The document discusses optimal speed traffic flow models, which aim to more realistically model driver behavior compared to previous car following models. It describes several generations of optimal speed models that have been developed over time to address limitations. The models incorporate factors like desired optimal speed that is independent of the leader's speed, safe distance between vehicles, and asymmetric acceleration and deceleration behavior. The latest models presented in the document aim to produce realistic traffic dynamics like spontaneous jam formation and recover better delay time and kinematic wave properties.

1. 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
BASHIRU ABDU
SPS/16/MCE/00006
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

2. INTRODUCTION
• The development of mathematical relationship to describe
traffic flow began in 1920s, with an analysis of traffic
equilibrium and later transformed or modified in to first
and second principles of equilibrium in the 1950s.
• However, there is no satisfactory general theory that can
be consistently applied to real flow situation even with the
advancement in computer processing power.
• The existing traffic flow models, consists of empirical and
theoretical approach which are developed in to traffic
forecasting and identification of traffic congestion for
network adjustment.
• The behaviour of vehicular traffic can be described as a
complex and also nonlinear, therefore the laws of
mechanics cannot be applied to explain vehicular
interaction as a result of individual drivers’ reaction but
rather display cluster formation and shock wave
propagation.

3. • The study of traffic behaviour to solve the problem of traffic
congestion led to the development of three major traffic flow
models namely:
• Microscopic traffic flow models: these models consider every
vehicle as an individual entity and the behaviour of each
vehicle is governed by an ordinary differential equation.
• Macroscopic traffic flow models: it described the dynamics of
traffic in terms of aggregated traffic flow parameters such as
density of the vehicles k(x, t), mean speed v(x, t) and or flow
rate q(x, t) as a function of location and time.
• Mesoscopic traffic flow models: these are categorized between
the microscopic and macroscopic models, as their aggregation
level is in between those of micro and macroscopic models.
TRAFFIC FLOW MODELS

4. MICROSCOPIC TRAFFIC FLOW MODELS
• The microscopic traffic flow modelling describes the dynamics of
traffic flow at the level of each individual vehicle. It also describes
the car following behaviour as well as the lane changing behaviour of
each individual vehicle in a traffic stream (Kesting et al., 2008). They
are based on the assumptions that drivers adjust their speed with
respect to the speed of the leader (Darbha et al., 2008).
• An example of microscopic traffic flow models is the car following
models also known as the follow the leader models which described
the processes in which drivers follow each other in the traffic stream
(Pipes, 1953). It consists of the safe-distance models, stimulus
response models, psycho-spacing models and optimal speed models.
• Safe-distance or collision avoidance models assumed that the
follower respect a certain safe distance at least the length of the car
between him and the leading vehicle ahead for every 16.1km/hr in
order to avoid rear end collision. From these model the required gross
distance headway dn is given by the equation below;
dn (v) = 𝑙 𝑛 1 +
𝑣
16.1
… … … … … … … … … … … … … . . (1)

5. • The stimulus–response model is based on the assumption
that the driver of the following vehicle perceives and
reacts appropriately to the spacing and the speed difference
between the following and the lead vehicles (Jabeena,
2013). The response is taken as the acceleration of the
following vehicle, the stimulus taken as the speed while
sensitivity is a function of speed differential and spacing.
• Leutzback,(1988) first proposed the psycho-spacing model
in order to remedy the unrealistic behavioural aspect in the
stimulus- response model and its rules are described as
follows; at large distance headways, the follower is not
influenced by the speed difference and at small
disturbance headways the alertness of driver is increased.
• Car following process is one of the main processes in all
microscopic models as well as in modern traffic flow
theory. The model assumes that, in the absence of the
leader the follower’s acceleration tends to zero, these does
not reflect the real traffic observation and has led to the
introduction of another model called the optimal speed
traffic flow model (Bando et al, 1995).
MICROSCOPIC TRAFFIC MODELS (Cont.)

