Stochastic modeling of the dynamics of incident-induced lane traffic states for incident-responsive local ramp control

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Physica A 386 (2007) 365–380 www.elsevier.com/locate/physa

Stochastic modeling of the dynamics of incident-induced lane traffic states for incident-responsive local ramp control Jiuh-Biing Sheu Institute of Traffic and Transportation, National Chiao Tung University, 4F, 114 Chung Hsiao W. Road, Section 1, Taipei, Taiwan 10012, ROC Received 17 March 2007; received in revised form 22 July 2007 Available online 9 August 2007

Abstract Incident-induced traffic congestion has been recognized as a critical issue to solve in the development of advanced freeway incident management systems. This paper investigates the applicability of a stochastic optimal control approach to real-time incident-responsive local ramp control on freeways. The architecture of the proposed ramp control system embeds two primary functions including (1) real-time estimation of incident-induced lane traffic states and (2) dynamic prediction of ramp-metering rates in response to the changes of incident impacts. To accomplish the above two goals, a discrete-time nonlinear stochastic optimal control model is proposed, followed by the development of a recursive prediction algorithm. Based on the simulation data, the numerical results of model tests indicate that the proposed method permits relieving incident impacts particularly under low-volume and medium-volume conditions, relative to high-volume lane-blocking conditions. Particularly, the incident-induced queue lengths can be improved by 50.1% and 67.9%, compared to the existing ramp control and control-free strategies, respectively. r 2007 Elsevier B.V. All rights reserved. Keywords: Incident-induced lane traffic behavior; Stochastic optimal control; Incident management

1. Introduction Incident-induced traffic congestion has been recognized as a critical problem to solve in the development of advanced freeway traffic management systems. It is generally agreed that lane-blocking incidents on freeways interrupt traffic flows unexpectedly, and are also a major cause of over-congestion that contributes readily to bottlenecks. During a lane-blocking incident, the time-varying traffic demand continues to exceed the reduced link capacity, and meanwhile, the growing incident-induced lane changes and queue lengths upstream to the incident site significantly interrupt the traffic flows among adjacent lanes. As a consequence, the existing freeway traffic control and management systems may malfunction owing to the lack of capability to respond appropriately to incident impacts. Theoretically, traditional freeway traffic control strategies such as tactical reduction in on-ramp flow rate and breaking the platoons of the entering vehicle to facilitate vehicular merging Tel.: +886 2 2349 4963; fax: +886 2 2349 4953.

E-mail address: [email protected] 0378-4371/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2007.08.005

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in weaving areas can be ineffective under conditions of over-congestion which is a typical traffic phenomenon of lane-blocking incidents on freeways. Despite the awareness that ramp-metering control has been extensively used as a fundamental mechanism to improve freeway traffic operations, and various ramp-metering control technologies have been proposed for either local operations [1–8] or system coordination [9–15], there seems to remain a lack of research on exploring real-time technologies to address the issues of time-varying incident impacts on ramp-metering control. The investigations by Shaw and McShane [2] can be regarded as a pioneering study in incidentresponsive ramp control, where they formulated the problem of traffic jams at the incident site with a deterministic queuing theory-based model. Given the incident duration as well as some other ideal assumptions, Shaw et al. proposed to compare the predicted waiting time of any given on-ramp traffic arrival with a predetermined threshold to determine whether or not the new arrival is allowed to enter the freeway. The demand-capacity and percent-occupancy strategies [3,4] which are the two typical ramp control modes used in the USA, and claimed to be incident-responsive, are based essentially on the same fundamental that the measurements such as volumes and occupancies upstream to the ramp under control are compared to preset thresholds to determine the ramp-metering rate. While under conditions of lane-blocking incidents, the determination in terms of these fixed thresholds may turn out to be a critical issue remaining in the aforementioned two traffic-responsive control algorithms due to the variety of incident-induced traffic flow patterns. To deal efficiently with the variety of freeway traffic congestion conditions including nonrecurrent congestion cases [10] Chen et al. proposed an ingenious ramp control strategy employing fuzzy control theories. In comparison with the existing controllers at the study site under limited six incident scenarios, their test results implied that it is promising to gain higher ramp control efficiency with quick response to various incident cases via refined strategies. Wang [16] proposed to use a linear programming method together with a moving-average technique to formulate the problem of ramp-metering control in response to nonrecurring congestion situations, where the ramp control rate was updated per 5 min in his approach to reduce incident impacts on mainline capacities of freeways. Rooted in a three-phase traffic theory, Kerner [8] proposed a congested pattern control approach (ANCONA) to address the issue of congestion propagation upstream from a freeway bottleneck. Although the potential advantages of Kerner’s model were demonstrated in comparison with the well-known ALINEA1 approach, the suitability of the embedded lane-changing rules used for diverse lane-blocking incident cases and the resulting effects on real-time ramp control performance were not investigated in Kerner’s study. In addition, a number of ALINEA-based ramp control approaches were continuously proposed [5,7,14], and however, nonrecurrent congestion problems caused by lane-blocking incidents on freeways remained unsolved in these studies. Apparently, real-time incident-responsive ramp control warrants further research to accomplish. As pointed out in Sheu et al. [17], integrating the functions of real-time incident impact prediction and incident-responsive traffic control is a critical stage in freeway congestion management. Furthermore, the gap between ramp control and freeway incident management still remains in the previous literature for the lack of sophisticated incidentinduced traffic prediction models to dynamically characterize incident impacts in the process of ramp-metering control. For instance, the efficiency of the method by Wang stated previously relies to a great extent on the accuracy in the estimation of aggregated vehicular travel time spent to go through the mainline detection zone while the effects of incident-induced lane traffic maneuvers such as lane changing and queuing may be ignored in this approach. Similar problems can also be found in the other ramp control strategies noted above. Accordingly, rooted in the stochastic optimal control methodology, this paper presents a new approach to real-time incident-responsive local ramp control. Under conditions of lane-blocking incidents on freeway mainline segments, local ramp-metering control is formulated as a stochastic optimal control problem with the goal of minimizing the incident impacts upstream to the incident spot. The most distinctive feature of the proposed control method is that the dynamics of incident-induced inter-lane and intra-lane traffic states as well as incident impacts can be estimated in real time, and then used as the parameters in the time-varying objective function to serve specific control purposes during lane-blocking incidents. The details of the primary procedures involved in the development of the proposed methodology and preliminary tests are described in the following sections. 1

ALINEA is the acronym for ‘‘Asservissement LINEeaire d’entree Autoroutiere’’ French for ‘‘linear ramp metering control.’’

