Due to the changes of vehicle number on each lane caused by lane-changing, a source term is needed to add to the continuum equation. Furthermore, we introduce a “ viscous force” term to the original acceleration equation of Payne’ s model to demonstrate the impact of lane-changing on traffic flow dynamics. The general form of this model is

where ρ _l and q_l are the vehicle density and flow rate on the l-th lane, respectively; a and Tr are the sonic speed and time delay; Q_e (ρ ) = ρ U_e (ρ ), U_e (ρ ) is the equilibrium speed function; Φ _{l′ l} is the net lane-changing rate, which equals to the rate of vehicles that change from the l′ -th lane to the l-th lane minus the rate of vehicles that change from the l-th lane to the l′ -th lane per hour per kilometer; f_{viscous, l} is the “ viscous force” per unit length, with a unit of veh· h^{− 2}. In this paper, free flow speed is denoted by u_f; jam density is denoted by ρ _j.

In Laval’ s model, the source term Φ _{l′ l} is based on the speed difference Δ u_{l′ l} between adjacent lanes. Suppose Δ u_{l′ l} = u_l − u_l′ > 0, then Φ _{l′ l} is determined as follows:

where τ is a constant parameter; Δ t and Δ x are temporal step and spatial step, respectively, assuming Δ x /Δ t = u_f; Q is the maximum flow rate. Laval' s model consists of the first equation of Eq.  (1) and a triangular fundamental diagram instead of the acceleration equation

where w is the congestion wave speed (w ≈ u_f/4). We derive from this equation that Q ≈ ρ _ju_f / 5. Rearrange Eqs.  (2)– (5), a simplified form of source term can be obtained as follows:

In Zhu’ s model, ^{[10, 11]} another form of source term based on density difference between adjacent lanes is used. This form is simpler but discontinuous

where k_{l′ l} is a constant. To avoid the computation inconvenience caused by discontinuity, we can simplify Zhu’ s source term according to the simplification of Laval’ s source term. Suppose the density difference Δ ρ _{l′ l} = ρ _l′ − ρ _l > 0, then

The source terms of Laval’ s model and Zhu’ s model emphasized the differences between adjacent lanes, so both models can simulate well the phenomenon of density equalization in the lane-changing problem. However, they both have weaknesses. Laval’ s model cannot cover car-following phenomenon since it is a single-equation model, and Zhu’ s model does not include a “ viscous force” term driven by lanechanging to its acceleration equation, which is incomplete for lane-changing simulation. Besides, as the source terms are determined by either speed difference or density difference in both models, lane-changing behavior stops happening when speed difference is zero or density difference is zero, which means Laval’ s model implies an equilibrium state when adjacent lanes have the same speed but different densities, while Zhu’ s model implies an equilibrium state when adjacent lanes have the same density but different speeds. This contradicts real traffic phenomenon. In order to avoid these weaknesses, we combine Eqs.  (7) and (9), and introduce a new source term that considers both speed difference and density difference between adjacent lanes

where C₁ and C₂ are dimensional constants, C₁ has a unit of h· km^{− 2}, and C₂ has a unit of km· h^{− 1}· veh^{− 1}.

The form of “ viscous force” f_{viscous, l} in Eq.  (1) can be obtained as follows: Given that f_{viscous, l} is driven by the difference between adjacent lanes, we may assume an ideal state where the traffic condition is homogenous in each lane, that is, all terms of partial derivatives with respect to x equal to zero and traffic flow is close to the equilibrium state. Therefore, the term of equilibrium speed function can be dropped. Thus equation  (1) is simplified to

Applying the triangular fundamental diagram (6), we obtain

Note that f_viscous, l is a function of Φ _{l′ l}, which can be easily determined by Eqs.  (10)– (12).

