- Email: [email protected]
A Parking Equilibrium Model for Park Pricing and Park & Ride Planning
Copyright © IFAC Transportation Systems Chania, Greece, 1997
A PARKING EQUlLmRIUM MODEL FOR PARK PRICING AND PARK&RIDE PLANNING
Stefano Carrese, Stefano Gori, Tommaso Picano
Depanment of Science of Civil Engineering - Universitii di Roma Tre, Via C. Segre 60 00146 Roma - Fax +6155175032 - Email [email protected]
Abstract: The close relationship between parking aVailability and accessibility to an urban area leads to a unified framework of parking plan and transit service design. The occasion for the development of an integrated model, in order to represent this phenomenon, has been a research on the born of a new university in a semi peripheral area of Rome. Keywords: Urban System, Park&Ride, Park Pricing, DUE Assignment Model, Demand Management Systems
necessary to use a multiclass assignment model, in order to represent the behaviors of different classes: i.e. different sensitiveness to fares, as trip purpose or short / long time parking. 2) if parking demand exceeds the available supply, two different phenomena can occur. a) Some auto trip will frod convenience to shift to public transport or to P&R mode because of the high generalized cost of the auto mode (including parking cost). A multimodal assignment model can then be used to determine an equilibrium flow pattern of all the trips on the network. In this model, the congestion, caused by the parking representation, plays a fundamental role. b) The congestion between vehicles, due to the lack of parking supply, reflects such a great effect on network equilibrium, that every new parking lot causes consistent changes in auto demand. To determine these changes through an equilibrium that considers explicitly demand elasticity, a variable demand assignment model can be used. The model is characterized by an original representation of the auto network and new "ad hoc" cost-volume and time-volume functions for every kind of parking links.
1. INTRODUCTION
In this paper an innovative procedure for Park Pricing simulation and for Park&Ride (p&R) planning (PREM) is presented. The approach is based on a Parking Equilibrium Model (PEM) that has already been tested for parking planning in congested urban areas (Carrese, et al., 96). The original aspect of PEM is that it considers the parking phase as a further increase of generalized cost (times and costs) while calculating the path time during the auto assignment In fact, in this way, Wardrop's user optimal principle ensures the existence and the uniqueness of an equilibrium flow pattern on the network, that takes into account even the parking phase within the generalized travel cost. Three cases of network equilibrium can be considered depending on the degree of congestion due to the lack of parking places. 1) If parking demand is lower than the available supply, a deterministic user equilibrium with fixed demand can be considered to represent the flows pattern on the network follOwing the changes in parking capacity. In this way the most convenient (near and cheap) parking lots will be used till capacity. Afterwards in these zones only illegal parking (at higher cost) will be possible and other parking lots more expensive or distant will become convenient. To evaluate the effects of different fares for each parking lot, it is
2. DEMAND MODELING
The PREM makes it possible to simulate the choices
1235
of different user classes on a multimodal network. Each class is characterized by a typical value of the utility associated to every transportation alternative. The model deals with the problem of simulating a transportation system, operating for every time interval of the day (hour), with as many OlD matrices as the user classes. The user classes are not only characterized by trip purpose, but mostly by the stop duration after the trip and before taking the possible following trip. This stop time influences the utility perceived by the users in a decisive way, especially regarding auto alternatives because it conditions the value of parking cost, which the explicit modeling deals with. We assume the origin/destination matrices for several user classes because they contain complete information about demand on all transportation modes. The share of demand relative to the different transportation modes available on every OlD pair is calculated by the assignment to the network, according to the total performances of the transportation system.
subway links
--.....
