Dynamic Traffic Shockwave Speed Estimation in ...

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Dynamic Traffic Shockwave Speed Estimation in Connected Vehicle Environment Jiangchen Li1, Chenhao Wang1 1

Shuxian He2, Tony Z. Qiu1,2 2

Department of Civil and Environmental Engineering University of Alberta Edmonton, Canada jiangche, [email protected]

length), density and speed can be estimated based on the observations. Indeed, spatial observations are only available for the spots where detectors or cameras are installed which cause the generated shockwave estimations with low spatial resolutions. As a result, it is difficult to capture shockwave observations in a wide range using the fixed-point based method [1]. Another option for shockwave estimation is to use probe vehicle-based observations to get target information. As probe vehicles drive in the traffic flows, they can observe traffic status with data collection devices onboard such as devices with global positioning system (GPS) [8]–[10]. Using this method, probe vehicles can report their trajectories for estimating traffic status. However, probe vehicle is limited in terms of acquiring volume-related information unless strong exogenous assumptions (e.g., fundamental diagram (FD) applies or probe vehicle penetration rate is known) are used [1]. In addition to the above two methods, there are also other traffic status estimation-based methods such as using road microphones measuring cumulative road acoustics [11], compressed sensing framework [12]–[14] and advanced driver assistance systems measuring vehicle spacing [1], [2]. Such methods can be limited in terms of the requirement of special equipment or the cost of maintaining road pavements.

Abstract—Traffic state estimation is important for active traffic planning, management, and control. By utilizing traffic status over time and space, a key theoretical analysis tool, i.e., shockwave theory, can be adopted as efficient tools to solve bottleneck or congestion problems. Two methods, who are fixed point observations and mobile probe vehicle observations, are typical methods and popular in traffic analytical agencies and projects. But both of them has their problems and new insights are needed for current proactive and online traffic implementations. In this paper, a dynamic traffic shockwave estimation method with both high temporal resolution and high spatial resolution is proposed based on integrating vehicle detecting system data from fixed loop detectors and trajectory data from probe vehicles in a connected vehicle environment. The proposed method was validated in a prevailing dataset. Results showed that the proposed method combined the advantages of mobile and fixed data. As a result, the proposed method is effective in improving the spatial and temporal resolution of the shockwave estimation in the considered general scenario. Keywords — Dynamic shockwave estimation, Connected vehicle, Compressed sensing

I. INTRODUCTION Traffic status is critically important various transportation science and engineering, i.e., traffic operations, planning, and management. It is known that traffic status over time and at different locations can be monitored and then processed for inputs of transportation problems [1]–[3]. By processing the traffic status, a key theoretical analysis tool, i.e., shockwave theory, can be adopted as efficient tools to solve bottleneck or congestion problems[4], [5]. Further applications can traffic scheduling and routing [6], and ramp metering strategies developing [7]. Moreover, recent emerging reactive traffic control such as signal optimization and ramp metering control requires higher resolution of either spatial or temporal information of the road network. Therefore, dynamic shockwave estimation with high temporal and spatial resolutions is of significant interest in recent transportation practices.

Since the advantages of fixed-point and vehicle-based estimation methods compensate the disadvantages of each other, it is of interest to combine the two methods to acquire more observations both temporally and spatially for higherresolution shockwave estimation. Recently, connected vehicle (CV) has been developing rapidly and receiving tremendous attention due to its potential for facilitating intelligent transportation systems (ITS). With its low latency [15] and built-in GPS sub-system, the trajectory data collected from the vehicle can be forwarded to transportation management center frequently. Therefore, it is reasonable and practical to take advantage of CV for collecting data from probe vehicles. A hybrid approach of collecting observations based on the fixedpoint method, specifically loop detectors, combined with the probe vehicle method, specifically CV enabled probe vehicles, is further exploited in this paper to improve the spatialtemporal estimation of density in this paper. The main contributions of this work are listed as follows:

For shockwave estimation in a target road segment, some methods have been proposed. A typical method is to use the fixed-point observations to get the traffic status estimation over that segment, which acquires the inputting traffic state variables at fixed spots, like using loop detectors or video cameras. With certain assumptions (e.g., on average vehicle

