Network Equilibrium Models: Varied and Ambitious

Transcription

1 Network Equilibrium Models: Varied and Ambitious Michael Florian Center for Research on Transportation University of Montreal INFORMS, November

2 The applications of network equilibrium models are varied: They range from simple to the very complex - single mode, single class equilibrium assignment - multi-class equilibrium assignment - generalized cost on road and transit networks - equilibrium assignment on congested transit networks - path analyses - complex multi-modal equilibration - combined mode trips - dynamic network equilibrium models INFORMS, November

3 The single mode single class network equilibrium model min a A v a 0 s a ( ) x dx subject to h = g, i I, k K c i k h 0, k K, i I k ( v = δ h, a A) a ak k k K i i i Numerous algorithms have been developped for its solution; As is well known the arc flows, v a are unique, but the path flows, are not unique. h k INFORMS, November

4 Some selected applications The presentation includes examples of the application of various network equilibrium models carried out around the world The first set of applications was carried out with static models of increasing complexity both for road and transit networks The second set of applications presents new results with a dynamic network equilibrium model INFORMS, November

5 Some Straightforward Applications Madrid, Spain Pretoria, South Africa Auckland, New Zealand INFORMS, November

6 Madrid- AM Peak Flows and Speeds INFORMS, November

7 Regional Study of the Province of Gauteng, South Africa Study carried out by Vela VKE Pretoria, South Africa, INFORMS, November

8 Scenario Without New facility INFORMS, November

9 Scenario with New facility INFORMS, November

10 Scenario comparison INFORMS, November

11 Planning Transport in Auckland Study carried out by Auckland Regional Council, Auckland, NZ INFORMS, November

12 AM Vehicle Flows, 2001 Motorways Arterials Local 5% 36% 59% INFORMS, November

13 Complex Variable Demand Network Equlibrium Models Demand Model Equilibrium Trip Assignment Impedance Step size Mechanism The Step Size Mechanism may take different forms depending on the knowledge that one has of the underlying model. Does the model have an equivalent convex cost optimization formulation? Can the model be formulated as a variational inequality? The model is very complicated and one carries out ad-hoc feedback by some averaging scheme. INRO

14 Complex Variable Demand Network Equilibrium Models: Equivalent Convex Cost Optimization Formulations Some well known variants of such models are the Combined Distribution- Assignment Model, Combined Distribution-Assignment-Mode Choice Model, Equilibrium Assignment Model with Variable Demand,... One can use adaptations of nonlinear programming algorithms to obtain the solution of such models The Step Size Mechanism may be trivially stated to be the result of a line search on the objective function 1 min ( k ( k k )), k + F x + λ d x x = x k + λ( d k x k ) 0 λ 1 where x k is a current solution and d k is a direction of descent INFORMS, November INRO

15 Network EquilibriumModels: Variational Inequality Formulations It is well known that models are with asymmetric cost functions, such as intersection delays, transit travel time depending on auto travel times in multimode models can be formulated as : s v v v * * a( )( a a) 0 subject to h = g, i I, k K c i k h 0, k K, i I k ( v = δ h, a A) a ak k k K i i i INFORMS, November

16 Network EquilibriumModels: Variational Inequality Formulations Such models may be solved by a variety of algorithms. Often the sufficient conditions for convergence are impossible to verify A common heuristic method used in practice is the Method of Successive Averages The Step Size Mechanism may be related to the averaging of the link costs or the averaging of link flows k+ 1 k k+ 1 k k+ 1 k x = x + α ( x x ) ; x = T( x ) k 0< α < 1 ; α ( 1 α ) = + k = 1 k where T(x k ) is the computed procedure that is used to obtain the next iterate k INFORMS, November

17 A Complex Model: Rigorous Formulation-Heuristic Solution Algorithm Santiago, Chile Complex demand model Road Network Equilibrium Transit Network Equilibrium Combined modes: road-transit; transit-transit INFORMS, November

