Evaluating Efficiency and Safety of Mixed Traffic with Connected and Autonomous Vehicles in Adverse Weather

Abstract

Connected and autonomous vehicles (CAVs) are expected to significantly improve traffic efficiency and safety. However, the overall impacts of CAVs on mixed traffic have not been clearly studied because most previous research focused on one subset of the performance of mixed traffic. This study aims to provide complete information for the policymakers to make better decisions on future CAV implementation strategies with a comprehensive evaluation of the overall performance of mixed traffic. With this purpose, this study develops an integrated framework to evaluate the efficiency and safety of mixed traffic with CAVs under adverse weather conditions, which is composed of a traffic simulation, multi-vehicle crash model, single-vehicle crash model, and performance assessment. For the first time, a unified performance index is introduced to reflect the overall efficiency and safety performance of mixed traffic. The proposed framework is demonstrated with an evaluation of the performance of mixed traffic on a highway segment. Traffic efficiency and safety under different weather conditions are investigated. The impact of reaction time of human-driving vehicles (HDVs) and CAVs are also studied. Simulation results show that the overall traffic performance in terms of traffic efficiency, multi-vehicle safety, and single-vehicle safety increases with the increase in the market penetration rate (MPR). In addition, it is found that CAVs have a greater impact on improving overall traffic performance under rainy and snowy weather than in clear weather. Moreover, a shorter reaction time of HDVs and CAVs can lead to better overall traffic performance.

1. Introduction

Connected and autonomous vehicles (CAVs) are expected to make a significant contribution to traffic efficiency and safety in the future [1,2]. The vehicle-to-vehicle (V2V) communication technology enables vehicles to communicate with each other in traffic to improve their perception of the driving environment. In addition, vehicles equipped with autonomous vehicle (AV) technology have a much shorter reaction time than human-driving vehicles (HDVs). These characteristics make CAVs very advantageous in improving traffic efficiency and safety. However, it will take decades before the market penetration rate (MPR) of CAVs reaches 100% [3]. This means CAVs and HDVs will coexist for many years in the early stage.

Unlike the traffic with pure HDVs, mixed traffic is much more complicated because of the different driving logics of HDVs and CAVs and the interactions between them. Although both efficiency and safety of mixed traffic improve with the MPR, according to previous studies, they may change in different patterns. It is documented in previous studies that the increase in traffic capacity is very limited when the MPR is less than 20% [4], whereas the improvement of traffic safety can be significant at this MPR level [5]. Hence, both efficiency and safety of mixed traffic need to be investigated thoroughly. Due to the limitation of available data and simulation models, most current studies focus on one subset of the performance of CAVs, such as traffic efficiency or safety. Because traffic data of CAVs are often site-specific, the outcome in different studies may be scenario specific. In addition, different assumptions adopted in different simulation models may lead to different simulation results. Therefore, the pure compilation of findings from different studies that focus on separate subsets (e.g., efficiency, safety) may provide incorrect information about the overall impact of CAVs. Given that the overall impacts of CAVs on mixed traffic have not been clearly studied, a comprehensive evaluation of traffic efficiency and safety of mixed traffic is needed in order to provide complete information for the policymakers to make better decisions on future CAV implementation strategies.

Different indicators have been used to quantify the performance of connected and autonomous traffic in previous studies. Traffic capacity is a common indicator of the traffic efficiency of a single road segment [6]. At the network level, aggregated indicators such as travel rate (i.e., the rate of motion in minutes per kilometer) [7] and maximum density in the network fundamental diagram [8] are used to measure the traffic efficiency. Widely used traffic safety indicators include time-to-collision (TTC), number of conflicts, and probability of accidents. Because most current studies of connected and autonomous traffic focus on either traffic efficiency or safety and very little attention has been paid to the broader impacts of CAVs, only one of the above-mentioned indicators is often used. However, when there are multiple performance aspects of CAVs such as traffic efficiency, safety, economics, and environmental impact to be considered, not only individual performance indicators but also a unified performance index that can integrate the performance of CAVs in different aspects is needed. A unified performance index that reflects the overall economic impact or social welfare impacts can provide a clear optimal pathway about how to implement and regulate CAVs [9].

Adverse weather conditions, such as high winds, rain, snow, rain, sleet, hail, fog, flooding, and extreme temperatures, have a dramatic impact on traffic efficiency and safety. In the United States, each year about 5000 people are killed and 418,000 are injured in more than 1 million weather-related crashes [10]. Snow, ice, and fog are responsible for an estimated 23% of the non-recurrent delay on highways across the United States [10]. It was reported that heavy rain and heavy snow can reduce the free capacity by an average of 14% and 22%, respectively [11]. There have been many studies on the impact of inclement weather conditions on traffic with only traditional HDVs [11,12,13,14]. However, it is still unclear how adverse weather affects mixed traffic with CAVs. It is necessary to quantify the weather impact on mixed traffic to help highway agencies develop effective management strategies and operating policies to minimize this impact.

To address those challenges, this study develops an integrated framework to comprehensively evaluate the efficiency and safety of mixed traffic with CAVs: first, the mixed traffic flow is simulated with a cellular automaton (CA)-based model by considering weather conditions and MPRs; second, based on time-dependent vehicle information from the traffic simulation, multi-vehicle crash (MVC) and single-vehicle crash (SVC) risk of the mixed traffic is assessed with the MVC and SVC simulation model, respectively; third, a unified performance index is introduced for comprehensively evaluating the traffic efficiency and safety. The proposed framework is demonstrated with an example in which the mobility and safety performance of mixed traffic on a one-lane highway segment under different weather conditions is evaluated; the effect of the reaction time of HDVs and CAVs on traffic performance is also investigated.

