An asymmetric cellular automata model for heterogeneous traffic flow on freeways with a climbing lane

Physica A 535 (2019) 122277

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An asymmetric cellular automata model for heterogeneous traffic flow on freeways with a climbing lane ∗

Liu Yang a,b , , Jianlong Zheng a , Yang Cheng b , Bin Ran b a

Key Laboratory of Highway Engineering of Ministry of Education, Changsha University of Science & Technology, Changsha 410114, China b Department of Civil and Environmental Engineering, University of Wisconsin-Madison, Madison 53706, United States

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Article history: Received 4 April 2019 Received in revised form 19 June 2019 Available online 17 August 2019 Keywords: Freeway Uphill Climbing lane Heterogeneous traffic flow Cellular automata Lane-changing model

a b s t r a c t Traffic congestions frequently occur on uphill segments of four-lane freeways in China, which have become typical bottlenecks. Therefore, this paper focuses on the analysis, modeling, and simulation of heterogeneous traffic flow on uphill, in order to understand and eliminate such bottlenecks. The traffic characteristics were obtained from the realistic data, and a cellular automata model for longitudinal driving and lane changing was proposed and validated. The longitudinal driving rules were established based on the Nagel–Scheckenberg model. Lane changing was classified into active, inactive, and mandatory types which were used to clearly describe asymmetric lane-changing rules on two-lane segments and uphill with a climbing lane. The expressions of lane-changing motivation and safety were established. The measured results show that cars and 6-axle articulated trucks are the main types, and the speed difference between them is large. For normal slopes with a high truck ratio, even the total traffic is light, cars are unable to run freely. The simulated results prove that the realistic lane changing is asymmetric. The effects of uphill and climbing lanes on traffic flow are related to density. Setting up a climbing lane can alleviate or eliminate the uphill bottleneck effect. A critical density for distinguishing free flow from non-free flow does not exist on two-lane segments but exists on the uphill with a climbing lane. Vehicle segregation is significant under asymmetric lane-changing rules. The segregation degree is related to the traffic flow state. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In recent years, the number of heavy trucks has increased rapidly on freeways in China. Traffic congestions are not frequent on uphill segments with at least three lanes, but frequent on two-lane uphill segments which have become traffic bottlenecks. In China, the main heavy truck on freeways is 6-axle articulated trucks with a maximum mass of 49 tons and a power-mass ratio of 5.1 W/kg [1,2]. In the United States and Europe, the power-mass ratio of similar trucks is 10.0 and 8.3 W/kg, respectively [2]. Because of the huge difference in the performance of heavy trucks, the uphill traffic problem is much more serious in China than in the United States and Europe. At present, this problem has not been widely concerned. In the past, the researches focused on the uphill of two-lane highways rather than four-lane freeways. Guo et al. [3] measured the equilibrium speeds of 6-axle articulated trucks on different slopes in field experiments. Shim ∗ Corresponding author. E-mail address: [email protected] (L. Yang). https://doi.org/10.1016/j.physa.2019.122277 0378-4371/© 2019 Elsevier B.V. All rights reserved.

