- Email: [email protected]
Delay-based passenger car equivalents for trucks at signalized intersections
Transportation Research Part A 34 (2000) 437±457
www.elsevier.com/locate/tra
Delay-based passenger car equivalents for trucks at signalized intersections Rahim F. Benekohal a
a,*
, Weixiong Zhao
b
University of Illinois at Urbana-Champaign, Newmark Civil Engineering Laboratory, MC-250, 205 North Mathews Avenue, Urbana, IL 61801, USA b Transportation Engineer, Houston, TX, USA Received 8 January 1998; received in revised form 16 March 1998; accepted 23 March 1999
Abstract This paper presents a new methodology for computing passenger car equivalents at signalized intersections that is based on the delay concept. Unlike the commonly used headway-based methods that consider only the excess headway consumed by trucks, the delay-based approach fully considers the additional delay heavy vehicles cause on trac stream. Delay-based passenger car equivalents are not constant, but depend on trac volume, truck type and truck percentage. The ®eld data indicated that the passenger car equivalents increase as the trac volume and the percentage of heavy vehicles increase. The ®eld data were used to calibrate TRAF-NETSIM simulation model that was used to cover a broad range of trac conditions. Mathematical models to estimate the equivalencies were developed. The passenger car equivalent for single unit trucks vary from 1.00 to 1.37, and for combination trucks 1.00±2.18 depending on trac volume and truck percentage. The passenger car equivalents are highly correlated with trac volume and, to some degree, with percentage of heavy vehicles. Although the PCE of 1.5 recommended in the 1985 HCM seems to be more reasonable than the 2.0 recommended in the 1994 and 1997 HCM, both overestimate the impact of single unit trucks. For combination trucks, the 1997 HCM overestimates the capacity reduction eects of the trucks in most cases. Ó 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction The passenger car equivalents (PCE) concept has been used to account for the adverse eects of heavy vehicles on trac operations. Heavy vehicles, due to their size and lower acceleration/ deceleration capabilities, may adversely aect trac ¯ow performance at intersections. PCE
*
Corresponding author. Tel.: +1-217-244-6288; fax: +1-217-333-1924. E-mail address: [email protected] (R.F. Benekohal)
0965-8564/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 5 - 8 5 6 4 ( 9 9 ) 0 0 0 2 6 - 9
438
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
indicate the number of passenger cars that would have the same eects on trac operations as one truck. Stopped heavy vehicles maintain longer headway than cars when crossing intersections; thus causing delay to the vehicles behind. The delay comes from not only the longer headway, but also from the increased headway by vehicles that are behind the heavy vehicle (Molina, 1987). In most cases the only adverse eect considered is the additional time headway the heavy vehicles consume, however, the eects on other variables such as travel time, vehicular delay and queue length have not been adequately accounted for. The headway-based approach suggests using a constant value that is independent of, heavy vehicle type, trac volume and percentage of heavy vehicles. For instance, the 1985 highway capacity manual (HCM) (TRB, 1985) used a constant PCE of 1.5 and the 1994 and 1997 HCM (TRB, 1994, 1997) used a factor of 2.0 for signalized intersections. These PCE are based on the headway concept and do not fully consider the delay heavy vehicles cause at intersections. This paper introduces a new methodology for computing PCE for signalized intersections that is based on delay instead of time headway. The Delay-based PCE, called D_PCE, concept not only uses delay as the main criteria, but also considers trac volume and percentage of heavy vehicles in ®nding the equivalency factors. The D_PCE is simply de®ned as the ratio of delay caused by a heavy vehicle to the delay of a car in an all-passenger car trac stream. The paper presents development of mathematical models to estimate passenger car equivalents as a function of vehicle type, trac volume and percentage of trucks. Field data collected at seven intersections were used to determine the D_PCE. The ®ndings from ®eld data needed to be generalized to other trac conditions. To establish the general relationship between D_PCE, trac volume and percentage of trucks, the TRAF-NETSIM simulation model was calibrated and used. The simulation results were compared to ®eld data to assess the appropriateness and accuracy of the TRAF-NETSIM simulation model for computing D_PCE. Mathematical models were ®t to the data and general relationships were developed. The paper also compares the capacity reduction eects of using D_PCE and the PCE values used in HCM. 2. Past research on PCE for signalized intersections 2.1. Webster's method Webster (1958) performed a controlled ``track'' experiment to compute PCE. Vehicles were classi®ed into two groups: light vehicles and goods vehicles. The goods vehicles included medium and heavy commercial vehicles. The PCE value for a heavy vehicle was determined by grouping data from 12 successive signal cycles. The average number of heavy vehicles per cycle ng Rng =N was plotted against the average number of light vehicles per cycle n1 Rn1 =N . ng and n1 are the number of departing heavy and light vehicles in individual cycles and N is the number of cycles in the set. There was very little scatter in the data and a straight line was drawn through the ng and n1 values. The PCE value was estimated as the reciprocal of the slope of the line. In another study, Webster and Cobbe (1966) recommended using a PCE of 1.75 for heavy truck and 2.25 for buses in computing saturation ¯ow for through movements.
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
439
2.2. Regression method Branston and van Zuylen (1978) and Branston and Gipps (1981) presented a multiple linear regression method to estimate trac parameters, including PCE, at trac signals. The green time given to an approach was divided into consecutive ``®rst'', ``middle'' and ``last'' counting periods. The period covered the buildup to saturation ¯ow rate at the start of green, the period of constant saturation ¯ow and the fall-o of saturation ¯ow during the amber period. The vehicle departures were recorded over a period T, which began and ended at arbitrary instants. Then, the number of passenger cars n1 was regressed on the number of vehicles ni of other classes i to obtain the estimates of the coecients B0 and Bi in the model: X n1 B0 T ÿ Bi ni e; 1 i61
where B0 is the saturation ¯ow rate in passenger cars per unit time, Bi the PCE value of vehicles of class i and e the error term. The recording of the departure times of the vehicles was ``synchronous'' or ``asynchronous''. The former terminated the observation the instant a vehicle departed, but the latter terminated the observations at an arbitrary time. The PCE for the asynchronous case was 1.74 and for the synchronous case it was 1.59. The regression methods were applied to through trac in main stream and excluded the start-up time and transition lost time. 2.3. Headway ratio method The headway ratio (HR) method was pioneered by Greenshields et al. (1947). The method is based on the ratio of the average time headway for the vehicles of interest to the average time headway for passenger cars PCEt
ht ; hc
2
where PCEt is the PCE for vehicles of class t, ht the eective average headway of vehicles of class t, and hc the average headway of passenger cars. The HR method is most commonly used to ®nd PCE at signalized intersections. Although the HR method is simple and straightforward, it does not consider the adverse eects of heavy vehicles on delay. Evans et al. (1981) reported that large trucks are over-represented in queue leader positions. Using the HR method, the PCE for heavy truck (de®ned as a vehicle with more than three axles) in a non-leader position was found to be 2.63. Whereas, the PCE for a heavy truck in the lead position in the queue was 4.05. Molina (1987) proposed a modi®ed headway ratio method that considered the increase in headway for vehicles queued behind the heavy vehicle. The PCE was calculated as: PCEt or
ht DH hc
3
440
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
TTt ÿ TTc ; 4 hc where PCEt is the PCE for a vehicle of type t, ht the headway of vehicle of type t, hc the saturation ¯ow headway of passenger car, DH the total increased headway of the queue caused by the truck, TTt the total discharge time of a truck queue, and TTc the total discharge time of a passenger car queue. Molina found that the position of a truck in the queue did not have a signi®cant eect on the PCE for the two- and three-axle single-unit trucks. However, the position of a vehicle in the queue had a pronounced eect on PCE for ®ve-axle combination trucks. The scope of Molina's study was limited to through-movement trac, and only a single truck in the queue. Other factors, such as trac volume and percentage of trucks were not considered. Molina recommended PCE of 3.7 and 1.7 for heavy and light vehicles, respectively. Although Molina's method considers the incremental headway, it does not fully consider the additional delay the vehicles queued behind the truck experience. The method takes into account the headway increases up to the eighth-positioned vehicle in the queue. If there are vehicles at farther than 8th position, they do not aect the PCE. West and Thurgood (1995) employed Molina's method to generate PCE for various left-turning light and heavy trucks at compressed diamond interchange. From a set of sample data, composite PCE for standard light and heavy trucks are reported as 1.7 and 4.4, respectively. Kimber et al. (1985) examined Webster's, HR, and regression methods for obtaining PCE. It was concluded that the PCE factors depend on the method of derivation. PCEt 1
2.4. HCM methodology The HCM (TRB, 1985, 1994, 1997) applies a heavy vehicle adjustment factor fHV to reduce the ideal saturation ¯ow rate at intersections. The fHV is computed from the following equation: fHV
1 ; 1 PH PCE ÿ 1
5
where PCE is the Passenger car equivalents for heavy vehicles, and PH the percentage of heavy vehicles. For intersections, the 1985 HCM recommended a PCE of 1.5 while the 1994 and 1997 HCM increased the PCE to 2.0. HCM did not provide the reasons for such an increase. It is important to note that for steep or long grades (de®ned as speci®c grade) on highways HCM recommends PCE that depend on the type and percentage of heavy vehicle in trac stream. However, for intersections HCM uses a constant PCE regardless of the type of heavy vehicle and volume of trac. 2.5. Other methods Keller et al. (1984) used a macroscopic trac simulation model (TRANSYT-7) on an urban network to derive PCE for large vehicles as a function of vehicle size, signal timing and trac volume. PCE were estimated as the ratio of the total travel times of heavy vehicles to that of passenger cars when travelling through an urban network
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
PCEt
TTt ; TT0
441
6
where PCEt is the PCE value of the given vehicle type t, TTt the total travel time of vehicle of type t over the network and TT0 the total travel time of the base vehicle over the network. Results for urban arterials are presented for each of the seven vehicle classes, six dierent levels of service and three types of signal settings. As a result, PCE were found ranging from 1.09 for a single unit truck to 1.53 for a tractor-trailer truck under level of service C operation. Sumner et al. (1984) employed the NETSIM trac simulation model to derive passenger car equivalent values between consecutive signalized intersections on urban arterial roads. These values were cumulative over the length of road between intersections and re¯ect the vehicle-hours of road utilization that were added when large commercial vehicles were introduced into the trac stream. The PCE were derived using the product of vehicle number and time spent by those vehicles in the section. The authors found PCE that varied from 1.10 for single unit trucks to 1.45 for 5-axle combination trucks for two-lane streets under level of service D; and from 1.18 for single unit trucks to 1.53 for 5-axle combination trucks for two-lane streets under level of service B. The authors' work was focused on segments and did not isolate the heavy vehicles' eect at signalized intersections. Kimber et al. (1985) proposed that the PCE as used for adjusting the saturation ¯ow rate should not be the equivalent number of passenger cars a heavy vehicle would physically displace. Rather it should be a number that yields the minimum delay for the intersection. The procedure included the following steps: (1) Trac parameters such as the mean arrival rate, mean and standard deviation of headways were chosen. (2) Signal settings were chosen using Webster's method, and a PCE value for heavy vehicles was assumed. (3) A simulation run was made, and the mean vehicle delay was measured using Webster's delay formula. (4) Steps 2 and 3 were repeated using dierent PCE values. (5) The PCE value corresponding to the setting that gave minimum delay was determined by plotting mean delay against PCE value. An example was given for ®nding the PCE from simulation of a two-phase signal. When the mean headway of 2.0 and 4.56 s were assumed for cars and heavy vehicles, the PCE was 2.3. In summary, using a constant PCE indicates that trac volume or percentages of heavy vehicles do not aect PCE. On the other hand, a variable PCE indicates that truck type, volume or position of truck in queue aects PCE. Although some of the above-mentioned studies resulted in non-constant PCE, they did not establish direct relationships among PCE, trac volume and percentage of heavy vehicles. This study will attempt to ®nd such relationships.