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2. Specification of system states The system scope studied here aims at any given detection zone which is defined as the mainline segment bounded by the pair of detector stations located upstream and downstream to a given on-ramp, as shown in Fig. 1. To collect real-time raw traffic data used as the input of the proposed method, specific point detector layouts are proposed, mainly involving two types of detector stations: (1) the mainline detection stations and (2) the gate detectors implemented at the corresponding on-ramp of the target freeway system. Now, let us specify the problem background. Suppose that a lane-blocking incident occurs on the mainline segment of a given detection zone, and the lane traffic data, e.g., traffic counts, speeds, and occupancies, collected from the specified point detectors are available in each time interval. Before removing the incident, the local on-ramp metering nearby is the only one measure available to dynamically regulate the number of on-ramp vehicles entering into the detection zone in response to the time-varying incident impacts on the lane traffic flows upstream to the incident site. Therein, the lane traffic loads referring to the numbers of vehicles present in given lanes of given subsystems (i.e., d1i ðkjkÞ, d1j ðkjkÞ, d1l ðkjkÞ, d2i ðkjkÞ, d2j ðkjkÞ, and d2l ðkjkÞ) are used as indexes to characterize the space-based incident impact, where d1i ðkjkÞ, d1j ðkjkÞ, and d1l ðkjkÞ represent the time-varying lane traffic loads present in blocked lane i, adjacent lane j (i.e., the lane adjacent to block lane i) and independent lane l (i.e., the lane being neither blocked lane i nor adjacent lane j) of subsystem 1 (i.e., upstream to the incident site) at the end of time interval k, respectively, and similarly, d2i ðkjkÞ, d2j ðkjkÞ, and d2l ðkjkÞ represent the corresponding lane traffic loads of subsystem 2 (i.e., downstream to the incident site) observed at the end of time interval k. In reality, these space-based incident impacts are the derivatives of lane traffic states under ramp-metering control. For instance, d1i ðkjkÞ may include the new traffic arrivals entering into subsystem 1 and the preceding vehicles remaining in blocked lane i without conducting lane changing maneuvers during time interval k, and thus can be expressed as " # X   d1i ðkjkÞ ¼ a1i ðkÞ þ li  b1 ðkÞ þ d1i ðk  1jk  1Þ  1  p1ij ðkÞ , (1) 8j2J

point detector incident vehicle

ramp metering

downstream detector station

upstream detector station

Subsystem 1

Subsystem 2 detection zone

Fig. 1. Specification of detector configurations.

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where a1i ðkÞ represents the time-varying lane traffic counts collected from the detector installed in blocked lane i of subsystem 1 in time interval k; b1 ðkÞ is the time-varying lane traffic counts measured from the on-ramp detector station close to the mainline segment of the freeway in time interval k; p1ij ðkÞ represents the mandatory lane-changing fraction from blocked lane i to adjacent lane j in subsystem 1 in time interval k; for a given lane ‘‘s’’, the parameter ls is given by  1 if lane s is an outside lane; (2) ls ¼ 0 otherwise: Similarly, we can readily derive the other incident-impact indexes (d1j ðkjkÞ, d1l ðkjkÞ, d2i ðkjkÞ, d2j ðkjkÞ, and by h i   d1j ðkjkÞ ¼ a1i ðkÞ þ li  b1 ðkÞ þ d1i ðk  1jk  1Þ  p1ij ðkÞ  1  r1ij ðkÞ h i h i þ a1j ðkÞ þ lj  b1 ðkÞ þ d1j ðk  1jk  1Þ  1  r1j ðkÞ , ð3Þ

d2l ðkjkÞ)

    d1l ðkjkÞ ¼ a1l ðkÞ þ ll  b1 ðkÞ þ d1l ðk  1jk  1Þ  1  r1l ðkÞ , d2i ðkjkÞ ¼

Xn

(4)

½½a1j ðkÞ þ lj  b1 ðkÞ þ d1j ðk  1jk  1Þ  r1j ðkÞ þ d2j ðk  1jk  1Þ  p2ji ðkÞ

8j2J

o þd2i ðk  1jk  1Þ  ½1  r2ji ðkÞ ,

d2j ðkjkÞ ¼

8 < ½½a1i ðkÞ þ li  b1 ðkÞ þ d1i ðk  1jk  1Þ  p1ij ðkÞ  r1ij ðkÞ

ð5Þ 9 =

: ½½þa1j ðkÞ þ lj  b1 ðkÞ þ d1j ðk  1jk  1Þ  r1j ðkÞ þ d2j ðkÞ  ½1  p2ji ðkÞ ;

d2l ðkjkÞ ¼ f½a1l ðkÞ þ ll  b1 ðkÞ þ d1l ðkjkÞ  r1l ðkÞ þ d2l ðk  1jk  1Þg  ½1  r2l ðkÞ,

 ½1  r2j ðkÞ,

(6)