There are two main contributions of this study as follows:

• An integrated framework for comprehensive performance assessment of mixed traffic with CAVs is proposed. This framework not only can be used for evaluating the performance of mixed traffic in several aspects, such as traffic efficiency, multi-vehicle safety, and single-vehicle safety, but also has the capability of assessing the impact of hazardous driving conditions on mixed traffic, such as strong winds, slippery road surfaces, and road curves. Therefore, it can serve as a promising tool for the planning and management of mixed traffic;
• A unified performance index that reflects the overall traffic performance of mixed traffic is introduced for the first time. Such a unified index can provide a big picture for the policymakers about mixed traffic and support them to make better decisions on future CAV implementation strategies.

The rest of this paper is organized as follows. Section 2 presents the literature review. Section 3 describes the methodology and simulation framework. In Section 4, the simulation framework is demonstrated with a numerical simulation, and simulation results are provided and discussed. In Section 5, conclusions are given along with possible future work.

2. Literature Review

A lot of effort has been put to model the mixed traffic flow with CAVs. Different simulation software has been used for CAV simulation in previous studies, such as VISSIM, AIMSUN, PARAMICS, and SUMO [15,16,17]. Compared to other traffic simulation approaches, the CA model has its own advantages, such as high efficiency and remarkable simplicity. Accordingly, there are many studies on CAVs using the CA traffic simulation model. To reflect different driving logics of different types of vehicles, different CA update rules were adopted for HDVs and CAVs [18,19,20]. Researchers have modeled mixed traffic flow with CA models by considering some important factors such as drivers’ character differences, aggression levels of AVs, differentiated per-lane speed limit policy, and dedicated lane policy [21,22,23,24,25].

Extensive effort has been devoted to the study of the impact of CAVs on traffic efficiency and safety. Existing literature indicates that traffic capacity can be significantly enhanced by the CAVs in mixed traffic [4,24,26]. For example, Adebisi et al. [4] found that CAVs can increase the roadway capacity by as much as 35 to 40% and developed capacity adjustment factors for CAV traffic on freeways. Ye and Yamamoto [24] studied the impact of dedicated CAV lanes on traffic capacity and found that setting a dedicated CAV lane can improve traffic flow throughput when the MPR exceeds 40%. It was found by Liu et al. [26] that traffic capacity can be improved by 46% when the MPR is 100% and the impact of smart lane-changing behaviors of CAVs on traffic capacity improvement is insignificant. Similarly, traffic safety can be improved by the existence of CAVs by reducing traffic conflicts according to previous studies [2,5,27,28,29,30,31]. For example, Zhang et al. [2] investigated the impact of trucks on the safety of mixed traffic and concluded that traffic safety can be improved by setting exclusive lanes in high truck proportion scenarios. Ye and Yamamoto [5] found that traffic safety of mixed traffic with CAVs, in terms of the frequency of dangerous situations, improves with the increase in the MPR. Morando et al. [27] investigated the safety impact of AVs with a simulation-based surrogate safety measure approach and found that AVs can improve safety significantly at signalized intersections and roundabouts at high MPRs. Papadoulis et al. [28] evaluated the safety of mixed traffic with a decision-making CAV control algorithm. According to their results, traffic safety can be improved significantly by CAVs even at low MPRs. Virdi et al. [30] studied the impact of CAVs on traffic safety with microsimulation. Their results showed that CAVs can improve traffic safety by reducing conflicts at priority-controlled intersections but worsen the safety at signalized intersections when the MPR is low. However, a high MPR would lead to a global increase in traffic safety for all intersection types.

Some research investigated the efficiency and safety of the highway segments [4,5,24,26] while others focused on the large-scale network [7,8,30]. The impact of factors such as MPR levels, truck ratios, exclusive lane policy, and CACC technologies on CAV traffic have been investigated intensively. However, most of these studies focus on either traffic efficiency or safety. There are very limited studies that focus on both traffic efficiency and safety. For example, Guériau and Dusparic [7] studied the impact of CAVs on traffic efficiency and safety of three types of networks (i.e., urban, national, and motorway). Their study results showed that the CAVs impact traffic efficiency and safety in a gradual and nonlinear way and the near-maximum benefits can be reached at an MPR of 20% to 40%. Li et al. [21] investigated the overall traffic efficiency and safety of mixed traffic with AV fleets with various aggression levels. Significant improvement in traffic efficiency and safety brought by the introduction of AVs was not found in their research. They also found that overly cautious AVs may have a negative influence on traffic efficiency and safety. Zheng et al. [32] analyzed the influences of cooperative driving by V2V on traffic efficiency. It was found in their study that traffic efficiency can benefit from cooperative driving. In addition, aggressive driving behaviors in the V2V environment could reduce safety performance. However, in these studies, traffic efficiency and safety were evaluated and quantified separately and a unified index that can reflect the overall impact of CAV is still missing. Another limitation in current research on the safety of mixed traffic is that only MVC risk is evaluated, and no attention has been given to SVC risk.

