A RAMP-METERING ALGORITHM BASED ON PROBABILISTIC BREAKDOWN by Peng Lin Office of Highway Development Maryland State Highway Administration 707 North Calvert Street Baltimore, MD 21202 U.S.A Phone: (410) 545-8016 Email:
[email protected]
Lily Elefteriadou, Ph.D. Transportation Research Center University of Florida 512 Weil Hall Gainesville, FL 32611 U.S.A Phone:(352) 392-9537, Ext. 1452 Fax: (352) 392-3394 Email:
[email protected] Alexandra Kondyli1 Transportation Research Center University of Florida 511 Weil Hall Gainesville, FL 32611 U.S.A Phone:(352) 392-9537, Ext. 1537 Fax: (352) 392-3394 Email:
[email protected]
Submitted for presentation and publication to the Transportation Research Board July 2005 Number of words: 4,466 + (6 Tables + 6 Figures =) 3,000 = 7,466
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Corresponding Author
Lin, P., Elefteriadou, L., and Kondyli, A.
ABSTRACT This paper proposes a new algorithm for ramp-metering that considers the probabilistic nature of the freeway breakdown occurrence. The procedure is based on obtaining the optimum ramp-metering rate based on the probability of breakdown occurrence. An illustrative algorithm was developed for a freeway-merge site in Toronto, Canada, and was tested using the CORSIM simulation model. First, speed and volume data were obtained and analyzed to develop the breakdown probability model for the study site. Next, a methodology was developed to estimate the minimum ramp-metering rate that would prevent the ramp queue from spilling back to the surface street. Finally, to safely explore the algorithm effectiveness, simulation was used to obtain preliminary indications on the capabilities and potential drawbacks of the algorithm developed. The CORSIM simulation model was selected to conduct the necessary experiments. It was tested and found to adequately replicate the process of breakdown at the study site. The simulator was then used to test the effectiveness of the probabilistic ramp-metering algorithm, and to compare it to several other algorithms. It was concluded that the proposed algorithm is very promising both in reducing the probability of breakdown and congestion, and in reducing the overall delay to motorists.
Thorough testing and evaluation of the algorithm however would be
required before the algorithm can be implemented in the field, including its effectiveness along an entire freeway facility, or freeway/arterial corridor.
1.
INTRODUCTION
Ramp metering is a method of restricting traffic flow from the on-ramp to the freeway, and it has been used to help reduce traffic congestion and minimize delay on the freeway mainline. Ramp metering is usually employed during the peak period. During this period, ramp control restricts the ramp flow by imposing periodic delays onto vehicles entering from the ramp. Traffic operations along the adjacent surface streets may be adversely affected as a result of queues forming on the ramp.
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Lin, P., Elefteriadou, L., and Kondyli, A.
The ramp-metering algorithms developed to-date rely on a deterministic estimation of capacity and transition to congested flow. Implied in the definition and understanding of deterministic freeway capacity is the notion that the facility will "break down" (transition from an uncongested state to a congested state) when demand exceeds a specified capacity value. Elefteriadou et al (1995) showed however that breakdown does not necessarily occur always at the same demand levels, but can occur when flows are lower or higher than those traditionally accepted as capacity. In that research it was observed that, at the same site and for the same ramp and freeway flows, breakdown may or may not occur, and a model was developed to predict the probability of breakdown as a function of the ramp and freeway demand distributions. Similarly to that research, Evans et al (2001) also developed a model for predicting the probability of breakdown at ramp freeway junctions, which was based on Markov chains. In a subsequent paper, Lorenz and Elefteriadou (2001) conducted an extensive analysis of speed and flow data collected at two freeway-bottleneck-locations in Toronto, Canada, to investigate whether the probabilistic models previously developed replicated reality. The data confirmed that there is wide variability in the flows under which breakdown and the transition to congested flow occurs. Persaud et al (2001) also investigated the breakdown phenomenon, and quantified the probability of breakdown as an increasing function of volume at a ramp-metered site. They also investigated, using simulation, the use of fixed and variable-rate metering in increasing the throughput for a freeway system in Canada. The research described in this paper builds upon the research conducted by Persaud et al (2001), and further investigates the topic of ramp-metering as a function f the probability of breakdown. The main objective of this paper is to develop, and test using simulation, a ramp-metering algorithm for a single ramp junction, that considers the probabilistic nature of breakdown. Such an algorithm has the potential to increase the freeway and ramp throughput by postponing or avoiding breakdown and/or decreasing travel delays on the freeway mainline, without providing unnecessary restrictions on the ramp flow. The algorithm considers the available ramp storage to prevent ramp queue spilling back to the surface street. This paper describes and simulates the algorithm for a single ramp, however the methodology described can relatively easily be extended to consider several ramps.