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Intelligent Transportation Systems Research Center Wuhan University of Technology Hubei, China [email protected], zhijunqiu @ualbert.ca

First, the proposed method uses trajectories of CV, which is commonly accessible in CV with its built-in GPS devices. The proposed method features require no additional devices other



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loop detectors are used for generating the cumulative counts and corresponding vehicle index data (fused with trajectory data [18],.). The loop detectors are connected to RSE (examples of RSE connected to traffic infrastructure can be found in [19]), and as a result, the loop detector data can be forwarded to the RSE timely and then immediately broadcast to the OBE in probe vehicles with time synchronization approach [20]. Due to the connections between RSE and the detectors and the connections between RSE and OBE, the above two types of data can be integrated into the probe vehicles to improve accuracy in density estimation with higher spatial and temporal resolution.

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sl7

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... sln-1

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ρn-1,qn-1 ρn,qn

u1,2= (ρ2-ρ1)/(q2-q1)

un-1,n= u6,7= (ρ7-ρ6)/(q7-q6) (ρn-ρn-1)/(qn-qn-1) (a) A general scenario

Loop detector data

Connected Vehicle Data

Without loss of generality, one direction of a common twoway segment with one lane in the mainline is considered and shown in Fig. 1(a). It worthy to mention that this method is not limited to be only suitable for one lane and can be extended to the multi-lane case by choosing further complex random sensing matrix ( which will be introduced in following subchapter III.B2). Considering that there are n vehicles including m CV probe vehicles in the target segment. In Fig. 1(a), there are two CVs used as examples. Solid blue vehicles are CV probe vehicles and dashed gray vehicles are non-CV vehicles. The target segment is divided into n+1 sub-links from sub-link sl0 to sub-link sln. Sub-link sln-1 is denoted as a segment between two adjacent vehicles, i.e., (n-1)th vehicle and its preceding nth vehicle. Two special sub-links are sl0 and sln, which are the segment between VDS1 and the first vehicle, and the segment nth vehicle and VDS2, respectively. The corresponding properties of sub-link sln-1 are the point density (i.e., ρn-1) and the point flow (i.e., qn-1). Then, the point density and point flow of all vehicles can be represented as two vectors, which are point density vector ρ(ρ0,ρ1, ρ2,…, ρn) and point flow vector q(q0,q1,q2,…,qn). Thus, the shockwave speed at different points can be obtained from the point density and point flow vector. One OBE is deployed in each connected vehicle. VDSs are deployed in the upstream (VDS1) and the downstream (VDS2) of the mainline as shown in Fig. 1. The corresponding flow (qu,qd) can be obtained by these VDSs. And the trajectories of m CV probe vehicles can be obtained from connected vehicle technology.

Vehicle index/Density/Flow

Measured sparse shockwave speed of CV

Compressed Sensing Framework

Estimated shockwave speeds of flow

(b) The proposed idea of estimations for high spatial shockwave speed estimations Fig. 1. Problem definition: (a) general scenario (b) proposed idea of estimations for high spatial shockwave speed estimations

than a basic CV environment, and is ready to implement with the development of CV [16], [17] . Second, a shockwave estimation using compressed framework is proposed to achieve instantaneous shockwave estimations with high temporal and spatial resolutions. To our Compared to previous works using fixed and mobile observations, the proposed method provides both more temporal and more spatial observations from the probe vehicle trajectory data collected. The result shows that the position data collected from CV probe vehicles can effectively improve both spatial and temporal resolution in the shockwave estimation. II. PROBLEM DEFINITION In this paper, we want to solve a problem that how to improve spatial and temporal resolutions of the shockwave estimation for a road segment based on the integration of Lagrangian data ( i.e., the position samples from the CV probe vehicle trajectories ) and Euler data ( i.e., fixed location based flows from loop detectors ).

