12.2 Traffic Signal Coordination: Bandwidth, Offsets, & Progression Models

12.2 Traffic Signal Coordination: Bandwidth, Offsets, & Progression Models

PTOE Exam Focus: Traffic signal coordination is a core topic in Domain 4. Exam questions frequently require calculating optimal system cycle lengths using Webster's equation ($C_{\text{opt}} = \frac{1.5L + 5}{1 - \sum Y_i}$), computing link travel times and offsets, evaluating Bandwidth Efficiency and Attainability percentages, determining optimal signal spacing for alternate progression ($d = \frac{v \cdot C}{2}$), and interpreting time-space diagrams.

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1. Principles & Objectives of Signal Coordination

When traffic signals are spaced within $\frac{1}{2}\text{ mile}$ ($2,640\text{ ft}$) of one another—or where platoon dispersion is minimal—operating signals in isolation causes severe stop-and-go delays, excessive fuel consumption, elevated emissions, and rear-end crash risks. Signal coordination synchronizes adjacent intersections along an arterial corridor or network to allow platoons of vehicles to travel through multiple intersections without stopping.

Primary Coordination Objectives:

1. Establish Green Bands (Progression Bands): Provide continuous time-space windows through which platoons travel at design progression speeds ($v$).
2. Reduce Arterial Delay & Stops: Concentrate stops and queues on minor cross streets while maintaining smooth arterial flow.
3. Manage Queue Spillback: Prevent queues at downstream intersections from spilling back and blocking upstream intersection boxes.

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2. System Cycle Length Selection & Webster's Formula

All coordinated intersections within a coordinated subsystem must share a common system cycle length ($C$) (or an integer multiple/fraction such as half-cycling $C/2$ or double-cycling $2C$) to maintain stationary time relationships.

The system cycle length is dictated by the critical intersection (the most heavily congested intersection requiring the longest cycle length to satisfy capacity demands).

Webster's Optimum Minimum Delay Cycle Length Formula:

$C_{\text{opt}} = \frac{1.5 L + 5}{1 - \sum_{i=1}^{n} Y_i}$

Where:

• $L$ = Total intersection lost time per cycle (seconds), $L = \sum (t_L) = \sum (l_1 + l_2) = \sum (Y_i + R_{ci} - e_i)$
• $Y_i = \frac{v_i}{s_i}$ = Critical flow ratio for phase $i$ (design volume $v_i$ divided by saturation flow rate $s_i$)
• $\sum Y_i$ = Sum of critical flow ratios for all conflicting phase stages (must be $< 1.0$ for undersaturated conditions)

+-----------------------------------------------------------------------------------+
|                         WEBSTER'S DELAY VS CYCLE LENGTH                           |
|                                                                                   |
|  Average Delay                                                                    |
|  (sec/veh) ^                                                                      |
|            |    \                                    /                            |
|            |     \   Unsaturated                   /   Oversaturated Delay        |
|            |      \  Lost Time Penalty           /     (Excessive Red Time)       |
|            |       \                           /                                  |
|            |        \__                     __/                                   |
|            |           \___             ___/                                      |
|            |               \___ C_opt _/                                          |
|            +--------------------+----------------------------------------->       |
|            0                   C_opt                       Cycle Length (s)       |
+-----------------------------------------------------------------------------------+

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3. Time-Space Diagrams & Progression Geometry

A Time-Space Diagram is a two-dimensional plot representing signal coordination along an arterial:

• Y-Axis (Vertical): Cumulative distance along the arterial corridor (feet or miles).
• X-Axis (Horizontal): Time elapsed in seconds (spanning several cycle lengths).
• Signal Timing Bands: Horizontal green, yellow, and red intervals plotted for each intersection along the corridor.
• Vehicle Trajectories: Sloped lines representing platoons moving along the corridor. The slope of the trajectory line equals progression speed: $\text{Slope} = \frac{\Delta x}{\Delta t} = v$ (in $\text{ft/s}$). A steeper line represents higher vehicle speed.

  Distance (ft) ^
  Int 3 (2640') |=== [ Red ] ===[  Green  ]===[ Red ]===[  Green  ]===[ Red ]===
                |              /             /         /             /
                |             /   GREEN     /         /   GREEN     /
  Int 2 (1320') |=== [   Green   ]===[ Red ]===[   Green   ]===[ Red ]===[  Green  ]
                |           /             /         /             /
                |          /     BAND    /         /     BAND    /
  Int 1 (0')    |=== [  Green  ]===[ Red ]===[  Green  ]===[ Red ]===[  Green  ]===
                +----+-----------+-----------+----+-----------+-----------+-------->
                0    20          50          90   110         140         180 Time (s)
                     |<------ Bandwidth ---->|

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4. Offsets & Coordinated Phase Parameters

A. Offset Definition

Offset is the time difference (in seconds or percentage of cycle length) between a master system time reference point (the system sync pulse / Master Zero) and a defined local intersection milestone.

• Reference Milestones: The local reference point is most commonly defined as the Start of Coordinated Phase Green (or yellow onset, or center of green).

