- Email: [email protected]
A Low Cost Tool for Freeway Ramp Metering
Copyright © IFAC Low Cost Automation, Buenos Aires, Argentina, 1995
A LOW COST TOOL FOR FREEWAY RAMP METERING
M. Papageorgiou
I)
I),
H. Haj-Salem Z)
Dynamic Systems and Simulation Laboratory, Technical University o/Crete, 73100Chania, Greece 2) INRETS-DART, 2 QV. du Gl. Mal/eret Joinville, 94114 Arcueil, France.
Abstract: Local, traffic-responsive ramp metering is a wellknown control measure aiming at ameliorating traffic conditions on interurban or metropolitan freeways and motorways. A particular control strategy (ALINEA), the only local ramp metering strategy to be based on rigorous application of classical regulator theory, is presented, along with a low cost implementation concept. Field application of ALINEA and evaluation of its perfonnance in France and The Netherlands confinn its clear superiority as compared to other popular approaches. Keywords: Closed-loop control; control equipment; disturbance rejection; implementation; integral control; ramp metering; sample-data control; traffic control.
1. INTRODUCTION
of the local, traffic-responsive type. Hundreds of ramp metering installations, mainly in U.S.A. but increasingly also in Europe and elsewhere, have provided valuable experience about the benefits of this kind of control measure.
Freeways and motorways have been originally conceived so as to provide the possibility of quick travel without delays, both in metropolitan and in interurban areas. However, in the last decades, the number and extent of freeway congestions have been steadily increasing, leading to considerable delays, increasing fuel consumption and environmental pollution, and decreasing road safety. Congestions are caused either by high demand that exceeds the freeway capacity (daily recurrent congestions) or by capacity reducing incidents (nonrecurrent congestions). In presence of congestion, the freeway traffic flow becomes lower than capacity, thus rendering the utilization of the expensive infrastructure non optimal. One proposed way of ameliorating this situation is ramp metering by use of traffic lights at the freeway on-ramps. This control measure aims at limiting access to the freeway mainstream so as to achieve and maintain capacity flow .
From a control engineering point of view, the control laws implemented in the vast majority of existing ramp metering systems may be characterized as naive or heuristic. More precisely, these control laws attempt a sort of feedforward disturbance rejection that makes them particularly sensitive to a variety of changing conditions. The first local ramp metering control strategy that has been based on straightforward application of classical feedback control theory is ALINEA (Papageorgiou, et aI. , 1991). This paper presents in some detail the derivation, the features and possible extensions of ALINEA, along with an implementation concept and field results. 2
Ramp metering control strategies have been proposed at several levels of sophistication, (see Papageorgiou, 1991, for classified reviews of methods) but the high majority of implemented and operating systems are Fig. 1. The ramp metering process. 49
2. THE BASIC PROBLEM
2.1 The ramp metering problem Figure depicts schematically the motorway mainstream and the on-ramp. The following quantities are defmed: qout and '!in are the measurable mainstream traffic volumes or flows (veh/h) downstream and upstream of the ramp respectively. 00ut and 0in are the measurable mainstream occupancy rates downstream and upstream of the ramp respectively. An occupancy rate measures the time occupancy (in %) of a detector placed below the highway pavement. r is the measurable on-ramp traffic volume or flow (veh/h). r may be controlled using ordinary traffic lights, either on an one-car-per-green basis, or on a n-cars-per-green basis (with n=2, 3 or more), or on the basis of a fixed traffic cycle subdivided into a green and a red phase of controllable duration. 8 is the distance between sites 1 and 2. A simple model relating q and 0 at a given site is provided by the wellknown fundamental diagram q=Q(o) (Fig. 2) having the typical shape of an inverse D, where o=ocr is the critical occupancy resulting in maximum (or capacity) flow qcap= Q(ocr)' For 0 close to zero or close to a maximum occupancy 0max' the corresponding flow approaches zero. It is the main aim of a ramp metering installation to
control r so as to keep the downstream mainstream flow qout near a set value 4 , e.g. 4 =qcap· Alternatively, one may attempt to regulate 00ut to a set value 6 , e.g. 6 =ocr'
2.2 Constraints All control strategies calculate suitable ramp volumes r. In the case of traffic cycle realisation, r is converted to a green phase duration g by use of
100 oeeupwu:y 0
Fig. 2. The fundamental diagram .
