1. Bibliographic Information

1.1. Title

扇形覆冰特高压八分裂导线舞动特性分析 (GALLOPING BEHAVIORS OF SECTOR-SHAPE ICED EIGHT BUNDLE CONDUCTORS)

1.2. Authors

CAI Mengqi*,2), XU Qian†, ZHOU Linshu**, LIU Xiaohui††,3), YAN Bo***

• (*School of Architecture & Civil Engineering, Chengdu University, Chengdu 610106, China)
• (†School of Architecture & Environment, Sichuan University, Chengdu 610044, China)
• (**Sichuan Electric Power Test & Research Inst, Chengdu 610016, China)
• (††School of Civil Engineering & Architecture, Chongqing Jiaotong University, Chongqing 400074, China)
• (***School of Aeronautics & Astronautics, Chongqing University, Chongqing 400044, China)

1.3. Journal/Conference

The paper is published in a Chinese journal, likely related to mechanics or engineering, given the doi:10.6052/1000-0879-18-061. The abstract also states the journal name in Chinese: 力学与实践, which translates to "Mechanics & Practice". This journal is a reputable publication in the field of mechanics in China, focusing on the application of mechanics principles in engineering.

1.4. Publication Year

The publication year can be inferred from the DOI or context as 2018 (from 18-061).

1.5. Abstract

This paper investigates the galloping characteristics of sector-shape iced eight-bundle conductors in heavy ice regions of UHV (Ultra-High Voltage) transmission lines, which are more prone to forming such ice shapes. The research combines wind tunnel experiments and numerical simulations. Aerodynamic parameters of individual iced sub-conductors, varying with wind attack angle, are obtained through wind tunnel tests on a segment model. A finite element model (FEM) of a single-span sector-shape iced eight-bundle transmission line is established in ABAQUS software. Aerodynamic parameters are input via the UEL (User Element) subroutine. Galloping trajectories and amplitudes are obtained from numerical simulation results. Finally, the paper discusses the influence of wind speed, span length, and initial wind attack angle on the galloping characteristics of the eight-bundle conductors. The results indicate that sector-shape iced eight-bundle conductors can effectively represent the galloping characteristics of eight-bundle conductors under heavy ice conditions.

1.6. Original Source Link

/files/papers/694a2a2c07f8689679b7d087/paper.pdf (Publication status: Officially published)

2. Executive Summary

2.1. Background & Motivation

The core problem addressed by this paper is the galloping phenomenon in UHV transmission lines, especially those in heavy ice regions. Galloping refers to low-frequency, large-amplitude self-excited vibrations of iced conductors under lateral wind loads, caused by asymmetric ice shapes. This phenomenon can lead to severe consequences such as power outages, transmission line damage, and even tower collapse.

The problem is particularly important in China, where an increasing number of $\pm 800 \mathrm { k V }$ DC and $1000 \mathrm { k V }$ AC UHV transmission lines have been built to balance energy supply and demand across the country. Recent large-scale galloping incidents, like those in Anhui and Hubei provinces in early 2018, highlight the urgent need for research into galloping mechanisms and prevention.

Prior research has extensively studied the aerodynamic characteristics of iced conductors, initially focusing on single conductors and later expanding to multi-bundle conductors (e.g., two-split, three-split, four-split, eight-split). Common ice shapes studied include crescent-shape and sector-shape. While crescent-shape iced eight-bundle conductors have been studied, there is a recognized gap in comprehensive aerodynamic parameter data and galloping analysis for sector-shape iced eight-bundle conductors, particularly under heavy ice conditions. Sector-shape ice is noted to be more representative of heavy ice conditions than crescent-shape ice.

The paper's innovative idea is to conduct a detailed study on sector-shape iced eight-bundle conductors, combining wind tunnel experiments to obtain crucial aerodynamic parameters and nonlinear finite element numerical simulations to analyze galloping characteristics under various environmental and structural conditions. This aims to fill the research gap and provide valuable insights for practical engineering.

2.2. Main Contributions / Findings

The paper makes several primary contributions:

• Comprehensive Aerodynamic Data: It provides a comprehensive set of aerodynamic parameters (lift, drag, and torque coefficients) for all eight sub-conductors of a sector-shape iced eight-bundle conductor through wind tunnel experiments, varying with wind attack angle. This data is crucial for accurate galloping simulations.

• Numerical Simulation Methodology: It establishes a finite element model for sector-shape iced eight-bundle transmission lines in ABAQUS using UEL to incorporate the complex, measured aerodynamic loads, enabling realistic galloping simulations.

• Influence of Key Parameters: It systematically investigates the effects of wind speed, span length, and initial wind attack angle on the galloping characteristics (trajectories and amplitudes) of sector-shape iced eight-bundle conductors.

  Key conclusions and findings reached by the paper include:

• Galloping Susceptibility: Wind tunnel tests indicate that sector-shape iced eight-bundle conductors have Nigol coefficients (related to torsional instability) that are negative over a significantly wider range of wind attack angles compared to Den Hartog coefficients (related to vertical instability). This suggests that torsional-vertical coupling is a critical factor in galloping for this ice shape and should be avoided in design. Specifically, the range of $90^\circ \sim 150^\circ$ for Den Hartog coefficients and $20^\circ$ (with other local regions) for Nigol coefficients are identified as high-risk zones.

• Wind Speed Effects: Increased wind speed leads to a more pronounced increase in horizontal galloping amplitude compared to vertical amplitude. The galloping trajectory evolves from an ellipse to a circle as wind speed increases, indicating stronger in-plane and out-of-plane mode coupling.

• Span Length Effects: Longer span lengths result in increased horizontal galloping amplitude. At larger spans (e.g., $500 \mathrm { m }$), the galloping trajectory can transform into an "8-shaped" pattern, further confirming in-plane and out-of-plane mode coupling.

• Initial Wind Attack Angle Effects: The initial wind attack angle significantly influences galloping amplitude. An initial wind attack angle of $140^\circ$ results in substantially larger vertical and horizontal galloping amplitudes compared to an angle of $20^\circ$.

  These findings contribute to a better understanding of the galloping mechanism for UHV eight-bundle conductors under heavy ice conditions and provide valuable guidance for anti-galloping design in practical engineering.

