A Multi-Regime Car-Following Model Capturing Traffic Breakdown

Abstract

Traffic breakdown refers to the first-order transition from free flow to synchronized flow. It is characterized by a rapid decrease in speed, suddenly increasing density, and abruptly plummeting capacity in most of the relevant observations. To understand its cause and model its empirical observations, a multi-regime car-following model is proposed, which classifies the car-following state into four regimes, i.e., Free driving, High-speed following, Low-speed following, and Emergency. Simulation results demonstrate that traffic breakdown and spontaneous jam formation can be reproduced simultaneously by the new model. Experimental verification has shown that the new model can successfully simulate the observed concave growth pattern of traffic oscillations.

1. Introduction

Traffic breakdown from free flow to congested flow and spontaneous formation of traffic jams are two of the most complex highway traffic flow phenomena that have long puzzled traffic researchers. Traffic breakdown, whose mechanisms and quantitative properties are arguably the oldest and most central question of traffic flow theory [1,2], is characterized by an abrupt decrease in speed, a sharp increase in density, and, in particular, a plummeting drop in capacity [3,4]. The identification of traffic breakdown can be traced back to the finding of the reverse-lambda-shaped structure of the flow-density data [5,6]. The peak of the “lambda” represents the high free flow states, which are unstable and may rapidly drop into congested flow marked by the other leg of the “lambda” and thus cause traffic breakdown.

Many observations have shown that traffic breakdown happens at merges and lane drops, which are the bottlenecks where flow is non-conserving (i.e., the flow rate per lane differs significantly upstream and downstream of the bottleneck). Athol and Bullen [7] suggested that the probability of the transition from uncongested to congested flow depends on the flow rate and that the expected time to breakdown is a declining function of flow. Elefteriadou et al. [8] also reported the stochastic nature of breakdown. Banks [9] investigated four metered bottlenecks and discovered that when data were aggregated over 12-min intervals before and after breakdown, the flow drop was approximately 3 percent across all lanes. Hall and Agyemang-Duah [10] found that the flow drop was of the order of 5 to 6 percent, averaged across all lanes. The duration of this transition period ranged from 6 min to 31.5 min, with an average of 16 min and a median of 15 min. Persaud et al. [11,12] found that substantial gains can be achieved, (i.e., increased flow rate of more than 10 percent) if flows can be maintained at the levels before the breakdown. Cassidy and Bertini [4] reported that the average flow rate after the breakdown can be 10% lower than the flow before the breakdown based on the observations from two freeway bottlenecks in and near Toronto, Canada. Matt and Elefteriadou [13] made an extensive analysis of the speed and flux of more than 40 congestion events and confirmed that breakdown has a probabilistic nature, and its occurrence probability increases with increasing flow rate. Zhang and Levinson [14] examined 27 active bottlenecks and discovered that the proportion of the flow drop after breakdown ranges from 2% to 11%. Yuan et al. [15] discovered that the discharging rates after breakdown perhaps can be managed by the transformation of the traffic flow states. They argued that turning the stop-and-go traffic into a standing fleet at a bottleneck may enhance the bottleneck capacity. Arnesen and Hjelkrem [16] proposed a new estimator for calculating the probability of traffic breakdown. Furthermore, Baer et al. [17] introduced a novel threshold queue with two service phases to capture the capacity drop.

Traffic breakdown also occurs at flow conservation bottlenecks. Koshi et al. [18,19] reported that some of the tunnels, vertical alignment sags, and S curves on Japanese motorways were recognized as bottlenecks and congestion queues were frequently observed. Speed reduction of vehicles in a platoon happened at the upstream of these bottlenecks. Related observations were also reported in other places or countries, such as in Edie and Foote [5], Newell [20], Iwasaki [21], Tadaki, et al. [22], Brilon and Bressler [23], and Sun et al. [24]. Bekiaris-Liberis et al. [25] proposed a model-based approach for estimating vehicle density and flow in mixed traffic using connected vehicles’ data.

