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

Although the model in Eq. (11) is much simpler than the models in the literature (Tang and Huang, 2004, 2005; Huang et al., 2006), the inclusion of an acceleration equation that incorporates the effect of lane-changing on a two-lane highway also means that it can be used to analytically study the dynamic properties that are caused by lane-changing. Eq. (11) reduces to the SG model of Jiang et al. (2001a, 2002a) without lane-changing. To study the wave properties that result from lane-changing, we first examine the eigenvalues of matrix A in Eq. (11) by setting

det([lambda]I - A) = 0, (12)

where I is a 2 x 2 identity matrix. Eq. (12) can then be rewritten as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

The eigenvalues are

[[lambda].sub.1] = v - c([rho]), [[lambda].sub.2] = v, (14)

and the corresponding eigenvectors are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Eq. (11) is thus a strictly hyperbolic system, the characteristic speeds of which are no greater than the traffic speed. This ensures the anisotropy of the traffic flow, and addresses the criticism of Daganzo (1995). To illustrate, we use the following Riemann problem with the initial values

[rho](x,0) = [[rho].sub.jam]H(x), t = 0, (16)

v(x,0) = 0, t = 0, (17)

where H{x) is a Heaviside function with H{x) = 0, [for all]x < 0 and H(x) = 1,[for all]x[greater than or equal to]0, and [[rho].sub.jam] is the jam density. Under these conditions, the correct solution is that all of the vehicles remain stationary, that is, dv/d/ = 0. Substituting dv/dt = 0 into Eq. (10) therefore gives us

v(x, t) = [v.sub.e] ([rho] [rho][epsilon]([rho])) = [v.sub.e]([rho]) [greater than or equal to] 0. (18)

As all of the vehicles are at a standstill, no lane-changing occurs, and thus [epsilon]([rho]) = 0. Eq. (18) shows that no backward movement appears in our model, even though lane-changing is allowed.

3. Linear Stability Analysis

To study the effect of lane-changing on the stability of traffic flow, we denote [rho]* and v* as the steady-state solution to Eq. (11) and let [rho] = [[rho].sub.*] [xi], and v = [v.sub.*] [eta] be the perturbed solution, where [xi](x,t) and [eta](x, t) represent small smooth deviations from the steady-state solution. Substituting [rho] = [[rho].sub.*] [xi] and v = [v.sub.*] [eta] into Eq. (11), taking the Taylor's series expansion at point ([[rho].sub.*],[v.sub.*]) , and neglecting the higher-order terms then gives us the following equations:

[partial derivative]/[partial derivative]t [v.sub.*] [partial derivative][xi]/[partial derivative]x [[rho].sub.*] [partial derivative][eta]/[partial derivative]x = 0, (19)

[partial derivative][eta]/[partial derivative]t ([v.sub.*] - c([[rho].sub.*])) [partial derivative][eta]/[partial derivative]x = A[xi] - [eta]/[tau], (20)

where A = [V'.sub.e]([[rho].sub.*](1 [epsilon]([[rho].sub.*]))(l [epsilon]([[rho].sub.*]) [[rho].sub.*][epsilon]'([[rho].sub.*])). Eliminating r| from Eqs (19) & (20) gives

([[partial derivative].sub.t] [C.sub.0] [[partial derivative].sub.x]) [tau] (([[partial derivative].sub.t] [C.sub.1] [[partial derivative].sub.t]) ([[partial derivative].sub.t] [C.sub.2] [[partial derivative].sub.t]))[xi] = 0, (21)