Traffic Flow Fundamentals and Transport Data

Fundamental Diagram (FD) and Greenshields Model

• FD describes graphical relationships for uninterrupted traffic flow (speed, flow, density) and is used to identify special points and model relationships.

• Special points in the FD:

  • Free-flow speed: $v_f$

  • Capacity (maximum flow): $q_c$

  • Critical density: $k_c$

  • Speed at capacity: $v_c$

  • Jam density: $k_j$

• Greenshields model (a foundational mathematical model for the FD):

  • Linear speed–density relationship: $v = v<em>f[1 - \frac{k}{k</em>j}]$

  • Flow (throughput) relationship: $q = v k$

  • Substituting #1 into #2 gives a parabolic flow–density relation:
    

    $q = v<em>f k - \frac{v</em>f}{k_j} k^2$

  • Expressing density in terms of speed:
    

    $k = k<em>j \left(1 - \frac{v}{v</em>f}\right)$

  • Substituting #3 into $q = v k$ yields a parabolic speed–flow relation in terms of speed:
    

    $q = k<em>j v - \frac{k</em>j}{v_f} v^2$

• Capacity, critical density, and speed at capacity derived from Greenshields:

  • Critical density: $k<em>c = \frac{k</em>j}{2}$

  • Capacity (maximum flow): $q<em>c = \frac{v</em>f k_j}{4}$

  • Speed at capacity: $v<em>c = \frac{v</em>f}{2}$

• Key interpretations:

  • At density below $k<em>c$ the flow increases with density; after $k</em>c$, flow decreases as density increases.

  • The FD is often visualized as a backward-C or parabolic curve in q–k space.

• Parabolic speed–flow relationship (derived form):

  • $q = k<em>j v - \frac{k</em>j}{v_f} v^2$

  • This form can be used to compute capacity and speed-at-capacity by locating the maximum of q with respect to v.

• Practical example (from the notes): Given a flow–density relation of Greenshields form, one can compute:

  • Free-flow speed: $v_f = 80 \mathrm{km/h}$

  • Jam density: $k_j = 200 \mathrm{veh/km}$

  • Capacity (maximum flow): $q<em>c = \frac{v</em>f k_j}{4} = \frac{80\times 200}{4} = 4000 \mathrm{veh/h}$

  • Critical density: $k<em>c = \frac{k</em>j}{2} = 100 \mathrm{veh/km}$

  • Speed at capacity: $v<em>c = \frac{v</em>f}{2} = 40 \mathrm{km/h}$

  • Check: $q<em>c = v</em>c k_c = 40\times100 = 4000 \mathrm{veh/h}$

• Derivations and relationships (overview):

  • The model yields three primary relationships for the FD:
    

    1) Flow–density: $q(k) = v<em>f k - \frac{v</em>f}{k<em>j} k^2$ 2) Speed–density: $v(k) = v</em>f\left(1 - \frac{k}{k<em>j}\right)$ 3) Speed–flow: $q(v) = k</em>j v - \frac{k<em>j}{v</em>f} v^2$

• Points of caution and limitations:

  • Greenshields predicts speed always decreases with increasing density; field data on modern motorways show a plateau of near free-flow speed up to a relatively high density before decline begins. This motivates more advanced or modified models.

  • Modified Greenshields and other models were proposed to address discrepancies at low density.

• Occupancy as a surrogate for density: in practical traffic data, occupancy (the percentage of time a detector is occupied by a vehicle) is often used as a proxy for density, though it is not a direct measure of veh/km.

• Summary of fundamental relationships:

  • Speed–Density: $v = v<em>f\left(1 - \frac{k}{k</em>j}\right)$

  • Flow–Density: $q = v<em>f k - \frac{v</em>f}{k_j} k^2$

  • Speed–Flow: $q = k<em>j v - \frac{k</em>j}{v_f} v^2$

Special Points in the FD (revisited)

• Free-flow speed: $v_f$

• Jam density: $k_j$

• Capacity: $q_c$

• Critical density: $k_c$

• Speed at capacity: $v_c$

• Relationship: at capacity, density equals $k<em>c = k</em>j/2$, speed equals $v<em>c = v</em>f/2$, and capacity is $q<em>c = v</em>f k_j/4$

Data Relationships and Graph Interpretations

• Speed–Density: plotting speed vs density yields a downward trend with Greenshields; more complex real-world curves may flatten at low densities for multi-lane motorways with high capacity.

