Continuum modelling of pedestrian flows: From microscopic principles to self-organised macroscopic phenomena

Physica A xx (xxxx) xxx–xxx

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

Continuum modelling of pedestrian flows: From microscopic principles to self-organised macroscopic phenomena Q1

Serge P. Hoogendoorn ∗ , Femke van Wageningen-Kessels, Winnie Daamen, Dorine C. Duives Delft University of Technology, Stevinweg 1, Delft, The Netherlands

highlights • • • • •

State-of-the-art macroscopic model with global and local route choice behaviour. Multi-class continuum modelling based on microscopic modelling principles (i.c. social forces model). Derivation of explicit relation for equilibrium speed and walking direction. Reproduction of self-organised phenomena, such as dynamic lane formation, and formation of diagonal stripes. Preliminary results show model’s ability to also reproduce phase transitions.

article

info

Article history: Received 5 May 2014 Received in revised form 1 July 2014 Available online xxxx Keywords: Pedestrian flow model Crowd dynamics Continuum model Self-organisation

abstract The dynamics of pedestrian flows can be captured in a continuum modelling framework. However, compared to vehicular flow, this is a much more challenging task. In particular the integration of flow propagation and path choice are known to be problematic. Furthermore, pedestrian flow is characterised by different self-organised phenomena, such as the formation of dynamic lanes and diagonal stripes, which have not yet been captured in a continuum modelling framework. This contribution puts forward a novel multi-class continuum model that captures some of the key features of pedestrian flows. It considers path choice behaviour on both the strategic (pre-trip) and tactical (en-route) level. To achieve this, we present a methodology to derive a continuum model from a microscopic walker model, in this case the social forces model. In doing so, we show that the interaction term present in the social forces model introduces a local path choice component in the equilibrium velocity. Having derived the model, we analyse its properties both by means of mathematical analyses and simulation studies. This reveals the general behaviour of the model, as well as the ability of the model to reproduce self-organised structures, and phase transitions. To the best of our knowledge, this is the first continuum model that is able to reproduce these self-organised structures. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Understanding, reproducing and predicting pedestrian flows is an important challenge that requires our attention in order to solve urgent societal problems in station design, pedestrian flow routing in airports, optimal evacuation planning

∗

Corresponding author. Tel.: +31 15 278 5475. E-mail address: [email protected] (S.P. Hoogendoorn).

http://dx.doi.org/10.1016/j.physa.2014.07.050 0378-4371/© 2014 Elsevier B.V. All rights reserved.

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and operations in sport stadiums, or in large vessels, crowd management, etc. The field of pedestrian flow or crowd simulation is, however, not as mature as its vehicular flow counterpart. On the one hand, this is because the field is relatively young. On the other hand, pedestrian flow operations are much more complex. This is not only due to the fact that a pedestrian flow is – at least – two-dimensional, while vehicular flows are by and large one-dimensional, but also due to the fact that in the flow dynamics operations and route choice levels are much more interwoven. On top of this, empirical and experimental investigations show that pedestrian flows feature complex dynamics, such as the self-organisation of spatio-temporal structures in bi-directional flows (lane formation) and crossing flows (diagonal stripes). The pedestrian flow modelling playing field – in particular those models that have made it to the engineering practise – has been dominated by microscopic models, such as the social-forces model of Helbing and Molnar [1]. Few macroscopic or continuum models have been put forward that capture the core properties of pedestrian flows, which implies that large scale applications have not benefited from the advantages of macroscopic models. Furthermore, with the advent of new technology to monitor pedestrians and pedestrian flows, the need for analytical (macroscopic) frameworks allowing for state-estimation by, for instance, Kalman filters, is increasing. This contribution puts forward a new continuum model that captures some of the key characteristics of a pedestrian flow, while including only few parameters. The article continues with a literature review (Section 2) The multi-class model is derived from a microscopic model (Section 3), and reproduces plausible state-space solutions (equilibrium speeds), and selforganised spatial temporal patterns (Section 4). Having presented the results of the mathematical analyses and simulations, we review the key findings of this contribution (Section 5).

