1. Robust Synchronization of Actuated Signals on Arterials Project #2008-003: Simulation-Based Robust Optimization for Actuated Signal Timing and Setting Lihui Zhang Yafeng Yin

2. Outline Background Introduction Background Introduction Bandwidth Maximization Bandwidth Maximization Robust Coordination Model Robust Coordination Model Numerical Example Numerical Example Concluding Remarks Concluding Remarks

3. Background -- Coordination Synchronize traffic signals to provide smooth progression for major traffic movements along arterials Synchronize traffic signals to provide smooth progression for major traffic movements along arterials Offset + Cycle length Offset + Cycle length

4. Background Actuated signals Actuated signals Current practice performing coordination Current practice performing coordination Uncertainty of start and end of the green Uncertainty of start and end of the green Uncertainty of start and end of the green Uncertainty of start and end of the green Semi-Actuated signal coordination Semi-Actuated signal coordination

5. Background Two approaches: Two approaches: by maximizing green bandwidth by maximizing green bandwidth (MAXBAND and PASSER-II ) (MAXBAND and PASSER-II ) by minimizing total delay by minimizing total delay (TRANSY-7F ) (TRANSY-7F )

6. Coordination Time Distance Signal h Signal i Red Interval Trajectory Bandwidth Offset Cycle length Speed

7. Little’s Bandwidth Maximization s.t. s.t. Offset Cycle length Red interval

8. Bandwidth Maximization Bandwidth Maximization Geometry of the Green Bands

9. Background --Early Return to Green

10. Synchronization of Actuated System Stochastic optimization to generate robust synchronization plan Stochastic optimization to generate robust synchronization plan Scenario based Scenario based Scenario set K={k: R k =(r 1 k, r 2 k ……r n k )} Scenario set K={k: R k =(r 1 k, r 2 k ……r n k )} Represent traffic uncertainty Represent traffic uncertainty Loss function Loss function Deal with 10% worst case scenarios Deal with 10% worst case scenarios

11. Model Formulation

12. Numerical Example Network (Corsim: ActCtrl Example) Network (Corsim: ActCtrl Example) Plan Generation Plan Generation Plan Evaluation Plan Evaluation Macro-simulation Macro-simulation Micro-simulation Micro-simulation Modeling System: GAMS Solver: CPLEX

14. Numerical Example --Plan Generation Robust Plan Robust Plan 250 scenarios 250 scenarios Red times of the sync phases: Red times of the sync phases: independently normally distributed with independently normally distributed with specific mean and same variance. specific mean and same variance. Scenarios have equal probability to occur Scenarios have equal probability to occur Confidence level 0.90 Confidence level 0.90 Nominal Plan Nominal Plan Red times: use fixed mean red times Red times: use fixed mean red times

15. Numerical Example --Plan Evaluation 2000 samples generated from independent normal distributions 2000 samples generated from independent normal distributions 2000 samples generated from uniform distributions 2000 samples generated from uniform distributions Under Robust Plan and Nominal Plan Under Robust Plan and Nominal Plan Performance measure: Performance measure: 1. Mean bandwidth 2. Worst case bandwidth 3. 90% CVaR 4. 90 th percentile minimum bandwidth bandwidth Macro-SimulationSimulationRobustNominalNormalXX UniformYY

16. Monte Carlo Simulation--Normal Distribution BandwidthMeanWorstcase 90 th percentile90%CVaR Change MeanWorstcase percentile90%CVaR Nominal Plan 0.3730.2210.3090.521 RobustPlan0.4480.2720.3830.43120.0%23.1%23.9%-17.3%

17. Monte Carlo Simulation --Uniform Distribution BandwidthMeanWorstcase 90 th percentile90%CVaR Change MeanWorstcase percentile90%CVaR Nominal Plan 0.4750.2310.3160.702 RobustPlan0.5540.2820.3870.58616.7%22.1%22.5%-16.5%

18. Micro-Simulation and Hypothesis Test Multi-run result Control delay Travel time Stop ratio Robust plan Mean96.190318.3500.148 SD12.24112.7670.0077 Nominal plan Mean122.6346.840.176 SD9.87812.4330.010 Hypothesis test F value 1.541.051.32 T value 5.3095.0567.076

19. Conclusions & Remarks Robust synchronization model: MILP Robust synchronization model: MILP Perform better against high-consequence Perform better against high-consequence scenarios scenarios Can be applied to design coordination plans as Can be applied to design coordination plans as well as fine-tune plans well as fine-tune plans Current research Formulate and solve a deterministic integrated model for signal optimization, simultaneously optimizing cycle length, green splits, offsets, phase sequences Formulate and solve a robust counterpart of the deterministic model considering day-to-day demand variations

20. Thank you ! Question? Thank you ! Question?

21. Numerical Example --Plan Generation Computational time and plan difference

22. Robust synchronization --CVaR