plane four-half wave; the natural frequency of the twist three-half wave is 0.83 Hz. It is nearly equal to the natural frequency of 0.80 Hz for the out-of-plane three-half-wave; for a line with a span of 400 m, the natural frequency of the in-plane single half-wave of 0.14 Hz is less than the natural frequency of the in-plane double half-wave 0.29 Hz, and the frequency of the in-plane single half-wave is nearly half of the natural frequency of the double half-wave. The natural frequency of the out-of-plane three-half-wave 0.43 is almost equal to the natural frequency of the torsional three-half-wave 0.48 Hz. The natural frequency of the in-plane double half-wave 0.52 Hz is the same as the natural frequency of the out-of-plane double half-wave 0.52 Hz. It is not difficult to find that in this case, there are 1:2 and 1:1 internal resonance conditions by analyzing the modal and natural frequencies of the 200- and 400-m spans.

It can be observed from the figure that a relatively dense natural frequency is concentrated in the frequency range of 1.043–1.138 Hz, with three directions of in-plane, out-of-plane and torsion, single half-wave, double half-wave, three-half-wave, four-half-wave wave, and several other modes [67]. For example: f = 1.043 Hz and f = 1.045 Hz are single half-waves, f = 1.048 Hz, f = 1.052 Hz, f = 1.138 Hz are double half-waves, f = 1.058 Hz and f = 1.097 Hz are three half-waves, and f = 1.081 Hz is four half-waves.

In the wind tunnel experiment, it can be found that different turbulence intensities will lead to different aerodynamic coefficients of UHV transmission lines, and the Den Hartog and Nigol coefficients determined by the aerodynamic coefficients are also different (Den Hartog 1932; Nigol et al., 1977). The effect of different parameters on the galloping of UHV transmission lines is discussed, and the law of influence of wind speed, angle of wind attack, and span length on conductor galloping under the action of the turbulent wind is obtained.

The sub-span oscillation of iced eight-bundle conductors during galloping under different wind velocities, span length, initial angle of wind attack, and turbulence intensity is simulated and analyzed by using the FEM. The following conclusions can be obtained as follows:

1) The subconductors of the iced eight-bundle transmission line gallop in the same direction under different wind velocities, and the main oscillation direction is the vertical direction. Meanwhile, there is an oscillation in the sub-span during the galloping of the iced eight-bundle conductor, which may cause the subconductor to vibrate.