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An extended cost potential field cellular automata model considering behavior variation of pedestrian flow
Physica A 462 (2016) 630–640
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Physica A journal homepage: www.elsevier.com/locate/physa
An extended cost potential field cellular automata model considering behavior variation of pedestrian flow Fang Guo a , Xingli Li a,c,∗ , Hua Kuang b , Yang Bai a , Huaguo Zhou c a
School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, 030024, China
b
College of Physical Science and Technology, Guangxi Normal University, Guilin 541004, China
c
Department of Civil Engineering, Auburn University, Auburn, AL 36849, USA
highlights • • • •
The extended cost potential CA model describing behavior variation is presented. The quantitative formula between behavior variation and level of tension is given. The influences of different factors on evacuation efficiency are explored. Some important results are obtained from the numerical simulations.
article
info
Article history: Received 14 December 2015 Received in revised form 18 May 2016 Available online 24 June 2016 Keywords: Pedestrian flow Cost potential function field Behavior variation Cellular automaton Evacuation time
abstract The original cost potential field cellular automata describing normal pedestrian evacuation is extended to study more general evacuation scenarios. Based on the cost potential field function, through considering the psychological characteristics of crowd under emergencies, the quantitative formula of behavior variation is introduced to reflect behavioral changes caused by psychology tension. The numerical simulations are performed to investigate the effects of the magnitude of behavior variation, the different pedestrian proportions with different behavior variation and other factors on the evacuation efficiency and process in a room. The spatiotemporal dynamic characteristic during the evacuation process is also discussed. The results show that compared with the normal evacuation, the behavior variation under an emergency does not necessarily lead to the decrease of the evacuation efficiency. At low density, the increase of the behavior variation can improve the evacuation efficiency, while at high density, the evacuation efficiency drops significantly with the increasing amplitude of the behavior variation. In addition, the larger proportion of pedestrian affected by the behavior variation will prolong the evacuation time. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Recently, the number of accidents caused by crowded people or emergencies has increased year by year. Pedestrian modeling has become one of the most exciting fields in traffic science and engineering [1–4]. Understanding pedestrian flow characteristics beforehand is extremely important in emergency management to improve evacuation procedures and relevant regulations. The dynamic properties of pedestrian crowds, including various self-organization phenomena, have
∗
Corresponding author at: School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, 030024, China. E-mail address: [email protected] (X. Li).
http://dx.doi.org/10.1016/j.physa.2016.06.103 0378-4371/© 2016 Elsevier B.V. All rights reserved.
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been observed and successfully reproduced by various physical methods. However, it is generally known that evacuation exercise under emergency may be difficult than normal pedestrian flow because of the danger and possible crowd disaster caused by incidents. To solve this problem, various modeling approaches to study evacuation behavior have been proposed [5–36]. Generally speaking, pedestrian flow models can be classified into two categories, macroscopic [5–8] and microscopic [9–36]. In macroscopic model, pedestrians are treated as a fluid with the use of partial differential equation, where the dynamic characteristics of the crowd flow are described by average speed, density, location and time. Microscopic model can be mainly divided into three groups: continuous model with a representative of the social force (SF) model [9–14] and discrete model represented by the cellular automata (CA) model [15–32] and lattice gas model [33–35]. The floor field (FF) model first proposed by C. Burstedde et al. [15], which is more flexible to simulate individual pedestrian motion and can reproduce collective behavior of pedestrian dynamics observed experimentally and described by other more complex models, has become one of the most widely used CA models in evacuation research. In general, there are two types of floor fields: the static floor field S and the dynamic floor field D. The static field depends only on the distance measure (from a cell to the destination), and thus S remains unchanged in the evolution. The dynamic field reflects the virtual tracks left by moving pedestrians. Since then, lots of extended or modified FF models [16–32] are proposed by researchers in different aspects, such as the interaction among pedestrians or pedestrians and obstacles [21–24], layout effects of exit or door [25–27], behavioral or group effects [28–30]. In 2012, Zhang et al. established a cellular automata model of pedestrian flow that defines a cost potential field, which takes into account the costs of travel time and discomfort in the journey, for a pedestrian to move to an empty neighboring cell [31]. Without the discomfort term, the resulting cost potential in a cell would simply measure the distance between the cell and the destination, which is independent of time and similar to the static field in the floor field CA model (see Ref. [15]). With the discomfort term, the cost distribution increases with the local density, which is reconstructed at each time step and, thus, is time dependent. In this case, the resulting cost potential field is similar to the dynamic field in the floor field CA model. In 2014, Jian et al. proposed a perceived potential field cellular automata model with an aggregated force field to simulate the pedestrian evacuation in a walking domain with poor visibility or complex geometries [32]. These above mentioned models can generally reproduce pedestrian dynamics under normal situation. However, Helbing et al. [36] found that in situations of escape panics, individuals are getting nervous, i.e., they tend to develop blind actionism. Furthermore, people try to move considerably faster than normal, etc. They think that the transition between the ‘‘rational’’ normal behavior and the apparently ‘‘irrational’’ panic behavior is controlled by a single parameter, the ‘‘nervousness’’, which influences fluctuation strengths, desired speeds, and the tendency of herding. Here it should be noted that there is no precise accepted definition of panic although in the media usually aspects like selfish, asocial or even completely irrational behavior and contagion that affects large groups are associated with this concept [37]. In 2008, Rogsch et al. performed a detailed discussion about ‘‘panic’’ and based on the difficulties of different definitions, they used the following definition to investigate the real origins of some mass-emergencies, namely, ‘‘Panic: People flight based on a sudden subjective or ‘infected’ fear; People are moving imprudently; The cause of this movement cannot be recognized by an outsider’’ [38]. Up to now, the terminology ‘‘panic’’ is highly controversial and usually avoided. In this manuscript, we use ‘‘fear’’ to describe the above mentioned pedestrians’ psychological characteristics when they confront an emergency. Therefore, the previous cost potential field CA model is used in normal situation is insufficient to describe pedestrian emergency evacuation. In this paper, we mainly focus on how to extend the original cost potential field CA model for pedestrian flow to emergency situations. The paper is organized as follows. In Section 2, we formulate an extended cost potential field cellular automata model considering behavior variation from nervousness. Section 3 gives simulation results and corresponding discussion, followed by conclusions in the final section.
2. Model The model follows the basic rules of the cost potential field CA model. We consider a room divided into L × L finite twodimensional grids. The outer lattices denote the room wall. Each of the other grids can be empty or occupied by a pedestrian and is equivalent to a cell whose size can be regarded as 0.4 × 0.4 m2 [15], where L and W are the width of the room and the exit, respectively. As shown in Fig. 1, the black part and the red solid circle denote the wall and pedestrian, respectively. The exit is located at the edge of the wall. In this paper, the Moore neighborhood [15] is adopted and each occupied cell has eight neighboring cells, corresponding to nine probabilities for the pedestrian in the occupied cell to update his or her position (see Fig. 2). Define the neighborhood set of (i0 , j0 ) as Si,j = {(i, j)| |i0 − i| ≤ 1, |j0 − j| ≤ 1}, in which an empty cell set is S i,j = {(i′ , j′ )|(i′ , j′ ) ∈ Si,j , (i′ , j′ ) is an empty cell}.
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Fig. 1. Sketch of model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. (a) An occupied cell (0, 0) and its eight neighboring cells; (b) the corresponding probabilities for occupation in the next update step.
2.1. The original cost potential field The cost potential in a cell is defined as the minimal cost for traveling from this cell to the destination [31]. We assume that each pedestrian is fully aware of the surroundings and the destination, so a pedestrian can choose one or more empty neighbors as his or her target cell, which is expected to minimize the traveling cost. The cost potential function includes two terms: one is the cost of traveling time determined by the pedestrian’s walking speed and the distance from the present position to the target position, the other is the cost of discomfort depending on the surrounding density. Define the cost distribution in a cell (x, y) as
τ (x, y, t ) =
1
ve (ρ(x, y, t ))
+ g (ρ(x, y, t )).
