CIE4801 Transportation and spatial modelling Trip distribution

Transcription

1 CIE4801 ransportaton and spatal modellng rp dstrbuton Rob van Nes, ransport & Plannng 17/4/13 Delft Unversty of echnology Challenge the future

2 Content What s t about hree methods Wth specal attenton for the frst Specal ssues 2

3 1. rp dstrbuton: what s t about 3

4 Introducton to trp dstrbuton Zonal data rp producton / rp attracton rp frequency choce ransport networks rp dstrbuton Destnaton choce ravel resstances Modal splt Mode choce Perod of day me choce Assgnment Route choce ravel tmes network loads etc. 4

5 What do we want to know? 5 5

6 Introducton to trp dstrbuton Gven: Productons and attractons for each zone (.e. departures and arrvals) Determne: he number of trps from each zone to all other zones o: zone 1 zone total producton From: zone 1 zone total attracton 6

7 Introducton to trp dstrbuton to: B from: A B A 7

8 rp dstrbuton: 3 methods Analogy based Extrapolaton or growth Choce modellng 8

9 2.1 Method 1: Analogy 9

10 he gravty model m Newton: 1 G = g m m 2 d m d G = gravtatonal force between and g = gravtatonal constant m, m = mass of planet ( respectvely) d = dstance between and m 10

11 he gravty model Assumptons: Number of trps between an orgn and a destnaton zone s proportonal to: a producton ablty factor for the orgn zone an attracton ablty factor for the destnaton zone a factor dependng on the travel costs between the zones Mathematcal formulaton: = ρq X F ρ Q X F = # trps from zone to zone = measure of average trp ntensty = producton potental of zone = attracton potental of zone = accessblty of from Possble nterpretatons of Q and X : populatons, producton & attracton, 11

12 rp dstrbuton usng the gravty model Dependng on the amount of nformaton avalable, dfferent models result Producton and attracton are both unknown Drect demand model Only producton or attracton s known Sngly constraned model (orgn or destnaton based) Both producton and attracton are known Doubly constraned model 12

13 Drect demand model Basc gravty model: = ρq X F = ρ Q X F = # trps from zone to zone = measure of average trp ntensty = producton potental of zone = attracton potental of zone = accessblty of from he producton potental and attracton potental have to be derved from populaton, area, obs (see also OD-matrx estmaton) he producton numbers and attracton numbers are unknown. ρ = Q X F 13

14 Sngly constraned model Basc gravty model: = ρq X F ρ Q X F = # trps from zone to zone = measure of average trp ntensty = producton potental of zone = attracton potental of zone = accessblty of from If the trp productons P are known: = P If the trp attractons A are known: = A 14

15 Sngly constraned: orgn = ρq X F = P ( ρ ) ρ ( ) = Q X F = Q X F = P Q = ρ P XF P = ρ X F = apx F ρ XF ( a = balancng factor) Sngly constraned orgn based model: = apx F 15

16 Sngly constraned: destnaton = ρq X F = A ( ρ ) ρ ( ) = Q X F = X QF = A A X = ρ QF A = ρq F = bq AF ρ QF ( b = balancng factor) Sngly constraned destnaton based model: = bq AF 16

17 Doubly constraned model Basc gravty model: = ρq X F ρ Q X F = # trps from zone to zone = measure of average trp ntensty = producton potental of zone = attracton potental of zone = accessblty of from rp productons P and trp attractons A are known: = P and = A 17

18 Doubly constraned model = ρq X F = A = P ( ρ ) ρ ( ) = Q X F = Q X F = P ( ρ ) ρ ( ) = Q X F = X QF = A Q = and ρ P ( XF ) X A = ρ ( QF ) 18

19 Doubly constraned model P A 1 = ρ F = ab PA F ρ XF ρ QF ρ a b = balancng factor = balancng factor Doubly constraned model: = ab PA F 19

20 2.1b Method 1b: Entropy based 20

21 Maxmsng entropy gven constrants Analogue to the thermodynamc concept of entropy as maxmum dsorder, the entropy- maxmzng procedure seeks the most lkely confguraton of elements wthn a constraned stuaton. he obectve can be formulated as: ( ) Max w = = P A c = C =!! 21

22 Illustraton entropy prncple How many ways can you dstrbute 4 people? C H Let s assume we use a con to decde where a person wll go to: thus flp a con 4 tmes In total there are 16 sequences leadng to 5 optons: 4H,0 (1) 3H,1 (4), 2H,2 (6), 1H,3 (4), 0H,4 (1) Weght of each opton s determned by!! 22

23 Dervaton (1/2) Lagrangan of maxmsaton obectve:! λ P + + λ A β C c! + Usng as approxmaton ( N ) ln! ln ( N! ) N ln ( N) N ln N N ( ) = ln λ λ β c = 0 λ λ β c λ λ β c = e = e e e ( ) Settng dervatves equal to zero and solvng the equaton Note that the other dervatves lead to the orgnal constrants 23

24 Dervaton (2/2) Substtute result n constrants = e λ P = λ λ β c e = e λ e 1 e λ β c = a λ β c = P Smlar for destnatons, and substtute n λ λ β c β c = e e e = a P b A e Whch s equvalent to the doubly constraned model hus dfferent routes lead to smlar formulaton 24