In particular, traffic breakdown was also observed on single-lane roads. For example, Shiomi et al. [26] reported a single-lane road breakdown, where the tunnel acts as the bottleneck. In these observations, a capacity drop was reported when a traffic breakdown occurred. Jin et al. [27] analyzed the empirical vehicle platoon data on a single-lane highway section. They found that an abrupt speed drop can occur in the platoon when the speed of the platoon leader is quite high (~70 km/h), which illustrated that traffic breakdown may be the process by which free flow changes into synchronized flow on the single-lane road.

There were also observations showing that throughput might increase after breakdown, see [28,29,30]. The origin remains unclear, but some possible explanations include the random nature of the breakdown process, a burst of higher flow, and aggressive merging behavior. This paper focuses on modeling the capacity drop phenomenon. The throughput increases phenomenon needs to be further studied in future work.

Spontaneous formation of jams, also known as “phantom traffic jams”, refers to jam formation without obvious origin such as bottlenecks. The phenomenon was first reported by Treiterer and Myers [31], in which vehicle trajectories were obtained by arterial photography. The traffic experiment on a circular road [32] further demonstrated that jams form spontaneously at high densities.

To simulate traffic breakdown from free flow to congested flow and spontaneous formation of jams, numerous traffic flow simulation models have been established. In traditional models, the mechanisms of the two phenomena are the same. In these models, the unique speed density (or spacing) relationship is adopted in the steady traffic. The steady traffic flow is unstable or metastable in a certain density range. Small disturbances can grow in unstable or metastable traffic, which gives rise to spontaneous jam formation, as demonstrated in [33,34,35].

However, those traditional models have been questioned in the last two decades. Based on the empirical data [36,37,38,39,40,41], Kerner claimed that jam (J) should be regarded as an independent phase, in which the average speed is very low (~0 km/h) and the density is very high (~ρ_jam). Congested flow excluding jam is named as synchronized flow (S). Figure 1 shows an example of the NGSIM trajectory data on Lane 1 (the leftmost lane) and Lane 4 on the US 101 highway. On Lane 1 (Figure 1a), the formation and development of six jams in the synchronized flow can be clearly observed, although it is difficult to precisely define the quantitative speed and density criteria of the jam, while, on Lane 4 (Figure 1b), the synchronized flow emerges and no jam occurs. More empirical observations of synchronized flow can be found in various countries, such as Germany [40,42], the UK [43], Singapore [44], Iran [45], and China [46,47].

Accordingly, there are three kinds of traffic flow transitions: (i) the transitions from free flow (F) to S (F → S), i.e., the state of traffic flow changes from free flow to synchronized flow; (ii) S to J (S → J), i.e., jams emerge in the synchronized flow; (iii) F to J (F → J). F → J seldom happens, except if bottleneck strength is very strong [48]. Therefore, traffic breakdown corresponds to F → S, and spontaneous formation of jams corresponds to S → J.

Recently, we have conducted several experimental studies of the car-following behavior on an open road section [46,49,50] and found that (i) the spacing between a leader-follower car pair can change remarkably, even if their speeds are nearly constant and their speed difference is quite small; (ii) platoon length changes significantly among different experiments with the same platoon average speed; (iii) the car Speed Standard Deviations (SSTD) along the platoon increase in a concave mode in traffic oscillations. These findings do not support the basic assumption of traditional car-following models in which the SSTDs grow in a convex pattern in the initial stage. Later, the concave growth mode was validated by the empirical trajectory data [51].

We proposed two car-following models based on different mechanisms [49,50]. Both can reproduce the spontaneous jam formation and the concave growth of oscillations. Furthermore, Treiber and Kesting [52] showed that adding external stochasticity to the Intelligent Driver Model (IDM) results in a minimal model to reproduce the concave growth pattern. Moreover, the car-following models proposed by Laval et al. [53] and Xu and Laval [54] can also generate a concave growth pattern due to the consideration of stochasticity. Karimi et al. [55] proposed a hierarchical control framework and cooperative trajectory optimization for CAVs in highway merging areas with mixed traffic. They defined different phases for different merging patterns and proposed corresponding movement algorithms for controlling vehicles.