• Speed–Flow: parabolic shape where maximum flow occurs near the capacity point.

• Flow–Density: classic upside-down U-shaped curve; maximum at the critical density.

• Occupancy as a surrogate for density: occupancy often used in FD plots, but it is not a direct density and depends on the detector and traffic mix.

• Visual interpretation: the same traffic state can map to different points on the three FD plots depending on which variable is held constant and which is varied.

Data Collection Methods for Traffic Flow (Data Types and Tools)

• Three broad data collection categories:

  • Point measurement (road-side sensors): measures traffic at a fixed point; typically provides aggregated flow, speed, and density data.

  • Probe data collection (vehicle tracking sensors): follows a sample of vehicles along routes to obtain travel times and trajectories.

  • Video-based data collection (cameras, drones, rooftops): captures detailed movement trajectories over space and time; enables extraction of trajectories of all vehicles in a scene.

• Manual Traffic Counting:

  • Hand counters and intersection traffic volume collectors; useful for short-term studies; captures turning movements at intersections and roundabouts.

• Pneumatic Tubes (Intrusive Methods):

  • Rubber tubes that count axle pairs and classify vehicles by axle spacing using two detectors.

• Loop Detectors (Inductive Loops) (Intrusive Methods):

  • The most prevalent detector; detects vehicle presence and measures counts (volume), speed, and density.

• Radar Sensors (Non-Intrusive Methods):

  • Microwave radar measures speed via Doppler effect; can be deployed as handheld radar guns or fixed road-side sensors.

• Video Cameras (Non-Intrusive Methods):

  • Video image processing enables car counts and trajectory extraction; modern tools can track multiple vehicles and extract time–space trajectories.

• Video Image Processing (three-part process):

  • Car counts and vehicle detections from video; example figures show counts like 64 vehicles identified in a scene.

• Video-based Trajectory Data Collection:

  • Extracts detailed vehicle trajectories from video data using automatic image processing; yields time–space diagram representations of vehicle paths.

• Drone-based Trajectory Data Collection:

  • Uses drone-mounted cameras and computer vision to cover larger spatial areas; can capture diverse road users (vehicles, pedestrians, cyclists).

• Summary of data collection capabilities (technology trend):

  • Point measurement provides aggregated macroscopic data with high temporal resolution but limited spatial coverage.

  • Probe data provides point-to-point travel times and full trajectories for sample vehicles; improves travel-time estimates and route-level measurements.

  • Video-based data provides detailed microscopic trajectories over larger spatial areas; high spatial detail but requires processing power and can have sampling limitations.

  • Drone-based data extends spatial coverage further, enabling wide-area trajectory data, but may introduce more logistical considerations.

• Practical note: data collection technologies continue to evolve, improving spatial and temporal coverage and trajectory resolution (e.g., from loop detectors to drone-based imaging).

Volume, Demand, and Capacity

• Definitions:

  • Volume (flow): the actual number of vehicles passing a point per unit time (e.g., veh/h).

  • Demand: the number of vehicles that would like to pass a point if there were no constraints; often higher than observed volume during congestion.

  • Capacity: the physical limit or maximum rate at which vehicles can pass a point under current roadway conditions (intrinsic road property).

• The three concepts are related but distinct; typical relationships along a road segment depend on the region of the fundamental diagram in operation (uncongested vs congested).

• In the diagrams:

  • Capacity is the maximum achievable flow when the system is at its operating limit.

  • Volume is what actually passes given current conditions.

  • Demand is what would pass in the absence of capacity constraints; it can exceed capacity leading to queues.

Traffic Volume Studies and Metrics

• Volume characteristics commonly used in practice:

  • Annual Average Daily Traffic (AADT): the average 24-hour traffic over a full year (veh/day).

  • Average Daily Traffic (ADT): the average daily traffic over a sampling period (veh/day).

  • Vehicle Kilometres Travelled (VKT): sum over road segments of the product of daily traffic and segment length (veh-km per day).

  • Peak Hour Factor (PHF): a measure of demand variation within an hour; captures how peaky the hourly volume is.

• AADT and ADT calculations (examples from the notes):

  • ADT from a monthly volume:

  • ADT = (Monthly Volume) / (Days in month)

  • Example: January monthly volume = 425,000 vehicles; days = 31 ⇒ $ADT_{Jan} = \frac{425{,}000}{31} \approx 13{,}710 \mathrm{veh/day}$

  • AADT: average daily traffic over the year:

  • AADT = (Total yearly volume) / 365

  • Example: Total yearly volume = 5{,}445{,}000; AADT = (5{,}445{,}000 / 365 \approx 14{,}918) veh/day

• Vehicle Kilometres Travelled (VKT):

  • Daily VKT on segment i: $VKT<em>i = AADT</em>i \times L<em>i$ where $L</em>i$ is the segment length.

