A REVIEW OF OPTIMUM SPEED MODEL An Assignment On Advanced Traffic Engineering (CIV8329) by Sanusi Dauda SPS/16/MCE/00027   Submitted to Prof. H. M. Alhassan  Highway and Transportation Engineering (Option) Department of Civil Engineering Faculty of Engineering Bayero University, Kano 19TH May, 2017

1. A REVIEW OF OPTIMUM SPEED MODEL
An Assignment On Advanced Traffic Engineering (CIV8329)
by
Sanusi Dauda
SPS/16/MCE/00027
Submitted to
Prof. H. M. Alhassan
Highway and Transportation Engineering (Option)
Department of Civil Engineering
Faculty of Engineering
Bayero University, Ka n o
1 9 T H M a y, 2 0 1 7

2. INTRODUCTION
Microscopic models describe traffic flow dynamics in terms of single vehicles.
The microscopic car-following model is a favorite type of traffic flow dynamics to describe the individual
behavior of drivers (individual driver-vehicle units).
In a strict sense, car-following models describe the driver’s behavior only in the presence of interactions with
other vehicles, and the model involves phase transition from the uniform flow to traffic jam. A car following
model is complete if it’s able to describe all situations including acceleration and cruising in free traffic,
following other vehicles in stationary and non-stationary situations, approaching slow or standing vehicles and
red traffic lights.

3. REVIEWS
The Optimal Velocity Model (OVM) is a time-continuous model whose acceleration function is of the form
amic(s,v). The acceleration equation is given by:
𝒗 =
𝒗 𝒐𝒑𝒕 𝒔 −𝒗
𝝉
Optimal Velocity Model (1.0)
This equation describes the adaption of the actual speed v =vα to the optimal velocity vopt(s) on a time scale given by
the adaptation time τ.
Comparing the acceleration equation (1.0) with the steady-state condition, it becomes evident that the optimal
velocity(OV) function vopt(s) is equivalent to the microscopic fundamental diagram ve(s). It should obey the
plausibility conditions
𝑣opt(s) ≥ 0, vopt(0) = 0, lim
𝑠→∞
𝑣opt(s) =v0, (1.1)
The acceleration equation (1.0) defines a whole class of models whose members are distinguished by their
respective optimal velocity functions. The OV function originally proposed by Bando et al., uses a hyperbolic
tangent

4. vopt(s) = 𝑣 𝑜
tanh
𝑠
∆𝑠
− 𝛽 +tanh β
1 +𝑡𝑎𝑛ℎ 𝛽
(1.2)
Besides the parameter τ which is relevant for all optimal velocity models, the OVM of Bando et al. has three
additional parameters, the desired speed v0, the transition width Δs, and the form factor β. A more conscious
(spontaneous) OV function can be derived by characterizing free traffic by the desired speed v0, congested
traffic by the time gap T in car-following mode under stationary conditions, and standing traffic by the minimum
gap s0. In analogy to the Section-Based Model, we obtain
vopt(s) = max 0, min 𝑣 𝑜
𝑠−𝑠 𝑜
𝑇
(1.3)
The simulation results are similar to that of the hyperbolic tangent OV function. Typical parameter values are
given in Table 1.0.

5. Parameter Typical value
Highway
Typical value city
traffic
Adaptation time τ
Desired speed v0
Transition width Δs [vopt according to Eq.(1.2)]
Form factor β [vopt according to Eq.(1.2)]
Time gap T [vopt according to Eq.(1.3)]
Minimum distance gap s0 [vopt according to Eq.(1.3)]
0.65 s
120 km/h
15 m
1.5
1.4 s
3 m
0.65 s
54 km/h
8 m
1.5
1.2 s
2 m
Table 1.0 Parameter of two variants of the Optimal Velocity Model (OVM)
Model properties. For the simulations of the model, it was concluded that on a :
• quantitative level, the OVM results are unrealistic.
• qualitative level, the simulation out come has a strong dependency on the fine tuning of the model parameters, i.e.,
the OVM is not robust (vigorous or strong).

