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.

2010.10.27

OUTLINEOUTLINE

Introduction Literature Review Methodology

– Research Framework– Model Formulation– Optimization Process

Numerical Experiments– A test network– A real city network

Concluding Comments

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Introduction

Literature Review

Methodology

Numerical Experiment

Concluding Remarks

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BackgroundBackground Basically, a traffic corridor includes three

major parts:Mainline Freeway segmentsOn-ramps and off-rampsOne or more parallel surface streets

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MotivationMotivation

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• 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.

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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.

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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

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Introduction

Literature Review

Methodology

Numerical Experiment

Concluding Remarks

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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.

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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.

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Introduction

Literature Review

Methodology

Numerical Experiment

Concluding Remarks

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Research FrameworkResearch Framework

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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.

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Model FormulationModel 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.

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Notations:xij(k) is the queue length of movement from i to j at

time interval k.qi(k) is the inflow of section i at time interval k.

ui(k) is the outflow of section i at time interval k.

ri(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 JD = τ × Σ Σ xij(k)

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Concept of the modelConcept of the model

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Mainstream of FreewayFlow conservation

qH2(k) = uH1(k) + uR1(k)

qH3(k) = uH2(k) - qR2(k)

Queue lengthxHi(k+1) = xHi(k) + τ[qHi(k) - uHi(k)]

xmax,Hi = βHi(ρmax,Hi – ρcr,Hi)

0 x≦ Hi(k) x≦ max,Hi

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qi(k):inflowui(k):outflow

qi(k):inflowui(k):outflowxi(k):queue lengthβHi :length of section

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On-ramp ControlALINEA

ri(k+1) = ri(k) + H[oi* - oout,i(k)]

oout,i(k) = (βv + βd) × ρcr,Hj(k) / 1000

ρcr,Hj(k) = qHj(k) / (βHj × nHj)Outflow - on-ramp & off-ramp

uRi(k) ≦ α × ri(k)

uRj(k) u≦ sat,Rj

Queue lengthxRi(k+1) = xRi(k) + τ[qRi(k) - uRi(k)]

0 x≦ Ri(k) x≦ max,Ri17th ITS WORLD CONGRESS

qi(k):inflowri(k):metering rateβHi :length of sectionβv :length of vehicleβd :length of detectorni:number of lanesui(k):outflowri(k):metering rate

qi(k):inflowui(k):outflowxi(k):queue length

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Urban StreetsCycle, Green time, Lost time

Σ gγ,μ = c – Lγ

Exit flow of a section.sUi(k) = tij × qUi(k)

Queue lengthxUi(k+1) = xUi(k) + τ[(1-tij)qUi(k) + dUi(k) - uUi(k)]

0 x≦ Ui(k) x≦ max,Ui

Inflow & OutflowqUi(k) = Σ tUi,UjuUj(k)

uUi(k) = Sui × gUi(k) / c17th ITS WORLD CONGRESS

qi(k):inflowui(k):outflowtuiuj :turning rateS :saturation flow rateg:green timec:cycle time

qi(k):inflowsi(k):exit flowtij :exit rateqi(k):inflowui(k):outflowxi(k):queue lengthdi(k):demand

c:cycle timeg:green timeL:lost time

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route guidance : VMS

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Optimization ProcessOptimization Process

Formulation Construction

Use CPLEX to optimize the problem

2009/12/3

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Introduction

Literature Review

Methodology

Numerical Experiments

Concluding Remarks

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The Test Network

includes a mainstream of freeway, ramps , and urban networks

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Experimental Design

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• 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

percentageSignal cycle time

Initial metering rate

1 100 50% 60s 20

2 120 60% 60s 20

3 140 70% 60s 20

4 160 80% 60s 20

5 180 90% 60s 20

6 200 100% 60s 20

7 220 110% 60s 20

8 240 120% 60s 20

9 260 130% 60s 20

10 280 140% 60s 20

11 300 150% 60s 20

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Change of Objective ValuesChange of Objective Values

It is obvious that objective values increase with respect to the demand level.

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Differences of objective values between consecutive iterations

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under saturation

lowLevel

Median Level

High Level

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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 : 14290 vehs-min• Average values 3.308 min

• High demand level (case 8)• Number of vehicles 5760 vehicles• Total delay : 29636 vehs-min• Average values 5.145 min

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Low LevelLow LevelLow LevelLow Level Median LevelMedian LevelMedian LevelMedian Level High LevelHigh LevelHigh LevelHigh Level

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Results of Green Time Results of Green Time AllocationsAllocations

Link Interval 0Interval

1Interval

2Interval

3Interval

4Interval

5

S-N4,19

10 10 10 10 10 10

E-W 45 45 45 45 45 45

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Link Interval 0Interval

1Interval

2Interval

3Interval

4Interval

5

S-N4,19

20 20 20 42 20 10

E-W 35 35 35 13 35 45

Link Interval 0Interval

1Interval

2Interval

3Interval

4Interval

5

S-N4,19

30 30 30 30 24 10

E-W 25 25 25 25 31 45

lowLevel

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 ratesResults of Metering rates

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lowLevel

Median Level

High Level

Link Interval

0Interval 1 Interval 2 Interval 3 Interval 4 Interval 5

4 20 15.8 11.6 7.3 3.1 1

29 20 15.8 11.6 7.3 3.1 1

Link Interval

0Interval 1 Interval 2 Interval 3 Interval 4 Interval 5

4 20 7.7 1 1 3.67 1

29 20 7.7 1 1 3.67 1

Link Interval

0Interval 1 Interval 2 Interval 3 Interval 4 Interval 5

4 20 10.2 1 3.1 1 1

29 20 10.2 1 3.1 1 1

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A Real Network – Taoyuan Network A Real Network – Taoyuan Network

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Freeway No. 1

Freeway No. 2

No. 31 No. 4

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NO.flow

(veh/region/10mins)

Flow percentage

Signal cycle timeInitial metering rate

1 6746 50% 60s 20

2 8095 60% 60s 20

3 9444 70% 60s 20

4 10793 80% 60s 20

5 12142 90% 90s 20

6 13491 100% 90s 20

7 14840 110% 90s 20

8 16189 120% 120s 20

9 17538 130% 120s 20

10 18887 140% 120s 20

11 20237 150% 120s 20

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LowLow LowLow mediummediummediummedium highhighhighhigh

The interchange is a critical point in the network

Vehicles accessing airport also cause traffic congestion

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Introduction

Literature Review

Methodology

Numerical Experiment

Concluding Remarks

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Concluding CommentsConcluding 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.

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Future DevelopmentsFuture 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

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Thank You for Your Attention.

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