Crowd Dynamics and Networks

CrowdDynamics&Networks

Engineering perspective on Theory, Modelling and Applications Prof. dr. Serge Hoogendoorn

2

3

Engineeringchallenges foreventsorregularsituations…• Canweforacertainevent/situationpredictifasafetyorthroughputbottleneckoccurs?

• Canwedevelopmodels&methodstosupportorganisation,planninganddesign?

• Canwedevelopapproachestoreal-timemanagelargepedestrianflowssafelyandefficiently?

Deepknowledgenetworkcrowddynamicsessentialtoanswerthesequestions!

Pedestrianflowoperations…

Simple case example: how long does it take to evacuate a room? • Consider a room of N people• Suppose that the (only) exit has capacity of C Peds/hour• Use a simple queuing model to compute duration T• How long does the evacuation take?

• Capacity of the door is very important• Which factors determine capacity?

4

T =N

C

Npeopleinarea

Doorcapacity:C

N

C


Simple case example: how long does it take to evacuate a room? • Wat determines capacity?• Experimental research on behalf of Dutch Ministry of

Housing• Experiments under different circumstances and

composition of flow

• Empirical basis to express the capacity of a door (per meter width, per second) as a function of the considered factors:

6

Increaseinfrictionresultinginarcformationbyincreasingpressurefrombehind(force-

Pedestriancapacitydropandfaster-is-slowereffect• Capacitydropalsooccursinpedestrianflow

• Faster=slowereffect

• Pedestrianexperiments(TUDresden,TUDelft)haverevealedthatoutflowreducessubstantiallywhenevacueestrytoexitroomasquicklyaspossible(rushing)

• Capacityreductioniscausedbyfrictionandarc-formationinfrontofdoorduetoincreasedpressure

• Capacityreductioncausessevereincreasesinevacuationtimes

HowoldDutchtraditionsmayactuallybeofsomeuse…

8

• Real-lifesituationsin(public)spacesoftenmorecomplex

• Limitedempiricalknowledgeonmulti-directionalflowsmotivatedfirstwalkerexperimentsin2002

• Worldpremiere,manyhavefollowed!

• Resultedinauniquemicroscopicdataset

Firstinsightsintoimportanceofself-organisationinpedestrianflows

Fascinatingself-organisation

• Example efficient self-organisation dynamic walking lanes in bi-directional flow• High efficiency in terms of capacity and observed walking speeds• Experiments by Hermes group show similar results as TU Delft experiments,

but at higher densities

9

Fascinatingself-organisation

• Relatively small efficiency loss (around 7% capacity reduction), depending on flow composition (direction split)

• Same applies to crossing flows: self-organised diagonal patterns turn out to be very efficient

• Other types of self-organised phenomena occur as well (e.g. viscous fingering)

• Phenomena also occur in the field…

10

Bi-directionalexperiment

Studyingself-organisationduringrockconcertLowlands…


So with this wonderful

self-organisation, why do

we need to worry about

crowds at all?

12

Break-downofefficientself-organisation• Whenconditionsbecometoocrowded(densitylargerthancriticaldensity),efficientself-organisation‘breaksdown’causing

• Flowperformance(effectivecapacity)decreasessubstantially,potentiallycausingmoreproblemsasdemandstaysatsamelevel

• Importanceof‘keepingthingsflowing’,i.e.keepingdensityatsubcriticallevelmaintainingefficientandsmoothflowoperations

• Hassevereimplicationsonthenetworklevel

ANewPhaseinPedestrianFlowOperations

• When densities become very large (> 6 P/m2) new phase emerges coined turbulence

• Characterised by extreme high densities and pressure exerted by the other pedestrians

• High probabilities of asphyxiation

14

Intermezzo:TheSAILtallshipevent• BiggestpubliceventintheNederland,organisedevery5yearssince1975

• OrganisedaroundtheIJhaven,Amsterdam

• Thistimearound600tallshipsweresailingin

• Around2,3millionnationalandinternationalvisitors

• ModellingsupportofSAILprojectinplanningandbydevelopmentofacrowdmanagementdecisionsupportsystem

Microscopicmodelsforplanningpurposes

Application of differential game theory: the NOMAD model • Pedestrians minimise predicted walking cost (effort), due

to straying from intended path, being too close to others / obstacles and effort, yielding:

• This simplified model is similar to Social Forces model of Helbing

Face validity? • Model results in reasonable macroscopic flow characteristics (capacity

values and fundamental diagram)• What about self-organisation?

