2.1. Impact of Adverse Weather on Traffic Flow

Research on the impact of adverse weather on traffic flow mainly focuses on analyzing the impact of adverse weather on traffic safety and the impact on macrotraffic flow characteristics. Some scholars also study how to modify the parameters of the simulation model.

The severity of the impact of adverse weather on traffic safety can be qualitatively and quantitatively analyzed based on historical accident data. Ghasemzadeh et al. identified the influencing factors of crash severity in work areas under different space, time, and environmental conditions and found that weather and lighting conditions are one of the most important influencing factors [5]. Pennelly et al. used Canadian actual traffic collision data and related weather data, analyzed the relationship between severe weather and the number and severity of traffic accidents, and found that severe weather affects traffic conflicts in various ways [6]. Fior et al. quantified the impact of severe weather on vehicle safety based on the relationship between historical accident data and meteorological data in specific dangerous areas [7].

The operating efficiency of traffic flow decreases under bad weather, and changes in macroscopic traffic flow characteristics are manifested in reduced flow and speed. Zhang et al. quantitatively studied the impact of rainfall on traffic flow characteristics of urban freeways. The results showed that the Greenshields model is the most suitable macroscopic basic graph model for rainfall weather. Under light rain, moderate rain, and heavy rain, the road capacity decreases by 7.26%, 10.87%, and 17.09%, respectively, and the free flow velocity decreases by 3.07%, 5.29%, and 6.64%, respectively [8]. Rijavec et al. studied the impact of severe weather conditions on lane flow distribution and vehicle speeds in different lanes. The results showed that lane flow distribution and lane speed distribution at a specific location depend on weather conditions [9]. Ivanovic et al. introduced a method to analyze the impact of adverse weather on the urban transportation system, studied the changes in transportation demand and transportation supply characteristics caused by adverse weather, and quantified the impact of weather on the transportation system [10]. Roh et al. established a method based on a dummy variable regression model. The winter weather model was used to quantify the impact of snowfall and temperature changes on traffic flow in Alberta, Canada [11].

The revised simulation model can better describe the traffic flow characteristics under adverse weather conditions. Hammit et al. evaluated the driver’s car-following behavior in various adverse weather conditions (rain, snow, and fog) and calibrated the Gipps car-following model (Gipps model [GM]). The results showed that under different adverse weather conditions, there is an intradriver heterogeneity [12]. In order to effectively simulate traffic under snowy road conditions, Asamer et al. calibrated the car-following model parameters [13]. Gong et al. developed a mixed traffic cellular automaton model considering limited visual distance and explored the impact of visibility level and AV market penetration on traffic efficiency. The results showed that when visibility is low, the introduction of AV driving is conducive to improving the efficiency of mixed traffic. 100% AVs can increase traffic capacity by 182% in dense fog environments [14].

2.2. Freeway Traffic Flow Characteristics

Research on the characteristics of freeway traffic flow mainly focuses on the distribution model of traffic flow parameters. Some scholars also study the flow–speed–density basic diagram of traffic flow in different scenarios. Some scholars also focus on the flow, speed, and delay of traffic flow study of macroscopic characteristics.

The research on the distribution characteristics of traffic flow parameters under normal weather is relatively mature. In recent years, there have been more and more studies on the parameter distribution characteristics under bad weather. Based on actual observed traffic flow data on freeways, Wang et al. studied the probability distribution and cumulative probability distribution of microscopic traffic flow parameters (such as speed, acceleration, headway, and lateral offset) in rainy weather environments [15]. Wang et al. relied on freeway traffic flow and rainfall data to study the speed and density distribution of traffic flow under different rainfall conditions, thereby calibrating the speed and density distribution model of traffic flow in different rainy days [16]. Regarding the study of basic traffic flow diagrams, Guan et al. explored the basic speed–density relationship of urban freeways by analyzing the correlation between speed distribution and density. Regarding the research on the macroscopic characteristics of traffic flow [17], Yue et al. proposed a two-lane freeway vehicle microsimulation model to evaluate the delay, speed, flow, and density of the traffic flow. The experimental results verified the effectiveness of the model [18].