18 Santiago, Chile Strategic Planning Model Base network 409 centroids including 49 parking locations 1808 nodes, 11,331 directional links 1116 transit lines and line segments 11 modes, including 4 combined modes (bus-metro, txc-metro, auto-metro and auto passenger-metro) The demand subdivided into 13 socio-economic classes 3 trip purposes ( work, study, other ) driving license holders can access to 11 modes no license holders can access to 9 modes

19 Base Network of Santiago, Chile

20 Variational Inequality Formulation * * Find ( h, T ) Ω such that pn ( ij ) g G m g m g p g p p pnm * * * 1 png* png png* [ φgcr ( h, T )( hr hr) + ln T pn ij ( Tij Tij )) + β r φ ln( T / T )( T T ) 0, ( h, T) Ω. pnm* png* pnm pnm* ij ij ij ij

21 Trip Ends and Conservation of Flow Constraints png pn ij = i g j pn T O, i, p, n ( α i ) png p ij = j g n i p T D, j, p ( ξ i ) T pnm png ij Tij = 0, j, p, n, g ( L png ij ) m g r pnm ij pnm pnm 0 ( µ ij ) φ p g ( h T ) =, j, p, n, g r R m h T T pnm r pnm ij > ij pnm 0, r, p, n, m ( γ r ) 0, ij, p, n, m png > 0, ij, p, n, g

22 Network Equilibrium Models: car and transit Multi-class network equlibrium model Multi-class transit network equlibrium model Heuristic equilibration that resorts to averaging of flows and travel impedances INFORMS, November

23 Solution Procedure Start and Initialize Trip Distribution and Mode Choice Equil. Auto Assignment MSA Auto Volume Auto Skims Convergence? end Transit Auto equivalent flow Congested time Auto Impedance Multiple Transit Assignments Standard Transit Assignment for buses Bus Impedance Equilibrium Transit Assignment for metro Metro MSA Impedance Park-and-Ride Model for auto-metro and bus-metro All Impedances

24 Santiago, Chile Strategic Planning Model The next slide shows the convergence of the MSA algorithm that uses link flow averaging for the car network and travel time averaging for the transit network. The convergence of both the car demand and link flows are given for two variants: uncongested transit assignment and equilibrium transit assignment. INFORMS, November

25 Convergence of equilibration Normalized gap (%) Normalized Gap vs. Iterations congested vs. non-congested metro assignment with metro capacity Auto demand Auto link volume cong. Link vol. cong. demand non-cong. Link vol. non-cong. demand Iteration

26 Metro Volume (non-congested vs. congested version)

27 Metro Volume Changes

28 Auto-metro volume (non-congested vs. congested version) Congested Non-congested

29 Metro Volume - metro 5A, non-congested version

30 Metro Volum2 - metro 5A, congested version

31 Some Complex Applications Los Angeles, California Toll Highways Poznan, Poland Toll Bridge, Montreal, INFORMS, November

32 The SCAG Regional Transportation Planning Model A complex and very large scale model Lack of rigorous formulation; network equilibrium sub-models A multi-class multi-mode network equilibrium model with asymmetric cost function is part of the model Heuristic solution algorithm based on an outer averaging scheme INFORMS, November

33 START auto skims for PK transit skims for PK transit skims for OP auto skims for OP Trip generation HBW Logsums for PK (mc) HBW Logsums for OP (mc) trip distribution for PK (gravity) trip distribution for OP (gravity) SCAG MODEL FLOW CHART: mode choice model for PK demand computations (time of-day model) mode choice model for OP auto-truck assignments for AM auto-truck assignments for MD successive average link volume for outer loop of AM successive average link volume for outer loop of MD is the number of outer loops satisfied? auto-truck assignments for PM auto-truck assignments for NT transit assignments for AM transit assignments for MD END INFORMS, November

34 Network Overview - Highway Network by facility type INFORMS, November

35 Network Overview - Highway Network with parking lots INFORMS, November

36 Equilibration Algorithm Convergence Results SCAG Model Convergence (AM peak) link relative difference loops inner-loop outer-loop INFORMS, November