6. OPTIMAL SPEED TRAFFIC FLOW MODELS
• The optimal speed model evolved in order to remedy the
assumptions of the car following model, and it is based on the
assumptions that each driver has a safe speed which depends on the
spacing with the leading vehicle (Bando et al, 1995).
• According to these models the driver adapts a desired speed called
the optimal speed, which is not influenced by the leaders speed.
But under certain conditions, small disturbances are amplified and
results to jams. Therefore, these models are able to replicate stop
and go waves in traffic flows. The OS model is given by;

7. 𝑎 𝑛(𝑡) = 𝑘(𝑉𝑜𝑝𝑡(𝑆𝑛(𝑡)) − 𝑉𝑛(𝑡) … … … … … … … … … … … . (2)
• Where an is the acceleration of the nth car, k is the model calibration
parameters.
• Helbing and Tilch (1998) extend the OS model by incorporating l in
the equation as the length of the car and the parameters V1, V2 and
C1, C2 as follows;
Vopt (Sn(t)) = V1 + V2 tan ℎ 𝐶1 (𝑆𝑛 𝑡 − 1 − 𝐶2 … … … … … … … … … … … (3)
• The OS model is more practical because it is able to produce
spontaneous traffic jam as in real traffic situations with few
parameters.
• Few years later Bando et al, (1998) extend the OS model by
introducing the explicit delay time. This was done in order to produce
a realistic traffic flow models and is included in the previous
dynamical equation of OS models and the equation becomes;
OPTIMAL SPEED TRAFFIC FLOW MODELS (Cont.)

8. 𝑎 𝑛 𝑡 + 𝜏 = 𝑘 𝑉 𝑜𝑝𝑡 𝑆 𝑛 𝑡 − 𝑉𝑛 𝑡 … … … … … … … … … … … . (4)
• It was revealed from the analysis; that no effect was
observed when the explicit delay time is minimal.
• Furthermore, a new pattern of OS model traffic jam
appears when the introduced explicit delay time is large.
However, the OS model has encountered the problems of
high acceleration and unrealistic deceleration.
• In order to remedy the problems of high acceleration and
deceleration encountered by optimal speed model a
generalised force model was developed by adding new
term to the right of the equation (Helbing and Tilch, 1998).
𝑎 𝑛 𝑡 + 𝜏 = 𝑘(𝑉𝑜𝑝𝑡 𝑆 𝑛 𝑡 − 𝑉𝑛 𝑡 + 𝜆Θ −Sn t Sn t … … … … … … … … … … … . (5)
OPTIMAL SPEED TRAFFIC FLOW MODELS (Cont.

9. • The above equation take cares of the difference in speed between the
vehicle behind and that in front. The disadvantage of the GF model is
that it fails to consider the effect of positive speed difference on
traffic dynamics into account.
• Jiang et al. (2001) proposed a more realistic model known as the full
velocity difference model which account for both negative and
positive speed difference. The FVD model eliminates unrealistic high
acceleration and predicts a correct delay time of car motion and
kinematic wave speed at jam density.
• Zhao and Gao (2005) believe that the above model does not consider
driver’s behaviour under certain conditions and the velocity
difference is not enough to avoid accident as such the FSD model is
extended by incorporating acceleration difference and a new model
OPTIMAL SPEED TRAFFIC FLOW MODELS (Cont.)

10. called full velocity acceleration difference model is formed and the
equation is given by;
𝑎 𝑛 𝑡 = 𝑘(𝑉𝑜𝑝𝑡 𝑆 𝑛 𝑡 − 𝑉𝑛 𝑡 + λSn t + 𝛽𝑔(𝑆𝑛 𝑡 − 1 𝑎 𝑛+1 𝑛 + 1 𝑡 𝑆 𝑛 𝑡 − 1 … … … … . . (6)
and the acceleration difference between the leading vehicle and the
following vehicle is
given by:
𝑆 𝑛 𝑡 = 𝑎 𝑛+1 𝑡 − 𝑎𝑛 𝑡 … … … … … … … … … … … … … (7)
and g is a function which determine the sign of the acceleration difference
term.
OPTIMAL SPEED TRAFFIC FLOW MODELS (Cont.)

