Paper No.

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Title: A Series of New Local Ramp Metering Strategies

Author(s):

E. Smaragdis, and M. Papageorgiou

Transportation Research Board 82nd Annual Meeting January 12-16, 2003 Washington, D. C.

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A Series of New Local Ramp Metering Strategies

Emmanouil Smaragdis1, and Markos Papageorgiou2

1

Dynamic Systems and Simulation Laboratory, Technical University of Crete, 73100 Chania, Greece, Tel: +30-8210-37309, Fax: +30-8210-69568, E-mail: [email protected]

2

Dynamic Systems and Simulation Laboratory, Technical University of Crete, 73100 Chania, Greece, Tel: +30-8210-69324, Fax: +30-8210-69568/69410, E-mail: [email protected]

Number of words (incl. Figures): 7710

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ABSTRACT The local ramp metering strategy ALINEA is the only available feedback strategy that is based on powerful and robust Automatic Control methods. A number of modifications and extensions of ALINEA are proposed in this paper, so as to address specific issues and needs that are not covered by ALINEA. More specifically, the following new local ramp metering strategies are presented: FL-ALINEA is a flow-based strategy; UP-ALINEA is an upstreamoccupancy-based version; UF-ALINEA is an upstream-flow-based strategy. X-ALINEA/Q is the combination of any of the above strategies with efficient ramp-queue control to avoid interference with surface street traffic. The new local ramp metering strategies are discussed with respect to their features, their limitations and their relation to available strategies. Macroscopic simulation investigations demonstrate the capabilities and limitations of the new strategies.

KEYWORDS: Ramp metering, Freeway control.

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INTRODUCTION Ramp metering, when properly applied, is a valuable tool for efficient traffic management on freeways and freeway networks. According to a recent overview (1), ramp metering strategies may be classified into: •

Reactive strategies (tactical level) aiming at maintaining the freeway traffic conditions close to pre-specified, set (desired) values by the use of real-time measurements and

•

Proactive strategies (strategic level) aiming at specifying optimal traffic conditions for a whole freeway or a freeway network based on demand and model predictions over a sufficiently long time horizon.

Both kinds of metering strategies may be combined within a hierarchical control structure, whereby a proactive network-wide strategy delivers optimal traffic conditions to be used as set (desired) values by sub-ordinate reactive strategies. Reactive ramp metering strategies may be local or coordinated. Local strategies make use of traffic measurements in the vicinity of each ramp, to calculate the corresponding individual ramp metering values, while coordinated strategies may use available traffic measurements from greater portions of a freeway. Local strategies are far more easy to design and implement; nevertheless, they have proved non-inferior to more sophisticated coordinated approaches under recurrent traffic congestion conditions (2). The most well-known local ramp metering strategies are the demand-capacity (DC) strategy (3), the occupancy (OCC) strategy (3) and ALINEA (4). The DC and OCC strategies are feedforward disturbance-rejection schemes that are based on mainstream measurements of flow and occupancy, respectively, upstream of the ramp; on the other hand ALINEA is a feedback regulator that is based on mainstream measurements of occupancy downstream of the ramp. It is probably due to this feedback structure, that ALINEA was found to lead to

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significantly better performance as compared to DC and OCC strategies in several comparative field-evaluations (2). According to a recent report (5), ALINEA has been implemented in various sites in five European countries. This paper suggests a number of modifications and extensions of ALINEA that enable several new aspects to be taken into account: •

Use of upstream (rather than downstream) measurements.

•

Use of flow-based (rather than occupancy-based) set values and measurements.

•

Efficient ramp-queue control to avoid interference with surface street traffic.

Moreover, development work related to the automatic real-time adaptation of set values so as to maximize the downstream freeway flow is in a final phase of investigation and will be reported in the near future. These new features may be used either separately or in combination. It is expected that the resulting variety of control laws will substantially broaden the spectrum of applications of local ramp metering and improve its performance. The paper presents the details of the corresponding modifications and extensions along with preliminary simulation investigations employing a macroscopic traffic flow model.

AVAILABLE LOCAL RAMP METERING STRATEGIES Local ramp metering strategies make use of real-time traffic measurements in the vicinity of a freeway ramp, to calculate suitable ramp metering flows. The strategies are activated at each time interval T, whose value is typically selected from the range 20s…60s. More specifically, at the end of each running period T, time-averaged measurements of traffic volume (flow) and occupancy from the ending period are used to calculate (via the corresponding strategy) the ramp flow to be implemented in the next period.

