Unsteady-state Traffic Flow Models for Urban Regulation Strategy

ARCHITECTURE AND URBAN DESIGN
In Press
ARCHITECTURE AND URBAN DESIGN
Unsteady-state Traffic Flow Models for
Urban Regulation Strategy Planning
Maria N. Smirnova1,2 *, Aleksandra I. Bogdanova1 , Nickolay N. Smirnov1,3 ,
Alexey B. Kiselev1 , Valeriy F. Nikitin1,3 , Aleksandra S. Manenkova1
1
Moscow M.V. Lomonosov State University, Moscow 119992, Russia
Saint Petersburg State Polytechnical University, 29 Politechnicheskaya Str., 195251 St. Petersburg Russia
3 Scientific Research Institute for System Studies of the Russian Academy of Sciences, 36-1 Nakhimovskiy pr., Moscow 117218,
Russia
*Corresponding author: [email protected]; [email protected]
2
Abstract:
The present research aims mathematical modeling of essentially unsteady-state traffic flows on
multilane roads, wherein massive changing of lanes produces an effect on handling capacity
of the road segment. The model uses continua approach. However, it has no analogue with
the classical hydrodynamics, because momentum equations in the direction of a flow and
in orthogonal directions of lane- hanging are different. Thus, velocity of small disturbances
propagation in the traffic flow is different depending on direction: counter flow, co-flow, orthogonal
to the flow. Numerical simulations of traffic flows in multilane roads were performed and their
results are presented. Massive changing of lanes produces an effect on handling capacity of the
road segment. Mathematical models allow describing the conditions for maximal road handling
capacity, the origination and propagation of traffic humps and jams, the influence of various
traffic control devices, and transverse evolution of traffic flows on multi-lane roads.
Keywords:
Traffic; Control; Continua Model; Multilane Road; Handling Capacity
1. INTRODUCTION
First attempts of developing the mathematical models for traffic flows were by Lighthill and Whitham
[1], Richards [2], Greenberg [3], Prigogine [4]. Whitham [5] provided detailed analysis of the obtained
results. The authors regarded the traffic flow from the position of continuum mechanics applying empirical
relationships between the flow density and velocity of vehicles. Development of computer technique gave
birth to continua models application not only for analytical solutions [6] but also in numerical simulations
of traffic flows [7–10]. Another approach using discrete approximation for public traffic and passengers’
interaction is illustrated by [11, 12].
The present research aims resolving arising difficulties in rational traffic organization the Moscow
Government was faced in the recent years. The decrease of the roads handling capacity in the centre of
the city caused by the increase of traffic density brought to necessity of re-organizing the traffic flows and
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ARCHITECTURE AND URBAN DESIGN
constructing new roads in and around the city.
2. THE CONTINUA MODEL OF TRAFFIC FLOWS.
We introduce the Euler’s co-ordinate system with the Ox axis directed along the autoroute and the Oy
axis directed across the traffic flow. Time is denoted by t. The average flow density ρ (E, y, t) is the ratio
of the surface of the road occupied by vehicles to the total surface of the road considered.
ρ=
hn` n`
Str
=
= ,
S
hL
L
where h is the lane width, L is the sample road length, ` is an average vehicle’s length plus a minimal
distance between jammed vehicles, n is the number of vehicles on the road. With this definition, the
density is dimensionless changing from zero to unit.
The flow velocity denoted V (x, y,t) = (u(x, y,t), v(x, y,t)) where u can vary from zero up to its maximal
value Umax , where Umax is maximal permitted road velocity. From definitions, it follows that the maximal
density ρ = 1 relates to the case when vehicles stay bumper to bumper. It is naturally to assume that the
traffic jam with V = 0 will take place in this case.
Determining the “mass” distributed on a road sample of the area S as:
Z
m=
ρdσ ,
S
one can develop a “mass conservation law” in the form of continuity equation:
∂ ρ ∂ (ρu) ∂ (ρv)
+
+
= 0.