15

FROM MICROSCOPIC TO MACROSCOPIC INTERACTIONMODELING

SERGE P. HOOGENDOORN

1. Introduction

This memo aims at connecting the microscopic modelling principles underlying thesocial-forces model to identify a macroscopic flow model capturing interactions amongstpedestrians. To this end, we use the anisotropic version of the social-forces model pre-sented by Helbing to derive equilibrium relations for the speed and the direction, giventhe desired walking speed and direction, and the speed and direction changes due tointeractions.

2. Microscopic foundations

We start with the anisotropic model of Helbing that describes the acceleration ofpedestrian i as influence by opponents j:

(1) ~ai

=~v0i

� ~vi

⌧i

�Ai

X

j

exp

�R

ij

Bi

�· ~n

ij

·✓�i

+ (1� �i

)1 + cos�

ij

2

◆

where Rij

denotes the distance between pedestrians i and j, ~nij

the unit vector pointingfrom pedestrian i to j; �

ij

denotes the angle between the direction of i and the postionof j; ~v

i

denotes the velocity. The other terms are all parameters of the model, that willbe introduced later.

In assuming equilibrium conditions, we generally have ~ai

= 0. The speed / directionfor which this occurs is given by:

(2) ~vi

= ~v0i

� ⌧i

Ai

X

j

exp

�R

ij

Bi

�· ~n

ij

·✓�i

+ (1� �i

)1 + cos�

ij

2

◆

Let us now make the transition to macroscopic interaction modelling. Let ⇢(t, ~x)denote the density, to be interpreted as the probability that a pedestrian is present onlocation ~x at time instant t. Let us assume that all parameters are the same for allpedestrian in the flow, e.g. ⌧

i

= ⌧ . We then get:(3)

~v = ~v0(~x)� ⌧A

ZZ

~y2⌦(~x)

exp

✓� ||~y � ~x||

B

◆✓�+ (1� �)

1 + cos�xy

(~v)

2

◆~y � ~x

||~y � ~x||⇢(t, ~y)d~y

Here, ⌦(~x) denotes the area around the considered point ~x for which we determine theinteractions. Note that:

(4) cos�xy

(~v) =~v

||~v|| ·~y � ~x

||~y � ~x||1

Level of anisotropy reflected by this parameter

~vi

~v0i

~ai

~nij

~xi

~xj

• Simplemodelshowsplausibleself-organisedphenomena

• Modelalsoshowsflowbreakdownincaseofoverloading

• Presentedmodelishoweverincompleteasitrequiresspecificationofa(desired)route…

• Generalassumptionofcostminimisationreasonable?

• Whatdoesdatasay?

Completingthemodel?

• The NOMAD / social-forces model requires information about the desired walking direction

• General assumption is that pedestrians choose path / route that minimises generalised cost (time or more generally effort or disutility)

• Different studies in pedestrian route choice show how cost definition depends on walking purpose

• Example: pedestrian route choice during SAIL (can we find a cost definition?)

17



1. Introduction




(1) ~ai

=~v0i

� ~vi

⌧i

�Ai

X

j

exp

�R

ij

Bi

�· ~n

ij

·✓�i

+ (1� �i

)1 + cos�

ij

2

◆

where Rij



ij


i




(2) ~vi

= ~v0i

� ⌧i

Ai

X

j

exp

�R

ij

Bi

�· ~n

ij

·✓�i

+ (1� �i

)1 + cos�

ij

2

◆


i


~v = ~v0(~x)� ⌧A

ZZ

~y2⌦(~x)

exp

✓� ||~y � ~x||

B

◆✓�+ (1� �)

1 + cos�xy

(~v)

2

◆~y � ~x

||~y � ~x||⇢(t, ~y)d~y


(4) cos�xy

(~v) =~v

||~v|| ·~y � ~x

||~y � ~x||1

18

Routechoicebehaviouratevents…• Studyintostatedandrevealedchoicebehaviourshowswhichcostcomponentsarerelevant

• Attributesincludeattractionsalongroute,walkingnearwater,roadwaywidth,andcrowdedness

• Dependenceoncontextishuge!

