Adaptive Speed Estimation Using Transfer ... - Semantic Scholar

Transportation Research Record 1783 ■ Paper No. 02-2753

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Adaptive Speed Estimation Using Transfer Function Models for Real-Time Dynamic Traffic Assignment Operation Nhan Huynh, Hani S. Mahmassani, and Hossein Tavana The application of a transfer function model (TFM) in a real-time dynamic traffic assignment system is investigated. The motivation is to improve the speed estimation method to enable better system consistency with reality in real-time operation. The study is conducted by adopting a TFM derived from actual detector data in San Antonio, Texas. This model is then used in the traffic simulation module of the DYNASMART-X dynamic traffic assignment system to update the network link speeds. A nonlinear least-squares optimization algorithm, implemented for this study, is coupled with DYNASMART-X to enable adaptive estimation of the TFM parameters. Simulation-based experiments are carried out on the Fort Worth test network. These experiments are designed to evaluate the TFM performance and to gain insight into its operational properties under different conditions. The results show that the TFM, both adaptive and nonadaptive, can consistently approximate the true underlying speed–density dynamics. Of significant importance is the transferability and robustness of TFMs in different settings. The outcome of this research substantiates the premise that good speed estimation can be achieved through the use of TFMs.

Dynamic traffic assignment (DTA) model systems are recognized as potential tools to support the operation of advanced traffic management and information systems (ATMIS) (1–4). A primary concern with real-time DTA systems is their ability to estimate and predict actual network traffic conditions over time. The inconsistency problem, defined as the divergence of the predicted system status from the actual network conditions, stems from the inability of the system to estimate the actual unfolding conditions and subsequent propagation of discrepancies introduced by fluctuations in the actual system and imperfect model representation. Maintaining consistency is of significant importance because it directly affects the quality and effectiveness of the advisories and control strategies supplied by the real-time DTA system. Several inherent sources of error, both structural and random in nature, can prevent any system from perfectly estimating the actual conditions: 1. 2. 3. 4. 5.

Time-dependent origin–destination (O-D) demands, Incidents and other disruptions, Path selection and other assumptions concerning user behavior, Traffic modeling and propagation, and Online measurement errors.

These sources of error, although not exhaustive, exemplify the multitude of factors that can cause the performance of any real-time sysN. Huynh and H. Mahmassani, Department of Civil Engineering, University of Texas at Austin, ECJ 6.2, Austin, TX 78712. H. Tavana, Continental Airlines, Operations Research Group, 1600 Smith St., Suite 918D HQSRT, Houston, TX 77002.

tem to deteriorate. Consistency checking and updating of the model values and underlying model parameters are essential functions in an online estimation and control process and are therefore needed in order to maintain the ability of the real-time DTA system to estimate actual conditions to a certain degree of accuracy. Several approaches have been proposed and investigated to a limited extent in this area of online consistent DTA. Peeta and Bulusu sought to develop consistency in terms of minimizing the deviations of the predicted time-dependent number of users on each path from the corresponding actual number of users (5). He et al. addressed calibration of three components for the analytical DTA (6): 1. Dynamic link travel time functions, 2. Route choice model, and 3. Flow propagation. Kang examined both short-term and long-term adjustment methods (7). The short-term method was intended to reduce the discrepancy between the estimated and actual link density by changing the simulator link speed. The long-term method aimed at adjusting the O-D demand on the basis of the differences between the estimated and actual number of vehicles in the network. Doan et al. explored an alternative approach to correcting the simulator link speed (8). This updating methodology is based on the widely used proportionalintegral-derivative (PID) control scheme. A second method for ensuring online consistency was also proposed by minimizing the errors in the time-dependent demand used by the simulator in the previous roll period [in a rolling-horizon framework (9)]. Although the aforementioned studies individually and collectively make important contributions for this emerging class of problems and provide useful approaches to the application of online simulation and operational dynamic network models, several alternative and complementary avenues are available to address online consistency issues. The previous approaches have primarily relied on the numerical correction of the discrepancies, which is appropriate when the discrepancies are due to essentially random fluctuations. However, there are other approaches that recognize explicitly structural deficiencies of the underlying model representations because of incomplete specification of the complex dynamic traffic and behavioral phenomena under way in the actual system. One such shortcoming is the utilization of a static speed–density relation, in particular, the modified Greenshields model. Since the modified Greenshields model (and, by definition, any equilibrium speed– density relation) does not explicitly capture serial correlation and other dynamic effects in the speed–density time series data, it will overestimate or underestimate the actual speeds (10). The divergence between the estimated and actual speeds will require the system to make continual correction. Such periodic correction is an

