The new proposed model is defined on a lattice of length L, where each cell can be empty or occupied by just one vehicle. The system consists of N vehicles moving in one direction in a single lane, therefore overtaking and lane changing are not permitted. The speed of each vehicle can take on one of the (V _max+1) allowed integer values . For simplicity, we only consider one type of vehicle in this paper and therefore, the same maximum velocity v _max will be used for all vehicles. denotes the maximum acceleration rate in quick start (emergency braking). denotes the comfortable acceleration (deceleration) rate. t _s denotes the time step. and denote the location of the vehicle n and n+1 respectively at time t. and denote the velocity of the vehicle n and n+1 respectively at time t. denotes the acceleration of vehicle n at time t. denotes the actual distance between vehicles n and n+1, that is the distance from the front bumper of vehicle n to the rear bumper of vehicle n+1. , where l _vehicle denotes the vehicle length. It is assumed that the position of a vehicle is denoted by the cell that contains its rear bumper.

A definition of safe distance d _safe is introduced before considering the micro-level step updating mechanism of the model. Suppose there are N vehicles in total, denoted by . The safe distance between vehicle n and n+1, denoted by , can be calculated by

According to many previous studies, if a driver decides to accelerate or decelerate they prefer to use the comfortable acceleration .^[28] However, there is also a possibility that the rate of acceleration will deviate . Considering that a Gaussian distribution describes the distribution of the acceleration well,^[33] a deformed Gaussian distribution is introduced to describe the distribution of the acceleration because the upper limit (lower limit) of the acceleration distribution is a specific integer value in our CA model, instead of positive infinity (negative infinity) in normal Gaussian distribution functions.

If , the distribution function for the random variable X is

In the above equation, the terms and are constant adjustment terms to ensure that the integral of the probability density function for a specified interval equals one. When , the mathematical expectation of the distribution is , which illustrates that the vehicle tends to accelerate at a comfortable rate. Of course, there is a small possibility that the vehicle will decelerate, since some drivers do not like to accelerate even if the gap between his/her vehicle and the preceding vehicle is large enough. When , the mathematical expectation of the distribution is , which illustrates that the vehicle tends to decelerate at a comfortable rate. The vehicle is not permitted to accelerate in this situation to avoid collision.

The value of acceleration should be a discrete integer in CA models. Thus, the probability of acceleration, , can be calculated by Eq. (6). Suppose that the upper limit (lower limit) of the interval of acceleration is ( ). From the above analysis we can see that or while . Suppose that X is an integer in the interval .

One update of the system consists of the following consecutive steps.

Step 1 Calculate the interval of acceleration . In this step, we should obtain the values of , , and . Determine by comparing the value of with , , and .

Step 2 Determine the acceleration of each vehicle . First, we calculate the probability density function of the acceleration and then determine the acceleration of each vehicle according to the probability of acceleration .

Step 3 Change the velocity of vehicles by calculating ,

Step 4 Realize vehicle movement. Each vehicle moves forward using its new velocity determined in Step 3:

Simulation results for a single-lane road segment with periodic boundary conditions are shown here. The injecting cells and the extracting cells are coincident in the periodic boundary conditions. The global density ρ is defined as

Initially, the vehicles stop near the injecting cells in a line and then start up one by one based on the proposed model. In order to get to a stable state of the system, each simulation, with different values for density ρ, is conducted for 3.6×10³ time steps after a spin-up period of the first 10³ time steps. Flow, velocity, and density are the commonly used parameters which represent the most important macroscopic traffic characteristics.^[38] If the flow is measured in “vehicles/h”, the relationship between density ρ and flow q, and the correlation between vehicles’ average velocity and density ρ are shown in Fig. 2.

Fig 2.

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	Fig 2. The fundamental diagram and the correlation between density and velocity of the proposed model.

