## 1 Introduction  
Previous ice disasters have illustrated that ice can cause electrical issues, including short circuits on transmission lines, and mechanical failures such as conductor breakage or even tower collapse, posing a significant threat to the stable operation of the electrical power system [1; 2; 3]. To prevent the damage of transmission lines during ice disasters, scholars have conducted extensive research on predicting ice loads and analyzing the mechanical response of transmission tower-line systems under icing conditions. Based on field measurement data, [PERSON] et al. [4] proposed a correction coefficient to enhance the accuracy of uneven icing assessment on transmission lines. [PERSON] et al. [5; 6] investigated the growth mechanism of ice on transmission lines and introduced a novel median volume diameter of the droplet distribution, which was derived from microscopic meteorological data and utilized for calculating the collision coefficient. [PERSON] et al. [7] proposed a method to calculate the equivalent ice thickness of transmission lines based on conductor tension by establishing the state equations of the conductor. [PERSON] et al. [8; 9] studied the mechanical response of a tower-line system subjected to ice shedding and conductor breakage. In the aforementioned studies, the researchers investigated the ice loads on transmission lines from a microscopic perspective and analyzed the specific mechanical responses of the transmission lines. However, the relationship between ice load and the failure probability of transmission lines from a macroscopic perspective has not been fully explored. There is a lack of assessment on the failure probability of transmission tower-line systems under varying ice loads. Regarding probabilistic approaches, [PERSON] et al. [10] and [PERSON] et al. [11] adopted the approach based on the traditional Wohler concept and the direct optimized probabilistic computation method to study the influence of fatigue damage on the steel support truss during the whole service life, respectively. [PERSON] et al. [12] used the two structural limit analysis algorithms they previously proposed [13; 14] to predict the elastoplastic response of the infrastructure to evaluate the collapse state of the truss-frame structure. Subsequently, [PERSON] et al. extended the spirit of these algorithms to spatial cable-rib structures, and proposed a complementarity problem formulation for simulating the mechanical behavior of cable-rib structures [15; 16]. In addition, fragility analysis is another primary method for calculating the failure probability of structures, which can evaluate the bearing capacity of the structure from a probabilistic perspective [17; 18]. [PERSON] et al. [19] conducted an incremental dynamic analysis (IDA) on a specific transmission tower project under different wind attack angles, determined the fragility curve of the tower, and evaluated the failure probability of the tower. The results showed that the most unfavorable wind attack angle for this engineering case was 45\({}^{\circ}\). [PERSON] et al. [20] investigated the dynamic response of transmission towers subjected to near-field earthquakes, constructed an earthquake probability model utilizing the IDA method, and studied the influence of seismic incident angle on structural fragility. [PERSON] and [PERSON] [21] proposed a moment-matching method for calculating the vulnerability of transmission towers under wind loads through wind tunnel tests and nonlinear buckling analysis. The above studies on the fragility of transmission lines focused on single loads, yet the influence of wind load during ice disasters cannot be ignored. In addition, ice on transmission lines also increases the windward area of the structure, resulting in an increased wind load. Therefore, both ice and wind loads should be considered simultaneously for the analysis of icing transmission lines. A fragility analysis of the structure under combined ice and wind loads should be conducted [22]. [PERSON] et al.