(1)
The first term in Eq. (1) is related to the traveling time, with ve (ρ) being the velocity–density relationship and ve′ (ρ) < 0; the second term represents the discomfort of pedestrians, with g (0) = 0 and g ′ (ρ) > 0. Based on the basic rules of CA model, ve (ρ(x, y, t )) = vmax , thereby assuming that 1
+ g0 ρ γ vmax here we set g0 = 0.075 and γ = 2. τ (x, y, t ) =
(2)
The total cost of walking from cell (i, j) to (i0 , j0 ) is
φ(x, y, t ) =
(x0 ,y0 ) (x,y)
(τ (x, y, t )x′ (s)dx + τ (x, y, t )y′ (s)dy).
(3)
We assume that the integral is independent of the path, i.e.
φx (x, y, t ) = −τ (x, y, t )x′ (s),
φy (x, y, t ) = −τ (x, y, t )y′ (s).
2.2. Quantitative description of behavior variation Here, we introduce a new parameter µ(0 ≤ µ ≤ 1) similar to ‘‘nervousness’’ parameter in Ref. [36] to describe the level of tension of pedestrian under emergency. This parameter will influence the intensity of individual behavior variation and herd behavior, etc. During the evacuation process, the level of tension at time t can be quantitatively described as [39]
µ(t ) =
v0 (t ) − v0 (0) vmax (t ) − v0 (0)
(4)
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Fig. 3. The relationship between behavior variation η and level of tension µ.
where v0 (t ) represents the current speed, v0 (0) is the normal speed before the fear, and vmax denotes the maximum ideal speed. When the level of tension is larger than the threshold of the critical value, the pedestrian will be completely controlled by the fear, causing a huge behavior variation. According to the above analysis, we introduce behavior variation parameter η(0 ≤ η ≤ 1), which is defined as follows: 0 k1 µ + a1 exp(k2 µ) + a2
η=
0 ≤ µ ≤ 0.05 0.05 < µ ≤ 0.6 0.6 < µ ≤ 1.
(5)
Eq. (5) shows that when the level of tension is very small (µ ≤ 0.05), the behavior variation caused by the level of tension can be neglected, which means pedestrians are in a rational state and correspondingly they will select the shortest route to the exit reasonably, i.e., the empty cell with the largest Si,j will be the target cell. When there are many target cells, the selection probability is same. When the level of tension is 0.05 < µ ≤ 0.6, it will lead to a linear increase of the behavior variation. At this time, some pedestrians have priority over others, the movement probability towards the diagonal direction will increase and pedestrians will update their positions according to the rules of the cost potential field. When the level of tension is µ > 0.6, namely, the level of tension exceeds the critical value of psychological endurance, the behavior variation will have an exponential growth with the level of tension. We set the parameters in Eq. (5) as k1 = 0.247, k2 = 0.98, a1 = −0.012, a2 = −1.66 and then obtain the relationship between the behavior variation and the level of tension (see Fig. 3). Compared to normal evacuation, behavior variation under an emergency can also affect the cost of considering the comfort, which will lead to an increase in the cost of discomfort, so we extend the original cost of discomfort to two parts: one is related to the density, the other is caused by the behavior variation. The cost function (2) is redefined as
τ (x, y, t ) =
1
vmax
+ aρ α + b η β
(6)
here we choose the corresponding parameters as a = 0.0125, b = 0.0625, α = 2, β = 3. 2.3. Moving probability Assume (0, 0) is the coordinate of each occupied cell, then the procedure for determining the probabilities pi,j is specified as follows: If (0, −1) is empty, using the forward difference quotient, the difference quotient is φ0′ ,−1 = |(φ0n,−1 − φ0n,0 )/di,j |; if (0, 1) is empty, using the backward difference quotient, the difference quotient is φ0′ ,1 = |(φ0n,0 − φ0n,1 )/di,j |. Compute the
difference quotient φi′,j = |(φin,j − φ0n,0 )/di,j | for (i, j) ∈ S i,j , where di,j = (i − 0)2 + (j − 0)2 is the distance between cell (i, j) and cell (0, 0), φin,j is the potential field values of cell (i, j) in the nth step. Define the set
Sm = {(i, j)|φi′,j = min φi′,j } (i,j)∈S i,j
and probability
pi,j =
0 1 |S | m 1
(i, j) ̸∈ Sm (i, j) ∈ Sm (i, j) = (0, 0),
where |Sm | is the number of elements in Sm .
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Fig. 4. Resolve occupation conflict.
2.4. Resolve conflict Every empty cell (i, j) may be a target cell of m(0 ≤ m ≤ 8) in the kth step. As shown in Fig. 4, which gives rise to an occupation conflict if m ≥ 2. To resolve a possible conflict, pedestrian will select a large probability to move, while when the value is equal, they will move randomly. When the two pedestrians simultaneously choose the same target grid (see Fig. 4(a)), the corresponding moving probability determines which pedestrian will occupy the lattice. If pa > pb , cell a moves to the target location, while cell b still remains static (see Fig. 4(b)); if pa < pb , cell b moves to the target location, while cell a still remains static (see Fig. 4(c)). 2.5. Updating rules (1) When the level of tension is µ ≤ 0.05, walkers will select the shortest route to the exit reasonably. Here, static field is introduced to describe the distance between each cell and the exit. It can be obtained according to the following function [16]: S(i,j) = max ( min
(i′ ,j′ )∈Ω (i0 ,j0 )∈Γ0
(i0 − i′ )2 + (j0 − j′ )2 − min
(i0 ,j0 )∈Γ0
(i0 − i)2 + (j0 − j)2 )
(7)
where the first term is the minimum distance between the neighbor cell (i′ , j′ ) and cell (i0 , j0 ) of exit; the last term is the minimum distance between the cell (i, j) and cell (i0 , j0 ) of exit; Ω and Γ0 denote walking area set and exit set, respectively. Assuming that each element is empty or occupied and each pedestrian walks in a cell length in a time step or is static; (2) When the level of tension is 0.05 < µ ≤ 0.6, pedestrians will update their positions according to the rules of the potential function field. Under the influence of behavior variation, some pedestrians will have priority over others, the movement probability towards the diagonal direction of the cell will increase, and the moving speed along the diagonal direction also increases to v = 1.5 m/s; (3) When the level of tension is µ > 0.6, the level of tension exceeds the psychological threshold, and pedestrians’ behavior becomes blind, such as obviously increasing walking speed, walking backward far away from the exit, etc. Therefore, we adopt the extended Moore neighborhood during the moving process (see Fig. 5), here AB denotes the exit, and the arrow represents the possible moving directions. Different from previous Moore neighborhood [15], the cell’s length along diagonal direction is roughly approximated by a cell length and the moving speed is v = 1.5 m/s. Besides, moving two cells in one time step is considered and the corresponding velocity is v = 2 m/s. At the same time, pedestrians can walk backward directly or diagonally, for example, (0, 0) → (0, 2) or (0, 0) → (1, 1) or (0, 0) → (−1, 1). The specific rules of the model are as follows: (1) Give the random initial distribution of pedestrians in a room; (2) Calculate the moving probability according to the neighbor states, judge the next target cell, then determine the target location; (3) Update the location. Move if the target location is empty and remain static if the target is selected; (4) Remove the pedestrian arriving at the exit automatically from the calculation; (5) Repeat the above rules until the room is empty. 3. Simulation results and discussions Initially, all walkers are distributed randomly at the sites on the square lattice with the density ρ = N /(L − 2) × (L − 2), here note that considering the outer lattices are occupied by the wall. Following the above rules, all the cells’ states are parallel updated. In the simulations, in order to remove the transient effect, the evacuation time was obtained by averaging over 20 runs. The related parameters are taken as L = 20 and W = 3 unless otherwise mentioned. Firstly, we study the influence of the behavior variation magnitude on evacuation time. Fig. 6 gives the relationship between the evacuation time T and density ρ under different level of tensions µ = 0, 0.1, 0.3, 0.5, 0.7, 0.9. From Fig. 6, it
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Fig. 5. An extended Moore neighborhood.