3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

To fully understand this paper, a beginner should be familiar with the following foundational concepts:

• Galloping: A low-frequency, large-amplitude self-excited vibration of overhead transmission lines. It occurs when ice accumulation on the conductor creates an asymmetric aerodynamic shape, which, under certain wind conditions, generates aerodynamic forces that drive and sustain the vibration. It's distinct from aeolian vibration (high-frequency, low-amplitude, wind-induced) and sub-span oscillation (medium-frequency, medium-amplitude, wind-induced for bundled conductors).
• Iced Conductors: Transmission line conductors that have accumulated ice or sleet. The shape, thickness, and distribution of this ice are crucial because they alter the conductor's aerodynamic profile, making it asymmetric and susceptible to galloping.
• UHV (Ultra-High Voltage) Transmission Lines: Power transmission lines operating at very high voltages, typically $1000 \mathrm { k V }$ AC or $\pm 800 \mathrm { k V }$ DC, for long-distance, large-capacity power transmission. These lines often use bundled conductors (multiple individual conductors grouped together) to reduce corona losses and electromagnetic interference, and to increase power transmission capacity.
• Eight-Bundle Conductors: A specific configuration of bundled conductors where eight individual sub-conductors are arranged in a circular or square pattern and held together by spacers. This configuration is common in UHV transmission lines.
• Aerodynamic Coefficients (Lift, Drag, Torque): Dimensionless quantities used to quantify the aerodynamic forces and moments acting on an object in a fluid flow.
  • Lift Coefficient ($C_L$): Relates the lift force (perpendicular to the direction of flow) to the fluid density, flow velocity, and reference area. In galloping, negative lift slope ($\partial C_L / \partial \alpha < 0$) is often associated with instability.
  • Drag Coefficient ($C_D$): Relates the drag force (parallel to the direction of flow) to the fluid density, flow velocity, and reference area. Drag generally dissipates energy and tends to inhibit galloping.
  • Torque Coefficient ($C_M$): Relates the aerodynamic torque or moment (tendency to cause rotation) to the fluid density, flow velocity, reference area, and reference length. Negative torque slope ($\partial C_M / \partial \alpha < 0$) can lead to torsional instability and torsional galloping.
• Wind Tunnel Test: An experimental facility used to study the effects of air moving past solid objects. Objects (models) are placed in a controlled stream of air, and forces, pressures, and flow patterns are measured. In this paper, segment models (short sections of the conductor) are used to measure aerodynamic coefficients.
• Finite Element Method (FEM): A numerical technique for finding approximate solutions to boundary value problems for partial differential equations. It subdivides a large problem into smaller, simpler parts called finite elements. In structural mechanics, it's widely used to analyze the behavior of complex structures under various loads.
• ABAQUS: A suite of finite element analysis (FEA) software used for modeling, visualization, and implicit and explicit dynamics applications. It's commonly used in engineering for simulating physical phenomena.
• UEL (User Element) Subroutine: A feature in ABAQUS that allows users to define custom finite elements or load applications by writing their own code (e.g., in Fortran). This is crucial for applying complex, empirically derived aerodynamic loads that vary with wind attack angle and conductor motion, as done in this paper.
• Den Hartog Criterion: A classical galloping stability criterion proposed by J.P. Den Hartog in 1932. It states that vertical galloping is likely to occur if the lift force decreases with increasing wind attack angle, specifically when $\partial C_L / \partial \alpha + C_D < 0$. This condition indicates that the aerodynamic forces are destabilizing and can sustain vertical oscillations.
• Nigol's Criterion: An extension to the Den Hartog criterion, focusing on the role of torsional motion in galloping. Proposed by O. Nigol in 1981, it suggests that torsional galloping (which can couple with vertical motion) can be induced when the torsional aerodynamic moment decreases with increasing wind attack angle, i.e., $\partial C_M / \partial \alpha < 0$, especially when the torsional frequency is close to the vertical vibration frequency. This highlights the importance of torsional stiffness and aerodynamic torque in galloping phenomena.

3.2. Previous Works

The paper extensively reviews prior research on iced conductor galloping, primarily focusing on aerodynamic characteristics and numerical simulation methods.

• Early Research on Single Conductors:

  • Nigol et al. [4] conducted wind tunnel tests on iced single conductors with various ice thicknesses and shapes (e.g., $1.8 \mathrm { m m } { \sim } 7 . 0 \mathrm { m m }$ ice thickness) to measure aerodynamic coefficients. This established foundational understanding of how ice geometry affects aerodynamic behavior.
  • Alonso et al. [5] used segment models of triangular and rhomboidal cross-sections to measure aerodynamic coefficients in a wind tunnel and studied galloping mechanisms using a two-degree-of-freedom model.
  • Gu Ming et al. [6, 7] studied quasi-elliptical iced single conductors using wind tunnel tests, analyzing aerodynamic parameters and the influence of turbulence and ice thickness.

• Transition to Multi-Bundle Conductors:

  • Li Wanping et al. [8-10] pioneered wind tunnel measurements of static aerodynamic parameters for crescent-shape and sector-shape iced three-bundle conductors. This marked a shift towards more complex bundled conductor configurations.
  • Zhang Hongyan et al. [11] investigated aerodynamic parameters of crescent-shape and sector-shape iced four-bundle conductors, including individual sub-conductor characteristics and effects of wind speed and ice thickness.
  • Wang Xin et al. [12] further explored the impact of turbulence on lift, drag, and torque coefficients for crescent-shape and D-shape iced conductors.
  • Xiao Zhengzhi et al. [13] studied ice accretion processes and aerodynamic parameters for crescent-shape and sector-shape iced eight-bundle conductors in a wind tunnel, although their reported aerodynamic parameters were noted as incomplete.
  • Zhou et al. [14] and Zhou Linshu et al. [15] specifically researched aerodynamic parameters and galloping characteristics of crescent-shape iced eight-bundle conductors under stable wind conditions.

• Numerical Simulation Methods for Galloping:

  • Yan Bo et al. [16, 17] developed numerical methods for iced twin-bundle conductors and three-bundle conductors, using penalty functions to constrain sub-conductor movements at spacer locations and apply different aerodynamic loads. They also considered wake effects.
  • Hu et al. [18] developed a method in ABAQUS using UEL to define aerodynamic load application for iced quad-bundle conductors and studied galloping characteristics under various wind speeds and line parameters.
  • Yan Bo et al. [19] investigated nonlinear galloping of iced single conductors, identifying internal resonance phenomena under specific conditions.
  • Liu Xiaohui et al. [20] studied the influence of span length and number of spans on galloping characteristics in multi-span transmission lines.
  • Zhou et al. [14] and Zhou Linshu et al. [15] also applied numerical methods to crescent-shape iced eight-bundle conductors to study galloping and the effects of line parameters.

3.3. Technological Evolution

The research on iced conductor galloping has evolved significantly over time:

1. Early Focus on Single Conductors (1930s-1980s): Initial studies, spearheaded by Den Hartog, focused on understanding the fundamental aerodynamic instability of single conductors with simplified ice shapes. Wind tunnel experiments were crucial for measuring basic aerodynamic coefficients. The Den Hartog criterion emerged from this era.

