CEE 320 Fall 2008 Course Logistics HW7 due today (9 total) Midterm next Friday (Wednesday review) Signalized Intersections (Chapter 7 of text) Last material before midterm Final set of topics: transportation planning

CEE 320 Fall 2008 Signalized Intersections CEE 320 Anne Goodchild

CEE 320 Fall 2008 Outline 1.Key Definitions 2.Baseline Assumptions 3.Control Delay 4.Signal Analysis a.D/D/1 b.Random Arrivals c.LOS Calculation d.Optimization

CEE 320 Fall 2008 Key Definitions (1) Cycle Length (C) –The total time for a signal to complete a cycle Phase –The part of the signal cycle allocated to any combination of traffic movements receiving the ROW simultaneously during one or more intervals (a consistent period) Green Time (G) –The duration of the green indication of a given movement at a signalized intersection Red Time (R) –The period in the signal cycle during which, for a given phase or lane group, the signal is red

CEE 320 Fall 2008 Key Definitions (2) Change Interval (Y) –Yellow time –The period in the signal cycle during which, for a given phase or lane group, the signal is yellow Clearance Interval (AR) –All red time –The period in the signal cycle during which all approaches have a red indication

CEE 320 Fall 2008 Key Definitions (3) Start-up Lost Time (l 1 ) –Time used by the first few vehicles in a queue while reacting to the initiation of the green phase and accelerating. 2 seconds is typical. Clearance Lost Time (l 2 ) –Time between signal phases during which an intersection is not used by traffic. 2 seconds is typical. Lost Time (t L ) –Time when an intersection is not effectively used by any approach. 4 seconds is typical. –t L = l 1 + l 2

CEE 320 Fall 2008 Key Definitions (4) Effective Green Time (g) –Time effectively utilized for movement –g = G + Y + AR – t L Effective Red Time (r) –Time during which a movement is effectively not permitted to move. –r = R + t L –r = C – g

CEE 320 Fall 2008 Red Green YellowRed C YRG AR l1l1 headways typically longer saturation headway l2l2 time space end of intersection

CEE 320 Fall 2008 Key Definitions (5) Saturation Flow Rate (s) –Maximum flow that could pass through an intersection if 100% green time was allocated to that movement. –S (vehicles/hour) = 3600/headway (seconds per vehicle) Approach Capacity (c) –Saturation flow times the proportion of effective green –c = s × g/C

CEE 320 Fall 2008 Key Definitions (6) Flow Ratio –The ratio of actual flow rate (v) to saturation flow rate (s) for a lane group at an intersection Lane Group –A set of lanes established at an intersection approach for separate analysis Critical Lane Group –The lane group that has the highest flow ratio (v/s) for a given signal phase Critical Volume-to-Capacity Ratio (X c ) –The proportion of available intersection capacity used by vehicles in critical lane groups –In terms of v/c and NOT v/s

CEE 320 Fall 2008 Baseline Assumptions D/D/1 queuing Approach arrivals < departure capacity –(no queue exists at the beginning/end of a cycle)

CEE 320 Fall 2008 Quantifying Control Delay Two approaches – Deterministic (uniform) arrivals (Use D/D/1) – Probabilistic (random) arrivals (Use empirical equations) Total delay can be expressed as –Total delay in an hour (vehicle-hours, person-hours) –Average delay per vehicle (seconds per vehicle)

CEE 320 Fall 2008 D/D/1 Signal Analysis (Graphical) Arrival Rate Departure Rate Time Vehicles Maximum delay Maximum queue Total vehicle delay per cycle Red Green Queue dissipation

CEE 320 Fall 2008 D/D/1 Signal Analysis – Numerical Time to queue dissipation after the start of effective green Proportion of the cycle with a queue Proportion of vehicles stopped

CEE 320 Fall 2008 D/D/1 Signal Analysis – Numerical Maximum number of vehicles in a queue Total delay per cycle Average vehicle delay per cycle Maximum delay of any vehicle (assume FIFO)

CEE 320 Fall 2008 Definition – Level of Service (LOS) Chief measure of “quality of service” –Describes operational conditions within a traffic stream –Does not include safety –Different measures for different facilities Six levels of service (A through F) Based on control delay measure