However, to our knowledge, none of the above-mentioned car-following models can reproduce the F → S traffic breakdown. Motivated by this fact, this paper proposed a new multi-regime car-following model. The competition of the speed difference and the spacing has been considered in the high-speed following regime, which enables the model to reproduce synchronized traffic. To our knowledge, it is the first car-following model that can simulate traffic breakdown, spontaneous jam formation, and the concave growth mode of oscillations simultaneously.

The remainder of the paper is structured as follows. Section 2 describes the new model. Section 3 explores its properties concerning fundamental diagrams and spatiotemporal diagrams in the standard test scenario of circular roads. Simulation results on an open road with a bottleneck were reported as well. Section 4 shows that the new model can reproduce the concave growth of oscillations consistent with field observations. Finally, Section 5 concludes the paper.

2. Car-Following Model

Empirical observations have reported that driving behavior differs by regime and may change discontinuously between regimes, see e.g., Koshi et al. [19] and Dijker et al. [56]. Due to this reason, some well-known multi-regime car-following models were built, such as the Wiedemann model [57,58], and the Fritzsche model [59]. The two models have been used in the famous software VISSIM 11, and PARAMICS 27, respectively, which have been widely used in traffic studies and applications.

In the Wiedemann model, five traffic regimes are classified, including Free Driving, Following, Closing in, Emergency, and Collison regimes. In the Fritzsche model, the Free Driving, Following, Closing In, and Danger regimes are distinguished. Moreover, the Following regime has been further classified into Following I and Following II. A similar framework is used in other model, such as MITSIMLab [60], which differentiates the Free driving, Car-following, and Emergency regimes.

We propose a car-following model following the same framework. Four different regimes are classified in our model, including: Free driving, High-speed following, Low-speed following, and Emergency regimes. Definitions of parameters and variables are shown in

Table 1. Definition of parameters and variables.

Symbol	Definition
dn	Actual space gap of car n
dsa	The minimum safe space gap
dn,de	Desired space gap
dfr	Space gap boundary between free driving state and car-following state
s0	Bumper-to-bumper distance in jam
vc	Critical speed of traffic breakdown occurs
vmax	The maximum speed of vehicles
Δvn	Speed difference
T	Time gap between two vehicles
Tsa	Safe driving time gap
Tfr	Free driving time gap
Tn,de	Desired time gap
xn	Location of the following car
xn+1	Location of the leading car
a	The maximum acceleration
b	The comfortable deceleration
bmax	The maximum comfortable deceleration
Lcar	Vehicle length
λ1	Spacing determined component for driver
λ2	Speed difference determined component for driver

, and a detailed explanation will be presented below.

Figure 2 illustrates definition of the four regimes. In the Figure 2, the y-axis d represents the gap between two adjacent vehicles, and the x-axis v represents the speed of the following vehicle.

The four regimes are divided according with a relation between gap and speed of vehicles, as higher speed requires longer distance to emergently brake. The parameter d_sa is the minimum safe space gap; if the actual gap between two vehicles is smaller than d_sa, the following vehicle is required to brake for preventing collision. The parameter s₀ is bumper-to-bumper distance in jam, d_fr is the space gap boundary between the free driving state and car-following state, v_c is the critical speed that divides traffic states as free-flow and traffic jam, and traffic flow with a speed larger than the critical speed represents free-flow; otherwise, it represents traffic jam. The critical speed can be derived from the fundamental diagram [61].

Emergency regime: when d_n ≤ d_sa, the car will decelerate for safety.

$$a_n^{\mathrm{sa}}=-a{\left({\displaystyle \frac{d_{n,de}}{d_n}}\right)}^2$$

(1)

where:

$$d_{sa}=\max \left(v_n{T}_{\mathrm{sa}}-{\displaystyle \frac{v_n\Delta v_n}{2\sqrt{ab}}},0\right)+s_0$$

(2)

$$d_{n,de}=\max \left(v_n{T}_{n,\mathrm{de}}-{\displaystyle \frac{v_n\Delta v_n}{2\sqrt{ab}}},0\right)+s_0$$

(3)

The actual space gap of car n is denoted as d_n, and d_n,de is the desired space gap.