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The Demand-Capacity Strategy The demand-capacity (DC) strategy reads (3) qcap − qin (k − 1) if oin (k − 1) ≤ ocr r (k ) =  else rmin

(1)

where (Figure 1a) k=1,2,… is the discrete time index; r(k) is the ramp flow (in veh/h) to be implemented during the new period k; qin(k–1) is the last measured upstream freeway flow (in veh/h) (all lanes); oin(k–1) is the last measured upstream freeway occupancy (in %) (averaged over all lanes); qcap is the downstream freeway capacity; rmin is a minimum admissible ramp flow; ocr is the downstream critical occupancy (where the freeway flow becomes maximum, see Figure 1c). The DC strategy (Equation 1) attempts to add to the upstream flow qin(k–1) as much ramp flow r(k) as necessary to reach the known downstream freeway capacity. If, however, for some reason the last upstream measured occupancy oin(k–1) becomes overcritical (i.e. a congestion may form), the ramp flow r(k) is reduced to the minimum flow rmin, in order to avoid or to dissolve the apparent congestion. Finally, in order to avoid ramp closure, the ramp flow r(k) resulting from Equation 1 is truncated, if it is smaller than rmin. What we actually wish to control, however, are the freeway traffic conditions downstream of the ramp. This means that a genuinely feedback control strategy should involve (feed back) a measurement of the traffic conditions under control, else the strategy is blind with regard to the control outcome. Because Equation 1 is essentially based on the upstream flow qin(k–1), the DC strategy is not a feedback but a feedforward disturbancerejection scheme, which is generally known to be quite sensitive to various further nonmeasurable disturbances (e.g. a slow vehicle, a short shock wave, merging difficulties, etc.), see (4).

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The Occupancy (OCC) Strategy If the left-hand side of the fundamental diagram (Figure 1c) is approximated via a straight line, we have qin =

ν f ⋅ oin

(2)

g

where ν f is the free speed of the freeway and g is the g-factor. Replacing Equation 2 in the upper part of Equation 1, we get r (k ) = K1 − K 2 ⋅ oin (k − 1)

(3)

where K 1 = qcap , K 2 = ν f g and r(k) is truncated if it exceeds a range [rmin, rmax], where rmax is the ramp’s estimated flow capacity. Thus the OCC strategy (Equation 3) is an occupancybased feedforward strategy, which is even more inaccurate than the DC strategy due to the linearity assumption for the fundamental diagram and the uncertainty in the values of ν f and g (9).

The ALINEA Strategy ALINEA is a feedback ramp metering strategy (2,4) r (k ) = r (k − 1) + K R [o − oout (k − 1)] )

(4) )

where KR>0 is a regulator parameter and o is a set (desired) value for the downstream )

occupancy. Typically, but not necessarily, o = ocr may be selected, in which case the downstream freeway flow becomes close to qcap (see Figure 1b). ALINEA is an I-type regulator, hence it is easily seen that at a stationary state (i.e. if qin is constant), )

o = oout (k − 1) results from Equation 4, independently of the inflow value qin that is not

explicitly used in the strategy. Note that the same value of KR has been used in all known simulation or field applications of ALINEA without any need for fine-tuning.

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The ramp flow r(k) resulting from Equation 4 is truncated if it exceeds a range [rmin, rmax], where rmax is the ramp’s flow capacity. This truncated ramp flow value is used as r(k–1) in Equation 4 in the next time step, to avoid the well-known wind-up effect of I-type regulators. )

ALINEA reacts smoothly even to slight errors o − oout (k − 1) , thus stabilizing the traffic flow around the set value. Comparative field-evaluations in various countries demonstrated the clear superiority of ALINEA as compared to the DC and OCC strategies (2).

SIMULATION TESTING This paper suggests a number of modifications and extensions of ALINEA that enable several new aspects to be taken into account. The series of new strategies are tested by the use of the macroscopic simulator METANET (6), in order to investigate and demonstrate their basic impact and characteristics. METANET employs a validated second-order macroscopic traffic flow model (7), which is deemed sufficient for a first investigation of the proposed ramp metering strategies; further testing using microscopic simulation and field implementation at the Autobahn A94 near Munich, Germany, are planned for the near future within the European project RHYTHM (8). The simulation investigations aim at testing and demonstrating the control impact of the various strategies according to their specific control goal. To avoid interference of various phenomena that may deter attention from the specific control tasks, the simulation is conducted for a long, homogeneous three-lane freeway axis of 3 km with a single on-ramp located at km 2. The mainstream capacity (resulting from the utilised model parameter values) is about 6000 veh/h and is obtained at density values in the range of 35-40

