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Capacity maximising traffic signal control policies

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Presentation from NORTHMOST - a new biannual series of meetings on the topic of mathematical modelling in transport.

Hosted at its.leeds.ac.uk, NORTHMOST 01 focussed on academic research, to encourage networking and collaboration between academics interested in the methodological development of mathematical modelling applied to transport.

The focus of the meetings will alternate; NORTHMOST 02 - planned for Spring 2017 - will be led by practitioners who are modelling experts. Practitioners will give presentations, with academic researchers in the audience. In addition to giving a forum for expert practitioners to meet and share best practice, a key aim of the series is to close the gap between research and practice, establishing a feedback loop to communicate the needs of practitioners to those working in university research.

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Capacity maximising traffic signal control policies

  1. 1. NORTHMOST (12 December 2016) Capacity-maximising traffic signal control policies “If you eliminate the impossible, whatever remains, however improbable, must be the truth.”
  2. 2. NORTHMOST (12 December 2016) Capacity-maximising traffic signal control policies “If you eliminate the impossible, whatever remains, however improbable, must be the truth.” (Sherlock Holmes, Sign of the Four, 1890)
  3. 3. NORTHMOST (12 December 2016) Capacity-maximising traffic signal control policies Mike Smith, Ronghui Liu, Takamasa Iryo, Tung Le, Hai Vu The University of York, UK ITS, University of Leeds, UK Kobe University, Japan Swinburne University of Technology, Melbourne, Australia University of Monash , Melbourne, Australia
  4. 4. RELEVANCE / IMPORTANCE / AIM TO REDUCE CONGESTION / POLLUTION IN CITIES
  5. 5. RELEVANCE / IMPORTANCE / AIM TO REDUCE CONGESTION / POLLUTION IN CITIES IN PART AUTOMATICALLY
  6. 6. Traffic Control and Route Choice Traffic Control Route Choice
  7. 7. Modelling Signal Control and Route Choice Allsop, Dickson, Gartner, Smith, Van Zuylen, Meneguzzer, Gentile, Noekel, Taale, Cantarella, Mounce, Ke Han, Viti, Schlaich, Haupt, Lo, Rinaldi, Cantelmo, Cascetta, Tung Le, Hai Vu, . . . ● ● ● Previous Work
  8. 8. Origin Destination Route-swaps C s2 s1
  9. 9. Origin Destination Route-inflow swaps and Stage green-time swaps C s2 s1
  10. 10. Origin Destination Route flow and stage green-time swaps motivated by route costs and “stage pressures” C s2 s1
  11. 11. HOPE: Stability V(x) = 4 V(x) = 2 V(x) = 1 V(x) = 3 EQUILIBRIUM: V(x) = minimum
  12. 12. HOPE: Stability V(x) = 4 V(x) = 2 V(x) = 1 V(x) = 3 EQUILIBRIUM: V(x) = minimum ●
  13. 13. HOPE: Stability V(x) = 4 V(x) = 2 V(x) = 1 V(x) = 3 EQUILIBRIUM: V(x) = minimum ● ● ●
  14. 14. HOPE: Stability V(x) = 4 V(x) = 2 V(x) = 1 V(x) = 3 EQUILIBRIUM: V(x) = minimum ● ● ● ● ●
  15. 15. WHAT CAN GO WRONG DYNAMICALLY STABLE WITH STANDARD POLICIES ???
  16. 16. WHAT CAN GO WRONG DYNAMICALLY STABLE WITH STANDARD POLICIES ??? NO!
  17. 17. (Poisson traffic + equi-saturation) ORIGIN SIGNAL DESTINATION
  18. 18. (Poisson traffic + equi-saturation) • , . 0 X1 s s X2 = equilibria Non-unique DEMAND ORIGIN SIGNAL DESTINATION
  19. 19. (Poisson traffic + equi-saturation) • , . 0 X1 s s X2 = equilibria Non-unique DEMAND ORIGIN SIGNAL DESTINATION
  20. 20. (Poisson traffic + equi-saturation) • , . 0 X1 s s X2 = equilibria Non-unique DEMAND ORIGIN SIGNAL DESTINATION
  21. 21. (Poisson traffic + equi-saturation) • , . 0 X1 s s X2 = equilibria Non-unique DEMAND ORIGIN SIGNAL DESTINATION
  22. 22. (Poisson traffic + equi-saturation) • , . 0 X1 s s X2 Bold lines = equilibria Non-unique T T ORIGIN SIGNAL DEMAND DESTINATION
  23. 23. PITCHFORK (Poisson traffic + equi-saturation) • , . 0 X1 s s X2 Bold lines = equilibria Non-unique PITCHFORK BIFURCATION ORIGIN SIGNAL DESTINATION
  24. 24. SHORT AND LONG ROUTES • , 0 X1 s 2s X2 Bold lines = equilibria Non-unique BIFURCATION ORIGIN DESTINATION
  25. 25. P0: BASIC IDEA DEMAND SET D -C b SUPPLY – FEASIBLE SET S n(D)