3. Data collection and reduction Data were collected on ten approaches of seven intersections in Central Illinois. The ten approaches are labeled A though J. Trac volume varied from 80 to 467 vph per lane and percentage of heavy vehicles varied from 2% to 30%. The sites were selected such that as many ideal features as possible were met. The collected data included headway, delay for individual vehicles in all-passenger trac stream, the number of queued and non-queued vehicles, vehicle type,
442
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
position of heavy vehicles in queue, signal timing information and geometric data. A summary of the data is shown in Table 1. The predominant heavy vehicle types were large single unit trucks and ®ve-axle combination trucks. The headway was measured as the rear axles of the vehicles passed over the stop line. The headway of the ®rst vehicle was de®ned as the elapsed time between the initiation of the green signal and the time the rear wheels of the vehicle passed the stop line. This way of measuring headway was consistent with previous studies (Molina, 1987; TRB, 1994; Cuddon and Ogden, 1992) and ensured that headways were measured for those vehicles in front of the queue that stopped with their front wheels past the stop line. Two computer programs were used in data reduction: one for headway measurement and the other for delay measurement. The programs were designed to extract trac data, including: the time a vehicle joined the queue, the time a vehicle left the queue, position of the vehicle in the queue, total number of vehicles in the queue, vehicle types, and so forth. Delay used in computation of D_PCE can be stopped delay or queue delay. The stopped delay accounts for the time a vehicle is stopped. Queue delay includes stopped delay as well as part of the delay due to acceleration/deceleration. Using stopped delay would underestimate the eects of heavy vehicles because it does not fully represent the poor performance of heavy vehicles. Therefore, queue delays were measured in ®eld analysis. Data were collected from through lanes, except at Locations A, D and H where the through lanes were shared with right turning trac. The performance of shared right/through may be aected when the percentage of the right turning trac is high. Data indicated that the proportion of right turning trac at those three locations were small (3%, 4% and 9%, respectively). Most of the right-turn vehicles at Locations A and H use the wide shoulder to make the right turn. Moreover, no right-turning heavy vehicles were counted at these locations. Thus, the eects, if any, were negligible.
Table 1 Summary of the data collected Location
A
B
C
D
E
F
G
H
I
J
Total
In the stream Time period (h) All vehicles No. of cars No. of trucks % truck
6 1765 1472 293 16.6
6 1175 908 267 22.7
2 444 377 67 15.1
3.3 1442 1376 66 4.6
3.3 1568 1533 35 2.2
1.8 341 332 9 2.6
1.8 328 320 8 2.4
3.6 1402 1361 41 2.9
3.6 984 933 51 5.2
4 299 208 91 30.4
35.4 9748 8820 928 9.5
690 571 119 39 75 5
486 356 130 24 101 5
593 565 28 16 5 5 2
551 537 14 8 2 3 1
248 241 7 4 3
239 231 8 4 4
321 302 19 2 14 3
199 138 61 19 38 4
4158 3717 441 137 272 28 4
In queue All vehicles No. of cars No. of trucks No. of sus No. of CTs No. of PTs No. of buses
322 271 51 13 35 2 1
Notes: SU Single unit truck; CT combination truck; PT passenger car trailer.
504 492 12 8 3 1
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
443
The data from adjacent lanes of the same approach were combined to increase the number of observations. The combined locations were D and E, F and G, and H and I. Thus, a total of seven data sets, namely A, B, C, D/E, F/G, H/I and J, each representing one intersection were used. 4. Introduction of delay-based passenger car equivalent To quantify the eect of heavy vehicles on delay, a new term called delay-based passenger car equivalent (D_PCE) is introduced. The D_PCE is simply de®ned as the ratio of delay caused by a heavy vehicle to the delay of a car in an all-passenger car trac stream. Mathematically, D_PCE for a heavy vehicle type i is expressed as: Ddi ; 7 d0 where D_PCEi is the D_PCE for a heavy vehicle type i, Ddi the additional delay caused by a vehicle of type i and d0 the average vehicle delay when the trac is composed of all-passenger cars. The D_PCE factor gives the number of passenger cars that would replace a heavy vehicle in the trac stream and result in the same amount of delay as the original trac stream. For example, assume a platoon of 10 cars causes 50 s of delay; then, the average delay d0 5 s/veh. However, if a platoon of eight cars and two trucks (still a total of 10 vehicles) causes 70 s of delay, the additional delay caused by each truck is Dd 70 ÿ 50=2 10 s. Then, the D_PCE factor for the truck is computed as 1 10=5 3. Thus, the computed D_PCE are directly related to the delay caused by heavy vehicles. D PCEi 1
4.1. Delay-based PCE at signalized intersections When a trac stream is composed of all passenger cars, it is referred to as ``base case'' and its delay is referred to as ``base delay''. When there is at least one heavy vehicle in the stream, it is referred to as a ``truck case or mixed trac case'' and a queue containing one or more trucks is referred to as a ``truck queue or mixed trac queue''. When total delays for the base and truck cases are known, D_PCE are obtained as Dt ÿ D0 d0 : 8 D PCE 1 Vt The above equation can be written as: dt ÿ d0 ; 9 pH Xd0 where Dt is the total delay for the truck case (s), D0 the total delay for the base case (s), d0 the average delay for the base case (s/veh), dt the average delay for the truck case (s/veh), pH the percentage of trucks V =Vt , V the total trac volume (vph) and Vt the total truck volume (vph). Conceptually, the queuing diagram in Fig. 1 shows that the delay for an all-passenger trac is the area of triangle ABC, AABC . When heavy vehicles appear in the queue, the triangular shape changes to a polygon of ABDEF. The area of polygon BDEFC, ABDEFC , represents the additional delay due to the presence of heavy vehicles. Thus, the average additional delay for each truck would be D PCE 1
444
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
Fig. 1. Are part of this paper, but not included in this document due to their size.
P
Ddi ABDEFC ; 10 nt nt P where Dd is the average additional delay due to heavy vehicles, Ddi the total additional delay caused by heavy vehicles, ABDEFC the area of BDEFC, which is the additional delay caused by heavy vehicles, nt the number of heavy vehicles in that cycle.For the base case, when the arrival rate, saturation ¯ow rate and signal information are known, the average queue length may be computed as Dd
nq R
qs ; sÿq
11
where q is the arrival rate (veh/s), s the saturation ¯ow rate (veh/s), nq the average number of passenger cars in the queue and R the eective red time for the study approach (s). For the base case, the total queue delay in one cycle is 1 1 qs : D0 AABC Rnq R2 2 2 sÿq
12
The average delay per passenger car (base delay) is computed as d0
D0 1 R2 qs ; n0 2 n0 s ÿ q
13
where d0 is the average delay for passenger cars, n0 the total number of passenger cars in the cycle.The additional delay, the area of BDEFC, can be viewed as the summation of delays to individual vehicles, dj , which are behind the heavy vehicle
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
X
Ddi
m X dj ;
445
14
jp
where dj is the additional delay for a vehicle at the jth position in the mixed queue, p the position of the heavy vehicle in the mixed queue and m the position of the last vehicle in the queue.The value of dj is determined by: dj TTtj ÿ TT0j ;
15
TTtj
is the travel time of the jth vehicle in the mixed queue and where vehicle in all passenger car queues.
TT0j
the travel time of the jth
5. Computing D_PCE from ®eld data D_PCE were computed from ®eld data for single unit trucks and combination trucks. Queue delay from ®eld data was used because it was planned to use TRAF-NETSIM queue delay for comparison. Queue delay includes not only the stopped time, but also some of the delay due to accelerating and decelerating. To compute average base delay from ®eld data, queue delays for individual cars were averaged. A summary of base delays obtained from ®eld measurement is given in Table 2. The base delays were also estimated using the following equations: 1 d0q R; 2 d0
16
qs R2 ; 2n0 s ÿ q
17
where d0q is the estimated base delays for queued cars (s), d0 the estimated delay for all cars in the stream (s), R the red time (s), q the arrival rate (veh/s), s the saturation ¯ow rate (veh/s), n0 the qXC and C the cycle length (s) The estimated and ®eld measured base delays were very close to each other. Paired-t tests indicated that there were no signi®cant dierences between the measured base delays and the
Table 2 Comparison of measured and estimated base delays (s/veh) Data set A B C D/E F/G H/I J
Measured in ®eld
Estimated
Queued cars
All cars
Queued cars
All cars
15.50 15.85 32.95 18.34 34.38 12.08 19.29
7.50 8.06 23.28 7.78 24.72 6.53 17.41
14.5 15.5 31.0 16.5 35.0 11.5 24.5
7.30 7.97 22.79 8.05 25.22 6.69 16.95
446
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
estimated base delays. The D_PCE values when a single unit truck was in the lead position showed that the D_PCE values are highly correlated with the number of vehicles queued behind the truck (Zhao, 1996). The queue length is an indication of trac volume, when other signal timing variables are kept constant. When two consecutive trucks were present in the same queue, the total additional delay caused by the two trucks was computed and its average was used in computing the D_PCE. Average D_PCE were computed to re¯ect the eects of trucks at dierent queue positions. Although the queued trucks are the main sources of additional delays, all of the trucks in the trac stream should to be considered in order to simplify the computation of D_PCE. In normal trac conditions, some trucks are stopped in queue and some are not. Total additional delay caused by the queued trucks is averaged over the entire truck volume to re¯ect the average D_PCE. It should be noted that when the entire truck volume is used (instead of the volume of queued trucks), the base delay for the entire car volume is used (instead of base delay for queued cars). A summary of the D_PCE computed using the ®eld data is presented in Table 3. Further discussion of the ®eld data analysis is given elsewhere (Zhao, 1996; Benekohal and Zhao, 1996). Data Set J is unique because of low volume and high percentage of heavy vehicles (10.7% single unit and 18.1% combination trucks). In most signal cycles, the queue dissipated even before the saturation headway was reached. As a result, the D_PCE factor for combination trucks was the lowest, 1.19, even though it has the highest percentage of combination trucks. Similarly, the D_PCE for single unit trucks, 1.07, were also the lowest of the data sets. The plots of D_PCE at dierent volumes and truck percentages are shown in Fig. 2. The ®gure shows that the D_PCE change with trac volume and percentage of heavy vehicles. For example, Data Set F/G and H/I both had 1.2% single unit trucks, but trac volumes were 256 and 374 vph, respectively. The D_PCE for single unit trucks in Data Set F/G was 1.11 compared to 1.20 for Data Set H/I. On the other hand, Data Sets B and F/G had practically the same trac volumes, but percentages of single unit trucks were 4.6 and 1.2 and percentages of combination trucks were 16.1 and 1.2. The D_PCE factor for single unit trucks was 1.26 for Data Set B compared to 1.11 for data set F/G. Similarly, the D_PCE factor for combination trucks was 1.49 for data set B and 1.38 for data set F/G. A general trend can then be described qualitatively: at the same trac volume level, D_PCE increases with percentage of heavy vehicles; similarly, D_PCE increases with trac volume when percentage of heavy vehicles remains the same. For under saturated trac conditions, the average D_PCE from the ®eld data are found to be in the range of 1.07±1.47 for single unit trucks, and 1.19±1.81 for the combination trucks.
Table 3 D_PCE from ®eld data averaged over all trucks Data set Trac volume (vph) Single unit truck Combination truck
% trucks D_PCE % trucks D_PCE
A
B
C
D/E
F/G
H/I
J
302 5.4 1.30 10.3 1.55
246 4.6 1.26 16.1 1.49
286 4.5 1.27 9.7 1.50
467 1.9 1.47 0.5 1.81
256 1.2 1.11 1.2 1.38
374 1.2 1.20 1.0 1.62
80 10.7 1.07 18.1 1.19
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
447
Fig. 2. Are part of this paper, but not included in this document due to their size.
