Embed Size (px)
DESCRIPTION
Application of Accelerated User Equilibrium Traffic Assignments. Howard Slavin Jonathan Brandon Andres Rabinowicz Srinivasan Sundaram Caliper Corporation May 2009. Traffic Assignment Convergence. Most traffic assignments not sufficiently converged and give semi-random results - PowerPoint PPT Presentation
Citation preview
Application of Accelerated User Equilibrium Traffic Assignments
Howard SlavinJonathan BrandonAndres RabinowiczSrinivasan Sundaram
Caliper CorporationMay 2009
Traffic Assignment Convergence
• Most traffic assignments not sufficiently converged and give semi-random results
• At low convergence-Counter-intuitive or error-prone forecasts and noise all over the network for even minor local changes
• More rapid convergence is now readily available and very helpful in modeling.
• Congested speeds are key model inputs as well as a primary benefit measure.
11th TRB Transportation Planning Applications Conference, Daytona Beach, FL
Empirical Testing of Faster Algorithms & Convergence Impacts
• Examination of algorithmic sales claims from the transportation science literature
• Speed enhancements through distributed processing and multi-threading
• Benefits of tighter convergence• Use of more realistic and appropriate
test cases
Two approaches now proven for faster traffic assignment convergence
• Multi-threaded (and/or distributed) FW
• Dial’s Algorithm B (“OUE” in TransCAD)
• B is a significant innovation
• Both improvements provided in TransCAD 5
Test Case • Well-calibrated regional model for Washington DC that
Caliper developed for MNCPPC-Prince George’s County• 2500 zones, 6 purposes, 3 time periods, 5 assignment
classes• Feedback through distribution, mode choice, &
assignment• Calibrated to Relative Gap of .001, Skim matrix root
mean square error < .1%, Close match to ground counts.• 80 – 170 Assignment iterations and 4 feedback loops• HCM planning BPR coefficients that vary by road class• Subsequent more accurate traffic assignments
performed• Primary test computer-3 year old 3GHz dual Xeons
Washington Regional NetworkNodes 20343
Links 57374
OD Pairs
6365529
Trips 2977171
Extent 92 x 109 mi
PGC Assignment RunTime (FW and OUE)
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000
Time (min)
Gap
FW - Wcrest FW - Ocho OUE - Wcrest OUE- Ocho
Cold Start Convergence Rates PM Assignment
Cold Start Convergence Times (Min) PM Assignment
GapFW – 4 core Woodcrest
FW -8 core Ocho
OUE – 4 core Woodcrest
OUE – 8 core Ocho
1.00E-02 8.3 4.3 16.4 15.5
1.00E-03 28.5 15 33.7 29.6
1.00E-04 108.1 56.1 60.4 53.5
1.00E-05 715.1 369.7 93.7 82.8
1.00E-06 191.5 172.5
1.00E-07 431.2 406.7
Warm Start Convergence with Random Trip Table Perturbations RG=10-5
Time to converge
Cold Start 01:28:02
+/-5% perturbation run 1 00:08:51
+/-5% perturbation run 2 00:08:53
+/-5% perturbation run 3 00:08:45
+/-10% perturbation run 1 00:11:10
+/-10% perturbation run 2 00:11:18
+/-10% perturbation run 3 00:10:00
Cold Start and Warm Start Convergence Rate Comparison
Warm start with perturbations
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
00:00.0 14:24.0 28:48.0 43:12.0 57:36.0 12:00.0 26:24.0 40:48.0
Time
Gap
Cold Start
+/- 5% 1
+/- 5% 2
+/- 5% 3
+/- 10% 1
+/- 10% 2
+/- 10% 3
Feedback loop run times with OUE assignment (0.001 RG)
Model Steps Loop 1 Loop 2 Loop 3 Loop 4
All other Steps 25 min 16 min 16 min 16 min
AM Assn 18 min 5 min 6 min 5 min
PM Assn 34 min 8 min 6 min 5 min
MD Assn 20 min 7 min 6 min 5 min
Loop Time 1 hr 37 min 36 min 34 min 31 min
Comparison of OUE and FW solutions at the same Relative Gap
GAP %RMSE between UE and OUE
Max Link Flow Difference
Objective FunctionValue UE
Objective FunctionValue OUE
0.01 7.87 1663.55 52,329,996.9 52,196,755.6
0.001 2.95 776.15 51,959,104.1 51,946,295.8
0.0001 0.90 378.3 51,922,622.2 51,916,312.2
0.00001 0.36 225.9 51,918,464.7 51,914,194.1
Model run times with OUE assignment (0.0001 RG)
Model Steps Loop 1 Loop 2 Loop 3 Loop 4
All other Steps 25 min 16 min 16 min 16 min
AM Assn 31 min 6 min 5 min 5 min
PM Assn 1 hr 1 min 8 min 7 min 6 min
MD Assn 35 min 9 min 6 min 6 min
Loop Time 2 hr 32 min 39 min 34 min 33 min
How much error is there in the link flows in an unconverged assignment?