Fig. 1
3. SUPPLY MODELING
i
P&R links
auto links
Cl
centroids
access links
•
park nodes
park links
0
auto nodes
pedestrian
C
P&Rnodes
Window of the graph
that even for low volume of parking users, everyone takes to minutes for parking operations always. For increasing volumes, parking time tsp slowly increases because more time l!J is needed to find a free place (usually more time means longer walking distance, too) tsp =to +l!J . When parking volume is equal to the capacity of the parking lot Cl> tsp =tc ; this value is obtained by adding the value of the fme M , multiplied by the probability to be fmed P to the default value to. In formulas:
To define transport supply, a detailed description of the multimodal auto and transit network is required. In particular the following items have to be specified: main road network (auto); centroids-network connections (access); parking links (on and off street); pedestrian parking-destination links; transit links (subway); transfer links (Park&Ride). In order to implement the Park&Ride mode, a particular structure of the graph for transfers from auto to transit mode has been considered. A window of the graph is shown in figure 1. The functions, representing the different links described, assume different patterns that can be generally expressed as the sum of i constant components and j variable ones depending on the flows if~ m with 1 = links and m = modes): Cl.m = LCLm + alm * LCj(flm)
~
.-.
tsp
= le =to + (M * P) Ir,
(2)
where r is the conversion rate Lit/minutes. The model, through the parking functions, is able to represent every change in the parking system: e.g. the construction of a toll parking lot is represented by a step F (fare), starting from the default value, to. c. r-4.--'..:.;.,--------...:....
(1)
I
j
i
For example, the different components of a P&R trip, are access time to car mode t ac , travel time by car tre, parking search time tsp , park&ride cost Cp&r, transfer time tx , waiting time tWo transit travel time tu and walking time to destination tp. In this case tac , Cp&r , tx , tw ,tp do not depend on traffic volumes, while tte , tsp ,tu are results of the multimodal assignment model. For the determination of the parking search time and the parking cost, the ad hoc developed parking timevolume functions are shown in Figure 2. It shows
. . ... ., .... I ~.
!
'. I ._ 0' _._ .- -_ .- _.- .-
"' -
- -- .-. -~~--
"' -
'0
Fig. 2
1236
Parking time-volume functions
4.1 A multiclass User Equilibrium Assignment Model
In this case ~ =tf=
(3)
to +Llt + FIr,
If the new parking lot is to be considered free of toll, the parking function has just a different value of Cl . In the same figure a time-volume function for a secondary link is shown for comparison. Further it can be noted that network equilibrium models traditionally rely on the assumption of continuous travel cost function. However congestion pricing schemes make use of discontinuous step function tolls and the existence of network equilibria in the presence of discontinuities has been explored. In particular it has been shown that when such costs are at least lower semicontinuous, a behaviorally meaningful notion of user equilibrium can still be defined that reduces to Wardrop equilibrium in the continuous case (Bemstein, Smith 1994). In addition it is shown that such equilibria are guaranteed to exist under fairly general condition.
In the general case different categories of travelers on a given link a e A experience different cost functions. Let m e M denote the set of distinct user classes. For each user class m the cost on link a depends on the class volume v~' of all classes M,
(4) The problem is equivalent to an equilibrium assignment with generalized costs and corresponds to a variational inequality problem, which, in general, has no unique solution and also cannot be transformed into a convex optimization problem. A simpler form of the multiclass assignment can be carried out. It is assumed that the cost of a link a perceived by a user of class m can be written as
ae A,
This implies that the different user classes are all subject to the same congestion effect. but each user class perceives a different constant bias bam . It is easy to show that this simplified multiclass user equilibrium assignment problem is equivalent to a convex optimization problem of the form
4. A PARK&RIDE EQUILIBRIUM MODEL The flow chart of the Park&Ride Equilibrium Model (PREM) is shown in figure 3. It can be seen that the multiclass and multimodal DUE assignment model performs the split between auto and P&R mode.
Min I..r;aSa(v)dv+ I. I.v~b~ aeA meMaEA
Representation of the study area through a Park&Ride base network
!
r
me M (5)
t
(6)
subject to the usual conservation of flow constraints.