B. Ideal One-Way Offset Formulation

For ideal progression in a single direction over link distance $d$ (ft) at speed $v$ (mph):

$\text{Travel Time } t = \frac{d}{1.467 \cdot v}$ $\text{Offset}_{i} = \left( \text{Offset}_{i-1} + \frac{d_{i-1, i}}{1.467 \cdot v} \right) \pmod C$

C. Coordinated Controller Constraints (Force-Offs & Yield Points)

In semi-actuated coordinated controllers:

• Yield Point: The exact point in the cycle where the coordinated phase green is allowed to terminate only if a valid call exists on an actuated non-coordinated phase.
• Force-Off Point: A fixed point in the cycle where an actuated non-coordinated phase is forcibly terminated, guaranteeing that the controller returns to the coordinated main street phase on time.
• Permissive Window: The time window following the yield point during which an actuated call can be accepted to service a minor phase without disrupting coordination.

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5. Bandwidth Efficiency & Attainability

Progression quality is measured using two dimensionless efficiency ratios:

1. Bandwidth Efficiency ($E$)

Measures the proportion of the total cycle length dedicated to unobstructed platoon progression:

$E = \left( \frac{B}{C} \right) \times 100\%$

Where:

• $B$ = Progression bandwidth duration (seconds)
• $C$ = Cycle length (seconds)
• Evaluation Scale: $E < 30%$ is poor/marginal; $30%\text{ to }40%$ is good; $E > 40%$ is excellent.

2. Bandwidth Attainability ($A$)

Measures how effectively the available coordinated green time is converted into progression bandwidth:

$A = \left( \frac{B}{g_{\text{coord\_min}}} \right) \times 100\%$

Where:

• $g_{\text{coord_min}}$ = Minimum through green time displayed at any critical intersection along the corridor.
• Evaluation Scale: $A = 100%$ indicates that the progression band captures the entirety of the most restrictive green interval.

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6. Progression Systems: Alternate vs. Simultaneous Systems

+-----------------------------------------------------------------------------------+
|                         PROGRESSION NETWORK TYPOLOGIES                            |
|                                                                                   |
|  1. Alternate System (Single Alternate):                                          |
|     • Signals show alternating green / red indications (Offset = 0.5 C = 180°).   |
|     • Ideal Spacing: d = (v * C) / 2                                              |
|     • Provides 100% two-way progression bandwidth with equal green splits!        |
|                                                                                   |
|  2. Double Alternate System:                                                      |
|     • Signals operate in pairs with identical indications (Offset = 0.5 C pairs). |
|     • Ideal Spacing: d = (v * C) / 4                                              |
|                                                                                   |
|  3. Simultaneous System:                                                          |
|     • All signals along the corridor turn green and red simultaneously (Offset=0).|
|     • Ideal Spacing: Very short block lengths d < (v * C) / 4 (e.g., downtowns).  |
+-----------------------------------------------------------------------------------+

Mathematical Proof for Ideal Alternate Spacing:

To achieve bidirectional two-way progression with equal bandwidth on an arterial with uniform spacing $d$ and speed $v$, the travel time between adjacent intersections must equal half the cycle length ($t = \frac{C}{2}$):

$t = \frac{d}{1.467 \cdot v} = \frac{C}{2} \implies d = \frac{1.467 \cdot v \cdot C}{2} = 0.7335 \cdot v \cdot C$

Comparison Matrix: Arterial Signal Progression Systems

7. Worked Calculation Example: Webster's Optimum Cycle & Bandwidth Attainability

Problem Statement:

A critical 4-phase intersection on an arterial has total phase lost time of $L = 16\text{ seconds}$ per cycle. The critical lane group volume-to-saturation flow ratios ($Y_i = v_i / s_i$) are:

• Phase 1 (EB Left): $Y_1 = 0.12$
• Phase 2 (WB Thru): $Y_2 = 0.28$
• Phase 3 (SB Left): $Y_3 = 0.08$
• Phase 4 (NB Thru): $Y_4 = 0.17$

Along the arterial corridor, the coordinated green band achieved by SYNCHRO optimization is $B = 24\text{ seconds}$, and the minimum coordinated through green displayed at any intersection is $g_{\text{coord_min}} = 40\text{ seconds}$.

1. Calculate the optimum cycle length ($C_{\text{opt}}$) using Webster's formula.
2. Assuming a design system cycle length of $C = 90\text{ seconds}$, calculate the Bandwidth Efficiency ($E$).
3. Calculate the Bandwidth Attainability ($A$).

Step-by-Step Solution:

1. Webster's Optimum Cycle Length ($C_{\text{opt}}$): $\sum Y_i = Y_1 + Y_2 + Y_3 + Y_4 = 0.12 + 0.28 + 0.08 + 0.17 = 0.65$ $C_{\text{opt}} = \frac{1.5 L + 5}{1 - \sum Y_i} = \frac{1.5(16) + 5}{1 - 0.65} = \frac{24 + 5}{0.35} = \frac{29}{0.35} = 82.86\text{ seconds} \approx 83\text{ seconds}$

2. Bandwidth Efficiency ($E$): $E = \left( \frac{B}{C} \right) \times 100\% = \left( \frac{24\text{ s}}{90\text{ s}} \right) \times 100\% = 26.67\%$

3. Bandwidth Attainability ($A$): $A = \left( \frac{B}{g_{\text{coord\_min}}} \right) \times 100\% = \left( \frac{24\text{ s}}{40\text{ s}} \right) \times 100\% = 60.0\%$