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inefficient use of the system’s computational resources and does not guarantee accurate predictions into the future. If, however, the actual speeds can be more accurately estimated, little or no correction is needed to administer the traffic modeling errors. Moreover, such capability would help stabilize the system and improve the accuracy of its prediction and the quality of the resulting control actions. The focus here is specifically on the traffic modeling aspect of the DTA system. The application of a transfer function model (TFM) in place of the simple first-order continuum model is investigated to better estimate the prevailing speeds in the network. This research is motivated by the promising results found in the initial exploratory work of Tavana and Mahmassani, in which it was shown that the derived TFM outperformed a variety of equilibrium speed–density relations in different settings (10). In addition, there are several advantages of the TFM over the widely used simple first-order continuum model. First, its specification explicitly recognizes the time-series nature of the data stream and the underlying dynamic characteristics of the process. Second, it circumvents the need to derive and calibrate a speed–density relation for each individual link in the network. Last, its apparent transferability and robustness would be attractive for real-time operational support systems that manage multiple networks. This research extends the previous work (10) in several facets. First, the TFM is incorporated into a DTA simulation-based framework. Second, the estimation of speeds using the TFM is implemented as an adaptive process, in which the model parameters are updated online on the basis of prevailing traffic conditions. Last, a nonlinear least-squares optimization procedure is incorporated into the DTA system to enable the estimation of the TFM parameters online. The TFM adopted in this study is described in the next section. The methodology section describes the algorithm implemented and the simulation framework used to carry out the experimentation. The experimental design is explained, and then the results are presented and analyzed. Finally, the paper concludes with a summary, discussion of limitations, and remarks about ongoing and future work.

BACKGROUND In this study, the functional specification of the TFM derived using actual detector data in San Antonio, Texas, in the previous work (10) is adopted. A discussion of the model is given below. The reader is encouraged to consult other work (10–12) for information on TFMs, nonlinear least-squares estimation, and Gauss–Newton methods. Some rudimentary understanding of these concepts is helpful in understanding the material presented in this section and the next. The functional form of the TFM identified in the previous study (10), which is also used in this research, is as follows. It defines the relationship between the inputs to a system and its outputs. In this context, the input is density (denoted x) and the output is speed (denoted y). yt +1 = ωxt +1 + at +1 − θat

(1)

where xt +1 = ∇Xt +1 = ∇k = kt +1 − kt yt +1 = ∇Yt +1 = ∇uˆ = uˆt +1 − uˆt In terms of û and k, which denote speed and density, respectively, Equation 1 becomes

uˆt +1 = uˆt = ω( kt +1 − kt ) + at +1 − θat

(2)

where at+1 at ût+1 ût ut

= = = = =

ût+1 − ut+1, ût − ut, estimated speed for current interval, estimated speed for previous time interval, and measured speed for previous time interval.

The noise term at+1 is typically set at its expected value, zero. Equation 2 implies that the estimated speed at the current time interval (ût+1) depends on the estimated speed in the previous time interval (ût), the change in density (kt+1 − kt), and the difference between the estimated speed and its corresponding actual value in the previous time interval (ût − ut). Essentially, the estimated speed adjusts itself to compensate for the rise and fall in the density level on the link and the discrepancy between the previously estimated speed and its actual value. Moreover, the change in the estimated speed depends on the parameters ω and θ, which can be obtained using nonlinear least-squares estimation. The idea is to fit Equation 2 to a set of timeseries speed–density data with the objective of minimizing the noise. The values of ω and θ that yield the minimal noise are called least-squares estimates. To solve for ω and θ in Equation 2, the Gauss–Newton algorithm suggested by Box and Jenkins (11) is implemented. This algorithm enables the estimation of ω and θ in a real-time implementation. In the work by Tavana and Mahmassani, the estimation was performed offline using a spreadsheet’s optimization utility (10).

