A Study of Freeway Volume-to-Capacity Ratio Based Travel ...web.cecs.pdx.edu/~maf/Under Review - Working Papers... · 26 Submitted for presentation and publication to the 27 90th

A Study of Freeway Volume-to-Capacity Ratio Based Travel Time Approximations Using Archived 1 Loop Detector Data 2

3

Meead Saberi K. 4

Graduate Student 5

Department of Civil and Environmental Engineering 6

Portland State University 7

Email: [email protected] 8

9

Miguel A. Figliozzi (*) 10

Associate Professor 11

Department of Civil and Environmental Engineering 12


Email: [email protected] 14

15

(*) Corresponding Author 16

17

18

Mailing Address: 19


Civil and Environmental Engineering 21

PO Box 751 22

Portland, OR 97207 23

24

25

Submitted for presentation and publication to the 26

90th Annual Meeting of the Transportation Research Board 27

January 23–27, 2011 28

29

Submission Date: 30

August 1, 2010 31

32

Number of words: 4,747 + 8 Figures + 2 Tables = 7,247 33

34

Saberi and Figliozzi 2

1

Abstract. Developments in high resolution traffic sensors over the past decades are providing a wealth of 2 empirical speed-flow data. However, existing travel time and speed-flow relationships have not been 3 revalidated against new detailed traffic data. In addition, existing travel time approximations do not take 4 full advantage of the detailed traffic data provided by loop detectors. This research evaluates existing 5 approximations. The key contributions of this research are twofold: (a) it analyzes and compares the 6 goodness of fit of existing approximations not only at the link level but also at the lane level and (b) it 7 proposes a new approximation to estimate travel times under congested conditions. The proposed 8 approximation not only takes advantage of the wealth of loop detector data already archived but also 9 explicitly accounts for the impacts of queuing and the variability of key traffic parameters in a 10 parsimonious and intuitive manner. 11

12

Keywords: travel time estimation, congestion, loop-detector data 13

14


1 1. INTRODUCTION 2

The estimation of freeway segments travel times has many applications in transportation planning and 3 policy analysis. Applications of travel time estimation range broadly from assignment models to the 4 estimation of performance measures, emissions, and congestion costs. It is not surprising that the 5 relationship between travel time (or speed) and flow has been a topic of intense research in the 6 transportation community. 7

This research deals with travel time approximations that take advantage of abundant loop 8 detectors data. Although there have been many research efforts regarding travel time approximations in 9 freeways since that 1960‟s, approximations have not been widely validated or updated. Hence, there is a 10 mismatch between the fast development of high resolution traffic sensors archives in the early 2000‟s and 11 the lack of new travel time approximations that take advantage of the relatively new wealth of empirical 12 speed-flow data. This research reexamines speed-flow relationships based on actual measured conditions 13 in congested and uncongested freeways utilizing archived loop detector data. 14

The contributions of this research utilizing archived traffic data are twofold: (a) it analyzes and 15 compares the goodness of fit of existing approximations not only at the link level but also at the lane level 16 and (b) it proposes a new approximation to estimate travel times under congested conditions. The 17 proposed approximation not only takes advantage of the wealth of loop detector data already archived but 18 also explicitly accounts for the impacts of queuing and the variability of key traffic parameters in a 19 parsimonious and intuitive manner. 20

This paper is organized as follows. Section two will provide a literature review of existing travel 21 time/speed-flow approximations. Section three will present a description of the data and case study used 22 in this research. Section four will discuss the methodology utilized to separate congested and uncongested 23 traffic conditions and estimate key traffic variables. Section five will analyze the variability associated to 24 key traffic parameters. Section six will report goodness of fit results for most commonly used 25 approximations. Section seven introduces a new approximation to estimate travel times in congested and 26 uncongested conditions. Section eight ends with conclusions. 27

28

2. TRAVEL TIME APPROXIMATIONS LITERATURE REVIEW 29

Assuming that traffic conditions on a given road segment are stationary and drivers behave the same 30 way, on average, under the same average conditions, a relationship between speed and flow exists (1). In 31 free-flow traffic conditions, prior to queue formation, detected flow equals demand flow at the detection 32 point. As soon as a queue forms, the usual counting procedures (by loop detectors, tubes, or etc.) are only 33 able to count the served demand, and not the demand flow (2). The type of flow commonly used in travel 34 time approximations is demand flow. Hence, utilizing loop detectors, it is more difficult to estimate 35 demand flow than detected flow. 36

