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Project supported by the National Natural Science Foundation of China (Grant No. 71471163).
The social force model has been widely used to simulate pedestrian evacuation by analyzing attractive, repulsive, driving, and fluctuating forces among pedestrians. Many researchers have improved its limitations in simulating behaviors of large-scale population. This study modifies the well-accepted social force model by considering the impacts of interaction among companions and further develops a comprehensive model by combining that with a multi-exit utility function. Then numerical simulations of evacuations based on the comprehensive model are implemented in the waiting hall of the Wulin Square Subway Station in Hangzhou, China. The results provide safety thresholds of pedestrian density and panic levels in different operation situations. In spite of the operation situation and the panic level, a larger friend-group size results in lower evacuation efficiency. Our study makes important contributions to building a comprehensive multi-exit social force model and to applying it to actual scenarios, which produces data to facilitate decision making in contingency plans and emergency treatment.
Emergency management is an important social issue about people’s safety and wellbeing. The primary thing is to evacuate the people involved efficiently when an emergency occurs. Given the importance and complicacy of evacuation, it is critical to complement existing theoretical and practical approaches to ensure a smooth evacuation.
Most studies use modeling and programming to simulate pedestrians’ behavior when emergencies occur. Pedestrian simulation models in the existing literature can be categorized into two categories: discrete model and continuous model. The majority of discrete models are based on cellular automata models, with space and time discrete during the evacuation while the continuous models simulate everyone’s position at every second in succession. The fluid-dynamic model,[1–3] cost function model,[4,5] and social force model[1,6,7] are certain examples of continuous models. The social force model that considers the physical force of the environment and introduces the socio-psychological force has been widely used by researchers.
Lu et al.[8] developed an extended floor field cellular automaton model to test the influence of walking behaviors of pedestrian groups on pedestrian flow dynamics in reality, which needs five basic rules to support. You et al.[9] gave a calculation method of the probability of the pedestrian movement by using a three-dimensional cellular automata evacuation model with dynamic variation of the exit. Li et al.[10] found that heterogeneous behavioral tendencies in humans, that is, familiarity and aggressiveness, impacted the evacuation time by an extended cellular automata model. Although much research has advanced the simulation progress of pedestrians’ evaluation with discrete models, we find that a discrete model must establish a series of rules to meet the discrete requirement for time and space, which makes pedestrian behavior simulations not as accurate as a continuous model since it could simulate the behaviors of pedestrians more vividly. Therefore, the social force model is preferred in this study.
With the development of research on pedestrian evacuation, more and more researchers have improved the social force model and applied it to practice. Li et al.[11] developed a four-stage transfer model and noted that the congestion risk was higher in transfer aisles than on escalators. Yang et al.[12] built a social force model involving the factor of guidance to simulate the evacuation issues in the Beijing South Railway Station and found that the guidance enhanced evacuation efficiency. Wang et al.[13] introduced friendship attraction and attractive forces of an exit into the initial social force model and simulated the pedestrian flow in a station hall during the Chinese spring festival. Jiang et al.[14] found the optimal design of architectural entities based on social force. Yang et al.[15] explored the pedestrian evacuation under fire in a subway station on the foundation of the social force model, using the software FDS + Evac.
In addition, lots of factors, which affect the evacuation efficiency, have also been discussed in the existing literature. The attributes of exits, such as the height and the width, have large impacts on the progress of an evacuation. Wei et al.[16] modified the static floor field model and conducted a regression analysis to explore the relationship between the reference point and the exit width. Lin et al.[17] proposed a granular dynamic model on the basis of the discrete element method, to simulate the characteristics of crowd flow through an exit. They also suggested exploring multi-exit evacuation in future studies. However, multi-exit choice may be the most difficult and complex aspect of an actual scenario. There are few studies discussing individual decisions regarding multiple exits, particularly in terms of combining a continuous model with a multi-exit model.[18] In addition, not only physical factors but also psychological factors exhibit large impacts on evacuation efficiency. Moussaid et al.[19] analyzed the crowd dynamics by building an individual escape model and found that social interactions among group members significantly influenced crowd dynamics. Hu et al.[18] proposed a novel three-dimensional cellular automata model to simulate evacuation from a lecture theatre and found that a larger group size resulted in longer evacuation time. Though behavioral models have been used to describe features of pedestrian flow in normal operation situations,[11,12,16,18,19] few studies analyze the interaction among friends when an emergency occurs.