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veh/km/lane. The space and time discretisation intervals for the macroscopic simulation model amount to 250 m and 5 s, respectively. “Upstream” measurements used by some ramp metering strategies correspond to the macroscopic variables of the segment just upstream of the ramp, while “downstream” measurements correspond to the macroscopic traffic variables of the segment where the on-ramp is merging (Figure 2a). Traffic density ρ is used in the controllers in place of traffic occupancy, because density is the variable closest to occupancy that may be directly provided by the macroscopic simulator. The time interval T used for all controllers in all investigations amounts to 30 s. The minimum admissible ramp flow rmin is set equal to 400 veh/h while the ramp’s flow capacity is rmax = 1600 veh/h. The same trapezoidal demand scenario displayed in Figure 2b for the mainstream and the on-ramp is used in all investigations (unless otherwise noted) over a time horizon of 4 h. More specifically, the demand starts with low values and increasing tendency; about 70 min later the total demand of mainstream and on-ramp exceeds the freeway capacity and remains at over-capacity levels for about 100 min before returning to low values. This demand scenario, which is quite typical for a peak period, was selected in order to test the strategy performance under different traffic conditions. Each investigation is performed twice: i. With deterministic demands (as in Figure 2b) and modelling equations in order to assess the control performance under visually clear conditions. ii. With stochastic noise added to both demands of Figure 2b and to some modelling equations, in order to test the control behaviour in a typical noisy environment; the same series of stochastic numbers is generated for all reported investigations. Figures 3 displays the simulation results obtained in the cases of no control and ALINEA application for the deterministic (i) and stochastic (ii) scenarios. In the no-control case the density (Figure 3a) reaches the critical range at about t=75 min, at which time the traffic volume (Figure 3c) reaches the capacity level of 6000 veh/h; as

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the density continues to increase (congestion) due to the uncontrolled high demand, the outflow drops to values around 5500 veh/h; the created congestion persists until about t=210 min, after which density and flow return to low values in accordance with the low demand. In the ALINEA case the density trajectory is identical to the one of no control until )

about t=85 min, after which it gets stabilised on the set value of ρ =37.5 veh/h, leading to a stable capacity flow of 6000 veh/h (Figure 3c). Figures 3b, 3d display similar control and no control behaviour as above, albeit only with regard to average values due to the introduced noise. For comparison, Figure 4 presents a sample of field results obtained via ALINEA application (in Glasgow, Scotland) based on occupancies. The resemblance with the simulation results of Figure 3b is apparent. Note that )

the occupancy values of Figure 4 start drifting from the set value o =26% after 17:50 h due to lack of demand, similarly to Figure 3b after t=170 min. Clearly the ramp metering action leads to the formation and dissipation of a ramp queue that will be explicitly addressed later in this paper. Note that due to the higher outflow the density under ALINEA becomes undercritical much earlier (at t=175 min), i.e. the demand is being served earlier, than in the no-control case. Thus the ramp metering action leads to a corresponding reduction of the total travel time spent by all vehicles.

NEW LOCAL RAMP METERING STRATEGIES Flow-based ALINEA (FL-ALINEA) )

ALINEA (Equation 4) is occupancy-based, i.e. both the set value o and the real-time measurements o(k) relate to traffic occupancies. A basic reason for using occupancies rather than flows, is that, as evidenced by Figure 1c, traffic volumes do not uniquely characterise the traffic state. More specifically, the same traffic volume may appear for non-congested and

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for congested traffic conditions due to the inverse U-shape of the fundamental diagram (Figure 1c). On the other hand, occupancy measurements may not be readily related to the classic traffic flow variables (density, traffic volume, mean speed) due to the uncertainties in the gfactor (see Equation 2) that is known to depend on the traffic composition (e.g. cars vs. trucks), the measurement device sensitivity and different installation conditions. Moreover in case of a central network-wide specification of set values (strategic level) for the local ramp metering (1), it may be easier to specify set values for flows than for occupancies. For these reasons it may be useful under certain conditions to apply the following flow-based version of ALINEA (FL-ALINEA)

r (k − 1) + K F [q − qout (k − 1)] if oout (k − 1) ≤ ocr  r (k ) =   else. rmin )

(5) )

The upper part of Equation 5 attempts to stabilise the flow qout around the set value q ; this will be possible without ambiguity, as long as the prevailing occupancy oout is undercritical, because under this condition, the measurement of flow characterises the traffic state in a unique way, and hence the course of action of the regulator is also uniquely determined. If, however, due to a multitude of possible reasons, the occupancy becomes overcritical, the action of the regulator may become counter-productive, hence r(k)=rmin should be set, so as to reach the undercritical area the sooner possible. Clearly, this lower part of Equation 5 should not be activated too often, else the frequent abrupt switches to r(k)=rmin may irritate the drivers at the on-ramp and, moreover, will lead to a bursty rather than smooth traffic )

volume trajectory; to avoid this, the set value q should not be close or equal to qcap (the boundary value of the undercritical region), hence FL-ALINEA is only recommended, if the )

set value is sufficiently smaller than qcap, e.g. q ≤ 0.9qcap .