6. These deficiencies are mainly due to the fact that the OVM acceleration function does not contain the speed difference as
exogenous variable, i.e., the simulated driver reaction depends only on the gap, but is the same whether the leading
vehicle is slower or faster than the subject vehicle. This corresponds to an extremely short-sighted driving style.
Full Velocity Difference Model
By extending the OVM with an additional linear stimulus for the speed difference, the Full Velocity Difference Model
(FVDM) is obtained:
𝑣 =
𝑣 𝑜𝑝𝑡 𝑠 −𝑣
𝜏
– γΔv Full Velocity Difference Model. (1.4)
As in the OVM, the steady-state equilibrium is directly given by the optimal velocity function vopt. When assuming
suitable values for the speed difference sensitivity γ of the order of 0.6s−1, the FVDM remains accident-free for speed
adaptation times of the order of several seconds. Furthermore, the fact sheet shows that the waves in the congested
region of the freeway scenario are more realistic than that of the OVM, although the wave lengths remain too short. And,
the accelerations remain in a realistic range. However, in contrast to the OVM, it is not able to describe all traffic
situations.

7. The reason is that the term γΔv describing the sensitivity to speed difference in Eq.(1.4) does not depend on the gap.
Consequently, a slow vehicle (or a red traffic light corresponding to a standing virtual vehicle) leads to a significant
decelerating contribution even if it is miles away. Thus, simulated vehicles do not reach their desired speed even on a
long road with no other vehicles.
Improved Full Velocity Difference Model. In the following, the sensitivity to speed differences must decrease with
the gap s and tend to zero as s → ∞. This can be realized by replacing the contribution −γΔv of Eq.(1.4) by a
multiplicative term − 𝛾Δv/s. However, now the sensitivity diverges for s → 0 which is unrealistic. Furthermore, 𝛾
has a different unit compared to γ (and, consequently, a different numerical value) which should be avoided if
possible. The simplest approach to resolve these new problems consists in applying the inverse proportionality only if
the gap is larger than the interaction length v0T. Hence, the resulting acceleration equation of the new “complete”
variant of the FVDM is given by
𝑣 =
𝑣 𝑜𝑝𝑡 𝑠 −𝑣
𝜏
−
𝛾 𝛥𝑣
𝑚𝑎𝑥 1, 𝑠/ 𝑣 𝑜 𝑇
(1.5)

8. It turns out that model (1.5) is able to realistically simulate the cruising phase, in contrast to the original model (1.4),
and produces realistic accelerations, in contrast to the OVM. However, the robustness problem is not resolved.
Newell’s Car-Following Model
Newell’s car-following model is the arguably simplest representative of time-discrete models. Its speed function is
directly given by the optimal speed function.
v(t + T) = vopt(s(t)), vopt(s) = min 𝑣 𝑜,
𝑠
𝑇
Newell’s Model. (1.6)
Newell’s model has two parameters: The time gap or reaction time T, and the (effective) vehicle length leff. Since in
this regime the kinematic wave velocity is constant and is given by
w = ccong =
− 𝑙 𝑒𝑓𝑓
𝑇
The set of model parameters can alternatively be expressed by {T,w} or by {leff,w}.

9. Relation to the Optimal Velocity Model. Newell’s model is mathematically equivalent to the OVM in the car-
following regime (bound traffic) if one sets τ = T and updates the OVM speed according to the explicit integration
scheme and the vehicle positions by the simple Euler scheme.
xα(t +Δt) = xα(t)+vα(t +Δt)Δ t. (1.7)
As a consequence, the parameter T of Newell’s model has the additional meaning of a speed adaptation time τ.
Relation to the macroscopic Section-Based Model. When disaggregating the solutions of the Section-Based
Model with the function by generating trajectories from the density and speed fields of congested traffic using the
macro-micro relation, these trajectories are simultaneously solutions of Newell’s model. Interpretation from the
driver’s point of view.
The trajectories corresponding to the solution of Newell’s model for congested traffic which are given by the
recursive relations.

10. xα(t +T) = xα−1(t) + wT = xα−1(t) − leff,
vα(t +T) = vα−1(t). (1.8)
This means that the trajectory of the follower is completely determined by the trajectory of the leading vehicle. In
Newell’s car-following model, the position of a vehicle following another vehicle at time (t + T) is given by the position
of the leader at time t minus the (effective) vehicle length leff. As a corollary, the speed profile of a vehicle exactly
reproduces that of its leader with a time delay T.
From the above considerations, one simply conclude that the parameter T of Newell’s model can be interpreted in four
different ways:
1. As the reaction time when interpreting Eq.(1.6) as a time-delay differential equation or when considering the
trajectories (1.8).
2. As the time gap of the microscopic fundamental diagram (1.3).
3. As the speed adaptation time following from the equivalence between Newell’s model and the OVM combined with
speed update rule.
4. And as the numerical update time T = Δt when interpreting Eq.(1.6) as a discrete-time model.