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et al. [4] compared crash occurrences before and after 30 climbing lanes on expressways were changed from pocket to overtaking types to evaluate the safety. Choi et al. [5] compared the performance differences between the pocket and overtaking climbing lanes on freeways by simulation, using total delay and collision time to evaluate efficiency and safety, respectively. Ko et al. [6] developed a dynamic control method for the climbing lane on expressways based on real-time traffic conditions and proved that it could improve mobility and safety by simulation. Hou et al. [7] collected data of collision, flow rate, and geometric parameters of 11 climbing lanes on three freeways, and analyzed the impact of setting up a climbing lane on collision frequency. Traffic flow is usually represented and simulated by traffic flow models which are classified into macroscopic and microscopic ones. It is difficult for macroscopic models to describe multi-lane heterogeneous traffic flow. Wang et al. [8] classified microscopic models into traffic engineering and statistical physics ones, and defined the latter as a class of models which use concise driving rules to simulate complex traffic phenomena. Cellular automata models belonging to statistical physics models are widely used because of simplicity, flexibility, and efficiency. Based on the 184th rule created by Wolfram [9], Nagel and Scheckenberg [10] created the famous one-lane cellular automata model named NS or NaSch model, which consists of four steps: accelerating, decelerating, moderating randomly and updating the location. By random moderation, it can reproduce some traffic flow phenomena such as spontaneous jams and stop-and-go traffic waves. Subsequently, the NS model was constantly modified by researchers to simulate more complex phenomena. Chowdhury et al. [11] and Maverivoet et al. [12] classified and summarized those modified NS models. Huang et al. [13] introduced a speed-dependent variable safe gap into the NS model. Jin et al. [14] calibrated a revised NS model by video trajectories and GPS data, and found that it is impossible to simulate all traffic flow phenomena with one model and fixed parameters. Guzmán et al. [15] revised the NS model based on the kinetics theory, vehicle capability, and driver reaction, which can avoid impulsive acceleration and reproduce car–truck interaction. At the same time, two-lane models were proposed continuously. Nagatani [16] created a two-lane model with deterministic driving rules. Rickert et al. [17] proposed a set of lane-changing rules based on the NS model, in which lane changing is expressed as motivation, safety, and randomness. Based on the Rickert’s rules, Chowdhury et al. [18] established a symmetric two-lane NS model named STNS model, in which both fast and slow vehicles are considered. It is used to reproduce and explain moving bottleneck phenomena proposed by Gazis et al. [19] for queuing after slow vehicles on multi-lane highways. Nagel et al. [20] summarized the two-lane traffic rules in the previous models. In order to solve the problem that the moving bottleneck effect is overestimated in two-lane models, Knospe et al. [21] pointed out that drivers are able to predict the front vehicle speed and accept a smaller gap than that in the NS model, which will reduce the probability and duration of moving bottlenecks. Jia et al. [22] added a honk effect to the STNS model. The honk effect is that a fast vehicle will remind a slow vehicle to give way by honking when blocked by the slow vehicle. Li et al. [23] added an aggressive lane-changing behavior of fast vehicles to the STNS model, which means that a fast vehicle will reduce the safety to increase the lane-changing probability when blocked by a slow vehicle. Kerner et al. [24] considered the influence of various driving behaviors and vehicle parameters on heterogeneous traffic flow. Li et al. [25] studied segregation phenomena based on the STNS model and found that the segregation is significant in free flow. Li et al. [26] took into account the mechanical restriction differences between cars and heavy trucks, as well as the active deceleration of trucks for eliminating moving bottlenecks. Yang et al. [27] modified the deceleration and lane-changing rules of the STNS model and established different models for cautious, normal and aggressive drivers. Xiang et al. [28] introduced dynamic lane-changing probability into the brake-light STNS model. In addition, Jiang et al. [29] studied the two-lane heterogeneous traffic flow composed of cars and buses based on the field data of urban roads. The above studies show that the STNS model has been widely improved in the past. However, the realistic lane changing on freeways could be asymmetric. Fukuda et al. [30] found that the lane changing from slow to fast lane differs from that from fast to slow lane on freeways in Japan. Furthermore, a cellular automata model containing a climbing lane has not been established in the existing literature. In addition, the freeway and vehicle characteristics are different in different countries. Consequently, it is necessary to investigate, analyze, model, and simulate the heterogeneous traffic flow on uphill based on China’s reality, which is helpful for understanding and eliminating the uphill bottleneck effect. 2. Analysis of freeway and vehicle characteristics in China 2.1. Freeway characteristics in China In most cases, the left lane is a fast lane, and the right is slow on four-lane freeways. Cars should run on an appropriate lane according to their speed. Trucks should run on the slow lane but are allowed to overtake on the fast lane. After overtaking, they should change back to the slow lane. The freeway longitudinal slopes in China are the same as those in the United States and Europe, which are all designed for 8∼9-W/kg trucks [2]. The speed limit and longitudinal slope of freeways with a design speed 120 km/h are as follows: the maximum speed limit of cars and trucks is 120 and 100 km/h, respectively, the minimum speed limit is 60 km/h, the maximum slope is 3%, the maximum length for 3% slope is 900 m, and there is no maximum length limit where the slope is less than 3% [2]. In this paper, uphill means a continuous and steep upslope. Both uphill and normal slopes have no exact definitions in the literature [2]. We define uphill as an upslope where the equilibrium speed of heavy trucks is lower than the allowable

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Fig. 1. Plane and profile of four-lane freeways with a climbing lane in China.

