State-dependent pricing for real-time freeway management: Anticipatory versus reactive strategies

Transportation Research Part C 19 (2011) 644–657

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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

State-dependent pricing for real-time freeway management: Anticipatory versus reactive strategies Jing Dong a,1, Hani S. Mahmassani a,⇑, Sevgi Erdog˘an b, Chung-Cheng Lu c a

Transportation Center, Northwestern University, 600 Foster Street, Evanston, IL 60208, USA Department of Civil & Environmental Engineering, University of Maryland, 1173 Glenn L. Martin Hall, College Park, MD 20742, USA c Graduate Institute of Information and Logistics Management, National Taipei University of Technology, 1 Section 3, Chung-Hsiao East Road, Taipei 106, Taiwan b

a r t i c l e

i n f o

Article history: Received 25 June 2008 Received in revised form 12 August 2010 Accepted 11 October 2010

Keywords: Dynamic pricing Managed lanes Toll lanes Freeway corridor management Anticipatory congestion pricing Reactive congestion pricing Predictive strategies Lane control Dynamic traffic assignment

a b s t r a c t This paper proposes the notion of anticipatory (dynamic) pricing, and investigates the advantages of using predicted traffic conditions over the use of prevailing and/or historical conditions in setting time-varying link tolls along a freeway corridor to maintain target level of service (LOS) and avoid traffic breakdown on toll links. This is accomplished through an anticipatory toll generator intended to operate in tandem with a real-time traffic estimation and prediction system. Using a calibrated network model of the Baltimore – Washington, DC corridor as test bed, simulation experiments are performed to compare the proposed anticipatory pricing strategies to reactive as well as static pricing schemes. The results indicate that setting prices on the basis of predicted conditions can make a substantial difference in terms of achieving the objectives of pricing in managed-lane situations.  2010 Elsevier Ltd. All rights reserved.

1. Introduction Road pricing, such as road tolls and cordon (area) tolls, is increasingly considered as an effective demand management strategy to reduce traffic congestion and improve system performance during peak periods in many metropolitan areas. In particular, several value pricing applications (e.g. I-15 in San Diego, I-394 in Minnesota and SR-167 in Seattle) that allow users to choose between two adjacent roadways – one tolled but free-flowing, and another free but congested (Small and Yan, 2001) – have been deployed in the United States. These programs apply dynamic pricing strategies, using real-time information collected from loop detectors. For example, in the MnPASS lanes on I-394, tolls are adjusted, as frequently as every three minutes, based on the spacing of vehicles. Specifically, when roadway sensors detect platoons of vehicles, it is assumed that a drop in average speed has occurred or will soon occur. The rate is accordingly adjusted upward to discourage the usage of the toll lanes. Similarly, the I-15 pilot program determines toll values by comparing aggregated volumes (on the tolled facility) obtained from two observation intervals against volume thresholds prescribed in a look-up table. The tolls are then displayed on variable message signs. An evaluation study of this pilot dynamic and state-dependent pricing application for the US Department of Transportation has concluded that it was successful (Perez et al., 2003). Recognizing that these

⇑ Corresponding author. Tel.: +1 847 491 7287; fax: +1 847 491 3090. E-mail addresses: [email protected] (J. Dong), [email protected] (H.S. Mahmassani), [email protected] (S. Erdog˘an), jasoncclu@ gmail.com (C.-C. Lu). 1 Tel.: +1 847 491 7287; fax: +1 847 491 3090. 0968-090X/$ - see front matter  2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.trc.2010.10.001

