Adaptive demand-diversion prediction for integrated control of freeway corridors

Transpn. Res.-C. Vol. IC, No. I, pp. 23-42. 1993 Printed in Great Britain.

096&090X/93 16.00+ .oO 0 1993 Per6amon Press Ltd.

ADAPTIVE DEMAND-DIVERSION PREDICTION FOR INTEGRATED CONTROL OF FREEWAY CORRIDORS YORGOS J. STEPHANEDES and EIL KWON Department of Civil and Mineral Engineering, University of Minnesota, Minneapolis, MN 55455 U.S.A. (Received 8 February 1991; in revisedform 18 November 1991) freeway corridor control, which includes efficient real-time management of freeway traffic diversion onto less congested arterials, is one of the most cost-effective ways to cope with freeway congestion. Because traffic diversion is influenced by ramp metering and intersection signal timing, the effectiveness of an integrated corridor control strategy draws on its ability to predict the diversion resulting from the control in real time. An adaptive demanddiversion predictor is developed that reflects the drivers’ choice behavior in a rapidly changing traffic environment. The new method explicitly treats the time-variant effects of control on the traffic demand to be predicted by combining behavioral modeling with filtering. Behavioral demand-diversion models and an extended Kalman filter are developed, with the filter continuously updating the model parameters with the most recent prediction error. The method was applied in several freeway entrance ramps of the Minneapolis-St. Paul metropolitan area freeway system to predict the demand-diversion of traffic flow approaching the ramp area in real time. Following extensive testing and evaluation, the method was incorporated in a new demandresponsive control logic for the online control of freeway corridors. Abstract-Integrated

1. INTRODUCTION

Real-time diversion of freeway traffic onto less congested arterials is recognized as one of the most cost-effective integrated control schemes to cope with congestion in freeway corridors. Because corridor control policies such as ramp metering and intersection signal timing influence diversion, the effectiveness of such policies largely depends on their ability to estimate and predict the diversion resulting from the control in real time. Although the importance of integrated control has been growing, the state of the art in signal systems in the United States has not reached the point where freeways and arterials are coordinated routinely (Rutherford, Schroder, Jacobson, & Hallenbeck, 1990). A major difficulty lies in the lack of reliable demand-diversion models that can describe the time-dependent interaction between freeway and arterial flows in real time. To be sure, a major integrated U.S. system in operation is adopting oversimplified assumptions regarding diversion, such as driver omniscience, user-optimized equilibrium flow and uniform reaction of drivers to the control (Gartner and Reiss, 1987). Further, no online demand predictor developed to date has explicitly considered the effect of control on the demand to be predicted. Existing control strategies are mostly empirical and can address only known or anticipated peak period loads. Development of reliable models that can predict traffic demand diversion in response to a control or management action is an essential element of integrated management schemes for freeway corridors. This paper presents a new adaptive demand-diversion predictor that reflects the drivers’ choice behavior in a rapidly changing traffic environment. Unlike previous work, the new method explicitly treats the time-variant effects of control on the traffic demand to be predicted by combining behavioral modeling with filtering. In particular, behavioral demand-diversion models and an extended Kalman filter are developed, with the filter continuously updating the model parameters with the most recent prediction error. The method was applied in several freeway entrance ramps of the Minneapolis-St. Paul metropolitan area freeway system to predict the demand diversion of traffic flow approaching the ramp area in real time. Following extensive testing and evaluation, the new method was incorporated in a new demand-responsive control logic now under development at the University of Minnesota for adaptive control of freeway corridors. 23

24

Y.

J.

STEPHANEDES

Fig. 1. Entrance-ramp

and E.

KWON

area in freeway corridors.

2. BACKGROUND

Drivers approaching the freeway experience two major opportunities for diversion at the ramp area (for a typical entrance-ramp area consisting of an intersection, a ramp and a frontage road, see Fig. 1). The first diversion opportunity occurs when drivers approach the intersection directly upstream from the ramp. At that time, depending on intersection and ramp traffic conditions, a fraction of the original freeway demand may divert to the arterial. The traffic continuing toward the ramp may further divert to the frontage road as the entrance-ramp traffic condition deteriorates. Such diversion directly affects the demand entering the freeway and, concurrently, the traffic volume on the arterials in the corridor. Figure 2 summarizes the interrelationship between traffic diversion and control at the ramp area.

FREEWAY

I

i+l

i

i-l

Total Volume Approaching Ramp Area I

a Fr0tWbWy Traftic

Traffic Condition at Ramp Area i

RAMP AREA CONTROL Ramp Metering

lntorrrction Signal

’

L Fig. 2. Diversion-control

interrelationship at a ramp area.