Competingthemodel:routechoicetheory

Use of dynamic programming: • Let W(t,x) denote the minimum cost of getting

from (t,x) to the destination area A• We can then show that this value function W(t,x)

satisfies the HJB equation

• Optimal velocity follows steepest descent towards destination A:

• Solution schemes (Fleming & Soner,1993) 0 20 40 60 80x1-axis (m)

0

20

40

60

x2-a

xis

(m)

16

2024

28

2828

28

32

323236

40

4044

4448

48

48

52 52

52

56

56

60

64

72

�@W

@t

= L(t, ~x,~v⇤) + ~v

⇤ ·rW +�

2

2�W

~v⇤ = �rW


Numerical approaches (which we will skip…) other than finite differences • Discretise area using any type of meshing (e.g. triangular mesh, rectangles)• Interpret mesh as network with nodes and links • Probability of jumping from to is equal to

• Using these, we can compute the value function by solving the following controlled stochastic Markovian jump process:

⌦

~x

~y p

~v(~x, ~y) =�t

||~y � ~x||

✓~v · ~y � ~x

||~y � ~x||

◆+

W (t, ~x) = inf~v2�

2

4�tL(t,~v, ~x) +X

~y2

p

~vW (t+�t, ~y)

3

5


• Approach can be used to solve route choice problem for generic cost definitions without pre-specifying network structure

• Generalisation to destination choice and activity-scheduling is straightforward (Hoogendoorn and Bovy, 2004)

• Drawbacks? Approach is computationally demanding…

• Different heuristics to circumvent issues proposed in literature…

• Applicationforplanningpurposes(e.g.SAIL)

• Questionableifforreal-timeandoptimisationpurposessuchamodelwouldbeuseful,e.g.duetocomplexity

• Coarsermodelsproposedsofarturnouttohavelimitedpredictivevalidity,andareunabletoreproduceself-organisedpatterns

Macroscopicmodelling

23

Multi-class macroscopic model of Hoogendoorn and Bovy (2004): • Kinematic wave model for pedestrian flow for each destination d

• Here V is the (multi-class) equilibrium speed; the optimal direction:

• Stems from minimum cost Wd(t,x) for each (set of) destination(s) d• Is this a reasonable model? • No, since there is only pre-determined route choice, the model will have unrealistic features

!γd (t,!x)= − ∇Wd (t,

!x)||∇Wd (t,

!x) ||

∂ρd∂t

+∇⋅!qd = r − s with !qd =

!γd ⋅V (ρ1,...,ρD )

!qd =!γd ⋅ρd ⋅V (ρ1,...,ρD )

Macroscopicmodelling

24Solution? Include a term describing local route / direction choice

6 S. P. Hoogendoorn, F. L. M. van Wageningen-Kessels, W. Daamen, and M. Sarvi / Transportation Research Procedia 00 (2015) 000–000

Ω

y

∂ρ∂t

+ ∂(ρU(ρ))∂x1

= 0Boundary conditions: ρ(t,0, x2 ) = ρ0 (t, x2 )

′y

Fig. 1. Considered walking area ⌦

Fig. 2. Numerical experiment showing the impact of only considering global route choice on flow conditions.

To remedy these issues, in this paper we put forward a dynamic route choice model, that takes care of the fact thatthe global route choice behaviour is determined pre-trip and does not include the impact of changing flow conditionsthat may result in additional costs. To this end, we introduce a local route choice component that reflects additionallocal cost 'd (e.g. extra delays, discomfort) caused by the prevailing flow conditions. We assume that these aredependent on the (spatial changes in the) class-specific densities.

In the remainder, we will assume that the local route cost function 'd can be expressed as a function of the class-specific densities and density gradients. By di↵erentiating between the classes, we can distinguish between local routecosts incurred by interacting with pedestrians in the same class (walking into the same direction), and between classinteractions. As a result, we have for the flow vector the following expression:

~qd = ~�d(⇢1, ..., ⇢D,r⇢1, ...,r⇢D) · ⇢d · U(⇢1, ..., ⇢D) (4)

This expression shows that we will assume that the equilibrium speed a function of the local densities of the respectiveclasses, while the equilibrium direction is a function of both the densities and the density gradients.

Finally, note that in assuming that initial and boundary conditions are known, eqn. (1) and (4) together givea complete mathematical model allowing us to determined the dynamics. Furthermore, the function U should beknows, as discussed before. The local route choice function ~� will be discussed in the next Section.FvW: Hierboven ref naar functie U en ~� toegeveogd.