Fig. 2. Statistics of truck types on freeways in China in 2016 [1]. (Type 1 is 2-axle trucks, type 2 is 3-axle trucks, type 3 is 4-axle trucks, type 4 is 3-axle articulated trucks, type 5 is 4-axle articulated trucks, type 6 is 5-axle articulated trucks, and type 7 is 6-axle articulated trucks.) Table 1 Equilibrium speeds of 49-ton and 5.1-W/kg 6-axle articulated trucks on different slopes [3]. Longitudinal slope Equilibrium speed (km/h)

0.0% 104.4

0.5% 88.3

1.0% 75.0

1.5% 64.1

2.0% 55.3

2.5% 48.1

3.0% 42.3

3.5% 37.6

4.0% 33.8

4.5% 30.7

5.0% 28.1

minimum speed, which is 60 km/h when the design speed is 120 km/h. A slope that does not meet this definition is a normal slope. In the literature [2], it is advisable to set up a climbing lane on uphill of four-lane freeways, as shown in Fig. 1. The starting and ending points of the climbing lane are set at the locations where the truck speed damps and recovers to the allowable minimum speed. 2.2. Vehicle characteristics on freeways in China Fig. 2 [1] shows that the number, driven distance and freight turnover of 6-axle articulated trucks corresponding to type 7 account for 39.5%, 46.8% and 81.3% of trucks, respectively. Although the 2-axle trucks corresponding to type 1 account for 42.4% of trucks, the driven distance and freight turnover of them are far less than those of 6-axle articulated trucks. Moreover, they have a stronger climbing ability than 6-axle articulated trucks. Table 1 [3] shows that the equilibrium speed of fully-loaded 6-axle articulated trucks decreases rapidly with the increase of slope. When the slope is greater than 0.8% or 1.7%, the equilibrium speed is less than 80 or 60 km/h, respectively. In order to understand the vehicle characteristics, we investigated a two-lane segment of the G4 Freeway in Changsha, where the maximum speed limit is 120 km/h and the slope is 1.0%. The investigation period was from 8:00 to 11:00 a.m. when the weather was good. The average headway measured is 179 m which indicates a low traffic density. The statistics of vehicle types are shown in Table 2. Cars and 6-axle articulated trucks account for 79.7%. The average speed of cars is only 87.1 km/h, which indicates that most of them run in a non-free state. Other 2-axle vehicles consist of minibuses, buses, and light trucks. The last four types are heavy trucks which have at least 3 axles and account for 27.5%. The average speed of 6-axle articulated trucks is only 70.0 km/h, which is less than the equilibrium speed 75.0 km/h in Table 1, indicating that some of them do not run at full speed.

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L. Yang, J. Zheng, Y. Cheng et al. / Physica A 535 (2019) 122277 Table 2 Statistics of vehicle types on G4 Freeway. Statistical parameter

2-axle cars

Other 2-axle vehicles

3-axle trucks

4-axle trucks

3∼5-axle articulated trucks

6-axle articulated trucks

Flow ratio Average wheelbase (m) Average speed (km/h)

61.5% 2.71 87.1

11.0% 4.47 75.6

3.8% 6.66 67.8

3.9% 7.70 70.2

1.6% 10.89 71.1

18.2% 14.70 70.0

The above results show that cars and 6-axle articulated trucks are the main types. The speed difference between the two types is large. The power-mass ratio of fully-loaded 6-axle articulated trucks is 5.1 W/kg, but the longitudinal slopes are designed for 8∼9-W/kg trucks, which leads to their extremely low climbing speed. Even on normal slopes with a low density and high truck ratio, cars are unable to run freely because of the disturbance and blockage of slow trucks. 3. Modeling 3.1. Longitudinal driving model on normal slopes In order to make the longitudinal driving rules more realistic, we revised the NS model as follows: (1) The two steps for accelerating and decelerating are combined to one step for adjusting the speed; (2) The realistic maximum acceleration is considered; (3) The moderating deceleration is random. A vehicle will run according to the following rules. Step 1: Adjusting the speed. vexpect = min(vt + amax , vmax ),

(1)

vt+1 = min(vexpect , g).