Free driving regime: when d_n ≥ d_fr, the car will accelerate freely. The free acceleration process of the IDM [33] is adopted here:

$$a_n^{\mathrm{fr}}=a\left(1-{\left({\displaystyle \frac{v_n}{v_{\max }}}\right)}^4\right)$$

(4)

where:

$$d_{\mathrm{fr}}=\max \left(v_n{T}_{\mathrm{fr}}-{\displaystyle \frac{v_n\Delta v_n}{2\sqrt{ab}}},0\right)+s_0$$

(5)

Here, T_fr is the free driving time gap.

When d_sa < d_n < d_fr, the vehicle is in the car-following state. We further classify two different car-following states. Empirical observations indicate that traffic breakdown happens in the free flow and, meanwhile, triggers the emergence of the synchronized traffic flow that can steadily exist when the speed of the traffic flow is greater than some critical value. When the speed of traffic flow is lower than the critical value, jams will occur in the synchronized traffic flow. Therefore, a critical speed v_c is proposed.

High-speed following regime: here, we introduce two parameters, λ₁ and λ₂, to provide a measure of how close the current car-following state is to the desired state, in which λ₁ represents space gap and λ₂ represents relative speed. The two parameters will determine whether the driver should accelerate or decelerate within the following regime.

When v_n > v_c:

$${\lambda }_1=\left\{\begin{array}{l}-{\displaystyle \frac{d_n-d_{n,de}}{d_{sa}-d_{n,de}}}, d_{sa}<d_n<d_{n,de}\\ {\displaystyle \frac{d_n-d_{n,de}}{d_{fr}-d_{n,de}}}, d_{fr}>d_n>d_{n,de}\end{array}\right.$$

(6)

$${\lambda }_2=\min \left(\max \left({\displaystyle \frac{\Delta v_n}{\gamma v_n}},-1\right),1\right)$$

(7)

where λ₁ corresponds to the deviation of a car’s actual space gap d_n from its desired space gap d_n,de. To illustrate the physical meaning of λ₁, we consider the first term in Equation (6) as an example. When d_n is slightly larger than d_sa, λ₁ approaches −1, which represents that the ego car will almost reach the minimum safety space gap. The ego car should brake immediately to avoid collision with the front car. Conversely, when d_n is close to d_n,de, this represents that the ego car is close to the desired space gap and only slight deceleration is required. In this situation, λ₁ will approach 0. This parameter enables drivers to smoothly accelerate or decelerate to achieve the desired space gap.

Further, λ₂ corresponds to the car’s speed difference Δv_n from the preceding car over its own speed, and γ is a sensitivity parameter. The parameter λ₂ represents the ratio of the relative speed to the front car and the speed of the ego car. A negative value of Δv indicates that the ego car is approaching the front car, resulting in λ₂ being negative.

Both λ₁ and λ₂ are bounded in the range [−1, 1] to guarantee that the strength of λ₁ and λ₂ are at the same level.

The car-following behaviors, namely the decision to accelerate or decelerate, depends on the larger absolute value of the λ₁ and λ₂, as shown in Equations (8) and (9) below:

$$a_n^{\mathrm{hf}}={\displaystyle \frac{a_{hf}}{2}}\left({\lambda }_1+{\lambda }_2\right)$$

(8)

where,

$$a_{hf}=\left\{\begin{array}{l}a, \mathrm{If} {\lambda }_1+{\lambda }_2>0,\\ b, \mathrm{Otherwise}.\end{array}\right.$$

(9)

Equations (6)–(9) are based on the following behavioral assumptions. The driver n hopes to keep the desired spacing d_n,de and zero speed difference from the driver’s leader. When the real spacing d_n is smaller/larger than d_n,de, the driver has a motivation to accelerate/decelerate. We quantify the motivation by spacing-dependent parameter λ₁. Similarly, when the speed difference is positive/negative, the driver has a motivation to accelerate/decelerate. We quantify the motivation by speed-difference-dependent parameter λ₂. When the sum of the two parameters is positive/negative, the driver will accelerate/decelerate, and the acceleration/deceleration is proportional to the sum.

For instance, if the front car is far ahead the ego car, but the ego car travels faster than the front car, λ₁ will be positive, while λ₂ will be negative. When the actual space gap is significantly smaller than the desired space gap, the absolute value of λ₁ exceeds that of λ₂. Consequently, a positive acceleration is derived from Equation (8), indicating that the driver should accelerate at this moment.