Modellingforplanningandreal-timepredictions

• NOMAD / Social-forces model as starting point:

• Equilibrium relation stemming from model (ai = 0):

• Interpret density as the ‘probability’ of a pedestrian being present, which gives a macroscopic equilibrium relation (expected velocity), which equals:

• Combine with conservation of pedestrian equation yields complete model, but numerical integration is computationally very intensive

25



1. Introduction




(1) ~ai

=~v0i

� ~vi

⌧i

�Ai

X

j

exp

�R

ij

Bi

�· ~n

ij

·✓�i

+ (1� �i

)1 + cos�

ij

2

◆

where Rij



ij


i




(2) ~vi

= ~v0i

� ⌧i

Ai

X

j

exp

�R

ij

Bi

�· ~n

ij

·✓�i

+ (1� �i

)1 + cos�

ij

2

◆


i


~v = ~v0(~x)� ⌧A

ZZ

~y2⌦(~x)

exp

✓� ||~y � ~x||

B

◆✓�+ (1� �)

1 + cos�xy

(~v)

2

◆~y � ~x

||~y � ~x||⇢(t, ~y)d~y


(4) cos�xy

(~v) =~v

||~v|| ·~y � ~x

||~y � ~x||1



1. Introduction




(1) ~ai

=~v0i

� ~vi

⌧i

�Ai

X

j

exp

�R

ij

Bi

�· ~n

ij

·✓�i

+ (1� �i

)1 + cos�

ij

2

◆

where Rij



ij


i




(2) ~vi

= ~v0i

� ⌧i

Ai

X

j

exp

�R

ij

Bi

�· ~n

ij

·✓�i

+ (1� �i

)1 + cos�

ij

2

◆


i


~v = ~v0(~x)� ⌧A

ZZ

~y2⌦(~x)

exp

✓� ||~y � ~x||

B

◆✓�+ (1� �)

1 + cos�xy

(~v)

2

◆~y � ~x

||~y � ~x||⇢(t, ~y)d~y


(4) cos�xy

(~v) =~v

||~v|| ·~y � ~x

||~y � ~x||1



1. Introduction




(1) ~ai

=~v0i

� ~vi

⌧i

�Ai

X

j

exp

�R

ij

Bi

�· ~n

ij

·✓�i

+ (1� �i

)1 + cos�

ij

2

◆

where Rij



ij


i




(2) ~vi

= ~v0i

� ⌧i

Ai

X

j

exp

�R

ij

Bi

�· ~n

ij

·✓�i

+ (1� �i

)1 + cos�

ij

2

◆


i


~v = ~v0(~x)� ⌧A

ZZ

~y2⌦(~x)

exp

✓� ||~y � ~x||

B

◆✓�+ (1� �)

1 + cos�xy

(~v)

2

◆~y � ~x

||~y � ~x||⇢(t, ~y)d~y


(4) cos�xy

(~v) =~v

||~v|| ·~y � ~x

||~y � ~x||1


• First-order Taylor series approximation: yields a closed-form expression for the equilibrium velocity , which is given by the equilibrium speed and direction:

with:• Check behaviour of model by looking at isotropic flow ( ) and homogeneous flow

conditions ( ) • Include conservation of pedestrian relation gives a complete model…

26

2 SERGE P. HOOGENDOORN

From this expression, we can find both the equilibrium speed and the equilibrium direc-tion, which in turn can be used in the macroscopic model.

We can think of approximating this expression, by using the following linear approx-imation of the density around ~x:

(5) ⇢(t, ~y) = ⇢(t, ~x) + (~y � ~x) ·r⇢(t, ~x) +O(||~y � ~x||2)

Using this expression into Eq. (3) yields:

(6) ~v = ~v0(~x)� ~↵(~v)⇢(t, ~x)� �(~v)r⇢(t, ~x)

with ↵(~v) and �(~v) defined respectively by:

(7) ~↵(~v) = ⌧A

ZZ

~y2⌦(~x)

exp

✓� ||~y � ~x||

B

◆✓�+ (1� �)

1 + cos�xy

(~v)

2

◆~y � ~x

||~y � ~x||d~y

and

(8) �(~v) = ⌧A

ZZ

~y2⌦(~x)

exp

✓� ||~y � ~x||

B

◆✓�+ (1� �)

1 + cos�xy

(~v)

2

◆||~y � ~x||d~y

To investigate the behaviour of these integrals, we have numerically approximatedthem. To this end, we have chosen ~v( ) = V · (cos , sin ), for = 0...2⇡. Fig. 1 showsthe results from this approximation.

0 1 2 3 4 5 6 7−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

angle

valu

e

α

1

α2

β

Figure 1. Numerical approximation of ~↵(~v) and �(~v).

For the figure, we can clearly see that � is independent on ~v, i.e.

(9) �(~v) = �0

FROM MICROSCOPIC TO MACROSCOPIC INTERACTION MODELING 3

Furthermore, we see that for ~↵, we find:

(10) ~↵(~v) = ↵0 ·~v

||~v||

(Can we determine this directly from the integrals?)From Eq. (6), with ~v = ~e · V we can derive:

(11) V = ||~v0 � �0 ·r⇢||� ↵0⇢

and

(12) ~e =~v0 � �0 ·r⇢

V + ↵0⇢=

~v0 � �0 ·r⇢

||~v0 � �0 ·r⇢||

Note that the direction does not depend on ↵0, which implies that the magnitude ofthe density itself has no e↵ect on the direction, while the gradient of the density does

influence the direction.