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There is yet another issue worth mentioning, in case q is selected close or equal to qcap. Because qcap is an estimated value of the freeway capacity, there is a risk that the real capacity is actually lower or that it reduces, e.g. due to darkness or rain or other adverse environmental conditions. In these cases, Equation 5 would target a value that is not attainable in real traffic. Hence FL-ALINEA is definitely not recommended as a flowmaximising ramp metering strategy. The value of the positive regulator parameter KF may be chosen equal to 1, or slightly less for a more damped control behaviour. FL-ALINEA is also an I-type regulator, hence, )

under stationary conditions qout(k) becomes automatically equal to q through the action of Equation 5. The ramp flow resulting from Equation 5 must be truncated, if it exceeds rmax, and the truncated value must be used as r(k–1) in the next time step, to avoid wind-up. Figure 5 presents some simulation results using FL-ALINEA. Figure 5a presents the )

flow trajectory qout(k) in the case q =5400 veh/h, i.e. a set value 10% less than capacity. The flow qout(k) is seen to reach the set value at about t=75 min and eventually to be stabilised on )

q so long as the demand is sufficiently high. The undercritical region is not exceeded, hence

the lower part of Equation 5 is never activated. Figure 5b demonstrates a similar average behaviour under noisy simulation conditions. Note that the peak values of the demand profiles of Figure 2b were lowered by 6% in the cases of Figures 5a, 5b due to the lower set values. )

On the other hand, Figures 5c, 5d address the case q =6000 veh/h, i.e. a set value equal to capacity. Figure 5c depicts the resulting flow trajectories (deterministic scenario), when FL-ALINEA is applied with and without switching to the lower part of Equation 5. A bursty trajectory results (as expected), when the switching is included in the controller, because even slight departures from the undercritical region activate the lower part of Equation 5. When the switching is not included, the flow qout(k) appears to be stabilised on

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)

the set value q at the edge of the undercritical region. However, this stabilisation on capacity flow cannot be retained under the more realistic noisy scenario as demonstrated in Figure 5d on the basis of the resulting density trajectories ρ out (k ) . It may be seen, that, if the switching is not included in FL-ALINEA, the density departs from the undercritical region and a congestion is formed (leading to flow values around 5500 veh/h as in the no-control case) as a result of the noise and the flawed course of action of the regulator in the overcritical region. In contrast, when FL-ALINEA with switching is employed, the density remains in the critical area, leading to near-capacity flow, albeit with frequent switches to r(k)=rmin (as in Figure 5c) which may not be desirable in many cases.

Upstream-Occupancy Based ALINEA (UP-ALINEA) In some known cases, there has been an interest to implement or test ALINEA in the field, but this plan is hindered due to the lack of measurement devices downstream of the on-ramp, while measurement devices are available on the freeway just upstream of the on-ramp. This situation may be present due to previous installation of feedforward strategies (DC or OCC strategies) or due to other operational reasons. The issue to be addressed here, is the sensible application of the feedback ramp metering strategy ALINEA based on freeway measurements collected upstream of the ramp. Although it may sound paradoxical to apply a feedback strategy based on upstream measurements, this is in fact possible, if suitable estimates for oout(k) can be made available for use in Equation 4. To produce an estimate for oout(k), we start with the known relationship (the time index k is omitted for brevity)

ρ out =

qout

ν out

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where ν out is the freeway mean speed downstream of the ramp. An estimate q~out of the downstream flow may be obtained from q~out = qin + q ramp

(7)

where qramp is the measured ramp flow entering the freeway, which may sometimes be different from the ordered ramp flow r. For an estimate ν~out of the mean speed we assume

α ⋅ qin ~ νout = α ⋅ νin = ρin

(8)

where ν in is the freeway mean speed upstream of the ramp. Although ν out may sometimes be slightly lower than ν in due to the merging process, it is reasonable to assume that the factor α is roughly equal to 1. Replacing Equations 7, 8 in Equation 6 and recalling that o = ρ ⋅ g , we obtain the following estimate o~out (k ) of the downstream occupancy, based on measurements of the upstream occupancy, the upstream flow and the ramp flow qramp (k )  λin  o~out (k ) = oin (k ) 1 +  qin (k )  λout 

(9)

where λin resp. λout denote the number of mainstream lanes upstream resp. downstream of the ramp. The term λ in λ out results from the accordingly different g-factor for the two respective locations in this case. Thus UP-ALINEA is given by (compare with Equation 4) ) r (k ) = r (k − 1) + K R [o − o~out (k − 1)]

(10)

with o~out (k − 1) provided by Equation 9. All further issues and procedures, such as the values )

of o and KR, the truncation of r(k) etc., remain exactly the same as in ALINEA. Note that application of ALINEA on the basis of upstream occupancy measurements was already considered in (10) based on an assumed relationship o~out (k ) = A ⋅ oin (k ) , where A was supposed to be a constant parameter; an attempt was made in (10) to calibrate A using

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(simulated) time-series of oout and oin. In view of Equation 9, however, A depends on the ratio qramp/qin, hence it cannot be considered as a constant parameter. )

Another issue worth noting here is the way of selecting the set value o . For the ordinary ALINEA, a fundamental diagram, like the one of Figure 1c, may be obtained before ramp metering application by plotting (qout, oout)-measurement points, e.g. in order to visually )

estimate the value of ocr and set o = ocr . In absence of a downstream measurement device, these direct measurement points are not available, but they can be replaced by estimates

(q~out , o~out )

according to Equations 7, 9, respectively. Note that in this case, if α in Equation 8

is slightly less than 1, the corresponding small error is conveyed in the same way to the )

specified set value o and hence the regulation error in UP-ALINEA (Equation 10) becomes consistent, albeit with a slightly higher regulator gain. Figure 6 displays the UP-ALINEA simulation results for the deterministic and noisy scenarios. Comparing with Figure 3 it may be seen that there is hardly any difference visible between ALINEA and UP-ALINEA control results.

