ASSIGNMENT BY ADAMU MUHAMMAD GYAMBAR

1. REVIEW OF OPTIMAL SPEED
MODEL
MASTERS CLASS ASSIGNMENT
BY
ADAMU MUHAMMAD GYAMBAR
BAYERO UNIVERSITY KANO, NIGERIA.
DEPARTMENT: CIVIL ENGINEERING
COURSE: ADVANCE TRAFFIC
ENGINEERING
SUMMITED TO
PROF. H M ALHASSAN
29 MAY, 2017.

2. INTRODUCTION
 Just like normal blood circulation necessitate a healthy body, a well or smooth traffic
flow is necessary for healthy business, works, trips community development in a city .
 Traffic congestions haunt, displeased cities and communities from various perspectives.
 It inflicts uncertainties, drains resource, reduction in productivity, stress commuters
and harm environment due to it continues deterioration of urban traffic condition.
 Speed is one of the most relevant factors that characterize road operation, directly and
decisively influencing its evaluation by all its users, having several effects, either positive
or negative.

3. HISTORY
In describing traffic dynamic, most useful tool is car following model (Gazis–Herman–
Rothery (GHR) model 1958) developed by Chandler et, of microscopic simulation model
base on two objective which comply with the first but failed to describe the second objective
 Reducing the speed difference,
 maintain an appropriate spacing between the following vehicle and the leading vehicle.
(Newell 1961) proposed a model capture the characteristic of CF behaviors in maintaining an
optimal spacing corresponding but is not suitable in traffic simulation. Linear models (Pipes
1967) were the mainstream models in the beginning of the traffic simulation history
The Gipps CF model (1981), the most successful collision avoidance model,
Fritzsche model 1994 and the Wiedemann model and Reiter 1992) have a two dimension
zone in the “spacing-relative speed” diagram
(Bando et al. 1995 & 1998) developed a new model called Optimal Speed Model (OVM)
Thirty years later

4. THEORETICAL MODEL OF OPTIMUM SPEED MODEL
The theoretical models that support the optimum speed model are as follows
 Microscopic model
 Mesoscopic model
 Macroscopic model
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 car-following models.
Car-following models describe the processes in which drivers follow each other in the traffic
stream. The car-following process is one of the main processes in all microscopic models as well
as in modern traffic flow theory. each vehicle is considered separately and its behavior is
modeled as it reacts and anticipates to vehicles in front by its own dynamic equation having the
following form:
T is the reaction-time, dn is the distance headway with respect to the vehicle in front,
and Vn is the speed of the considered vehicle.
•Different functions of f result in various types of car-following models. These are:
(1)

5. Safe-distance models, stimulus-response models, psycho-spacing models and optimal
speed models. We also have microscopic traffic models based on cellular automata (CA).
Mesoscopics:
There are three types of mesoscopic models: headway distribution models, cluster models
and gas-kinetic models . In gas-kinetic models, vehicles and drivers’ behavior are described
in more aggregate terms than in microscopic models, by means of probability distribution
functions.
Macroscopic (continuum) traffic models: This deal with traffic flow in terms of aggregate
variables as a function of location and time. They describe the dynamics of the traffic
density k (x, t), mean speed v(x, t) and or flow rate q(x, t).
Macroscopic models often require less information input than microscopic models. This
simplifies the calibration and validation process, which make this type of model very
suitable for control applications.
(2)

6. PROBLEMS STATEMENT
Optimal Speed model : Ever since, the development of this model in traffic stream by
Bando et al (1995 and 1998) . The short comings of the model such as:
 Unrealistic
 instability,
 changes in traffic congestion
 formation of stop-and-go waves
Due to this factors necessities the frequent review of the model, so as to improved on the
models.

7. Definition of Optimal (optimum) speed limit: can be define as process or situation where
by a moving vehicle attain steady and maintain maximum legally permitted design
pavement speed limit on a free way or maintain a maximum desire legally permitted speed
limit of a moving vehicle over a period of time on an free way (Flow stream) without
accelerating further of the permitted speed limit.