11. The interpretation in terms of a reaction time or a time gap can only be applied for congested traffic. In contrast, the
interpretation as a speed adaptation time or a numerical update time is generally valid.
Relation of Newell’s model with anticipation to the FVDM. In order to compensate for at least part of the
reaction time delay described by Newell’s model, a driver would try to predict the distance gap (the only exogenous
stimulus of Newell’s model) by a certain time interval (Ta) into the future.
Using the rate of change 𝑠 = − Δv for an estimate of the gap at this time, 𝑠(t + Ta) = s(t) − TaΔv, this results in the
generalized Newell’s model
v(t +T) = vopt (s(t) − TaΔv) ≈ vopt(s(t)) − 𝑣opt(s)TaΔv (1.9)
This is equivalent to a time-continuous model given by
𝑑𝑣
𝑑𝑡
=
𝑣 𝑜𝑝𝑡 𝑠 −𝑣
𝑇
−
𝑇𝑎 𝑣 𝑜𝑝𝑡(𝑠)
𝑇
𝛥𝑣 (2.0)
This corresponds to a Full Velocity Difference Model with a gap dependent sensitivity γ(s) = Ta 𝑣 𝑜𝑝𝑡(s)/T.

12. From the OV plausibility conditions (1.1) it follows that lim
𝑠→∞
𝑣 𝑜𝑝𝑡(𝑠) = 0, i.e., the sensitivity tends to zero when the
gap becomes sufficiently large. This means, the resulting FVDM-like model is complete and similarly to the
“improved” FVDM.
Generalized Force Model (GFM)
Jiang et al. (2001) pointed out that when the preceding vehicle is much faster, the following vehicle may not apply
break, even though the spacing is smaller than the safe distance. The basis of GFM and taking the positive factor
𝑆 𝑛 𝑡 into account. The main drawback of GFM is that the effect of positive velocity difference on traffic dynamics
was not taken into account. The model only considered the case where by the following vehicle velocity is more than
that of the leading vehicle. The dynamics equation is given.
𝑎 𝑛 𝑡 = 𝑘 𝑉𝑜𝑝𝑡 𝑆 𝑛(𝑡) − 𝑣 𝑛(𝑡) + 𝜆 𝑆 𝑛 𝑡 (2.1)

13. Full Velocity and Acceleration Difference Model (FVADM)
Previous models OVM, GFM and FVDM does not described the driver’s behavior under an urgent case which
maybe: “A situation that the preceding car decelerates strongly, if two successive cars move forward with much
small headway-distance, 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”, the
velocity difference is not enough to avoid an accident under such urgent case. Zhao and Gao (2005), extend the
FVDM by incorporating the acceleration difference, and the new model was called the full velocity and acceleration
difference model (FVADM). The model is as follow:
𝑎 𝑛 𝑡 = 𝑘 𝑉𝑜𝑝𝑡 𝑆 𝑛(𝑡) − 𝑣 𝑛(𝑡) + 𝜆 𝑆 𝑛 𝑡 + 𝛽𝑔 𝑆 𝑛 𝑡 − 1 , 𝑎 𝑛+1 𝑡 𝑆 𝑛 𝑡 − 1 (2.2)
With 𝑆 𝑛 𝑡 = 𝑎 𝑛+1 𝑡 − 𝑎 𝑛(t) is the acceleration difference between the preceding vehicle n + 1 and the following
vehicle α. Function g(∙) is to determine the sign of the acceleration difference term.

