Network Equilibrium Models: Varied and Ambitious Michael Florian Center for Research on Transportation University of Montreal INFORMS, November 2005 1
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 2005 2
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 2005 3
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 2005 4
Some Straightforward Applications Madrid, Spain Pretoria, South Africa Auckland, New Zealand INFORMS, November 2005 5
Madrid- AM Peak Flows and Speeds INFORMS, November 2005 6
Regional Study of the Province of Gauteng, South Africa Study carried out by Vela VKE Pretoria, South Africa, INFORMS, November 2005 7
Scenario Without New facility INFORMS, November 2005 8
Scenario with New facility INFORMS, November 2005 9
Scenario comparison INFORMS, November 2005 10
Planning Transport in Auckland Study carried out by Auckland Regional Council, Auckland, NZ INFORMS, November 2005 11
AM Vehicle Flows, 2001 Motorways Arterials Local 5% 36% 59% INFORMS, November 2005 12
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
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 2005 14 INRO
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 2005 15
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 2005 16
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 2005 17
Santiago, Chile Strategic Planning Model Base network 409 centroids including 49 parking locations 1808 nodes, 11,331 directional links 1116 transit lines and 52468 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
Base Network of Santiago, Chile
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
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
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 2005 22
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
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 2005 24
Convergence of equilibration Normalized gap (%) 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 Normalized Gap vs. Iterations congested vs. non-congested metro assignment with metro capacity Auto demand Auto link volume 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 cong. Link vol. cong. demand non-cong. Link vol. non-cong. demand Iteration
Metro Volume (non-congested vs. congested version)
Metro Volume Changes
Auto-metro volume (non-congested vs. congested version) Congested Non-congested
Metro Volume - metro 5A, non-congested version
Metro Volum2 - metro 5A, congested version
Some Complex Applications Los Angeles, California Toll Highways Poznan, Poland Toll Bridge, Montreal, INFORMS, November 2005 31
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 2005 32
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 2005 33
Network Overview - Highway Network by facility type INFORMS, November 2005 34
Network Overview - Highway Network with parking lots INFORMS, November 2005 35
Equilibration Algorithm Convergence Results SCAG Model Convergence (AM peak) 1.4 1.2 link relative difference 1 0.8 0.6 0.4 0.2 0 1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 1 5 9 13 17 21 25 loops inner-loop outer-loop INFORMS, November 2005 36
Assignment Results AM peak volume INFORMS, November 2005 37
Assignment Results AM link speed INFORMS, November 2005 38
Assignment Results AM truck volume by class INFORMS, November 2005 39
Assignment Results AM HOV VMT Grid Map INFORMS, November 2005 40
Assignment Results HCM Level of Service INFORMS, November 2005 41
Scenario Comparison VMT changes INFORMS, November 2005 42
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 2005 43
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 2005 44
Base year No Toll INFORMS, November 2005 45
Base year Medium Cost Toll INFORMS, November 2005 46
Base year High Toll INFORMS, November 2005 47
Year 2010 High Cost Toll INFORMS, November 2005 48
Year 2020 Medium Cost Toll INFORMS, November 2005 49
Year 2010 Low Cost Toll INFORMS, November 2005 50
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 2005 51
The Montreal Region INFORMS, November 2005 52
The new proposed bridge INFORMS, November 2005 53
Zones around Bridge INFORMS, November 2005 54
Demand for Current and Future Year INFORMS, November 2005 55
The Current Bridge Flows INFORMS, November 2005 56
Bridge Flows with New facility INFORMS, November 2005 57
Toll Income with Various Toll Levels INFORMS, November 2005 58
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 2005 59
The PLANET Model British Rail Zones Rail Road INFORMS, November 2005 60
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 2005 61
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 2005 62
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 2005 63
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 2005 64
fundamental diagram Q V = free-flow speed flow (veh/hr/lane) 2000 1600 1200 800 400 1 V -W 1 Q K = = 1 LV 1 L + R 0 0 30 60 90 120 K density (veh/km/lane) W = L R INFORMS, November 2005 65
An application in Stockholm INFORMS, November 2005 66
An application in Stockholm INFORMS, November 2005 67
An application in Montreal INFORMS, November 2005 68
An application in Auckland,NZ INFORMS, November 2005 69
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 2005 70