METHODOLOGY Most algorithms for the least-squares estimation of nonlinear parameters have followed one of two main approaches. In the first approach, the model is expanded as a Taylor series and the corrections to the parameters are made (at each iteration) on the basis of the assumption of local linearity (13). The second approach consists of various modifications of the steepest-descent method. The algorithm implemented in this study belongs to the former approach. Its main difference from the traditional Gauss–Newton method is that the derivatives are not solved for explicitly. In the interest of readability, the mathematical details are not presented; these can be found in a standard reference (11). The general structure of the algorithm, along with the mathematical notation, is given in Figure 1. The notations shown in Figure 1 are adopted from Box and Jenkins (11). Note (ω) (φ) (θ) that the terms d (δ) i,t, d j,t, d g,t, and d h,t denote derivatives, whereas di denotes step size. To evaluate the effectiveness of the TFM in estimating the actual speeds, Equation 2 is augmented to the traffic simulation module of a DTA system, coupled with the algorithm shown in Figure 1. This research employs the DYNASMART-X real-time DTA system to conduct the experimentation (1, 2). For the study, only the traffic simulation module of the DYNASMART-X system and its realtime architecture are retained. Figure 2a shows the structure of the simulation environment. There are two algorithmic modules in the system. One is the traffic simulation and the other is the nonlinear least-squares estimation algorithm (NLSEA). The traffic simulation module is primarily the DYNASMART simulator that aims to estimate the current traffic status of the network (9, 14, 15). The one modification made to

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Let

β 0' = (δ1,0 ,..., δr ,0 ;ω0,0 ,...,ωs ,0 ;φ1,0 ,..., φp ,0 ;θ1,0 ,..., θq ,0 ) be the initial guess

Compute [at δ1,0 ,..., δr , 0 ;ω 0, 0 ,...,ωs , 0 ;φ1,0 ,..., φp , 0 ;θ1,0 ,...,θq ,0 ] [at δ1,0 + d1 ,..., δr ,0 ;ω 0, 0 ,...,ωs , 0 ;φ1,0 ,...,φ p ,0 ;θ1,0 ,...,θ q ,0 ] [at δ1, 0 ,..., δr ,0 + d r ;ω 0,0 ,...,ω s ,0 ;φ1,0 ,..., φp , 0 ;θ1, 0 ,..., θq , 0 ] and so on ...

Calculate derivatives: d i(,tδ ) , d (jω,t ) , d g(φ,t) and d h(θ,t )

d i(,δt ) =

(

[a δ [a δ

For example:

t

1, 0 ,..., δi , 0 ,..., δ r , 0 ;ω 0 , 0 ,...,ω s , 0 ;φ1, 0 ,..., φp , 0 ;θ1, 0 ,..., θq , 0

t

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]

−

,..., δi , 0 + d i ,..., δr ,0 ;ω 0,0 ,...,ωs , 0 ;φ1, 0 ,..., φp ,0 ;θ1, 0 ,...,θq , 0

Update parameters:

]

)

/ di

δi , ω j , φg , and φ k

For example:

Σ [a ]d n

δ i δ i ,0 = −

t =0 n

t ,0

(δ ) i ,t

Σ (d )(d ) t =0

(δ ) i ,t

(δ ) i ,t

Convergence of all parameters? No

For example: δ i − δi ,0 ≤ 0.001

Yes Stop

FIGURE 1 Nonlinear least-squares estimation algorithm (11).

DYNASMART for this study is the replacement of the modified Greenshields model by the TFM (Equation 2) to update the speed on a chosen link. Note that Equation 2 is used in the manner of shortterm forecasting to predict the real-time speeds (ût+1): the incoming measured speed (ut) is actually an average speed for the past time interval. For comparison, the speeds on all other links remain to be updated every simulation interval (6 s) by the modified Greenshields model, which has the following form, as discussed extensively by Jayakrishnan et al. (15): ut = (u f − u0 )(1 − kt k j ) + u0 α

where ut = link speed at time t; kt = prevailing link density at time t; uf, u0 = parameters corresponding to free mean speed and minimum speed, respectively; kj = jam density; and α = parameter that captures sensitivity of speed to density. The two algorithmic modules are set up to run independently of one another. Their execution is triggered by the cyclic executive, which executes each module on the basis of a prespecified schedule. The

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Cyclic Executive Algorithms

State Estimation

Nonlinear Least Squares Estimation Algorithm

Data Broker

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(b) FIGURE 2 (a) Simulation environment; (b) illustration of system’s execution.