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2. Literature review

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In this section, we review some of the key studies that form the foundation of the model presented in this manuscript. While not trying to be complete, we briefly present relevant empirical studies, as well as the main modelling approaches that have been put forward, with an emphasis on continuum models. In particular the latter type of models will be discussed in more detail, focusing on their ability to reproduce aforementioned pedestrian flow phenomena.

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2.1. Empirical features

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In the past decades, characteristics of pedestrian flow have been studied empirically and experimentally. These studies revealed many interesting features, including the existence of a fundamental relation between density and flow and selforganised structures. In this section, we briefly discuss some of these characteristics relevant for the topics discussed in this contribution. The fundamental relation reflects the statistical relation between density ρ and (absolute) flow Q or speed V , i.e. V = V (ρ) or Q = ρ · V = Q (ρ); see Ref. [2]. Many factors influence the shape of the fundamental diagram: for instance, [3] shows how cultural differences influence the shape of the fundamental relation; the capacity, the jam density and the shape of the fundamental diagram are also influenced by factors such as trip purpose and the heterogeneity of the pedestrians [4]. Next to the fundamental relation between density and flow, pedestrian dynamics are characterised by different selforganised phenomena. Self-organisation is defined as the spontaneous occurrence of qualitatively new behaviour through the non-linear interaction of many objects or subjects [5] without the intervention of external influences [6]. The most common self-organisation movement is lane formation [7]. During this process a number of lanes of varying width form dynamically in a corridor. Next to lane formation in bi-directional flows, diagonal stripe formation in crossing flows has been observed; e.g. see Ref. [7]. During their research at the Jamarat bridge Helbing et al. [4] found stop & go waves. These are temporarily interrupted longitudinally flows that appear at higher densities in uni-directional crowds. In an even more dense flow regime, turbulent flows were found. In this regime a pedestrian has no control over its own movements anymore. Local force based interactions between pedestrian bodies are seen. Three other effects have been described at an operational level, namely herding [8], the zipper effect [9] and the fasteris-slower effect [5]. The first effect describes the case where unclarity of the situation causes individuals to follow each other instead of taking the optimal route. This behaviour is predominantly seen during stressful evacuation situations. The zipper effect describes the situation in which individuals allow others within the territorial space diagonally in front of them, as long as the direct space in front of their feet is still empty. It allows for narrower lanes in a bottleneck than expected based on the width of a pedestrians territorial zone. The third effect (Faster-is-Slower) describes a situation where the density in a queue upstream of a bottleneck is increasing, due to the fact that people keep heading forward while the bottleneck is clogged. The higher densities cause coordination problems since a large number of individuals is competing for a few small gaps. Bodily interaction and friction slow down the total crowd motion.

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2.2. Modelling approaches

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In a recent study, a comparison of pedestrian flow modelling approaches were cross compared, focusing on how these models are able to simulate the key phenomena indicated in the previous paragraphs [10]. The paper discusses different types of models, such as cellular automata, social force models, velocity-based models, continuum models, hybrid models,

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behavioural models and network models. The comparison shows that ‘‘the models can roughly be divided into slow but highly precise microscopic modelling attempts and very fast but behaviourally questionable macroscopic modelling attempts’’. Pedestrian flow literature often distinguishes microscopic and macroscopic models. Microscopic models represent pedestrian flow at the level of individual pedestrians, and generally aim to describe the individual behaviour and interactions. On the contrary, macroscopic models describe the flow dynamics in more aggregate terms, using quantities such as flows, densities, and speeds.