(7)

where a1j ðkÞ, and a1l ðkÞ represent the time-varying lane traffic counts collected from the upstream detectors in adjacent lane j and independent lane l of the mainline segment, respectively; p2ji ðkÞ is the discretionary lanechanging fraction from adjacent lane j to blocked lane i in subsystem 2 in time interval k; r1j ðkÞ and r1l ðkÞ represent the proportions of vehicles present in adjacent lane j and independent lane l in subsystem 1 which pass by the incident site in time internal k; r1ij ðkÞ refers to the proportion of vehicles conducting lane changes from blocked lane i to adjacent lane j which can pass by the incident site through adjacent lane j in time interval k; similarly, r2j ðkÞ and r2l ðkÞ represent the proportions of vehicles present in adjacent lane j and independent lane l in subsystem 2 which can pass the downstream detector station in time interval k; r2ji ðkÞ corresponds to the proportion of vehicles conducting lane changes from adjacent lane j to blocked lane i in subsystem 2 which pass the downstream detector station in time interval k; lj and ll are also the lane-based parameters which can be determined by Eq. (2). In reality, all the above lane traffic loads are derivable subject to the condition that the values of all the variables shown on the right hand side of Eqs. (1)–(7) are available. Therein, the variables of lane traffic counts (i.e., a1i ðkÞ, a1j ðkÞ, a1l ðkÞ, and b1 ðkÞ) are measurable using traffic detectors; however, the proportionrelated variables (r1ij ðkÞ, r1j ðkÞ, r1l ðkÞ, r2j ðkÞ, ðr2l ðkÞÞ, and r2ji ðkÞ) and lane-changing fractions (i.e., p1ij ðkÞ and p2ji ðkÞ) remain unknown, and need to be further estimated. In addition, the effect of ramp-metering control should be taken into account as it determines the number of on-ramp vehicles entering the freeway system (i.e., b1 ðkÞ), which may influence not only the lane traffic loads but also the dynamics of inter-lane and intra-lane traffic states exhibited upstream to the incident site. Here, we introduce a time-varying control variable (OðkÞ) to determine the length of the on-ramp green time in each unit time interval in the process of real-time ramp optimal control. Note that instead of employing the ramp-metering rate, which is broadly used as a control variable in the early literature, here OðkÞ is defined as a time-varying ratio of the green time to the length of a unit time interval, and thus

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is given by GðkÞ , (8) t where G(k) represents the length of the on-ramp green time in time interval k. Therefore, it is clear that to carry out real-time incident-responsive ramp-metering control, the aforementioned unknown lane traffic states (termed as the basic lane traffic states in the following) and ramp control variable must be predicted in real time in response to the time-varying incident impacts in each time interval toward the goal of dynamically alleviating the incident-induced traffic congestion to the greatest extent. OðkÞ ¼

3. Stochastic system modeling In the following development, we propose a discrete-time, nonlinear stochastic model to reproduce the timevarying relationships among the aforementioned basic lane traffic states, control variables, and traffic measurements under conditions of real-time ramp control for real-time prediction. The proposed stochastic model is composed of three groups of dynamic equations, including (1) state equations, (2) measurement equations, and (3) boundary constraints. These equations are respectively described below. 3.1. State equations The state equations denote the change patterns of the basic lane traffic states revealed in the temporal domain. Rooted in the early research of Sheu et al. [17] which utilized the features of random walk models to conceptualize the time-varying patterns of traffic states exhibited in the presence of an incident, we postulate the assumption that the time-varying basic lane traffic states may follow Gaussian–Markov processes to facilitate characterizing the time-varying relationships of the basic lane traffic states and their one-step-ahead projections in a stochastic system. That is, if there is no disturbance existing, the projections of the timevarying lane traffic states tend to merely follow the change patterns of the existing states, thus contributing to a deterministic system; otherwise, the next-time-interval state projections may oscillate around the existing system states by following Gaussian processes, where the magnitude of the oscillation depends primarily on the pattern of the disturbance as illustrated in Fig. 2. Based on the aforementioned postulation, we have the generalized form of the state equations given by X ðk þ 1Þ ¼ F ½xðkÞ; k þ L½xðkÞ; OðkÞ; kW ðkÞ.

(9)

prediction of system state Gaussian process

0

1

2

3……………………. time interval

Fig. 2. Illustration of the change pattern of a system state.

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In Eq. (9), X ðk þ 1Þ is a ½6nj þ 2nl   1 time-varying vector of basic lane traffic states (denoted by xðkÞ) in time interval k+1, where nj and nl represent the numbers of adjacent lanes and independent lanes, respectively; F ½xðkÞ; k represents a deterministic term of the state equations with a ½6nj þ 2nl   1 dimension of timevarying states estimated in time interval k; L½xðkÞ; OðkÞ; k is a ½6nj þ 2nl   ½6nj þ 2nl  state-dependent noise matrix; and W ðkÞ corresponds to a ½6nj þ 2nl   1 state-independent Gaussian noise vector. The mathematical forms of X ðk þ 1Þ, F ½xðkÞ; k, L½xðkÞ; OðkÞ; k, and W ðkÞ are given by Eqs. (10), (11), (12), and (13), respectively. 2

p1ij ðk þ 1Þ

3

7 6 1 6 rj ðk þ 1Þ 7 7 6 6 r1 ðk þ 1Þ 7 7 6 ij 7 6 6 r1 ðk þ 1Þ 7 7 6 l 7 6 6    7 7 6 X ðk þ 1Þ ¼ 6 p2 ðk þ 1Þ 7 , 7 6 ji 7 6 2 6 r ðk þ 1Þ 7 7 6 j 7 6 2 6 rji ðk þ 1Þ 7 7 6 7 6 2 6 rl ðk þ 1Þ 7 5 4 .. . ð6nj þ2nl Þ1 2

p1ij ðkÞ

(10)