∂t
∂x
∂y
(1)
Then, we derive the equation for the traffic dynamics. The traffic flow is determined by different factors:
drivers’ reaction on the road situation, drivers activity and vehicles response, technical features of the
vehicles. The following assumptions were made in order to develop the model:
• We model the average motion of the traffic described and not the motion of the individual vehicles.
Consequently, the model deals with the mean features of the vehicles not accounting for variety in power,
inertia, deceleration way length, etc.
• We assume the “natural” reaction of all the drivers is assumed. For example, if a driver sees red
lights, or a velocity limitation sign, or a traffic hump ahead, he decelerates until full stop or until reaching
a safe velocity, and does not keep accelerating with further emergency braking.
• It is assumed that the drivers are loyal to the traffic rules. In particular, they accept the velocity
limitation regime and try to maintain the safe distance depending on velocity.
The velocity equation for the x-component is the following:
du
= a; a = max −a− ; min a+ ; a0 ;
dt
2
Unsteady-state Traffic Flow Models for Urban Regulation Strategy Planning
a0 = σ aρ + (1 − σ )
Z Y
0
ω (y) aρ (t, s) ds +
U (ρ) − u
τ
k2 ∂ ρ
,
ρ ∂x
aρ = −
Here, a is the acceleration of the traffic flow; a+ is the maximal positive acceleration, a− is the
emergency braking deceleration; a+ and a− are positive parameters, which correspond to technical
features of vehicles. The parameter k > 0 is the small disturbances propagation velocity (“sound velocity”),
as shown in [13–15]. The parameter τ is the delay time, which depends on the finite time of a driver’s
reaction on the road situation and the vehicle’s response. This parameter is responsible for the drivers’
tendency to keep the vehicles velocity as close as possible to the safe velocity depending on the traffic
density U(ρ) [16–18]:
(
U (ρ) =
−k ln ρ, u < Umax ,
Umax , u ≥ Umax .
The velocity U(ρ) is determined from the dependence of the traffic velocity on density in the “plane
wave” when the traffic is starting from the initial conditions ρ0 = 1 and u = 0, with account of velocity
limitation from above (u ≤ Umax ). The value of τ could be different for the cases of acceleration or
deceleration to the safe velocity U(ρ):
(
τ=
τ + , U (ρ) < u
τ − , U (ρ) ≥ u
In the formula for a0 the first term describes an influence on the driver’s reaction in a local situation, the
second – a situation ahead the flow and the third – a driver’s tendency to drive a car with the velocity,
which is the safest in each case. If we assume σ = 0, then the expression for a0 will be:
1
a =
∆
0
Z x+∆
x
aρ (t, s) ds +
U (ρ) − u
.
τ
In this case acceleration is not a local parameter, but depends on its values in the region of length ∆
ahead of vehicle, where ∆- is the distance each driver takes intro account on making decisions. This
distance depends on road and weather conditions. The last term of the relaxation type takes into account
tendency to reach optimal velocity.
If we assume σ = 1, then the expression for a0 will be:
a0 = aρ +
U (ρ) − u
.
τ
Now acceleration depends on local situation only.
We will consider a case when σ = 1, τ = ∞, so the equation of motion for the x-component is:
ax = −
k2 ∂ ρ
.
ρ ∂x
(2)
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ARCHITECTURE AND URBAN DESIGN
The equation of motion for the y-component we can write in such form as for the x direction:
ay = −
A2 ∂ ρ
.
ρ ∂x
(3)
The description for the parameter A will be given below.
In order to understand the physical meaning of parameter A we shall consider the following model
problem. One car is changing it’s lane with the density ρ 6= 0 to the lane with the density ρ = 0. In this
case a car has the maximum acceleration a max. So by putting ∂∂ρy = 0−ρ
h to the velocity equation for the
2 ∂ρ
y-component we can obtain ρ dv
dt = −A ∂ y and then
A2
h
= a max.
√
Figure 1 illustrates the diagram for v(y) According to it we have: v max = amax h, that is a max =
v2 max
or A2 = v2 max. But in this model v max = 2vav(vavis an average speed of car’s changing a lane),
h
2
so A = 4v2 av, where vmax
av - the average speed for v max.