Travel time approximations have been widely studied since the 1960s. The well known 37 approximation proposed by the Bureau of Public Roads (BPR) in 1964 (3) relates free-flow speed and v/c 38 ratios (volume-to-capacity) as follows: 39

b

c

vatt )(10 (1)

where: 40 t = average travel time per unit distance, 41


0t = free-flow travel time per unit distance, 1

v = demand volume (vph), 2 c = practical capacity (vph) (80 percent of the maximum capacity), 3 a = 0.15, and b = 4. 4 5

A clear definition of capacity or how to measure capacity was not provided by the BPR (3). The 6 parameter a determines the ratio of free-flow travel time to the travel time at capacity and the parameter 7

b determines from the rate of change of travel time. The standard BRP function is defined by equation 8 (1) with a = 0.15 and b = 4. 9

Dowling et al. (4) and Dowling and Skabardonis (5) empirically evaluated the standard BPR 10 function and concluded that equation (1) tend to overestimate travel times at v/c ratios between 0.8 and 1 11 and underestimate travel times if queuing takes place. Additional limitations of the standard BPR function 12 are identified in Skabardonis and Dowling (6): the BPR function is based on data that do not reflect 13 today‟s traffic operating conditions and it does not take into account individual characteristics of different 14 facilities. 15

The standard BPR function was used in the 1965 Highway Capacity Manual (HCM) (7). The 16 HCM 1985 (8) described the same speed-flow relationship but with higher sensitivity to low traffic flows. 17 The HCM 1994 (8) described a different shape for the speed-flow relationship based on empirical 18 observations. Most recently, the HCM 2000 (9) describes speed-flow relationships for different classes of 19 facilities. One of the major drawbacks of the relationship proposed by the HCM 2000 is its inability to 20 estimate speeds for v/c>1 ratios. A review of different speed-flow functions can be found in Branston 21

(10). Calibrated versions of BPR function with different values of parameters a and b are proposed in 22 Dowling et al. (4), Skabardonis and Dowling (6), Singh (11, 12), Kurth et al. (13), and Hansen et al. (14). 23 Many highway or planning agencies have calibrated the BPR function. For example, the Portland 24

metropolitan planning organization (METRO) uses a version of the BPR function where a = 0.15, and b 25 = 7 (14): 26

b

c

vatt )

75.0(10 (2)

To improve upon some of the limitations of the BPR function, Spiess (15) introduced the conical 27 approximation: 28

)(.0 xftt

)1()1(2)( 222 xxxf

(3)

(4)

where: 29

t = average travel time per unit distance, 30


x = v/c ratio (no clear definition of capacity was provided), 32

22

12

, and is a number greater than 1. 33

34


Spiess (15) argues that the conical function has a better computational efficiency than the BRP function in 1 assignment algorithms. METRO also uses its own calibrated version of the conical function where 2

7 and 0833.1 (14). 3

Davidson (16, 17) proposed a travel time function based on concepts of queuing theory: 4

)1(10

X

XJtt D

(5)



X = degree of saturation (volume-to-capacity ratio; no clear definition of capacity was provided), and 8

DJ = delay parameter (time per unit distance). 9

10

The Davidson equation has a definitional inconsistency which was first raised by Golding (18) and then 11 further discussed by Akçelik (19). The inconsistency of the Davidson equation is due to its implication 12 that the capacity can be defined as the inverse of free-flow travel time and the degree of saturation as 13 volume times free-flow travel time. In addition, the Davidson function estimates finite travel times for 14 v/c<1, negative travel times for v/c>1, and infinite travel times for v/c=1. 15

Akçelik (19) developed a time-dependent version of Davidson‟s function using the steady-state 16 delay equation for a single channel queuing system: 17

cT

XJXXTtt A8

)1()1(25.0 2

0 (6)



X = degree of saturation (volume-to-capacity ratio; no clear definition of capacity was provided), 21 T = duration of analysis period (h), 22 c = capacity (vph), and 23

AJ = delay parameter (unitless). 24

25

The delay parameter AJ corresponds to the quality of service provided by the road section and is 26

independent of the traffic flow but sensitive to the value of travel time at capacity (19, 20). Chapter 30 of 27 the HCM 2000 (9) suggests the following modified speed-flow equation: 28

2

22

0

16)1()1(25.0

T

XJLXXTDtt q

(7)

where: 29 t = average travel time (h), 30

0t = free-flow travel time (h), 31

qD = delay due to leftover queue from prior hour (h), 32


X = degree of saturation (volume-to-capacity ratio), 1 T = duration of analysis period (h), 2 L = segment length (mi), and 3

J = delay parameter (22 mih ). 4

Details about calculating the delay caused by leftover queue can be found in Dowling et al. (20). Dowling 5 and Skabardonis (2) proposed a simplified version of the Akçelik equation as follows: 6