Now the control equations of our new model are fully developed as follows:

where the lane changing source term demonstrates the impact of speed difference and density difference between adjacent lanes on free lane-changing behavior, and the viscosity term addresses the momentum exchange induced by the lane-changing behavior. Other factors that may influence lanechanging behaviors in real traffic situations can be added to the proposed model.

Let L be the characteristic length, and let q* = q/Q, u* = u/u_f, ρ * = ρ /ρ _j, x* = x/L, t* = t/(L / u_f), a* = a/u_f, Tr* = Tr/(L / u_f), , Φ * = Φ /(ρ _ju_f / L), which are the dimensionless flow rate, speed, density, length, time, sonic speed, time delay, equilibrium flow function, lane-changing source term, and lanechanging viscous term. Substituting these to Eq.  (1), we obtain the dimensionless equations as follows (“ * ” is omitted):

Accordingly, equations  (11), (12), and (14) change to

where

The ranges of values of C₁ and C₂ were estimated according to the empirical traffic data from an 81-meter long section on Yan’ an Elevated Road, Shanghai, China. The data were extracted from a video recorded on July 21, 2008, from 7:40 a.m. to 11:00 a.m. and were grouped every ten minutes, including density and speed on each lane, and the number of lane-changing to the left (this can be regarded as free lanechanging, in contrast to compulsive lane-changing, for example, lane changing to the right triggered by a ramp ahead). Density and speed were calculated according to the coordinates of vehicles on each video frame.^{[15, 16]} The number of lane changes was manually counted from the video.

We determined the upper limits of C₁ and C₂ through data fitting. Generally, we have C₁ ≤ 0.3  h· km^{− 2} and C₂ ≤ 25  km· h^{− 1}· veh^{− 1}, and dimensionless parameters and The values of C₁ and C₂ depend on road conditions and vehicle types. These upper limits proposed are generally practical, because firstly, we omitted the free lane changing number that changed from left to right, and secondly, cars make up a large proportion with medium speed (40  km· h^{− 1} approximately) in our sample. Our ongoing research shows that vehicles with this speed may change lane more often.

In order to validate the new model and the discretization scheme, we study a simple two-lane situation: lane 1 is the fast lane with lower density; lane 2 is the slow lane with higher density; speed and density on both lanes are evenly distributed, so variables will not change with respect to x direction. Hence, the dimensionless control equations are simplified as follows:

where lane-changing function

equilibrium flow Q_e (ρ ) = 5ρ (1 − ρ ), and the initial values are ρ ₁₀, ρ ₂₀, q₁₀, and q₂₀, respectively. The characteristic values are set as L = 15  km, u_f = 90  km· h^{− 1}, ρ _j = 143  veh· km^{− 1} (the same as that in Section  4.2). Equation  (23) is ordinary differential equations. Its accurate results can be achieved through the fourth-order Runge– Kutta method, which will be regarded as analytical results hereafter.

Numerical results and analytical results are shown in Figs.  1 and 2 (with the dimensional units indicated on the axes). Numerical results are obtained as described in Section  3. Constants are set as Tr = 0.02, a = 0.4, C₁ = 0.25,

Fig.  1.

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	Fig.  1. Numerical results against analytical results at low density (ρ 10 = 14.3  veh· km− 1, ρ 20 = 25.74  veh· km− 1). (a) Density evolution in lane 1, (b) density evolution in lane 2, (c) speed evolution in lane 1, and (d) speed evolution in lane 2.

Fig.  2.

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	Fig.  2. Numerical results against analytical results at medium density (ρ 10 = 50.05  veh· km− 1, ρ 20 = 71.5  veh· km− 1). (a) Density evolution in lane 1, (b) density evolution in lane 2, (c) speed evolution in lane 1, and (d) speed evolution in lane 2.