In the case of n-cars-per-time realization, one typically has a constant-duration green light that permits exactly n vehicles to pass. The ramp volume r is controlled by varying the red phase duration between a minimum and a maximum value. If the queue of vehicles on the ramp becomes excessive, interference with surface street traffic may occur. This may be detected by suitably placed detectors and leads to an override of the regulator decisions, so as to allow more cars to enter the motorway and hence the queue to diminish. Above specifications and constraints apply in the same way to all control strategies.
2.3 Popular control strategies All known strategies but ALINEA are based on a feedforward disturbance rejection principle, that makes them particularly sensitive and not sufficiently accurate.
The demand-capacity strategy, that is extensively used in the USA (Masher, et aI. , 1975; Koble, et aI., 1980), is based on measuring qin and comparing with qc,p' However, because the value of traffic volume alone is insufficient to determine whether the motorway is congested or free flowing (see Fig. 2), the occupancy 00ut is also used according to the following scheme applying at each period k= 1, 2, 3, ... q cap -q in (k) r(k) =
{
(2) rmin
otherwise
where rmin is a minimum ramp volume value and othres is a threshold of occupancy.
(1)
g = (r/rsaJC
where C is the fixed traffic cycle duration and r sat is the ramp capacity flow (or saturation flow) that may be fixed or estimated in real time, based on ramp flow measurements filtered over some past cycles. g is constrained by gE[gmin, gmax] , where gmin>O to avoid ramp closure, and gmax:$;C.
The occupancy strategy (Masher, et al. , 1975; Koble, et al., 1980), that is in use in the USA, is essentially based on the same philosophy as the demandcapacity strategy, i.e. a) measure qin and add r so as to maintain downstream capacity and b) in case of congestion switch to a minimum value rmin' The difference between percent-occupancy and demand-capacity strategies is the following: a) Upstream flow CLn is estimated using occupancy measurements. The main reason for this is that only one detector (e.g. in a middle lane) may be used for estimation of qin' 50
b) Congestion is detected by the upstream detector (thus the method needs only one mainstream detector station).
3.2 Regulator design An appropriate feedback law for the process (7) is given by the following I-regulator
3 DERIV AnON OF ALINEA
3.1 Modelling
(8)
The conservat;on of vehicles between sites I and 2 gives
pet) = [Cfu,(t)+r(t) -
qout(t»)/O
where KR>O is a regulator parameter. Applying (8) to (7), we obtain the z-transfer function of the closedloop system
(3)
where the traffic density p (veh/km) is defmed as the number of cars included in the stretch, devided by the length 8 , t being the time argument. Because traffic density is not readily measurable (it requires video detectors), it is convenient to replace pet) in (3) by 0out(t), using the approximate relationship p = a 0out, where a = ~(lOOA), ~ being the number of lanes of the mainstream and A being the mean effective vehicle length (as "seen" by the underground electromagnetic loop detector). Thus, substituting also qout=Q(oouJ in (3), a nonlinear first-order dynamic model is obtained. This model may be linearized around a nominal steady-state (oout, Cl in , f), such that
A time-optimal deadbeat regulator is obtained by choosing KR=Q ' . Because Q' >0, the linearisation of (7) is strictly valid only on the left-hand side of the fundamental diagram. However, even for congested traffic, the feedback law (8) leads to reduction of traffic occupancy and can thus be applied in the same way. If site 2 is located at a bottleneck further downstream and/or T is chosen short, the entering traffic may not reach site 2 at the end of each time interval, in which case ~ cannot be neglected. Applying the regulator (8) to the original model (6), the closed-loop ztransfer function becomes
00ut = 0; Clout = Q(oout); f = Clin- Clout . (4) With the notation A·(t)=·(t)--:- , the linearisation yields
(10) with the eigenvalues ~1/2 = ~±j[~{I-a»)1I2. Thus the closed-loop system remains always stable, since ~< I is always verified. Nevertheless, the transient behaviour for high ~-values may be slow. In these cases, an amelioration may be achieved by application of a PI-regulator
where ()' =dQ( 6 )/do out , i.e. er is the slope of the tangent of the fundamental diagram at 6 , and hence its value is positive on the left-hand side of the fundamental diagram. Because the control input r is updated every T (typically T = C = 40 ... 80s), time discretisation of (5) with sample T yields
r(k) = r(k-I) - KR[6 -ooutCk») - Kp[oout(k)-ooutCk-I») (11)
AOoulk+ I) = ~Aooulk) + [(I-In! Q' ][A'Ln(k)+ Ar(k») (6)
where Kp>O is a further regulator parameter. Deadbeat regulation is achieved with KR=Q' , Kp = ~Q' l(l-a) .
