Macroscopic modeling of pedestrian flow based on a second-order predictive dynamic model

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Macroscopic modeling of pedestrian flow based on a second-order predictive dynamic model Yan-Qun Jiang a,b,∗, Ren-Yong Guo c, Fang-Bao Tian d, Shu-Guang Zhou e a

Department of Mathematics, Southwest University of Science and Technology, Mianyang, Sichuan, China Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing, China c College of Computer Science, Inner Mongolia University, Hohhot, China d School of Engineering and Information Technology, University of New South Wales Canberra, ACT 2600, Australia e China Aerodynamics Research and Development Center, Mianyang, Sichuan, China b

a r t i c l e

i n f o

Article history: Received 18 November 2015 Revised 8 June 2016 Accepted 20 June 2016 Available online xxx Keywords: Second-order model Pedestrian flow Predictive dynamic user-equilibrium Non-equilibrium phenomena Stop-and-go waves

a b s t r a c t This work presents a second-order predictive dynamic model for pedestrian flow to investigate movement patterns and non-equilibrium phenomena in pedestrian traffic. This model is described as a system of nonlinear hyperbolic conservation laws with relaxation under the hypothesis that a group of pedestrians are regarded as a continuous anisotropic medium. The desired or preferred walking direction of pedestrians is assumed to minimize the total actual walking cost based on predictive traffic conditions, which satisfies the predictive dynamic user-equilibrium assignment. To solve this model, a cell-centered finite volume method for hyperbolic conservation laws coupled with a self-adaptive method of successive averages for an arisen discrete fixed point problem is adopted. The proposed model and algorithm are validated by comparing the results carried out by the model with experimental observations under non-congested conditions. Numerical examples are designed to investigate macroscopic features and path-choice behaviors of pedestrian flow. Numerical results indicate that the proposed model is able to reproduce some complex nonlinear phenomena in pedestrian traffic, such as the formation of congestions and stopand-go waves. © 2016 Published by Elsevier Inc.

1. Introduction In any large crowd, people could be injured and even lose lives because of the dynamics of the crowd’s behavior [1]. The importance of understanding pedestrian crowd dynamics and simulating behaviors and movements of the crowds has regained a significant level of interest. There are typically three types of crowd motion simulation models based on different descriptive details: microscopic models (like social force models [2–4], cellular automaton models [5–8] and agent-based models [9–11]), mesoscopic (kinetic) models [12,13] and macroscopic models [14–20]. None of the three types of models is completely satisfactory since various technical and conceptual advantages and drawbacks are linked to modeling at each scale [13]. In the case of a large crowd, macroscopic characteristics of pedestrian flow (e.g., speed, density and flow) are of

∗

Corresponding author at: Department of Mathematics, Southwest University of Science and Technology, Mianyang, Sichuan, China. Tel.: +86 8166900676. E-mail addresses: [email protected], [email protected] (Y.-Q. Jiang), [email protected] (R.-Y. Guo), [email protected] (F.-B. Tian), [email protected] (S.-G. Zhou). http://dx.doi.org/10.1016/j.apm.2016.06.041 0307-904X/© 2016 Published by Elsevier Inc.

Please cite this article as: Y.-Q. Jiang et al., Macroscopic modeling of pedestrian flow based on a second-order predictive dynamic model, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.041

