A Bi-level Formulation for the Combined Dynamic Equilibrium based Traffic Signal Control

Satish V. Ukkusuri, Associate Professor, Purdue University (sukkusur@purdue.edu), School of Civil Engineering,

Purdue University

Kien Doan, Purdue University (kdoan@purdue.edu), Ph.D. Student, School of Civil Engineering, Purdue University

H. M. Abdul Aziz, Purdue University (haziz@purdue.edu), Ph.D. Student, School of Civil Engineering, Purdue

University

AbstractThis paper provides an approach to solve the system optimal dynamic traffic assignment

problem for networks with multiple O-D pairs. The path-based cell transmission model is

embedded as the underlying dynamic network loading procedure to propagate traffic. We

propose a novel method to fully capture the effect of flow perturbation on total system cost

and accurately compute path marginal cost for each path. This path marginal cost pattern is

used in the projection algorithm to equilibrate the departure rate pattern and solve the

system optimal dynamic traffic assignment. We observe that the results from projection

algorithm are more reliable than those from method of successive average algorithm

(MSA). Several numerical experiments are tested to illustrate the benefits of the proposed

model.

Dynamic Signal Control Optimization

Solution method

Numerical results

Departure rate pattern

Departure rate pattern and corresponding cost for O-D 1-12

Departure rate pattern and corresponding cost for O-Ds 21-29, 41-46

Departure rate pattern and corresponding cost for O-D 31-36

Optimal signal phasing and timing

Convergence of the algorithm

Total-cost comparisons with base case for different departure rate variations

DUESC problem formulated as a Stackelberg game

Related Works•Allsop (1974), Allsop and Charlesworth (1977), Heydecker (1987), Meneguzzer (1995),

Lee and Machemehl (2005), etc

Static networks and cannot capture traffic dynamics

•Gartner and Stamatiadis (1998), Chen and Ben-akiva (1998), Ceylan and Bell

(2004), Taale and Van Zuylen (2003), Taale (2004), Sun et al (2006), etc

Do not incorporates departure time choice and is not based on a realistic traffic

flow

Dynamic Network Loading (DNL)•Path-based cell transmission model (Ukkusuri et al, 2012; Daganzo, 1995) to propagate

traffic in multiple OD networks.

•Incorporate signalized intersections in the DNL.

•It includes 1) Cell update constraints for ordinary, merging, diverging, and intersection

merging cells; and 2) flow update constraints for ordinary, merging, diverging, and

intersection links.

•Travel cost is based on the average travel time computation method (Ramadurai, 2009; Han

et al, 2011; Ukkusuri et al, 2012).

Dynamic User EquilibriumIn Dynamic User Equilibrium assignment, no traveler has an incentive to unilaterally shift

her route of departure time.

Signal operatorsDesign signal settings to

optimize system performance

Road usersChoose routes and departure time to

minimize travel cost

Contributions1.Using a spatial queue based dynamic network loading model that incorporates both route

choice and departure time choice in the integrated DUESC model,

2.Handling the DUESC problem for general multiple O-D networks,

3.Considering dynamic sequence and duration of phases in signal setting,

4.Including cycle length constraint and handling all possible turning behaviors to address

all possible phases,

5.Formulating the DUESC problem as Nash-Cournot game and Stackelberg game,

6.Solving the formulation by iterative method and exploring the robustness of the signal

control solution under different traffic conditions through several numerical experiments.

Problem definitionGiven:

•A traffic network with signalized intersections (in cell-based form)

•Each O-D pair with multiple paths

•Fixed OD demand

•Predefined phases

Output:

•Path flow (departure rate) at equilibrium condition

•Optimal signal timing plan

•Phase sequence and duration

(Dynamic Equilibrium)

(Early and late schedule delay)

(Demand satisfaction)

(Minimizing system travel time)

(Constraint for signal timing)

(Traffic flow propagation constraints)

The problem formulated as a Stackelberg game

Iterative Optimization and Assignment (IOA) algorithm

Solution ExistenceWe show:

•Solution existence for the upper level

•Solution existence for the lower level

Conclusions•Propose a combined signal control and traffic assignment in dynamic contexts

•Use advanced traffic flow model (CTM and path-based CTM)

•Formulate the problem as a Nash-Cournot game and a Stackelberg game

•Develop a heuristic algorithm based on iterative optimization and assignment

•Solve upper level by mixed integer programing and lower level by projection

algorithm

•Perform sensitivity analysis to confirm the robustness of the optimal solution

30% variation in demand make less than 7% change in total system cost, which demonstrates the robustness of the DUESC model.

The network inefficiency goes from 1.26 to 1.01, which illustrates the effectiveness of the proposed DUESC model

Majority of travelers

choose path 1 and depart at

these time

Some of them use the second path

OD 1-12 contains three paths: Path 1 includes cells: 1,2,3,4,5,6,102,7,8,9,10,11,12Path 2 includes cells: 1,2,3,61,62,63,64,65,66,67,68,69,70,71,11,12Path 2 includes cells: 1,2,3,4,5,6,105,44,45,51,52,53,54,11,12There is a fixed demand for each OD.

Leader: signal operator who optimizes the network performance

is the decision variable for signal setting

G(; r()) is the function of total travel cost

r() the rational response of the road users to a given signal setting

Follower: travelers who minimize their own costs

r() is a solution of VI(R(); F) for a given

R() is set of feasible solution r corresponding to certain

F is the cost function that map departure rate r and given signal setting to cost vector c

In this Stackelberg game, the road users always optimize their utilities based on the signal

settings controlled by the signal operator. The leader knows how road users will response to

their signal settings.

The pair (*; r*) is a Stackelberg equilibrium if and only if:

1) the follower has no incentive to shift their decisions because it is the best solution

based on *, and

2) the leader has no incentive to deviate from * because if he/she does so, the follower

will change their decision as well, which makes the leader worst-off.

.

Some of them use the second path

1: green, 0: redEach time interval: 10s

Presented at ISTTT 20th, Noordwijk, the Netherlands