where k=O, I, 2, .. .. Thus 0out(k) is the occupancy at time kT, and A'Ln(k), ruCk) are assumed constant over the interval [(k-I)T, kT) . The parameter ~ results from the discretization to be ~ = exp(-Q ' . TI
3.3 Elimination of constant disturbances Regulator (8) is capable of elinifuating constant disturbances qin when applied to (7). In fact we have
(a·o)). ~ may be neglected, if the ratio ofT is sufficiently small. This will be the case if traffic volume entering the motorway reaches site 2 during the time interval T. In this case, the effect of entering traffic r(k) will be visible at site 2 before the end of the corresponding time interval. Setting ~=O in (6)
0out(Z) = qin(z)
(z-I)/Q'
(12)
z(z-I+KR /en
which is zero in the steady-state, i.e. for z=l. 51
What happens if the actual ramp volume is biased (i.e. equal to r+d) compared with the ramp volume r ordered by the regulator (e.g. due to inaccurate rsat in (1) or due to red light violations)? Such a bias d acts in (7) as a disturbance, in the same way as ctn does. Hence, the bias is automatically eliminated by the control system. This statement does not hold true, if the actual (measured) ramp volume is used as a retarded value r(k-l) in the feedback law (8) (as will be recommended in the next section for anti-wind up purposes). In this case, r(k-l)+d(k-l) takes the place of r(k-l) in (11), and we obtain the z-transfer function 00ut ( z )
d(z)
l/Q ~'
= -----''----
(13)
volumes, be visible in the measurements. A distance of 40m in Paris (Haj-Salem, et al., 1990) was found to be adequate as well as a distance of 400m in Amsterdam (Middelbam and Smulders, 1991). Measurement of r(k) may also be necessary, e.g. for real-time estimation of rsat, or for use in (8). However, ALINEA is also applicable directly to the green or red phase duration, which circumvents the need of estimating rsat and measuring r. In fact, combining (1) and (8), we obtain instead of (8) the feedback law g(k) = g(k-l) - K~ [6 -oout(k)]
(14)
where K~ =KRC/rsat• Regulator (14) has not been tested extensively in the field as yet.
z-I+(KR/Q')
Thus, for constant d, (13) yields a steady-state error of d/KR.
4. DISCUSSION OF ALINEA FEATURES The simple regulator (8) has been proposed by Papageorgiou, et al. (1991) for local, traffic responsive ramp metering under the name ALINEA (Asservissement LIneaire d' Entree Autoroutiere). Note that both the demand-capacity and the occupancy strategies react to excessive occupancies oout only after a threshold value is reached, and in a rather crude way, whereas ALINEA reacts smoothly even to slight differences 6 -oout(k) and may thus prevent congestion in an elegant way, stabilizing traffic flow at a high throughput level. It is interesting to note that the feedback structure of ALINEA appears to some members of the Traffic Engineering community, with little or no knowledge of Automatic Control theory, to be peculiar or even counter-intuitive. In order to avoid wind-up phenomena of the 1regulator, the measured actual ramp volume r(k-I) should be used in (8), any time one of the constraints mentioned in section 2.2 becomes active. In all other cases, r(k-I) should be equal to the last calculated ramp volume in order for the regulator to be able to eliminate possible deviations between the ordered and actual ramp volumes, as discussed in section 3.3. In field experiments, it was found that ALINEA is not very sensitive w.r.t. the choice of the regulator parameter KR. A value of K R=70 veh/h was found to yield excellent results in many different sites. ALINEA requires only one mainstream detector station for 00ut downstream of the ramp entrance. The measurement location should be such that a congestion, originating from excessive on-ramp 52
It was demonstrated in section 3.3 that, if 'lin is constant, then ALINEA leads to zero offset, whereas for time varying ctn ALINEA acts according to (12) as a smoothing filter.