internal data interchange between the two modules is handled by the data broker. That is, the data broker assists in the storing and retrieving of data. For example, once the newly estimated TFM parameters are available, they can be passed to the data broker. These parameters can then be retrieved from the data broker by the traffic simulation module. Figure 2b shows the operation of the system. In this illustration, the traffic simulation module is set to run every 30 s, whereas the NLSEA is set to reestimate the TFM parameters every 5 min. Note that the NLSEA cannot run at time zero because no data are available. Thus, in the first 5 min, the traffic simulation module will use some initial model parameters. Starting at Minute 5, the NLSEA will estimate the model parameters using the gathered past 5 min of data. In reality, 5 min of data is usually insufficient; it is used here simply for illustration purposes. This issue will be addressed later in the paper. These newly estimated parameters are then passed on to the data broker, where they will remain until being overridden with new data. So the parameters estimated at Minute 5 will be used by the traffic simulation module until they are changed at Minute 10. The process continues until it reaches the end of a specified simulation horizon.

EXPERIMENT DESCRIPTION To evaluate the performance of the TFM, three sets of simulation experiments are conducted on the Fort Worth test network (Figure 3). The physical network is made up of a freeway, which is a segment of I-35W (between I-20 and I-30), surrounded by major arterials and local streets. The network consists of 178 nodes and 441 links, dis-

tributed over 13 zones. In regard to the load on the network, the typical morning traffic pattern is used, which predominantly flows in the northwest direction (16). The O-D demand depicts a 35-min peak period. The simulation horizon is set to be 90 min to sufficiently capture the evolution of traffic buildup and dissipation. One of the requirements in using a TFM is real-time speed–density data. However, in the absence of real-time detector data for the Fort Worth test network, pseudo-real-time data are generated using DYNASMART. It is assumed that the modified Greenshields model represents the true underlying speed–density relation as observed in reality. It is important to recognize that the method by which the pseudo-real-time data are generated is not critical to the study objectives. Alternative models like Greenberg’s, Underwood’s, or even higher-order continuum models could have been used. The objective is to have a process that represents reality so that an evaluation can be made on how well the TFM adapts and captures the underlying speed–density dynamics. The experiments are designed to obtain insight into the effectiveness and properties of the TFM. The first set of experiments investigates the performance of the TFM under the condition of no updating of the model parameters. This condition represents the case in which a real-time DTA system would use the model parameters estimated with the previous day’s data without further online updating of the parameter values. So the idea is to study if such parameters can be applied under substantially different situations, such as higher demand and with occurrence of incidents. Also examined is the applicability of model parameters estimated using data from one link to another link. The second set of experiments examines the performance of the TFM when the model parameters can be

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FIGURE 3 Fort Worth network.

updated. This situation represents the case in which a real-time DTA system would utilize the incoming speed–density data stream to update the estimated model parameters online and regularly send those parameters to the traffic simulation module. The first two sets of experiments deal primarily with the effectiveness of the TFM. In the third and last set of experiments, the attributes of the model are analyzed. The factors examined are the frequency of updating the model parameters and the number of past observations to use.

EXPERIMENT RESULTS To train the TFM (i.e., to estimate the ω and θ parameters in Equation 2), DYNASMART (with 9,449 vehicles generated) is run 15 times with different random number seeds to obtain the mean speeds and densities for a chosen link (Link 99, see Figure 3). Note that the random number seed affects only the generation and loading of vehicles onto the network (i.e., when and where a vehicle

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leaves its destination) and not the mechanism governing its propagation along the network (17). The speeds and densities are realized over 1-min intervals. Tests show that 15 runs is sufficient since the averaged results are virtually the same beyond 15 runs. This set of data is referred to as the base case. The parameters ω and θ estimated using the base case data yield values of −0.3803 and 0.2606, respectively. The identification and estimation procedures are described in the previous paper by Tavana and Mahmassani (10). The evaluation of the estimated TFM under the no-updating scenarios is accomplished by testing if the parameters estimated using the base case data are applicable to other instances. To do this, first DYNASMART is run with the modified Greenshields model to generate the pseudo-real-time data. To control for other factors that might affect the results, all the vehicle-trip information such as origin, destination, departure time, arrival time, and path from this run are recorded. In the second run, the vehicle-trip information from the first

run is used as the input. The difference in the second run is that the speed on a selected test link is updated using the TFM; the speeds on all other links are updated as usual, with the modified Greenshields model. This procedure is repeated 15 times with different random number seeds. The technique outlined here allows for the evaluation of the TFM performance by observing how close its estimated speed is to the assumed actual value for the selected test link. The reason for focusing on a single test link is to simplify the scope of the problem and to evaluate the performance of the TFM while other factors are unchanged. First Set of Experiments The results for Link 99 with the base case data are shown in Figure 4a and b. Figure 4a shows the performance of the TFM using the parameters estimated from the base case data (−0.3803 and 0.2606 for

