Trip Cost Estimation of Connected Autonomous Vehicle Mixed Traffic Flow in a Two-Route Traffic Network

With the advancement of connected autonomous vehicle (CAV) technology, research on future traffic conditions after the popularization of CAVs needs to be resolved urgently. Bounded rationality of human drivers is essential for simulating traffic flow precisely, but few studies focus on the traffic flow simulation considered bounded rationality in CAV mixed traffic flow. In this study, we introduce random bounded rationality into the hybrid feedback strategy (HFS) under CAVmixed traffic flow to explore the impacts of CAV penetration rate on the trip cost of vehicles. First, we investigated the bounded rationality of drivers, and we found that it follows normal contribution. +en, we proposed HFS considering random bounded rationality and the CAV penetration rate to simulate the traffic condition. +e numerical results show that the enhancement of the CAV penetration rate could reduce total trip cost. +e research could help us to simulate the CAVs mixed traffic flow more precisely and realistically.


Introduction
Along with the development of the Internet of vehicle technology and the autonomous vehicle technology, mixed traffic conditions of human-driven vehicles (HDVs) and connected autonomous vehicles (CAVs) should be studied [1,2]. Simulation of the mixed traffic condition is essential for addressing the future development of intelligent transportation systems.
In recent years, researchers have proposed many traffic flow models in previous studies. Traffic flow models could be generally divided into macromodels [3][4][5] and micromodels [6][7][8][9]. Macromodels aim to study the relationship between flow, speed, and density. In contrast, micromodels focused on driving behaviors. However, the researches above did not study the traffic flow models based on the mixed traffic conditions of HDVs and CAVs. Most studies focused on the traffic flow models of HDVs. Many researchers proposed car-following models to simulate the traffic flow based on HDVs, but they ignored the impacts of humans' bounded rationality on the traffic flow operation. erefore, Ou et al. [10] incorporated bounded rationality into two feedback strategies to study vehicle trip cost under the feedback strategies. Moreover, several researchers focused on carfollowing models based on CAVs.
However, the researches above did not study the traffic flow model considering the mixed traffic conditions of HDVs and CAVs, and their feedback strategies ignored the penetration rate of CAVs. Furthermore, they did not consider the randomness of HDV drivers' bounded rationality. However, under the background of the popularity of CAVs in the future, the phenomenon of mixed traffic between CAVs and HDVs will inevitably appear. erefore, the study of mixed traffic flow between CAVs and HDVs is imperative.
To solve the problem, we propose a hybrid feedback strategy (HFS) considering the penetration rate of CAVs and random bounded rationality. en, we explore the regulation of random bounded rationality of HDV drivers based on the questionnaire. Next, we study the impacts of the penetration rate of CAVs on each vehicle's travel time in the two-route traffic network. Additionally, in this study, our basic hypotheses are as follows: (1) HDVs and CAVs randomly pass the two-route traffic network; (2) two routes of the two-route traffic network are homogeneous; (3) CAVs make behavioral choices under absolutely rational circumstances. is paper has three main contributions: (1) we fit the distribution of values of random bounded rationality based on investigations; (2) the proposed HFS considered the randomness of drivers' bounded rationality and the CAV penetration rate to simulate the traffic flow; (3) we explore the impacts of the CAV penetration rate on the trip cost in a two-route traffic network. Furthermore, the research is valuable for promoting the use of CAVs in the future, and it can help us solve the route guidance problem in the context of the mixed traffic flow between CAVs and HDVs.
is paper is organized as follows: literature review is included in Section 2, and Section 3 describes the data we used in this study. e method is indicated in Section 4, and the results of the case study are included in Section 5. Finally, Section 6 contains the discussion, the conclusion, and future research.

Car-Following Model of HDVs.
Many studies focused on the microtraffic flow models of HDVs. In terms of carfollowing models, several researchers improved the methods. ese researches could be divided into two types: (1) calibrating the parameters of existing car-following models; (2) learning a new car-following model from the historical trajectory data based on the machine learning methods.
e first kind of researches aims to calibrate the parameters of existing car-following models. e studies generally test the macroscopic features consistency between simulation result and physical truth to improve the carfollowing model while the data only contain loop detector data [11][12][13][14]. Additionally, to calibrate the parameters of car-following models, researchers also focused on the time headway and spatial headway distributions of car-following models [15,16] and phase diagrams [17,18]. e second kind of study learns a new car-following model from the historical trajectory data based on the machine learning methods. Generally, the methods input features such as velocities, velocity differences, and position differences; then, the velocity and acceleration in the next time interval are output [19][20][21]. e data-driven methods could reduce the human interference and better fit the complex driving behavior observed.