3

7 6 1 6 rj ðkÞ 7 7 6 6 r1 ðkÞ 7 7 6 ij 7 6 6 r1 ðkÞ 7 7 6 l 7 6 6  7 7 6 F ½xðkÞ; k ¼ 6 p2 ðkÞ 7 , 7 6 ji 7 6 2 6 r ðkÞ 7 7 6 j 7 6 2 6 rji ðkÞ 7 7 6 7 6 2 6 rl ðkÞ 7 5 4 .. . ð6nj þ2nl Þ1 2 6 6 6 6 6 6 6 6 6 L½xðkÞ; OðkÞ; k ¼ 6 6 6 6 6 6 6 6 4

‘11 ðkÞ

0

0

‘22 ðkÞ

0 0

0 0

0 0

0 0

0

0

0 .. .

0 .. .

(11)

0

0

0

0

0

0



3

7 7 7 ‘33 ðkÞ 0 0 0 0 0 7 7 0 ‘44 ðkÞ 0 0 0 0 7 7 7 0 0 ‘55 ðkÞ 0 0 0 7 , 7 7 0 0 0 ‘66 ðkÞ 0 0 7 7 0 0 0 0 ‘77 ðkÞ 0 7 7 7 0 0 0 0 0 ‘88 ðkÞ    7 5 .. .. .. .. .. .. .. . ð6nj þ2nl Þð6nj þ2nl Þ . . . . . . 0

0

0

0

0

0

(12)

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2

wp1ij ðkÞ

371

3

7 6 6 wr1 ðkÞ 7 j 7 6 7 6 6 wr1ij ðkÞ 7 7 6 6 w 1 ðkÞ 7 7 6 rl 7 6 6    7 7 6 7 W ðkÞ ¼ 6 , 6 wp2ji ðkÞ 7 7 6 6 w 2 ðkÞ 7 7 6 rj 7 6 6 w 2 ðkÞ 7 7 6 rji 7 6 6 wr2 ðkÞ 7 7 6 l 5 4 .. . ð6nj þ2nl Þ1

(13)

where the elements of L½xðkÞ; OðkÞ; k take the following forms: " # X 1 ‘11 ðkÞ ¼ 1  ð1 þ lj Þ  OðkÞ  pij ðkÞ  ð1 þ lj Þ  OðkÞ  r1ij ðkÞ,

(14)

8j2J

" ‘22 ðkÞ ¼ 1 

X

# ð1 þ lj Þ  OðkÞ 

p1ij ðkÞ

 ð1 þ lj Þ  OðkÞ  p1ij ðkÞ þ ½1  ð1 þ lj Þ  OðkÞ  r1j ðkÞ,

8j2J

(15) " ‘33 ðkÞ ¼ 1 

X

# ð1 þ lj Þ  OðkÞ 

p1ij ðkÞ

 ð1 þ lj Þ  OðkÞ  p1ij ðkÞ,

(16)

8j2J

‘44 ðkÞ ¼ 1  ð1 þ ll Þ  OðkÞ  r1l ðkÞ, X ‘55 ðkÞ ¼ ½1  p2ji ðkÞ  r2ji ðkÞ,

(17) (18)

8j2J

‘66 ðkÞ ¼ ½1  p2ji ðkÞ  ½1  r2j ðkÞ, ‘77 ðkÞ ¼

X

½1  p2ji ðkÞ  p2ji ðkÞ,

(19) (20)

8j2J

‘88 ðkÞ ¼ 1  r2l ðkÞ.

(21)

Eqs. (10) and (11) indicate the change patterns of the basic lane traffic states under the condition of a deterministic system. As stated previously, without the disturbance of noise terms, the basic lane traffic states are assumed to merely follow the existing states to change one step ahead, exhibiting the Markov properties. Eqs. (12) and (13) characterize the noise terms that may divert the basic lane traffic states from Markovbased deterministic to Gaussian–Markov stochastic conditions. Essentially, our rationale of formulating these noise terms is rooted in the postulation that in a stochastic traffic flow system, the deterministic conditions of time-varying traffic states could be affected by two disturbance sources, state-dependent and stateindependent, as indicated by the elements of L½xðkÞ; OðkÞ; k, and W ðkÞ, respectively. In the study, the statedependent noise terms may originate from three potential disturbance sources, including (1) the amount of vehicles remaining in given lanes lasting for more than one time interval, (2) the ease with which vehicles pass a given subsystem in a given time interval, and (3) the effects of the upstream ramp-metering control (see Eqs. (14)–(21)). On the other hand, the magnitude of the state-dependent noises shown in L½xðkÞ; OðkÞ; k may also depend on the diversity of mainline traffic arrivals which are independent of the lane traffic states.

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Thus, the state-independent Gaussian noise vector (W ðkÞ) is involved to account for such an effect in the real-time state estimation. 3.2. Measurement equations The measurement equations denote the time-varying relationships between the measured lane traffic counts and the basic lane traffic states. In the proposed methodology, the measured lane traffic counts serve to update the prior predictions of the basic lane traffic states in real-time based on the specified time-varying relationships, and the corresponding update procedures are detailed in the next section. Here, we have the generalized form of the measurement equations given by ZðkÞ ¼ H½xðkÞ; k þ V ðkÞ,

