A REVIEW OF OPTIMAL SPEED MODEL

1. A REVIEW OF OPTMAL SPEED MODEL
Master Of Engineering Class
Assignment
BY:
ABUBAKAR IBRAHIM
(SPS/16/MCE/00037)
SUBMITTED TO
PROF. H.M. ALHASSAN
BAYERO UNIVERSITY KANO
DEPARTMENT OF CIVIL ENGINEERING
MAY 10TH, 2017

2. INTRODUCTION
To understand traffic behavior, we require a thorough
knowledge of traffic stream parameters and their mutual
relationships, this relationship between the traffic parameters
results, many researches yielded many mathematical models
named Traffic flow models, With the rapid quantitative increase
of cars, the traffic jam becomes more and more serious.
The researcher’s activity had its beginnings from the 1920’s, by
describing the propagation of traffic flows by means of
dynamic macroscopic and microscopic models.

3. MACROSCOPIC TRAFFIC MODEL
Macroscopic description is used when the state of the system is
described by averaging gross quantities, namely, density k, speed
v, flow q, regarded as variables dependent on time and space.
This model shows better than Greenshield and Greenberg
Models for uncongested condition but not good in congested
condition, meanwhile, in this model the speed becomes zero only
when density reaches infinity. Hence this cannot be used for
predicting speeds at high densities (Jabeena, 2013).
Macroscopic traffic flow models make use of the picture of
traffic flow as a physical flow of a fluid. The goal of these
models is to be able to characterize the global behavior of the
traffic

4. MICROSCOPIC TRAFFIC MODEL
A microscopic model of traffic describes the car following behavior as
well as the lane changing behavior of every vehicle in the traffic. The
most famous one is the Car-Following models (Bando et al., 1995;
Helbing and Tilch, 1998; Jiang et al., 2001), where the driver adjusts his
or her acceleration according to the conditions in front and each vehicle
is governed by an ordinary differential equation (ODE) that depends on
speed and distance of the car in front (Darbha et al., 2008). In
microscopic models, cars are numbered to indicate their order: n is the
vehicle under consideration, n – 1 its leader, n + 1 its follower, etc., as
shown below. The behavior of each individual vehicle is modeled in
terms of the position of the front vehicle x, velocity v = dx/dt ,
acceleration a = d2x / d2t
Car-following notation

5. CONT.
Several theories have been proposed to model car following behavior, which
can be divided into three classes based on behavioral assumptions, namely,
Safe-distance models, stimulus-response models, optimal velocity models.
Safe-distance or collision avoidance models try to describe simply the
dynamics of the only vehicle in relation with his predecessor, so as to respect
a certain safe distance
Microscopic traffic models use different approaches for car following in order
to describe the dynamics of individual vehicles interaction with neighboring
vehicles (Chowdhury, et al., 2000; Helbing, 2001; Nagel, et al., 2003). Most
of them are based on the fact that each driver reacts with certain sensitivity to
a stimulus via the acceleration input after a specific time delay, the stimulus
may be a change in the relative speed or the driver visual angle. Others
models use safety distance or collision avoidance based approaches and
Psycho-spacing (Rekersbrink, 1994).

6. CONT.
Few years ago, Bando, et al., (1995) modified the models by making the
driver desired speed called “OPTIMAL SPEED MODEL” as a function of
the distance headway to the leader. With few parameters, this model is able to
describe and to interpret several traffic flow situations, optimal speed model
takes into account longitudinal driver behavior, it introduces the driver
perception of the risk of rear-end collision, it is a function of both the
distance headway to the leader and the relative speed the driver adapts to a
certain optimal value, rather than to the leaders speed.
When controlling his car, the driver behavior changes according to traffic
situation, neighboring vehicles and infrastructure characteristics. Leutzback,
(1988) has first proposed a psychophysical spacing models with perceptual
thresholds for situations classification.
Driver’s parameters such as the desired speed, safety need and reaction time
are used in order to determine the drivers’ level of perception for four
different driving situations describe below:

7. CONT.
 Free driving: This situation occurs generally in free flow traffic, the driver is
uninfluenced by the others vehicles. The driver desired speed is rather
constant and is determined as a compromise between need of safety (road
geometry and adhesion, visibility, e.t.c) and travel time minimization.
 Approaching: The relative speed is positive and the driver is closing the front
vehicle, he has thus to slow down and to adjust his speed to the speed of the
preceding vehicle, meanwhile the driver leads a headway distance or time
according to his desire of safety.
 Braking: when headway distance or time to collision is under a minimal
value, the driver initiates a braking maneuver until stoppage or recovering of
the desired safety level.
 Car-following: The driver follows the leading vehicle and tries to regulate his
speed and to maintain a desired headway.

