An improved lattice hydrodynamic model considering the “backward looking” effect and the traffic interruption probability

Abstract

By incorporating the “backward looking” effect and traffic interruption probability, we put forward an improved lattice model. Applying linear stability analysis, linear stability criterion is derived. The mKdV equation is deduced through nonlinear theory, which demonstrates that the solution of mKdV equation can describe traffic congestion. Furthermore, numerical simulation shows that the two factors can enhance traffic flow stability in the driving process.

1 Introduction

Recently, due to the serious traffic problems with the growing number of vehicles on the road, traffic jam problem has increasingly captured the attention of scholars with different backgrounds. Consequently, plenty of traffic flow models have been constructed [1,2,3,4,5,6,7] to solve the traffic problems, including car-following models [8,9,10,11,12,13,14,15,16], cellular automation models [17,18,19,20,21], macro-traffic models [22,23,24,25,26] and lattice hydrodynamic models [27, 28]. Nagatani [29] firstly developed a lattice hydrodynamic model. What’s more, the mKdV equation is derived by Nagatani’s original lattice hydrodynamic model. Later, an increasing number of novel lattice hydrodynamic models [30,31,32,33,34,35,36,37,38,39,40,41,42,43] have been proposed by taking the other factors into consideration such as “backward looking” effect, driver’s memory during a period of time, driver’s desired average flux and traffic interruption probability.

In recent years, many scholars have taken into account the effect of the traffic interruption probability. Tang et al. [44, 45] considered this factor in traffic flow models. Subsequently, by considering the factor, Redhu and Gupta [46] derived a novel lattice hydrodynamic model. Furthermore, Peng et al. [47] and Sun et al. [48] researched this factor in lattice hydrodynamic model. However, the “backward looking” effect wasn’t taken into consideration in these models.

Fig. 1

Phase diagram in parameter space $(\rho , a)$

In this paper, the extended lattice hydrodynamic model is formulated by incorporating the “backward looking” effect and the influence of traffic interruption probability. The new model will be presented in the following section. The linear and nonlinear analyses of the new model are carried out in Sects. 2 and 3, respectively. Section 4 explores numerical simulations, and Sect. 5 draws conclusions.

2 The extended lattice hydrodynamic model

On the basis of original lattice hydrodynamic model, an extended model is put forward by incorporating the “backward looking” effect and traffic interruption probability as follows:

$$\begin{aligned}&\rho _j (t+\tau )v_j (t+\tau )\nonumber \\&\quad =\rho _o V_\mathrm{F} (\rho _{j+1} (t))+\lambda _1 p_{j+1} \cdot H(\bar{{\rho }}-\rho _{j-1} (t))\nonumber \\&\qquad \times H(\rho _{j-1} (t)-\rho _c )\cdot \rho _o V_\mathrm{B} (\rho _{j-1} (t)) \nonumber \\&\qquad +\,(1-p_{j+1} )\lambda _2 (\rho _{j+1} v_{j+1} -\rho _j v_j ), \end{aligned}$$

(1)

$$\begin{aligned}&\partial t\rho _j +\rho _o (\rho _j v_j -\rho _{j-1} v_{j-1} )=0, \end{aligned}$$

(2)

where $H\left( \cdot \right)$ represents the Heaviside function, $\bar{{\rho }}$ is approaching to 1 and $\rho _C$ indicates safety density. If $\bar{{\rho }}$ is less than the density, the “backward looking” influence will not play its role. Furthermore, as the safety density is greater than the density, the “backward looking” influence will also not play its role. $\tau$ represents the delay time and $a=\frac{1}{\tau }$ is the sensitivity. $\lambda _1$ and $\lambda _2$ are two reactive coefficients and $p_{j+1}$ represents the interruption of lattice $j+1$.

Table 1 Coefficients $k_i$ of the model

Table 2 Coefficients $g_i$ of the model

Here the coefficients $g_i$ are given in Table 2.

