TSP at isolated intersections: Some advances under simulation environment Zhengyao Yu Vikash V. Gayah Eleni Christofa TESC 2018 December 5, 2018
Overview Motivation Problem introduction Assumptions Formation Delays Flexible cycle length Phase rotation Numerical experiments Concluding remarks TESC 2018 2
Motivation Planning Office, the city of Muenster Boston Globe GOAL: optimize signal plan to reduce total passenger delay TESC 2018 3
Earlier problem assumptions Assumptions in earlier work by Christofa et al. (2013) Under-saturated traffic conditions Fixed capacities for each lane group Fixed phase sequence, fixed cycle length Constant passenger occupancies and uniform arrivals for cars Known passenger occupancies and deterministic arrival times for buses TESC 2018 4
Earlier problem formation Mixed-integer non-linear programming with quadratic objective function Global optimum guaranteed with a positive definite Hessian matrix Objective Function: Minimize total passenger delay [dddddddddd oooooooooooooooooo] Decision Variables gg ii,tt : green times allocated to each phase ii in the design cycle TT Constraints Max/min green times Max/min cycle length Under-saturated traffic conditions for all lane groups TESC 2018 5
Car delay calculation DD jj,tt = oooooouu cccccc qq jj ss jj RR jj 2 + RR jj 1 2 2 CL T Aggregated for each lane group jj, assuming the same occupancy on each car Queueing diagram for lane group j Christofa et al., 2013 Delay from two cycles are included TESC 2018 6
Bus delay calculation ww bb ββ ww bb αα dd bb1 ww bb γγ DD bb = oooooouu bb dd bb Calculated for each individual bus CL T ddbb2 Queueing diagram for lane group j Christofa et al., 2013 ww αα bb : passes in current cycle with some delay ββ ww bb : passes in current cycle without delay ww γγ bb : passes in next cycle TESC 2018 7
Optimization process Before the beginning of each cycle, the following information is collected and used as inputs in the mathematical programming problem: General lane and signal settings (saturation flow, lost time, etc.) Signal timing in the previous cycle (green splits) Traffic demand on each lane Bus information (for all buses estimated to arrive within next cycle) Lane group Occupancy Estimated arrival time TESC 2018 8
Flexible cycles Queueing diagram for lane group j Yu et al., 2017 Fixed planning horizon equal to CCCC mmmmmm + CCCC mmaaaa Design cycle length can fluctuate between CCCC mmmmmm and CCCC mmaaaa Buses arriving within CCCC mmmmmm are included Car delays from three cycles are included, two are estimated Application in a rolling horizon TESC 2018 9
Phase rotations Possible solutions Adding binary variables Enumeration Enumeration is found more efficient in our setting Assume background phase sequence for future What about future cycles? Assume optimized phase sequence for future TESC 2018 10
Optimization process updated Before the beginning of each cycle, the following information is collected and used as inputs in the mathematical programming problem: General lane and signal settings (saturation flow, lost time, etc.) Signal timing in the previous cycle (green splits and phase sequence) Traffic demand on each lane Bus information (for all buses estimated to arrive within CLL mmmmmm ) Lane group Occupancy Estimated arrival time TESC 2018 11
Numerical experiment set-up Platform: MATLAB with YALMIP (fmincon and B&B) Atherton St and College Ave in State College, PA Min/max green times: 15/60 s Bus demands: 30 veh/hr Bus occupancy: 10 to 60 Car occupancy: 1.2 College Ave Atherton St Intersection lane groups 1 2 3 4 Phase sequences allowed TESC 2018 12
Results with flexible cycle length Total delay Car delay Bus delay Cycle length: TESC 2018 13
Results with flexible cycle length and phase rotation Optimized sequence Background sequence Total delay change Car delay change TESC 2018 14
Results with flexible cycle length and phase rotation cont. Optimized sequence Background sequence Bus delay change Phase rotation frequency TESC 2018 15
Concluding remarks Flexible cycle length significantly cuts down bus delays Minimal impacts to cars and 30% to 55% delay reductions for buses when compared to a fixed-cycle-length strategy More delay reduction when bus demands are higher It is worthwhile to identify ways to balance between decreasing passenger delays and minimizing phase rotations Assuming optimized phase sequence: more frequent rotations; works well under all demands; higher delay savings (4% - 8% compared to a no-rotation strategy at 0.8 intersection flow ratio ) Assuming background phase sequence: few rotations under high demands; lower delay savings (under 3% compared to a no-rotation strategy at 0.8 intersection flow ratio); avoids excessive disruption and confusion to road users TESC 2018 16
Zhengyao Yu Pennsylvania State University zuy107@psu.edu Vikash V. Gayah Pennsylvania State University gayah@engr.psu.edu Eleni Christofa University of Massachusetts, Amherst christofa@ecs.umass.edu Thank you for listening.