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tolled facilities’ success depends on their ability to provide time savings and reliable travel times, the present study aims at developing a systematic approach (pricing strategy) for efficiently determining time-varying and state-dependent tolls so as to maintain high level of service (LOS) on toll lanes. Although freeways were initially intended to provide uninterrupted high LOS to travelers, experience in metropolitan areas indicates that they frequently fail to provide the desired service quality because of recurrent and/or nonrecurrent congestion. Various freeway control measures, such as ramp metering, lane use control, traveler information and guidance systems have been developed to reduce congestion and prevent/delay the onset of breakdown (Papageorgiou et al., 2003). However, imposing tolls on designated lanes has only recently been viewed as a possible control, with objectives similar to those of control methods, such as increasing efficiency (or reducing travel time), reliability, mobility, and throughput. This motivates the present study, where dynamic pricing is used as a control method to maintain high LOS on designated lanes, with the purpose of managing demand for toll lanes by setting the toll values effectively to provide uninterrupted flow conditions. Given the non-linear dynamic properties of traffic flow, and the instabilities associated with flow levels approaching nominal capacity levels, achieving the desired control objectives through dynamic prices poses several challenges. Current operation, which reacts to measured flow levels, may result in prices that are ‘‘too little, too late”—no matter how high they are set, flow breakdown may have already occurred, resulting in stop-and-go conditions. This means that customers who have may have paid the maximum allowed toll charges may in fact be experiencing unacceptably poor traffic conditions—with high and highly unreliable travel times. Furthermore, users may not respond to the prices in exactly the intended manner— especially because their response may not depend only on conditions prevailing on the tolled lane, but rather on the differential with the free lanes. This may result in fluctuations in the demand level for the tolled facility that could further exacerbate flow instabilities. The principal challenge is to set the price at a sufficiently high level before, and not after the onset of congestion/breakdown. In fact the proactive (anticipatory) measure has been introduced and proves to be more effective than reactive measure in travel information provision (e.g. Dong et al. 2006), ramp metering (e.g. Paesani et al., 1997), and some fields of business management (e.g. Schweitzer, 2004) as it could avert a crisis as opposed to repair its damage. Therefore the application of proactive management in dynamic pricing calls for the use of prediction in conjunction with sensor measurements in setting prices, resulting in anticipatory pricing strategies. To set prices as a control, we adapt logic similar to ramp-metering control strategies. Reactive ramp-metering strategies, such as ALINEA (Asservissement Linéaire d’Entrée Autoroutière), have been used at a tactical level to keep freeway traffic conditions close to target values, based on traffic measurements (Papageorgiou et al., 1991). ALINEA is a closed-loop ramp-metering strategy aimed at maintaining maximum throughput in the mainline. It is relatively simple to apply, but has been shown to be quite effective in several field applications (Papageorgiou et al., 1991, 1997) and simulation-based evaluation studies (Papageorgiou et al., 1997; Chu et al., 2004). Several variations of ALINEA have been proposed to address specific issues and requirements not covered by the initial implementations of the logic (Smaragdis and Papageorgiou, 2003). On the other hand, anticipatory ramp metering schemes have also been designed for real-time application. As noted, the rationale of anticipatory control strategies is to prevent traffic breakdown before it occurs, by relying on predicted traffic conditions. For example, System Wide Adaptive Ramp Metering algorithm (SWARM) used linear regression and Kalman filtering processes to forecast system evolution (Paesani et al., 1997); Advanced Real-time Ramp Metering System (ARMS) employed an optimal self-learning congestion predictor based on pattern recognition technique to predict short-term breakdown (Liu et al., 1994). Bellemans et al. (2004) proposed a macroscopic simulation program, METANET (Messmer and Papageorgiou, 1990), as a prediction model. Although comparable performance was reported in the test results of ALINEA and SWARM algorithms, potential benefits of the anticipatory ramp metering schemes were expected to be greater when traffic predictions are accurate (Zhang et al., 2001). Inspired by the logic of the above ramp-metering control methods, this study proposes and compares dynamic link pricing strategies which determine link tolls based on prevailing measures and anticipatory measures. Prevailing measures depict current network conditions (or so-called instantaneous conditions). Anticipatory measures are derived from on-line forecasts of future network states. In the proposed methods, the differences between link concentrations (or alternatively, occupancies) and a given set of target concentrations (occupancies) on toll links are obtained, and then link tolls are determined as a function of these differences (given some control parameters). Under a reactive strategy, the link tolls are based exclusively on link concentrations extracted from prevailing network states. On the other hand, the strategy is anticipatory when the link tolls are determined using predicted link concentrations. As explained, a potential drawback of using prevailing traffic measures for setting prices (i.e. reactive pricing) is that network conditions may change significantly during the time that drivers who pay that toll are actually traveling, resulting in experienced service levels that are not appropriate for the price charged. More important, once breakdown occurs, significant and often irretrievable loss of throughput is experienced at a time when potential demand is highest. Alternatively, the anticipatory pricing strategy proposed in this paper builds on recent research findings on advanced traveler information systems (ATIS), which showed the effectiveness of using a real-time simulation-based traffic estimation and prediction system to provide anticipatory travel time information, compared to prevailing travel time information (Mahmassani et al., 2005b; Dong et al., 2006). The conceptual basis for anticipatory information provision was articulated by Kaysi (1992), before developments in real-time traffic estimation and prediction enabled actual application of this concept. When the predictions are accurate or within an acceptable range, anticipatory measures are generally expected to be more effective than the prevailing measures because they can account for the rapid changes in traffic conditions spatially and

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temporally, and are as such based on the traffic conditions predicted to prevail at the time the trip-maker reaches a particular location. One of the main contributions of this paper is to investigate the potential effectiveness of utilizing predicted traffic measures in setting the toll values in order to prevent breakdown on toll lanes before it occurs. This paper investigates and compares the performance of two different dynamic pricing strategies, namely reactive and anticipatory, applied in a freeway corridor to maintain a target level of service on toll links. A third strategy, for static pricing, is used as a benchmark. The design of the link toll generators is similar to the logic of local ramp-metering control methods. To examine and compare the performance of static, reactive and anticipatory link pricing strategies, the toll generator is embedded in a real-time traffic estimation and prediction system (TrEPS), which continually publishes prevailing and predicted link concentrations. Road users’ route choices in response to toll charges are taken into account through the mechanism, based on bounded-rationality, embedded in the user behavior component of the network state estimation/prediction module in TrEPS. The Baltimore – Washington, DC corridor network is used as the test bed network with two lanes of the I-95 southbound converted to be the toll lanes. A series of simulation experiments is conducted and results are analyzed. The paper is organized as follows. First, the second-best pricing problem of interest is defined. In the third section, the methodologies used to implement the dynamic pricing strategies (or link toll generators) are described, followed by the experimental design and the simulation results. Finally, concluding remarks and discussion of possible future research are stated. 2. Problem statement The following notation is used to represent the variables and parameters in this paper. N A Td pmin(l) pmax(l) m m0 p(l, t) ~ðl; tÞ p P(t) ~ PðtÞ P c(l, t) ~cðl; tÞ C(t) ~ CðtÞ cðlÞ  C ^cðlÞ

a r

v(l) vf(l) uf(l) v0(l) cjam(l)