With the development of AVs, some scholars have analyzed and compared the differences in basic diagrams and macroscopic characteristics of AV and HV traffic flows, and some scholars have studied the improvement of traffic efficiency brought about by AVs.

Compared to HVs, the operating efficiency of AV traffic flow has been significantly improved. As the penetration rate of AV increases, its advantages in terms of traffic capacity, travel speed, delays, etc. will become more and more significant. In order to study the impact of commercial AVs on traffic flow, Shi et al. proposed a general method that combines empirical experiments and theoretical models. They used high-resolution trajectory data of multiple commercial AVs to conduct simulation experiments and found that the traditional basic diagram is still suitable for describing the traffic flow characteristics of AV traffic, and the road capacity is significantly improved [19]. Guo et al. considered the situation of vehicle on-ramp merging and off-ramp separation and studied the impact on mixed traffic flow as the market penetration of connected AVs increases [20]. Through VISSIM microsimulation, Beza et al. studied the impact of AVs and connected AVs on traffic capacity, average traffic speed, average travel time per vehicle, and average delay per vehicle. The simulation results show that under the 100% market penetration rate of the basic freeway section, the traffic rate of connected AVs has increased more than that of AVs [21]. Obai et al. studied the expected impact of AVs on road traffic parameters from a macrolevel. The traffic parameters studied include daily kilometers traveled, daily driving hours, daily total delay, and average driving speed of the road network [22], Alozi et al. employed the microsimulation of the Dubai Freeway section in the United Arab Emirates, and quantified the impact of AV penetration on the average delay, average speed, and total travel time of traffic flow [23]. Kavas et al. constructed AV and non-AV′s model based on simulation andstudied the impact of AV on the average speed and fluidity of traffic flow on the road [24].

2.3. Freeway Speed Limit Strategy

Research on freeway speed limit strategies focuses on variable speed limit control, combined with ramp and mainline control to improve traffic capacity.

Variable speed limits are the most common control strategy for freeways, and can achieve good results when combined with ramp control. In recent years, changes in visibility, road adhesion coefficient, etc. caused by adverse weather have been considered in speed limit strategies. In order to cooperatively control the traffic flow on ramps and mainlines, Carlson et al. proposed two types of variable speed limits (VSL) local mainstream traffic flow control (MTFC) feedback controllers. The results show that the feedback controller has satisfactory control behavior and is close to the optimal control result [25]. Jiang et al. studied the effectiveness of the weather response management strategy based on connected vehicles on freeway corridor safety issues under adverse weather through analysis, modeling and simulation. Among them, the emergency response management strategy based on connected vehicles was applied to forward collision warning, Three aspects advance lane change suggestions and variable speed limits [26]. Guo et al. proposed a variable speed limit control strategy under severe weather conditions. According to the impact of rainfall intensity under rainy conditions, visibility under foggy conditions, and road adhesion coefficient under snowy conditions on traffic capacity, severe weather conditions are determined. The results show that the variable speed limit control strategy under severe weather conditions effectively improves road traffic capacity and reduces vehicle driving risks [27]. Tian et al. studied the classification of severe weather and the identification of dangerous sections in Inner Mongolia, used the evaluation point method to calculate the risk value under adverse weather conditions, and proposed control measures such as traffic diversion, traffic organization, speed limit, and road closure [28].

3.1. Construction of the Dynamic Speed Limit Model on Rainy Days

The safe distance between vehicles directly determines driving safety. In rainy weather, the driver’s visual distance and the perception range of AV are affected. If the visual distance and perception range are smaller than the safe distance between vehicles, the vehicle may be affected by braking. Insufficient moving distance leads to traffic accidents. Therefore, safe driving distance should be considered when setting the safe speed limit on freeways in rainy days.

In rainy weather, as the rainfall intensity, friction coefficient, and the slope of the road where the vehicle is located change, the safe speed limit’s value will also change [29]. In order to ensure that the vehicle always drives safely, it needs to be based on the current rainfall intensity, friction coefficient, water film thickness, road visibility, etc., to calculate the safe speed limit value in real time, so that the safe speed limit value can dynamically adapt to changes in the driving environment.

