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Traffic signal optimization and traffic assignment Traffic signals - - PowerPoint PPT Presentation

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Dagstuhl, October 2015 Traffic Signals and User Equilibria Martin Strehler , Ekkehard K ohler BTU Cottbus-Senftenberg { martin.strehler,ekkehard.koehler } @b-tu.de Traffic signal optimization and traffic assignment Traffic signals Traffic

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Dagstuhl, October 2015

Traffic Signals and User Equilibria

Martin Strehler, Ekkehard K¨

  • hler

BTU Cottbus-Senftenberg

{martin.strehler,ekkehard.koehler}@b-tu.de

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Traffic signal optimization and traffic assignment

Traffic signals

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Traffic signal optimization and traffic assignment

Traffic signals Traffic assignment traffic signals affect travel times road users may change routes combined traffic signal optimization traffic assignment problem

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Traffic signal optimization and traffic assignment

Traffic signals Traffic assignment traffic signals affect travel times road users may change routes combined traffic signal optimization traffic assignment problem traffic signals change over time need for dynamic traffic assignment

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Challenges

  • traffic signal optimization is NP-hard
  • pseudo-polynomial time algorithms for

multi-commodity flows over time (weakly NP-hard)

  • no constant-factor approximation

algorithm for combined problem

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Challenges

  • traffic signal optimization is NP-hard
  • pseudo-polynomial time algorithms for

multi-commodity flows over time (weakly NP-hard)

  • no constant-factor approximation

algorithm for combined problem

  • n-going project (DFG grant) Optimization and network wide analysis of traffic signal control

together with Kai Nagel, TU Berlin Primary objectives:

  • develop a linear model
  • focus on system optima
  • solve it with mixed integer

programming for medium-sized scenarios

  • validate solutions with traffic simulation

Recent objectives

  • system optimum vs. user equilibrium
  • robustness, e.g., major events
  • public transport
  • . . .
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The cyclically time-expanded model

Simple network

x y t = 3 t = 3 t = 1 t = 2

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The cyclically time-expanded model

Time expansion (infinite time horizon)

x y t

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 . . . . . . . . . . . .

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The cyclically time-expanded model

Time expansion (infinite time horizon)

x y t

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 4 5 . . . . . . . . . . . .

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The cyclically time-expanded model

Periodic traffic signals → finite and cyclic expansion modulo T

x y t

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 4 5 3 4 5

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The cyclically time-expanded model

connect consecutive copies → waiting possible

x y t

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 4 5 3 4 5

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The cyclically time-expanded model

connect consecutive copies → waiting possible

x y t

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 4 5 3 4 5 semi-dynamic or hybrid model

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Modeling a traffic signal

1 2 3 4 5 6 7

t = 3 t = 1

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Modeling a traffic signal

1 2 3 4 5 6 7

c0 = b0c c1 = b1c c2 = b2c c3 = b3c c4 = b4c c5 = b5c c6 = b6c t = 3 t = 1

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Modeling a traffic signal

1 2 3 4 5 6 7

t = 3 t = 1

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Modeling signalized intersections

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Modeling signalized intersections

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Modeling signalized intersections

  • collision free

b1,i +b2,i ≤ 1 ∀i ∈ {0,...,T −1}

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Modeling signalized intersections

  • collision free

b1,i +b2,i ≤ 1 ∀i ∈ {0,...,T −1}

  • green length limitations

tmin ≤ ∑T−1

i=0 b1,i ≤ tmax

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Modeling signalized intersections

  • collision free

b1,i +b2,i ≤ 1 ∀i ∈ {0,...,T −1}

  • green length limitations

tmin ≤ ∑T−1

i=0 b1,i ≤ tmax

  • switching only once per cycle

– new variables bon 1,i – ∑T−1 i=0 bon 1,i = 1 – b1,i −b1,i−1 ≤ bon 1,i

∀i ∈ {0,...,T −1}

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1 2 3 6 7 4 5

Figure: The Brunswick network with 5 signalized intersections (1–5) and two pedestrian signals (6,7). The three commodities under consideration are visualized with arcs in different colors. The underlying map was created with www.openstreetmap.org.