The density–flow relationship and the density–velocity relationship obtained from the simulations of this model are consistent with the classical and empirical findings obtained in some other prevalent models such as the BJH model,^[37] the Slow-to-start model,^[39] and the comfortable driving (CD) model.^[40] Our proposed model is also able to reproduce the hysteresis phenomenon and the three phases (free flow, synchronized flow, and jammed flow) observed in real traffic, as is shown in Fig. 2. When the density , the traffic flow is in free-flow mode. The critical density , which is consistent with various findings in the real world (∼25 (1/km)).^[41] When , phase separation is observed, and the traffic flow comprises free flow and congested flow at the same time. As is well known, congested flow includes synchronized flow and jammed flow. When , free flow disappears and traffic flow is congested.

The spatiotemporal shapes of different traffic phases (free flow, synchronized flow, and jammed flow) for different simulation scenarios are depicted in Fig. 3. In these scenarios, the density ρ increases from 20 vehicles/km to 180 vehicles/km, in steps of 20 vehicles/km.

Fig 3.

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	Fig 3. The spatiotemporal shapes of different traffic phases in different densities: (a) 20 veh/km, (b) 40 veh/km, (c) 60 veh/km, (d) 80 veh/km, (e) 100 veh/km, (f) 120 veh/km, (g) 140 veh/km, (h) 160 veh/km, and (i) 180 veh/km.

From Fig. 3, we see that the traffic flow shown in panel (a) is free, where the flux can be increased along with an increase in car density ρ. In free flow, the space–time trajectories corresponding to different vehicles moving forward can be easily found. Figure 3(b)–Figure 3(i) show congested traffic flow with jams moving backwards. With increasing densities, jammed flow gradually pushes out synchronized flow.

To depict the three traffic states (free flow, synchronized flow, and jammed flow) more clearly, Figure 4 describes the one minute average flux against the density of the proposed model under periodic boundary conditions. The simulation data are obtained by a virtual probe. This virtual probe can record the flux and velocity of the vehicles which pass through a vertical section perpendicular to the road. We record the flux and average velocity using the virtual probe and then calculate the value of the local density by

Fig 4.

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	Fig 4. One minute average flux against density.

The most important difference between Fig. 2 and Fig. 4 is that Figure 2 describes overall traffic conditions using data points of average values in one hour, while Figure 4 describes the local traffic conditions based on the simulation results in one minute. It means that data points in Fig. 4 only represent the situation of one phase or two consecutive phases for the traffic flow has only moved a short distance. Therefore, Figure 4 can discern between the different traffic phases (free flow, synchronized flow, and jammed flow) more clearly. The upper scatter boundary with an average velocity of 90 km/h represents the free-flow mode while the lower scatter boundary with an average velocity of 3.6 km/h forms the split between synchronized flow, and jammed flow. Figure 4 shows that the new proposed model can simulate free flow, light synchronized flow (where the average velocity of vehicles is close to the free-flow velocity), and heavy synchronized flow at the same time. As we know, vehicle velocity can vary greatly in real traffic flow in cities. Vehicles can advance at maximum velocity when traffic flow is sparse, but can be stopped for periods of time in jammed traffic. Thus the field of application of the model needs to be broad enough so as to be capable of simulating all the scenarios of real traffic flow. In this respect, the performance of our proposed model is superior to some of the prevalent models such as CD and MCD models.^{[40, 42]} In CD model, the average velocity of the upper scatter boundary is 60 km/h which means that the model is not suitable for light synchronized flow simulation. While in MCD model, the average velocity of the lower scatter boundary is 24 km/h which means the model is not suitable for heavy synchronized flow simulation.

We also find that the performance of this proposed model is more realistic by analyzing the correlations between one vehicle’s different state parameters. A series correlation analysis of a vehicle’s state parameters was conducted to illustrate the performance of the proposed model in a more detailed way.