11. • The FVADM is advantageous over FVDM because it describe the
driver’s behaviour under an urgent case, where collision and
unrealistic deceleration does not occur as compared to the previous
car-following models where vehicle collide after only a few
seconds.
• Zhi-Peng and Yui-Cai (2006) conducted a thorough analysis of
FVDM and found out that no allowance was made in the above
equation to take care of the effect of the inter-car spacing
independently of the relative speed. Therefore, they formulate a
new model known as velocity-difference-separation model
(VDSM) which takes care of the spacing between cars and the
equation becomes:
𝑎 𝑛 𝑡 = 𝑘(𝑉𝑜𝑝𝑡 𝑆 𝑛 𝑡 − 𝑉𝑛 𝑡 + 𝜆𝜃 𝑆 𝑛 𝑡 𝑆 𝑛(𝑡)(1 + tan ℎ 𝐶 − 1(𝑆𝑛 𝑡 − 1) − 𝐶2)^3 +
OPTIMAL SPEED TRAFFIC FLOW MODELS (Cont.)

12. • The VDSM is more realistic in predicting the dynamical evolution of
congestion induced by a small disturbance, as well as predicting
the correct delay time of car motion and kinematic wave speed at jam
density.
• In 2010 a new car following model based on FVDM with
acceleration of the front car considered was proposed by Lijuan and
Ning (2010). They discover that when FVDM simulate the car
motion all the vehicle accelerate until the maximal speed and when
the speed reach maximal the acceleration and deceleration appeared
simultaneously. The dynamic equation of the system is modified to
take care of the influencing factor of the following car by adding up
to the acceleration of the leader as:
𝑎 𝑛 𝑡 = 𝑘(𝑉𝑜𝑝𝑡 𝑆 𝑛 𝑡 − 𝑉𝑛 𝑡 + 𝑎 − 𝑛 + 1 𝑡 … … … … … … … … … … . . (9)
OPTIMAL SPEED TRAFFIC FLOW MODELS (Cont.)

13. • The function γ explained the intensity of the response of the
following car to leading acceleration. They proved that their
new model is useful for establishment of Intelligent Transport
Systems (ITS). Previous models used only one type of ITS
information, headway, velocity, or acceleration difference of
other cars to stabilize the traffic flow.
• Li et al. (2011) proposed a new car-following model based on
the idea that traffic flow can be more stable by introducing all
the three types of ITS information which takes into account the
effects of the acceleration difference of the multiple preceding
vehicles which affects to the behaviour of the following vehicle
just as the headway and the speed difference, called multiple
head- way, velocity, and acceleration difference (MHVAD). The
mathematical equation is given by:
𝑉 𝑜𝑝𝑡 𝑆 𝑛 𝑡 = [ 𝑡𝑎𝑛ℎ 𝑆 𝑛 𝑡 − ℎ𝑐 + 𝑡𝑎𝑛ℎ ℎ 𝑐
𝑣𝑚𝑎𝑥
2
… … … … … … … … … … … … (10)
OPTIMAL SPEED TRAFFIC FLOW MODELS (Cont.)

14. • Where vmax is the maximal velocity of the vehicle, and hc is the safe
distance.
• The main advantage of MHVAD Compared with the other existing
models is that the proposed model does not only take the headway,
velocity, and acceleration difference information into account, but
also considers more than one vehicle in front of the following
vehicle. The model improved the stability of the traffic flow and
restrains the traffic jams. Others category of car-following models
inspired their idea to modify or to propose a new model via optimal
speed function. Among them, Jing et al. (2011) introduced a new
optimal velocity function and modified the additional term of FVDM.
In the first time, they proposed the modified full velocity difference
model (MFVDM I) taking into account a new optimal velocity
function proposed by (Helbing and Tilch, 1998):
𝑎 𝑛 𝑡 = 𝑘 𝑉 𝑜𝑝𝑡 𝑆 𝑛, 𝑉𝑛 − 𝑉𝑛 𝑡 + 𝜆𝑆𝑛 𝑡 … … … … … … … … … … . . (11)
𝑉 𝑜𝑝𝑡 𝑆, 𝑉𝑛 = 𝑉𝑛[1 −
𝑒Λ((𝑆𝑛−𝑆 𝑉 𝑛
)
𝑅 𝑛
)] … … … … … … … . (12)
OPTIMAL SPEED TRAFFIC FLOW MODELS (Cont.)

