Office of Planning, Environment, & Realty (HEP)

The signalized intersection delay specification, described in Appendix A, was implemented in a travel forecasting model (a specially modified version of QRS II) and tested. An attempt was made to extract the implied delay/volume relationship while letting the model determine the phasing and green times. Since green times are no longer exogenous variables, the possibility exists for a simpler means of calculating delay.

Figure 3 shows three delay/volume curves for the same intersection. The curves show the delay on all approaches (subject, opposing, and conflicting) when the volume on just one subject approach is varied. This intersection has a high percentage of turns (25% lefts and 25% rights at all approaches). It is readily seen that the delay on any approach depends on the volumes for the other approaches. For instance, the delay for both the subject and conflicting approaches are nearly the same, even though the conflicting volume was held fixed at 800 vph. The delay on the opposing approach is more complex - first rising gradually, peaking at 2400 vph on the subject approach, and then declining. The reason for the declining delay is the increasingly ample green time available to handle the 800 vph on the opposing approach.

**Figure 3: Delay on All Approaches of a Signalized Intersection as a Function of Volume on a Single Approach(25% Right Turns, 25% Left Turns, 800 VPH at Opposing and Conflicting Approaches, No Exclusive Lanes, 3600 VPH Ideal Saturation Flow Rate, 20 mph speed)**

Figure 4 is similar to Figure 3, except that there are no turning vehicles. The subject and conflicting delay curves have similar shapes, but do not coincide. It is again seen that the delay on the opposing approach declines, in this case after 800 vph on the subject approach. Figure 4 also shows that the delay on the subject approach is not necessarily monotonic (i.e., steadily increasing with volume). The delay rises to a local maximum at 800 vph (the fixed volume on the conflicting and opposing approaches), then declines to a local minimum at 1600 vph, before increasing again.

The delay curves of Figures 3 and 4 are very consistent with the theory and procedures of Chapter 9 of the Highway Capacity Manual. Consequently, it can be concluded that the results are realistic. However, these results could cause difficulties for traditional travel forecasting models. Delay cannot be a declining function of volume without introducing the possibility of multiple, equally valid, equilibrium solutions. Whether multiple equilibria could occur in real, full-scale networks has not yet been established.

**Figure 4: Delay on All Approaches of a Signalized Intersection as a Function of Volume on a Single Approach(0% Right Turns, 0% Left Turns, 800 VPH at Opposing and Conflicting Approaches, No Exclusive Lanes, 3600 VPH Ideal Saturation Flow Rate, 20 mph speed)**

The signalized intersection delay specification was extensively exercised, varying the percentage of turns, the cycle length, the approach type, the presence or absence of exclusive lanes, and the levels of opposing and conflicting volumes. A selection of these delay/volume curves are shown in Appendix C. A review of these curves indicate that no simple relationship, such as the BPR formula, can accurately estimate intersection delay.

*Flow Ratio Method. *The best that can be offered for models dependent on the BPR formula is a weak approximation to these simulation results. Assumptions must be made about the amount of traffic at all approaches, the cycle length, the number of phases, and the saturation flow rate of all approaches, including the effects of turns. A capacity, c, for the approach is approximately,

A practical use of Equations 6 and 7 would require capacities to be computed after volumes have been assigned to the network, rather than given as data.

*Equal Greens Method. *In the absence of information about opposing and conflicting volumes, it would be necessary to assume that the flow ratios are identical at all approaches. Under such a situation the green times would be approximately equal on all approaches. Equations 6 and 7 reduce to a single equation,

Equation 8 is similar to methods currently used by planners prior to network calibration. Because Equation 8 ignores signal timing, it is not a desirable method for estimating capacity.

*Graphical Method. *A related method of calculating the capacity of an approach is to use the information such as that contained in Appendix C and in Figures 3 and 4. The first parameter of the BPR formula would be set so that delay at capacity is exactly twice delay at zero volume (α = 1.0). As seen previously, this setting for a is approximately correct for most uncontrolled road segments. The capacity would then be defined at the volume on the subject approach that exactly doubles delay. This capacity can be directly read from one of the graphs, or interpolated from two or more graphs.

For example, in Figure 3 the delay for the subject approach at zero volume is 18 seconds. "Capacity" would therefore be slightly less than 1200 vph (Figure 3 shows the delay at 1200 vph to be about 38 seconds). In Figure 4, "capacity" is seen to be slightly more than 2400 vph. This result can be compared with Equation 21, assuming Vs 2400 and L = 6,

c = 3600 [0.667/(O.667 + 0.222) ] (90 - 6)/90 = 2524The results of these methods appear to be reasonably consistent. The graphical method could best be viewed as an aid to hand calibration of networks.

*Drawbacks. *All three methods are clumsy. They require prior assumptions about volumes and require a considerable amount of user intervention, especially for the calculation of saturation flow rates. Furthermore, the three methods deviate to varying extents from the HCM.

Once capacity has been calculated, it is possible to estimate delay from the BPR or a related function. Figure 5 shows the best fits of the BPR, Spiess' and Overgaard's functions to the subject approach delay from Figure 4 (Ss = 3600, 0% turns). As described in the last section, capacity was taken to be the volume that doubles delay. Therefore, the value of a was set to 1.0 in the BPR function; no changes were required of Spiess' function. It is seen that the BPR and Spiess' functions fit well; the Overgaard function misses badly at volumes exceeding capacity. The best fit of the BPR curve was obtained with = 5.3; the best fit of Spiess' curve was obtained with a = 7.4.

**Figure 5: Least-Squares Fit to Signalized Intersection Specification**