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Capacity Analysis of Interchange Ramp Terminals: Final Report (1997)

Chapter: APPENDIX D Closely-Spaced Intersection Flow Models

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Suggested Citation:"APPENDIX D Closely-Spaced Intersection Flow Models." Transportation Research Board. 1997. Capacity Analysis of Interchange Ramp Terminals: Final Report. Washington, DC: The National Academies Press. doi: 10.17226/6350.
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Suggested Citation:"APPENDIX D Closely-Spaced Intersection Flow Models." Transportation Research Board. 1997. Capacity Analysis of Interchange Ramp Terminals: Final Report. Washington, DC: The National Academies Press. doi: 10.17226/6350.
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Suggested Citation:"APPENDIX D Closely-Spaced Intersection Flow Models." Transportation Research Board. 1997. Capacity Analysis of Interchange Ramp Terminals: Final Report. Washington, DC: The National Academies Press. doi: 10.17226/6350.
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Suggested Citation:"APPENDIX D Closely-Spaced Intersection Flow Models." Transportation Research Board. 1997. Capacity Analysis of Interchange Ramp Terminals: Final Report. Washington, DC: The National Academies Press. doi: 10.17226/6350.
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Suggested Citation:"APPENDIX D Closely-Spaced Intersection Flow Models." Transportation Research Board. 1997. Capacity Analysis of Interchange Ramp Terminals: Final Report. Washington, DC: The National Academies Press. doi: 10.17226/6350.
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Suggested Citation:"APPENDIX D Closely-Spaced Intersection Flow Models." Transportation Research Board. 1997. Capacity Analysis of Interchange Ramp Terminals: Final Report. Washington, DC: The National Academies Press. doi: 10.17226/6350.
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Suggested Citation:"APPENDIX D Closely-Spaced Intersection Flow Models." Transportation Research Board. 1997. Capacity Analysis of Interchange Ramp Terminals: Final Report. Washington, DC: The National Academies Press. doi: 10.17226/6350.
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Suggested Citation:"APPENDIX D Closely-Spaced Intersection Flow Models." Transportation Research Board. 1997. Capacity Analysis of Interchange Ramp Terminals: Final Report. Washington, DC: The National Academies Press. doi: 10.17226/6350.
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APPENDIX D CLOSELY-SPACED INTERSECTION FLOW MODELS D.1 OVERVIEW The following appendix presents research that addresses traffic operations on the arterial street connections (links) between He traffic signals at signalized interchange ramp terminals, and also on the connecting links to adjacent intersections downstream from the interchange. Various Arc models are applied to Be complex traffic operational conditions. The fundamental problems of queue spilIback arid flow blockage are addressed. Guidelines for identifying "closely-spaced" intersections are provided. A detailed computer-based aigorithm,basedon the Prosser-Dunn model, for assessing the impacts of queue spilIback was developed arid applied to a wide range of traffic conditions, including oversaturation. D.2 LINK FLOW CONDITIONS Traffic flow on a signalized cross arterial link traveling through an interchange is very complex due to several factors. The downstream traffic signal routinely interrupts the flow forming queues behind the signal which must be subsequently dissipated Junng the next cycle. The amount of queue formation depends on the amount of upstream mining traffic, the quality of signal progression, and the resulting total traffic demand that arrives on the movement. Traffic operations on the link ultimately depend on whether the link's arrival demand volume for any movement exceeds the link's capacity to service it. If the link becomes oversaturated, due to demand exceeding output capacity, then the link urine initially experience severe queue spilIback and soon watt become flooded with cars. The degree of resulting traffic flow and congestion depends almost entirely on the ability of the link to discharge vehicles downstream, otherwise total stoppage (grid lock) mI} occur. During heavily congested conditions, signal coordination primarily determines which feeding movements get to proceed, but usually not the Overall quantity of flow on the link. Assuming that traffic conditions on a link are urldersaturated, then flows on the link may be assumed to cycle through a series of states shown in Figure D-} where the arrival flows from the upstream movements proceed downstream and some are routinely stopped, accumulated in queue behind the downstream signal displaying red, and then subsequently serviced on the following green. Whereas arrival flows to the interchange ramps and some minor cross streets may be random at some average arrival flow rate, most arrival demands to the head of a link are, in reality, the output flow profile from an upstream signal modified by some platoon dispersion, depending on the distance "raveled downstream. Platoons are usually not dispersed Critic they travel about two (2.0) minutes, so platooned flow along the crossing arterial is common for urban interchange operations. D - ~

L u TV ~ q.= Flow, q (vph) _ Vm (qm+ql+qr) Saturation q -S Arrival Flow qm = V "d" qm= V No Flow qm= 0 Green Red Green Time in Cycle Figure D-1. Traffic Flow Conditions on a Link. D - 2

D.2.l HCM Arrival Flow Profile Anival flow to a downstream closely-spaced arterial signal is inherently dependent on upstream traffic and signal conditions which may not be well known. Modeling assumptions may have to be made to provide tractable solutions. For Highway Capacity Manual fly level of analysis, one assumption usually made is Hat He anival flow to an arterial signal is composed of two component flows: one arriving during the downstream green and one during He red, or Va = Vg (g/CJ + Vr (r/C) where: v = a Vg = vr g/C = r/C = = average arrival volume during cycle, vph, anival volume on green, vph; arrival volume on red, vph; green ratio; and red ratio. D.2.2 PASSER I! Arrival Flow Profile (D-~) The next higher level of arrival flow profile found in traffic signal timing software is in the PASSER I} - 90 software developed by IT] for arterial signal timing optimization (29. PASSER I! assumes a two-flow model somewhat like the HCM except that one flow, the larger arrival flow, is defined as being the flow of the larger arriving platoon, in time, space, and rate of flow, and all other flows are combined into a single secondary flow region. This flow model is va = vp (s/CJ + vnp (C p)/C (D-2) where: v = a Vp = ~= no average arrival volume during cycle, vph, maximum arrival volume in largest platoon, vph, and average arrival volume during remainder of cycle, vph. Flow profiles depend on the progression arid are calculated based on the resulting time-space diagram for the arterial. Delay is calculated as being the area of the queue polygon resulting from the piece-w~se integration of the input-output flow profiles over a representative cycle. D.2.3 PASSER Ill Arrival Flow Profile PASSER Ill is a computer model developed by TT! that is widely used to evaluate arid optimized traffic operations at signalized diamond interchanges (3~. Its arrival flow model provides the next higher level of sensitivity to variation in predictable arrival patterns. PASSER Ill models D-3