The distributions of the velocity with different headway distances are shown in Fig. 5. When the headway distance is 5 m, the distribution of the velocity is asymmetrical with a small proportion of vehicular velocities exceeding 5 m/s. The highest velocity in this distribution is 10 m/s, which is considerable since the maximum acceleration in this study is set to be . The velocity changes before the location, so the vehicle with a velocity of 10 m/s will move forward with a velocity of 5 m/s only if its acceleration is −5 m/s². The velocity will decrease to 0 m/s as long as it maintains that acceleration (−5 m/s²). In other words, this vehicle will theoretically only move forward 5 m when using the rules in the proposed model, and a potential collision would be avoided under this assumption. Distributions of the velocity with other headway distances can be similarly understood.

Fig 5.

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	Fig 5. The distribution of velocity with different headway distances.

The distribution becomes approximately symmetrical and the axis of symmetry moves gradually in the positive direction along the “velocity” axis, together with headway distance increasing (headway=10 m, 15 m, 20 m). When the headway distance is 25 m or 30 m, the velocity distributions become asymmetrical again and present distributions with two peaks. The second peak is at “velocity=25 m/s”. The second peak becomes bigger with increasing headway distances which means that with a bigger headway distance, more vehicles can move ahead with maximum velocity.

We use nonlinear curve fitting in OriginPro 9.1 to fit the obtained data scatter points. In general, the distribution curves (without the second peak if it exists) fit a Lorentz distribution well in a discrete state with the following probability density function

Table 1.

Table 1.

Table 1. The values and standard errors of the parameters in Eq. (12). . Headway/m 5 10 15 20 25 30 p 0 value Standard error v c value 2.53 7.27 11.9 16.0 19.8 21.6 Standard error 0.118 0.101 0.173 0.188 0.143 0.135 w value 6.20 4.15 7.04 5.23 5.93 3.27 Standard error 0.747 0.468 1.03 0.908 0.676 0.510 φ value 2.21 1.21 1.70 1.14 1.23 0.797 Standard error 0.354 0.145 0.313 0.218 0.155 0.110 value 0.972 0.959 0.930 0.896 0.952 0.915

Table 1. The values and standard errors of the parameters in Eq. (12). .

From Table 1 we find that v _c and headway distance have a strong positive correlation. v _c increases along with an increase in the headway distance d. This correlation can be represented with the following equation, where d denotes the headway distance

Fig 6.

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	Fig 6. The correlation between the headway distance in front of a vehicle and the average velocity of that vehicle.

We next study the correlation between the headway distance and the average velocity with the simulation results grouped according to the acceleration of vehicles. The corresponding results are shown in Fig. 6(b). These results show that the acceleration is greater when the average velocity is relatively slower for a certain headway distance. This is in accordance with reality since in the same environmental condition, slower vehicles generally have greater motivation to accelerate, to both save themselves some time and because of interactions with adjacent cars. For example, the horn of a rear vehicle will promote the acceleration of the front vehicle. The results show that this proposed model performs reasonably and is in general realistic.

From Fig. 6, it can be seen that when the headway distance is known, the value of the average velocity will be slower if the velocity difference is smaller or the acceleration is greater. It seems that a relatively greater (lesser) acceleration and a relatively lesser (greater) velocity difference both correspond to the same average velocity. Furthermore, we notice that previous studies have found that the effect of velocity difference on acceleration is significant from different field observation data and experiments.^[43] For this reason, we looked at the detailed correlation between velocity difference and acceleration. Considering the coupling effect of a vehicle’s state parameters, including the velocity difference, headway distance, and velocity, we first constructed a set of state parameter groups to present the possible discrete values of headway distance and velocity for a certain velocity difference. We then calculate the acceleration for this velocity difference using a weighted algorithm. The weighting factor is the probability of an occurrence of each state parameter group in the simulation.

Suppose that denotes the velocity difference of vehicle n and vehicle n+1; d(n) denotes the headway distance of vehicle n to vehicle n+1; v(n) denotes the velocity of vehicle n; denotes the acceleration of vehicle n. A state parameter group is defined as . The number of times that has appeared in the simulation is denoted by . Thus, denotes a state parameter group where , and , . It is important to note that and are both related to , here . Therefore, the possible values of i and j are both related to . Suppose that ( ) and ( ) denote the maximum (minimum) of the set of i and j. denotes the probability of the occurrence of ,

Fig 7.