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prime interest. Therefore, to better understand macroscopic features of pedestrian crowd motion in especially large crowds, pedestrian dynamic models presented at a macroscopic level are necessary and significative to study this topic due to relatively lower computational complexity and fewer model parameters. Basically, macroscopic pedestrian flow models can be derived from the underlying mesoscopic (kinetic) description for active particles [13]. Based on the analogy between the traffic flow and fluid flow, Lighthill and Whitham [21] first proposed the well-known first-order LWR model. Since then many other methods and models have been developed based on the mathematical modeling of crowd dynamics within the framework of continuum mechanics [14–20,22–27]. In these models, a crowd of pedestrians is generally treated as a continuum, provided the characteristic distance scale between pedestrians is much less than the characteristic distance scale of the region where the pedestrians move. Analogous to macroscopic vehicular traffic flow models, macroscopic pedestrian flow models are classified as first-order models that only involve the mass conservation equation and second-order models or dynamic models that involves both the mass conservation equation and linear momentum equation. Hughes [14] presented a continuum theory for the first-order pedestrian flow and established equations of motion that govern both single and multiple pedestrian types caused by differences among pedestrian walking patterns. Huang et al. [18] revisited Hughes’ model and verified that the developed model can capture the formation of shockwaves. However, due to the assumption that the traffic flow is always in an equilibrium state, the first-order models are not capable of explaining other complex phenomena, e.g., clogging at bottlenecks and localized clusters or stop-and-go waves [22,26]. For instance, qualitatively similar stop-and-go traffic that corresponds to regions with relatively high density and relatively low speed which propagate backward the flow has been experimentally observed in high density crowd scenes [28–30]. At even higher densities, a sudden transition from stop-and-go waves to “crowd turbulence” could arise before crowd disasters [31,32]. To resolve this problem in the first-order models, several second-order models [17,22,26,33,34] have been proposed. In these models, the equilibrium speed-density relationship is replaced by a dynamic equation with respect to the average velocity of pedestrian flow. A typical model is the one presented by Bellomo and Dogbé [17] which consists of the equations of conservation of mass and equilibrium of linear momentum involving an acceleration term. However, the desired direction of motion in this model is fixed at any point of the facility and does not change with the time-varying traffic conditions. Therefore, it can’t quite express the real path-choice behavior of pedestrians. For the issue mentioned above, in this paper a second-order predictive dynamic model for pedestrian flow is presented to investigate macroscopic characteristics of pedestrian movement and some non-equilibrium phenomena, such as the experimentally observed stop-and-go waves [29,30]. This model is described as a two-dimensional (2D) hyperbolic system of nonlinear conservation laws with source terms under the hypothesis that a group of pedestrians move like a continuous anisotropic medium. In the model, the pedestrian path-choice strategy is determined based on predictive traffic information gained through experience. Specifically, the desired or intended walking direction of pedestrians is to minimize the total actual walking cost from/at the current position/time to the destination based on predictive travel cost information gained through experience, which satisfies the predictive dynamic user-equilibrium (PDUE) assignment [35,36]. A linear stability analysis of the presented model shows that non-equilibrium phase transitions, such as the transition from stable to unstable flow, can be described by this model. A numerical method used to solve the predictive dynamic model is designed as a cell-centered finite volume method coupled with a self-adaptive method of successive averages (MSA) for an arisen fixed point problem. To test the validity of the proposed model and algorithm, we compare the results carried out by the model with experimental observations. The proposed model and algorithm are also applied to study crowd movement in a 2D continuous walking facility scattered with a square obstruction to demonstrate their applicability and effectiveness. The outline of this paper is as follows. In the next section, the mathematical formulation for the movement of pedestrians is described in detail. Section 3 gives a numerical algorithm for the model. The numerical results are presented in Section 4. Finally, some concluding remarks are given in Section 5.

2. Problem formulation We regard a group of pedestrians walking in a domain denoted by  as a compressible continuum fluid medium. The boundary of  consists of inflow boundary  i , outflow boundary  o , and solid wall boundary  w . T = [0, tend ] (in s) is the modeling period. For the sake of simplicity, tend is considered to be fixed and is large enough to make sure that all pedestrians can leave the modeling domain  within the modeling period T. In the context of continuum mechanics, the dynamic model for pedestrian flow, which is composed of equations of conservation of mass and equilibrium of linear momentum [17], is written as follows.

⎧ ∂ρ ⎨ + ∇ · ( ρ v ) = 0, ∂t ⎩ ∂ v + ( v · ∇ )v = F ( ρ , v ), ∂t

(1)

where ρ = ρ (x, y, t ) (in ped/m ) and v = (u, v ) (in m/s) denote the density and velocity of pedestrian flow, respectively. Here, the force F = [Fx , Fy ] in Eq. (1) models the local acceleration and characterizes the internal driving force or motivation of pedestrian flow. 2

Please cite this article as: Y.-Q. Jiang et al., Macroscopic modeling of pedestrian flow based on a second-order predictive dynamic model, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.041

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2.1. Second-order dynamic model = We assume that pedestrians expect to move with a desired speed Ue (ρ ) (in m/s) in a desired direction ν (νx (x, y, t ), νy (x, y, t )) and therefore there exists an optimal velocity Ue (ρ )ν for the pedestrian velocity v. Pedestrians tend to adapt their actual velocity of motion to achieve their expectation. In addition, pedestrian-pedestrian interactions exist. Similar to the microscopic representation in the social force model [2,3], F in Eq. (1) consists of two terms: the relaxation  − v )/τ , where τ is a constant (taking 0.5 s in this paper) representing the relaxation time of v toward the term (Ue (ρ )ν  , and the anticipation term c¯ ∇ · v (representing the process that pedestrian flow reacts to the traffic optimal velocity Ue (ρ )ν conditions ahead). Here, c¯ (in m/s) expresses the propagation velocity of small disturbance in crowd flow. We define P(ρ ) as a monotonically increasing function of the pedestrian density ρ , i.e. P > 0. Here, P(ρ ) can be treated as the “traffic pressure” (or the anticipation factor) by analogy with gas dynamics. Then the propagation velocity of small disturbance c¯ is expressed by c¯ = (c¯1 , c¯2 ) = ρ P  1 where 1 = (1, 1 ). Note that P has different expressions in different models and plays the role of an anticipation factor. Eq. (1) is thus rewritten as