6. Generalizing results to other trac conditions The ®ndings from ®eld data need to be generalized to other trac conditions. TRAF-NETSIM (FHWA, 1989) simulation model was calibrated to meet the needs of the D_PCE computation. It should be noted that, without such calibration and validation TRAF-NETSIM could not be used for this purpose. TRAF-NETSIM is a microscopic simulation model and has been used to simulate trac performance on a variety of trac conditions (Benekohal and Abu-Lebdeh, 1994). The TRAF-NETSIM output does not provide detailed information for individual vehicles, but does provide enough information for computing average D_PCE for trucks. First, the simulation
448
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
results were compared to ®eld data to assess the appropriateness and accuracy of the TRAFNETSIM simulation model for computing D_PCE. Field trac and geometric data were used as input to the simulation models. Then, delay estimations and D_PCE values from simulation were compared with those from ®eld data. Mathematical models were ®t to the data and general relationships were provided. 6.1. Comparison of simulation delay to ®eld data for cars In this study queue delays from ®eld were compared to queue delays from TRAF-NETSIM. Queue delay in TRAF-NETSIM is de®ned as time spent when the acceleration of a vehicle is less than 2 ft/s2 and its speed is less than 9 ft/s. Queue delays measured in the ®eld were consistent with this de®nition. Queue delay estimations from the simulation runs were compared to the ®eld data for the base case (all cars). The results are summarized in Table 4. To examine the dierences, t-tests were used with 95% con®dence level. For all-car trac, there were no signi®cant dierences between the queue delays from TRAF-NETSIM and ®eld data a 0:05. 6.2. Calibration of TRAF-NETSIM for mixed trac Trac conditions on the study sites were simulated using TRAF-NETSIM with the default values for trucks. TRAF-NETSIM uses only one heavy vehicle type and provides default values such as length and headway factors for it. When the default values were used, the D_PCE computed from the simulation results were signi®cantly lower than those from ®eld data (Zhao, 1996). This was true for both single unit and combination trucks on all study sites. Thus, calibration of the truck factors in TRAF-NETSIM model was necessary. Default values for vehicle length, maximum acceleration rate and headway was calibrated to provide accurate results. A vehicle length 30 ft for SUT and 55 ft for combination was used instead of the 35 ft default. The maximum acceleration rate of 3.5 and 2.1 was used for SUT and combination trucks instead of 3.2 mph/s/s. Headway factors were calibrated based on the ®eld data collected in this study. They re¯ect the passenger car and truck headways measured in the ®eld.
Table 4 Comparison of TRAF-NETSIM queue delay to ®eld data for cars (base case) Data A B C D/E F/G H/I J a
Simulation (®ve replications) 1
2
3
4
5
Avg
6.9 7.5 21.8 7.6 23.8 5.2 16.1
7.3 7.5 23.3 7.3 24.3 6.9 16.3
7.6 7.5 23.3 8.7 23.2 6.5 15.2
8 6.8 22.7 7.9 23.5 6.9 18.0
6.5 8.2 23.3 7.5 25.3 6.5 16.8
7.26 7.5 22.88 7.8 24.02 6.4 16.48
ta0:05 4 2:776.
a
Field
Di.
t-value
7.50 8.06 23.28 7.78 24.72 6.53 17.41
0.24 ÿ0.56 ÿ0.4 0.02 ÿ0.7 ÿ0.13 ÿ0.93
0.92 ÿ2.53 ÿ1.36 0.08 ÿ1.9 ÿ0.42 ÿ2.02
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
449
A headway factor for each vehicle type was calculated using the following equation: P10 PK j1 k1 nijk X hf ijk hfi P10 PK j1 k1 nijk
18
and t
hf ijk
hijk c
hjk
;
19
where hfi is the headway factor for vehicle type i, hf ijk the average headway factor for vehicle type i at site j in queue position k, nijk the number of observations of headway of heavy vehicle type i at t site j in queue position k, hijk the mean headway of heavy vehicle type i at site j in queue position c k, hjk the mean headway of a passenger car at site j in queue position k, i the vehicle type, 1 for single unit truck, 2 for combination truck, j the sites ®eld data were collected, j 1; 2; . . . ; 10, k the queue position, k 1; 2; . . . ; K and K the maximum number of vehicles in the queue (observed K was 9). The headway factors computed from ®eld data for single unit and combination trucks are shown in Table 5. The headway factor was 1.64 for SUT and 2.58 for combination trucks. Thus, in the calibrated TRAF-NETSIM, headway factors of 160% and 260% were used for single unit and combination trucks, respectively. 6.3. Comparison of calibrated TRAF-NETSIM delays to ®eld data for mixed trac The comparison of queue delays when trucks were present was a little bit more complicated because TRAF-NETSIM does not directly report the additional delay a heavy vehicle causes. Queue delays from TRAF-NETSIM for mixed trac (truck case) are listed in Table 6. These values were used in the following equation to calculate the additional delay caused by a heavy vehicle: Dd
dt ÿ d0 ; PH
20
where Dd is the additional delay caused by a truck, d0 the average delay for base case (s/veh), dt the average delay for truck case (s/veh), pH the percentage of trucks. For each data set, ®ve replications of TRAF-NETSIM were made. Then the average values for Dd were computed. Table 6 shows that the additional queue delay estimations from the Table 5 Headway factor computation from ®eld data k Single unit trucks Combination trucks
hf1k n1k hf2k n2k
1
2
3
4
5
6
7
8
9
All
1.53 40 2.50 86
1.61 27 2.53 58
1.67 19 2.57 49
1.77 12 2.65 24
1.80 10 2.69 18
1.68 6 2.72 6
1.72 7 ÿ ÿ
1.67 1 2.73 4
1.67 2 ÿ ÿ
1.64 124 2.58 245
450
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
Table 6 Comparison TRAF-NETSIM additional queue delay to ®eld data for mix trac Data
Simulation (®ve replications) 1
2
3
SUT
A B C D/E F/G H/I J
0 1.6 4 0 10 0 0
2 2 6 5 0 0 0.9
2 0 8 5 0 3 0.9
CT
A B C D/E F/G H/I J
3 2.5 8 10 10 10 0.56
4 3.75 9 4 0 0 1.11
5 4.38 12 6 10 0 0
a
Field 4
5 2 2 2 0 0 3 0
7 5.63 8 7 0 10 2.22
Di.
t-value
Avg
2 2 2 5 10 1 0.9
1.6 1.6 4.4 3.0 4.0 1.4 0.55
2.27 2.08 6.22 3.68 2.75 1.33 1.15
ÿ0.67 ÿ0.48 ÿ1.82 0.68 1.25 0.07 ÿ0.60
ÿ1.67 ÿ1.2 ÿ1.56 ÿ0.56 0.51 0.10 ÿ2.71
3 3.13 14 5 20 0 2.22
4.4 3.88 10.2 6.4 8.0 4.0 1.22
4.1 3.96 11.52 6.32 9.31 4.04 3.28
0.3 ÿ0.09 ÿ1.32 0.08 ÿ1.31 ÿ0.04 ÿ2.06
0.40 ÿ0.16 ÿ1.1 0.08 ÿ0.35 ÿ0.02 ÿ4.63 a
Indicates statistically dierent with ®eld measurement.
simulation were not signi®cantly dierent than ®eld data. The only exception was the queue delay estimation for the combination truck case for data set J. The simulation slightly underestimated the additional delay in this case. One should note that the queue delays in TRAFNETSIM are reported to 1/10 of a second accuracy. Thus, the additional delays due to trucks are accurate at this level. 6.4. Computing D_PCE from simulation results To compute D_PCE for a given heavy vehicle type, two sets of TRAF-NETSIM runs were made: one for base case, in which trac was composed of all-passenger cars, and the other for the truck case, in which trac stream had some trucks. To create identical trac streams and to reduce the variance of the simulation output, the common random number (CRN) technique was employed. Application of CRN technique to TRAF-NETSIM for paired simulation runs was discussed by Benekohal and Abu-Lebdeh (1994). Simulation time was 3600 s and ®ve replications were for each set. First, simulation runs were made when trac was composed of all passenger cars to obtain the base delays (D0 and d0 ). Then, simulation runs were made for mixed trac (truck case) with and delays for the truck case (Dt and dt ) were obtained. Then, D_PCE were computed as: D PCE 1
Ddt Dt ÿ D0 =Vt 1 ; d0 d0
21
D PCE 1
dt ÿ d0 ; pH Xd0
22
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
451
where Dt is the total delay for the truck case (s), D0 the total delay for the base case (s), dt the average delay for the truck case (s/veh), d0 the average delay for the base case (base delay) (s/veh), V the total trac volume (vph), Vt the total truck volume (vph) and pH the percentage of trucks. It should be noted that the number of vehicles processed in the two simulation runs should be the same in order for Eqs. (21) and (22) to be valid. Each set of simulation runs was checked to be sure that the number of vehicles processed was equal. When the numbers were not equal, the runs were not used. 6.5. Comparison of D_PCE from queue delay and overall delay D_PCE can be computed using queue delays or overall delays. Queue delays take into account the stopped delay and most part of acceleration/deceleration delay. The overall delays include all of the delay due to traveling at a lower speed than the desired speed. Let D_PCEod denote D_PCE computed using overall delays, and D_PCEqd denote D_PCE computed using queue delays. Both D_PCEod and D_PCEqd were not signi®cantly dierent than D_PCE from ®eld data, see Table 7. The only exception was combination trucks in data set J. This data set had very low volume (80 vph) and very high truck percent (30%), thus simulation delays were expected to be lower than ®eld data. This agreement is because the dierence between overall delay and queue delay for trucks are similar to those for passenger cars. Since the D_PCE values computed based on queue delay and overall delay were similar, it was decided to compute D_PCE based on the overall delay from the simulation model. Unless speci®ed, D_PCE computed from simulation refers to D_PCEod . 7. Prediction models for D_PCE Trac-related factors that aect vehicular delay at signalized intersections might also in¯uence D_PCE values. These factors may include trac volume, truck percentage and signal timing parameters (cycle length and green splits). Table 7 Comparisons of D_PCE
a
Data Single unit trucks set D_PCE D_PCEod Dierence ®eld (a) (b) (b) ÿ (a) t-value
D_PCEqd Dierence (c) (c) ÿ (a) t-value
D_PCE D_PCEod Dierence D_PCEqd Dierence ®eld (d) (e) (f) (e) ÿ (d) t-value (f) ÿ (d) t-value
A
1.30
1.27
1.22
1.55
1.60
B
1.26
1.20
1.49
1.53
C
1.27
1.20
1.50
1.45
D/E
1.47
1.44
1.81
1.80
F/G
1.11
1.15
1.38
1.37
H/I
1.20
1.25
1.62
1.60
J
1.07
1.04
1.19
1.07
a b
ÿ0.03
ÿ0.62
ÿ0.06
ÿ0.88
ÿ0.07
ÿ1.29
ÿ0.03
ÿ2.45
0.04
0.58
0.05
0.61
ÿ0.02
ÿ1.52
Combination trucks
1.21 1.19 1.40 1.16 1.21 1.03
ta0:05 4 2:776. Indicates it is statistically signi®cant.