8. According to Green shield :
When the density is zero, the flow is zero because there are no vehicles on the roadway.
 As the density increases, the flow also increases to some maximum flow conditions.
 When the density reaches a maximum, generally called jam density, the flow must be zero
because the vehicles tend to line up end to end
REVIEW OF OPTIMAL SPEED MODEL

9. According H. Ez-Zahraouy, et al June 2004. considering a one-dimensional road of length
L with open boundary conditions the particles are injected with a rate probability α and β at
one end of the road opposite to each other. car following models with the optimal speed
function with an explicit delay time τ . Base on Newell and Whitham description analysis
on the traffic model, the following equation of motion of car j: V
Where xj(t) is the position of the vehicle j at time t, ∆xj(t)=xj+1(t)-xj(t) is the headway of
vehicle j at time t, and τ is the delay time ( how is allows for the time lag that it takes the car
speed to reach the optimal speed V(∆xj(t)) when the traffic flow is varying. V(∆xj(t)) is the
optimal speed of vehicle j and is given by
Where hc is the safety distance and Vmax is the maximal speed of vehicle j when other
vehicles do not exist.
Ẍj(t)=a{V(Xj+1(t)−Xj(t))−Ẋj(t)} (3)

10. In the original car following models, the optimal speed is the same for all the vehicles.
Thus, the optimal speed function of each vehicle is different from each other. Generally, it
is necessary that the optimal speed function has an upper bound (maximal speed). Also,
it is important that the optimal speed function has the turning point.
The OVM can explain behaviors of traffic flow, for example, the transition from a free flow
to a congested flow, a density-flow relationship, a kind of effective delay of car motion.
Consider a pair of cars, a leader and a follower. Assume the leader changes the velocity
according to vl = v0(t) and the follower duplicates the leader’s velocity but with some
delay time τ , that is, vf = v0(t−T). Under such a situation we can clearly define the delay
time of car motion by T. It is known that the observed delay time τ of car motion is of the
order of 1 sec, but the known physical or mechanical response time τ is of the order of
0.1 sec. it has been confirmed that the equation (2) really produces T of order 1 sec

11. optimal speed function most has an upper bound (maximal speed).
optimal speed function most has the turning point.
The idea of the above car following model is that a driver adjusts the vehicle speed
According to the observed headway ∆xj(t). The delay time τ allows for the time lag that it
takes the speed of each vehicle to reach the optimal speed V (∆xj(t)) of each
vehicle when the traffic flow is varying. By taylor expanding, Eq.(1), one obtains the
differential equation model
(5)
Where a=1/τ is the sensitivity of a driver Furthermore, by transforming the time derivative
to the difference in Eq (1), one can obtain the difference equation model
Xj(t+2τ) = xj(t+τ)+τV(xj(t)) (6)
(4)

12. considering such a case that the dimensionless delay time of vehicle j is uncorrected with other
vehicles and is given by τj=<τ>+∆τ[2rnd(j)-1.0 (7)
Where rnd(j) is the random number between zero and unity, <τ> is the average value of the
dimensionless delay time, and ∆τ is the strength of the variation of the dimensionless delay
time.
Properties of Optimal Speed Model are :
 Linear Analysis
 Numerical Simulation,
 Unrealistic
 instability,
 changes in traffic congestion
 formation of stop-and-go waves
 upper boundary (maximum speed)
 Turning point

13. According Hao Wang et al August, 2014, by introducing the DBOVF into the original OVM, the
new model allows drivers to reach their steady states within a wide region instead of a specific
optimal solution in a steady state.
(8)

14. Considering the facts that drivers would like to accept a range of spacing instead of an
optimal one, we assume that the steady state occupies a two-dimension area in the
speed-spacing diagram. As shown in Figure 1, there are two boundaries in the steady
state region. Each boundary can be formulated by a certain type of optimal velocity
function. The two boundaries of the steady state divide the speed-spacing diagram into
three regions