(2)

Step 2: Moderating randomly. vt+1 = max(vt+1 − dmod , 0), dmod = randInt (1, amax )

if pm ≥ rand (0, 1).

(3)

Step 3: Updating the location. st+1 = st + vt+1 .

(4)

where, vexpect is the expected speed at time t + 1, vt and vt+1 are the speeds at time t and t + 1, respectively, amax is the maximum acceleration, vmax is the maximum speed, g is the front gap at time t, dmod is the moderating deceleration, randInt(1, amax ) is a function to take a random integer in the interval [1, amax ], pm is the moderating probability, rand(0,1) is a function to take a random decimal in the interval (0, 1), and st and st+1 are the locations at time t and t + 1, respectively. 3.2. Longitudinal driving model on uphill The maximum speed and acceleration of vehicles climbing on uphill are less than those running on normal slopes. The longitudinal driving rules on uphill are the same as Eqs. (2)–(4), but the expected speed is determined by the following equation.

vexpect

⎧ ( ) ⎨max vt − ddamp , vclimb = vclimb ⎩min (v + a t climb , vclimb )

if vt > vclimb if vt = vclimb if vt < vclimb

where, ddamp is the damping deceleration when climbing, vclimb is the maximum climbing speed, and aclimb is the maximum climbing acceleration.

(5)

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Table 3 Asymmetric lane-changing rules on two-lane segments. Vehicle type Cars Trucks

Lane-changing direction

Lane-changing type

Change Change Change Change

Active Inactive Inactive Active

to to to to

the the the the

fast lane slow lane fast lane slow lane

3.3. Free lane-changing model for active and inactive lane changing Lane changing is generally classified into free and mandatory types. According to the motivation, we further classified free lane changing into active and inactive types: (1) Active lane changing Active lane changing is that as long as the expected speed can be achieved on the adjacent lane or the driving condition on the adjacent lane is not worse than that on the current lane, the driver will have an active motivation to change to the adjacent lane; (2) Inactive lane changing Inactive lane changing is that only if the expected speed cannot be achieved on the current lane and the driving condition on the adjacent lane is better than that on the current lane, the driver will have an inactive motivation to change to the adjacent lane. The driving condition refers to the front gap on the current or adjacent lane. A larger gap means a better driving condition. A vehicle will change lanes successfully only if the following expressions of motivation, safety, and randomness are all satisfied. Step 1: Motivation. Active motivation: gother ≥ min(g , vexpect ).

(6)

Inactive motivation: g < min(gother , vexpect ).

(7)

Step 2: Safety. gotherBack ≥ gsafe , expect otherBack

(8)

.

(9)

pc ≥ rand (0, 1).

(10)

gsafe = v

Step 3: Randomness.

where, gother is the front gap on the adjacent lane at time t , gotherBack is the back gap on the adjacent lane at time t , gsafe is the lane-changing safety gap, expect votherBack is the expected speed of the back vehicle on the adjacent lane at time t + 1, and pc is the lane-changing probability. vexpect in Eqs. (6) and (7) is the same as Eq. (1). The speed and location of the vehicle should be updated according to Eqs. (2)–(4). gsafe should ensure that the vehicle does not collide with the back vehicle on the adjacent lane after lane changing. In the STNS model, gsafe is equal to the maximum speed of fast vehicles, which is conservative and reduces the lane-changing feasibility. In reality, drivers can predict the next speed of adjacent vehicles according to the vehicle type and running state. Therefore, gsafe in this model is equal to the expected speed of the back vehicle on the adjacent lane, which ensures no collision after lane changing, improves lane-changing feasibility, and maintains consistency with the longitudinal driving rules. 3.4. Asymmetric lane-changing rules on two-lane segments According to the realistic traffic rules, we assumed that the lane changing is asymmetric on two-lane segments, including normal slopes and uphill. The lane-changing rules can be expressed by active and inactive lane changing based on the vehicle type and lane-changing direction, as shown in Table 3. Cars tend to run on the fast lane, so it is active for them to change to the fast lane and inactive to the slow lane, whereas trucks do the opposite. We named the cellular automata model based on the rules as an asymmetric two-lane NS model.

