review of optimal speed models assignment. submitted to Engr. Prof. H. Alhassan.

1. REVIEW OF OPTIMUM SPEED
TRAFFIC FLOW MODEL
MASTERS CLASS ASSIGNMENT(HIGHWAY AND TRANSPORT OPTION, 2016/2017)
DEPARTMENT OF CIVIL ENGINEERING, BAYERO
UNIVERSITY KANO. NIGERIA
Ibrahim Tanko Abe
SPS/16/MCE/00028
SUBMITTED TO
Engr. Prof. H.M. Alhassan

2. INTRODUCTION
Previous years have experienced a considerable
development in traffic flow theory. A large number
of traffic models have been recommended by
scientists. Generally speaking, there are two
types of traffic models:
1.macroscopic models and
2.microscopic models.

3. TYPES OF TRAFFIC MODELS
 Macroscopic models regard the whole traffic
flow as a flow of continuous medium based on a
continuum approach.
 Microscopic models, including the car-following
models and cellular automata models, pay
attention to each individual vehicle.

4. CAR FOLLOWING MODELS

5. CAR FOLLOWING MODELS
 In this paper, we focus on the car-following models and
optimum velocity model(OVM) in particular. The car-following
models describe the motion of vehicles following each other
on a single lane, the earliest model was proposed by
Reuschel and Pipes
dt
𝑑𝑣 𝑛
(t) = 𝜏
1
𝛥𝑣 𝑛 ……………… (1)
where τ is the reaction time.
Δ𝑣 𝑛 = 𝑣 𝑛+1(t) − 𝑣 𝑛(t), ………(1a)
𝑣 𝑛(t) is the speed of the following car n at time t, and n + 1 is
the leading car.

6. One can see that when the speed of the following car is higher
than that of the leading car, the following car will slow down,
vice versa. Later, Chandler found drivers always adjust their
speed through the speed difference with the leading car during
the reaction time, so he proposed another model:
dt
𝑑𝑣 𝑛
(t + τ) = 𝜆𝛥𝑣 𝑛 …………………………… (2)
where λ is the sensitivity,
𝜆 =
𝑎: 𝛥𝑥 𝑛(𝑡) < 𝑥 𝑐
𝑏: 𝛥𝑥 𝑛(𝑡) ≥ 𝑥 𝑐
………………….(3)
And 𝛥𝑥 𝑛(t) = 𝑥 𝑛+1(t) − 𝑥 𝑛(t), ……(3a) is the space headway,
𝑥 𝑛(t) is the position of car n,
a, b, 𝑥 𝑐 are constants.

7. HERMAN
Moreover, Herman found drivers always like to pay attention
to more vehicles ahead, so he proposed a model with
considering the next nearest vehicle ahead.
However, aforementioned models mainly considered the
influence of the speed of the car ahead to the following car.
Their defects are obvious, such as they cannot describe the
acceleration of a single vehicle correctly.

8. NEWEL
In 1961, Newell proposed a different model. He thought
𝑣 𝑛(t + τ) has connection with the space headway Δ𝑥 𝑛(t),
and drivers adjust their speed during the reaction time to
achieve the optimal velocity V (Δ𝑥 𝑛(t)), which is
determined by Δxn(t), i.e.
𝑣 𝑛(t + τ) = V (Δ𝑥 𝑛(t)) ………………….. (4)
Nevertheless, this model is unsuited to describe the
behavior of the acceleration circumstance when the traffic
light just turns green.

9. In 1995, Bando et al presented the optimal velocity model
(OVM), which is also based on the idea that each vehicle
has an optimal velocity, and the optimal velocity also
depends on the following distance with the preceding
vehicle.
dt
𝑑𝑣 𝑛
(t) = k[V (Δ𝑥 𝑛(t)) − 𝑣 𝑛(t)], ……………………..(5)
where k is a sensitivity constant and
V is the optimal velocity.
BANDO et al

10. OPTIMAL SPEED MODEL
The concept of this model is that each driver tries to
achieve an optimal velocity based on the distance to the
preceding vehicle and the speed difference between the
vehicles. The formulation is based on the assumption that
the desired speed 𝑣 𝑛 𝑑𝑒𝑠𝑖𝑟𝑒𝑑
depends on the distance
headway of the nth vehicle.
OVM is a time-continuous model whose acceleration
function is of the form 𝑎mic(s, v), i.e., the speed difference
exogenous variable is missing.