The set value 6 in (8) is provided by the user. This set value may be changed any time, and hence ALINEA may be directly embedded into a hierarchical control system with set values of the individual ramps being specified in real time by a superior coordination level or by an operator. The main reason for regulating occupancy, rather than volume, is due to the fact that traffic volume may have the same values both for light and congested traffic, see Fig. 2. An additional advantage in the case 6 =oc.. i.e. regulation to capacity flow, is that the critical occupancy ocr seems to be less sensitive w.r.t. weather conditions and other influences compared with the capacity qcap of a motorway stretch, see Keen, et aI., 1986. As a consequence, considering the set value 6 =ocr is a more robust and reliable way of achieving capacity flow than considering Cl. =qcap, because variations of qcap, caused by environmental or other conditions, are stronger compared with variations of ocr. Two possible extensions may be suggested if b!T»KR/a. The first extension is to use the PI regulator (11) with Kp=(ablT)-KR' as discussed in section 3.2. The second extension is to add to the feedback laws (8) or (11) a term Y['lin(k)-ctn(k-I)], if the value ctn(k) can be predicted accurately by use of further upstream measurements. This second extension corresponds to a feedforward disturbance rejection, with a positive smoothing parameter y5, I. These extensions were not tested in the field. Finally, it should be noted that, if the output site 2 is located far downstream of the on-ramp (at a downstream bottleneck), there is a risk of congestion, due to incidents or other disturbances, in the interior of the stretch (1 ,2), that will not be visible at the
output measurement site 2. In such cases, the feedback (or any other) control system may be of little help for eliminating the congestion, if no additional measurement stations are provided.
5. A LOW COST IMPLEMENTATION CONCEPT For the implementation of the ramp metering strategy, we can distinguish three main processes: - Real data collection - Calculation of the metering rate - Activation of the signal light. In order to minimize the implementation cost, the used hardware corresponds to a well-known IBM-PC or Compatible including a simple and specific hardware board which was developed at INRETS and commercialised since 1988 by a private company BECR. The cost is around 250$ US. This interface is called MASTER-PC2 (see Haj-Salem, 1986).
Fig. 4. Occupancies and constraints activation. Signs (VMS). MASTER-PC2 has been used for the implementation of ramp metering strategies on the Corridor Peripherique of Paris. The overall architecture can be summarised as follows: The signal loop data are transmitted to the central room though phone lines. The receiver, located at the central room, identifies each signal loop, which are considered as an input of MASTERPC2 board. The developed software performs data collection, data screening, application of the selected strategy, and command of the signal light according to the output of the ramp metering strategy (metering rate). For the Corridor Peripherique, the ramp metering system covers around 12 km of the motorway axis including 60 loop detectors and three signal lights (for three on-ramps).
In general the traffic data collection process using
loop detectors is based on the software elementary signal loop scrutinising. The scrutinise method is simple but it needs a powerful Central Processor Unit (CPU) and optimised software. On the other hand, the accuracy of the measurement is very much related to the used scrutinise time slice (e.g. 1 msec). The approach for data collection used in MASTERPC2 board is based on the hardware sampling loop detector signal with a hardware aggregation of the collected traffic variables. The start of loop detector sampling is done automatically using a very simple hardware logic. The time slice sampling is fixed by the user using a MASTER-PC2 internal and programmable clock. Three programmable internal clocks are available, one for the signal sampling (see Fig. 3) and two others for programming the data collection time slice (1 sec or more) and the signal light duration. All the counters are mapped in the central memory. This means that the data process is limited to one memory instruction read from the overall counters at each time slice in order to keep the aggregated information (traffic volume, occupancy) issued from MASTER-PC2 board. MASTER-PC2 is able of generating 16 binary control signals (0-12 Volts), that can be used for the control of several signal lights or Variable Message
6. FIELD RESULTS ALINEA was first implemented and tested at the onramp Branc:;:ion of the Boulevard Peripherique in Paris. On-ramp volumes are realised on the basis of a traffic cycle C=40sec with a minimum green phase of 10 sec. The set value is 6 =29%, and the regulation parameter KR=70 vehlh. Override tactics are applied to avoid interference of the on-ramp queue with the surface street traffic. The conversion. of r(k), ordered by ALINEA, into green phase durations was done using (1) with real-time estimates for rsat . An arising average bias of -2 veh/cycle or -180 vehlh (which is independent of ALINEA) between ordered and actual ramp volumes was automatically eliminated by the regulator. Typical results of a 30-min period are shown in Figs. 4, 5. Figure 4 shows the upstream and downstream occupancies and the periods of active constraints. Figure 5 shows the ordered and actual on-ramp volumes, and the green phase dQIation. Results of Fig. 4 indicate a proper functioning of ALINEA that keeps 00ut near 6 in the average, whenever constraints are not activated.