Figure 1.
For description of lane change dynamics, we approximately assume that the trajectory of maneuver of
lane changing incorporates two parts of identical circles (Figure 2). On the Figure 2 the bold line is a
trajectory of lane’s changing and it begins in the middle of the lane on which the car is going and ends in
the middle of next lane.
2
The centripetal force which acts on the car is: F = RmV
, where Rturn is the radius of the turn, V - is the
turn
velocity of the car, directed by the tangent to the car’s trajectory. Let F∗ be the maximum possible flank
force under which car is drivable (not skidding), F ≤ F∗ . Then we may calculate the radius of the turn
Rturn = Fm∗ V 2 . vmax
av = V sin θ . From the similar triangles ∆ABC and ∆OAD (Pic.2) derive that AD = DB,
labeling AD = b will get:
b
h/4
Rh
=
or b2 =
,
R
b
4
r
√
h
b
Rh 1 h
sin θ =
= =
=
.
4b R
2R
2 R
1
Then vmax
av = V sin θ = 2 V
4
q
h
R
= 12 V
q
hF∗
mV 2
=
1
2
q
hF∗
m
Unsteady-state Traffic Flow Models for Urban Regulation Strategy Planning
Figure 2.
that is the average speed for v max is vmax
av =
for A is A2 =
1
2
q
hF∗
m .
We obtained above A2 = 4v2 av, so the expression
hF∗
m .
Thus we derived parameterA2 being dependent on the force F∗ , assignable to the lane width, mass of
the car and the traction of tires.
The equations (1),(2),(3) provide a system describing traffic flows on multilane roads:
∂ ρ ∂ (ρu) ∂ (ρv)
+
+
= 0,
∂t
∂x
∂y
∂u
∂u
∂u
k2 ∂ ρ
+u +v
=−
,
∂t
∂x
∂y
ρ ∂x
∂v
∂v
∂v
A2 ∂ ρ
+u +v
=−
;
∂t
∂x
∂y
ρ ∂y
3. ONE DIMENSIONAL MODEL APPLICATION FOR STUDYING TRAFFIC
CONTROL DEVICES AND STRATEGIES
We consider two most popular devices used to control automobile traffic on the urban roads: street
traffic lights and “laying policemen”. The last is a set of low jumps across the road surface, which are
placed at some distance from each other and force the drivers to decrease velocity essentially in the zone
of their placement. The “laying policemen” are often used to provide friendly conditions for pedestrians
crossing roads near schools, institutions, marketplaces, etc.
We should consider the following boundary conditions on the endings of the motorway section0 ≤ x ≤ L.
Two variants of the conditions at the inflow x = 0 are possible:
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ARCHITECTURE AND URBAN DESIGN
1) No traffic hump: the flow density is given, and the velocity is equal to the maximal safe for this
density:
ρ (0, t) = ρ0 ; v (0, t) = v (ρ0 ) .
2) Traffic hump or jam at the inflow ending: zero density gradient is assumed, and the velocity is
equal to the maximal safe velocity at the density at the inflow:
∂ ρ = 0; v (0,t) = V (ρ) .
∂ x x=0
Presence or absence of the hump at the inflow boundary of the domain is determined each time after a
time step calculation using the following criterion. The hump takes place, if two conditions are true at
x = 0:
∂ ρ > 0 and ρ > ρ0 .
∂ x x=0
The “free outflow” condition is set on the outflow ending of the domain at x = L:
∂ρ
∂v
= 0;
= 0.
∂x
∂x
As for the initial conditions, we assume that the domain portion at 0 ≤ x ≤ x0 is occupied with vehicles
with density ρ0 moving at the velocityV (ρ0 ), and the rest part x0 < x ≤ L is free from the vehicles
(ρ = 0, v = 0).
3.1 Street Traffic Lights.
The main parameters of this device are the green, yellow and red signals duration, denoted tg , ty , tr
correspondingly. The following algorithm is proposed to model the street lights work.
1.