11. The acceleration equation is given by:
v =
𝒗 𝒐𝒑𝒕 𝒔 −𝒗
𝝀
Optimal Velocity Model……1
amic(s, v, v) = 0,
or ] Steady state condition ……2
ā 𝑚𝑖𝑐(s, v, 0) = 0
This equation describes the adaption of the actual speed v
= 𝑣 𝛼 to the optimal velocity 𝑣 𝑜𝑝𝑡(s) on a time scale given by
the adaptation time τ.

12. Comparing the acceleration equation (1) with the steady-
state condition (2) it becomes evident that the optimal
velocity (OV) function 𝑣 𝑜𝑝𝑡(s) is equivalent to the
microscopic fundamental diagram 𝑣𝑒(s). It should obey the
plausibility conditions but is arbitrary, otherwise.
ὐ 𝑜𝑝𝑡(s) ≥ 0,
𝑣 𝑜𝑝𝑡(0) = 0,
lim
𝑠→∞
𝑣 𝑜𝑝𝑡 𝑠 = 𝑣0……………….(3)

13. The OV function originally proposed by Bando et al.,
𝑣 𝑜𝑝𝑡 𝑠 = 𝑣0
tanh
𝑠
𝛥𝑠
− 𝛽 +𝑡𝑎𝑛ℎ𝛽
1+𝑡𝑎𝑛ℎ𝛽
…….(4)
uses a hyperbolic tangent. 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.

14. A more intuitive 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. we obtain
𝑣 𝑜𝑝𝑡 𝑠 = 0, min 𝑣0,
𝑠− 𝑠0
𝑇
…..(5)

15. OPTIMUM SPEED MODEL PROPERTIES
1. The OVM could reproduce many properties of real traffic
flow, such as :
a. the instability of traffic flow,
b. the evolution of traffic congestion, and
c. the formation of stop-and-go waves,
2. On a quantitative level, the OVM results are unrealistic.
3. On a qualitative level, the simulation outcome has a
strong dependency on the fine tuning of the model
parameters, i.e., the OVM is not robust

16. Full velocity difference model (FVDM).
Jiang et al. found the generalized force model (GFM) by Helbing and Tilch
(2008) is poor in anticipating the delay time of car motion and kinematic wave
speed, so they improved the GFM and proposed the Full velocity difference
model (FVDM).
𝑑𝑡
𝑑𝑣 𝑛
(𝑡) = k [V (Δ𝑥 𝑛(t)) − 𝑣 𝑛 (t)] + λΘ(−Δ𝑣 𝑛(t)) Δ𝑣 𝑛(t) …………(GFM)
𝑑𝑡
𝑑𝑣 𝑛
(𝑡) = k [V (Δ𝑥 𝑛(t)) − 𝑣 𝑛(t)] + λΔ𝑣 𝑛(t) …………….(FVDM)
Since the empirical accelerations and decelerations are usually limited to the
range between −3 and +4 m/𝑠2, Simulation results show that there also exists
unrealistic deceleration in OVM and FVDM. Furthermore, all of them could not
avoid collisions in urgent braking cases.

