Modelling and Optimisation of Dynamic Motorway Traffic Ying Li
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Modelling and Optimisation of Dynamic Motorway Traffic Thesis submitted to University College London for the degree of Doctor of Philosophy by Ying Li Department of Civil, Environmental & Geomatic Engineering Centre for Transport Studies University College London December 2015 Declaration I, Ying Li, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. Signed: Date: iii Abstract Ramp metering, variable speed limits, and hard shoulder running control strategies have been used for managing motorway traffic congestion. This thesis presents a modelling and optimisation framework for all these control strategies. The optimal control prob- lems that aim to minimise the travel delay on motorways are formulated based upon a macroscopic cell transmission model with piecewise linear fundamental diagram. With the piecewise linear nature of the traffic model, the optimal control problems are for- mulated as linear programming (LP) and are solved by the IBM CPLEX solver. The performance of different control strategies are tested on real scenarios on the M25 Mo- torway in England, where improvements were observed with proper implementation. With considering of the uncertainties in traffic demand and characteristics, this thesis also presents a robust modelling and optimisation framework for dynamic motorway traffic. The proposed robust optimisation aims to minimise both mean and variance of travel delays under a range of uncertain scenarios. The robust optimisation is formulated as a minimax problem and solved by a two stage solution procedure. The performances of the robust ramp metering are illustrated through working examples with traffic data collected from the M25 Motorway. Experiments reveal that the deterministic optimal control would outperform slightly the robust control in terms of minimising average delays, while the robust controller gives a more reliable performance when uncertainty is taken into account. This thesis contributes to the development and validation of dynamic simulation, and deterministic and robust optimisation. v Acknowledgements First and foremost, I would like to present my deep gratitude to my principal supervisor Andy Chow for his incredible support, insightful instructions and detailed comments and suggestions on the thesis. I am forever grateful for his help. I would also like to thank my secondary supervisor Professor B.G. Heydecker for his time to provide sound advice and valuable criticism for completing the thesis. I am also grateful to both examiners Professor Richard Gibbens and Dr Ke Han for their useful comments and suggestions on the thesis. Many staff at the Centre for Transport Studies have assisted me during my stay at University College London. I am particularly grateful to Catherine Holloway, Taku Fujiyama, Nicola Christie, and Helena Titheridge. Special thanks to Angela Cooper (in UCL Centre for Languages & International Education) who supports me on the thesis writing. I would like to convey my warm and profound thanks to my dearest friends in the United Kingdom and China for their patience and invaluable support throughout my study period. I am grateful to my friends and fellow colleagues RuNing Ye, ChienPang Liu, Nuo Duan, Simrn Gill, Shuai Li, Rui Sha, Fang Xu, Aris Pavlides, Matthew Tsang, HuaiDong Wang, SiDi Sun, ZiJia Wang, Kun Liu, MengYang Qin, and Li Zhao for standing beside me along this path and for all the time, joy and sorrow we shared together. I am tremendously indebted to my family members, especially my parents XiaoJuan Zhang and ZhongXian Li, and my master for all their support, care and encouragement. They are the sources of confidence and strength for me to overcome difficulties in study and life. Finally, I would also like to acknowledge Highways England for providing the MIDAS traffic data, and IBM Academic Initiative for providing the software and license for using the CPLEX Optimisation Studio. The contents do not reflect the official views or policies of Highways Agency, and other organisations. vii viii Contents Abstract v Acknowledgements vii List of Figures xiii List of Tables xv Abbreviations xvi Symbols xix 1 Introduction1 1.1 Background and Motivation..........................1 1.2 Review of Traffic Models............................2 1.2.1 Microscopic models...........................3 1.2.2 Macroscopic models..........................6 1.2.3 Discussion................................ 10 1.3 Current Practice of Traffic Control...................... 12 1.3.1 Ramp metering............................. 12 1.3.2 Variable speed limits.......................... 17 1.3.3 Hard shoulder running......................... 19 1.4 Research Contributions............................ 