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293 A P P E N D I X D On-Ramp Queue Spillback Check Queue spillback into an arterial intersection may occur when the freeway merge segment has insufficient capacity to process the rampâs demand. Spillback may also occur in cases of ramp metering. This Appendix presents the methodology for determining whether spillback will occur from an on-ramp into the upstream intersection. The methodology considers signalized intersections, two-way stop controlled intersections, all-way stop controlled intersections, and roundabouts. The first step of the proposed procedure estimates the demand approaching the on-ramp (determined based on the upstream intersectionâs configuration), while the second step estimates the capacity of the off-ramp. The existing methodology for oversaturated conditions along freeway facilities (HCM Chapter 10) can estimate the resulting queue length, however, the user must input the on-ramp demand flow rate. The methodology framework for conducting this spillback check is presented in Figure D-1 and described in more detail in the remainder of this appendix.
294 Figure D-1. Procedure for detecting spillback occurrence at an on-ramp
295 Step 1 â Demand Estimation The first step in the methodology calculates the entering demand flow rate at the onramp (vR), as a function of the upstream intersection configuration and operations. Under low demand conditions, the on- ramp demand flow rate is calculated as the sum of the demands on each of the intersection approaches that discharge into the ramp. However, if any of these movements is operating over capacity, the total throughput to the ramp will be constrained by the capacity of these oversaturated movements. Hence, this check ensures that the on-ramp demand is not overestimated. The analysis approach for each of four intersection types is presented next. Case A â Signalized intersections The throughputs of a signalized intersection are highly dependent on several parameters such as phasing sequences, actuation, cycle lengths, and permitted-protected phasing, among others. The methodology developed identifies the movements that discharge to the on-ramp and their operational characteristics (permitted or protected). Typical diamond interchanges will include a left-turn movement, a right-turn movement and a through movement (which will typically have negligible flow). Protected movements are analyzed on an individual cycle basis using the Queue Accumulation Polygon (QAP) described in HCM Chapter 31 (Signalized Intersection Supplemental). A typical QAP for a protected movement is illustrated in Figure D-2 and can be divided into three discharging patterns: Effective red time (r). During this period, no vehicles are discharged and queue is building at a rate qr (arrival flow rate during the effective red time). 1. Queue service time (gs). During this period, the queue previously built discharges at the saturation flow rate (sadj). Therefore, the total number of discharged vehicles for movement i during the queue service time is given by: ð , , = ð ð¥ ð (Equation D-1) Where: NR,i,gs = total number of vehicles discharged for movement i during the queue service time si = saturation flow rate (veh/h/ln), as defined in HCM Equation 19-8, of movement i gs = green service time (s) = Qr/(s â qg) 2. Extension green time (ge). Corresponds to the remaining portion of the effective green when the queue has been completely discharged. During that time vehicles are discharged at the same rate they arrive at the intersection. The extension green time calculation is only applicable to undersaturated approaches, as its duration will be zero when the effective green time is insufficient to clear the queue. The total number of discharged vehicles for movement i during the extension green time is given by: ð , , = ð ð¥ ð (Equation D-2) Where: NR,i,ge = total number of vehicles discharged for movementi during the green extension time qg = arrival flow rate during the effective green time = P q C/g (veh/s) ge = green extension time (s) = g - gs The total number of vehicles discharged for a protected movement i during a cycle is then given by:
296 ð , = ð , , ð , , (Equation D-3) Figure D-2. Discharging patterns for a typically protected movement using a queue accumulation polygon (QAP) The next step is to estimate the permitted movementsâ flow into the ramp. Right-turning movements often operate as permitted (RTOR), however their effect is not addressed in the HCM Chapter 31 (Signalized Intersections Supplemental). Calculations to account for permitted movement flow are increasingly more complex, given that each phasing combination requires the development of a specific queue polygon. Finally, the total on-ramp demand during a single cycle can be calculated as: ð , = â ð , â ð , (Equation D-4) Where: NR,i = total number of vehicles discharged from each protected movement i NR,k = total number of vehicles discharged from each permitted movement The last step is to convert the total demand to the on-ramp into an hourly flow rate. For a pre-timed signal, the aggregated on-ramp flow can be estimated by Equation D-5: ðð¹ = ðµð¹,ððððð ð ðððððª (Equation D-5) In the case of a semi-actuated or fully actuated intersection with unknown cycle length, the procedure described in section 2 of HCM Chapter 31 can be applied to estimate phase durations and cycle lengths. These can then be aggregated into an hourly flow rate using the same procedure. Case B â Two-Way Stop Controlled (TWSC) intersections. The current HCM methods for TWSC intersection analysis are based on the calculation of the potential capacities of each movement, based on factors such as priority order, conflicting flow, and critical gap.
