Effect of driver’s anticipation in a new two-lane lattice model with the consideration of optimal current difference

Abstract

In this paper, a new lattice hydrodynamic traffic flow model is proposed by considering the driver’s anticipation effect (DAE) in sensing optimal current difference (OCD) for two-lane system. The effect of anticipation parameter on the stability of traffic flow is examined through linear stability analysis and shown that it can significantly enlarge the stability region on the phase diagram. Nonlinear analysis is conducted, and mKdV equation is derived to describe propagation behavior of a density wave near the critical point. The driver’s physical delay in sensing optimal current difference effect is also investigated and found that it has different effect on two-lane traffic based on whether lane changing is allowed or not. Simulation results are found in good agreement with the theoretical findings, which confirms that traffic jam can be suppressed efficiently by considering the DAEOCD effect in a two-lane traffic system.

Appendix 1

In this appendix, we give the expansion of each terms in Eq. (8) using Eqs. (17) and (18) to the fifth order of $\epsilon$.

$$\begin{aligned} \rho _j(t+\tau )= & {} \rho _c +\epsilon R+\epsilon ^2 (b\tau )\partial _X R \nonumber \\&\quad +~\frac{\epsilon ^3}{2}(b\tau )^2 \partial ^{2}_X R+\frac{\epsilon ^4}{6}(b\tau )^3 \partial ^{3}_X R+\epsilon ^4\tau \partial _T R \nonumber \\&\quad +~ \frac{\epsilon ^5}{24}(b\tau )^4 \partial ^{4}_X R +\epsilon ^{5}b\tau ^2 \partial _T \partial _X R. \end{aligned}$$

(27)

$$\begin{aligned} \rho _j(t+2\tau )= & {} \rho _c +\epsilon R+\epsilon ^2 (2b\tau )\partial _X R \nonumber \\&\quad +~\frac{\epsilon ^3}{2}(2b\tau )^2 \partial ^{2}_X R \nonumber \\&\quad +~\frac{\epsilon ^4}{6}(2b\tau )^3 \partial ^{3}_X R+\epsilon ^4(2\tau ) \partial _T R\nonumber \\&\quad + \,\frac{\epsilon ^5}{24}(2b\tau )^4 \partial ^{4}_X R +\epsilon ^{5}(4b\tau ^2) \partial _T \partial _X R. \end{aligned}$$

(28)

$$\begin{aligned} \rho _{j+1}(t)= & {} \rho _c+\epsilon R+\epsilon ^2 \partial _X R \nonumber \\&\quad +~\frac{\epsilon ^3}{2} \partial ^2_X R+\frac{\epsilon ^4}{6}\partial ^3_X R+\frac{\epsilon ^5}{24}\partial ^4_X R. \end{aligned}$$

(29)

$$\begin{aligned}&\rho _{j+1}(t+\tau )-2\rho _{j}(t+\tau )+\rho _{j-1}(t+\tau ) \nonumber \\= & {} \epsilon ^3\partial ^2_X R +~\epsilon ^4(b\tau )\partial ^3_X R\nonumber \\&\qquad +~\frac{\epsilon ^5}{12}(1+6b^2\tau ^2)\partial ^4_X R. \end{aligned}$$

(30)

The expansion of optimal velocity function at the turning point is

$$\begin{aligned} V(\rho _j)= & {} V(\rho _c)+V'(\rho _c)(\rho _j-\rho _c) \nonumber \\&\quad +~\frac{V'''(\rho _c)}{6}(\rho _j-\rho _c)^3. \end{aligned}$$

(31)

$$\begin{aligned} V(\rho _{j+1})= & {} V(\rho _c)+V'(\rho _c)(\rho _{j+1}-\rho _c) \nonumber \\&\quad +~\frac{V'''(\rho _c)}{6}(\rho _{j+1}-\rho _c)^3. \end{aligned}$$

(32)

Using Eqs. (31) and (32), we get

$$\begin{aligned}&V(\rho _{j+1})-V(\rho _j)\nonumber \\&\quad = V'(\rho _c)\left[ \epsilon ^2\partial _XR +\frac{\epsilon ^3}{2}\partial _X^2R+\frac{\epsilon ^4}{6}\partial _X^3R+\frac{\epsilon ^5}{24}\partial _X^4R\right] \nonumber \\&\qquad \quad +\frac{V'''(\rho _c)}{6}\left[ \epsilon ^4\partial _XR^3+\frac{\epsilon ^5}{2}\partial ^2_XR^3\right] . \end{aligned}$$

(33)

Some other important expansions are also computed using Eqs. (27)–(33) and are given as

$$\begin{aligned}&V'(\rho _{j+2}(t))\tilde{{\varDelta }}\rho _{j+2}(t)-2V'(\rho _{j+1}(t))\tilde{{\varDelta }}\rho _{j+1}(t) \nonumber \\&\quad \quad +~V'(\rho _{j}(t))\tilde{{\varDelta }}\rho _{j}(t)=\epsilon ^3 (b\tau )V'(\rho _c)\partial ^2_X R \nonumber \\&\quad \quad +~\epsilon ^4 (b\tau )V'(\rho _c)\partial ^3_X R \nonumber \\&\quad \quad +~\frac{\epsilon ^5 }{2} (b^2\tau ^2+2b\tau )V'(\rho _c)\partial ^4_X R. \end{aligned}$$

(34)

$$\begin{aligned}&V(\rho _{j+2})-2V(\rho _{j+1})+V(\rho _j)\nonumber \\&\quad =\epsilon ^3 V'(\rho _c) \partial ^2_X R +\epsilon ^4 V'(\rho _c)\partial ^3_X R \nonumber \\&\quad \quad +~\frac{7\epsilon ^5 }{12}V'(\rho _c)\partial ^4_X R. \end{aligned}$$

(35)

$$\begin{aligned}&V'(\rho _{j+1}(t))\tilde{{\varDelta }}\rho _{j+1}(t) -V'(\rho _{j}(t))\tilde{{\varDelta }}\rho _{j}(t)\nonumber \\&\quad =\frac{\epsilon ^3}{2} (b\tau )V'(\rho _c)\partial ^2_X R\nonumber \\&\quad \quad +~\frac{\epsilon ^4}{2} (B^2\tau ^2+b\tau )V'(\rho _c)\partial ^3_X R+\epsilon ^5 V'(\rho _c)\nonumber \\&\quad \quad \times ~\left[ \tau \partial _T \partial _X R +\frac{(4b\tau +6b^2\tau ^2+4b^3\tau ^3)}{24}\partial ^4_X R\right] .\nonumber \\ \end{aligned}$$

(36)

By inserting (27), (28), (30), (33), (34), (35) and (36) into Eq. (8), we obtain Eq. (19).