2. Introduction of Torsional Mechanisms (1980s): Nigol's work highlighted the critical role of torsional motion and its coupling with vertical motion in galloping, leading to the Nigol criterion. This recognized that iced conductors are not merely rigid bodies but can twist.

3. Expansion to Multi-Bundle Conductors (1990s-2000s): With the increasing use of high-voltage and UHV transmission lines, multi-bundle conductors became prevalent. Research shifted to studying two-, three-, and four-bundle conductors, considering wake effects between sub-conductors and the more complex aerodynamic interactions.

4. Advanced Numerical Simulations (2000s-Present): The rise of powerful finite element analysis (FEA) software like ABAQUS and the development of user-defined subroutines (UEL) enabled more sophisticated numerical modeling of galloping. This allowed for the simulation of nonlinear, large-amplitude vibrations of flexible, multi-span conductor systems under complex aerodynamic loads derived from wind tunnel tests.

5. Focus on UHV and Specific Ice Shapes (Present): Current research, including this paper, addresses the challenges of UHV lines and increasingly complex bundled conductor configurations (like eight-bundle). There's a growing need to investigate more realistic heavy ice shapes (e.g., sector-shape), moving beyond simpler crescent-shapes, and to understand the combined effects of aerodynamic parameters, structural parameters, and environmental conditions.

   This paper's work fits into the latest stage of this evolution by focusing on sector-shape iced eight-bundle conductors for UHV lines, combining state-of-the-art wind tunnel testing with advanced nonlinear FEM to provide a comprehensive analysis of galloping characteristics under various influential factors.

3.4. Differentiation Analysis

Compared to the main methods in related work, this paper's approach has several core differences and innovations:

• Specific Ice Shape and Conductor Type: While previous studies have looked at iced multi-bundle conductors (e.g., three-split, four-split, crescent-shape eight-split), this paper specifically focuses on sector-shape iced eight-bundle conductors. The authors emphasize that sector-shape ice is more representative of heavy ice conditions in high-altitude or severe icing areas compared to the crescent-shape often studied. This addresses a critical gap for UHV lines in heavy ice regions.
• Comprehensive Aerodynamic Data for Eight-Bundle: The paper claims that existing literature on sector-shape iced eight-bundle conductors is scarce and often incomplete (referencing Xiao Zhengzhi et al. [13] for incomplete parameters). This study aims to provide a complete and detailed set of aerodynamic parameters for all eight sub-conductors, including lift, drag, and torque coefficients across a full range of wind attack angles ($0^\circ \sim 180^\circ$). The wake effects between sub-conductors are implicitly captured by measuring each sub-conductor within the bundle configuration.
• Integrated Experimental and Numerical Approach: The methodology rigorously combines wind tunnel experimental data with nonlinear finite element simulations. The wind tunnel tests are used to directly measure the aerodynamic coefficients specific to the sector-shape eight-bundle configuration. These empirically derived coefficients are then directly fed into a nonlinear finite element model in ABAQUS via a UEL subroutine. This integrated approach provides a more realistic and accurate simulation of galloping compared to purely theoretical or simplified aerodynamic models.
• Detailed Parameter Influence Analysis: Beyond just simulating galloping, the paper systematically investigates the influence of critical parameters such as wind speed, span length, and initial wind attack angle on galloping trajectories and amplitudes. This provides practical insights for transmission line design and anti-galloping mitigation strategies.
• Emphasis on Torsional Effects: The analysis of Den Hartog and Nigol coefficients specifically highlights that for sector-shape iced eight-bundle conductors, Nigol coefficients (related to torsional instability) are negative over a significantly wider range of wind attack angles than Den Hartog coefficients. This finding underscores the importance of considering torsional-vertical coupling in anti-galloping design for this specific ice shape, differentiating it from previous studies that might have focused more on vertical instability.

4. Methodology

4.1. Principles

The core idea of the method used in this paper is a hybrid approach that combines physical wind tunnel experiments with numerical simulations to comprehensively study the galloping characteristics of sector-shape iced eight-bundle conductors.

The theoretical basis and intuition behind this approach are as follows:

1. Complexity of Aerodynamics: Galloping is initiated and sustained by complex aerodynamic forces acting on an iced conductor. These forces (lift, drag, and torque) are highly dependent on the ice shape, wind attack angle, and interactions between sub-conductors in a bundle. It is extremely difficult to accurately model these aerodynamic coefficients purely theoretically due to the irregular and asymmetric nature of ice. Therefore, wind tunnel experiments are indispensable for obtaining realistic aerodynamic parameters.

2. Structural Dynamics Simulation: Once the aerodynamic forces are known, they act upon the flexible transmission line structure. Transmission lines are large-span, flexible cable structures that undergo nonlinear, large-amplitude vibrations during galloping. Simulating this dynamic behavior accurately requires a robust structural analysis tool capable of handling geometric nonlinearity and time-dependent loads. Nonlinear finite element analysis (FEA) is well-suited for this task.

3. Integration of Data and Simulation: The key challenge is to seamlessly integrate the experimentally derived aerodynamic parameters into the structural dynamics simulation. This is achieved by defining aerodynamic loads as functions of the instantaneous wind attack angle (which changes as the conductor moves and twists) and applying these loads dynamically within the FEA software.

   In essence, the wind tunnel experiment provides the aerodynamic "fingerprint" of the iced conductor, while the numerical simulation uses this fingerprint to predict the dynamic response (galloping) of the entire transmission line under various conditions.

4.2. Core Methodology In-depth (Layer by Layer)

The methodology is divided into two main parts: wind tunnel experiments to obtain aerodynamic parameters and numerical simulations using these parameters to analyze galloping.

4.2.1. Sector-Shape Iced Eight-Bundle Conductor Wind Tunnel Experiment

This phase focuses on creating realistic iced conductor models and measuring their aerodynamic characteristics in a controlled wind tunnel environment.

4.2.1.1. Iced Model Fabrication

• Ice Shape Selection: The paper selects a sector-shape ice profile, as observed in actual heavy ice zones (Figure 1a provides a visual reference, showing a lighter and a sector-shaped ice model). This specific shape is chosen because it closely resembles ice accumulation under heavy ice conditions.

• Ice Dimensions: The ice thickness is set to $18 \mathrm { m m }$, the outer arc angle to $120^\circ$, and the inner arc angle to $140^\circ$.

• Model Material and Construction: The ice is simulated using lightweight wood carvings. The conductor model itself is made from a hollow aluminum tube with a diameter of $30 \mathrm { m m }$ and a wall thickness of $2 \mathrm { m m }$. The outer surface of the aluminum tube is uniformly wrapped with fine rubber wire to mimic the actual stranded structure of steel-cored aluminum stranded conductors (LGJ-500/35).