In rainy weather, the visual distance of both manual and AVs will be affected. For manually driven vehicles, road visibility can be considered as the driver’s visual distance, that is, S_c = S_v, and the maximum value is 300 m. Generally speaking, when the rainfall intensity is relatively small, road visibility does not affect the driver’s safe driving speed. The safe driving speed depends on the following status and the road speed limit. However, when the rainfall intensity increases to a certain value, the vehicle’s visual distance will decrease, which directly affects the safe speed limit of the vehicle.

For AVs, the visual distance of the vehicle in rainy weather depends on the sensing distance of the sensor and communication equipment. First of all, for millimeter-wave radar, the performance will not be greatly affected. According to Table 1, the detection distance under normal circumstances is 250 m; second, for cameras, on rainy days, due to raindrops blocking the lens or interfering with imaging, there will be a larger high missing rate and low confidence, but the rain removal algorithm can be used to alleviate this situation. According to Table 1, under normal circumstances, the detection distance of the camera is 250 m. After the rain removal algorithm, its performance is almost unaffected [31]; again, for radar, rainfall has little impact on detection distance, but due to rain coverage, the solid surface is smooth, resulting in a reduction in reflectivity when the laser reaches obstacles and vehicle body surfaces, reducing to ensure the accuracy of the measurement; finally, for dedicated short range communications (DSRCs), Lee tested the performance of the vehicle-mounted DSRC on rainy days through real vehicle experiments, and the performance was stable and good within 150 feet [32].

Generally speaking, cameras and millimeter-wave radar can capture obstacles and vehicle information even on rainym days and are almost unaffected by changes in rainfall intensity. Although lidar has some effects, it is feasible to use it as an information auxiliary means. The short-distance sensing performance of DSRC is very stable on rainy days. Therefore, through analysis, it can be seen that the sensing distance of an AV in light or moderate rain is determined by millimeter-wave radar, camera, and lidar. In order to ensure safety, it is set to 200 m. In heavy rain and heavy rain, the sensing distance depends on DSRC, set to 47.5 m.

According to the rainfall intensity, the relationship with road visibility is to further refine the existing rainfall levels, using the visual distance of 200, 100, and 50 m as the boundaries to find the rainfall intensity interval corresponding to each level. Among them, the transverse slope gradient is taken as 0.02°, and the longitudinal slope is taken as the most unfavorable case of −0.03°. The results are shown in Table 1.

Through analysis, it was found that AVs have a larger safe speed limit than HVs under any rainfall level.

For HVs, when there is light rain or moderate rain, that is, when the visual distance exceeds 200 m, a higher safe speed limit can be maintained, with a value range of 93–120 km/h. However, when the weather is heavy rain or heavy rain, that is, when the visual distance is less than 200 m, considering the restrictions of traffic regulations, its safe speed limit value drops sharply.

For AVs, when there is light rain or moderate rain, the vehicle’s visual distance is almost unaffected and can maintain the expected speed of 110–120 km/h. However, when the weather is heavy rain or heavy rain, the sensor and communication equipment are affected and the visual distance drops sharply, causing the expected speed to also drop sharply, but the expected speed can still be maintained at around 64 km/h.

In this model, “dynamic” is reflected in the adaptability to rainfall conditions and road geometric design. Specifically, the setting of the safe speed limit value of the freeway in this study should be determined based on the current rainfall intensity and road longitudinal slope, that is, the speed limit value under each rainfall intensity of each road section is changing, dynamically adapting to the longitudinal slope and rainfall intensity of the current road section.

3.2. Improvement of the IDM–Following Model

The microscopic driving dynamics of the vehicle are determined by the interaction of the car-following model, the intersection model, and the lane-changing model. Specifically, the car-following model determines the vehicle’s speed relative to the vehicle in front; the intersection model determines the vehicle’s behavior in terms of right-of-way rules, gap acceptance, and congestion avoidance at the intersection; and the lane-changing model determines the vehicle’s behavior on multilane roads, lane selection, and lane-changing speed adjustment.

