8. macroscopic traffic modelingsite.iugaza.edu.ps/emasry/files/2011/02/macroscopic-modelling.pdfdr....

Dr. Essam almasriTraffic Management and Control (ENGC 6340)

8. Macroscopic

Traffic ModelingTraffic Modeling


Introduction

• In traffic stream characteristics chapter we learned that the fundamental relation (q=k.u) and the fundamental diagrams enable us to describe the traffic state of stationary and homogeneous traffic. stationary and homogeneous traffic.

• Thus we can calculate the two remaining variables for a given value of a macroscopic variable.


Introduction

• When traffic is stationary and

homogeneous, we know that the values

for these variables will remain constant

along the entire road and for some along the entire road and for some

extended period.

• However, real traffic is neither

homogeneous nor stationary


Introduction

• In doing so, we define the dynamic

relation between q(x,t), u(x,t) and k(x,t).

• We assume, therefore, that we are

dealing with point variables: variables that dealing with point variables: variables that

are singularly defined at any moment and

at every location.

• By doing this we can show these three

variables as functions in the t-x plane.


Macroscopic Traffic Flow Modeling

• Tries to describe the aggregate behaviour of the

flow characteristics

• Various analogies use to describe traffic flow• Various analogies use to describe traffic flow

– Fluid flow analogy

– Heat flow analogy

– Gas flow analogy


Macroscopic Traffic Flow Modeling

• Hydrodynamic analogy

• Law of conservation of number of vehicles

• Balance continuity (conservation) equation• Balance continuity (conservation) equation

0),(),(=

∂∂

+∂

∂x

txq

t

txk

k and q are unknowns.


Derivation of conservation equation

• The conservation equation can easily be derived

by considering a unidirectional continuous road

section with two counting Stations 1 and 2

(upstream and downstream, respectively)



• Let N be the number of cars (volume) passing Station i

during time ∆t and qi , the flow passing station i; ∆t is the

duration of simultaneous counting at Station 1 and 2.

• Without loss of generality, suppose that N1 >N2 . Because

there is no loss of cars ∆ x (i.e., no sink), this assumption there is no loss of cars ∆ x (i.e., no sink), this assumption

implies that there is a buildup of cars between Station 1 and

Station 2.



• Let (N2 – N1 ) = ∆N; for a buildup ∆N will be negative. Based

on these definitions we have then the build-up of cars

between stations during ∆t will be (- ∆q) ∆t:

B ACD

Situation at time t1



• Let (N2 – N1 ) = ∆N; for a buildup ∆N will be negative. Based

on these definitions we have then the build-up of cars

between stations during ∆t will be (- ∆q) ∆t:

B ACD

Situation at time t2



B ACD


Solution of the conservation equation

• The conservation equation is a state equation that can be used to determine the flow at any section of the roadway.

• The attractiveness of this equation is that it relates two fundamental dependent variables, density and two fundamental dependent variables, density and flow rate, with the two independent ones (i.e., time t, and space x).

• Solution of the equation is impossible without an additional equation or assumption.

0)),((),(=

∂∂

+∂

∂x

txkq

t

txk


Solution of the conservation equation

• Lighthill and Whitham solved the equation by

assuming flow rate q is a function of local density k.

• k(x,t) and q(x,t) are dependent, so q(k(x,t))• k(x,t) and q(x,t) are dependent, so q(k(x,t))

• Therefore the continuity equation takes the form:

0)),((),(=

∂∂

+∂

∂x

txkq

t

txk


Daganzo simplification of the solution

• Daganzo further simplified the solution scheme

of Lighthill and Whitham by adopting the

following relationship between q and k:

{ }min , , ( )f jamq v k Q w k k= −

• Daganzo named his model as: Cell

Transmission Model (CTM)


Time • is discretized into equal intervals of ∆t (1, 2, 3, 5s)

Networ

k

• is divided into segments called cells

Cell Transmission Model (CTM)

i-1 i i+1

• Cell length = ∆x = vf . ∆t

Cell properties

ni ni+1ni-1

qiqi+1

ni: Cell occupancy (actual number of vehicles)