2.1. Homogeneous flow conditions. Note that in case of homogeneous conditions,i.e. r⇢ = ~0, Eq. (11) simplifies to

(13) V = ||~v0||� ↵0⇢ = V 0 � ↵0⇢

i.e. we see a linear relation between speed and density. For the direction ~e, we then get:

(14) ~e =~v0

V + ↵0⇢= ~e0

In other words, in homogeneous density conditions the direction of the pedestrians isequal to the desired direction.

2.2. Isotropic walking behaviour. Let us also note that in case � = 1 (isotropicflow), and assuming that ⌦ is symmetric around ~x, we get:

(15) ~↵(~v) = ⌧A

ZZ

~y2⌦(~x)

exp

✓� ||~y � ~x||

B

◆~y � ~x

||~y � ~x||d~y = ~0

which means ↵0 = 0. In this case, we have:

(16) V = ||~v0 � �0 ·r⇢||

This expression shows that in this case, the speed is only dependent on the densitygradient. If a pedestrian walks into a region in which the density is increasing, the speedwill be less than the desired speed; and vice versa. Also note that in case of homogenousconditions, the speed will be constant and equal to the free speed. Note that this isconsistent with the results from Hoogendoorn, ISTTT-2003.

For the direction, we find:

(17) ~e =~v0 � �0 ·r⇢

||~v0 � �0 ·r⇢||

FROM MICROSCOPIC TO MACROSCOPIC INTERACTION MODELING 3

Furthermore, we see that for ~↵, we find:

(10) ~↵(~v) = ↵0 ·~v

||~v||

(Can we determine this directly from the integrals?)From Eq. (6), with ~v = ~e · V we can derive:

(11) V = ||~v0 � �0 ·r⇢||� ↵0⇢

and

(12) ~e =~v0 � �0 ·r⇢

V + ↵0⇢=

~v0 � �0 ·r⇢

||~v0 � �0 ·r⇢||

Note that the direction does not depend on ↵0, which implies that the magnitude ofthe density itself has no e↵ect on the direction, while the gradient of the density does

influence the direction.

2.1. Homogeneous flow conditions. Note that in case of homogeneous conditions,i.e. r⇢ = ~0, Eq. (11) simplifies to

(13) V = ||~v0||� ↵0⇢ = V 0 � ↵0⇢

i.e. we see a linear relation between speed and density. For the direction ~e, we then get:

(14) ~e =~v0

V + ↵0⇢= ~e0

In other words, in homogeneous density conditions the direction of the pedestrians isequal to the desired direction.

2.2. Isotropic walking behaviour. Let us also note that in case � = 1 (isotropicflow), and assuming that ⌦ is symmetric around ~x, we get:

(15) ~↵(~v) = ⌧A

ZZ

~y2⌦(~x)

exp

✓� ||~y � ~x||

B

◆~y � ~x

||~y � ~x||d~y = ~0

which means ↵0 = 0. In this case, we have:

(16) V = ||~v0 � �0 ·r⇢||

This expression shows that in this case, the speed is only dependent on the densitygradient. If a pedestrian walks into a region in which the density is increasing, the speedwill be less than the desired speed; and vice versa. Also note that in case of homogenousconditions, the speed will be constant and equal to the free speed. Note that this isconsistent with the results from Hoogendoorn, ISTTT-2003.

For the direction, we find:

(17) ~e =~v0 � �0 ·r⇢

||~v0 � �0 ·r⇢||

α 0 = πτAB2 (1− λ) and β0 = 2πτAB3(1+ λ)

4.1. Analysis of model properties

Let us first take a look at expressions (14) and (15) describing the equilibrium290

speed and direction. Notice first that the direction does not depend on ↵0, which

implies that the magnitude of the density itself has no e↵ect, and that only the

gradient of the density does influence the direction. We will now discuss some

other properties, first by considering a homogeneous flow (r⇢ = ~0), and then

by considering an isotropic flow (� = 1) and an anisotropic flow (� = 0).295

4.1.1. Homogeneous flow conditions

Note that in case of homogeneous conditions, i.e. r⇢ = ~0, Eq. (14) simplifies

to

V = ||~v0||� ↵0⇢ = V 0 � ↵0⇢ (16)

i.e. we see a linear relation between speed and density. The term ↵0 � 0

describes the reduction of the speed with increasing density.300

For the direction ~e, we then get:

~e =~v0

||~v0|| = ~e0 (17)

In other words, in homogeneous density conditions the direction of the pedestri-

ans is equal to the desired direction. Clearly, the gradient of the density yields

pedestrians to divert from their desired direction.