15. • In real driving behaviours, keeping a safe distance reflects the
drivers’ driving intention and accordingly affects vehicle
manoeuvres. Based on this study, Liu et al. (2012) targeted at
developing a new car-following model that takes the impact of a
desired following speed and safe distance as part of driving
behaviour modelling.
• In car-following approach, the efforts are more and more dedicated to
the development of models with a high performance.
• In this regard, Xu et al. (2013) presented an asymmetric full velocity
difference approach, in which take into account the effect of
asymmetric acceleration and deceleration in a car-following. The
most existing car-following models have not sufficiently taken the
equality of acceleration and deceleration behaviours into
OPTIMAL SPEED TRAFFIC FLOW MODELS (Cont.)

16. consideration. The GFM were extended to an asymmetric full velocity
difference (AFVD) approach in which two sensitivity coefficients are defined
to separate the model to positive and negative velocity. The AFVD model can
be expressed as:
𝑎 𝑛 𝑡 = 𝑘 𝑉 𝑜𝑝𝑡 𝑆 𝑛 𝑡 − 𝑉𝑛 𝑡 + 𝜆1 𝐻 −𝑆𝑛 𝑡 𝑆 𝑛 𝑡 + 𝜆2 𝐻 𝑆 𝑛 𝑡 𝑆 𝑛 𝑡 … … … … … . . (13)
OPTIMAL SPEED TRAFFIC FLOW MODELS (Cont.)

17. • The purpose of the analysis of AFVDM pointed out that the positive
velocity difference term is significantly higher than the negative
velocity difference term, which agrees well with the results from
studies on vehicle mechanics.
• In 2015, the authors (Xu et al., 2015) interested in taking the
asymmetric characteristic of the velocity differences of vehicles and
they proposed an asymmetric optimal velocity model for a car-
following theory (AOV). They based on the assumption that the
relationship between relative speed and acceleration (deceleration) is
in general nonlinear as demonstrated by actual observations
(Shamoto et al., 2011). They formulated FVDM to get an asymmetric
optimal velocity (AOV) car- following model as follows:
𝑎 𝑛 𝑡 = 𝑘 𝑉 𝑜𝑝𝑡 𝑆 𝑛 𝑡 − 𝑉𝑛 𝑡 + 𝑆𝑛 𝑡 exp −𝜇𝑠𝑛 𝑡 … … … … … … … … … . (14)
OPTIMAL SPEED TRAFFIC FLOW MODELS (Cont.)

18. • The main advantages of AOV model are avoiding the unrealistically
high acceleration appearing in previous models when the speed
difference becomes large, however, the asymmetry of AOV model
between acceleration and deceleration depends nonlinearly on the
speed difference with the asymmetrical factor μ.
• Recently, Yi-Rong et al. (2015) proposed a new car- following model
with consideration of individual anticipation behaviour. However, the
effect of anticipation behaviour of drivers has not been explored in
existing car-following models. In fact, they suggested a new model
including two kinds of typical behaviour, the forecasting of the future
traffic situation and the reaction-time delay of drivers in response to
traffic stimulus. The main idea of this model is that a driver adjusts
his driving behaviour not only according the observed velocity vn(t)
but also the comprehensive anticipation information of headway and
speed difference. The dynamics equation is as follows:
OPTIMAL SPEED TRAFFIC FLOW MODELS (Cont.)

19. 𝑎 𝑛 𝑡 = 𝑘 𝑉 𝑜𝑝𝑡 𝑆 𝑛 𝑡 + 𝑝1 𝜏 − 𝑉𝑛 𝑡 + 𝜆𝑆 𝑛 𝑡 + 𝑝2 𝜏 … … … … … … … … (15)
• The effect of individual anticipation behaviour has an important influence
on the stability of the model and this effect should be considered in the
modelling of traffic flow (Lazar et al., 2016).
OPTIMAL SPEED TRAFFIC FLOW MODELS (Cont.)