Research approaches for quantifying the impact of inclement weather on traffic can be broadly classified into two categories: data-based and simulation-based. In data-based studies, linear and logistic regression models are often used to estimate the relationship between traffic performance indicators (e.g., travel time, operating speed, traffic capacity, crash rate, and injury rate) and weather factors [13,33,34]. However, the data-based approaches are critically dependent on the sufficiency of traffic and weather data. One primary challenge is that historical traffic and weather data are usually unavailable, especially for some extreme weather conditions. Therefore, simulation-based approaches become the primary option because of their ability to model traffic flow and evaluate traffic performance under various weather conditions. Driving simulators and microscopic simulation models are common tools employed in simulation-based studies [12,35,36,37]. Chen et al. [35] studied the impact of adverse weather on traffic efficiency by using a driving simulator and the microscopic traffic simulation program VISSIM. Firstly, driving behaviors under different weather conditions were simulated with the driving simulator. Then, the obtained driving behaviors were integrated into the traffic simulation with VISSIM. Finally, traffic performance such as travel speed, traffic volume, and road capacity was evaluated with the calibrated VISSIM model. Hou and Chen [12] developed a simulation-based framework for the evaluation of work-zone traffic safety under adverse weather conditions. The work-zone safety was assessed with a risk index with which both MVC and SVC risk can be considered. However, these studies focused on traffic with only traditional HDVs. To my best knowledge, there are very limited studies that have been performed to investigate the impact of inclement weather conditions on mixed traffic with CAVs. Yang et al. [38] evaluated the safety performance of connected vehicle pilots under winter snowy weather conditions with a microsimulation approach. Their results indicated that crash risk under adverse weather conditions can be reduced by CVs due to increased drivers’ situational awareness. In addition, traffic safety improvement was found to increase linearly with the MPR of CVs.

In summary, despite the extensive efforts made to study the impact of CAVs, several research gaps still need to be addressed: (1) a unified index to reflect the overall performance of mixed traffic is missing; and (2) the impact of adverse weather conditions on mixed traffic has not been investigated. There is a need to develop an integrated framework with which these gaps can be addressed. This study presents an effort to develop such a framework to comprehensively evaluate the efficiency and safety of mixed traffic with CAVs under adverse weather conditions.

3. Methodology and Simulation Framework

An integrated simulation framework is developed to comprehensively assess the efficiency and safety of mixed traffic with connected and autonomous vehicles. As shown in Figure 1, the framework is comprised of four main components: traffic simulation model, MVC model, SVC model, and performance assessment. First, the traffic simulation model is used to simulate the mixed traffic flow with CAVs by considering MPR, vehicle density, speed limit, and road surface conditions (e.g., icy, snowy, or dry). Second, the traffic simulation model transfers vehicle trajectory data to the MVC and SVC models. With the time-dependent vehicle information and hazard data, the TET (time exposed time-to-collision) is obtained through the MVC model, and the SVC probability is obtained through the SVC model. Finally, based on obtained TET, SVC probability, as well as traffic capacity from the traffic simulation, a unified performance index is calculated for the performance assessment of mixed traffic. The detail of each component of the framework is introduced as follows.

3.1. Traffic Simulation Model

A one-lane CA-based model developed by Jiang et al. [18] is used to simulate the mixed traffic flow with CAVs. In this model, three car-following modes are considered, namely, human-driven vehicle (HDV), adaptive cruise control (ACC), and cooperative adaptive cruise control (CACC). The HDV car-following mode applies to two situations: (1) when both the preceding and the following vehicles are HDVs; and (2) when the preceding vehicle is a CAV, and the following vehicle is an HDV. The ACC mode applies when a CAV follows an HDV, while the CACC mode applies when a CAV follows another CAV. The reaction time of drivers in the HDV mode usually ranges from 1 to 2 s while CAVs equipped with a sensing system in the ACC mode have a short reaction time of around 0.5 s [18,20]. The reaction time in the CACC mode comes from the communication and braking delay of the CACC, which is very short and can be neglected [18,20]. In addition, because of uncertain factors associated with human drivers, random deceleration occurs in the HDV mode. In this model, the velocity and movement of a vehicle can be updated with the CA rules according to its current car-following mode. The CA rules, which include acceleration, deceleration, randomization, and movement, can be described as follows:

(1)

Acceleration

$$\begin{array}{cc}{v}_n\left(t+\Delta t\right)=\hfill & \left({\alpha }_n+{\beta }_n\right)min\left(v_n\left(t\right)+a\Delta t,v_{max},\frac{d_n}{\Delta t}\right)\hfill \\ & +{\gamma }_n\min \left(v_n\left(t\right)+a\Delta t,v_{max},\frac{d_n}{\Delta t},\frac{d_n}{\Delta t}+v_{n-1}\left(t+\Delta t\right)-\frac{d_{n,safe}}{\Delta t}\right), if d_n>d_{n,safe}\hfill \end{array}$$

(1)

$$d_{n,safe}=v_n\left(t\right)\left({\alpha }_n{\tau }^{HDV}+{\beta }_n{\tau }^{CAV}\right)+\left({\alpha }_n+{\beta }_n\right)\left(\frac{v_n{\left(t\right)}^2-v_{n-1}{\left(t\right)}^2}{2B}\right)+{\gamma }_n{d}_s^{CACC}$$

(2)

(2)

Deceleration

$$v_n\left(t+\Delta t\right)=\left({\alpha }_n+{\beta }_n\right)min\left(v_n\left(t\right),\frac{d_n}{\Delta t}\right)+{\gamma }_n{v}_{n-1}\left(t+\Delta t\right), if d_n\le d_{n,safe}$$

(3)

(3)