(22)

where ZðkÞ is a ðni þ nj þ nl Þ  1 time-varying measurement vector which is composed of the elements in terms of the lane traffic counts collected from the downstream detector station in time interval k; similar to nj and nl defined previously, ni corresponds to the number of blocked lanes; H½xðkÞ; k is a ðni þ nj þ nl Þ  1 timevarying vector in which each element associates a specific combination of the basic lane traffic states and measured lane traffic arrivals with a given element shown in ZðkÞ to indicate the components of the measured downstream lane traffic counts; V ðkÞ is a ðni þ nj þ nl Þ  1 Gaussian noise vector which represents the measurement errors of the collected traffic counts in time interval k. ZðkÞ, H½xðkÞ; k and V ðkÞ are given, respectively, by 2 2 3 zi ðkÞ 6 2 7 6 zj ðkÞ 7 7 6 ZðkÞ ¼ 6 2 7 , (23) 6 zl ðkÞ 7 5 4 .. . ðni þnj þnl Þ1 2 Pn

½ða1j ðkÞ þ d1j ðk  1ÞÞ  r1j ðkÞ þ d2j ðk  1Þ  p2ji ðkÞ þ

d2i ðk1Þ nj

6 8j2J 6 6 fða1 ðkÞ þ d1 ðk  1ÞÞ  p1 ðkÞ  r1 ðkÞ þ ½ða1 ðkÞ þ d1 ðk  1ÞÞ 6 i ij ij j i j 6 2 1 2 2 H½xðkÞ; k ¼ 6 r ðkÞ þ d ðk  1Þ  ð1  p ðkÞÞg  r ðkÞ 6 j ji j j 6 6 1 1 2 6 f½al ðkÞ þ dl ðk  1Þ  r1l ðkÞ þ dl ðk  1Þg  r2l ðkÞ 4 .. . 2

vi ðkÞ

o

 r2ji ðkÞ

3 7 7 7 7 7 7 7 7 7 7 5

,

(24)

ðni þnj þnl Þ1

3

6 vj ðkÞ 7 7 6 7 V ðkÞ ¼ 6 6 vl ðkÞ 7 5 4 .. .

,

(25)

ðni þnj þnl Þ1

z2i ðkÞ,

z2j ðkÞ,

and z2l ðkÞ represent the lane traffic counts measured from the downstream detectors in where blocked lane i, adjacent lane j and independent lane l, respectively, in time interval k; vi ðkÞ, vj ðkÞ, and vl ðkÞ are the Gaussian measurement errors associated with these measured downstream lane traffic counts. 3.3. Boundary constraints The boundary conditions are specified in consideration of the requirement of feasible optimal solutions of the basic lane traffic states and control variables. In the proposed model, three boundary conditions associated with the estimates of the basic lane traffic states (xðkÞ), the predictions of ramp control variables (OðkÞ), and the minimum ramp-metering rate are formulated, respectively, and their generalized forms

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are given by 0pxðkÞp1

for all k ,

(26)

0pOðkÞp1

for all k ,

(27)

ny P

½Oðk þ Þ  t XT g;min

¼0

for all k and y ,

(28)

where ny is the maximum number of the sequential time intervals which belong to a given cycle y; T g;min represents the minimum on-ramp green time in a given cycle y. 4. Stochastic control algorithm Although there are diverse objective functions proposed in published ramp control strategies in an attempt to optimize specific performance measures such as delay, throughput, travel time, and speed, in this study we attempt to minimize the differences between the ideal and the estimated values of the basic lane traffic states in the presence of a lane-blocking incident in the detection zone. The ideal basic lane traffic states mentioned above are defined as the desired values of the basic lane traffic states that facilitate vehicular movement to the greatest extent under conditions of incident-induced traffic congestion, and herein, the upper bounds of the basic lane traffic states are considered to use. Therefore, we have the objective function (x) given by ( ) N X  T  T   x ¼ min E ½X ðkÞ  X ðkÞ QðkÞ½X ðkÞ  X ðkÞ þ ½OðkÞ  OðkÞ  RðkÞ½OðkÞ  OðkÞ  , (29) k¼0

where X(k) is a ½6nj þ 2nl   1 time-varying state vector containing the estimates of the time-varying basic lane traffic states; X  ðkÞ is a ½6nj þ 2nl   1 unit vector representing the target vector associated with X ðkÞ; QðkÞ represents the covariance matrix with respect to W ðkÞ, which is a ½6nj þ 2nl   ½6nj þ 2nl  time-varying diagonal, positive-definite matrix; N corresponds to the total number of time intervals in terms of incident duration; RðkÞ is the covariance matrix of V ðkÞ, which is a ½ni þ nj þ nl   ½ni þ nj þ nl  time-varying diagonal, positive-definite matrix; OðkÞ represents a ½ni þ nj þ nl   1 time-varying control-variable vector which involves the identical control variables associated with lanes i, j and l, respectively; O ðkÞ represents the target vector associated with OðkÞ. Note that each element in X  ðkÞ represents the ideal value of a given basic lane traffic state that can relieve incident-induced traffic congestion to the greatest extent, and conveniently, it is set to be 1 in this study; the elements of O ðkÞ are set to be the same as the elements of Oðk  1Þ to serve the purpose of minimizing the cost caused by ramp-control signal switching. Coincidentally, the aforementioned ramp control fundamental serves to generate the results similar to those under the ramp control strategies with the goals of maximizing the mainline throughput in case of low-volume incidents and minimizing the total delay in case of high-volume incidents. In addition, it is noted that here we incorporate both the covariance matrixes QðkÞ and RðkÞ into the objective function to quantify the time-varying weights associated with the estimation deviations of state and control variables. From the theoretical point of view, such a treatment is arguably agreeable as it may infer that a higher value of covariance may contribute to significant influence on the estimation performance, relative to a lower value of covariance. That’s the reason why these covariance matrixes are also taken into in the following stochastic optimal control process using Kalman filtering techniques. To execute the mechanism of real-time incident-responsive traffic control toward the aforementioned goals, a stochastic optimal control-based algorithm is developed using the extended Kalman filtering technique. The primary computational stages involved in the proposed algorithm include (1) system initialization, (2) prior prediction of system states, (3) stochastic optimal estimation of traffic states, and (4) determination of the time-varying ramp control variable. In order to obtain the minimum mean square estimates of the basic lane traffic states through the aforementioned stages (2) and (3), the fundamentals of an extended Kalman filter are applied. The concepts and distinctive features of extended Kalman filtering technologies can be readily found elsewhere [18,19], thus being omitted in this content. In addition, it is also worth mentioning that except the