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2.2.1. Microscopic and mesoscopic walker models In the last few decades several streams of microscopic and mesoscopic walker models have been developed. The first stream of microscopic models was Cellular Automata [11,12], which describe the movement of agents through a simulated environment using a discrete representation of both space and time. A second stream of microscopic models, Social Force models, was introduced by Helbing and Molnar [1]. These models simulate movements based on deterministic force-based interactions and have a continuous representation of space. The effects modelled might be physical in appearance (collisions) or used to model interaction with elements or humans within the movement space (human interaction, light effects, attraction zones, etc.). Also within this model stream numerous adaptations have been proposed, among which, vision fields, collision bias and group formation forces. From this type of model we will derive a macroscopic model in the next section. A third, quite new, stream of models was proposed by the gaming industry. Velocity-based models simulate local operative movements (collision avoidance) by means of the optimisation of time to collision with other pedestrians and objects at any point in time [13–15]. 2.2.2. Macroscopic modelling approaches Within macroscopic modelling approaches three modelling streams can be distinguished, being network models, hybrid models and continuum models. The first stream, network models, solves crowd movement problems by means of mathematical approaches developed in graph theory [16–18]. A second stream tries to incorporate the advantages of both microscopic and macroscopic modelling approaches (a.o. [19,20]). The two levels are either used in different spaces or time periods within the overall simulation, or combined sequentially (i.e. the output of the first level gives input for the second level). Models in the third stream of macroscopic modelling approaches, continuum models, simulate the global movement effects during which pedestrians are interpreted as particles of flow and PDE models are used to compute the solution. Hughes was the first to describe crowd movements by means of a continuous potential field approach, which is closely related to fluid dynamics [21]. Several studies have used hydrodynamic principles as the foundation for their simulation models (a.o. [22–24]). In recent years, also macroscopic continuum models based on other principles have been proposed. Interactions of individuals among animal societies were used as inspiration of the Self-Organised Hydrodynamics model [25]. Additionally, Cristiani et al. (2010) proposed measure-based macroscopic model, and Rosini et al. (2011) a model based on the Lighthill–Whitham and Richards (LWR) model [26,27]. Schwandt et al. (2013) based their model on the principles of diffusion and convection [28]. The microscopic principles of velocity-obstacle based models provided the basis for Ref. [29]. Each of these models uses the law of conservation of mass. 2.3. Modelling challenges Microscopic modelling approaches, by definition, are very capable of simulating heterogeneous crowds. Yet, the computational effort required to operate these models for large crowds in any sizeable infrastructure is generally great. Therefore, computing optimal strategies remains a tedious endeavour. Macroscopic models are not aimed at simulating the behaviour of individual pedestrians. However, when one requires a good estimation of the aggregate characteristics of crowd movements (i.e. velocity, density and flow), these models might provide useful predictions. Since the computational effort of macroscopic models is considerably lower than that of microscopic models, especially in situations where simulations need to be fast and not necessarily highly accurate, macroscopic models provide a good option. As such, the computation of optimal evacuation strategies and the operational management of large-scale pedestrian movement systems can benefit from the ongoing development of macroscopic models. Yet, as discussed previously, the macroscopic models proposed into this moment are not capable of simulating all relevant behavioural processes and characteristics that crowds show. Only a limited number of macroscopic models are capable of modelling bi-directional lane-formation. The more sophisticated forms of (self-organising) crowd movements, such as diagonal stripe formation in crossing flows, have not yet been simulated using macroscopic models, nor are we aware of macroscopic models that can capture the different phase transitions relevant for pedestrian flows. 3. Methodology This section presents the model derivation approach and the resulting continuum pedestrian model. We also present briefly the numerical solution approach that will be used for the case studies in the next section.

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The model that will be derived is a first-order continuum model. This implies among other things that we assume that the flow is in equilibrium (accelerations are equal to zero). This assumption is crucial for the derivation of the model, as shown in the ensuing of the section. While we assume that the global path choice is known a priori, we will use the interaction term of the social-forces model to derive the local path choice of the pedestrians. By means of a first-order Taylor series approximation around the considered location, we are able to derive a closed-form relation for the velocity (speed and direction).