14. g 𝑆 𝑛 𝑡 − 1 , 𝑎 𝑛+1 𝑡 =
−1, 𝑆 𝑛 𝑡 − 1 > 0
𝑎𝑛𝑑 𝑎 𝑛+1 𝑡 ≤ 0
1, 𝑜𝑡ℎ𝑒𝑟𝑠
The main advantage of FVADM compared to previous models is that it describes the driver’s behavior under an urgent
case, where no collision occurs and no unrealistic deceleration appears.
Velocity Difference Separation Model (VDSM)
A detailed analysis of FVDM was investigated by Zhi-Peng and Yui-Cai in 2006 and was discovered that second term in
the right side of Eq. (2.1) gives no allowance for the effect of the inter-vehicle spacing independently of the relative
velocity. A model called velocity-difference-separation model (VDSM) which takes the separation between cars into
account was proposed and the dynamics equation becomes
an t = k Vopt Sn(t) − vn(t)
+ λΘ Sn(t) Sn(t) 1 + tanh C1 Sn t − l − C2
3
+ λΘ −Sn(t) Sn(t) 1 − tanh C1 Sn t − l − C2
3
(2.3)

15. VDSM has the capacity to predicting more realistically the dynamical evolution of congestion induced by a small
disturbance to normal traffic flow and accurately the delay time of car motion and kinematic wave speed at jam density.
A new car following model based on FVDM in which the leading vehicle acceleration is considered gets suggested by
Lijuan and Ning in 2010. With detailed study, it was observed that in FVDM simulation, all the vehicle accelerate until
the maximal velocity is reached. At the point of maximal velocity, repeated acceleration and deceleration was observed.
So Eq. (2.1) can then be modify to take into account the factor influencing the following car by adding the leading
vehicle acceleration. The dynamic equation of the system is obtained as
𝑎 𝑛 𝑡 = 𝑘 𝑉𝑜𝑝𝑡 𝑆 𝑛(𝑡) − 𝑣 𝑛(𝑡) + 𝜆 𝑆 𝑛 𝑡 + 𝛾𝑎 𝑛−1 𝑡 (2.4)
Where 𝛾 which is the sensitivity, expressing the response intensity of the follow vehicle to leading acceleration. They
proved that their new model has certain enlightenment significance for traffic control, and is useful for establishment of
Intelligent Transport Systems (ITS). Previous models used only one type of ITS information, either headway, velocity, 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.

16. Multiple Headway Velocity and Acceleration Difference (MHVAD)
Based on idea above, Li et al. (2011) proposed a new car-following model taking into account the effects of the
acceleration difference of the multiple preceding vehicles which affects the behavior of the following vehicle just as the
headway and the velocity difference, called multiple headway velocity and acceleration difference (MHVAD). It is
mathematical expressed as:
an t = k Vopt j=1
q
βjSn+j−1 t − vn t + λ j=1
q
ζjSn+j−1 t + γ j=1
q
ξjSn+j−1 t (2.5)
Taking q as the preceding vehicles and 𝛽𝑗, 𝜁𝑗, 𝜉𝑗 ∈ 𝑅, 𝑎𝑛𝑑 𝛽𝑗 ≥ 0, 𝜁𝑗 ≥ 0, 𝜉𝑗 ≥ 0 are different weighting value
coefficients, respectively. The βj, satisfies two conditions:
1. 𝛽𝑗 is a monotone decreasing function with 𝛽𝑗, 𝛽𝑗−1 , Because the effect of the preceding vehicle to the current car
reduces with the increase of the headway distance.
2. j=1
q
βj = 1, βj = 1 for q = 1 so as to 𝜁𝑗, 𝜉𝑗, and 𝛽𝑗 is defined as follows

17. 𝛽𝑗 =
𝑞−1
𝑞 𝑗 𝑓𝑜𝑟 𝑗 ≠ 𝑞
1
𝑞 𝑗−1 𝑓𝑜𝑟 𝑗 = 𝑞
j = (1, 2, …, q)
The optimal velocity function Vopt used here as form:
𝑉𝑜𝑝𝑡 𝑆 𝑛(𝑡) = tanh 𝑆 𝑛 𝑡 − ℎ 𝑐 + tanh ℎ 𝑐
𝑣 𝑚𝑎𝑥
2
(2.6)
Where vmax is the maximal speed 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
velocity function Eq. (2.3). Among them, Jing et al. (2011) introduced a new optimal velocity function and modified
the additional term of FVDM (Eq. (2.1)). In the first time, they proposed the modified full velocity

