REVIEW OF OPTIMAL SPEED TRAFFIC FLOW MODEL

BAYERO UNIVERSITY, KANO
FACULTY OF ENGINEERING
DEPARTMENT OF CIVIL ENGINEERING
ASSIGNMENT ON:
REVIEW OF OPTIMAL SPEED TRAFFIC FLOW MODELS
COURSE TITLE & CODE: ADVANCE TRAFFIC ENGINEERING (CIV8329)
SUBMITTED BY
AKILU BOYI BAITI
SPS/16/MCE/00025
PROGRAMME: M.ENG. HIGHWAY & TRANSPORTATION ENGINEERING
TO:
ENGR. PROF. H.M ALHASSAN B.ENG., M.ENG., Ph.D., MNSE, R.ENGR. (COREN)
(COURSE FACILITATOR)
APRIL, 2017

INTRODUCTION
• According to Surbhi, S., The term speed is defined as the
distance an object travels in a definite time. It can be
understood at the rate at which a body travels some distance
in the unit time. Therefore, speed determines the quickness
of an object, i.e. how fast an object is going. An object’s
whose moving speed is high covers large distance in less
time, unlike an object with a low moving speed which covers
small distance in the same amount of time. When an object
does not travel any distance. Its speed will be zero.
• In traffic engineering, speed is used to measure the
quality of traffic flow. Basically, speed is the total
distance traversed divided by the time of travels
(Alhassan, 2017).

• Speed is not consistent, the speed of any vehicle depends on
many factors like the location of the vehicles, the design of
the roadway, the purpose for which the person is driving the
vehicle, the time in which the person is riding, the
congestion condition and the visibility on the road e.t.c.
Because all these factors are varying and are very complex to
tackles, the speed is always changing in very small amount of
time.
INTRODUCTION (Cont.)

OPTIMAL SPEED
• The term optimal refers to the greatest degree or best result
obtained (or obtainable) under specific conditions. It also
refers to as the most favorable/desirable (or in simple term
“the best”). Therefore, optimal speed is defined as the best
speed obtainable under specific condition.
• In traffic engineering, the term optimal speed is that
favorable or best speeds obtainable under specific
conditions of the roadway.
• In order to understand optimal speed traffic flow model, it is
better to study it from the traffic point of view, then study (in
detail) the traffic flow models that employed the use of
optimal speed model in detail.

TRAFFIC FLOW
• Traffic flow is the study of individual drivers and vehicles between
two points and the interactions they make with one another.
• Unfortunately, studying traffic flow is difficult because driver
behavior is something that cannot be predicted with one hundred
percent certainty, fortunately, drivers tend to behave within a
reasonably consistent range and, thus, traffic streams tends to have
some reasonable consistency and can be roughly represented
mathematically. ( Alhassan, 2017)
• In order to understand traffic behavior it is required to have a
thorough knowledge of traffic stream parameters and their mutual
relationships. This relationship between the traffic parameters
results many researchers yielded many mathematically models
named traffic flow models (Hajar Et al, 2016).

TRAFFIC FLOW MODELS
• Traffic flow theory and modeling started in 1093s, pioneered
by the US-America Bruce D. Green shields. However, since
1930s, the field has gained considerable attraction as overall
traffic demand has increased and more data as well as easy
access to computing power has become available. There are
three types of traffic flow models; they are;
• Microscopic models
• Mesoscopic models
• Macroscopic models

• Microscopic traffic flow models describe the dynamics of traffic flow
at the level of each individual vehicle. They have existed since the
1960s with the typical ear-following models.
• In the case of mesoscopic traffic flow models, the behavioural rules
are still describe at an individual level. But the dynamics of these
models are generally governed by various processes, such as
acceleration, interactions between vehicles, and lane-changing,
describing the individual driver’s behavior.
• The macroscopic traffic flow models deal with traffic flow in terms of
aggregate variables as a function of location and time. They describe
the dynamic of the traffic density K(x,t), mean speed V(x,t) and /or
rate q(x,t). Macroscopic models have a number of advantages over
others, such as better agreement with real data, suitability for
analytical investigations, simple treat of inflow from ramps e.t.c.
TRAFFIC FLOW MODEL (Cont.)