As shown in Fig. 1(b), the proposed idea fully utilizes the raw data created by loop detectors and connected vehicles and to generate secondary data, like vehicle index, density, and flow. After that, the measured sparse shockwave speeds will be generated and these data together with generated secondary data are used as inputs for the compressed sensing framework for recovering the high spatial shockwave speeds. Finally, the subsequent part describes the methodology to obtain an accurate estimation of shockwave vectors at different points using the density and flow vectors (i.e., ρ and q) under compressed sensing framework.

The problem considered in this work is to estimate the shockwave of the road segment in which probe vehicles are driving among hybrid traffic flows (q). The architecture consists of both CV technology and existing vehicle detecting system (VDS, or called loop detectors). The real-time trajectory data of CV-enabled probe vehicles gathered from the GPS in the OBE are used for spot speed observations, space observations, and travel time observations. The flow data from

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III. DYNAMIC TRAFFIC SHOCKWAVE ESTIMATION THROUGH COMPRESSED SENDING FRAMEWORK



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Sensing matrix based on vehicle index (III.B2)

Triangular FD assumption (III.A)

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Sparsity assumption of original shockwaves (III.B1)

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In this section, a traffic shockwave estimation through compressed sensing framework is proposed to recover the complete intermediate traffic shockwave speed over one segment. The proposed approach for traffic shockwave estimation is introduced by following several steps: A) triangular fundamental diagram assumption, B) shockwave recovery via compressed sensing framework. Without loss of generality, the on-ramp flow can be integrated with the upstream flow as an additional lane flow of mainline. Similarly, the off-ramp flow emerges with downstream flow the as an additional lane flow of mainline.

Compressed sensing framework

umCVu1

Reconstructions of original shockwaves (III.B4)

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all )CV mun * unu1

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l1

Fig. 3. Flow chart of compressed sensing framework for shockwave estimations

observations from a few limited measurements within acceptable error levels. The measurements for recovering original observations in CS theory can be far fewer than what Nyquist-Shannon sampling theory requires. Because of the CS theory’s breakthrough for sampling, it is possible to use it for shockwave reconstructions. However, there were only a few attempts [12]–[14] in the previous literature applying CS theory in the transportation area, and the low-rank or sparsity assumptions in these CS frameworks were based on empirical analysis of historical data which is not so solid. Instead, the sparsity assumption in this methodology is based on fundamental traffic theory, i.e. shockwave theory.

A. Triangular Fundamental Diagram Assumption The traffic flow is assumed to follow a widely used triangular fundamental diagram [12] as shown in Fig. 2. Here qc denotes the capacity of this segment; ρc and ρj are critical density and jam density, respectively; vf and wb denote the freeflow speed and backward shockwave speed, respectively. From the triangular fundamental diagram, we can see that if two adjacent sub-links are free-flow or congestions conditions, they share the same shockwave speed which is either free-flow speed vf or backward wave speed wb.

1) Derived sparsity assumption of shockwaves based on fundamental diagram assumption We define the number of generated shockwaves as sparse coefficient k, which demonstrates the sparsity level of shockwave; the lower value of k means higher sparsity level. As shown in Fig. 2 (previous section III.A), if the traffic status is in free-flow condition or congestion condition, the shockwave in the whole segment should be the same. If few bottlenecks appear (like an accident happens) among the segment, or upstream demand exceeds segment capacity, or flow exceeds downstream capacity, shockwaves will be generated and be propagated to the upstream or downstream. Also for usual equilibrium traffic flows, the sparse coefficient k is still not so much high when compared to the number of n vehicles. When k is much less than n, we call this condition as a high sparse condition. This means the original shockwave speed vector u can be sparse for the usual equilibrium traffic flows. It is straight forward to find the sparsity level of the shockwave vector u for different conditions. This makes the low-rank assumption (i.e., high sparse condition) for CS theory solid and practical.