Time Interval Tn =Tn+ 1+dt
4.2 Multimodal Equilibrium Assignment Model
OD Matrices relatives to the auto mode and characterized by trip purpose and time interval Tn
Supply and demand outlined above are inputs of the multimodal equilibrium assignment model, which consists of: a transit assignment model, based on the concept of "optimal" (Spiess and Florian 89). In this model the traveler's choice set consists of all feasible strategies. It is clear that. due to the asymmetry inherent to the transit trip, this model can only be solved efficiently by assigning demands from all Origins to a single destination. A non negative time (or cost) is usually used to quantify each trip component. with the exception of "waiting for a vehicle"; this is quantified by using the statistical distribution of waiting times for the arrival of the first vehicle of a given transit line at a given stop. The model may be solved by an algorithm which consists of two parts. In a flfSt pass, from the destination node to all origins, the optimal strategy and the expected total travel times from each node to the destination node are computed. In a second pass,
DATA: subway and parking fares, number of free parking spaces and load factor at the end of Tn-1
y
multiclass and multimodal DUE
..
1.J
Modal Split between auto and Park&Ride Parking demand distribution in Tn
•
N last interval ? YES
NO
N=N+1
~
l
Evaluation of results and measures
Fig. 3
Flow chart of the PREM
1237
-
from all origins to the destination, the demand is assigned to the network according to the optimal strategy. a variable demand auto assignment model. This model computes the equilibrium auto demand, link flows and travel times by the Partial Linear Approximation Method PLAM (Florian 86), which is an improvement of the Linear Approximation Method LAM (Frank and Wolfe 56) used for Fixed Demand. Given a current feasible solution, the LAM finds a descent direction by solving a sub problem where the objective function is a linear approximation of the nonlinear objective function of the problem to be solved. Moreover in the PLAM the algorithm may be implemented by requiring only evaluations of the inverse demand function. As only the auto demand function is required, the inverse demand is obtained numerically, by using the secant method. Given a current solution, the PLAM finds a descent direction, by solving a sub problem where the delay functions are linearized, but the inverse demand functions are not.
5. APPLICATION OF PREM IN THE CASE STUDY OF ''ROMA TRE" An application of the model in the area of the University of Roma Tre has been performed. The transport plan includes the design of road traffic, mass transit and parking system. The parking plan has been simulated with PEM.
students and workers . In this case we supposed that only subway service could attract a P&R user. The multiclass user assignment model is used to separate student trips from worker ones. In fact the survey results indicate strong differences between the two user classes in the distribution of times arrival, duration of stay and willingness to use public transport. These results have been represented through different sensitivity to the conversion rate r, that is rs = 7000 lirelbour for students and rw = 14000 lirelbour for workers. Further strong differences in origin distribution of trips between the two user classes have been outlined. In fact while workers present a quite uniform distribution in the urban land, students are concentrated in the south of the town. Stating the split of auto demand in the forecasting matrix with 30000 students and 11800 workers, the simulation of three scenarios of parking fares have been performed, with the subway ticket including the cost of P&R. The differences in the three scenarios consider only the cost of the destination parking: scenario A) toll free access; scenario B) 800 lire per hour; scenario C) 1600 lire per hour. Fares have been chosen with comparison of the actual parking fares of CBD that "Comune di Roma" has been fixed in 2000 lirelbour. Unlimited capacity for P&R and destination parking lots has been initially considered, because in the preliminary design the PREM splits the quote of auto demand between the auto mode and the P&R mode as a function of travel times and travel cost
5.1 The Parking Plan
The present number of parking spaces has been surveyed in the study area, which has been divided in three wnes: Ostiense, ex Alfa Romeo and Valco S: Paolo. The great part of parking supply is available on street and mostly used by residents. This situation causes decrease of capacity and speed on main streets. For each wne the number of parking spaces has been represented by one or more "parking links", which join different nodes of the road network with a parking node. From this node it is possible to reach the trip destination only through a pedestrian link. This representation permits to weight every single phase or restriction of parking activity (search time, walk distance, toll parking, etc.). The future parking plan has been provided by "Comune di Roma". It includes three parking typologies: Park&Ride, residents and Roma Tre. The total number of forecasted parking spaces is 5000.