Moreover, the desired time gap T_n,de is assumed to change over time:

$$T_{n,de}\left(t+\mathsf{\Delta }t\right)=\min \left(\max \left(T_{n,de}\left(t\right)+\xi ,T_{sa}\right),T_{fr}\right)$$

(10)

Here, ξ is a uniformly distributed random number between –δ and δ, where δ is a constant and it is the changing step-size of the desired time gap. Δt is the time step and is set to 0.1 s in the model. Thus, the spacing d_n can change, even if the speed difference Δv_n = 0. This describes the experimental observation in Jiang et al. [49,50], that the spacing between two consecutive cars can change remarkably, even if the speeds of the two cars are almost identical and approximately remain constant. The random walk (11) of the desired time gap is active in all the car-following regimes during the simulations, although it has not been used to calculate the acceleration in some regimes.

Equation (8) shows that the maximum deceleration in this regime is b. Figure 3 shows the vehicles’ 1th percentile of deceleration (we use 1th percentile to suppress effect of randomness of data) increases with speed when the speed is greater than 7 m/s. Therefore, we assume that b is a speed dependent function:

$$b=b_{\max }-\left(b_{\max }-a\right){\displaystyle \frac{v_n}{v_{\max }}}$$

(11)

Here, b_max is the maximum comfortable deceleration, v_max is maximum speed of cars.

Low-speed following regime: when v_n ≤ v_c, the car moves following the 2D-IDM, due to the good performance of the 2D-IDM in describing the movement of vehicles in low-speed regime [49], in which Equation (10) is applied for the calculation of the desired time gap.

$$a_n^{\mathrm{lf}}=a\left(1-{\left({\displaystyle \frac{v_n}{v_{\max }}}\right)}^4-{\left({\displaystyle \frac{d_{n,de}}{d_n}}\right)}^2\right)$$

(12)

It should be mentioned that when leaving the High-speed following regime to the emergency regime, the model may display strong discontinuities if the dynamic desired time gap T has a value near the time gap T_fr. Fortunately, this happens very rarely in the simulation results.

3. Model Simulation

For evaluating the proposed model, two cases of simulation are conducted. The simulation road is a circular road, which is frequently utilized in the research of the car-following model. The first case shown in Section 3.1 simulates vehicles driving without any disturbance. And the second case simulates bottleneck with the rubbernecking. The detailed simulation settings and results are presented in Section 3.2.

3.1. Simulation of Traffic Flow on a Circular Road

Firstly, simulations on a circular single-lane road with the length L_road = 3500 m are conducted. Initially, cars with density k on the road are populated on the road with different initial distributions: (1) all cars are distributed homogeneously on the road with speed v_max; (2) all cars are in a mega-jam with speed zero. The simulation environment is shown in Figure 4.

Traffic flow speed v_t is set as the average speed of all cars on the road. Then traffic flow rate f = kv_t. The parameter values used are shown in Table 1. The car length is set to 5 m.

In

Table 2. Model Parameter Values for the Circular Road Simulation.

Parameter	a	bmax	s0	vmax	δ	γ	vc	Tsa	Tfr
Value	0.8	2.5	2.0	33.33	0.2	0.06	10	0.5	2.0
Unit	m/s2	m/s2	m	m/s	s	\	m/s	s	s

, s₀ = 2 m and v_max = 33.33 m/s are the same as in [33]. v_c is set to 10 m/s, corresponding to the critical speed separating stable traffic from unstable traffic, which is found to be between 8.33 and 11.11 m/s, see [45]. T_fr is set to 2 s, which leads to the maximum flow of about 2400 veh/h, consistent with the empirical finding, see [62].

Figure 5 shows the flow-density and speed-density diagrams. The empirically discovered reverse-lambda-shaped structure of the flow-density diagram can be observed. The free flow and the synchronized traffic flow are obtained from the initial homogenous distribution. The jam is obtained from the initial mega-jam distribution.