Upstream-Flow Based ALINEA (UF-ALINEA) For similar reasons that lead to the design of UP-ALINEA, one might be interested to implement and operate the flow-based ALINEA (FL-ALINEA) based on measurements collected upstream of the ramp. In this case, Equation 5 (FL-ALINEA) includes two downstream measurements qout(k) and oout(k) that need to be estimated by use of upstream measurements. This is readily accomplished via Equations 7 and 9, respectively, and thus UF-ALINEA is given by (compare with Equation 5) ) r (k − 1) + K F [q − q~out (k − 1)] if o~out (k − 1) ≤ ocr  r(k) =  r else   min

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with q~out (k − 1) and o~out (k − 1) provided by Equations 7 and 9, respectively. All further )

issues and procedures, such as the values of q and KF, the truncation of r(k) etc., remain exactly the same as in FL-ALINEA. It is interesting to note that if: •

KF is exactly equal to 1 (which is a reasonable choice) and

•

qramp(k–1) is equal to r(k–1) (which will often be the case) and

•

q is equal to qcap (which is not recommended as discussed earlier),

)

then UF-ALINEA becomes identical to the DC strategy (Equation 1) except for o~out (k − 1) replacing oin (k − 1) in the switching condition. We remark the following: i. The use of o~out instead of oin in the switching condition is deemed quite essential because o~out is likely to track earlier than oin any departure of the downstream occupancy from the undercritical region, hence the reaction of UF-ALINEA to such departures is expected to be quicker compared to the DC strategy. )

ii. For reasons detailed earlier, UF-ALINEA is not recommended for set values q

equal or close to qcap, while the DC strategy is typically employed with this set value. In summary, this discussion reveals some intrinsic relationships between UF-ALINEA and the DC strategy but also some inherent weaknesses of the latter. Figure 7 displays the UF-ALINEA simulation results for exactly the same cases as in Figure 5 for FL-ALINEA. Figures 5a, 5b are seen to be virtually identical to their counterparts, Figures 7a, 7b, which indicates a proper performance of UF-ALINEA for )

sufficiently low set value q . Regarding Figure 7c, the main difference compared to Figure 5c is observed for the case of UF-ALINEA including switching; more specifically, the observed number of switches to the lower part of Equation 11 is lower, compared to Equation 5 (Figure

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5c); this is attributed to the fact that, due to the assumption a =1 in Equation 8, the estimates o~out slightly underestimate the real oout values, hence the switching threshold in Equation 11

is reached less frequently than in Equation 5. In the more realistic noisy environment (Figure 7d) UF-ALINEA results are virtually equivalent to the FL-ALINEA results of Figure 5d when switching is included. When switching to the lower part of Equation 11 is not included, the density is seen to depart from the undercritical region leading to a congestion. The fact that the congestion forms later than in Figure 5d is attributed to the stochastic noise impact. Repeated simulation runs with different random number realisations lead always to a congestion, albeit at different (stochastic) times.

Ramp-Queue Control (X-ALINEA/Q) Ramp metering has a side effect, which needs to be addressed. This is the formation of a queue on the ramp, as a result of the fact that not all the vehicles which enter the ramp, are allowed to enter the freeway. If the queue exceeds a certain length, it will interfere with the adjacent street traffic. A popular countermeasure is to place a detector at the ramp entrance and release ramp metering when the loop occupancy exceeds a threshold (e.g. 50%). This is a rather rough logic, which could lead to an oscillatory override behaviour and under-utilisation of the ramp storage space (11). This could be avoided via a tighter ramp-queue control under the assumption that either a good estimate of the queue length or a measurement device such as a video sensor is available. The idea is to design a genuine ramp-queue controller whose set )

value w is chosen to be the maximum permissible queue length. To this end, we start with the conservation equation describing the ramp-queue dynamics w(k + 1) = w(k ) + T [d (k ) − r ′(k )]

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where w(k) is the current queue length, d(k) is the demand flow entering the ramp, and r ′( k ) is the flow of vehicles entering the freeway. Note that, in contrast to all other variable definitions in this paper, w(k) denotes the queue length at time-instant k ⋅ T , not the average value over the last time period. A so-called deadbeat controller results directly from Equation )

