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INTRODUCTIONVehicular trajectory provides massive spatial‐temporal traffic information, which has been used in many transportation research, for example, transportation fuel consumption and emissions [1–3], traffic signal optimization [4–6], traffic flow modelling, validation, and calibration [7–11], driving behaviour analysis [12, 13], traffic state estimation [14–17] in terms of volume [18, 19], queue length [20–22], and fundamental diagram [23, 24]. Therefore, it is of great significance to obtain more vehicle trajectory data [25, 26].The acquisition of vehicle trajectory data is generally based on sensors, such as video, aerial photography, GPS, or connected vehicles (CVs). These methods have the following disadvantages: (i) Fixed detectors are expensive to install and maintain, and they can only collect vehicle trajectory data within a limited detection range; (ii) Due to the low penetration rate (PR) of CVs, only limited trajectory data with measuring error can be obtained. With the development of connected vehicles and automated driving technologies, connected automated vehicles (CAVs) can not only provide their own trajectories, but also obtain the exact trajectories of vehicles within their detection range [27]. Therefore, all trajectories can be obtained theoretically if the PR of CVs and CAVs is enough. However, the deployment and upgrade of CAVs will take an extended period [28–30]. The latest research [31] showed that the PR of Level‐4 CAVs was expected to reach 24.8% in 2045. Consequently, the mixed traffic flow will be composed of regular vehicles (RVs), CVs, and CAVs for an extended period in the future. In the mixed traffic flow, most trajectories cannot be obtained directly except for CVs and CAVs. There is still challenging to obtain a full‐sample vehicle trajectory due to the low PRs of CVs and CAVs. Therefore, it is more important to develop the trajectory reconstruction method based on limited trajectory data of CVs and CAVs [32]. Wang et al. [26] proposed a trajectory reconstruction method for the mixed traffic flow with CAVs and human‐driven vehicles (HDVs). They assumed that all HDVs were RVs, which cannot provide the vehicle trajectory. However, if some HDVs are equipped with some apps, such as Uber, Didi, Google Maps, and so on, they can be regarded as CVs, sharing their trajectory data. Therefore, the mixed traffic flow can contain three types of vehicles, RVs, CVs, and CAVs.To fill the gap, this study aims to propose a trajectory reconstruction method for the mixed traffic flow with RVs, CVs, and CAVs. In this study, the impact of lane‐changing behaviour will not be considered, so we only reconstruct the single‐lane trajectories. Firstly, the car‐following behaviour and the data environment of mixed traffic flow are analysed. Then, the trajectory data provided by CAVs and CVs are fully utilized, and a trajectory reconstruction method is developed for the mixed traffic flow based on the general car‐following model. Finally, numerical simulation is designed to investigate the influence of traffic density and PR of CAVs and CVs. Therefore, the main contributions of this study are as follows:The number of RVs within the detected range of the CAV is estimated based on different traffic densities. Unlike the existing studies [26], it is assumed that the number of RVs obtained within the detected range of the CAV is fixed and is not affected by the traffic density.On the bias of the car‐following behaviour, a non‐linear optimization model is developed to estimate the number of inserted RVs.To minimize the impact on other vehicle trajectories, a non‐linear optimization model is proposed to estimate the position of inserted RVs.The performance of the proposed method is tested under different traffic densities and PRs of CAVs and CVs.The remainder of this paper is organized as follows. The literature review of the vehicle trajectory reconstruction method is presented in Section 2. Section 3 illustrates the methodology's details for reconstructing trajectories in mixed traffic flow with RVs, CVs, and CAVs. The influence of traffic density and PRs of CAVs and CVs on the proposed method's performance based on the simulation experiment is discussed in Section 4. Section 5 further verifies the performance of the proposed model with NGSIM data. Finally, Section 6 ends the study with conclusions and future work.LITERATURE REVIEWMany recent studies have carried out vehicle trajectory reconstruction, mainly based on classical traffic flow theory and statistical probability models. Moreover, the data source of these methods can be divided into three categories: fixed traffic detectors, mobile detectors, and multi‐source data.Coifman [33] reconstructed the vehicular trajectory by estimating the space‐time diagram using data from dual loop detectors based on the traffic flow theory and the postulate of the triangular flow density relationship. The result showed that the method was suitable for freeways with uninterrupted traffic flow and was not ideal for the interrupted flow of urban roads with traffic signals. Yu et al. [34] proposed a model to reconstruct incomplete vehicle trajectories through depth‐first traversal and TOPSIS algorithms based on vehicle identification data. Rao et al. [35] used automatic license plate recognition (ALPR) data to construct a vehicle path reconstruction method in a transportation network, which can further estimate the network's historical OD pattern based on the path flow. The result showed that this method had been verified in urban traffic networks and micro‐simulation software. However, they did not discuss the influence of the coverage rate and effective recognition rate of the automatic vehicle identifier. Zhang et al. [36] developed a High‐angle Spatial‐Temporal Diagram Analysis (HASDA) model to reconstruct