three flow regions of the cycle, having two flows for each of the three upstream protected phases (i.e., A, B. C). The resulting downstream flow profiles are va VAP (Ap/C) + VAa fA a/C) + VBp (Bp/C) + Vga (Ba/C) + vcp (Ca/C) + VCa (Ca/C) (D-3) where: average arrival volume during cycle, vph; platoon flow downstream from Phase A, vph; average arrival flow to/fiom Phase A, vph; platoon flow downstream from Phase B. vph; average arrival flow to/from Phase B. vph; platoon flow downstream from Phase C, vph; and average arrival flow to/from Phase C, vph. For diarnondinterchanges,arrival flows comingirom an upstream Phase C signal are normally zero because this phase is the outbound left turn phase to the on-ramp. D.2.4 TRANSYT 7F Arrival Flow Profile An even higher-level arrival flow model is employed in TRANSYT 7F, another signal timing optimization and analysis program supported by Federal Highway Administration (4). A discretized output flow profile is developed for each upstream input flow which, when adjusted for platoon dispersion, becomes a component to the arrival flow profile of interest. The individual arrival flow profiles are summed to generate the total arrival flow profile. As in the above programs, the upstream phasing sequence must be defined together with all elements ofthe time-space diagram, signal progression, and platoon dispersion The discretizationtime slice in TRANSYT 7F can be as small as I/60 of a cycle. Thus, there can be up to sixty discrete arriva] flows calculated for a movement, which are the sums of all upstream feeding flows for that time slice (T+K), or v&(T+K&) = ~vm(T+K=) (D-4) where T is the travel time between the intersections. This high level of detail can only be achieved by automating the analytics to make the volume estimation process practically feasible. D.2.5 Arr~val Flow Mode! Recommended The selection of arrival flow modeling sets the standard for the level of precision for all ~e level of service modeling to follow. It is presumed that the average flows that would occur during the cycle, peek period, and wi~in the hour of interest can be estimated with suff~cient accuracy. For external approaches or isolated approaches that do not have a signal within two minutes travel time to the subject approach, the assumption of random arrival flows (Poisson) is sufficient. For traditional coordinated arterial signal systems having modest turning traffic (say 20%) and generous D -4

intersection spacings, a two-flow model (flow during red, flow during green) as employed in Chapter 1 1 (Arterials) of the HCM is probably sufficient in most cases. An obvious improvement would be to transition to a two-flow model like used in PASSER II (flow during platoon, flow not in platoon). For interchanges operating with high turning traffic and within a system of closely- spaced intersections, arrival modeling should be at least as detailed as used in PASSER III wherein six arrival flows are possible, two (platoon and non platoon) for each of three feeding movements during the cycle. While coordination in interchange systems is probably not as effective (due to higher turning traffic and more balanced flows) as in arterial systems having predominate through traffic, at least the features of platoons arriving at interchanges could be readily identified so that queuing and delays could be reduced somewhat over solutions assuming random flow. D.3 MACROSCOPIC FLOW MODELING Models of traffic flow routinely used to describe continuous flow traffic facilities, like freeways, can also be used to describe the dyna~nicsof flow attraff~c signals (5). These models can be used to describe the nature of operational problems experienced. Traffic signal operation routinely creates brief interruptions to continuous flow, forming bottlenecks where the arrival demand exceeds output capacity during the red interval. Shock waves form behind the signal due to the queuing and spillback that occurs (6). D.3.! Flow Models When traffic flow can be assumed to be in a steady-state condition, even for a fairly brief period of time (t) and space (x) such as for random arrival flow or platoon flow at saturation, then the average flow rate during this period, v (vpt) can be thought of as being produced by a traffic stream having an average density k (vpx) traveling at an average speed u (xpt)' or v =ku where: v = average traffic flow rate' vpt, k = average traffic stream density, vpx; and z' = average traffic speed, xpt. (D-5) In undersaturated,signalized arterial operations,the flow rate on a short section of roadway just upstream of a traffic signal is usually in one of three states: v = v, v = 0, or v = s. That is, the flow is characterized as being either the arrival flow, stopped in queue, or the queue has been transformed into a platoon with saturation flow, s. These three flow states were identified in Figure D-1 . Whether these changes from one state to another occur almost instantaneously(creating shock waves) or transition over a brief period of time (forming characteristic waves) is more of a theoretical issue and presumed herein to be of minimal importance. For convenience, rapid response to signal change is assumed so that conventional shock wave analysis can be employed. D - 5

Basic macroscopic traffic flow modeling for steady-state conditions presumes (hoary car- follow~ng laws and empirical observations) that the speed of operation is a function of the average density of the section of road (immediately ahead) and the current speed (5) such that ~U~-m i- = 1 u k f q and solving Equation D-6 for u =f(k) and substituting into v = k u, ~ = kf~J yields v = k Uf t1 _ ~ k yI~1~1/~1-m) q for the section x of interest, where: v = traffic flow in section x at t, vpt; k = traffic density, vpx; Uf = free speed, xpt; kq = jammed queue density, vpx; and I, m = shape coefficients. (D-6) (D-7) The traffic flow graphs (for Case I) shown in Figure D-2 were drawn for the following assumed arterial operating conditions: Uf = 64 km/in (40 mph); kq = 143 vpkmpI, (7m/veh, 230 vpmpI, 23 It /veh), s = 1800 vphp} at a saturation flow speed of 37 km/in (23 mph). Under these assumptions,] = 2.645 and m = 0.666. The three flow vanables (v, k and u) are all interrelated and can be calculated given one of them together with the section parameters arid coefficients. D.3.2 Saturation Flow Saturation flow can be assumed to be the "capacitor" flow region of the flow curves, suggesting that the saturation flow may increase as the platoon speeds up, or flows faster if the platoon can travel faster for a given vehicular spacing (density). Research shows that saturation flow increases with increases roadway quality and operating speed. Over several editions of the HCM, freeway capacity has increased from about ~ 800 vphpT at 50 km/in to 2000 vphp! at 70 kmAl, and is now approaching 2300 vphp! at 90 larch. This NCHRP research combined with the results of NCHRP 3-40 (~) suggests similar bends for sianalizedintersections end interchanges when flowing - - - ~ ~ ~ - -7 ~ ~ ~ - ~ ~ ~ o ~ ~ . . ~ . ~ ~ ~ a, 1 at equal loadings and pressure. Figure ~-;z also Sates now the ~low-censlty curves would expand with increasing quality of operations. Case I] assumes a saturation flow of 2,000 vphp} at 40 km/in (25 mph). The loci to the family of curves would represent the expected saturation flow at the signal when We platoon is flowing at its maximum flow for increasing quality of operating conditions. D - 6

Flow vs Density 2000 1800 1600 1400 - ~ 1200 - 1000 0 800 - E~ - 600 400 200 O 2000 - . 18~ , _ , '. \ Saturation Flow stopped Quelle \' Case I ~ \ "+ Case II :'.. 1 1 1 11~ 2S SO 7S 10012S lSO Density (vpkm) Speed vs Flow 1600 1400 1200 1000 800 600 400 200 o ~ ~ . ~ ' ~ ,"~ """ ;;~/ Samration Flow \ / \ ' .'/ \ '% ~HI ~ ~II ~ it" ~ ERIC ~ lk ~'` ~ 1 1 1 1 1 1 ~1 ~ 0 10 20 30 40 50 60 70 80 90 Speed (kmph) S=1800 vph S=2000 vph Uf= 64 kInph Uf= 88 kmph . Figure D-2. Characteristics of Traffic Flow for Two Capacity Conditions. D-7