In the moment when the signal changes to yellow, the following distance is calculated:
v0
xr = max
2ar
2
,
where ar is the usual deceleration less than the deceleration of the emergency braking a− . The vehicles
which are closer to the street lights at this moment than this distance, cannot stop and therefore they pass
the street lights, this yields the traffic rules.
2. When the yellow signal is on, the maximal permitted velocity at the point x` (t) is set to be the
following:
tess
vemax (x` ) = v0max ·
te
where tess is the time left for the yellow signal to be on; x` (t) moves towards the street lights like:
x` = L1 − xr ·
6
tess
,
ty
Unsteady-state Traffic Flow Models for Urban Regulation Strategy Planning
and L1 is the co-ordinate of the street lights device. As the result, by the moment of red lights all the
traffic flow is blocked at the streetlights.
3. In the moment when the streetlights change to green, the maximal permitted velocity change back
to v0max like in the other portions of the domain.
3.2 Laying Policemen
This system of velocity limitation is modelled by maximal permitted velocity the position of the device,
which is much lower than one for the other portions of the roadv0max . The present work deals with a system
of two “policemen” laying at a given distance between them; this case is the most often one. The position
of the first device is denotedL1 . Then, the maximal permitted velocity along the domain 0 ≤ x ≤ L is as
follows:
(
vmax =
vp ,
x ∈ {L1 , L1 + d} ,
v0max , x ∈ [0, L] {L1 , L1 + d},
where vp < v0max , and the parameter v p (maximal velocity of passage) is one of the governing parameters.
4. RESULTS OF NUMERICAL INVESTIGATIONS OF TRAFFIC REGULATION STRATEGIES
The numerical simulations of the problems used the TVD method with second order of approximations
(Van Leer’s flux limiter). The mesh had N = 201 grid nodes.
The following parameters values were used in simulations:
L = 1000 m is the length of the domain; x0 = 100 m is the length of the domain portion with vehicles
at the first instance;
L1 = 500 m is the position of the traffic control devices (either street lights or the first “laying policemen”);
d = 50 m is the distance between the “laying policemen”;
ρ0 = 0.1 ÷ 0.5 is the inflow density, v0max = 25 m/s is the maximal permitted velocity on the main
portion of the road;
vp = 3 m/s is the maximal velocity of the “laying policemen” passage;
k = 7.9 m/s is the velocity of small disturbances propagation in the automobile traffic flow;
a+ = 1.5 m/s2 is the maximal traffic flow acceleration;
a− = 5 m/s2 is the maximal (emergency) deceleration of the flow;
ar = 1.5 m/s2 is the standard deceleration;
Y0 = 100 m is the characteristic visibility ahead;
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ARCHITECTURE AND URBAN DESIGN
σ0 = 0.7 is the local situation “weight”; τ + = 3.3 s,
τ − = ∞ are the delay times responsible to maintaining the safe velocity;
tg = 40 ÷ 300 s , ty = 5 s, tr = 30 s are durations of the street lights cycle.
Thus, the varying parameters were the inflow density ρ0 (together with the inflow velocity) and the
green lights durationtg .
The results are presented at the Figure 3 – Figure 6 and in the Table 1.
The Figure 3 and Figure 4 illustrate traffic flow density distribution for the different time moments
which are shown on the figs, for the case of street lights traffic controlling. The green lights duration
was taken tg = 50 s, and the inflow density was ρ0 = 0.18 (Figure 3) and ρ0 = 0.3 (Figure 4). The time
moments on those figures correspond to the following street lights cycles: Figure 3(a) – first cycle, end
of the green period; Figure 3(b) – first cycle, yellow period; Figure 3(c) – first cycle, end of the red
period; Figure 3(d) – second cycle, green period; Figure 3(e) – second cycle, green period; Figure 3(f)
– second cycle, end of the red period. Figure 4(a) corresponds to the first cycle, yellow period; Figure
4(b) – first cycle, end of the red period; Figure 4(c) – second cycle, end of the green period; Figure 4(d)
– fifth cycle, end of the green period; Figure 4(e) – fifth cycle, end of the red period; Figure 4(f) – sixth
cycle, end of the green period. It can be seen from the figures that at ρ0 = 0.18 the traffic hump does not
occur (Figure 3), and at ρ0 = 0.3 the traffic hump arises and propagates counter-flow decreasing vehicles
velocity essentially.
