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
Presented at ISTTT 20th,
Noordwijk, the Netherlands
H. M. Abdul Aziz, Purdue University (haziz@purdue.edu), Ph.D. Student, School of Civil Engineering, Purdue University
Dynamic Signal Control Optimization
Abstract
This paper provides an approach to solve the system optimal dynamic traffic assignment
Numerical results
(Minimizing system travel time)
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
(Constraint for
signal timing)
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.
Signal operators
Design signal settings to
optimize system performance
(Traffic flow
propagation
constraints)
Road users
Choose routes and departure time to
minimize travel cost
Departure rate pattern
Majority of
travelers
choose path 1
and depart at
these time
Related Works
•
OD 1-12 contains three paths:
Path 1 includes cells: 1,2,3,4,5,6,102,7,8,9,10,11,12
Path 2 includes cells: 1,2,3,61,62,63,64,65,66,67,68,69,70,71,11,12
Path 2 includes cells: 1,2,3,4,5,6,105,44,45,51,52,53,54,11,12
There is a fixed demand for each OD.
Some of
them use the
second path
Allsop (1974), Allsop and Charlesworth (1977), Heydecker (1987), Meneguzzer
(1995), Lee and Machemehl (2005), etc
Departure rate pattern and corresponding cost for O-D 1-12
The problem formulated as a Stackelberg game
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
Departure rate pattern and corresponding cost for O-Ds 21-29, 41-46
Contributions
1.
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
DUESC problem formulated as a Stackelberg game
Leader: signal operator who optimizes the network performance
all possible phases,
 is the decision variable for signal setting
5.
Formulating the DUESC problem as Nash-Cournot game and Stackelberg game,
G(; r()) is the function of total travel cost
6.
Solving the formulation by iterative method and exploring the robustness of the signal
r() the rational response of the road users to a given signal setting 
control solution under different traffic conditions through several numerical
Some of
them use the
second path
Departure rate pattern and corresponding cost for O-D 31-36
Follower: travelers who minimize their own costs
r() is a solution of VI(R(); F) for a given 
experiments.
R() is set of feasible solution r corresponding to certain 
1: green, 0: red
Each time interval: 10s
Optimal signal phasing and timing
F is the cost function that map departure rate r and given signal setting  to cost vector c
Problem definition
In this Stackelberg game, the road users always optimize their utilities based on the signal
Given:
settings controlled by the signal operator. The leader knows how road users will response to
• A traffic network with signalized intersections (in cell-based form)
their signal settings.
• Each O-D pair with multiple paths
The pair (*; r*) is a Stackelberg equilibrium if and only if:
• Fixed OD demand
• Predefined phases
1) the follower has no incentive to shift their decisions because it is the best solution based
on *, and
Output:
• Path flow (departure rate) at equilibrium condition
2) the leader has no incentive to deviate from
*
Convergence of the algorithm
because if he/she does so, the follower
• Optimal signal timing plan
will change their decision as well, which makes the leader worst-off.
• Phase sequence and duration
.
The network inefficiency goes from 1.26 to 1.01, which illustrates the effectiveness
of the proposed DUESC model
Solution method
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).
Total-cost comparisons with base case for different departure rate variations
30% variation in demand make less than 7% change in total system cost, which
demonstrates the robustness of the DUESC model.
Iterative Optimization and Assignment (IOA) algorithm
Dynamic User Equilibrium
In Dynamic User Equilibrium assignment, no traveler has an incentive to unilaterally shift
Conclusions
her route of departure time.
• Propose a combined signal control and traffic assignment in dynamic contexts
Solution Existence
(Dynamic Equilibrium)
(Early and late schedule
delay)
(Demand satisfaction)
• Use advanced traffic flow model (CTM and path-based CTM)
We show:
• Formulate the problem as a Nash-Cournot game and a Stackelberg game
• Solution existence for the upper level
• Develop a heuristic algorithm based on iterative optimization and assignment
• Solution existence for the lower level
• Solve upper level by mixed integer programing and lower level by projection algorithm
• Perform sensitivity analysis to confirm the robustness of the optimal solution