JXXXTtt 2

0 )1()1(25.0 (8)

where all variables are the same as defined before in equation (7). Parameter J in equation (8) is unitless. 7

In this equation, the constant multiplier 8 for the Akçelik AJ parameter is subsumed within the J 8

calibration parameter itself and the variable capacity is dropped. An hour-long analysis period is assumed. 9 Dowling and Skabardonis (2) argue that the Akçelik curve is consistent with the theoretical delay due to 10 queuing for v/c>1. The parameter T in the Akçelik equation is the time interval during which a constant 11 flow rate persists. The Akçelik equation assumes no initial queue at the start of the flow period. However, 12 the extra delay caused by the leftover queue from the prior period can be calculated and added to the 13 Akçelik equation as in Dowling et al. (20) and HCM 2000 (9). 14

Figure 1(a) shows a comparison of the standard BPR, METRO updated BPR, METRO conical, 15 Davidson, Akçelik, and the HCM 2000 speed-flow curves with a free-flow speed of 60 mph, link length 16 of 1 mile, 1 hour long analysis period, and a capacity of 2,300 vehicle per hour per lane (vphpl) for 17

v/c<1. Delay parameters 1.0AJ and 04.0J22 mihr are used for the Akçelik and HCM 2000 18

equations, respectively. The Akçelik curve closely matches the HCM 2000 curve. The Davidson and 19 METRO conical models predicts higher average travel times when compared to the other models. The 20 travel times predicted by the standard BPR and METRO BPR models are constantly higher than those 21 predicted by Akçelik and HCM 2000 except for v/c close to 1. Figure 1(b) compares the same travel time 22 estimation functions for v/c>1. The Davidson function is excluded from this figure since it estimates 23 negative times. The Akçelik curve closely matches the HCM 2000 curve. Travel times estimated by the 24 Akçelik and HCM2000 models increase almost linearly for v/c>1 (the delay parameter J can be used to 25 adjust the slope) and the BPR curve increases nonlinearly. In METRO BPR formulation travel times 26 increase rapidly for v/c>1. The conical function behaves as a quasi-linear function for v/c>1. 27

Summarizing, there are few studies utilizing empirical data to validate, calibrate, or compare 28 goodness of fit among different approximations. Although there have been some attempts to improve 29 travel time estimations under congested conditions, existing approximations do not take advantage of 30 archived loop detector data or take into account the variability that is found in key traffic parameters. In 31 addition, although many formulas have been proposed, most research efforts do not provide clear 32 guidelines to estimate the key traffic parameters or inputs, e.g. how to estimate the appropriate value of 33 capacity. The rest of the sections aim to provide a method to estimate key parameters and a new 34 approximation that takes into account variability and key traffic parameters under congested conditions. 35

36


0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Avera

ge T

ravel

Tim

e (

min

/mil

e)

Volume/Capacity

Standard BPR (a=0.15, b=4)

METRO BPR (a=0.15, b=7)

METRO Conical (a=7, b=1.0833)

Davidson (J=0.1)

Akçelik (J=0.1)

HCM 2000 (J=0.04)

1

(a) 2

0

20

40

60

80

100

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Avera

ge T

ravel

Tim

e (

min

/mil

e)

Volume/Capacity

Standard BPR (a=0.15, b=4)

METRO BPR (a=0.15, b=7)

METRO Conical (a=7, b=1.0833)

Akçelik (J=0.1)

HCM 2000 (J=0.04)

3

(b) 4

5

FIGURE 1 Comparison of Travel Time Estimation Functions for (a) v/c> 1 and (b) v/c> 1. 6

7


3. STUDY LOCATION AND DATA SOURCES 1

The study area includes two locations on southbound OR-217 in Portland, Oregon at mileposts 1.92 2 (Beaverton-Hillsdale Hwy) and 3.12 (Denney Rd). Typically, queues form on the OR-217 southbound 3 during morning and afternoon peak periods. In the morning peak period, a recurrent bottleneck is located 4 between Scholls-Ferry Road and Greenburg Road, and the resulting queue propagates over 4–5 mi 5 upstream. The bottleneck activates as a result of a large inflow from the onramp at Scholls-Ferry Road 6 and remains active from 7 a.m. to 9 a.m. (21) (See figure 2). At both selected locations the freeway has 7 two main lanes. At milepost 1.92, a two-lane off-ramp to Allen Blvd is located 0.25 mile downstream of 8 the Beaverton-Hillsdale Hwy on-ramp. At milepost 3.12, a one-lane off-ramp to Hall Blvd is located 0.32 9 mile downstream of the Denney Rd on-ramp. Figure 2 shows a sketch of the study locations. 10