-
PC IIuI
ALINEA was compared with other ramp metering strategies at the on-ramp Branc:;:ion over a period of one year, see Haj-Salem, et al., 1990, for detailed results. Evaluation criteria include:
Fig. 3. MASTER-PC2 data collection principle 53
On-....,_tw-nt4000c) 40 35
30
25
!r-\\
\::\,
\
f
•.•
L/\/:I'-~v'
15
'0
~~~-r-r~~~,~,~,~,-.,-.,-.,-.,-.,-.-.~~(~~'
'0
30
Fig. 5. On-ramp volume and green-phase duration. Total time spent (TTS), in veh ' h, equals total travel time on the mainstream stretch (1,2), plus total waiting time on the ramp. - Total travel distance (TTD), in veh' km, corresponds to the total served demand. - Mean speed (MS), in krn/h, equals TTD/TTS. Congestion duration (CD), in min, is the accumulated time with 0ou?'0cr during a peak period from 7:00 a.m. to 10:00 a.m.
- presented along with a low cost implementation concept. ALINEA is well funded on classical regulator theory and proved in extended field tests to be very efficient in ameliorating traffic conditions on motorways. Comparative evaluation with popular naive approaches at different sites demonstrated its clear superiority. ALINEA is recommended for any motorway site experiencing recurrent and/or nonrecurrent congestion. Particularly the numerous sites currently applying naive strategies would probably have much to gain through switching to the ALINEA concept.
-
Table 1 depicts the evaluation results for the no control, ALINEA, demand-capacity (DC), and occupancy (0) strategies. These results demonstrate the high efficiency and clear superiority of ALINEA. ALINEA is currently operational at three on-ramps of the Boulevard Peripherique in Paris. More recent evaluations considering a 6 km long motorway stretch and the parallel arterial have shown that it has a very beneficial impact also on the adjacent street network traffic, see Haj-Salem and Papageorgiou, 1995. Moreover, ALINEA was tested at the on-ramp Coen Tunnel on Al 0 near Amsterdam and was compared to a demand-capacity-type strategy with similar results as reported here, see Middelham and Smulders, 1991. More recently, ALINEA was implemented on four consecutive ramps of Al 0 near Amsterdam and was compared to a demandcapacity-type strategy. Once more, results have demonstrated the ability of ALINEA to reduce ITS in the order of 15 ... 20%, as well as its clear superiority compared with naive strategies, see Middelham, et al., 1995. Implementation of ALINEA is currently in preparation in various sites around Europe.
7. CONCLUSIONS ALINEA, a recently proposed local trafficresponsive strategy for ramp metering has been Table 1 Comparative evaluation results for the peak period 7:00 a.m to 10:00 a.m.
REFERENCES Haj-Salem, (1986). Interface d'acquisition des donnees de trafic et de commande de processus en temps reel. INRETS Research Report No. 11, Paris, France. Haj-Salem, H., J.M. Blosseville and M. Papageorgiou (1990). ALINEA: A local feedback control law for on-ramp metering-A real life study. In: Proc. 3rd lEE Intern. Con! on Road Traffic Control, London, U.K., pp. 194-198. Haj-Salem, H. and M. Papageorgiou (1995). Ramp metering impact on urban corridor traffic: Field results. Transportation Research A, 29, to appear. Keen, K.G., M.J. Schofield and G.C. Hay (1986). Ramp metering access control on M6 motorway. In: Proc. 2nd lEE Intern. Con! on Road Traffic Control, London, U.K. Koble, H.M., T.A. Adams and V.S. Samant (1980). Control strategies in response to freeway incidents. Report FHWA/RD-801005, Federal Highway Administration, Washington, DC. Masher, D.P., D.W. Ross, P.J. Wong, P.L. Tuan, H.M. Zeidler and S. Petracek (1975). Guidelines for design and operation of ramp control systems. Stanford Research Institute, Menid Park, Calif. Middelham, F. and S. Smulders (1991). Isolated ramp metering: Real-life study in The Netherlands. Del. No. 7a, DRIVE Project CHRlSTIANE (V 1035), Brussels, Belgium. Middelham, F., H. Taale and B.V. Velzen (1995). An assessment of multiple ramp metering on the Amsterdam ring-road. Ministry of Transport, Public Works and Water Management, Rotterdam, The Netherlands. Papageorgiou, M. (Ed.)(l99 I). Concise Encyclopedia of Traffic and Transportation Systems. Pergamon Press, London, U.K Papageorgiou, M., H. Haj-Salem and J.M. Blosseville (1991). ALINEA: A local feedback control law for on-ramp metering. Transportation Research Record, 1320, pp. 58-64.