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stage of system initialization which is conducted only when the algorithm is triggered at the beginning of the ramp control period, the other three computational stages are executed in sequence in each time interval until the incident impacts no longer exist. Fig. 3 presents the block diagram of the overall stochastic optimal control system, and the following summarizes the major computational steps of the proposed ramp control logic. Step 0: Initialize system states and the input raw traffic data. Given k ¼ 0, system states including (1) the basic lane traffic states X ð0j0Þ, (2) the covariance matrix of the state estimation error Fð0j0Þ, and (3) the covariance matrix Qð0Þ are initialized. In addition, let the time-varying ramp control-variable vector Oð0Þ ¼ 0 to ensure that the on-ramp vehicles do not contribute durably to deterioration on the incident impacts at the onset of the incident. Step 1: Compute prior predictions of basic lane traffic states (X ðk þ 1jkÞ) and the covariance matrix of the state estimation error (Fðk þ 1jkÞ), respectively, by X ðk þ 1jkÞ ¼ F ½xðkÞ; OðkÞ; k,

(30)

T Fðk þ 1jkÞ ¼ F_ ðkÞFðkjkÞF_ ðkÞ þ L½xðkÞ; OðkÞ; kQðkÞL½xðkÞ; OðkÞ; kT ,

(31)

T

where F_ ðkÞ is the transpose matrix of F_ ðkÞ; F_ ðkÞ is given by  qF ½xðkÞ; OðkÞ; k . F_ ðkÞ ¼  qX ðkÞ xðkÞ¼xðkjkÞ

(32)

Step 2: Calculate the Kalman gain by T _ þ 1ÞFðk þ 1jkÞH_ T ðk þ 1Þ þ Rðk þ 1Þ1 , Kðk þ 1Þ ¼ Fðk þ 1jkÞH_ ðk þ 1Þ½Hðk

(33)

_ þ 1Þ is denoted by where Hðk  qH½xðk þ 1Þ; k þ 1 _ Hðk þ 1Þ ¼ .  qX ðk þ 1Þ xðkþ1Þ¼xðkþ1jkÞ W(k)

(34)

L[x(k), Ω(k), k]

Ω (k)

+

B(k)

V (k) +

X (k)

+ X (k + 1)

H (k)

+

Z (k)

+ F (k)

Measurement subsystem

State subsystem

+ K(k) Controller

_

Kalman filter

H(k) ˆ k〉 X〈k

_

+

E(k) +

Xˆ k k − 1

+ +

F (k)

η (k)

Ω(k)

ˆ X〈k+1 k〉

+

B(k)

Fig. 3. The block diagram of the proposed stochastic optimal control system.

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Step 3: Update the prior estimates of the basic lane traffic states (X ðk þ 1jk þ 1Þ) by X ðk þ 1jk þ 1Þ ¼ X ðk þ 1jkÞ þ Kðk þ 1ÞDZðk þ 1jkÞ,

(35)

where DZðk þ 1jkÞ is given by DZðk þ 1jkÞ ¼ Zðk þ 1Þ  H½xðk þ 1jkÞ; k þ 1.

(36)

Step 4: Truncate the estimates of the basic lane traffic states (X ðk þ 1jk þ 1Þ) with the conditions of state boundaries, and normalize the estimated lane-changing fractions in a given blocked lane upstream to the incident site such that X p1ij ðk þ 1Þp1. (37) 8j2J

Step 5: Update the covariance matrix of the state estimation error (Fðk þ 1jk þ 1Þ) as _ þ 1ÞFðk þ 1jkÞ. Fðk þ 1jk þ 1Þ ¼ ½I  Kðk þ 1ÞHðk

(38)

Step 6: Update the states of the space-based incident impacts at the end of time interval k+1. In this step, the formulae of the space-based incident impacts, i.e., the lane traffic loads specified previously, are employed. Step 7: Calculate the control-variable vector Oðk þ 1Þ. According to the fundamentals of stochastic optimal control theories [18,19], the estimates of the basic lane traffic states (X ðk þ 1jk þ 1Þ) are fed back through the control gain matrix Eðk þ 1Þ by Oðk þ 1Þ ¼ Eðk þ 1ÞX ðk þ 1jk þ 1Þ þ Zðk þ 1Þ.

(39)

Herein, Eðk þ 1Þ and Zðk þ 1Þ are given respectively by Eðk þ 1Þ ¼ ½BT ðk þ 1ÞSðk þ 2ÞBðk þ 1Þ þ Rðk þ 1Þ1 BT ðk þ 1ÞSðk þ 2ÞF_ ðk þ 1Þ,

(40)

Zðk þ 1Þ ¼ ½BT ðk þ 1ÞSðk þ 2ÞBðk þ 1Þ þ Rðk þ 1Þ1 ½Bðk þ 1ÞQðk þ 1ÞX ðk þ 1Þ þ Rðk þ 1ÞOðk þ 1Þ, (41) where the matrix Sðk þ 2Þ should satisfy the Riccati equation shown as follows: T T Sðk þ 1Þ ¼ Qðk þ 1Þ þ F_ ðk þ 1ÞSðk þ 2ÞF_ ðk þ 1Þ  F_ ðk þ 1ÞSðk þ 2ÞBðk þ 1ÞEðk þ 1Þ