6. RESULTS OF PREM APPLICATION Results of the P&R equilibrium model are shown in table 1, where the number of trips, split between auto mode and P&R mode for every zone provides an estimation of design data for the parking plan. Table 1 Results of the PREM application
Zone
Seen. A Class Mode toll free Trips (split %)
Stud. P&R Auto Ostiense W k P&R or . Auto
Via
5.2 The Park Pricing arui Park&Ride Plan
For the simulation of P&R lots, the PREM has elaborated data available from a survey to 800
1238
2129 (42) 2964 (58) 4216 (46) 4984 (54)
Seen. B 1I2$1b Trips (split %)
Seen. C toll1$1b Trips (split %)
2321 (46) 2772 (54) 4233 (46) 4966 (54)
3227 (63) 1866 (37) 4855 (53) 4344 (47)
P&R ex Alfa Stud. Auto Romeo W k P&R or . Auto
278 795 173 337
(26) (74) (34) (66)
469 604 250 260
(44) (56) (49) (51)
921 152 269 241
(86) (14) (53) (47)
P&R Valco Stud. Auto S. Paolo W k P&R or . Auto
263 3871 224 982
(6) (94) (19) (81)
1135 (27) 3000 (73) 237 (20) 969 (80)
2351 1783 465 741
(57) (43) (39) (61)
demand. The future situation is even worst. The public network, even with the addition of a new line for university service, does not attract the greatest quote of new transport demand. - The parking plan provides a strong increase of number of spaces, which is not able to balance the modal split between transit and auto. Only with the introduction of specific parking fares for each zone, it is possible to obtain a trip distribution between user classes, so that the transportation system can be optimized. Consequently three major characteristics of the PREM can be outlined: - good representation of split between auto and transit modes, depending on parking characteristics; - strong sensitivity to parking fares, travel times and walking distances in parking choice problems; the complexity of network representation is balanced by the facility of cbanging design parameters.
The Ostiense zone has been loaded with trips of 9200 workers and 5100 students. There is no limitation to parking availability and so only the restriction due to network capacity splits these trips into 58% on auto and 42% on P&R. The introduction of the parking fare brings small variations in modal split because of the weight of travel times, stronger than the weight of costs. This effect is more evident in students class, because it is more sensitive to the fare The ex Alfa Romeo zone instead is conditioned by its very close position to the subway station "Marconi". In fact, even in case of toll free parking, the percentage of P&R users is high for both classes (1070 students and 510 workers), although the good accessibility with the road network. The Valco S. Paolo zone is characterized by a minimum walking distance of 800 meters from the subway stop "S . Paolo". In fact the 4150 students do not use P&R if parking is toll free (only 6%). The 1200 workers present the same behavior but less evident It has to be outlined that the variations in modal split between student and workers are due not only to the different money value but mostly to the different distribution of origins of trips. In this case the use of a DUE model has provided a good representation of congestion in auto volumes and parking saturation. Finally, the fare which optimize traffic conditions has been chosen for every zone. It is then possible to verify the fmal scenario with capacity constraints, which provides: total flows on network links; users in residents, P&R and Roma Tre parking lots; incomes from parking fares and every component of travel times for every mode.
-
REFERENCES Bemstein, D., T.E. Smith (1994). Equilibria for Networks with Lower Semicontinous Costs: With an Application to Congestion Pricing. Transp. Science Vol. 28, No. 3, Aug. 1994, pp. 221-235 Carrese, S., S. Gori and T. Picano (1996). Relationship between parking location and traffic flows in urban area. In: Advanced Methods in Transportation Analysis (L. Bianco and P. Toth Ed.), pp. 183-214. Springer&Verlag Transp. Analysis. Florian, M. (1986). Non linear Cost Network Models in Transportation Analysis. Math. Programming Study 26, pp. 167-196. Frank, M. and P. Wolfe (1956). An Algorithm for Quadratic Programming. Naval Res. Logist. Q., 3, pp. 95-110 Spiess, H. and M. Florian (1989) . Optimal Strategies: A New Assignment Model for Transit Networks. Transp. Res. B, Vol. 23B, pp. 83-102.