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Fig. 3. Zones for lane-changing rules on the uphill with a climbing lane.

3.5. Mandatory lane-changing model Mandatory lane changing means that a vehicle must change lanes before arriving at a closed point. If the vehicle does not change lanes in time, it must decelerate, stop and wait. As long as the gap between the vehicle and the closed point is less than a critical gap, and the expected speed can be achieved on the adjacent lane or the driving condition on the adjacent lane is not worse than that on the current lane, the driver will have a mandatory motivation to change to the adjacent lane, which is expressed as follows. vexpect = max(vt − dman , 0),

(11)

gclosed < gcritical gother ≥ min(g , vexpect ).

(12)

∧

where, dman is the mandatory lane-changing deceleration, gclosed is the gap between the vehicle and the closed point at time t, and gcritical is the gap between the mandatory lane-changing starting point and the closed point. vexpect in Eq. (11) for mandatory lane changing is lower than that in Eq. (1) for free lane changing. Drivers expect to accelerate to the maximum speed in free lane changing, but they are willing to decelerate or even stop for changing lanes successfully in mandatory lane changing. Safety and randomness expressions are the same as Eqs. (8)–(10). The speed and location should be updated according to Eqs. (2)–(4). If mandatory lane changing occurs, the expected speed for updating the speed is the same as Eq. (1); otherwise, it is determined by the following equation. max (vt − dman , vmin )

{

vexpect = vmin

min (vt + amax , vmin )

if vt > vmin if vt = vmin lf vt < vmin .

(13)

where vmin is the minimum speed limit. This equation means that the vehicle will decelerate to, accelerate to or keep at vmin on its current lane. Decelerating to vmin can ensure that the vehicle has sufficient time and opportunities for lane changing before arriving at the closed point, and accelerating to or keeping at vmin can avoid that it blocks the traffic flow because of a very low speed. For example, it is mandatory for a truck to change from the converging transition to the slow lane. The closed point is the end of the converging transition. When the truck arrives at the starting point of the converging transition, it will try to change to the slow lane. If failing, it will continue to run on the converging transition at an appropriate speed and look for an opportunity to change lanes. When arriving at the end, it will stop and wait. 3.6. Asymmetric lane-changing rules on the uphill with a climbing lane We divided the uphill segment into 7 zones shown in Fig. 3 to describe the lane-changing rules, which are expressed by active, inactive and mandatory lane changing based on the zone, vehicle type, and lane-changing direction, as shown in Table 4. The rules on zones 1, 2 and 7 are the same as those in Table 3. On zone 3, it is active for trucks to change to the diverging transition and inactive to the fast lane, and cars are forbidden to change to the diverging transition. Because the lane marking between zones 4 and 5 is a solid line, all vehicles are forbidden to change lanes between the two zones. On zone 4, cars and trucks are forbidden to change to the right. On zone 5, trucks are forbidden to change to the left. On zone 6, it is mandatory for trucks to change to the slow lane. 4. Experiments and results Based on the Visual C++ platform, we developed a cellular automata program to simulate the traffic flow of four-lane freeways. The program can be used to set experimental parameters conveniently, simulate traffic flow efficiently, output statistical results perfectly, and display spatiotemporal patterns in real-time.

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Table 4 Asymmetric lane-changing rules on the uphill with a climbing lane.

Table 5 Vehicle settings in the validation. Vehicle type

Length

Maximum speed

Maximum acceleration

Cars Trucks

4 cell (5.6 m) 13 cell (18.2 m)

24 cell/s (121.0 km/h) 15 cell/s (75.6 km/h)

5 cell/s2 (7.0 m/s2 ) 2 cell/s2 (2.8 m/s2 )

Table 6 Comparison between the measured and simulated results on G4 Freeway. Statistical result

Flow rate (veh/h)

Measured result 933 Simulated result 943 Relative error 1.1%

Speed (km/h)

Speed of cars (km/h)

Speed of trucks (km/h)

Speed of fast lane (km/h)

