Modeling and analyses of driver’s characteristics in a traffic system with passing

Abstract

This paper investigates the effect of aggressive or timid characteristics of driver’s behavior with passing by means of lattice hydrodynamic traffic flow model. The effect of driver’s characteristic on the stability of traffic flow is examined through linear stability analysis. It is shown that for both the cases of passing or without passing the stability region significantly enlarges (reduces) as the proportion of aggressive (timid) drivers increases. To describe the propagation behavior of a density wave near the critical point, nonlinear analysis is conducted and mKdV equation representing kink–antikink soliton is derived. It is observed that jamming transition occurs between uniform flow and kink jam phase with increase in aggressive driver’s characteristics for smaller values of passing. When passing constant is greater than a critical value, jamming transitions occur among uniform traffic flow and kink-Bando traffic wave through chaotic phase. Numerical simulation is carried out to validate the theoretical findings which confirm that traffic jam can be suppressed efficiently by considering the driver’s characteristics in a single-lane traffic system with or without passing.

Appendix

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

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

(32)

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

(33)

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

(34)

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

(35)

The expansion of optimal velocity function at the turning point is

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

(36)

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

(37)

Using Eqs. (36) and (37), we get

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

(38)

Some other important expansions are also computed using Eqs. (32)–(38) and are given as

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

(39)

By inserting (32), (33), (35), (38), and (39) into Eq. (8), we obtain Eq. (19).