7. CONCLUSION The application of PREM to the area of the university of Roma Tre brings to the following general considerations. - The road network, although the realization of the forecasted infrastructures, does not present a satisfying level of service for the present traffic
1239
A PARKING EQUlLmRIUM MODEL FOR PARK PRICING AND PARK&RIDE PLANNING
Stefano Carrese, Stefano Gori, Tommaso Picano
Depanment of Science of Civil Engineering - Universitii di Roma Tre, Via C. Segre 60 00146 Roma - Fax +6155175032 - Email [email protected]
Abstract: The close relationship between parking aVailability and accessibility to an urban area leads to a unified framework of parking plan and transit service design. The occasion for the development of an integrated model, in order to represent this phenomenon, has been a research on the born of a new university in a semi peripheral area of Rome. Keywords: Urban System, Park&Ride, Park Pricing, DUE Assignment Model, Demand Management Systems
necessary to use a multiclass assignment model, in order to represent the behaviors of different classes: i.e. different sensitiveness to fares, as trip purpose or short / long time parking. 2) if parking demand exceeds the available supply, two different phenomena can occur. a) Some auto trip will frod convenience to shift to public transport or to P&R mode because of the high generalized cost of the auto mode (including parking cost). A multimodal assignment model can then be used to determine an equilibrium flow pattern of all the trips on the network. In this model, the congestion, caused by the parking representation, plays a fundamental role. b) The congestion between vehicles, due to the lack of parking supply, reflects such a great effect on network equilibrium, that every new parking lot causes consistent changes in auto demand. To determine these changes through an equilibrium that considers explicitly demand elasticity, a variable demand assignment model can be used. The model is characterized by an original representation of the auto network and new "ad hoc" cost-volume and time-volume functions for every kind of parking links.
1. INTRODUCTION
In this paper an innovative procedure for Park Pricing simulation and for Park&Ride (p&R) planning (PREM) is presented. The approach is based on a Parking Equilibrium Model (PEM) that has already been tested for parking planning in congested urban areas (Carrese, et al., 96). The original aspect of PEM is that it considers the parking phase as a further increase of generalized cost (times and costs) while calculating the path time during the auto assignment In fact, in this way, Wardrop's user optimal principle ensures the existence and the uniqueness of an equilibrium flow pattern on the network, that takes into account even the parking phase within the generalized travel cost. Three cases of network equilibrium can be considered depending on the degree of congestion due to the lack of parking places. 1) If parking demand is lower than the available supply, a deterministic user equilibrium with fixed demand can be considered to represent the flows pattern on the network follOwing the changes in parking capacity. In this way the most convenient (near and cheap) parking lots will be used till capacity. Afterwards in these zones only illegal parking (at higher cost) will be possible and other parking lots more expensive or distant will become convenient. To evaluate the effects of different fares for each parking lot, it is
2. DEMAND MODELING
The PREM makes it possible to simulate the choices
1235
of different user classes on a multimodal network. Each class is characterized by a typical value of the utility associated to every transportation alternative. The model deals with the problem of simulating a transportation system, operating for every time interval of the day (hour), with as many OlD matrices as the user classes. The user classes are not only characterized by trip purpose, but mostly by the stop duration after the trip and before taking the possible following trip. This stop time influences the utility perceived by the users in a decisive way, especially regarding auto alternatives because it conditions the value of parking cost, which the explicit modeling deals with. We assume the origin/destination matrices for several user classes because they contain complete information about demand on all transportation modes. The share of demand relative to the different transportation modes available on every OlD pair is calculated by the assignment to the network, according to the total performances of the transportation system.
subway links
--.....