• Scaling: The conductor model has a 1:1 diameter ratio to the prototype conductor, ensuring accurate representation of flow around the conductor's main cross-section.

• Sub-conductor Consistency: All eight sub-conductors in the bundle are covered with identical sector-shape ice models of the same size and orientation, reflecting the assumption that in real heavy icing conditions, ice shapes on individual sub-conductors are similar.

• Model Length: Each iced sub-conductor segment model has a length of $700 \mathrm { m m }$, a length chosen to fit the dimensions of the wind tunnel.

  The following figure (Figure 1 from the original paper) shows the sector-shape icing experiment model:

  该图像是扇形覆冰试验中的两部分，(a)显示了扇形覆冰的特写，(b)展示了用于实验的导线模型，为研究八分裂导线的舞动特性提供了具体的实验素材。

4.2.1.2. Experimental Setup and Procedure

• Wind Tunnel Facility: The experiment is conducted in the 1.4m x 1.4m low-speed wind tunnel at the China Aerodynamics Research and Development Center.

• Bundle Configuration: Eight identical iced conductor segment models, each $0.7 \mathrm { m }$ long, are fixed in a bundle arrangement. The spacing between adjacent sub-conductors is $0.4 \mathrm { m }$, which matches the typical spacing in actual high-voltage transmission lines.

• Mounting and Measurement: The eight sub-conductors are fixed at both ends to circular end plates (Figure 2a). Force sensors (specifically, strain gauge balances) are installed inside the conductors to measure the lift force ($F_L$), drag force ($F_D$), and torque ($M$) acting on each sub-conductor as air flows past.

• Rotation and Wind Speed: To obtain aerodynamic parameters across different wind attack angles, the iced eight-bundle conductor model is rotated in the wind tunnel. The rotation range is $0^\circ \sim 180^\circ$ with 5-degree intervals for data acquisition. The wind speed ($U$) for the tests is set to $18 \mathrm { m / s }$.

  The following figure (Figure 2 from the original paper) gives the aerodynamic characteristic measurement scheme of the iced eight-split conductor:

  该图像是插图，展示了覆冰八分裂导线的气动特性测量试验方案。图中(a)部分显示了导线的布局，包括上端板、下端板和子导线；(b)部分展示了扇形覆冰导线的截面，标有气动力和攻角等重要参数。此图为理解试验方案和气动特性提供了清晰的视觉辅助。

4.2.1.3. Aerodynamic Coefficient Calculation

The measured aerodynamic loads ($F_L$, $F_D$, $M$) are converted into dimensionless aerodynamic coefficients ($C_L$, $C_D$, $C_M$) using the following formulas:

$ C _ { \mathrm { L } } = \frac { 2 F _ { \mathrm { L } } } { \rho U ^ { 2 } L d } $ $ C _ { \mathrm { D } } = \frac { 2 F _ { \mathrm { D } } } { \rho U ^ { 2 } L d } $ $ C _ { \mathrm { M } } = \frac { 2 M } { \rho U ^ { 2 } L d ^ { 2 } } $

Where:

• $C_L$: Lift coefficient.
• $C_D$: Drag coefficient.
• $C_M$: Torque coefficient.
• $F_L$: Lift force measured by the sensor (in Newtons).
• $F_D$: Drag force measured by the sensor (in Newtons).
• $M$: Aerodynamic torque or moment measured by the sensor (in Newton-meters).
• $\rho$: Air density during the experiment (in $\mathrm{kg/m^3}$).
• $U$: Air flow velocity in the wind tunnel (i.e., wind speed) (in $\mathrm{m/s}$).
• $L$: Length of the sector-shape iced sub-conductor segment model (here, $0.7 \mathrm { m }$).
• $d$: Diameter of the bare conductor model (in meters).
  • Note: The diameter $d$ in the torque coefficient formula is squared, indicating that torque is typically normalized by a reference area times a reference length (which, for a circular cross-section, often involves $d^2$).

4.2.1.4. Aerodynamic Parameter Analysis (Results from Section 2.1)

The wind tunnel tests yield the aerodynamic coefficient curves for all eight sub-conductors as functions of wind attack angle.

• Lift Coefficient ($C_L$): Generally exhibits a W-shape trend. Maximum $C_L$ near $90^\circ$, minimum (negative) $C_L$ near $30^\circ$ and $150^\circ$. Negative lift slope regions ($\partial C_L / \partial \alpha < 0$) are observed in $0^\circ \sim 30^\circ$ and $90^\circ \sim 150^\circ$, indicating potential for galloping according to Den Hartog theory.

• Drag Coefficient ($C_D$): Shows significant variation with wind attack angle and notable differences between sub-conductors due to wake effects. A sharp decrease in $C_D$ for sub-conductors 3 and 8 is observed around $130^\circ$ due to the wake from leading conductors. Low drag regions might facilitate galloping or wake-induced oscillation.

• Torque Coefficient ($C_M$): Less affected by wake effects overall. Generally positive in the middle range of wind attack angles and negative at the ends. Negative torque slope regions ($\partial C_M / \partial \alpha < 0$) are primarily found in $0^\circ \sim 50^\circ$ and $130^\circ \sim 180^\circ$, with local negative slopes due to wake effects. These regions suggest potential for torsional galloping if torsional and vertical frequencies are coupled, as per Nigol's theory.

  The following figure (Figure 3(a) and 3(b) from the original paper) shows the lift coefficient and drag coefficient of each sub-conductor of the sector-shape iced conductor as a function of wind attack angle:

  该图像是图表，展示了扇形覆冰八分裂导线在不同风攻角 $\alpha$ 下的升力系数 $C_L$ 和阻力系数 $C_D$ 的变化。图中的两部分分别标注为“(a) 升力系数”和“(b) 阻力系数”，横坐标为风攻角 $\alpha$（单位为度），纵坐标分别为$c_L$和$c_D$。不同的曲线对应不同的导线编号，显示了气动参数的变化特征。

The following figure (Figure 3(c) from the original paper) shows the torque coefficient of each sub-conductor of the sector-shape iced conductor as a function of wind attack angle:

该图像是图表，展示了扇形覆冰各子导线气动参数随风向角变化的曲线。横轴表示风向角 $\alpha$ (°)，纵轴为气动系数 $C_M$。不同符号和颜色代表不同的子导线，在风向角变化范围内，气动系数呈现出相应的变化趋势，反映了各子导线在不同风向条件下的气动特性。图中标注了多个子导线的编号。

4.2.1.5. Galloping Stability Analysis (Results from Section 2.2)

Based on the measured aerodynamic coefficients, the Den Hartog criterion and Nigol's criterion are applied to assess galloping susceptibility.