⎧ ∂ρ ⎨ + ∇ · ( ρ v ) = 0, ∂t ⎩ ∂ v + (v · ∇ )v = Ue (ρ )ν − v + c¯ ∇ · v. ∂t τ

(2)

The component forms of the momentum equations are

  ∂u ∂u ∂ u Ue (ρ )νx − u ∂ u ∂v +u +v = + ρ P + , ∂t ∂x ∂y τ ∂x ∂y   ∂v ∂v ∂v Ue (ρ )νy − v ∂ u ∂v  +u +v = + ρP + . ∂t ∂x ∂y τ ∂x ∂y

(3)

(4)

It is clear that the model (2) is reduced to the 1D AW model for vehicular traffic flow [37] if the crowd marches in 1D space. 2.2. Linear stability analysis Assume that (ρ 0 , v0 ) are the steady-state solutions of the system (2) satisfying

, v0 = Ue (ρ0 )ν and

ρ = ρ0 + ξ ( x , t ) , v = v 0 + w ( x , t ) ,

(5)

are perturbed solutions of the system, where ξ (x, t) and w(x, t ) = (w1 , w2 ) are small perturbations of density and velocity, respectively. Next we analyze the qualitative properties of the model (2) with a linear stability method [38]. Substituting the perturbed solutions (5) into (2), and then taking Taylor series expansions of the perturbed equations and neglecting higher order terms of ξ and w, we have the following linearized equations.

⎧ ⎪ ⎨ ∂ξ + v0 · ∇ξ + ρ0 ∇ · w = 0, ∂t  ⎪ ⎩ ∂ w + (v0 · ∇ )w − c¯ 0 ∇ · w = ξ (Ue ν )0 − w , ∂t τ

(6)

−τ (∂t + c1 · ∇ )(∂t + c2 · ∇ )ξ = (∂t + c · ∇ )ξ ,

(7)

where c¯ 0 = ρ0 P0 1. By eliminating w from the second vector equation of the system (6), we obtain  )0 is the kinematic wave velocity, and c1 = v0 , c2 = v0 − c¯ 0 . Substituting ξ (x, y, t ) = ξ0 expi(k · x − ωt ) where c = v0 + ρ0 (Ue ν into (7), where ξ 0 = 0 is amplitude, k = (k1 , k2 ) is the wave number vector with the components k1 and k2 in x- and ydirections, respectively, and ω is the frequency, we have an equation for frequency

τ ω2 − [τ (c1 + c2 ) · k − i]ω + [τ (c1 · k )(c2 · k ) − (c · k )i] = 0.

(8)

We replace ω with a + bi in the term on the left-hand side of (8) and obtain the imaginary and real parts of the term:

a − c · k + τ [2ab − b(c1 + c2 ) · k] = 0,

τ [(a − b ) − (c1 + c2 ) · ka + (c1 · k )(c2 · k )] − b = 0. 2

2

(9) (10)

It is obvious that the solutions are stable if and only if the imaginary parts of both of the roots ω are non-positive, i.e.  )0 | ≤ P0 , which is the linear stability criterion b ≤ 0 for given k. It is easily verified that the requirement for this is |(Ue ν of the system (2). When the condition is violated or when traffic disturbances are large enough, traffic becomes unstable, which generates complex traffic phenomena, such as the formation of congestions and stop-and-go waves. Traffic instability of pedestrian flow is also investigated analytically based on a 2D optimal velocity model at a microscopic level in [39]. Please cite this article as: Y.-Q. Jiang et al., Macroscopic modeling of pedestrian flow based on a second-order predictive dynamic model, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.041

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2.3. Conservation form and eigenvalues Multiplying the first equation of (2) by P and then adding the result to (3) and (4), respectively, yields the following system

∂ (u + P ) ∂ (u + P ) ∂ (u + P ) Ue (ρ )νx − u +u +v = , ∂t ∂x ∂y τ ∂ (v + P ) ∂ (v + P ) ∂ (v + P ) Ue (ρ )νy − v +u +v = . ∂t ∂x ∂y τ

(11) (12)

Multiplying the first equation of (2) by u + P and v + P, respectively, we have

∂ρ ∂ (ρ u ) ∂ (ρv ) + (u + P ) + (u + P ) = 0, ∂t ∂x ∂y ∂ρ ∂ (ρ u ) ∂ (ρv ) (v + P ) + (v + P ) + (v + P ) = 0. ∂t ∂x ∂y (u + P )

(13) (14)

Multiplying (11) and (12) by ρ , respectively, and combining (13) and (14), we obtain the conservation laws of (2) as follows.

where

∂Q ∂F ∂G + + = S, (x, y ) ∈ , t ≥ 0, ∂t ∂x ∂y

(15)

Q (·, 0 ) = Q0 ,

(16)