Based on evacuations under multi-exit scenarios in reality, this paper aims to propose a comprehensive model by combining the behavioral model with a multi-exit utility function. We develop a model based on the modified social force model indicated in our previous work.[13] In this paper, significant improvements include the following four aspects: (i) visual range and relative friend attractiveness of the corresponding velocity are incorporated into the social force model; (ii) a multi-exit choice mechanism is added into the advanced model; (iii) the congestion effect and design of exits are taken into account; (iv) the emergency case is discussed and the difference between different levels of panic is compared. By considering all these aspects, a series of computer simulation experiments are conducted to further quantify the impact of these factors on pedestrian evacuation.
In order to explore individual decision mechanisms under multi-exit conditions and identify the factors that affect evacuation efficiency, the comprehensive model and initial processing of the simulation of evacuation in Wulin Square Subway Station in Hangzhou are presented in the next section. Then we program to modify the simulation of a pedestrian crowd in the subway station waiting hall to determine how the pedestrian density, panic level and friend-group size affect evacuation efficiency in both normal operation cases and emergency cases in Section 3. Finally, we conclude the limitations and future research avenues of our study in Section 4.
The initial social force model does not consider the effect of visual range on the psychological force. Something within one’s visual range can easily draw one’s attention. In terms of the pedestrians’ psychological characteristics, we define the visual range as the 90 degrees to the left and to the right of pedestrian i’s moving direction where φi j(t) in Fig.
In conclusion, we define
Although many studies have already considered interactions among pedestrians, they have seldom considered relative speed. Thus, we consider the relative speed and define the following expression:
![](cpb_25_11_118901/cpb160928ieqn1.gif)
To consider the visual range, we modify the initial social force model as follows:
![](cpb_25_11_118901/cpb160928ieqn4.gif)
![](cpb_25_11_118901/cpb160928ieqn5.gif)
![](cpb_25_11_118901/cpb160928ieqn6.gif)
The interaction between individuals and walls or obstacles is computed analogously using the psychological force, repulsive force and friction force. The corresponding expression is as follows:
Finally, the desired velocity is modified by adding a panic parameter[13]
![](cpb_25_11_118901/cpb160928ieqn7.gif)
![](cpb_25_11_118901/cpb160928ieqn8.gif)
In conclusion, we extend the acceleration equation as follows:
Previous research indicated that the width of an exit influenced the attraction of the exit significantly, that is, a wider exit would exert a greater attractive force on pedestrians.[21] In addition, the height of an exit is also a factor considered in our study. Thus, we apply the dynamic floor field model to compute the effect of the height and width of the exit
Congestion occurs when people gather toward an exit. The physiological features and behaviors of a congested crowd affect the outcome of an evacuation apparently.[22] The following advanced model proposed by Zainuddin and Shuaib is used to describe the utility of the effect of the distance between individual i and exit k:[23]
The distance between individual i and exit k can be divided into two parts: the uncongested distance diuk and the congested distance rk, which is the radius of the semicircle area. According to Eq. (
Under multi-exit circumstances, the exit choice depends on the individual’s perception of congestion. If the preceding exit is too crowded to escape, a pedestrian must change his current direction. Helbing[22] and Senevarante[24] noted that a pedestrian would head for a new exit if the utility difference between the new exit and the current exit was larger than the threshold. In this manner, the multi-exit choice function
Combining the above functions, we can obtain the multi-exit choice utility function as follows:
The best_exit(t) represents the current optimal exit;
Finally, we create a modified model of the multi-exit social force model as the combination of Eqs. (
Social force model:
Multi-exit choice model:
This study chooses the passenger evacuation hall of the Wulin Square Station, which is the largest subway station in Hangzhou, China, as a basic research background. Given the hall (Fig.