• Den Hartog Criterion: Vertical galloping is possible if the Den Hartog coefficient ($\partial C _ { \mathrm { L } } / \partial \alpha + C _ { \mathrm { D } }$) is less than zero. $ \partial C _ { \mathrm { L } } / \partial \alpha + C _ { \mathrm { D } } < 0 $ Where:

  • $\partial C_L / \partial \alpha$: Slope of the lift coefficient with respect to the wind attack angle $\alpha$.
  • $C_D$: Drag coefficient.
  • This condition indicates that as the conductor moves vertically, the change in lift force and drag force combine to provide a net force that amplifies the motion.

• Nigol's Criterion: Torsional galloping (potentially coupled with vertical motion) is possible if the Nigol coefficient ($\partial C _ { \mathrm { M } } / \partial \alpha$) is less than zero, especially if torsional frequency matches vertical vibration frequency. $ \partial C _ { \mathrm { M } } / \partial \alpha < 0 $ Where:

  • $\partial C_M / \partial \alpha$: Slope of the torque coefficient with respect to the wind attack angle $\alpha$.

  • This condition indicates that as the conductor twists, the change in aerodynamic torque provides a net moment that amplifies the torsional motion, which can then couple into vertical movement.

    Analysis of the calculated coefficients shows:

• Den Hartog Coefficients: Mostly positive, but consistently negative in the $90^\circ \sim 150^\circ$ range for all eight sub-conductors, indicating this as a high-risk range for vertical galloping.

• Nigol Coefficients: Significantly more negative regions than Den Hartog coefficients, especially around $20^\circ$ (where it reaches values close to -5) and other areas like $90^\circ$ and $140^\circ$. This suggests that torsional-vertical coupling is a dominant factor for galloping in sector-shape iced eight-bundle conductors.

  The following figure (Figure 4 from the original paper) shows the Den Hartog coefficient and Nigol coefficient curves for the iced eight-split conductor as a function of wind attack angle:

  该图像是图表，展示了不同导线在风攻角变化下的邓哈托系数和尼格尔系数。图中包含六个子图，分别对应于六条导线的气动参数变化，清晰地呈现了各导线气动系数的波动趋势。

  该图像是图表，展示了覆冰八分裂导线上子导线 7 和 8 的邓哈托系数与尼格尔系数随风攻角的变化曲线。X轴为风攻角 heta / (°)，Y轴为对应的系数值，数据体现了不同风攻角对各导线气动特性的影响。

4.2.2. Sector-Shape Iced Eight-Bundle Conductor Galloping Simulation

This phase uses the aerodynamic parameters from the wind tunnel tests to simulate the dynamic response of the transmission line under wind loads.

4.2.2.1. Finite Element Model (FEM) Setup

• Software: ABAQUS finite element software is used for the numerical simulation.
• Conductor Modeling: The transmission line is modeled as a large-span flexible structure. The bending stiffness of the conductor is neglected, which is a common and reasonable assumption for flexible cable structures undergoing large displacements.
• Element Type: Cable elements are used. Critically, these elements are modified to have torsional degrees of freedom, which is essential for simulating galloping that involves twisting motions (especially given the Nigol criterion findings). This is achieved by releasing the bending degrees of freedom at the spatial beam element nodes and retaining the torsional degrees of freedom. The material properties are set as incompressible.
• Line Parameters: A single-span transmission line is simulated with a span length of $400 \mathrm { m }$. The initial tension is $31.25 \mathrm { k N }$, and the sag is $8.75 \mathrm { m }$.

4.2.2.2. Aerodynamic Load Application via UEL

• Dynamic Load Calculation: The aerodynamic loads (drag force $F_D$, lift force $F_L$, and torque $M$) acting on the iced conductor are dynamically calculated based on the instantaneous wind speed ($U$) and the current wind attack angle ($\alpha$) experienced by the moving conductor. The wind attack angle changes as the conductor vibrates and twists.

• Formulas for Aerodynamic Loads: The magnitude of these loads is determined by the following formulas, which are directly derived from the aerodynamic coefficients obtained from the wind tunnel tests:

  $ \left. \begin{array} { r } { F _ { \mathrm { D } } = \frac { 1 } { 2 } \rho U ^ { 2 } L d C _ { \mathrm { D } } } \ { F _ { \mathrm { L } } = \frac { 1 } { 2 } \rho U ^ { 2 } L d C _ { \mathrm { L } } } \ { M = \frac { 1 } { 2 } \rho U ^ { 2 } L d ^ { 2 } C _ { \mathrm { M } } } \end{array} \right} $

Where:

• $F_D$: Drag force (in Newtons).

• $F_L$: Lift force (in Newtons).

• $M$: Aerodynamic torque (in Newton-meters).

• $\rho$: Air density (in $\mathrm{kg/m^3}$). This is the actual air density in the natural environment for the simulation.

• $U$: Wind velocity (in $\mathrm{m/s}$). This is the incoming wind speed for the simulation.

• $L$: Length of the conductor element for which the load is calculated (in meters).

• $d$: Diameter of the bare conductor (in meters).

• $C_D$: Drag coefficient, obtained from wind tunnel data as a function of the instantaneous wind attack angle.

• $C_L$: Lift coefficient, obtained from wind tunnel data as a function of the instantaneous wind attack angle.

• $C_M$: Torque coefficient, obtained from wind tunnel data as a function of the instantaneous wind attack angle.

• UEL Implementation: The ABAQUS User Element (UEL) subroutine is utilized to implement the application of these aerodynamic loads. The UEL allows the user to define the element's behavior, including its stiffness, mass, and load contribution, which in this case involves calculating the aerodynamic forces and moments based on the element's current position, velocity, and orientation (determining the wind attack angle). As the conductor moves, the wind attack angle changes, and the UEL dynamically updates the applied aerodynamic loads based on the input $C_L$, $C_D$, $C_M$ curves. This allows the simulation to capture the self-excited nature of galloping.

4.2.2.3. Galloping Trajectory and Amplitude Analysis

The numerical simulation provides the dynamic response of the iced conductor over time. From these results, galloping characteristics such as vibration trajectories (e.g., at the mid-span of sub-conductors) and maximum vertical and horizontal amplitudes are extracted and analyzed. The paper specifically investigates the influence of:

• Wind speed (e.g., $6 \mathrm { m/s }$, $8 \mathrm { m/s }$, $10 \mathrm { m/s }$).
• Span length (e.g., $300 \mathrm { m }$, $400 \mathrm { m }$, $500 \mathrm { m }$).
• Initial wind attack angle (e.g., $20^\circ$, $140^\circ$).