vehicle trajectories from infrastructure traffic surveillance videos, and the method was examined with NGSIM data. The result showed that more than 90% of vehicle trajectories could be constructed on average.There are many trajectory reconstruction methods based on mobile detectors. Hao et al. [37], Wan et al. [38], and Shan et al. [39] respectively used a random model, expectation‐maximization (EM) algorithm, and maximum likelihood estimation (MLE) to reconstruct the motion trajectories of floating‐car from a microscopic perspective based on the probability theory. In addition to probability theory, the classic traffic wave theory can describe the changes in traffic flow in time and space, which is the basis for reconstructing the full sample vehicle trajectories. Newell [40–42] extended the two‐dimensional traffic flow theory to the 3D traffic flow theory by increasing the cumulative queuing length of road network nodes in the space‐time coordinates. Then, a variational formulation (VF) of kinematic waves was developed [43, 44]. Based on the VF of the kinematic wave, Sun et al. [45] assumed that the arriving vehicles between floating cars obeyed a uniform distribution and reconstructed the full sample trajectory of the straight traffic flow at a single‐point intersection. Then, the approach was calibrated and verified based on microscopic simulation and NGSIM data. However, they only reconstructed short trajectories around an intersection and cannot extend to the arterial. Li et al. [46] expanded the sample size using incomplete trajectories based on K‐nearest neighbour regression. This method used a similar complete trajectory to represent the incomplete trajectory, and then expanded the sample size of trajectories to estimate the travel time. Wang et al. [47] developed a piecewise deceleration and acceleration model and estimated the distribution parameters of the acceleration data for each travel mode by a new EM algorithm. Then, the trajectories were reconstructed by the acceleration statistics.Some studies also try to reconstruct trajectories based on multi‐source data sources. Based on the modified timings of traffic signals, vehicle passing times, and vehicle travel time, Xie et al. [48] reconstructed the vehicle trajectory of the urban arterial in a microscopic simulation environment. Considering the differences in the format, sampling frequency, and precision of the data collected by different types of detectors, more and more researchers tend to fuse multi‐source data to reconstruct trajectories. Feng et al. [49] proposed a vehicle trajectory reconstruction method for a large‐scale network by using AVI and traditional detector data, whose framework was particle filter theory. Mehran et al. [50] developed a method for estimating the vehicles' trajectory in arterial based on the variational theory by fusing second‐level floating car data, AVI data, and traffic signal parameters. Verification and error analysis was carried out by the measured data of a signalized arterial in Tokyo with five intersections. The results showed that although the traffic wave variational theory can be used to reconstruct the trajectories of the vehicle. The proposed method is based on the ‘first‐in‐first‐out’ criterion and the triangular fundamental diagram, which was only suitable for single‐lane scenarios. When the roadside entrance and exit traffic flow is high, the PRs of floating‐car are low, and the phenomena of lane changes and overtaking on the road section are frequent, the accuracy of the full sample trajectory reconstruction will be reduced significantly.The above studies can effectively reconstruct the vehicular trajectories, but they only focus on the human‐driven environment with RVs and CVs and do not consider the mixed traffic flow with CAVs. With the development of technologies related to CAVs, CAVs can provide their trajectories and other vehicle trajectories within a detection range through sensing devices (i.e., cameras and radars). Therefore, the mobile sensor data collected by CAVs provides new data sources for trajectory reconstruction. Wang et al. [26] reconstructed the trajectories of RVs using the data detected by CAVs on the freeway under the mixed CAVs environment based on the car‐following and cellular automaton models. However, this method only considered the mixed traffic flow with two types of vehicles: CAVs and RVs. In the future, the mixed traffic flow will generally consist of RVs, CVs, and CAVs. In addition, Wang et al. [26] and Jiang et al. [51] set the number of detected vehicles by CAVs to a fixed value. This means that they do not consider the difference under different traffic flow densities and occasions in which some vehicles are blocked by a front vehicle. Therefore, this study proposes a new trajectory reconstruction method that takes the detection range of CAVs into account for mixed traffic flow with RVs, CVs, and CAVs. To sum up, the detailed comparison of studies on trajectory reconstruction is summarized in Table 1.1TABLEThe comparison of selected literature on trajectory reconstructionReferenceVehicle typesTraffic sceneData sourceTheoretical basis[33](1) RVsFreewayDual loop detectorsTriangular flow density relationship[34](1) RVsNetworkALPR dataDFS and TOPSIS[35](1) RVsNetworkALPR dataParticle filter[36](1) RVsFreewayTraffic surveillance videosHASDA[37](1) RVs, (2) CVsUrban arterialCVsRandom model[38](1) RVs, (2) CVsUrban arterialCVsEM algorithm[39](1) RVs, (2) CVsFreewayCVsMaximum likelihood estimation[45](1) RVs, (2) CVsIntersectionCVsVariation formulation[46](1) RVs, (2) CVsUrban arterialCVsK‐nearest neighbour regression[47](1) RVs, (2) CVsIntersectionLow‐frequency CVsNew EM algorithm[48](1) RVsUrban arterialLoop detectors and traffic control dataGeneric particle filter framework[49](1) RVsNetworkAVI and traditional detectorParticle filter theory[50](1) RVs, (2) CVsIntersectionCVs, AVI, and