# Evaluation of Subway Emergency Evacuation Based on Combined Theoretical and Simulation Methods
## 2 Establishment of Emergency Evacuation Time of Subway
Furthermore, it is assumed that the number of individuals disembarking from the subway cars is synchronized, with no interference occurring between the cars. The average value is utilized as a theoretical reference for the actual evacuation time, with the calculation formula provided in reference [11] as follows:  
\[t_{1}=\frac{s^{2}n^{2}}{2 Kw}\times\frac{2.2+\frac{\varepsilon}{2}}{2.2-\frac{ \varepsilon}{2}}\ln\frac{4.4}{e} \tag{1}\]  
\[n=\frac{N}{f} \tag{2}\]  
In the first equation, \(t_{1}\) is the average time consumed for each subway car to evacuate all people from the subway to the platform (s); \(s\) is the average floor area of each person, generally 0.2 m\({}^{2}\); \(N\) is the total number of people in all cars of the subway; \(n\) is the average number of people per car; \(e\) is the door width, which is a fixed value. For example, the door width of Xi'an Metro No. 1 is 1.3 m. \(K\) is a model parameter and its value varies with the subway station; w is the width of the waiting area in the subway car (m). \(f\) is the number of cars in a subway train. The time model for movement from the platform to the stairway entrance (\(t_{2}\)):  
The correlation between passenger flow velocity and passenger flow density is examined through a dynamic framework, leading to the conclusion that the velocity of crowd flow adheres to a specific equation in a horizontal plane:  
\[v_{0}=v_{m}(\alpha A+\beta B+\gamma) \tag{3}\]  
\[A=1.32-0.82\ln\rho_{i} \tag{4}\]  
\[B=3.0-0.76\rho_{i} \tag{5}\]  
In these equations, \(v_{0}\) is the horizontal movement speed of the evacuees on the platform level (m/s), \(v_{m}\) is the maximum average movement speed of personnel (m/s). The maximum average movement speed of a person in an emergency is 1.2 m/s. \(A\) is the influence coefficient of front and rear congestion on speed. \(B\) is the influence coefficient of left and right congestion on speed. \(\rho_{i}\) is the passenger flow density of \(i\)-th layer. \(\alpha\), \(\beta\), and \(\gamma\) are the contribution weights to personnel speed under the combined effects of front, back, left, and right congestion and other factors, respectively, where \(\alpha\) varies in 0.25-0.44, \(\beta\) ranges from 0.014 to 0.088, and \(\gamma\) varies in the range of 0.15-0.26. In this paper, the average value of \(\alpha\), \(\beta\), and \(\gamma\) is taken as the final influence weight, namely \(\alpha\) = 0.35, \(\beta\) = 0.05, and \(\gamma\) = 0.20. Hence, the time to evacuate the platform and reach the stairway entrance is calculated as follows:  
\[t_{2}=\frac{L}{v_{0}} \tag{6}\]  
where \(L\) is the maximum distance between the crowd and the stairway entrance (m). The time taken to move from the stairs entrance to the station hall (\(t_{3}\)):  
The assessment of the time taken by a crowd on the stairs can be segmented into two distinct components: initially, the duration required for the crowd to traverse the stairs, followed by the congestion time experienced by individuals waiting at the stairway entrance. Utilizing the calculation formula referenced in the literature [21], the congestion time for all individuals at the stairway entrance is denoted as \(t_{31}\). \[\frac{b}{m}=\frac{8.04}{t_{31}^{1.37}} \tag{7}\]\[t_{31}=1.37\sqrt{\frac{8.04m_{1}}{b_{1}}} \tag{8}\]  
Here, the variable \(b\) represents the effective width of the staircase, defined as the portion of the evacuation passage that is actively utilized (m). The variable \(m\) denotes the number of individuals present on each floor of the platform. The term \(b_{1}\) refers to the effective width of the platform floor stairs, which indicates the width that is actually employed for pedestrian movement (m). Additionally, \(m_{1}\) signifies the total number of individuals located on the platform floor.