Arterial Traffic

Adaptive demand-diversion

prediction

25

Because diversion is substantially affected by the traffic conditions resulting from the control in the ramp area, that is, intersection signal timing and ramp metering, control action should be based on the predicted diversion behavior of traffic flow responding to the control to be applied. However, no online demand predictor developed to date is comprehensive enough to reflect the effect of the control on driver decisions. To be sure, existing online predictors, mostly developed for intersection control, adopt a statistical trend-tracking approach, that is, prediction is based primarily on the past trend of flow measurements. The original demand predictors in the second and third Urban Traffic Control System (UTCS) generations (Federal Highway Administration [FHWA], 1973; Liberman et al., 1974) provide good examples of traffic forecasting tools that have been widely tested in the field. They predict turning volumes at intersections without considering the effects of signal timing on the turning movement, based only on historical information and the recent prediction error. Time-series models assume demand prediction is a point process and use purely statistical techniques to identify the stochastic nature in the observed data. Whereas such models have been developed for short-term traffic demand forecasting (Moorthy and Ratcliffe, 1988) and freeway volume/occupancy estimation (Ahmed and Cook, 1979; Ahmed, 1983; Kyte, Marek, & Frith, 1989), they do not explain the interrelationship between the control and the resulting flow patterns. More important, the vast majority of models use constant parameters computed offline, significantly restricting model adaptability to the ever-changing traffic environment. An effort (Okutani and Stephanedes, 1984) has been made to update model parameters on the basis of the most recent traffic demand prediction error; however, this approach, as most of those existing, did not explicitly treat the effects of control policies on the predicted volume. The lack of reliable online predictors for freeway demand diversion has been one of the major problems in developing integrated strategies that can coordinate control of the freeway with other corridor elements in real time. For example, the most sophisticated control strategies to date use equilibrium assignment to predict demand diversion for a 15 30-min time horizon under the assumption that demand will not change appreciably over this time period (Gartner and Reiss, 1987). However, it has been shown that, even when hourly volume is very high, traffic flow exhibits substantial minute-to-minute variations (Keen, Scofield, & Hay, 1989) that can cause smooth flow to break down. Further, it has been argued that the underlying assumption in the equilibrium assignment approach- Wardrop’s principle (Wardrop, 1952) -is not applicable to the dynamic traffic environment mainly because of the human nature of drivers, that is, drivers are not well informed or are not sufficiently skilled to choose the best route (Hall, 1983.) Responding to the need for prediction that explicitly treats the time-varying effects of control on corridor demand in real time, this paper develops adaptive predictors for traffic demand and diversion at entrance ramp areas in freeway corridors. The new method predicts the ramp-approaching flow at the intersection upstream from the ramp, and the ramp-entering proportion, that is, the ratio of vehicles entering the ramp over the total number of vehicles approaching the ramp from the intersection. Prediction is performed continuously for every time interval, usually every 5 min. The proposed method is applied to the online prediction of freeway demand diversion at entrance ramps and nearby intersections in the I-35W freeway corridor in Minneapolis, Minnesota. 3. PREDICTION FRAMEWORK

The predictor developed in this research combines behavioral modeling and filtering. In particular, for predicting demand and diversion at the ramp area the extended Kalman filter updates the parameters of models that reflect driver diversion behavior. The behavioral characteristics of commuter route choice that represent the basis on which these models are developed were earlier identified from an extensive questionnaire survey conducted by the authors (Stephanedes, Kwon, & Michalopoulos, 1989). The earlier route-choice findings indicate that freeway demand can be treated as a decision-making entity that makes diverting decisions based on maximization of trip

Y. J. STEPHANEDES and

26

E.KWON

utility determined by traffic conditions. More specifically, diversion is primarily influenced by the perceived congestion delay and is not affected by drivers’ socio-economic characteristics, such as income. Further, the diverting decision is reflected in the proportion of freeway users in the total traffic flow approaching the ramp area. In modeling demand diversion, U(x,c,O) represents the trip utility of freeway users under given ramp traffic conditions, x, and control, c-number of cars on the ramp and ramp metering- with unknown parameters, 0, that summarize the ways in which drivers trade off the ramp traffic condition for the expected delay. The total flow, Qk, crossing the upstream intersection stopline during green time, responds to the ramp condition as reflected by the ramp-approaching proportion, PF,k, and the ramp-entering proportion, PR,k, of Qk. For each time interval k

PF,k P R.k =

g[uR,k(x,c,e)l

where u,k iS the freeway ramp-approaching utility and uRBkis the ramp-entering utility of Qk. Two logit models are formulated to estimate the u,k and uR,k for every time interval k. The unknown logit parameters, 8, are considered as state variables following the random walk process (Young and Jakeman, 1984) and estimated in real time by applying the extended Kalman filter with the most recent demand-diversion information. Further, historical and current flow data are employed to predict Qk, which is also a function of intersection green time. The resulting predictor provides, for every time interval k, the freeway ramp-approaching flow, QF,kr the ramp-entering flow, QR,k, and the diverting flow to arterials in real time. Figure 3 illustrates the framework of the proposed adaptive demand-diversion predictor. 4. MODEL FORMULATION

