Presentation on theme: "A Bi-level Formulation for the Combined Dynamic Equilibrium based Traffic Signal Control Satish V. Ukkusuri, Associate Professor, Purdue University (firstname.lastname@example.org),"— Presentation transcript:
1 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 UniversityKien Doan, Purdue University Ph.D. Student, School of Civil Engineering, Purdue UniversityH. M. Abdul Aziz, Purdue University Ph.D. Student, School of Civil Engineering, Purdue UniversityPresented at ISTTT 20th, Noordwijk, the NetherlandsAbstractThis 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 OptimizationNumerical results(Minimizing system travel time)(Constraint for signal timing)OD 1-12 contains three paths:Path 1 includes cells: 1,2,3,4,5,6,102,7,8,9,10,11,12Path 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.(Traffic flow propagation constraints)Signal operatorsDesign signal settings to optimize system performanceRoad usersChoose routes and departure time to minimize travel costDeparture rate patternMajority of travelers choose path 1 and depart at these timeSome of them use the second pathRelated WorksAllsop (1974), Allsop and Charlesworth (1977), Heydecker (1987), Meneguzzer (1995), Lee and Machemehl (2005), etcStatic networks and cannot capture traffic dynamicsGartner and Stamatiadis (1998), Chen and Ben-akiva (1998), Ceylan and Bell (2004), Taale and Van Zuylen (2003), Taale (2004), Sun et al (2006), etcDo not incorporates departure time choice and is not based on a realistic traffic flowDeparture rate pattern and corresponding cost for O-D 1-12The problem formulated as a Stackelberg gameDeparture rate pattern and corresponding cost for O-Ds 21-29, 41-46ContributionsUsing 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 gameSome of them use the second pathLeader: signal operator who optimizes the network performance is the decision variable for signal settingG(; r()) is the function of total travel costr() the rational response of the road users to a given signal setting Follower: travelers who minimize their own costsr() 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 cIn 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 *, and2) 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 31-361: green, 0: redEach time interval: 10sOptimal signal phasing and timingProblem definitionGiven:A traffic network with signalized intersections (in cell-based form)Each O-D pair with multiple pathsFixed OD demandPredefined phasesOutput:Path flow (departure rate) at equilibrium conditionOptimal signal timing planPhase sequence and durationConvergence of the algorithmThe network inefficiency goes from 1.26 to 1.01, which illustrates the effectiveness of the proposed DUESC modelSolution methodDynamic 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 variations30% variation in demand make less than 7% change in total system cost, which demonstrates the robustness of the DUESC model.Dynamic User EquilibriumIn Dynamic User Equilibrium assignment, no traveler has an incentive to unilaterally shift her route of departure time.Iterative Optimization and Assignment (IOA) algorithmConclusionsPropose a combined signal control and traffic assignment in dynamic contextsUse advanced traffic flow model (CTM and path-based CTM)Formulate the problem as a Nash-Cournot game and a Stackelberg gameDevelop a heuristic algorithm based on iterative optimization and assignmentSolve upper level by mixed integer programing and lower level by projection algorithmPerform sensitivity analysis to confirm the robustness of the optimal solutionSolution ExistenceWe show:Solution existence for the upper levelSolution existence for the lower level(Dynamic Equilibrium)(Early and late schedule delay)(Demand satisfaction)
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