aVOT

set of nodes in the network (–) set of directed links (i, j), " i 2 N, j 2 N in the network (–) planning horizon in terms of time intervals (min) minimum toll on link l (USD) maximum toll on link l (USD) number of links in the network (–) number of links that can be tolled for a certain amount within the range of [pmin(l), pmax(l)] (–) toll value (price) charged for traveling on link l during time interval t (USD) anticipatory toll for traveling on link l during time interval t (USD) vector of link tolls during time interval t, consisting of p(l, t), l = 1, . . ., m0 (–) ~ðl; tÞ, l = 1, . . ., m0 (–) vector of predictive link tolls for time interval t, consisting of p 0 vector of static link tolls, consisting of p(l), l = 1, . . ., m traffic concentration on link l at time interval t (vpmpl) predicted traffic concentration on link l during time interval t (vpmpl) vector of link concentrations at time interval t, consisting of c(l, t), l = 1, . . ., m0 (–) vector of predicted link concentrations in time interval t, consisting of ~cðl; tÞ, l = 1, . . ., m0 (–) historical traffic concentration on link l averaged over the peak period (vpmpl) vector of average link concentration, consisting of cðlÞ, l = 1, . . ., m0 (–) preset target (nominal) concentration for link l (vpmpl) control factor (a parameter) (USD/vpmpl) prediction horizon in terms of number of time intervals (min) speed on link l (mph) speed-intercept (mph) free-flow speed on link l (mph) minimum speed on link l (mph) jam density on link l (vpmpl) value of time (USD/min)

This problem considers a traffic network G = (N, A). The entire period of interest (planning horizon) is discretized into small departure time intervals t = 1, 2, . . ., Td during which no perceptible changes in traffic conditions occur. The time-dependent OD trip desires for the planning horizon are known a priori. Further, a subset of links subject to tolls (i.e. toll links) in the 0 0 0 network, A , A # A, is given. A feasible continuous range of toll values for link l," l 2 A is defined as [pmin, pmax]. A constant value of time (VOT) is assumed for all users in the network, though user heterogeneity in VOT could be addressed by adapting the approach recently proposed by Lu et al. (2008). Given a set of preset target concentrations ^cðlÞ on toll links, the problem is to find a vector of time dependent link toll values P(t) (or P in the static case), so as to maintain high level of service on the toll links. Since only a subset of links is subject to pricing, the problem is considered as a second-best pricing problem.

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3. Methodology This section describes the theoretical rationale and simulation-based approaches for the dynamic pricing problem. Two dynamic pricing strategies are presented, namely reactive and anticipatory, for maintaining high LOS on toll lanes, in addition to the static strategy used as a benchmark in the comparison. The three strategies are implemented and evaluated in a simulation–assignment based real-time traffic estimation and prediction system (TrEPS), described following the strategies. 3.1. A theoretical analysis on a two-route network In previous studies, static second best price has been solved analytically and represented in a closed form for a simplified two-route network (e.g., Verhoef, 2002). As explained, this research is intended to solve for dynamic second best prices. Consider a simplified two-route network: one toll road, one alternative road. Without loss of generality, assume both routes have the same attributes. Time-varying travel demand is assigned to the network, which causes congestion (or, demand exceeds capacity) on one or both routes. Therefore, a set of time-varying prices is derived to maintain free-flowing condition (or some minimum level of service expressed in terms of speed, maximum density or flow) on the toll road as well as minimize total travel time. As there are only two alternative routes in this special case, the target flow assignment can be derived easily: when demand is low, traffic is distributed equally to maintain free-flow conditions on both routes; when demand increases, assign the maximum allowed flow (beyond which the service level would degrade below the target level) to toll road and the rest of the flow to the alternative road. In order to achieve the target flow assignment, time-varying tolls need to satisfy the following equilibrium condition, for each departure time interval t = 1, 2, . . ., Td:

T 2 ðtÞ ¼ T 1 ðtÞ þ

1

aVOT

 pðtÞ

ð1Þ

Assume a closed form representation of travel time as a function of flow rate (i.e. link performance function). Given the target flow assignment, the optimal dynamic tolls can be obtained as follows:

pðtÞ ¼ aVOT  ðT 2 ðtÞ  T 1 ðtÞÞ

ð2Þ

The result cannot be readily extended to a general network, because of the simplified topology considered here, with only one alternative to the toll road, and the simplistic flow-travel time relation. In addition, complete knowledge of demand variation is assumed to known a priori. Nonetheless, this example offers two important insights for developing practical dynamic pricing methodologies. First, as shown in Eq. (2), the dynamic prices can be determined based on traffic conditions on the toll road and the alternative road under the target assignment. However, noting that T 1 ðtÞ can be obtained as a function of the target flow q ðtÞ assigned to the toll road, and T 2 ðtÞ can be obtained as a function of the target flow assignment ½Q ðtÞ  q ðtÞ, where Q ðtÞ is the known total O–D demand, it becomes clear that the dynamic price depends primarily on the target assignment to the toll road. This is reflected in the procedure for setting reactive and anticipatory prices as a function of link concentration in the following sections. Second, the seemingly circularity that travel time at time t is an outcome of implementing current price but also determines the price at time t, motivates the development of an anticipatory pricing scheme, which evaluates the outcome (e.g. speed or concentration at time t) of the proposed price through prediction in an attempt to resolve the circularity. 3.2. Reactive pricing strategy Reactive pricing is set by continuously comparing the prevailing link concentrations, extracted from field measurements of detectors or simulation estimates of current network state, with the preset target link concentration values and adjusting the current link tolls accordingly so as to react to the prevailing network condition. As illustrated in Fig. 1, dynamic toll values are adjusted through the reactive link toll generator and communicated to travelers via local VMS (variable message sign) at the entry point, which could also be disseminated via radio, in-vehicle equipment, mobile, internet etc. We assume that the prices charged on toll links depend only on the link concentrations during the current time interval and the toll values in the last time interval, and can be expressed in the following general form:

PðtÞ ¼ F r ½Pðt  1Þ; CðtÞ

ð3Þ

Note that in this theoretical presentation, and with no loss of generality, we ignore processing and/or communication delays that may introduce time lags in computing the prevailing concentrations, and/or communicating the prices to tripmakers. As such, ‘‘current” concentration should be understood as the latest-available measured concentration values.

Toll generator

Toll values

Real-world traffic/Estimation

Traffic data Fig. 1. Reactive pricing framework.

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As a variant on Eq. (3), the current toll value for each link could be determined based on the previous tolls and the current concentrations on all or several related links, e.g. upstream links, downstream links, parallel links on alternative routes and the like. For simplicity, this study only considers a local toll generator, which assumes that the toll value on one link is independent of traffic states on other links. If the toll generator is assumed to be linear, a relatively simple closed-loop control mechanism can be applied, shown in Eq. (4):

pðl; tÞ ¼ pðl; t  1Þ þ a½cðl; tÞ  ^cðlÞ;

for l ¼ 1; . . . ; m0

ð4Þ

where a > 0 is a control parameter and the ^cðlÞ is the target concentration for link l, which is typically, but not necessarily set as cut-off concentration. In addition, the toll values are subject to a feasibility constraint pmin ðlÞ 6 pðl; tÞ 6 pmax ðlÞ. The toll generator uses current link concentration and adjusts the toll value according to the target concentration: raise the price if the concentration exceeds the target value, lower the price otherwise. The model reacts smoothly even to slight differences in concentration, and adjusts the current price based on the previous value. Alternatively, an S-curve (e.g. a logistic function) may be considered for adjusting toll values in a non-linear fashion. The rationale is to react swiftly to changes in concentration around the target point, so as to pull the system back to the steady state. The adjustment to the price follows this expression:

SðcÞ ¼ p1

eaðc^cÞ  p0 1 þ eaðc^cÞ

ð5Þ

where a, ^c, p0 and p1 are parameters. Specifically, a > 0 is a control parameter that determines the shape of the curve, ^c is the location parameter that could be set as the cut-off (or target) concentration, p0 is the offset value to set the price adjustment to zero when the concentration reaches the target value and p1 is the multiplier to keep the adjustment within the preset toll value range in order to satisfy the feasibility constraint pmin ðlÞ 6 pðl; tÞ 6 pmax ðlÞ. Note that a is the only control factor in the non-linear toll generator, while other parameters could be pre-determined based on the traffic flow model and pricing assumptions. The local non-linear toll generator is defined as follows:

pðl; tÞ ¼ pðl; t  1Þ þ Sðcðl; tÞÞ;

for l ¼ 1; . . . ; m0

ð6Þ

3.3. Anticipatory pricing strategy Unlike the reactive approach, anticipatory pricing takes into account not only current network conditions but also predicted conditions that reflect the evolution of traffic demand and flows in the network over the near future. As such, it offers the potential to prevent the occurrence of extreme congestion and breakdown, and hence to use the available freeway capacity more efficiently. Since the anticipatory pricing strategy calls for predicted link concentrations, a state prediction module is introduced, as shown in Fig. 2. The prediction module takes current (and past) network states as input, along with the previously predicted prices determined by the link toll generator (at the previous time interval), to predict the future network states. The resulting predicted link concentrations are fed back to the anticipatory toll generator, which adjusts the previously predicted toll values according to the future network states, and generates prices to be implemented in real-time (for the current time interval) as well as the anticipatory prices for future intervals, for use in the state prediction module. The procedure is shown in Fig. 2. The anticipatory pricing strategy can be expressed in the following form:

~ ~ þ 1Þ; . . . ; Cðt ~ þ rÞ ~ þ 1Þ; . . . ; Pðt ~ þ rÞ ¼ F a ½P ~ old ðtÞ; CðtÞ; ½PðtÞ; Pðt Cðt

ð7Þ

~ old ðtÞ denotes the previously predicted prices for the first interval; these are modified to obtain In the above expression, P the new (current) prices for the first time interval, P(t), which are then implemented in real-time. In general, the current and predicted toll values for each link could be determined jointly for all or several links (i.e. in an integrated manner, as discussed previously). In this study, a local anticipatory toll generator is defined, which uses a time-weighted combination of predicted link concentrations to fine tune the previous prediction. The linear anticipatory toll generator is expressed as follows:

~ðl; tÞ þ a pðl; tÞ ¼ p

r X

wi ½~cðl; t þ iÞ  ^cðlÞ;

for l ¼ 1; . . . ; m0

ð8Þ

i¼0

Toll values

Toll generator Predictive data

Traffic prediction

Real-world traffic/Estimation

Traffic data

Fig. 2. Anticipatory pricing framework.