signal control parametersVariational theory[26](1) RVs, (2) CAVsFreewayCAVsCellular automationThis study(1) RVs, (2) CVs, (3) CAVsFreewayCAVs and CVsCar‐following modelMETHODOLOGYThe framework of the proposed methodIn the mixed CAVs environment, there are three types of vehicles considered, RVs, CVs, and CAVs. In addition to providing their trajectory data, CAVs can detect the real‐time position and behaviour of the surrounding vehicles through the sensing equipment. Thus, they can also provide trajectory data of vehicles within the detection range. As shown in Figure 1, trajectory data of CAVs with their surrounding vehicles and CVs themselves could be obtained. Since the CVs have no sensing equipment, they can only provide their own GPS trajectory data. Moreover, RVs have no communication function or the communication function is not turned on, so they can provide neither their own trajectory data nor the trajectory data of surrounding vehicles. Thus, the schematic diagram of trajectories is shown in Figure 2. The trajectories represent the positions of the vehicles. Since this paper aims to study single‐lane trajectories, the trajectory curves cannot cross. The red dashed lines represent the trajectories of CAVs, the blue dotted lines represent the trajectories of CVs, and the solid black lines represent the trajectories of RVs detected by CAVs.1FIGUREThe composition of different types of vehicles in mixed traffic flow2FIGUREThe trajectory of mixed traffic flowAs shown in Figure 2, this study aims to reconstruct the trajectory of RVs in the undetected range of the CAVs. Namely, the trajectories of the vehicles involved in the shaded areas A and B. In the car‐following model, the behaviour of the following vehicle is mainly related to the leading vehicle. This means the trajectory of the leading vehicle directly affects the trajectory of the following vehicle. Taking the shaded area A as an example, the last vehicle detected by the previous CAV can be defined as Vn−1${V_{n - 1}}$, and the CV2 can be defined as Vn${V_n}$. If there are no other vehicles in area A, Vn${V_n}$ will follow Vn−1${V_{n - 1}}$ to drive. The car‐following behavior of Vn${V_n}$ and Vn−1${V_{n - 1}}$ can be described by the car‐following model. However, if there are unobserved RVs in area A, the acceleration of Vn${V_n}$ calculated by the car‐following model will be quite different from the actual acceleration. To this end, the proposed algorithm finds the optimal number and position of trajectories in area A to minimize the impact on other vehicle trajectories. The algorithm of trajectory reconstruction can be determined, as described in Table 2.2TABLESteps of the trajectory reconstruction algorithmStep1. Find the area where it is needed to insert RVsFind out the undetected range of CAVs.Step2. Estimate the number of inserted RVsFor m = 1 :1: NCalculate the theoretical acceleration of Vn${V_n}$ after inserting m cars.Endm∗${m^*}$ will be the optimal number of inserted RVs when the RMSE between theoretical acceleration and actual acceleration is minimal.Step3. Estimate the speed of the inserted RVsEstimate the speed of the inserted RVs based on the local average speed.Step4. Estimate the positions of the inserted RVsCalculate the inserted positions of the RVs with optimization problems based on the car‐following model.Step5. Reconstruct trajectoriesDraw the vehicular trajectories based on Step1 to Step4.The flowchart of the proposed method is shown in Figure 3. Figure 3 shows that the proposed method includes two main parts, that is, insert RVs in the undetected ranges and reconstruct trajectories of inserted RVs. The details of the method will be discussed later.3FIGURESchematic of trajectory reconstructionCar‐following behaviourIn this study, the impact of lane‐changing behaviour will not be considered. A more straightforward scene is selected to illustrate the proposed method. In the microscopic traffic model, the car‐following model is generally used to describe the car‐following behaviour between adjacent vehicles. The model indicates that the following vehicle's acceleration and deceleration are related to its own speed, speed difference, and position difference between the leading vehicle and the following vehicle. Therefore, when the part of the trajectories is known, the car‐following model can be used to analyse the possibility of other vehicle trajectories in the undetected range. The general car‐following model form fn${f_n}$ is shown in Equation (1).1an(t+Δt)=fn(vn(t),vn−1(t)−vn(t),xn−1(t)−xn(t)),\begin{eqnarray} {a_n}( {t + \Delta t}) = {f_n}({{v_n}(t),{v_{n - 1}}(t) - {v_n}(t),{x_{n - 1}}(t) - {x_n}(t)}), \end{eqnarray}where an(t+Δt)${a_n}( {t + \Delta t} )$ represents the acceleration of the vehicle n at time t+Δt$t + \Delta t$; xn(t)$\;{x_n}( t )$ and xn−1(t)$\;{x_{n - 1}}( t )$ represent the position of the vehicle n and n−1$n - 1$ at time t, respectively; vn(t)$\;{v_n}( t )$ and vn−1(t)$\;{v_{n - 1}}( t )$ represent the speed of the vehicle n and n−1$n - 1$ at time t, respectively.Inserting RVs based on CAVs and CVs' trajectoriesData environment analysisThe trajectory data provided by the CVs can be divided into two types. The trajectory data within the detection range of CAVs, such as CV1 and CV3 in Figure 2, which can be directly observed by CAVs. In the real world, the trajectory of CVs is provided by their GPS. Therefore, the two copies of trajectory data of CVs can be used for mutual validation of the accuracy of CAVs in detecting the positions of surrounding vehicle trajectories. In this paper, because the real detection data of CAVs cannot be obtained, we assume that the trajectory data obtained by the two methods are the same. The difference between the two is not considered. Besides, if the CVs are not within the detection range of the CAVs, as shown in CV2 in Figure 2, the trajectory of CV2 can be used directly to reconstruct the vehicular trajectory in the shadow areas A and B.Estimating the number of inserted RVsAs shown in Figure 4, the last following RV within the detection range of CAV1 is defined as Vn−1${V_{n - 1}}$, and CV2 is defined as Vn${V_n}$. Considering the minimum safety distance between the adjacent vehicles, let N (veh) represents the maximum possible number of the inserted RVs, which can be estimated by Equation (2).2N=minxn−10−xn0s0+l,xn−1Δt−xnΔts0+l,…,xn−1kΔt−xnkΔts0+l,\begin{eqnarray} N &=& \left\lfloor\min\left(\frac{{{x_{n - 1}}\left( 0 \right) - {x_n}\left( 0 \right)}}{{{s_0} + l}}, \frac{{{x_{n - 1}}\left( {\Delta t} \right) - {x_n}\left( {\Delta t} \right)}}{{{s_0} + l}},\right.