A Continuum Modeling Approach to Congestion Management of Transportation System in an Urban City

Transcription

1 The University of Hong Kong A Continuu Modeling Approach to Congestion Manageent of Transportation Syste in an Urban City S.C. Wong Departent of Civil Engineering The University of Hong Kong E-ail: hhecwsc@hkucc.hku.hk URL: Supported by a grant RGC HKU 7183/08E fro the Hong Kong Research Grants Council

2 Introduction Continuu transportation syste in an urban city Model forulation and solution algorith Soe applications Conclusions and further works

3 Continuu Transportation Syste Two approaches for odeling traffic equilibriu probles Discrete odeling approach Patriksson (1994); Gendreau and Marcotte (2002); Lee (2003); etc. Continuu odeling approach Idealized city configuration (e.g. circular city) La and Newell (1967); Zitron (1967); D Este (1987); Wong (1994); etc. General city configuration Beckann (1952), one of the earliest works in this area; Wardrop (1971); Willias and Ortuzar (1976); Puu (1977); Buckley (1979); Daferos (1980); Sasaki et al. (1990); Yang et al. (1994); Wong (1998) etc.

4 Continuu Transportation Syste Basic assuptions in continuu odeling approach The difference in land use pattern and network configuration between adjacent areas is relatively sall, copared with the variation over the entire city The characteristics of the land use and network can be represented by piecewise sooth atheatical functions (Vaugha, 1987) The interrelation aong these atheatical functions are governed by soe appropriate fors of differential or integral equations The user equilibriu conditions that are coonly adopted in discrete network odeling can be generalized to the continuu case in a two-diensional plane

5 Continuu Transportation Syste Rationale for using continuu odeling approach For the initial phase of planning and odeling in broad-scale regional studies, the focus ay be on the general trend and pattern of the distribution and travel choice of users, and their changes in response to policy changes at the acroscopic level Insufficient data for setting up the dense transportation network for detailed analysis Conceptual plan for the catchent regions of copeting facilities (such as airports, ports), locations of cordons for congestion charging, location of an additional CBD, strategic expansion of transportation syste in different parts of the city, etc. Travel deand is inherently continuously spread over the city

6 Model forulation and solution algorith The odeled city Boundary, CBD 1 CBD 2 Destination 1 Destination 2, Class 2 Destination 2, Class 1 y x Single center (destination) ~ Multiple centers (destinations) Single class ~ Multiple classes (k)

7 Model forulation and solution algorith Travel deand Boundary, CBD 1 CBD 2 q (x, y) y dy (x,y) dx Travel deand per unit area for class and destination n (heading CBD n) x (k)

8 Model forulation and solution algorith Traffic flows Boundary, Flux in the y-direction per unit width (dx) for class and destination n CBD 1 f (x, y) 2 CBD 2 y x dy (x,y) dx f (x, y) 1 Flux in the x-direction per unit width (dy) for class and destination n (k)

9 Model forulation and solution algorith Conservation of flows Boundary, f (x, y) x f (x, y) y 1 2 q (x, y) CBD 1 f (x, y) 2 CBD 2 dy (x,y) dx f (x, y) 1 y x (k)

10 Model forulation and solution algorith Local cost function Boundary, c (x,y, f) a (x,y) b (x,y) f rs r s CBD 1 f (x, y) 2 CBD 2 f 2 1 f f 2 2 y dy (x,y) dx f (x, y) 1 Flow intensity for class and destination n x (k)

11 Model forulation and solution algorith Path travel cost Boundary, CBD n, with a congestion cost of C CBD 1 Integrate along the path CBD 2 (strealine) fro the location concerned to the CBD p c ds y u (x, y) C p c ds x (k)

12 Model forulation and solution algorith User equilibriu Boundary, CBD 1 CBD 2 u unused u used Any used routes would have equal and iniu cost, u used y x (x,y) Unused path (not along a strealine) Any unused route would have greater or equal cost, i.e. u unused u used (k)

13 Model forulation and solution algorith Elastic deand Boundary, q (x, y) D (x, y,u (x, y)) CBD 1 CBD 2 q (x, y) dy (x,y) dx y x (k)