D.3.3 Shock Wave Speed When arterial arrival flow passing through a green signal is suddenly stopped by the onset of red, the output flow suddenly drops from ~ = v to v = 0. When this charge in output flow of the section occurs, the storage begins to queue behind the signal at a storage density k,, the queuing Clam density'' of about 7.0 meters per vehicle (23 feet/vehicle). The speed at which the storing queue propagates (spills back) upstream can be estimated from shock wave theory 66) as W l~v v (D-8) q where: Wr = shock wave spillback speed due to red onset, xpt, v = arrival volume (dunng red), vpt; and k = traffic stream density, vpx. The speed at which the shock wave propagates upstream increases with increasing arrival volume. For an approach having average flow conditions described as Case ~ in Figure D-2, and a green ratio of 30% which yields a phase capacity of 540 vphpl, then for arrival volumes of 20, 60 and 100 % of signal capacity, the shock wave speed progressing upstream during red would be estimated by Equation D-8 to be 0.22, 0.67, and 1.12 mps, respectively. However, if the signal became oversaturated or poorly timed such Mat the start of red ended platoon motion while at saturation flow v = s, then the shock wave speed would rise to 5.35 mps. However, as long as the signal is undersaturated, the maximum queue length per cycle would remain the same. - 7 .A '' ~ D.3.4 Platoon Wave Speed When the signal turns green following an extended red time and subsequent queue buildup, the platoon responds and begins to move forward, reaching saturation flow conditions in a few seconds after green onset. For HCM-level of analysis, the platoon is assumed to reach saturation flow almost immediately once the queue-platoon transformation begins at any point in the queue. Under these simplifying assumptions, the platoon's green wave speed would be given by W = where: Wg k - h _ Ant = REV S g Ak k - k q s platoon start-up green wave, apt; saturation flow during green, vptpl; platoon density during saturation flow, vpxpl; and queue storage density, vpxpl. D - 8 (D-9)

The queue's transformation into a moving platoon would propagate upstream following green onset at 5.35 mps (12 mph). If the output saturation flow is unimpeded on the downstream link, then the green wave speed is fairly constant over all arrival volume conditions, simplifying the analysis. The main problem would be to determine how far upstream queue spilIback has progressed each cycle before it begins clearing. D.3.5 Clearing Wave Speed Once the platoon begins to move forward on green, flow in the platoon is assumed to be saturation until the platoon clears the stopline. The arrival volume, v, is entering the upstream end ofthe platoon; whereas, saturation flow, s, presumably is occurring downstream to the stopline, or downstream boundary. During this cleanng penod, the back of the platoon is traveling downstream from its maximum backup at the platoon clearing wave speed of W = Av = s - v C Ak ks - k (D-10) Continuing with the data of Figure D-2 where kq = 143 vpkmpI, s = I 800 vphpI, us = 37 km/il, then ks = 48.6 vpkmp] from k = v/u. An examination of the above wave speeds follows. Figure D-3 presents the resulting wave speeds for the above conditions for volume-to- capacityratiosofO.2to I.O for aselectedg/C ratio of 0.3 end a LOO-second cycle. The speed ofthe shock wave, W., is very slow (about ~ nips) and only increases slightly with increasing v/c ratios of the signal. The platoon start-up wave, Wg, is noted to be a constant of 5.35 mps, and the platoon clearing wave is high (about ~ 6 mps) and only decreases slightly with increasing arrival volumes. Thus, because the wave speeds are fairly insensitive to arrival volumes at traffic signals, analyses based on the wave speeds are relatively stable as long as traffic flow on the link is undersaturated. D.3.6 Queue SpilIback The duration and extent of queue spilIback determines whether an upstream intersection wall be severely affected by downstream operations. In essence, the characterization of adjacent intersections being too "closely spaced" can be defined for undersaturated conditions where X<= 1 . Using the above shock wave theory, the maximum length of queue spilIback can be determined in time arid space by algebraic solution of the Wr and Wg wave intercepts for a given red time. r The elapsed time following green onset before Wg catches Wr is ~7 W T = r r (D-1 1) g W- W where Tg is the elapsed time since the onset of (effective) green. D-9

we we 2s 1 = 2.645 20 - . m = 0.666 E ~-- - IS C ~ 10 3 s- O- ~1 0 0.2 0.4 0.6 - Wr o.g l 1.2 Phase Volume-t - Capacity Ratio, X Figure D-3. Wave Speeds at Traffic Signals During Undersaturated Conditions. The maximum queue backup for undersaturated conditions, [n'' is equal to A, = Wg*T,, or W W L = g r r m W ~ W g r subject to the restriction that the downstream phase is not oversaturated. (D-12) Figure D-4 presents the queue spilIback for an approach having random flow-(uniformly distributed over the cycle) for v/c ratios up to ~ .0, or saturation. The traffic and control conditions are as above (s = ~ 800 vphpI, C = ~ 00 see, g/C = 0.31. The capacity of the approach is 540 vphp! (c = 0.3 * ~ 800~. The maximum queue spilIback distance upstream from the stopline would be S2 meters and 99 meters for v/c ratios of 0.6 and ~ .0, respectively, using Equation D-12. These results indicate that storage links less than ~ 00 m may often experience spilIback problems on entry flows where good progression is not provided, even when the downstream flow is undersaturated. D- 10

100 90 So - as - - An 70 60 SO 40 30 20 10 o . ~1 C = 100 see g/C = 0.3 Random Flow _ ~ / I 1 1 0 0.2 0.4 0.6 0.8 1 1.2 Phase Volume - Capacity Ratio, X Figure Dot. Affects of Volume on Queue SpilIback for Undersaturated Coalitions. D.3.7 Two-Flow Arrival Models The ~ 994 Highway Capacity Manual (HCM) and its proposed arterial enhancements (l, 79 assume that the arrival volume along an arsenal is composed of two arrival flows: a flow arriving on He red, ~,~ and a flow arriving on the green, vg, as noted in Equation Dot. The HCM's two-flow arrival mode} cart also be applied to the above queue spilIback equations with little change in form. Defining the arrival volume on green, vg, to be v = R^v g -'P where harp is the platoon ratio, and the arrival volume on red, or, to be v = r (D-13) C - g R p v r D- 11 (D-14)

then the maximum queue backup during undersaturated conditions can be calculated from W W L = gvg Or r m W ~ W gag ~vg (D-15) where the wave speeds are based on the arrival volumes estimated for the respective red/green signal indications (r, g). Two wave speeds for the initial shock wave, W., should be computed: one based on arrival volume vie and one based on vg Should the volume arriving on red be eliminated by great progression, then the shock wave speed on red, We,,, would be zero (0.0) and no queue spillback would occur. This routinely occurs on the outbound phase at diamond interchanges with good four- phase with two-overlap timing. The extremely bad progression case where all the arrival volume arrives on the red is described below. D.3.S Maximum SpilIback It is useful to examine the worst-case scenario for undersaturated conditions where all the arrival traffic to a phase arrives on red due to extremely bad progression. The main issue here is "how far upstream would the queue back up, storing at 7.0 meters per vehicle when arrival volumes have reached but not exceeded capacity levels?" Applying the equation of continuity to this fairly simple problem during red results in the following equation for maximum queue backup: LmaX = 7-0 Vc Vc < Ct where: (D-16) Lnzax TIC Cal C = s = maximum queue backup, meters, average arrival volume per cycle per lane, vpCpl; phase capacity per cycle per lane, stC/n -11/3600; cycle length, see, and saturation flow, vphgpl. Figure D-5 provides graphs ofthe maximum spillback for two high-volume cases (400 and 600 vphpl). Also depictedin Figure D-5 is the queue spillbackat saturation when the average arrival volume (on red) is equal to the phase capacity. Phase capacity presumes three equal phases each having 4.0 seconds lost time with saturation flows of 1,800 and 2,OOO vphgpl. Maximum arrival volumes of 520 and 580 vphpl can be handled for a 100-second cycle at capacity. Maximum queue spillback is noted to vary with several factors. Maximum spillback increases with increasing arrival volume and cycle length (due to longer reds). Capacity also increases with increasing cycle due to reductions in the proportions of lost time per cycle. A maximum queue spillback of 166 meters (545 feet) is estimated for a 140 second cycle. ~. . . , . ~. . .. D- 12

