Macroscopic modeling of lane-changing for two-lane traffic flow

Journal of Advanced Transportation, Fall, 2009 by Tie-Qiao Tang, S.C. Wong, Hai-Jun Huang, Peng Zhang

[rho](x,0) = [[rho].sub.0] [DELTA][rho]{[cosh.sup.-2](160/L(x - 5L/16)) - 1/4 [cosh.sup.-2] (40/L(x - 11L/32))}, (33)

where the first term is the initial density, the second term is the disturbance, and L is the length of the road section. We assume that L = 32.2 km, and that the following periodic boundary conditions apply.

[rho](0,t) = [rho](L,t), v(0,t) = v(L,t). (34)

We use the equilibrium speed-density function (32), and assume that the initial speed is v(x,0) = [v.sub.e]([rho](x,0) [rho](x,0)[epsilon]([rho](x,0))) and that [DELTA][rho] = 0.01. The space discretization interval is 100 m, and the time interval is 1 s. The other parameters are identical to those that are used in Section 4.1.

In principle, we can obtain the unstable region of traffic flow analytically by substituting these parameters into Eq. (22). However, it is very difficult, if not impossible, to obtain the exact region, because the consideration of lane-changing makes condition (22) very complicated. We therefore determine the unstable region by simulation, where the unstable region is 0.037 < p < 0.067 when lane-changing is considered and is 0.04 < p < 0.077 without lane-changing (Jiang et al., 2002a). This illustrates that lane-changing reduces the size of the stable region when traffic is light and increases the size of the stable region when traffic is heavy. These simulation results are consistent with the analytical results that are obtained in Section 3.

The evolution of the small perturbation is shown in Fig. 4, from which the following observations can be made.

(a) When traffic is light or heavy, that is, when the initial density is less than the lower critical density value or greater than the upper critical density value, the small perturbation is dissipated without any amplification in magnitude (see Figs 4 (a) and (e)). When the initial density is in the unstable region, the small perturbation is amplified and eventually leads to traffic instability (see Figs 4 (b)-(d)). These results are similar to those obtained by Jiang et al. (2002a), which shows that our model effectively reproduces the evolution of a small perturbation and the properties of non-equilibrium flow.

(b) The analytical results in Section 3 show that lane-changing has a great effect on the propagation speeds of the second-order and first-order waves when the traffic flow is unstable, but has little effect on the propagation speeds of these waves when the traffic flow is stable. This implies that lane-changing will easily produce more vibrant stop-and-go waves than has been posited in the literature (Jiang et al., 2002a) when the initial density falls within the unstable region. This in turn demonstrates that lane-changing worsens traffic congestion when traffic is unstable (shown in Fig. 4 (b)-(d)), but has little effect on traffic conditions when the initial density is within the stable region (shown in Figs 4 (a) and (e)).

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

We further investigate the relationship between the actual flow rate and the lane-changing rate in the presence of a small perturbation. Figs 5 (a)-(e) and Table 2 show the time-space evolution of the actual flow rate and lane-changing rate with the same initial values, where q and [q.sub.c] denote the actual flow rate and lane-changing rate, respectively, if lane-changing is allowed, and [q.sub.p] denotes the potential flow rate (the flow rate that would have occurred had lane-changing not been allowed). Indices i and j in