Speed of slow lane (km/h)

80.6 81.5 1.1%

84.7 84.3 0.5%

69.7 74.0 6.2%

88.0 86.1 −2.2%

74.2 75.9 2.3%

4.1. Basic experimental settings The cell length is 1.4 m. The experimental freeway is 7000 m long, with a 120 km/h design speed, a 120 km/h maximum speed limit, and a 60 km/h minimum speed limit. The freeway boundary is periodic. Only cars and 6-axle articulated trucks were considered in experiments. We assume that all cars do not exceed the speed limit and all trucks are fully-loaded, so the maximum speed of cars is 120 km/h, and that of trucks is the equilibrium speed in Table 1. The initial distribution of vehicles is uniform in space, and no vehicles are initially distributed on the climbing lane. The initial speed of vehicles is 0. The moderating probability is 0.2, and the lane-changing probability is 0.8. The simulation time step is 1 s. We conducted 200 simulations runs. Each simulation ran for 3000 s, and the data from 2000 to 3000 s were collected and used for the following analysis. 4.2. Validation The model was validated by the field data of the G4 Freeway. The slope is 1.0%. The 2-axle vehicles and the other vehicles in Table 2 are classified as cars and trucks, respectively. Their parameters are shown in Table 5. The traffic conditions are set as ρ = 5.8 veh/km and R = 0.275 where ρ is the single lane density and R is the truck ratio. In reality, the freeway boundary is open, and each vehicle of the same type has different parameters, such as the maximum speed, maximum acceleration, moderating probability, and lane-changing probability. In the simulation, the freeway boundary is periodic, and each vehicle of the same type has the same parameters. The difference between

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simulation and reality will lead to errors. Table 6 shows that the relative errors of flow rates and average speeds are small, which prove that the longitudinal driving model, lane-changing model, asymmetric lane-changing rules, and parameter settings are valid. 4.3. Simulations and discussions We simulated three freeways, one containing only a normal slope segment, one containing a normal slope segment and an uphill segment without a climbing lane, and another containing a normal slope segment and an uphill segment with a climbing lane. The normal slope is 0.8%. The uphill length and slope are 2800 meters and 2.3%, respectively. The climbing lane is 2800 meters long. The other geometric parameters are set based on the literature [2], as shown in Fig. 4. The vehicle parameters are shown in Table 7. The mandatory lane-changing deceleration in Eq. (11) is equal to the maximum acceleration.

Fig. 4. Geometric settings of the freeway with a climbing lane in the simulation. Table 7 Vehicle settings in the simulation. Vehicle type

Length

Maximum speed

Maximum acceleration

Maximum climbing speed

Maximum climbing acceleration

Damping deceleration

Cars

4 cell (5.6 m)

24 cell/s (121.0 km/h)

5 cell/s2 (7.0 m/s2 )

22 cell/s (110.9 km/h)

4 cell/s2 (5.6 m/s2 )

1 cell/s2 (1.4 m/s2 )

13 cell (18.2 m)

16 cell/s (80.6 km/h)

2 cell/s2 (2.8 m/s2 )

10 cell/s (50.4 km/h)

1 cell/s2 (1.4 m/s2 )

1 cell/s2 (1.4 m/s2 )

Trucks

The traffic conditions are set as ρ = 0∼25 veh/km and R = 0.2. It is generally believed that R ≥ 0.2 represents a high truck ratio [2] and 25 veh/km is close to the critical density corresponding to the maximum flow rate. The statistical space of the three freeways is the segments of the normal slope, two-lane uphill and three-lane uphill, respectively. It is noted that three-lane uphill refers to uphill with a climbing lane. The differences between the three segments in the flow rate, speed, lane-changing frequency, and conflict frequency were analyzed as follows. Figs. 5 and 6 show that the flow rate and average speed of cars on the two-lane uphill are significantly lower than those on the normal slope. On these two segments, the flow rate curves of cars are approximately linear, and the average

Fig. 5. Flow rates of cars on different segments (R = 0.2).

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Fig. 6. Average speeds of cars on different segments (R = 0.2).