Fig. 6. The relationship between evacuation time T and density ρ under different level of tensions µ = 0, 0.1, 0.3, 0.5, 0.7, 0.9.
can be found that with the increase of density in the room, the evacuation time is prolonged, but under different densities, the effect of level of tension µ on the evacuation time is different. As ρ < 0.2, with the increase of µ, the evacuation time gradually decreases. The evacuation time of µ = 0 is longer than that of µ ̸= 0. This is because at low density, on the one hand, the collision and interaction among pedestrians can be ignored; on the other hand, the behavior variation caused by nervousness can also lead to the increase of pedestrian speed, which makes pedestrian leave the room quickly. As ρ > 0.2, the evacuation time µ ̸= 0 is larger than that of µ = 0, but the change of evacuation time is different in different density ranges. For example, as 0.2 < ρ < 0.5, the evacuation time of µ = 0.7, 0.9 is less than that of µ = 0.1, 0.3, 0.5, while with the increase of density, especially ρ > 0.6, the evacuation time of µ = 0.7, 0.9 is more than that of µ = 0.1, 0.3, 0.5, i.e., the difference of evacuation time under different level of tensions becomes significant. This means that at higher density, the cost potential from discomfort becomes large, which reflects the strong interaction, the more serious competition and congestion among pedestrians caused by the behavior variation inevitably results in the reduction of the evacuation efficiency. In addition, under different densities, the distribution of evacuation time appears to the obvious range under different level of tensions. Typically, the curves of evacuation time at ρ = 0.5 can be roughly divided into two parts: µ = 0 and µ > 0, which means that the evacuation time under different level of tensions (µ > 0) shows little difference; the curves of evacuation time at ρ = 0.7, 0.9 obviously belong to three parts: µ = 0, µ = 0.1, 0.3, 0.5 and µ = 0.7, 0.9, which gives the key value of level of tension influencing evacuation time. A valuable conclusion can be drawn that during the evacuation, to control the pedestrians behavior variation reasonably to a certain range is significant to improve evacuation efficiency. In order to explore the macroscopic behavior characteristics, Fig. 7 gives the typical spatiotemporal patterns during the evacuation process at initial density ρ = 0.1. We can see easily that with the evolution, pedestrians quickly move towards the exit and gather near the exit. This is consistent with the results of pedestrian evacuation [10]. Compared with the normal situation (µ = 0), it can be directly seen that pedestrians under the emergency situation can move away from the room more quickly. From Fig. 7(d) and (f), we can find the number of non-evacuated people of µ = 0.9 is significantly less than that of µ = 0.3 at t = 10. Pedestrian can move more flexibly and more quickly at a higher level of tension. Therefore, at low density, the increase of the level of tension can improve evacuation efficiency. In order to gain a deeper insight into the microscopic mechanism why a larger level of tension could reduce evacuation efficiency at high density obviously, Fig. 8 gives the typical spatiotemporal patterns obtained at ρ = 0.7. At the early stage,
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Fig. 7. The typical spatiotemporal patterns obtained at ρ = 0.1. µ = 0: (a) t = 5 and (b) t = 10; µ = 0.3: (c) t = 5 and (d) t = 10; µ = 0.9: (e) t = 5 and (f) t = 10.
pedestrians move towards the exit, form a jamming blockage at the exit (see Fig. 8(a) and (b)). Under the normal situation, they gather at the exit, displayed as a jamming pattern similar to a vase, and dissolve gradually (see Fig. 8(a), (c) and (e)). This is because pedestrians join a queue rationally ahead of the exit and escape in turn, which decreases the interaction among pedestrians and weakens the effect of blockage, and thus improves the evacuation efficiency considerably. Being a larger level of tension (µ = 0.9), pedestrians are in an extremely nervous state and then friction, as a kind of an internal local pressure between the pedestrians, becomes remarkable in regions of high density. The nervous behaviors among pedestrians and the rapid expansion of friction easily cause a variety of congestion ‘‘bottleneck’’, and thus arching and clogging near the exit can be observed. Moreover, alternative cells are often overlooked or not efficiently used in escape situations and a few empty cells in the crowed pedestrians can be found (see Fig. 8(b), (d) and (f)). All these reflect pedestrians with exceeding nervous try to move considerably faster than normal, but in fact they get the opposite of what they expect. This also indirectly reproduced the common phenomenon ‘‘faster-is-slower effect’’ [10]. Next, we study the influence of the proportion with different behavior variation on the evacuation time. As we know, when encountering unexpected events, some people are very calm, while some people seem to be in an exceeding nervous state. In the above simulation, we assume that all pedestrians have the same tension. Now, 22 Cases with different types of behavior variations are listed in Table 1, here we call one proportional distribution as one Case. Fig. 9 shows the relationship between evacuation time and different cases at ρ = 0.3, 0.5, 0.8. It can be found that: (1) Case 1–3: At low and middle density, evacuation efficiency of Case 2 is lowest, different from that of Case 3 at high density; (2) Case 4–13: The variation range of the evacuation time from Case 4 to 8 is not obvious, which denotes the pedestrian proportion with different level of tensions has little effect on the evacuation time. But from Case 9 to 13, there is a gradual increase in evacuation time. All these imply that without the people of exceeding nervous in the crowd, i.e., µ = 0.9, to a certain extent, the evaluation efficiency will not be affected significantly;
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Fig. 8. The typical spatiotemporal patterns obtained at ρ = 0.7. µ = 0: (a) t = 10, (c) t = 30 and (e) t = 60; µ = 0.9: (b) t = 10, (d) t = 30 and (f) t = 60.
Fig. 9. The relationship between evacuation time T under different cases at ρ = 0.3, 0.5, 0.8.
(3) Case 14–18: The larger the proportion of pedestrian with high level of tension, the longer the evacuation time; (4) Case 19–22: The reduction of the proportion with µ = 0.9 will decrease the evacuation time sharply. At higher density, the change tendency is more obvious. In a word, at the same density, if the crowd contains normal pedestrian and one kind of behavior variation, proportion of pedestrian has little effect on evaluation time; if the crowds include two kinds of behavior variation, the larger the pedestrian proportion with exceeding nervous, the lower the evaluation efficiency.
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F. Guo et al. / Physica A 462 (2016) 630–640 Table 1 The proportion of pedestrians under different level of tensions at µ = 0, 0.6, 0.9.
µ=0 Case 1 Case 2 Case 3
100%
Case 4 Case 5 Case 6 Case 7 Case 8
90% 80% 70% 60% 50%
Case 9 Case 10 Case 11 Case 12 Case 13
90% 80% 70% 60% 50%
µ = 0.9
100% 100%
Case 14 Case 15 Case 16 Case 17 Case 18 Case 19 Case 20 Case 21 Case 22
µ = 0.6
50% 50% 50% 50%
10% 20% 30% 40% 50% 10% 20% 30% 40% 50% 90% 80% 70% 60% 50%
10% 20% 30% 40% 50%
10% 20% 30% 40%
40% 30% 20% 10%
Fig. 10. The relationship between the logarithm of evacuation time T and density ρ under different system sizes L = 10, 30, 50, 100, 300 with
µ = 0, 0.5, 0.7.