⎤ ⎡0 ⎡ ⎤ ⎡ ⎤ ρ q1 ⎢ Ue νx − u ⎥ ⎥, Q = ⎣ρ (u + P (ρ ))⎦ = ⎣q2 ⎦, S = ⎢ρ τ ⎣ ⎦ U ν −v ρ (v + P (ρ )) q3 ρ e y τ ⎡ ⎤ ⎡ ⎤ ρu ρv F = ⎣ρ u(u + P (ρ ))⎦, G = ⎣ρv(u + P (ρ ))⎦. ρ u(v + P (ρ )) ρv(v + P (ρ ))

The quasi-linear system can be written in a vector form

∂Q + A(Q )Qx + B(Q )Qy = S, (x, y ) ∈ , ∂t

(17)

where the flux Jacobian matrices A(Q) and B(Q) are respectively expressed by

⎡

−P − ρ P 

⎢ q2 ∂F ⎢ − 2 − q2 P  A (Q ) = = ⎢ q2 1 ∂Q ⎢ ⎣ q2 q3  −

and

⎡

q21

− q3 P

−P − q1 P  ⎢ q2 q3 − 2 − q2 P  ∂G ⎢ q1 B (Q ) = =⎢ ⎢ ∂Q ⎣ q2 − 32 − q3 P  q1

1 2

q2 −P q1 q3 q1

0 q2 −P q1

0

1 q2 q1

q3 −P q1 0

⎤

0

2

q3 −P q1

⎥ ⎥ ⎥, ⎥ ⎦

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

By solving the eigenvalue equation with respect to λ: |A(Q )nx + B(Q )ny − λI| = 0, the eigenvalues of the composite Jacobian matrix of the system (17) are obtained as

λ¯ 1 = unx + vny − ρ P (nx + ny ), λ¯ 2 = λ¯ 3 = unx + vny .

(18)

The system is hyperbolic since all the eigenvalues of the composite Jacobian matrix A(Q )nx + B(Q )ny are real in an arbitrary spatial direction (nx , ny ). It also preserves the anisotropic property, which means that in the fluid field, information that affects the current position depends on the current and ahead information only. Please cite this article as: Y.-Q. Jiang et al., Macroscopic modeling of pedestrian flow based on a second-order predictive dynamic model, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.041

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2.4. Constraint-based path choice  describes pedestrians’ path-choice behavior. From Eq. (15), the desired or intended walking direction of pedestrian flow ν Generally speaking, the behavior is affected by several factors, such as the cost (or time) of walking, “discomfort” and outer environmental conditions. However, the path-choice strategies of pedestrians depend mainly on the cost (or time) of walking and “discomfort” [14,18]. Let C(x, y, t) (in s) be the local travel cost distribution, which is defined as

C (x, y, t ) =

1 + g( ρ ) , Ue (ρ )

(19)

where the discomfort field g(ρ ) is an increasing function of the pedestrian density and expresses that pedestrians try to avoid congestions when making decisions. Let (x, y, t) (in s) be cost potential that describes the lowest cost incurred by pedestrian flow walking from the origin (x, y) ∈  at time t to the destination  o where (x, y, t ) = 0. We assume that pedestrians have predictive information about traffic conditions over time and the desired direction of pedestrian movement is always to minimize the actual travel cost based on predictive travel cost information gained through experience, which results in a PDUE assignment [15,16,35,36]. For the PDUE condition, the optimum motion trajectory can be uniquely determined by [36]

ν () = −

∇ (x, y, t ) . |∇ (x, y, t )|

(20)

Here, (x, y, t) satisfies a time dependent Hamilton–Jacobi (HJ) equation:

∂ = Ue (|∇ | − C ). ∂t

(21)

See [36] for the derivation process of the PDUE condition composed of Eqs. (20) and (21). For the PDUE pattern, the total walking cost (x, y, t) to the destination  o only depends on the traffic conditions that will occur in the future, and has nothing to do with those that happened in the past. Without loss of generality, all pedestrians are supposed to have left the modeling domain at tend , i.e. ρ (x, y, tend ) = 0. Consequently, the traffic state is static and the travel cost to the destination at tend is regarded as the instantaneous cost, i.e. (x, y, tend ) = 0 (x, y ). Here, 0 (x, y) satisfies an Eikonal equation:

|∇ 0 (x, y )| = C (x, y, tend ), (x, y ) ∈ ,

(22)

0 (x, y ) = 0, ∀(x, y ) ∈ o.