17. Comprehensive optimal velocity model (COVM).
To overcome these shortcomings abovementioned, they proposed a new car-
following model, whose optimal velocity function not only depends on the following
distance of the preceding vehicle, but also depends on the velocity difference with the
preceding vehicle
𝑣 𝑜𝑝𝑡 = v(Δ𝑥 𝑛(𝑡), Δ𝑣 𝑛(t))……………………...(1)
𝑑𝑡
𝑑𝑣 𝑛
(𝑡) = k [V (Δ𝑥 𝑛(t), Δ𝑣 𝑛(t)) − 𝑣 𝑛(t)] ………..(2)
For simplicity,
v(Δ𝑥 𝑛(𝑡), Δ𝑣 𝑛(t)) = 𝑣1(Δ𝑥 𝑛(t) + αΔ𝑣2(t)) …………(3)
where, α = reaction coefficient to relative velocity, 0 < α < 1, so
𝑑𝑡
𝑑𝑣 𝑛
(𝑡) = k [𝑣1(Δ𝑥 𝑛(t)) − 𝑣 𝑛(t)] + kα𝑣2(Δ𝑣 𝑛(t))……..(4)
Taking λ= kα, equation (4) above becomes
𝑑𝑡
𝑑𝑣 𝑛
(𝑡) = k [𝑣1(Δ𝑥 𝑛(t)) − 𝑣 𝑛(t)] + λ𝑣2(Δ𝑣 𝑛(t))………..(5)
both λ and α are sensitivity.

18. Optimal velocity forecast model
optimal velocity forecast model (for short, OVFM) is presented as follows:
𝑥 𝑛 (t) = α[v(𝛥𝑥 𝑛(t)) − 𝑣 𝑛(t)] + k𝛥𝑣 𝑛 + γ[v(𝛥𝑥 𝑛(t + τ)) − v(𝛥𝑥 𝑛(t))] …………….(1)
where:
𝑥 𝑛 (t) is the position of car n at time t;
𝛥𝑥 𝑛 (t) = 𝑥 𝑛−1 (t) − 𝑥 𝑛 (t) ………(a) and 𝛥𝑣 𝑛 (t) = 𝑣 𝑛−1 (t) − 𝑣 𝑛 (t) ………(b) are the headway and
the velocity difference between the preceding vehicle n+1 and the following vehicle n, respectively;
α is the sensitivity of a driver;
V is the optimal velocity function (OVF);
γ[v(𝛥𝑥 𝑛(t + τ)) − v(𝛥𝑥 𝑛(t))]…………(2) is the optimal velocity difference term,
γ is the response forecast coefficient of the optimal velocity difference between,
v(𝛥𝑥 𝑛(t + τ)) and v(𝛥𝑥 𝑛(t)) ,
τ is the forecast time.
The new model conforms to the FVDM if γ=0. The optimal velocity function is adopted calibrated with
observed data by Helbing below:
v(x) = 𝑣1+ 𝑣2tanh(𝐶1(x −𝑙 𝑐) − 𝐶2)
where 𝑙 𝑐 = length of vehicles = 5m

19. Conclusion
The model considers cars alone
which cannot be easily adopted in
Nigeria.

20. Recommendation
 There is need for inclusion of tricycles and the behaviors
of Nigerian drivers.

21. References
 Bando. M, Hasebe. K, Nakanishi. K, Nakayama. A, (1998)
Analysis of optimal velocity model with explicit delay. Phys. Rev. E vol.58, No.5, pp. 5429- 5435.
 Ez-Zahraouy. H, Benrihane. Z, Benyoussef. A, (2004)
The Optimal Velocity Traffic Flow Models With Open Boundary. M. J. condensed matter vol. 5, No. 2, .
….pp. 140-146
 Jun-Fang. T, Bin. J, Xing-Gang. L, (2010)
A New Car Following Model: Comprehensive Optimal Velocity Model. Phys. Vol. 55, No. 6, pp. 1119–
…1126.
 Nakayama. A et al (2015)
Scaling from Circuit Experiment to Real Traffic based on Optimal Velocity Model. TGF15.
 Yang. D, Jin. P, Pu. Y, Ran. B, (2014)
Stability analysis of the mixed traffic flow of cars and trucks using heterogeneous optimal velocity
…car-following model. Phys. A 395, pp. 371–383.