Finally, we further explore the system size effect on evacuation efficiency. Here we set one parameter D = W /L as constant 0.15. Fig. 10 gives the relationship between the logarithm of evacuation time T and density ρ under different system sizes L = 10, 30, 50, 100, 300 with µ = 0, 0.5, 0.7. It can be seen easily that at the same density, the evacuation time increases with the increase of the system size L, but the change tendency of curves is almost the same, which indicates that the system size has no obvious effect on evacuation efficiency. 4. Conclusion In order to investigate how the behavior variation under an emergency influences evacuation efficiency, based on the original cost potential model, we have presented an extended cost potential cellular automaton model describing behavior
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variation through introducing level of tension function. The numerical simulations are performed to investigate the effects of the magnitude of level of tension and the different pedestrian proportion affected by behavior variation on the evacuation process in a room. The evacuation mechanism at higher density caused by level of tension is explored. Through investigating the evacuation time and typical spatiotemporal patterns, we have drawn the following conclusions: (1) Compared with the normal evacuation, the behavior variation caused by the fear under the sudden situation has a greater impact on the evacuation efficiency; (2) The effect of level of tension on the evacuation efficiency is different in different density ranges. At low density, a moderate level of tension can improve the evacuation efficiency; however, at high density, the increasing tension will lead to an obvious reduction in evacuation efficiency; (3) During emergency evacuation, the pedestrians’ proportion with different behavior variation has a huge impact on the evacuation process, and the larger the pedestrian proportion with larger behavior variation, the lower the evacuation efficiency; (4) The system size has no obvious effect on evacuation efficiency. Although we have already attained some important conclusions, more considerations and investigations still need to be done. In this paper, simulations are only done in simple assumption from the psychological viewpoint and we have not simulated some complex surroundings. In the future, we will perform further corresponding research. Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 11262005 and 11562020), the Natural Science Foundation of Shanxi Province (201601D011013), the 131 Leading Talents Project of Shanxi Province (201509) and the Guangxi Natural Science Foundation (No. 2014GXNSFAA118007). References [1] D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys. 73 (2001) 1067–1141. [2] T. Nagatani, The physics of traffic jams, Rep. Progr. Phys. 65 (2002) 1331–1386. [3] D. Helbing, L. Buzna, A. Johansson, T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transp. Sci. 39 (2005) 1–24. [4] A. Schadschneider, W. Klingsch, H. Klupfel, T. Kretz, C. Rogsch, A. Seyfried, Evacuation dynamics: empirical results, modeling and applications, in: R.A. Meyers (Ed.), Encyclopedia of Complexity and System Science, Springer, Berlin Heidelberg, 2009, pp. 3142–3176. [5] L.F. Henderson, The statistics of crowd fluids, Nature 229 (1971) 381–383. [6] R.L. Hughes, A continuum theory for the flow of pedestrians, Transp. Res. B 36 (2002) 507–535. [7] S.P. Hoogendoorn, F. van Wageningen-Kessels, W. Daamen, D.C. Duives, M. Sarvi, Continuum theory for pedestrian traffic flow: Local route choice modelling and its implications, Transp. Res. C 59 (2015) 183–197. [8] Y. Jiang, W. Zhang, S. Zhou, Comparison study of the reactive and predictive dynamic models for pedestrian flow, Physica A 441 (2016) 51–61. [9] D. Helbing, P. Molnr, Social force model for pedestrian dynamics, Phys. Rev. E 51 (1995) 4282–4286. [10] D. Helbing, I. Farkas, T. Vicsek, Simulating dynamical features of escape panic, Nature 407 (2000) 487–490. [11] M. Chraibi, A. Seyfried, A. Schadschneider, Generalized centrifugal-force model for pedestrian dynamics, Phys. Rev. E 82 (2010) 046111. [12] G. Köster, F. Treml, M. Gödel, Avoiding numerical pitfalls in social force models, Phys. Rev. E 87 (2013) 063305. [13] T. Kretz, On oscillations in the social force model, Physica A 438 (2015) 272–285. [14] Y. Ma, R.K.K. Yuen, E.W.M. Lee, Effective leadership for crowd evacuation, Physica A 450 (2016) 333–341. [15] C. Burstedde, K. Klauck, A. Schadschneider, J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A 295 (2001) 507–525. [16] A. Kirchner, A. Schadschneider, Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics, Physica A 312 (2002) 260–276. [17] Y. Chen, N. Chen, Y. Wang, Z. Wang, G. Feng, Modeling pedestrian behaviors under attracting incidents using cellular automata, Physica A 432 (2015) 287–300. [18] Z. Fu, X. Zhou, K. Zhu, Y. Chen, Y. Zhuang, Y. Hu, L. Yang, C. Chen, J. Li, A floor field cellular automaton for crowd evacuation considering different walking abilities, Physica A 420 (2015) 294–303. [19] C. Feliciani, K. Nishinari, An improved Cellular Automata model to simulate the behavior of high density crowd and validation by experimental data, Physica A 451 (2016) 135–148. [20] S. Li, C. Zhai, L. Xie, Occupant evacuation and casualty estimation in a building under earthquake using cellular automata, Physica A 424 (2015) 152–167. [21] Q. Zhang, B. Han, Simulation model of pedestrian interactive behavior, Physica A 390 (2011) 636–646. [22] A. Kirchner, K. Nishinari, A. Schadschneider, Friction effects and clogging in a cellular automaton model for pedestrian dynamics, Phys. Rev. E 67 (2003) 056122. [23] H.J. Huang, R.Y. Guo, Static floor field and exit choice for pedestrian evacuation in rooms with internal obstacles and multiple exits, Phys. Rev. E 78 (2008) 021131. [24] D. Yanagisawa, A. Kimura, A. Tomoeda, R. Nishi, Y. Suma, K. Ohtsuka, K. Nishinari, Introduction of frictional and turning function for pedestrian outflow with an obstacle, Phys. Rev. E 80 (2009) 036110. [25] H. Yue, H. Guan, C. Shao, X. Zhang, Simulation of pedestrian evacuation with asymmetrical exits layout, Physica A 390 (2011) 198–207. [26] Z. Zainuddin, E.A. Lim, Intelligent exit-selection behaviors during a room evacuation, Chin. Phys. Lett. 29 (2012) 018901. [27] H. Tian, L. Dong, Y. Xue, Influence of the exits’ configuration on evacuation process in a room without obstacle, Physica A 420 (2015) 164–178. [28] Z. Wang, B. Song, Y. Qin, W. Zhu, L. Jia, Effect of vertical grouping behavior on pedestrian evacuation efficiency, Physica A 392 (2013) 4874–4883. [29] J. Hu, L. You, J. Wei, M.S. Gu, Y. Liang, The effects of group and position vacancy on pedestrian evacuation flow model, Phys. Lett. A 378 (2014) 1913–1918. [30] D. Li, B. Han, Behavioral effect on pedestrian evacuation simulation using cellular automata, Saf. Sci. 80 (2015) 41–55. [31] P. Zhang, X.X. Jian, S.C. Wong, K. Choi, Potential field cellular automata model for pedestrian flow, Phys. Rev. E 85 (2012) 021119. [32] X.X. Jian, S.C. Wong, P. Zhang, K. Choi, H. Li, X. Zhang, Perceived cost potential field cellular automata model with an aggregated force field for pedestrian dynamics, Transp. Res. C 42 (2014) 200–210.
640 [33] [34] [35] [36]
F. Guo et al. / Physica A 462 (2016) 630–640
Y. Tajima, T. Nagatani, Scaling behavior of crowd flow outside a hall, Physica A 292 (2001) 545–554. D. Helbing, M. Isobe, T. Nagatani, K. Takimoto, Lattice gas simulation of experimentally studied evacuation dynamics, Phys. Rev. E 67 (2003) 067101. H. Kuang, X.L. Li, T. Song, S.Q. Dai, Analysis of pedestrian dynamics in counter flow via an extended lattice gas model, Phys. Rev. E 78 (2008) 066117. D. Helbing, I. Farkas, P. Molnar, T. Vicsek, Simulation of pedestrian crowds in normal and evacuation situations, in: M. Schreckenberg, S.D. Sharma (Eds.), Pedestrian and Evacuation Dynamics, Publisher Springer, 2002, pp. 21–58. [37] J.P. Keating, The myth of panic, Fire J. 76 (1982) 57–62. [38] C. Rogsch, M. Schreckenberg, E. Tribble, W. Klingsch, T. Kretz, Was it panic? An overview about mass-emergencies and their origin all over the world for recent years, in: W.W.F. Klingsch, C. Rogsch, A. Schadschneider, M. Schreckenberg (Eds.), Pedestrian and Evacuation Dynamics 2008, Publisher Springer, 2010, pp. 743–755. [39] X. Wang, Z.M. Xie, X.J. Guan, Study on micro-simulation of occupant evacuation in panic, in: Proceedings of 2010 International Conference on Future Information Technology and Management Engineering, 2010, pp. 40–43 (in Chinese).