297 With very few adjustments, estimating the on-ramp throughput from this intersection type is a relatively straightforward task. The first step is to identify the movements that discharge to the on-ramp and their respective ranks (priority orders). The proposed methodology on freeway-arterial interactions assumes that, for TWSC interchanges, the arterial will always be the major street. Figure D-3 illustrates a typical TWSC interchange, where movements discharging into the on-ramp are numbered according to their ranks, using the default numbering of the HCM methodology (Chapter 20, Exhibit 20-1). Figure D-3. Schematic of movements turning to an on-ramp from a TWSC intersection Similarly to signalized intersections, there are three movements turning into the ramp, and their respective flows are discussed below: 1. Rank 1 movement (right turn from the major street). This movement is considered unimpeded, experiencing zero delay. The only physical constraint able to limit the throughput of this movement is its saturation flow rate if demand is very high. Therefore, the maximum throughput Î»RT (veh/h) for this right-turn movement is given by: ð = min ð£ , ð ) (Equation D-6) Where: Î»RT = departure rate from major street right turn into the on-ramp (veh/h) vRT = demand flow rate for the major street right turn sRT = saturation flow rate for a right-turn movement (veh/h) 2. Rank 2 movement (left turn from the major street). The maximum throughput for this movement is limited by its potential capacity (cp,j), as defined in HCM Equation 20-36. Therefore, the maximum throughput Î»LT (veh/h) for this left-turn movement is given by: ð = min ð£ , ð , ) (Equation D-7)
298 Where: Î»LT = departure rate from major street left turn into the on-ramp (veh/h) vLT = demand flow rate for the major street left turn cp,j = potential capacity for the major street left turn (veh/h) 3. Rank 3 movement (through from the minor street). Similar to rank 2 movements, the maximum throughput for this movement is limited by its potential capacity (cm,k), as defined in HCM Equation 20-47. Therefore, the maximum throughput Î»Th (veh/h) for this through movement is given by: ð = min ð£ , ð , ) (Equation D-8) Where: Î»Th = departure rate from the minor street through into the on-ramp (veh/h) vTh = demand flow rate for the minor street through cm,k = potential capacity for the minor street through (veh/h) Finally, the total on-ramp demand flow rate vR can be estimated as follows: ð£ = ð ð ð (Equation D-9) Case C â All-Way Stop Controlled (AWSC) intersections The current AWSC methodology already addresses departure headways (hd) for each approach, making the calculation of the on-ramp flow straightforward. Figure D-4 illustrates the movements discharging into an on-ramp from an AWSC intersection. Figure D-4. Schematic of movements turning to an on-ramp from an AWSC intersection The onramp demand flow rate can be obtained directly from the departure headways of the three movements combined:
299 ð£ = , + , + , (Equation D-10) Where: vR = on-ramp flow rate (veh/h) hd,RT = departure headway for the major street right turn (s) hd,LT = departure headway for the major street left turn (s) hd,Th = departure headway for the minor street through(s) Case D â Roundabouts The current HCM methodology for roundabouts is based on the calculation of the potential capacities of each approach, based on three main variables: the critical and the follow-up headways, and the circulating flow (Equation 22-21 through Equation 22-23). Both critical and follow-up headway values can be obtained from HCM Chapter 33. The procedure to evaluate the occurrence of queue spillback into roundabouts is highly integrated to the evaluation of the impacts of queue spillback, given the interdependence of entering flows and the capacities at the roundabout. Therefore, the methodology for this case is discussed in Appendix E - On-Ramp Queue Spillback Analysis.