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3.1. Macroscopic modelling framework

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The basis for the proposed continuum model is the pedestrian conservation equation, which describes the dynamics of the density ρ(t , ⃗ x) :

∂ρ + ∇(ρ · v⃗ ) = r − s. (1) ∂t Here, v ⃗ = v⃗ (t , ⃗x) denotes the two-dimensional velocity vector, describing the speed and direction of the pedestrian flow; r

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and s denote the sink and source terms, where pedestrians are generated and leave the area respectively. Clearly, the conservation equation alone does not constitute a complete model, since the velocity is not specified. In the following of this section, we will derive a relation for the equilibrium velocity v ⃗ = ⃗e · V as a function of the density ρ and the density gradient ∇ρ , that will be given by expressions (15) and (16) respectively.

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3.2. Model derivation

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⃗i of pedestrian i as We will start with the anisotropic social force model of Helbing [1] that describes the acceleration a influenced by the pedestrians j near pedestrian i:    − Rij v⃗i0 − v⃗i 1 + cos φij Bi ⃗i = e a − Ai · n⃗ij · λi + (1 − λi ) τi 2 j

⃗ij is the unit vector pointing from pedestrian i to j; φij denotes where Rij denotes the distance between pedestrians i and j; n the angle between the direction of i and the position of j; v ⃗i denotes the velocity. The vector v ⃗i0 denotes the desired velocity of pedestrian i. It describes both the preferred walking speed, and the desired walking direction. We will assume that the desired velocity is a function of time and space, i.e. v ⃗i0 = v⃗i0 (t , ⃗x), and is given a priori. That is, we assume that there is no direct influence of the prevailing traffic conditions on the desired velocity and it is, for example, only based on the shortest route to the destination. For approaches to determine the desired velocity, we refer to Refs. [21,30]. The parameter τi describes the relaxation time, reflecting the time needed to accelerate to the desired velocity; Ai denotes the interaction strength; Bi denotes a scaling parameter, describing the rate at which the interaction force reduces as a function of distance; 0 ≤ λi ≤ 1 denotes the anisotropy parameter, reflecting the relative influence from pedestrians in front of i compared to those at the back. Note that λi = 1 implies isotropy, i.e. pedestrians reacting as strongly to stimuli from the back as to stimuli from the front. ⃗i = 0. Under this assumption, the As mentioned, we will assume that the system is in equilibrium, that is, we assume a velocity satisfies: v⃗i = v⃗ − τi Ai 0 i



R

e

− Bij i

  1 + cos φij ⃗ . · nij · λi + (1 − λi )

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(3)

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j

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(2)

3.2.1. Macroscopic interaction model derivation Let us now make the transition from microscopic to macroscopic interaction modelling. Consider a location ⃗ x. Let ρ(t , ⃗ x) denote the density, to be interpreted as the probability that a pedestrian is present on location ⃗ x at time instant t. We assume that all parameters are the same for all pedestrians in the flow, e.g. τi = τ . We then get the following expression for the equilibrium velocity v ⃗ (see Ref. [26]):

v⃗ = v⃗ − τ A 0

 e

x∥ − ∥⃗y−⃗ B



⃗∈Ω (⃗x) y

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1 + cos φxy (⃗ v) 2

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⃗ − ⃗x y ρ(t , y⃗)dy⃗. ∥⃗y − ⃗x∥

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Here, Ω (⃗ x) denotes the area around the considered point ⃗ x for which we determine the effect of the interactions. Note that: cos φxy (⃗ v) =

⃗ − ⃗x v⃗ y · . ∥⃗v ∥ ∥⃗y − ⃗x∥

(5)

From this expression, we can find both the equilibrium speed and the equilibrium direction, which in turn can be used in the macroscopic model. Using this expression in conjunction with the conservation eq. (1) yields a complete model specification, given that correct initial and boundary conditions are provided.