The data used in this study comes from PORTAL (http://portal.its.pdx.edu/). PORTAL archives 11 data from more than 670 inductive loop detectors in the Portland region. These detectors were initially 12 deployed as part of a comprehensive ramp metering system and dual mainline loops are located just 13 upstream of on-ramp locations and the on-ramps themselves. At 20-second intervals, each loop detector 14 records vehicle count, average speed of these vehicles, and occupancy (percentage of the sample period 15 when a vehicle was over the detector) (22). This study employs archived loop detector data aggregated 16 over 20-second intervals over 30 weekdays from September 2009 to February 2010. Since adverse 17 weather conditions can add noise to the data (23-26) and for the sake of consistency with previous speed-18 flow studies the data analysis is restricted to days with favorable weather conditions (i.e. days with no 19 rain or snow). The analysis is for morning peak periods (6 a.m. to 10 a.m.). 20

21

FIGURE 2 Schematic Map of the Study Locations: Milepost 1.92 (Beaverton-Hillsdale Hwy) and 22 Milepost 3.12 (Denney Rd). 23

24

MP 3.12Denney Rd

ToHall Blvd

MP 1.92Beaverton-Hillsdale Hwy

ToAllen Blvd

http://portal.its.pdx.edu/


4. METHODOLOGY TO IDENTIFY TRAFFIC STATES AND KEY TRAFFIC 1 PARAMETERS 2

It was mentioned that the type of flow commonly used in travel time approximations is demand flow. 3 However, under congested queuing conditions loop detectors estimate detected flow. It is usually 4 assumed in travel time approximations that v/c<1 represent uncongested traffic conditions without clearly 5 indicating how to quantify capacity or identify uncongested conditions. 6

Different algorithms have been proposed to separate congested and uncongested traffic regimes. 7 Several methods utilize speed threshold-based algorithms. Chen et al. (27) utilizes vehicle speed to 8 indentify traffic states. Zhang and Levinson (28) developed an algorithm based on occupancy 9 differentials. The ASDA/FOTO algorithm uses Kerner‟s three-phase traffic theory and speed and flow 10 thresholds to identify different traffic states (29). The rescaled cumulative curve method proposed by 11 Cassidy and Bertini (30) is another tool to track congested traffic features. A recent study by Li and 12 Bertini (31) has tested the rescaled cumulative curve method with speed thresholds and with 13 ASDA/FOTO rules; results indicate that the rescaled cumulative curve with speed thresholds works well 14 when only two traffic states (congested and uncongested) are defined. The Chen algorithm was optimized 15 for use in Portland by Wieczorek et al. (32); results suggested that a speed threshold of 35 mph is 16 appropriate for Portland freeways. Thus, in order to separate congested and uncongested data this research 17 utilizes the rescaled cumulative plots with a speed threshold of 35 mph using 20-second aggregated data. 18 Figure 3 presents a rescaled cumulative curve for ORE 217 SB, milepost 4.35, on January 22, 2010 (for 19 details on rescaled cumulative curves construction see (31). From figure 4(b), bottleneck activation and 20 deactivation times can be identified. Between the bottleneck activation and deactivation times, the 21 average speed of 22 mph, which is below the speed threshold of 35 mph, is observed. Therefore, from 22 7:33:20 to 8:35:00, the traffic state is congested and for the rest of the time period (from 6:00:00 to 23 7:33:20 and from 8:35:00 to 10:00:00), the traffic state is uncongested. 24

25

FIGURE 3 Rescaled Cumulative Curve Construction, ORE 217 SB, Milepost 4.35, January 22, 26 2010 27

Counts were multiplied by a factor of 180 to convert the reported 20-second estimates to hourly 28 flows. A travel time rate (min/mile) was determined for each data point by inversing the reported speed. 29 Because the 20-second volume figures are integer values the hourly flows are also integer numbers 30


multiples of 180. The resulting flow–travel time pairs can therefore be naturally sorted into several „bins‟. 1 For each of these bins, the average of the reported travel times was calculated and plotted. 2

Capacity is defined as the 5-minute breakdown flow. Breakdown points are identified when the 3 speed drop between two consecutive time intervals exceeds a threshold of 10 mph and low speed (lower 4 than 55 mph) is sustained for some time as defined in Dong and Mahmassani (33). For each detector on 5 each day the breakdown flow using 5-minute aggregated data was measured (5-minute aggregation level 6 is used to be consistent with the literature). 7

Free-flow speeds are defined as the average speed under flow rates per lane that are equal or 8 smaller than 360 vph. This definition is congruent with the HCM 2000 (9) definition that states that free-9 flow speed is “the mean speed of passenger cars under low to moderate flow rates that can be 10 accommodated on a uniform roadway under prevailing roadway and traffic conditions.”. 11