37 Assignment Results AM peak volume INFORMS, November

38 Assignment Results AM link speed INFORMS, November

39 Assignment Results AM truck volume by class INFORMS, November

40 Assignment Results AM HOV VMT Grid Map INFORMS, November

41 Assignment Results HCM Level of Service INFORMS, November

42 Scenario Comparison VMT changes INFORMS, November

43 Toll highway analysis Poznan, Poland It involves the following models: -Multi-class equilibrium assignment with generalized costs -Demand models for toll-no toll choice and future year demands -Equilibration of the demand for toll highway and network performance INFORMS, November

44 The multi-class equilibrium model with tolls min v a ( ) s x dx + v θ t c c c a a a a A 0 c C a A c subject to h = g, i I, c C k K c i k c h 0, k K, i I k c ( v = δ h, a A, c C) a ak k k K The numerical solution of this model is well known; It is worthwhile to point out that the flows by class, are not unique, nor are the path flows, but the h k arc flows are unique. c i i i c v a v a INFORMS, November

45 Base year No Toll INFORMS, November

46 Base year Medium Cost Toll INFORMS, November

47 Base year High Toll INFORMS, November

48 Year 2010 High Cost Toll INFORMS, November

49 Year 2020 Medium Cost Toll INFORMS, November

50 Year 2010 Low Cost Toll INFORMS, November

51 Toll Bridge Study Montreal, Canada Study carried out by the Ministry of Transportation of Quebec The analysis relied heavily on the analysis of paths generated by the assignment algorithm INFORMS, November

52 The Montreal Region INFORMS, November

53 The new proposed bridge INFORMS, November

54 Zones around Bridge INFORMS, November

55 Demand for Current and Future Year INFORMS, November

56 The Current Bridge Flows INFORMS, November

57 Bridge Flows with New facility INFORMS, November

58 Toll Income with Various Toll Levels INFORMS, November

59 National Models These are very large multi-modal multi-class models The underlying demand models are rather complex and the running times are very high An example of a national model is the PLANET model developed for British Rail INFORMS, November

60 The PLANET Model British Rail Zones Rail Road INFORMS, November

61 Dynamic Network Equilibrium Model Solved by a hybrid optimization-simulation model a discretized version of a variational inequality formulation of a dynamic network equilibrium model The theoretical properties of the model are difficult to establish Wardrop s user equilibrium in a temporal framework is a basis for the model INFORMS, November

62 Dynamic equilibrium variables ( 0, T ) = I = set of OD pairs K k i demand period = set of paths for i () = () h t flow t on path k i k () () ( 1, ) t = assignment interval t T i= OD pair, i I g t = demand for OD pair i s t = travel time ( t) on path k constraints h () t = g ( t) k K () t i k i hk ( t) 0 equilibrium conditions { } u () t = min s () t i k k K i = u () t if h () t > 0 i k s () t k u () t otherwise i INFORMS, November

63 Dynamic assignment model Network definition Time-dependent OD matrices Traffic Control Data choose initial paths calculate path flows (t) run traffic simulation modify path sets: add a new path or remove an existing path (each O-D pair & each interval) determine path travel times (t) STOP YES convergence achieved? NO INFORMS, November

64 Traffic simulation model simplified model of vehicle interactions allows for an efficient event-based simulation car following lane changing gap acceptance sophisticated lane selection heuristics local lane selection rules stochastic look-ahead strategy INFORMS, November

65 fundamental diagram Q V = free-flow speed flow (veh/hr/lane) V -W 1 Q K = = 1 LV 1 L + R K density (veh/km/lane) W = L R INFORMS, November

66 An application in Stockholm INFORMS, November

67 An application in Stockholm INFORMS, November

68 An application in Montreal INFORMS, November

69 An application in Auckland,NZ INFORMS, November

70 Ending Remarks The equilibrium model of route choice is here to stay for both static and dynamic models We have a lot to thank to the landmark contribution of 1956 INFORMS, November