According to the classical articles,[6,23] we adjust certain parameters, while it should be accepted that the calibration of these parameters needs more future work because of the shortage of experimental data. The key parameters of the present model are exhibited in Table
![]() | Table 1. Parameters of the modified social force model. . |
Pedestrian flow simulations in the waiting room of the station hall were conducted under two different situations: the situation under normal operations, and the situation under an emergency (or congestion). Under normal operations, we considered the impacts of the pedestrian density and the friend-group size. When emergencies occurred, we took the friend-group size and different levels of panic into account.
There also exist two states no matter under what the operation case: typical state and holiday state. In the typical state, the number of passengers is rather different from that in the holiday state. We regarded less than 400 people in the hall (typical state) as low pedestrian density, while more than 600 people (holiday state) as high. We defined a high friend-group density for holiday states (the proportion of pedestrians in groups to total pedestrians was as high as 80%) but a low friend-group density for the typical state (the proportion was only 20%). Besides, according to Moussaid’s study, people always gathered 2 to 4 people in a small group when they hung out,[19] therefore, we set the size of friend-group ranging from 2 to 4.
In the typical state of normal operation situation, there are approximately 400 pedestrians in the waiting room. Thus, the total pedestrian flow is set to 400 with a regular speed of 1.5 m/s. No matter on workdays or in the holidays, parts of pedestrians hang out in groups. In this way, the pedestrians were partitioned into two parts: one part represents friend-group (green ones in Fig.
The self-organization phenomena can be clearly observed in the simulation process. As shown in Figs.
The phenomena that arose in the simulation process (e.g., self-organization phenomena and “arch” effect) were in line with those in the classical articles,[6,23] which proved that the modified model regarding these adjusted parameters is effective to some degree.
Evacuation efficiency is highly related to the pedestrian density (the number of passengers). The desired velocity will increase if pedestrian density is low.[19,26] We simulated the evacuation process with passenger numbers varying from 100 to 1000, set the normal speed to 1.5 m/s in each condition, performed each condition more than 50 repetitions, and obtained the additional average evacuation time (see Table
![]() | Table 2. Case I: evacuation time with different numbers of evacuated people. . |
The results in Fig.
There is sufficient space for individuals to keep a certain distance from each other when the pedestrian density is low (or in the typical state: less than 400 people in the hall). With weak interaction among each other,[25] an individual’s behavior mainly depends on the individual himself. Thus, fewer constraints lead to lower efficiency. When pedestrian density increases,[25] the interaction force increases synchronously, but the mechanisms become increasingly complex, which may result in an un-organizational phenomenon or chaotic flow. When the pedestrian density reaches to high (or holiday state: more than 600 people in the hall), first the increasing interaction improves evacuation efficiency at the early stage while the efficiency decreases and then reaches a second optimal point in high pedestrian density when the number of pedestrians evacuated is 900. Therefore, we found the different thresholds of pedestrian density in typical state and holiday state respectively, which was of importance to emergency management.
The simulation results in the “low friend-group density” (or the typical state) and the “high friend-group density” (or the holiday state) are shown in Table
![]() | Table 3. Average evacuation steps in all conditions of different friend-group sizes. . |
On one hand, a 4 (friend-group size: single, two, three and four) × 2 (friend-group density: high and low) between subject ANOVA analysis reveals significant main effects on both friend-group size (F = 154.256; p < 0.001) and friend-group density (F = 218.501; p < 0.001). The interaction effect of friend-group size and friend-group density is significant (F = 59.781; p < 0.001) as well.
On the other hand, it is interesting to find that in the low friend-group density level there is no significant difference between group size 3 and group size 2 (p = 0.378), but all different friend-group sizes are significantly different from each other in high friend-group density level.
What we can observe from Fig.
Once an emergency occurs, the waiting hall will be filled with panic and people will push each other and run towards the exits in order to escape from the waiting hall, thus leading to congestion and impacting the desired velocity of pedestrians. We suppose that the parameter of a low panic level is 0.3 while 0.7 represents a high panic level. Meanwhile, we set the situation as in the holiday state where the pedestrian density is 600 and the friend-group density is high (the proportion of pedestrians in groups to total pedestrians is 80%). Then we simulate each condition 50 times.