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3.2.2. Closed form approximation by Taylor series expansion Eq. (4) provides little insight into the properties of the equilibrium relation. In fact, determining the equilibrium velocity would require solving an integral equation which is not straightforward. However, by using the following linear approximation of the density around ⃗ x:

ρ(t , y⃗) = ρ(t , ⃗x) + (⃗y − ⃗x) · ∇ρ(t , ⃗x) + O(∥⃗y − ⃗x∥ ) 2

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(6)

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we will be able to determine a closed-form expression for the equilibrium velocity. Ignoring the second-order term and substituting expression (6) into (4) yields the following expression for the equilibrium velocity:

v⃗ = v⃗ 0 − α⃗ (⃗v ) · ρ − β(⃗v ) · ∇ρ

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(7)

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with α ⃗ (⃗v ) and β(⃗v ) defined respectively by:

α⃗ (⃗v ) = τ A



β(⃗v ) = τ A

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e−

∥⃗y−⃗x∥

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B

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λ + (1 − λ)

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1 + cos φxy (⃗ v)

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⃗ − ⃗x y ⃗ dy ∥⃗y − ⃗x∥

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and

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∥⃗y−⃗x∥

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λ + (1 − λ)

1 + cos φxy (⃗ v)



∥⃗y − ⃗x∥dy⃗.

(9)

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Using the assumption that the area of influence Ω (⃗ x) is a circle1 with radius W , we can prove that β is independent of

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β(⃗v ) = β0 (W ).

(10)

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β0 = lim β0 (W ) = 2πτ (1 + λ)AB . 3

W →∞

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In this case, β0 expresses the resulting influence of the gradient of the density in this situation. We can furthermore show that for α ⃗ the following holds:

α⃗ (⃗v ) = α0 (W ) ·

v⃗ . ∥⃗v ∥

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For W very large, we can conclude:

α0 = lim α0 (W ) = π τ (1 − λ)AB2 . W →∞

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(13)

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As for β0 , we can thus show that as the area Ω is chosen sufficiently large, the influence of W vanishes and α0 (W ) = α0 . We can now easily determine a closed-form expression for the equilibrium velocity. By substituting v ⃗ = ⃗e · V into Eq. (7) and reordering, we find:

⃗e(V + α0 ρ) = v⃗ − β0 ∇ρ. 0

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On both sides, we take the absolute value and reorder to find the equilibrium speed V as a function of the density and the density gradient: V = V (ρ, ∇ρ) = ∥⃗ v − β0 · ∇ρ∥ − α0 ρ. 0

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We divide both sides of Eq. (14) by V + α0 ρ to find the following closed-form expression for the equilibrium direction:

⃗e = ⃗e(ρ, ∇ρ) =

v⃗ 0 − β0 · ∇ρ v⃗ 0 − β0 · ∇ρ = 0 . V + α0 ρ ∥⃗v − β0 · ∇ρ∥

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(16)

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Since v ⃗ 0 is generally determined via a global optimal walking cost or potential function ψ (e.g. see Refs. [21,30]), that is v⃗ 0 = ∇ψ , we can interpret the term ∇θ = −β0 · ∇ρ as a local cost or potential θ that is superimposed on the global cost ψ to reflect the impact of local density differences, i.e.:

⃗e =

∇ψ + ∇θ . ∥∇ψ + ∇θ ∥

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(17)

In the next section, we will discuss some of the implicaties of this expression.

1 Note that this does not mean that pedestrians will also react equally to all pedestrians within this circle: the relative interaction strength is reflected by λ.

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3.3. Multi-class modelling In the model presented so far, the desired walking behaviour including speed and direction is described by the desired velocity. In doing so, the model does not allow describing multidirectional flows. To accommodate this, we here present a simple generalisation of the model, which essentially entails distinguishing groups of pedestrians that are characterised by the same desired velocity functional v ⃗d0 (t , ⃗x); see Ref. [31] for more details. Let ρd (t , ⃗ x) denote the density belonging to the pedestrian class d having desired velocity v ⃗d0 . For each class d, the conservation eq. (1) holds. The equilibrium speed and direction can be determined in the same way as for the single class case. Keeping matters as general as possible, we distinguish interaction parameters A and B to describe interactions between the current pedestrian class d and the opponent class d′ . For the equilibrium speed Vd we get:

      0   Vd =  v βd′ →d · ∇ρd′  − ⃗d − αd′ →d · ρd′   ′ ′

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where

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αd′ →d = π τ (1 − λ)Ad′ →d B2d′ →d

(19)

βd′ →d = 2π τ (1 + λ)Ad′ →d B3d′ →d .