Ni: Maximum possible cell occupancy

qi : Actual inflow

Qi : Inflow capacity


Q

q

Approximation of q-k relationship

q = min { vf . k ; Q ; w . (kmax – k) }


qi = min { ni-1 ; Qi ; w/vf . (Ni – ni) }

Waiting

vehicles

Inflow

capacity

Available space

vf-w

kmax ki-1 i i+1

ni ni+1ni-1

qiqi+1

Vf k

w k

∆x ∆x ∆x

With k = n/∆x = n/(vf * ∆t)

for ∆t = 1 =>


Continuity equation

tk

xq

∂∂−=∂

∂

i-1 i i+1

ni ni+1ni-1

q q


=+1tin

t

int

iq+t

iq 1+−

1

1

t t t t

i i i iq q n n++ − = − −

qiqi+1

∆x ∆x ∆x


Simulation of Traffic Signal

i-1 i i+1

ni ni+1

qiqi+1

ni-1

= 0iq (t )For t ∈ red phase

For t ∈ green phase As previous equation

qi = min { ni-1 ; Q ; w/vf . (Ni – ni) }

qiqi+1


dI(t)

∆∆∆∆ t where: kI

i

q (t )v (t )

k (t )=

iI

n (t )k (t )

x=

∆

Derivation of Delay Equation

By Almasri 2006i i

f

td (t ) t v (t )

v

∆= ∆ − ×

For one

vehicle

in (t )×

No. of

vehicles∆∆∆∆ t

vf

vI(t)vf

f

xv

t

∆=∆

niqk

Once delay is determined at cell level, it can be determined at link or

network level, by summing up the delays for all cells.

[ ]I i kd (t) t n (t) q (t) t= ∆ − ×∆

[ ] t=1 I I kFor d (t) n (t) q (t)∆ ⇒ = −


Cell Representation and

Boundary Conditions

• When the free flow speed is

50 kph and the simulation

step is 1s, then the length of 28m

step is 1s, then the length of

each cell in the network is

13.89m ((50/3.6) × 1).

• Therefore, the input and exit

sections should have 3 and

2 cells respectively

42 m 28m

42 m


Cell Representation and Boundary Conditions

5 4 3 6 7

15

14

1 8

16

2 5 4 3 6 7

13

12

11

1

10

9

8 2

Origin Cell

Gate Cell

Destination Cell

Normal Cell


Cell Representation and Boundary Conditions

16

Origin, destination, and gate cells are used to specify boundary

conditions as follows:

• Origin cells (e.g. cells 1and 9) must

have an infinite number of vehicles (n

=infinity ) that discharge into empty “gate”

cells.

∞

5 4 3 6 7

13

12

11

15

14

1

10

9

8 2

Origin Cell

Gate Cell

Destination Cell

Normal Cell

cells.

• Gate cells (e.g. cells 2 and 10 in) must

have an infinite size (N =infinity) and the

inflow capacities of the cells Q(t) are set

equal to the desired section input flow for

time interval t .

• Destination cells (e.g. cells 8 and 16),

where traffic flows terminate and exit the

network, should have infinite sizes (N =

infinity).


Calculation Procedure

∞

1. Initialize the cell occupancy (n) based on the input data of initial

occupancy proportion (n/N) either with zeros when the network is

empty or with definite values when the network is preloaded.

2. Calculate the flows q(t) for the first time step in all cells using the

already described flow equations without any restriction of cell

order. qi = min { ni-1 ; Qi ; w/vf . (Ni – ni) }order.

3. Calculate the delays d(t) for the first time step in all cells using the

already derived delay equations also without any restriction of cell

order.

4. Calculate the cell occupancies n(t+1) for the next time steps in all

cells.

5. Repeat steps 2-4 for each time step till the end of the time horizon.

i i-1 i f i i

[ ] t=1 I I kFor d (t) n (t) q (t)∆ ⇒ = −

=+1tin

t

int

iq+t

iq 1+−


Pseudo-code for the calculation steps

∞

// Initialization

Initialize nI for all cells with 0 or percentages of NI// Loop

For time t = 0 to the time horizon

For cell I = 1 to number of cells

Calculate flow q(t) at time t for all cells ( No order of calculation)

End loopEnd loop


Calculate delays d(t) for all cells ( No order of calculation)

End loop


Calculate number of vehicles n(t+1) for all cells ( No order of calculation)

End loop

End loop

8. macroscopic traffic modelingsite.iugaza.edu.ps/emasry/files/2011/02/macroscopic-modelling.pdfdr....

Documents