  • Total daily VKT for the network: $\text{Daily VKT} = \sum<em>{i\in I} VKT</em>i$

  • Yearly VKT: $\text{Yearly VKT} = \text{Daily VKT} \times 365$

• Peak Hour Factor (PHF):

  • Definition: PHF measures how the total hourly volume relates to the peak 15-minute flow within the hour in a given direction.

  • Formula: $\text{PHF} = \frac{V}{q<em>{15}} = \frac{V}{4 V</em>{15}}$ where:

  • $V$ = hourly volume in the direction of interest (veh/h)

  • $V_{15}$ = maximum 15-minute volume within the hour (veh/15 min)

  • $q<em>{15} = 4 \times V</em>{15}$ (peak 15-minute flow rate within the hour, veh/h)

  • Interpretation:

  • PHF = 1 implies uniform flow within the hour (no peaking).

  • PHF = 0.25 would imply all flow occurs in a single 15-minute interval (highly peaked).

  • Realistic values typically range from ~0.70 to ~0.98 depending on the site and period.

• Example calculations from the notes:

  • Example 1: If hourly directional volume is 2500 veh/h and the maximum 15-minute interval within that hour is 650 veh in 15 minutes, then:

  • $q<em>{15} = 4\times V</em>{15} = 4\times 650 = 2600 \mathrm{veh/h}$

  • $\text{PHF} = \frac{V}{q_{15}} = \frac{2500}{2600} \approx 0.962$

  • Example 2: If hourly volume is 2400 veh/h and the maximum 15-minute volume is 650 veh, then as above PHF ≈ 0.923; if the max 15-minute interval were 2000 veh, PHF ≈ 0.3.

• Variations in traffic volume:

  • Variations occur cyclically and repeat over days, weeks, and seasons (daily, weekly, seasonal patterns).

  • Data sources (e.g., Austroads) discuss these variations for planning and analysis.

Data Sources and Examples (Key Points)

• AADT and ADT data are useful for funding and planning decisions, maintenance scheduling, and long-term capacity planning.

• Data collection methods provide different types of data useful for different analyses:

  • Point measurement provides high-frequency data at fixed locations (volume, speed, density).

  • Probe data provides travel-time and trajectory information by tracking selected vehicles along routes.

  • Video-based data provides detailed movement trajectories from captured imagery; can extract time-space trajectories for all vehicles in view.

• The choice of data source depends on the required spatial/temporal resolution and the specific traffic analysis objective.

Practical Example: Greenshields Model (Worked Section)

• Given a highway section with Greenshields form:

  • Free-flow speed: $v_f = 80 \mathrm{km/h}$

  • Jam density: $k_j = 200 \mathrm{veh/km}$

• Compute:

  • Capacity: $q<em>c = \frac{v</em>f k_j}{4} = \frac{80\times200}{4} = 4000 \mathrm{veh/h}$

  • Critical density: $k<em>c = \frac{k</em>j}{2} = 100 \mathrm{veh/km}$

  • Speed at capacity: $v<em>c = \frac{v</em>f}{2} = 40 \mathrm{km/h}$

  • Verify: $q<em>c = v</em>c k_c = 40\times100 = 4000 \mathrm{veh/h}$

• Alternative parabolic form relating flow and speed:

  • $q = k<em>j v - \frac{k</em>j}{v_f} v^2$

• If given a velocity, you can find the corresponding density via: $k = k<em>j\left(1 - \frac{v}{v</em>f}\right)$ and then compute the flow as $q = v k$.

• Example problem included in the notes: Given a flow–density relationship $q = 80 k - 0.4 k^2$ (with units as indicated), identify free-flow speed, jam density, capacity, critical density, and speed at capacity. Solved values (from the notes):

  • $v_f = 80 \mathrm{km/h}$

  • $k_j = 200 \mathrm{veh/km}$

  • $k_c = 100 \mathrm{veh/km}$

  • $q_c = 4000 \mathrm{veh/h}$

  • $v_c = 40 \mathrm{km/h}$

Limitation of Greenshields Model

• Major limitation: discrepancy at low density where actual speed remains near free-flow over a range of densities, especially on high-capacity motorways.