Fig. 1
3. SUPPLY MODELING
i
P&R links
auto links
Cl
centroids
access links
•
park nodes
park links
0
auto nodes
pedestrian
C
P&Rnodes
Window of the graph
that even for low volume of parking users, everyone takes to minutes for parking operations always. For increasing volumes, parking time tsp slowly increases because more time l!J is needed to find a free place (usually more time means longer walking distance, too) tsp =to +l!J . When parking volume is equal to the capacity of the parking lot Cl> tsp =tc ; this value is obtained by adding the value of the fme M , multiplied by the probability to be fmed P to the default value to. In formulas:
To define transport supply, a detailed description of the multimodal auto and transit network is required. In particular the following items have to be specified: main road network (auto); centroids-network connections (access); parking links (on and off street); pedestrian parking-destination links; transit links (subway); transfer links (Park&Ride). In order to implement the Park&Ride mode, a particular structure of the graph for transfers from auto to transit mode has been considered. A window of the graph is shown in figure 1. The functions, representing the different links described, assume different patterns that can be generally expressed as the sum of i constant components and j variable ones depending on the flows if~ m with 1 = links and m = modes): Cl.m = LCLm + alm * LCj(flm)
~
.-.
tsp
= le =to + (M * P) Ir,
(2)
where r is the conversion rate Lit/minutes. The model, through the parking functions, is able to represent every change in the parking system: e.g. the construction of a toll parking lot is represented by a step F (fare), starting from the default value, to. c. r-4.--'..:.;.,--------...:....
(1)
I
j
i
For example, the different components of a P&R trip, are access time to car mode t ac , travel time by car tre, parking search time tsp , park&ride cost Cp&r, transfer time tx , waiting time tWo transit travel time tu and walking time to destination tp. In this case tac , Cp&r , tx , tw ,tp do not depend on traffic volumes, while tte , tsp ,tu are results of the multimodal assignment model. For the determination of the parking search time and the parking cost, the ad hoc developed parking timevolume functions are shown in Figure 2. It shows
. . ... ., .... I ~.
!
'. I ._ 0' _._ .- -_ .- _.- .-
"' -
- -- .-. -~~--
"' -
'0
Fig. 2
1236
Parking time-volume functions
4.1 A multiclass User Equilibrium Assignment Model
In this case ~ =tf=
(3)
to +Llt + FIr,
If the new parking lot is to be considered free of toll, the parking function has just a different value of Cl . In the same figure a time-volume function for a secondary link is shown for comparison. Further it can be noted that network equilibrium models traditionally rely on the assumption of continuous travel cost function. However congestion pricing schemes make use of discontinuous step function tolls and the existence of network equilibria in the presence of discontinuities has been explored. In particular it has been shown that when such costs are at least lower semicontinuous, a behaviorally meaningful notion of user equilibrium can still be defined that reduces to Wardrop equilibrium in the continuous case (Bemstein, Smith 1994). In addition it is shown that such equilibria are guaranteed to exist under fairly general condition.
In the general case different categories of travelers on a given link a e A experience different cost functions. Let m e M denote the set of distinct user classes. For each user class m the cost on link a depends on the class volume v~' of all classes M,
(4) The problem is equivalent to an equilibrium assignment with generalized costs and corresponds to a variational inequality problem, which, in general, has no unique solution and also cannot be transformed into a convex optimization problem. A simpler form of the multiclass assignment can be carried out. It is assumed that the cost of a link a perceived by a user of class m can be written as
ae A,
This implies that the different user classes are all subject to the same congestion effect. but each user class perceives a different constant bias bam . It is easy to show that this simplified multiclass user equilibrium assignment problem is equivalent to a convex optimization problem of the form
4. A PARK&RIDE EQUILIBRIUM MODEL The flow chart of the Park&Ride Equilibrium Model (PREM) is shown in figure 3. It can be seen that the multiclass and multimodal DUE assignment model performs the split between auto and P&R mode.
Min I..r;aSa(v)dv+ I. I.v~b~ aeA meMaEA
Representation of the study area through a Park&Ride base network
!
r
me M (5)
t
(6)
subject to the usual conservation of flow constraints.