25 2.1c Method 1: Dstrbuton functons and example 25

26 Dstrbuton or deterrence functons F = f( c ) c = travel cost from zone to zone f dstrbuton functon descrbes the relatve wllngness to make a trp as a functon of the travel costs. bke tran car 10km 50km dstance 26

27 Requrements for dstrbuton functons Decreasng wth travel costs Integral should be fnte Fracton ( ) ( ) F a c F c depends on value of c Fxed changes should have a dmnshng relatve mpact: ( ) ( ) ( ) ( + ) F c +Δ c F c + A+Δc > F c F c A 27

28 Dstrbuton functons f Power functon: f( c ) = c α Exponental functon: f( c) = α exp( βc) op-exponental functon: f ( c ) = αc β exp( γc ) 2 Lognormal functon: f( c ) = α exp( β ln ( c + 1)) 2 op-lognormal functon: f ( c ) = αc β exp( γ ln ( c + 1)) Log-logstc functon: f( c ) = α/(1 + exp( β + γ log( c ))) travel cost 28

29 Example doubly constraned model rp balancng 3 P A

30 Example doubly constraned model P rp balancng ravel costs c A

31 Example doubly constraned model P 100 rp balancng ravel costs c Accessbltes F A F = f( c ) = 5 exp( 0.5 c ) 31

32 Example doubly constraned model A rp balancng 3 P ravel costs c Accessbltes F Balancng factors a, b

33 Example doubly constraned model rp balancng 3 P ravel costs c Accessbltes F Balancng factors a, b A

34 rp dstrbuton usng the gravty model rp balancng 3 P ravel costs c Accessbltes F Balancng factors a, b A

35 Example doubly constraned model rp balancng 3 P ravel costs c Accessbltes F Balancng factors a, b Repeat untl there are no changes: =>OD matrx A

36 2.2 Method 2: Extrapolaton or growth 36

37 Growth factor models base matrx trp observatons outcomes of gravty model traffc counts Growth factors outcomes future trp generaton model outcomes future gravty model predcted matrx 37

38 Growth factor models Advantages Network specfc peculartes can be captured by observatons A base matrx s more understandable and verfable than a model Dsadvantages New resdental zones are dffcult to capture Hstorcal patterns may change over tme 38

39 Growth factor models 0 g = g 0 base matrx cell growth factor predcted matrx cell = g 0 Network ndependent, general factor = g 0 or = g 0 Network ndependent, orgn or destnaton specfc factor = g g 0 Network ndependent, factors for orgns and destnatons = g Network dependent, OD-specfc factors 0 39

40 Common applcaton Gven expected spatal development Future producton (departures) Future attracton (arrvals) Fll n new areas by copyng columns and rows of nearby zones from the base year matrx Adapt ths base year matrx usng approprate factors a and b Iteraton process sldes

41 2.3 Method 3: Choce modellng 41

42 Dscrete choce model = P exp( µ V ), exp( µ V ) k k V = β X β c 1 2 = β1, β2 = µ = P = X = c = number of trps from to parameters scalng parameter trp producton at zone trp attracton potental at zone travel cost from zone to zone 42

43 Explanatory varables? Inhabtants Households Jobs Retal obs Students Denstes Locaton types Etc. Mnus travel costs More suted for trps or for tours? 43

44 Dervaton of the gravty model (reprse) zone Observed utlty for actvtes n zone and zone : V = N N β c 2 Subectve utlty: U = V + ε Number of people travelng from to : N P c N zone µ V e p = µ Vrs e rs = exp( µ N ) exp( µ N ) exp( µβ2c ) exp( µ V ) rs rs = ρq X F 44

45 3. Specal ssues 45

46 Specal ssues Dstrbuton functon and trp length dstrbuton Intrazonal trps External zones: through traffc All trps or sngle mode? 46

47 Dstrbuton functon and trp length dstrbuton Smlar or dfferent? Smply put: dstrbuton functon s nput and trp length dstrbuton s output! 47

48 Intrazonal trps What s the problem? Intrazonal travel costs? Rule of thumb: (or?) of lowest cost to neghbourng zone rue for publc transport? Alternatve: rp generaton for ntrazonal only How? 48

49 External zones wo possble ssues Sze ssue Very large zones => hgh values for producton and attracton => ntrazonal trps? => small errors lead to large dfferences Cordon models hrough traffc follows from other source, e.g. lcense plate survey or other model => through traffc s thus fxed nput and should not be modelled usng trp dstrbuton models 49

50 Approach for cordon model Determne producton and attracton for nternal zones usng e.g. regresson Determne producton and attracton for external zones usng e.g. counts Derve matrx for through traffc (.e. from cordon zone to cordon zone) from e.g. a regonal model Subtract through traffc from producton and attracton of the external zones Apply gravty model wth the resultng producton and attracton, whle makng sure that there s no through traffc, e.g. by settng the travel costs between cordon zones equal to Add matrx for through traffc to the resultng matrx of the gravty model 50

51 All trps or a sngle mode? Check the sldes Whch parts consder a sngle mode? 51