\right.\nonumber\\ &&\left.\left. \ldots ,\frac{{{x_{n - 1}}\left( {k\Delta t} \right) - {x_n}\left( {k\Delta t} \right)}} {{{s_0} + l}}\right) \right\rfloor, \end{eqnarray}where ⌊.⌋$\lfloor.\rfloor$ represents round down; l represents the length of vehicle; s0 represents the minimum space gap for completely stopping for vehicle; k represents the number of time steps.4FIGUREThe trajectory of the inserted RVsLet m$m\;$represents the number of inserted RVs. Definition {Vn1,Vn2,Vn3,Vn4,…,Vnm$V_n^1,V_n^2,V_n^3,V_n^4, \ldots ,V_n^m$} is the RVs that are inserted between Vn−1${V_{n - 1}}$ and Vn${V_n}$, where Vnm$V_n^m$ is the vehicle closest to Vn${V_n}$. This means the vehicle Vnm$V_n^m$ is the leading vehicle of Vn${V_n}$. After inserting those vehicles, the speed and position of the vehicle Vnm$V_n^m$ can be approximately estimated by Equations (3) and (4).3vnm(t)=vn(t)−vn(t)−vn−1(t)m+1,\begin{equation} v_n^m (t) = {v_n} (t) - \frac{{{v_n}(t) - {v_{n - 1}} (t)}}{{m + 1}},\end{equation}4xnm(t)=xn(t)−xn(t)−xn−1(t)m+1,\begin{equation} x_n^m (t) = {x_n} (t) - \frac{{{x_n} (t) - {x_{n - 1}} (t)}}{{m + 1}},\end{equation}where vnm(t)$v_n^m( t )\;$represents the speed of the vehicle Vnm$V_n^m$ at timet$\;t$; xnm(t)$x_n^m( t )\;$represents the position of the vehicle Vnm$V_n^m$ at timet$\;t$;vn(t)$\;{v_n}( t )$ represents the speed of the vehicle Vn${V_n}$ at time t;vn−1(t)$\;{v_{n - 1}}( t )$ represents the speed of Vn−1${V_{n - 1}}$ at time t.After estimating the speed and position of the vehicle Vnm$V_n^m$, the trajectory of the vehicle Vnm$V_n^m$ can be obtained. Based on the car‐following model, the theoretical acceleration of the vehicle Vn${V_n}$ at each time step can be calculated from the trajectory of the vehicle Vnm$V_n^m$. The calculation equation is shown in Equation (5).5ânm(t)=fn(vn(t−Δt),vn−1(t−Δt)−vn(t−Δt),xn−1(t−Δt)−xn(t−Δt)),\begin{eqnarray} \hat a_n^m (t) &=& {f_n} ({v_n}({t - \Delta t}),{v_{n - 1}} ({t - \Delta t})\nonumber\\ && -\, {v_n} ({t - \Delta t}),{x_{n - 1}} ({t - \Delta t}) - {x_n} ({t - \Delta t})), \end{eqnarray}where ânm(t)$\hat a_n^m( t )$ is the theoretical acceleration of the vehicle Vn${V_n}$ at time t when m vehicles are inserted.Here, the root mean square error (RMSE) between the theoretical and actual acceleration per time step is selected as the performance index. m∗${m^*}$ will be the optimal number of inserted RVs when the RMSE is the minimum, which can be calculated by Equation (6).6m∗=argminm1k∑t=ΔtkΔtânmt−ant2,m=0,1,2,…,N,\begin{eqnarray} {\rm{\;}}{m^*} = \mathop {{\rm{argmin}}}\limits_m \sqrt {\frac{1}{k}\mathop \sum \limits_{t = \Delta t}^{k\Delta t} {{\left[ {\hat a_n^m\left( t \right) - {a_n}\left( t \right)} \right]}^2}} \;,m = 0,1,2, \ldots ,N,\nonumber\\ \end{eqnarray}where an(t)${a_n}( t )$ is the actual acceleration of the vehicle Vn${V_n}$ at time t.In summary, the estimation of the number of inserted RVs is a non‐linear optimization problem. The objective function is Equation (6), and the constraints are Equations (2)–(5). This model is a typical non‐linear optimization model, which can be solved directly with the built‐in function FMINCON of MATLAB.Reconstructing trajectories of inserted RVsEstimating the speed of the inserted RVsTake the shaded area in Figure 5 as an example, the inserted RV can be defined as Vi${V_i}$. When there are no special circumstances on the road, the speed difference between the leading and following vehicles is minimal. Furthermore, small‐scale fluctuations in speed have little effect on the outcome of the car‐following model. Therefore, referring to [26], the speed s(At)$s( {{A_t}} )$ in the time‐space area At${A_t}$ can be estimated as follows.7s(At)=d(At)t(At)=[xn−1(t+Δt)−xn−1(t)]+[xn(t+Δt)−xn(t)]2Δt,\begin{eqnarray} s( {{A_t}}) &=& \frac{{d( {{A_t}})}}{{t({{A_t}})}}\nonumber\\ &=& \frac{{[ {{x_{n - 1}}( {t + \Delta t} ) - {x_{n - 1}} (t)}] + [{{x_n}({t + \Delta t}) - {x_n} (t)}]}}{{2\Delta t}},\nonumber\\ \end{eqnarray}where At${A_t}\;$represents the time‐space area of the vehicle Vi${V_i}$ at time t; k represents the number of time steps; Δt$\Delta t$ represents the length of the time step.5FIGURETime‐space area At${A_t}$Since the inserted RV is in area A, its speed at any time can be approximated as s(At)$s( {{A_t}} )$.Estimating the positions of the inserted RVsAt any moment, the position of inserted RVs must satisfy the safe distance between the leading and the following vehicle. Thus, the location of the vehicles can only be within a specific range. As shown in Figure 4, the position is the shaded area, which can be denoted by Equation (8).8xnt+s0+l≤xit≤xn−1t−s0−l\begin{equation}{x_n}\left( t \right) + {{\rm{s}}_0} + l \le {x_i}\left( t \right) \le {x_{n - 1}}\left( t \right) - {s_0} - l\end{equation}When the speed and the possible position range of the inserted RVs are obtained, Vi${V_i}$ and Vn${V_n}\;$can be regarded as the leading vehicle and following vehicle. Then, the insertion position xi(t)${x_i}( t )$ of the vehicle Vi${V_i}$ will have an impact on the vehicle Vn${V_n}$. This effect will be reflected in the acceleration of the vehicle Vn${V_n}$. Thus, the acceleration of the vehicle Vn${V_n}$ can be calculated based on the car‐following model.9ânxit=fnvnt−Δt,vit−Δt−vnt−Δt,xit−Δt−xnt−Δt\begin{eqnarray} {\hat a_n}\;\left( {{x_i}\left( t \right)} \right) &=& {f_n}\;\left({v_n}\left( {t - \Delta t} \right),{v_i}\left( {t - \Delta t} \right) - {v_n}\left( {t - \Delta t} \right),\right.