15. ASSUMPTIOMS MADE
 In region I, the spacing is too small for the driver to accept, and the driver will reduce the
speed towards the optimal speed indicated by the left boundary optimal velocity function.
 In region Ⅱ, the driver is satisfied with current conditions, and will not change the speed
until the vehicle moves out of this steady region.
 In region Ⅲ, the spacing is too large, and the driver will accelerate towards the optimal
speed indicated by the right boundary optimal velocity function.
Simple Example of DBOVF and types of OVF over the history in traffic flow studies
are:
The three types which were most widely used by researchers in Left boundary and Right
boundary namely,
 (Newell 1961), the convex type represented by the exponential function.
 (Daganzo 1994), the piecewise linear function represented by triangle fundamental
diagram model.
 (Bando et al. 1995), the S shape function In order to make comparison with the original
OVM.

16. The second requirement is from the consideration that the deceleration is usually stronger 
than the acceleration at the margin of steady state. Suppose a vehicle moves a small
distance.
(Michaels 1961; Evans and Rothery 1977), the first requirement comes from the studies on
psychophysical car following models. These studies indicated that drivers perceive spacing
changes through changes on visual angle subtended by the vehicle ahead.
The S shape function In order to make comparison with the original OVM, we use the
Bando’s S-shape function as the boundary function to build the DBOVF. Before modeling
the DBOVF, two requirements are considered as
 (i ) The range of spacing in the steady state increases with the speed increasing;
 (ii ) For a given speed Ve , the smallest and largest spacing in steady state are

17. Features of DBOVM:
The feature of DBOVM is the local stability
Considered 3 vehicles in the local stability studies. State that All vehicles are in steady
state at the beginning of the simulation:
 (ⅰ) initial state on the right boundary of steady region,
 (ⅱ) initial state on the left boundary of steady region,
 (ⅲ) initial state satisfying the optimal function in original OVM.

18. Stability Features of the Basic DBOVM
The DBOVM does not have a uniform model expression as a multiphase car following
model, which makes it difficult for the analytical stability analysis.
Treiber and Kesting (2013) gave a detailed theoretical analysis on traffic stability. It is
pointed out that all time-continuous car-following models with a negative derivative of
acceleration with respect to speed are unconditionally locally stable, if there are no
explicit reaction times in models. if there are no explicit reaction times in models.
Recall the dual-boundary-optimal-velocity-function displayed in Figure 1, the basic
DBOVM satisfies the criterion suggested by Treiber and Kesting when the local traffic
state is located outside of the dual boundary steady region. However, as the driver
does not perform any acceleration within the steady region, it delays the driver’s
response to the leading vehicle when the traffic state moves through the dual
boundary region in the speed spacing phase diagram. Therefore, the basic DBOVM is
analogous to the OVM with explicit delay (Bando et al. 1998) in some extent. The
simulation studies on the local stability of the basic DBOVM are as follows.

19. Three vehicles are considered in the local stability studies. All vehicles are in steady state at
the beginning of the simulation. For the studies on the basic DBOVM, the initial state of
vehicles should satisfy either the left or the right boundary optimal velocity function.
Otherwise, the following vehicles may not respond to the perturbation from the leading
vehicle according to the law of the basic DBOVM.
Therefore, three simulations are conducted for three different scenarios respectively, which
are;
 (ⅰ) initial state on the right boundary of steady region,
 (ⅱ) initial state on the left boundary of steady region,
 (ⅲ) initial state satisfying the optimal function in original OVM

20. Advantage
 It reduce fuel consumption or expenditure
 It reduce accident rate
 It makes driving safer
 It helps in maintaining the design life span of the pavement.
 It reduces pollution.
 It increase traffic flow speed and its stability
 It prevent over speeding
 It reduce fuel consumption or expenditure
(Bando et al. 1995), all three simulations begin with the driving speed of 10 m/s for all
vehicles. Such a speed ensures that the initial condition satisfies the string stability criterion
of the original OVM Then a small perturbation is added on the leading vehicle by giving its
position an instantaneous change (either increasing by 1 m or reducing by 1m). The
simulations are conducted with the time step of 0.1 s, and results of the simulations.