Demand-diversion at intersection Freeway-bound commuters approaching the upstream intersection make real-time decisions regarding diversion, that is, whether to approach the ramp or divert to arterial streets, depending on the ramp traffic condition. These decisions, usually made within a short time due to the limited storage capacity of the intersection, are directly reflected in the value of the ramp-approaching proportion of the total flow crossing the intersection

I

I

Updated Model Parameters

-

Intersection Croroing Flow

I

Ramp Approaching Proportion

Ramp Entering Proportion L

PREDICTED DEMAND-DIVERSION

EXTENDED KALMAN FILTER

Fig. 3. Framework of proposed adaptive predictor.

I

Adaptive demand-diversion

prediction

27

stopline during green time. Two models are formulated to predict the ramp-approaching flow from the intersection for every time interval k. The first model predicts Qj,k, the number of vehicles crossing stopline j of a given intersection during the kth time interval, where

Qj,k = QGj.k * Gj.k and Gj,k is the green time for approach j during the /cth time interval. Assuming the flow per green time during the kth time interval, QGj.k, is a function of historical information and the current day measurement, the model equation follows:

Qj,k =

ej,k

{E[QGjIk + E[QGjIk-1+ QGj,k-1lGj.k

(2)

where E[QGj]k is the past average flow crossing stopline j per green time during the kth time interval and ej,, unknown model parameters to be identified in real time. The above model assumes that flow per green at the present time interval is a weighted average of past and current day measurements; 0 is continuously updated in real time with the most recent information on the demand fluctuation. The underlying assumption is that the total number of vehicles approaching the intersection stopline is not affected by ramp traffic conditions during a short time interval. This assumption is based on the Minneapolis survey findings indicating that, for every lo- to 15-min peak interval, an overwhelming portion of traffic consists of the same commuters following the same route every weekday (Stephanedes, et al., 1989). The assumption can be modified during incidents and special events, when driver behavior is highly sensitive to real-time information on changing traffic conditions, an issue currently under investigation by the authors. The second model predicts the freeway ramp-approaching proportion, Pf.k, of the total (i.e. overall approaches) flow Qk crossing the intersection during the kth time interval, where P F,k

-

Po.k

*

(3)

Pn.k

freeway-bound = (original freeway-bound demand),/&, Pn,k = Q,,/(original demand),, Qk = CjQj,kQFek= (original freeway-bound demand), - (diverted demand),the total flow approaching the ramp from the intersection. Freeway-bound traffic demand is assumed to respond to traffic conditions following the utility principle and Weibull distribution of errors as expressed in a logit formulation. In the KSUlthg model, PF,kis a function of PO,&the original freeway-bound proportion of the intersection flow Qk and of freeway utility U,(X,,C,,~,) that is primarily determined by the condition at the ramp:

and Po.k

PF,k

=

Po,k/{i

+

w[%,k

+

%k(xk/ck)i)

(4)

where ei,k are model parameters to be identified in real time. In the above formulation, with @,k expected to be positive, the ramp condition is represented by the composite variable, x&k; as the form of the composite variable implies, when x,, the number of vehicles on the ramp and in the area between it and the intersection, is low or metering rate, Gk, is high, the ramp condition does not have a substantial effect on the diversion. Further, the value of Pockis identified in real time, following the same process as that of the other model parameters, ek. Whereas prediction of PF,k employs only current ramp information and no historical data, any available information on the PO,krsuch as historical data, could improve the accuracy of prediction. Demand-diversion at entrance ramp Following the diversion decision at the intersection upstream from the ramp, drivers approaching and entering the ramp face a second decision regarding diversion-whether to enter and remain on the ramp or to divert to the frontage road. For predicting the

Y. J. STEPHANEDES and E. KWON

28

ramp diversion, we formulate the model of the ramp-entering proportion, PR,k, the ratio of vehicles entering the ramp to those approaching the ramp from the intersection at every time interval k. We model this decision as a function of the traffic condition of the ramp. In particular, we assume that two major factors represent the entrance-ramp traffic condition to these drivers-the size of the queue on the ramp as a driver approaches and the speed of queue reduction while the driver is waiting at the ramp. We represent the size of the queue by considering the total number of vehicles using the ramp during the previous time interval; similarly, we represent speed of queue reduction via the current ramp metering rate. Therefore, the ramp-entering utility at time k, UR,k, is a function of the current freeway-entering rate, C,, and the total number of vehicles that used the ramp in the previous time interval, &_ , + Rk_ , . Following these assumptions, a Jogit model describing the ramp-entering proportion, P R,k, of the total ramp-approaching flow, QF,k, is formulated with the ramp utility UR.k(Xk_,,Rk_I,Ck,ek): P R,k = l/(1

