Fig. 5.3 Schematic representation of the total travel time during the interval (tn-1, tn) under congested-flows
(Source: Nam and Drew 1999)
Nam and Drew stipulated the traffic condition as congested when the value of the variable m(tn) is either zero or negative. Under such conditions, none of the vehicles that enter the link during the interval tn-1 to tn exit the link during the same interval. Then, the travel time is calculated based on all the vehicles that enter during the interval under consideration,and thevalue corresponding to m (tn) for congested condition is calculated as
1 1 1
''( ) ( , ) ( , )
Thus, under the congested-flow conditions, the travel time is calculated as the shaded area in Figure 5.3, and this is equal to
(
1)
1 '' ' '' ( ) ( ) 2 t −tn− + t −tn m tn
, (5.17) where,t′ = expected time of departure from the link of the last vehicle that enters the link during the interval (tn-1 , tn), and
t′′ = expected time of departure from the link of the first vehicle that enters the link during the same time interval.
After interpolating the values of t′and t′′, and substituting them in Equation 5.17, the travel time T(tn) and is calculated as shown in Equation 5.18:
( )
( )
(
1 1)
( ) 2.
k t k t ∆x n- n T tn q xi ,tn + = +
(5.18)
After the calculation of travel time, exponential averaging was applied to smooth the dynamic travel time estimates. This numerical technique favored the most recent estimate by assigning weight factors. Thus, the instantaneous travel time estimate at the next time interval (tn-1 , tn) was calculated as
1 1
( ) ( ) ( ) ( )
T tn =T tn− +αT tn −T tn− ≤Tf
,
(5.19) where,α = exponential weighing factor
t
T
∆
, ∆t is the aggregation interval, and T is the smoothing time interval,
T(tn) = instantaneous travel time estimates by time tn,
T
(tn-1) = exponentially smoothed travel time estimates by time tn-1, and Tf = free-flow link travel time.Toronto, Canada. The ILD data were reported every 30 seconds and was accumulated to 2- minute intervals before analysis. Data from all the lanes for a given ILD were aggregated and were treated as single lane data. At the end of the 4-hour study period, the data showed a 3 percent difference in traffic counts, which indicated a violation of the conservation of vehicles principle. A volume adjustment factor was determined for each 30-minute period, and the measurements were adjusted to correct this discrepancy in the data. The models were then validated using the corrected data.
The main criticisms of the above model were the necessity to know the number of vehicles in the link at the start of data collection and the high sensitivity of the travel time estimates with respect to the errors in the measurements from the detectors (Son 1996; Petty et al. 1998; Dhulipala 2002; Oh et al. 2003). Another drawback of this method was related to the calculation of densities from cumulative flow measurements. Here, the accuracy of the estimation solely depends on the accuracy of the flow values. This would be an efficient method to find out the true density in a section, if the ILDs were working perfectly and an automatic initialization process could be employed frequently, on the order of every few minutes (May 1990). However, in reality the detectors may not be working perfectly. Many researchers, such as May (1990), have raised this concern pointing out that the calculated density from input-output counts getting frequently affected even with a low level of detector errors. Petty et al. (1998) expressed their concern in the following manner: “Loop detectors are notorious for over- and under-counting vehicles. Hence the cumulative flow lines that Drew and Nam were relying upon can
systematically drift over time (indeed they might even cross).” Issues related to the use of cumulative flow curves for the calculation of density was raised by Oh et al. (2003): “A simple subtraction of cumulative arrivals at the two detectors would yield the number of vehicles in between the detectors and thus the density of the section between them. If such a density is known at any point in time, we can make a very good estimate of the true section travel time using simple fluid model relations for traffic. The reality however is different, in that the detectors in the field are not perfect and each detector has its own tendency to undercount vehicles and thus the cumulative arrival counts at two detectors (with their own “cumulative count drift”) cannot be used to find the density at any time.” Son (1996) pointed out another
drawback of this model as the calculation of travel time under normal-flow considering only the vehicles that entered and exited in the same time interval. To address all these concerns, a number of modifications are proposed in this dissertation and are detailed in the following section.