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Transportation Research Procedia 9 (2015) 149 - 163

Transportation

Procedía

www.elsevier.com/locate/procedia

21st International Symposium on Transportation and Traffic Theory

Probe vehicle-based traffic flow estimation method without

fundamental diagram

Toru Seo*, Takahiko Kusakabe

Department of Civil Engineering, Tokyo Institute of Technology, 2-12-1-M1-20, O-okayama, Meguro, Tokyo 152-8552, Japan

Abstract

This paper proposes a method of estimating the traffic state based on new probe vehicle data that contain the spacing and position of probe vehicles. In this study, the probe vehicles were assumed to observe spacing by utilizing an advanced driver assistance system, which has been implemented in practice and is expected to spread in the near future. The proposed method relies on the conservation law of traffic flow but is independent of the fundamental diagram. The conservation law is utilized for reasonable aggregation of the spacing data to acquire the traffic state, namely flow, density and speed. Its independence from the fundamental diagram means that the method does not require any predetermined assumptions with regard to the traffic flow model parameters. The estimation performance was validated through a field experiment conducted under actual traffic conditions. The results confirmed that the proposed method can estimate the traffic state precisely, even if the probe vehicle penetration rate is quite low.

©2015TheAuthors.PublishedbyElsevierB.V. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Scientific Committee of ISTTT21

Keywords: probe vehicle; traffic flow; traffic state estimation; conservation law; spacing measurement; advanced driver assistance system

1. Introduction

Probe vehicles are one of the most effective methods for collecting road traffic data because of their wide coverage area over time and space. In particular, global positioning system (GPS)-equipped probe vehicles that report their position and speed are commonly used at present (e.g., Herrera et al., 2010).

Traffic states are represented by the flow, density and speed. These variables can be estimated by using partially observed traffic data. This is known as traffic state estimation (TSE) and has been incorporated with GPS-equipped probe vehicles (e.g., Nanthawichit et al., 2003; Herrera and Bayen, 2010; Yuan et al., 2012; Mehran et al., 2012). In order to estimate the states, traffic flow models such as the LWR model (Lighthill and Whitham, 1955; Richards, 1956) and its successors have often been assumed and utilized. The LWR model is based on two assumptions: the flow-density relationship and the conservation law (CL). The flow-density relationship, which is also known as the fundamental diagram (FD), has a significant role in probe vehicle-based TSEs with regard to acquiring the flow or density from the observed speed. Most existing TSE methods assume some exogenous conditions on an FD (e.g.,

* Corresponding author. Tel.: +81-3-5734-2575 E-mail address: t.seo@plan.cv.titech.ac.jp

2352-1465 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the Scientific Committee of ISTTT21

doi:10.1016/j.trpro.2015.07.009

its functional form and parameters). However, an FD is a complicated phenomenon that involves various factors (e.g., road and road user conditions) and cannot be described or predicted completely. Therefore, those exogenous assumptions for an FD may negatively affect the estimation robustness. On the other hand, the CL should always be satisfied in the estimation model unless discontinuity points such as merging/diverging sections exist. Our interest is the traffic state estimation by using probe vehicle data without FD. This requires, however, additional observed data.

Recently, several technologies for acquiring information on the surrounding environment from a running vehicle have been developed. While their original and current purposes are for advanced driver assistance systems (ADAS), such as adaptive cruise controls and autonomous driving (National Highway Traffic Safety Administration, 2013), they are also valuable for TSE because the spacing and speed of the vehicle must be measured precisely in order to enable effective ADAS. A few researchers have proposed such TSE methods by relying on simple aggregations based on the relation between density and spacing (e.g., Seo et al., 2015). One of the deficiencies of the method proposed by Seo et al. (2015) is that its precision in high resolution may not be practically well because of its independency from traffic flow dynamics. For more sophisticated and precise estimation, traffic flow models need to be explicitly considered. The problem is that almost all existing traffic flow models utilize an exogenously given FD, even though an FD can be directly observed by ADAS-equipped probe vehicles.

The objective of this study was to develop and validate a method that estimates traffic state based on the observed spacing and position data of probe vehicles. For represent dynamics of traffic, the method utilizes a CL. On the other hand, it does not use any predetermined FD in order to exploit the advantages of spacing measurement technology. The developed method was verified with using actual data taken from a field experiment performed on an urban expressway. Section 2 describes the development of the method. Section 3 and 4 describes the validation of the proposed method.

2. Estimation method

This section describes the method of estimating the traffic state based on the observed spacing and position data from probe vehicles.