(23)

The second-order predictive dynamic model of pedestrian flow can now be formulated as two parts. The one part is conservation laws with source terms (15) with the initial conditions (16). The another part is the time dependent HJ Eq. (21) with the initial condition (22) and the boundary condition (x, y, t ) = 0, ∀(x, y ) ∈ o. Therefore, a forward-backward structure in time will be dealt with in this dynamic model. Such forward-backward problems which couple a conservation law and a Hamilton–Jacobi–Bellman (HJB) equation are also known in the mathematical literature as mean field games [40–43]. 3. Numerical method The difficulty about the numerical computation of the predictive dynamic model described in Section 2 is that the initial condition of the HJ Eq. (21) is obtained at t = tend , as opposed to that of Eq. (15) which is given at t = 0. To solve this problem, we use a first-order cell-centered finite volume (FV) scheme to solve Eqs. (15) and (21), a fast sweeping method (FSM) to solve Eq. (22), and a self-adaptive MSA to solve the arisen discrete fixed point problem. The computational domain   is divided into I × J non-overlapping cells by a partition of  = 1≤i≤I,1≤ j≤J Ii j . Here, Ii j = Ii × I j , Ii = [xi− 1 , xi+ 1 ] and J j = [y j− 1 , y j+ 1 ]. The centers of Ii and Jj are denoted by xi = 2

2

mesh size in both dimensions is h.

1 2 (xi− 12

+ xi+ 1 ) and y j = 2

1 2 (y j− 12

2

2

+ y j+ 1 ), respectively, and the 2

3.1. Finite volume method A popular method for treating inhomogeneous hyperbolic equations of the form (15) is to use a fractional step (splitting) method, by which each time step t is split into three steps. Eq. (15) can be split into the ordinary differential equations (ODEs) and the partial differential equations (PDEs), i.e.

dQ  ()), = S(Q, ν dt

(24)

∂Q ∂F ∂G =− − , ∂t ∂x ∂y

(25)

and a numerical method appropriate for each separate system is applied independently. Please cite this article as: Y.-Q. Jiang et al., Macroscopic modeling of pedestrian flow based on a second-order predictive dynamic model, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.041

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The solution values of Q n+1 at the n+1th time step are obtained from Qn at the nth time step tn as follows. First, we update Qn with Q˜ by applying a classical implicit Euler method to Eq. (24) with the time increment 12 t. Second, we update Q˜ with Q¯ by applying the Lax–Friedrichs scheme to Eq. (25) with the time increment t. Finally, we update Q¯ with Q n+1 by applying the implicit Euler method to Eq. (24) with the time increment 12 t. The first-order cell-centered FV scheme for Eq. (15) can be written as

1 tS(Q˜i j , ν inj ), 2 t ˆ [(Fi+ 1 j − Fˆi− 1 j ) + (Gˆ i j+ 1 − Gˆ i j− 1 )], Q¯ i j = Q˜i j − 2 2 2 2 h 1 n+1 n +1 n  i j ), Qi j = Q¯ i j + tS(Qi j , ν 2 Q˜i j = Qinj +

(26)

where Fˆi± 1 j and Gˆ i± 1 j are the Lax–Friedrichs numerical fluxes in the x- and y-direction, respectively, and are expressed by 2

2

Fˆi+ 1 j 2

1 = [F (Q˜i j ) + F (Q˜i+1 j ) − αx (Q˜i+1 j − Q˜i j )], 2

Gˆ i j+ 1 = 2

(27)

1 [G(Q˜i j ) + G(Q˜i j+1 ) − αy (Q˜i j+1 − Q˜i j )]. 2

(28)

Here, αx = max{ui j , (u − ρ P  )i j } and αy = max{vi j , (v − ρ P  )i j }. ij

ij

We define H = Ue (|∇ | − C ) in (21). From (22), the initial condition of the time dependent HJ Eq. (21) is given at t = tend and thus the first-order cell-centered FV scheme for Eq. (21) is given by

ni j = ni j+1 − t Hˆ [(x )−i j , (x )+i j , (y )−i j , (y )+i j ],

(29)

where

(x )−i j = (y )−i j =

+1 ni j+1 − ni−1 j



n+1 ij

h +1 − ni j−1 h

, (x )+ = ij , (y )+ = ij

+1 ni+1 − ni j+1 j



n+1 i j+1

h − ni j+1 h

, .

Here, Hˆ in (29) is the Lax–Friedrichs numerical flux and is defined by



Hˆ (u− , u+ , v− , v+ ) = H

u− + u+ , 2

v− + v+ 2



−

1 H + [α (u − u− ) + αyH (v+ − v− )], 2 x

where αxH = max ∂∂H and αyH = max ∂∂H . ij

x

ij

y

3.2. Method of successive averages Note that the discrete system composed of Eqs. (26) and (29) cannot be solved together as the initial conditions of the two equations are given at different times. We define the solutions of the discrete system at each grid point and each time level as two vectors.

ρ = {ρinj , i = 1, 2, . . . , I, j = 1, 2, . . . , J, n = 1, 2, . . . , N},  = {n , i = 1, 2, . . . , I, j = 1, 2, . . . , J, n = 1, 2, . . . , N},  ij  old , we can solve Eq. (15) by (26) from t = 0 where N is the number of grid points in the t-direction. With a given vector   old ). With the obtained vector  . This computation step can be described as ρ  = g ( to t = tend and thus obtain the vector ρ  old with   new based on (29) from t = t ρ , we can update the vector  to t = 0 . This computation step can be described end  new = h(ρ  ). We consider the two computation steps as one iteration and denote it as as 

 new = h(g(  old )) = f(  old ). 