Car-Following Model of CAVs.
Recently, several researchers focused on the car-following model of CAVs. Along with the future introduced CAV technology, the CAV car-following model should be proposed to simulate the future traffic flow, whereas CAVs have different car-following features from HDVs because CAVs can obtain the traffic condition information of the entire network.
Treiber et al. first developed the intelligent driver model (IDM) [22]. e IDM is commonly used in intelligentvehicle situation because it can output the realistic acceleration on a single lane. rough adjusting the parameters, the revised IDM can be applied into CAV simulation [23]. Yet, the deceleration of IDM could be unrealistically high while the current spatial headway is significantly smaller than the expected spatial headway. us, Treiber et al. considered that the current spatial headway is smaller than the expected spatial headway as a relatively mild-critical condition [24]. To solve the problem, Kesting et al. reduced the unrealistic deceleration by combining the IDM and the constant acceleration heuristics (CAH), and the adaptive cruise control (ACC) model was proposed [25]. Next, Zhou et al. improved the cooperative intelligent demand model based on the IDM and tested the CAV system performance [23].

Bounded Rationality Applied in the Car-Following Model.
In traffic flow models, bounded rationality is essential for simulating HDVs' driving behaviors more realistic, whereas the above models did not consider bounded rationality [26].
To consider the bounded rationality in the car-following model, Tang et al. [27][28][29][30] took bounded rationality into account and developed two car-following models to study the driving behavior and vehicles' emissions. Di et al. [31] stimulated the impacts of bounded rationality on the drivers' route choice behavior. en, Li et al. [32,33] explored the relationship between bounded rationality and commuting behaviors. Jiang and Deng [34] proposed a Bayesian network to describe travel time distribution based on bounded rationality. In the frame of the nonlinear pairwise swapping rule, Zhang et al. [35] used absolute bounded rationality and relatively bounded rationality to study the route choice behaviors of drivers. Guo et al. [36] proposed a dynamic pricing policy based on bounded rationality and studied the impacts of the policy on traffic system optimization. Li et al. [37] developed a Markov Logic Network method to combine the bounded rationality principle with traffic choice and quantify characteristics and choice models of travelers. Batista et al. [38] used a Lighthill-Whitham-Richards mesoscopic traffic simulator to explore the impacts of bounded rationality on a traffic network performance. Xue et al. [39] explored the impacts of bounded rationality on commuter departure time choice. en, Fujino and Chen [40] incorporated the bounded rationality of drivers into the traffic network and studied the performance of the traffic network. Moreover, Tang et al. [27][28][29][30]41] and Leng et al. [42,43] developed a car-following model to study vehicles' trip costs in different situations. en, Tang et al. [44], respectively, proposed two feedback strategies and studied each vehicle's trip cost under the feedback strategies. Based on the research of Tang et al. [44], Ou et al. [10] incorporated bounded rationality into two feedback strategies to study each vehicle's trip cost under different feedback strategies. e researchers above, respectively, focused on the carfollowing models based on HDVs and CAVs, and they introduced the bounded rationality into HDVs' carfollowing models to simulate the driving behaviors more realistic. However, they did not study the traffic flow model considering the mixed traffic conditions of HDVs and CAVs. Furthermore, they did not consider the randomness of HDVs' drivers' bounded rationality. To bridge these research gaps, the random bounded rationality is introduced into the car-following model of HDVs and study the impact of random bounded rationality on the mixed traffic conditions of HDVs and CAVs.

Data Collection.
is research aims to study the status of mixed traffic flow between HDVs and CAVs, and the random bounded rationality of HDVs should be considered. To find out the distribution laws of drivers' bounded rationality, we did some investigations about the bounded rationality of HDVs in April 2020.
For investigating the bounded rationality of HDVs, we designed a questionnaire. e detailed questionnaire design is shown in Table 1.
We choose respondents from different ages and different genders to do a questionnaire survey through by stratified sampling method (see Table 2). en, we delete the wrong questionnaires, such as incomplete questionnaires, and questionnaires containing error information.
e survey contains 1,000 questionnaires, including 979 valid questionnaires. e purpose of the questionnaire is to investigate the ε n (ε n is a parameter which reflects the nth driver's bounded rationality) and v AB (v AB is a parameter that reflects the tolerance of speed difference between the route a and route b, where the lengths are, respectively, L a and L b , and the routes are as for Figure 1). Referring to the research of Ou et al. [10], reference values of ε n and v AB are 0.1 and 2 (see Table 2). To test the internal consistency reliability of the questionnaire, we use SPSS to calculate the value of Cronbach's alpha (see Table 2). e value of Cronbach's alpha indicates that the questionnaire is reasonable.