2.1. Concepts

The method estimates traffic states in a road section where some of the vehicles in the flow are probes that measure the geographic position and spacing of the vehicle ahead without any errors. The road section's schematics are assumed to be known to analysts. The probe vehicles' driving behavior is assumed to be the same as that of non-probe vehicles, i.e., the probes are randomly sampled from all of the vehicles. In addition, the traffic flow is assumed to be single-lane traffic that satisfies the first-in first-out (FIFO) condition in order to simplify the situation.

The spacing observed by a vehicle at a time point depends on microscopic vehicular phenomena that depend on macroscopic traffic flow phenomena. For example, spacing of a vehicle takes volatile value which is mainly determined by whether the vehicle was the leader of a platoon or not; meanwhile, development of the platoons is determined by the global traffic state. Therefore, aggregation is needed in order to estimate the traffic state from the observed vehicular variables. In this method, the observed vehicular variables are aggregated based on the CL. Specifically, the number of vehicles between two specific probe vehicles is a constant along a section where flow discontinuity points (e.g., a node in a road network) do not exist. This number of vehicles is to be estimated by aggregating data observed by the two boundary probe vehicles. The estimation procedure is as follows:

Step 1 The number of vehicles between two consecutive probe vehicles and two consecutive discontinuity points is estimated based on the observed data of the two probe vehicles.

Step 2 The cumulative count at the probe vehicle trajectories is calculated from the estimated number of vehicles.

Step 3 The continuous cumulative count over the entire time-space is estimated by interpolating one at the probe vehicle trajectories.

Step 4 The traffic state (i.e., flow, density and speed) is derived by partially differentiating the continuous cumulative count.

Fig. 1. Three-dimensional representation of traffic flow; the height is the continuous cumulative count N, the contour line of N is a vehicle's trajectory, the slope of N in the t direction (dN/dt) is the flow q, and the slope of NN in the -x direction (-dN/dx) is the density k.

( j + 1)-th discontinuity point, xj+i

Space x

region

j-th discontinuity point, x j

probe m - 1 probe m

'region am (Am )

'number of/Vehicles |N(Am)| - 2

Time t

Fig. 2. Concept of the method as a time-space diagram.

Figure 1 visualizes the relationships between the cumulative count, vehicle trajectory, and traffic state (for details, see Makigami et al., 1971; Daganzo, 1997). This method can be regarded as based on rectangles on the n-x coordinate plane, similar to the n-t Lagrangian coordinates of Leclercq et al. (2007); van Wageningen-Kessels et al. (2013), because the primary subject of estimation is the number of vehicles in the n-x rectangles. The step 1 and 2 is described in section 2.2; then the step 3 and 4 is described in section 2.3.

2.2. Estimation method for cumulative count at probe vehicle's trajectory

Consider the time-space region AJm (c.f., Figure 2) surrounded by

• trajectory of the m-th probe vehicle,

• trajectory of the leading vehicle of the (m - 1)-th probe vehicle,

• position of the j-th discontinuity point, and

• position of the ( j + 1)-th discontinuity point.

Because of this definition and the FIFO assumption, the number of vehicles in the region A}m is always identical to the number of vehicles between the m-th probe vehicle and the leading vehicle of the (m - 1)-th probe vehicle. The following notations are employed to represent the time and space coordinates: x'm is the position of the m-th probe vehicle at the time point t, tm is the time point of the m-th probe vehicle at the position x, and xj is the position of the j-th discontinuity point.

The cumulative count N(t, x) at a probe vehicle's trajectory—with the 0th probe vehicle1as a reference point—is described by using the number of vehicles between two probe vehicles as follows:

N(t0, x) = 0, (1a)

N(tm, x) = N(tm-1, x) + |N(Am)| - 2 (m > 1, xj < x < xj+i), (1b)

where N(A) is the set of all the vehicles in the time-space region A and |N(A)| is the number of vehicles of N(A). The equation (1) means that the number of vehicles between the m-th and (m - 1)-th probe vehicles at the position x, namely, N(tm, x) - N(t^_p x), is identical to the number of vehicles in the region N(A^) minus two2, namely, |N(Am)| - 2, if the position x is between the j-th and (j + 1)-th discontinuity points.