5. Experimental Setup

5.1. Datasets

The paper does not use pre-existing public datasets in the conventional sense. Instead, the "dataset" for the numerical simulation is generated directly from the wind tunnel experiments.

• Source: The aerodynamic parameters (lift coefficient $C_L$, drag coefficient $C_D$, and torque coefficient $C_M$) for the sector-shape iced eight-bundle conductor are obtained from wind tunnel tests conducted at the China Aerodynamics Research and Development Center low-speed wind tunnel.
• Characteristics: These parameters are measured for each of the eight sub-conductors as a function of wind attack angle (ranging from $0^\circ$ to $180^\circ$ in $5^\circ$ increments). The ice thickness is $18 \mathrm { m m }$, and the wind speed during the wind tunnel test is $18 \mathrm { m/s }$. These curves, shown in Figure 3, represent the unique aerodynamic "fingerprint" of the iced conductor configuration under specific ice conditions.
• Domain: The data is specific to sector-shape ice on LGJ-500/35 type conductors arranged in an eight-bundle configuration.
• Purpose: These experimentally derived curves are crucial input for the ABAQUS finite element model, allowing the numerical simulation to accurately apply aerodynamic loads that vary dynamically with the conductor's movement and orientation during galloping. This approach ensures that the simulated galloping characteristics are based on realistic aerodynamic behavior.

5.2. Evaluation Metrics

The paper evaluates the galloping characteristics primarily by analyzing the vibration trajectories and amplitudes of the iced conductors. While not explicitly defined by mathematical formulas in the paper's main text as standard evaluation metrics, their conceptual definitions are clear:

1. Vibration Trajectory:

   • Conceptual Definition: This refers to the path traced by a specific point on the vibrating conductor (typically the mid-span point of a sub-conductor) in a plane perpendicular to the conductor's axis over time. It visually represents the combined vertical and horizontal motion during galloping. The shape of the trajectory (e.g., elliptical, circular, 8-shaped) provides insight into the coupling between different vibration modes.
   • Mathematical Formula: Not a single formula, but typically represented by a parametric plot of horizontal displacement (x(t)) versus vertical displacement (y(t)) over time $t$.
   • Symbol Explanation: x(t) represents the horizontal displacement of the conductor point at time $t$, and y(t) represents the vertical displacement at time $t$.

2. Vertical Amplitude ($A_V$):

   • Conceptual Definition: This quantifies the maximum vertical displacement (up or down) of the conductor during galloping. It represents the extent of the vertical oscillation. Large vertical amplitudes are critical for assessing potential flashover risks or clearances to ground/structures.
   • Mathematical Formula: A_V = \max(|y(t)|), where y(t) is the vertical displacement from the equilibrium position.
   • Symbol Explanation: $A_V$ is the maximum vertical amplitude, y(t) is the vertical displacement at time $t$.

3. Horizontal Amplitude ($A_H$):

   • Conceptual Definition: This quantifies the maximum horizontal displacement (side-to-side) of the conductor during galloping. It represents the extent of the horizontal oscillation. Large horizontal amplitudes can lead to clashing between adjacent conductors or phases.

   • Mathematical Formula: A_H = \max(|x(t)|), where x(t) is the horizontal displacement from the equilibrium position.

   • Symbol Explanation: $A_H$ is the maximum horizontal amplitude, x(t) is the horizontal displacement at time $t$.

     These metrics are effective for validating the method's performance by directly quantifying the severity and nature of the galloping phenomenon under different conditions.

5.3. Baselines

The paper does not compare its proposed method against other galloping simulation models as baselines in the traditional sense. Instead, it uses its established wind tunnel data-driven FEM to perform a parametric study. The "baselines" for comparison are different operating conditions and structural parameters of the same sector-shape iced eight-bundle conductor system. These include:

• Different Wind Speeds: Comparing galloping characteristics at $6 \mathrm { m/s }$, $8 \mathrm { m/s }$, and $10 \mathrm { m/s }$.

• Different Span Lengths: Comparing galloping characteristics at $300 \mathrm { m }$, $400 \mathrm { m }$, and $500 \mathrm { m }$ spans.

• Different Initial Wind Attack Angles: Comparing galloping characteristics at $20^\circ$ and $140^\circ$.

  These internal comparisons allow the authors to analyze the influence of these critical factors on galloping behavior, which is the primary objective of the numerical simulation part of the study. The paper's earlier discussion of crescent-shape iced eight-bundle conductors (Zhou et al. [14]) serves as a qualitative comparison for aerodynamic characteristics, highlighting the differences in galloping susceptibility between the two ice shapes, but not as a direct quantitative baseline for the numerical simulations presented.

6. Results & Analysis

6.1. Core Results Analysis

The paper's core results stem from the numerical simulations that investigate the influence of wind speed, span length, and initial wind attack angle on the galloping characteristics of sector-shape iced eight-bundle conductors.

6.1.1. Effect of Wind Speed

The simulations compared galloping trajectories at wind speeds of $6 \mathrm { m/s }$, $8 \mathrm { m/s }$, and $10 \mathrm { m/s }$, with a fixed span length of $400 \mathrm { m }$ and initial wind attack angle of $140^\circ$.

• Amplitude: Table 1 shows that as wind speed increases from $6 \mathrm { m/s }$ to $10 \mathrm { m/s }$ (for $140^\circ$ initial attack angle), the vertical amplitude increases from $2.75 \mathrm { m }$ to $7.44 \mathrm { m }$, and the horizontal amplitude increases from $0.98 \mathrm { m }$ to $5.89 \mathrm { m }$. This clearly indicates that higher wind speeds lead to significantly larger galloping amplitudes.

• Trajectory Shape: As seen in Figure 5, at $6 \mathrm { m/s }$, the trajectory is generally elliptical. As wind speed increases to $8 \mathrm { m/s }$ and $10 \mathrm { m/s }$, the trajectory gradually transitions towards a circular shape. This shift indicates an increasing coupling between the in-plane (vertical) and out-of-plane (horizontal) vibration modes of the conductor. The circular trajectory suggests that the horizontal motion becomes as significant as the vertical motion at higher wind speeds. This observation is consistent with findings in other galloping studies [20].