+ exp[&& +

%,kCk

+

%,k(xk-I

+

Rk-,)I)

(3

where xk is the number of vehicles on the ramp in the beginning of the kth time interval, Rk is the number of vehicles entering the ramp during the kth time interval and ei,k the unknown parameters to be identified in real time. The parameter, @k, represents the effects of the unknown variables not captured in this for ulation. ln 5. ADAPTIVE

PREDICTION

in the previous section reflect the The parameters, (i+&, of the models formulated response of flow to given traffic conditions and control for every time interval. Although these parameters could be estimated offline, the inherent uncertainty in traffic flow and the factors not captured in the models could result in substantial error, reducing the effectiveness of the prediction of demand diversion. For effective real-time traffic management, the prediction should be adaptive to the continuously changing traffic environment. To address the need for adaptive prediction, the model parameters are identified in real time by incorporating the extended Kalman filter algorithm (Jazwinski, 1970) into the prediction models. This approach considers the model parameters as state variables representing the behavioral state of the traffic flow at a given time interval. Further, the extended Kalman filter algorithm identifies the optimal unbiased estimates of the behavioral state in real time by using the most recent prediction error. The state evolution of the model parameters, 8, can be represented by 0 k+l

=

Gek +

wk

(6)

where G represents the fixed characteristics of the process with noise, wk. The model parameters, ek, are assumed to follow the nonstationary random walk process. Random walk has been successfully applied to model physical systems that are subject to rapid variation (Young and Jakeman, 1984). Using the prediction models as observation equations, the extended Kalman filter continuously updates the model parameters by recursively determining the minimum variance estimates of the prediction parameters. The extended Kalman filter is based on the well-known theory developed by Kalman (Kalman, 1960) and was originally intended for the state identification of a nonlinear dynamic system. It relinearizes the system dynamics about each new state as they become available. As a consequence of relinearization, large initial estimation errors are not allowed to propagate through time, and the linearity assumptions are less likely to be violated. The procedure for updating the model parameters via the extended Kalman filter is summarized below and in Fig. 4:

29

Adaptive demand-diversion prediction

Qk

%

+

+ Kk

i_

'k/k *

1:

pf,k PR,k T

Delay

Prediction Qk ,

Parameter

Model

Pf,kr

h,k ek-llk-l

‘k/k-l

Fig. 4. Parameter update

process.

1. Initialize algorithm (k = 0) with any prior knowledge of model parameters

c k/k

Gl

=

where &k = E[(@k ek,k)(ek ek/k)rl* Set the model parameters &+r,k = &k. proportion 3. Predict flow per intersection approach Qj,k+1, freeway ramp-approaching by using prediction model eqns (2), (4) and P F,k+I and ramp-entering proportion PR,k+, (5) with the parameters e)k+i/k’ 4. Measure actual Qj.k+1,PF,k+, and PR,k+I and obtain prediction error vector ek+,, which is defined as ek = (measured value) - (predicted value). 5. Update model parameters 6 k+,,k by using the gain and prediction error

2.

e K k+l

k+l/k+l

=

e k+l/k

=

+

Kk+,

eks where

c k+l/k~k+,TISk+ICk+I/kSk+lT

+

Sk]-’

is the gain vector,

ck+l/k

=

ck

+

qk

the covariance matrix, Sk+l

=

[aQ/W,

aP,/ae,

aP,/a@]‘I 8 = &+,,k

the sensitivity vector,

ck+l/k+l

=

(1

-

Kk+,&+,)

ck+,,k

the updated covariance matrix, defined by the covariance matrices

and w and v are zero-mean Gaussian white noise sequences for the state eqn (6) and the observation eqns (2), (4) and (5), respectively. 6. Let k = k + 1 and return to step 2. 6. TESTING AND VALIDATION

Intersection prediction The adaptive predictor developed in the previous section is tested and validated with real traffic data collected from the I-35W freeway corridor in Minneapolis, Minnesota. The I-35W corridor, which, as Fig. 5 suggests, crosses the twin cities in a north-south

Y.

30

!

J.

STEPHANEDESand

MINNEAPOLIS

E.

KWON

35th

St.

I

CBD

46th St.

Minnehaha Creek Diamond Lake St.

