Signalized Intersections 3l591k

CEE 320 Winter 2006

Signalized Intersections

CEE 320 Steve Muench

Outline 1. 2. 3. 4.

Key Definitions Baseline Assumptions Control Delay Signal Analysis

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a. b. c. d.

D/D/1 Random Arrivals LOS Calculation Optimization

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

• Green Time (G) – The duration of the green indication of a given movement at a signalized intersection

• Red Time (R) CEE 320 Winter 2006

– The period in the signal cycle during which, for a given phase or lane group, the signal is red

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)

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– All red time – The period in the signal cycle during which all approaches have a red indication

Key Definitions (3) • Start-up Lost Time (l1)

– 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 (l2)

– Time between signal phases during which an intersection is not used by traffic. 2 seconds is typical.

• Lost Time (tL)

– Time when an intersection is not effectively used by any approach. 4 seconds is typical. – tL = l1 + l2

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• Total Lost Time (L) – Total lost time per cycle during which the intersection is not used by any movement.

Key Definitions (4) • Effective Green Time (g) – Time actually available for movement – g = G + Y + AR – tL

• Extension of Effective Green Time (e) – The amount of the change and clearance interval at the end of a phase that is usable for movement of vehicles

• Effective Red Time (r)

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– Time during which a movement is effectively not permitted to move. – r = R + tL – r=C–g

Key Definitions (5) • Saturation Flow Rate (s) – Maximum flow that could through an intersection if 100% green time was allocated to that movement. – s = 3600/h

• Approach Capacity (c) – Saturation flow times the proportion of effective green – c = s × g/C

• Peak Hour Factor (PHF)

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– The hourly volume during the maximum-volume hour of the day divided by the peak 15-minute flow rate within the peak hour; a measure of traffic demand fluctuation within the peak hour.

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

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• Critical Volume-to-Capacity Ratio (Xc) – The proportion of available intersection capacity used by vehicles in critical lane groups – In of v/c and NOT v/s

from Highway Capacity Manual 2000

Baseline Assumptions

• D/D/1 queuing • Approach arrivals < departure capacity

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– (no queue exists at the beginning/end of a cycle)

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)

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– Average delay per vehicle (seconds per vehicle)

D/D/1 Signal Analysis (Graphical)

Vehicles

Departure Rate

Arrival Rate

Queue dissipation Total vehicle delay per cycle Maximum delay

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Maximum queue

Time Red

Green

Red

Green

Red

Green

D/D/1 Signal Analysis – Numerical 

 

  1.0

• Time to queue dissipation after the start of effective green

r t0  1    • Proportion of the cycle with a queue

r  t0 Pq  c

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• Proportion of vehicles stopped

 r  t0  r  t0 Ps    Pq  r  g  c

 r  t0  t0 t0 Ps     r  g  c c

D/D/1 Signal Analysis – Numerical 

• Maximum number of vehicles in a queue

Qm  r • Total delay per cycle

r 2 Dt  21    • Average vehicle delay per cycle

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r 2 1 r2 Dt    21    c 2c1    • Maximum delay of any vehicle (assume FIFO)

dm  r

 

  1.0

Signal Analysis – Random Arrivals •

Webster’s Formula (1958) - empirical d' d 

1/ 3

x  c   0.65 2  x 2  5( g / c ) 2 1  x    2

d’ = avg. veh. delay assuming random arrivals

CEE 320 Winter 2006

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)

Signal Analysis – Random Arrivals •

Allsop’s Formula (1972) - empirical 9  x2  d '  d   10  2 1  x  

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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)

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

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• Six levels of service (A through F)

Signalized Intersection LOS • Based on control delay per vehicle

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– How long you wait, on average, at the stop light

from Highway Capacity Manual 2000

Typical Approach • Split control delay into three parts – Part 1: Delay calculated assuming uniform arrivals (d1). This is essentially a D/D/1 analysis. – Part 2: Delay due to random arrivals (d2) – Part 3: Delay due to initial queue at start of analysis time period (d3). Often assumed zero.

d  d1 PF   d 2  d3 d = Average signal delay per vehicle in s/veh CEE 320 Winter 2006