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Time (min) Estimated Speed Using TFM Run # 1 Estimated Speed Using TFM Run # 2 Estimated Speed Using TFM Run # 3 Estimated Speed Using TFM Run # 4 Estimated Speed Using TFM Run # 5 Pseudo Real-Time Speed (a)

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(b) FIGURE 4 Performance of TFM (a) with base case condition on selected runs; (b) averaged over 15 TFM runs with base case condition (RMSE 0.5292); (continued)

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(e) FIGURE 4 (continued) (c) averaged over 15 TFM runs with higher demand (RMSE 0.9509); (d) averaged over 15 TFM runs with incident (RMSE 1.1253); and (e) averaged over 15 TFM runs with different link (RMSE 0.6535).

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ω and θ, respectively) relative to the pseudo-real-time speed for individual runs. Only five randomly selected runs are presented, to avoid clutter. Note that although the results from individual runs are presented in this scenario, the remaining analysis of the results will focus solely on the average results, shown in Figure 4b. It can be observed in Figure 4a that in each realization the TFM performs quite well when there is a gradual change in the pseudo-real-time speed, but its performance is not as accurate when there is a drastic change. Clearly, in all cases, the TFM speed is able to capture the trend of the pseudo-real-time speed. As shown in Figure 4b, the average estimated speed (i.e., the average speed over 15 TFM runs) is almost an exact match of the pseudo-real-time speed. The root-mean-square error (RMSE) is used as a basis for evaluating the performance of the estimation methods and is computed using the following expression, the unit for which is miles per hour (mph): RMSE =

1 n

n

∑ (u

i

i =1

2 − uˆi ) =

1 n

n

∑e

2 i

i =1

where n is the number of observations. Under the base case condition, the RMSE of the TFM average performance for Link 99 is 0.5292. The results presented in Figure 4a and b correspond to the case in which the unfolding traffic condition is identical to that used to estimate the model parameters. However, traffic conditions will likely vary from day to day. Therefore, it is desirable to evaluate the robustness of the estimated parameters under other settings. One likely scenario is that the network is more congested (approximately 15% higher demand). Figure 4c shows the performance of the TFM for Link 99 in this case. As seen in the results of the base case condition, the average speed is close to the pseudo-real-time speed. A notable merit in this scenario is that the RMSE is only 0.9509; though higher compared with that in the base case, it remains relatively low. The higher RMSE is expected because the TFM is tested on a data set that is very different from the data set on which it was trained. In addition to uniformly higher or lower network loads, incidents might occur in the network, which could drastically affect the traffic flow pattern. In this study, a single incident is placed on Link 82 (see Figure 3). The incident starts at Minute 10 and lasts for 30 min. It reduces the link capacity by 75%. The result for Link 99 for this case is shown in Figure 4d. The merit of the TFM pointed out previously is also true in this scenario; its RMSE (1.1253), higher compared with the previous traffic conditions, remains relatively low. It can be seen in Figure 4d that the pseudo-real-time speed changes abruptly after the termination of the incident; the TFM speed is able to capture the change but over a longer period. The three previous experiments show the performance of the TFM under different conditions for Link 99. Given its effectiveness, an extension is to apply the TFM to all links in the network or have different TFMs for different subsets of links (e.g., freeways, ramps, arterials). Although such extension is beyond the scope of this paper, an attempt is made to determine whether the parameters estimated using the base case data on Link 99 can be applied to another link. Link 86 (see Figure 3) is chosen for this study; it is a freeway link, like Link 99, but serves traffic moving in the opposite direction. The result of this case is shown in Figure 4e. Despite a rather different speed evolution on Link 86, the effectiveness of the TFM is again evident in this scenario. The promising result is that the RMSE in this case is 0.6535, which is comparable with the base case condition (0.5292). Note that this RMSE is lower than the RMSEs in the higher-demand and with-incident scenarios. This finding sug-

gests that the model parameters estimated using data from one freeway link can be applied to other freeway links. A generalization of this result is that a single TFM can be used to update speeds on all freeway links; however, this hypothesis should first be tested using more extensive experiments and real-world data.