200 17S lSO C, ~ 12S D 100 - U] a so 7S 2S o 60 v = 400 ~php1 v = 600 ~1 - --- S = lSOO ~pl - - lo- - S = 2000 vphp1 80 100 120 140 160 Cycle Length (see) Figure D-5. Maximum Queue SpilIback for Capacity Conditions. D.5 OVERSATURATION The period of oversaturationis the time when the arrival (demand) volume on a signal phase exceeds its capacity. Oversaturation may occur because the arrival demand has grown to a level greater than the capacity, or the phase capacity may have been reduced due to downstream spillback impeding the saturation flow during green. In any case, queue spillback and congestion on the link are eminent unless control conditions are rapidly changed. Once the signal becomes oversaturated, then traffic operational response to signal control inputs is often the reverse of prior undersaturated response. This error can only feed the growing congestion unless corrections are made quickly. D.5.! Transition Time The time a link ~ is in transition from undersaturated to oversaturated operating conditions is brief. The time can be calculated from the input-output modelderived from the basic equation of continuity of vehicular flow on the link. The input-output model for time periods longer than one cycle is D- 13

N (L, t) = No ~ ~ Vm t - ~ Cm t N(L, t) < New (D- 1 7) where: N(L,t) = number of vetches operating on Me him of length ~ at time t, vehicles; No = number of vehicles operating on the link at start of period, vehicles; vm = total arrival volume to head of link destined to movement m, vph; cm Nan,,` output capacity of link serving movement m, vph; and maximum number of vehicles that can store on link, vehicles kq ~ with a typical storage density of 143 ~kmpI, or storage spacing of 7.0 m/vein (23 flc/ vein). 1' When it can be assumed for analysis of a phase that the arrival volume and capacity flow are constants for a period of cycles, Men Equation D- 1 7 can be solved directly for t. Letting N(L, T7 = lima,` such that Tf is the time to completely fill the link when vm T = Nmax NO v - c m am 2 Cm, then Tf is (D-1 8). Letting vm be described in terms of the v/c ratio, X = v/c, and substituting the above relationships (k - k) l T = q (D-19) f c (X - 1) Examination oftypical situations is aided by letting c = (g/C) s, X = 1/PHF where PHF is the peak hour factor (PHF = 0.9), s = 1 800 vphpl, k = km = 9 vpkmpl, and kq = 143 vpkmpl so that T = (143 - 9) L glC 1800(1.1 - 1.0) 180g/C = 134 L (D-20) where Tf is in hours and ~ in km. Assuming that the green ratio, g/C, is 0.3 results in link fill times dunug an X= ~ . ~ oversaturationof ~ S. 30, and 45 minutes for available storage lengths of ~ 00, 200 and 300 meters, respectively. As Equation D-19 shows, there are some obvious advantages to having long spacings between signals, longer links can tolerate proportionally longer periods of oversaturation before filling and spilling back into an upstream intersection. By defining ~ to be the available storage space on the link, it can also be seen for the above example that a 200-meter link having a maximum queue spilIback of 100 meters during undersaturated conditions would begin to experience spilIback into its upstream intersection 15 D- 14

minutes into the oversaturationpenod and become filthy saturated and congested 30 minutes into the period. Smart signal progression could forestall the onset of major congestion at the upstream intersection by up to ~ 5 minutes in this case. Obviously, the larger the rate of overload, X>l, the shorter the time before major congestion begins et the impeded upstream intersection, presumining minimal upstream metenug of input flow. D.5.2 Oversaturated Operations Once the subj ect link, [, becomes flooded with vehicles due to oversaturation, flow into the link from the upstream signal depends almost entirely on the output capacity of the downstream signal of the subject link, which then may depend on its downstream signal, etc. The dependence increases with the increasing percentage of upstream traffic bourns for the downstream bottleneck phase. Essentially, the above equation of continuity (Equation D-17) requires this result because N(L, T7 must equal No' so that the volume in must equal the vogue out, or v = cat (D-21) An example of this constrained input process is shown in Figure D-6 where two upstream flows having a combined demand volume of ~ .6 times the downstream capacity load the link. The microscopic traffic simulation program NETSIM was used to develop these observations which represent the average value of ~ O replications of the conditions shown. The offset between the upstream arid downstream signals was varied while the total link throughput and the individual feeding flows were observed for both upstream movements. The total throughput volume was observed to be essentially constant (the downstream capacity) over all offsets. However, the input flows that did occur were related to the signal offset (~ ofthe link, but the sum was constant, or Vua (~' + Vub (By cat (D 22) In this case when all upstream turning movements are so large that each can keep the link field when the phase ends, then no downstream storage space remains at the end of any phase, N(L, T7 = No Actual cases like this are not rare at urban interchanges where lush hour turning volumes can be high on all feeding movements to the link. As Figure D-6 shows,the selection of offset determines the share each competing movement receives ofthe total available downstream stream capacity. That is, the offset allocates the limited resource, output capacity, among competing and excessive demands. What one upstream movement receives, another loses, as this situation becomes a "zero sum" game. As Figure D-6 demonstrates, signal offsets that are based on undersaturate~progression analysis, say using PASSER Il. are often designed to favor the cross arterial's operation, usually do not provide the same favored status when periods of oversaturation occur. In fact, the PASSER IT offsets may actually wrongly favor the cross-s~eet turning traffic in some cases of oversaturation. D- 15

2000 1600 as 1200 - os ~ 8~ ~ 1 400 O 1 1 _~% at-' I 1 1 1 ~ ~ ~ 1 0 10 20 30 f ~ I i · To" ~ LAIR 40 SO 60 70 80 90 Offset (see) Figure D-6. Traffic Performance on an Oversaturated Link Related to Offset. In another case of oversaturation, where some input phases at the upstream signal cannot keep the subject link ~ filled to capacity at the end of the phase, some link storage capacity becomes available to subsequent movements. Offset analysis algorithms, such as the PDX Mode! to follow, should be able to determine how to optimally use this available storage space. Another problematic case cart arise during oversaturation. The condition is called "demand starvation", which seems like an oxymoron during periods of high demand volumes. Demand starvation may routinely occur on short links when an upstream phase feeding the link (plus link storage) cannot keep the downstream bottleneck signal (say Phase c) flowing at saturation during its green. This lost output per cycle cannot be recovered, and the maximum throughput flow is reduced below the phase capacity cat, even though the total demand on the downstream phase may exceed car Demand starvation can be a problem for some interchanges when the outbound left turn Phase c closes off flow into the interchange. Signal coordination should be provided to ensure that the cut-off phase does not empty when it is (or becomes) oversaturated. In the following sections, two traffic models wall be presented that describe operating conditions dunng oversaturation. One simple mode! estimates the maximum delay that might be incurred on an oversaturatedlink. Another more complex mode! estimates all performance measures for a link after the mode! determines whether the link is experiencing spilIback and oversaturation. D- 16

