Route Assignment

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Introduction

The current method SwashSim uses to move vehicles from an entry link to an exit link is based on user-specified turning percentages for each link the vehicle travels to after exiting the previous link. SwashSim cannot manage the path through a network a vehicle will take after entering. It only can adjust the turning percentages at each link, but is unable to show the route each vehicle is on. The route assignment method uses an origin-destination (OD) matrix with user equilibrium (UE) traffic assignment to specify the full route through the network each vehicle will take as soon as it enters.

User Equilibrium Route Assignment

UE Route Assignment is the method our team is using to create the OD network. The UE Route Assignment creates an OD trip network that is equal for all links being used. This assignment is calculated using an iterative process using the given link travel time and continually adjusting it based on the calculated flow on each link. The flow on each link is calculated from the previous known link travel time. This iterative process is known as the Frank-Wolfe Approach. Below shows the process' steps and formulas.


 * Step 0. Initialization. Perform all-or-nothing assignment based on $$t_a = t_a(0), ∀a$$. This yields x1. Set counter n:=1
 * Step 1. Update. Set $$t_a^n=t_a(x_a^n),∀a. $$
 * Step 2. Direct finding. Perform all-or-nothing assignment based on $$ t_a^n,∀a$$. This yields (direction) flows yn
 * Step 3. Line Search. Find $$ \alpha_n $$ that solves
 * $$min_{0\le\alpha_n\le1} \sum_\alpha \int_0^{x_a^n+\alpha_n(y_a^n - x_a^n)} t_a (\omega)d\omega $$


 * Step 4. Move. Set $$ x^{n+1}=x^n+\alpha_n(y^n-x^n)$$
 * Step 5. Convergence test. If $$ \frac{\sqrt{ \sum_\alpha (x_a^{n+1} - x_a^n)^2}}{ \sum_\alpha x_a^n} < \epsilon $$, stop; otherwise
 * n:=n+1, and go to Step 1.

There are links in the UE network that will have zero traffic flow because it has an undesirable travel time. If a user were to choose to use an unused link, they would not be able to increase their travel time. The Frank Wolfe Approach uses the Bureau of Public Roads (BPR) function shown below. A flow vs. travel time plot is also shown below.



$$ t_\alpha = t_\alpha^0(1 + 0.15(\frac{v_\alpha}{c_\alpha})^{4}) $$

XXE

“XXE is a traffic assignment program based on the standard user equilibrium principle, which is defined as: "The travel time between a specified origin and destination on all used routes is the same and is less than or equal to the travel time that would be experienced by a traveler on any unused route." With a defined network and a given origin-destination matrix, XXE assigns traffic flows by setting up the user equilibrium problem as a mathematical program as shown in Equation 8.8 in Mannering and Washburn (2012).” (http://swashware.com/XXE) XXE is a program developed by Scott Washburn and Fred Mannering in 2007. Our group plans to connect with XXE. Our project will call XXE with the input to perform UE traffic assignment. We will modify the code to save and update the path flow during the Frank-Wolfe approach. 



Input

The input for our project is all the SwashSim network properties. This is information that our group has come up with and used in our project. This data is adjustable and can be modified for different simulations. The properties include the link and nodes we have created in the SwashSim network. Each of the links have properties of length, free-flow travel time, capacity, and traffic demand. We hope to use a spreadsheet to allow users to specify link-node structure and OD demand.

Output

With the UE traffic assignment method, the volume of each link and the associated travel times are outputs. The desired output for our project is the path assignment for each vehicle in the network. The path assignment is based on the XXE outputs. The path assignment outputs include the shortest (travel time) path(s) for each OD pair. The output would be able to assign volume to path(s) for each OD pair, vehicles to OD pairs (random), and vehicles to shortest paths.

Test Network

The test network scheme shown below has four traffic analysis zones, twelve OD pairs, four network nodes, and eight network links. The traffic analysis zones are shown as triangles on each corner of the network. This is where vehicles can enter and exit the network. The twelve OD pairs are also detailed in the table below. The table lists the origin zones, destination zones, and respective number of trips. The origin zone is where vehicles enter the network and destination zone is where the vehicle will exit the network. The four network nodes are shown as circles on each corner of the rectangle network. The eight network links are shown as lines connecting each of the four links together as a rectangle. Each link has an adjacent and parallel link that flows in the opposite direction.



The network shown below is the original concept from the scheme shown above as it is entered into SwashSim. SwashSim requires entry and exit nodes to be attached to through lanes rather than turning lanes. This requirement forced the network to expand an extra link between each entry and exit link and turning links (nodes). This network uses a horizontal distance between entry and exit links of 16,632 ft. The four intersections are perfect squares with an area of 1296 ft^2 (36 ft sides). Each entry and exit link has a length of 500 ft. The added links between each intersection and entry/exit links also has a length of 500 ft. The second figure below shows an updated scheme of this network similar to the one detailed above. This scheme has 20 network nodes and 24 network links. This scheme still only has four traffic analysis zones and twelve OD pairs, the same as the first scheme shown above.







References

http://swashware.com/XXE