Second Set of Experiments All the results discussed up to this point involve no online updating of the model parameters. The second set of experiments explores the possible benefits of updating the model parameters by using the prevailing traffic conditions. To effectively show such benefits, the experiments are constructed to show the difference between updating (i.e., adaptive) versus no updating (i.e., nonadaptive). The number of past observations to use is chosen to be 45; this is equivalent to 45 min since the pseudo-real-time speeds and densities are output at one observation per minute. The frequency in updating the model parameters is chosen to be 5 min. Since the length of past observations is 45 min, no updating can be performed before this time. So the actual updating times are at Minutes 45, 50, 55, and so on. To provide the initial model parameters for the traffic simulation module in the first 45 min, the parameters ω and θ are estimated using the first 45 min of base case data. Their values are −0.3776 and 0.08685, respectively. The result for Link 99 under the base case condition, with updating, is shown in Figure 5a, which shows the performance of adaptive and nonadaptive TFM versions relative to the pseudo-real-time speed. Note that there is no difference in performance between the two versions of TFM in the first 45 min because both versions use the same initial parameters. From Minute 45 on, the adaptive version updates the model parameters every 5 min until the end of the simulation. On the other hand, the nonadaptive version continues to use the same initial model parameters. Both versions of TFM performed well. It makes sense intuitively that the adaptive version would outperform the nonadaptive because the recent variation of traffic flow conditions is incorporated in the estimation of the parameters. The RMSE of the adaptive TFM is 0.7935 and is lower than its counterpart RMSE (1.4327). In order to better appreciate the advantage of updating the model parameters, it is useful to look at the results in Figure 5a in two parts. The first part is the results from Minutes 1 to 45, and the second part is the results from Minutes 46 to 90. In the first part, there is no updating of model parameters. This case is similar to the scenario examined in the first set of experiments. Hence, it is expected that the RMSE in the first part of the updating case will be comparable with the RMSE of the no-updating base case condition; indeed, this is the case, 0.5836 compared with 0.5292. In the second part, the difference between the two TFM versions becomes sharper. The RMSE of the nonadaptive model (1.940) is substantially higher than the RMSE (0.9586) of the adaptive model. It can be concluded that under the case of no updating, the performance of the TFM deteriorates beyond the time frame of the data used to estimate the model parameters. The effectiveness of the TFM with updating is also investigated under the conditions of higher demand and with incident. The respective results for Link 99 are shown in Figure 5b and c. These results confirm the benefit of updating; in both cases, the RMSE of the adaptive TFM is lower than the RMSE of the nonadaptive TFM. Another advantage of using the prevailing traffic conditions to update model parameters is that the TFM speed is able to respond to abrupt changes, as can be seen in Figure 5c. In the incident scenario,

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Time (min) Estimated Speed Using TFM With Updating; RMSE = 0.7935 Estimated Speed Using TFM Without Updating; RMSE = 1.4327 Pseudo Real-Time Speed

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(c) FIGURE 5 Performance of adaptive versus nonadaptive TFM (a) averaged over 15 TFM runs with base case condition; (b) averaged over 15 TFM runs with higher demand; and (c) averaged over 15 TFM runs with incident.

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the pseudo-real-time data experience an almost instantaneous jump in speed. The adaptive TFM took a shorter time to react to the rise in speed and therefore is able to maintain a closer approximation than the nonadaptive version (which is clearly shown with the segmentation). In all three cases considered in the second set of experiments, it can be seen that the adaptive TFM consistently matches the trend of pseudo-real-time speed, whereas the nonadaptive TFM tends to either underestimate or overestimate the pseudo-real-time speed over a long period.

Third Set of Experiments To gain an understanding of the effect of updating frequency on the TFM performance, a 45-min length of past observations is chosen. The frequencies examined are 1, 5, 10, 15, and 20 min. Since the length of past observations is 45 min, updating occurs at Minute 45 and after. For example, updating in the case of 1 min would occur at Minutes 45, 46, 47, and so on till Minute 90. The base case condition is used. The result for Link 99 is shown in Figure 6a. The curve resembles a quadratic function with a minimum at 10-min updates. If the model parameters are estimated too frequently, the TFM constantly adjusts its parameters to compensate for the difference in density and the noise in the previous time interval. The result