Randomization

$$v_n\left(t+\Delta t\right)=\max \left(v_n\left(t\right)-{\alpha }_n{\eta }_n^{t}b\Delta t,0\right)$$

(4)

$${\eta }_n^t=\{\begin{array}{cc}1,\hfill & \hfill if rand\le p_{slow} and t mod {\tau }^{HDV}=0\\ 0,\hfill & \hfill if rand>p_{slow} and t mod {\tau }^{HDV}=0\\ {\eta }_n^{t-\Delta t},\hfill & \hfill other\end{array}$$

(5)

(4)

Movement

$$x_n\left(t+\Delta t\right)=x_n\left(t\right)+v_n\left(t+\Delta t\right)·\Delta t$$

(6)

where ${\alpha }_n$, ${\beta }_n$, and ${\gamma }_n$ are 0–1 variable representing the car-following mode of vehicle $n$, namely, HDV, ACC, and CACC, respectively: the value of ${\alpha }_n$, ${\beta }_n$, and ${\gamma }_n$ will be 1 if vehicle $n$ has the corresponding car-following mode; otherwise, it will be 0; $x_n\left(t\right)$ and $v_n\left(t\right)$ are the longitudinal position and velocity of vehicle $n$ at time $t$, respectively; $d_n$ is the space gap of vehicle $n$, which is the clear distance between vehicle $n$ and its preceding vehicle $n-1$, $d_n=x_{n-1}\left(t\right)-x_n\left(t\right)-l_{n-1}$; $l_{n-1}$ denotes the length of vehicle $n-1$; $\Delta t$ is the time step; $d_{n,safe}$ is the safe distance of vehicle $n$; $a$ is the acceleration rate; $b$ is the randomization deceleration rate; $v_{max}$ is the maximum velocity; $v_{n-1}\left(t+\Delta t\right)$ is the speed of vehicle $n-1$ (preceding vehicle) at time $t+\Delta t$; ${\tau }^{HDV}$ and ${\tau }^{CAV}$ are the reaction time of HDVs and CAVs, respectively; $B$ is the maximum deceleration rate; $d_s^{CACC}$ is the safety distance of vehicles with CACC mode; ${\eta }_n^t$ is the indicator of the randomization status of vehicle $n$ at time $t$; $p_{slow}$ is the randomization probability.

3.2. MVC Model

Rear-end crashes are one of the most frequently occurring types of collisions, so the risk of rear-end crashes is used to represent the MVC risk in this study. Time-to-collision (TTC) is one of the most important indexes to measure rear-end crash risk [31]. Assuming that two successive vehicles on the same lane keep their current velocity, TTC indicates the time required for the vehicles to collide if the following vehicle is faster than the preceding one. TTC is given by

$$TTC\left(t\right)=\frac{d_n}{v_n\left(t\right)-v_{n-1}\left(t\right)},\hspace{1em}\hspace{1em}{v}_n\left(t\right)>v_{n-1}\left(t\right)$$

(7)

The time-exposed time-to-collision (TET) was developed based on TTC, which indicates the total time during conflicts when the TTC is below a certain threshold value [39]. Because the calculation of TET requires microscopic traffic data (i.e., vehicle position and velocity), it is suitable for microscopic traffic simulation [40]. Therefore, TET is used in this study as the MVC risk indicator. TET can be described as

$$TET={{\displaystyle \sum }}_{i=1}^N{{\displaystyle \sum }}_{t=1}^T\Delta t, 0<TTC\left(t\right)<TT{C}^{*}$$

(8)

where $N$ is the total number of vehicles; $T$ is the total simulation time; $TT{C}^{*}$ is the threshold value of $TTC$.

3.3. SVC Model

An SVC model previously developed by the authors is used to evaluate the SVC risk under hazardous driving conditions [41]. To provide some essential background information, the SVC model is briefly introduced below [41]. In this model, a vehicle is modeled with three rigid bodies that represent a sprung mass and two unsprung masses of the front and rear axles. Five differential equations are used to describe the balance of lateral force and yaw moment of the entire vehicle, and the roll motion of the sprung and unsprung masses. These equations and related parameters can be found in the reference [42]. The external loads acting on vehicles such as wind loads, tire forces, and force due to superelevation can be considered in the dynamic equations. By solving the dynamic equations, the response of vehicles under hazardous driving conditions (e.g., strong crosswinds, complex road geometry, slippery road surfaces) can be obtained. A vehicle under hazardous conditions may roll over or sideslip over a certain time, which will be assessed against the “critical sustained time” (CST) [42]. CST means the minimum time required to sustain the specific combination of adverse environments and driving conditions to enable an accident to occur. In this study, a rollover or sideslip crash is assumed to occur when the CST of a vehicle is less than the reaction time of the vehicle. With the time-variant information vehicle information (i.e., vehicle position, vehicle speed, and vehicle type) from the traffic flow simulation, as well as the hazard data, the crash occurrence of each vehicle in the traffic flow will be checked at a defined time interval (e.g., 1 min). Then, a vulnerable vehicle ratio, which is the ratio of the number of vehicles that experience rollover or sideslip crashes to the total number of vehicles in the traffic flow, is obtained. The vulnerable vehicle ratio can represent the overall SVC risk during this time. In order to consider the stochastic nature of traffic flow, the same experiment will be repeated over time continuously by evaluating the passing vehicles through the same observation window. Finally, based on the statistical analyses of the results from the repeated experiments, the probability of SVC can be expressed with the median value of the vulnerable vehicle ratios throughout the entire simulation time.