20. APPLICATIONS OF OPTIMAL SPEED
TRAFFIC FLOW MODELS
• The study of traffic flow dynamics have led to the formulation of
optimal speed traffic flow models which are important in order to
design comfortable and safe roads, to solve road congestion
problems as well as designing adequate traffic management
measures.

21. CONCLUSION
• The OS model is one of the microscopic car-following models
which describe the dynamics of traffic flow based on the
assumptions that a driver adopts a desired safe speed according to
the spacing of the leading vehicle.
• The review of optimal speed model, have shown that the model is
more realistic because it is able to produce spontaneous traffic jam
as in real traffic situations with few parameters.
• The OS model is simple to use and calibrate but gives unrealistic
large acceleration in some situations. These attract the attention of
researchers to remedy the aforementioned problem and some new
OS models were proposed in order to realistically describe the
dynamical nature of traffic by modifying the dynamical equation
of OS model.

22. • The equations were extended by incorporating a new optimal
speed functions; the explicit delay time or speed difference, or
acceleration difference, while the recent model account for drivers
behaviour in relation to certain situations.
• Therefore the genealogy of traffic flow models have shown that
each of the models have their advantages and pitfalls, these led to
evolutions of more realistic models and paved ways for more
research in studying the dynamics of traffic in order to minimised
traffic congestion problems as it affects major cities of the world as
well as predicting the real traffic flow.
CONCLUSION (Cont.)

23. RECOMMENDATIONS
• Based on the above review of the optimal speed traffic flow model,
it is therefore recommended that researchers should consider the
effect of individual anticipation behaviour in modelling of traffic
flow models and those factors which have a great effect on
individual anticipation behaviour such as the size of vehicles, the
age, the experience, and the physical fitness level of drivers, as
well as the environment of the road and so on. All these have a
great effect on individual anticipation behaviour.

24. REFERENCES
• Bando, M., Hasebe, K., Nakayama, A., Shibata, A., Sugiyama, Y. (1995)
Dynamical model of traffic congestion and numerical simulation.
Physical Review E. 51(2), pp. 1035-1042.
• Bando, M., Hasebe, K., Nakanishi, K., Nakayama, A. (1998) Analysis of
optimal velocity model with explicit delay. Physical Review E. 58(5), pp.
5429-5435.
• Darbha, S., Rajagopal, K., Tyagi, V. (2008) A review of mathematical
models for the flow of traffic and some recent results. Nonlinear
Analysis: Theory, Methods & Applications. 69(3), pp. 950-970.
• Helbing, D., Tilch, B. (1998) Generalized force model of traffic
dynamics. Physical Review E. 58, pp. 133-138.
• Jabeena, M. (2013) Comparative Study of Traffic Flow Models and Data
Re- trieval Methods from Video Graphs. International Journal of
Engineering Research and Applications. 3(6), pp. pp. 1087-1093.

25. • Kesting, A., Treiber, M., Helbing, D. (2008) Agents for Traffic
Simulation. In: Multi-Agent Systems: Simulation and Applications.
(Uhrmacher, A., Weyns, D. (ed.)), Chapter 11, pp. 325-356. CRC
Press, Boca Raton, Florida, USA. 2009.
• Pipes, L. A. (1953) An Operational Analysis of Traffic Dynamics.
Journal of Applied Physics. 24(3), pp. 274-281.
• Zhao, X., Gao, Z. (2005) A new car-following model: full velocity
and acceleration difference model. The European Physical Journal
B. 47(1), pp. 145-150.
• Zhi-Peng, L., Yun-Cai, L. (2006) A velocity-difference-separation
model for car-following theory. Chinese Physics. 15(7), pp. 1570-
1576.
REFERENCES (Cont.)

26. THANK YOU FOR READING