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Lin, P., Elefteriadou, L., and Kondyli, A.
Section 2 of the paper presents an overview of the proposed algorithm, while section 3 discusses the breakdown distributions obtained from field data, and the breakdown probability model developed to be used in the algorithm. Section 4 includes the results of the investigation into CORSIM to assess its capability to replicate the process of breakdown, the results of the simulation testing of the algorithm and comparisons to other algorithms. Section 5 presents the conclusions of the research as well as recommendations for further enhancement and testing of the algorithm.
2. OVERVIEW OF THE ALGORITHM
The main characteristic of the proposed Probabilistic Ramp-Metering Algorithm (PRMA) is that it can select appropriate ramp-metering rates as a function of a preselected probability of breakdown. The algorithm determines the optimum metering rates in real time as a function of traffic demands and a pre-selected acceptable level of breakdown probability. It estimates the appropriate metering rate every 20-seconds, based on the demand in the previous 1-minute interval. The PRMA flowchart is shown in Figure 1. The first step in the algorithm is to select an acceptable probability of breakdown, P(B), and use the breakdown probability model for the subject ramp to select the corresponding ramp metering rate. The breakdown probability model provides P(B) as a function of freeway and ramp flows; the ramp flow can be estimated when the approaching freeway demand and desired breakdown probability are known. For each time interval t, the algorithm selects the ramp-metering rate, Vr (t) to maintain a given P(B), and compares it to the minimum rate. The minimum ramp-metering rate, Vr (Min), is defined as the rate that prevents the ramp queue from spilling back to surface streets. If Vr (t) > Vr (Min), the algorithm implements Vr (t). If Vr (t) < Vr (Min), then Vr (Min) would be implemented, and the corresponding P(B) would be estimated. A warning may be provided to the operator if P(B) exceeds a pre-specified level. The concept of breakdown probability distributions and the method for estimating the minimum ramp-metering rate are discussed in more detail in the remainder of this section.
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Lin, P., Elefteriadou, L., and Kondyli, A.
2.1 Concept of Freeway Breakdown Probability Distributions
The probability of breakdown at a site is a function primarily of the freeway and ramp demands, but also drivers’ characteristics and actions, and other prevailing conditions, such as weather, etc. (Elefteriadou et al, 1995). The breakdown probability is defined in this paper as the frequency of breakdown at a certain flow level compared to all occurrences of that flow during the studied time period (Lorenz and Elefteriadou, 2001). Figure 2 shows a series of hypothetical breakdown distributions for a site. Each curve indicates the probability of breakdown for a given ramp-metering rate as a function of the freeway demand. The probability of breakdown increases with increasing freeway demand, and with increasing ramp-metering rate. After selecting an acceptable breakdown probability and obtaining the freeway demand, the ramp-metering rate can be determined for time interval t. For example, using the hypothetical curves of Figure 2, for a desired breakdown probability of 0.1 and a freeway demand of 40 veh/min, the corresponding ramp-metering rate is 10 veh/min. A series of curves such as the ones shown in Figure 2 are developed for the study site and used as part of the proposed algorithm. The development of these curves using field data is discussed in Section 3.
