A cellular automata model for high-density crowd evacuation using triangle grids

Physica A 509 (2018) 1034–1045

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Physica A journal homepage: www.elsevier.com/locate/physa

A cellular automata model for high-density crowd evacuation using triangle grids ∗

Ji Jingwei a , , Lu Ligang b , Jin Zihao a , Wei Shoupeng a , Ni Lu a a b

Jiangsu Key Laboratory of Fire Safety in Urban Underground Space, China University of Mining and Technology, China School of Civil Engineering, Xuzhou University of Technology, China

highlights • • • •

Maximum crowd density of 8 people/m2 can be simulated. The triangular mesh makes the model have better spatial adaptability. Pedestrians have more movement directions in the model. Experiments proved that the simulation results have high reliability.

article

info

Article history: Received 7 April 2018 Received in revised form 10 June 2018 Available online xxxx Keywords: Evacuation Cellular automata High density crowd Triangle grid

a b s t r a c t In this paper, according to the characteristics of evacuation in high-density crowd, a new triangular grid cellular automata model is proposed. In this model, the maximum density of crowd can reach to 8 person/m2 which is measured by the experiment. And pedestrians can move to 14 directions if there are no obstructions around them. Meanwhile, this paper proposes the concept and calculation rules of moving potential in the moving field. The moving potential provides reference for the movement of a pedestrian. Through the comparison of the measured values and the simulated values of an evacuation process in a building, it is proved that the model can accurately simulate the evacuation process of high density crowd. © 2018 Elsevier B.V. All rights reserved.

1. Introduction In recent years, hundreds of casualties have been caused as a result of stampede caused by high density crowd, such as Madhya Pradesh in 2013, the bund of Shanghai in 2014 and Saudi during the 2014 Hajj. Therefore, pedestrian evacuation dynamics (PED) has become a hot research issue, and the relevant research mainly focuses on the basic data or phenomenon of evacuation based on experimental or observation methods and the evacuation model of crowd [1–9]. In experimental and observational studies, Helbing analyzed video recordings of the crowd disaster in Mina/Makkah during the Hajj on January 12, 2006 [10], and pointed out that the highest density of pedestrian is up to 9 person/m2 in the disaster. Marija Nikolić analyzed data from the underground station in Lausanne [11], Switzerland and Daamen conducted large-scale evacuation experiments [12]. Their research results show that the density of pedestrian ranges from 0 to 7 person/m2 . However, time and resources spent on evacuation experiment and observation are relatively large, and it is difficult to extract relevant observational data, and it has certain limitations in repeatability and operability [1,13]. Computer simulation technology can safely and effectively visualize the movement process in complex conditions and reproduce the simulation. ∗ Corresponding author. E-mail address: [email protected] (J. Ji). https://doi.org/10.1016/j.physa.2018.06.055 0378-4371/© 2018 Elsevier B.V. All rights reserved.

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Fig. 1. The cellular automata neighbor model of triangular mesh in four positions.