D.6 DELAY MODEL DURING FLOODED CONDITIONS D.601 Flooded Conditions In "flooded" conditions of oversaturation, the upstream demands keep the link full at all times. Flow conditions on the link are either saturation flow or stopped in queue, but are never the arrival flow. The spillback constantly engulfs the link in either a stopped queue or a moving fully saturated platoon. Figure D-7 presents both the undersaturated and oversaturated flow conditions upstream of a traffic signal. In oversaturated, fully flooded flow, only two flow states exist (saturation and no flow), resulting in only two "moving" waves (shock wave and platoon wave). We/ we \ Wr\ \ We 1 _ J ~ , _ _ ·w~ __ 1 1 u row D ~1 I Queue | | 1 - - Wc g Wg _ _ 1_ r wr g - Wg _ _ W. 1 d _ ,_ , 1 U '~' D ~ _ -- ~1 1r 1 I Queue Figure D-7. Traffic Flow Regions for Undersaturated and Oversaturated Conditions. D- 17

D.6.2 Delay Mode! In the flooded condition having only two flow states, the speed of the shock wave at red onset, Wr, is equal to the speed of the flow wave following green onset, Wg, and they are = W = W = ~ v s ~ O k- k q s (D-23) The average link density over a representative cycle of time is assumed to be given by the fraction of time any point on the lirkp(x,t) is stopped or moving. Because the two wave speeds are equal, We fractions are equal to the fractions of time the cycle (C) is effectively (g) green and red (r), respectively, for dedicated movements (lane groups). In this case, the average density becomes k = g k + - k C s C q (D-24) Assuming that the average flow on the link is given by the average downstream capacity cat where V = Cd = s d then the average travel speed on the link can be calculated to be v u, = - ~-25) (g/C) Sd k (g/C) k5 ~ (r/C) kq (D-26) Dividing We above equation by g/C and then defining ~ = r/g for the controlling downstream signal phase, the average travel speed on the link can be found to be u, = sd k + ark s q (D-27) Since the speed at saturation flow is us = s/kS, then dividing the above equation by kS/kS yields u, = us ' (1 + p) (D-28) D- 18

where: kq ks g r (D - 29) For example, if kq = 143 vpkmpl' k s = 51.3 vpkmpl' and s = 1900 vphpl, so that the nominal saturation speed is 3 7 Elmer, then for a green split at the downstream bottleneck intersection of 0.46, kq r 143 r = ~2 79 . r = 2.79 - = 3 27 (D-30) ks g 51.3 g g 46 or the average link travel speed, u,, is Us 37 = g.66 km/hr (D-31) 1 ~ ,8 1 ~ 3.27 The link average travel speed dunng flooded conditions will always be less than the speed at saturation and will be highly dependent on the throughput capacity of Me bottleneck phase. Link delay can now be calculated as the difference between the overall link travel time and the baseline running time at the approach running speed us, d (sec/veh) = l _ _ (D-32) u u I a Should demand starvation also occur during the cycle, then the delay should be reduced in proportion to We reductionin effective queue length. Assuming that the link length was 100 meters and the nominal link running speed was 56 km/hr (35 mph), then the delay on the oversaturatedlink would be d = 3600 ~ 0 TOO - 0.~00: = 35.1 sec/veh (D-33) D- 19

Do63 Delay Mode! Results Several simulation studies were conducted to further test and verify Equations D-31 and D- 32. A range of split and offsets were examined using the NETSIM microscopic traffic simulation model. Three downstream green splits were tested so that the over a cycle of offsets to verify the model results. Figure D-8 presents the simulation study for a 100 meter (328 foot) link length. 120 100 80 - ~S Q 60 40 20 o 1 ~ ___ I i I I ~r 10 20 30 40 50 60 Offset (seconds) To so so cots SAC ~ 033 SAC = 0.2S Figure D-~. NETSIM Simulation Results of Oversaturated Delay on 100 m Link. A summary of these study findings is presented in Table D- ~ O for the 1 00 meter (328) foot link length for the three green splits. It should be noted that the projected v/c ratio (degree of saturation) on the downstream is 2.0, 3.1 and 4.3 for the three green splits, but the delays predicted by Equation D-32 are not extremely large. However, large delays would be estimated by the Highway Capacity Manual delay estimation model. D -20

Table D.10 Estimated and Simulated Link Speec! and Delay for Flooded OversaturatedL'nks Study ~Downstrea3n | Travel Speed | Model | NETSIMa | HCMb Case No. Green Splits km/hrLink Delay Speed Delays Speed Delayb sec/veh ~ km/hr sec/veh ~ km/hr sec/veh 1 1 0.46 1 8.661 35.1 l8 70 34.7 1 2.22 155.7 0.29 1 4.73 1 69.7 14.51 72.9r2.13 162.7 ~ 3 ~ 0.21 ~3.22~ 105.4 1 3.14 107.7 1 2.09 166.0 a Averaged over all onsets for ~ 0 iterations. b Arbitrarily restricted to v/c < ~ .2, overwise, HCM mode] delays would have been extremely large. Other findings can be observed from these studies for oversaturated, Filly flooded conditions from Equations D-31 and D-32. The speed and delay on these links depend on the downstream bottleneck r/g ratio and resulting throughput rate. The delay on the link also depends on the length of the link, [, as given by Equation D-32. Longer links cart have proportionally higher delays, but the average travel speed and delay rate per kilometer do not charge for oversaturated links. These latter findings suggest that the level of service for an extended section of an arsenal based on average travel speed provides a constant assessment of the qualtiy of operations provided. However, methods based on point measures of queuing delay are not likely to reliably estimate the level of service being provided, nor the true congestion experienced. Additionalimprovementsin traffic performance Will arise as the link length increases. The length is not as likely to flood with increasing length, particularly in mesosaturation conditions, and for short-term oversaturation conditions noted earlier. If minor movements feed the link, then the link will partially clear dunug these phases permitting increased link speed and lower delays for the following phases, depending on the relative offset being emoloYed for the subject link. D.7 PROSSER-DUNNE-EXTENDED TRAFFIC MODEL ~J This section describes the theoretical framework for a macroscopic traffic model developed to solve operational problems associated with high-volume operations on urban arterials having closely-spaced signalized intersections. As may occur, the vehicle discharge flow rate at an upstream intersection may be seriously affected by operational conditions on the downstream link. Operational impacts may include (~) reduced saturation flow due to limited travel distance to the back of a stopped queue on the downstream link and (2) total signal blockages that reduce the elective green time due to queue spilIbackinto the upstreamintersection.These issues are carefully evaluated in this traffic mode] to provide accurate prediction of the discharge flow rate, capacity and delay at the upstream intersection. Other related outputs from this model include the queue dynamics informationthroughoutthe cycle and its interaction with the estimated discharge flow rate. D-21

The underlying theory ofthis paired-intersectionmode} is based on an analytic traffic mode! originally presented by Prosser and Dunne in 1994 (by. The Prosser-Durme Mode} used a graphical technique to estimate the reduced effective green time based on the assumption that no vehicle can be discharged from the upstream intersection whenever there is a queue spilIback blocking the intersection. The Pro sser-Dunne Mode! assumed that the downstream congested link was completely oversaturated, or flooded. D.7.! Mode! Enhancements Texas Transportation Institute (TTI) has extended this mode} to a wide garage of operating conditions, increased the modeling to three feeding movements, and added the effects of saturation flow variation with travel distance (as described in Appendix C) to the mode} structure. In addition, TT! has coded Me upgraded mode] into the FORTRAN language for implementation and testing. A second-by-second time-step approach has been unplemented to increase the qualtiy of the analysis. Extensive NETSIM simulation studies have be conducted to calibrate and verify the overall model's capabilities with field data collected within this project. The following sections describe the design, logic and test results of Me resulting aIgonthms and computer program, which is referred to as the PDX Model. Do7~2 PDX Mode! The PDX Mode! addresses the operational problem depicted in Figure D-9. Three upstream turning movements (m) can possibly feed a downstreamintersection along the link u-a,. The link can be of any length, L. These fuming movement may be the upstream through, left and right turns onto link L. Both intersections must operate on a pretimed phasing and have the same cycle length for the period of interest. The algorithm seeks to determine the effective green, phase capacity and delay for the upstream movement and the delay on He downstream link, [, for a given set of signal timings and relative offset (0 between the signals. The current program does not seek to determine an optimal offset Abut this feature could be provided by existing optimizationprograrns like PASSER IlI. PDX is only an analysis program like the HCM/HCS software. M1 M2 Ma U 1 J L D Figure D-9. Paired-latersection Operational Problem Analyzed by PDX Model. D -22