Time Interval Tn =Tn+ 1+dt
4.2 Multimodal Equilibrium Assignment Model
OD Matrices relatives to the auto mode and characterized by trip purpose and time interval Tn
Supply and demand outlined above are inputs of the multimodal equilibrium assignment model, which consists of: a transit assignment model, based on the concept of "optimal" (Spiess and Florian 89). In this model the traveler's choice set consists of all feasible strategies. It is clear that. due to the asymmetry inherent to the transit trip, this model can only be solved efficiently by assigning demands from all Origins to a single destination. A non negative time (or cost) is usually used to quantify each trip component. with the exception of "waiting for a vehicle"; this is quantified by using the statistical distribution of waiting times for the arrival of the first vehicle of a given transit line at a given stop. The model may be solved by an algorithm which consists of two parts. In a flfSt pass, from the destination node to all origins, the optimal strategy and the expected total travel times from each node to the destination node are computed. In a second pass,
DATA: subway and parking fares, number of free parking spaces and load factor at the end of Tn-1
y
multiclass and multimodal DUE
..
1.J
Modal Split between auto and Park&Ride Parking demand distribution in Tn
•
N last interval ? YES
NO
N=N+1
~
l
Evaluation of results and measures
Fig. 3
Flow chart of the PREM
1237
-
from all origins to the destination, the demand is assigned to the network according to the optimal strategy. a variable demand auto assignment model. This model computes the equilibrium auto demand, link flows and travel times by the Partial Linear Approximation Method PLAM (Florian 86), which is an improvement of the Linear Approximation Method LAM (Frank and Wolfe 56) used for Fixed Demand. Given a current feasible solution, the LAM finds a descent direction by solving a sub problem where the objective function is a linear approximation of the nonlinear objective function of the problem to be solved. Moreover in the PLAM the algorithm may be implemented by requiring only evaluations of the inverse demand function. As only the auto demand function is required, the inverse demand is obtained numerically, by using the secant method. Given a current solution, the PLAM finds a descent direction, by solving a sub problem where the delay functions are linearized, but the inverse demand functions are not.
5. APPLICATION OF PREM IN THE CASE STUDY OF ''ROMA TRE" An application of the model in the area of the University of Roma Tre has been performed. The transport plan includes the design of road traffic, mass transit and parking system. The parking plan has been simulated with PEM.
students and workers . In this case we supposed that only subway service could attract a P&R user. The multiclass user assignment model is used to separate student trips from worker ones. In fact the survey results indicate strong differences between the two user classes in the distribution of times arrival, duration of stay and willingness to use public transport. These results have been represented through different sensitivity to the conversion rate r, that is rs = 7000 lirelbour for students and rw = 14000 lirelbour for workers. Further strong differences in origin distribution of trips between the two user classes have been outlined. In fact while workers present a quite uniform distribution in the urban land, students are concentrated in the south of the town. Stating the split of auto demand in the forecasting matrix with 30000 students and 11800 workers, the simulation of three scenarios of parking fares have been performed, with the subway ticket including the cost of P&R. The differences in the three scenarios consider only the cost of the destination parking: scenario A) toll free access; scenario B) 800 lire per hour; scenario C) 1600 lire per hour. Fares have been chosen with comparison of the actual parking fares of CBD that "Comune di Roma" has been fixed in 2000 lirelbour. Unlimited capacity for P&R and destination parking lots has been initially considered, because in the preliminary design the PREM splits the quote of auto demand between the auto mode and the P&R mode as a function of travel times and travel cost
5.1 The Parking Plan
The present number of parking spaces has been surveyed in the study area, which has been divided in three wnes: Ostiense, ex Alfa Romeo and Valco S: Paolo. The great part of parking supply is available on street and mostly used by residents. This situation causes decrease of capacity and speed on main streets. For each wne the number of parking spaces has been represented by one or more "parking links", which join different nodes of the road network with a parking node. From this node it is possible to reach the trip destination only through a pedestrian link. This representation permits to weight every single phase or restriction of parking activity (search time, walk distance, toll parking, etc.). The future parking plan has been provided by "Comune di Roma". It includes three parking typologies: Park&Ride, residents and Roma Tre. The total number of forecasted parking spaces is 5000.