ÿ0.08
ÿ1.51
ÿ0.04
ÿ0.83
ÿ0.08
ÿ1.52
ÿ0.08
ÿ0.48
0.05
0.52
0.01
0.06
ÿ0.03
ÿ2.31
0.05
0.65
0.04
0.39
ÿ0.04
ÿ1.18
ÿ0.01
ÿ0.08
ÿ0.01
ÿ0.07
ÿ0.02
ÿ0.20
ÿ0.12
ÿ3.31
1.60 1.53 1.44 1.82 1.33 1.67 b
1.07
0.05
0.61
0.03
0.39
ÿ0.05
ÿ1.04
0.01
0.07
ÿ0.05
ÿ0.33
0.06
0.14
ÿ0.12
ÿ4.62
b
452
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
7.1. Factors aecting delay-based PCE Two factors that had major impact on D_PCE were trac volume and percentage of heavy vehicles. Eects of green split and cycle length were examined using simulation results and twoway analysis of variance tests. It was found that signal timing had no signi®cant eect on D_PCE (Zhao, 1996). They did not have signi®cant eects mainly because D_PCE are computed based on the ratio of delays. This conclusion supports, although a dierent approach is used, the ®ndings by Cuddon and Ogden (1992) that signal timing parameters have no signi®cant correlation with the headway-based PCE. In order to obtain D_PCE values for various trac volumes and percentages of trucks, the calibrated TRAF-NETSIM was used for pre-timed signal. The simulated intersections had through-lanes only and were operating in under-saturated conditions. An equal volume of trac was assigned to the opposite approach, but for crossing streets the volumes were lower. Trac volume per lane varied from 200 to 600 vph, and percentage of trucks varied from 1% to 20%. These conditions cover most of the typical under-saturated trac conditions. D_PCE were computed for each simulation run. Then averages of ®ve replication runs were computed. The plots of the average computed D_PCE are shown in Fig. 3. For combination trucks, with trac volume at 600 vph and percentage of trucks at 18% or higher, saturation conditions were reached. At a given trac volume, the D_PCE for single unit trucks rapidly increased at low truck percentages (4±8%), but ¯attened at higher percentages. Fig. 3 clearly show a non-linear relationship between D_PCE and percentage of trucks. PCE for combination trucks showed a similar trend. From the plot it is clear that D_PCE for trucks is reaching a constant value at higher percentages. The trend indicates that the combined adverse eect of X trucks is less than the sum of adverse eects of X individual trucks. This is a logical phenomenon because more than one truck may be in the same queue. D_PCE increased rapidly when saturation levels were reached (about 600 vph and over 18% trucks in our case). At saturation levels, it is not possible to isolate the eects of trucks from the eects of congestion itself. For a given percentage of trucks, D_PCE increased with trac volume, see Fig. 3. The increase seemed to be linear for single unit trucks than combination trucks. This indicates that the combination trucks had much more adverse eects than the single unit trucks. The above discussions indicate that both trac volume and percentage of trucks aect D_PCE values and the eects of trac volume are more pronounced than the percentage of trucks. Considering either factor alone may result in an inappropriate D_PCE value. 7.2. D_PCE as a function of trac volume and percentage of trucks A mathematical formulation of D_PCE as a function of trac volume and percentage of trucks was sought. Dierent models were ®t to the data and regression analysis technique was used to examine the goodness of ®t (Zhao, 1996). 7.2.1. D_PCE for SUT as a function of trac volume and truck percent To ®nd mathematical relationship between D_PCE, trac volume and percentage of single unit trucks (SUT) a plot of dt /d0 (ratio of average delay for truck case to that of base case) versus percentage of heavy vehicles was made (Zhao, 1996). It showed a linear relationship between dt /d0
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
453
Fig. 3. (a) and (b) are part of this paper, but not included in this document due to their size.
and PH . For a ®xed trac volume, the ratio increased linearly with percentage of trucks. It indicated that vehicular delay increases proportionally with the relative number of heavy vehicles in the trac. The relationship between dt /d0 and PH can be expressed as: dt =d0 a bPH ; where a and b are the parameters.
23
454
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
The regression lines ®tted the observed points very well. The R2 values were 0.99 or higher. The intercepts were very close to one at all volume levels. However, the slope of the lines, b, increased with trac volume. In general, b seemed to be a function of volume. Thus, a linear relationship in the form of b dV was assumed. Thus, dt a dVPH d0 :
24
Substituting dt yielded the following equation: D PCE 1
dt ÿ d0 a dVPH d0 ÿ d0 1 1 1 a ÿ 1 dV : PH PH d0 PH d0
25
Rewriting the above equation, an expression for D_PCE was obtained D PCE 1 a1 1=PH a2 V :
26
The intercept for this model is 1, which is a very desirable feature for the model. This model ®t the data very well and resulted in an R2 value of 0.925. Thus, the D_PCE prediction model for single unit trucks is D PCE maxf1; 1 ÿ 0:232204X 1=PH 0:000763XV g:
27
The above model gives a D_PCE value of one or greater than one. From the practical point of view, a D_PCE value of less than 1 for SUT is not meaningful. The plot of the ®tted curves is shown in Fig. 3. It should be noted that we wanted to have a single model to represent all of the data points. This would result in a less complicated relationship. Of course it is possible to ®t a dierent model for each trac volume, but this would cause discontinuity and make the calculations unnecessarily more complicated. Thirteen other models to predict D_PCE for single unit trucks were examined (Zhao, 1996). However, they did not show signi®cant improvements so they were not recommended. 7.2.2. D_PCE for combination trucks as a function of trac volume and % trucks The procedure for determining D_PCE for SUT was used for combination trucks, as well. The recommended model for D_PCE for combination trucks is: PCE maxf1; 1 ÿ 0:532664X 1=PH 0:00201XV g:
28
The R2 for this model was 0.893. The plot of D_PCE versus percentage of trucks is shown in Fig. 3. At higher volume the predicted values are not as high as the simulation results because at high volumes the intersection was reaching the saturation level. At saturated conditions not all of the additional delay is due to trucks. Part of it is due to heavy vehicles and part of is due to high trac volume. It should be noted that trac volume and percentage of trucks are the two variables in the D_PCE models. As the plot of the models clearly show, trac volume in¯uences PCE much more that percentage of truck. The eect of percentage of truck levels o at higher percentages, as is expected. The truck percentage and trac volume ranges used in this study would cover the typical trac conditions; thus, D_PCE are for such typical cases. For unusual cases, the models may be extrapolated with care.
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
455
7.3. Eects of heavy vehicles on capacity and delay The adverse eects of heavy vehicles on trac ¯ow are usually handled by applying adjustment factors to ideal conditions. Using D_PCE, the heavy vehicle adjustment factor would be computed from the following equation: fHV
1
1 ; P D PCEi ÿ 1 i Hi
P
29
where PHi is the percentage of heavy vehicle type i and D PCEi the D_PCE value for heavy vehicle type i. 7.4. Comparison of HCM and D_PCE adjustment factors Heavy vehicle adjustment factors (fHV ) were computed using D_PCE and HCM values and are compared directly. The HCM uses a constant value for PCE at signalized intersections. The PCE was 1.5 in the 1985 HCM and 2.0 in the 1994 and 1997 HCM. In contrast, D_PCE vary with trac volume and percentage of heavy vehicles. Fig. 4(a) shows the plot of the adjustment factor versus percentage of trucks for SUT. It clearly indicates that the 1994 HCM overestimate the reduction for single unit trucks. The lower the trac volume, the greater the overestimation. Also, the higher the percent trucks the higher the overestimation. For example, when trac had 10% single unit trucks, the 1994 HCM over-adjusted by 7.8%, 7.1%, 6.4%, 5.6% and 4.9% at trac volumes of 200, 300, 400, 500 and 600 vph, respectively. The over-adjustment could be as high as 14% for volume of 200 vph and SUT of 20%. The adjustment factors for combination trucks are shown in Fig. 4(b). When trac volume is 500 vph or under, the 1994 HCM overestimated the capacity reduction. At a 500 vph, the 1994 HCM curve was close to D_PCE. When trac volume was 600 vph and the percentages of trucks were 3% or higher, the 1994 HCM underestimated the reduction. The 1985 HCM overestimated when trac volume was low, 200 vph or under. It overestimated at 300 vph level when the percentages of combination trucks were 4% or lower. At higher trac volume levels, 400 vph or higher, the 1985 HCM underestimated the capacity reduction. 8. Conclusions and recommendations New methodologies for computing PCE factors for signalized intersections were developed. The proposed D_PCE method more accurately considers the adverse eects of heavy vehicles on delay. The ®eld data indicated that D_PCE increased as the trac volume and the percentage of heavy vehicle increased. Mathematical models to estimate D_PCE as a function of the two variables were developed. The D_PCE values were not constant, but varied from 1.00 to 1.37 for single unit trucks, and from 1.00 to 2.18 for combination trucks depending on trac volume and truck percentage. D_PCE were highly correlated with trac volume, but to a lesser degree with percentage of heavy vehicles. The constant PCE of recommended in HCM overestimated the impact of single unit trucks in all cases. For combination trucks, the 1994 and 1997 HCM overestimated the capacity reduction eects of the trucks in most cases, but not all.
456
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
Fig. 4. (a) and (b) are part of this paper, but not included in this document due to their size.
D_PCE computations were limited to single unit trucks and combination trucks. The ®eld data for this study were for through movements at signalized intersections in under-saturated trac conditions. Further studies are needed to cover other trac and geometric conditions as well as other heavy vehicle types. The trac volume in the ®eld data varied from 80 to 467 vph and percentage of trucks varied from 1% to 18%. It is recommended to verify the D_PCE with ®eld data for trac conditions outside of the ranges used in developing the D_PCE prediction models.
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
457
References Benekohal, R.F., Abu-Lebdeh, G., 1994. Variability analysis of trac simulation outputs: practical approach for TRAF-NETSIM. Transportation Research Record 1457, TRB, NRC, pp. 198±207. Benekohal, R.F., Zhao, W., 1996. Eects of heavy vehicles on intersection delay and passenger car equivalents. Moving Forward in a Scaled-Back World, ITE International Conference, Dana Point, California, March. Branston, D., Gipps, P., 1981. Some experience with a multiple linear regression method of estimating parameters of the trac signal departure process. Transportation Research 15, 445±458. Branston, D., van Zuylen, H., 1978. The Estimation of saturation ¯ow, eective green time and passenger car equivalents at trac signals by multiple linear regression. Transportation Research 12, 47±53. Cuddon, A.P., Ogden, K.W., 1992. The eect of heavy vehicles on saturation ¯ows at signalised intersections, Congestion Management, 16th ARRB Conference, Perth, Western Australia. Evans, L., Rothery, R., 1981. In¯uence of vehicles size and performance on intersection saturation ¯ow, Proceedings of the Eighth International Symposium on Transportation and Trac Theory. Federal Highway Administration, 1989. TRAF-NETSIM User's Manual, FHWA, Washington DC. Greenshields, B.D., Shapiro, D., Erickson, E.L., 1947. Trac performance at urban intersection, Bureau of Highway Trac, Technical Report No. 1, Yale University, New Haven, Conn. Keller, E.L., 1984. Passenger car equivalents from network simulation. ASCE Journal of Transportation Engineering 110, July. Kimber, R.M. et al., 1985. Passenger car units in saturation ¯ows: concept, de®nition, derivation. Transportation Research B 19, 39±61. Molina, C.J., 1987. Development of passenger car equivalencies for large trucks at signalized intersections. ITE Journal, November. Sumner, R., Hill, D., Shapiro, S., 1984. Segment passenger car equivalent values for cost allocation on urban arterial roads. Transportation Research A 18, 399±406. Transportation Research Board, 1985. National Research Council, Highway Capacity Manual, Special report 209, Washington DC. Transportation Research Board, 1994. National Research Council, Highway Capacity Manual, Special report 209, 3rd ed., Washington DC. Transportation Research Board, 1997. National Research Council, Highway Capacity Manual, Special report 209, 3rd ed., Washington DC. Webster, F.V., 1958. A controlled experiment on the capacity of junctions with trac signals, Road Research Lab. Res. Note No. RN/3313/FVW. BR587. Webster, F.V., Cobbe, B.M., 1966. Trac Signals. Road Research Technical Paper No. 56, HMSO, London. West, J.E., Thurgood, G.S., 1995. Developing passenger car equivalents for left-turning trucks at compressed diamond interchanges, Presented at the 74th TRB annual meeting, Washington DC. Zhao, W., 1996. Development of a methodology for measuring delay-based passenger car equivalent for heavy vehicles at signalized intersections. PhD Dissertation, Department of Civil Engineering, the University of Illinois, Urbana IL.