A LOW COST TOOL FOR FREEWAY RAMP METERING
M. Papageorgiou
I)
I),
H. Haj-Salem Z)
Dynamic Systems and Simulation Laboratory, Technical University o/Crete, 73100Chania, Greece 2) INRETS-DART, 2 QV. du Gl. Mal/eret Joinville, 94114 Arcueil, France.
Abstract: Local, traffic-responsive ramp metering is a wellknown control measure aiming at ameliorating traffic conditions on interurban or metropolitan freeways and motorways. A particular control strategy (ALINEA), the only local ramp metering strategy to be based on rigorous application of classical regulator theory, is presented, along with a low cost implementation concept. Field application of ALINEA and evaluation of its perfonnance in France and The Netherlands confinn its clear superiority as compared to other popular approaches. Keywords: Closed-loop control; control equipment; disturbance rejection; implementation; integral control; ramp metering; sample-data control; traffic control.
1. INTRODUCTION
of the local, traffic-responsive type. Hundreds of ramp metering installations, mainly in U.S.A. but increasingly also in Europe and elsewhere, have provided valuable experience about the benefits of this kind of control measure.
Freeways and motorways have been originally conceived so as to provide the possibility of quick travel without delays, both in metropolitan and in interurban areas. However, in the last decades, the number and extent of freeway congestions have been steadily increasing, leading to considerable delays, increasing fuel consumption and environmental pollution, and decreasing road safety. Congestions are caused either by high demand that exceeds the freeway capacity (daily recurrent congestions) or by capacity reducing incidents (nonrecurrent congestions). In presence of congestion, the freeway traffic flow becomes lower than capacity, thus rendering the utilization of the expensive infrastructure non optimal. One proposed way of ameliorating this situation is ramp metering by use of traffic lights at the freeway on-ramps. This control measure aims at limiting access to the freeway mainstream so as to achieve and maintain capacity flow .
From a control engineering point of view, the control laws implemented in the vast majority of existing ramp metering systems may be characterized as naive or heuristic. More precisely, these control laws attempt a sort of feedforward disturbance rejection that makes them particularly sensitive to a variety of changing conditions. The first local ramp metering control strategy that has been based on straightforward application of classical feedback control theory is ALINEA (Papageorgiou, et aI. , 1991). This paper presents in some detail the derivation, the features and possible extensions of ALINEA, along with an implementation concept and field results. 2
Ramp metering control strategies have been proposed at several levels of sophistication, (see Papageorgiou, 1991, for classified reviews of methods) but the high majority of implemented and operating systems are Fig. 1. The ramp metering process. 49
2. THE BASIC PROBLEM
2.1 The ramp metering problem Figure depicts schematically the motorway mainstream and the on-ramp. The following quantities are defmed: qout and '!in are the measurable mainstream traffic volumes or flows (veh/h) downstream and upstream of the ramp respectively. 00ut and 0in are the measurable mainstream occupancy rates downstream and upstream of the ramp respectively. An occupancy rate measures the time occupancy (in %) of a detector placed below the highway pavement. r is the measurable on-ramp traffic volume or flow (veh/h). r may be controlled using ordinary traffic lights, either on an one-car-per-green basis, or on a n-cars-per-green basis (with n=2, 3 or more), or on the basis of a fixed traffic cycle subdivided into a green and a red phase of controllable duration. 8 is the distance between sites 1 and 2. A simple model relating q and 0 at a given site is provided by the wellknown fundamental diagram q=Q(o) (Fig. 2) having the typical shape of an inverse D, where o=ocr is the critical occupancy resulting in maximum (or capacity) flow qcap= Q(ocr)' For 0 close to zero or close to a maximum occupancy 0max' the corresponding flow approaches zero. It is the main aim of a ramp metering installation to
control r so as to keep the downstream mainstream flow qout near a set value 4 , e.g. 4 =qcap· Alternatively, one may attempt to regulate 00ut to a set value 6 , e.g. 6 =ocr'
2.2 Constraints All control strategies calculate suitable ramp volumes r. In the case of traffic cycle realisation, r is converted to a green phase duration g by use of
100 oeeupwu:y 0
Fig. 2. The fundamental diagram .