3.4. Performance Assessment

With the help of the traffic simulation model, MVC model, and SVC models, the traffic capacity, TET, and SVC probability can be obtained, respectively. Based on these results, a unified performance index, which can simultaneously consider traffic efficiency and safety, is calculated to comprehensively evaluate the overall performance of mixed traffic flow. The method for calculating the unified performance index of mixed traffic flow is presented as follows.

Firstly, crash vulnerability indicators are introduced to reflect the safety performance under a wide range of traffic conditions and hazard conditions. Here, traffic density is used as the measure of traffic conditions. High winds, together with icy or snowy road conditions, are one of the main causes of single-vehicle crashes. The SVC risk during high winds will be evaluated, so wind speed is used as the hazard measure in this study. Therefore, the MVC vulnerability is defined as the area between the TET curve and the traffic density axis, while the SVC vulnerability is defined as the volume between the crash probability surface and the traffic density-wind speed plane [12]. The MVC and SVC vulnerability can be expressed with Equations (9) and (10), respectively.

$$V_m={\displaystyle \int }TET\left(\rho \right)d\rho$$

(9)

$$V_s={\displaystyle \iint }{P}_s\left(\rho ,u\right)d\rho du$$

(10)

where $V_m$ and $V_s$ are the MVC and SVC vulnerability, respectively; $TET\left(\rho \right)$ is the probability function of $TET$; $P_s\left(\rho ,u\right)$ is the probability function of SVC; $\rho$ is the traffic density; $u$ is the wind speed. It is noted that both traffic density $\rho$ and wind speed $u$ can be either a continuous range of value or a discrete value, depending on the needs and preferences of the stakeholders.

Secondly, the individual performance index of traffic efficiency, multi-vehicle safety, and single-vehicle safety of traffic flow for a given MPR are defined as the change in traffic capacity, MVC vulnerability, and SVC vulnerability compared to the traffic flow with pure HDVs, which can be expressed by Equations (11)–(13), respectively. Here, a positive index means a gain in performance while a negative one means a reduction of performance.

$$P{I}_c=\frac{C-C_0}{C_0}$$

(11)

$$P{I}_m=\frac{V_{m,0}-V_m}{V_{m,0}}$$

(12)

$$P{I}_s=\frac{V_{s,0}-V_s}{V_{s,0}}$$

(13)

where $P{I}_c$, $P{I}_m$, and $P{I}_s$ are individual performance indexes of traffic efficiency, multi-vehicle safety, and single-vehicle safety, respectively. $C_0$, $V_{m,0}$, and $V_{s,0}$ are traffic capacity, MVC vulnerability, and SVC vulnerability when the MPR is 0 under a specific weather condition.

Finally, a unified performance index is proposed to evaluate the overall performance of mixed traffic flow for a given MPR by considering both traffic efficiency and safety, which is defined in Equation (14). ${\theta }_c$, ${\theta }_m$, and ${\theta }_s$ are the weight parameters to define the contributions from traffic capacity, multi-vehicle safety, and single-vehicle safety, respectively. The sum of these three weight parameters is 1.0. The value of each parameter can be decided based on site-specific data or the preference of the stakeholders based on specific circumstances and priorities [12].

$$UPI={\theta }_{c}P{I}_c+{\theta }_{m}P{I}_m+{\theta }_{s}P{I}_s$$

(14)

4. Simulation and Results

The proposed framework will be used to evaluate the mobility and safety performance of mixed traffic on a one-lane highway segment with a length of 1000 m. The length of both CAVs and HDVs is set as 5 m. According to Jiang et al. [18], the safe distance of vehicles with CACC mode $d_s^{CACC}$ is assumed to be constant, which is set as 0.5 m. A maximum string length of 10 to 20 was proved to be appropriate for highway traffic by Liu et al. [43]. Therefore, the maximum string length of CACC vehicles is set as 10 in this study. Under clear weather, the free-flow speed is assumed to be 33.5 m/s; the reaction time of CAVs and HDVs are set as 0.6 s and 1.7 s, respectively [18]. In the CA model, periodic boundary conditions are used. A small cell length of 0.01m is adopted to provide high simulation accuracy, and the simulation time step is 0.1 s. Drivers’ behavior varies differently under different weather conditions. For example, adverse weather conditions usually lead to decreased acceleration and deceleration rates of vehicles, decreased free-flow speed, and increased reaction time [10,12,44]. Values of these parameters under different weather conditions (i.e., clear, rainy, and snowy) used in the CA model are given in

Table 1. Parameter values of the CA model under different weather conditions.

Parameter	Weather Condition
	Clear	Rainy	Snowy
$v_{max}$ (m/s)	33.5	30	23.5
$a$ (m/s2)	2	1.5	1
$b$ (m/s2)	2	1.5	1
$B$ (m/s2)	7	4	2
$p_{slow}$	0.1	0.2	0.3
${\tau }^{HDV}$ (s)	1.5	1.7	1.9
${\tau }^{CAV}$ (s)	0.6	0.6	0.6
$d_s^{CACC}$ (m)	0.5	0.5	0.5

. Each simulation run lasts 4000 time steps. Simulation results for the first 2000 time steps are discarded in order to capture steady traffic flow characteristics. In the MVC model, the value of the threshold of $TTC$, $TT{C}^{*},$ is assumed to be 1.5 s, since this value has been widely used in identifying severe conflicts previously [45]. In the SVC model, detailed parameters of the vehicle dynamic model can be found in the reference [41]. The weight parameters ${\theta }_c$, ${\theta }_m$, and ${\theta }_s$ are assumed to be 0.5, 0.25, and 0.25 in this study, respectively. These parameter values listed above will be used throughout this study unless otherwise specified.