12

5. VARIABILITY OF KEY TRAFFIC PARAMETERS 13

Utilizing the methodology presented in section four, it was possible to obtain the distribution of key 14 traffic parameters such as bottleneck activation and deactivation times, vehicles‟ speed within the queue, 15 breakdown flow, and maximum flow. For example, figure 4 shows the distribution of breakdown flow per 16 lane of travel. 17

A statistically significant difference at 99% level (chi-square test) is obtained between capacity 18 distributions at each lane and between mileposts. At both locations, the right lanes experience lower 19 values of capacities which indicate the possible impact of the ramps and weaving sections on breakdown 20 flow. The lower breakdown flow values at milepost 1.92 can be explained by the significantly higher 21 flows on the adjacent on-ramp. These results confirm findings in the literature (34-45) that the magnitude 22 of the freeway capacity is not a single or constant value and flow breakdown (and maximum flow) can 23 occur over a wide range of flow rates. 24

25

0

10

20

30

40

50

60

70

80

90

100

1000 1250 1500 1750 2000 2250 2500

Em

pir

ical C

um

ula

tive D

istr

ibu

tio

n

Capacity (Breakdown Flow, vphpl)

Lef t Lane MP 1.92 Lef t Lane MP 3.12

Right Lane MP 1.92 Right Lane MP 3.12

26

FIGURE 4 Empirical Cumulative Distribution of 5-minute Breakdown Flows. 27


Existing travel time estimation models do not take into account capacity variability although 1 empirical studies have shown that flow breakdown is not a constant unchanged value. Also literature 2 shows that flow breakdown does not necessarily occur at the nominal maximum flow (34, 38, 41, 42). 3 Breakdown flow may be lower or higher than the maximum flow. Several studies have explored the 4 stochastic characteristics of capacity (34, 35, 38, 43-45) and probabilistic queuing models have been 5 developed (46). A travel time approximation that explicitly takes into account capacity variability is 6 presented in section seven. 7

8

Table 1 Statistical Description of Measured Traffic Parameters 9

min mean median max stdv

Milepost 1.92

Free-flow speed (mph) 58 60 60 61 1

Speed at capacity (mph) 46 53 53 57 3

Speed within the queue (mph) 15 21 21 34 4

Breakdown flow (vph) 2,940 3,234 3,222 3,732 201

Maximum flow (vph) 3,192 3,429 3,444 3,732 164

Bottleneck activation time 7:28:00 7:39:22 7:39:40 7:50:20 0:05:45

Bottleneck deactivation time 8:00:00 8:26:06 8:21:20 9:07:40 0:21:17

Bottleneck duration time 0:15:40 0:46:44 0:38:30 1:31:00 0:20:54

Milepost 3.12

Free-flow speed (mph) 58 60 60 63 1

Speed at capacity (mph) 48 52 52 55 2

Speed within the queue (mph) 23 32 32 39 4

Breakdown flow (vph) 3,420 4,008 3,990 4,596 271

Maximum flow (vph) 3,720 4,151 4,140 4,596 191

Bottleneck activation time 7:17:00 7:30:42 7:30:50 7:43:20 0:05:52

Bottleneck deactivation time 8:03:40 8:33:40 8:26:30 9:07:20 0:20:40

Bottleneck duration time 0:33:40 1:02:58 0:51:30 1:37:20 0:19:20

10

The distributions of key traffic parameters including free-flow speed, speed at capacity, speed 11 within the queue, breakdown flow, maximum flow, bottleneck activation time, bottleneck deactivation 12 time, and bottleneck duration time at each lane separately and over lanes were obtained and analyzed. 13 Table 1 provides key statistical descriptors of traffic parameters measured over the lanes at each location. 14 The measured median free-flow speed in the left lane was 4 mph larger than the measured median free-15 flow speed in the right lane at both locations. Results of t-tests show that the observed speed differences 16 between lanes were statistically significant at the 99% level. The measured median speed at capacity in 17 the left lane was 2-4 mph larger than the measured median speed at capacity in the right lane. Results of t-18 tests show that the observed capacity differences between lanes were statistically significant at the 95% 19 level. It is also observed that the measured median speed at capacity was 9-12 mph lower than the 20 measured median free-flow speed. 21


6. UNCONGESTED GOODNESS OF FIT ANALYSIS 1

The BPR, Akçelik, and conical approximations were calibrated. Figure 5(a) shows the estimated travel 2 time vs. v/c ratio curves at the left lane of milepost 1.92. The curves fall closely into the cloud of the field 3 data except for the conical function. To explore error trends residuals are analyzed. A residual is the 4 difference between the observed value and the predicted model value. For all models, except the 5 calibrated BPR function, a systematic error was observed. Table 2 details the goodness of fit of tested 6 models: bias, root mean square error (RMSE), and mean absolute error (MAE). The calibrated BPR 7 approximation shows the best performance. If the v/c ratios are broken down into three categories: (a) 8