14 Model forulation and solution algorith Trip distribution function Boundary, q q s exp exp u u s CBD 1 CBD 2 dy (x,y) dx y x (k)

15 Model forulation and solution algorith Boundary conditions Boundary, n (noral vector) f ( f1, f2) CBD 1 CBD 2 f n g n f n d Q y x (k)

16 Model forulation and solution algorith Matheatical progra Miniize f, q, Q subject to z Q f, q, Q Q Sn 1 2 n 1 b ~ n r s q 0 D f 1 n f rs 0 d q ln q q d 1 b n d 1 b q ln q n ~ a f q f q f q q n 0, (x, y), n N, M 0, (x, y), M 0, (x, y),n N, M nc f n d Q 0, n N, M

17 Model forulation and solution algorith Equivalent set of partial differential equations f q 0, (x,y),n N, M Conservation equation ~ c f f 0, (x,y),n N, M Cost potential function lnq b 0, (x,y),n N, M Distribution function ~ C q 1 b n D q 1 lnq q 0, (x,y), M 0, 0, b n N, M (x,y), M Elastic deand Boundary conditions f 0, (x,y),n N, M 0, (x,y) nc,nn, M nc f nd Q 0, n N, M

18 Model forulation and solution algorith Finite eleent ethod The finite eleent ethod (FEM) (Zienkiewicz and Taylor, 1989) is used to approxiate the continuous variables in the odeled city V (x,y) y k V(x, y) v N (x, y) i i N vknk (x, y) Continuous variables v j j (x, y) i e j Discretized Nodal variables: (v i, v j,v k ) x N i (x, y) 1 i k j i k N j (x, y) 1 j i k 1 N k (x, y) j

19 Model forulation and solution algorith Finite eleent ethod ( (x, y)) 0 A set of partial differential equations ( x, y) ( f, u, ) The Galerkin forulation of the weighted residual technique (Cheung et al., 1996; Zienkiewicz and Taylor, 1989) ( ) 0 A set of nonlinear algebraic equations ( f, q, )

20 Model forulation and solution algorith Newtonian algorith 0 ( ) ( ) ( )( ) 0 0 Residual vector, R Jacobian atrix, J Neglect higher order ters Set for a stationary point ( ) R [ J]( 0 ) 0 Set a recursive procedure Convergence criterion, R 0 (k1) (k) [ J( (k) )] 1 R( (k) )

21 Soe applications Facility copetition proble Airports copetition within Pearl River Delta Region There are four ajor airports Hong Kong, Guangzhou, Shenzhen and Zhuhai within the PRD region serving both international and doestic flights Passengers within this region ake their choice of airport based on the geographical location, congestion, and costs Other factors such as the cross boundary penalty affect passengers choice of airport A acroscopic odel is used to study the route and airport choice of passengers

22 Soe applications Facility copetition proble The odel region 1 2 Legend: 1. Qingyuan Municipality 2. Zhouqing Municipality 3. Foshan Municipality 4. Jiangen Municipality 5. Guangzhou 6. Zhongshan Municipality 7. Zhuhai 8. Huizhou Municipality 9. Dongguan Municipality 10. Shenzhen 11. Hong Kong 12. Fictitious Boundary Layer Distance (k)

23 Soe applications Facility copetition proble Finite eleent esh Guangzhou Shenzhen Zhuhai Hong Kong Distance (k)

24 Soe applications Facility copetition proble Flow pattern for doestic service Catchent boundary (k)

25 Soe applications Facility copetition proble Cost potential for doestic service Total Cost (k)

26 Soe applications Facility copetition proble Flow pattern for international service Catchent boundary (k)

27 Soe applications Facility copetition proble Cost potential for international service Total Cost (k)

28 Soe applications Facility copetition proble Market shares aong airports in the region Airport Deand (passengers/ year) Estiated arket share (%) Hong Kong 5,618, Shenzhen 5,467, Zhuhai 2,661, Guangzhou 10,993, Cross Boundary flow fro Hong Kong Doestic Base Case 1,773,651 passengers/year Airport International Base Case Estiated arket share (%) Deand (passengers/ year) Hong Kong 26,971, Shenzhen 0 0 Zhuhai 0 0 Guangzhou 671, Cross Boundary flow fro Mainland 609,170 passengers/year