(20)

and

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(18)

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Furthermore, we can determine the following closed-form expression for the equilibrium direction:

 βd′ →d · ∇ρd′ v⃗d0 − d′ ∈D . ⃗ed =    0 v⃗ −  βd′ →d · ∇ρd′    d

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(21)

d′ ∈D

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For now, we will assume that the equilibrium speed Vd and direction ⃗ed can be determined using Eqs. (15) and (16) using d ρd . By doing so, reductions in speeds as a result of different flow compositions are not explicitly considered.

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ρ=

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3.4. Numerical solution approach

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In the remainder of the manuscript, we will illustrate the characteristics of the model both by simple mathematical analysis and by showing the results of simulation. In the latter cases, we will use a newly developed Godunov-based numerical scheme. In our simulations, the area is divided into cells of 0.5 × 0.5 m, time is divided into time steps of 0.075 s. The number of pedestrians in each cell is calculated for each time step. The scheme determines for class d the fluxes at the cell interfaces, by taking the minimum of the maximum outflow from the cell and the maximum inflow into an adjacent cell at their interface. For details we refer to Ref. [32].

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4. Results

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This section describes the characteristics of the model. We will briefly discuss some of the analytical properties (Section 4.1), after which we will show some features of the model by means of numerical simulation (Sections 4.2–4.4).

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4.1. Analysis of model properties

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Let us first take a look at expressions (15) and (16) describing the equilibrium speed and direction. Notice first that the direction (16) does not depend on α0 ρ , which implies that the magnitude of the density itself has no effect on the walking direction, and that only the gradient of the density influences the direction. We will now discuss some other properties, first ⃗ and then by considering an isotropic flow (λ = 1) and an anisotropic flow by considering a homogeneous flow (∇ρ = 0), (λ = 0).

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4.1.1. Homogeneous flow conditions

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⃗ Eq. (15) simplifies to the following expression: Note that in case of homogeneous conditions, i.e. ∇ρ = 0, V = ∥⃗ v 0 ∥ − α0 ρ = V 0 − α0 ρ

(22)

i.e. we see a linear relation between speed and density. The term α0 ≥ 0 describes the reduction of the speed with increasing density.

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Fig. 1. Impact of β0 for uni-directional pedestrian flow. Left picture shows result for β0 = 0; right picture shows result for β0 = 4, other parameters are the same in both simulations.

For the direction ⃗ e, we then get:

⃗e =

v⃗ 0 = ⃗e0 . ∥⃗v 0 ∥

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(23)

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In other words, in homogeneous density conditions the direction of the pedestrians is equal to the desired direction. Clearly, it is the gradient of the density that yields pedestrians to divert from their desired direction in homogeneous flows.

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4.1.2. Isotropic walking behaviour Let us now consider the situation where λ = 1 (isotropic flow). Substituting this in the expression for α0 , Eq. (13), we find:

α0 = 0.

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(24)

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We thus get: V = ∥⃗ v 0 − β0 · ∇ρ∥.

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(25)

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This expression shows that in this case, the speed is only dependent on the density gradient. If a pedestrian walks into a region in which the density is increasing, the speed will be less than the desired speed, but also vice versa. This implies that the speed can be higher than the desired speed, due to pedestrians pushing from the back. This is possible due to the fact that the model is not fully anisotropic, i.e. there are not only stimuli in front of a pedestrian, but also in the back. Also note that in case of isotropy and homogeneous conditions, the speed (22) will be constant and equal to the free speed. Note that this is consistent with the results from Ref. [33]. The walking direction (23) is equal to the desired direction.

