Mobility Modelling

Lecture 4 - Vehicle Mobility Modelling 

27 Feb 2023

Mozhgan Pourmoradnasseri, Ph.D.

  1. Planning and design: evaluate the performance of existing transportation systems and to design new transportation infrastructure.
  2. Management: monitoring, forecasting and network optimization.
  3. Environment: assess the environmental impacts of transportation, such as air pollution, noise pollution, and greenhouse gas emissions.

APPLICATIONS OF TRAFFIC FLOW MODELS

4. Economic analysis: evaluate the economic impacts of transportation.

5. Emergency response: plan for and respond to disasters or other emergency situations, by providing information on traffic flow patterns, evacuation routes.

source: https://github.com/TianzhenLi/Intelligent-Intersection
  • The model only has a few parameters.
  • Parameters are (easily) observable and have realistic values.
  • Not require more input data than what is available for calibration.
  • Relevant phenomena are reproduced and predicted by the model.
  • Captures the most important characteristics of the underlying system.
  • The model allows fast computations for state estimation or prediction. 
  • Not require more time and memory than what available hardware permit.

A USEFUL MODEL

Real World

Model World

Scenario & Metrics

Encode

Decode & Evaluate

Model specification

Source: Traffic Flow Modelling, Femke Kessels

Relation between spacing and speed?

A simple example is the observations by Greenshields of vehicles passing his camera in the 1930s.

Plotting the distance between the vehicles (spacing) and their change in position in consecutive photographs leads to a theory that spacing and speed are related. Subsequently, this leads to a model with a linear relationship between spacing and speed.

A HISTORICAL EXAMPLE

General pattern: start with a model which is simple, transparent, insightful... and also wrong. This simple model can then be improved in ways to make it more correct and useful.

A model is a set of mathematical equations that tries to describe (or replicate) a physical process (ex. evolution of traffic congestion over time and space).

…all models are approximations. Essentially, all models are wrong, but some are useful. However, the approximate nature of the model must always be borne in mind…

George E. P. Box

4-step model in transportation modeling

1. Trip generation: estimated number of trips made by each zone. 

4-step model in transportation modeling

1. Trip generation: estimated number of trips made by each zone. 

2. Trip distribution:  trips are assigned to different destinations, such as homes, workplaces, or shopping centers.

3. Mode Choice: determining how people will travel to their destinations, such as by car, bus, or train.

4-step model in transportation modeling

1. Trip generation: estimated number of trips made by each zone. 

2. Trip distribution:  trips are assigned to different destinations, such as homes, workplaces, or shopping centers

3. Mode Choice: determining how people will travel to their destinations, such as by car, bus, or train.

4. Route Assignment: the trips are assigned to specific routes or transportation facilities, such as highways, transit lines, or bike paths.

Limitations of 4-step model

  1. Limited scope: Focuses primarily on passenger travel

  2. Simplistic assumptions: Relies on assumptions about travel behavior and decision-making that may not accurately reflect real-world behavior. 

  3. Data availability: Requires a large amount of data to be effective, including data on population, employment, land use, and transportation infrastructure.

  4. Lack of sensitivity: Not sensitive enough to capture small changes in travel behavior or transportation infrastructure.

  5. Difficulty in modeling congestion: May not accurately predict the impacts of congestion on travel behavior or the transportation network.

What mode of transport did you use today?

What influenced your choice?

Simple example

Every choice has some consequences; spent time, cost, discomfort, ...

\(U_i = a_i c +b_i t\)

Utility of choice for person \(i\)

weights of attributes

cost        time

Mode Cost Time
Bus 1 5
Car 3 1

\(a_3=-1, b_3=-2\)

\(a_1=-3, b_1=-1\)

\(a_2=-1, b_2=-1\)

\(U_1^B=-8,  U_1^C=-10\)

\(U_2^B=-6,  U_2^C=-4\)

\(U_3^B=-11,  U_3^C=-5\)

Plan

Plan

Plan

Plan

Plan

Plan

Plan

Plan

Plan

Plan

Plan: Sequence of activities, Activity timing, route choice, Mode choice

In reality

Agent-based modeling

Agents are 

  • autonomous;
  • interact with each other;
  • interact with the environment.  

1. Every agent has a set of day plans.

Each agent selects a plan from its memory.

2. Mobility plans are simulated.

3. Plan scores are computed after each mobsim run.

4. A certain share of the agents (often 10%) are allowed to replan.

5. The iterative process is repeated until the average population score stabilizes.

Source:  The Multi-Agent Transport Simulation MATSim

CLASSIFICATION OF TRAFFIC FLOW MODELS

  • Microscopic (mainly used for simulation)
    • models car-following and lane-changing behavior.
  • Mesoscopic
    • captures vehicles with similar characteristics in aggregated terms. 
  • Macroscopic
    • similar to fluid mechanics, variables like flow and density are used to describe the state of the network.
Smith, J., Blewitt, R. at al: Traffic Modelling
Guidelines. Transport for London (2010)

Agent-based models

  • intuitively describe the behavior of individual travelers.
  • Based on assumptions on human behavior.
  • The behavior of each vehicle is modeled based on the position, velocity, and acceleration of the leading vehicle.