29 Soe applications Cordon-based congestion-charging proble Coon charging ethods: point-based, tie-based, distancebased, and area-based Cordon-based charging schees have been ipleented in Singapore, Oslo, Trondhei, Bergen, and London Easy ipleentation, socially acceptable Location of cordons? Charging levels? Social benefits: first or second best? Exhaustive evaluations by discrete network optiization approach

30 Soe applications Cordon-based congestion-charging proble Cordon-based toll for each cordon layer Layer j T j+1 T j Flat toll charged for entering Layer j, j T j1 T j Charging levels for the ulti-layer cordons: { 0, 1, 2, 3,...}

31 Soe applications Cordon-based congestion-charging proble The exaple city Y Lake X CBD Local travel cost function: CBD travel cost function: Deand function: c f C Q u D 100e (k)

32 Soe applications Cordon-based congestion-charging proble Traffic flow pattern (k)

33 Soe applications Cordon-based congestion-charging proble Flow intensity Flow Intensity (k)

34 Soe applications Cordon-based congestion-charging proble Location-dependent toll rate Toll rate ($/k) CBD flat toll = $ (k)

35 25 Soe applications Cordon-based congestion-charging proble Distance-based toll paid by a user $5 20 $5 $0 $ $5 $10 For cordons were deterined at $10, $15, $20 and $ (k)

36 Soe applications Housing proble It is well recognized that land use and transportation systes interact strongly with each other. For exaple, if the accessibility of a place increases, residents and investors will be attracted to this place. Thus, the land use will be changed. Also, if a large residential estate is built, the accessibility and deand of travel will be altered. Arnott (1995) and Boyce and Matsson (1999) studied the interaction of the land use and transportation systes by incorporating the concept of rent This study ais to develop a continuu user equilibriu odel for the land use and transportation proble.

37 Soe applications Housing proble Optiu housing allocation using bi-level prograing approach For a given budget, find an optiu housing allocation pattern that axiizes the total utility of the syste users Upper-level odel Optiized housing provision pattern, subject to Maxiu housing developent Budgetary constraint Meeting housing needs Non-negativity constraint Sensitivity inforation of the lower level variables at optiu point Lower-level odel For a given housing provision pattern, find the distributions of different classes of syste users such that all users travel in an user optial anner

38 Soe applications Housing proble Nuerical exaple Total deand: Class 1 couters: 60,000 units (ore sensitive to rent) Class 2 couters: 80,000 units (ore sensitive to travel cost) Cost ipedance functions: Class 1 couters: c x, y vx, f f v y Class 2 couters: 1. 2 c v x, y v x, y 2 2 where vx, y (x 14) (y 20) Housing rent functions: Class 1 couters: h q f 350 q Class 2 couters: 801 q 350 q h f 2 2

39 Soe applications Housing proble Nuerical exaple The finite eleent esh adopted (k)

40 Soe applications Housing proble Distribution of the additional housing units Higher housing provision cost near CBD Added housing unit (units/sq. k) Pressure to build ore housing near CBD to reduce the travel cost of couters (k)

41 Soe applications Housing proble Deand contour for class 1 users Deand contour for class 2 users Deand of class 1 users (users/sq. k) Deand of class 2 users (users/sq. k) (k) (k) Class 1 couters are ore seriously affected as they are ore sensitive to rent

42 Soe applications Housing proble Housing rent of class 1 user (before upper-level optiization) Housing rent of class 1 users (HKD) Housing rent of class 1 user (after upper-level optiization) Housing rent of class 1 users (HKD) (k) (k)

43 Soe applications Housing proble New towns CBD

44 Conclusions and Further Works The continuu odeling approach to traffic equilibriu probles in an urban city has been briefly described The odel foration and finite eleent solution algorith for a typical continuu odel have been discussed Soe potential applications of this continuu approach, such as facility copetition, cordon-based congestion-pricing, and housing probles, have been given Directions of future research Extension to discrete/continuous odel, in which the ajor freeway are odeled by discrete links, and surface streets by continuu Extension to dynaic probles Evaluation of environental ipacts, such as greenhouse gas and air pollutions