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

Satish V. Ukkusuri, Associate Professor, Purdue University School of Civil Engineering, Purdue University Kien Doan, Purdue University Ph.D. Student, School of Civil Engineering, Purdue University H. M. Abdul Aziz, Purdue University Ph.D. Student, School of Civil Engineering, Purdue University Presented at ISTTT 20th, Noordwijk, the Netherlands Abstract This 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 Numerical results (Minimizing system travel time) (Constraint for signal timing) 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. (Traffic flow propagation constraints) Signal operators Design signal settings to optimize system performance 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 Some of them use the second path 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 Departure rate pattern and corresponding cost for O-D 1-12 The problem formulated as a Stackelberg game Departure rate pattern and corresponding cost for O-Ds 21-29, 41-46 Contributions Using a spatial queue based dynamic network loading model that incorporates both route choice and departure time choice in the integrated DUESC model, Handling the DUESC problem for general multiple O-D networks, Considering dynamic sequence and duration of phases in signal setting, Including cycle length constraint and handling all possible turning behaviors to address all possible phases, Formulating the DUESC problem as Nash-Cournot game and Stackelberg game, Solving the formulation by iterative method and exploring the robustness of the signal control solution under different traffic conditions through several numerical experiments. DUESC problem formulated as a Stackelberg game Some of them use the second path 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. . Departure rate pattern and corresponding cost for O-D 31-36 1: green, 0: red Each time interval: 10s Optimal signal phasing and timing Problem definition Given: 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 Convergence of the algorithm 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. Dynamic User Equilibrium In Dynamic User Equilibrium assignment, no traveler has an incentive to unilaterally shift her route of departure time. Iterative Optimization and Assignment (IOA) algorithm 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 Solution Existence We show: Solution existence for the upper level Solution existence for the lower level (Dynamic Equilibrium) (Early and late schedule delay) (Demand satisfaction)

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