An Optimal Control Model for Traffic Corridor Management Ta-Yin Hu Tung-Yu Wu Department of Transportation and Communication Management Science, National Cheng Kung University, Taiwan, R.O.C

OUTLINE  Introduction  Literature Review  Methodology –Research Framework –Model Formulation –Optimization Process  Numerical Experiments –A test network –A real city network  Concluding Comments 17th ITS WORLD CONGRESS 2

 Introduction  Literature Review  Methodology  Numerical Experiment  Concluding Remarks 17th ITS WORLD CONGRESS 3

Background  Basically, a traffic corridor includes three major parts: Mainline Freeway segments On-ramps and off-ramps One or more parallel surface streets 17th ITS WORLD CONGRESS 4

Motivation Traffic jams occur in many traffic corridors because of increasing number of vehicles and insufficient traffic infrastructure. Under ITS, the intelligent corridor management can utilize route guidance, ramp control and signal control, to improve the efficiency and enhance the service quality of corridors. 5

 Papageorgiou (1995) developed a linear optimal control model to optimize the traffic corridor, and the model takes freeways, on-ramps and parallel arterial streets into consideration.  The concept of the model is based on the store-and-forward model (Gazis and Potts, 1963)  The advantage of the store-and-forward model is that a single performance index is used to evaluate the system. 17th ITS WORLD CONGRESS 6

 Objectives –to develop a linear mathematical model for the ICM based on the store-and-forward model –to explicitly consider route guidance strategies –to optimize related decision variables 17th ITS WORLD CONGRESS 7

 Introduction  Literature Review  Methodology  Numerical Experiment  Concluding Remarks 17th ITS WORLD CONGRESS 8

 Moreno-Banos et al. (1993) proposed an integrated control strategy addressing both route guidance and ramp metering.  Diakaki et al. (1997) described a feedback approach with consideration of the overall network.  Mehta (2001) integrated DynaMIT with the Traffic Management Center and MITSIMLab especially toward Boston’s Central Artery Network. 17th ITS WORLD CONGRESS 9

 Kotsialos et al. (2002) proposed a generic formulation for designing integrated traffic control strategies for traffic corridor.  Kotsialo and Papageorgiou (2004) provided an extensive review for the methods used for the design of freeway network control strategies.  Papamichail et al. (2008) presented a non- linear model-predictive hierarchical control approach for coordinated ramp metering of freeway networks. 17th ITS WORLD CONGRESS 10

 Introduction  Literature Review  Methodology  Numerical Experiment  Concluding Remarks 17th ITS WORLD CONGRESS 11

Research Framework 17th ITS WORLD CONGRESS Collect the information, such as flow data Establish the mathematical model for the traffic corridor including urban streets, ramp, and freeway. Route Guidance Strategies Solved the Problem by CPLEX Results analysis for different traffic situations. 12

Model Formulation  Assumptions: 1. Discrete time interval, time-dependent problem 2. The operation of traffic corridor is under the same management level; therefore, data and information can be exchanged 3. For signalized intersection:  The cycle time is fixed.  Based on a fixed number of phases.  The total lost time of intersection is given. 17th ITS WORLD CONGRESS 13

 Notations: x ij (k) is the queue length of movement from i to j at time interval k. q i (k) is the inflow of section i at time interval k. u i (k) is the outflow of section i at time interval k. r i (k) is the metering rate of section i at time interval k. τ is the time interval.  Objective Function: Minimize the total queue length. Min J D = τ × Σ Σ x ij (k) 17th ITS WORLD CONGRESS 14

Concept of the model 2010/1/18 15

 Mainstream of Freeway Flow conservation q H2 (k) = u H1 (k) + u R1 (k) q H3 (k) = u H2 (k) - q R2 (k) Queue length x Hi (k+1) = x Hi (k) + τ[q Hi (k) - u Hi (k)] x max,Hi = β Hi (ρ max,Hi – ρ cr,Hi ) 0 ≦ x Hi (k) ≦ x max,Hi 17th ITS WORLD CONGRESS q i (k):inflow u i (k):outflow q i (k):inflow u i (k):outflow q i (k):inflow u i (k):outflow x i (k):queue length β Hi :length of section q i (k):inflow u i (k):outflow x i (k):queue length β Hi :length of section 16

 On-ramp Control ALINEA r i (k+1) = r i (k) + H[o i * - o out,i (k)] o out,i (k) = (β v + β d ) × ρ cr,Hj (k) / 1000 ρ cr,Hj (k) = q Hj (k) / (β Hj × n Hj ) Outflow - on-ramp & off-ramp u Ri (k) ≦ α × r i (k) u Rj (k) ≦ u sat,Rj Queue length x Ri (k+1) = x Ri (k) + τ[q Ri (k) - u Ri (k)] 0 ≦ x Ri (k) ≦ x max,Ri 17th ITS WORLD CONGRESS q i (k):inflow r i (k):metering rate β Hi :length of section β v :length of vehicle β d :length of detector n i :number of lanes q i (k):inflow r i (k):metering rate β Hi :length of section β v :length of vehicle β d :length of detector n i :number of lanes ui(k):outflow ri(k):metering rate ui(k):outflow ri(k):metering rate qi(k):inflow ui(k):outflow xi(k):queue length qi(k):inflow ui(k):outflow xi(k):queue length 17