The only unknown variable in (1) is N(A^). This can be estimated from the probe vehicle data as follows:

NAm )| = -———, (2)

(|am-1(Am )| + |am(Am )|)/2

where am(A) is the time-space region between the m-th probe vehicle and its leading vehicle in region A, which should be observed by the probe vehicle, and |A| is the area of the region A. The equation (2) means that all the vehicle in time space region A^ are distributed with "time space interval" of (|am-1(A^)| + |am(A^)|)/2, which is an average of the two probe vehicle's ones. This is consistent with the generalized definitions of Edie (1963). It also implies that a number of vehicle between two vehicle is determined by the two vehicle's spacing experienced at a road section without discontinuity points. By using the estimated value, the cumulative count at a probe vehicle's trajectory (1) can be calculated as follows:

N(t0, x) = 0, (3a)

N(£, x) = N(t*m_1, x) + |N(Am)| - 2 (m > 1, xj < x < xj+1). (3b)

2.3. Estimation method for traffic state

The flow, which is defined as q = dNN/dt, can be estimated as

mm, x) - Ntmx) -1

The continuous cumulative count N is estimated from

q(t, x) = m':_tx- (tm-i < t < tm). (4)

f q(T, Jtx

N(t, x) = q(r, x)dr. (5)

In fact, (4) and (5) mean that N is linearly interpolated from the discrete NN of (3) over the t-axis (Figure 3 (a)) where uniform distribution of the vehicles between tm and tx . is assumed.

Once the continuous cumulative count is obtained, the density and speed can be estimated as

1 This vehicle is to be specified by an analyst

2 the (m - 1)-th probe vehicle and its leading vehicle

Toru Seo and Takahiko Kusakabe / Transportation Research Procedia 9 (2015) 149 - 163 Cumulative count N

N(tm, x) N(tm, x)

N(f_ x)

probe m

■N(t, x) x)

Cumulative count N

Time t

(a) n-t cumulative curve at a certain point x in space (xj, xj+j).

Cumulative count N

N(t, xtm) N(t, xtm)

N(t, xtm_1) N(t, xtm-1)

probe m

k(x, t

|N(Am )|-2

'-N(t; x)

N(t, xtm) N(t, xtm)

N(t, x)

xj+i Space x

probe m

■k(x, t)*f\s

■obe'm - 1

N(t, x) N(t, x)

xj+1 xtm_iSpace x

(b) n-x cumulative curve at a certain time point t where a condition (c) n-x cumulative curve at a certain time point t where a condition

xj < xtm < x'm_j < xj+i is satisfied. xj < x'm < xj+i < x'm_j is satisfied.

Fig. 3. Illustrated examples of estimation method on cumulative curves.

% dN(t, x)

k(t, x) =----, (6)

v(t, x) ^^—-, (7)

k(t, x)

based on their definitions, k = -dN/dx and v = q/k, except at the discontinuity points.

The estimation procedure can also be visualized as shown in Figure 3. First, the number of vehicles between two convective probe vehicles |N(A^)| - 2 is estimated based on the observed data from probe vehicles. Then, the cumulative count p x), N(tm, x) and continuous cumulative count N(t, x) are estimated along with its slope on the n-t plane, which means the flow q(x, t) between 1 and m, as shown in Figure 3(a). By using the continuous cumulative count N(t, x), the density k(x, t) can be estimated as the slope of N(t, x) on the n-x plane, as shown in Figure 3(b) and (c).

2.4. Discussions on the proposed method

The method can derive the flow, density, and speed without any assumptions for an FD. Therefore, the method can easily be applied to various traffic conditions, especially where an FD is not known or not predictable. The independence from an FD also means that the method can be utilized to estimate an FD itself. In fact, estimating the

cumulative count based on (2) is identical to estimating the area-flow (i.e., MFD; Geroliminis and Daganzo, 2008; Daganzo and Geroliminis, 2008) between two discontinuity points and two consecutive probe vehicles.

The method utilizes the CL in order to estimate traffic state from a vehicle's spacing. This means that the traffic state of a certain time point and position is estimated based on spacings observed by the same vehicle in wider area of time and space. Therefore, a precision of an estimated traffic state is relatively robust against fluctuations in microscopic vehicular phenomena (e.g., vehicle platoons, lane-change) by comparing the estimation method without considering the CL proposed by Seo et al. (2015). As mentioned, equation (2) is identical to estimating the area-flow from the probe vehicle data based on Edie's generalized definition. Therefore, the accuracy and precision of |N(A4)| can be analytically approximated based on the results of Seo et al. (2015), as follows:

r. ; ; A | T {kn(A]m))2

4|N(Am)|] - |N(Am)| --v '. ,, (8)

2(xj+i - Xj)ii{hn(Am))

RMSE-|N(Am )|) --J ^tj, (,)

(Xj+1 - Xj)p (hn(AJm))

where p() represents mean, t()2 represents variance, hn(A}m) means mean headway of a vehicle n in region A^, and RMSE is root mean square error, an index for precision. The equations (8) and (9) means that the accuracy and precision of the estimated cumulative count N will be improved as mean of mean headway of vehicles increases and variance of that decreases; therefore, the accuracy and precision of estimated traffic state will have similar tendencies.