21 1.5 Thesis Outline................................. 23 2 Modelling of Motorway Traffic 25 2.1 Introduction................................... 25 2.2 Cell Transmission Model............................ 25 2.3 Traffic Data................................... 29 ix Contents x 2.3.1 Flow................................... 30 2.3.2 Occupancy............................... 31 2.3.3 Speed.................................. 31 2.3.4 Density................................. 32 2.4 Model Calibration............................... 33 2.4.1 Step 1: Free flow line.......................... 34 2.4.2 Step 2: Capacity............................ 35 2.4.3 Step 3: Congested line......................... 36 2.4.4 Review of procedure.......................... 41 2.5 Summary.................................... 43 3 Optimisation of Ramp Metering 45 3.1 Introduction................................... 45 3.2 Objective Function............................... 46 3.3 Ramp Metering Formulation......................... 50 3.4 Effect of Off-ramp Position.......................... 57 3.5 Sensitivity Analysis of Balance Parameter η ................. 60 3.6 Working Example................................ 63 3.6.1 Without ramp metering........................ 68 3.6.2 With ramp metering.......................... 71 3.7 Summary.................................... 76 4 Optimisation of Variable Speed Limits and Hard Shoulder Running 77 4.1 Introduction................................... 77 4.2 Variable Speed Limits............................. 78 4.2.1 Changes in fundamental diagram under VSL............ 78 4.2.2 Optimisation of VSL.......................... 80 4.2.3 The effect of fundamental diagram specifications under VSL.... 89 4.3 Hard Shoulder Running............................ 94 4.3.1 Changes in fundamental diagram under HSR............ 94 4.3.2 Optimisation of HSR.......................... 95 4.3.3 Optimisation of integrated control strategy............. 97 4.4 Working Example................................ 100 4.4.1 Variable speed limits.......................... 100 4.4.2 Hard shoulder running......................... 102 4.4.3 Integrated control strategy...................... 104 4.5 Summary.................................... 107 Contents xi 5 Robust Optimisation of Ramp Metering 109 5.1 Introduction................................... 109 5.2 Characteristic of Road Traffic Flow...................... 110 5.2.1 Set-valued fundamental diagram................... 110 5.2.2 Demand uncertainty.......................... 114 5.3 Robust Optimisation of Ramp Metering................... 115 5.3.1 Review of robust optimisation..................... 115 5.3.2 Likelihood region............................ 117 5.3.3 Robust ramp metering formulation.................. 122 5.4 Working Example................................ 125 5.5 Summary.................................... 134 6 Concluding Remarks 135 6.1 Summary.................................... 135 6.2 Future Work.................................. 138 A Calculate Density from the Occupancy Measurements 141 B Constrained Regression 143 References 145 List of Figures 1.1 Fundamental diagram.............................6 1.2 Configuration of responsive ramp metering system............. 13 1.3 Demand-capacity ramp metering strategy.................. 14 1.4 ALINEA ramp metering strategy....................... 15 1.5 Variable speed limits in operation....................... 18 1.6 Operation of hard shoulder running on the M42 Motorway......... 20 2.1 Discretisation of a road section........................ 26 2.2 Triangle shaped fundamental diagram.................... 28 2.3 Empirical scatter plot of traffic flow and density.............. 34 2.4 Estimation of free flow speed......................... 36 2.5 Estimation of capacity............................. 37 2.6 Trained regression............................... 41 2.7 Calibrated fundamental diagram for CTM.................. 42 3.1 Specification of outflow ratio......................... 51 3.2 Performance of ramp metering........................ 59 3.3 Delays of ramp metering with η lower than one............... 61 3.4 UK M25 Traffic Speed............................. 64 3.5 UK M25 Motorway map - section between Junctions 10 and 16...... 65 3.6 Observed density count plot - section between Junctions 10 and 16.... 66 3.7 Modelling result between Junctions 12 and 16 over one day........ 70 3.8 Comparison of main road densities (Three Junctions)............ 73 3.9 Ramp queues under metering at Junction 15................ 74 3.10 Comparison of main road delays....................... 75 3.11 Comparison of ramp delays.......................... 75 4.1 Changes in fundamental diagram under VSL with real data........ 79 4.2 Changes in fundamental diagram under VSL...............