 Urban Streets Cycle, Green time, Lost time Σ g γ,μ = c – L γ Exit flow of a section. s Ui (k) = t ij × q Ui (k) Queue length x Ui (k+1) = x Ui (k) + τ[(1-t ij )q Ui (k) + d Ui (k) - u Ui (k)] 0 ≦ x Ui (k) ≦ x max,Ui Inflow & Outflow q Ui (k) = Σ t Ui,Uj u Uj (k) u Ui (k) = S ui × g Ui (k) / c 17th ITS WORLD CONGRESS q i (k):inflow u i (k):outflow t uiuj :turning rate S :saturation flow rate g:green time c:cycle time q i (k):inflow u i (k):outflow t uiuj :turning rate S :saturation flow rate g:green time c:cycle time q i (k):inflow s i (k):exit flow t ij :exit rate q i (k):inflow s i (k):exit flow t ij :exit rate q i (k):inflow u i (k):outflow x i (k):queue length d i (k):demand q i (k):inflow u i (k):outflow x i (k):queue length d i (k):demand c:cycle time g:green time L:lost time c:cycle time g:green time L:lost time 18

 route guidance : VMS 17th ITS WORLD CONGRESS 19

Optimization Process  Formulation Construction  Use CPLEX to optimize the problem 2009/12/3 20

 Introduction  Literature Review  Methodology  Numerical Experiments  Concluding Remarks 17th ITS WORLD CONGRESS 21

The Test Network  includes a mainstream of freeway, ramps, and urban networks 17th ITS WORLD CONGRESS 22

Experimental Design 17th ITS WORLD CONGRESS 23 Objectives: To observe the system performance in terms of objective values To observe the variation of decision variables, such as green time and ramp metering rates Experimental factor Demand levels: 11

The Virtual Network Experiment NO. flow (veh/region/10mins) Flow percentage Signal cycle time Initial metering rate %60s %60s %60s %60s %60s %60s %60s %60s %60s %60s %60s20 17th ITS WORLD CONGRESS 24

Change of Objective Values  It is obvious that objective values increase with respect to the demand level. 2010/1/18 25

 Differences of objective values between consecutive iterations 17th ITS WORLD CONGRESS under saturation low Level Median Level High Level 26

Comparisons of Different demand level. Low demand level (case 1) Number of vehicles 2400 vehicles Total delay : 218vehs-min Average values 0.091min Median demand level (case 5) Number of vehicles 4320 vehicles Total delay : vehs-min Average values min High demand level (case 8) Number of vehicles 5760 vehicles Total delay : vehs-min Average values min 17th ITS WORLD CONGRESS 27

17th ITS WORLD CONGRESS Low Level Median Level High Level 28

Results of Green Time Allocations Link Interval 0 Interval 1 Interval 2 Interval 3 Interval 4 Interval 5 S-N 4,19 10 E-W 45 17th ITS WORLD CONGRESS Link Interval 0 Interval 1 Interval 2 Interval 3 Interval 4 Interval 5 S-N 4, E-W Link Interval 0 Interval 1 Interval 2 Interval 3 Interval 4 Interval 5 S-N 4, E-W low Level Median Level High Level In low and median level, more green time is allocated for the E-W. In high level, more green time is allocated for the S-N. 29

Results of Metering rates 17th ITS WORLD CONGRESS low Level Median Level High Level Link Interval 0 Interval 1Interval 2Interval 3Interval 4Interval Link Interval 0 Interval 1Interval 2Interval 3Interval 4Interval Link Interval 0 Interval 1Interval 2Interval 3Interval 4Interval

A Real Network – Taoyuan Network 2010/1/18 Freeway No. 1 Freeway No. 2 No. 31No. 4 31

NO. flow (veh/region/10min s) Flow percentage Signal cycle time Initial metering rate %60s %60s %60s %60s %90s %90s %90s %120s %120s %120s %120s /1/18 32

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2010/1/18 LowLowmediummediumhighhigh The interchange is a critical point in the network Vehicles accessing airport also cause traffic congestion 34

 Introduction  Literature Review  Methodology  Numerical Experiment  Concluding Remarks 17th ITS WORLD CONGRESS 35

Concluding Comments  The optimal control model is developed based on the concept of the store-and- forward, thus a linear model could be formulated to solve the problem.  The total queue length increases with respect to demand levels.  As the traffic is getting congested, the ramp metering rate drops dramatically  For the VMS applications, acceptance percentages need to be determined in advance. 17th ITS WORLD CONGRESS 36

Future Developments  Evaluate the optimal strategies through simulation models  Relax the cycle time constraints in the formulation –More variables –More constraints –Difficult to solve for the signal optimization problems 17th ITS WORLD CONGRESS 37

 Thank You for Your Attention. 17th ITS WORLD CONGRESS 38