5.0/0 cv , (b) 8.0/5.0 cv , and (c) 1/8.0 cv , then the calibrated conical and Akçelik 9

values of RMSE and MAE are clearly higher for the range 1/8.0 cv . For the calibrated BPR 10 function, the RMSE and MAE remain roughly constant for all the v/c ratio ranges. Figure 5(b) shows the 11 bias values by v/c intervals for the selected calibrated approximations. As can be seen, the bias for the 12 calibrated BPR approximation is almost zero while other calibrated approximations have higher values of 13 bias. 14

In summary, results show that the standard BPR and METRO updated functions overestimate 15 travel times for v/c ratios close to 1. The conical function highly overestimates travel times for v/c<1. 16 The Akçelik and HCM 2000 models underestimate travel times for v/c< 0.8 and overestimate travel times 17 for v/c close to 1. The results also show that for v/c<1, the calibrated BPR function has the best overall 18 performance. 19

20


1

0.8

0.9

1.0

1.1

1.2

1.3

0.0 0.2 0.4 0.6 0.8 1.0

Avera

ge T

ravel

Tim

e (

min

/mi)

Volume/Capacity

Field Data

Calibrated BPR (a=0.07, b=1.60)

Calibrated Conical (a=12.61, b=1.04)

Calibrated Akcelik (J=0.0045)

2

(a) Comparison of Calibrated Travel Time Approximations against Field Data for v/c< 1 for the 3 Left Lane at Milepost 1.92. 4

Right Lane MP 1.92

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

BPR

Conical

Akceli

k

Bia

s

0≤v/c≤0.5

0.5<v/c≤0.8

0.8<v/c≤1

Right Lane MP 3.12

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

BPR

Conical

Akceli

k

Bia

s

0≤v/c≤0.5

0.5<v/c≤0.8

0.8<v/c≤1

5

(b) Bias Values of Calibrated Approximations by v/c Intervals for the Right Lanes at both 6 Locations 7

FIGURE 5 Fit and Bias of Calibrated BPR, Conical, and Akçelik Approximations 8


TABLE 2 Comparison of bias, RMSE, and MAE for Uncalibrated and Calibrated Travel Time Estimation Functions 1

2

Parameters Unit BPR Conical Akçelik

Uncalibrated Calibrated Uncalibrated Calibrated Uncalibrated Calibrated

Left Lane

MP 1.92

Bias min/mi -0.0063 0.0004 -0.1713 -0.1178 -0.0035 0.0142

MAE min/mi 0.0255 0.0150 0.1733 0.1214 0.0382 0.0388

RMSE min/mi 0.0329 0.0188 0.2970 0.2728 0.1643 0.1577

Right

Lane MP

1.92

Bias min/mi -0.0143 -0.0010 -0.1901 -0.1340 -0.0166 0.0047

MAE min/mi 0.0264 0.0169 0.1912 0.1363 0.0420 0.0424

RMSE min/mi 0.0374 0.0215 0.3171 0.2913 0.1723 0.1605

Left Lane

MP 3.12

Bias min/mi -0.0056 -0.0011 -0.1584 -0.1084 0.0017 0.0145

MAE min/mi 0.0200 0.0165 0.1595 0.1104 0.0399 0.0443

RMSE min/mi 0.0649 0.0590 0.2821 0.2598 0.1837 0.1803

Right

Lane MP

3.12

Bias min/mi 0.0041 -0.0014 -0.1662 -0.0711 0.0180 0.0260

MAE min/mi 0.0210 0.0166 0.1674 0.0790 0.0415 0.0437

RMSE min/mi 0.0268 0.0211 0.2755 0.2144 0.0908 0.0891


7. PROPOSED TRAVEL TIME APPROXIMATION FOR CONGESTED CONDITIONS 1

The results of the previous section indicate that existing travel time approximations such as the Akçelik 2 and conical model do not fit the data well for v/c close to one. On the other hand, the calibrated BPR 3 function has the best overall performance for v/c<1 but the Akçelik and HCM 2000 model has the highest 4 consistency with queuing theory. The Akçelik and HCM 2000 model for congested conditions is based on 5

the average delay in the queue. As shown in figure 6(a), the average delay ( AD ) for stationary arrival and 6

departure rates for oversaturated conditions can be calculated as: 7

)1(2

1 XTDA

(9)

where X is the v/c ratio and T is the time period where the demand rate persists. The actual delay 8 experienced by individual vehicles in a queue depends on the vehicle‟s departure time or the time when 9 the vehicle joins the queue. Figure 6(a) illustrates a graphical representation of the delay experienced by 10

vehicle i in a queue with stationary arrival and departure rates where it is the time when the vehicle joins 11

the queue, id is the delay experienced by the vehicle i, and T is the time interval during which an average 12

demand flow rate persists. Different graphical representations of delay in a queue can be plotted and 13 therefore different expressions, instead of equation (9), can be proposed. 14