The dependence of critical inflow density ρ0∗ under which the traffic hump does not occur, on the
duration of the green period of the street lights tg was investigated, and the results are placed in the Table
1. All the other parameters were fixed as shown above. The dependence of ρ0∗ on tg can be described by
the following formula:
ρ0∗ = a ln (btg )
(4)
where a, b are parameters depending on many factors including the red lights duration tr . For the
parameters used, we have got a = 0, 054, b = 0, 87 . The table 1 contains also the difference of the critical
inflow density obtained by traffic flow simulation and the value obtained using the formula (4). The mean
square deviate is equal to 0.011629, and the maximal difference is 0,021566.
The Figure 5,Figure 6 illustrate the profiles of ρ(x) for different time moments in the case of “laying
policemen”. The Figure 5 corresponds to the inflow density ρ0 = 0.1, and the Figure 6 – to ρ0 = 0.3.
The case ρ0 = 0.1 corresponds to the free traffic passage via the zone regulated by two “laying
policemen”, and the case ρ0 = 0.3 corresponds to origination of the traffic hump and its propagation
counter-flow. It can be seen from both Figure 5 and Figure 6 that two locations with increased density
take place. In case ρ0 < 0.2 this does not prevent the traffic flow from the free passage. If the inflow
density is higher than 0.2, then the traffic hump originates in front of the first “laying policeman”; it
propagates towards the inflow portion and finally the density at the inflow increases thus decreasing the
inflow velocity, and the traffic turns to be very slow (Figure 6).
5. SOLVING A MODEL TWO-DIMENSIONAL PROBLEM
The considered problem is the nature of car’s behavior on a multilane rectangular road.
8
Unsteady-state Traffic Flow Models for Urban Regulation Strategy Planning
Table 1.
Green lights duration, Critical density deter- Critical density calcu- Table 1.
s
mined numerically
lated using formula (4.3)
Divergence
40
0,18
0,191679
0,011679
50
0,19
0,203729
0,013729
60
0,21
0,213574
0,003574
70
0,22
0,221899
0,001899
80
0,23
0,229109
-0,00089
0,00547
90
0,23
0,23547
100
0,23
0,241159
0,011159
110
0,25
0,246306
-0,00369
120
0,25
0,251004
0,001004
130
0,26
0,255327
-0,00467
140
0,26
0,259329
-0,00067
150
0,27
0,263054
-0,00695
160
0,27
0,266539
-0,00346
170
0,28
0,269813
-0,01019
180
0,28
0,2729
-0,0071
190
0,29
0,275819
-0,01418
200
0,29
0,278589
-0,01141
210
0,3
0,281224
-0,01878
220
0,3
0,283736
-0,01626
230
0,3
0,286136
-0,01386
240
0,31
0,288434
-0,02157
250
0,31
0,290639
-0,01936
260
0,31
0,292757
0,01724
270
0,31
0,294795
-0,01521
280
0,31
0,296758
-0,01324
290
0,31
0,298653
-0,01135
300
0,31
0,300484
-0,00952
The traffic flow is divided into two parts on the left boundary by y f ix . For 0 ≤ y ≤ y f ix the flow has the
bigger value then for y f ix < y ≤ H.
Boundary conditions for x=0:
v = 0, u = u1 , ρ = ρ1 , u1 ρ 1 = q1
v = 0, u = u2 , ρ = ρ2 , u2 ρ 2 = q2
)
u>k
and
(
q = ρu =
q1 ,
q2 ,
y > y f ix > 0
y ≤ y f ix ≤ H
Boundary conditions for x=L:
∂ ρv
∂ ρu
= 0,
= 0,
∂x
∂x
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ARCHITECTURE AND URBAN DESIGN
The numerical calculations of the problems were processed using the AUSM method. The mesh had
Nx = 201and Ny = 21 grid nodes.