Looking further at the expressions for ↵0 and �0, we can see the influence of305

the various parameters on their size; ↵0 scales linearly with A and ⌧ , meaning

that the influence of the density on the speed increases with increasing values

of A and ⌧ . At the same time, larger values for B imply a reduction of the

influence of the density. Needs to be revised!

The same can be concluded for the influence of the gradient: we see linear310

scaling for A and ⌧ , and reducing influence with larger values of B. This holds

for the equilibrium speed and direction. Needs to be revised!

13

4.1. Analysis of model properties

Let us first take a look at expressions (14) and (15) describing the equilibrium290

speed and direction. Notice first that the direction does not depend on ↵0, which

implies that the magnitude of the density itself has no e↵ect, and that only the

gradient of the density does influence the direction. We will now discuss some

other properties, first by considering a homogeneous flow (r⇢ = ~0), and then

by considering an isotropic flow (� = 1) and an anisotropic flow (� = 0).295

4.1.1. Homogeneous flow conditions

Note that in case of homogeneous conditions, i.e. r⇢ = ~0, Eq. (14) simplifies

to

V = ||~v0||� ↵0⇢ = V 0 � ↵0⇢ (16)

i.e. we see a linear relation between speed and density. The term ↵0 � 0

describes the reduction of the speed with increasing density.300

For the direction ~e, we then get:

~e =~v0

||~v0|| = ~e0 (17)

In other words, in homogeneous density conditions the direction of the pedestri-

ans is equal to the desired direction. Clearly, the gradient of the density yields

pedestrians to divert from their desired direction.

Looking further at the expressions for ↵0 and �0, we can see the influence of305

the various parameters on their size; ↵0 scales linearly with A and ⌧ , meaning

that the influence of the density on the speed increases with increasing values

of A and ⌧ . At the same time, larger values for B imply a reduction of the

influence of the density. Needs to be revised!

The same can be concluded for the influence of the gradient: we see linear310

scaling for A and ⌧ , and reducing influence with larger values of B. This holds

for the equilibrium speed and direction. Needs to be revised!

13

!v = !e ⋅V


• Uni-directional flow situation• Picture shows differences between

situation without and with local route choice for two time instances

• Model introduces ‘lateral diffusion’ since pedestrians will look for lower density areas actively

• Diffusion can be controlled by choosing parameters differently

• Model shows plausible behaviour 27

6 S. P. Hoogendoorn, F. L. M. van Wageningen-Kessels, W. Daamen, and M. Sarvi / Transportation Research Procedia 00 (2015) 000–000

Ω

y

∂ρ∂t

+ ∂(ρU(ρ))∂x1

= 0Boundary conditions: ρ(t,0, x2 ) = ρ0 (t, x2 )

′y

Fig. 1. Considered walking area ⌦

Fig. 2. Numerical experiment showing the impact of only considering global route choice on flow conditions.

To remedy these issues, in this paper we put forward a dynamic route choice model, that takes care of the fact thatthe global route choice behaviour is determined pre-trip and does not include the impact of changing flow conditionsthat may result in additional costs. To this end, we introduce a local route choice component that reflects additionallocal cost 'd (e.g. extra delays, discomfort) caused by the prevailing flow conditions. We assume that these aredependent on the (spatial changes in the) class-specific densities.

In the remainder, we will assume that the local route cost function 'd can be expressed as a function of the class-specific densities and density gradients. By di↵erentiating between the classes, we can distinguish between local routecosts incurred by interacting with pedestrians in the same class (walking into the same direction), and between classinteractions. As a result, we have for the flow vector the following expression:

~qd = ~�d(⇢1, ..., ⇢D,r⇢1, ...,r⇢D) · ⇢d · U(⇢1, ..., ⇢D) (4)

This expression shows that we will assume that the equilibrium speed a function of the local densities of the respectiveclasses, while the equilibrium direction is a function of both the densities and the density gradients.

Finally, note that in assuming that initial and boundary conditions are known, eqn. (1) and (4) together givea complete mathematical model allowing us to determined the dynamics. Furthermore, the function U should beknows, as discussed before. The local route choice function ~� will be discussed in the next Section.FvW: Hierboven ref naar functie U en ~� toegeveogd.

