Feedback Loops Guy Rousseau Atlanta Regional Commission

ARC Historical Perspective Old feedback model iterated with simple iteration (naïve approach) Looked at the Suzanne Evans algorithm of optimal convex combination formulation Looked at the MSA (method of successive averages). Included the inter-relationship between model formulation, system-wide convergence statistics, link volume convergence and output accuracy as determined by traffic counts. Results were quantified by looking at link volume differences using GIS plots and statistical indicators.

ARC Historical Perspective Allowed to estimate relationship between system and link level convergence. System-wide criteria estimated the degree of convergence to the optimal/equilibrium solution for purposes of validating the output. Relationships between system statistics and convergence for individual links were established, and system level convergence (or relative gap). Inclusion of trip table estimation into the equilibrium process was investigated, and looked at the elasticity of trip tables with respect to congestion, in establishing the accuracy of the assignment.

ARC Background and Findings Common method for doing feedback is to average link flows (using weighted successive averages) rather than averaging times. Nothing technically wrong with using averaged times, as long as the closure checks are achieved, but one could get slightly different answers using averaged link flows. Averaging times may not always lead to convergence, but ARC has not had problems reaching closure using averaged times based on the criteria set.

ARC Feedback Loop Definition Skims are calculated using congested travel times Gravity model uses gamma functions for 4 income groups. Iteratively executes the primary steps in the travel demand model to insure that trip generation, trip distribution and mode choice are based on accurate input speeds and travel times, then creates the person and vehicle trip tables to be used in the assignment process. Congested speeds and travel times from each iteration serve as inputs to the next iteration. 1st loop begins with fixed seed AM trip table, and each successive loop is a feedback from the preceding run.

Method of Successive Averages MSA built into equilibrium highway assignment procedures Travel times on each link from the previous iteration are averaged with the travel times from the most recent assignment based on the number of the current iteration, and new input speeds are determined for the next iteration. The process uses 2 criteria to test for closure / convergence. Travel times are within +/- 5% for 95% of the network links between each iteration. Link volumes are within +/-5% for 95% of the network links between each iteration. Travel times and volumes are within 5% between the last 2 iterations for 95% of the network links. Convergence occurs if less than 5% of O-D pairs have peak hour travel times that change by more than 5% & average change in peak hour link volumes is less than 5%.

ARC Feedback Loop Description Run AM Peak Assignment Input Files Skims 1 st loop – Built from Lookup Table of Congested Skims 2-+ Loops – Built from Congested Highway Times from assignment Check for Method of Successive Averages Check between iteration Travel times within +/-5% for 95% of network Link volumes within +/-5% for 95% of network

Number of Feedback Loops This prescribes the number of iterative loops the model will go through during the Feedback Loop Module, for instance 15. The first loop begins with a fixed seed AM trip table, and each successive loop is a feedback from the preceding run. Build peak skims creates matrices of toll, distance and time for every zone-to-zone combination within the peak network. Initially the peak travel time skims are based on lookup tables, but they are updated during the feedback loop.

ARC Feedback Loop Closure The feedback loop closure criteria were modified. Previously, there were cases where the closure check for difference in O-D volumes would produce slightly erroneous results. An additional midday highway assignment has been incorporated within the feedback loop. Two midday highway assignments are now performed within the feedback loop regardless of the number of assignments or loops required for the AM peak period assignment.

ARC Feedback Loop Module This module runs an AM peak period assignment and builds congested highway travel time skims. The module includes functions that build transit support links, calculate TAZ transit walk accessibility, and estimates transit walk times. It also runs trip generation, trip distribution and mode choice and then creates the person and vehicle trip tables to be used in the assignment process.

Feedback Loop & Trip Tables While merging trip tables, we combine the output trip tables from the mode choice model into a final trip table containing truck, SOV, and HOV trips. This trip table is used in the highway assignment procedure. It creates a new AM trip table that is fed back into the AM assignment procedure in the feedback loop process.