T = 240 s - calculation time;
L = 500 m – the length of the domain;
H = 12 m– the width of the domain;
ρ0 = 0, 01 – the density of traffic flow on the bounder x = 0;
u0 = 7, 0 m/s – the x-velocity of traffic flow on the bounder x = 0;
v0 = 0, 0 m/s – the y-velocity of traffic flow on the bounder x = 0;
Umax = 20 m/s– maximal permitted road velocity;
k = 9 m/s– the small disturbances propagation velocity (“sound velocity”) on x axis;
A = 3 m/s– the analogy of k coefficient on y axis;
y f ix = 3, 75 m- the devisor of the road on 2 parts;
q0 = 7- a flow on the left boundary for 0 ≤ y ≤ y f ix ;
q0 = 3- a flow on the left boundary for y f ix < y ≤ H;
The obtained results are presented in Figure 7 in the form of maps for components of velocity and for
density for different moments of time.
10
Unsteady-state Traffic Flow Models for Urban Regulation Strategy Planning
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3. Traffic flow density in the zone traffic regulation by “lights”. Time in seconds correspond to second cycle,
initial traffic density ρ0 = 0, 18: a - the end of “green period”, b - “yellow period”, c - “yellow period”, d
- “green period”, e – end of “green period”, f - end of the “red period”.
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ARCHITECTURE AND URBAN DESIGN
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4. Traffic flow density in the zone traffic regulation by “lights”. Time in seconds correspond to sixth cycle,
initial traffic density ρ0 = 0.3, a - “yellow period”, b - end of the “red period”, c - green period, d - end of
the green period, e - end of the red period, f - end of the “green period ”.
(a)
(b)
Figure 5. Traffic flow density in the zone traffic regulation by a “laying policemen”. Time in seconds is provided
on top of the figure; inflow traffic density ρ0 = 0.1.
12
Unsteady-state Traffic Flow Models for Urban Regulation Strategy Planning
(a)
(b)
(c)
(d)
Figure 6. Traffic flow density in the zone traffic regulation by a “laying policemen”. Time in seconds is provided
on top of the figure; inflow traffic density ρ0 = 0.3.
13
ARCHITECTURE AND URBAN DESIGN
Density
T=5
Density
T=30
Density
T=120
Velocity x comp.
T=5
Velocity x comp.
T=30
Velocity x comp.
T=120
14
Unsteady-state Traffic Flow Models for Urban Regulation Strategy Planning
Velocity y comp.
T=5
Velocity y comp.
T=30
Velocity y comp.
T=120
Figure 7. Maps for flow density and velocity components for different moments of time.
6. CONCLUSIONS
The mathematical model for traffic flows simulations in multi-lane roads is developed.
Analysis of traffic regulation strategies design shows that in case of low inflow density of the automobile
traffic the “laying policemen” devices allow to maintain traffic control without disturbing the free traffic
flow. However, the increase of the inflow density leads to origination of the traffic jam and its propagation
counter-flow. The regulation by the traffic lights allows improving the road handling capacity essentially
when the duration of the different parts of the traffic lights cycle is properly chosen.
A model problem for traffic evolution in multi-lane road with non-uniform flux in different lanes was
regarded. The results show that on entering the road segment high orthogonal fluxes occur in the direction
of less dense lanes, which brings to slowing down flow velocity in that lanes and increasing density. That
motion brings to formation of inverse flow from those lanes back. In time flow evolution brings to a
more smooth transition zone, however, on coming in contact flow zones of different fluxes always cause
formation of an orthogonal flow in the direction of less density lane in the nearest vicinity followed by an
inverse flow at some distance.
Mathematical models developed allow describing the conditions for maximal road handling capacity,
the origination and propagation of traffic humps and jams, the influence of various traffic control devices,
and transverse evolution of traffic flows on multi-lane roads.
15
ARCHITECTURE AND URBAN DESIGN
ACKNOWLEDGMENTS
Russian Foundation for Basic Research (grant 13-01-12056) is acknowledged for financial support.
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