\nonumber\\ &&\left.{x_i}\left( {t - \Delta t} \right) - {x_n}\left( {t - \Delta t} \right)\right) \end{eqnarray}where ân(xi(t))${\hat a_n}( {{x_i}( t )} )$ represents the estimated acceleration of the vehicle Vn${V_n}$ with given xi(t)${x_i}( t )$ at timet${\rm{\;}}t$.According to Equation (9), the acceleration of the vehicle Vn${V_n}\;$can be expressed as a function of variable xi(t)${x_i}( t )$. The acceleration of vehicle Vn${V_n}$ will change with different xi(t)${x_i}( t )$ in the feasible region. If the estimated acceleration ân(xi(t))${\hat a_n}( {{x_i}( t )} )$ is closest to the actual acceleration an(t)${a_n}( t )$, the corresponding xi(t)${x_i}( t )$ is the optimal position of the vehicle Vi${V_i}$.Thus, an objective function can be defined to find the optimal position of the inserted RV. Considering the trajectory contains the positions set of the vehicle at different times, the minimum root mean square error (RMSE) between the actual and the estimated acceleration of the vehicle Vn${V_n}$ can be selected as the optimization objective, which is shown in Equation (10).10minfxit=1k∑t=ΔtkΔtânxit−ant2\begin{equation}{\rm{\;}}\min f\left( {{x_i}\left( t \right)} \right) = \sqrt {\frac{1}{k}\mathop \sum \limits_{t = \Delta t}^{k\Delta t} {{\left( {{{\hat a}_n}\left( {{x_i}\left( t \right)} \right) - {a_n}\left( t \right)} \right)}^2}} \;\end{equation}In summary, the objective function of the optimization model is Equation (10), and the constraints are Equations (8) and (9). This model is a typical non‐linear optimization model that can be solved directly with the built‐in function FMINCON of MATLAB.The vehicle position at each moment is calculated by the optimization model, the smoothness of the vehicular trajectory is not considered. Therefore, the maximum acceleration and minimum acceleration limits are considered here to increase the smoothness of the trajectory.11xit=xit−Δt+vit−ΔtΔt+amaxΔt22,ait>amaxxit−Δt+vit−ΔtΔt+aminΔt22,ait<amin,{\fontsize{9.2}{11.2}{\selectfont{ \begin{eqnarray} {x_i}\;\left( t \right) = \left\{ \def\eqcellsep{&}\begin{array}{@{}*{1}{c}@{}} {\;{x_i}\left( {t - \Delta t} \right) + {v_i}\left( {t - \Delta t} \right)\Delta t + \dfrac{{{a_{max}}\Delta {t^2}}}{2},\;{a_i}\left( t \right) > {a_{max}}}\\[15pt] {{x_i}\left( {t - \Delta t} \right) + {v_i}\left( {t - \Delta t} \right)\Delta t + \dfrac{{{a_{min}}\Delta {t^2}}}{2},\;{a_i}\left( t \right) < {a_{min}}} \end{array} \right.,\nonumber\\ \end{eqnarray}}}}where amin${a_{min}}$ and amax${a_{max}}$ represent the minimum and maximum accelerations, respectively. The maximum and minimum acceleration are determined by the vehicle's performance. Normally, the maximum acceleration is set to 2.87 m·s−2${\rm{m}} \cdot {{\rm{s}}^{ - 2}}$, and the maximum deceleration is set to 4.33 m·s−2${\rm{m}} \cdot {{\rm{s}}^{ - 2}}$ [52].NUMERICAL EXPERIMENTSParameter settingsReferring to Wang et al. [26], a road segment with an exit and an entrance ramp is carried out in this study, as shown in Figure 6. This means no vehicles exit or enter the middle of the road segment. Considering that CAV technology has not been widely used, it is very challenging to obtain actual data. To verify the effectiveness of the proposed method, cellular automata rules were designed based on the car‐following model to simulate the mixed traffic flow of a single‐lane freeway. The cell length is 0.1 m, the simulation time is 1000 s, and the simulation step is 1 s. A random deceleration was set for the RVs and CVs during the simulation process to make the simulation data closer to the actual data.6FIGUREThe road segment of the simulation experimentMoreover, we use an anthropomorphic design for CAVs; that is, all car‐following behaviours for different vehicles (i.e. RVs, CVs, and CAVs) were described by the same car‐following model [53, 54]. In the existing research, the car‐following model can be divided into five types: stimulus‐response model [55], safe distance model [56], social force model [57], optimal velocity model [58, 59, 56] and low‐order linear model [55]. As a kind of social force model, the Intelligent Driver Model (IDM) [60, 61] can well capture the driving habits of experienced drivers and has a wide range of applications. Thus, the IDM model is adopted to examine the performance of the method proposed in this study. The form of the IDM model is shown as Equation (12).12ant=α1−vntvf4−snt∗xn−1t−xnt−l2,\begin{eqnarray} {a_n}\;\left( t \right) = \;\alpha \left[ {1 - {{\left( {\frac{{{v_n}\left( t \right)}}{{{v_f}}}} \right)}^4} - {{\left( {\frac{{{s_n}{{\left( t \right)}^*}}}{{{x_{n - 1}}\left( t \right) - {x_n}\left( t \right) - l}}} \right)}^2}} \right],\nonumber\\ \end{eqnarray}where13snt∗=s0+max0,Tvnt+vntvn−1t−vnt2αβ12,\begin{eqnarray} {s_n}{\left( t \right)^{\rm{*}}} = {s_0} + \max \left[ {0,{\rm{\;}}T{v_n}\left( t \right) + \frac{{{v_n}\left( t \right)\left[ {{v_{n - 1}}\left( t \right) - {v_n}\left( t \right)} \right]}}{{2{{\left( {\alpha \beta } \right)}^{\frac{1}{2}}}}}} \right],\nonumber\\ \end{eqnarray}where vf${v_{f\;}}$ represents the desired speed in free‐flow traffic conditions;α${\rm{\;}}\alpha $ is the maximum acceleration; β is the desired deceleration;T$\;T$ represents the safe time headway. Referring to Treiber et al. [61], the parameter values for the IDM model are α=1m·s−2,β=2m·s−2,s0=2m,l=5m,vf=33.3m·s−1,$\alpha = 1\;{\rm{m}} \cdot {{\rm{s}}^{ - 2}},\beta = 2\;{\rm{m}} \cdot {{\rm{s}}^{ - 2}},\;{s_0} = \;2\;{\rm{m}},l = 5\;{\rm{m}},\;{v_f} = \;33.3\;{\rm{m}} \cdot {{\rm{s}}^{ - 1}},$ and T=1.5s.$T = 1.5\;{\rm{s}}.