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4.1.3. Anisotropic walking behaviour The previous section showed that in case of isotropic flow, the influence of the density vanished and only the density gradient plays a role in equilibrium relations. When λ is chosen smaller, we see that α˜ 0 will increase, reaching its maximum value for λ = 0. At the same time, we see a decreasing value for β˜ 0 , although it will not vanish for physically realistic values of λ, i.e. λ ∈ [0, 1].

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4.2. Case 1: single class pedestrian flows We illustrate the behaviour of the model for a simple, uni-directional flow, using simulations. We consider an area in which pedestrians walk from left to right. We aim to show the impact of β0 on the dynamics of the flow, in particular showing the impact of local interactions of flow dispersion. We consider a 30 by 30 m region. Pedestrians are generated on the left side of the considered area, uniformly between −5 and 5 m. Fig. 1 shows simulation results, just as two videos submitted as electronic supplementary material (see Appendix A) (mmc8.mp4 and mmc7.mp4). From these, we see that if β0 = 0, there is no diffusion of the density and the pedestrians adhere to the planned route described by v ⃗ 0 . This is because pedestrians do not react to local variations in density. For β0 > 0, we see that the local densities cause diffusion: as expected, pedestrians divert from their intended path to avoid high density regions. This shows that the inclusion of the interaction term yields qualitatively valid simulation results.

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Fig. 2. Formation of diagonal stripes for crossing flows, with α0 = 0.25, βd→d = 2 and βd→d′ = 6 for d ̸= d′ , with demand equal to 0.145 P/m/s. 1

4.3. Case 2: crossing flows

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In the second and third case, we show that the model is able to reproduce qualitatively the self-organised patterns that are observed in empirical studies. The second case focuses on crossing flows. We again consider a 30 by 30 m area. We consider two classes of pedestrians: one class entering from the left and moving to the right, and one entering from the bottom and moving to the top. For both classes, pedestrians are generated between −5 and 5 m. Note that we have assumed that the influence of the own class on local (direction and speed) choice behaviour (reflected by βd→d ) is smaller than the influence of the other class (reflected by βd′ →d ). Fig. 2 shows the simulation results, just as two videos submitted as electronic supplementary material (mmc4.mp4 and mmc3.mp4, see Appendix A). The results reveal that after some time patterns are self-organised resembling the diagonal stripes reported in literature. The stripes are formed for both classes, and essentially move through each other resulting in limited interaction between the different pedestrians classes. Qualitatively, the model appears able to reproduce self-organised diagonal patterns. The demands (entry flows) have a strong influence on the shape of the patterns which are formed. Fig. 3 shows the result for a higher demand scenario, just as two videos submitted as electronic supplementary material (mmc1.mp4 and mmc2.mp4, see Appendix A). In this case, the patterns which are formed are less structured, showing occasional very high densities in the cells. The demand is however sufficiently low for the flow conditions to recuperate, restoring the structured diagonal stripe patterns as can be seen for the different snap shots. It is important to note that for higher demands, flow breakdown occurs. Analysis of these properties in more detail will be addressed in future papers on the model.

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4.4. Case 3: bi-directional flows

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In the final case, we consider a bi-directional pedestrian flow. As with the previous cases, we consider a 30 by 30 m area. Pedestrians are generated at either side of this area, and move to the opposite direction. Again, we assume that the interaction is asymmetrical, meaning that pedestrians try harder to avoid the other class then their own (reflected by the choices of the parameters βd→d′ ). As can be seen from Fig. 4 and two videos submitted as electronic supplementary material (mmc6.mp4 and mmc5.mp4, see Appendix A), lanes are formed dynamically. Note that we only have included the flow from left to right in the figure. The lane formation process is very dynamic and could even be coined chaotic, also in the sense that the number of lanes