4.1. Performance Evaluation of Mixed Traffic under Different Weather Conditions

In this section, the performance of mixed traffic including traffic efficiency, multi-vehicle safety, and single-vehicle safety under different weather conditions will be studied. Both the individual and overall performance indexes will be calculated with the above-mentioned approach. It should be noted that the baseline scenario for each weather case is the traffic flow with 0% MPR under the current weather condition. In addition, to investigate the overall traffic performance under various traffic conditions and wind speeds, the traffic density used in the evaluation ranges from 0 to 180 veh/km/lane and the wind speed ranges from 0 to 30 m/s.

Firstly, the traffic efficiency under clear, rainy, and snowy weather conditions is investigated. For each weather condition, the flow-density diagrams of 6 cases with different MPRs (i.e., 0%, 20%, 40%, 60%, 80%, and 100%) are compared in Figure 2. Traffic capacity and performance index $P{I}_c$ under different weather conditions are given in

Table 2. Traffic capacity and performance index $P{I}_c$ under different weather conditions.

Weather Condition	MPR = 0%		MPR = 20%		MPR = 40%		MPR = 60%		MPR = 80%		MPR = 100%
	CAP	$\mathit{P}{\mathit{I}}_{\mathit{c}}$	CAP	$\mathit{P}{\mathit{I}}_{\mathit{c}}$	CAP	$\mathit{P}{\mathit{I}}_{\mathit{c}}$	CAP	$\mathit{P}{\mathit{I}}_{\mathit{c}}$	CAP	$\mathit{P}{\mathit{I}}_{\mathit{c}}$	CAP	$\mathit{P}{\mathit{I}}_{\mathit{c}}$
Clear	2375	0	2536	0.07	3019	0.27	3774	0.59	5595	1.36	14,386	5.06
Rainy	2089	0	2151	0.03	2614	0.25	3454	0.65	5279	1.53	13,500	5.46
Snowy	1689	0	1985	0.18	2401	0.42	3060	0.81	4709	1.79	11,421	5.76

Note: MPR = market penetration rate; CAP = capacity (Unit: veh/hour/lane).

. It can be seen from Figure 2 that, for each MPR case under a specific weather condition, traffic flow increases with the increase in traffic density, and then decreases after reaching the traffic capacity. Meanwhile, it is found that higher MPR leads to higher traffic capacity under all three weather conditions. The capacity increase is nonlinear with respect to the increase in the MPR. For example, as shown in Table 2, under clear weather conditions, as the MPR increases from 60% to 80%, the traffic capacity increases from 3774 to 5595 veh/hour/lane; when the MPR further increases to 100%, the traffic capacity increases to 14,386 veh/hour/lane. The nonlinearity can also be reflected in the performance index $P{I}_c$, which are 0.59, 1.36, and 5.06 when MPR is 60%, 80%, and 100%, respectively, representing 59%, 136%, and 506% increase compared to the 0% MPR case. Adverse weather conditions are found to lead to reduced traffic capacity. For example, when the MPR is 0%, traffic capacity under rainy and snowy weather reduces by 12% and 29%, respectively, as compared to clear weather. This result is consistent with previous findings [11,46]. The results in Table 2 also show that the performance index $P{I}_c$ for traffic under rainy and snowy weather generally increases faster with MPR than clear weather. This indicates that CAVs may bring even greater traffic efficiency benefits to mixed traffic under adverse weather conditions.

Secondly, the multi-vehicle safety performance of mixed traffic flow is investigated under different weather conditions. Figure 3 shows the TET-density relationship under different weather conditions.

Table 3. Performance index $P{I}_m$ under different weather conditions.

Weather Condition	MPR
	0%	20%	40%	60%	80%	100%
Clear	0	0.16	0.36	0.58	0.80	1.00
Rainy	0	0.13	0.32	0.55	0.79	1.00
Snowy	0	0.13	0.32	0.55	0.80	1.00

lists the performance index $P{I}_m$ under different weather conditions. As shown in Figure 3, TET generally increases as the traffic density increases. Meanwhile, with the increase in MPR, the TET decreases under all three weather conditions. When the MPR is 100%, the TET is zero under all traffic conditions, indicating no rear-end crash risk. As shown in Table 3, under clear, rainy, and snowy weather conditions, the multi-vehicle safety performance index $P{I}_m$ increases with MPR in a similar pattern.