$ This section designs two benchmark experiment schemes. The first scheme is to discuss the influence of different traffic densities via setting the PR of CAVs and CVs at 8% and 20%, respectively. The second scheme examines the impact of different PRs of CAVs and CVs, when the traffic density is 70 veh/km.The impact of traffic densityTo study the impact of traffic density on the proposed method, the PRs of CAVs and CVs are set up to be constant values, for example, 8% and 20%. Then, the volume‐density‐speed relationship of the road segment can be obtained based on the simulation experiment, as shown in Figure 7.7FIGURERelationship between volume‐density‐speedFigure 7 indicates that the optimal traffic density in the simulated road segment is 20 veh/km. This means that when the traffic density is more than 20 veh/km, the vehicles on the road segment begin to be congested. Besides, Figure 7 shows that when the traffic density is less than 20 veh/km, most vehicles can run freely with considerable space headway, which makes it difficult for the car‐following model to describe the relationship between vehicles accurately. The errors of the positions of the inserted RVs will be significant. Therefore, traffic density may have a great influence on the proposed method. To further discuss this influence, we set some different traffic densities, such as 20, 30, 40, 50, 60, 70, 80, and 90 veh/km.Referring to [62, 26], the detection range of the CAV is set as 100 m in this study. In this case, the number of vehicles detected by the CAV fluctuates over time. To address this issue, the reconstruction area can be partitioned according to the number of known trajectories. The trajectories of each sub‐area can be reconstructed respectively, and then the trajectories of the whole area can be obtained by connecting. For simplicity, the number of vehicles detected by CAV is assumed to be a fixed value in this paper [62, 26]. If the vehicles were evenly distributed on the road, the number of vehicles detected by CAV under different traffic densities is shown in Table 3. Considering the occlusion between vehicles and the error of sensing equipment, the trajectories near the edge of the detection range may not be fully recorded, so the two edge trajectories were removed. Given wireless communication's fragility, when the traffic density is high, data transmission will be delayed and restricted [63]. Thus, the upper limit of the number of trajectories detected by CAV was set to 6 even at high density. The schematic is in Figure 8.3TABLEThe number of vehicles detected by CAVDensity (veh/km)Average headway (m)Theoretical value (veh)Revised value (veh)2050.0223033.3224025.0425020.0446016.7647014.3668012.5869011.1868FIGUREThe vehicles were detected by CAV under different traffic densitiesTo avoid the influence of random factors, different random seeds are used for simulation. The average value under the same traffic density with different random seeds is taken as the result. The reconstructed trajectories under different traffic densities are shown in Figure 9. The number of inserted RVs is shown in Table 4.9FIGUREReconstructed trajectories under different traffic densities4TABLEEstimation of the number of inserted RVs under different traffic densitiesThe number of inserted RVs*Density (veh/km)Number of CAVsEstimated numberActual numberMAE/vehMAPE/%20858.8459.160.320.54301289.9889.520.540.604016119.40120.380.980.815020126.80127.921.120.886024152.07153.491.541.007028144.82151.208.985.948032146.96171.6028.5216.629036167.64193.2441.2021.32*The number of inserted RVs is the average value of multiple simulation experiments with different random seeds, so it may not be an integer.Table 4 shows that the errors of the number of inserted RVs increase with the increase in traffic density. This is because the space headway decreases with the increase in traffic density, which increases the difficulty in inserting the correct number of vehicles. Besides, as the traffic density increases, the stability of traffic flow is more sensitive to sudden deceleration, and the error of trajectory reconstruction increases. The superposition of the two kinds of errors shows the current pattern.When traffic density is more than 70 veh/km, the vehicles' average speed is less than 20 km/h, which rarely occurs on the freeway. On the other hand, when traffic density is less than 70 veh/km, the average absolute percentage error (MAPE) of the number of inserted RVs is less than 5.94%. Therefore, in a congested state, the errors of the number of inserted RVs obtained by the proposed method are within the acceptable range. This indicates that the proposed method is suitable for various traffic densities on the freeway.To further evaluate the performance of the proposed method, the average absolute error (MAE), MAPE, and root mean square error (RMSE) between the actual and estimated positions of the inserted RVs are selected to capture the accuracy of the reconstructed trajectories. The calculation equations of related indicators are as follows.14MAE=1k∑t=0kΔtx̂t−xt\begin{equation}{\rm{MAE}} = \frac{1}{k}\;\mathop \sum \limits_{t = 0}^{k\Delta t} \left| {\hat x\left( t \right) - x\left( t \right)} \right|\end{equation}15MAPE=100%k∑t=0kΔtx̂t−xtxt\begin{equation}{\rm{MAPE}} = \frac{{100\% }}{k}\;\mathop \sum \limits_{t = 0}^{k\Delta t} \left| {\frac{{\hat x\left( t \right) - x\left( t \right)}}{{x\left( t \right)}}} \right|\end{equation}16RMSE=1k∑t=0kΔtx̂t−xt2\begin{equation}{\rm{RMSE}} = \sqrt {\frac{1}{k}\mathop \sum \limits_{t = 0}^{k\Delta t} {{\left( {\hat x\left( t \right) - x\left( t \right)} \right)}^2}} \;\end{equation}where x̂(t)$\hat x( t )$ and x(t)$x( t )$ represent the estimated and actual position of the inserted RV at time t.According to Equations (14), (15), and (16), the errors under different traffic densities can be obtained, as shown in Table 5.5TABLEThe errors of trajectory reconstructionThe