## 1 Introduction  
With the expansion of urban areas, shopping malls, parking lots, roads, and subways are increasingly developed in underground spaces. However, these underground spaces are vulnerable to flooding, especially during extreme rainfall events driven by climate change, raising significant concerns about potential loss of life. As a critical component of public transportation, which often operates in crowded conditions, subways must be prepared for such flood events. Notably, in July 2021, a heavy rainfall event in Zhengzhou, China, led to subway flooding and subsequent fatalities. Similarly, in August 2022, subway operations in Seoul, Republic of Korea, were suspended due to flooding. It has been reported that when floodwater levels exceed 0.3 m, doors may become difficult to open, and rising water levels in confined spaces could result in fatalities [1]. Thus, rapid evacuation before flood conditions worsen is imperative. However, the presence of floodwaters significantly impacts pedestrian walking speed and safety, necessitating careful consideration during evacuation. This is particularly critical for the elderly, who are more vulnerable to instability when walking in flood conditions, underscoring the need for an assessment of walking stability on flooded subway platforms. Underground spaces, such as subways, are more vulnerable to inundation than above-ground structures, even in similarly low-lying areas. Furthermore, due to their enclosed nature, inundation depth can rise more quickly in these spaces, increasing the risk of flooding. Moreover, the limited evacuation routes in underground environments further heighten the danger for evacuees [2]. Research has been conducted to assess the flood risk in subway systems, taking these unique characteristics into account. For instance, [PERSON] et al. [3] evaluated the flood risk in the Beijing subway system by categorizing it into hazard, exposure, and vulnerability factors, identifying land subsidence as a significant contributor to the increased flood risk. [PERSON] et al. [4] emphasized the importance of considering the network between stations, passenger exposure, and flooding risk when analyzing the flood risk of subway stations. [PERSON] et al. [5] developed a combined model linking extreme rainfall, surface runoff, and water intrusion into the subway, analyzing how changes in rainfall affect the rate of flood inflow. While these studies provide valuable insights for managing the flood vulnerability of subway systems from a macro perspective, there is a need for an evacuation risk assessment that considers the characteristics of flood flow and the structural features of subway platforms. Given that flooding can significantly impede evacuation speed, the importance of conducting thorough pedestrian risk assessments is underscored [6; 7]. Flood risk assessment indices for evaluating pedestrian instability in flood conditions have been developed using human body models and pedestrian evaluation experiments. These experiments have measured hydraulic conditions that define the walking limits in floodwaters, proposing empirical formulas for flood intensity (\(hU\)), calculated using the product of water depth (\(h\)) and flow velocity (\(U\)), and adjusted for human body weight and height [8; 9]. Based on these findings, a flood hazard index that evaluates walking stability within specific ranges of \(h\) and \(U\) has been proposed [10; 11; 12; 13]. Further theoretical analyses of instability mechanisms in simplified human body shapes have been conducted, which suggest moment and friction instability boundaries [14]. [PERSON] et al. [15] developed incipient velocity formulas that trigger sliding and toppling based on experimental results from human body models and theoretical analyses of instability mechanisms. Using the incipient velocity formulas proposed by [PERSON] et al. [15], [PERSON] et al. [16] compared the results of the flood hazard risk analysis results with the flood hazard indices based on \(hU\). [PERSON] et al.