Therefore, with the development of computer technology, the research of evacuation models has become an important research direction. Up to now, dozens of evacuation models have been published, some of which have been commercially available, such as Simulex, Steps, BuildingExdous and so on [9]. In general, these PED models can be classified into two categories: continuous model and discrete model [14]. Continuous models are based on functions or differential equations to describe the evacuation process of pedestrian, such as fluid mechanics model, magnetic field force model, social force model, etc. Discrete models discretize time and space, and people move in this discretized space–time based on certain rules. The discrete model can be divided into two categories: rough grid model and fine grid model. Rough grid models, like Evacnet4 and TimtexI are based on the idea that building structure is compared to pipes according to its characteristic and pedestrians are compared to water flow in pipes. The fine grid model divides the building plane into grids, and the pedestrians move in these grids according to the rules established in advance, such as CA models and lattice-gas models. At present, continuous model has been used to study the evacuation of high density crowd and some achievements have been achieved. Qu studied approaches of modeling crowd evacuation process and dynamic behavior characteristics based on the heuristic force-based model and user optimal criterion [15,16]. Zhao presented a kind of high-density crowd evacuation model based on swarm intelligence theory [17]. Mohamed H, Dridi applied an evacuation simulation software called PedFlow based on microscopic model to high-density crowd evacuation study [18,19]. However, continuous model has a large amount of calculation, which is more suitable for the study of the micro behavior of the pedestrian movement under the smaller scene. Compared with the continuous model, the discrete model can also describe the individual’s motion characteristics, and can only consider the impact of small-scale environment on the movement of people [20], thus significantly reducing the amount of computation and having unique advantages in large scene simulation. Many researchers applied the discrete model to the simulation of large scenes. Ansgar Kirchner presented simulations of evacuation processes using a cellular automaton model for pedestrian dynamics [21]; Xiaoping Zheng applied cellular automaton model to simulate the evacuation process in a square [22]; Ahmed Abdelghany presented a hybrid simulation-assignment modeling framework for studying crowd dynamics in large-scale pedestrian facilities [23]. These discrete models greatly reduce the amount of calculation, but the accuracy is relatively low. The reason is that the movement direction of the pedestrian is restricted to the grid. Usually, they divide the space into rectangular grids with a certain density. The pedestrian can only move from a grid to another adjacent grid according to the rule of motion, while the maximum number of adjacent grids is 8, so the movement direction of the pedestrian is generally no more than 8. In real evacuation scenarios, the movement directions of a pedestrian are actually more than 8. This affects the accuracy of the simulation, especially in high-density crowds. To solve this problem, we developed a new type of evacuation model based on CA models. The main advantage of this model is that it divides the simulated space into triangular grids. So that the movement direction of the pedestrian is extended from 8 to 14, which is more suitable for pedestrian movement. In the model, the maximum density of crowd can reach to 8 person/m2 . According to the comparison with the experimental value, the model can simulate the evacuation process accurately. 2. Establishment of evacuation model for high density crowd 2.1. Basic characteristics of the model The model studied in this paper is based on two-dimensional cellular automaton. The spatial grid division of the traditional two-dimensional cellular automata usually consists of three types: regular triangle, square and hexagonal. And neighbor models are usually divided into Von Neumann types, extensional Von Neumann, Moore and extensional Moore [24]. In this research, cellular space is divided into several triangle grids as shown in Fig. 1 which are different from traditional models. Every isosceles right triangle can become an independent cell. As shown in Fig. 1, the cell is divided into 4 types according to the different direction of the triangle, the black triangle represents the current cell, and the gray triangles around it are its neighbors. It is observed that each cell that is not on the boundary is connected to the surrounding 14 cells. Compared with the traditional CA model, the new cellular automata model of triangular mesh has three characteristics and advantages.

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Fig. 2. The flow diagram of algorithm for evacuation movement in the model.

(1) The new model can generate two triangular cells in a common square grid with size of 0.5 m × 0.5 m, that is, the pedestrian density can reach 8 person/m2 . Of course, the size of the grid can also be redefined by the user or developer according to actual needs. (2) The new model combines the features of triangular and rectangular grid. Cells have more neighbors in the new model. As shown in Fig. 1, each cell has 14 neighbors, which can make pedestrians move to more directions. It can simulate the phenomenon of pedestrians crowding into the crowd with a smaller insertion angle, which occurs frequently in high-density crowd evacuation. Obviously, the simulation of this phenomenon helps to obtain more accurate and true results about the evacuation of high-density crowd. (3) The new model can use three-dimensional array for programming to improve the computational efficiency. The computational process of the model is similar to the classic cellular automata evacuation model, as shown in Fig. 2. Some of these key steps and parameters are discussed in the following sections. 2.2. Pre-processing of the model 2.2.1. Grid size and meshing The architectural space in the model is meshed into triangle grids as shown in Fig. 1. Each grid represents a position in the building space which can only accommodate one person or be occupied by obstacle or wall. The grid size is mainly according to the shoulder width of human bodies. We measured the shoulder width of 100 Chinese adults. The average shoulder width of men and women was 46.5 cm and 40.5 cm respectively, while the average shoulder width was 43.5 cm. The body thickness of typical Chinese people is 20.1 cm which can be approximated to 20 cm. From the viewpoint of geometric, the human body is generally regarded as an ellipse. The long and short axis of the ellipse is the shoulder width and thickness of the human body. The ellipse can be completely surrounded by a isosceles right-angled cell as shown in Fig. 3. The right-angled side of isosceles right-angled triangle is 0.5 m. The actual measurement also confirmed the feasibility of the grid size, as shown in Fig. 4.