(42)

and B(k+1) is determined by Bðk þ 1Þ ¼

qF ½X ðkÞ; OðkÞ; k . qOðkÞ

(43)

Note that Zðk þ 1Þ is viewed as a modification term of the control gain matrix accounting for the aggregate weighting effect oriented from the estimation deviations toward the goal of stochastic optimal control. Step 8: Check the estimate of the time-varying ramp control variable to satisfy both the conditions of state boundaries and the minimum ramp-metering rate shown in Eqs. (27) and (28), respectively. Step 9: Check incident status by conducting the following rules: If the incident is removed and the queues in blocked lanes no longer exist, then stop the incident-responsive ramp control algorithm, and resume the normal ramp control mechanism. Otherwise, input the next-timeinterval raw traffic data; let the time interval index k ¼ k þ 1, and then go back to Step 1 to continue the control algorithm. 5. Numerical study The numerical study serves two major purposes: (1) testing the validity of the estimated system states and (2) evaluating the performance of the proposed incident-responsive ramp control model. They are described below. The scenario of testing the validity of the estimated system states is considerably important in this study. According to the proposed methodology, the time-varying control variables are dynamically determined based on the goal of minimizing the differences between the ideal and the estimated values of the basic lane traffic

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states. Therefore, the accuracy in the basic lane traffic estimation must be verified to ensure the validity of the proposed methodology. Considering the difficulty in gathering enough real incident-related traffic data for diverse incident cases, in this test scenario, we aimed to compare the estimated basic lane traffic states with simulation data generated from Paramics, Version 3.0 which is a microscopic traffic simulator was used in the test scenario. The Paramics simulator was calibrated prior to this study, and tasks related to evaluating, qualitatively and quantitatively can also be found in the early related research [20]. To simulate diverse lane-blocking incidents in a given detection zone, a simplified 3-lane freeway mainline segment which is 3 km in length, and comprises one signalized on-ramp was built using Paramics. The key parameters preset for simulation are summarized in Table 1. Lane-blocking incidents were mainly generated on the mainline segment of the study site, and then the output data simulated from Paramics were collected in each 10-s time interval. Here, 27 types of laneblocking incidents associated with diverse incident attributes including incident duration, incident location in a given lane, the lane blocked, and traffic flow condition were simulated, and a total of 1620 simulated data sets were gathered. Two types of statistics including (1) the chi-square (w2) statistics and (2) the mean absolute percentage error (MAPE) statistics were used as the performance measures which are given, respectively, by w2 ¼

N 1 X

~ ½xðkÞ  xðkjkÞ2 , xðkjkÞ k¼0

(44)

N1 P

 ðxðkÞ ~ ~   xðkjkÞÞ=xðkÞ

MAPE ¼

k¼0

 100%, (45) N ~ where xðkÞ is the simulation value of a given lane traffic state measured in time interval k; N represents the sample size associated with the given lane traffic state. In this test scenario, the significance level a ¼ 0.99 was used in the w2 tests, where a means the area or probability in the upper tail of the w2 distribution. Using the aforementioned performance measures, we compared the time-varying lane-changing fractions and lane traffic loads of the blocked lane upstream to the incident site with the corresponding simulation data as these two lane traffic states, herein, are regarded as two important variables indicating incident effects on inter-lane and intra-lane traffic flows in the study. Table 2 summarizes the corresponding numerical results which provide several generalizations depicted below. Overall, the results of w2 tests imply that the estimated system states are valid for further applications in incident-responsive ramp control. It can be found that all the estimates of w2 statistics shown in Table 2 for the estimation of either lane changing or queuing are less than the predetermined critical point. This indicates the acceptability of the goodness-of-fit test results. Moreover, the MAPE estimates support the generalization mentioned above since all the MAPE measures are less than 50% which is viewed broadly as a reasonable Table 1 Preset characteristics of the simulated study site Freeway mainline segment Geometric characteristics

Traffic characteristics

On-ramp segment Geometric characteristics

Number of lanes Lane width Detector spacing

3 4m 2 km

High-volume Medium-volume Low-volume Speed limit

6000 vph 3000 vph 1500 vph 100 kph (62.5 mph)

Number of lanes Ramp length Length of buffer lane for traffic merging

1 250 m 150 m

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Table 2 Summary of the results associated with the w2 and RMSE tests Type of incident case

High-outside-up Medium-outside-up Low-outside-up High-central-up Medium-central-up Low-central-up High-inside-up Medium-inside-up Low-inside-up High-outside-middle Medium-outside-middle Low-outside-middle High-central-middle Medium-central-middle Low-central-middle High-inside-middle Medium-inside-middle Low-inside-middle High-outside-down Medium-outside-down Low-outside-down High-central-down Medium-central-down Low-central-down High-inside-down Medium-inside-down Low-inside-down

w2 statistics (critical point: 37.48)

MAPE statistics

Lane-changing

Traffic load

Lane-changing

Traffic load

2.71 3.58 4.86 2.01 2.17 3.05 2.83 3.80 5.95 9.84 1.72 5.86 1.92 2.10 3.10 0.51 1.91 6.83 1.81 1.58 2.09 0.46 1.87 1.75 0.55 2.91 6.02

3.26 2.05 5.83 3.17 2.84 4.87 1.98 2.62 6.77 8.45 7.82 5.73 1.98 1.87 2.46 1.21 1.84 9.78 2.65 1.90 3.48 0.76 1.90 3.05 1.78 2.64 6.51

35.12 32.97 39.52 32.69 33.54 35.72 38.33 48.62 34.84 44.91 48.94 49.28 33.68 35.61 36.25 11.67 45.63 38.03 47.29 33.94 40.69 20.32 31.98 32.60 21.32 48.87 49.70