Fig. 7. Flow rates of fast lane on different segments (R = 0.2).

speed of cars decreases rapidly with the increase of density when ρ < 10 veh/km, which indicates that cars are unable to run freely even at a low density. After the climbing lane is set up, the flow rate curve of cars is nonlinear; the flow rate and average speed of cars increase dramatically; the average speed is greater than 108.0 km/h when ρ < 10 veh/km, which indicates that cars can run freely at a low density. The experimental results show that the average speed of trucks is constant at about 79.0 km/h on the normal slope and 50.0 km/h on the uphill when ρ < 25 veh/km. The flow rate-density curve of the fast lane is shown in Fig. 7 and its average speed-density curve is very similar to that in Fig. 6. The average speed-density curve of the slow lane is shown in Fig. 8. The flow rate and average speed of the fast and slow lanes on the two-lane uphill are significantly lower than those on the normal slope. On these two segments, the flow rate curves of the fast lane and the average speed curves of the slow lane are approximately linear. After the climbing lane is set up, the flow rate curve of the fast lane and the average speed curve of the slow lane are nonlinear; the flow rate of the fast lane and the average speeds of the fast and slow lanes increase dramatically; the average speeds of the fast and slow lanes are about 107.0 and 106.0 km/h when ρ = 10 veh/km, respectively. The results indicate that cars can run freely on the two lanes at a low density.

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Fig. 8. Average speeds of slow lane on different segments (R = 0.2).

Fig. 9. Lane-changing frequency of cars on different segments (R = 0.2).

Fig. 9 shows that the lane-changing frequency of cars on the two-lane uphill is higher than that on the normal slope when ρ > 3 veh/km, and the maximum value appears at ρ = 5 veh/km on these two segments. Fig. 10 shows that the lane-changing frequencies of trucks on these two segments are the same. After the climbing lane is set up, the lanechanging frequency of cars reaches the maximum at ρ = 15 veh/km, and that of trucks decreases dramatically. On the three segments, the lane-changing frequency of cars is much higher than that of trucks. We defined an event that a vehicle decelerates at least twice its maximum acceleration as a conflict. The conflict frequency can reflect the collision risk between the front and back vehicles. Figs. 11 and 12 show that both cars and trucks have more conflicts on the two-lane uphill than on the normal slope. On these two segments, the conflict frequency of cars increases rapidly with the increase of density when ρ < 10 veh/km. After the climbing lane is set up, the conflict frequencies of cars and trucks decrease dramatically and approach 0 when ρ < 10 veh/km. On the three segments, the conflict frequency of cars is much higher than that of trucks.

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Fig. 10. Lane-changing frequency of trucks on different segments (R = 0.2).

Fig. 11. Conflict frequency of cars on different segments (R = 0.2).

Fig. 13 shows that trucks run mostly on the climbing lane and rarely on the fast lane. When ρ = 10 veh/km, 0.5%, 1.5% and 98.0% of trucks run on the fast lane, slow lane, and climbing lane, respectively. As the density increases, more trucks run on the fast and slow lanes. From the above analysis, it can be concluded that when R = 0.2, the critical density for distinguishing free flow from non-free flow does not exist on the two-lane segments, but it is near 10 veh/km on the uphill with a climbing lane. In order to compare the spatiotemporal characteristics of traffic flow on the three segments, we studied the spatiotemporal patterns when ρ = 10 veh/km and R = 0.2. Fig. 14 is the typical spatiotemporal patterns of traffic flow on the two-lane uphill. Although trucks run mostly on the slow lane, their frequent overtaking blocks cars and brings about moving bottlenecks. The formation and dissipation of moving bottlenecks are random in space and time, and there is a huge gap in front of them. The traffic flow is in an unstable state, changing irregularly between unblocked and blocked states. The experiment results show that the traffic flow on the normal slope also presents the same feature, as in Fig. 14.

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Fig. 12. Conflict frequency of trucks on different segments (R = 0.2).

Fig. 13. Distribution of trucks on the uphill with a climbing lane (R = 0.2).