Another limitation of existing travel time approximations is that key traffic parameters that 15 describe queuing conditions and characteristics of the bottleneck are not incorporated into the 16 approximations. This research proposes an approximation that takes advantage of loop detector data and 17 incorporates key (intuitive) bottleneck traffic parameters to the travel time estimation formula. Herein and 18 without loss of generality, this research adopts the linear model assumed in figure 6(a) to illustrate how 19 bottleneck characteristics from loop detector data can be used in travel time approximations. 20

21

22

(a) (b) 23

FIGURE 6 The Delay Experienced by Vehicles in A Queue with Stationary Arrival and Departure 24 Rates 25

26

27


The delay experienced by vehicle i, assuming that arrival rates and departure rates are proportional to 1 demand flow and capacity respectively, can be calculated as follows: 2

( 1)i i

vd t

c , Tti 0 and i=1 to n (10)

The queuing analysis presented in figure 6 assumes that vehicles are stacked on top of each other and 3 form a “point” queue next to the bottleneck. However, in the real world, a queue takes considerable 4 physical space and backs up. Using a “virtual arrival curve” described by Daganzo (47), it is assumed that 5

the queue forms directly upstream of the bottleneck and vehicles travel at an average speed qV within the 6

queue of length qd . Denoting qt as the queuing time of a vehicle and w as the delay of this vehicle then 7

the following relationships exist (if it is assumed that vehicles approach the queue at a free-flow speed): 8

q

q

qV

dt (11)

f

q

qV

dtw (12)

Equation (12) expresses that the delay of any vehicle is the difference between its travel time within the 9 queue and the time that it would take to travel the same distance at free-flow speed. Therefore, by 10

substituting q

q

V

d with qt , the queuing time ( qt ) is calculated as follows: 11

)1(f

q

q

V

V

wt

(13)

Equation 13 can be simplified as: 12

wtq

)1(

1

f

q

V

V

1

(14)

(15)

Since parameter is only related to the free-flow speed and the average speed of vehicles within the 13

queue, it is the same for all the vehicles within the queue (47). Therefore, assuming that the queue forms 14 directly upstream of the bottleneck the delay experienced by vehicle i is calculated as follows: 15

( 1)i i

vd t

c (16)


Figure 6(b) illustrates the time dependent relationship between travel time and v/c ratio 1

considering vehicle‟s arrival time at the queue. The first vehicle (i=1, 0it ) that joins the queue 2

experiences a minimum delay of close to zero and the last vehicle (i=n, Tti ) that joins the queue 3

experiences a maximum delay equal to ( 1)v

Tc

where T is bounded by zero and bottleneck duration 4

time. 5

The distributions of the input data ( qV and c ) can be estimated using the methods described in 6

section 4. For example, in the rescaled cumulative curve of speed shown in figure 3 the interpolated lines 7 with constant slope represent stationary traffic conditions. Between the bottleneck activation and 8 deactivation times the slope of the interpolated line represents the average speed within the queue which 9 is 22 mph. Figure 7 shows the empirical cumulative distribution of vehicle‟s speed within the queue at the 10 study locations. As can be seen, higher speeds within the queue are observed on the left lanes compared to 11 the right lanes at both locations. 12

13

0

0.2

0.4

0.6

0.8

1

10 15 20 25 30 35 40

Emp

iric

al C

um

ula

tive

Dis

trib

uti

on

Speed (mph)

Right Lane MP 1.92 Left Lane MP 1.92

Right Lane MP 3.12 Left Lane MP 3.12

14

FIGURE 7 Empirical Cumulative Distribution of Vehicles’ Speed within the Queue. 15

16

As discussed earlier, capacity and other traffic parameters have been treated as constant values or 17 not considered explicitly. Lorenz and Elefteriadou (34) suggested a definition of capacity as the rate of 18 flow (expressed in pcphpl) along a uniform freeway segment corresponding to the expected probability of 19 breakdown deemed acceptable under prevailing traffic and road conditions in a specified direction. 20 Following this conceptual idea, this research incorporates the concept of “expected probability of 21 breakdown” into the travel time estimation as follows: 22