This study chooses freeways as the road scene for research. The scope of road scenes is limited to freeway sections. The impact of on- and off-ramps on the mainline is not considered. The lane change selection is mainly free lane change, and because the freeway intersections are all in the form of interchanges, so there is no complex right-of-way issue of intersection conflict. Therefore, this study mainly improves the car-following model and uses a suitable car-following model to describe the microscopic driving behavior of vehicles in rainy freeway scenes.

In the expression of expected speed, the static safety distance s₀ is only meaningful for slow-moving vehicles when the traffic flow is in a congested state; Tv means that in a stable traffic flow, the vehicle follows the vehicle in front with a constant safe headway. vΔv(2∗ab) is mainly suitable for unstable traffic flow. This item ensures that the vehicle can brake under any circumstances, even in emergency braking conditions. In order to make the model model more intuitive and easier to measure, this item is usually set to 0.

The IDM model after calibration using the traffic flow data of the Xi’an Freeway section under normal and rainy weather is shown in Table 2. By comparing the parameter changes under normal and rainy weather, it is found that rainfall has the most obvious impact on the expected speed, while the maximum acceleration, comfortable deceleration, and safe time interval change slightly. Therefore, it is considered that the expected speed is the main affected parameter, and the maximum acceleration, comfortable deceleration, and safe time interval are the secondary affected parameters. At the same time, the stationary safety distance is the distance between the vehicle and the vehicle in front when it is stationary, and it is considered to remain unchanged on rainy days.

This model is suitable for HV and AV car-following models; however, the parameters are not exactly the same. For maximum acceleration, comfortable deceleration, and safe time intervals, it can be considered as the impact of vehicle performance on rainy days. The parameter values of the two are the same, and the values change with the change in rainfall level.

However, the expected speeds of human and automatic car-following models are not the same. First, the main control body of artificially driven vehicles is human drivers, and human drivers vary in age, gender, driving experience, and other physiological characteristics, resulting in different driving habits between drivers, so there are differences between vehicles. Qualitatively, the expected speed should obey the normal distribution; and the control of AVs is completely based on intelligent algorithms, and the driving behavior is highly consistent [36]. Therefore, it can be considered that AVs belong to the same driving style. In the IDM model, expected speed is a constant value. Second, compared to HV, AV has a shorter reaction time and shorter reaction distance. Therefore, under the same rainfall intensity and visual distance conditions, the expected speed of AV is greater.

3.3. Improved Expected Speed Optimization of Car-Following Models

3.3.2. Expected Speed Considering AV’s Time Headway

The functions that can be achieved include using onboard sensors to detect the position of the vehicle ahead, using communication equipment to share information such as location and speed, and controlling vehicle movement trajectories in real time. This paper designs an expected speed optimization method based on the headway based on the vehicle’s motion status and on a rainy day, aiming to keep the driving speed of AV in the best state.

From a microscopic perspective, the driving states of vehicles on the road can be divided into three categories: free state, following state, and braking state.

• 1.

  The free state means that there are no other vehicles within a certain range in front of the lane where the vehicle is located, the driving state is not restricted, and the vehicle can drive freely on the road at the expected speed.

• 2.

  The following state means that there are other vehicles in front of the vehicle in the lane, which restricts the driving state of the vehicle. The vehicle needs to adopt the corresponding speed based on the position, speed, and other information of the vehicles in front and behind.

• 3.

  The braking state refers to a state in which the vehicle is following the vehicle in front and the vehicle in front suddenly decelerates or the speed difference between the front and rear vehicles is too large. In order to avoid a collision, the vehicle has to apply emergency braking.

This study uses headway as the standard to judge the vehicle motion state [29]. The critical points of the three states are free critical headway t_max and braking critical headway t_min. Specifically,

• 1.

  If the headway is equal to the headway t > t_max, the vehicle is in a free state

• 2.

  If, then the vehicle is in the following state

• 3.