Thirdly, the single-vehicle safety of mixed traffic flow is investigated under different weather conditions. Unlike multi-vehicle safety, single-vehicle safety is affected by more factors, such as traffic conditions, wind speed, road surface conditions, and reaction time of vehicles. Generally, low traffic density, high wind speed, rainy and snowy road surfaces, and longer reaction time lead to a higher SVC risk. Due to the page limitation, only a part of the simulation results is presented in this paper. Figure 4 shows the SVC probability under several combinations of hazardous driving conditions. Four types of SVC are considered, namely, rollover crash of HDV (HDV-rollover), sideslip crash of HDV (HDV-sideslip), rollover crash of CAV (CAV-rollover), and sideslip crash of CAV (CAV-sideslip). A bar in Figure 4 provides the probability of each type of SVC under a specific driving condition (i.e., traffic density, wind speed, weather, and MPR). For example, the 2nd bar from left in Figure 4b shows that the probabilities of HDV-rollover, HDV-sideslip, CAV-rollover, and CAV-sideslip are 0.715, 0.058, 0.134, and 0.014 under clear weather when $\rho$ = 25 veh/km/lane, $U$ = 25 m/s, and MPR = 20%. It can be seen from Figure 4a,c,e that single-vehicle crashes (SVC) under rainy and snowy weather conditions occur at lower wind speeds than in clear weather. This indicates that adverse weather can cause a higher SVC risk. It is also found that in clear weather, the most dominant crash type is rollover. In rainy and snowy weather, sideslip dominates under low wind speeds while rollover prevails under high wind speeds. As shown in Figure 4a,c,e, under low wind speeds, as the MPR increases, the SVC probability generally increases first and then decreases after reaching the maximum value at the MPR of 40%. When the wind speed is relatively low, both vehicle speed and reaction time are critical factors. When the MPR is 40%, the mean vehicle speed in the traffic flow is relatively high. Meanwhile, the proportion of HDVs is also relatively high. As a result, the maximum SVC probability is reached when the MPR is 40%. In contrast, under high wind speeds, as shown in Figure 4b,d,f, the SVC probability decreases as the MPR increases. This is because both wind speed and reaction time are two dominant factors; HDVs and CAVs with the ACC mode have high SVC risk because of relatively long reaction time. It is interesting to find that even for traffic with 100% MPR, the SVC probability is about 10%, which is the proportion of CAVs with the ACC mode.

By using Equation (13), the single-vehicle safety performance index $P{I}_s$ under different weather conditions is calculated and given in

Table 4. Performance index $P{I}_s$ under different weather conditions.

Weather Condition	MPR
	0%	20%	40%	60%	80%	100%
Clear	0	0.05	0.14	0.29	0.54	0.83
Rainy	0	0.08	0.16	0.37	0.61	0.87
Snowy	0	0.05	0.15	0.38	0.60	0.88

. As shown in Table 4, with the increase in MPR, the safety performance index $P{I}_s$ increases. Moreover, the increase in MPR leads to a slightly faster increase for mixed traffic in rainy and snowy weather than in clear weather.

Finally, the unified performance index $UPI$ under different weather conditions is calculated with Equation (14), which is plotted in Figure 5. It can be seen from Figure 5 that the $UPI$ increases with the increase in the MPR under all three weather conditions. The increase in $UPI$ is more significant when the MPR is greater than 80%. This is mainly due to the contribution of the traffic efficiency performance index $P{I}_c$. As shown in Table 2, as the MPR increases from 80% to 100%, $P{I}_c$ for clear, rainy, and snowy weather conditions increases by 272%, 257%, and 222%, respectively. By comparing the traffic under different weather conditions, $UPI$ under snowy weather conditions is the highest whereas $UPI$ under clear weather conditions is the lowest. This indicates that CAVs may have a greater impact on improving the overall performance of traffic under adverse weather conditions.

4.2. Effect of the Reaction Time of HDVs and CAVs

The reaction time of vehicles does not only affect traffic efficiency but also traffic safety. However, the reaction time of HDVs and CAVs varies because of different driving behaviors and external environmental factors. Sensitivity analysis on the reaction time of HDVs and CAVs is necessary to understand their impact on mixed traffic. In this section, the effect of the reaction time of HDVs and CAVs on the overall traffic performance is investigated. Nine cases with different reaction times of HDVs and CAVs for mixed traffic under rainy weather conditions are studied: Case 1 (${\tau }^{HDV}$ = 1.2 s, ${\tau }^{CAV}$ = 0.4 s), Case 2 (${\tau }^{HDV}$ = 1.2 s, ${\tau }^{CAV}$ = 0.6 s), Case 3 (${\tau }^{HDV}$ = 1.2 s, ${\tau }^{CAV}$ = 0.8 s), Case 4 (${\tau }^{HDV}$ = 1.7 s, ${\tau }^{CAV}$ = 0.4 s), Case 5 (${\tau }^{HDV}$ = 1.7 s, ${\tau }^{CAV}$ = 0.6 s), Case 6 (${\tau }^{HDV}$ = 1.7 s, ${\tau }^{CAV}$ = 0.8 s), Case 7 (${\tau }^{HDV}$ = 2.2 s, ${\tau }^{CAV}$ = 0.4 s), Case 8 (${\tau }^{HDV}$ = 2.2 s, ${\tau }^{CAV}$ = 0.6 s), and Case 9 (${\tau }^{HDV}$ = 2.2 s, ${\tau }^{CAV}$ = 0.8 s). The unified performance index $UPI$ for the 9 cases is presented in Figure 6. It is noted that the baseline scenario for the performance index calculation of these cases is the 0% MPR traffic under rainy weather with the reaction time of HDVs and CAVs of 1.7 s and 0.6 s, respectively. It is observed in Figure 6 that the $UPI$ increases with the decrease in ${\tau }^{HDV}$ and ${\tau }^{CAV}$ except when the MPR is 100%. ${\tau }^{HDV}$ is found to have a greater impact on the $UPI$ than the ${\tau }^{CAV}$, especially when the MPR is not very high. When the MPR is 100%, the $UPI$ is nearly not affected by ${\tau }^{HDV}$ and ${\tau }^{CAV}$. It is interesting to find that when the MPR is 0% the $UPI$ of Cases 7 to 9 are negative. This indicates that the increased reaction time of HDVs and CAVs can lead to a reduction in overall traffic performance.