(30)

With the definitions of one iteration and the function f, we obtain a fixed point problem

 = f()  , 

(31)

which can be solved by the MSA [36].  1 of the fixed point problem (31) is chosen as a vector of the minimum instantaneous walking cost The initial value  which yields a reactive dynamic user-equilibrium (RDUE) assignment. As a consequence, the walking cost (x, y, t) in (20) satisfies the following Eikonal equation [18]

|∇ (x, y, t )| = C (x, y, t ), ∀(x, y ) ∈ ,

(32)

Please cite this article as: Y.-Q. Jiang et al., Macroscopic modeling of pedestrian flow based on a second-order predictive dynamic model, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.041

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Fig. 1. Walkway configuration for the experiment (WA: Waiting area; ROI: Region of interest).

(x, y, t ) = 0, ∀(x, y ) ∈ o,

(33)

which is solved by the FSM [44]. With the calculation of Eq. (32) at each time step

tn ,

we can obtain

ni j

and then compute

 1. ρinj (1 ≤ i ≤ I, 1 ≤ j ≤ J) by (26) from t = 0 to t = tend . Therefore, we obtain the density vector ρ 1 and the initial value  1  stores the solution of the model composed of Eqs. (15) and (32) from t = 0 to t = tend . By computing (29) from Note that ρ 1 = h (ρ  1 , we obtain a temporary vector y  1 ). t = tend to t = 0 with ρ 3.3. Solution procedure The solution procedure is described as follows.  2 is calculated by 1 . The total actual walking cost   1 and the temporary vector y Step 1. Compute the density vector ρ

 2 = (1 − λ1 )  1 + λ1 y 1 . 

(34)

 k (k = 2, 3, . . . ), compute ρ k by  k and y Step 2. Given the solution vector 

 k ) and y  k ), k = h (ρ  k ) = f ( ρ k = g(

(35)

 k+1 by and then update the solution vector 

 k+1 = (1 − λk )  k + λk y k .  Step 3. Stop the iteration process if

place k by k + 1 and go to Step 2.

 k+1 

(36) k −

2 ≤ ε with a given convergence threshold value ε =

10−2 .

Otherwise, re-

Here, λk is the step size determined as [36] for the self-adaptive MSA. 4. Experiments and simulations 4.1. Experiment results In the experimental scenario, students were recruited and asked to walk down a 14-m-long and 3-m-wide corridor bounded by traffic cones in a normal manner. The dimensions and configuration of the region of interest (ROI) are shown in Fig. 1. The movements of students were recorded with a digital video camera and the camera was positioned to overlook the ROI. At the start of each experiment, a group of students were arranged in a waiting area at the left or right end of the walkway (see Fig. 1). The pedestrian population ranged from 24 to 90. To achieve the fundamental flow-density or velocity-density diagram, the coordinate trajectory of each individual had to be known in the experiments. From the experimental data, 653 sample points for the average density ρ (in ped/m2 ) and speed U (in m/s) are obtained. Here, the density was measured as the number of pedestrians in the ROI divided by the area of the ROI denoted as SROI (in m2 ), and the average speed was determined as the average distance moved by the pedestrians between consecutive frames divided by the sample interval 0.2s. The details of the experimental setup can be found in [45,46]. Fig. 2 shows the fundamental flow-density relationship among the pedestrians, which is obtained by regressing the experimental data in the ROI. Here, the flow q (in ped/m/s) is equal to ρ × U. We define the desired walking speed Ue in Eq. (2) as q(ρ )/ρ . Therefore, the desired walking speed Ue is specified as

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(37)

where γ = 0.075 and v f = 1.034 m/s are obtained from regression analysis [45]. Apparently, alternative speed-density relationships [47–49] can also be applied for different walking scenarios. Consequently, the  critical density ρ c that divides the non-congested and congested density regions can be solved from q (ρ ) = 0 as ρc = 1/ 2γ ≈ 2.58 ped/m2 . The functional relationship between |Ue | and density ρ is shown in Fig. 3, where |Ue | reaches a maximum value 0.24 at ρ = ρc . The discomfort function in (19) is defined as

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(38)

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P (ρ ) = σ ρ ,

(39)

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Fig. 2. The fundamental flow-density relationship.