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Fig. 3. Formation of diagonal stripes for crossing flows, with α0 = 0.25, βd→d = 2 and βd→d′ = 6 for d ̸= d′ , with demand equal to 0.193 P/m/s.

that are formed and their spatio-temporal characteristics vary during the simulation (different steady-state solutions). In illustration, Fig. 4 shows at t = 50 s the initial formation of lanes; 20 s later, we see clearly that more stable patterns have formed. As in reality, we see that lanes occasionally split up into to lanes (e.g. the top lane at t = 130 s). Furthermore, we see occasional localised congestion occurring (see t = 220 s), which generally recuperate since the demand is relatively low (see t = 240 s and beyond). As for the crossing flow case, the demands have a strong impact on the patterns that are formed: low demands yield neat, structured patterns; high demand yield more dynamic patterns, but even phase transitions from laminar to jammed flow. The conditions under which such phase transitions occur will be the focus of future research. In sum, from the mathematical analysis and the simulations we conclude that the model qualitatively captures some of the key phenomena that we observe in pedestrian flows, including self-organisation of structured spatial patterns such as dynamic lanes and diagonal stripes. 5. Discussion and conclusions Using the microscopic social forces model as a foundation, we have derived and analysed a first-order (equilibrium) pedestrian flow model. The interaction term present in the social forces model introduces a local path choice component in the equilibrium velocity for the continuum model presented in this contribution; see Eq. (16). At the same time, the interaction term causes a reduction of the equilibrium speed as shown in Eq. (15). Given that the global path choice is known, the model has two independent parameters only (α0 and β0 ), making it parsimonious compared to many (microscopic) models presented in literature. 5.1. Face validity and plausibility The results presented in this contribution are plausible, compared to empirical findings. Increasing the density causes a reduction in the average speed. Noting that empirical studies have shown jam densities of around 5 P/m2 and maximum speeds of around 1.3 m/s would allow us to determine α0 ≈ 3.8 m3 /(s P). The mechanism describing the local route choice behaviour is plausible as well: pedestrians have a natural aspiration to walk to regions where the density is relatively small. At the same time, they may be inclined to reduce their speeds

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Fig. 4. Formation of lanes for bi-directional flows, with α0 = 0.25, βd→d = 2 and βd→d′ = 6 for d ̸= d′ , with demands equal to 0.113 P/m/s.

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when walking into a region with increasing density. The influence of the parameter β0 is as expected: stronger interactions between pedestrians (reflected by large values for A, τ , etc.) yield a stronger impact of the density gradient. The simulation results show that the model has desirable features and is able to reproduce key characteristics of pedestrian flows, including self-organisation, and as such, this paper is, to the authors best knowledge, the first to report a macroscopic model to have this ability. Future work will in particular focus on the conditions under which the selforganisation occurs and when transitions take place, as well as on the relation with the parameter values and height of the demand. The simulation results presented here provide evidence that the model can not only capture self-organisation, but is also able to reproduce phase transitions from neat self-organised, laminar flow conditions to jammed conditions.

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5.2. Construct and predictive validity

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Although the results are very promising, improvements to the model are of course possible. These in particular pertain

Q5 to the more quantitative characteristics of the model. For instance, the equilibrium speed V (ρ, ∇ρ) is linear, which given

the empirical relations presented in literature is not plausible. A more general formulation, in which the linear relation is replaced by a more realistic relation between speed and density, can however be easily formulated.

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That said, calibration and validation of the model clearly need further attention. Next to the speed–density relation, this entails investigating quantitatively the impact of the density gradient and choices for β that would result in simulation results that are construct valid (such as the correct number of lanes, and the correct lane width). This is an important future research direction.

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Acknowledgements This research is performed as part of the NWO-VICI project ‘‘Travel Behavior And Traffic Operations In Case Of Exceptional Events’’ (Hoogendoorn, Duives), and as part of the NWO Aspasia grant of Daamen (Daamen, Van Wageningen-Kessels).

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Appendix A. Supplementary material

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Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.physa.2014.07.050.

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