Fig. 5. I-35W freeway corridor, Minneapolis.

direction, often experiences severe congestion that extends through the off-peak period. Three entrance-ramp areas of the corridor were selected for testing the new predictor. The selected areas, illustrated in Figs. 6a through 6c, represent major geometric-control configurations of freeway corridors; each ramp area includes a frontage road that can accommodate traffic diverting from the ramp (for the location of each ramp along I-35W, see Fig. 5). Of the three selected areas, the first, Diamond Lake Street, represents the typical entrance-ramp area in the corridor with a three-way intersection at a two-way arterial, a frontage road and a metered ramp. The second ramp area, 35th Street, includes a twoway intersection at a one-way arterial and is similarly ramp controlled. The third, 46th Street, includes a three-way intersection at a two-way arterial but no metering. For each entrance-ramp area, the following traffic movements were measured manually for every four cycles (4-6 min.) during the morning rush hour over a period of 2 to 14 weekdays, depending on the location, from October 1988 to September 1989: Number of vehicles crossing the intersection stopline during green time, Qj,k 1. 2. 3. 4. 5.

Number Number Number Number Number

of of of of of

vehicles vehicles vehicles vehicles vehicles

approaching entering the diverting to entering the on the ramp

Using these data, each component

the ramp from the intersection, QF,k ramp, Rk the frontage road from the ramp, D, freeway from the ramp, C, in the beginning of each time interval, X,. model of the adaptive predictor

is tested and three

Adaptive demand-diversion

II

prediction

I-35W

N.

31

_.

Fig. 6a. Diamond Lake Street entrance ramp.

error indexes are calculated to evaluate the performance tions. OPE, one-step prediction error (percentage): 100 * 1[measured],

of the predictor,

- [predicted],] /[measured],

MOPE, mean one-step prediction error (percentage): 100/N * Ck( [measured],

- [predicted], ( /[measured],

MAE, mean absolute error: l/N * C, 1[measured],

- [predicted], I

I-35W

N.

,_.

Fig. 6b. 46th Street entrance-ramp area: not metered. la(C)

1-l-C

for N predic-

32

Y.

J.

STEPHANEDES

3 0

and E.

I-35W

KWON

N. 1

.

Jii?ii!L

=~,I~ Fig. 6c. 35th Street entrance ramp.

Of the above error indexes, the one-step prediction error indicates the adaptability of the new predictor in the continuously changing traffic environment, and MAE is a conventional error measurement. The following sections summarize the test results from each location. Ramp control, two-way arterial: Diamond Lake Street. The entrance-ramp intersection predictor was first tested at the Diamond Lake Street entrance-ramp area over a two-week period in March 1989. The prediction results for the total flow crossing the intersection, Qk, and its ramp-approaching proportion, P F,k,are illustrated in Figs. 7 and 8 for every four-cycle (6-min) interval during two weekdays. The following models and initial parameter values were used to perform the prediction:

QA. = Cej,Q(HQGjlk-l + E[QGjl, + QG~--I)G~,~ PF.k= Op,/{l ej.Q

+ expPR + %(-WG)ll

= 0.33, ep, = 0.8, epz = -1.0, ep3 = 1.0

The performance of the intersection prediction method was tested through application of both prediction models, that is, intersection flow and ramp-approaching proportion predictors. The results from the prediction of the ramp-approaching proportion indicate that MAE ranges from 0.053 to 0.069. For a typical flow Qk = 100 vehicles per 6 min crossing the intersection, eqn (3) suggests that such a MAE range would correspond to an error of 5.3 - 6.9 vehicles per 6 min in the total flow, Qf,k, approaching the ramp from the upstream intersection. Further, mean OPE lies between 8.3% and 19.8%. For predicting the flow crossing the intersection, two types of historical information were employed-the measurements from the previous day and those from the same day of the previous week. The prediction results indicate that the accuracy of the prediction of total flow based on the previous week’s data is slightly higher than that based on the previous day’s data. Test results from the flow prediction lead to a MAE range of 7.5 to 14.8 vehicles per 6 min and a MOPE range of 7.9% to 12.6%. The autocorrelations of flow and proportion-prediction error sequences do not indicate any significant trend at the 0.05 level. Ramp control, one-way arterial: 35th Street. In order to evaluate the performance of the models developed for the Diamond Lake Street intersection, the Diamond Lake

Adaptive demand-diversion

33

prediction

models, with the same initial parameter values, were used for flow prediction at 35th Street. The tests were performed over a two-day peridd in September 1989. Figure 9 illustrates the results from performance evaluation on 13 September 1989, a weekday, for every four-cycle (Cmin) interval. The evaluation includes tests of both intersection models, that is, the intersection flow and the ramp-approaching proportion predictors. Prediction of the ramp-approaching proportion over a two-day period leads to a MAE range of 0.048 to 0.049 and a MOPE of 8.9% to 10.9%. Further, the performance of the flow predictor is tested by implementing it to predict total flow crossing the intersection on 13 September 1989, using September 12 data. This test results in a MAE of 7.1 vehicles per 4 min and a MOPE of 12.8%. The autocorrelations of flow and proportion-prediction error sequences indicate no significant trend at the 0.05 level. No ramp control, two-way arterial: 46th Street. Because ramp control does not affect the diversion at this location, the prediction model of the ramp-approaching proportion is modified by deleting the control variable from the original form: PFvk= %.K41

+ exp(%.k)l.