Fig. 3. The functional relationship between |Ue | and ρ .

where σ is a positive constant and represents the anticipation behavior of pedestrians. Based on the linear stability analysis  ) | > σ , traffic instability will occur. in Section 2.2, when |(Ue ν The camera-based observations are used for calibration of the proposed dynamic model and improve the acceptability and accuracy of the data carried by the simulation under non-congested conditions. We assume that a group of pedestrians enter (leave) from the left-hand (right-hand) end of the corridor shown in Fig. 1 and take the vertex on the underneath left side of the corridor as the coordinate origin. Pedestrian movement is simulated based on the proposed dynamic model (2). The anticipation parameter σ in Eq. (39) ranges from 0.01 to 5. The initial and boundary conditions for the dynamic model are described here. Initially, the walking facility is empty with ρ (x, y, 0 ) = 0 ped/m2 and v(x, y, 0 ) = 0 m/s. At the entrance  i , a free and equilibrium inflow is given by

ρ

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where ρ m (in ped/m ) is the peak value of density. The total pedestrian population Np can be calculated as N p =  60  0 i ρUe d sdt. For example, Np is about 30, 58, 82 and 110 for ρm = 0.5, 1.0, 1.5 and 2.5, respectively. The density of pedestrians in the ROI can be controlled by adjusting the value of ρ m . At the solid walls  w , the free-slip and non-permeable boundary conditions are applied, i.e. ∂ ρ /∂ n = 0 and v · n = 0. At the outflow boundary  o , (u, v ) = (v f , 0 ) and ∂ ρ /∂ n = 0.   The average density and speed of pedestrian flow in the ROI are computed as ROI ρ dxdy/SROI and ROI |v|dxdy/SROI , respectively. Fig. 4 shows that the simulated flows obtained with the proposed dynamic model are consistent with the equilibrium flow given by the fundamental diagram at a relatively low density, whereas the simulated flows are slightly larger than the equilibrium flow nearby the critical density. Therefore, the credibility of the data produced by the proposed dynamic model can be validated through comparing with experimental data of pedestrian flow collected in normal conditions. Moreover, pedestrian flow is always in an equilibrium state for different values of the anticipation parameter σ only under conditions of low-population density. With the increase of density, pedestrian flow is slightly out of equilibrium and the deviate degree is influenced by the anticipation parameter σ (see Fig. 4(d)). However, it must be noted that the behavior of 2

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4.2. Numerical examples In this simulation, the validated model is adopted to simulate a large group of pedestrians moving through a 100 m × 50 m rectangular walking facility, which is divided into 100 × 50 cells (see Fig. 5). A square obstacle with side of length 20 m is located at (60 m, 20 m ). An entrance is set at x = 0 for 0 ≤ y ≤ 50 and two exits are set at x = 100 for |y − 10| ≤ 5 and |y − 40| ≤ 5, respectively. Initially, the walking facility is empty. At the entrance  i , a free and equilibrium inflow is Please cite this article as: Y.-Q. Jiang et al., Macroscopic modeling of pedestrian flow based on a second-order predictive dynamic model, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.041

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with ρm = 2.5 ped/m2 . The other boundary conditions are the same as those previously described.  k+1 −   k 2 . To demonstrate the effectiveness of the numerical We denote the errors produced by the MSA as E :=  algorithm described in Section 3, Fig. 6 plots the errors on a logarithmic scale (i.e., lg(E)), which indicates a very good convergence of the MSA for different values of the anticipation parameter σ . From this figure, we observe that the total numbers of iterations obtained with different values of σ are no more than 24. After about 7 iterations, each error lg(E) declines quickly as the number of iterations increases and the error obtained with σ = 5 falls at a faster rate during the whole iterative process. Fig. 7 shows the density distributions at different times for the dissipation of pedestrian flow in six phases with the parameter σ = 0.1. In the first phase (t = 60), pedestrians walk around the square obstacle and a triangular region that is hardly occupied by pedestrians is formed on the left-hand of the obstruction (see Fig. 7(a)). This is caused by pedestrians’ optimal path-choice strategies to minimize their actual walking cost in the moving process. In the second phase (t = 90), two Please cite this article as: Y.-Q. Jiang et al., Macroscopic modeling of pedestrian flow based on a second-order predictive dynamic model, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.041

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high-density regions, where the density exceeds the critical density ρc = 2.58 ped/m , are scattered around the upper-left and lower-left corners of the obstruction (see Fig. 7(b)). This means traffic congestions occur because of the sharp reduction in the width of the walking facility. In the third phase (t = 120), these high-density regions are expanding quickly, which reveals more serious traffic congestions due to more and more pedestrians assembled together on the left-hand side of the obstruction (see Fig. 7(c)). Multiple clusters that correspond to stop-and-go waves are caused by traffic instability, when  ) | > σ works with σ = 0.1 < 0.24 (the maximum value of |Ue |). In the fourth phase (t = 150), traffic the formula |(Ue ν congestions extend over larger regions near the obstruction (see Fig. 7(d)). At this time, no pedestrians enter the facility and there is a large empty region on the left-hand of the obstruction left by the departing pedestrians. Arching and clogging are observed near the northeast exit of the facility. To save the total actual travel cost, a few pedestrians near the northeast exit who have perfect information about the facility select the southeast exit to leave the facility in response to dynamic changes in their surroundings. In the fifth phase (t = 180), local cluster effects are also observed around the obstruction and the pile-up phenomenon lasts a long time near the northeast exit (see Fig. 7(e)). In the sixth phase (t = 240), the rest of pedestrians divide spontaneously into multiple sub-streams and queue up in front of the two exits to leave the facility (see Fig. 7(f)). The flow vector ρ v plotted in Fig. 8 illustrates the movement of pedestrians passing through the walking facility based on predictive traffic information gained through experience. The actual movement direction of pedestrians to walk from a given 2