Data Processing.
In this paper, we investigated ε n and v AB of drivers, and we used ORIGIN to fit the data. Statistical results and the fitting curves are shown in Figures 1 and 2. Moreover, fitting parameters of ε n and v AB are shown in Table 3. Figures 2 and 3 show the fitting condition of ε n and v AB . Additionally, R 2 is the goodness of fit, which could represent the fitting degree of data. R 2 of ε n and v AB demonstrate that they follow the normal distribution, and their distribution is as follows: (1)

Car-Following Model of HDVs.
e basic car-following model is as follows: where v n , Δx n , and Δv n are, respectively, the nth vehicle's velocity, headway, and relative speed; f is the stimulus function.
We can obtain different car-following models from equation (2), e.g., the optimal velocity (OV) model, the generalized force (GF) model, and the full velocity difference (FVD) model [45]. Comparing with the OV model and GF model, the FVD model can produce the desired result and disappear unrealistically high acceleration [42]. Since the model could simulate many complex traffic phenomena, we use the FVD model to simulate vehicles' behavior. e FVD model [45] is as follows: where k and c are two reactive coefficients; V is the optimal velocity; Δx n and Δv n are, respectively, the nth headway and relative speed. Jiang et al. [45] defined k, c, and V(Δx n ) as where l c is the vehicles' average length; V 1 , V 2 , C 1 , C 2 , and l c are parameters which were calibrated by Helbing and Tilch [46], and they indicated optimal parameter values. V 1 , V 2 , C 1 , C 2 , and l c are defined as [45] V 1 � 6.75 m/s, To study the impacts of bounded rationality, Tang et al. [27] proposed a car-following model with bounded rationality as where ε n is a parameter that reflects the nth driver's bounded rationality; k and c are two reactive coefficients; v n , Δx n and Δv n are, respectively, the nth vehicle's velocity, headway, and relative speed; V is the optimal velocity; and dv n /dt indicates the nth vehicle's accelerated velocity.

Car-Following Model of CAVs.
e PATH Laboratory of the University of California, Berkeley, has carried out longterm research on the cooperative adaptive cruise control (CACC) model and proposed a CACC car-following model based on a constant headway [47][48][49][50]. e model structure is as a n (t) � k 1 a n− where a n (t) represents the acceleration of vehicle n at time t; x n (t) indicates the location of vehicle n at time t; v n (t) is the velocity of vehicle n at time t; l represents the length of the vehicle; s 0 indicates the safe distance between vehicles; t g is the expected time headway of CACC; k 1 , k 2 , and k 3 are coefficients of the model, and we take k 1 , k 2 , and k 3 as 1.0, 0.2, and 3.0, respectively [51]. e CACC model proposed by PATH is clear in the physical meaning and is currently commonly used as the CACC car-following model. However, the expected spatial headway under the constant spatial headway carfollowing strategy is a linear function of speed, which makes the impact of traffic flow characteristics with different spatial headway different. erefore, this paper proposes the nonlinear dynamics-expected time headway strategy of the CACC car-following model. Based on the traffic flow following characteristics, the lower vehicle speed makes expected spatial headway smaller. Based on the results of [52], the dynamic expected spatial headway is as where h d (t) represents the expected spatial headway; v 0 is the velocity of free traffic flow. According to [53], s 0 and l are, respectively, 2 m and 5 m.     [54][55][56], whereas the feedback strategies did not consider random bounded rationality and the penetration rate of CAVs. erefore, we proposed HFS in a microsimulation, which takes random bounded rationality and penetration rate of CAVs into account. In HFS, because drivers usually choose the route based on the speed of traffic flow, we take the average speed of vehicles on the route as the reference index. en, we give the following underlying assumptions: (1) e traffic network includes routes a and b (see Figure 3), where the lengths are, respectively, L a and L b ; N vehicles enter/leave the network at the origin/ destination, where each vehicle's no. on the route a/b is n a /n b , and its time headway at the origin is Δt 0 ; and the routes are homogeneous. (2) When n ≤ N 0 , the nthe driver randomly selects route; when n > N 0 , the driver selects a route based on the HFS. (3) M a/b is the number of vehicles on route a/b; v j,a/b is the jth vehicle's speed on route a/b. (4) v AB > 0, ε n > 0 are parameters that represent random bounded rationality. (5) e HFS with random bounded rationality consists of HVFS (HDV feedback strategy) and CVFS (CAV feedback strategy); pr CAV is the penetration rate of CAVs. In this strategy, the differences between HDVs and CAVs are reflected in ε n and v AB . erefore, we proposed HVFS and CVFS to reflect the differences in the mobility characteristics of these vehicles. (6) e HVFS with random bounded rationality is defined as follows: , the vehicle randomly chooses route (7) e CVFS is defined as follows: Finally, we define the calculation of trip cost as where T n is the nth vehicle's trip cost; t n is the nth vehicle's travel time; α is the value of time (the unit is Yuan/s); T a , T b is the total trip cost on the route a/b, respectively; T k,a , T k,b is the kth vehicle's trip cost on the route a/b, respectively; and T total is the traffic network's total trip cost. In the numerical test, parameters [27,44] are as equation (14). e numerical test indicates that the first ten vehicles randomly choose route, and then, other vehicles choose the route based on HFS. Moreover, we assume that the time value is 0.01 Yuan/s [27,44]: We use the numerical test to study the impacts of random bounded rationality, and penetration rate of CAVs Journal of Advanced Transportation 5 on vehicles' trip cost under HFS in the traffic network, and we use the Euler difference to discretize equations (7) to (15) and (16): where Δt � 0.1 s is the time-step length and x n is the nth vehicle's position.