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	Fig 7. The correlation between velocity difference and acceleration of vehicles.

The fitting equation in Fig. 7 is shown in the following:

We use data extracted from real traffic flow by other researchers to verify our proposed model from both macroscopic and microscopic perspectives, as shown in Table 2. The data are a type of single-vehicle processed data obtained using Autoscope Rackvision Terra, in which velocities and time-headways of all vehicles passing the equipment can be recorded. Error correction was performed on the data by checking the original videos using Tracker software. The error of every single-vehicle velocity was found to be no more than 1 m/s (3.6 km/h).^[44]

Table 2.

Table 2.

Table 2. Two datasets extracted from real traffic flow. . Group Data type Flux/(veh/h) Average velocity by detector/(m/s) Density/(veh/km) 1 1st real dataset 1780 13.1 37.7 2 2nd real dataset 2044 17.0 33.4

Table 2. Two datasets extracted from real traffic flow. .

The first dataset was derived from empirical video data on the Nanjing Airport Highway.^[45] This highway is the main road connecting Nanjing urban areas with Nanjing Lukou Airport. There were 40 cameras along the 28-km-long highway which has two lanes in each direction. The second dataset was obtained from a large-scale construction project also situated on the Nanjing Airport Highway. Every day one of the two lanes was temporarily closed on a 5–10-km-long section of road. This enabled the collection of empirical single-lane traffic flow data.^{[46, 47]} The speed limit in both traffic scenarios was 120 km/h ( m/s).

We adjusted the maximum velocity in our proposed model from 25 m/s to 33 m/s to make the simulation conditions same as real traffic scenarios in datasets. A virtual detector was set at the end of the road. Each simulation, using a certain global density lasted for one minute, and sixty groups of simulations with the same global density were conducted. The sum of the fluxes from these sixty groups and the average velocity of the vehicles that passed by the virtual detector were recorded. Then the local densities for different fluxes were calculated using The correlation between average velocity and flux, along with increasing global density in the simulation group is depicted in Fig. 8(a). The correlation between local density and flux along with increasing global density in the simulation group is depicted in Fig. 8(b). The results show that the real data sets are consistent with the macroscopic performance of the proposed model.

Fig 8.

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	Fig 8. The comparison of real data sets and the macroscopic performance of the proposed model.

The above validation proves that the proposed model is a competitive CA model. The proposed model can simulate traffic flow ensuring the authenticity of the features, and is able to reproduce the empirical findings. Moreover, we found that the acceleration of a vehicle is mostly influenced by the velocity difference when the absolute value of acceleration is not large. This finding is consistent with other studies which showed that acceleration always has a linear correlation to the velocity difference when the absolute value of acceleration is not large.^{[31, 32]}

We compare the results produced by our proposed model with those produced by the SVR driving behavior model. Wei and Liu depicted the correlation between relative velocity and acceleration for headway distances of 9 m and 12 m.^[31] The relative velocity is in an interval of [–10, 10] km/h ([–2.78, 2.78] m/s). We therefore analyzed the coupled distribution of velocity and velocity difference in a case study using our proposed model for headway distances of 9 m and 12 m. The results are shown in Table 3 and Table 4.

Table 3.

Table 3.