13 by demanding the queue length at the next time step to be equal to its set value w (and using the approximation d (k ) ≈ d (k − 1) ) as follows r ′(k ) = −

1 ) [w − w(k )] + d (k − 1) . T

(14)

This ramp queue regulator makes use of real-time measurements w(k) and d (k − 1) . Clearly, it does not make sense to apply the ramp-queue regulator when the freeway demand is low, as this would provoke an unnecessary queue formation. In fact the proposed ramp metering rate R(k) to be finally applied is given by

{

}

R (k ) = max r (k ), r ′(k )

(15)

where r(k) is the ramp metering rate decided by any of the preciously described ramp metering strategies. By selecting the maximum between both values, this combined strategy works as follows: If the freeway demand is low, the ramp metering strategy will allow for a high ramp flow r(k), since the freeway is far from being congested. In contrast to this, the )

queue regulator will deliver a low ramp flow r ′(k ), so that the queue grows towards w . By selecting the maximum ramp flow, we allow ramp metering to work as desired, that is without being obstructed by ramp queue considerations. On the other hand, if the freeway cannot accommodate the whole ramp demand without getting congested, the ramp metering strategy will calculate low ramp flows r(k), leading to a growing ramp queue, while the queue )

regulator will deliver higher ramp flows r ′( k ) in order to avoid for the queue to exceed w . Thus, when r ′( k ) becomes actually higher than r(k), the ramp-queue control will take over according to Equation 15, that is a queue override will take place, but only to the extent and

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for the duration necessary to avoid an over-long ramp queue, after which (via the same Equation 15) control will be automatically handed over to the ramp metering strategy, and so forth. This procedure is deemed preferable to a simultaneous ramp metering and ramp-queue control (12) because it sets clear priorities: Ramp-queue control is activated only when necessary and only to the extent necessary thus guaranteeing full utilisation of the ramp storage space and a proper operation without oscillations. Obviously, Equation 15 can be combined with any ramp metering strategy producing r(k), hence we add the attribute /Q to its usual name, e.g. ALINEA/Q, UP-ALINEA/Q, etc., to denote the resulting combined control strategy. Figures 8 and 9 display simulation results for ALINEA/Q, that is for queue control used in combination with the classical ALINEA. The ramp queue is seen to start growing around t=80 min (Figure 9) as a result of ramp metering. Until t=130 min the queue is not sufficiently long and the ALINEA ramp metering control prevails. Beyond that point, rampqueue control is seen to take over until the demand profiles decrease sufficiently. Note that )

the ramp queue is held exactly equal to w (Figure 9) so long as the queue regulator prevails, which guarantees a maximum exploitation of the available ramp storage. Note also that, when the queue regulator is activated, the density ρ out becomes overcritical (Figures 8a, b) and the flow qout drops (Figures 8c, d) due to the occurring congestion. This is the price to pay for the protection of the adjacent street traffic; the developed queue control method aims at minimizing this price.

CONCLUSIONS Ramp metering is a very efficient traffic control tool, provided it is designed and applied with a clear understanding of its impact, specific goals and limitations. The local ramp metering

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strategy ALINEA has been the only available feedback strategy that is based on application of powerful and robust Automatic Control methods. A number of modifications and extensions of ALINEA were proposed in this paper, so as to address specific needs that are not covered by ALINEA. More specifically, •

FL-ALINEA is a flow-based version of ALINEA.

•

UP-ALINEA is an upstream-occupancy-based version of ALINEA.

•

UF-ALINEA is an upstream-flow-based version of ALINEA.

•

X-ALINEA/Q is the combination of any of the above strategies with efficient ramp-queue control to avoid interference with surface street traffic.

A further extension related to the automatic real-time adaptation of set values that maximize the downstream freeway flow, is in a final stage of investigation and will be reported in the near future. The new local ramp metering strategies are discussed with respect to their features, their limitations and their relation to available strategies. Moreover, preliminary simulation investigations using the macroscopic simulator METANET demonstrate the capabilities and limitations of the new strategies. Further testing by use of microscopic simulators as well as field implementation at the Autobahn A94 near Munich, Germany, is planned within the European project RHYTHM.

ACKNOWLEDGEMENT This research was partly supported by the European Commission’s IST (Information Society Technologies) Program under the project RHYTHM (IST-2000-29427). All statements of the paper are under the sole responsibility of the authors and do not necessarily reflect European Commission’s policies or views.