Continuum models

  • consider traffic as a compressible fluid and replicate its behavior on a macroscopic level through aggregated state variables such as flow and density.

MAIN MODEL TYPES FOR TRAFFIC FLOW

WATER VERSUS GRANULAR FLOW

Water: outflow equals inflow, up to a maximum.

Granular: at low inflows, outflow equals inflow. When the inflow is too high, particles ‘get stuck’ in bottleneck, and outflow is lower than with a medium inflow.

High Densities, Low Speeds and Vice Versa

The precise relationship between variables such as density, headway, speed, and flow, is an important subarea of traffic flow research.

A simple human behaviour: drivers tend to choose a speed that is as high as possible, while still safe. Therefore, traffic flow models commonly use a decreasing—or at least nonincreasing—relationship between density and speed.

Thus, maximum flow (density × speed) occurs at some intermediate density and speed values.

MACROSCOPIC TRAFFIC FLOW MODELS

Given a road segment, space is discretized (\(dx\)). We have:

  • density \(\rho(x,t)\): number of vehicles per length unit (veh/km)
  • traffic flow \(q(x,t)\): number of vehicles per time unit (veh/h)
  • mean speed \(v(x,t)\) in km/h

 

This is a mathematical abstraction inspired by fluid mechanics.

We assume in \(dx\) space we have homogeneous conditions for density and speed.

FUNDAMENTAL DIAGRAM OF TRAFFIC FLOW

Greenshields fundamental diagram (1934)

Daganzo fundamental diagram (1994)

FUNDAMENTAL DIAGRAM; 2 SIMPLE MODELS

Linear in the density-velocity plane and thus parabolic in the density-flow.

Most widespread due to its simplicity. It is bi-linear (triangular) in the density-flow plane.

flow = density × speed

MICROSCOPIC TRAFFIC FLOW MODELS

  • Based on the assumption that drivers adjust their behavior to the leading vehicle.
  • Car-following and lane-changing behavior are modeled.
  • We get the detailed trajectories of vehicles following each other.
  • Many software are available for microscopic simulation.

Gipps, P. G. (1981). A behavioural car-following model for computer simulation. Transportation Research Part B: Methodological, 15(2), 105-111.

Gipps, P. G. (1986). A model for the structure of lane-changing decisions. Transportation Research Part B: Methodological, 20(5), 403-414.

MICROSCOPIC TRAFFIC FLOW MODELS: CAR-FOLLOWING

The earliest car-following models include a car-following rule based on the safe following distance by expressing the position of the leader as a function of the position of its follower.

Timid drivers would keep a longer distance when the leader decelerates into congestion, while aggressive drivers tend to keep shorter following distances.

   We expect in a rational model:

  1. If velocity decreases, the vehicle accelerates less.
  2. If headway increases, the vehicle accelerates more.
  3. If relative velocity increases, the vehicle accelerates more.   

MICROSCOPIC TRAFFIC FLOW MODELS: LANE-CHANGING

  • Mandatory lane change because of their route choice, e.g. from on-ramp to main road to enter the freeway, from main road to off-ramp to leave the freeway, from one lane to the next because the first lane will end or is blocked.
  • Voluntary lane change occurs when a driver seeks a speed advantage, this often includes overtaking.
  • Random lane change (without apparent reason) or forced merging (when a driver creates a gap to enter a congested lane).

 Lane-change models consist of three steps:

  1. Decide about the necessity of lane change
  2. Choose target lane
  3. Decide whether to accept gap

MICROSCOPIC VS MACROSCOPIC MODELING

  • Macroscopic models are analytical, not just simulators. We can write down the analytical form of PDEs and use them to develop model-based control strategies.
  • Computation time: Macroscopic models are much faster than microscopic ones. However, microscopic models contain details.
  • We need to think about
    • Required resolution
    • Traffic control measures
    • Potential accuracy vs validation efforts

HOW DO WE VALIDATE OUR MODEL?

  • Logical and mathematical consistency.
  • Qualitative testing.  Does the model reproduce all the essential phenomena of the process?
    If I have a traffic network and some bottlenecks that create the congestion, my model should be able to reproduce this congestion and equivalently, my model should have no contradiction with the essential phenomenon of the physical process.
  • Quantitive accuracy. Estimating the parameters and checking the sensitivity and transferability of the model. 
  • Computational complexity.

SOFTWARE FOR SIMULATION OF TRAFFIC

Tutorial

https://traffic-simulation.de/