Speed–density functional relationship for heterogeneous traffic data: a statistical and theoretical investigation
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摘要
This study is an attempt to establish a suitable speed–density functional relationship for heterogeneous traffic on urban arterials. The model must reproduce the traffic behaviour on traffic stream and satisfy all static and dynamic properties of speed–flow–density relationships. As a first attempt for Indian traffic condition, two behavioural parameters, namely the kinematic wave speed at jam(Cj) and a proposed saturation flow(k), are estimated using empirical observations. The parameter Cjis estimated by developing a relationship between driver reaction time and vehicle position in the queue at the signalised intersection. Functional parameters are estimated using Levenberg–Marquardt algorithm implemented in the R statistical software.Numerical measures such as root mean squared error, average relative error and cumulative residual plots are used for assessing models fitness. We set out several static and dynamic properties of the flow–speed–density relationships to evaluate the models, and these properties equally hold good for both homogenous and heterogeneous traffic states.From the numerical analysis, it is found that very few models replicate empirical speed–density data traffic behaviour.However, none of the existing functional forms satisfy all the properties. To overcome the shortcomings, we proposed two new speed–density functional forms. The uniqueness of these models is that they satisfy both numerical accuracy and the properties of fundamental diagram. These new forms would certainly improve the modelling accuracy, especially in dynamic traffic studies when coupling with dynamic speed equations.
This study is an attempt to establish a suitable speed–density functional relationship for heterogeneous traffic on urban arterials. The model must reproduce the traffic behaviour on traffic stream and satisfy all static and dynamic properties of speed–flow–density relationships. As a first attempt for Indian traffic condition, two behavioural parameters, namely the kinematic wave speed at jam(Cj) and a proposed saturation flow(k), are estimated using empirical observations. The parameter Cjis estimated by developing a relationship between driver reaction time and vehicle position in the queue at the signalised intersection. Functional parameters are estimated using Levenberg–Marquardt algorithm implemented in the R statistical software.Numerical measures such as root mean squared error, average relative error and cumulative residual plots are used for assessing models fitness. We set out several static and dynamic properties of the flow–speed–density relationships to evaluate the models, and these properties equally hold good for both homogenous and heterogeneous traffic states.From the numerical analysis, it is found that very few models replicate empirical speed–density data traffic behaviour.However, none of the existing functional forms satisfy all the properties. To overcome the shortcomings, we proposed two new speed–density functional forms. The uniqueness of these models is that they satisfy both numerical accuracy and the properties of fundamental diagram. These new forms would certainly improve the modelling accuracy, especially in dynamic traffic studies when coupling with dynamic speed equations.
This study is an attempt to establish a suitable speed–density functional relationship for heterogeneous traffic on urban arterials. The model must reproduce the traffic behaviour on traffic stream and satisfy all static and dynamic properties of speed–flow–density relationships. As a first attempt for Indian traffic condition, two behavioural parameters, namely the kinematic wave speed at jam(Cj) and a proposed saturation flow(k), are estimated using empirical observations. The parameter Cjis estimated by developing a relationship between driver reaction time and vehicle position in the queue at the signalised intersection. Functional parameters are estimated using Levenberg–Marquardt algorithm implemented in the R statistical software.Numerical measures such as root mean squared error, average relative error and cumulative residual plots are used for assessing models fitness. We set out several static and dynamic properties of the flow–speed–density relationships to evaluate the models, and these properties equally hold good for both homogenous and heterogeneous traffic states.From the numerical analysis, it is found that very few models replicate empirical speed–density data traffic behaviour.However, none of the existing functional forms satisfy all the properties. To overcome the shortcomings, we proposed two new speed–density functional forms. The uniqueness of these models is that they satisfy both numerical accuracy and the properties of fundamental diagram. These new forms would certainly improve the modelling accuracy, especially in dynamic traffic studies when coupling with dynamic speed equations.
引文
1.Lighthill MJ,Whitham GB(1955)On kinematic waves.II.Atheory of traffic flow on long crowded roads.Proc R Soc London A 229(1178):317-345
2.Papageorgiou M,Jean-marc B,Hadj-salem H(1989)Macroscopic modelling of traffic flow on the Boulevard Peripherique in Paris.Transp Res Part B Methodol 236(1):29-47
3.Daganzo CF(1994)The cell transmission model:a dynamic representation of highway traffic consistent with the hydrodynamic theory.Transp Res Part B Methodol 28(4):269-287
4.Khan S,Maini P(1999)Modeling heterogeneous traffic flow.Transp Res Rec 1678(1):234-241