Fig. 2

Phase diagram of the model according to different values of parameter p

For convenience, $p_{j+1}$ is chosen as a constant p. The “forward looking” optimal velocity function is chosen as

$$\begin{aligned} V_\mathrm{F} (\rho _{j+1})= & {} \tanh \left( {\frac{2}{\rho _o }-\frac{\rho _{j+1} (t)}{\rho _o^2 }-\frac{1}{\rho _c }} \right) \nonumber \\&+\tanh \left( {\frac{1}{\rho _c }} \right) , \end{aligned}$$

(3)

and the “backward looking” optimal velocity function is as follows:

$$\begin{aligned} V_\mathrm{B} (\rho _{j-1} )= & {} r\left[ -\tanh \left( {\frac{2}{\rho _o }-\frac{\rho _{j+1} (t)}{\rho _o^2 }-\frac{1}{\rho _c }} \right) \right. \nonumber \\&\left. +\,\tanh \left( {\frac{1}{\rho _c }} \right) \right] , \end{aligned}$$

(4)

where $\rho _o$ represents the initial density and r represents a positive constant. We set $r=0.15,0.25,0.4$, respectively. Eliminating speed v in Eqs. (1) and (2), we will derive

$$\begin{aligned}&\rho _j (t+2\tau )-\rho _j (t+\tau )+\tau \rho _o^2 \nonumber \\&\quad \left[ V_\mathrm{F} (\rho _{j+1} (t))-V_\mathrm{F} (\rho _j (t))+\lambda _1 p(V_\mathrm{B} (\rho _{j-1} (t))\right. \nonumber \\&\qquad \left. -VB(\rho _{j-2} (t))) \right] \nonumber \\&\qquad +\,(1-p)\lambda _2 \left[ \rho _{j+1} (t)-\rho _{j+1} (t+\tau )\right. \nonumber \\&\left. \qquad +\,\rho _j (t+\tau )\rho _j (t) \right] . \end{aligned}$$

(5)

We investigate the BL-LH model with the linear stability analysis. The steady-state solution for Eqs. (1) and (2) is given as:

$$\begin{aligned} \rho _j (t)= & {} \rho _o , \end{aligned}$$

(6)

$$\begin{aligned} v_j (t)= & {} V_\mathrm{F} (\rho )+V_\mathrm{B} (\rho ), \end{aligned}$$

(7)

$y_j$ is a small perturbation of traffic flow from the steady-state flow as

$$\begin{aligned} \rho _j (t)=\rho _o +y_j (t). \end{aligned}$$

(8)

Fig. 3

Headway profile at $t=10{,}300$ under the different value of p

Fig. 4

Phase diagram of the model according to different values of parameter r

Fig. 5

Headway profile at $t=10{,}300$ under the different value of r

Inserting Eq. (6) into (5), we will deduced

$$\begin{aligned}&y_j (t+2\tau )-y_j (t+\tau )+\tau \rho _o^2 \nonumber \\&\quad \left[ V_\mathrm{F} (\rho _o)'(y_{j+1} (t)-y_j (t))\right. \nonumber \\&\quad \left. +\,\lambda _1 pV_\mathrm{B} (\rho _o)'(y_{j-1} (t)-y_{j-2} (t)) \right] \nonumber \\&\quad +\,(1-p)\lambda _2 \left[ y_{j+1} (t)-y_{j+1} (t+\tau )\right. \nonumber \\&\quad \left. +\,y_j (t+\tau )-y_j (t) \right] . \end{aligned}$$

(9)

Expanding $y_j =\exp (ikj+zt)$, it reads:

$$\begin{aligned}&e^{2z\tau }-e^{z\tau }+\tau \rho _o^2 \left[ V_\mathrm{F} (\rho _o)'(e^{ik}-1)\right. \nonumber \\&\quad \left. +\,\lambda _1 pV_\mathrm{B} (\rho _o)'(e^{-ik}-e^{-2ik}) \right] \nonumber \\&\quad +\,(1-p)\lambda _2 (e^{ik}-e^{ik+z\tau }+e^{z\tau }-1)=0, \end{aligned}$$

(10)

where $V(\rho _o )'=\frac{\mathrm{d}V_\rho }{\mathrm{d}\rho \left| {\rho =\rho _o } \right. }$. Let $z=z_1 (ik)+z_2 (ik)^{2}+\cdots$, then the first-order and second-order terms of ik are:

$$\begin{aligned} z_1= & {} -\rho _o^2 \left[ {V_\mathrm{F} (\rho _o )'+\lambda _1 pV_\mathrm{B} (\rho _o)'} \right] , \end{aligned}$$

(11)

$$\begin{aligned} z_2= & {} \rho _O^2 \left[ \frac{V_\mathrm{F} (\rho _o)'-3\lambda _1 pV_\mathrm{B} (\rho _o)'}{2}+\lambda _2 (V_\mathrm{F} (\rho _o)'\right. \nonumber \\&\left. +\,\lambda _1 pV_\mathrm{B} (\rho _o )')) \right] \nonumber \\&-\frac{3}{2}\tau \left[ {V_\mathrm{F} (\rho _o )'+\lambda _1 pV_\mathrm{B} (\rho _o )'} \right] ^{2}. \end{aligned}$$