CEE 320 Fall 2008 Control Delay Applies to both signalized and not signalized intersections Referred to as signal delay for a signalized intersection Total delay experienced by the driver as a result of the control Includes deceleration time, queue move- up time, stop time, and acceleration time

CEE 320 Fall 2008 Signal Analysis – Random Arrivals Webster’s Formula (1958) - empirical d ’ = avg. veh. delay assuming random arrivals d = avg. veh. delay assuming uniform arrivals (D/D/1) x = ratio of arrivals to departures ( c/  g) g = effective green time (sec) c = cycle length (sec)

CEE 320 Fall 2008 Signal Analysis – Random Arrivals Allsop’s Formula (1972) - empirical d ’ = avg. veh delay assuming random arrivals d = avg. veh delay assuming uniform arrivals (D/D/1) x = ratio of arrivals to departures ( c/  g)

CEE 320 Fall 2008 Signalized Intersection LOS Based on control delay per vehicle –How long you wait, on average, at the stop light from Highway Capacity Manual 2000

CEE 320 Fall 2008 Typical Approach Split control delay into three parts –Part 1: Delay calculated assuming uniform arrivals (d 1 ). This is essentially a D/D/1 analysis. –Part 2: Delay due to random arrivals (d 2 ) –Part 3: Delay due to initial queue at start of analysis time period (d 3 ). d=Average signal delay per vehicle in s/veh PF=progression adjustment factor d 1, d 2, d 3 =as defined above

CEE 320 Fall 2008 Uniform Delay (d 1 ) d1d1 =delay due to uniform arrivals (s/veh) C=cycle length (seconds) g=effective green time for lane group (seconds) X=v/c ratio for lane group

CEE 320 Fall 2008 Incremental Delay (d 2 ) d2d2 =delay due to random arrivals (s/veh) T=duration of analysis period (hours). If the analysis is based on the peak 15-min. flow then T = 0.25 hrs. k=delay adjustment factor that is dependent on signal controller mode. For pretimed intersections k = 0.5. For more efficient intersections k < 0.5. I=upstream filtering/metering adjustment factor. Adjusts for the effect of an upstream signal on the randomness of the arrival pattern. I = 1.0 for completely random. I < 1.0 for reduced variance. c=lane group capacity (veh/hr) X=v/c ratio for lane group

CEE 320 Fall 2008 Initial Queue Delay (d 3 ) Applied in cases where X > 1.0 for the analysis period –Vehicles arriving during the analysis period will experience an additional delay because there is already an existing queue When no initial queue… –d 3 = 0

CEE 320 Fall 2008 Control Optimization Conflicting Operational Objectives – minimize vehicle delay Fuel consumption Air quality – minimize vehicle stops – minimize lost time – major vs. minor service (progression) – pedestrian service – reduce accidents/severity

CEE 320 Fall 2008 The “Art” of Signal Optimization Long Cycle Length – High capacity (reduced lost time) – High delay on movements that are not served – Less efficient if uneven demand Short Cycle Length – Reduced capacity (increased lost time) – Reduced delay for any given movement – More efficient if equal demand “snappy” operations

CEE 320 Fall 2008 Minimum Cycle Length C min =estimated minimum cycle length (seconds) L=total lost time per cycle (seconds), 4 seconds per phase is typical (v/s) ci =flow ratio for critical lane group, i (seconds) XcXc =critical v/c ratio for the intersection

CEE 320 Fall 2008 Minimum cycle length set X c = 1.0 critical v/c will be 1 –you can just squeeze all the vehicles through on that phase’s green time However, if you set X c = 1 –there will be times when more arrivals than your assumed v will show up and the cycle will fail Therefore, often values less than 1 are assumed for X c (such as 0.90).