In the density range k > k₂, where k₂ is critical density bound for traffic breakdown, one can observe traffic breakdown from free flow to synchronized flow, see Figure 6a for a typical example. One can see that the free flow is maintained for about 5 min. Then traffic breaks down and traffic flow evolves into the state that free flow and synchronized flow coexist. As density increases, the synchronized flow region expands and the free flow region shrinks, see Figure 7a. However, at the density, if the traffic starts from an initial mega-jam distribution, traffic flow will develop into the coexistence state of free flow and jam, see Figure 7b. When the density further increases, synchronized flow becomes unstable and transition to traffic jams is observed; see Figure 6b. Moreover, in the density range k₁ < k < k₂, the initial mega-jam distribution of vehicles will develop into the state that free flow and jam coexist.

Furthermore, we study the traffic breakdown probability, which is defined as follows. In the simulation, traffic data are collected by a virtual detector on the road. We record the average speed of vehicles passing the detector every 10 s. Traffic breakdown occurs when the average speed decreases below 27.78 m/s and maintains for more than 100 s. At each flow rate, we perform 100 runs of the simulations. The simulation time interval is set as 1000 s in each run. We denote the number of runs that traffic breakdown happens as N_br. Thus, traffic breakdown probability is calculated by N_br/100 × 100%. Figure 8 shows that the breakdown probability monotonously increases with flow rate. This feature is consistent with the empirical characteristics of traffic breakdown.

3.2. Simulation of Traffic Flow on an Open Road with a Bottleneck

It is well known that traffic breakdown is usually induced by bottlenecks. Now we demonstrate that the new car-following model can reproduce traffic breakdown at the bottleneck. To this end, we simulate the rubbernecking bottleneck, since Chen et al. [63] have demonstrated that the rubbernecking bottleneck causes the traffic breakdown in Figure 1a.

The rubbernecking bottleneck is set as follows: when cars enter the rubbernecking zone, at each simulation time step, the drivers rubberneck with a probability p_rub, which will cause them to decelerate with the deceleration d_rub for h_rub second. Rubbernecking can occur at most once for each car in this zone. In the simulation, the bottleneck is located at [0.9L_road, 0.9L_road + L_bottleneck] with the length L_bottleneck = 100 m.

Initially, it is assumed that the single-lane road with the length L_road = 3500 m is filled with uniformly distributed cars with the density k and speed v_max. We would like to mention that when the density is below the threshold k₁, the traffic speed in the stationary state is very close to the maximum speed. For more details, see Figure 4. For the leading car, it will be removed when going beyond L_road. Its follower becomes the new leading one and moves freely. In the simulation, a virtual detector is placed at the location x_upstream = (0.9L_road − 300) m to collect the traffic data.

Figure 9 shows that the breakdown probability also monotonously increases with flow rate, which agrees with the empirical observation. Figure 10 shows an example of the traffic breakdown. In Figure 10a, after the breakdown, the synchronized flow forms and no jam emerges. In contrast, jams emerge from the synchronized flow in Figure 10b. Therefore, the empirical congested pattern in Figure 1 can be simulated by the new model.

4. Model Calibration and Validation by Car-Following Field Data

We verify the concave growth mode of speed standard deviation in the model with car-following experimental data collected by Jiang et al. [49]. With a platoon of 25 cars, the experiments were conducted on a 3200 m road section. There is no traffic light in this section, which has three lanes in each direction. The section was not closed to traffic at the time of the experiments. Nonetheless, there was little interference from other vehicles since the traffic flow was very low and no other vehicles ever cut into our platoon during the experiments. During the experiments, each car is equipped with high-precision difference GPS devices, whose measurement errors are less than ±1 m for position and ±1 km/h for speed. In each run of the experiments, all vehicles moved along the same lane and lane-changing was forbidden. The leading car of the platoon was required to drive at a constant speed. Other drivers are asked to follow their leader and drive as usual. They do not know how the leading car moves. When the platoon arrived at the end of section, the cars made a U-turn and a new run of experiment started. For more details, one can refer to Jiang et al. [50].

In the experiments, driver of the leading car was required to move with different constant speed v_leading. Within each v_leading, several runs were performed. To calculate the speed standard deviation, we extracted the stationary state data extracted, and averaged over the runs. Jiang’s experimental results are illustrated with blue dotted line in Figure 10. One can see the concave growth of the speed standard deviation.