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where the weights (wi’s) decay exponentially over future prediction time intervals in the following fashion, wi = q  wi–1, ~ þ qÞ; q = 1, . . ., r are determined in a similar manner, albeit over i = 1, . . ., r and 0 < q < 1. The future anticipatory prices Pðt a progressively shorter horizon, as follows:

~ðl; t þ qÞ ¼ p ~old ðl; t þ qÞ þ a p

rq X

for l ¼ 1; . . . ; m0 ; q ¼ 1; . . . ; r

wi ½~cðl; t þ q þ iÞ  ^cðlÞ;

ð9Þ

i¼0

~old ðl; t þ qÞ denotes the previous anticipatory price for the same interval. where p The non-linear toll generator for the anticipatory pricing strategy takes the following form, where the function S(.) is given by Eq. (5):

~ðl; tÞ þ S pðl; tÞ ¼ p

r X

! wi ½~cðl; t þ iÞ  ^cðlÞ ;

for l ¼ 1; . . . ; m0

ð10Þ

i¼0

and

~ðl; t þ qÞ ¼ p ~old ðl; t þ qÞ þ S p

rq X

! wi ½~cðl; t þ q þ iÞ  ^cðlÞ ;

for l ¼ 1; . . . ; m0 ; q ¼ 1; . . . ; r

ð11Þ

i¼0

3.4. Static pricing strategy The static pricing strategy sets fixed/flat link-specific tolls for a particular period of the day, typically derived off-line  and possibly other considerations. In this study, static pricing is used based on average link concentrations, i.e. P ¼ F s ½C, for benchmarking purposes, and is as such considered parametrically, with high and low values intended to illustrate the range of response behaviors. The approach assumes that the toll value on each link depends only on the concentration of that link, and the price-concentration relationship follows a linear function given in the following form:

pðlÞ ¼ minfpmax ðlÞ; max½pmin ðlÞ; aðcðlÞ  ^cðlÞÞg;

for l ¼ 1; . . . ; m0 :

ð12Þ

Hence, if the average concentration on link l exceeds the target concentration ^cðlÞ, a price up to pmax(l) is set proportionally to the difference between ^cðlÞ and cðlÞ, and pmin(l) is set otherwise. 3.5. Traffic estimation and prediction The real-time anticipatory pricing schemes require predicted concentrations (and/or associated traffic state descriptors) as an essential input. This capability is enabled through recent development of simulation assignment-based real-time traffic estimation and prediction systems (TrEPS), such as DYNASMART-X (Peeta and Mahmassani, 1995; Mahmassani, 1998, 2001; Mahmassani et al., 2004, 2005a; Mahmassani and Zhou, 2005) and DynaMIT-R (Ben-Akiva et al., 2002), which could be used for estimation/prediction of network states. The structure of TrEPS includes the following modules (Fig. 3): OD estimation, OD prediction, real-time network state simulation, consistency checking, updating and resetting functions, and network state prediction. An anticipatory toll generator is added to implement the anticipatory pricing strategy described above. The OD estimation and prediction module is first activated to predict the time-dependent OD pattern for the next (OD prediction) stage using historical OD information and observed/archived information from the surveillance system and/or vehicle probes (Mahmassani and Zhou, 2005; Zhou and Mahmassani, 2007). Simultaneously, the real-time traffic simulator, supported by a series of consistency checking and updating functions, estimates current network traffic conditions. The supporting consistency checking functions compare measured values of selected state variables in the actual system with the corresponding values in the simulator and update the internal representation within the simulator to ensure consistency

ATIS/ATMS Historical Database and Surveillance Data

OD Estimation

Consistency Checking/Updating Network State Estimation

OD Prediction

Network State Prediction

Anticipatory Link Toll Generator

Fig. 3. Traffic estimation and prediction system structure.

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with actual conditions (Doan et al., 1999; Zhou and Mahmassani, 2005). At the start of each state prediction stage, the predictor reads the current network conditions from the real-time simulator and uses the anticipatory time-varying O–D demands to predict network conditions over the prediction horizon. New predictions could be computed in real-time when a major disruption, such as an incident, is detected. The predicted states provide the basis for setting the anticipatory prices. In the experiments presented in the next section, comparison of the anticipatory to the reactive and static strategies is conducted using the same network simulation–assignment modeling platform. Essentially, the network state estimation module, which provides a mesoscopic simulation–assignment capability, is used to evaluate user responses to the prices and associated network states, including network performance measures, such as throughput and path travel times. In that framework, travelers receive the updated information and respond to it according to the behavioral rules built into the assignment–simulation procedures. Assume that travelers have access to up-to-date travel time and pricing information for all the feasible routes in their choice set. The criteria for the driver’s route choice include monetary cost (p) and travel time (T). Accordingly a generalized cost (GC) is specified in Eq. (13) to reflect these attributes.