2.2 Method for Determining the Minimum Ramp-Metering Rate The minimum ramp-metering rate required to avoid queue spillback on the ramp junction is estimated as a function of the prevailing ramp traffic demand and the available storage, based on the M/G/1 queuing system (Gross and Harris, 1998). Note that the selection of the M/G/1 queuing system does not affect the methodology and the algorithm proposed; the most appropriate queuing system could be selected and applied based on the prevailing conditions of each ramp. Table 1 gives the minimum ramp-metering rates estimated for various ramp traffic demands and maximum allowable queue lengths. The shaded area of the table indicates non-feasible ramp metering rates. The maximum rampmetering rate that can be implemented is approximately 15 veh/min (900 vph) because the minimum departure time headways are approximately 4 sec/veh.
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3. DEVELOPMENT OF THE PROBABILISTIC BREAKDOWN MODEL USING FIELD DATA
Detailed speed and volume data were obtained through the Ministry of Transportation of Ontario for a non-metered freeway-ramp junction along the Highway 401 freeway system in Toronto, Canada. A total of 30 days of data (which included 38 breakdown events) were obtained and analyzed. The site is a fairly common freewayramp junction configuration with a one-lane ramp merging to a three-lane freeway mainline. The distances from this ramp to the downstream on-ramp and upstream offramp are greater than 1500 ft, thus it can be considered to be an isolated freeway-ramp junction according to the HCM definition (HCM, 2000). The ramp junction is considered to be an active bottleneck as in this location there are observations of flow breakdowns and upstream queue formations (Daganzo et al., 1999). A sketch of the study site including the location of data detectors is shown in Figure 3. The posted speed limit on the 401 Freeway is 100 km/h (approximately 62 mph). Free-flow speed during the off-peak time periods was found to range between 100 km/h and 120 km/h (approximately 62 mph to 75 mph). Traffic volumes on the 401 Freeway are generally heavy during the weekday morning (about 7:00am to 9:00am) and afternoon peak periods (about 4:00 pm to 6:00 pm), and breakdowns in traffic flow occur often during these times. Paired detectors installed on the freeway and ramp, provide vehicle counts and speed estimates continuously at 20-second intervals.
3.1 Definition of Breakdown Event and Breakdown Flow
This paper uses the following freeway breakdown definition, as developed in Lorenz and Elefteriadou (2001): Breakdown occurs when the speed of the freeway drops below 90 km/h for a time period of at least 10 minutes. This definition was developed based on time-series plots of speed and flow data at two sites along the 401 Freeway in Toronto, Canada, to identify breakdowns and to distinguish between short-term “recoverable” speed drops and speed drops that resulted in congested operations.
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The breakdown freeway and ramp flow rates are defined as the per-lane, equivalent hourly freeway flow rate and ramp flow rate that occur during the one-minute time interval immediately prior to the breakdown occurrence on the freeway. As an example of a breakdown event, the one-minute period and the corresponding breakdown volumes and ramp flow rate for the Oct. 29, 1997 data are shown in Figure 4. Based on the time series speed plot of Figure 4(a), the three data points that represent the breakdown volumes are highlighted in Figure 4(b). These points represent 20-second intervals that occur immediately prior to the sustained speed drop to under 90 km/h. The observed equivalent hourly flow rate (averaged across all three lanes) corresponding to all three data points is noted in Figure 4(b) as well.