  The following figure (Figure 5 from the original paper) shows the vibration trajectories of the midpoint of the sector-shaped ice-covered eight-stranded conductor at a wind speed of $6 \mathrm { m/s }$:

  该图像是图5，展示了风速为 $6 ext{ m/s}$ 时，扇形覆冰八分裂子导线中点的振动轨迹。X轴表示水平振幅，Y轴表示垂直振幅，各轨迹标记了不同的子导线。

The following figure (Figure 5 (continued) from the original paper) shows the vibration trajectories of the midpoints of the sector ice-covered eight-strand conductor at different wind speeds (8 m/s and 10 m/s):

该图像是图表，展示了在不同风速（8 m/s 和 10 m/s）下，扇形覆冰八分裂子导线中点的振动轨迹。横轴表示水平振幅（m），纵轴表示垂直振幅（m），通过不同颜色的线条描绘出各子导线的振动特征。

6.1.2. Effect of Span Length

The study analyzed span lengths of $300 \mathrm { m }$, $400 \mathrm { m }$, and $500 \mathrm { m }$ with wind speed at $6 \mathrm { m/s }$ and initial wind attack angle at $140^\circ$.

• Amplitude: Figure 6 illustrates that increasing span length leads to a noticeable increase in horizontal amplitude. While not explicitly tabulated for all spans, the visual trend from the trajectories confirms this.

• Trajectory Shape: As span length increases, the galloping trajectory changes significantly. At $300 \mathrm { m }$, the trajectory is somewhat elliptical. As span length increases to $500 \mathrm { m }$, the trajectory can evolve into an "8-shaped" pattern (Figure 6, span $500 \mathrm { m }$). This "8-shaped" trajectory is characteristic of strong coupling between in-plane and out-of-plane vibration modes and has been observed in studies of crescent-shape iced conductors as well [14].

  The following figure (Figure 6 from the original paper) shows the vibration trajectories of the midpoint of the ice-covered eight-split sub-conductor at different spans for 300m and 400m:

  该图像是图表，展示了不同档距下覆冰八分裂子导线中点的振动轨迹。图中包含两个子图：(a) 档距为 300 m 的振动轨迹，上方围绕原点的椭圆形轨迹；(b) 档距为 400 m 的振动轨迹，呈现出较为紧凑的线性运动模式。风速设置为 $6 \mathrm{ m/s}$，风攻角为 $140^{\circ}$。

The following figure (Figure 6 (continued) from the original paper) shows the vibration trajectories of the midpoint of the ice-covered eight-split conductor at different spans for 500m:

该图像是图表，展示了在不同档距下，覆冰八分裂子导线中点的振动轨迹。图中横轴表示水平振幅（单位：米），纵轴表示垂直振幅（单位：米），并标注了不同的振动轨迹序号。该图对应的试验条件为风速 $6 \mathrm{ m/s}$ 和风攻角 $140^\circ$，档距为 $500 \mathrm{ m}$。

6.1.3. Effect of Initial Wind Attack Angle

The paper investigated the influence of initial wind attack angles of $20^\circ$ and $140^\circ$, keeping span length at $400 \mathrm { m }$ and wind speed at $6 \mathrm { m/s }$.

• Amplitude: Both Figure 7 and Table 1 clearly demonstrate a significant impact of initial wind attack angle on galloping amplitude.

  • At $6 \mathrm { m/s }$, vertical amplitude is $1.79 \mathrm { m }$ at $20^\circ$ vs. $2.75 \mathrm { m }$ at $140^\circ$. Horizontal amplitude is $0.16 \mathrm { m }$ at $20^\circ$ vs. $0.98 \mathrm { m }$ at $140^\circ$.
  • This trend holds for higher wind speeds as well (Table 1): at $8 \mathrm { m/s }$, amplitudes are ($2.96 \mathrm { m }$ vertical, $2.52 \mathrm { m }$ horizontal) for $20^\circ$ vs. ($7.16 \mathrm { m }$ vertical, $7.44 \mathrm { m }$ horizontal) for $140^\circ$.
  • Similarly, at $10 \mathrm { m/s }$, amplitudes are ($3.49 \mathrm { m }$ vertical, $4.06 \mathrm { m }$ horizontal) for $20^\circ$ vs. ($7.44 \mathrm { m }$ vertical, $5.89 \mathrm { m }$ horizontal) for $140^\circ$.
  • In all cases, the galloping amplitudes for an initial wind attack angle of $140^\circ$ are significantly larger than for $20^\circ$. This correlates with the aerodynamic coefficient analysis in Section 2.2, where the Den Hartog coefficient was found to be consistently negative in the $90^\circ \sim 150^\circ$ range, indicating higher galloping susceptibility in this region.

• Trajectory Shape: Figure 7 shows that while both angles can induce galloping, the vibration pattern and extent are different. The larger amplitudes at $140^\circ$ result in more pronounced and potentially more complex trajectories.

  The following figure (Figure 7 from the original paper) shows the vibration trajectories of the midpoints of the iced eight-split conductor at different wind attack angles ($140^\circ$ is displayed here):

  该图像是图7，展示了不同风攻角下覆冰八分裂导线中点的振动轨迹。横轴为水平方向的振幅，纵轴为垂直方向的振幅，风攻角设定为140°。

The following figure (Figure 7 (continued) from the original paper) is a characteristic analysis graph of the flutter behavior, showing the relationship between amplitude and horizontal direction of the sector-shaped ice-covered split conductors at different angles of attack ($20^\circ$ is displayed here):

该图像是舞动特征分析图，展示了不同攻角下扇形覆冰八分裂导线的振幅与水平方向的关系，是通过风洞实验和数值模拟得到的结果。图中各曲线代表不同的导线状态。

6.2. Data Presentation (Tables)

The following are the results from Table 1 of the original paper:

6.3. Ablation Studies / Parameter Analysis

The paper's investigation into the effects of wind speed, span length, and initial wind attack angle serves as a comprehensive parameter analysis rather than a typical ablation study (which usually removes components of a model). This analysis is crucial for understanding how different environmental and structural factors contribute to or mitigate galloping.

• Wind Speed Analysis: The increase in galloping amplitude and the transition of trajectory shape from elliptical to circular with increasing wind speed (from $6 \mathrm { m/s }$ to $10 \mathrm { m/s }$) highlight the critical role of wind energy input in driving galloping. Higher wind speeds provide more energy to overcome damping and sustain larger oscillations, while also promoting mode coupling.

• Span Length Analysis: The observation that longer spans (e.g., $500 \mathrm { m }$) lead to increased horizontal amplitudes and "8-shaped" trajectories indicates that the structural characteristics of the line itself significantly influence galloping behavior. Longer spans generally have lower natural frequencies and less stiffness, making them more susceptible to large-amplitude vibrations and mode coupling.