In this case, we set the size of friend-group to be integers ranging from 2 to 4, the desire velocity 3 m/s, and then discuss the variation in evacuation efficiency with the increase of the panic parameter [0, 0.9] (see Fig.
The simulation results in the “low panic” and the “high panic” are shown in Fig.
The evacuation time, similar to a U-shaped curve, first decreases and then increases with an increasing panic parameter (see Fig.
![]() | Fig. 11. Case II: evacuation time curve with different panic parameters (fifty trials in each condition). |
Figure
As for different friend-group sizes when there is an emergency in the holiday state, the different panic level (high and low) is manipulated. In each condition, we set different friend-group sizes (single, two, three, and four) and controlled the density of friend-group in the high level.
A 4 (friend group size: single, two, three and four) ×3 (panic level: low, high and no) between-subject analysis of variance (ANOVA) test was conducted to analyze the evacuation time. The ANOVA analysis for evacuation time shows that both friend-group size and panic level have significant main effects (F > 100, P = 0.000, and F > 100, P = 0.000, respectively). Meanwhile, the friend-group size has a significant interaction with the panic level (F = 62.821, p < 0.001).
Figure
![]() | Table 4. Average evacuation time: 600 passengers with different panic parameters and group sizes. . |
Finally, we draw four conclusions based on the above results. First, we found the evacuation thresholds in different operation states. Pedestrians with a high density require more time to escape, but the evacuation efficiency fluctuates with the density increasing, reaching the highest level when the pedestrian number is approximately 200 in low pedestrian density (in typical state) and 900 in high density (in the holidays) under normal operation cases. Second, the evacuation efficiency in the holidays is lower than that in the typical state. In addition, the friend-group size has a positive impact on the evacuation time apparently, that is, a larger size results in longer evacuation time. However, there is no significant difference between sizes of 2 and 3 in the typical state. Third, the panic parameters significantly influence the evacuation efficiency. As the panic parameter increases, the evacuation time first decreases and then increases, thus presenting a U-shaped curve. 0.23 is the most efficient panic level to evacuate pedestrians. In addition, if the panic level is overly high, it will lead to congestion, which also influences the evacuation time inversely. Finally, a larger friend-group size leads to lower evacuation efficiency no matter what the level of panic is in the holidays under emergency cases. The evacuation efficiency is the highest in the low panic level, which also confirms the third conclusion. Therefore, smaller friend-groups would be preferred in an evacuation process. In addition, the evacuation efficiency in a high panic condition reduces faster than that in a low panic condition with friend-group size increasing.
This study makes two main contributions. First, the study represents an advancement of researches on the social force model and multi-exit choice model. We further develop the social force model and combine it with a utility function to implement evacuation decisions, which is of importance in extending previous works. The modifications include not only the visual range and the relative friend attractiveness but also the congestion effect and design of multi-exit. These factors make the advanced model valid by describing the main observations of pedestrian dynamics in both normal operation cases and the emergency cases. Second, simulations are conducted in different situations of the actual subway station, which provides a safety threshold of pedestrian density and panic levels in the waiting hall. The model could be extended to other scenarios to find different thresholds for each scenario, which is critical to emergency management. The results regarding the relationship among evacuation time, pedestrian density, panic level and friend-group size are also useful for decision making with regard to contingency plans and emergency treatments.
This paper indeed has some limitations. First, our research focuses only on the waiting room in the subway station hall, ignoring the ticket checking machines and other equipment, which may be obstacles to evacuation. Second, we just assume that everyone could bear the external forces from outside and that no one falls down when the force surpasses his tolerance or no confliction occurs during the evacuation. However, if confliction occurs, people who get hurt will become obstacles. Thus, we will consider these factors and the effect of obstacles and the confliction on evacuation time in our future research. Third, many researchers have considered the calibration part recently while the calibration is not discussed in our current work. Zeng et al.[28] accomplished a more accurate calibration based on the observed dataset or the maximum likelihood estimation, which offers us useful information to make the results much more precise in further study.
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