• For these last few decades, the development of various theories
concerning traffic phenomena has received considerable attention. An
increasing number of investigators with different backgrounds and
points of view have considered various aspect of traffic phenomena
with very gratifying results.
• Car- following models were developed to model the motion of
vehicles following each other on a simple lane without any overtaking.
It is based on the assumption that each driver reacts in some specific
fashion to stimulus from the vehicles ahead of him. All car-following
models have in common that they are defined by ordinary differential
equations describing the complete dynamics of the vehicle’s position
Xα and the velocities Vα. it is assumed that the input stimuli of the
drivers are restricted to their own velocity Vα, the net distance
(bumper-to-bumper) Sα=Xα-1-Xα-Lα-1 to the distance vehicle (where
Lα-1 denotes the vehicle length), and the velocity Vα-1 of the leading
vehicle. The equation of motion of each vehicle is characterized by an
acceleration function that depends on those input stimuli:

𝛼 𝑡 = 𝛼 𝑡 = 𝐹 𝑉𝛼 𝑡 , 𝑆 𝛼 𝑡 , 𝑉𝛼−1 𝑡 … … … … … … … … … … … (1)
• In general, the driving behavior of a single driver-vehicle unit α
might not merely depend on the immediate leader α - 1 but on the
na vehicles in front. The equation of motion in this more generalized
form reads:
𝛼(𝑡)= 𝑓 𝑋 𝛼 𝑡 , 𝑉𝛼 𝑡 , 𝑋 𝛼−1 𝑡 , 𝑉𝛼−1 𝑡 … … … 𝑋 𝛼−𝑛𝑎 𝑡 , 𝑉𝛼−𝑛𝑎 𝑡 … … … 2
• There are two main objectives in the car-following process: (i)
reducing the speed difference and (ii) maintain an appropriate
spacing between the following vehicle and the leading vehicle. Most
early models were defined based on the first objective, but failed to
describe the second one.

• Newell (1961) proposed a different model which successfully
captures the characteristics of car following behaviors in
maintaining an optimal spacing corresponding to the driving
speed. However, due to the speed expression of Newell’s
model, it is not convenient to be used traffic simulations.
Thirty years later, a new model called Optimal Speedy
Model (OSM) was developed (Bando et al. 1995 & 1998).

• The optimal speed model was introduced to remedy the
problem faced by car-following model, in which the
followers accelerations tends to zero in the absence of the
leader. The OSM was introduced based on the assumption
that each driver has a safe speed which depends on the
distance headway to the leader. According to this approach,
the driver adapts its speed to a certain optimal value, rather
than to the leaders speed. It has been shown by these
models that under certain conditions, small disturbances are
amplified and lead to jams. Therefore, these models are able
to replicate stop and go waves in traffic flows. (Al Hassan,
2017)

THE OPTIMAL SPEED MODEL
• Car-following theories describe how one vehicle follows another
vehicle in an uninterrupted flow. Various models were formulated
to represent how a driver reacts to the changes in the relative
positions of the vehicle ahead. Models like pipes, forbes, general
motors and optimal speed model are worth discussing. However,
the focus of this work is on Optimal Speed Model.
• The concept of optimal speed model is that each driver tries to
achieve an optimal speed based on the distance to the preceding
vehicles and the speed difference between the vehicles. This was
an alternative possibility explored recently in car following models.
The formulation is based on the assumption that the desired speed
depends on the distance headway of the nth vehicle, i.e. =, where
is the optimal speed function which is a function of the
instantaneous distance headway therefore is given by

𝛼 𝑛
𝑡
=
1
𝑇
𝑣 𝑜𝑝𝑡
∆𝑥 𝑛
𝑡
− 𝑣 𝑛
𝑡 1
𝑇
---------------------------------------
------3
• Where is called the sensitivity coefficient. In short, the driving
strategy of nth vehicle is that, it tries to maintain a safe speed which
in turn depends on the relative position, rather than relative speed,
(Mathew, 2014).
• Similar to Newell Model, the OSM contains the optimal speed
functions which allow the following vehicle to adjust its speed
towards the optimal one, and consequently maintaining the
appropriate spacing. Moreover, the OSM does not have a time
delay in its model expression, which makes it convenient for
theoretical analysis; the OSM has drawn widespread attention
during the last twenty years (Helbing and Tilch 1999; Jiang et al.
2001)
THE OPTIMAL SPEED MODEL (Cont.)