  If t < t_min, the vehicle is in a braking state

In addition, when a vehicle is in a car-following state, in order to pursue the maximum driving speed, it often chooses to keep the desired time t_des distance. The driving status division of the vehicle is shown in Figure 1.

Vehicles will pursue driving with the highest efficiency under different motion states, that is, pursuing driving at the expected speed. However, in order to ensure driving safety, the expected speed cannot exceed the maximum speed. In rainy weather, the expected speed of the vehicle depends on the intensity of the rainfall, the vehicle’s following status, and the road design speed, which, respectively, correspond to the safe speed limit under the corresponding rainfall intensity V_R and the maximum expected time distance in the following mode. The safe speed V_F, road design speed V_L, and expected speed should be based on the minimum of all speed limits.

In rainy weather, for AVs, the dynamic control process of the vehicle is shown in Figure 1.

• 1.

  The vehicle detects the position of the vehicle in front based on the rainfall intensity and the sensing distance under the rainfall intensity. If the vehicle in front is not within the sensing distance, the vehicle is in a free state. V_L The expected speed takes V_R the smaller value of and. If the vehicle in front is within the sensing distance, we calculate the headway between your vehicle and the vehicle in front t.

• 2.

  We determine whether the headway t is greater than the free critical headway t_max. If so, t > t_max, the vehicle is still in the free state. If so t < t_max, continue to compare with the braking critical headway.

• 3.

  We determine t whether the headway is less than the critical braking headway. If t < t_min, the vehicle is in the braking state, V_L and V_F the expected speed is V_R the minimum V_F; if t > t_min, the vehicle is in the following state.

  $$ ()

•  

  where V_fo is the following vehicle speed (m/s) and V_ld is the speed of the leading vehicle (m/s).

• 4.

  When the vehicle is in the following state, it is judged whether the headway t reaches the expected headway t_des. If it has reached the expected headway, we return to the next calculation step. Otherwise, the expected speed of the vehicle V_R is V_L taken as V_F minimum value.

3.4. Freeway Speed Limits on Rainy Days

Since the speed limit value lacks adaptability to changing rainfall intensity and road gradient, it cannot be dynamically adjusted according to objective driving conditions, resulting in reduced reliability of the speed limit value. In order to achieve refined management of freeway speed limits on rainy days, it is necessary to design speed limits for different road sections based on real-time rainfall intensity and road slope.

The freeway is divided into sections according to the longitudinal slope, as shown in Figure 2. The speed limit value of each section is different. Among them, the maximum speed limit is the design speed of 120 km/h, and the speed limit value should be 10 km/h, which is an integer multiple of the speed limit difference between two adjacent road sections, which should not be greater than 20 km/h.

4.1. Simulation results of the Speed Limiting Method

In order to verify the effectiveness of the speed limit method in improving traffic safety in rainy weather, simulation experiments need to be conducted to compare the safety of the speed limit scheme and the unlimited speed scheme. At the same time, in order to put forward reasonable safe speed limit recommendations for freeways in rainy weather, simulation experiments are needed to analyze how to set reasonable speed limit values under different longitudinal slopes and different rainfall intensities.

4.1.2. Speed Limit Value Sensitivity

Simulation experiments are used to analyze how to set reasonable speed limit values under different longitudinal slopes and different rainfall intensities. According to the classification of rainfall levels and vehicle types, the safe speed limit value curves under different longitudinal slopes and different rainfall intensities are drawn, as shown in Figure 3. It can be found from the Figure 3 that

• 1.

  During light/moderate rain, the speed limit for manually driven vehicles remains above 90 km/h, while the speed limit for AVs remains above 110 km/h.

• 2.

  During heavy/torrential rains, the speed limit of manually driven vehicles has a larger span, dropping from 100 to 35 km/h, while the safe speed limit of AVs has a smaller span, remaining at 64 km/h, within the range of 67 km/h.

• 3.

  When the longitudinal slope is 0.01°, both HVs and AVs have the highest safe speed limit. When the longitudinal slope is −0.03°, both the safe speed limits are the smallest, which is the most unfavorable driving condition.