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84 84 84 1 2 3A 6 7 4A 3B 6 7 4A 4B 5A 5B Figure: Optimized green bands for the Brunswick scenario. Time and state of the signals is shown on the vertical axis. Signals are labelled with intersection numbers and signal groups, e.g., signal 3A and

3B must not be green at the same time. Distances on the horizontal axis are chosen with respect to

transit time, the slope of the parallelograms is chosen with respect to free speed. Note that the purple commodity is travelling from right to left and the cyclic overflow is visualized by a negative slope.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 River Spree

0−100 100−200 200−300 300−400 400−500 500−600 600−800 > 800

Figure: Traffic assignment for the Cottbus scenario calculated with the cyclically time-expanded model (cars per hour and lane). The traffic on the outer road is increased in clockwise direction.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 River Spree

Figure: Splitting of two commodities in the Cottbus scenario.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 River Spree

≤ 5 5−50 50−100 100−150 150−200 200−250 250−300 300−400 > 400

Figure: This diagram shows the waiting time integrated over all cars during one cycle at each traffic signal after optimization with our model.

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Flow-independent travel times?

The cyclically time expanded model includes

  • constant transit time on each arc
  • cost = flow · transit time
  • objective: minimize total travel time
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Flow-independent travel times?

The cyclically time expanded model includes

  • constant transit time on each arc
  • cost = flow · transit time
  • objective: minimize total travel time

Is this too simplifying?

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Flow-independent travel times?

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Flow-independent travel times?

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Flow-independent travel times?

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Flow-independent travel times?

max capacity 10 11 12 13 14 15 16 17

flow average travel time

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Platoons

Traffic signals cause varying traffic density, i.e., platoons of cars:

  • arrival time at intersections is important ⇒ coordination
  • traffic signals may create, split, merge, densify, or loosen platoons
  • actual travel time depends an various parameters, e.g., position within the platoon
  • average travel time also depends an several parameters
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Platoons

Traffic signals cause varying traffic density, i.e., platoons of cars:

  • arrival time at intersections is important ⇒ coordination
  • traffic signals may create, split, merge, densify, or loosen platoons
  • actual travel time depends an various parameters, e.g., position within the platoon
  • average travel time also depends an several parameters

Link performance functions etc. are not suitable for capturing platoons.

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Platoons in the cyclically expanded network

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Computed travel time of a platoon

10 20 30 40 50 60 30 20 10 50 40 30 10 15 20 25 30 35 40 Platoon length Arrival time Average travel time

average travel time of a car in a platoon with respect to platoon length (in flow units) and arrival time at the traffic signal (in seconds after beginning of green phase)

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Simulation with VISSIM

scenario in VISSIM

  • 30 s green, 30 s red
  • platoons of different lengths (i.e., different traffic flow) and low density arrive
  • narrow road becomes wider (more lanes)
  • first car may wait longest if it arrives at beginning of red
  • platoon is densified by signal
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Simulation with VISSIM

10 20 30 40 50 60 5 10 15 20 25 30 Platoon Length Average Travel Time

computed travel time (first car arrives 10 seconds before green)

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Simulation with VISSIM

10 20 30 40 50 60 5 10 15 20 25 30 Platoon Length Average Travel Time

computed travel time (first car arrives 10 seconds before green)

10 20 30 40 50 60 5 10 15 20 25 30 Platoon Length Average Travel Time

simulated travel time (100 runs, 5 hrs simulated time each)

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User equilibria in the cyclically time-expanded model

in theory:

  • system optimum is user equilibrium (constant travel-times, no road user can switch to

faster route → Wardrop condition fulfilled)

  • capacitated network → value of user equilibria not unique
  • price of anarchy unbounded

in practice:

  • simulate user equilibria with MATSim
  • there is a difference between our solutions and MATSim equilibria
  • difficult to compare (structure of) solutions
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What is a realistic equilibrium?

‘projected’ travel times

  • more cars may imply lower travel

times

– traffic signals setter – adaptive signals – ...

ue with capacities

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What is a realistic equilibrium?

‘projected’ travel times

  • more cars may imply lower travel

times

– traffic signals setter – adaptive signals – ...