S. P. Hoogendoorn, F. L. M. van Wageningen-Kessels, W. Daamen, and M. Sarvi / Transportation Research Procedia 00 (2015) 000–000 11

with other classes results in di↵erent local costs than with other classes). Thirdly, we investigate to which extent thenew model can capture the breakdown of self-organisation.

Next to this, we will investigate the impacts of the di↵erent terms in the model. In particular, we will look at theimpact of di↵erences in the weights �� and ⌘� that describe the relative impact of the densities of the other groupscompared to the density of the own group. Finally, we will look at the relative impacts of the crowdedness terms andthe delay term via changing the parameter ↵d.

5.1. Introducing the base model and the scenarios

Our base model is defined by using a simple linear speed-density relation U(⇢) = v0 ·(1�⇢/⇢ jam) = 1.34·(1�⇢/5.4).We will use �� = ⌘� = 1 for � = d and �� = ⌘� = 4 otherwise, meaning that the impact of the densities of the othergroups are substantially higher. Finally, in the base model we will only consider the impact of the crowdedness factor,i.e. ↵d = 0.

To test the base model, and the di↵erent variants on it, we will use three scenarios:

1. Unidirectional flow scenario, mostly used to show flow dispersion compared to the example presented in section3.2. In this scenario, pedestrians enter on the left side of the forty by forty meter area and walk towards the left.

2. Crossing flow scenario, where the first group of pedestrians is generated on the left, and walks to the right, whilethe second group is generated at the bottom of the area, and moves up.

3. Bi-directional flow scenario, where two groups of pedestrians are generated at respectively the left and the rightside of the forty by forty meter area.

In all scenarios, pedestrians are generated on the edges of the area between -5 and 5.

5.2. Impact of local route choice of flow dispersion

Let us first revisit the example presented in section 3.2, shown in Fig. 1. The example considered a uni-directionalpedestrian flow with fixed (global) route choice. The resulting flow operations showed no lateral dispersion of pedes-trians, which appears not realistic since it caused big di↵erences in speeds and travel times for pedestrians that arespatially very close.

Including the local choice term 'd causes dispersion in a lateral sense: since pedestrians avoid high density areas,densities will disperse and smooth over the area, see Fig. 3. The resulting flow conditions appear much more realisticthat the situation that occurred in case the local route choice model was not included.

Fig. 3. Numerical experiment showing the impact of only considering global route choice on flow conditions. For the simulations, we have used alinear fundamental diagram U(⇢) = v0 � ↵0⇢. The simulations where performed using a newly developed Godunov scheme for two dimensionaltra�c flows.

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Macroscopicmodelyieldsplausibleresults…• Firstmacroscopicmodelabletoreproduceself-organisedpatterns(laneformation,diagonalstripes)

• Self-organisationbreaksdownsincaseofoverloading

• Continuummodelseemstoinheritpropertiesofthemicroscopicmodelunderlyingit

• Formssolidbasisforreal-timepredictionmoduleindashboard

• Firsttrialsinmodel-basedoptimisationanduseofmodelforstate-estimationarepromising

29

CrowdManagementforEvents• Uniquepilotwithcrowdmanagementsystemforlargescale,outdoorevent

• FunctionalarchitectureofSAIL2015crowdmanagementsystems

• Phase1focussedonmonitoringanddiagnostics(datacollection,numberofvisitors,densities,walkingspeeds,determininglevelsofserviceandpotentiallydangeroussituations)

• Phase2focussesonpredictionanddecisionsupportforcrowdmanagementmeasuredeployment(model-basedprediction,interventiondecisionsupport)

Data fusion and

state estimation: hoe many people are there and how

fast do they move?

Social-media analyser: who are

the visitors and what are they talking

about?

Bottleneck inspector: wat

are potential problem

locations?

State predictor: what will the situation look like in 15

minutes?

Route estimator:

which routes are people

using?

Activity estimator: what are people doing?

Intervening: do we need to apply certain

measures and how?

Exampledashboardoutcomes

• Density estimates based on data fusion by means of data fusion and filtering (see figure)

• Other examples show volumes and OD flows • Results used for real-time intervention, but also for

planning of SAIL 2020 (simulation studies)

1988

1881

4760

4958

2202

1435

6172

59994765 4761

4508

3806

3315

2509

17523774

4061

2629

13592654

21391211

1439

2209

1638

2581

311024653067

2760

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

11 12 13 14 15 16 17 18 19

Dichtheden)Veemkade

dichtheid2(ped/m2)

Work in coming years will

be about use macroscopic

models for real-time prediction

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UseofSocialMediadata• Nexttotrafficdatacollection,socialmediadatawereanalysedtotestitsusabilityforcrowdmonitoring

• ExampleshowsTwitteractivityrelatedtoSailand“Crowdedness”

• Infindingcorrelationsbetweenthesedataandtrafficdata,improvementsinstateestimationsareforeseen!