www.elsevier.com/locate/tra
Delay-based passenger car equivalents for trucks at signalized intersections Rahim F. Benekohal a
a,*
, Weixiong Zhao
b
University of Illinois at Urbana-Champaign, Newmark Civil Engineering Laboratory, MC-250, 205 North Mathews Avenue, Urbana, IL 61801, USA b Transportation Engineer, Houston, TX, USA Received 8 January 1998; received in revised form 16 March 1998; accepted 23 March 1999
Abstract This paper presents a new methodology for computing passenger car equivalents at signalized intersections that is based on the delay concept. Unlike the commonly used headway-based methods that consider only the excess headway consumed by trucks, the delay-based approach fully considers the additional delay heavy vehicles cause on trac stream. Delay-based passenger car equivalents are not constant, but depend on trac volume, truck type and truck percentage. The ®eld data indicated that the passenger car equivalents increase as the trac volume and the percentage of heavy vehicles increase. The ®eld data were used to calibrate TRAF-NETSIM simulation model that was used to cover a broad range of trac conditions. Mathematical models to estimate the equivalencies were developed. The passenger car equivalent for single unit trucks vary from 1.00 to 1.37, and for combination trucks 1.00±2.18 depending on trac volume and truck percentage. The passenger car equivalents are highly correlated with trac volume and, to some degree, with percentage of heavy vehicles. Although the PCE of 1.5 recommended in the 1985 HCM seems to be more reasonable than the 2.0 recommended in the 1994 and 1997 HCM, both overestimate the impact of single unit trucks. For combination trucks, the 1997 HCM overestimates the capacity reduction eects of the trucks in most cases. Ó 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction The passenger car equivalents (PCE) concept has been used to account for the adverse eects of heavy vehicles on trac operations. Heavy vehicles, due to their size and lower acceleration/ deceleration capabilities, may adversely aect trac ¯ow performance at intersections. PCE
*
Corresponding author. Tel.: +1-217-244-6288; fax: +1-217-333-1924. E-mail address: [email protected] (R.F. Benekohal)
0965-8564/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 5 - 8 5 6 4 ( 9 9 ) 0 0 0 2 6 - 9
438
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
indicate the number of passenger cars that would have the same eects on trac operations as one truck. Stopped heavy vehicles maintain longer headway than cars when crossing intersections; thus causing delay to the vehicles behind. The delay comes from not only the longer headway, but also from the increased headway by vehicles that are behind the heavy vehicle (Molina, 1987). In most cases the only adverse eect considered is the additional time headway the heavy vehicles consume, however, the eects on other variables such as travel time, vehicular delay and queue length have not been adequately accounted for. The headway-based approach suggests using a constant value that is independent of, heavy vehicle type, trac volume and percentage of heavy vehicles. For instance, the 1985 highway capacity manual (HCM) (TRB, 1985) used a constant PCE of 1.5 and the 1994 and 1997 HCM (TRB, 1994, 1997) used a factor of 2.0 for signalized intersections. These PCE are based on the headway concept and do not fully consider the delay heavy vehicles cause at intersections. This paper introduces a new methodology for computing PCE for signalized intersections that is based on delay instead of time headway. The Delay-based PCE, called D_PCE, concept not only uses delay as the main criteria, but also considers trac volume and percentage of heavy vehicles in ®nding the equivalency factors. The D_PCE is simply de®ned as the ratio of delay caused by a heavy vehicle to the delay of a car in an all-passenger car trac stream. The paper presents development of mathematical models to estimate passenger car equivalents as a function of vehicle type, trac volume and percentage of trucks. Field data collected at seven intersections were used to determine the D_PCE. The ®ndings from ®eld data needed to be generalized to other trac conditions. To establish the general relationship between D_PCE, trac volume and percentage of trucks, the TRAF-NETSIM simulation model was calibrated and used. The simulation results were compared to ®eld data to assess the appropriateness and accuracy of the TRAF-NETSIM simulation model for computing D_PCE. Mathematical models were ®t to the data and general relationships were developed. The paper also compares the capacity reduction eects of using D_PCE and the PCE values used in HCM. 2. Past research on PCE for signalized intersections 2.1. Webster's method Webster (1958) performed a controlled ``track'' experiment to compute PCE. Vehicles were classi®ed into two groups: light vehicles and goods vehicles. The goods vehicles included medium and heavy commercial vehicles. The PCE value for a heavy vehicle was determined by grouping data from 12 successive signal cycles. The average number of heavy vehicles per cycle ng Rng =N was plotted against the average number of light vehicles per cycle n1 Rn1 =N . ng and n1 are the number of departing heavy and light vehicles in individual cycles and N is the number of cycles in the set. There was very little scatter in the data and a straight line was drawn through the ng and n1 values. The PCE value was estimated as the reciprocal of the slope of the line. In another study, Webster and Cobbe (1966) recommended using a PCE of 1.75 for heavy truck and 2.25 for buses in computing saturation ¯ow for through movements.
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
439
2.2. Regression method Branston and van Zuylen (1978) and Branston and Gipps (1981) presented a multiple linear regression method to estimate trac parameters, including PCE, at trac signals. The green time given to an approach was divided into consecutive ``®rst'', ``middle'' and ``last'' counting periods. The period covered the buildup to saturation ¯ow rate at the start of green, the period of constant saturation ¯ow and the fall-o of saturation ¯ow during the amber period. The vehicle departures were recorded over a period T, which began and ended at arbitrary instants. Then, the number of passenger cars n1 was regressed on the number of vehicles ni of other classes i to obtain the estimates of the coecients B0 and Bi in the model: X n1 B0 T ÿ Bi ni e; 1 i61
where B0 is the saturation ¯ow rate in passenger cars per unit time, Bi the PCE value of vehicles of class i and e the error term. The recording of the departure times of the vehicles was ``synchronous'' or ``asynchronous''. The former terminated the observation the instant a vehicle departed, but the latter terminated the observations at an arbitrary time. The PCE for the asynchronous case was 1.74 and for the synchronous case it was 1.59. The regression methods were applied to through trac in main stream and excluded the start-up time and transition lost time. 2.3. Headway ratio method The headway ratio (HR) method was pioneered by Greenshields et al. (1947). The method is based on the ratio of the average time headway for the vehicles of interest to the average time headway for passenger cars PCEt
ht ; hc
2
where PCEt is the PCE for vehicles of class t, ht the eective average headway of vehicles of class t, and hc the average headway of passenger cars. The HR method is most commonly used to ®nd PCE at signalized intersections. Although the HR method is simple and straightforward, it does not consider the adverse eects of heavy vehicles on delay. Evans et al. (1981) reported that large trucks are over-represented in queue leader positions. Using the HR method, the PCE for heavy truck (de®ned as a vehicle with more than three axles) in a non-leader position was found to be 2.63. Whereas, the PCE for a heavy truck in the lead position in the queue was 4.05. Molina (1987) proposed a modi®ed headway ratio method that considered the increase in headway for vehicles queued behind the heavy vehicle. The PCE was calculated as: PCEt or
ht DH hc
3
440
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
TTt ÿ TTc ; 4 hc where PCEt is the PCE for a vehicle of type t, ht the headway of vehicle of type t, hc the saturation ¯ow headway of passenger car, DH the total increased headway of the queue caused by the truck, TTt the total discharge time of a truck queue, and TTc the total discharge time of a passenger car queue. Molina found that the position of a truck in the queue did not have a signi®cant eect on the PCE for the two- and three-axle single-unit trucks. However, the position of a vehicle in the queue had a pronounced eect on PCE for ®ve-axle combination trucks. The scope of Molina's study was limited to through-movement trac, and only a single truck in the queue. Other factors, such as trac volume and percentage of trucks were not considered. Molina recommended PCE of 3.7 and 1.7 for heavy and light vehicles, respectively. Although Molina's method considers the incremental headway, it does not fully consider the additional delay the vehicles queued behind the truck experience. The method takes into account the headway increases up to the eighth-positioned vehicle in the queue. If there are vehicles at farther than 8th position, they do not aect the PCE. West and Thurgood (1995) employed Molina's method to generate PCE for various left-turning light and heavy trucks at compressed diamond interchange. From a set of sample data, composite PCE for standard light and heavy trucks are reported as 1.7 and 4.4, respectively. Kimber et al. (1985) examined Webster's, HR, and regression methods for obtaining PCE. It was concluded that the PCE factors depend on the method of derivation. PCEt 1
2.4. HCM methodology The HCM (TRB, 1985, 1994, 1997) applies a heavy vehicle adjustment factor fHV to reduce the ideal saturation ¯ow rate at intersections. The fHV is computed from the following equation: fHV
1 ; 1 PH PCE ÿ 1
5
where PCE is the Passenger car equivalents for heavy vehicles, and PH the percentage of heavy vehicles. For intersections, the 1985 HCM recommended a PCE of 1.5 while the 1994 and 1997 HCM increased the PCE to 2.0. HCM did not provide the reasons for such an increase. It is important to note that for steep or long grades (de®ned as speci®c grade) on highways HCM recommends PCE that depend on the type and percentage of heavy vehicle in trac stream. However, for intersections HCM uses a constant PCE regardless of the type of heavy vehicle and volume of trac. 2.5. Other methods Keller et al. (1984) used a macroscopic trac simulation model (TRANSYT-7) on an urban network to derive PCE for large vehicles as a function of vehicle size, signal timing and trac volume. PCE were estimated as the ratio of the total travel times of heavy vehicles to that of passenger cars when travelling through an urban network
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
PCEt
TTt ; TT0
441
6
where PCEt is the PCE value of the given vehicle type t, TTt the total travel time of vehicle of type t over the network and TT0 the total travel time of the base vehicle over the network. Results for urban arterials are presented for each of the seven vehicle classes, six dierent levels of service and three types of signal settings. As a result, PCE were found ranging from 1.09 for a single unit truck to 1.53 for a tractor-trailer truck under level of service C operation. Sumner et al. (1984) employed the NETSIM trac simulation model to derive passenger car equivalent values between consecutive signalized intersections on urban arterial roads. These values were cumulative over the length of road between intersections and re¯ect the vehicle-hours of road utilization that were added when large commercial vehicles were introduced into the trac stream. The PCE were derived using the product of vehicle number and time spent by those vehicles in the section. The authors found PCE that varied from 1.10 for single unit trucks to 1.45 for 5-axle combination trucks for two-lane streets under level of service D; and from 1.18 for single unit trucks to 1.53 for 5-axle combination trucks for two-lane streets under level of service B. The authors' work was focused on segments and did not isolate the heavy vehicles' eect at signalized intersections. Kimber et al. (1985) proposed that the PCE as used for adjusting the saturation ¯ow rate should not be the equivalent number of passenger cars a heavy vehicle would physically displace. Rather it should be a number that yields the minimum delay for the intersection. The procedure included the following steps: (1) Trac parameters such as the mean arrival rate, mean and standard deviation of headways were chosen. (2) Signal settings were chosen using Webster's method, and a PCE value for heavy vehicles was assumed. (3) A simulation run was made, and the mean vehicle delay was measured using Webster's delay formula. (4) Steps 2 and 3 were repeated using dierent PCE values. (5) The PCE value corresponding to the setting that gave minimum delay was determined by plotting mean delay against PCE value. An example was given for ®nding the PCE from simulation of a two-phase signal. When the mean headway of 2.0 and 4.56 s were assumed for cars and heavy vehicles, the PCE was 2.3. In summary, using a constant PCE indicates that trac volume or percentages of heavy vehicles do not aect PCE. On the other hand, a variable PCE indicates that truck type, volume or position of truck in queue aects PCE. Although some of the above-mentioned studies resulted in non-constant PCE, they did not establish direct relationships among PCE, trac volume and percentage of heavy vehicles. This study will attempt to ®nd such relationships.