0 (1 ( ) )( )

( ) ( 1)( )

b

c

c q i

c

vt a

c pt

vt p t

c p

1( )

1( )

c

c

v

c p

v

c p

(17)

1 where: 2

1( )

( )(1 )

qq q

f

pV p

V

3

t = average travel time (h), 4

0t = free-flow travel time (h), 5

ct = travel time at capacity (h); )1(0 attc , 6

it = time when the vehicle i joins the queue (h), 7

v = demand volume (vph), 8

( )cc p the value of capacity obtained from the quantile function of capacity for a given probability cp , 9

( )q qV p the value of vehicles‟ speed within the queue obtained from the quantile function of vehicles‟ 10

speed within the queue for a given probability qp , and 11

fV = free-flow speed (mph). 12

13 For v/c<1, equation (17) presents a probabilistic calibrated BPR function; for v/c>1 a function 14

that accounts for the stochastic nature of traffic parameters and key bottleneck characteristics is 15 suggested. It is important to highlight that the empirical data shows a weak linear correlation between 16

the observed data pairs ( , )qV c . The validity of equation (17) is limited by natural boundary 17

conditions, for example: 18

i q

i w

t V l

t V l

19

where: 20

wV the value of the average shockwave speed, and

21

l the length of the link where travel time is being estimated. 22 23

Shockwave speeds can also be estimated using data from series of loop detectors by subtracting the 24 bottleneck activation/deactivation times of consecutive locations (see Table 1). 25 26

To calculate the average delay assuming demand and capacity conditions as described in figure 27

(6), it can be replaced by / 2T in equation (17). Figure 8 shows the volume vs. travel time relationship 28

using average delay for the left lane at milepost 1.92 where a = 0.07, b = 1.6, T = 60 min, 0t = 0.9524 29

min/mi ( fV = 63 mph), ct = 1.0190 min/mi ( cV = 58.88 mph), and qV = 21.8 mph ( = 1.53) against field 30

data from 5 days. The empirical quantile function of capacity of the selected lane is used with 31


probabilities of cp = 0.1,

cp = 0.5, and cp =0.9 to plot different curves. In figure (8), the probability

qp 1

is simply the median value or qp = 0.5 (median value). Hence, the slope of the curves would increase 2

(decrease) for a larger (smaller) value of qp . 3

4

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

0 500 1000 1500 2000

Tra

vel

tim

e (

min

/mi)

Volume (vph)

p = 0.1

p = 0.5

p = 0.9

0

5

10

15

20

25

30

35

40

45

50

1500 2500 3500

Tra

vel

tim

e (

min

/mi)

Volume (vph)

p = 0.1

p = 0.5

p = 0.9

5

v/c< 1 v/c> 1 6

FIGURE 8 Probabilistic Volume vs. Travel Time Relationship for the Left Lane at Milepost 1.92 7 against Field Data from 5 Days. 8

9

8. CONCLUSIONS 10

Different existing travel time approximations were evaluated empirically and theoretically. It was found 11 that for v/c<1, the calibrated BPR function has the best overall performance among tested models for 12 studied locations. It was also found that different freeway lanes and segments may have substantially 13 different traffic characteristics and parameters such as free-flow speed, speed at capacity, and capacity. 14

A new travel time approximation formula that is a function of key traffic parameters is proposed. 15 The proposed approximation takes full advantage of available loop detector data. Another benefit of the 16 proposed approximation is that it is possible to use available loop detector data to adjust the values of the 17 key traffic parameters as a function of one (or more) probabilities p. Hence, the analyst is given tools to 18 understand or simulate the sensitivity of travel time as a function of the variability of key traffic 19 parameters such as capacity and speed within the queue. The shape and slope of the travel time 20 approximation is a function of intuitive and readily quantifiable parameters. 21

There are several research directions in which the ideas presented in this paper can be continued. 22 The empirical evaluation presented in this paper used field data only for v/c<1 conditions and a 23 theoretical approach was developed for v/c>1. Further research is required to validate the proposed travel 24 time approximations under congested conditions. In addition, to better understand the effects of freeway 25 lane and segment characteristics on key traffic parameters more locations should be studied. 26


ACKNOWLEDGMENTS 1

The authors thank the Oregon Department of Transportation (ODOT) and the Portland Oregon 2 Transportation Archive Listing (PORTAL) for providing the loop-detector data and OTREC (Oregon 3 Transportation Research and Education Consortium) for providing the necessary financial support to 4 carry out this research. The authors also acknowledge Robert Bertini for recognizing the need to study 5 travel time approximations and facilitating this study. However, any error, mistake, or omission related to 6 this research paper is the sole responsibility of the authors. 7


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