GC ¼ p þ aVOT  T

ð13Þ

In this case, the response follows rules based on bounded-rationality, proposed by Simon (1955) in business decisions and subsequently adapted by Mahmassani and Stephan (1988) for modeling commuter route choice behavior (Mahmassani and Liu, 1999). Under this assumption, travelers switch routes if the improvement in the generalized cost exceeds a certain threshold (e.g. 20% savings compared to the original route) or minimum level (e.g. 1 USD savings). Feeding the users’ route choices to the simulator, the model generates the link concentrations and feeds back to the link toll generator for updating toll values. For the reactive strategies, the prevailing concentrations are used to set the prices, as shown in Fig. 4. Note that although in reality the calculated reactive link tolls may not be put in effect at the same time when prevailing link concentrations are collected due to some delays resulting from sensing, communication and state estimation times, this study assumes (with no loss of generality) that the pricing module could react to current network state immediately (i.e. those delays could be ignored). For the anticipatory strategies, the prediction module is activated to produce predictions of the network states, which are in turn provided to the anticipatory toll generator, as shown in Fig. 5. As network state forecast is subject to prediction errors and user response to updated prices might also invalidate the prediction, predicted traffic conditions can be viewed as a projection of likely future traffic evolution based on the current pricing scheme. However, the online nature of the price setting process in the rolling horizon framework provides frequent updates that continuously reflect measured conditions in generating next period forecasts, thereby reducing the impact and propagation of errors. Naturally, the technology used for estimation and prediction will likely improve as such systems are deployed in practice. Moreover, user interaction with dynamic tolls could be accounted for by introducing an iterative procedure between user evaluation and toll adjustment, though this will increase the computational burden for real-time implementation. The next section describes application of the above framework in the comparison of the proposed pricing strategies. 4. Experimental design A series of simulation experiments are designed and conducted to compare the performance of freeway managed lanes, especially the toll lanes, under the three pricing strategies described in the previous section. As explained, the evaluation of the pricing strategies is conducted with the DYNASMART-X simulation–assignment platform, with the addition of reactive and anticipatory link toll generators to the basic system structure. Specifically, to evaluate the static pricing strategy, the state estimation module is activated to simulate traffic conditions taking static link tolls as input. To emulate the reactive pricing strategy, the state estimation module and the reactive toll generator are activated and executed interactively. Finally,

Cˆ

Link toll generator

Route info.

En-route switching

Real-time C (t ) simulator

Fig. 4. Reactive pricing framework.

Fig. 5. Anticipatory pricing framework.

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for the anticipatory pricing strategy, the prediction modules are activated, in conjunction with the anticipatory toll generator, and the resulting prices are supplied to the simulation–assignment model (embedded in the state estimation module). As explained, a key motivation for implementing state-dependent pricing strategies is to maintain high LOS (or uncongested regime) along freeway toll lanes. The link concentrations obtained from the traffic flow model of the simulator are used as the main measures in determining the toll values. The calibrated dual-regime modified Greenshield’s model is applied in the traffic simulator to represent the freeway density–speed relationship.

v ðlÞ ¼ uf

if 0 6 cðlÞ 6 ^cðlÞ

v ðlÞ  v 0 ðlÞ ¼ ðv f ðlÞ  v 0 ðlÞÞ 

 a cðlÞ 1 cjam ðlÞ

ð14Þ

if ^cðlÞ < cðlÞ 6 cjam

where a is the power term and ^cðlÞ is the cut-off concentration which can be used as preset target concentration. When the traffic concentrations on toll links are kept below the cut-off value, vehicles are able to move at free-flow speed and therefore maintain high level of service on the freeway. Under this assumption, the traffic concentration on the links are defined as the observed variables (traffic state indicators) of the system and the toll values charged on the links are the decision variables to be determined or optimized so as to maintain the target LOS. 4.1. The test bed network Fig. 6 shows the test network used in this study, which is constructed based on the CHART (Maryland, United States) network. DYNASMART-X had been calibrated for this network using real world observations, obtained from multiple-day detector data (Fei et al., 2005; Mahmassani et al., 2005a). The network consists, primarily, of the I-95 corridor between Washington, DC and Baltimore, MD, and is bounded by two beltways (I-695 Baltimore Beltway to the north and I-495 Capital Beltway to the south). The network has 2241 nodes, 3459 links and 111 traffic analysis zones (TAZ). A 2-h morning peak dynamic O–D demand table estimated for the network is used in the experiments. In order to evaluate different pricing strategies, two of the 20-mile long southbound lanes of the I-95 corridor are converted to hypothetical toll lanes. The other two lanes of I-95 remain free. The major merge points to the I-95 corridor are from I-195, MD-100, MD-32 and MD-198, where the toll lanes access/egress points are specified.

Toll Lane Start I-95-MD-166 Junction

Toll Lane End I-95-I-495 Junction Intermediate Access/Egress Locations

Fig. 6. The CHART test bed network.

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4.2. Experimental factors and scenario design The simulation experiments conducted in this study focus on two primary experimental factors: (1) the pricing method and (2) the control factor a, a parameter which determines how the toll generator responds to the change in concentration. 4.2.1. Pricing method Three pricing strategies are tested: (1) static pricing, (2) reactive pricing, and (3) anticipatory pricing, against no pricing case. As discussed in the methodology section, dynamic toll values (both reactive and anticipatory) are determined according to real-time network state measurements (estimates) and/or predictions. Two static toll values are tested for benchmarking purposes. For the reactive and anticipatory pricing strategies, two types of toll generators are implemented: (1) linear toll generator and (2) non-linear toll generator, as specified in the previous section. Therefore, a total of six pricing scenarios are tested and compared against the base case in which no toll is applied on the freeway; that is, static-low, static-high, reactive-linear, anticipatory-linear, reactive-non-linear, and anticipatory-non-linear. 4.2.2. Control factor (a) The control factor a determines the rate at which link tolls are adjusted according to (prevailing or anticipatory) concentration deviations from the target ones. Although the control factor can be set specifically for different links, this study applies the same control factor for all links and time intervals. Moreover, similarly to what is suggested by the ramp-metering control literature, the control factor needs to be calibrated/optimized in order to achieve the desired nominal conditions, for example, link concentrations should be lower than the cut-off values. However, optimizing this control parameter is beyond the scope of this paper; instead, a range of a values were tested for each pricing strategy so as to produce toll values within the determined toll range, and the values that provided the best performance were selected. With the selected control factors, the pricing strategies could then be compared in terms of traffic performance on both toll lanes and regular freeway lanes. For all the experiments, we assume that all the users have access to pricing information posted on variable message signs and hence are able to evaluate their route at each toll road entry point (on-ramp) and to switch routes based on the specified route choice mechanism (based on boundedly rational behavior in this case). In order to compare the trip travel times the simulation is continued until the network is empty, that is, till all vehicles reach their destinations.