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Fig. 3. Two pedestrians in triangular grids.

Fig. 4. Pedestrian occupancy experiment.

2.2.2. Field of moving potential In physics, the voltage at a certain point in the electric field indicates the potential of the positively charged unit in the electric field. The positive charge always flows from the high potential to the low potential under the action of the electric field force. Pedestrians have two kinds of potential in the moving field: local moving potential and global moving potential. In this model, we can draw on the above ideas and put forward the concept of ‘‘pedestrian moving field’’. Local moving potential is given by the distance between a grid and an exit of the building. The moving potential of grids connected to the exit is 0, and the moving potential of its neighbor grid increases by 1, and so on, until it covers the entire cellular space. Take a 4 m × 4.5 m building with two evacuation exits as an example. The distribution of the local moving potential of exit A and exit B is shown in Fig. 5. In Fig. 5, the grid marked by a red circle is defined as M, and the local moving potential of the grid relative to the exit A and the exit B are M (A) = 2 and M (B) = 5, respectively. If a pedestrian occupies grid M, according to the shortest path principle, the pedestrian will choose exit A as the exit for evacuation. The global moving potential refers to the moving potential of the grid when buildings have multiple exits, and the value is the minimum value of each local moving potential of the grid. Suppose there are n exits (E1, E2, . . . , En) in a space, and the local moving potential of grid M relative to the exits E1, E2, . . . , En are M (E1), M (E2), . . . , M (En) respectively, then take the minimum of these potential as the global moving potential of grid M. Fig. 6 shows the distribution of the global moving potential in a space with two exits. In practice, there are various obstacles inside the building, and pedestrians need to avoid obstacles when walking. Therefore, the distribution of obstacles has important implications for both exit selection and evacuation routes. In the model,

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Fig. 5. The local moving potential distribution diagram relative to different exits.

Fig. 6. The global moving potential distribution diagram for a multi-exit space.

Fig. 7. The local moving potential and the global moving potential of a space with obstacles.

we set the moving potential of grids occupied by a building or a wall to a large value, like 99 999, and then calculate moving potentials at other grids. Fig. 7(a) and (b) show the values of the local moving potential and the global moving potential of a space with obstacles, respectively. As shown in Fig. 7, in the model, persons evacuated to the outside of the exit still have an impact on the evacuation of internal groups near the exit of the building. To avoid this effect, some layers of grid are added around the exterior space of the building. The principle of setting moving potential of these external grids is the same as that of the internal grids, but just set the moving potential of the external grids adjacent with exits to −1, and the moving potential of the other external grids decreased by −1 in turn. Pedestrians in internal grids with the moving potential of 0 can only move to grids whose moving potential is −1.

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Fig. 8. Pedestrians’ movement directions and four occupancy status.

Fig. 9. Evacuation angles used for path decision . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2.3. Setting pedestrians The model can set pedestrians in the evacuation scenario by three ways, that is global, local and click chosen. The types of pedestrians are classified into seven categories based on gender and age. 2.3.1. Moving directions As mentioned above, pedestrians have four kinds of occupancy status and can move to one of the 14 neighbor grids around them randomly in one time step, as shown in Fig. 8. In considering the moving potential, according to the shortest path principle, pedestrians always move closer to the evacuation exit. The size of evacuation angle can be used to determine the nearest direction to the evacuation exit as shown in Fig. 9. One edge of the evacuation angle is the connection between the current grid and the adjacent grid, and the other edge is the connection between the current grid and the evacuation exit. As shown in Fig. 9, the red dot represents the center of the current grid and the two blue dots represent the center of two grids to be compared. α and β represent the angles used for the decision. Smaller angle means the path along this direction is closer to the evacuation exit. α is less than β in Fig. 9, so pedestrians select the lower grid as the next grid to move into. If there are obstacles on the route the pedestrian is moving, the pedestrian selects the next moving grid according to the value of the global moving potential. If there are several grids in the unoccupied grids around the person with the same minimum moving potential, the pedestrian will randomly select one of the grids to move in by using formula (1) Pi = ∑14

j=1

Occi

(1)