may be a see-saw oscillation of the estimated speed. On the other hand, if the model parameters are estimated too infrequently, it may fail to capture the continual change in the speed-density dynamics. The 10-min updating frequency appears sufficient to allow the TFM to capture the continual change of speed–density dynamics, but it is long enough to allow for smooth adjustment of speed. Similarly, to study the effect of the length of past observations on TFM performance, the updating frequency of 10 min is fixed and the length of past observations is varied. Five different lengths are examined: 15, 25, 35, 45, and 55. Updating, in the case of 15, occurs at Minutes 15, 25, 35, . . . , 90, and updating, in the case of 55, occurs at Minutes 55, 65, 75, and 85. Before updating occurs, the initial parameters used are −0.3776 and 0.08685. The result for Link 99, using the base case condition, is shown in Figure 6b. The RMSE decreases to a minimal value at Minute 45, at which point it starts to increase. It appears that a length of 15 min is inadequate. There is a sharp drop in RMSE from Minute 15 to Minute 25. The 45-min length yields the lowest RMSE (0.7162); lengths longer than 45 min have higher RMSEs. This finding corresponds to the expectation that the length of past observations should strike a balance between obtaining a sufficient number of past observations to allow robust parameter estimation and at the same time being responsive to rapidly unfolding changes in the speed– density dynamics.

0.82 0.80 RMSE (mph)

0.80

0.79

0.78 0.76

0.76 0.74

0.73

0.72

0.72

0.70 1

5

10

15

20

Updating Frequency (min) (a)

14.00 12.00

RMSE (mph)

12.00 10.00 8.00 6.00

5.55

5.43

4.00 2.00 0.94

0.7162

0.00 15

25

35

45

55

Length of Past Observations (min) (b) FIGURE 6 Performance of TFM (a) with different updating frequency and (b) with different length of past observations.

Huynh et al.

SUMMARY AND CONCLUSIONS The application of a TFM in a real-time dynamic traffic assignment system has been investigated. The motivation is to improve the speed estimation method to enable better system consistency with reality in real-time operation. The study is conducted by adopting a particular form of TFM derived from actual detector data in San Antonio, Texas. This model is then used in the traffic simulation module of the DYNASMART-X dynamic traffic assignment system to update the network link speeds. A nonlinear least-squares optimization algorithm, implemented for this study, is coupled with DYNASMART-X to enable online adaptive estimation of the TFM parameters. Simulation-based experiments are carried out on the Fort Worth test network. These experiments are designed to evaluate the TFM performance and to gain insight into its operational properties under different conditions. Three sets of experiments are performed. The first set evaluates the model performance under the condition of no online updating (i.e., nonadaptive), whereas the second set assesses the model performance with online updating (i.e., adaptive). The third and final set aims to gain an understanding of the effect of updating frequency and length of past observations on the model performance. The results show that the employed TFM, both adaptive and nonadaptive, can consistently estimate the true underlying speed– density dynamics. Naturally, the model performs better with updating. That is, the adaptive model outperforms the nonadaptive model. Results from the third set of experiments suggest that an updating frequency of 10 min coupled with a time series spanning 45 min of data yields the most favorable model performance. Of significant importance is the transferability and robustness of TFMs to different settings. The results of this study have confirmed the usefulness of TFMs in the context of real-time traffic estimation and prediction. The scope of this study is limited to updating speeds on a single link. The intent was to investigate in some depth the premise that TFMs can provide good speed estimation in a controlled environment. A natural extension of this research would be to consider the effect at the network level when TFMs are applied to two or more links. Another extension would be to account for the typically sparse detector coverage. Though this study has shown that a TFM calibrated from one detector site can be used at another site (without detector), further investigation is necessary to determine if a single TFM could be used for all links in the network or several TFMs are needed, one for each different type of link (e.g., freeways, ramps, arterials). Moreover, research is needed to study how TFMs could be adapted to estimate speed on links with intersection delays created by signals. These questions are best answered in the context of a real-world operational study using actual field data; the drawback is that only a limited set of experimental conditions would be observed.

ACKNOWLEDGMENTS This paper is based on a project titled “Development of a Deployable Real-Time Dynamic Traffic Assignment System,” sponsored by FHWA and managed by Oak Ridge National Laboratory (ORNL). This work has benefited from the collective effort of several former and current graduate researchers who helped to develop DYNASMART-X, namely, Akmal Abdelfatah, Yi-Chang Chiu, Khaled Abdelghany, and Travis Waller. The overall effort has also

Paper No. 02- 2753

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benefited from fruitful interaction with Rekha Pillai of ORNL and Henry Lieu of FHWA. The authors are of course responsible for all results and opinions expressed in this paper.

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