• The optimal speed function assumes that there is one-to-
one correspondence between the spatial headway and the
optimal driving speed in steady traffic state. However, such
assumptions may be too ideal from the driver’s perspective
(Boer, 1999). Experience tells us that drivers are satisfied with
a range of conditions instead of an accurate optimal
performance.
• Newell’s optimal speed model is one of the first models
learning on an analysis of the trajectories of vehicles, the
model equation is thus

• The optimal speed function assumes that there is one-to-one
correspondence between the spatial headway and the optimal
driving speed in steady traffic state. However, such
assumptions may be too ideal from the driver’s perspective
(Boer, 1999). Experience tells us that drivers are satisfied with a
range of conditions instead of an accurate optimal
performance.
• Newell’s optimal speed model is one of the first models
learning on an analysis of the trajectories of vehicles, the
model equation is thus:
𝑉𝑛 𝑡 + 𝜏 = 𝑉 𝑆 𝑛(𝑡) ………………………………….4

• Where V (Sn (t) is the optimal speed under the headway Sn(t).
This model has directly given the speed of n-th car by the
optimal speed function
• Based on this model, (Bando et al, 1995; Nugra hani, 2013)
introduced on optimal speed model (OSM), which is given by
Where k is the sensitivity.
𝛼 𝑛 𝑡 = 𝑘 𝑉𝑜𝑝𝑡 𝑆 𝑛 𝑡 − 𝑉𝑛(𝑡) … … … … … … … … … … … … … .5

• Helbing and Tilch (1998) give the function of OSM model as
follows
𝑉𝑜𝑝𝑡 𝑆 𝑛(𝑡) = 𝑉1 + 𝑉2 𝑡𝑎𝑛ℎ 𝐶1 𝑆 𝑛 𝑡 − 𝑙 − 𝐶2 … … … 6
• where l is the length of vehicle, and V1, V2, C1, C2 are
calibrated parameters.

• However, the same authors (Bando et al., 1998) analyzed the
OSM with the explicit delay time. They proposed to introduce
the explicit delay time in order to construct a realistic models
of traffic flow for that it’s included in the dynamical equation of
OSM (Eqn (5) therefore become as follows
𝛼 𝑛 𝑡 + 𝜏 = 𝑘(𝑉𝑜𝑝𝑡 𝑆 𝑛 𝑡 − 𝑉𝑛(𝑡)……………………………….7

• In their analysis, they found that the small explicit delay time
has almost no effects. Unlike, where the explicit delay time
introduced a new phase of the congestion pattern of OSM
seems to appear. However, the OSM has encountered the
problems of high acceleration and unrealistic deceleration.
• However, Helbing and Titch add new term to the right of
eqn. (5). They called it generalized force model GFM. This
new term represents the impact of the negative difference in
speed on condition that the speed of the front vehicle is
lower than that of the follower. The GFM formula is

𝛼 𝑛 𝑡 = 𝑘(𝑉𝑜𝑝𝑡 𝑆 𝑛 𝑡 − 𝑉𝑛(𝑡) −𝑆 𝑛(𝑡) 𝑆 𝑛 𝑡 + 𝜆Θ … … … … … … … … 8
• Where  is the Heaviside function GFM has the same form as
OSM, and the difference lies in that they have different values of
sensitivity K. The main drawback of GFM is that it doesn’t take the
effect of positive speed difference n(t) on traffic dynamics into
accounts and only considers the case where the speed of the
following vehicle is larger than that of the leading vehicle. In Jiang,
et al. (2001), they pointed out that when the preceding car is much
faster, the following vehicle may not break even though the
spacing is smaller than the safe distance. The basis of GFM and
taking the positive factor Sn (t) into account, Jiang et al. (2001)
obtained a more systematic model, one whose dynamics equation
is as

𝛼 𝑛 𝑡 = 𝑘 𝑉𝑜𝑝𝑡(𝑆 𝑛 𝑡 ) −𝑉𝑛 (𝑡) + 𝜆 𝑆 𝑛 𝑡 … … … … … … … . . 9
• The proposed model takes both positive and negative
velocity difference into account, they call it a full speed
difference model (FSDM). The main advantage of FSDM is
eliminating unrealistically high acceleration and predicts a
correct delay time of car motion and kinematic wave speed
at Jam density. Then, Zhao and Gao (2005) argued that
previous models OSM, GFM and FSDM does not describe the
driver’s behavior under and urgent case where they can be
defined as:

• “A situation that the preceding car decelerates strongly, if
two successive cars move forward with much small headway-
distances e.g. a freely moving car decelerates drastically for
an accident in front or the red traffic light at an intersection,
the following car is freely moving and the distance between
the two cars is quite small.’’
• However, they found out that speed difference is not enough
to avoid an accident under such urgent case in previous
models for that, they extend the FSDM by incorporating the
acceleration difference and then got a new model called the
full speed and acceleration difference model (FSADM) as
follows;

𝛼 𝑛 𝑡 = 𝑘 𝑉𝑜𝑝𝑡 𝑆 𝑛 − 𝑉𝑛(𝑡) + 𝜆 𝑆 𝑛 𝑡 + 𝛽𝑔 𝑆 𝑛 𝑡 − 1 , 𝑎 𝑛+1 𝑡 𝑆 𝑛 𝑡 − 1 … .10
• With n (t) = is the acceleration difference between the preceding
vehicle n+1 and the following vehicle . Function g (.) is to
determine the sign of the acceleration difference term.

𝑆 𝑛 𝑡 − 1 , 𝑎 𝑛+1(𝑡) = 𝑎 𝑛+1 𝑡 ≤0
1,𝑜𝑡ℎ𝑒𝑟
−1, 𝑆 𝑛 𝑡−1 >0
… … … … … … … … 11

• The main advantage of FSADM compared to previous
models that can describe the driver’s behavior under an
urgent case, where no collision occurs and no unrealistic
deceleration appears while vehicles determined by the
previous car-following models collide after only a few
seconds. In 2006, Zhi-Peng and Yui-Cai (2006) conducted a
detailed analysis of FSDM and found out that second term in
the right side of Eq. (9) makes no allowance of the effect of
the inter-car spacing independently of the relative speed. For
that, they proposed a difference-separation model (DSM)
which takes the separation between cars into account and
the dynamics equation becomes

𝑎 𝑛 𝑡 = 𝐾(𝑉𝑜𝑝𝑡 𝑆 𝑛 𝑡 − 𝑉𝑛 𝑡 + 𝜆Θ 𝑆 𝑛 𝑡 1 + tanh (𝐶1(𝑆 𝑛 𝑡 − 1 − 𝐶2
3
+
𝜆Θ − 𝑆 𝑛(𝑡) 𝑆 𝑛(𝑡)(1 − tanh (𝐶1 𝑆 𝑛 𝑡 − 1 − 𝐶1)
3
…………………………12

• The strong point of SDSM that the model can perform more
realistically in predicting the dynamical evolution of
congestion induced by a small perturbation, as well as
predicting the correct delay time of car motion and
kinematic wave speed at jam density Lijuan and Ning (2010)
suggested a new car following model based on FSDM with
acceleration of the front car considered. With detailed study,
they observed than when FSDM simulate the car motion all
the vehicle accelerate until the maximal and when the reach
maximal velocity the acceleration and deceleration appeared
repeatedly. For that, they modified the Eq ( 9) to take into
account the influencing factor of the following car by adding
up to Eq (9) the leading acceleration. The dynamic equation
of the system is obtained as

𝑎 𝑛 𝑡 = 𝐾 𝑉𝑜𝑝𝑡(𝑆 𝑛 𝑡 − 𝑉𝑛(𝑡) + 𝜆 𝑆 𝑛 𝑡 + 𝛾 𝑆 𝑛 𝑡 + 𝛾𝑎 𝑛−1 𝑡 … … … … … … … . . 13
• Where λ is the sensitivity, expressing the response intensity of the
follow car to leading acceleration. They proved that their new
model has certain enlightenment significance for traffic control,
and is useful for establishing of Intelligent Transport Systems (ITS).
Previous models used only one type of ITS information, either
headway speed or acceleration difference of other cars to stabilize
the traffic flow. However, traffic flow can be more stable by
introducing all the three types of ITS information.