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
An extended cost potential field cellular automata model considering behavior variation of pedestrian flow Fang Guo a , Xingli Li a,c,∗ , Hua Kuang b , Yang Bai a , Huaguo Zhou c a
School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, 030024, China
b
College of Physical Science and Technology, Guangxi Normal University, Guilin 541004, China
c
Department of Civil Engineering, Auburn University, Auburn, AL 36849, USA
highlights • • • •
The extended cost potential CA model describing behavior variation is presented. The quantitative formula between behavior variation and level of tension is given. The influences of different factors on evacuation efficiency are explored. Some important results are obtained from the numerical simulations.
article
info
Article history: Received 14 December 2015 Received in revised form 18 May 2016 Available online 24 June 2016 Keywords: Pedestrian flow Cost potential function field Behavior variation Cellular automaton Evacuation time
abstract The original cost potential field cellular automata describing normal pedestrian evacuation is extended to study more general evacuation scenarios. Based on the cost potential field function, through considering the psychological characteristics of crowd under emergencies, the quantitative formula of behavior variation is introduced to reflect behavioral changes caused by psychology tension. The numerical simulations are performed to investigate the effects of the magnitude of behavior variation, the different pedestrian proportions with different behavior variation and other factors on the evacuation efficiency and process in a room. The spatiotemporal dynamic characteristic during the evacuation process is also discussed. The results show that compared with the normal evacuation, the behavior variation under an emergency does not necessarily lead to the decrease of the evacuation efficiency. At low density, the increase of the behavior variation can improve the evacuation efficiency, while at high density, the evacuation efficiency drops significantly with the increasing amplitude of the behavior variation. In addition, the larger proportion of pedestrian affected by the behavior variation will prolong the evacuation time. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Recently, the number of accidents caused by crowded people or emergencies has increased year by year. Pedestrian modeling has become one of the most exciting fields in traffic science and engineering [1–4]. Understanding pedestrian flow characteristics beforehand is extremely important in emergency management to improve evacuation procedures and relevant regulations. The dynamic properties of pedestrian crowds, including various self-organization phenomena, have
∗
Corresponding author at: School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, 030024, China. E-mail address: [email protected] (X. Li).
http://dx.doi.org/10.1016/j.physa.2016.06.103 0378-4371/© 2016 Elsevier B.V. All rights reserved.
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been observed and successfully reproduced by various physical methods. However, it is generally known that evacuation exercise under emergency may be difficult than normal pedestrian flow because of the danger and possible crowd disaster caused by incidents. To solve this problem, various modeling approaches to study evacuation behavior have been proposed [5–36]. Generally speaking, pedestrian flow models can be classified into two categories, macroscopic [5–8] and microscopic [9–36]. In macroscopic model, pedestrians are treated as a fluid with the use of partial differential equation, where the dynamic characteristics of the crowd flow are described by average speed, density, location and time. Microscopic model can be mainly divided into three groups: continuous model with a representative of the social force (SF) model [9–14] and discrete model represented by the cellular automata (CA) model [15–32] and lattice gas model [33–35]. The floor field (FF) model first proposed by C. Burstedde et al. [15], which is more flexible to simulate individual pedestrian motion and can reproduce collective behavior of pedestrian dynamics observed experimentally and described by other more complex models, has become one of the most widely used CA models in evacuation research. In general, there are two types of floor fields: the static floor field S and the dynamic floor field D. The static field depends only on the distance measure (from a cell to the destination), and thus S remains unchanged in the evolution. The dynamic field reflects the virtual tracks left by moving pedestrians. Since then, lots of extended or modified FF models [16–32] are proposed by researchers in different aspects, such as the interaction among pedestrians or pedestrians and obstacles [21–24], layout effects of exit or door [25–27], behavioral or group effects [28–30]. In 2012, Zhang et al. established a cellular automata model of pedestrian flow that defines a cost potential field, which takes into account the costs of travel time and discomfort in the journey, for a pedestrian to move to an empty neighboring cell [31]. Without the discomfort term, the resulting cost potential in a cell would simply measure the distance between the cell and the destination, which is independent of time and similar to the static field in the floor field CA model (see Ref. [15]). With the discomfort term, the cost distribution increases with the local density, which is reconstructed at each time step and, thus, is time dependent. In this case, the resulting cost potential field is similar to the dynamic field in the floor field CA model. In 2014, Jian et al. proposed a perceived potential field cellular automata model with an aggregated force field to simulate the pedestrian evacuation in a walking domain with poor visibility or complex geometries [32]. These above mentioned models can generally reproduce pedestrian dynamics under normal situation. However, Helbing et al. [36] found that in situations of escape panics, individuals are getting nervous, i.e., they tend to develop blind actionism. Furthermore, people try to move considerably faster than normal, etc. They think that the transition between the ‘‘rational’’ normal behavior and the apparently ‘‘irrational’’ panic behavior is controlled by a single parameter, the ‘‘nervousness’’, which influences fluctuation strengths, desired speeds, and the tendency of herding. Here it should be noted that there is no precise accepted definition of panic although in the media usually aspects like selfish, asocial or even completely irrational behavior and contagion that affects large groups are associated with this concept [37]. In 2008, Rogsch et al. performed a detailed discussion about ‘‘panic’’ and based on the difficulties of different definitions, they used the following definition to investigate the real origins of some mass-emergencies, namely, ‘‘Panic: People flight based on a sudden subjective or ‘infected’ fear; People are moving imprudently; The cause of this movement cannot be recognized by an outsider’’ [38]. Up to now, the terminology ‘‘panic’’ is highly controversial and usually avoided. In this manuscript, we use ‘‘fear’’ to describe the above mentioned pedestrians’ psychological characteristics when they confront an emergency. Therefore, the previous cost potential field CA model is used in normal situation is insufficient to describe pedestrian emergency evacuation. In this paper, we mainly focus on how to extend the original cost potential field CA model for pedestrian flow to emergency situations. The paper is organized as follows. In Section 2, we formulate an extended cost potential field cellular automata model considering behavior variation from nervousness. Section 3 gives simulation results and corresponding discussion, followed by conclusions in the final section.
2. Model The model follows the basic rules of the cost potential field CA model. We consider a room divided into L × L finite twodimensional grids. The outer lattices denote the room wall. Each of the other grids can be empty or occupied by a pedestrian and is equivalent to a cell whose size can be regarded as 0.4 × 0.4 m2 [15], where L and W are the width of the room and the exit, respectively. As shown in Fig. 1, the black part and the red solid circle denote the wall and pedestrian, respectively. The exit is located at the edge of the wall. In this paper, the Moore neighborhood [15] is adopted and each occupied cell has eight neighboring cells, corresponding to nine probabilities for the pedestrian in the occupied cell to update his or her position (see Fig. 2). Define the neighborhood set of (i0 , j0 ) as Si,j = {(i, j)| |i0 − i| ≤ 1, |j0 − j| ≤ 1}, in which an empty cell set is S i,j = {(i′ , j′ )|(i′ , j′ ) ∈ Si,j , (i′ , j′ ) is an empty cell}.
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Fig. 1. Sketch of model. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 2. (a) An occupied cell (0, 0) and its eight neighboring cells; (b) the corresponding probabilities for occupation in the next update step.
2.1. The original cost potential field The cost potential in a cell is defined as the minimal cost for traveling from this cell to the destination [31]. We assume that each pedestrian is fully aware of the surroundings and the destination, so a pedestrian can choose one or more empty neighbors as his or her target cell, which is expected to minimize the traveling cost. The cost potential function includes two terms: one is the cost of traveling time determined by the pedestrian’s walking speed and the distance from the present position to the target position, the other is the cost of discomfort depending on the surrounding density. Define the cost distribution in a cell (x, y) as
τ (x, y, t ) =
1
ve (ρ(x, y, t ))
+ g (ρ(x, y, t )).