300 Step 2 â Capacity Estimation As shown in Figure D-1, capacity at the on-ramp must be estimated in order to predict the occurrence of queue spillback. Three cases may occur: Case 1 â Ramp metering is active In this case, the metering rate is a required user input (veh/h), and it is used as the ramp capacity. Regarding merging operations, there is a modification required for the Freeway Facilities Oversaturated methodology where the maximum output flow rate value (ONRO) is set to be equal to the ramp metering rate. Case 2 â No ramp metering, oversaturated merge segment In this case the ramp capacity can be obtained from the Freeway Facilities Oversaturated methodology (HCM Chapter 25, parameter ONRQ). If a merge segment operates above capacity, the current methodology is able to estimate the resulting queue across the on-ramp for every 15s time step. Case 3 â No ramp metering, undersaturated merge segment This case does not require any adjustments to the existing methodology.
301 Example Problem â Signalized intersection with ramp metering This example is based on the configuration used in the HCM Chapter 34 (Ramp Terminals and Alternative Intersections Supplemental), as shown in Figure D-5. Figure D-5. Schematic of the study interchange for the example problem The diamond interchange has two closely-spaced signalized intersections, spaced 500 ft from each other. The on-ramp connecting Intersection 2 to the freeway is metered at a rate of 650 veh/h and is being evaluated for queue spillback. The signal controller is set to pre-timed operation, and the phasing sequence and timings are presented in Figure D-6. Figure D-6. Phasing sequences and timing for the study interchange
302 There are two movements that discharge directly to the on-ramp: WBR (Î¦16 â Phase I) and EBL (Î¦5 â Phase III). The WBR movement is a protected-only movement. Additional parameters used in the analysis are presented in Table D-1. Table D-1 - Input parameters for the two movements discharging to the on-ramp Phase I (WB R) Phase III (EB L) Green (G) 63 39 Y+R 5 5 Demand flow rate (veh/h) 520 450 Effective green (g) 64 40 Effective red (r) 96 120 Platoon Ratio (Rp) 1.00 1.33 Prop veh arriving on green (P) 0.4 0.3325 Available queue storage (veh) 24 20 Sat flow rate (veh/h) 1818.5 1703 Sat flow rate (veh/s) 0.505 0.473 Table D-2 - Additional input parameters for the example problem Ramp metering rate (veh/h) 650 Ramp metering rate (veh/s) 0.181 Average length of vehicle (ft) 25 Ramp length (ft) 1200 Cycle length (s) 160 Number of cycles per 15-min 5.625 For the spillback check procedures, a 15-minute aggregation is recommended. Though the intersection throughput volumes are calculated for every individual movement and on a cycle-by-cycle basis, they must be aggregated to a 15-minute time period. Given a cycle length of 160s, the analysis can consider 900/160 = 5.6 cycles. The calculations to estimate on-ramp throughput for the WBR movement are presented in Table D-3. For each cycle, 24 vehicles are discharged considering the sum of queue serving time and green extension. Therefore, for a 15-minute period it is expected that a total of 24 x 5 = 120 vehicles enter the onramp from the WBR movement:
303 Table D-3 â On-ramp throughput for WBR movement Movement WBR demand flow rate (veh/h) WBR arrival rate (veh/s) qr (veh/s) qg (veh/s) Qr (veh/ cycle) gs (s) ge (s) Vehicles discharged - queue serving time (veh) Vehicles discharged - green extension (veh) Total vehicle discharge (veh) WBR 520 0.144 0.144 0.144 13.87 38.44 25.56 109 21 135 EBL 520 0.144 0.144 0.144 13.87 38.44 25.56 116 0 118 Finally, the on-ramp demand (sum of the two contributing movements) is compared to the freeway discharge capacity (given by the ramp metering rate), and any unserved vehicles in the cycle are stored for the start of the next cycle. As shown in Table D-4, during the fourth cycle the ramp storage ratio exceeds one, indicating that spillback into the intersection is expected to occur at that time. Table D-4 â Queue storage analysis for the onramp Cycle # Hourly demand to ramp - vR (veh/h) Total demand to onramp (veh) Freeway discharge capacity (veh) Ramp queue (veh) Ramp queue (ft) Ramp storage ratio (RQ) Spillback expected? 1 1012.5 45 29 16 400 0.33 No 2 1012.5 45 29 32 800 0.67 No 3 1012.5 45 29 48 1200 1.00 No 4 1012.5 45 29 64 1600 1.33 Yes 5 1012.5 45 29 80 2000 1.67 Yes