Nevertheless, the original model is used for predicting the flow crossing the intersection

INTERSECTION-CROSSING

FLOW PREDICTION

Car8 I 6 mln

*O”l 160 -

100 -

6:60

7:20

I:60

6:20

Time -+

Obrervod

RAMP-APPROACHING

*

Predlchd

PROPORTION

PREDICTION

0.8

0.6 i

lu I\

0.4

0.2

0.0 6:60

7:20

7:60

6:20

Tlmo

+ Fig.

7.

Obrorvcd

+

Predicted

Prediction results at Diamond Lake Street intersection (21 March 1989).

and E. KWON

Y. J. STEPHANEDES

34

INTERSECTION-CROSSING Cars I

FLOW PREDICTION

4 min

01”” 7:20

7:oo

7:40

8:OO

Time -

Observed

RAMP-APPROACHING

*

Predicted

PROPORTION

PREDICTION

4 7:oo

7:40

7~20

8:OO

Time +

Fig. 8. Prediction

Observed

results at Diamond

+

Predlcted

Lake Street intersection

(23 March

1989).

as in the previous two cases. The tests at this intersection were performed over a 2-week period in December 1988. Prediction results from two weekdays, for every four-cycle (6-min) interval, are illustrated in Figs. 10 and 11. Both intersection predictors are tested. From the tests, the ramp-approaching proportion has a MAE ranging from 0.028 to 0.05 and a MOPE range of 4.1% to 7.2%. Prediction of total flow crossing the intersection leads to a MAE range of 13.8 to 20.9 vehicles per 6 min and a MOPE range between 6.1% and 10.5%. The autocorrelations of the prediction errors indicate no significant trend at the 0.05 level. Similar to the Diamond Lake Street test results, prediction of total flow based on the previous week’s data indicates slightly better performance than prediction with the previous day’s data. Ramp diversion The ramp Diamond Lake PR,k of the total

prediction diversion prediction model was tested in the two metered and 35th Street ramps. The model predicts the ramp entering ramp-approaching flow for every time interval k P R.k = l/(1

+ exp(8

I,k

+

@Z,kCk

+

%,k(Xk-I

+

Rk-l)).

locationsproportion,

Adaptive demand-diversion

35

prediction

The tests were first performed with real data collected at the 35th Street ramp in October 1988 during the morning rush period. To avoid a large initialization error, the initial values of the model parameters were determined by fitting the model with the past observations, leading to the following set of initial parameter values for this ramp: 0, =

-5.0,82

= 0.05, cl3 = 0.05.

Predicted and observed values are illustrated in Fig. 12 for two typical weekdays. The results indicate that the MOPE ranges between 5.5% and 8.8% and the MAE ranges from 0.04 to 0.06. Further, the OPE distribution indicates that approximately 75% of errors have values < 10% (see Fig. 13). As illustrated in Fig. 12, the occasional largerthan-usual error does not propagate through time, indicating the adaptive nature of the predictor. Further, the investigation of residual autocorrelation indicates (see Fig. 14) that there is no significant trend (0.05 significance level) in the error pattern. The above ramp diversion model, with the same initial parameter values, was transferred to the Diamond Lake Street ramp, where it was tested with a second set of data collected in March 1989. Prediction results from these data are illustrated in Fig. 15 over

INTERSECTION-CROSSING

100

Cars I

FLOW

PREDICTION

4 min

7212

7~32

762 Time

-+

Observed

RAMP-APPROACHING

0.0

+

Predicted

PROPORTION

PREDICTION

II 7:12

7:32

7:S2

a:12

Time -& Fig. 9.

Obaerved

+

Predlcted

Prediction results at 35th Street intersection (13 September 1989).

