Modeling Transportation Network Dynamics as a Complex System
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Modeling Transportation Network Dynamics as a Complex System David Levinson Bhanu Yerra Objectives Why do networks expand and contract? Do networks self-organize into hierarchies? Roads - an emergent property? Can investment rules predict location of network improvement?

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Presentation Transcript

Objectives l.jpg
Objectives

  • Why do networks expand and contract?

  • Do networks self-organize into hierarchies?

  • Roads - an emergent property?

  • Can investment rules predict location of network improvement?

  • How can this improved knowledge help in planning transportation networks?


How networks change with time l.jpg
How networks change with time?

  • State of a network node changes

  • Travel time of a link changes

  • Capacity of a link changes

  • Flow on a link changes

  • New links and nodes are added

  • Existing links are removed

  • System properties, like congestion, change over time


How to model a network as a complex system l.jpg
How to model a network as a complex system?

  • Links and nodes are agents

  • Agent properties

  • Rules of interaction that determine the state of agents in the next time step

  • Spatial pattern of interaction between agents

  • External forces and variables

  • Initial states


Transportation network as a complex system l.jpg
Transportation network as a complex system

  • Network is split into two layers

    • Network layer

    • Land use layer

  • Network is modeled as a directed graph

  • Land Use Layer has small land blocks as agents that determine the populations and land use


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Network Layer

Land Use Layer

Figure 1: Splitting the system into network layer and land use layer


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Models Required

  • Land use and population model

  • Travel demand model

  • Revenue model

  • Cost model

  • Network investment model



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Network

  • Grid network

  • Random networks will also be tested

  • Networks under modeling stage

    • Cylindrical network

    • Torus network

  • Initial speed distribution

    • Every link with same initial speed

    • Uniformly distributed speeds


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Land Use and Demography

  • Small Land Blocks are agents

  • Population, business activity, and geographical features are attributes

  • Uniformly distributed and bell shaped land use are modeled

  • Land use is assumed to be static

  • Dynamics of land use will be added later


Travel demand model l.jpg
Travel Demand Model

  • Trip Generation

    • Using Land use model trips produced and attracted are calculated for each cell

    • Cells are assigned to network nodes using voronoi diagram

    • Trips produced and attracted are calculated for a network node using voronoi diagram



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Travel Demand Model(Contd.)

  • Trip Distribution

    • Calculates trips between network nodes

    • Gravity model

      Where,

      trs is trips from origin node r to destination node s,

      pr is trips produced from node r,

      qsis trips attracted to node s,

      drsis cost of travel between nodes r and s along shortest path


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Travel Demand Model(Contd.)

  • Traffic Assignment

    • Path with least cost of traveling

    • Cost of traveling a link is

      Where, la is length of link a

      va is speed of link a

       is value of time

      o is tax rate

      1, 2 are coefficients


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Travel Demand Model(Contd.)

  • Traffic Assignment ( Contd.)

    • Dijkstras Algorithm

    • Flow on a link is

      Where,

      Krsis a set of links along the shortest path from node r to node s,

      a,rs = 1 if a  Krs, 0 otherwise


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Revenue Model

  • Toll is the only source of revenue

  • Annual revenue generated by a link is total toll paid by the travelers

    Where, coefficients are same as coefficients used in traveling cost function

  • A central revenue handling agent can be modeled


Cost model l.jpg
Cost Model

  • Assuming only one type of cost

  • Cost of a link is

    Where, c is cost rate,

    1, 2, 3 are coefficients.

  • Introducing more cost functions makes the model more complicated and probably more realistic


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Network Investment Model

  • A link based model

  • Speed of a link improves if revenue is more than cost of maintenance, drops otherwise

    Where, vat is speed of link a at time step t,

    • is speed reduction coefficient.

  • No revenue sharing between links: Revenue from a link is used in its own investment


  • Examples l.jpg
    Examples

    • Base case

      • Network - speed ~ U(1, 1)

      • Land use ~ U(10, 10)

      • Travel cost da { = 1.0, o = 1.0, 1 = 1.0, 2 = 0.0}

      • Cost model {c =365, 1 = 1.0, 2 = 0.75, 3 = 0.75}

      • Improvement model {  = 1.0}

      • Speeds on links running in opposite direction between same nodes are averaged

      • Symmetric route assignment


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    Initial network

    Equilibrium network state after 8 iterations

    Slow

    Fast

    Figure 4 Equilibrium speed distribution for the base case on a 10x10 grid network

    Base Case


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    Base Case

    Initial network

    Equilibrium network state after 9 iterations

    Slow

    Fast

    Figure 5 Equilibrium speed distribution for the base case on a 15x15 grid network


    Case 2 same as base case but initial speeds u 1 5 l.jpg

    Initial network

    Equilibrium network state after 8 iterations

    Slow

    Fast

    Figure 6 Equilibrium speed distribution for case 2 on a 10x10 grid network

    Case 2: Same as base case but initial speeds ~ U(1, 5)


    Case 2 same as base case but initial speeds u 1 523 l.jpg

    Initial network

    Equilibrium network state after 8 iterations

    Slow

    Fast

    Figure 7 Equilibrium speed distribution for case 2 on a 15x15 grid network

    Case 2: Same as base case but initial speeds ~ U(1, 5)


    Case 3 base case with a downtown l.jpg

    Initial network

    Equilibrium network state after 7 iterations

    Slow

    Fast

    Figure 8 Equilibrium speed distribution for case 3 on a 10x10 grid network

    Case 3: Base case with a downtown


    Case 3 base case with a downtown25 l.jpg

    Initial network

    Equilibrium network state after 7 iterations

    Slow

    Fast

    Figure 9 Equilibrium speed distribution for case 3 on a 15x15 grid network

    Case 3: Base case with a downtown


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    Conclusions

    • Succeeded in growing transportation networks

    • Sufficiency of simple link based revenue and investment rules in mimicking the network structure

    • Hierarchical structure of transportation networks is a property not entirely a design

    • Future work will help in better understanding of network dynamics


    Future work l.jpg
    Future Work

    • Trip distribution - An agent based combined trip distribution and traffic assignment

    • Traffic assignment - Deterministic and Stochastic user equilibrium

    • Revenue sharing between links

    • Modeling construction of new links

    • Maintenance and construction costs

    • Spatial dependent revenue and investment models



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