3.2 Data Analysis and Breakdown Probability Distributions
To develop the breakdown probability distributions for the ramp-metering algorithm, the breakdown freeway and ramp flow rates were extracted from the field data. Table 2 presents the breakdown probabilities estimated for 5 groups of ramp volumes. The breakdown probabilities were determined for groups of ramp volumes and freeway volumes because the number of breakdowns available in the database didn’t allow for consideration of each individual volume level. The probability of breakdown for a given combination of freeway and ramp flow rates is the relative frequency of these breakdown flow rates compared to all occurrences of these flow rates (column 6 in Table 2). These later frequencies of each pair of freeway and ramp flow rate combination were obtained for pre-breakdown conditions during the 30 days of data analyzed. The breakdown frequency for a given pair of freeway and ramp flow rates is the number of breakdowns occurring at this pair of flow rates divided by the total number (breakdown plus no breakdown) of this pair of flow rates. This can be expressed as:
N
(V , V ) Breakdown fi ri P(B ) = i N (V , V ) + N No − Breakdown fi ri Breakdown
where: 6
(V
fi
,V
ri
)
Lin, P., Elefteriadou, L., and Kondyli, A.
P(Bi) ------Breakdown probability of ith pair of freeway and ramp flow rate (Vfi ,Vri ), i = 1,2, … ,n Vfi ,Vri ---- The ith pair of freeway flow rate and ramp flow rate. N Breakdown (V fi , Vri ) ----Observed number of breakdowns at ith pair of freeway and ramp flow rate (Vfi ,Vri ) during pre–breakdown traffic conditions. N No − Breakdown (V fi , Vri ) ----Observed number of ith pair of freeway and ramp flow rate (Vfi ,Vri ) that didn’t result in breakdown, obtained during pre– breakdown traffic conditions.
The estimated breakdown frequencies (column 6 in Table 2) may at first glance appear erroneous. For example, when the ramp volume is 12-14 veh/min, the breakdown probability for freeway flow rate Vf = 2300veh/h is less than the breakdown probability of Vf = 2100veh/h (2.5% vs. 1.79%), which seems unreasonable. This seeming inconsistency however reflects the fact that the probabilities obtained as described above, are conditional probabilities, i.e., they were observed and occur on the condition that breakdown didn’t already occur at a lower flow rate. The fact that lower frequencies of breakdown are reported for the higher flows reflects that flows seldom reach these levels without having reached breakdown. These observed breakdown frequencies should thus be used and interpreted as conditional breakdown probabilities.
The conditional breakdown probability is further illustrated in the tree diagram of Figure 5. The following variables are defined: (Vfi , Vri) ---- The ith group (or pair) of freeway and ramp flow rates, where i represents the time interval. Bi ---- The event that breakdown occurs at ith group (or pair) of freeway and ramp
flow rates.
NBi
---- The event that no breakdown occurs at ith group (or pair) of freeway and ramp flow rates.
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Lin, P., Elefteriadou, L., and Kondyli, A.
Figure 5 shows that during each time interval there is a certain probability of breakdown associated with the freeway and ramp flows. It also shows that breakdown would occur only on the condition that it didn’t occur previously. The following probabilities are defined:
P(Bi|NB1∩NB2∩…∩NBi-1)
-----The conditional breakdown probability that breakdown occurs at ith group of freeway and ramp flow rates, given that no breakdown occurred previously.
P(NBi|NB1∩NB2∩…∩NBi-1) -----The conditional no-breakdown probability that no
breakdown occurs at ith group of freeway and ramp flow rates, given that no breakdown occurred previously. Thus, the observed breakdown frequencies at ith group of freeway and ramp flow rates as defined above and recorded in Table 2, column 6 represent conditional probabilities:
P(Bi|NB1∩NB2∩…∩NBi-1)
The numerical values of the probabilities of breakdown for each interval shown in Figure 5 cannot realistically be obtained from field data, as the observed breakdown and its associated probability are a function of the occurrence and frequency of previous combinations of freeway and ramp demands, which are too numerous to consider. Simulation may be a more feasible alternative for obtaining these probabilities. If we assume that the flow levels at the study site follow reasonably similar trends prior to reaching breakdown (a reasonable assumption based on the data plots shown in Figure 4), and also that the flow is only increasing in time, i.e.:
(Vf1 , Vr1) < (Vf2 , Vr2) < …< (Vfi , Vri) < …