3. Data collection and reduction Data were collected on ten approaches of seven intersections in Central Illinois. The ten approaches are labeled A though J. Trac volume varied from 80 to 467 vph per lane and percentage of heavy vehicles varied from 2% to 30%. The sites were selected such that as many ideal features as possible were met. The collected data included headway, delay for individual vehicles in all-passenger trac stream, the number of queued and non-queued vehicles, vehicle type,
442
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
position of heavy vehicles in queue, signal timing information and geometric data. A summary of the data is shown in Table 1. The predominant heavy vehicle types were large single unit trucks and ®ve-axle combination trucks. The headway was measured as the rear axles of the vehicles passed over the stop line. The headway of the ®rst vehicle was de®ned as the elapsed time between the initiation of the green signal and the time the rear wheels of the vehicle passed the stop line. This way of measuring headway was consistent with previous studies (Molina, 1987; TRB, 1994; Cuddon and Ogden, 1992) and ensured that headways were measured for those vehicles in front of the queue that stopped with their front wheels past the stop line. Two computer programs were used in data reduction: one for headway measurement and the other for delay measurement. The programs were designed to extract trac data, including: the time a vehicle joined the queue, the time a vehicle left the queue, position of the vehicle in the queue, total number of vehicles in the queue, vehicle types, and so forth. Delay used in computation of D_PCE can be stopped delay or queue delay. The stopped delay accounts for the time a vehicle is stopped. Queue delay includes stopped delay as well as part of the delay due to acceleration/deceleration. Using stopped delay would underestimate the eects of heavy vehicles because it does not fully represent the poor performance of heavy vehicles. Therefore, queue delays were measured in ®eld analysis. Data were collected from through lanes, except at Locations A, D and H where the through lanes were shared with right turning trac. The performance of shared right/through may be aected when the percentage of the right turning trac is high. Data indicated that the proportion of right turning trac at those three locations were small (3%, 4% and 9%, respectively). Most of the right-turn vehicles at Locations A and H use the wide shoulder to make the right turn. Moreover, no right-turning heavy vehicles were counted at these locations. Thus, the eects, if any, were negligible.
Table 1 Summary of the data collected Location
A
B
C
D
E
F
G
H
I
J
Total
In the stream Time period (h) All vehicles No. of cars No. of trucks % truck
6 1765 1472 293 16.6
6 1175 908 267 22.7
2 444 377 67 15.1
3.3 1442 1376 66 4.6
3.3 1568 1533 35 2.2
1.8 341 332 9 2.6
1.8 328 320 8 2.4
3.6 1402 1361 41 2.9
3.6 984 933 51 5.2
4 299 208 91 30.4
35.4 9748 8820 928 9.5
690 571 119 39 75 5
486 356 130 24 101 5
593 565 28 16 5 5 2
551 537 14 8 2 3 1
248 241 7 4 3
239 231 8 4 4
321 302 19 2 14 3
199 138 61 19 38 4
4158 3717 441 137 272 28 4
In queue All vehicles No. of cars No. of trucks No. of sus No. of CTs No. of PTs No. of buses
322 271 51 13 35 2 1
Notes: SU Single unit truck; CT combination truck; PT passenger car trailer.
504 492 12 8 3 1
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
443
The data from adjacent lanes of the same approach were combined to increase the number of observations. The combined locations were D and E, F and G, and H and I. Thus, a total of seven data sets, namely A, B, C, D/E, F/G, H/I and J, each representing one intersection were used. 4. Introduction of delay-based passenger car equivalent To quantify the eect of heavy vehicles on delay, a new term called delay-based passenger car equivalent (D_PCE) is introduced. The D_PCE is simply de®ned as the ratio of delay caused by a heavy vehicle to the delay of a car in an all-passenger car trac stream. Mathematically, D_PCE for a heavy vehicle type i is expressed as: Ddi ; 7 d0 where D_PCEi is the D_PCE for a heavy vehicle type i, Ddi the additional delay caused by a vehicle of type i and d0 the average vehicle delay when the trac is composed of all-passenger cars. The D_PCE factor gives the number of passenger cars that would replace a heavy vehicle in the trac stream and result in the same amount of delay as the original trac stream. For example, assume a platoon of 10 cars causes 50 s of delay; then, the average delay d0 5 s/veh. However, if a platoon of eight cars and two trucks (still a total of 10 vehicles) causes 70 s of delay, the additional delay caused by each truck is Dd 70 ÿ 50=2 10 s. Then, the D_PCE factor for the truck is computed as 1 10=5 3. Thus, the computed D_PCE are directly related to the delay caused by heavy vehicles. D PCEi 1
4.1. Delay-based PCE at signalized intersections When a trac stream is composed of all passenger cars, it is referred to as ``base case'' and its delay is referred to as ``base delay''. When there is at least one heavy vehicle in the stream, it is referred to as a ``truck case or mixed trac case'' and a queue containing one or more trucks is referred to as a ``truck queue or mixed trac queue''. When total delays for the base and truck cases are known, D_PCE are obtained as Dt ÿ D0 d0 : 8 D PCE 1 Vt The above equation can be written as: dt ÿ d0 ; 9 pH Xd0 where Dt is the total delay for the truck case (s), D0 the total delay for the base case (s), d0 the average delay for the base case (s/veh), dt the average delay for the truck case (s/veh), pH the percentage of trucks V =Vt , V the total trac volume (vph) and Vt the total truck volume (vph). Conceptually, the queuing diagram in Fig. 1 shows that the delay for an all-passenger trac is the area of triangle ABC, AABC . When heavy vehicles appear in the queue, the triangular shape changes to a polygon of ABDEF. The area of polygon BDEFC, ABDEFC , represents the additional delay due to the presence of heavy vehicles. Thus, the average additional delay for each truck would be D PCE 1
444
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
Fig. 1. Are part of this paper, but not included in this document due to their size.
P
Ddi ABDEFC ; 10 nt nt P where Dd is the average additional delay due to heavy vehicles, Ddi the total additional delay caused by heavy vehicles, ABDEFC the area of BDEFC, which is the additional delay caused by heavy vehicles, nt the number of heavy vehicles in that cycle.For the base case, when the arrival rate, saturation ¯ow rate and signal information are known, the average queue length may be computed as Dd
nq R
qs ; sÿq
11
where q is the arrival rate (veh/s), s the saturation ¯ow rate (veh/s), nq the average number of passenger cars in the queue and R the eective red time for the study approach (s). For the base case, the total queue delay in one cycle is 1 1 qs : D0 AABC Rnq R2 2 2 sÿq
12
The average delay per passenger car (base delay) is computed as d0
D0 1 R2 qs ; n0 2 n0 s ÿ q
13
where d0 is the average delay for passenger cars, n0 the total number of passenger cars in the cycle.The additional delay, the area of BDEFC, can be viewed as the summation of delays to individual vehicles, dj , which are behind the heavy vehicle
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
X
Ddi
m X dj ;
445
14
jp
where dj is the additional delay for a vehicle at the jth position in the mixed queue, p the position of the heavy vehicle in the mixed queue and m the position of the last vehicle in the queue.The value of dj is determined by: dj TTtj ÿ TT0j ;
15
TTtj
is the travel time of the jth vehicle in the mixed queue and where vehicle in all passenger car queues.
TT0j
the travel time of the jth
5. Computing D_PCE from ®eld data D_PCE were computed from ®eld data for single unit trucks and combination trucks. Queue delay from ®eld data was used because it was planned to use TRAF-NETSIM queue delay for comparison. Queue delay includes not only the stopped time, but also some of the delay due to accelerating and decelerating. To compute average base delay from ®eld data, queue delays for individual cars were averaged. A summary of base delays obtained from ®eld measurement is given in Table 2. The base delays were also estimated using the following equations: 1 d0q R; 2 d0
16
qs R2 ; 2n0 s ÿ q
17
where d0q is the estimated base delays for queued cars (s), d0 the estimated delay for all cars in the stream (s), R the red time (s), q the arrival rate (veh/s), s the saturation ¯ow rate (veh/s), n0 the qXC and C the cycle length (s) The estimated and ®eld measured base delays were very close to each other. Paired-t tests indicated that there were no signi®cant dierences between the measured base delays and the
Table 2 Comparison of measured and estimated base delays (s/veh) Data set A B C D/E F/G H/I J
Measured in ®eld
Estimated
Queued cars
All cars
Queued cars
All cars
15.50 15.85 32.95 18.34 34.38 12.08 19.29
7.50 8.06 23.28 7.78 24.72 6.53 17.41
14.5 15.5 31.0 16.5 35.0 11.5 24.5
7.30 7.97 22.79 8.05 25.22 6.69 16.95
446
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
estimated base delays. The D_PCE values when a single unit truck was in the lead position showed that the D_PCE values are highly correlated with the number of vehicles queued behind the truck (Zhao, 1996). The queue length is an indication of trac volume, when other signal timing variables are kept constant. When two consecutive trucks were present in the same queue, the total additional delay caused by the two trucks was computed and its average was used in computing the D_PCE. Average D_PCE were computed to re¯ect the eects of trucks at dierent queue positions. Although the queued trucks are the main sources of additional delays, all of the trucks in the trac stream should to be considered in order to simplify the computation of D_PCE. In normal trac conditions, some trucks are stopped in queue and some are not. Total additional delay caused by the queued trucks is averaged over the entire truck volume to re¯ect the average D_PCE. It should be noted that when the entire truck volume is used (instead of the volume of queued trucks), the base delay for the entire car volume is used (instead of base delay for queued cars). A summary of the D_PCE computed using the ®eld data is presented in Table 3. Further discussion of the ®eld data analysis is given elsewhere (Zhao, 1996; Benekohal and Zhao, 1996). Data Set J is unique because of low volume and high percentage of heavy vehicles (10.7% single unit and 18.1% combination trucks). In most signal cycles, the queue dissipated even before the saturation headway was reached. As a result, the D_PCE factor for combination trucks was the lowest, 1.19, even though it has the highest percentage of combination trucks. Similarly, the D_PCE for single unit trucks, 1.07, were also the lowest of the data sets. The plots of D_PCE at dierent volumes and truck percentages are shown in Fig. 2. The ®gure shows that the D_PCE change with trac volume and percentage of heavy vehicles. For example, Data Set F/G and H/I both had 1.2% single unit trucks, but trac volumes were 256 and 374 vph, respectively. The D_PCE for single unit trucks in Data Set F/G was 1.11 compared to 1.20 for Data Set H/I. On the other hand, Data Sets B and F/G had practically the same trac volumes, but percentages of single unit trucks were 4.6 and 1.2 and percentages of combination trucks were 16.1 and 1.2. The D_PCE factor for single unit trucks was 1.26 for Data Set B compared to 1.11 for data set F/G. Similarly, the D_PCE factor for combination trucks was 1.49 for data set B and 1.38 for data set F/G. A general trend can then be described qualitatively: at the same trac volume level, D_PCE increases with percentage of heavy vehicles; similarly, D_PCE increases with trac volume when percentage of heavy vehicles remains the same. For under saturated trac conditions, the average D_PCE from the ®eld data are found to be in the range of 1.07±1.47 for single unit trucks, and 1.19±1.81 for the combination trucks.
Table 3 D_PCE from ®eld data averaged over all trucks Data set Trac volume (vph) Single unit truck Combination truck
% trucks D_PCE % trucks D_PCE
A
B
C
D/E
F/G
H/I
J
302 5.4 1.30 10.3 1.55
246 4.6 1.26 16.1 1.49
286 4.5 1.27 9.7 1.50
467 1.9 1.47 0.5 1.81
256 1.2 1.11 1.2 1.38
374 1.2 1.20 1.0 1.62
80 10.7 1.07 18.1 1.19
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
447
Fig. 2. Are part of this paper, but not included in this document due to their size.