• Initial Wind Attack Angle Analysis: The marked difference in galloping amplitudes between initial wind attack angles of $20^\circ$ and $140^\circ$ directly validates the importance of aerodynamic instability criteria. The $140^\circ$ angle falls within the region where the Den Hartog coefficient is significantly negative, as identified in the wind tunnel tests (Section 2.2). This confirms that galloping is highly dependent on the effective aerodynamic profile presented to the wind, which is determined by the ice shape and the wind attack angle. The larger negative Den Hartog and Nigol coefficients in certain regions directly translate to more severe galloping in simulations.

  This comprehensive parameter analysis effectively verifies the applicability and accuracy of the wind tunnel-derived aerodynamic data within the numerical simulation framework to predict galloping under various realistic scenarios.

7. Conclusion & Reflections

7.1. Conclusion Summary

This paper successfully investigated the galloping characteristics of sector-shape iced eight-bundle conductors for UHV transmission lines by combining wind tunnel experiments and nonlinear finite element numerical simulations. The study yielded several important conclusions:

• Aerodynamic Parameter Insight: Wind tunnel tests revealed that sector-shape iced eight-bundle conductors exhibit negative Nigol coefficients (indicating torsional instability) over a much wider range of wind attack angles compared to Den Hartog coefficients (indicating vertical instability). This emphasizes that torsional-vertical vibration coupling is a dominant mechanism for galloping in this type of iced conductor, urging designers to prevent such coupling.

• Wind Speed Impact: Wind speed significantly influences galloping. As wind speed increases, the horizontal amplitude grows more substantially than the vertical amplitude, and the galloping trajectory transitions from an elliptical shape to a circular one, highlighting increased mode coupling.

• Initial Wind Attack Angle Influence: The initial wind attack angle plays a crucial role in galloping severity. An angle of $140^\circ$ consistently resulted in significantly larger galloping amplitudes (both vertical and horizontal) compared to an angle of $20^\circ$, aligning with the aerodynamic instability regions identified from the wind tunnel data.

  The findings provide valuable guidance for the anti-galloping design of eight-bundle conductors in practical engineering applications, particularly in heavy ice regions.

7.2. Limitations & Future Work

The paper implicitly suggests some limitations and potential avenues for future research:

• Ice Shape Randomness: The paper acknowledges that ice shapes in nature are random and varied. While sector-shape is a good representation of heavy ice, studying other irregular or randomly generated ice shapes could provide a more complete picture.
• Turbulence Effects: While some related works (e.g., Wang Xin et al. [12]) studied the influence of turbulence on aerodynamic coefficients, this paper's wind tunnel tests were likely conducted under smooth flow conditions (standard for initial aerodynamic parameter derivation). Future work could incorporate turbulent flow conditions in the wind tunnel or numerical simulations to better reflect natural wind environments.
• Multi-Span Effects: The numerical simulations focused on a single-span transmission line. Real UHV lines consist of multiple spans. As noted by Liu Xiaohui et al. [20], continuous spans can influence galloping characteristics. Extending the numerical model to multi-span configurations would offer more comprehensive insights.
• Wake-Induced Oscillation: The paper mentions that the sudden reduction in drag coefficient in the wake region might lead to another dynamic phenomenon: wake-induced oscillation. While the wake effects are implicitly captured in the individual sub-conductor aerodynamic parameters, a dedicated study on wake-induced oscillation for sector-shape eight-bundle conductors could be valuable.
• Dynamic Aerodynamic Coefficients: The current approach uses static aerodynamic coefficients that vary with attack angle. In reality, aerodynamic forces can also be affected by the conductor's velocity and acceleration (i.e., dynamic aerodynamic effects). Incorporating dynamic aerodynamic models could enhance simulation accuracy.
• Mitigation Strategies: The paper discusses galloping characteristics. Future work could extend to evaluating the effectiveness of various anti-galloping devices or design modifications using this established numerical framework.

7.3. Personal Insights & Critique

This paper presents a rigorous and practical approach to studying galloping in UHV eight-bundle conductors. The combination of wind tunnel experiments to capture complex aerodynamics and nonlinear FEM for structural dynamics is a well-established and effective methodology, particularly when analytical aerodynamic models are insufficient.

Strengths:

• Relevance to Real-World Problems: Focusing on sector-shape ice in heavy ice regions for UHV eight-bundle conductors directly addresses a significant engineering challenge with high economic and safety implications.
• Robust Methodology: The detailed description of the wind tunnel setup and FEM implementation (especially the use of UEL in ABAQUS for dynamic load application) demonstrates a strong technical foundation.
• Comprehensive Parameter Analysis: The systematic investigation of wind speed, span length, and initial wind attack angle provides valuable insights into the sensitivity of galloping to these factors, which is directly applicable to design.
• Highlighting Torsional Instability: The emphasis on the prevalence of negative Nigol coefficients for sector-shape ice is a crucial finding, underscoring the importance of torsional considerations in anti-galloping design, which might be overlooked if only vertical instability is considered.

Potential Issues / Areas for Improvement (from a critical perspective):

• Uncertainty Quantification: The paper does not discuss uncertainties associated with wind tunnel measurements or numerical model assumptions. In real-world scenarios, ice shapes, wind conditions, and material properties all have inherent variability. Incorporating probabilistic analysis or sensitivity studies to quantify the impact of these uncertainties on galloping predictions would enhance the robustness of the findings.
• Model Validation: While the results are qualitatively consistent with existing literature (e.g., "8-shaped" trajectories), a more direct validation of the numerical model against field observations or other benchmarking data (if available) would strengthen confidence in the quantitative predictions.
• Detailed Mode Coupling Analysis: The paper mentions in-plane and out-of-plane mode coupling and internal resonance. A more in-depth analysis of the modal characteristics of the eight-bundle system and how these modes couple under aerodynamic excitation could provide deeper theoretical understanding.
• Damping: The paper implicitly considers structural damping in the numerical simulation but does not explicitly discuss its magnitude or influence. Damping is a critical factor in galloping amplitude and stability.

Transferability: The methodology of combining wind tunnel testing for aerodynamic coefficients with finite element analysis for structural response is highly transferable. This approach can be applied to:

• Other Icing Problems: Studying galloping or other wind-induced vibrations for different ice shapes, conductor types (e.g., compact conductors), or bundle configurations.

• Other Cable Structures: Adapting the framework to analyze wind-induced vibrations in other flexible cable structures like bridge cables, guyed masts, or offshore risers, where complex fluid-structure interactions are present.

• Structural Optimization: The insights gained from such simulations can inform the optimization of transmission line design (e.g., spacer designs, damping devices, conductor arrangements) to improve galloping resistance.

  Overall, this paper provides a valuable contribution to the field of transmission line aerodynamics and structural dynamics, offering practical guidance for engineers dealing with galloping challenges in UHV systems.