(1)
The first term in Eq. (1) is related to the traveling time, with ve (ρ) being the velocity–density relationship and ve′ (ρ) < 0; the second term represents the discomfort of pedestrians, with g (0) = 0 and g ′ (ρ) > 0. Based on the basic rules of CA model, ve (ρ(x, y, t )) = vmax , thereby assuming that 1
+ g0 ρ γ vmax here we set g0 = 0.075 and γ = 2. τ (x, y, t ) =
(2)
The total cost of walking from cell (i, j) to (i0 , j0 ) is
φ(x, y, t ) =
(x0 ,y0 ) (x,y)
(τ (x, y, t )x′ (s)dx + τ (x, y, t )y′ (s)dy).
(3)
We assume that the integral is independent of the path, i.e.
φx (x, y, t ) = −τ (x, y, t )x′ (s),
φy (x, y, t ) = −τ (x, y, t )y′ (s).
2.2. Quantitative description of behavior variation Here, we introduce a new parameter µ(0 ≤ µ ≤ 1) similar to ‘‘nervousness’’ parameter in Ref. [36] to describe the level of tension of pedestrian under emergency. This parameter will influence the intensity of individual behavior variation and herd behavior, etc. During the evacuation process, the level of tension at time t can be quantitatively described as [39]
µ(t ) =
v0 (t ) − v0 (0) vmax (t ) − v0 (0)
(4)
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Fig. 3. The relationship between behavior variation η and level of tension µ.
where v0 (t ) represents the current speed, v0 (0) is the normal speed before the fear, and vmax denotes the maximum ideal speed. When the level of tension is larger than the threshold of the critical value, the pedestrian will be completely controlled by the fear, causing a huge behavior variation. According to the above analysis, we introduce behavior variation parameter η(0 ≤ η ≤ 1), which is defined as follows: 0 k1 µ + a1 exp(k2 µ) + a2
η=
0 ≤ µ ≤ 0.05 0.05 < µ ≤ 0.6 0.6 < µ ≤ 1.
(5)
Eq. (5) shows that when the level of tension is very small (µ ≤ 0.05), the behavior variation caused by the level of tension can be neglected, which means pedestrians are in a rational state and correspondingly they will select the shortest route to the exit reasonably, i.e., the empty cell with the largest Si,j will be the target cell. When there are many target cells, the selection probability is same. When the level of tension is 0.05 < µ ≤ 0.6, it will lead to a linear increase of the behavior variation. At this time, some pedestrians have priority over others, the movement probability towards the diagonal direction will increase and pedestrians will update their positions according to the rules of the cost potential field. When the level of tension is µ > 0.6, namely, the level of tension exceeds the critical value of psychological endurance, the behavior variation will have an exponential growth with the level of tension. We set the parameters in Eq. (5) as k1 = 0.247, k2 = 0.98, a1 = −0.012, a2 = −1.66 and then obtain the relationship between the behavior variation and the level of tension (see Fig. 3). Compared to normal evacuation, behavior variation under an emergency can also affect the cost of considering the comfort, which will lead to an increase in the cost of discomfort, so we extend the original cost of discomfort to two parts: one is related to the density, the other is caused by the behavior variation. The cost function (2) is redefined as
τ (x, y, t ) =
1
vmax
+ aρ α + b η β
(6)
here we choose the corresponding parameters as a = 0.0125, b = 0.0625, α = 2, β = 3. 2.3. Moving probability Assume (0, 0) is the coordinate of each occupied cell, then the procedure for determining the probabilities pi,j is specified as follows: If (0, −1) is empty, using the forward difference quotient, the difference quotient is φ0′ ,−1 = |(φ0n,−1 − φ0n,0 )/di,j |; if (0, 1) is empty, using the backward difference quotient, the difference quotient is φ0′ ,1 = |(φ0n,0 − φ0n,1 )/di,j |. Compute the
difference quotient φi′,j = |(φin,j − φ0n,0 )/di,j | for (i, j) ∈ S i,j , where di,j = (i − 0)2 + (j − 0)2 is the distance between cell (i, j) and cell (0, 0), φin,j is the potential field values of cell (i, j) in the nth step. Define the set
Sm = {(i, j)|φi′,j = min φi′,j } (i,j)∈S i,j
and probability
pi,j =
0 1 |S | m 1
(i, j) ̸∈ Sm (i, j) ∈ Sm (i, j) = (0, 0),
where |Sm | is the number of elements in Sm .
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Fig. 4. Resolve occupation conflict.
2.4. Resolve conflict Every empty cell (i, j) may be a target cell of m(0 ≤ m ≤ 8) in the kth step. As shown in Fig. 4, which gives rise to an occupation conflict if m ≥ 2. To resolve a possible conflict, pedestrian will select a large probability to move, while when the value is equal, they will move randomly. When the two pedestrians simultaneously choose the same target grid (see Fig. 4(a)), the corresponding moving probability determines which pedestrian will occupy the lattice. If pa > pb , cell a moves to the target location, while cell b still remains static (see Fig. 4(b)); if pa < pb , cell b moves to the target location, while cell a still remains static (see Fig. 4(c)). 2.5. Updating rules (1) When the level of tension is µ ≤ 0.05, walkers will select the shortest route to the exit reasonably. Here, static field is introduced to describe the distance between each cell and the exit. It can be obtained according to the following function [16]: S(i,j) = max ( min
(i′ ,j′ )∈Ω (i0 ,j0 )∈Γ0
(i0 − i′ )2 + (j0 − j′ )2 − min
(i0 ,j0 )∈Γ0
(i0 − i)2 + (j0 − j)2 )
(7)
where the first term is the minimum distance between the neighbor cell (i′ , j′ ) and cell (i0 , j0 ) of exit; the last term is the minimum distance between the cell (i, j) and cell (i0 , j0 ) of exit; Ω and Γ0 denote walking area set and exit set, respectively. Assuming that each element is empty or occupied and each pedestrian walks in a cell length in a time step or is static; (2) When the level of tension is 0.05 < µ ≤ 0.6, pedestrians will update their positions according to the rules of the potential function field. Under the influence of behavior variation, some pedestrians will have priority over others, the movement probability towards the diagonal direction of the cell will increase, and the moving speed along the diagonal direction also increases to v = 1.5 m/s; (3) When the level of tension is µ > 0.6, the level of tension exceeds the psychological threshold, and pedestrians’ behavior becomes blind, such as obviously increasing walking speed, walking backward far away from the exit, etc. Therefore, we adopt the extended Moore neighborhood during the moving process (see Fig. 5), here AB denotes the exit, and the arrow represents the possible moving directions. Different from previous Moore neighborhood [15], the cell’s length along diagonal direction is roughly approximated by a cell length and the moving speed is v = 1.5 m/s. Besides, moving two cells in one time step is considered and the corresponding velocity is v = 2 m/s. At the same time, pedestrians can walk backward directly or diagonally, for example, (0, 0) → (0, 2) or (0, 0) → (1, 1) or (0, 0) → (−1, 1). The specific rules of the model are as follows: (1) Give the random initial distribution of pedestrians in a room; (2) Calculate the moving probability according to the neighbor states, judge the next target cell, then determine the target location; (3) Update the location. Move if the target location is empty and remain static if the target is selected; (4) Remove the pedestrian arriving at the exit automatically from the calculation; (5) Repeat the above rules until the room is empty. 3. Simulation results and discussions Initially, all walkers are distributed randomly at the sites on the square lattice with the density ρ = N /(L − 2) × (L − 2), here note that considering the outer lattices are occupied by the wall. Following the above rules, all the cells’ states are parallel updated. In the simulations, in order to remove the transient effect, the evacuation time was obtained by averaging over 20 runs. The related parameters are taken as L = 20 and W = 3 unless otherwise mentioned. Firstly, we study the influence of the behavior variation magnitude on evacuation time. Fig. 6 gives the relationship between the evacuation time T and density ρ under different level of tensions µ = 0, 0.1, 0.3, 0.5, 0.7, 0.9. From Fig. 6, it
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Fig. 5. An extended Moore neighborhood.