In addition, we briefly introduce how to calculate the shockwave speed based on the traditional shockwave theory. Recalling if all status of the n vehicles is known in target segment, we can calculate the shockwave speed between any two adjacent sub-links as shown in Fig. 1. Because there are n+1 sub-links, there are n shockwave speeds exist. For the shock wave speed un-1,n between sub-link sln-1 and sub-link sln, the following equation can be used, (1)

Through Equation (1), n shockwave speeds can be denoted as a n*1 vector u (u0,1, u1,2, u2,3,…, un-1,n ). As shown in Fig. 1(a), two shockwave speeds u1,2 and u6,7 are calculated as examples, respectively. B. Shockwave Recovery via Compressed Sensing Framework In this section, a compressed sensing (CS) framework is developed to recover shockwave speed vector and the flow chart of this framework is shown in Fig. 3. To achieve zero information loss, the sampling rate should be at least twice of the signal bandwidth which is stated by Nyquist-Shannon sampling theory [12]. Recently, a new sampling method, called compressed sensing [21], was proposed to simultaneously capture and compress observations. The CS theory proves that if original observations are sparse themselves or sparse under certain basis, it is highly possible to reconstruct original

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Measured sparse shockwave CV speeds of CVs (III.B3) umu1

Randomness of sensing matrix based on CVs distribution (III.B2)

Compressed sensing equation (III.B4)

Fig. 2. A triangular fundamental diagram [12]

un-1,n = (qn-qn-1) / (ρn-ρn-1)

unallu1

) CV mun

2) Sensing Matrix Generation using Vehicle Index In CS theory, the sensing matrix should be carefully selected for reconstructions when applying sensing process to sparse observations u. Only if the restricted isometry property [21] condition is satisfied can the original observations be stably and precisely reconstructed. Different from previous attempts [12]–[14], a random binary matrix [21] generated from vehicle index was applied for practical implementations here. Because the CVs are strictly randomly distributed (randomness feature) among the whole traffic flow, which is satisfying restricted isometry property (RIP) condition.



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Then we employed our previous work [18] to generate the vehicle index based on the integration of the cumulative counts from loop detectors and the positions from connected vehicles. In previous work [18], a practical problem was solved when we considered the consistency of sampling periods of the VDSs and the connected vehicle, especially consistency in a discrete time domain. If considered in a continuous time domain, the cumulative vehicle counts n(i,t) in a widely used macroscopic traffic flow model (i.e., Newell’s three-detector model, TDM [22] ), at any location i within a homogenous freeway segment can be derived from the boundary inputs nu(t) and nd(t), which is shown as following equations,

li l l (2) ), nd (t  d i )  U jam (ld  li )} vf vb where ld denotes the downstream boundary of a freeway segment. li is the length between location i and upstream boundary. n(i, t )

min{nu (t 

The vehicle index for each connected vehicles and the total number of vehicles on the freeway segment can be calculated through the equations in work [18] or equation (2). We denote the total number of both non-CVs and CVs as n (be consistent to the description in section 2, please note for different time instant the actual number n is different. Then, a random binary CV matrix ) mun is generated as a sensing matrix for reconstructions. This random binary measurement matrix has only one ‘1’ per row and at most one ‘1’ per column, while other places are all zeros (‘0’). For each row, the index of the element ‘1’ means the index of one connected vehicle in the traffic flow. Because there is m connected vehicle, there are m rows in the sensing matrix. The matrix below is a general CV description of this binary matrix ) mun based on the vehicle index,

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shockwave speed vector umu1 and this vector can be mathematically shown in the below,

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there are m CVs in the traffic low. 4) Reconstructions of Original Shockwave Vector all Based on CS theory, the shockwave vector unu1 , the sensing matrix ) mun and the measured shockwave speed CV

CV

vector umu1 can be described by the following equation, (3)

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all ) CV mu n * unu1

(5)

Because unu1 is sparse and ) mun is satisfied the restricted isometry property condition, the original shockwave vector all

CV

unallu1 can be recovered with high probability by using the orthogonal matching pursuit (OMP) algorithm to solve the following underdetermined linear systems:

unallu1

argmin umCVu1

l1

s.t. umCVu1

all ) CV mun * u nu1

(6)

Though the OMP algorithm is a widely used method, some other similar algorithms can also be used.

and flow q i . Then using the

IV. RESULTS AND DISCUSSIONS In this section, the simulation setup is introduced and then analysis of corresponding results are shown as follows.

average density U i and flow q i replaces real status in sublink sli-1 and sub-link sli to calculate the shockwave speed uiCV 1,i for connected vehicle i. By this way, we get the measured