36

Y. J. STEPHANEDESand E. KWON INTERSECTION-CROSSINQ 360

FLOW PREDICTION

Car8 f 6 mln

300 260

200 160

Time -

Obaerwd

RAMP-hPPROACHINQ

-

PredIcted

PROPORTION

PREDICTION

0.4 0.2 -

0.01

’

’

’

I

a:40

:

’

1

t

7:lO

I

I

’

7:40

’

I

3

II:’ IO

Time

-

Obrrrvmd

+

Prsdlchd

Fig. 10. Prediction results at 46th Street intersection (7 December 1989).

two weekdays. From the figure, MAE ranges from 0.048 to 0.052, and mean OPE lies between 5.0% and 5.8%. These findings indicate the potential for transferability of the prediction algorithm along the freeway corridor. 7. DISCUSSION AND CONCLUSIONS

An adaptive method was developed for real-time prediction of demand and diversion of traffic flow at entrance-ramp areas of freeway corridors. The prediction method explicitIy treats the time-variant effects of control on the traffic demand to be predicted by combining behavioral modeling with filtering. In particular, behavioral demand-diversion models and an extended Kalman filter are developed, with the filter continuously updating the model parameters with the most recent prediction error. The new method treats the entrance-ramp area as one system consisting of the freeway entrance ramp and the upstream intersection. Three dynamic predictors are developed for predicting, at 4- to 6-min intervals, the flow crossing the intersection, the proportion of flow that approaches the freeway and the proportion of freeway-bound flow entering the entrance ramp. The predictor of flow crossing the intersection employs historical information, that

Adaptive

demand-diversion

INTERSECTION-CROSSING

37

prediction

FLOW PREDICTION

Cara I 6 mln 300

250 200 160

6:40

7:lO

7:40

8:lO

Time -+-

Observed

RAMP-APPROACHING

6:40

i(-

Predicted

PROPORTION

7:lO

7:40

PREDICTION

8:lO

Time +

Observed

+

Predicted

Fig. 11. Prediction results at 46th Street intersection (13 December 1989).

is, past average flow crossing the intersection and green time. The predictor of the rampapproaching proportion is built around a logit specification as a function of vehicles in the ramp area past the intersection and ramp metering rate. The predictor of the rampentering proportion is also built around a logit specification as a function of vehicles entering the ramp, ramp queue and ramp-metering rate. The prediction method was applied in several freeway entrance-ramp areas of the Minneapolis-St. Paul metropolitan freeway system with encouraging results. Findings from three typical ramp areas that include both metered and control-free ramps and both two-way and one-way adjacent arterials along the I-35W corridor indicate MAE of the freeway- and ramp-approaching proportions in the 0.03 to 0.07 range and 75% of OPE have values lower than lOolo. Prediction models are tested with datasets different from those with which they were developed without error degradation indicating the potential for transferability of the prediction algorithm along the freeway corridor. By combining behavioral modeling with filtering, the new method can be applied to reflect the drivers’ choice behavior in a rapidly changing environment. In particular, because the prediction operates in real time and does not allow errors to propagate, the new predictor can be implemented in conjunction with adaptive control. Further, by treating the entrance ramp and the nearby intersection together, the predictor can be

Y. J. STEPHANEDES~~~

October

o3 7:oa

E.KWON

6, 1988

7:36 Time +

Observed

-X-

October

8:08

Predicted

7, 1988

J

0

7:oo

730

a:00 Time

+

Obrerved

-I+

Predicted

Fig. 12. Ramp-entering proportion prediction at 35th Street ramp.

applied in the development of an integrated corridor control system. In such a system, the optimal corridor control problem could be summarized as that of determining an intersection signal-timing plan and ramp-metering rates for each ramp area i during time interval k; the solution should satisfy the corridor management objective, such as maximizing total travel or minimizing total delay, subject to freeway and arterial operating conditions. Figure 16 illustrates the modified hierarchical control structure by using the proposed prediction method. Unlike existing multi-layer control structures, the parameters of the prediction and optimization models are continuously updated in real time with the most recent prediction error, so that the control system realistically reflects current traffic conditions. Further, a prediction occurring every 5 min can lead to a more robust nominal control solution that lightens the load of the direct controller. Although this research focused on the demand diversion at the entrance-ramp area, it is expected that the behavioral principles underlying the models and the adaptive prediction algorithm should also be applicable to other diversion points in the corridor, such as freeway exit ramps and major arterial intersections. Such extensions of this research would include modeling the effect of mainline freeway congestion on exit volume of the nearby off-ramp and, further, modeling the effect of local congestion on turning movements at major arterial intersections.

Adaptive demand-diversion

39

prediction

October 5, l988 ERROR

DISTRISUTION

100%

60%

O-6

6-10 Error

October ERROR

W

7,

1988

DISTRIBUTION

1001

60%

Olb O-6

6-10

lo-16

Fig. 13. One-step prediction error (OPE) distribution: ramp-entering proportion,

35th Street.

Y.

J.

STEPHANEDES

and E.