PF = progression adjustment factor d1, d2, d3 = as defined above

Uniform Delay (d1)  g 0.5C 1   C  d1  g  1  min 1, X   C  d1 = delay due to uniform arrivals (s/veh) C = cycle length (seconds)

g = effective green time for lane group (seconds) CEE 320 Winter 2006

X = v/c ratio for lane group

Incremental Delay (d2)  d 2  900T  X  1  

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d2 =

8kIX   X  1   cT  2

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

Initial Queue Delay (d3) • 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…

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– d3 = 0

Control Optimization •

Conflicting Operational Objectives –

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minimize vehicle delay – minimize vehicle stops – minimize lost time – major vs. minor service (progression) – pedestrian service – reduce accidents/severity – reduce fuel consumption – Air pollution

The “Art” of Signal Optimization •

Long Cycle Length –

High capacity (reduced lost time) – High delay on movements that are not served – Pedestrian movements? Number of Phases? •

Short Cycle Length –

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Reduced capacity (increased lost time) – Reduced delay for any given movement

Minimum Cycle Length C min

L Xc  n v X c    i 1  s  ci

Cmin = 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) CEE 320 Winter 2006

Xc = critical v/c ratio for the intersection

Optimum Cycle Length Estimation C opt

1.5L   5  n v 1   i 1  s  ci

Copt = estimated optimum cycle length (seconds) to minimize vehicle delay L = total lost time per cycle (seconds), 4 seconds per phase is typical

CEE 320 Winter 2006

(v/s)ci = flow ratio for critical lane group, i (seconds)

Green Time Estimation  v   C  gi       s i  X i  g = effective green time for phase, i (seconds) (v/s)i = flow ratio for lane group, i (seconds) C = cycle length (seconds)

CEE 320 Winter 2006

Xi = v/c ratio for lane group i

Pedestrian Crossing Time N ped L  G p  3.2    2.7 Sp  WE

  for WE  10 ft. 

L G p  3.2   0.27 N ped  for WE  10 ft. Sp Gp = minimum green time required for pedestrians (seconds) L = crosswalk length (ft) Sp = average pedestrian speed (ft/s) – often assumed 4 ft/s

WE = effective crosswalk width (ft) CEE 320 Winter 2006

3.2 = pedestrian startup time (seconds) Nped = number of pedestrians crossing during an interval

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Effective Width (WE)

from Highway Capacity Manual 2000

Example An intersection operates using a simple 3-phase design as pictured.

SB

WB

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EB Phase

Lane group

Saturation Flows

1

SB

3400 veh/hr

2

NB

3400 veh/hr

3

EB

1400 veh/hr

WB

1400 veh/hr

NB

Example What is the sum of the flow ratios for the critical lane groups? 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? SB 150

30 400

EB

30

200 300

20 1000 50

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100

NB

WB

Example Calculate an optimal signal timing (rounded up to the nearest 5 seconds) using Webster’s formula.

Copt 

1.5L   5 n

1   v s ci

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i 1

Example Determine the green times allocation using v/c equalization. Assume the extension of effective green time = 2 seconds and startup lost time = 2 seconds.

v   C  i 1  s  i Xc  CL n

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 v   C  gi       s i  X i 

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).

d v  v

dA

i

i

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i

d v  v k

i i

dI

k

k

k

k

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 Winter 2006

L G p  3.2   0.27 N ped  for WE  10 ft Sp

FYI – NOT TESTABLE

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Signal Installation: “Warrants” •

Manual of Uniform Traffic Control Devices (MUTCD)

•

Apply these rules to determine if a signal is “warranted” at an intersection

•

If warrants are met, doesn’t mean signals or control is mandatory

•

8 major warrants

•

Multiple warrants usually required for recommending control http://mutcd.fhwa.dot.gov/

FYI – NOT TESTABLE

CEE 320 Winter 2006

Intersection Control Type

from Highway Capacity Manual 2000

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Primary References •

Mannering, F.L.; Kilareski, W.P. and Washburn, S.S. (2003). Principles of Highway Engineering and Traffic Analysis, Third Edition (Draft). Chapter 7

•

Transportation Research Board. (2000). Highway Capacity Manual. National Research Council, Washington, D.C.