6. Generalizing results to other trac conditions The ®ndings from ®eld data need to be generalized to other trac conditions. TRAF-NETSIM (FHWA, 1989) simulation model was calibrated to meet the needs of the D_PCE computation. It should be noted that, without such calibration and validation TRAF-NETSIM could not be used for this purpose. TRAF-NETSIM is a microscopic simulation model and has been used to simulate trac performance on a variety of trac conditions (Benekohal and Abu-Lebdeh, 1994). The TRAF-NETSIM output does not provide detailed information for individual vehicles, but does provide enough information for computing average D_PCE for trucks. First, the simulation
448
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
results were compared to ®eld data to assess the appropriateness and accuracy of the TRAFNETSIM simulation model for computing D_PCE. Field trac and geometric data were used as input to the simulation models. Then, delay estimations and D_PCE values from simulation were compared with those from ®eld data. Mathematical models were ®t to the data and general relationships were provided. 6.1. Comparison of simulation delay to ®eld data for cars In this study queue delays from ®eld were compared to queue delays from TRAF-NETSIM. Queue delay in TRAF-NETSIM is de®ned as time spent when the acceleration of a vehicle is less than 2 ft/s2 and its speed is less than 9 ft/s. Queue delays measured in the ®eld were consistent with this de®nition. Queue delay estimations from the simulation runs were compared to the ®eld data for the base case (all cars). The results are summarized in Table 4. To examine the dierences, t-tests were used with 95% con®dence level. For all-car trac, there were no signi®cant dierences between the queue delays from TRAF-NETSIM and ®eld data a 0:05. 6.2. Calibration of TRAF-NETSIM for mixed trac Trac conditions on the study sites were simulated using TRAF-NETSIM with the default values for trucks. TRAF-NETSIM uses only one heavy vehicle type and provides default values such as length and headway factors for it. When the default values were used, the D_PCE computed from the simulation results were signi®cantly lower than those from ®eld data (Zhao, 1996). This was true for both single unit and combination trucks on all study sites. Thus, calibration of the truck factors in TRAF-NETSIM model was necessary. Default values for vehicle length, maximum acceleration rate and headway was calibrated to provide accurate results. A vehicle length 30 ft for SUT and 55 ft for combination was used instead of the 35 ft default. The maximum acceleration rate of 3.5 and 2.1 was used for SUT and combination trucks instead of 3.2 mph/s/s. Headway factors were calibrated based on the ®eld data collected in this study. They re¯ect the passenger car and truck headways measured in the ®eld.
Table 4 Comparison of TRAF-NETSIM queue delay to ®eld data for cars (base case) Data A B C D/E F/G H/I J a
Simulation (®ve replications) 1
2
3
4
5
Avg
6.9 7.5 21.8 7.6 23.8 5.2 16.1
7.3 7.5 23.3 7.3 24.3 6.9 16.3
7.6 7.5 23.3 8.7 23.2 6.5 15.2
8 6.8 22.7 7.9 23.5 6.9 18.0
6.5 8.2 23.3 7.5 25.3 6.5 16.8
7.26 7.5 22.88 7.8 24.02 6.4 16.48
ta0:05 4 2:776.
a
Field
Di.
t-value
7.50 8.06 23.28 7.78 24.72 6.53 17.41
0.24 ÿ0.56 ÿ0.4 0.02 ÿ0.7 ÿ0.13 ÿ0.93
0.92 ÿ2.53 ÿ1.36 0.08 ÿ1.9 ÿ0.42 ÿ2.02
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
449
A headway factor for each vehicle type was calculated using the following equation: P10 PK j1 k1 nijk X hf ijk hfi P10 PK j1 k1 nijk
18
and t
hf ijk
hijk c
hjk
;
19
where hfi is the headway factor for vehicle type i, hf ijk the average headway factor for vehicle type i at site j in queue position k, nijk the number of observations of headway of heavy vehicle type i at t site j in queue position k, hijk the mean headway of heavy vehicle type i at site j in queue position c k, hjk the mean headway of a passenger car at site j in queue position k, i the vehicle type, 1 for single unit truck, 2 for combination truck, j the sites ®eld data were collected, j 1; 2; . . . ; 10, k the queue position, k 1; 2; . . . ; K and K the maximum number of vehicles in the queue (observed K was 9). The headway factors computed from ®eld data for single unit and combination trucks are shown in Table 5. The headway factor was 1.64 for SUT and 2.58 for combination trucks. Thus, in the calibrated TRAF-NETSIM, headway factors of 160% and 260% were used for single unit and combination trucks, respectively. 6.3. Comparison of calibrated TRAF-NETSIM delays to ®eld data for mixed trac The comparison of queue delays when trucks were present was a little bit more complicated because TRAF-NETSIM does not directly report the additional delay a heavy vehicle causes. Queue delays from TRAF-NETSIM for mixed trac (truck case) are listed in Table 6. These values were used in the following equation to calculate the additional delay caused by a heavy vehicle: Dd
dt ÿ d0 ; PH
20
where Dd is the additional delay caused by a truck, d0 the average delay for base case (s/veh), dt the average delay for truck case (s/veh), pH the percentage of trucks. For each data set, ®ve replications of TRAF-NETSIM were made. Then the average values for Dd were computed. Table 6 shows that the additional queue delay estimations from the Table 5 Headway factor computation from ®eld data k Single unit trucks Combination trucks
hf1k n1k hf2k n2k
1
2
3
4
5
6
7
8
9
All
1.53 40 2.50 86
1.61 27 2.53 58
1.67 19 2.57 49
1.77 12 2.65 24
1.80 10 2.69 18
1.68 6 2.72 6
1.72 7 ÿ ÿ
1.67 1 2.73 4
1.67 2 ÿ ÿ
1.64 124 2.58 245
450
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
Table 6 Comparison TRAF-NETSIM additional queue delay to ®eld data for mix trac Data
Simulation (®ve replications) 1
2
3
SUT
A B C D/E F/G H/I J
0 1.6 4 0 10 0 0
2 2 6 5 0 0 0.9
2 0 8 5 0 3 0.9
CT
A B C D/E F/G H/I J
3 2.5 8 10 10 10 0.56
4 3.75 9 4 0 0 1.11
5 4.38 12 6 10 0 0
a
Field 4
5 2 2 2 0 0 3 0
7 5.63 8 7 0 10 2.22
Di.
t-value
Avg
2 2 2 5 10 1 0.9
1.6 1.6 4.4 3.0 4.0 1.4 0.55
2.27 2.08 6.22 3.68 2.75 1.33 1.15
ÿ0.67 ÿ0.48 ÿ1.82 0.68 1.25 0.07 ÿ0.60
ÿ1.67 ÿ1.2 ÿ1.56 ÿ0.56 0.51 0.10 ÿ2.71
3 3.13 14 5 20 0 2.22
4.4 3.88 10.2 6.4 8.0 4.0 1.22
4.1 3.96 11.52 6.32 9.31 4.04 3.28
0.3 ÿ0.09 ÿ1.32 0.08 ÿ1.31 ÿ0.04 ÿ2.06
0.40 ÿ0.16 ÿ1.1 0.08 ÿ0.35 ÿ0.02 ÿ4.63 a
Indicates statistically dierent with ®eld measurement.
simulation were not signi®cantly dierent than ®eld data. The only exception was the queue delay estimation for the combination truck case for data set J. The simulation slightly underestimated the additional delay in this case. One should note that the queue delays in TRAFNETSIM are reported to 1/10 of a second accuracy. Thus, the additional delays due to trucks are accurate at this level. 6.4. Computing D_PCE from simulation results To compute D_PCE for a given heavy vehicle type, two sets of TRAF-NETSIM runs were made: one for base case, in which trac was composed of all-passenger cars, and the other for the truck case, in which trac stream had some trucks. To create identical trac streams and to reduce the variance of the simulation output, the common random number (CRN) technique was employed. Application of CRN technique to TRAF-NETSIM for paired simulation runs was discussed by Benekohal and Abu-Lebdeh (1994). Simulation time was 3600 s and ®ve replications were for each set. First, simulation runs were made when trac was composed of all passenger cars to obtain the base delays (D0 and d0 ). Then, simulation runs were made for mixed trac (truck case) with and delays for the truck case (Dt and dt ) were obtained. Then, D_PCE were computed as: D PCE 1
Ddt Dt ÿ D0 =Vt 1 ; d0 d0
21
D PCE 1
dt ÿ d0 ; pH Xd0
22
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
451
where Dt is the total delay for the truck case (s), D0 the total delay for the base case (s), dt the average delay for the truck case (s/veh), d0 the average delay for the base case (base delay) (s/veh), V the total trac volume (vph), Vt the total truck volume (vph) and pH the percentage of trucks. It should be noted that the number of vehicles processed in the two simulation runs should be the same in order for Eqs. (21) and (22) to be valid. Each set of simulation runs was checked to be sure that the number of vehicles processed was equal. When the numbers were not equal, the runs were not used. 6.5. Comparison of D_PCE from queue delay and overall delay D_PCE can be computed using queue delays or overall delays. Queue delays take into account the stopped delay and most part of acceleration/deceleration delay. The overall delays include all of the delay due to traveling at a lower speed than the desired speed. Let D_PCEod denote D_PCE computed using overall delays, and D_PCEqd denote D_PCE computed using queue delays. Both D_PCEod and D_PCEqd were not signi®cantly dierent than D_PCE from ®eld data, see Table 7. The only exception was combination trucks in data set J. This data set had very low volume (80 vph) and very high truck percent (30%), thus simulation delays were expected to be lower than ®eld data. This agreement is because the dierence between overall delay and queue delay for trucks are similar to those for passenger cars. Since the D_PCE values computed based on queue delay and overall delay were similar, it was decided to compute D_PCE based on the overall delay from the simulation model. Unless speci®ed, D_PCE computed from simulation refers to D_PCEod . 7. Prediction models for D_PCE Trac-related factors that aect vehicular delay at signalized intersections might also in¯uence D_PCE values. These factors may include trac volume, truck percentage and signal timing parameters (cycle length and green splits). Table 7 Comparisons of D_PCE
a
Data Single unit trucks set D_PCE D_PCEod Dierence ®eld (a) (b) (b) ÿ (a) t-value
D_PCEqd Dierence (c) (c) ÿ (a) t-value
D_PCE D_PCEod Dierence D_PCEqd Dierence ®eld (d) (e) (f) (e) ÿ (d) t-value (f) ÿ (d) t-value
A
1.30
1.27
1.22
1.55
1.60
B
1.26
1.20
1.49
1.53
C
1.27
1.20
1.50
1.45
D/E
1.47
1.44
1.81
1.80
F/G
1.11
1.15
1.38
1.37
H/I
1.20
1.25
1.62
1.60
J
1.07
1.04
1.19
1.07
a b
ÿ0.03
ÿ0.62
ÿ0.06
ÿ0.88
ÿ0.07
ÿ1.29
ÿ0.03
ÿ2.45
0.04
0.58
0.05
0.61
ÿ0.02
ÿ1.52
Combination trucks
1.21 1.19 1.40 1.16 1.21 1.03
ta0:05 4 2:776. Indicates it is statistically signi®cant.