The deficiencies of the method are as follows. First, single lane traffic and FIFO condition were assumed in order to estimate the number of a vehicle between two probe vehicles; they are not always satisfied in the real world. However, the method can be applied to traffic where these conditions are not satisfied if the probe vehicle's driving behavior is the same as other vehicles. This is because of the expected number of vehicles that overtake a probe vehicle and one that were overtaken by a probe vehicle is equal to each other. Note that if the road has multiple lanes, the cumulative number (3) is needed to be multiplied by the number of lanes. Second, the linear interpolation of the cumulative count over time have limitation in reproducing shockwave. Other deficiency is that real-time states cannot be estimated. Therefore, a short-term future prediction is required in order to get the real-time states3.

3. Simulation experiment-based validation of the proposed method

This section presents the validation of the proposed method based on a simulation experiment. First, section 3.1 summarizes the conditions of the field experiment. Section 3.2 describes the estimation results and the precision through a comparison with the ground truth. Section 3.3 presents the validation of the effect of incorporating the CL.

3.1. Simulation scenario

A microscopic traffic flow simulator named AIMSUN, which is developed by TSS-Transport Simulation Systems, S.L. and based on the car-following model developed by Gipps (1981), was employed for the simulation. The road section in the simulation had a single-lane, 5 km length and homogeneous geometry except a bottleneck at the end of the section. Note that this road setting always satisfy the assumptions on the proposed method4. The vehicles in the simulation had heterogeneity in the driving behavior, such as desired speed; the parameters are summarized at Table 1. The probe vehicles were randomly sampled from the entire vehicles to simulate a given penetration rate. Position and spacing of the probe vehicles with one second interval were used for estimating a traffic state.

We generated a traffic situation with a queue by setting the demand from the upstream of the section and the capacity at the bottleneck at the end of the section. A ground truth state was available since this was a simulation

3 This prediction can be based on the estimated traffic state and an FD which can be estimated by the method.

4 except the first vehicle that entered the road;because it do not have its leader. We removed it from candidates for the probe vehicles.

Table 1. Vehicle behavior parameters setting in the simulation.

Parameter name Mean Deviation

Length (m) 4 0.5

Desired speed (km/h) 80 10.0

Max acceleration (m/s2) 3 0.2

Normal deceleration (m/s2) 4 0.25

Max deceleration (m/s2) 6 0.5

Min spacing 1 0.3

Table 2. Error indices over the scenario of the proposed method.

Probe vehicle penetration rate Flow (veh/h) Density (veh/km) Speed (km/h)

P RMSPE(g) Bias(g) RMSPE® Bias® RMSPE(v) Bias(v)

0.2% 59% 151.0 107% -1.6 134% 1.1

1.0% 55% 134.8 87% 2.2 61% -2.1

3.5% 42% 83.6 52% 2.1 36% -0.7

5.0% 38% 69.6 48% 2.4 36% -0.2

10.0% 33% 63.8 45% 2.4 38% 0.7

experiment. Figure 4(a)-(d) show the ground truth state (flow q, density k, speed v) and cumulative count N as time-space diagrams. By applying the proposed method to the generated traffic situation, the characteristics of the proposed method in a various traffic situation, namely free flow, congested flow, flow with backward wave and flow with forward wave, can be investigated.

3.2. Results of traffic state estimation

The estimated variables were the traffic state at certain time-space resolutions (i.e., the intervals for the central differences) and the cumulative count in the section. The cumulative count was calculated at 1 min time interval and 100 m space intervals according to (5). The density k was calculated based on central differences at 100 m intervals according to (6).

Figure 4(e)-(h) show the estimated traffic state (q, k, V) and cumulative count N as a time-space diagram where a probe vehicle penetration is 3.5%. Through a comparison with the ground truth shown in Figure 4(a)-(d), the estimation can be used to acquire tendencies of the ground truth, such as queue propagation and dissolution. Some more detailed phenomena such as stop-and-go waves were partially captured, but with some errors. This is a deficiency of the interpolation method of interpolating cumulative count in the proposed method, namely liner interpolation over the time.

Figure 5 compares the estimated variables and ground truth as scatter diagrams. According to the figures, the estimated variables and ground truth correlated well although values were slightly overestimated, as analytically expected bias (9) suggested.

