Optimization
A parsimonious two-way shooting algorithm for connected automated traffic smoothing: Computational Issues and Optimization Fang Zhou1, Jiaqi Ma2, Xiaopeng Li1 1Department
of Civil and Environmental Engineering, Mississippi State University 2Transportation Solutions and Technology Applications Division, Leidos, Inc. Background
Advanced connected and automated vehicle technologies offer new opportunities for highway traffic smoothing by optimizing automated vehicle trajectories. Although controlling an individual or isolated object trajectory was not new to some other fields, optimizing trajectories of a stream of highway vehicles that constantly interact with each other has been seldom studied. This study proposes an efficient trajectory optimization algorithm that can simultaneously improve a range of performance measures for a stream of vehicles on a signalized highway section. This optimization is centered at a novel but parsimonious shooting heuristic (SH) for trajectory construction that considers realistic constraints including vehicle kinematic limits, traffic arrival patterns, car-following safety, and signal operations. SH is suitable for real-time applications when relevant technologies are in place in the near future.
• Smooth traffic flow • Mitigate environmental and safety impacts
Research Questions • Considering a highway segment leading to a signalized intersection with fixed signal timing, a more general case for freeway segment where red time is zero • Build smooth trajectories with a parsimonious Shooting Heuristic for all intersection vehicles • Develop robust numerical optimization algorithms for trajectory optimization
•Only five parameters are used: forward acceleration, forward deceleration, backward deceleration, backward deceleration; speed limit; •Repeat FSO and BSO for all arriving vehicles; •Consider realistic constraints including vehicle kinematic limits, traffic arrival patterns, carfollowing safety, and signal operations
Optimization Approach For any traffic arrival patterns, a series of trajectories for all arriving vehicles can be constructed to optimize traffic operations at signalized intersections. 1) Parsimonious algorithmic structure with small number of control variables 2) Very small computational complexity, suitable for real-time applications 3) Coordinated vehicle trajectory control for all equipped vehicles
Input
Stepwise trajectory function
Developed Numerical Gradient-based Method (NGM) with five key parameters, to optimize total system performance consisting of Travel Time, Fuel Consumption (VT-Micro), and Safety (time-to-collision): Step 1: Calculate numerical gradient with perturbation size ck; if solution not feasible, ck = ck * 0.5; gk = [y(xk+ck) – y(xk)] / ck Step2: Calculate next solution value with step size ak and calculate yk; check vehicle kinematic constraints, if violated, ak = ak * 0.5; xk+1 = xk – ak * gk Step 3-1: If relative improvement is positive and more than 1%, continue Step 2 with the same search direction - gk. (e.g., 10 trials) Step 3-2: If relative improvement is negative, reduce ak by half and continue Step 2. (1 trial)
(m/s2) (m/s2) Benchmark
Average Average Average Fuel Safety Travel Time Consumption Measure (m/s2) (sec/veh) (liter/veh) (sec/veh)
(m/s2)
Intelligent driver model
Total Cost ($/veh)
154.28
0.27
1.60
1.2197
Initial SH
2
-10
2
-10
68.62
0.30
2.05
0.7963
Optimal SH
0.76
-5.90
0.55
-6.42
69.16
0.24
0.10
0.6302
55%
11%
93%
48%
Improvement from Benchmark to Optimal SH 1500
Improvement from Benchmark to Optimal SH: 1. Significant improvement in total system performance for all three performance measures; 2. Optimized vehicle trajectories avoid stop-andgo patterns, a huge source of intersection delay 3. Intersection capacity is maximized due to a preacceleration of all vehicles resulted from our construction and optimization algorithm
Space (m)
• Improve the highway segment capacity and intersection efficiency
•The Algorithm applies Forward Shooting Operation (FSO) and Backward Shooting Operation (BSO) for each vehicles sequentially;
1000
Benchmark
500 0 0
50
100
150 T ime (s)
200
250
300
1500 Space (m)
With connected automation and advanced communication technologies, we aim to develop algorithms to construct and optimize trajectories for large number of vehicles at signalized intersections:
Results
Shooting Heuristic
1000
Initial SH
500 0 0
50
100 150 T ime (s)
200
250
1500 Space (m)
Motivations
1000
Optimal SH 500 0 0
50
100
150
200
250
Time (s)
NGM Optimization Results v.s. Genetic Algorithm (total system performance in $)
1
0.39
0.35
0.34
0.34
0.34
Comp. Time (sec) 9.27
2
0.39
0.36
0.35
0.34
0.33
9.33
14.09%
16.93%
3
0.43
0.39
0.38
0.37
0.36
9.94
13.15%
16.01%
4
0.47
0.43
0.42
0.42
0.41
10.24
11.01%
13.09%
5
0.52
0.49
0.48
0.48
0.47
9.37
9.30%
10.31%
6
0.55
0.50
0.49
0.49
0.48
9.54
11.52%
12.89%
7
0.59
0.55
0.54
0.54
0.52
10.08
8.72%
11.43%
8
0.63
0.59
0.58
0.58
0.57
10.86
8.71%
10.44%
11.22%
13.01%
Scenari Extreme Half extreme Good o# (2 -10 2 -10) (1 -2 1 -2) (1 -2 1 -2)
NGM
GA
Average
Imp.NGM
Imp.GA
13.22%
13.00%
Conclusions 1) The SH approach can build significantly improved vehicle trajectories, reducing stop-and-go, total delay, energy consumption and emissions by almost 50%. 2) The NGM optimization heuristic can further reduce total system performance by 11.22% on average – near optimal solution (only 2% Gap from GA optimal)
of Civil and Environmental Engineering, Mississippi State University 2Transportation Solutions and Technology Applications Division, Leidos, Inc. Background
Advanced connected and automated vehicle technologies offer new opportunities for highway traffic smoothing by optimizing automated vehicle trajectories. Although controlling an individual or isolated object trajectory was not new to some other fields, optimizing trajectories of a stream of highway vehicles that constantly interact with each other has been seldom studied. This study proposes an efficient trajectory optimization algorithm that can simultaneously improve a range of performance measures for a stream of vehicles on a signalized highway section. This optimization is centered at a novel but parsimonious shooting heuristic (SH) for trajectory construction that considers realistic constraints including vehicle kinematic limits, traffic arrival patterns, car-following safety, and signal operations. SH is suitable for real-time applications when relevant technologies are in place in the near future.