Occj × Mj n

where Pi represents the probability that a pedestrian moves in direction i; i and j are integers within the range of 1–14, representing a certain evacuation direction; Occ i and Occ j indicate occupation status of neighboring cells in direction i and j, respectively. 0 means the grid is occupied and 1 means the grid is free. Mj is a boolean variable. If the moving potential of the neighbor grid in direction j is smaller than the moving potential of current grid, Mj is 1, otherwise Mj is 0. If the neighboring grids with smaller moving potentials are all occupied, the pedestrian will stay in the original position or move toward to a grid with larger moving potential which means going back. In the model, a random number is used to determine what strategy is taken. The movement distance from a grid to an adjacent grid is different and depends on the occupancy status mentioned above. In essence, the four occupancy status of a pedestrian can be transformed into each other just by spatially rotating. So we only need to research one of the four status. The neighbor grids are marked by numbers from 1 to 14 as shown in Fig. 10 and the movement distance to corresponding direction is shown in Table 1.

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J. Ji et al. / Physica A 509 (2018) 1034–1045 Table 1 The movement distance to corresponding direction. Movement direction 1,13 2,14 3,7,10,11 √

Movement distance L (m)

2 2

√

4,9

10 6

5

6,12

√

√

5 3

1 3

2 3

8 √

2 3

2 6

Fig. 10. An occupancy status of pedestrian in the cellular space.

Fig. 11. Affected area for crowd density calculation.

2.3.2. The crowd density Crowd density greatly affects the movement speed of pedestrians. So, we defined an ‘‘affected area’’ to determine the crowd density. Only pedestrians, walls and obstacles in the 15 grids with shadow line are considered to have impact on moving speed of pedestrian i, as shown in Fig. 11. If the number of the surrounding grids which are occupied by obstacles, walls and other pedestrians is N, then the crowd density is derived from formula (2).

ρ (i) = format

(

N +1 2

) 0 ≤ N ≤ 15

(2)

where format is a rounding function; ρ (i) is the crowd density, person/m2 for an integer between 1 and 15. 2.3.3. Walking speed In evacuation models, pedestrian’s walking speed is a very important parameter which has a great impact on evacuation result, especially the total evacuation time. And in most cases, walking speed depends on crowd density. Fruin [25], Hankin, Wright [26], Predtechenskii, Milinskii [27], Ando [28] and other scholars had conducted many researches, but their conclusions had certain differences which largely depended on experimental environments and settings [29]. This model adopts an experience formula presented by Xie and Ji [6]. In their experiments, 1–8 individuals were bound in a square box with the size of 1 m × 1 m and walked together. In this case, the number of individuals represented the crowd density. On this condition, the relationship between the crowd density and the walking speed based on the measured data is, v (i) =

0.033 × D(i)2 − 0.636 × D (i) + 3.362

D (i) > 4

1.4

D (i) ≤ 4

{

(3)

where, v (i) is the walking speed of pedestrian i, m/s and D(i) is the crowd density at the location of pedestrian i, person/m2 . 3. Evacuation experiment and simulation results We conducted an evacuation experiment in a building and simulated the whole process by the model to verify its reliability.

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Fig. 12. The diagram of the evacuation scene.

Fig. 13. A video screenshot of evacuation.

Fig. 14. The simplified evacuation plan.

3.1. Evacuation experiment The experimental area was located at the first floor of a teaching building. The exit width is 1.5 m and in front of it is a short corridor with a width of 3.5 m and a length of 7.5 m. Pedestrian enter the short corridor from two directions, one is the corridor A, the other is the staircase B. Fig. 12 shows the diagram of the evacuation scene. The evacuation time, the number of evacuees and the evacuation phenomena were recorded by a camera. Fig. 13 is a video screenshot of the evacuation process. The whole evacuation time is 403 s and 943 evacuees walked through the exit. Among them, 335 people came from the east stair and 608 people came from the west corridor. The evacuation flow rate of unit width is 1.56 person/(m s) calculated by formula (4). Q=

P T ×W

(4)

where, Q is the flow rate per unit width at the exit; P is the total number of pedestrians; T is the evacuation time; W is the width of the exit. 3.2. Simulation results 3.2.1. Simulation settings Since the program does not have the function of simulating the evacuation of stair, the east stair is simplified as a corridor. The function of the corridor is only to provide a flow of pedestrians from the east stair into area A. The simplified evacuation scenario is shown in Fig. 14. In the figure, area A is the short corridor connected with the exit, Area B and Area C represent the east stair and the west corridor, respectively. Area D and area E are used to add pedestrians in the model.

