Review of Optimum speed model

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: macroscopic models and 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 = ……………… (1) where τ is the reaction time. Δ = (t) − (t), ………(1a) (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: = …………………………… (2) where λ is the sensitivity, ………………….(3)   And (t) = (t) − (t), ……(3a) is the space headway, (t) is the position of car n, a, b, 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 (t + τ ) has connection with the space headway Δ (t), and drivers adjust their speed during the reaction time to achieve the optimal velocity V (Δ (t)), which is determined by Δxn(t), i.e. (t + τ ) = V (Δ (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. = k[V (Δ (t)) − (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 depends on the distance headway of the nth vehicle. OVM is a time-continuous model whose acceleration function is of the form (s, v) , i.e., the speed difference exogenous variable is missing.  

11. The acceleration equation is given by: v = Optimal Velocity Model……1   (s, v, v) = 0 , or ] Steady state condition ……2 (s, v, 0) = 0 This equation describes the adaption of the actual speed v = to the optimal velocity (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 (s) is equivalent to the microscopic fundamental diagram (s) . It should obey the plausibility conditions but is arbitrary, otherwise. (s) ≥ 0, = 0, ……………….(3)  

13. The OV function originally proposed by Bando et al., = …….(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 , 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 , congested traffic by the time gap T in car-following mode under stationary conditions, and standing traffic by the minimum gap . we obtain = …..(5)  

15. OPTIMUM SPEED MODEL PROPERTIES 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). = k [V (Δ (t)) − (t)] + λΘ(−Δ (t)) Δ (t) …………(GFM)   = k [V (Δ (t)) − (t)] + λΔ (t) …………….(FVDM) Since the empirical accelerations and decelerations are usually limited to the range between −3 and +4 m/ , 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(Δ , Δ (t))……………………...(1) = k [V (Δ (t), Δ (t)) − (t)] ………..(2) For simplicity, v(Δ , Δ (t)) = (Δ (t) + αΔ (t)) …………(3) where, α = reaction coefficient to relative velocity, 0 < α < 1, so = k [ (Δ (t)) − (t)] + kα (Δ (t))……..(4) Taking λ= kα, equation (4) above becomes = k [ (Δ (t)) − (t)] + λ (Δ (t))………..(5) both λ and α are sensitivity.  

18. Optimal velocity forecast model optimal velocity forecast model (for short, OVFM) is presented as follows: (t) = α[v( (t)) − (t)] + k + γ[v( (t + τ)) − v( (t))] …………….(1) where: (t) is the position of car n at time t; (t) = (t) − (t) ………(a) and (t) = (t) − (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( (t + τ)) − v( (t))]…………(2) is the optimal velocity difference term, γ is the response forecast coefficient of the optimal velocity difference between, v( (t + τ)) and v( (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) = + (x − ) − ) where = 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.