Foreign tourists

Residents of Amsterdam

Bi-leveloptimisation• Findoptimalrouting/staging

• Formulatebi-leveloptimisationproblemcombiningroutechoiceandassignmentmodel

• Substantialimprovementsinevacuationtimes!

Optimal routing problem:

compute W(t,x)

First-order pedestrian flow model: compute

ρ(t,x)

ρ(t,x)

v*(t,x)~v

⇤(t, ~x)

⇢(t, ~x)

140

0

250

215

Remaining evacuees

Evacuation duration

NetworkFundamentalDiagram

• Much attention to network-scale models of vehicular (urban) network traffic

• Proof of existence of Network Fundamental Diagram • Yokohama example based on GPS data

• Recent work shows importance of spatial distribution of density (g-NFD)

• What about pedestrian networks?

MetamodelsandMacroFundamentalDiagrams

• NFD also exists for pedestrian networks! • Example shows relation between accumulation and

production (exit-rates)• Including spatial density

variation allows deriving accurate relations between average network flow, density and density variance:

34

Closingremarks

• Keeping pedestrian safety and comfort at high levels by means of crowd management leads to many scientific challenges in data collection, modelling & simulation, and control & management!

• Development of predictively valid models at microscopic and macroscopic scale, involving both operations and route choice modelling

• Accurate models to predict phase transitions • Efficient and accurate numerical solution approaches, in particular for macroscopic

models and for route choice modelling (e.g. via variational theory, Lagrangian formulation)

• Methods for state estimation, data fusion and (real-time) crowd management• Involving heterogeneity in behaviour and impacts thereof

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Moreinformation?

• Hoogendoorn, S.P., van Wageningen-Kessels, F., Daamen, W., Duives, D.C., Sarvi, M. Continuum theory for pedestrian traffic flow: Local route choice modelling and its implications (2015) Transportation Research Part C: Emerging Technologies, 59, pp. 183-197.

• Van Wageningen-Kessels, F., Leclercq, L., Daamen, W., Hoogendoorn, S.P. The Lagrangian coordinate system and what it means for two-dimensional crowd flow models (2016) Physica A: Statistical Mechanics and its Applications, 443, pp. 272-285.

• Hoogendoorn, S.P., Van Wageningen-Kessels, F.L.M., Daamen, W., Duives, D.C. Continuum modelling of pedestrian flows: From microscopic principles to self-organised macroscopic phenomena (2014) Physica A: Statistical Mechanics and its Applications, 416, pp. 684-694.

• Knoop, V.L., Van Lint, H., Hoogendoorn, S.P. Traffic dynamics: Its impact on the Macroscopic Fundamental Diagram (2015) Physica A: Statistical Mechanics and its Applications, 438, art. no. 16247, pp. 236-250.

• Campanella, M., Halliday, R., Hoogendoorn, S., Daamen, W. Managing large flows in metro stations: The new year celebration in copacabana (2015) IEEE Intelligent Transportation Systems Magazine, 7 (1), art. no. 7014395, pp. 103-113.

• Duives, D.C., Daamen, W., Hoogendoorn, S.P. State-of-the-art crowd motion simulation models (2014) Transportation Research Part C: Emerging Technologies, 37, pp. 193-209.

• Huibregtse, O., Hegyi, A., Hoogendoorn, S. Robust optimization of evacuation instructions, applied to capacity, hazard pattern, demand, and compliance uncertainty (2011) 2011 International Conference on Networking, Sensing and Control, ICNSC 2011, art. no. 5874936, pp. 335-340.

• Hoogendoorn, S.P., Daamen, W. Microscopic parameter identification of pedestrian models and implications for pedestrian flow modeling (2006) Transportation Research Record, (1982), pp. 57-64.

• Daamen, W., Hoogendoorn, S.P., Bovy, P.H.L. First-order pedestrian traffic flow theory (2005) Transportation Research Record, (1934), pp. 43-52.• Hoogendoorn, S.P., Bovy, P.H.L. Pedestrian travel behavior modeling (2005) Networks and Spatial Economics, 5 (2), pp. 193-216.• Hoogendoorn, S.P., Bovy, P.H.L. Dynamic user-optimal assignment in continuous time and space (2004) Transportation Research Part B: Methodological,

38 (7), pp. 571-592. 36

Crowd Dynamics and Networks

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