We calibrate the model using the standard deviation of speed under v_leading = 50, 30 and 7 km/h. For validation, we use the data under v_leading = 40 and 15 km/h. The Root Mean Square Percentage Error (RMSPE) is used as the fitness function:

$$RMSP{E}_{\sigma }=\sqrt{{\displaystyle \frac{1}{24}}{\displaystyle \sum _{n=2}^{25}{\left(\frac{{\sigma }_{v,n}^{\mathrm{simu}}-{\sigma }_{v,n}^{\exp }}{{\sigma }_{v,n}^{\exp }}\right)}^2}}$$

(13)

where n denotes the car number, ${\sigma }_{v,n}^{\mathrm{simu}}$ (${\sigma }_{v,n}^{\exp }$) denotes the speed standard deviation of car n in the stationary state in the simulations (experiments).

We list the calibrated parameter values in

Table 3. Model Parameter Values for the Platoon Simulation.

Parameter	a	bmax	s0	vmax	δ	γ	vc	Tsa	Tfr
Value	0.8	2.5	2.0	33.33	0.2	0.06	15	0.5	1.9
Unit	m/s2	m/s2	m	m/s	s	\	m/s	s	s

and show the calibration and validation results in

Table 4. Calibration and validation results.

	Calibration			Validation
vleading (units: m/s)	1.94	8.33	13.89	4.17	11.11
RMSPE	0.22	0.14	0.20	0.24	0.10
Average RMSPE	0.19			0.17

. The car length is set to 5 m in the simulations. All RMSPEs under different v_leading are quite small, and the average RMSPE of calibration is 0.19 versus 0.17, resulting from validation.

A comparison between the simulation results and the experimental ones is shown in Figure 11 and Figure 12. The simulated results agree with the experimental results quite well. Figure 12 shows that the spatiotemporal patterns of the experiments and simulations. One can see that although there is some difference between them, the stripe structures in the simulated spatiotemporal patterns agree with the experimental ones. Actually, it should be noted that since both the real car-following process and the model are stochastic; these diagrams cannot be exactly the same.

5. Conclusions

Traffic breakdown is characterized by quickly decreasing speed, suddenly increasing density, and abruptly plummeting capacity. It is arguably the oldest and most central problem in the traffic flow theory [2]. To simulate the traffic breakdown phenomenon, many traffic flow models have been proposed. Kerner empirically found that traffic breakdown occurs as a transition from free flow to synchronized flow, and spontaneous formation of jams is the transition from synchronized flow to jam [41]. The NGSIM empirical data also validate these facts. However, these phenomena cannot be reproduced by traditional car-following models.

To explain and simulate the empirical traffic breakdown, this paper proposed a multi-regime car-following model. Four different regimes are classified in our model, including Free driving, High-speed following, Low-speed following, and Emergency states. To test and verify the performances of the new model, simulations were performed on a circular single-lane road as well as on an open single-lane road with a traffic bottleneck. Then, calibration and validation were conducted with field data from car-following experiments. The simulation results show that the empirical properties of traffic breakdown (including the reverse-lambda-shaped structure of the flow-density diagram, the emergence of synchronized traffic flow as traffic breaks down, and the monotonous increasing function of traffic breakdown probability versus the flow rate) and the concave growth mode of traffic oscillations discovered from the experiments on the open road can be reproduced. However, one limitation is that simulation results on the circular single-lane road and on the open single-lane road with a traffic bottleneck lack validation with field data.

To address this limitation, in the future, on the one hand, more experiments need to be conducted to further explore the properties of car-following in traffic flow. Multiple types of driving behavior can be considered in further research, such as immediate brake, variable vehicle speed, heterogeneous driving styles, and cut-in. Moreover, the proposed model is required to be validated in multilane road scenarios. On the other hand, to compare with the empirical data collected on multilane roads, the model needs to be extended to consider lane-changing behavior. Finally, machine learning based on modeling provides an effective approach to describing car-following behaviors, see [64,65,66,67] for an example. It is worth to see that whether the traffic breakdown can be reproduced or not.