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REFERENCES 1. Papageorgiou, M., and A. Kotsialos. Freeway ramp metering: An overview. In Proc. 3rd IEEE Intern. Conference on Intelligent Transportation Systems, Dearborn, Michigan, 2000, pp. 228-239. 2. Papageorgiou, M., H. Haj-Salem, and F. Middleham. ALINEA local ramp metering: Summary of field results. In Transportation Research Record 1603, TRB, National Research Council, Washington D.C., 1997, pp. 90-98. 3. Masher, D.P., D.W. Ross, P.J. Wong, P.L. Tuan, H.M. Zeidler, and S. Petracek. Guidelines for Design and Operation of Ramp Control Systems. Stanford Research Institute, Menlo Park, California, 1975. 4. Papageorgiou, M., H. Haj-Salem, and J-M. Blosseville. ALINEA: A local feedback control law for on-ramp metering. In Transportation Research Record 1320, TRB, National Research Council, Washington D.C., 1991, pp. 58-64. 5. Kenis, E., and R. Tegebos. Ramp Metering Synthesis. Report of the Centrico project, European Commission (DG TREN), Brussels, Belgium, 2001. 6. Technical University of Crete, and A. Messmer. The Documentation of METANET: A Simulation Program for Motorway Networks. Technical University of Crete, Dynamic Systems and Simulation Laboratory, Greece, 2000. 7. Papageorgiou, M., Blosseville, J-M., and H. Hadj-Salem. Modelling and real-time control of traffic flow on the southern part of Boulevard Périphérique in Paris−Part Ι: Modelling. Transportation Research Vol. 24A, 1990, pp. 345-359. 8. RHYTHM project website: http://www.ist-rhythm.com/index.htm.

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9. Jia, Z., C. Chen, B. Coifman, and P. Varaiya. The PeMS algorithms for accurate, realtime estimates of g-factors and speeds from single-loop detectors. In Proc. 4th IEEE Conference on Intelligent Transportation Systems, Oakland, California, 2001, pp. 538543. 10. Oh, H.-U., and V.P. Sisiopiku. A modified ALINEA ramp metering model. 80th Annual Meeting TRB, Washington D.C., 2001, paper no. 01-3096. 11. Gordon, R.L. Algorithm for controlling spillback from ramp meters. In Transportation Research Record 1554, TRB, National Research Council, Washington D.C, 1996, pp. 162-171. 12. Kachroo, P., K. Özbay, and D.E. Grove. Isolated ramp metering feedback control utilizing mixed sensitivity for desired mainline density and the ramp queues. In Proc. IEEE Intelligent Transportation Systems Conference, Oakland, California, 2001, pp. 633638.

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FIGURE 1 Local ramp metering strategies: (a) Demand-capacity, (b) ALINEA, (c) the fundamental diagram. FIGURE 2 Simulation environment: (a) Freeway topology and discretisation, (b) the utilised demand scenario. FIGURE 3

ALINEA versus no control results: Downstream traffic density ρout (k) (a)

deterministic, (b) stochastic scenario; downstream traffic volume q out (k) (c) deterministic, (d) stochastic scenario. FIGURE 4

)

FL-ALINEA results: Traffic volume q out (k) for q = 5400 veh/h (a) )

deterministic and (b) stochastic scenario; the case of q = 6000 veh/h with and without switching: (c) Traffic volume q out (k) for deterministic and (d) traffic density ρout (k) for stochastic scenario. )

FIGURE 5 Sample of ALINEA field implementation results in Glasgow, Scotland: o = 26% and resulting trajectory o out (k) . FIGURE 6

UP-ALINEA results: Density ρout (k) (a) deterministic and (b) stochastic

scenario; traffic volume q out (k) (c) deterministic and (d) stochastic scenario. FIGURE 7

)

UF-ALINEA results: Traffic volume q out (k) for q = 5400 veh/h (a) )

deterministic and (b) stochastic scenario; the case of q = 6000 veh/h : (a) Traffic volume q out (k) for deterministic and (d) traffic density ρout (k) for stochastic scenario.

FIGURE 8 ALINEA/Q results: Density ρout (k) (a) deterministic and (b) stochastic scenario; traffic volume q out (k) (c) deterministic and (d) stochastic scenario. FIGURE 9 ALINEA/Q results: Ramp queue w(k ) (a) deterministic and (b) stochastic scenario.

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qin oin

qout oout

(a) r Demand-Capacity Strategy

qcap

FEEDFORWARD (open loop)

qin

qout

oin

oout

(b)

r

r(k) = r(k −1) + KR[o −oout(k −1)] )

)

o

qout

ALINEA

qcap (c)

ocr

oout

FIGURE 1 Local ramp metering strategies: (a) Demand-capacity, (b) ALINEA, (c) the fundamental diagram.

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upstream segment

ρin vin

merge segment

ρout vout

qin

qout

r w (queue)

(a)

d (demand)

ramp demand

freeway demand

8000

demand (veh/h)

7000 6000 5000 4000

(b)

3000 2000 1000 0 0

60

120

180

time (min)

FIGURE 2 Simulation environment: (a) Freeway topology and discretisation, (b) the utilised demand scenario.