Table 2 summarizes the error indices of the traffic state estimation for different combinations of the probe vehicle penetration rate P. The RMSPE and bias were employed as error indices, which were defined as RMSPE(0) =

yE[((0 - 8)/ff)2] and Bias(0) = E[0 - 0], respectively. The error indices of each scenario was calculated from 100 iterations under different samplings of probe vehicles. According to the results, the estimation accuracy and precision clearly improved, as the penetration rate was increased. Almost all the scenarios overestimated the states similar to Figure 5(d) and for the same reasons.

3.3. Effect of incorporating the CL

In order to investigate the effect of incorporating the CL, the flow was estimated using the method proposed by Seo et al. (2015) which does not consider the CL but uses the same probe vehicle data. It is defined as qwithoutCL in this paper; for details, see Appendix A or Seo et al. (2015). The time-space resolution for calculating qwithoutCL was identical to the intervals of the central difference in the proposed method, namely 1 min and 100 m. Table 3 shows results of the comparison. The Pol, which means percentages of improvement, is defined as (RMSPE(qwithoutCL) -

20 38 48 50 Time [min]

2500 2250 2000 1750 E 1500 S 1250 £ 1000 g 750 £ 500 250

20 30 40 Time [min]

(a) Ground truth flow q.

(b) Ground truth density k.

i i —

---- ---- *i 1 "i^i gd ----

20 30 40 Time [min]

(c) Ground truth speed v.

(d) Ground truth cumulative count N, the contour lines were drawn with interval of 100 veh.

10 20 30 40 Time [min]

(e) Flow q estimated by the proposed method.

20 30 40 Time [min]

(f) Density k estimated by the proposed method.

ll --- •• I ---- ---- ---- v

20 30 40 Time [min]

(g) Speed v estimated by the proposed method.

1600 1400 " 1200

20 30 40 Time [min]

(h) Cumulative count N estimated by the proposed method, the contour lines were drawn with interval of 100 veh.

Fig. 4. Time-space diagrams of the traffic state and the cumulative count.

2500 : 2000 ' 1500 1000 500

1 1 ' 1 / / f-

--- / / / r ✓ / -y--- ' 1 1 1

/ / / I^M I 1 1

/ / 1 1

350 i:

300 s^

250 S £ 200 ° | 150

---y - -

500 1000 1500 2000 2500 Ground truth flow [veh/h]

(a) Flow q.

20 40 60 80 Ground truth density [veh/km]

(b) Density k.

1 540 ■

360 s 300 o 240 Jj '

180 J . 120 '

✓ -----'/----- /\

/ / r^ 1 1 1 1

■ 640 560

320 °

240 160 " 80

I 36 32 28

16 fc XI

12 J 8 4

20 40 60 80 100 Ground truth speed [km/h]

0 500 1000 1500

Ground truth cumulative count [veh]

(c) Speed v. (d) Cumulative count NN.

Fig. 5. Scatter diagrams of estimated variables vs. ground truth variables.

Table 3. Comparison between the proposed methods and the method in Seo et al. (2015). P RMSPE(q) RMSPE(qwithoutCL)

0.2% 59% 66% 12%

1.0% 55% 63% 7%

3.5% 42% 59% 27%

5.0% 38% 53% 38%

10.0% 33% 52% 43%

RMSPE(q))/RMSPE(q) and is an index representing the positive effect of incorporating the CL. According to the results, incorporating the CL clearly improved the estimation precision.

4. Field experiment-based validation of the proposed method

In order to investigate the applicability of the proposed method to the real world, this section presents the validation of the proposed method based on data collected from a field experiment at an actual road.

4.1. Field experiment

A field experiment was conducted on September 24 (Tuesday), 2013, from 15:00 to 16:00 at an urban expressway in Tokyo. Only the minimum necessary information of the experiment is summarized here; a detailed description and evaluation of this experiment are provided in Seo et al. (2015).

The location was the Inner Circular Route (i.e., C1) of the Metropolitan Expressway, which is an urban expressway in Tokyo, Japan. Figure 6 shows the schematics of C1. In Figure 6, junction is abbreviated as JCT, the arrows on the upper side represent the merging/diverging section with other highway routes, and the arrows on the lower side

Tanimachi JCT Hamazakibashi JCT Ichinohashi JCT

Takebashi JCT Edobashi JCT Miyakezaka JCT

0 km 2km 4km 6km 8km 10km 11km 12km 14km

Fig. 6. Schematic of the experiment site: Inner Circular Route (counterclockwise direction).