Figure D-10 presents the basic time-space diagram for oversaturated links after Prosser- Dur~ne (99. The variables so identified are defined in the following mode} development. In the PDX Model, however, links can be analyzed that may not be oversaturated by all phases combined, or may not be flooded by some combinations of phase sequence and/or offset, as noted earlier for Remarry starved links. C+TI C+To CP T3 T2 Tt To We - - w8 J U HI Queue g D ' Figure D-10. Operational Time-Space Diagram on Congested Links. Figure D- ~ ~ presents a simplified flow chart for the enhaced PDX Model. Time Mung the cycle is sequentiallyincrementedin one-secondincrementsin the PDX Model. The Mode! is coded in FORTRAN computer code. The detailed step-by-step PDX process is summanzed in the following paragraphs. D -23

( Input signal thalag (C, G) and frame choline (V,) ) , ~ , ~Calculate saturation flow periods: TSj ~ Assign ~ saturation Bow Index to each movement: STATE' ~ O or 1 [or Dde~aturated or saturated conditions ' Detennine discharging new rates from upstream h~tersection second by second | Calculate clear and blocked periods | | Calculate effective green E"G' for each movement Reassign saturation indices SI ATE' saturation Boer tune and effective green TSi<E=Gi? YES ~ Undersaturated: STATE'= 0 it_ C; EFFG, = Go ? 1 YES UB u n blocked: EFFG,= G. Via= V" ~ r NO Saturated: STATE,= 1 _-L Is EFFGl= Gl? rho PB partially blocked: EFFG,< G. Via = V,, FB fu lly blocked: EFFG,< G. V64 < Vie Adjust phase lost time and saturation 110~r rates | Recalculate satu ration now tim e an d in d ices | No Do saturation in dices converge (ES l Output each movement's capacity and throughput | Figure D-~. Flow Chart of PDX Model. D -24 - YES , OS oversatu ration: EFFG, = G. Via = V,, 1

Step ]. Identify input Parameters The following parameters are identified as inputs to the program: Gu7n = Green time of upstream movements, see; Go = Downstream green time, see; C = Cycle time, sees, SL,EU = Start loss and end gain for each movement, see; = Offset between intersections, see; = Intersignal link length in meters, m, nmax = Number of vehicles that cart be stored in the intersignal link length; vein; vm = Arrival flow rate for upstream movement m, vph; tq = Blocking queue clearance time, see, if = Link travel time during saturation flow, see; and s = Unimpeded saturation flow rate, vpsg. The yellow end red clearance times are together taken to be four seconds for both intersections (u,d). Values of tq, t, and An, noted in Figure D-lO are calculated by the following formulas: If where: L kqd ksd Sd tq = L id - ks;/sd = L ksd/sd n =L *Ed Link length, km,. Queue density of downstream movement, vplun; Saturation flow density of downstream movement, vpkm, and Saturation flow rate of downstream movement, vphg. Conversion factors to the desired units are not shown in the above equations. Step 2. Calculate Star! andEnd of AllMovements , - . , , . . ~ (D-34) (D-35) (D-36) l he start Ed end ot any movement m (m=1, 2, 3) at the upstream intersection with respect to time To = 0 is referred to as SMm and EMm. These times can be calculated as follows: =0 SM, EM, SM2 EM2 = GUI ~ G2, etc. =GI =GU, (D-37) (D-38) (D-39) (D-40) where Gm is the displayed green time for movement m, m = 1,2,3, at the upstream intersection. D-25

The offset ~ between the intersections u-a7 is considered from the start of the green period of movement ~ at the upstream intersection until the start of the through green at the downstream intersection. Hence, Go starts a time ~ after To, equal to the offset with respect to the upstream intersection. The end of the downstream bottleneck phase, EM, is EM= To + B+ Go where ~ = offset in seconds from u to al. Step 3. Calculate To and T2 (D-41) The beginning of the clear period, T., at the upstream intersection starts when the blocking queue clears the upstream movement of interest after the start of through movement green time at the downstream intersection. Note that it has been assumed that a blocking queue exists. This may be determined later to not be true if volumes are not high enough for the length of link involved. If true, then T~=To +0 + tq (D-42) In order to evaluate whether the intersignal link length wait be completely blocked under different traffic conditions, some related factors have to be evaluated. One of them is the volume-to- -capaci~ ratio (X~) at the downstreamintersection. Another one is the critical link length (Lo) which determines the occurrence of queue spillback. The critical link length is calculated as follows: L~ = G~*Wr*Wg*6C- G~)/~(lYr+ W~)~6C/X~- G~)J where: Wr = Wq = Shock-wave speed (mps); and Platoon starting wave speed (mps). (D-43) When Xa. is greater than 1 .0, the downstream link will be oversaturated by vehicles coming from He upstream intersection. If the link length ~ is greater than critical link length [c, it is assumed that a proportion of the storage vehicles w~11 not be cleared from the link due to the limited capacity at the downstream intersection. Hence, some residual queue will remain on the link when the green ends. When the link length is shorter than the critical link length, all vehicles stored on the link will be cleared. If X~ is less than ~ .0, queue spillback may also occur due to an inappropriate signal timing plan for the downstream green time and offset. This problem can be solved by using He same assumption as the one for oversaturation and by checking the results after the loop calculation in the computer program. If some blockage is found to occur, then saturation flow adjusunents are made to account for either impelled or blocked flow using Equation C-46 in the · prevlous appenc 1X. D -26

The time at the upstream intersection, denoted as T2, when vehicles discharged from the upstream intersection ~} first begin queuing at the downstream intersection stop line on red, is calculated as follows: T2 = To + ~ + Go - If T2 2 To (Daft) Step 4. Calculate Input Flow Rates during the Cycle The potential flow rates from the upstream intersection during the cycle are calculated in this step. Each upstream movement's capacity is computed based on the elective green period of that movement (SMm + SL to EMm + KU) and the saturation flow rate s. The duration oftime, t9 that every other movement experiences saturation flow is given by Is = Vi (C - g; ~ / (Si - Vj ~ (D - S) Thus, from time SMm ~ AL to tS the flow is saturation and from tS to the end of movement green time (EMm + BL), the movement flows at its arrival flow rate, v. In the event Is is greater then the end of green of the movement, the flow continues at saturation flow for the entire period of green for that movement. The flow at all other times in the cycle (e.g., dunug yellow and red clearance time) is assumed to be zero. Step 5. FindEnd of Clear Period, T3 The next step in the process is to find T3, Me end of the clear period at the upstream intersection. The time T3 is defined as We time when the intersignal length would be once again completely filled with vehicles, thus blocking the upstream intersection. The related duration of time t3 iS needed to accumulate enough vehicles to fill the link ~ after time T2. The number of seconds required to completely fill the intersignal length depends on the output flow of vehicles from the upstream intersection after T2. The potential output flow for every second in the cycle is known from We previous steps. The summation of the output flows (vehicles per second) from the upstream intersection over time (seconds), gives the number of vehicles that may enter link ~ until the link completely fills to its storage capacity, An,, over We duration t3, or n~t3,l) = nO ~ ~`qum < nma~ o < t3 < C (D-46) The assessment of no is critical to the algorithm. no = 0 when T. < T2. However, if the link can store more vehicles than the downstream phase can serve (its capacity) (because the link may be long and/or the phase relatively short)then T < T. arid n0 = a,,,- cm > 0 (instead of zero, or some other value) at time T= To. When T2 < T ,, vehicles already on the link cannot clear the next downstream phase. Thus, the most vehicles that can enter the link during the next cycle cannot D-27