5. Simulation results The simulation results are presented in this section to illustrate and compare the effectiveness of the proposed pricing strategies, and test our motivating hypothesis that anticipatory pricing strategies have the potential to overcome the limitations identified in conjunction with static and reactive strategies. 5.1. Sensitivity analysis of control factor A sensitivity analysis with respect to the control factor a is conducted for each pricing strategy and is summarized in Table 1, where travel times for different pricing strategies and different control factors are presented. The performance of each strategy is evaluated on the basis of four measures: travel time on toll lanes (for the entire 20-mile stretch), travel time on regular lanes (for the entire 20-mile stretch), average OD (origin–destination) travel time for a selected major OD pair (Baltimore to Washington), and average travel time of all the vehicles in the network. The major OD pair selected not only accounts for heavy travel demand but also consists of the two areas located towards the end points of the studied freeway. For static pricing, a large a value generates a higher fixed toll on the corridor and hence leads to less congestion and lower travel time on the toll lanes. However, this is accompanied by severely degraded conditions on the regular freeway lanes, as well as inferior network performance due to under-utilization of the toll lanes capacity. On the other hand, a low toll encourages greater utilization of the toll lanes, resulting in better network-wide performance, but generating congestion along the toll road. Between these two extremes, dynamic pricing can regulate congestion levels on the toll facilities, provide better distribution of traffic and hence better overall performance. The fact that dynamic tolls generally yield much greater efficiency gains than static tolls has also been confirmed by Arnott et al. (1990) who analyzed various pricing regimes for a fixed demand on a network with parallel routes. For all dynamic pricing scenarios, no apparent relationship between a values and average travel times could be established in this network. Table 1 reveals that anticipatory pricing strategies always provide lower travel times on the toll lanes. Although low travel time on toll lanes is preferred, the other three travel time measures are also taken into consideration to determine the a value. In particular, the control factor (0.01 with linear controller) that minimizes the weighted summation of travel times on toll lanes (Ttoll), regular lanes (Tfree) and the network (Tnetwork) is selected for further comparison, that is,

a ¼ arg min ðx1  T toll þ x2  T free þ x3  T network Þ: a

ð15Þ

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J. Dong et al. / Transportation Research Part C 19 (2011) 644–657 Table 1 Sensitivity analysis results for various values of the control factor a.

a

Scenario

a

Toll

Regular

Major OD

Network

Free flowing No pricing

– –

18.2 37.45

18.8 35.49

20.0 46.10

– 21.68

Static (low) (high)

0.005 0.015

29.84 21.68

40.29 71.66

40.00 50.90

21.06 22.16

Reactive (non-linear)

0.01 0.011 0.012 0.013 0.014

28.22 25.25 27.31 27.35 27.09

36.09 43.94 37.17 36.61 36.23

38.10 41.50 36.90 38.10 35.60

22.70 21.80 22.57 22.45 23.13

Reactive (linear)

0.01a 0.02 0.03 0.04 0.05

26.36 26.34 28.02 28.31 27.27

36.74 37.70 36.00 41.26 44.73

38.10 42.80 38.50 42.80 54.60

22.70 21.36 20.81 20.82 21.43

Anticipatory (non-linear)

0.01 0.011 0.012 0.013 0.014

22.33 22.83 23.30 21.26 21.04

45.05 42.88 34.27 37.97 40.15

46.20 42.20 38.90 33.90 35.70

20.70 21.27 20.57 21.20 20.12

Anticipatory (linear)

0.01a 0.02 0.03 0.04 0.05

20.53 23.63 21.01 19.96 23.37

35.89 37.40 40.32 39.70 38.47

33.10 40.30 33.70 36.20 37.60

20.41 21.84 19.84 20.77 22.36

Travel time (min)

Indicates the chosen a value.

5.2. Sensitivity analysis of prediction error As explained, the benefit of anticipatory pricing strategy depends on the accuracy of the prediction of network states. This section explores the impacts of prediction errors on the performance of anticipatory pricing strategy. In the TrEPS framework, prediction error in demand largely determines the inaccuracy in link concentration prediction and thus affects the efficiency of the corresponding pricing scheme. Therefore a set of experiments is conducted to study the impacts of overestimated/underestimated demand. Table 2 shows that average network travel time is only slightly affected by the demand prediction error. More significant impact is observed on toll road performance. However, with ten percent of prediction error in demand, the anticipatory pricing strategy still outperforms the reactive one (as shown in Table 1).