represent on/off ramps. Most of C1 has two lanes, namely, cruising and passing lanes, with a speed limit of 50 km/h. Because of the presence of tunnels and law restrictions, the survey section was limited to an 11 km long section of the cruising lane, namely sections from 0 km point to 11 km point shown in Figure 6. C1 was equipped with many detectors at 250 m intervals for each lane with time aggregation at 1 min intervals. The traffic state observed by these detectors was compared with that estimated by the proposed method. Figure 7(a)-(d) show the state (flow q, density k, speed v) observed by detectors and cumulative count NN as time-space diagrams. The cumulative count was calculated from the flow observed by detectors with a probe vehicle's trajectory as a reference point. Discontinuities in the contour lines mean junctions (merging/diverging sections from/to other highway routes) in C1; these are represented as the arrows on the upper side in Figure 6. Note that the cumulative count NN did not always increase as t increased and x decreased where the discontinuities did not exist. This is because of on/off-ramps, lane-change by vehicles, and systematic errors in the measurements of the detectors.

Twenty standard-size passenger vehicles equipped with GPS loggers and video cameras and driven by non-professional drivers were employed as probe vehicles. The positions of the probe vehicles were identified from the GPS log. The spacings from each probe vehicle to its leading vehicle were estimated by analyzing the size of the leading vehicle in images captured by the camera. Most of the probe vehicles drove three laps on C1 during the experiment period, and a total of 59 laps were performed in the survey section. This corresponded to 42.1 veh/h/lane. Therefore, the probe vehicle's penetration rate roughly corresponded to 3.5% according to the average flow of the whole vehicles (1255.2 veh/h/lane) observed by the detectors.

4.2. Validation results

The estimation targets were the traffic state at certain time and space resolutions and the cumulative count in the survey section. The cumulative count was calculated at 1 min time intervals and 100 m space intervals according to (5). The density k was calculated based on central differences at 100 m intervals according to (6). In order to compare the estimated states with the detector data, the estimated states were averaged for time resolution of 5 min and space resolution of 500 m. The method requires the positions of the discontinuity points to be specified. In this estimation, only junctions were considered to be discontinuity points. Thus, on/off-ramps and lane-change were ignored.

Figure 7(e)-(h) show the estimated traffic state (q, k, v) with a 5 min x 500 m resolution and cumulative count NN as a time-space diagram. Through a comparison with Figure 7(a)-(d), the estimation can be used to acquire tendencies of the detectors' observed variables, for example, a queue propagation from the 8 km point. Figure 8 compares the estimated variables and detectors' observed variables as scatter diagrams. According to the figures, the estimated variables and detectors' observed variables correlated well. However, the values tended to be underestimated. One of the reasons was a bias in the probe vehicles' driving behavior (i.e., their driving behavior tended to be safer, which meant a slower speed and larger spacing than for other vehicles).

Table 4 summarizes the error indices of the traffic state estimation for different probe vehicle penetration rate P. The penetration rate of 0.2% was reproduced by randomly sampling all of the probe vehicles. This corresponded to roughly 2 probe veh/h. The error indices of scenarios with a 0.2% penetration rate were calculated from 25 iterations under different samplings of probe vehicles. According to the results, the estimation precision clearly increased as the penetration rate increased. Note that most of the scenarios underestimated the states, similar to Figure 8 and for the same reasons. Table 5 shows results of the comparison between the proposed method and a method without considering the CL; the precision was improved in the proposed method.

to „

1000 >,

3 _o u_

15:30 Time of Day [hh:mm]

----J-.i

\ _ L --

- — irtB

m 20«

15:30 Time of Day [hh:mm]

(a) Flow q observed by the detectors.

(b) Density k observed by the detectors.

u 10 -

a. 4 l/l

15:30 Time of Day [hh:mm]

60 „ .c

j._I_i_

15:30 Time of Day [hh:mm]

1200 „

1000 >,

600 (Ü >

400 J2

(c) Speed v observed by the detectors.

(d) Cumulative count NN observed by the detectors, the contour lines were drawn with interval of 200 veh.

1000 >.

3 o u-

15:30 Time of Day [hh:mm]

15:30 Time of Day [hh:mm]

(e) Flow q estimated by the proposed method.

(f) Density k estimated by the proposed method.

60 „ c

E u. 40"

Q. 20"

15:30 Time of Day [hh:mm]

9- 4 in

/ /• y / /

//////

11200 „ .c

1000 >,

800 §

400 200

15:30 Time of Day [hh:mm]

(g) Speed v estimated by the proposed method.

(h) Cumulative count NN estimated by the proposed method, the contour lines were drawn with interval of 200 veh.

Fig. 7. Time-space diagrams of the traffic state and the cumulative count in the field experiment.