exceed the downstream phase capacity, cm. If it is ultimately found that all the upstream flows can use the link for a time period T3 longer than To + C, then the link does not totally fill wing the cycle and the upstream signal is unblocked, but it may still have some reduced output flow. When queue blockage occurs, the end of the clear period for entry into the upstream intersection would be T3 = T2 + t3 T2 < T3 ' TO ~ C (D-47) In the PDX program, a simple DO-Ioop calculates t3. The loop increments the number of seconds after t2 and for each second adds the number of vehicles entering the link. No departing vehicles need to be considered here. When the link either fills completely or reaches Me next cycle (T~ + C) , We incrementation process stops. We number of time steps used gives We number of seconds elapsed after t2, which is t3. The saturation flow-queue interactionmodel described in Appendix C has been implemented in the PDX program. As each upstream phase begins to be analyzed dunug the cycle, the downstream queue length is calculated. The available travel distance to the back of the queue is then determ~ned,know~ngL. The adjusted saturation flow is determinedirom Equation C-8 for the upstream movement of interest. Preliminary analyses based on initial pointers and Equation D-43 estimate whether queue spilIback is likely. Should the completed t~me-step analysis not support the initial assumption, then the alternate equation is selected and the process repeated. Sfep 6. Identify Clear Period The clear period is defined as the time in the cycle from the end of queue blocking to the start of blocking in the next cycle at the upstream intersection. The clear period, CP, is the duration from T. to T3 when upstream input flow can occur, or CP = T3 T. (D-4g) The values of To and T3 have not been calculated module C. Thus, they can have values greater than the cycle length. While calculating the clear period, the values of SMm and EMm should be adjusted for start loss and end gun, respectively, at some convenient point in the program. Sfep 7. Compute the Modif edF Effective Green Period The modified effective green period (get) during which the upstream movement can discharge vehicles is the time overlap of the unblocked effective green (au) of the movement arid Me clear period. The clear period is calculated in the previous step. Table D- ~ ~ shows the modified effective green periods (aged) for Movement ~ for different positions of to and t3 with respect to the upstream signal time as calculated in the PDX program. Thus, the real effective green (g = geld is g = g rip CP D -28 (D-49)

Table D-~. Modified Effective Green Periods for Movement Value of I, ~Value of t3 ~gerf SMOG ~ t' < EM, ~EM t3>t, ~t3-t, | EM < t3 > it; t3 ~| EM' - I, | EM, < t3 > I,; mod(t3, ~ -)<t, | EM~-t, + t3-SM~ l | EM, < t3 > I,; mod(t3, ~ ~)>t, | EM - SM, EM, < to < C ~I, < t3 > C; mod(t3, C)< EM, ~t3 - SMOG I, < t3 > C; mod(t3, C) EM, ~EM, - SM, ~t3<2C; mod(t3,C)q,M, ~ mod(t3,C)-mod(t,,C) I, > C; ~t3 < 2C; C > mod(t3, C) ~EM, ~ EM, - mod(t,, C) SMOG< mod(t,,C) EM, ~ t3>2C;mod(t,,C)>mod(t3, -)<EM, ~ EM, - mod(t,, C) t3 > 2C; mod(t,, C)<mo. [(t3, C) ~ EM, - SM, A similar table can be developed for estimating the modified effective green periods ofthe other upstream movements. Step 8. Calculate the Capacity of file Movement The reduced capacity of the upstream movement is calculated using the updated effective green period by the following formula: c = Sa (ge~/c) (D-50) The adjusted saturation flow rate saincludes the queue interaction effects as given by Equation C-~. The above steps were coded in FORTRAN. The output of the PDX program was entered into a file which could be easily be loaded into a spreadsheet package for farther analysis. D.7.3 Mode! Testing and Verification The PDX Model has been tested for various traffic operating conditions. An expenmental testbed was designed for this purpose. To verify the test results, the microscopic traffic simulation program' TRAF-NETSIM, was used to provide comparative results under the same operating conditions. The following sections describe the procedures and results of this testing. D -29

Testbed Design. In order to test the above program's applicability,an experimental teethed was designed. An arbitrary paired intersection system was set up with all the traffic and signal timing variables affecting saturation flow being defined. This test system was analyzed using the Prosser-Dunne FORTRAN program for a range of conditions and the movement's capacities were computed. A design scheme of the study paired intersection was shown in Figure D-9. Required inputs such as traffic volumes, signal timing parameters such as green times, offsets, cycle lengths, and spacing between intersection were carefully prepared. Only a pretimed signal system was considered for this research. The spacing between the two intersections was considered to be 100, 200, and 300 meters, respectively. These spacings were assumed to be representative of most closely-spaced intersections within an interchange environment. Other parameters in the teethed are summarized in Table D-12. Table D-12. Testing Parameters for the Program and Simulation V/C Ratios 0.8, 0.9, 1.0, 1.1, 1.2, 1.S | Cycle length | 100 seconds Upstream Phase Splits ~ ?: | Downstream Phase Splits | 50-50, 60~O, 70-30 Offsets at ~ seconds Intervals (0 to C) | Spacing 7 100, 200, 300 meters l Total Cases Studied 900 1 ~ .. _. Total number of NETSIM simulations ~ Results and Verification. The computer program was run by using the above operating conditions to get Me data base. Each of the above cases was simulated 10 times using TRAF- NETSIM to obtain average simulation results. A total of 9000 NETSIM runs were performed during the testing process. The study results were categorized and evaluated according to different operating conditions and are summarized in the following sections. The throughput-offset relationship was examined in order to study the outputs from the PDX pro gram. The effects of different volume-to-capacity ratios and intersignal spacings were studied. A very close relationship between PDX program and NETSIM simulation was observed dunng the comparison process. The results are shown in Figures D-12 Trough D-15. In each figure, results from the PDX Model and TRAF-NETSIM are shown in the same dimension for comparison purposes. Figure D-16 is the regression plot between NETSIM arid PDX Model. The coefficient of regression between the two models was observed to be 0.85. Furler research studies are underway to improve the PDX Model. D - 30

. ~- 600 - ~ t _r ~ SOO - _ ~ · l~h~ugh-S'm ~ t I 1=~ _ X Tdel~im g300; - X ~oug~Model ~ ~ __ _ ~ · Righ~Model s 200 - ~ ~-M - l _ ~ Td a l-l Cod cl 100 O- l l l l 0 20 40 60 80 100 Offset (see) Figure D-12. Throughput-Offset Relationship between NETSIM and PDX Models for a spacing of 100 meters; v/c of 0.S and Saturation Flow of 1900 vphgpl. 2000 1600 -1200 800 400 o _ . _ ~-X ~ ~ X X X X ~ _ _^ _ _ ~ i it_ 0 20 40 60 Offset (see) 80 100 -Is Ritht-Sim ~ LcR-S~ - X Total~im X Throug~Motel let-Model I L~t-Model -- Total-Model Figure D-13. Throughput-Offset Relationship between NETSIM and PDX Models for a spacing of 100 meters; v/c of 1.5 and Saturation Flow of 1900 vphgpl. D -31