(12)

The stability of traffic flow is determined by the value of $z_2$ for long wavelengths. The stable traffic flow will become unstable when $z_2 <0$, but keep stable when $z_2 >0$. Therefore, one acquires the neutral stability condition as below:

$$\begin{aligned} \tau= & {} \frac{\frac{1}{2}\left[ {V_\mathrm{F} (\rho _o )'-3\lambda _1 pV_\mathrm{B} (\rho _o )'} \right] +(1-p)\lambda _2 \left[ {V_\mathrm{F} (\rho _o )'+\lambda _1 pV_\mathrm{B} (\rho _o )'} \right] }{-\frac{3}{2}\rho _o^2 \left[ {V_\mathrm{F} (\rho _o )'+\lambda _1 pV_\mathrm{B} (\rho _o )'} \right] ^{2}}.\nonumber \\ \end{aligned}$$

(13)

The stability condition is given:

$$\begin{aligned} \tau< & {} \frac{\frac{1}{2}\left[ {V_\mathrm{F} (\rho _o )'-3\lambda _1 pV_\mathrm{B} (\rho _o )'} \right] +(1-p)\lambda _2 \left[ {V_\mathrm{F} (\rho _o )'+\lambda _1 pV_\mathrm{B} (\rho _o )'} \right] }{-\frac{3}{2}\rho _o^2 \left[ {V_\mathrm{F} (\rho _o )'+\lambda _1 pV_\mathrm{B} (\rho _o )'} \right] ^{2}}.\nonumber \\ \end{aligned}$$

(14)

The phase diagram in the $(\rho , a)$—plane is described in Fig. 1 where $\rho$ is the density and a is sensitivity. The stability curves are shown as solid lines indicate. If the value of the parameter $\rho$ or r decreases, the critical points will rise. Different parameter indicates the simulation results. Thus, we can conclude that the of traffic stability is enhanced with consideration of the “backward looking” effect and the influence of the traffic interruption probability.

3 mKdV equation

The modified KdV equations will be derived to describe the slowly varying behaviors for long waves near the critical point $(\rho _c ,a_c )$. For the space variable n and the time t, define the slow variables X and T as follows:

$$\begin{aligned} X=\varepsilon (j+bt),T=\varepsilon ^{3}t,\rho _j =\rho _c +\varepsilon R(X,T),\nonumber \\ \end{aligned}$$

(15)

where $0<\varepsilon \ll 1$ and b is a constant to be determined. By inserting Eq. (13) into (5) and expanding each term in Eq. (5) to the fifth order of $\varepsilon$ by substituting, one obtains:

$$\begin{aligned}&\varepsilon ^{2}k_1 \partial _X R+\varepsilon ^{3}k_2 \partial _x^2 R \nonumber \\&\quad +\,\varepsilon ^{4}(\partial _T^R +k_3 \partial _x^3 R+k_4 \partial _X R^{3}) \nonumber \\&\quad +\,\varepsilon ^{5}(k_5 \partial _T \partial _R +k_6 \partial _x^4 R+k_7 \partial _x^2 R^{3})=0. \end{aligned}$$

(16)

By inserting $b=-\rho _o^2 (V_\mathrm{F} (\rho _o )'+\lambda _1 pV_\mathrm{B} (\rho _o )')$, $\tau =(1+\varepsilon ^{2})\tau _c$ into Eq. (14), one obtains:

$$\begin{aligned}&\varepsilon ^{4}(-g_1 \partial _X^3 R+g_2 \partial _X R^{3}+\partial _T R) \nonumber \\&\quad +\,\varepsilon ^{5}(g_4 \partial _X^4 R+g_5 \partial _X^3 R^{3}+g_3 \partial _X^2 R)=0. \end{aligned}$$

(17)

Here the coefficients $k_i$ are given in Table 1.