4.2. Analysis of Traffic Flow Characteristics of HV and AV Under the Speed Limit Method

Rainfall not only reduces the traffic efficiency but also increases the risk of traffic accidents and worsens traffic flow stability due to the heterogeneity of HVs. Therefore, this section starts from the three aspects of traffic efficiency, stability, and safety and analyzes the traffic flow characteristics of HV and AV under this speed limiting method based on the rainy days freeway speed limit method and the SUMO simulation platform.

In this experiment, the freeway in the simulation scenario was designed as a two-way 6 lane, with a lane width of 3.75 m, a design speed of 100 km/h, a cross slope of 0.02°, and the most unfavorable longitudinal slope of −0.03° on a sunny day. The friction coefficient is 0.05, the freeway sections are all set as rain areas, and the length is set as 5 km. The SUMO simulation warm-up time is 400 s, and the period for statistics of simulation results using the traffic control interface (TraCI) is from 400 to 4000 s.

4.2.1. Traffic Efficiency

The differences in visual distance, reaction time, and expected speed between manual and AVs will be reflected in the operating efficiency of traffic flow. AVs have higher expected speeds on rainy days, which will make the average travel speed of the entire traffic flow higher. High, average travel time is shorter. Therefore, the average travel speed of the road section, the average travel time of the road section, the average delay of the road section, and the proportion of delay to travel time are used as the efficiency evaluation indicators of traffic flow.

The average travel speed of a road section refers to the ratio of the sum of the average travel speeds of all vehicles passing through the road section to the number of vehicles on a certain road section as

$$ ()

where $$ is the average travel speed of the road section (km/h), $$ is the average travel speed of the i-th vehicle (km/h), and n is the number of all vehicles passing through the road segment (vehicles).

The average travel time of a road segment refers to the average of the actual travel time of all vehicles passing through the road segment, represented as

$$ ()

where $$ is the average travel time of road section (s) and t_i is the actual travel time of the i-th vehicle (s).

The average delay of a road section refers to the ratio of the total delay of a road section to the number of vehicles passing through the road section, see formula (22), where the total delay refers to the sum of the travel delays of all vehicles on a certain road section, that is, the total delay of all vehicles passing through the road section. The difference between actual and expected travel times for all vehicles on the road segment is summed. The expected travel time of the vehicle refers to the time it takes for the vehicle to always travel at the expected speed through a certain road section, as expressed in equation (23).

$$ ()

$$ ()

where $$ is the average delay of the road section (s), t_i,des is the expected travel time of the i-th vehicle (s), L₀ is the total length of the road section (m), v_i,des is the expected speed of the i- th vehicle (km/h) of delay to travel time, which refers to the ratio of the average delay of a road section to the average travel time of a road section, as shown in the following equation:

$$ ()

where P_D is the delay as a proportion of travel time.

The traffic efficiency index uses average travel speed, average travel time, average delay, and the ratio of delay to travel time. The simulation results of average travel speed and average travel time are shown in Table 6, where HV and AV represent HV and AVs.

Table 6. Average travel speed and travel time of HVs and AVs.

Rainfall intensity (mm/min)	HV		AV
	Travel speed (km/h)	Travel time (s)	Travel speed (km/h)	Travel time (s)
0	84.46	210.59	86.15	209.80
0.5	8 3.4 4	213.14	86.58	208.75
1	71.48	248.82	73.81	244.87
1.5	70.24	256.20	73.74	243.85
2	73.53	352.91	73.47	327.97
2.5	49.77	362.08	54.75	328.54
2.66	50.28	358.40	54.13	332.28
3	49.81	361.84	54.14	332.20
3.5	49.77	487.59	54.75	328.25
4	36.72	489.85	54.76	328.46
4.5	36.75	489.42	54.46	330.25
4.74	36.87	487.85	54.23	331.64
5	36.72	488.13	54.76	333.10

The average travel speed and average travel time are shown in Figure 4.

• 1.