ÿ0.08
ÿ1.51
ÿ0.04
ÿ0.83
ÿ0.08
ÿ1.52
ÿ0.08
ÿ0.48
0.05
0.52
0.01
0.06
ÿ0.03
ÿ2.31
0.05
0.65
0.04
0.39
ÿ0.04
ÿ1.18
ÿ0.01
ÿ0.08
ÿ0.01
ÿ0.07
ÿ0.02
ÿ0.20
ÿ0.12
ÿ3.31
1.60 1.53 1.44 1.82 1.33 1.67 b
1.07
0.05
0.61
0.03
0.39
ÿ0.05
ÿ1.04
0.01
0.07
ÿ0.05
ÿ0.33
0.06
0.14
ÿ0.12
ÿ4.62
b
452
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
7.1. Factors aecting delay-based PCE Two factors that had major impact on D_PCE were trac volume and percentage of heavy vehicles. Eects of green split and cycle length were examined using simulation results and twoway analysis of variance tests. It was found that signal timing had no signi®cant eect on D_PCE (Zhao, 1996). They did not have signi®cant eects mainly because D_PCE are computed based on the ratio of delays. This conclusion supports, although a dierent approach is used, the ®ndings by Cuddon and Ogden (1992) that signal timing parameters have no signi®cant correlation with the headway-based PCE. In order to obtain D_PCE values for various trac volumes and percentages of trucks, the calibrated TRAF-NETSIM was used for pre-timed signal. The simulated intersections had through-lanes only and were operating in under-saturated conditions. An equal volume of trac was assigned to the opposite approach, but for crossing streets the volumes were lower. Trac volume per lane varied from 200 to 600 vph, and percentage of trucks varied from 1% to 20%. These conditions cover most of the typical under-saturated trac conditions. D_PCE were computed for each simulation run. Then averages of ®ve replication runs were computed. The plots of the average computed D_PCE are shown in Fig. 3. For combination trucks, with trac volume at 600 vph and percentage of trucks at 18% or higher, saturation conditions were reached. At a given trac volume, the D_PCE for single unit trucks rapidly increased at low truck percentages (4±8%), but ¯attened at higher percentages. Fig. 3 clearly show a non-linear relationship between D_PCE and percentage of trucks. PCE for combination trucks showed a similar trend. From the plot it is clear that D_PCE for trucks is reaching a constant value at higher percentages. The trend indicates that the combined adverse eect of X trucks is less than the sum of adverse eects of X individual trucks. This is a logical phenomenon because more than one truck may be in the same queue. D_PCE increased rapidly when saturation levels were reached (about 600 vph and over 18% trucks in our case). At saturation levels, it is not possible to isolate the eects of trucks from the eects of congestion itself. For a given percentage of trucks, D_PCE increased with trac volume, see Fig. 3. The increase seemed to be linear for single unit trucks than combination trucks. This indicates that the combination trucks had much more adverse eects than the single unit trucks. The above discussions indicate that both trac volume and percentage of trucks aect D_PCE values and the eects of trac volume are more pronounced than the percentage of trucks. Considering either factor alone may result in an inappropriate D_PCE value. 7.2. D_PCE as a function of trac volume and percentage of trucks A mathematical formulation of D_PCE as a function of trac volume and percentage of trucks was sought. Dierent models were ®t to the data and regression analysis technique was used to examine the goodness of ®t (Zhao, 1996). 7.2.1. D_PCE for SUT as a function of trac volume and truck percent To ®nd mathematical relationship between D_PCE, trac volume and percentage of single unit trucks (SUT) a plot of dt /d0 (ratio of average delay for truck case to that of base case) versus percentage of heavy vehicles was made (Zhao, 1996). It showed a linear relationship between dt /d0
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
453
Fig. 3. (a) and (b) are part of this paper, but not included in this document due to their size.
and PH . For a ®xed trac volume, the ratio increased linearly with percentage of trucks. It indicated that vehicular delay increases proportionally with the relative number of heavy vehicles in the trac. The relationship between dt /d0 and PH can be expressed as: dt =d0 a bPH ; where a and b are the parameters.
23
454
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
The regression lines ®tted the observed points very well. The R2 values were 0.99 or higher. The intercepts were very close to one at all volume levels. However, the slope of the lines, b, increased with trac volume. In general, b seemed to be a function of volume. Thus, a linear relationship in the form of b dV was assumed. Thus, dt a dVPH d0 :
24
Substituting dt yielded the following equation: D PCE 1
dt ÿ d0 a dVPH d0 ÿ d0 1 1 1 a ÿ 1 dV : PH PH d0 PH d0
25
Rewriting the above equation, an expression for D_PCE was obtained D PCE 1 a1 1=PH a2 V :
26
The intercept for this model is 1, which is a very desirable feature for the model. This model ®t the data very well and resulted in an R2 value of 0.925. Thus, the D_PCE prediction model for single unit trucks is D PCE maxf1; 1 ÿ 0:232204X 1=PH 0:000763XV g:
27
The above model gives a D_PCE value of one or greater than one. From the practical point of view, a D_PCE value of less than 1 for SUT is not meaningful. The plot of the ®tted curves is shown in Fig. 3. It should be noted that we wanted to have a single model to represent all of the data points. This would result in a less complicated relationship. Of course it is possible to ®t a dierent model for each trac volume, but this would cause discontinuity and make the calculations unnecessarily more complicated. Thirteen other models to predict D_PCE for single unit trucks were examined (Zhao, 1996). However, they did not show signi®cant improvements so they were not recommended. 7.2.2. D_PCE for combination trucks as a function of trac volume and % trucks The procedure for determining D_PCE for SUT was used for combination trucks, as well. The recommended model for D_PCE for combination trucks is: PCE maxf1; 1 ÿ 0:532664X 1=PH 0:00201XV g:
28
The R2 for this model was 0.893. The plot of D_PCE versus percentage of trucks is shown in Fig. 3. At higher volume the predicted values are not as high as the simulation results because at high volumes the intersection was reaching the saturation level. At saturated conditions not all of the additional delay is due to trucks. Part of it is due to heavy vehicles and part of is due to high trac volume. It should be noted that trac volume and percentage of trucks are the two variables in the D_PCE models. As the plot of the models clearly show, trac volume in¯uences PCE much more that percentage of truck. The eect of percentage of truck levels o at higher percentages, as is expected. The truck percentage and trac volume ranges used in this study would cover the typical trac conditions; thus, D_PCE are for such typical cases. For unusual cases, the models may be extrapolated with care.
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
455
7.3. Eects of heavy vehicles on capacity and delay The adverse eects of heavy vehicles on trac ¯ow are usually handled by applying adjustment factors to ideal conditions. Using D_PCE, the heavy vehicle adjustment factor would be computed from the following equation: fHV
1
1 ; P D PCEi ÿ 1 i Hi
P
29
where PHi is the percentage of heavy vehicle type i and D PCEi the D_PCE value for heavy vehicle type i. 7.4. Comparison of HCM and D_PCE adjustment factors Heavy vehicle adjustment factors (fHV ) were computed using D_PCE and HCM values and are compared directly. The HCM uses a constant value for PCE at signalized intersections. The PCE was 1.5 in the 1985 HCM and 2.0 in the 1994 and 1997 HCM. In contrast, D_PCE vary with trac volume and percentage of heavy vehicles. Fig. 4(a) shows the plot of the adjustment factor versus percentage of trucks for SUT. It clearly indicates that the 1994 HCM overestimate the reduction for single unit trucks. The lower the trac volume, the greater the overestimation. Also, the higher the percent trucks the higher the overestimation. For example, when trac had 10% single unit trucks, the 1994 HCM over-adjusted by 7.8%, 7.1%, 6.4%, 5.6% and 4.9% at trac volumes of 200, 300, 400, 500 and 600 vph, respectively. The over-adjustment could be as high as 14% for volume of 200 vph and SUT of 20%. The adjustment factors for combination trucks are shown in Fig. 4(b). When trac volume is 500 vph or under, the 1994 HCM overestimated the capacity reduction. At a 500 vph, the 1994 HCM curve was close to D_PCE. When trac volume was 600 vph and the percentages of trucks were 3% or higher, the 1994 HCM underestimated the reduction. The 1985 HCM overestimated when trac volume was low, 200 vph or under. It overestimated at 300 vph level when the percentages of combination trucks were 4% or lower. At higher trac volume levels, 400 vph or higher, the 1985 HCM underestimated the capacity reduction. 8. Conclusions and recommendations New methodologies for computing PCE factors for signalized intersections were developed. The proposed D_PCE method more accurately considers the adverse eects of heavy vehicles on delay. The ®eld data indicated that D_PCE increased as the trac volume and the percentage of heavy vehicle increased. Mathematical models to estimate D_PCE as a function of the two variables were developed. The D_PCE values were not constant, but varied from 1.00 to 1.37 for single unit trucks, and from 1.00 to 2.18 for combination trucks depending on trac volume and truck percentage. D_PCE were highly correlated with trac volume, but to a lesser degree with percentage of heavy vehicles. The constant PCE of recommended in HCM overestimated the impact of single unit trucks in all cases. For combination trucks, the 1994 and 1997 HCM overestimated the capacity reduction eects of the trucks in most cases, but not all.
456
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
Fig. 4. (a) and (b) are part of this paper, but not included in this document due to their size.
D_PCE computations were limited to single unit trucks and combination trucks. The ®eld data for this study were for through movements at signalized intersections in under-saturated trac conditions. Further studies are needed to cover other trac and geometric conditions as well as other heavy vehicle types. The trac volume in the ®eld data varied from 80 to 467 vph and percentage of trucks varied from 1% to 18%. It is recommended to verify the D_PCE with ®eld data for trac conditions outside of the ranges used in developing the D_PCE prediction models.
R.F. Benekohal, W. Zhao / Transportation Research Part A 34 (2000) 437±457
457
References Benekohal, R.F., Abu-Lebdeh, G., 1994. Variability analysis of trac simulation outputs: practical approach for TRAF-NETSIM. Transportation Research Record 1457, TRB, NRC, pp. 198±207. Benekohal, R.F., Zhao, W., 1996. Eects of heavy vehicles on intersection delay and passenger car equivalents. Moving Forward in a Scaled-Back World, ITE International Conference, Dana Point, California, March. Branston, D., Gipps, P., 1981. Some experience with a multiple linear regression method of estimating parameters of the trac signal departure process. Transportation Research 15, 445±458. Branston, D., van Zuylen, H., 1978. The Estimation of saturation ¯ow, eective green time and passenger car equivalents at trac signals by multiple linear regression. Transportation Research 12, 47±53. Cuddon, A.P., Ogden, K.W., 1992. The eect of heavy vehicles on saturation ¯ows at signalised intersections, Congestion Management, 16th ARRB Conference, Perth, Western Australia. Evans, L., Rothery, R., 1981. In¯uence of vehicles size and performance on intersection saturation ¯ow, Proceedings of the Eighth International Symposium on Transportation and Trac Theory. Federal Highway Administration, 1989. TRAF-NETSIM User's Manual, FHWA, Washington DC. Greenshields, B.D., Shapiro, D., Erickson, E.L., 1947. Trac performance at urban intersection, Bureau of Highway Trac, Technical Report No. 1, Yale University, New Haven, Conn. Keller, E.L., 1984. Passenger car equivalents from network simulation. ASCE Journal of Transportation Engineering 110, July. Kimber, R.M. et al., 1985. Passenger car units in saturation ¯ows: concept, de®nition, derivation. Transportation Research B 19, 39±61. Molina, C.J., 1987. Development of passenger car equivalencies for large trucks at signalized intersections. ITE Journal, November. Sumner, R., Hill, D., Shapiro, S., 1984. Segment passenger car equivalent values for cost allocation on urban arterial roads. Transportation Research A 18, 399±406. Transportation Research Board, 1985. National Research Council, Highway Capacity Manual, Special report 209, Washington DC. Transportation Research Board, 1994. National Research Council, Highway Capacity Manual, Special report 209, 3rd ed., Washington DC. Transportation Research Board, 1997. National Research Council, Highway Capacity Manual, Special report 209, 3rd ed., Washington DC. Webster, F.V., 1958. A controlled experiment on the capacity of junctions with trac signals, Road Research Lab. Res. Note No. RN/3313/FVW. BR587. Webster, F.V., Cobbe, B.M., 1966. Trac Signals. Road Research Technical Paper No. 56, HMSO, London. West, J.E., Thurgood, G.S., 1995. Developing passenger car equivalents for left-turning trucks at compressed diamond interchanges, Presented at the 74th TRB annual meeting, Washington DC. Zhao, W., 1996. Development of a methodology for measuring delay-based passenger car equivalent for heavy vehicles at signalized intersections. PhD Dissertation, Department of Civil Engineering, the University of Illinois, Urbana IL.