1.225J J (ESD 205) Transportation Flow Systems

Transcription

1 .5J J ESD 5 Trasporaio Flow Sysems Lecre 3 Modelig Road Traffic Flow o a Li Prof. Ismail Chabii ad Prof. Amedeo Odoi Lecre 3 Olie Time-Space Diagrams ad Traffic Flow Variables Irodcio o Li Performace Models Macroscopic Models ad Fdameal Diagram Volme-Delay Fcio Microscopic Models: Car-followig Models Relaioship bewee Macroscopic Models ad Car-followig Models Smmary.5, // Lecre 3, Page

2 Time-Space Diagram: Aalysis a a Fied Posiio posiio L h h h 3 h 4 T ime.5, // Lecre 3, Page 3 Flows ad Headways m: mber of vehicles ha passed i fro of a observer a posiio drig ime ierval [,T]. e. m5 Flow rae: m T Headway h j : ime separaio bewee arrival ime of vehicles i ad i m h Average headway: h j m Wha is he relaioship bewee ad h? j.5, // Lecre 3, Page 4

3 If T is large, The, Flow Rae vs. Average Headway T m j T m h h j m j h j m h This is iiively correc. is also called volme i raffic flow sysem circles i.e..5 is also called freecy i schedled sysems circles i.e..4.5, // Lecre 3, Page 5 Time-Space Diagram: Aalysis a Fied Time posiio L s s ime.5, // Lecre 3, Page 6

4 Desiy ad Average Spacig : mber of vehicles i a road srech of legh L a ime Desiy: L s i : spacig bewee vehicle i ad vehicle i L s i i L i s i s Is his iiive? s.5, // Lecre 3, Page 7 Performace Models of Traffic o a Road Li Li: a represeaio of a highway srech, road from oe iersecio o he e, ec. Eample of measres of performace: Travel ime Moeary or eviromeal cos Safey Mai measre of performace: ravel ime 3 ypes of models: Macroscopic models: Fdameal diagram valid i saic saioary codiios oly. Log roads ad log ime periods Microscopic models: Car-followig models o lae chages Volme-delay fcios.5, // Lecre 3, Page 8

5 Macroscopic Flow Variables Three macroscopic flow variables of a li: Average desiy also deoed by ρ Average flow Average speed also deoed v Relaioships bewee variables:, crve: Fdameal diagram Fdameal diagram is a propery of he road, he drivers ad he evirome icy, sy, raiig 3 variables eaios oly oe variable ca be a idepede variable B oe of he variables,, ca o be idepede.5, // Lecre 3, Page 9 Daa Colleced from Hollad Tel Eddie, 63 Speed m/hr Average Spacig m Coceraio veh/m Nmber of Vehicles , // Lecre 3, Page

6 Desiy, Speed Diagram for he Field Daa Speed m/hr Desiy veh/m.5, // Lecre 3, Page Desiy, Speed Diagram wih a Fied Crve Speed m/hr y Desiy veh/m.5, // Lecre 3, Page

7 Desiy, Flow Diagram from he Field Daa Flow veh/hr Desiy veh/m.5, // Lecre 3, Page 3 Desiy, Flow Diagram wih a Fied Crve Flow veh/hr y Desiy veh/m.5, // Lecre 3, Page 4

8 Flow, Speed Diagram from he Field Daa Speed m/hr Flow veh/hr.5, // Lecre 3, Page 5 Spacig, Speed Diagram from he Field Daa 8 Speed m/hr Spacig m.5, // Lecre 3, Page 6

9 Flow, Pace Diagram from he Field Daa.4. Pace hr/m Flow veh/hr.5, // Lecre 3, Page 7 Relaioships bewee Flow Variables desiy, speed diagram desiy, flow diagram ma c Greeshiel d : 3 ma ma 3 3 c c : desiy he highway srech is lie a parig lo! a car legh ma c is he maimm flow, or li capaciy ma c c c.5, // Lecre 3, Page 8

10 Fdameal Diagram desiy, speed diagram Fdameal diagram ma ma 3 3 c 3 sable c sable c [ c, ]: arise whe flow is slower dow sream de o lae drops, a slow plowig-rc, ec c is criical, sice i mars he sar of a sable flow area where addiioal ip of cars decrease flow served by he highway, diagram is fdameal sice i represes he hree variable as compared o he oher diagrams.5, // Lecre 3, Page 9 Derived Diagrams flow, speed diagram flow, ravel ime diagram re relaioship c sable sable sable sable classical volme-delay fcio ma ma I geeral, cao be sed as a variable why? I he road ewor plaig area: is also called volme ravel ime is also called ravel delay I he case of volme-delay fcios, is sed as a variable.5, // Lecre 3, Page

11 Eamples of Classical Volme-Delay Fcios Noaio: is he li flow is he li ravel ime c is he pracical capaciy α ad β are calibraio parameers Davidso s fcio: [ α ] c US Brea of Pblic Roads β [ α ] c.5, // Lecre 3, Page Observaios o Classical Volme-Delay Fcios Eamples where he classical model may be accepable: Delay a a sigalized li < ma mild cogesio Wha maes he classical model ieresig? I is a fcio There is oly oe vale for a give Typical fcios sed are icreasig wih, ad heir derivaives are also icreasig i holds waer i is cove The above are aalyical properies ha have bee adoped o sdy he properies of, ad desig solio algorihms for, ewor raffic assigme models Lecres 4-6 A eample of radeoffs made bewee realism ad compaioal racabiliy.5, // Lecre 3, Page

12 Li Travel Time Models: Car-Followig Models Noaio: Follower Leader Flow L spacig space headway l d speed of vehicle : d accelerai o decceleraio of vehicle : l d d d car-followig regime: l is below a cerai hreshold l d.5, // Lecre 3, Page 3 Li Travel Time Models: Car-Followig Models Follower Leader Flow Simple car-followig model: T al a T : reacio ime T.5sec a : sesiiviy facor L l a.37s Qesios abo his simple car-followig model: Is i realisic? Does i have a relaioship wih macroscopic models?.5, // Lecre 3, Page 4

13 From Microscopic Models To Macroscopic Models Simple car-followig model: Fdameal diagram: Proof of eivalecy y a y y y dy a y y dy al y dy al dy a l l a T ma al al If l, he al dy.5, // Lecre 3, Page 5 From Microscopic Model o Macroscopic Model al a a a a If, he a Sice a a ma, he a ma Noe: if, he. Does his mae sese?.5, // Lecre 3, Page 6

14 .5, // Lecre 3, Page 7 No No-liear Car liear Car-followig Models followig Models 5. a T.5 l l a If T, he correspodig fdameal diagram is:.5 ma.5, // Lecre 3, Page 8 Flow Models Derived from Car Flow Models Derived from Car-Followig Models Followig Models l m T a T 3.5 Flow vs. Desiy m l.5 ma m ma ep ma ma ep c l

15 Lecre 3 Smmary Time-Space Diagrams ad Traffic Flow Variables Irodcio o Li Performace Models Macroscopic Models ad Fdameal Diagram Volme-Delay Fcio Microscopic Models: Car-followig Models Relaioship bewee Macroscopic Models ad Car-followig Models.5, // Lecre 3, Page 9