CEE 320 Fall 2008 Optimum Cycle Length Estimation C opt =estimated optimum cycle length (seconds) to minimize vehicle delay L=total lost time per cycle (seconds), 4 seconds per phase is typical (v/s) ci =flow ratio for critical lane group, i (seconds)

CEE 320 Fall 2008 Green Time Estimation g=effective green time for phase, i (seconds) (v/s) i =flow ratio for lane group, i (seconds) C=cycle length (seconds) XiXi =v/c ratio for lane group i

CEE 320 Fall 2008 Pedestrian Crossing Time GpGp =minimum green time required for pedestrians (seconds) L=crosswalk length (ft) SpSp =average pedestrian speed (ft/s) – assumed 4 ft/s WEWE =effective crosswalk width (ft) 3.2=pedestrian startup time (seconds) N ped =number of pedestrians crossing during an interval

CEE 320 Fall 2008 Effective Width (W E ) from Highway Capacity Manual 2000

CEE 320 Fall 2008 Examples Signalized Intersections

CEE 320 Fall 2008 Example At an intersection, saturation headways of 3 seconds are observed. What is the saturation flow rate? s=3600/3=1200 vehicles/hour NB SB EB WB

CEE 320 Fall 2008 Example If G NB = 20 seconds, Y NB = 3 seconds, R NB = 18 seconds, and AR= 2 seconds. What are the effective red and green times? The cycle time, and the lane- group capacity? g= = 29 seconds r=18+4=22 seconds C=22+29=51 seconds c=1200*29/51=682 vehicles/hour NB SB EB WB

CEE 320 Fall 2008 D/D/1 Signal Analysis (Graphical) Arrival Rate Departure Rate Time Vehicles Maximum delay Maximum queue Total vehicle delay per cycle Red Green Queue dissipation Assumes  g> C

CEE 320 Fall 2008 Determine total vehicle delay over 3 cycles if arrival rate is 500 veh/hr Arrival rate: 500/3600=.139 veh/sec Departure rate: 1200/3600=.333 vehicles/second Traffic intensity =.139/.333 =.417 Check capacity exceed arrivals: –.333x29=9.65 vehicles can get through on green –.139x51=7.09 vehicles arrive in cycle

CEE 320 Fall 2008 Total vehicle delay One cycle – 0.139*(22) 2 /2(1-.417)=57.5 vehicle seconds Three cycles = 173 vehicle seconds

CEE 320 Fall 2008 Example An intersection operates using a simple 3-phase design as pictured. NB SB EB WB PhaseLane group Saturation Flows 1SB3400 veh/hr 2NB3400 veh/hr 3EB1400 veh/hr WB1400 veh/hr

CEE 320 Fall 2008 Example SB NB EB WB What is the sum of the flow ratios for the critical lane groups (Y c )? What is the total lost time for a signal cycle assuming 2 seconds of clearance lost time and 2 seconds of startup lost time per phase?

CEE 320 Fall 2008 Key Definitions Flow Ratio: The ratio of actual flow rate (v) to saturation flow rate (s) for a lane group Critical Lane Group: The lane group that has the highest flow ratio (v/s) for a given signal phase Volume to Capacity ratio (X): v/c or C/  g Critical Volume-to-Capacity Ratio (X c ): The proportion of available intersection capacity used by vehicles in critical lane groups Sum of the Flow Ratios for the Critical Lane Groups (Y c ): sum of flow ratios for critical lane groups

CEE 320 Fall 2008 Example PHASE 1: SB T/left/right= ( )/3400 = PHASE 2: NB T/left/right = ( )/3400 = PHASE 3: –EB T/right = (200+20)/1400 = –WB T/right = (300+30)/1400 =  limiting since v/s is highest Y c = = Total lost time = 3(2+2) = 12 seconds

CEE 320 Fall 2008 Example Calculate an optimal cycle length using Webster’s formula, and a minimum cycle length. C opt = 1.5(12 seconds) + 5/( ) = 90.2 seconds C min = (12 seconds) (0.9)/ ( ) = 69.7 seconds

CEE 320 Fall 2008 Minimum cycle length set X c = 1.0 critical v/c will be 1 –you can just squeeze all the vehicles through on that phase’s green time However, if you set X c = 1 –there will be times when more arrivals than your assumed v will show up and the cycle will fail Therefore, often values less than 1 are assumed for X c (such as 0.90).

CEE 320 Fall 2008 Example Determine the green times allocation (assume C=95 seconds)

CEE 320 Fall 2008 DETERMINE X c X c = 0.745(95)/(95 – 12) = CALCULATE EFFECTIVE GREEN TIMES g SB = 0.171(95/0.853) = seconds g NB = 0.338(95/0.853) = seconds g EBWB = (95/0.853) = seconds CHECK = = 95 seconds

CEE 320 Fall 2008 Example What is the intersection Level of Service (LOS)? Assume in all cases that PF = 1.0, k = 0.5 (pretimed intersection), I = 1.0 (no upstream signal effects).