Fig. 6. The relationship between evacuation time T and density ρ under different level of tensions µ = 0, 0.1, 0.3, 0.5, 0.7, 0.9.
can be found that with the increase of density in the room, the evacuation time is prolonged, but under different densities, the effect of level of tension µ on the evacuation time is different. As ρ < 0.2, with the increase of µ, the evacuation time gradually decreases. The evacuation time of µ = 0 is longer than that of µ ̸= 0. This is because at low density, on the one hand, the collision and interaction among pedestrians can be ignored; on the other hand, the behavior variation caused by nervousness can also lead to the increase of pedestrian speed, which makes pedestrian leave the room quickly. As ρ > 0.2, the evacuation time µ ̸= 0 is larger than that of µ = 0, but the change of evacuation time is different in different density ranges. For example, as 0.2 < ρ < 0.5, the evacuation time of µ = 0.7, 0.9 is less than that of µ = 0.1, 0.3, 0.5, while with the increase of density, especially ρ > 0.6, the evacuation time of µ = 0.7, 0.9 is more than that of µ = 0.1, 0.3, 0.5, i.e., the difference of evacuation time under different level of tensions becomes significant. This means that at higher density, the cost potential from discomfort becomes large, which reflects the strong interaction, the more serious competition and congestion among pedestrians caused by the behavior variation inevitably results in the reduction of the evacuation efficiency. In addition, under different densities, the distribution of evacuation time appears to the obvious range under different level of tensions. Typically, the curves of evacuation time at ρ = 0.5 can be roughly divided into two parts: µ = 0 and µ > 0, which means that the evacuation time under different level of tensions (µ > 0) shows little difference; the curves of evacuation time at ρ = 0.7, 0.9 obviously belong to three parts: µ = 0, µ = 0.1, 0.3, 0.5 and µ = 0.7, 0.9, which gives the key value of level of tension influencing evacuation time. A valuable conclusion can be drawn that during the evacuation, to control the pedestrians behavior variation reasonably to a certain range is significant to improve evacuation efficiency. In order to explore the macroscopic behavior characteristics, Fig. 7 gives the typical spatiotemporal patterns during the evacuation process at initial density ρ = 0.1. We can see easily that with the evolution, pedestrians quickly move towards the exit and gather near the exit. This is consistent with the results of pedestrian evacuation [10]. Compared with the normal situation (µ = 0), it can be directly seen that pedestrians under the emergency situation can move away from the room more quickly. From Fig. 7(d) and (f), we can find the number of non-evacuated people of µ = 0.9 is significantly less than that of µ = 0.3 at t = 10. Pedestrian can move more flexibly and more quickly at a higher level of tension. Therefore, at low density, the increase of the level of tension can improve evacuation efficiency. In order to gain a deeper insight into the microscopic mechanism why a larger level of tension could reduce evacuation efficiency at high density obviously, Fig. 8 gives the typical spatiotemporal patterns obtained at ρ = 0.7. At the early stage,
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Fig. 7. The typical spatiotemporal patterns obtained at ρ = 0.1. µ = 0: (a) t = 5 and (b) t = 10; µ = 0.3: (c) t = 5 and (d) t = 10; µ = 0.9: (e) t = 5 and (f) t = 10.
pedestrians move towards the exit, form a jamming blockage at the exit (see Fig. 8(a) and (b)). Under the normal situation, they gather at the exit, displayed as a jamming pattern similar to a vase, and dissolve gradually (see Fig. 8(a), (c) and (e)). This is because pedestrians join a queue rationally ahead of the exit and escape in turn, which decreases the interaction among pedestrians and weakens the effect of blockage, and thus improves the evacuation efficiency considerably. Being a larger level of tension (µ = 0.9), pedestrians are in an extremely nervous state and then friction, as a kind of an internal local pressure between the pedestrians, becomes remarkable in regions of high density. The nervous behaviors among pedestrians and the rapid expansion of friction easily cause a variety of congestion ‘‘bottleneck’’, and thus arching and clogging near the exit can be observed. Moreover, alternative cells are often overlooked or not efficiently used in escape situations and a few empty cells in the crowed pedestrians can be found (see Fig. 8(b), (d) and (f)). All these reflect pedestrians with exceeding nervous try to move considerably faster than normal, but in fact they get the opposite of what they expect. This also indirectly reproduced the common phenomenon ‘‘faster-is-slower effect’’ [10]. Next, we study the influence of the proportion with different behavior variation on the evacuation time. As we know, when encountering unexpected events, some people are very calm, while some people seem to be in an exceeding nervous state. In the above simulation, we assume that all pedestrians have the same tension. Now, 22 Cases with different types of behavior variations are listed in Table 1, here we call one proportional distribution as one Case. Fig. 9 shows the relationship between evacuation time and different cases at ρ = 0.3, 0.5, 0.8. It can be found that: (1) Case 1–3: At low and middle density, evacuation efficiency of Case 2 is lowest, different from that of Case 3 at high density; (2) Case 4–13: The variation range of the evacuation time from Case 4 to 8 is not obvious, which denotes the pedestrian proportion with different level of tensions has little effect on the evacuation time. But from Case 9 to 13, there is a gradual increase in evacuation time. All these imply that without the people of exceeding nervous in the crowd, i.e., µ = 0.9, to a certain extent, the evaluation efficiency will not be affected significantly;
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Fig. 8. The typical spatiotemporal patterns obtained at ρ = 0.7. µ = 0: (a) t = 10, (c) t = 30 and (e) t = 60; µ = 0.9: (b) t = 10, (d) t = 30 and (f) t = 60.
Fig. 9. The relationship between evacuation time T under different cases at ρ = 0.3, 0.5, 0.8.
(3) Case 14–18: The larger the proportion of pedestrian with high level of tension, the longer the evacuation time; (4) Case 19–22: The reduction of the proportion with µ = 0.9 will decrease the evacuation time sharply. At higher density, the change tendency is more obvious. In a word, at the same density, if the crowd contains normal pedestrian and one kind of behavior variation, proportion of pedestrian has little effect on evaluation time; if the crowds include two kinds of behavior variation, the larger the pedestrian proportion with exceeding nervous, the lower the evaluation efficiency.
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F. Guo et al. / Physica A 462 (2016) 630–640 Table 1 The proportion of pedestrians under different level of tensions at µ = 0, 0.6, 0.9.
µ=0 Case 1 Case 2 Case 3
100%
Case 4 Case 5 Case 6 Case 7 Case 8
90% 80% 70% 60% 50%
Case 9 Case 10 Case 11 Case 12 Case 13
90% 80% 70% 60% 50%
µ = 0.9
100% 100%
Case 14 Case 15 Case 16 Case 17 Case 18 Case 19 Case 20 Case 21 Case 22
µ = 0.6
50% 50% 50% 50%
10% 20% 30% 40% 50% 10% 20% 30% 40% 50% 90% 80% 70% 60% 50%
10% 20% 30% 40% 50%
10% 20% 30% 40%
40% 30% 20% 10%
Fig. 10. The relationship between the logarithm of evacuation time T and density ρ under different system sizes L = 10, 30, 50, 100, 300 with
µ = 0, 0.5, 0.7.