KWON

October 6, ?!388 AUlOCORRELAlION 1.4 1.2 1 0.8 0.0 0.4 0.2 0 -0.2 -0.4 -0.0 -0.8 -1 -1.2 -1.4 0

1

2 Tlmr

Octobu

3

4

6

4

6

LDO

7, 1988

AUTOCORRELATION

-0.6

t

-1 -1.2 -1.4

i 0

1

2

3

Time Lag

Fig. 14. Autocorrelations

of residuals: ramp-entering proportion, 35th Street.

March 22,1989

(I:60

7:20

750

8:20

7dO

a:20

Time

March 28, 1989

0.6

0 6:60

7:20 Time

+

+

Observed

Predlctod

Fig. 15. Ramp-entering proportion prediction at Diamond Lake Street ramp. PREDICTION

DETECTION

I

J

UPDATED MODEL PARAMETERS

I SYSTEM OPTIMIZATION Nominal Control

I

I

I-

I-

FREEWAY

Fig. 16. Adaptive control frahework. 41

42

Y. J. STEPHANEDESand E. KWON

Additional areas for future research have been identified, dealing with improved modeling of the traffic process, improved accuracy of demand prediction and identification of the effects of traffic information/guidance on driver behavior. For instance, the effect of online traffic information by radio and changeable message signs on diversion and departure time variation can be at least as substantial as that of online control. Further, the accuracy of demand prediction could be improved by considering the upstream link volume (Okutani and Stephanedes, 1984) and determining suitable historical information via pattern recognition, classification and identification of demand patterns. Acknowledgemenrs-This study was supported by the National Science Foundation through project No. NSF/ CES-8713277. The Minnesota Supercomputer Institute provided partial support through project Nos. mg26701 and mg29901. The Center for Transportation Studies, Department of Civil and Mineral Engineering, University of Minnesota, is acknowledged for its support. The authors would also like to thank P. Michalopoulos for his cooperation and valuable suggestions and A. Chassiakos and Jane Govro for their help with the figures in this paper. Finally, we wish to acknowledge George Weiss for his helpful remarks.

REFERENCES Ahmed S. A. (1983) Stochastic processes in freeway traffic. TruJ Eng. Cont., 24, 306-310. Ahmed M. S. and Cook A. R. (1979) Analysis of freeway traffic time series data by using Box-Jenkins techniques. Trans. Res. Rec., 722, l-9. FHWA (1973) Urban traffic control system and bus priority system traffic adaptive network signal timing program: Software description. Federal Highway Administration, U.S. Department of Transportation, Washington, DC. Gartner N. H. and Reiss R. A. (1987) Congestion control in freeway corridors: The IMIS system. In A. R. Odoni et al. (eds.), Flow Control of Congested Networks, pp. 113-132. NATO AS1 Series F: Computer and Systems Science, Vol. 38, Springer-Verlag. Germany. Hall R. W. (1983) Traveler route choice: Travel time implications of improved information and adaptive decisions. Trans. Res., 17A, 201-214. Jazwinski A. H. (1970) Stochastic Process and Filtering Theory. Academic Press, New York. Kalman R. E. (1960) A new approach to linear filtering and prediction problems. J. &sic Eng., 82D, 35-45. Keen K. G., Scofield M. J. and Hay G. C. (1989) Ramp metering access control on M6 motorway. Proceedings of the Second International Conference on Road Traffic Control, IEE, England, London, April, 39-42. Kyte M., Marek J., and Frith D. (1989) Using multivariate time series analysis to model freeway traffic flow. Paper presented at Annual Transportation Research Board Meeting, Washington, D.C., January. Liberman E. B. et al. (1974) Variable cycle signal timing program: Vol. 4. Prediction algorithms, software and hardware requirements, and logical flow diagrams. FHWA, U.S. Department of Transportation, NTIS: PB 241720, Washington, DC. Moorthy C. K. and Ratcliffe B. G. (1988) Short-term traffic forecasting using time series methods. Trans. Plan. Tech., 12, 45-56. Okutani I. and Stenhanedes Y. J. (1984) Dynamic prediction of traffic volume through Kalman filtering theory. Trans. Res., 18’B,I-11. . _ _ Rutherford G. S., Schroder M., Jacobson L., and Hallenbeck M. E. (1990) Arterial control and integration. WA-RD 188.2, Washington State Transportation Center, Seattle, Washington. Stephanedes Y. J., Kwon E., and Michalopoulos P. G. (1989) Demand diversion for vehicle guidance, simulation and control in freeway corridors. Truns. Res. Rec., 1220, 12-20. Wardrop J. G. (1952) Some theoretical aspects of road traffic research. Proceedings of Institute of Civil Engineers. Road Paper N36, pp. 325-378. Young P. C. and Jakeman A. J. (1984) Recursive filtering and smoothing procedures for the inversion of ill-posed causal problems. Util. Moth., 25,351-376.