The individual performance indexes $P{I}_c$, $P{I}_m$, and $P{I}_s$ for cases with different reaction time are given in

Table 5. Performance indexes $P{I}_c$, $P{I}_m$, and $P{I}_s$ for cases with different reaction time.

Performance Index	Case	MPR
		0%	20%	40%	60%	80%	100%
$P{I}_c$	1	0.29	0.49	0.73	1.16	2.11	5.46
	2	0.29	0.40	0.71	0.98	1.89	5.46
	3	0.29	0.34	0.55	0.91	1.77	5.46
	4	0	0.03	0.34	0.71	1.63	5.46
	5	0	0.03	0.25	0.65	1.53	5.46
	6	0	0.03	0.26	0.54	1.34	5.46
	7	−0.24	−0.15	0.03	0.41	1.30	5.46
	8	−0.24	−0.19	0.01	0.39	1.22	5.46
	9	−0.24	−0.20	0	0.33	1.07	5.46
$P{I}_m$	1	−0.03	0.12	0.33	0.57	0.81	1.00
	2	−0.03	0.12	0.32	0.56	0.80	1.00
	3	−0.03	0.11	0.32	0.56	0.80	1.00
	4	0	0.14	0.33	0.57	0.80	1.00
	5	0	0.13	0.32	0.55	0.79	1.00
	6	0	0.12	0.31	0.55	0.79	1.00
	7	0.10	0.23	0.40	0.60	0.82	1.00
	8	0.10	0.21	0.38	0.59	0.81	1.00
	9	0.10	0.21	0.37	0.58	0.81	1.00
$P{I}_s$	1	−0.01	0.11	0.26	0.48	0.72	0.95
	2	−0.01	0.04	0.13	0.34	0.61	0.87
	3	−0.01	0.04	0.14	0.35	0.61	0.87
	4	0.02	0.13	0.28	0.49	0.72	0.95
	5	0	0.08	0.16	0.37	0.61	0.87
	6	0.02	0.07	0.17	0.36	0.62	0.87
	7	0.02	0.15	0.29	0.49	0.73	0.95
	8	0.02	0.08	0.19	0.37	0.62	0.87
	9	0.02	0.08	0.19	0.37	0.62	0.87

. Traffic efficiency performance index $P{I}_c$ is affected most by the reaction time while the multi-vehicle performance index $P{I}_m$ is affected least. At the MPR of 100%, all three individual performance indexes nearly keep the same with varying reaction time, which leads to a nearly constant $UPI$. Generally, $P{I}_c$ decreases with the increase in ${\tau }^{HDV}$ and ${\tau }^{CAV}$. When the MPR is less than 40%, $P{I}_c$ is mainly affected by ${\tau }^{HDV}$. When the MPR is higher than 40%, $P{I}_c$ is affected by both ${\tau }^{HDV}$ and ${\tau }^{CAV}$. $P{I}_m$ is also mainly affected by ${\tau }^{HDV}$ rather than ${\tau }^{CAV}$, although the impact is relatively insignificant. $P{I}_m$ slightly increases in general as ${\tau }^{HDV}$ increases. Similar to $P{I}_c$, $P{I}_s$ generally decreases with the increasing ${\tau }^{HDV}$ and ${\tau }^{CAV}$. However, it is found that ${\tau }^{CAV}$ has a greater impact than the ${\tau }^{HDV}$, especially when MPR is greater than 40%. Specifically, when the MPR is high enough (e.g., greater than 80%), ${\tau }^{HDV}$ nearly has no impact on the $P{I}_s$. Moreover, $P{I}_s$ can be further improved only by reducing ${\tau }^{CAV}$ to 0.5 s, which is the threshold of the rollover crash under rainy weather.

5. Conclusions and Future Work

This paper proposes a framework for comprehensively evaluating the performance of mixed traffic with CAVs. A unified performance index integrating the performance of CAVs in different aspects, i.e., traffic efficiency, multi-vehicle safety, and single-vehicle safety, is introduced for the first time. There are many potential applications of the proposed framework. For example, this framework can be used to evaluate the impact of different CAV policies, such as dedicated CAV lanes and traffic signal control. The framework is demonstrated with an evaluation of the performance of mixed traffic on a highway segment. In the demonstration example, traffic efficiency and safety of mixed traffic under clear, rainy, and snowy weather conditions are investigated first. Then, the impact of the reaction time of HDVs and CAVs on the performance of mixed traffic is also studied. The following main conclusions have been drawn:

• The increase in MPR leads to increased overall traffic performance in terms of traffic efficiency, multi-vehicle safety, and single-vehicle safety;
• CAVs have a greater impact on improving the overall performance of mixed traffic under adverse weather conditions, such as rainy and snowy weather;
• A shorter reaction time of HDVs and CAVs leads to better overall traffic performance by significantly improving traffic efficiency and single-vehicle safety.

Future work is needed to further improve the proposed framework. The one-lane traffic simulation model can be improved by incorporating lane-changing rules for HDVs and CAVs so that the framework can be extended for the multi-lane highway. Mixed traffic on urban arterials may be investigated in future studies as well. In addition, the simulation of the interaction mechanism between HDVs and CAVs needs to be improved in the future. Reduced visibility from fog, dust, and smoke might be a concern of CAVs. In future work, the author will also study the impact of reduced visibility on mixed traffic with CAVs.