2. 1500

12.0 10.5 , 9.0 ! 7.5 \ 6.0 I 4.5 i 3.0 : 1.5 0.0

3 500 1000 1500 2000 Flow from the detectors [veh/h]

■p 100

Density from the detectors [veh/km]

(a) Flow q.

(b) Density k.

Speed from the detectors [km/h]

> 1600

E 1200

0 400 800 1200 1600 Cumulative count from the detectors [veh]

(c) Speed v. (d) Cumulative count NN.

Fig. 8. Scatter diagrams of estimated variables vs. detectors' observed variables variables in the field experiment.

Table 4. Error indices over the scenario of the proposed method in the field experiment.

Probe vehicle penetration rate Flow (veh/h) Density (veh/km) Speed (km/h)

P RMSPE(q) Bias(q) RMSPE(k) Bias(k) RMSPE(v) Bias(v)

0.2% 30% -158.9 39% -4.9 60% 0.9

3.5% 23% -133.9 30% -2.3 24% -2.1

Table 5. Comparison between the proposed methods and the method in Seo et al. (2015) in the field experiment. P RMSPE(q) RMSPE(qwjthoutCL) P5T~ 0.2% 30% 43% 30%

3.5% 23% 26% 13%

4.3. Comparison between the simulation experiment and the field experiment based validations

Difference between the simulation experiment (Section 3) and the field experiment (Section 4) based validations is summarized in Table 6. The simulation experiment conducted under traffic conditions where the assumptions of the proposed method were always satisfied. On the other hand, the field experiment-based validation have some limitations such as slight violations of the assumption. However, it can validate the proposed method's characteristics under actual traffic conditions. This advantage would be important for validating the proposed method's practical

Table 6. Difference between the simulation experiment and the field experiment based validations.

The simulation experiment

The field experiment

FIFO condition The CL

Traffic flow model Traffic condition

Driving behavior of probe vehicles Ground truth

synthetic scenario not biased available

59% 42%

151.0 veh/h 83.6 veh/h

12% 27%

car-following model of Gipps (1981) actual traffic

always satisfied always satisfied

not always satisfied not always satisfied

actual traffic slightly biased detectors

30% 23%

-158.9 veh/h -133.9 veh/h

30% 13%

RMSPE(q) at P=0.2% RMSPE(q) at P=3.5% Bias(q) at P=0.2% Bias(q) at P=3.5%

Pol due to incorporating the CL at P=0.2% Pol due to incorporating the CL at P=3.5%

characteristics because of strong dependency of the method on the spacing, which was determined by a synthetic model (i.e., a car-following model) in the simulation experiment.

According to the validation results, it was suggested that the proposed method works well under an actual traffic environment, although the assumptions of the proposed method, namely the CL, were not exactly satisfied because of the on/off-ramps and lane-change. For example, the precision increases as the penetration rate increases in the both of the simulation (Table 2) and the field experiment (Table 4). Especially, in the field experiment, the precision was better than that in the simulation experiment. Ironically, one of the reasons is existence of lane-change. The lane-change would decrease the variance in mean headway; then the estimation precision of the cumulative count improved (c.f.,

The notable difference between the two validation was the sign of the biases. The simulation-based validation (Table 2) and analytical approximation (8) showed positive biases, while the field experiment-based validation showed negative biases. This may be due to biases in the probe vehicle's driving behavior. By comparing to the probe vehicle's speed and the traffic speed observed by detectors, the probe vehicles tended to be slower than other vehicles.

The incorporation of the CL improved the precision in the both of the validations (Tables 3 and 5). However, in the field experiment scenario with higher penetration rate, the PoI is less than that in lower penetration rate; this tendency is not the same as that in the simulation-based validation. This may be due to the violation of the assumptions, too. If a large amount of probe vehicle data is available, i.e., the traffic state can be estimated precisely even without incorporating the CL, the non-exact assumptions of the CL will negatively affect the precision.

5. Conclusion

This paper proposes a traffic state estimation method that depends on probe vehicle data only when probe vehicles can observe their spacing and position. In order to estimate the flow and density from the probe vehicle's spacing, an aggregation method based on the CL is employed. The method is based on rectangles on the n-x coordinate plane instead of the traditional x-t Eulerian coordinates or n-t Lagrangian coordinates. The proposed method does not consider or assume an FD, which is the other fundamental factor of a traffic flow model. Therefore, the proposed method does not require any predetermined assumptions for the traffic flow model parameters. This makes the method widely applicable to any traffic situation and robust against sudden change in a traffic situation such as an incident.