Results
First, we study the effects of CAV penetration rate on vehicle trip costs under the HFS (see Figures 4-6). In the HFS, we not only considered the random bounded rationality but also took the penetration rate of CAVs into account. In the HFS, ε n and v AB indicate the random bounded rationality, and pr CAV represents the penetration rate of CAVs. And in the numerical test, the value range of Δt 0 is in (0, 6) and the value range of pr CAV is in (0, 1). From Figures 4-6, we can obtain the following findings: (1) When pr CAV is relatively low (e.g., pr CAV � 0.25), the difference between the travel time of two routes is prominent. As pr CAV rises, the difference between the travel time of vehicles on the two routes drops. (2) When pr CAV is relatively high (e.g., pr CAV � 0.75), the difference between the travel time of two routes is inconspicuous. (3) When Δt 0 is short (e.g., Δt 0 � 0.5 s), vehicles' travel time rises with its number; each vehicle' travel time on each route is slightly longer than the corresponding one in the study [10], but each vehicle's travel time in our study is fluctuant. e result shows that random bounded rationality could increase the volatility of each vehicle's travel time. en, we analyze the effects of random bounded rationality and the penetration rate of CAVs on the total trip cost of the traffic network (see . Figures 7 and 8 show that the total trip cost drops rapidly while Δt 0 < 1.5 s, because of each vehicle's travel time on each route decreases rapidly with Δt 0 . Moreover, when Δt 0 ≥ 1.5 s, the total trip cost smoothly decreases. Furthermore, as the penetration rate of CAVs grows, the total trip   cost goes down consistently. It indicates that the penetration rate of CAVs effectively reduces the total trip cost in the traffic network. Impacts of Δt 0 and pr CAV are shown in Figure 9, and higher pr CAV and longer Δt 0 lead to less total trip cost (see Figure 9).

Discussion and Conclusions
6.1. Discussion. In this paper, we studied the effects of the CAV penetration rate on each vehicle's travel time and system total trip cost under the HFS considered the random bounded rationality, and we can obtain the following findings: (1) e bounded rationality of drivers follows the normal distribution, and the distribution is as equation (1). Before the investigation, we give the interviewees a reference value. It may influence their choices of bounded rationality.      (2) e CAV penetration rate has qualitative effects on the travel time of each vehicle and system total trip cost (see . As pr CAV increases, the travel time of each vehicle and total trip cost both smoothly decrease because of the CAVs' entry reduces the average hesitate time of all vehicles. (3) e total trip cost of the traffic network rapidly drops when Δt 0 < 1.5 s, because of each vehicle's travel time on each route declines rapidly with Δt 0 . Moreover, when Δt 0 ≥ 1.5 s, the system total cost smoothly decreases. And Figure 9 indicates that larger Δt 0 leads to less travel time and total trip cost.
Nevertheless, this research still has some limitations. For example, we did not test the numerical results by experiments on the road and a real medium-sized road network should be studied. Moreover, we will take more factors into account to simulate the traffic flow realistically, such as CAV market penetration, platooning intensity effects, single lane, and managed lane.

Conclusions.
Many feedback strategies were proposed to study the condition of the traffic flow in traffic networks [57], but the feedback strategies do not consider the randomness of bounded rationality and the penetration rate of CAVs in the mixed traffic flow of HDVs and CAVs. In this paper, we proposed HFS, which takes random bounded rationality and the penetration rate of CAVs into account. According to the numerical results, we found that drivers' bounded rationality follows the normal distribution (ε n ∼ N(0.101, 0.021 2 ), v AB ∼ N(1.897, 0.280 2 )). Based on the distribution of random bounded rationality, we found that the penetration rate of CAVs could effectively reduce the system total trip cost. Furthermore, Δt 0 under HFS has qualitative effects on the trip cost of vehicles.
In the future, more experiments can be concerned. We can test the numerical results by experiments on the road and simulate the traffic flow in a real medium-sized road network. Furthermore, we will take more factors into account to simulate the traffic flow realistically, such as CAV market penetration, platooning intensity effects, single lane, and managed lane, to make more contributions to the CAV study.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.