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Fig. 10. Densities along two lines with different values of σ .

origin to the exits of the facility can be found by locating the dynamic streamline based on the vectors in each sub-figure. The complex localized structure consisting of multiple local clusters caused by the nonlinear interactions of pedestrians can be seen near the obstruction and exits as the density becomes higher. We further take a larger value of σ to study the effect of the anticipation behavior of pedestrians in the dissipation of the crowd. Fig. 9 shows the density distributions of the dissipating crowd in the six phases with σ = 5. Based on the stability criteria |(Ue ν ) | ≤ σ , pedestrian flow is always in a stable state. The first phase (see Fig. 9(a)) demonstrates almost the same result as shown in Fig. 7(a). In the second phase, the density near the lower-left corner of the obstruction is relatively high (see Fig. 9(b)). In the third phase, a new high-density region appears near the upper-left corner of the obstruction (see Fig. 9(c)). In the fourth phase, traffic congestions occurring near the upper-left and lower-left corners of the obstruction become more serious (see Fig. 9(d)). In the fifth phase, the phenomenon of clogging appears near the northeast exit of the facility (see Fig. 9(e)). In the sixth phase, the rest of pedestrians are distributed equally near the two exits, which indicates full utilization of the two exits, and queue up to leave the facility (see Fig. 9(f)). A comparison of Figs. 7 and 9 clearly shows the difference in the density distributions caused by the effect of traffic pressure (or anticipation factor). Decreasing the anticipation parameter σ allows for smaller distances among pedestrians, which causes stronger interactions among pedestrians. Fig. 10 illustrates the densities of pedestrian flow along two lines, x = 48 at t = 120, 150 and x = 98 at t = 120, 210, with different values of σ ranging from 0.01 to 5. The density near the bottlenecks of the walking facility, such as the obstruction and exits, is greatly reduced with the increase of the value of σ . Therefore, increasing the traffic pressure has a stabilizing effect on pedestrian traffic due to full utilization of the existing space near the bottlenecks. This further demonstrates stronger interactions among pedestrians can cause more serious traffic congestions and even incur traffic instabilities in high-density regions. Please cite this article as: Y.-Q. Jiang et al., Macroscopic modeling of pedestrian flow based on a second-order predictive dynamic model, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.041

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5. Conclusions We have presented a second-order predictive dynamic model for pedestrian flow to investigate macroscopic features and path-choice behaviors of pedestrians walking in a facility scattered with an obstruction. This model is anisotropic, which means that pedestrians only react to the traffic state in front of them. The path-choice behavior of pedestrians is described by the fact that pedestrians always tend to choose a path with the minimum actual walking cost from/at the current position/time to the destination, which corresponds to the PDUE assignment. A cell-centered finite volume method coupled with a self-adaptive MSA is employed to solve the proposed model. This dynamic model and algorithm are validated through comparing with experimental data of pedestrian flow collected under non-congested conditions. Numerical results indicate that, when traffic instability occurs in dense regions, the model can reproduce some complex traffic phenomena, such as the formation of congestions and stop-and-go waves. Furthermore, the anticipation behavior of pedestrians has a significant effect on the density distributions and thus has a stabilizing effect on pedestrian traffic by increasing traffic pressure or weakening interactions among pedestrians. In this paper, the fundamental speed-density relationship, obtained from regression analysis of experimental observations, is applied to describe the desired or preferred walking speed of pedestrian movement. Alternative speed-density relationships [47–49] can be certainly applied for different walking scenarios. The experimental data of pedestrian flow is collected under non-congested conditions. Therefore, the proposed model and algorithm are very applicable to simulate non-congested pedestrian flow. For congested pedestrian flow, we investigate numerically the dynamics of pedestrian flow. However, the numerical results obtained with the proposed model are not validated by experimental observations, which is left to be resolved in the follow-up work. Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 11202175 and 11372294) and the Research Foundation of Southwest University of Science and Technology (No. 10zx7137). 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] [30] [31] [32]

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Please cite this article as: Y.-Q. Jiang et al., Macroscopic modeling of pedestrian flow based on a second-order predictive dynamic model, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2016.06.041