The proposed method was validated through probe vehicle data that were acquired from a simulation experiment and a field experiment. The simulation experiment was performed under the assumptions of the proposed method were always satisfied, while the field experiment was not. The results suggested that the method is capable of estimating the traffic state precisely under the both of synthetic and actual traffic situations. In particular, incorporating the CL clearly improved the estimation precision.

Several improvements can be suggested for future studies. First, advanced estimation method for the continuous cumulative count from the discrete cumulative count at probe vehicles trajectories is required. The estimation was treated by a linear interpolation over time ((4) and (5)) in this study. This caused some deficiencies in reproducing detailed traffic phenomena, such as shockwaves. We are now developing a method of detecting shockwaves from the probe vehicle data and a method for linear interpolation of the cumulative count over shockwaves. Second,

incorporating an endogenous estimation of an FD is another option to estimate the continuous cumulative count. Since the probe vehicle data contain spacing and headway, an FD can be estimated. Then, vehicle trajectory estimation methods based on traffic flow model with an FD and initial/boundary conditions (e.g., Laval and Leclercq, 2013) can be utilized for estimating. If it is combined with the second point, namely endogenous shockwave detection, this methodology will be also valuable for empirical investigation on a traffic theory; because dynamics of traffic change significantly when shockwaves go through (Laval, 2011). Third, the CL at nodes should be considered. If ADAS-equipped probe vehicles are spread over a road network, the CL at nodes can be explicitly described because of the presence of probe vehicles at all links. This will solve the problem of neglecting the flow of the on/off-ramps.

Acknowledgements

The traffic detector data were kindly provided by Metropolitan Expressway Co., Ltd. Part of this work was financially supported by the Research Fellow DC2 program of the Japan Society for the Promotion of Science (KAKENHI Grant-in-Aid for JSPS Fellows #26010218). We would like to thank an anonymous reviewer for his/her insightful comments on our work.

Appendix A: An estimation method without considering the CL (Seo et al., 2015)

This appendix briefly describes the method proposed by Seo et al. (2015), which estimates traffic state with probe vehicles' spacing and position data without considering the CL. The ignoring the CL implies both advantage and disadvantage comparing to the proposed method in this paper; the method can be applied for any traffic situation regardless the road geometry and the FIFO condition, though the method's estimation precision for high resolution traffic state is relatively low.

The method estimate traffic state in a time-space region, E, based on the Edie's generalized definition. For simplicity purposes, the region E was defined as an Eulerian rectangle as follows:

Ej = {(x, t) | xi < x < xi+1, tj < t < tj+i} i > 0, j > 0, (A.1)

xi+1 = x; + Ax, (A.2)

tj+1 = tj + At, (A.3)

where i and j are non-negative indices for space and time, respectively, (x0, t0) is the coordinates of a predetermined origin, (x;, tj) is the coordinates of the lower-left corner of the region Ej, Ax is a predetermined space resolution, and At is a predetermined time resolution. The estimators for traffic state were defined as follows:

: ^meP(E'j) dm(Ej

<7withoutCL(Ej) = -;-—, (A.4)

£m€P(Ej) |am(Ei )|

f ^meP(Ej) tm(Ej

kwithoutCL(Ej) = -;-—, (A.5)

£m€P(Ej) |am(E; )|

j ^m€P(Ei) dm(Ej ) VwithoutCL(Ei ) = -r, (A.6)

; " , tm(Ej)

■JmeP(Ej) tm(E

where dm(Ej) is the total distance traveled by the m-th probe vehicle in the region Ej and tm(Ej) is the total time spent by the m-th probe vehicle in the region Ej.

Figure A.1 shows estimated flow based on this method of (A.4) where P is 3.5% and (At, Ax) are the same to the intervals of central difference of the proposed method. By comparing to the proposed method's results shown in Figs 4(e) and 7(e)), the results of (A.4) were highly fluctuated in the detailed traffic state. This is due to the method of (A.4) estimate traffic state in a region using the observed data at the region only; the data strongly depends on

20 30 40 Time [min]

1750 -

1500 i a>

1250 >, 1000 § 750 E 500 250 0

_ 8 £

£ 6 <u u ro . a- 4 i/i

15:30 Time of Day [hh:mm]

(a) Result from the simulation experiment data

(b) Result from the field experiment data

Fig. A.1. Estimated flow based on the method without considering the CL.

microscopic vehicle behavior if the size of region is too small. On the other hand, the proposed method estimate traffic state in a point using the observed data from wider area, namely the area between two discontinuity points and two probe vehicle, by incorporating the CL; therefore, the results of the proposed method had less fluctuation and higher precision.

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