6. RESULTS OF PREM APPLICATION Results of the P&R equilibrium model are shown in table 1, where the number of trips, split between auto mode and P&R mode for every zone provides an estimation of design data for the parking plan. Table 1 Results of the PREM application
Zone
Seen. A Class Mode toll free Trips (split %)
Stud. P&R Auto Ostiense W k P&R or . Auto
Via
5.2 The Park Pricing arui Park&Ride Plan
For the simulation of P&R lots, the PREM has elaborated data available from a survey to 800
1238
2129 (42) 2964 (58) 4216 (46) 4984 (54)
Seen. B 1I2$1b Trips (split %)
Seen. C toll1$1b Trips (split %)
2321 (46) 2772 (54) 4233 (46) 4966 (54)
3227 (63) 1866 (37) 4855 (53) 4344 (47)
P&R ex Alfa Stud. Auto Romeo W k P&R or . Auto
278 795 173 337
(26) (74) (34) (66)
469 604 250 260
(44) (56) (49) (51)
921 152 269 241
(86) (14) (53) (47)
P&R Valco Stud. Auto S. Paolo W k P&R or . Auto
263 3871 224 982
(6) (94) (19) (81)
1135 (27) 3000 (73) 237 (20) 969 (80)
2351 1783 465 741
(57) (43) (39) (61)
demand. The future situation is even worst. The public network, even with the addition of a new line for university service, does not attract the greatest quote of new transport demand. - The parking plan provides a strong increase of number of spaces, which is not able to balance the modal split between transit and auto. Only with the introduction of specific parking fares for each zone, it is possible to obtain a trip distribution between user classes, so that the transportation system can be optimized. Consequently three major characteristics of the PREM can be outlined: - good representation of split between auto and transit modes, depending on parking characteristics; - strong sensitivity to parking fares, travel times and walking distances in parking choice problems; the complexity of network representation is balanced by the facility of cbanging design parameters.
The Ostiense zone has been loaded with trips of 9200 workers and 5100 students. There is no limitation to parking availability and so only the restriction due to network capacity splits these trips into 58% on auto and 42% on P&R. The introduction of the parking fare brings small variations in modal split because of the weight of travel times, stronger than the weight of costs. This effect is more evident in students class, because it is more sensitive to the fare The ex Alfa Romeo zone instead is conditioned by its very close position to the subway station "Marconi". In fact, even in case of toll free parking, the percentage of P&R users is high for both classes (1070 students and 510 workers), although the good accessibility with the road network. The Valco S. Paolo zone is characterized by a minimum walking distance of 800 meters from the subway stop "S . Paolo". In fact the 4150 students do not use P&R if parking is toll free (only 6%). The 1200 workers present the same behavior but less evident It has to be outlined that the variations in modal split between student and workers are due not only to the different money value but mostly to the different distribution of origins of trips. In this case the use of a DUE model has provided a good representation of congestion in auto volumes and parking saturation. Finally, the fare which optimize traffic conditions has been chosen for every zone. It is then possible to verify the fmal scenario with capacity constraints, which provides: total flows on network links; users in residents, P&R and Roma Tre parking lots; incomes from parking fares and every component of travel times for every mode.
-
REFERENCES Bemstein, D., T.E. Smith (1994). Equilibria for Networks with Lower Semicontinous Costs: With an Application to Congestion Pricing. Transp. Science Vol. 28, No. 3, Aug. 1994, pp. 221-235 Carrese, S., S. Gori and T. Picano (1996). Relationship between parking location and traffic flows in urban area. In: Advanced Methods in Transportation Analysis (L. Bianco and P. Toth Ed.), pp. 183-214. Springer&Verlag Transp. Analysis. Florian, M. (1986). Non linear Cost Network Models in Transportation Analysis. Math. Programming Study 26, pp. 167-196. Frank, M. and P. Wolfe (1956). An Algorithm for Quadratic Programming. Naval Res. Logist. Q., 3, pp. 95-110 Spiess, H. and M. Florian (1989) . Optimal Strategies: A New Assignment Model for Transit Networks. Transp. Res. B, Vol. 23B, pp. 83-102.
7. CONCLUSION The application of PREM to the area of the university of Roma Tre brings to the following general considerations. - The road network, although the realization of the forecasted infrastructures, does not present a satisfying level of service for the present traffic
1239