CEE 320 Fall 2008 Signalized Intersection LOS Based on control delay per vehicle –How long you wait, on average, at the stop light from Highway Capacity Manual 2000

CEE 320 Fall 2008 Typical Approach Split control delay into three parts –Part 1: Delay calculated assuming uniform arrivals (d 1 ). This is essentially a D/D/1 analysis. –Part 2: Delay due to random arrivals (d 2 ) –Part 3: Delay due to initial queue at start of analysis time period (d 3 ). d=Average signal delay per vehicle in s/veh PF=progression adjustment factor d 1, d 2, d 3 =as defined above

CEE 320 Fall 2008 Uniform Delay (d 1 ) d1d1 =delay due to uniform arrivals (s/veh) C=cycle length (seconds) g=effective green time for lane group (seconds) X=v/c ratio for lane group

CEE 320 Fall 2008 Incremental Delay (d 2 ) d2d2 =delay due to random arrivals (s/veh) T=duration of analysis period (hours). If the analysis is based on the peak 15-min. flow then T = 0.25 hrs. k=delay adjustment factor that is dependent on signal controller mode. For pretimed intersections k = 0.5. For more efficient intersections k < 0.5. I=upstream filtering/metering adjustment factor. Adjusts for the effect of an upstream signal on the randomness of the arrival pattern. I = 1.0 for completely random. I < 1.0 for reduced variance. c=lane group capacity (veh/hr) X=v/c ratio for lane group

CEE 320 Fall 2008 Determine the delay for each lane group SB lane group c = s (g/C) = 3200(19.04/95) = vehicles d 1 = (0.5)(95)(1 – 19.04/95) 2 /(1 – 0.853(19.04/95)) = seconds d 2 = 900(0.25)(( ) + sqrt((0.853 – 1) 2 + 8(0.5)(1.0)(0.853)/((641.35)(0.25))) = seconds d 3 = 0 (assumed) d = = seconds

CEE 320 Fall 2008 NB lane group c = s (g/C) = 3200(37.64/95) = vehicles d 1 = (0.5)(95)(1 – 37.64/95) 2 /(1 – 0.853(37.64/95)) = seconds d 2 = 900(0.25)(( ) + sqrt((0.853 – 1) 2 + 8(0.5)(1.0)(0.853)/(( )(0.25))) = 7.41 seconds d 3 = 0 (assumed) d = = seconds

CEE 320 Fall 2008 EB lane group c = s (g/C) = 1400(26.28/95) = vehicles d 1 = (0.5)(95)(1 – 26.28/95) 2 /(1 – 0.853(26.28/95)) = seconds d 2 = 900(0.25)(( ) + sqrt((0.853 – 1) 2 + 8(0.5)(1.0)(0.853)/((387.28)(0.25))) = seconds d 3 = 0 (assumed) d = = seconds

CEE 320 Fall 2008 WB lane group c = s (g/C) = 1400(26.28/95) = vehicles d 1 = (0.5)(95)(1 – 26.28/95) 2 /(1 – 0.853(26.28/95)) = seconds d 2 = 900(0.25)(( ) + sqrt((0.853 – 1) 2 + 8(0.5)(1.0)(0.853)/((387.28)(0.25))) = seconds d 3 = 0 (assumed) d = = secon

CEE 320 Fall 2008 Find the weighted average (by flow) of delay for the four lane groups d I = ((50.15)(580) (1150) (220) (330))/( ) = seconds From Table 7.4 this equates to LOS D (not very good)

CEE 320 Fall 2008 Example Is this signal adequate for pedestrians? A pedestrian count showed 5 pedestrians crossing the EB and WB lanes on each side of the intersection and 10 pedestrians crossing the NB and SB crosswalks on each side of the intersection. Lanes are 12 ft. wide. The effective crosswalk widths are all 10 ft.

CEE 320 Fall 2008 EB/WB G p = / (5) = seconds NB/SB G p = / (10) = seconds Shortest green time was 19 seconds, so OK

CEE 320 Fall 2008 Intersection Control Type from Highway Capacity Manual 2000 FYI – NOT TESTABLE