Jam and Fundamental Diagram in Traffic Flow on Sag and Hill

1. Jam and Fundamental Diagram in Traffic Flow on Sag and Hill K.Komada S.Masukura T.Nagatani Shizuoka Univ.Japan

2. Purpose of Study • Proposal of traffic model including the gravitational force 　　－We extend the optimal velocity model to study the jamming transition induced by the gravitational force. • Fundamental diagrams for the traffic flow on sag and hill 　　－We study the flow, traffic states ,and jamming transitions induced by sag and hill. • Jam induced by sag 　　－We clarify the relationship between densities before and after the jam from the theoretical current curves.

3. Traffic model Equation of motion on uphill About → 1 for → ∞ sensitivity → 0 for → 0 depends on the gradient of Extended Optimal velocity Function We extend the OV model and obtain the following

4. ① ① ③ ② ①OV function on normal section ② Extended OV function on uphill section ③Extended OV function on downhill section

5. Simulation method • Single lane • The periodic boundary condition • Forth-order Runge-Kutta method Values of parameters • LN1=LD1=LU1=LN2=L/4 • Time interval isΔｔ＝１/１２８ • Vf,max=2.0,xｃ＝4.0 • Number of cars N=200 • Length of road L=N×Δx

6. Fundamental diagram（ Xc=Xdown,b=Xup,b） Sensitivity:a=3.0>ac=2.0(critical value) Sensitivity:a=1.5<ac=2.0(critical value) Traffic jam induced by sag Velocity profile（ρ=0.17） Traffic jam induced by sag ＋oscillating jam at low sensitivity High sensitivity⇒3 traffic states Low sensitivity ⇒5 traffic states Velocity profile （ ρ=0.19）

7. Relationship between headway profile and theoretical current（Xｃ=Xup,b=Xdown,b） Headway profile(ρ=0.16) Steady state： Headways are the same. Velocities are Optimal Velocity. Theoretical current （ in the case of no jam at high sensitivity） Headway profile(ρ=0.20)

8. Fundamental diagram（ Xc=Xdown,b≠Xup,b） Velocity profile（ρ=0.16） 3 traffic states (3) of case2 is not consistent with that of case1 but (1) and (2) case 2 agree with those of case1. (1)Free traffic (2)Traffic with saturated current (3) Congested traffic Headway profile（ρ=0.16） xc=xup,b≠xdown,b：「the different case」（case1）xc=xup,b=xdown,b ：「the same case」（case2）

9. Relationship between headway profile and theoretical current （ Xc=Xdown,b≠Xup,b） Headway profile(ρ=0.16) In the case of Xc=Xdown,b≠Xup,b The length of jam shorten. Headway get narrow. Headway profile(ρ=0.20)

10. The dependence of traffic flow on the gradient Velocity profile(ρ=0.20) As the gradient is high, the maximum velocity become lower and higher on up- and down-hills respectively. The region of saturated flow extend. The maximum current is lower. Headway profile(ρ=0.20)

11. Fundamental diagram of traffic flow with two uphills Headway profile(ρ=0.20) The traffic jam occurs just before the highest gradient. Headway profile(ρ=0.20)

12. Summary ●We have extended the optimal velocity model to take into account the gravitational force as an external force. ● We have clarified the traffic behavior for traffic flow on a highway with gradients ●We have showed where, when, and how the traffic jams occur on highway with gradients. ● We have studied the relationship between densities before and after the jam from the theoretical analysis.