  The average travel speed of manually driven and AVs gradually decreases, and the average travel time gradually increases; among them, when the rainfall intensity q > 3.8 mm/min, the average travel speed and average travel time of manually driven vehicles tend to be stable, and when the rainfall intensity q > 2.2 mm/min, the average travel speed and average travel time of AVs tend to be stable.

• 2.

  During light/moderate rain, the traffic efficiency of HVs and AVs is not much different; when the rainfall intensity is q > 1.49 mm/min, the traffic efficiency of AVs is higher, especially during heavy rain, and the traffic efficiency of AVs is significantly better for HVs.

The delay proportion of HV and AVs is shown in Table 7 for the simulation results. As shown in Figure 5,

• 1.

  In rainy days, the average delays of manual and AVs are significantly higher than those in sunny weather.

• 2.

  When the rainfall intensity is 0 < q < 1 mm/min, the average delay of manual and AVs increases sharply; at that time, the delay of manual vehicles remains high, and, 1.0 mm/min < q < 3.0 mm/min 1.0 mm/min < q < 1.5 mm/min, at that time, the delay of AVs remains high.

• 3.

  When the rainfall intensity is, the average delay of self-driving vehicles is smaller, and when 2.0 mm/min < q < 3.0 mm/min 3.5 mm/min < q < 5.0 mm/min, the average delay and delay proportion of HVs are smaller.

Table 7. Average delays and delay proportions of HVs and AVs.

Rainfall intensity (mm/min)	HV		AV
	Delay (s)	Delay proportion (%)	Delay (s)	Delay proportion (%)
0	30.11	14	29.33	14
0.5	32.54	15	28.32	14
1	68.39	27	64.57	26
1.5	61.72	24	63.52	26
2	52.64	15	48.86	15
2.5	61.68	17	48.48	15
2.66	58.00	16	51.34	15
3	61.43	17	50.97	15
3.5	37.69	8	46.48	14
4	39.94	8	45.92	14
4.5	39.54	8	47.00	14
4.74	37.96	8	47.68	14
5	38.27	8	48.82	15

4.2.2. Stability

The standard deviation of the following distance can describe the continuity of traffic flow on the freeway, and the speed standard deviation represents the amplitude of vehicle speed changes and can describe the degree of discreteness of the traffic flow. Due to the influence of rainy weather, vehicles on the freeway reduce their speed. Upstream vehicles decelerating or braking can easily cause greater disturbance to the downstream traffic flow, causing the traffic flow to show an unstable trend, which is reflected in changes in following distance and speed superior. Therefore, the standard deviation of the following distance and the standard deviation of speed are selected to measure the traffic flow stability of HVs and AVs on rainy days.

The calculation method is shown in the following equations:

$$ ()

$$ ()

where σ_D is the standard deviation of following distance, $$ is the average following distance of the i- th vehicle (m), $$ is the average following distance of all vehicles passing through the road segment (m), σ_V is the speed standard deviation, $$ is the travel speed of the i-th vehicle (km/h), and $$ is the average travel speed of all vehicles passing through the road segment (km/h).

The stability index uses the standard deviation of the following distance to describe the continuity of the traffic flow, and the speed standard deviation to describe the degree of discreteness of the traffic flow. The following distance and speed difference of all vehicles on the road are collected every 10 s. The simulation results are shown in Table 8.

Table 8. Average following distance and average speed difference between HVs and AVs.

Rainfall intensity (mm/min)	HV		AV
	Following distance (m)	Speed difference (m/s)	Following distance (m)	Speed difference (m/s)
0	231.61	0.44	240.58	0.23
0.5	187.56	0.91	214.17	0.23
1	194.92	0.59	211.10	0.27
1.5	123.76	0.37	143.78	0.17
2	121.79	0.41	142.51	0.16
2.5	121.21	0.50	142.88	0.16
2.66	120.70	0.44	142.77	0.16
3	93.66	0.14	141.23	0.15
3.5	93.53	0.12	141.30	0.15
4	94.81	0.10	141.82	0.17
4.5	93.43	0.14	141.71	0.16
4.74	94.77	0.12	141.99	0.15
5	44.67	0.23	141.57	0.17

Using the rainfall intensity, average following distance and its standard deviation and average speed difference and its standard deviation of HVs and AVs, draw “rainfall intensity–average following distance-standard deviation” and “rainfall intensity–average speed difference,” standard deviation three-dimensional graph to visually display the magnitude of change, as shown in Figure 6. Through analysis, it was found that.