  • nearly no theory for non-convex,

non-monotone settings

  • easy to construct bad instances for

the general case

  • what can be done with additional

assumptions?

  • (but not suitable for optimization

purposes in our model) ue with capacities

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What is a realistic equilibrium?

‘projected’ travel times

  • more cars may imply lower travel

times

– traffic signals setter – adaptive signals – ...

  • nearly no theory for non-convex,

non-monotone settings

  • easy to construct bad instances for

the general case

  • what can be done with additional

assumptions?

  • (but not suitable for optimization

purposes in our model) ue with capacities

  • examples for unbounded price of

anarchy somewhat extreme/unnatural

  • small amount of blocking flow,

forcing long detours for other flow units

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What is a realistic equilibrium?

‘projected’ travel times

  • more cars may imply lower travel

times

– traffic signals setter – adaptive signals – ...

  • nearly no theory for non-convex,

non-monotone settings

  • easy to construct bad instances for

the general case

  • what can be done with additional

assumptions?

  • (but not suitable for optimization

purposes in our model) ue with capacities

  • examples for unbounded price of

anarchy somewhat extreme/unnatural

  • small amount of blocking flow,

forcing long detours for other flow units

  • dynamic flows and signals limit

blocking

  • Can one build a finer classification
  • f user equilibria?
  • cyclic time-expansion: who’s first?
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Does traffic signal optimization reduce the gap between SO and UE?

Conjecture: SO plain network UE plain network

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Does traffic signal optimization reduce the gap between SO and UE?

Conjecture: SO plain network SO cte network with signals UE plain network

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Does traffic signal optimization reduce the gap between SO and UE?

Conjecture: SO plain network SO cte network with signals UE optimized signals UE plain network ?

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Does traffic signal optimization reduce the gap between SO and UE?

Conjecture: SO plain network SO cte network with signals UE optimized signals UE plain network ? PoA without signals PoA with signals net PoA

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Do traffic signal reduce the gap between SO and UE?

Cottbus: scenario cyclically expanded network MATSim simulation Gap (%)

  • pt

1,103,380 1,140,077 3.33 best random 1,156,888 1,173,303 1.41 med random 1,197,666 1,210,419 1.06 avg random 1,198,688 1,236,851 3.18 worst random 1,285,766 1,291,993 1.00

Table: Comparison of system optimal solutions of the cyclically time-expanded network model and user equilibrium solutions of MATSim. All values represent total travel times in seconds for the morning peak from 6:30 to 9:30 am. Besides the optimized traffic signals (opt), we also compare optimized assignments for 100 random traffic signal settings.

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Do traffic signal reduce the gap between SO and UE?

Artificial Braess instance:

s v1 v2 t 5s 20s 5s 20s 5s

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Do traffic signal reduce the gap between SO and UE?

Artificial Braess instance:

s v1 v2 t 5s 20s 5s 20s 5s

coordinations Btu/60 #Z Matsim #Z minCoord 2143,2 3811 8 maxCoord 5550 30 6228 30 greenWaveZ 2850 48 4430 36 maxCoordEmptyZ 4275.6 5649 4 minCoordFullZ 4950 52.8 5934 38 maxCoordFullZ 5250 52.8 6137 34

Table: Total travel times and # of users on evil arc. (capacity 144 veh/min (≡ 8640 veh/h), time bin size 1s).

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Do traffic signal reduce the gap between SO and UE?

Questions/noticeable facts:

  • surprisingly high differences in this small scenario
  • difference of SO and UE in Cottbus scenario almost insignificant
  • starting time seem to have higher impact on travel times than route choice in Cottbus

scenario

  • signals have significant influence on assignment
  • Can signals be used for alternative road pricing approaches?
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How to track influences of signal optimization on surrounding area?

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How to track influences of signal optimization on surrounding area?

disattracted attracted

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How to track influences of signal optimization on surrounding area?

  • 1270 - -630
  • 630 - -310
  • 310 - -150
  • 150 - -70
  • 70 - -30
  • 30 - -10
  • 10 - 10

10 - 30 30 - 70 70 - 150 150 - 310 310 - 630 630 - 1270

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