On the uphill with a climbing lane, most trucks enter the climbing lane. As a result, the driving conditions on the fast and slow lanes are greatly improved. The traffic flow is usually in an unblocked state as shown in Fig. 15(a–c). Fig. 15(d–f) shows that the traffic flow is disturbed occasionally by the trucks running on the fast lane, but the spatiotemporal effect of the disturbance is local and transient, which will not cause a moving bottleneck, so the traffic flow is in a stable state. Figs. 14 and 15 show that the asymmetric lane-changing rules lead to obvious vehicle segregation, including that most cars run on the fast lane, most trucks run on the slow lane or climbing lane, cars mainly follow cars, and trucks mainly follow trucks. The traffic flow is composed of car groups and truck groups. The segregation of unblocked traffic flow is more significant than that of the blocked, which means that the segregation degree is closely related to the traffic flow state. The above results show that the concise model can reflect the essential characteristics of heterogeneous traffic flow, so it is valid. However, the real traffic is very complex, which determines that the model may not be perfect, and there are various errors in the simulated results. For example, the driving performance of trucks is different from each other in reality, which will affect the driving behaviors, but this difference is ignored in the simulation. It is necessary to analyze

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Fig. 14. Typical spatiotemporal patterns in different states on the two-lane uphill (ρ = 10 veh/km, R = 0.2, blue for cars, and magenta for trucks).

the errors by collecting more traffic data and to improve the model by considering more details. The qualitative simulated results have more generality than the quantitative results, which only reflect the case of R = 0.2. The impact of truck ratio on traffic flow needs further study.

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Fig. 15. Typical spatiotemporal patterns in different states on the uphill with a climbing lane (ρ = 10 veh/km, R = 0.2, blue for cars, magenta for trucks, and red for non-existent cells).

5. Conclusions and future work In this paper, we investigated and analyzed the realistic freeway and vehicle characteristics in China. The longitudinal driving rules on normal slopes and uphill were revised based on the NS model. According to the motivation, we classified lane changing into active, inactive, and mandatory types, and proposed their motivation expressions. According to the classification, the asymmetric lane-changing rules on two-lane segments and uphill with a climbing lane were clearly described. Lane-changing safety was improved to be consistent with the longitudinal driving rules. The model was validated by the field data. The differences in the heterogeneous traffic flow between a normal slope, an uphill without a climbing lane, and an uphill with a climbing lane were analyzed by simulation. Findings and conclusions are summarized as follows. (1) Cars and 6-axle articulated trucks are the main types on freeways in China. The speed difference between the two types is large. Because 6-axle articulated trucks have a weak climbing ability which is unfit for such longitudinal slopes, they are the main cause of congestions on uphill. (2) For normal slopes with a high truck ratio, even the total traffic is light, cars are unable to run freely because of the disturbance and blockage of slow trucks. (3) The classification, expressions, and rules of lane changing are valid, and the realistic lane changing is asymmetric. (4) The effects of uphill and climbing lanes on traffic flow are related to density. Setting up a climbing lane can dramatically improve the average speed of cars, decrease the lane-changing frequency of trucks, and reduce the conflicts between cars and trucks. The critical density for distinguishing free flow from non-free flow does not exist on two-lane segments, but it exists on the uphill with a climbing lane. (5) On two-lane segments, moving bottlenecks occur frequently and randomly, and the traffic flow is in an unstable state. Setting up a climbing lane can alleviate or eliminate moving bottlenecks obviously and keep the traffic flow in a stable state. Vehicle segregation is significant under asymmetric lane-changing rules, which presents that the vehicles of different types run on different lanes and the vehicles of the same type run in groups. The segregation degree is related to the traffic flow state.

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(6) The performance of heavy trucks or the longitudinal slope of freeways should be improved to coordinate with each other. Building or rebuilding a climbing lane should be demonstrated for the uphill with frequent congestions. In the future, we will investigate and analyze the field data on the uphill without and with a climbing lane to improve the model. In addition, we will consider the differences between the vehicles of the same type and study the impact of truck ratio on traffic flow. Acknowledgments This work is supported by the National Natural Science Foundation of China (Nos. 51338002 and 51678075), Changsha University of Science & Technology, China (No. 2019IC09), Education Department of Hunan Province, China (No. 16B008), Open Fund of the Key Laboratory of Highway Engineering of Ministry of Education (Changsha University of Science & Technology, China) (No. KFJ130101), and China Scholarship Council. 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