Finally, we further explore the system size effect on evacuation efficiency. Here we set one parameter D = W /L as constant 0.15. Fig. 10 gives the relationship between the logarithm of evacuation time T and density ρ under different system sizes L = 10, 30, 50, 100, 300 with µ = 0, 0.5, 0.7. It can be seen easily that at the same density, the evacuation time increases with the increase of the system size L, but the change tendency of curves is almost the same, which indicates that the system size has no obvious effect on evacuation efficiency. 4. Conclusion In order to investigate how the behavior variation under an emergency influences evacuation efficiency, based on the original cost potential model, we have presented an extended cost potential cellular automaton model describing behavior
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variation through introducing level of tension function. The numerical simulations are performed to investigate the effects of the magnitude of level of tension and the different pedestrian proportion affected by behavior variation on the evacuation process in a room. The evacuation mechanism at higher density caused by level of tension is explored. Through investigating the evacuation time and typical spatiotemporal patterns, we have drawn the following conclusions: (1) Compared with the normal evacuation, the behavior variation caused by the fear under the sudden situation has a greater impact on the evacuation efficiency; (2) The effect of level of tension on the evacuation efficiency is different in different density ranges. At low density, a moderate level of tension can improve the evacuation efficiency; however, at high density, the increasing tension will lead to an obvious reduction in evacuation efficiency; (3) During emergency evacuation, the pedestrians’ proportion with different behavior variation has a huge impact on the evacuation process, and the larger the pedestrian proportion with larger behavior variation, the lower the evacuation efficiency; (4) The system size has no obvious effect on evacuation efficiency. Although we have already attained some important conclusions, more considerations and investigations still need to be done. In this paper, simulations are only done in simple assumption from the psychological viewpoint and we have not simulated some complex surroundings. In the future, we will perform further corresponding research. Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 11262005 and 11562020), the Natural Science Foundation of Shanxi Province (201601D011013), the 131 Leading Talents Project of Shanxi Province (201509) and the Guangxi Natural Science Foundation (No. 2014GXNSFAA118007). References [1] D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys. 73 (2001) 1067–1141. [2] T. Nagatani, The physics of traffic jams, Rep. Progr. Phys. 65 (2002) 1331–1386. [3] D. Helbing, L. Buzna, A. Johansson, T. Werner, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transp. Sci. 39 (2005) 1–24. [4] A. Schadschneider, W. Klingsch, H. Klupfel, T. Kretz, C. Rogsch, A. Seyfried, Evacuation dynamics: empirical results, modeling and applications, in: R.A. Meyers (Ed.), Encyclopedia of Complexity and System Science, Springer, Berlin Heidelberg, 2009, pp. 3142–3176. [5] L.F. Henderson, The statistics of crowd fluids, Nature 229 (1971) 381–383. [6] R.L. Hughes, A continuum theory for the flow of pedestrians, Transp. Res. B 36 (2002) 507–535. [7] S.P. Hoogendoorn, F. van Wageningen-Kessels, W. Daamen, D.C. Duives, M. Sarvi, Continuum theory for pedestrian traffic flow: Local route choice modelling and its implications, Transp. Res. C 59 (2015) 183–197. [8] Y. Jiang, W. Zhang, S. Zhou, Comparison study of the reactive and predictive dynamic models for pedestrian flow, Physica A 441 (2016) 51–61. [9] D. Helbing, P. Molnr, Social force model for pedestrian dynamics, Phys. Rev. E 51 (1995) 4282–4286. [10] D. Helbing, I. Farkas, T. Vicsek, Simulating dynamical features of escape panic, Nature 407 (2000) 487–490. [11] M. Chraibi, A. Seyfried, A. Schadschneider, Generalized centrifugal-force model for pedestrian dynamics, Phys. Rev. E 82 (2010) 046111. [12] G. Köster, F. Treml, M. Gödel, Avoiding numerical pitfalls in social force models, Phys. Rev. E 87 (2013) 063305. [13] T. Kretz, On oscillations in the social force model, Physica A 438 (2015) 272–285. [14] Y. Ma, R.K.K. Yuen, E.W.M. Lee, Effective leadership for crowd evacuation, Physica A 450 (2016) 333–341. [15] C. Burstedde, K. Klauck, A. Schadschneider, J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton, Physica A 295 (2001) 507–525. [16] A. Kirchner, A. Schadschneider, Simulation of evacuation processes using a bionics-inspired cellular automaton model for pedestrian dynamics, Physica A 312 (2002) 260–276. [17] Y. Chen, N. Chen, Y. Wang, Z. Wang, G. Feng, Modeling pedestrian behaviors under attracting incidents using cellular automata, Physica A 432 (2015) 287–300. [18] Z. Fu, X. Zhou, K. Zhu, Y. Chen, Y. Zhuang, Y. Hu, L. Yang, C. Chen, J. Li, A floor field cellular automaton for crowd evacuation considering different walking abilities, Physica A 420 (2015) 294–303. [19] C. Feliciani, K. Nishinari, An improved Cellular Automata model to simulate the behavior of high density crowd and validation by experimental data, Physica A 451 (2016) 135–148. [20] S. Li, C. Zhai, L. Xie, Occupant evacuation and casualty estimation in a building under earthquake using cellular automata, Physica A 424 (2015) 152–167. [21] Q. Zhang, B. Han, Simulation model of pedestrian interactive behavior, Physica A 390 (2011) 636–646. [22] A. Kirchner, K. Nishinari, A. Schadschneider, Friction effects and clogging in a cellular automaton model for pedestrian dynamics, Phys. Rev. E 67 (2003) 056122. [23] H.J. Huang, R.Y. Guo, Static floor field and exit choice for pedestrian evacuation in rooms with internal obstacles and multiple exits, Phys. Rev. E 78 (2008) 021131. [24] D. Yanagisawa, A. Kimura, A. Tomoeda, R. Nishi, Y. Suma, K. Ohtsuka, K. Nishinari, Introduction of frictional and turning function for pedestrian outflow with an obstacle, Phys. Rev. E 80 (2009) 036110. [25] H. Yue, H. Guan, C. Shao, X. Zhang, Simulation of pedestrian evacuation with asymmetrical exits layout, Physica A 390 (2011) 198–207. [26] Z. Zainuddin, E.A. Lim, Intelligent exit-selection behaviors during a room evacuation, Chin. Phys. Lett. 29 (2012) 018901. [27] H. Tian, L. Dong, Y. Xue, Influence of the exits’ configuration on evacuation process in a room without obstacle, Physica A 420 (2015) 164–178. [28] Z. Wang, B. Song, Y. Qin, W. Zhu, L. Jia, Effect of vertical grouping behavior on pedestrian evacuation efficiency, Physica A 392 (2013) 4874–4883. [29] J. Hu, L. You, J. Wei, M.S. Gu, Y. Liang, The effects of group and position vacancy on pedestrian evacuation flow model, Phys. Lett. A 378 (2014) 1913–1918. [30] D. Li, B. Han, Behavioral effect on pedestrian evacuation simulation using cellular automata, Saf. Sci. 80 (2015) 41–55. [31] P. Zhang, X.X. Jian, S.C. Wong, K. Choi, Potential field cellular automata model for pedestrian flow, Phys. Rev. E 85 (2012) 021119. [32] X.X. Jian, S.C. Wong, P. Zhang, K. Choi, H. Li, X. Zhang, Perceived cost potential field cellular automata model with an aggregated force field for pedestrian dynamics, Transp. Res. C 42 (2014) 200–210.
640 [33] [34] [35] [36]
F. Guo et al. / Physica A 462 (2016) 630–640
Y. Tajima, T. Nagatani, Scaling behavior of crowd flow outside a hall, Physica A 292 (2001) 545–554. D. Helbing, M. Isobe, T. Nagatani, K. Takimoto, Lattice gas simulation of experimentally studied evacuation dynamics, Phys. Rev. E 67 (2003) 067101. H. Kuang, X.L. Li, T. Song, S.Q. Dai, Analysis of pedestrian dynamics in counter flow via an extended lattice gas model, Phys. Rev. E 78 (2008) 066117. D. Helbing, I. Farkas, P. Molnar, T. Vicsek, Simulation of pedestrian crowds in normal and evacuation situations, in: M. Schreckenberg, S.D. Sharma (Eds.), Pedestrian and Evacuation Dynamics, Publisher Springer, 2002, pp. 21–58. [37] J.P. Keating, The myth of panic, Fire J. 76 (1982) 57–62. [38] C. Rogsch, M. Schreckenberg, E. Tribble, W. Klingsch, T. Kretz, Was it panic? An overview about mass-emergencies and their origin all over the world for recent years, in: W.W.F. Klingsch, C. Rogsch, A. Schadschneider, M. Schreckenberg (Eds.), Pedestrian and Evacuation Dynamics 2008, Publisher Springer, 2010, pp. 743–755. [39] X. Wang, Z.M. Xie, X.J. Guan, Study on micro-simulation of occupant evacuation in panic, in: Proceedings of 2010 International Conference on Future Information Technology and Management Engineering, 2010, pp. 40–43 (in Chinese).