• Other models that come up as the extension of OSM are the
multiple headway speed and acceleration difference
(MHSAD) proposed by Li et al (2011), comprehensive optimal
speed model (COSM) proposed by Tian et al (2011) e.t.c.

APPLICATIONS OF OPTIMAL SPEED
• One scheme to control the societal cost of travel in traffic
systems is to set the speed limits based on the notion of
optimal speed with respect to societal costs. Determining
the optimal speeds and correct reinforcement of speed limits
in traffic systems will results in minimizing the unwanted
costs of travel such as accidents and the emission of
pollutants. It has been shown that rationalization of speed
limits applicable to each class of rural road and for each type
of vehicle, making the limits consistent with the optimal
speed in each case, has the potential to reduce casualty
crashes and crash costs substantially.

DISCUSSIONS AND SUGGESTION
FOR IMPROVEMENT (THE WAY
FORWARD FOR OSM)
• A review of optimal speed model was conducted the optimal
speed model belongs to the existing car-following models
(together with other classes like the stimulus response
models, and safe-distance or avoidance collision models). It
has the advantages that it is simple to use and calibrate. It
has the weakness or disadvantage of giving unrealistically
large accelerations in some circumstances. Most related
works on optimal speed model are those carried out by
Bando et al., 1995, Helbiny and Tilch, 1998, Zhao and Gao,
2005 e.t.c.

• The model has gain more and more attention from various
researchers. The main advancement of O.S.M is the extension to
full speed difference model (FSDM) which as mentioned above has
the advantage of eliminating unrealistically high acceleration and
predict a correct delay time of car motion and kinematic waves
speed jam density. Though, it was found out that the speed
difference is not enough to avoid an accident as in the case of
previous models, hence full speed difference model was extended
by incorporating the acceleration difference, and the a new model
called full speed and acceleration difference model (FSADM) which
has the advantage of describing the driver’s behavior under and
urgent case, where no collision occurs and no unrealistic
deceleration appearing while vehicles determined by the previous
car-following models collide after only a few seconds.
DISCUSSION AND SUGGESTION
FORWARD FOR OSM) (Cont.)

• Other advancement are the proposed multiple headway
speed and acceleration difference M H S A D, by Li et al.
(2011), which takes into account, the effects of the
acceleration difference of the multiple preceding vehicles
which affects to the behavior of the following just as the
headway and the speed difference. The main advantage of
MHSAD (as seen above) is that the model does not only take
the headway, velocity and acceleration difference
information into accounts, but also considers more than one
vehicle in front of the following vehicle. The model improved
the stability of the traffic and restrains the traffic jams.

• Other category of car-following models that comes up as
advancement to OSM are the comprehensive optimal speed
model (C O S M), Asymmetric full speed difference model (A
F S D M).
• The study of O S M is a good one and has yielded improved
performance on traffic analysis and therefore the research is
of great interest and advantageous to be continued.

CONCLUSION
• In this review, the most car-following model well known-the
optimal speed model (OSM) has been presented. The model
has successfully revealed the dynamical evolution of traffic
congestion on a simple way. Thereafter, inspired by the
optimal speed model, some car- following models were
successfully put forward to describe the nature of traffic
more realistically.

RECOMMENDATION
• This review has highlighted the drawbacks and advantages
of the existing car-following models (of which optimal speed
fell into). It therefore, recommend for the researchers to
develop the strong car-following model which will avoid the
collision and interpreted the traffic flow in a real manner.

REFERENCES
• A review analysis of Optimal Velocity models, Hajar Lazar,
Khadija Rhoulanu, Driss Rahmani, Periodica Polytechnica
Transportaing Engineering, 44(2), PP. 123-131, 2016.
• Bando M, Hasebe, K., Nakanishi, K., Nakayama, A. (1998)
Analysis of op-timal velocity model with explicit delay.
Physical Review E. 58(5), pp. 5429-5435. DOI:
10.1103physreve.58.5429
• Car-following Models, https: //en.Wikepedia.org/
• Full velocity difference model for a car-following theory, Rui
Jiang, Qingsong Wu, and Zuojin Zhu, Institute of Engineering
Science, University of Science and Technology of China,
Hefei, Anhui 230026, June 2001.

REVIEW OF OPTIMAL SPEED TRAFFIC FLOW MODEL

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