21. Disadvantage
 It wastes time for commercial drivers that are used to travel with high speed.
 It increase travel time
 It increase pollution
Application
 Capacity analysis.
 safety research
 Traffic simulation.
 Applicable in intelligent transportation systems, such as advanced vehicle control and
safety systems and autonomous cruise control systems.
 Optimal speed limit is applicable on moving vehicle such as car, truck, buses etc
 The parameters of general DBOVM are required to be calibrated by real traffic flow
data, and applications of the proposed model.

22.  It is used in various trains such as mono rail etc.
 We use real fuel-economy data to build a mathematical model for determining the
optimal speed. We'll solve the model graphically, and then analytically using calculus

23. LIMITATION
 The OVM does not have a time delay in its model expression, which makes it convenient for
theoretical analysis.
 The optimal speed function assumes that there is a one-to-one correspondence between the
spatial headway and the optimal driving speed in steady traffic state.
 The optimal velocity models have difficulty to avoid collisions in urgent braking cases. This
is mainly due to the fact that the phenomenon of anticipation is not explicitly taken into
account
CONCLUSION
 Generally, it is necessary that the optimal speed function has an upper bound (maximal
speed) Also, it is important that the optimal speed function has the turning point.
 The main contribution is the proposal of a simple car following model called general
Dual-Boundary-Optimal-Velocity-Model, which can describe the driving behavior of
accepting a range of satisfied conditions instead of an optimal one under steady traffic.

24.  The model is developed based on the Optimal Velocity Model with only two
additional parameters.
 Therefore, it is very convenient for both analytical and numerical analysis.
 A simple speed adjustment mechanism is introduced into the basic DBOVM, with
which traffic state can converge to steady state everywhere inside of the two
boundary steady region.
 Under the effect of speed adjustment, traffic states are transferred along some
specific paths with an approximately constant slope equal to the sensitivity parameter
of the speed difference term in general DBOVM.

25. CHARACTERISTIC OF OPTIMUM SPEED LIMIT
 Synthesize Present knowledge about speed and other factors that either influence
speed or are an outcome of speed such as Safety, Environmental Impacts, Road User
Costs
 To analyze speed data to determine the vehicle operating speed impacts from different
vehicle classifications, temporal factors, environmental factors, and road factors. Such as
Road Engineering, Regulatory and Enforcement Environment, Driver Attitude and
Behavior, Weather Factors, Temporal Factors, Vehicle Classification,
 The dual boundary steady region in DBOVM has the hysteresis effect, which is similar to
the effect of explicit delay in OVM.
 The wider the dual boundary steady region is, the stronger the hysteresis effect will be.
 In spite of the instability resulted from the hysteresis of dual boundary region, the speed
adjustment effect in general DBOVM restrains the hysteresis and improves the stability of
traffic.

26. RECOMMENDATION ON THE PROPOSED NEW MODEL
 The explicit delay time τ should be included in the dynamical equation in order to
construct realistic models of traffic flow.
 In open boundary optimal speed model variation of the delay time Δτ should be
introduce in new model so that the transition from unstable to meta stable and from meta
stable to stable state occur and determine it effects
 Dual Boundary Optimal Speed model (DBOVM) has Present a framework in general
and such dual boundary steady region can also be introduced into other well known car
following models.
 Dual Boundary Optimal Speed model (DBOVM) has Present a framework in general and
such dual boundary steady region can also be introduced into other well known car
following models.

27.  The basic DBOVM should be review or proposed a new DBOVM so that it can reach the
steady state inside of the dual boundary region during the dynamic process of traffic flow.
 Therefore, the amendment of the speed adjustment mechanism is necessary in the
general DBOVM.
 The DBOVM should be remodel so as to have uniform model expression as a multiphase
car following model, which will makes it simple for the analytical stability analysis

28. DEMOSTRATION OF NUMERICAL SIMULATION
http://www.popsci.com/sites/popsci.com/files/traffic_simulation.gifS

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Traffic flow model.( 2014)
G.F. Newell, Oper. Res. 9, 209 (1961).
Hashim.M. Alhassan, Advance Traffic Engineering Lecture Note
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31. HOPE IS EDUCATIVE
THANK YOU