18. difference model (MFVDM I ) taking into account a new optimal velocity function proposed by (Helbing
and Tilch, 1998) Eq. (2.8):
𝑎 𝑛(t) = k 𝑉𝑜𝑝𝑡 𝑆 𝑛, 𝑣 𝑛 − 𝑣 𝑛 𝑡 + 𝜆 𝑆 𝑛 𝑡 (2.7)
𝑉𝑜𝑝𝑡 𝑆 𝑛, 𝑣 𝑛 = 𝑣 𝑛
0
1 − 𝑒
𝑆 𝑛−𝑆 𝑣 𝑛
𝑅 𝑛 (2.8)
where Rn is the range of the acceleration interaction and S(vn) is a certain velocity-dependent safe distance. And
the improved optimal velocity Vopt (Sn,vn) is a function of the vehicle distances and the velocity of the following
vehicle which must satisfy these three conditions:
1. Vopt (Sn,vn) is monotonically increasing to Sn and vn.
2. The larger values of Vopt(Sn,vn) will be beneficial to make FVDM fit with the field data better.
3. lim
𝑆 𝑛→+∞
𝑉𝑜𝑝𝑡 𝑆 𝑛, 𝑣 𝑛 ≅ 𝑣 𝑛
0
and lim
𝑣 𝑛→𝑣 𝑛
0
𝑉𝑜𝑝𝑡 𝑆 𝑛, 𝑣 𝑛 ≅ 𝑣 𝑛
0
where 𝑣 𝑛
0
is the desired velocity of the following
vehicle.

19. For above analysis, they proposed a new optimal velocity function satisfies the above three conditions defined as forms
𝑉𝑜𝑝𝑡 𝑆 𝑛, 𝑣 𝑛 = 𝑣 𝑛
0
tanh
𝑆 𝑛−𝑆 𝑣 𝑛
𝑅 𝑛
(2.9)
In second time, substituting the Eq. (2.9) into Eq. (2.7), and they get the second modified full velocity difference model
(MFVDM II). Finally, they introduced a new optimal velocity function (Eq. (2.9)) and modified the additional term of
Eq. (2.1) to get a new model called the improved full velocity difference model (IFVDM) defined as follow
𝑎 𝑛 𝑡 = k 𝑉𝑜𝑝𝑡 𝑆 𝑛, 𝑣 𝑛 − 𝑣 𝑛 𝑡 + 𝜕 𝑆 𝑛 𝑡 (3.0)
The additional term ∂ defined as a form: ∂ =
1− tanh
𝑆 𝑛−𝑆 𝑣 𝑛
𝑅 𝑛
𝜇 𝑛
Where 𝜇 𝑛 is the reaction time of the addition term.
The author (Jing et al., 2011) pointed out that the new model can perform more realistically in predicting the correct
delay time of vehicle motion and kinematic wave speed at jam den- sity, as well as predicting the dynamical evolution of
congestion induced by a small disturbance

20. CONCLUSSION
From above, optimal velocity model has successfully revealed the dynamical evolution and process of
traffic as it transited from uniform flow phase to traffic congestion in a simple way. This model create
room for some new car-following models which were successively put forward to describe the traffic
phenomenal more realistically. Some were extended by incorporating a new optimal velocity function or
introducing multiple information of headway, velocity difference, or acceleration difference. The research
to develop a strong car-following model which avoid the possibilities of collision and interpreted the
traffic flow in a real manner is feasible.

21. REFERENCES
1. M. Treiber and C. Thiemann, (2013). Traffic Flow Dynamics - Data, Models and Simulation, Springer
Heidelberg New York Dordrecht London.
2. J. Treiterer and J. Myers, (1974). The hysteresis phenomena in traffic flow. In D. Buckley, editor,
“Proceedings of the Sixth Symposium on Transportation and Traffic Flow Theory”, pp. 13—38. Elsevier
3. P. Hoogendoorn, (2002). Traffic Flow Theory and Simulation. Transportation and Traffic Engineering
Section Faculty of Civil Engineering and Geosciences Delft University of Technology.
4. L. Hajar, R. Khadija, and R. Moulay, (2016). A Review Analysis of Optimal Velocity Models,
Transportation Engineering, Periodica Polytechnica, Research Gate Publication. Pp. 123 - 131

22. THANK YOU