Table 3. Coupled distribution (%) of velocity and velocity difference with headway distance of 9 m. . Velocity difference/(m/s) Velocity/(m/s) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Sum –6 — 0.12 — — — — — — — — — — — — — 0.12 –5 0.24 0.49 0.61 0.24 0.12 — — — — — — — — — — 1.70 –4 0.12 0.61 1.46 0.85 2.19 0.24 — — — — — — — — — 5.47 –3 — 0.12 1.95 2.19 3.78 4.02 1.46 — — — — — — — — 13.52 –2 — 0.24 — 0.85 2.07 5.36 10.72 1.10 — — — — — — — 20.34 –1 — — — 0.24 2.07 1.83 4.87 7.19 1.34 — — — — — — 17.54 0 — — 0.12 — — 0.97 1.95 3.17 4.99 0.12 — — — — — 11.32 1 — — — — — 0.37 0.85 1.95 2.56 1.46 — — — — — 7.19 2 — — — 0.12 0.12 0.12 0.24 0.85 1.58 1.10 0.85 — — — — 4.98 3 — — — — — — 0.12 0.49 1.46 1.58 1.22 0.85 — — — 5.72 4 — — — — — — — — 0.12 0.61 1.10 0.85 1.95 — — 4.63 5 — — — — — — — — 0.24 0.37 0.49 1.71 1.34 0.61 — 4.76 6 — — — — — — — — — — 0.24 0.37 0.61 0.49 0.49 2.20 7 — — — — — — — — — — — — 0.24 0.24 — 0.48 Sum 0.36 1.58 4.14 4.49 10.35 12.91 20.21 14.75 12.29 5.24 3.90 3.78 4.14 1.34 0.49 100

Table 3. Coupled distribution (%) of velocity and velocity difference with headway distance of 9 m. .

Table 4.

Table 4.

Table 4. Coupled distribution (%) of velocity and velocity difference with headway distance of 12 m. . Velocity difference/(m/s) Velocity/(m/s) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Sum –6 – – 0.17 0.34 – – – – – – – – – – – – 0.51 –5 0.17 0.17 0.17 0.34 0.69 0.34 – – – – – – – – – – 1.88 –4 – – – 1.20 0.51 1.03 1.54 0.51 – – – – – – – – 4.79 –3 – – – 0.17 1.20 1.20 2.06 3.95 1.54 0.34 – – – – – – 10.46 –2 – 0.17 – 0.17 0.34 0.69 1.72 3.43 5.49 6.86 1.37 – – – – – 20.24 –1 – – – – 0.17 0.17 0.17 0.69 4.63 6.17 9.78 0.34 – – – – 22.12 0 – – – – – 0.34 0.17 0.34 1.37 1.72 2.57 3.09 – – – – 9.58 1 – – – – – 0.17 – 0.34 0.51 0.86 1.72 2.06 0.51 – – – 6.17 2 – – – – – – 0.17 0.34 0.86 0.34 0.86 1.72 0.34 0.34 – – 4.97 3 – – – – – – – – – 0.17 0.86 0.51 2.06 0.51 0.69 – 4.80 4 – – – – – – – – – – 0.86 1.72 1.03 2.06 1.20 0.51 7.38 5 – – – – – – – – – – – 0.17 0.69 0.17 2.06 0.86 3.95 6 – – – – – – – – – – – – 0.17 0.17 1.03 1.20 5.354 7 – – – – – – – – – – – – – – – 0.17 0.17 8 – – – – – – – – – – – – – 0.17 0.17 – 0.34 Sum 0.17 0.34 0.34 2.22 2.91 3.94 5.83 9.60 14.40 16.46 18.02 9.61 4.80 3.42 5.15 2.74 100

Table 4. Coupled distribution (%) of velocity and velocity difference with headway distance of 12 m. .

Since, in the proposed model, and are the maximum values in the two groups and respectively, we chose state parameter groups and to be used as a contrast. The results are shown in Fig. 9.

Fig 9.

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	Fig 9. The model comparison of the proposed model and the SVR driving behavior model.

The values of acceleration are a little higher than the results of the SVR driving behavior model, mainly because this model’s acceleration strategy of allowing a driver to accelerate as long as there is a safe headway distance, is more radical than that of the SVR driving behavior model. The results also show that the proposed model reveals the law that acceleration has a linear correlation with the velocity difference when the absolute value of acceleration is not, in general, large. Thus, the results reflect the superiority of the proposed model, in that this model can present more detailed correlations between state parameters. The results indicate that the assumption that acceleration and deceleration rates can be considered to be constant in most empirical models, needs to be reevaluated.