5.Tiwari G,Fazio J,Gaurav S(2007)Traffic planning for nonhomogeneous traffic.Sadhana 32(4):309-328
6.Katz D(2009)Heterogeneous traffic mixes,[term paper].http://www.donaldkatz.com/CEE6603-TermPaperHeterogeneousTraffic.pdf.Accessed 22 July 2016
7.Greenshields BD,Channing WS,Miller HH(1935)A study of traffic capacity.In:Highway research board proceedings,1935.National Research Council(USA),Highway Research Board
8.May AD Jr,Harmut EMK(1967)Non-integer car-following models.Highw Res Rec 199:19-32
9.Drew DR(1968)Traffic flow theory and control.McGraw-Hill,New York
10.La Pipes(1967)Car following models and the fundamental diagram of road traffic.Transp Res 1(1):21-29
11.May AD(1990)Traffic flow fundamentals,2nd edn.Prentice Hall,Englewood Cliffs
12.Greenberg H(1959)An analysis of traffic flow.Oper Res7(1):79-85
13.Underwood RT(1961)Speed,volume,and density relationships:quality and theory of traffic flow.yale Bur Highw traffic 141-188
14.Drake JS,Schofer JL,May Jr AD(1967)A statistical analysis of speed-density hypotheses.In:Vehicular traffic science.Highw Res Rec(154):112-117
15.Newell GF(1961)Nonlinear effects in the dynamics of car following.Oper Res 9(2):209-229
16.Del Castillo JM,Benitez FG(1995)On the functional form of the speed-density relationship-I:general theory.Transp Res Part BMethodol 29(5):373-389
17.Lee HY,Lee HW,Kim D(1998)Origin of synchronized traffic flow on highways and its dynamic phase transitions.Phys Rev Lett 81(5):1130-1133.https://doi.org/10.1103/PhysRevLett.81.1130
18.Wang H,Li J,Chen QY,Ni D(2010)Representing the fundamental diagram:the pursuit of mathematical elegance and empirical accuracy.In:Transport Research Board 89th annual meeting,Washington,DC,USA
19.Thankappan A,Vanajakshi L(2015)Development and application of a traffic stream model under heterogeneous traffic conditions.J Inst Eng Ser A 96:267-275.https://doi.org/10.1007/s40030-015-0134-y
20.Heydecker BG,Addison JD(2011)Analysis and modelling of traffic flow under variable speed limits.Transp Res Part C Emerg Technol 19(2):206-217
21.Mallikarjuna C,Phanindra A,Rao KR(2009)Traffic data collection under mixed traffic conditions using video image processing.J Transp Eng 135(4):174-182.https://doi.org/10.1061/(ASCE)0733-947X(2009)135:4(174)
22.Elzhov TV,Mullen KM,Spiess AN,BolkerB,Mullen MKM(2012)R interface to the Levenberg-Marquardt nonlinear leastsquares algorithm found in MINPACK,plus support for bounds.Retrieved from CRAN:http://cran.rproject.org/web/packages/minpack.lm/minpack.lm.pdf.Accessed 10 July 2016
23.Hauer E(2015)The art of regression modeling in road safety.Springer,Berlin,p 38
2.Papageorgiou M,Jean-marc B,Hadj-salem H(1989)Macroscopic modelling of traffic flow on the Boulevard Peripherique in Paris.Transp Res Part B Methodol 236(1):29-47
3.Daganzo CF(1994)The cell transmission model:a dynamic representation of highway traffic consistent with the hydrodynamic theory.Transp Res Part B Methodol 28(4):269-287
4.Khan S,Maini P(1999)Modeling heterogeneous traffic flow.Transp Res Rec 1678(1):234-241
5.Tiwari G,Fazio J,Gaurav S(2007)Traffic planning for nonhomogeneous traffic.Sadhana 32(4):309-328
6.Katz D(2009)Heterogeneous traffic mixes,[term paper].http://www.donaldkatz.com/CEE6603-TermPaperHeterogeneousTraffic.pdf.Accessed 22 July 2016
7.Greenshields BD,Channing WS,Miller HH(1935)A study of traffic capacity.In:Highway research board proceedings,1935.National Research Council(USA),Highway Research Board
8.May AD Jr,Harmut EMK(1967)Non-integer car-following models.Highw Res Rec 199:19-32
9.Drew DR(1968)Traffic flow theory and control.McGraw-Hill,New York
10.La Pipes(1967)Car following models and the fundamental diagram of road traffic.Transp Res 1(1):21-29
11.May AD(1990)Traffic flow fundamentals,2nd edn.Prentice Hall,Englewood Cliffs
12.Greenberg H(1959)An analysis of traffic flow.Oper Res7(1):79-85
13.Underwood RT(1961)Speed,volume,and density relationships:quality and theory of traffic flow.yale Bur Highw traffic 141-188
14.Drake JS,Schofer JL,May Jr AD(1967)A statistical analysis of speed-density hypotheses.In:Vehicular traffic science.Highw Res Rec(154):112-117
15.Newell GF(1961)Nonlinear effects in the dynamics of car following.Oper Res 9(2):209-229
16.Del Castillo JM,Benitez FG(1995)On the functional form of the speed-density relationship-I:general theory.Transp Res Part BMethodol 29(5):373-389
17.Lee HY,Lee HW,Kim D(1998)Origin of synchronized traffic flow on highways and its dynamic phase transitions.Phys Rev Lett 81(5):1130-1133.https://doi.org/10.1103/PhysRevLett.81.1130
18.Wang H,Li J,Chen QY,Ni D(2010)Representing the fundamental diagram:the pursuit of mathematical elegance and empirical accuracy.In:Transport Research Board 89th annual meeting,Washington,DC,USA
19.Thankappan A,Vanajakshi L(2015)Development and application of a traffic stream model under heterogeneous traffic conditions.J Inst Eng Ser A 96:267-275.https://doi.org/10.1007/s40030-015-0134-y
20.Heydecker BG,Addison JD(2011)Analysis and modelling of traffic flow under variable speed limits.Transp Res Part C Emerg Technol 19(2):206-217
21.Mallikarjuna C,Phanindra A,Rao KR(2009)Traffic data collection under mixed traffic conditions using video image processing.J Transp Eng 135(4):174-182.https://doi.org/10.1061/(ASCE)0733-947X(2009)135:4(174)
22.Elzhov TV,Mullen KM,Spiess AN,BolkerB,Mullen MKM(2012)R interface to the Levenberg-Marquardt nonlinear leastsquares algorithm found in MINPACK,plus support for bounds.Retrieved from CRAN:http://cran.rproject.org/web/packages/minpack.lm/minpack.lm.pdf.Accessed 10 July 2016
23.Hauer E(2015)The art of regression modeling in road safety.Springer,Berlin,p 38