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Fig. 15. Distribution patterns of pedestrians at different time.

The number of people in each area at the beginning of the simulation is listed in Table 2. These figures are consistent with the number of people observed in the experiment. Among these people, 70% are male and 30% are female. The response time of pedestrians is set randomly from 0–10 s. 3.2.2. Simulation results and analysis Fig. 15 shows the changes in the number of people and the density of people during the evacuation process. It also presents the evacuation time and the number of people who is evacuated out. The simulation results show that the distribution of pedestrians is scattered at the early stage of evacuation and the density of pedestrian is less than 4 person/m2 . With the time goes by, the number of pedestrians in area A reaches to peak value at about 120 s. At this time, the density of pedestrian reaches to 6–8 person/m2 at the most part of the short corridor.

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Fig. 15. (continued).

Table 2 The number of people in each area. Area A Number of people 13

B 60

C 60

D 525

E 285

Subsequently, the number of pedestrians gathered in Area A is decreased, but the density near the exit is still at a high level until all pedestrians left area A. To avoid accidental error, the evacuation process was simulated for three times by the model. The comparison of simulated data with observed values is listed in Table 3. The absolute error between the observation value and the simulation value is 7.9 s, and the relative error is 1.96%. It indicates that the model presented in this paper can get reliable results.

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Table 3 Statistics of three simulation values compared with observation values. Number of simulation

Simulated evacuation time

Mean value of simulated evacuation time

Measured evacuation time

Absolute error

Relative error

1 2 3

411.4 s 385.7 s 388.2 s

395.1 s

403 s

7.9 s

1.96%

Fig. 16. Evacuation flow rate at the exit varies with time.

Fig. 17. The evacuation route of a pedestrian.

Fig. 18. The total number of evacuees varies with time.

Fig. 16 shows the flow rate at the exit varies with time. During the period of 20 s to 30 s, the crowd flow rate at the exit reaches its peak value, 2.0 person/(m s), which is a little greater than that given by the Green Guide [29] of 1.82 person/(m s). With the growing number of pedestrians goes into Area A and reaches the exit, they are seriously blocked at the exit, as shown in Figs. 13 and 15 (120s). During the period of 40 s to 370 s, the crowd flow rate keeps stable with a value ranges from 1.4 person/(m s) to 1.6 person/(m s). After 400 s, most of the pedestrians evacuate out of the Area A, the evacuation flow rate decreases to 0 at the end. Fig. 17 shows the evacuation route of a pedestrian coming from Area D. The walking route indicates that in his/her evacuation process, the pedestrian shows some behaviors, such as detour and retreat. In order to compare the simulation results of triangular grid model and traditional rectangular grid model, we use rectangular grid model to simulate the evacuation process again. The results show that the total evacuation time simulated

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by the traditional rectangular grid model is about 450 s which is significantly longer than the time simulated by the triangular grid with the same conditions, as shown in Fig. 18. The main reason is that the maximum density in the rectangular grid is 4 person/m2 and the movement direction is only 8. These conditions reduce the space occupancy rate and the selectivity of the moving direction. When the population density reaches the maximum, the number of people waiting or returning is increasing, which eventually leads to the reduced efficiency of simulated evacuation. 4. Conclusions This paper proposes a triangular grid cellular automaton evacuation model for crowd evacuation with high density. Based on geometric analysis and experimental verification, it is feasible to use a triangular mesh to increase the upper limit of the crowd density to 8 people/m2 . Meanwhile, the number of evacuation direction in this model can reach to 14, which increases the degree of freedom of pedestrian evacuation movement behavior and can express evacuation behavior more accurately and truthfully. The model developed in this paper realizes the establishment of evacuation scenarios and the simulation of high-density crowd evacuation behavior, and has the functions of displaying personnel density and paths. By comparing the simulated and evacuated characteristic parameters with the measured values, the model has good reliability. Compared with the rectangular grid evacuation model, the algorithm of the triangular grid is relatively simple and has higher computational efficiency. In future studies, more evacuation behaviors and related factors can be introduced into the model, such as leadership, following behavior and small-group behavior. Acknowledgment This paper is supported by ‘‘the Fundamental Research Funds for the Central Universities, China’’ (Grant No. 2018CXNL08). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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