8. CONT.
Different types of variables and threshold can be used for
transition from one situation to another. During free driving,
the simplest model of how the driver tries to approach the
desired speed is the use of a relaxation time T,
For car following, Pipes model uses as desired speed, the
speed of the preceding vehicle (Pipes, 1953). This is
motivated by the fact that speed of all vehicles is equal in
steady state. Chandler, et al., (1958) added a time delay ∆t
such that the vehicle acceleration becomes:

9. CONT.
Observing that the clustering effect of traffic cannot be reproduced, (Gazis, et
al., (1959) presented several model refinements where the relaxation time is
made dependent of the headway distance (s(t) = xp(t) – xf (t)) and the preceding
vehicle speed. The relative speed term (vp(t−∆t)−vf (t−∆t)) is called the stimulus
and the multiplicative term is the sensitivity. This final model is known as the
General Motor Nonlinear (GM) model. The model expression is:
The exponents β and ɣ are first proposed to be integers but they are now allowed
to be real values as shown by May and Keller, (1967), they are however difficult
to determine from real data. This equation can be integrated and a speed-density
relation for homogenous flow can be obtained. Few years ago, Bando, et al.,
(1994) modified model equation (1) above by making the driver desired speed
called ”OPTIMAL SPEED MODEL” as a function of spacing with the
preceding vehicle. With very few parameters, this model is able to describe and
interpret many traffic flow situations (Bando, et al., 1994) and (Bando, et al.,
1995).

10. CONT.
Where υref (s) is the spacing-dependent optimal speed that the driver attends
to achieve with a relaxation time T, this optimal speed has to vanish when the
spacing goes to zero and is bounded by the free speed when spacing goes to
infinity. This optimal speed can thus be chosen as an increasing but saturating
function of spacing. Assuming that the flow density is locally equal to the
spacing inverse, this gives a direct link to fundamental diagram, the optimal
speed is generally of the form:
Where parameters V1, V2, C1 and C2 are calibrated empirically using real
data measurements, an enhancement of this model is proposed, some
additional requirements are added in order to enhance the braking reactivity
of the model. The proposed model introduces a weighting factor of the
optimal speed that depends on the ratio of the relative speed to spacing,
which is the opposite of the inverse of time to collision (TTC)

11. CONT.
This reactivity is based on the excess of follower speed in comparison to that
of the leader, it also modulates this reactivity according to the actual spacing
with the leading vehicle, and this factor has to fulfill the following
requirements,
• It should maintain the reference speed υref remain unchanged when the
relative speed is positive.
• It has to be decreasing for negative decreasing relative speed and has to go
toward zero when the relative speed goes to −∞.
The optimal speed equation (5) is thus changed to given by
Where the weighting factor is set to:

12. CONT.
Obviously, the parameter A is necessarily equal to
1
2
, this can be
obtained by the limit (s˙/s → +∞).
The dynamic studies take into account the “STOP” and “GO”
(SAG) waves of the optimal speed model, which require the
number of cars in downstream-edges and upstream-edges
corresponding to an essential scale that controls the strength of
interaction between multiple SAG waves. Choosing the different
asymmetrical factor μ and repeating the numerical simulation of
the motion of cars moving along the circular load

13. CONCLUSION
The microscopic car-following model is a favorite type of traffic flow theory
that describe the individual behavior of drivers. Most car-following model
well-known the optimal speed model, which has successfully revealed the
dynamical evolution process of traffic congestion in a simple way. Thereafter,
inspired by the optimal speed model, some new car-following models were
successively put forward to describe the nature of traffic more realistically.
Some were extended by incorporating a new optimal velocity function or
introducing multiple information of headway or velocity difference, or
acceleration difference, whereas others considered the individual anticipation
behavior, the existing car following models and the recent one gives their
drawbacks and advantages to help the research to develop the strong car-
following model which avoid the collision and interpreted the traffic flow in
a real manner.
• This model has only two independent parameters and covers many
properties of the previous models, such as the correct delay time of car
motion, kinematic wave speed at jam

14. CONT.
density, the formation of stop-and-go waves and emergent traffic jams.
• This model avoids the unrealistic high acceleration when δvn becomes
infinite. The unrealistic negative velocity and headway never appear in
this optimal speed model. Numerical simulation shows that the velocity of
cars changes smoothly, this means that the stability of traffic flow is
improved.
• More over , from the review of the above models, it has been notice that
most of the model were developed base on Europe traffic data and
parameter, non of the models was develop using Nigerian traffic situation,
as such I suggest more research to be conducted using Nigeria traffic
situation, also the model do not consider the external factor of weather
condition on the driver behavior and nature of the tranverse road condition
which is important aspect in driving situation.

15. REFERENCES:
• Gazis DC, Herman R, Potts RB. Car following theory of steady state trafic
flow. Operations Research 1959; 7: 499-505.
• Guanghan Peng , Weizhen Lu , Hongdi He , Zhenghua Gu , Nonlinear
analysis of a new car-following model accounting for the optimal velocity
changes with memory, Com-munications in Nonlinear Science and
Numerical Simulation (2016).
• Hajar Lazar Mohammed, Khadija Rhoulami Mohammed, Moulay Driss
Rahmani, V University of Rabat, A review analysis of optimal velocity
models 2017.
• Nagel, K., P. Wagner and R. Woesler (2003). Still Flowing: Approaches to
Traffic Flow and Traffic Jam Modeling, Operations Research, 51, 5, 681-
710.
• Salim Mammar, Said Mammar, Habib Haj-Salem, A modified optimal
velocity model for vehicle following 2005 IFAC Elsevier publications.

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