• Greenshields predicts a continuous decrease in speed with density, which is not always observed in practice at low to moderate densities.

• Modifications to Greenshields (e.g., Modified Greenshields Model) have been proposed to better fit observed data.

• The notes show comparisons of actual data versus Greenshields predictions, highlighting the flat speed region at low densities and the improved fit of modified models.

Factors Influencing the Fundamental Diagram

• Road type and geometry: number of lanes, lane width, grades.

• Vehicle composition: proportion of cars vs trucks.

• Driver behavior: familiarity with the road, reaction patterns.

• Road conditions: lighting, weather (rain, snow, visibility).

• New technologies: Connected Autonomous Vehicles (CAVs) and their impact on flow, density, and overall traffic dynamics.

Additional Data Collection Details (Overview)

• Point measurement devices include loop detectors, pneumatic tubes, and radar sensors; mainly used for fixed-location, high-frequency data (volume, speed, density).

• Probe data collection uses tracking technologies like GPS, ANPR, toll tags, and Bluetooth detectors; useful for travel-time and trajectory analysis; provides point-to-point travel-time data and route-level estimates.

• Video-based data collection uses rooftop cameras, helicopters, or drones; enables extraction of detailed trajectories and macroscopic counts; provides rich spatial data and time-synchronised trajectories but requires substantial processing.

• Manual counting offers quick, low-cost supplemental data for short-term studies and turning movements.

Summary of Data Collection Methods (Quick Reference)

• Point measurement: continuous vehicle position tracking; detectors measure aggregated flow, density, and speed; high temporal resolution; limited spatial coverage.

• Probe data collection: GPS-equipped vehicles and other tracking technologies; provides headways, travel times, and whole trajectories; high temporal and moderate to high spatial coverage depending on penetration.

• Video-based data collection: mounted cameras and drones; yields high-resolution microscopic trajectories; wide spatial coverage with holistic traffic-state information; requires processing to extract trajectories.

Quick Reference Formulae (Greenshields and FD Concepts)

• Fundamental Diagram relationships:

  • $v = v<em>f\left(1 - \frac{k}{k</em>j}\right)$

  • $q = v k$

  • $q = v<em>f k - \frac{v</em>f}{k_j} k^2$

  • $k = k<em>j\left(1 - \frac{v}{v</em>f}\right)$

  • $q = k<em>j v - \frac{k</em>j}{v_f} v^2$

• Capacity and critical points:

  • $k<em>c = \frac{k</em>j}{2}$

  • $q<em>c = \frac{v</em>f k_j}{4}$

  • $v<em>c = \frac{v</em>f}{2}$

• PHF (Peak Hour Factor):

  • $\text{PHF} = \frac{V}{q<em>{15}} = \frac{V}{4 V</em>{15}}$

  • Where: $V_{15}$ is the maximum 15-minute volume within the hour; $V$ is the hourly volume (in the same direction).

• Example calculations (sanity checks):

  • Given $v<em>f = 80 \mathrm{km/h}$ and $k</em>j = 200 \mathrm{veh/km}$:

  • $q_c = \frac{80\times 200}{4} = 4000 \mathrm{veh/h}$

  • $k_c = 100 \mathrm{veh/km}$

  • $v_c = 40 \mathrm{km/h}$

• AADT and ADT definitions (from daily/annual volumes):

  • ADT = (Monthly Volume) / (Days in Month)

  • AADT = (Total Yearly Volume) / 365

• VKT definitions:

  • $VKT<em>i = \text{AADT}</em>i \times L_i$

  • $\text{Daily VKT} = \sum<em>{i} VKT</em>i$

  • $\text{Yearly VKT} = \text{Daily VKT} \times 365$

• Peak Hour Factor (PHF) example formula:

  • If hourly volume V = 2500 veh/h and max 15-min volume within that hour is 650 veh, then

  • $q<em>{15} = 4\times V</em>{15} = 4\times 650 = 2600 \mathrm{veh/h}$

  • $\text{PHF} = \frac{V}{q_{15}} = \frac{2500}{2600} \approx 0.962$

• Relationship to density and occupancy:

  • Density is often proxied by occupancy (%), but occupancy is detector-dependent and not a direct density measure (veh/km).

If you’d like, I can tailor these notes to focus on the exact topics your exam emphasizes (e.g., more on Greenshields derivations, or more on data collection methods and PHF calculations). You can also ask for a condensed quick-review version or a problem-set