CORE MODELING SYSTEM Module 1. Highway Network Building Build Highway Networks and Develop Highway Skims Module 2. Feedback Loops Trip Generation, Transit Networks, Trip Distribution, Trip table Manipulations, Mode Choice, AM Highway Assignment, Closure Checks Module 3. Transit Assignments Work & Non-Work Walk-to-Local / Walk-to-Premium / Drive-to-Transit Module 4. Time of Day Assignments Trip Table Preparation, AM, MD, PM, NT Highway Assignments Miscellaneous Post Processors (Value Pricing, Emissions, etc.) Miscellaneous Pre-Processors (Network Preparation, Data Formatting, etc.)

Recent Modifications Modifications were made to both the highway and transit skim building procedures. The model feedback loop was modified to reflect bus path building into every iteration. For highway, a correction was made to the HOV skims for both the AM and MD highway assignments in the feedback loop. Previously, the skims were calculated using un- congested travel times and have now been updated to use the congested travel times.

Other Modifications 2 modifications to the highway assignment Added a cost function for consistency in paths and equilibrium Changed the closure gap from 0.01 to The closure gap was set to be more stringent to try and minimize the fluctuation of equilibrium assignments between alternatives

TP+ Implications TP+ allows users to build paths using any variables, but unless specifically specified in a cost function, variables other than time will not be included in the equilibrium closure calculations. In the absence of a cost function, equilibrium closure is based on time alone. A cost function was added since we are adding a time penalty for tolls. By including the toll penalty in the cost function, we're trying to make the path building and equilibrium closure calculations more consistent.

Future Enhanements For the time being, ARC practically lets TP+ determine when assignments "converge" ARC will soon develop a TP+ code to manually do assignment/equilibrium gap closure determinations So that ARC could base closure on multiple vehicle class volumes & related time penalties.

More Enhancements Add a closure check for the maximum link flow change. Setup the feedback process to base feedback loop travel times on averaged link flows rather than averaging assignment times. Use the maximum link flow change as the primary feedback closure check.

Feedback Loop Criteria Scripting Must do at least 2 assignments to determine convergence of link volumes: Criteria 1: TRAVEL TIME / SPEEDS Determine new congested skims, using the highway load function to calculate travel time skims, then calculate and add intra zonal times, then add terminal time based on lookup table files, then create the AM peak period highway skims for trip distribution; Determine before and after assignment time; Calculate weighted average percent change in travel time for each O-D pair; Total percent change is calculated by weighting each O-D pair's, travel time relative to the O-D trips, using trip table from current iteration for weighted average.

Feedback Loop Criteria Scripting Criteria 2: LINK VOLUMES Calculate average total volume for all links, comparing the previous loaded network with the current loaded network; Calculate weighted average percent change in link volumes; Total percent change is calculated by weighting each link's % change

Convergence and Subsequent Steps If convergence criteria not met, produce skims for the next loop, using the average of times from the previous 2 assignments, with the first loop using the average of free-flow and congested. While building the peak skims, it creates matrices of toll, distance and time for every zone-to-zone combination within the peak network. Initially the peak travel time skims are based on lookup tables, but they are updated during the feedback loop. We also reflect bus path building into every iteration. An additional midday highway assignment has been incorporated within the feedback loop. Two midday highway assignments are performed within the feedback loop regardless of the number of assignments or loops required for the AM peak period assignment.

ARC Land Use Feedback Loop Composite Impedances: Linkage to Dram/Empal Land Use Model TP+ procedure uses a skim file with the HBW composite times from the AM peak period highway and transit assignment, the total person trip table from the distribution, and an equivalency file to prepare the composite impedance, by DRAM/EMPAL district (census tracts). This procedure is run after the model set has been run and fully evaluated. For instance, 2005 composite impedances feed into 2010 land use

ARC Land Use Feedback Loop HBW composite time by income group (4) This is a matrix of composite times using the AM peak period highway and transit assignment. Total Person Trips, a standard trip matrix file. The file has 7 tables, which are: Total Home Based Work Total Home Based Other Total Non Home Based Home Based Work: Income Group 1 Home Based Work: Income Group 2 Home Based Work: Income Group 3 Home Based Work: Income Group 4