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(4)

where ui 1,i is shockwave speed for connected vehicle i. And

3) Measured Sparse Shockwave Speed Vector This section gives a description how to get the measured shock wave speed for each CV. From the literature, several methods are introduced here. The first one is using advanced driver assistance systems as spacing equipment to measure the spacing information between subsequent and preceding vehicles, which is reasonable in the coming autonomous vehicle technology[1][2][23]. The second one [18][22], applied here, is using the cumulative counts and the positions to calculate the average density

CV CV CV CV (u0,1 , u1,2 , u2,3 ,.., uiCV 1,i ..., u m 1, m ),



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(a) Performance of one rum time for intuitive comparisons Fig. 6. Recovery accuracies under different penetrations

speed curve and the recovered shockwave curve cannot be ignored from one running time. The straightforward way here is to do the reconstruction for many times to get better performances. Fig. 5(b) shows different estimation results of the shockwave speed via running the reconstruction algorithm for different times. From Fig. 5(b), the errors are almost the same when the run time is over one hundred times, which means the errors will converge to a good level after several hundred run times. The convergence speed is important when used for online estimation to do some short-time predictions, like shortterm demand prediction.

(b) Convergence performance of difference run times Fig. 5. Convergence performances

C. Recovery accuracies under different penetrations After previous intuitive comparisons between the original curves and recovered curves, it is significant to investigate the quantitative performance of the proposed method when trying to recover the original shockwave speed. Fig. 6 give the recovery accuracies over different penetrations.

A. Experiments Setup Simulations for considered scenarios are conducted in MATLAB to evaluate the performance of the proposed methods. A prevailing dataset, i.e. the Next Generation Simulation ( NGSIM [24] ), is used here for data inputs and further results comparisons. The NGSIM data could provide ground truth trajectories for vehicles passing through study areas. These trajectories were extracted from video data mounted on the top of high buildings. Two sections located in California test sections, i.e., US-101 and I-80 sections, are involved in the data collections. In this paper, the US-101 segment is chosen for analysis which is shown in the Fig. 4. The virtual VDS1 and VDS2 are assumed to be located at the beginning and the ending of the target segment and used for flow estimations. Further, these flow estimations are processed to reveal the cumulative counts to estimate the vehicle index for each vehicle.

In Fig. 6, the blue solid curve is the absolute error between the original and the recovered data, where the root-meansquare deviation (RMSD) is used for quantifications. And the read dashed curve in Fig. 6 gives the relative errors of the differences between the original and the recovered data, where the percent root mean square difference (PRD) is calculated for comparisons. From the Fig. 6, we can see that the errors are decreased slowly when penetration rate is larger than 70%, which shows the efficiency of the proposed method. The errors of reconstructed shockwaves can be accepted when used for congestions or bottleneck identifications in practices. V. CONCLUSIONS In this paper, a dynamic traffic shockwave estimation method with both high temporal resolution and high spatial resolution is proposed based on integrating vehicle detecting system data from fixed loop detectors and trajectory data from probe vehicles in a connected vehicle environment. The proposed method was validated in a prevailing traffic dataset. Results showed that the proposed method combined the advantages of mobile and fixed data. As a result, the proposed

B. Convergence of the efficiency of the reconstruction The first thing needs to be investigated is the convergence performance of the reconstruction process. These results are shown in the following Fig. 5. In Fig. 5(a), the red dashed star line is the estimation curve of the shockwave speed via one time of reconstruction and the blue solid line is the original shockwave curve. From the Fig. 5(a), we can see that the errors between the original shockwave

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method is effective in improving the spatial and temporal resolution of the shockwave estimation in the considered general scenario.

[11]

In the future, more works can be further investigated by utilizing this proposed high temporal and spatial shockwave speed estimation method. One typical application is that the traffic status estimation for all sub-links can be calculated based on estimations of the original shockwave speed over that segment. It is worthy to mention that this method is not limited to be only suitable for one lane and can be extended to the multi-lane case by choosing further complex random sensing matrix.

[12] [13]

[14]

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