In the table, ${V}'=\mathrm{d}V(\rho )/\mathrm{d}\rho |\rho =\rho _c$, ${V}'''=\mathrm{d}^{3}V(\rho )/\mathrm{d}\rho ^{3}|\rho =\rho _c$. To obtain the regularized equation, transformation is made as follows:

$$\begin{aligned} T=\frac{1}{g_1 }{T}',R=\sqrt{\frac{g_1 }{g_2 }}. \end{aligned}$$

(18)

We can obtain the modified mKdV equation with an $O(\varepsilon )$ correction term as follows:

$$\begin{aligned} \partial _{{T}'} {R}'= & {} \partial _X^3 {R}'-\partial _X {R}'^{3} \nonumber \\&+\,\varepsilon \left[ {\frac{g_3 }{g_1 }\partial _X^2 {R}'+\frac{g_4 }{g_1 }\partial _X^4 {R}'+\frac{g_5 }{g_2 }\partial _X^2 {R}'^{3}} \right] . \end{aligned}$$

(19)

We get the mKdV equations with a kink solution after ignoring the $O(\varepsilon )$:

$$\begin{aligned} {R}'_o (X,{T}')=\sqrt{c}\tanh \sqrt{\frac{c}{2}}(X-c{T}'). \end{aligned}$$

(20)

Then, the $O(\varepsilon )$ correction is considered by assuming ${R}'_ (X,{T}')={R}'_o (X,{T}')+\varepsilon {R}'_1 (X,{T}')$. For the purpose of obtaining the propagation velocity c for the kink solution, the solvability condition should be satisfied. $({R}'_o ,M\left[ {{R}'_o } \right] \equiv \int _{-\infty }^{+\infty } {\mathrm{d}{X}'} {R}'_o M\left[ {{R}'_o } \right] )$, where $M\left[ {{R}'_o } \right] =\frac{g_3 }{g_1 }\partial _X^2 {R}'+\frac{g_4 }{g_1 }\partial _X^4 {R}'+\frac{g_1 g_5 }{g_1 }\partial _X^2 {R}'^{3}$. We get the general velocity c:

$$\begin{aligned} c=\frac{5g_2 g_3 }{2g_2 g_4 -3g_1 g_5 }. \end{aligned}$$

(21)

Subsequently, the general kink–antikink soliton solution of the mKdV equation can be obtained:

$$\begin{aligned} \rho _j (t)= & {} h_c \pm \sqrt{\frac{g_1 c}{g_2 }\left( {\frac{\tau }{\tau _c }-1} \right) }\times \tanh \sqrt{\frac{c}{2}\left( {\frac{\tau }{\tau _c }-1} \right) } \nonumber \\&\times \left[ {j+(1-cg_1 )\left( \frac{\tau }{\tau _c }-1\right) t} \right] \end{aligned}$$

(22)

4 Numerical simulation

In this section, the effect of “backward looking” and traffic interruption probability is investigated by numerical simulation. The boundary condition is chosen to be periodic. The initial condition is chosen as follows:

$\rho _j (0)=\rho _o =0.25$, $\rho _j (1)=\rho _o =0.25$, for $j\ne 50,51$, $\rho _j (1)=0.25-0.1$, for $j=50$, $\rho _j (1)=0.25+0.1$, for $j=51$. The total number of sites and the sensitivity are chosen as $N=100$ and $a=1.25$.

The simulation patterns of the density with different value of $r(p=0.2)$ are described after time $t=10^{4}$ s in Fig. 2. From pattern (a) with $r=0$, it confirms Peng’s model [47]. It is clear that the changes in the density become stranger than others. With a small disturbance, the initial steady traffic flow will turn into stop and go traffic waves, which is similar to the mKdV solution. With the increase in r, the scales of density waves is not large. The patterns of the density after $t=10^{4}$ s with different value of $p(r=0.2)$ are demonstrated in Fig. 4. With the increase in r, the changes in density reduce gradually. In conclusion, traffic flow stability can be improved efficiently with the consideration of the influence of the traffic interruption probability and “backward looking” effect.

Figures 3 and 5 reveal the density profile corresponding to Figs. 2 and 4 at $t=10{,}300$ s. So, the simulation results are in agreement with the theoretical analysis.

5 Conclusion

A novel lattice hydrodynamic model has been put forward by taking into account the “backward looking effect” and the influence of the traffic interruption probability. With the theory of the linear and nonlinear analyses, the traffic properties have been analyzed. We have obtained the stability of the model. To describe the traffic congestion, we have derived the mKdV equation. The numerical results is consistent with nonlinear analysis for the extended traffic model. Ge and Cheng [37] also considered the “backward looking” effect in lattice hydrodynamic model for the purpose of suppressing the traffic congestion. On the basis of Ge’s model, traffic interruption probability has been added in this paper. Compared with Ge’s model, we can see the changes in density reduce faster from the numerical simulation in this paper. This means traffic flow in this paper is more stable than Ge’s model. Therefore, the influence of the traffic interruption probability and “backward looking” effect can improve efficiently the stability of the traffic flow.