41.63 46.85 48.62 36.58 38.94 45.63 20.41 27.48 49.28 22.42 45.33 47.97 44.62 43.58 47.39 15.84 19.34 40.78 31.36 45.23 47.17 32.61 35.45 37.84 16.30 34.26 30.01

Note: high, high-volume (6000 vph); outside, outside-lane; medium, medium-volume (3000 vph); central, central-lane; low, low-volume (1500 vph); inside, inside-lane; up, upstream section; middle, middle-stream section; down, downstream section.

criterion to accept the estimation errors in MAPE tests. Nevertheless, it is worth noting that there may be the possibility of the proposed control method to suffer relatively high estimation error under the conditions of low-volume inside-lane incident events, as highlighted in Table 2. The major purpose of the second test scenario is to evaluate the performance of the proposed ramp control approach in comparison with that of two specific control strategies including pretimed control and controlfree modes which are widely implemented in the freeways of Taiwan. Given incident attributes the same as specified in the previous test scenario, diverse lane-blocking incidents under the control of the three specific ramp control strategies were simulated through Paramics, respectively. Therein, the proposed stochastic optimal control algorithm is coded with the C++ computer language. Using the simulated traffic data as the input, the time-varying control variables were estimated for a given time step, and then fed back to Paramics to obtain the simulated data for the next-time-step estimation. Such a routine was continuously conducted until the end of the simulated incident event. Note that as the corresponding computational time is quiet small, it is promising to implement the proposed algorithm for real-time applications. Measures including lane traffic loads, queue length, and the throughput downstream to the incident site were gathered for the use in comparison of the system performance. Table 3 lists the numerical results of this scenario by traffic flow conditions. Overall, the results summarized in Table 3 reveal a certain improvement in ramp control performance made by the proposed control method in contrast with the other two control strategies. Particularly, by comparing the results of the high-volume conditions, the system performance with respect to the reductions in queue lengths and lane traffic loads has been significantly improved under medium-volume conditions. One striking

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378 Table 3 Comparison of ramp control performance Incident case

Control strategy evaluation measure

Proposed method (mode-1)

Existing control (mode-2)

Free control (mode-3)

Improvement by mode-1 Mode-1 versus mode-2 (%)

Highvolume

Mediumvolume

Lowvolume

Lane traffic loads (veh/10-s) Queue length (veh/10-s) Throughput (veh) Lane traffic loads (veh/10-s) Queue length (veh/10-s) Throughput (veh) Lane traffic loads (veh/10-s) Queue length (veh/10-s) Throughput (veh)

Mode-1 versus mode-3 (%)

58

61

62

5.2

6.9

86

91

95

5.9

10.5

533

512

535

4.1

0.3

30

38

42

26.7

40

28

42

47

50.1

67.9

496

476

492

4.2

0.8

4

5

6

6

7

8

1.7

33.3

397

339

354

17.1

12.1

25

Fig. 4. Illustration of relative improvement by virtual samples of Paramics.

50

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379

example is that the relative improvement in terms of the queue lengths reaches up to 67.9% when compared with the control-free strategy in the medium-volume incident cases, as illustrated in Fig. 4, which is a virtual sample generated from Paramics in the process of simulation. As can be seen in Fig. 4, using the proposed incident-responsive ramp control strategy, the incident impact on queue lengths and lane densities exhibited on the mainline segment of the simulated freeway appear to be significantly improved in contrast with the performance of the control-free strategy. Such a consequence is understandable as explicated in formulating the objective function of the proposed stochastic model, the proposed ramp control strategy may contribute to minimizing the incident impacts under conditions of high-volume incident-induced traffic congestion. The aforementioned argument is also evidenced in medium-volume incident cases, and the improvement in control performance by the proposed method turns out to be more significant in these cases. In contrast, the proposed control method contributes to relatively significant effect in terms of improving the downstream throughput in response to low-volume incident cases. 6. Conclusions and recommendations This paper has presented a stochastic optimal control-based method in response to lane-blocking incidents on mainline segments of freeways. The proposed control approach performs incident-responsive local ramp control with the objective function of minimizing the time-varying function cost which is measured on the basis of comparing the real-time estimates of the inter-lane and intra-lane traffic states with their ideal values. To achieve the greatest reduction of incident impacts on traffic congestion in real-time via stochastic optimal control-based technologies, we specified three groups of time-varying lane traffic variables, and then proposed a discrete-time nonlinear stochastic model as well as a real-time ramp control algorithm. Our numerical results revealed the applicability of the proposed local ramp control method in terms of responding to incident-induced traffic congestion as well as estimating system states in real time under conditions of various incident cases on freeways. Results presented in the study also suggested the relative advantages of the proposed control method compared with two other specified ramp control strategies. More importantly, the proposed approach may indicate its potential in terms of characterizing incident-induced intra-lane and inter-lane traffic states together with incident impacts in real time in the procedure of real-time incident-responsive control, and by contrast, published advanced traffic control systems appear incomplete in providing such functionality to monitor the control performance in real time when an incident occurs. Nevertheless, more tests as well as comparisons with other advanced ramp control strategies warrant further research to verify the robustness of the proposed incident-responsive ramp control method, and its applicability to diverse incident cases. The extension of the proposed approach as well as system scope for the scenarios of corridor control will be undertaken in our further research. Moreover, efforts on either integrating the proposed ramp control method with other advanced traffic control and management technologies including variable message signs (VMS) seem to be needed in the development of incident management systems. Acknowledgments This research was supported by Grants NSC 95-2416-H-009-020-MY3 from the National Science Council of Taiwan. The author would also like to thank the editor Professor Gene Stanley and the referee for their constructive comments. Any errors or omissions remain the sole responsibility of the authors.

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