2000 1600 ~ ~=Through-Sim 1200 ~ LeB-Sim _ X Total-Sim S ~ Through-Modet g 800; _' \* \ · Right-Mod~t ' ~ ~i\. ~ ~-Total-Model ~ 400~ 0 20 40 60 80 100 Offset (see) Figure D-140 Throughput-Ofiset Relationship between NETSIM and PDX Models for a spacing of 200 meters; v/c of {.5 and Saturation Flow of 1900 vphgpl. 1' 2000 =~1600 ~_ , 47 ~+ ~|=T~ 41200- · Lcft-Sim ~X Total-Sim S _ ~-- : ~ lbrou0Model g 800- '~ ~ \ ~· Ritht-Modet ~ ~5 ~I -M-I ~I ~ ~ ~ am_ . ~Total-Model 40075~ ~ 0 20 40 60 80 100 Offset (see) Figure D-15. Throughput-Offset Relationship between NE:TSIM and PAX Models for a spacing of 300 meters; v/c of 1.5 and Saturation Flow of 1900 vphgpl. D-32

1200 ~ t ·. ~ .-.' I {~L6cc~ · i- 1 X t -~ :~r y=eg85~ 1 C Low at' ' ~, R2~0.8^ 1 0 200 400 600 Boo 1000 1200 NETSIM Throughput (vph) Figure D-16. Linear Regression between NETSIM and PDX Model Throughputs. D.7.4. Discussion of the PDX Mode! This computer program has been extensively calibrated against simulation results obtained from TRAF-NETSIM simulation program. The purpose of the calibration effort was to pinpoint the best fit parameter values used in this computer program to produce reasonable results. Traffic engineering judgement was also exercised during this process. Overall, the computer traffic program demonstrated good flexibility and accuracy in processing different types of traffic conditions based on the comparison results observed by the research team. It should be noted that in the original Prosser-Dunne Model, traffic operating conditions were always assumed to be oversaturated, therefore, blocking would always occur because of insufficient service capacity at the downstreamintersection. It was found, however, that blocking or queue spilIback may also occur during undersaturated conditions given the limited storage spacing and bad offsets. Project study results have shown the most important factors that affect the estimation of queue spilIback or blocking occurrence are downstream signal intersection's green time, the intersignal spacing (i.e., link length) and the volume-to-capacity ratio. Besides these factors, a critical spacing that defines the boundary condition of the occurrence of queue spiliback was identified as a function of the downstream intersection's green time and volume to capacity ratio and other parameters. This new methodology helps define different types of problems based D-33

on the varying nature of different operating conditions and renders the corresponding treatments. After applying these enhancements to the original Prosser-Dunne Model, a wide variety of real- worId operating conditions can now be categorized and evaluated systematically by their specific types of problems, such as queue spiliback due to inadequate storage spacing and/or oversaturation. Several key parameters used in the PDX computer program were calibrated extensively ureter different operating conditions. Sufficient attention has been given to the effects of the selected values of the saturation flow density on the subsequent calculation of other variables. Because of its direct impacts on the estimate of saturation flow speed and interacting traffic wave speed, any change to the saturation flow density would result in different mode] outputs. So far, the parameter values used in the model have been calibrated to produce reasonable outputs compared with the simulation results from TRAF-NETSIM. Other calibrated parameters include the phase lost time and the unimpeded saturation flow rate. A vehicle unit length of 7.0 meters (23 feet) at jam density was used as the result of our nationwide data collection and analysis effort. Another major Improvement made to the computer traffic mode] was the introduction of the Queue Estimation Model. As an important part of traffic signal operation, especially during saturation periods, the behavior and characteristics of queuing traffic and its evaluation methodology have been studied by various researchers recently. The study approach used by the research team during the mode! development end calibrationwas to obtain queue traffic information dynamically throughout the cycle. A queue calculation submodule was provided to achieve this purpose. Similarto Me capacity analysis methodology,the queue submodule performs second-by- second calculations to estimate the changing queue status at the downstream intersection. The useful information provided by this program can be subsequently analyzed or used in over traffic engineering models when studying the traffic operating conditions of a paired intersection. The effects of available downstream travel distance to the back of queue was also provided in the PDX Model. As presentedin Equation C-8, Me saturation flow rate on green may be reduced by insufficient clear distance at start of green that permits platoon vehicles from accelerating to nominal saturation flow speeds. Thus, the clear period may have impeded saturation flow, but not blocked flow. D.7.5 Existing Software Enhancements Implementation of Queue-Interaction Models into internationally recognized computer programs is highly recommended. Some work toward this objective is known to be already underway (10,11,129. D-34

REFERENCES 1. "Highway Capacity Manual." Special Report 209, Transportation Research Board, Washington, D.C., Third Edition, (1994). 2. Chang, E.C. and Messer, C.~. "PASSER Il-90 Users Manual." Texas Transportation Institute, College Station, Texas, (19901. Fambro, D. B., Chau&ary, N.A. and Messer, C.~. "PASSER IlI-90 User's Manual." Texas Transportation Institute, College Station Texas, (1991~. 4. Wallace, C.E. and Courage, K.G. "TRANSYT-7F User's Mar~ual." University of Florida, Gainesville, (1988~. May, A.D. Traffic Flow Fundamentals. Prentice-Hall, Englewood Cliffs, New Jersey, (1990) p.306. 6. Lighthill, M.~. and Whitham, G.B. "On Kinematic Waves: Part Il. A Theory of Traffic Flow On Long Crowded Roads." Proceeding of the Royal Society, A2239, No. INS, (1955~. . Fambro, D.B., Rouphail, N.M., SIoup. P.R., Daniel, I.R., Id, I., Anwar, M., and R.~. Engelbrecht. "Highway Capacity Revisions for Chapters 9 and ~ 1." Report No. FHWA- RD-96-088, Federal Highway Administration, Washington, D.C. (1996~. 8. Leiberman, E.B, McShane, M.R., and Messer, C.~. Traffic Signal Control For Saturated Conditions. KLD Associates, Inc. NCHRP Project 3-38~3) Report, Vol 2., (1992) p. 15. 9. Prosser, N. and Dunne, M. "A Procedure for Estimating Movement Capacities at Signalised Paired Intersections." 2nd International Symposium on Highway Capacity, Sydney, Australia, (1994~. 10. Rouphail, N.M. and Akcelik, R. "Paired Intersections: Initial Development of Platooned A~xivaland Queue Interaction Models." Australian Road Research Board. Working Paper WD TE91/010, Vermont South, Australia (19911. . Akcelik, R., Besley, M., and Shepherd, R. "SIDRA (Windows) Input Redesign for Paired Intersection Modelling." DiscussionNoteto WD 96/008. Australian Road Research Board, Vermont South, Australia (1996~. 12. Chaudhary, N.A. and Messer, C.~. "PASSERIV-96, Version2. l, User/ReferenceManu~." Texas Transportation Institute, College Station, Texas. (1996~. D-35

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