position of inserted RVsDensity (veh/km)Number of CAVsMAE/mMAPE/%RMSE/m20822.640.368623.45301212.890.295113.4140167.060.18607.4950203.380.09893.7860243.090.08504.3870285.540.19375.7080326.740.22327.0790367.500.23166.61Average value8.610.21038.99Considering that the distance between vehicles is not the same under different traffic densities, it is appropriate to use MAPE to evaluate the effect of trajectory reconstruction. Table 5 shows that when the traffic density is less than or equal to 60 veh/km, the MAPE of trajectory reconstruction gradually decreases with the increase of traffic density. When the traffic density is more than 60 veh/km, the number of vehicles detected by CAV is 6. The MAPE of trajectory reconstruction increases with the increase of the traffic density. This is because, in an undetected range, we reconstruct the first trajectory and then use this trajectory to reconstruct the previous trajectory. Finally, all the trajectories cyclically in the range can be reconstructed based on this method. Since the error of the first trajectory, when reconstructing the second trajectory, the error will accumulate. Therefore, the more the trajectory is inserted, the larger the reconstruction error is.The average MAE, MAPE, and RMSE of trajectory reconstruction are 8.61 m, 0.2103%, and 8.99 m, respectively, under different traffic densities. This means the error of the proposed method is relatively stable under different traffic densities.Furthermore, we analyse the sensitivity of the number of vehicles detected by CAV. When the traffic density is 60 veh/km, the results are shown in Table 6. It can be seen from the results that the number of vehicles detected by CAV has a small effect on the results of this model when the PRs of CAVs is 8%.6TABLEThe error of inserted RVs under different numbers of vehicles detected by CAVThe number of inserted RVsThe position of inserted RVsNumber of vehicles detected by CAVMAE/vehMAPE/%MAE/mMAPE/%RMSE/m22.111.163.100.08694.3941.541.003.090.08504.3861.280.993.050.08374.37The impact of PRs of CAVs and CVsThe trajectory provided by CAVs and CVs is the known data of the proposed method. Therefore, it is essential to study the influence of the PRs of CVs and CAVs on the performance of the proposed method.The impact of the PR of CAVsIn order to discuss the influence of the PR of CAVs, the traffic density is set as 60 veh/km, and the PR of CVs is set as 20%. The detection range of the CAV was set as 50 m. The PRs of CAVs are set as 2%, 4%, 6%, 8%, 10%, 12%, 14%, 16%, and 18% respectively. The error is shown in Table 7, and the reconstructed trajectories are shown in Figure 10, which are partially enlarged diagrams.7TABLEEstimation of the number of inserted RVs under different PR of CAVsThe number of inserted RVsPR of CAVsNumber of CAVsEstimated numberActual numberMAE/vehMAPE/%2%6206.64214.107.663.584%12188.28192.324.202.186%18169.68171.002.521.478%24152.07153.491.541.0010%30135.88136.721.040.7612%36119.78120.540.760.6314%42105.60106.200.600.5616%4894.0694.400.340.3618%5483.9684.180.300.3610FIGUREReconstructed trajectories under different PRs of CAVsIt can be seen from Table 7 that when the traffic density is 60 veh/km, the MAE and MAPE of the number of inserted RVs are gradually reduced with an increase in the PR of CAVs. This means the accuracy of the proposed method also increases with the PR of CAVs. This is because when the traffic density is constant, the number of inserted RVs decreases with the increase of PR of CAVs, and the difficulty of trajectory reconstruction decreases.Table 7 shows that the MAPE of the number of inserted RVs is between 0.36% and 3.58%. When the PR of CAVs is at a low level of 2%, the proposed method can also estimate the number of inserted RVs reasonably. Therefore, the proposed method can be well applied in the environment of different PRs in the future. Further, the position error of the reconstructed trajectories can be calculated, as shown in Table 8.8TABLEThe error of trajectory reconstructionThe position of inserted RVsPR of CAVsNumber of CAVsMAE/mMAPE/%RMSE/m2%63.840.10225.424%123.600.09565.126%183.450.09214.938%243.090.08504.3810%302.940.08314.1912%362.800.07624.0214%422.620.07303.7716%482.520.07233.5918%542.360.06743.36Table 8 shows that when the traffic density is 60 veh/km, the position error of trajectory reconstruction gradually decreases with the increase of PR of CAVs. This is because when the traffic density is constant, the number of inserted RVs decreases with an increase in PR of CAVs. Then, the cumulative error will decrease, and the total error of trajectory reconstruction will also decrease.The impact of PRs of CVsThe PR of CAVs is set at 8%. When the traffic density is 60 veh/km, the impact of different PRs of CVs is discussed, for example, 12%, 16%, 20%, 24%, and 28%. The errors of trajectory reconstruction are shown in Table 9.9TABLEThe error of inserted RVs under different PR of CVsThe number of inserted RVsThe position of inserted RVsPR of CVsNumber of CVsMAE/vehMAPE/%MAE/mMAPE/%RMSE/m12%365.963.504.100.10565.6316%482.821.753.390.09494.7520%601.541.003.090.08504.3824%721.080.732.700.07473.8328%840.320.232.550.07263.60Table 9 indicates that with the increase in the PR of CVs, the MAE and MAPE of the number of inserted RVs gradually decreases. When the PR of CVs is 28%, the MAE, MAPE, and RMSE of the position of inserted RVs are 2.55 m, 0.0726%, and 3.60 m, respectively.To further discuss the relationship between CAVs and CVs, the PR of CAVs is the X‐axis, and the PR of CVs is the Y‐axis, and the heat map of MAPE of the number and position of inserted RVs is plotted, as shown in Figure 11.11FIGUREHeat map of MAPEFigure 11 shows that the two errors are all reflected that the accuracy of trajectory reconstruction gradually improves as the PR of CAVs or CVs increases. Moreover, compared with the PR of CVs, the PR