• Easy to quantify with these tools• Using the lens of OUE, we compare
less converged solutions with more highly converged ones.
Average and Maximum Link Flow Differences between the OUE equilibrium solution and the solutions at lower relative gaps
0.0001
0.001
0.01
0.1
1
10
100
1000
10000
1.00E-10 1.00E-09 1.00E-08 1.00E-07 1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00
Gap
De
lta
Average Max
Differences from the equilibrium solution (RG 10-15)
Gap Number of Links with Abs_Flow_Diff>200 Avg Abs Diff Max Abs Diff RMSE %
1e-2 6,339 106.3 1,879 152.6
1e-3 960 35.7 1,148 60.6
1e-4 49 10.3 465 19.2
1e-5 1 3.1 278 7.3
1e-6 0 0.7 92 2.3
1e-7 0 0.13 26 0.6
1e-8 0 0.014 3 0.07
1e-9 0 0.00145 0.31 0.007
1e-10 0 0.000148 0.033 0.001
Flow Differences of OUE assignments at different relative gaps with the equilibrium OUE solution computed to a RG of 10-15
Convergence Levels & Project Impacts
• Three Examples Examined• An Irrelevant network change-
doubling the capacity of 2 links in rural VA
• New MD-5 and Beltway Interchange-Addition of a flyover ramp in PG County
• Woodrow Wilson Bridge Improvement-from 6 to 10 lanes.
Links with flow differences greater than 200 vehicles – Irrelevant Change Example
Links with flow changes greater than 200 vehicles – Interchange Project
Comparison of Base and Scenario – Interchange Project
Gap Number of Links with Abs_Flow_Diff > 200
VHTBase Case
VHT Scenario
VHT Saving (Veh-Hrs)
1e-2 162 1,105,726 1,106,134 -408
1e-3 56 1,091,153 1,091,247 -94
1e-4 44 1,090,136 1,090,116 +20
1e-5 45 1,090,090 1,090,088 +2
1e-6 47 1,090,084 1,090,079 +5
1e-7 45 1,090,087 1,090,085 +2
1e-8 45 1,090,089 1,090,086 +3
Links with flow differences greater than 200 vehicles – Bridge Project
Comparison of Base and Scenario – Bridge Project
Gap Number of Links with Abs_Flow_Diff > 200
VHTBase Case
VHT Scenario
VHT Saving (Veh-Hrs)
1e-2 1363 1,105,726 1,092,979 + 12,747
1e-3 1126 1,091,153 1,079,974 + 11,179
1e-4 1197 1,090,136 1,078,699 + 11,437
1e-5 1265 1,090,090 1,078,631 + 11,459
1e-6 1260 1,090,084 1,078,686 + 11,398
1e-7 1259 1,090,087 1,078,688 + 11,399
1e-8 1260 1,090,089 1,078,689 + 11,400
Other Findings
• Benefits estimated from FW were similar
• Our real problem was much tougher computationally than problems reported in the literature
• In these examples, a relative gap of 10-4 seems sufficient for impact analysis.
• Convergence levels should be tested for other assignment models
Conclusions
• Orders of magnitude greater convergence can be achieved with low computing times
• Greater convergence can reduce errors in models and estimated project impacts
• There is little risk in taking advantage of these developments