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no control

no control

ALINEA

70

8000

60

7000

40 30 20

5000 4000 3000 2000

10

1000 0

0 0

60

120

0

180

60

no control

180

(c)

(a) set value

120

time (min)

time (min)

ALINEA

no control

70

8000

60

7000

50

6000

flow (veh/h)

density (veh/km/lane)

ALINEA

6000

50

flow (veh/h)

density (veh/km/lane)

set value

40 30 20

ALINEA

5000 4000 3000 2000

10

1000

0

0 0

60

120

time (min)

180

0

60

120

180

time (min)

(d) (b) FIGURE 3 ALINEA versus no control results: Downstream traffic density ρout (k) (a) deterministic, (b) stochastic scenario; downstream traffic volume qout (k) (c) deterministic, (d) stochastic scenario.

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FIGURE 4 Sample of ALINEA field implementation results in Glasgow, Scotland: ô=26% and resulting trajectory oout(k).

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set value

FL-ALINEA

8000

8000

7000

7000

6000

6000

flow (veh/h)

flow (veh/h)

set value

5000 4000 3000

FL-ALINEA

5000 4000 3000

2000

2000

1000

1000

0

0

0

60

120

180

0

60

time (min)

120

180

time (min)

(c)

(a) set value

FL-ALINEA

FL-ALINEA 70

7000

60

density (veh/km/lane)

8000

6000

flow (veh/h)

FL-ALINEA without switching

5000 4000 3000 2000 1000 0

FL-ALINEA without switching

50 40 30 20 10 0

0

60

120

time (min)

180

0

60

120

180

time (min)

(b) (d) ) FIGURE 5 FL-ALINEA results: Traffic volume qout (k) for q = 5400 veh/h (a) deterministic and (b) stochastic scenario; the case )

of q = 6000 veh/h with and without switching: (c) Traffic volume qout (k) for deterministic and (d) traffic density ρout (k) for stochastic scenario.

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no control

UP-ALINEA

no control

70

8000

60

7000

40 30 20

5000 4000 3000 2000

10

1000

0

0 0

60

120

0

180

60

time (min)

set value

120

180

time (min)

(c)

(a) no control

no control

UP-ALINEA

70

8000

60

7000

UP-ALINEA

6000

50

flow (veh/h)

density (veh/km/lane)

UP-ALINEA

6000

50

flow (veh/h)

density (veh/km/lane)

set value

40 30 20

5000 4000 3000 2000

10

1000

0

0 0

60

120

time (min)

180

0

60

120

time (min)

(d) (b) FIGURE 6 UP-ALINEA results: Density ρout (k) (a) deterministic and (b) stochastic scenario; traffic volume qout (k) (c) deterministic and (d) stochastic scenario.

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UF-ALINEA

set value

8000

8000

7000

7000

6000

6000

flow (veh/h)

flow (veh/h)

set value

5000 4000 3000

UF-ALINEA

5000 4000 3000

2000

2000

1000

1000

0

0 0

60

120

0

180

60

time (min)

120

180

time (min)

(c)

(a) set value

UF-ALINEA

UF-ALINEA 70

7000

60

density (veh/km/lane)

8000

6000

flow (veh/h)

UF-ALINEA without switching

5000 4000 3000 2000 1000 0

UF-ALINEA without switching

50 40 30 20 10 0

0

60

120

time (min)

180

0

60

120

180

time (min)

(d) (b) ) FIGURE 7 UF-ALINEA results: Traffic volume qout (k) for q = 5400 veh/h (a) deterministic and (b) stochastic scenario; the )

case of q = 6000 veh/h : (a) Traffic volume qout (k) for deterministic and (d) traffic density ρout (k) for stochastic scenario.

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no control

ALINEA

ALINEA/Q

no control

70

8000

60

7000

50

6000

flow (veh/h)

density (veh/km/lane)

set point

40 30 20

ALINEA/Q

5000 4000 3000 2000

10

1000

0

0 0

60

120

180

0

120

60

time (min)

set point

180

time (min)

(c)

(a) no control

ALINEA

no control

ALINEA/Q

ALINEA

ALINEA/Q

8000

70

7000

60

6000

50

flow (veh/h)

density (veh/km/lane)

ALINEA

40 30 20

5000 4000 3000 2000

10

1000

0

0 0

60

120

time (min)

180

0

60

120

180

time (min)

(d) (b) FIGURE 8 ALINEA/Q results: Density ρout (k) (a) deterministic and (b) stochastic scenario; traffic volume qout (k) (c) deterministic and (d) stochastic scenario.

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set point

ALINEA

ALINEA/Q

500 450

queue (veh)

400 350 300 250 200 150 100 50 0 0

60

120

180

time (min)

(a) set point

ALINEA

ALINEA/Q

500 450

queue (veh)

400 350 300 250 200 150 100 50 0 0

60

120

180

time (min)

(b) FIGURE 9 ALINEA/Q results: Ramp queue w(k) (a) deterministic and (b) stochastic scenario.

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