• Smooth traffic flow • Mitigate environmental and safety impacts
Research Questions • Considering a highway segment leading to a signalized intersection with fixed signal timing, a more general case for freeway segment where red time is zero • Build smooth trajectories with a parsimonious Shooting Heuristic for all intersection vehicles • Develop robust numerical optimization algorithms for trajectory optimization
•Only five parameters are used: forward acceleration, forward deceleration, backward deceleration, backward deceleration; speed limit; •Repeat FSO and BSO for all arriving vehicles; •Consider realistic constraints including vehicle kinematic limits, traffic arrival patterns, carfollowing safety, and signal operations
Optimization Approach For any traffic arrival patterns, a series of trajectories for all arriving vehicles can be constructed to optimize traffic operations at signalized intersections. 1) Parsimonious algorithmic structure with small number of control variables 2) Very small computational complexity, suitable for real-time applications 3) Coordinated vehicle trajectory control for all equipped vehicles
Input
Stepwise trajectory function
Developed Numerical Gradient-based Method (NGM) with five key parameters, to optimize total system performance consisting of Travel Time, Fuel Consumption (VT-Micro), and Safety (time-to-collision): Step 1: Calculate numerical gradient with perturbation size ck; if solution not feasible, ck = ck * 0.5; gk = [y(xk+ck) – y(xk)] / ck Step2: Calculate next solution value with step size ak and calculate yk; check vehicle kinematic constraints, if violated, ak = ak * 0.5; xk+1 = xk – ak * gk Step 3-1: If relative improvement is positive and more than 1%, continue Step 2 with the same search direction - gk. (e.g., 10 trials) Step 3-2: If relative improvement is negative, reduce ak by half and continue Step 2. (1 trial)
(m/s2) (m/s2) Benchmark
Average Average Average Fuel Safety Travel Time Consumption Measure (m/s2) (sec/veh) (liter/veh) (sec/veh)
(m/s2)
Intelligent driver model
Total Cost ($/veh)
154.28
0.27
1.60
1.2197
Initial SH
2
-10
2
-10
68.62
0.30
2.05
0.7963
Optimal SH
0.76
-5.90
0.55
-6.42
69.16
0.24
0.10
0.6302
55%
11%
93%
48%
Improvement from Benchmark to Optimal SH 1500
Improvement from Benchmark to Optimal SH: 1. Significant improvement in total system performance for all three performance measures; 2. Optimized vehicle trajectories avoid stop-andgo patterns, a huge source of intersection delay 3. Intersection capacity is maximized due to a preacceleration of all vehicles resulted from our construction and optimization algorithm
Space (m)
• Improve the highway segment capacity and intersection efficiency
•The Algorithm applies Forward Shooting Operation (FSO) and Backward Shooting Operation (BSO) for each vehicles sequentially;
1000
Benchmark
500 0 0
50
100
150 T ime (s)
200
250
300
1500 Space (m)
With connected automation and advanced communication technologies, we aim to develop algorithms to construct and optimize trajectories for large number of vehicles at signalized intersections:
Results
Shooting Heuristic
1000
Initial SH
500 0 0
50
100 150 T ime (s)
200
250
1500 Space (m)
Motivations
1000
Optimal SH 500 0 0
50
100
150
200
250
Time (s)
NGM Optimization Results v.s. Genetic Algorithm (total system performance in $)
1
0.39
0.35
0.34
0.34
0.34
Comp. Time (sec) 9.27
2
0.39
0.36
0.35
0.34
0.33
9.33
14.09%
16.93%
3
0.43
0.39
0.38
0.37
0.36
9.94
13.15%
16.01%
4
0.47
0.43
0.42
0.42
0.41
10.24
11.01%
13.09%
5
0.52
0.49
0.48
0.48
0.47
9.37
9.30%
10.31%
6
0.55
0.50
0.49
0.49
0.48
9.54
11.52%
12.89%
7
0.59
0.55
0.54
0.54
0.52
10.08
8.72%
11.43%
8
0.63
0.59
0.58
0.58
0.57
10.86
8.71%
10.44%
11.22%
13.01%
Scenari Extreme Half extreme Good o# (2 -10 2 -10) (1 -2 1 -2) (1 -2 1 -2)
NGM
GA
Average
Imp.NGM
Imp.GA
13.22%
13.00%
Conclusions 1) The SH approach can build significantly improved vehicle trajectories, reducing stop-and-go, total delay, energy consumption and emissions by almost 50%. 2) The NGM optimization heuristic can further reduce total system performance by 11.22% on average – near optimal solution (only 2% Gap from GA optimal)