• 1.

  Overall, whether it is the average following distance or the average speed difference, the change range of AVs is smaller than that of HVs, indicating that the stability of AVs is better than that of manual vehicles.

• 2.

  The following distance of AVs is larger, but the speed difference is smaller and always remains low. Analysis believes that maintaining a larger following distance and a smaller speed difference is because the reaction time of AVs is shorter. It can adjust its own distance and speed to a reasonable range more quickly.

4.2.3. Safety

The safety indicator of traffic flow uses TTC, and the TTC of all vehicles on the road is collected every 10 s. If the TTC of the vehicle meets 0 < TTC ≤ 3, the s condition, the vehicle is considered to be in a conflict state and may collide. The number of conflicts and conflict rates data between HVs and AVs are shown in Table 9.

Table 9. Number of conflicts and conflict rates between HVs and AVs.

Rainfall intensity (mm/min)	HV		AV
	Number of conflicts (times)	Conflict rate (%)	Number of conflicts (times)	Conflict rate (%)
0	259	0.0545 _ _	38	0.0082
0.5	103	0.0321	6	0.0019
1	24	0.0076	1	0.0003
1.5	179	0.0392	1	0.0002
2	299	0.0639	2	0.0005
2.5	305	0.0658	2	0.0005
2.66	342	0.0731	1	0.0002
3	3	0.0005	0	0
3.5	3	0.0005	0	0
4	2	0.0003	0	0
4.5	9	0.0014	1	0.0002
4.74	1	0.0002	0	0
5	0	0	1	0.0002

The number and conflict rates of traffic flow conflicts are shown in Figure 7. It can be found that under this speed limit method, the number and conflict rate of AV are significantly smaller than that of HV.

For HVs, first, in light/moderate rain, the number of conflicts and conflict rate gradually decrease with the increase in rainfall intensity; second, in heavy rain, conflicts are the most serious, and the number of conflicts and conflict rates decrease with the increase of rainfall intensity. The number of conflicts gradually increases with the increase in rain intensity, exceeding the number of conflicts on sunny days; third, during heavy rain, almost no conflicts occur. Through analysis, it is believed that, first, during light/moderate rain, the safe distance is mainly affected by the driving speed. The safe speed limit decreases as the rainfall intensity increases, and the braking distance of the vehicle gradually decreases, so the conflict rate gradually decreases. Second, during heavy rain, the influence of water film thickness is intensified. As the rainfall intensity increases, the braking distance gradually increases. The insufficient braking distance causes the conflict rate to gradually increase. Third, during heavy rain, the speed limit value is very low. At this time, the braking distance of the vehicle is short, so there is less vehicle conflict.

For self-driving vehicles, whether it is sunny or rainy, the number of conflicts and conflict rates are significantly less than that of HVs. Among them, there are a small number of conflicts on sunny days and almost zero conflicts on rainy days. Through analysis, it is believed that the reaction time of AVs is shorter than that of manually driven vehicles. The braking distance of the former is shorter, so there are fewer conflicts under the same rainfall intensity. At the same time, compared to sunny days, the safe time of AVs on rainy days is larger, travels at a smaller speed, and can ensure a sufficient safe distance, so it is close to zero conflict.

In summary, the traffic flow characteristics of HV and AV under the speed limit method proposed in this study are analyzed through simulation means. The analysis results show that first, AVs are better than HVs in terms of efficiency, stability and safety. Second, for both HV and AV traffic flow, the travel speed decreases faster and delays are the highest during moderate, heavy, and heavy rains. Third, under this speed limit method, the number of conflicts between HVs is most obvious during heavy and heavy rains. Self-driving vehicles have close to zero conflict on rainy days.