C₂ = 1.5, Δ x = 0.1, and Δ t = 0.01; in Fig.  1, ρ ₁₀ = 0.1, ρ ₂₀ = 0.18, q₁₀ = 0.9, and q₂₀ = 0.82; in Fig.  2, ρ ₁₀ = 0.35, ρ ₂₀ = 0.5, q₁₀ = 0.65, and q₂₀ = 0.5. Note that the values adopted for C₁ and C₂ in this case are half of the upper limits determined in Section  2.3, which indicates that this case is assumed to have a medium lane changing intensity.

It is shown in this case that our model simulates the traffic with lane-changing behavior accurately both at low density and at medium density (similar results can be achieved for higher density). In Figs.  1 and 2, speed and density of the two lanes differ at the beginning, then monotonically converge towards each other, and reach an equilibrium state at last, which agrees with the real traffic process. It is also shown that the discretization scheme adopted ensures the convergence of the numerical method, because the numerical results agree with the analytical results not only in quality, but also in quantity, and they require the same amount of time to reach the equilibrium state.

Perturbation’ s evolution has been an important problem for the study of traffic models. Arising from perturbation problems, cluster effect is a typical phenomenon in nonlinear systems, as suggested by Kerner^[16] and Sun et al.^[14] We explored this problem using the same initial condition of small perturbation in density as adopted by Kerner and Sun et al.

Initial flow is the product of density and equilibrium speed, and the equilibrium speed is adopted from Ref.  [17]

We use homogeneous Neumann boundary conditions at upstream and downstream, the discretization of which is

where J is the total grid number in x direction. Constants are set to Tr = 0.02, a = 0.4, C₁ = 0.25, C₂ = 1.5, Δ x = 0.01, and Δ t = 0.002.

First, for the low-density situation, we set ρ ₁₀ = 0.1 and ρ ₂₀ = 0.14. Results are shown in Fig.  3. Perturbation propagates to downstream and then dissipates.

Then, we change the initial condition of lane 2 to medium-density, and set ρ ₁₀ = 0.1 and ρ ₂₀ = 0.21. Results show that without lane-changing terms, perturbation in lane 2 leads to congestion both upstream and downstream. However, results of our lane-changing model (Fig.  4) show that in lane 2, although breakdown still happens upstream, congestion does not appear downstream. Instead, lane-changing happens frequently, resulting in a gradual increase of the density at downstream in lane 1 and a cluster appearing in lane 2. Therefore, lane-changing behavior alleviates the traffic congestion.

Fig.  3.

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	Fig.  3. Dissipation of perturbation at low-density. (a) and (b) Density evolutions in lane 1 and lane 2, respectively. (c) The evolution of non-dimensional net lanechanging rate from lane 2 to lane 1.

When changing the initial condition of lane 1 to mediumdensity as well, that is, ρ ₁₀ = 0.19 and ρ ₂₀ = 0.21, we found that its ability to alleviate congestion reduces (Fig.  5), and perturbation turns to several clusters with different speeds in both lanes, which results in a stop-and-go wave. The net lanechanging rate profile also shows that, in this state, there are more vehicles that change lanes from lane 1 to lane 2 than their counterparts in front of the congestion wave. This phenomenon can also be observed in real traffic.

Finally, we change the initial conditions of both lanes to high-density, that is, ρ ₁₀ = 0.4 and ρ ₂₀ = 0.45. A perturbation wave propagates to upstream and dissipates shortly after, as shown in Fig.  6.

Our results are consistent with Kerner’ s simulations.^[17] However, they combined several lanes to one lane, whereas our viscous lane-changing model can provide the detailed evolution of perturbation on each lane. It is proved in this case

Fig.  4.

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	Fig.  4. Medium-density in lane 2 and low-density in lane 1. (a) and (b) Density evolutions in lane 1 and lane 2, respectively. (c) The evolution of non-dimensional net lane-changing rate from lane 2 to lane 1.

that lane-changing behavior helps to alleviate congestion. In Fig.  4, without lane-changing terms, lane 2 is congested while lane 1 remains clear; with lane-changing terms, both lanes are not congested. This conclusion agrees with real traffic phenomenon.