of CAVs has a significant impact on the results when the PRs of CAVs and CVs is high. This is because a CAV can obtain the trajectory of multiple vehicles on the road segment, while a CV can only obtain its own trajectory data. Therefore, when the PR of CAVs is high, there are more vehicles with known trajectories. Currently, the CVs have less influence on the experimental results.AN EMPIRICAL STUDYAfter using the simulation experiment for sensitivity analysis, vehicular trajectories from the Next‐Generation Simulation (NGSIM) are deployed to validate the proposed method. This paper uses the data set collected from the I‐80 highway, between 5:00 PM and 5:15 PM on 13 April, 2005. The time step is 0.1 s. There are only 20 continuous vehicle trajectories in the database, which meet the experimental requirements. Based on the simulation results (i.e. Tables 8 and 9), the penetration rate of CAVs and CVs are set to 8% and 20%, respectively. As a result, 8% and 20% of vehicles are randomly assigned as CAVs and CVs. Similarly, the IDM model is adopted to capture the car‐following of RVs, CVs, and CAVs. The parameter values are α=1.41m·s−2,β=2.23m·s−2,s0=2.17m,l=5m,vf=23.8m·s−1,$\alpha = 1.41\;{\rm{m}} \cdot {{\rm{s}}^{ - 2}},\beta = 2.23\;{\rm{m}} \cdot {{\rm{s}}^{ - 2}},\;{s_0} = \;2.17\;{\rm{m}},\;l = 5\;{\rm{m}},\;\;{v_f} = \;23.8\;{\rm{m}} \cdot {{\rm{s}}^{ - 1}},$ and T=1.27s,$T = 1.27\;{\rm{s}},$ which are calibrated by Kim et al. [64]. In addition, to verify the effectiveness of the proposed method, when the detection range of CAV is the smallest, the number of detected vehicles is set to 2. The errors of trajectory reconstruction are shown in Table 10, and the reconstructed trajectories are shown in Figure 12.10TABLEThe error of inserted RVsThe number of inserted RVsThe position of inserted RVsEstimated numberActual numberMAE/vehsMAPE/%MAE/mMAPE/%RMSE/m9.8010.101.5815.644.651.28994.1912FIGUREReconstructed trajectoriesFigure 12 presents the comparison between the actual trajectories and the reconstructed trajectories. It can be seen from Figure 12 that the actual trajectory is the same as the reconstructed trajectory. Table 10 shows that the MAE and MAPE of the number of inserted RVs are 1.58 vehicles and 15.64%, respectively. In addition, for the position of inserted RVs, the MAE, RMSE, and MAPE are 4.65 m, 4.19 m, and 2.97%, respectively. This suggests that the proposed method performs well in the empirical dataset.CONCLUSIONS AND FUTURE WORKBased on the car‐following model, this study proposes a vehicle trajectory reconstruction method in mixed traffic flow with RVs, CVs, and CAVs. Based on the simulation experiment, the following conclusions can be drawn:In a congested state, the number of inserted RVs obtained by the proposed method is within the acceptable range, for example, the MAPE of the number of inserted RVs is less than 5.94% when traffic density is less than 70 veh/km. Moreover, the trajectories can be reconstructed well, for example, the MAPE of the position of the inserted RVs is as low as 0.3686%.The reconstructed trajectories and the number of inserted RVs obtained by the proposed method are relatively accurate under the different PRs of CAVs and CVs, for example, the MAPE of the number and position of inserted RVs is less than 3.58% and 0.1022%, respectively. When the PR of CAV is low, the vehicle trajectories can also be reconstructed reasonably well.The accuracy of trajectory reconstruction gradually improves as the PR of CAVs or CVs increases. Compared with the PR of CVs, the PR of CAVs significantly impacts the results when the PRs of CAVs and CVs are high.In this study, we do not consider the lane‐changing behaviour of vehicles and the difference between the trajectory of CVs provided by their GPS and the trajectory obtained by nearby CAVs. However, we proved the effectiveness of the trajectory reconstruction method with limited data. In addition, we deeply discussed the influence factors of the proposed method, such as traffic density, the PRs of CAVs, and CVs. In the future, we will continue to verify the proposed method with other empirical datasets. And the lane‐changing model will be combined car‐following model to study vehicle trajectory reconstruction in multiple lanes freeway. Furthermore, cross‐checking between the two trajectories of CVs will be further explored to prove the effectiveness of the proposed methodology and optimization techniques.AUTHOR CONTRIBUTIONSZ.Y.: Conceptualization; Methodology; Writing – original draft; Writing – review & editing. M.L.: Validation; Visualization; Writing – original draft; Writing – review & editing. Y.J.: Conceptualization; Funding acquisition; Software; Supervision; Writing – review & editing. Y.T.: Conceptualization; Data curation; Funding acquisition; Writing – review & editing. B.R.: Conceptualization; Supervision; Writing – review & editing.ACKNOWLEDGEMENTSThe paper received research funding support from the National Natural Science Foundation of China under Grant 52002339, the Sichuan Science and Technology Program under Grant 2021YJ0535 and 2022YFG0152, the Fundamental Research Funds for the Central Universities under Grant 2682021CX058, and the Guangxi Science and Technology Program under Grant 2021AA01007AA.CONFLICT OF INTERESTThe authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.DATA AVAILABILITY STATEMENTThe data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCESLi, Z., Song, G., Yu, X., Yu, L., He, W.: Developing operating mode distributions from sparse trajectories for emission estimation. Transp. Res. Rec. J. Transp. 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IET Intelligent Transport Systems – Wiley
Published: Oct 14, 2022
Keywords: car‐following model; connected automated vehicles; freeway, mixed traffic flow; vehicle trajectory reconstruction
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