5.3. Comparison of performance measures for different pricing strategies To compare the different pricing strategies, time-dependent travel times, average concentration and throughput, on both the toll lanes and the regular lanes, are examined. First, Fig. 7 presents the time-varying toll values for the different pricing strategies (static pricing with high toll value, static pricing with low toll value, reactive pricing and anticipatory pricing). As shown, with the same control factor (a value), the anticipatory pricing strategy charges relatively lower and less volatile tolls compared to the reactive case, during most of the time period as it takes action before queue build up and maintains a relatively stable flow. In particular, at the beginning of the rush period (first 20 min), anticipatory prices foresee the congestion and discourage travelers from choosing toll lanes to prevent traffic breakdown, while reactive prices remain zero, resulting in congestion shortly after. Similarly, at the end of the rush period (after 140 min), the anticipatory price drops to zero to attract

Table 2 Sensitivity analysis results for various demand prediction errors. Demand prediction error Overestimate Demand Underestimate Demand

0 5% 10% 5% 10%

Travel time toll (min)

Increase (%)

Travel time network (min)

Increase (%)

20.53 21.18 23.21 21.80 22.42

– 3.15 13.04 6.20 9.18

20.41 20.68 20.13 20.86 21.55

– 1.32 1.40 2.22 5.61

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Fig. 7. Time-varying toll values for each pricing strategy.

Fig. 8. Time-varying path travel time on: (a) toll lanes and (b) regular lanes.

more travelers to use the toll lanes as congestion is expected to be eliminated by the time those travelers are actually traveling the toll lanes. Note that a damping element could be applied for real world implementation, which avoids changing the toll drastically for consecutive time intervals. Fig. 8 shows the time-varying travel times of the vehicles: (a) traveling on the toll lanes and (b) on the regular lanes. As shown in Fig. 8a, the anticipatory pricing strategy provides low and steady travel times on the toll lanes, indicating that the

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desired traffic flow conditions (uncongested flow) are attained and maintained to a large extent. Although static pricing with a high toll value also provides relatively high LOS (in terms of travel time) on the toll lanes, it dramatically degrades the LOS on the alternative freeway lanes, as shown in Fig. 8b. Furthermore, as seen in Figs. 9 and 10, the throughput of the toll lanes under the high static toll is quite low, reflecting low utilization. It is intuitive that the travel time on the toll lanes would be longer when no toll is charged than when pricing strategies are implemented, as is confirmed in Fig. 8a. One might also expect that the travel time on alternative regular lanes would be lower under the no pricing case than in pricing cases as pricing limits utilization of the toll road and diverts more traffic to regular lanes. This is indeed the case when static pricing strategies are implemented, especially with a high toll value. However, with dynamic pricing, traffic conditions on the regular lanes do not deteriorate compared with the base case, as shown in Fig. 8b. The explanation lies in the nature of traffic congestion dynamics. Without pricing, the freeways are congested and experience breakdown early on, resulting in highly unstable flow, and longer travel time on all lanes. Anticipatory dynamic pricing plays a metering effect on both the toll lanes and the regular lanes, as reflected in the traffic state descriptors discussed next. Fig. 9 depicts the average concentration and throughput on both toll lanes and regular lanes are examined as well. The concentrations are averaged over links along toll/regular lanes, weighted by the link length. Fig. 10 shows the throughput measured at a selected point along the freeways. When charging a fixed high toll, the toll lanes experience low concentration and low throughput, which indicates under-utilization of the toll road. This also results in relatively high concentration on the regular lanes, which explains the high travel time in Fig. 8b. Both dynamic pricing strategies and static pricing with a low toll value can maintain high utilization of toll lanes without worsening traffic conditions on the regular lanes. In particular, the anticipatory pricing strategy could detect the onset of congestion in advance and adjust prices accordingly so as to prevent the concentration from increasing dramatically, while maintaining a relatively higher and more stable throughput.

Fig. 9. Comparison of the average concentration on: (a) toll lanes and (b) regular lanes.

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Fig. 10. Comparison of the throughput on: (a) toll lanes and (b) regular lanes.

6. Conclusion This paper has proposed the notion of anticipatory (dynamic) pricing, and investigated the advantages of using predicted traffic conditions over the use of prevailing and/or historical conditions in setting time-varying link tolls along a freeway corridor to maintain target LOS and avoid traffic breakdown on toll links. This is accomplished through an anticipatory toll generator intended to operate in tandem with a real-time traffic estimation and prediction system. Simulation experiments were performed to compare the proposed anticipatory pricing strategies to reactive as well as static pricing schemes. The results confirmed the motivating hypothesis for this work, namely that setting prices on the basis of predicted conditions can make a substantial difference in terms of achieving the objectives of pricing in managed-lane situations. The results have also clearly highlighted the reasons that motivate the need for prediction, which are rooted in the nature of traffic congestion dynamics and the breakdown phenomenon. Through prediction, it is possible set prices at suitable levels to avoid or delay the onset of traffic breakdown into unstable regimes. The results presented here also indicate that in addition to preventing congestion from occurring and providing a stable LOS on the tolled lanes, anticipatory dynamic pricing in the cases considered did not noticeably worsen the LOS on regular lanes. The present paper has illustrated the basic concepts of anticipatory pricing, and its potential in light of current traffic and estimation prediction systems. However, improvement is possible in both the logic/specification used to set the prices given prediction output, as well as the quality of the prediction system itself. For instance, the process for determining the control parameters of the price mechanism could be refined, and eventually conducted online, adaptively. In addition, assumptions about user behavior and heterogeneity of user responses to pricing in the prediction process could be readily incorporated in the prediction framework. While such assumptions would undoubtedly improve the realism of the models and quality of

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