  26. 26. BASIC IDEA DEMAND SET D -C b SUPPLY – FEASIBLE SET S n(D) -C is normal to D∩S b is normal to SP0:
  27. 27. BASIC IDEA DEMAND SET D -C b SUPPLY – FEASIBLE SET S n(D) -C is normal to D∩S b is normal to S -C = n(D) + b P0:
  28. 28. BASIC IDEA DEMAND SET D -C b SUPPLY – FEASIBLE SET S n(D) -C is normal to D∩S b is normal to S -C = n(D) + b -(C+b) = n(D) P0:
  29. 29. BASIC IDEA DEMAND SET D -C b SUPPLY – FEASIBLE SET S n(D) -C is normal to D∩S b is normal to S -C = n(D) + b -(C+b) = n(D) -(C+b) is normal to D P0:
  30. 30. BASIC IDEA DEMAND SET D -C b SUPPLY – FEASIBLE SET S n(D) -C is normal to D∩S b is normal to S -C = n(D) + b -(C+b) = n(D) -(C+b) is normal to D EQUILIBRIUM consistent with P0 P0:
  31. 31. P0 STAGE PRESSURES sibi Basic P0
  32. 32. P0 STAGE PRESSURES sibi Basic P0 Qi/Gi P0+vertical queue
  33. 33. IS Q/G RIGHT ? sibi Basic P0 Qi/Gi P0 with vertical queue Qi p/Gi THIS TALK
  34. 34. Policy: At time “t” swap green-time towards the stage with the higher pressure IF Pressi(t) > Pressj(t) THEN swap some green from stage j to stage i
  35. 35. Policy: At time “t” swap green-time towards the stage with the higher pressure dGi (t)/dt = Pressi(t) - Pressj(t) dGj (t)/dt = Pressj(t) - Pressi(t)
  36. 36. Policy: At time “t” swap green-time towards the stage with the higher pressure dG1(t)/dt = Press1(t) – Press2(t) dG2(t)/dt = Press2(t) – Press1(t)
  37. 37. Exact Policy: At time “t” swap green-time to exactly equalise stage pressures Choose G(t) so that Press1(t)= Press2(t)
  38. 38. Origin Destination Exact p-policy: Green-times satisfy Q1 p/G1 = Q2 p/G2 C s2 = 2 s1 = 1
  39. 39. Origin Destination Exact p-policy: Green-times satisfy Q1 p/G1 = Q2 p/G2 C s2 = 2 s1 = 1
  40. 40. ONE possible p-dynamic dG1(t)/dt = Q1 p(t)/G1(t) - Q2 p(t)/G2(t) dG2(t)/dt = Q2 p(t)/G2(t) – Q1 p(t)/G1(t)
  41. 41. ONE possible p-dynamic dG1(t)/dt = Q1 p(t)/G1(t) - Q2 p(t)/G2(t) dG2(t)/dt = Q2 p(t)/G2(t) – Q1 p(t)/G1(t) dX1(t)/dt = Q2(t)/(2G2(t)) - Q1(t)/G1(t) dX2(t)/dt = Q1(t)/(G1(t)) – Q2(t)/2G2(t)
  42. 42. ONE possible p-dynamic dG1(t)/dt = Q1 p(t)/G1(t) - Q2 p(t)/G2(t) dG2(t)/dt = Q2 p(t)/G2(t) – Q1 p(t)/G1(t) dX1(t)/dt = Q2(t)/(2G2(t)) - Q1(t)/G1(t) dX2(t)/dt = Q1(t)/(G1(t)) – Q2(t)/2G2(t) dQ1(t)/dt = X1(t – c) – G1(t) dQ2(t)/dt = X2(t – c) – 2G2(t)
  43. 43. Green-time / route choice equilibria Equal p-pressures: Q1 p / G1= Q2 p / G2 Equal delays: Q1 / G1 = Q2 / (2G2) Eliminate G.
  44. 44. Green-time / route choice equilibria Equal p-pressures: Q1 p / G1= Q2 p / G2 Equal delays: Q1 / G1 = Q2 / (2G2) Eliminate G. To obtain: a constraint on the queue vector Q.
  45. 45. Q2 Q1
  46. 46. Q2 Q1 ROUTEING / p-POLICY EQUILIBRIA p = 2
  47. 47. Q2 Q1 ROUTEING / p-POLICY EQUILIBRIA p = 2
  48. 48. Q2 Q1 ROUTEING / p-POLICY EQUILIBRIA p = 2
  49. 49. X2 X1 ROUTEING / p-POLICY EQUILIBRIA p = 2
  50. 50. X2 X1 Throughput ROUTEING / p-POLICY EQUILIBRIA p = 2 ???
  51. 51. X2 X1 2 Throughput ROUTEING / p-POLICY EQUILIBRIA p = 2
  52. 52. X2 X1 1 2 Throughput ROUTEING / p-POLICY EQUILIBRIA p = 2
  53. 53. X2 X1 1 6/5 2 Throughput ROUTEING / p-POLICY EQUILIBRIA p = 2
  54. 54. X2 X1 p = 22 6/5 1 DYNAMICS
  55. 55. X2 X1 DYNAMICS p = 22 6/5 1
  56. 56. Q2 Q1 DYNAMICS p = 22 6/5 1
  57. 57. Q2 Q1 DYNAMICS p = 22 6/5 1
  58. 58. Q2 Q1 DYNAMICS p = 22 6/5 1
  59. 59. X2 X1 X-DYNAMICS p = 1 (P0)2 Throughput = 2 (MAX)
  60. 60. CONCLUSIONS: Stated a p-evolution eqn. showing how route-inflows, green-times and queues evolve for all future time.
  61. 61. CONCLUSIONS: Stated a p-evolution eqn. showing how route-inflows, green-times and queues evolve for all future time. p = 1 and p ≠ 1.
  62. 62. CONCLUSIONS: Stated a p-evolution eqn. showing how route-inflows, green-times and queues evolve for all future time. p ≠ 1: FAILS to maximise capacity p = 1: P0: maximises capacity.
  63. 63. CONCLUSIONS: p ≠ 1: FAILS to maximise capacity p = 1 (P0): maximises capacity. Holmes: All p ≠ 1 are eliminated, p = 1 or P0 remains; which must be the ONLY p such that “policy p is capacity maximising” is the truth.

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