In the case of n-cars-per-time realization, one typically has a constant-duration green light that permits exactly n vehicles to pass. The ramp volume r is controlled by varying the red phase duration between a minimum and a maximum value. If the queue of vehicles on the ramp becomes excessive, interference with surface street traffic may occur. This may be detected by suitably placed detectors and leads to an override of the regulator decisions, so as to allow more cars to enter the motorway and hence the queue to diminish. Above specifications and constraints apply in the same way to all control strategies.
2.3 Popular control strategies All known strategies but ALINEA are based on a feedforward disturbance rejection principle, that makes them particularly sensitive and not sufficiently accurate.
The demand-capacity strategy, that is extensively used in the USA (Masher, et aI. , 1975; Koble, et aI., 1980), is based on measuring qin and comparing with qc,p' However, because the value of traffic volume alone is insufficient to determine whether the motorway is congested or free flowing (see Fig. 2), the occupancy 00ut is also used according to the following scheme applying at each period k= 1, 2, 3, ... q cap -q in (k) r(k) =
{
(2) rmin
otherwise
where rmin is a minimum ramp volume value and othres is a threshold of occupancy.
(1)
g = (r/rsaJC
where C is the fixed traffic cycle duration and r sat is the ramp capacity flow (or saturation flow) that may be fixed or estimated in real time, based on ramp flow measurements filtered over some past cycles. g is constrained by gE[gmin, gmax] , where gmin>O to avoid ramp closure, and gmax:$;C.
The occupancy strategy (Masher, et al. , 1975; Koble, et al., 1980), that is in use in the USA, is essentially based on the same philosophy as the demandcapacity strategy, i.e. a) measure qin and add r so as to maintain downstream capacity and b) in case of congestion switch to a minimum value rmin' The difference between percent-occupancy and demand-capacity strategies is the following: a) Upstream flow CLn is estimated using occupancy measurements. The main reason for this is that only one detector (e.g. in a middle lane) may be used for estimation of qin' 50
b) Congestion is detected by the upstream detector (thus the method needs only one mainstream detector station).
3.2 Regulator design An appropriate feedback law for the process (7) is given by the following I-regulator
3 DERIV AnON OF ALINEA
3.1 Modelling
(8)
The conservat;on of vehicles between sites I and 2 gives
pet) = [Cfu,(t)+r(t) -
qout(t»)/O
where KR>O is a regulator parameter. Applying (8) to (7), we obtain the z-transfer function of the closedloop system
(3)
where the traffic density p (veh/km) is defmed as the number of cars included in the stretch, devided by the length 8 , t being the time argument. Because traffic density is not readily measurable (it requires video detectors), it is convenient to replace pet) in (3) by 0out(t), using the approximate relationship p = a 0out, where a = ~(lOOA), ~ being the number of lanes of the mainstream and A being the mean effective vehicle length (as "seen" by the underground electromagnetic loop detector). Thus, substituting also qout=Q(oouJ in (3), a nonlinear first-order dynamic model is obtained. This model may be linearized around a nominal steady-state (oout, Cl in , f), such that
A time-optimal deadbeat regulator is obtained by choosing KR=Q ' . Because Q' >0, the linearisation of (7) is strictly valid only on the left-hand side of the fundamental diagram. However, even for congested traffic, the feedback law (8) leads to reduction of traffic occupancy and can thus be applied in the same way. If site 2 is located at a bottleneck further downstream and/or T is chosen short, the entering traffic may not reach site 2 at the end of each time interval, in which case ~ cannot be neglected. Applying the regulator (8) to the original model (6), the closed-loop ztransfer function becomes
00ut = 0; Clout = Q(oout); f = Clin- Clout . (4) With the notation A·(t)=·(t)--:- , the linearisation yields
(10) with the eigenvalues ~1/2 = ~±j[~{I-a»)1I2. Thus the closed-loop system remains always stable, since ~< I is always verified. Nevertheless, the transient behaviour for high ~-values may be slow. In these cases, an amelioration may be achieved by application of a PI-regulator
where ()' =dQ( 6 )/do out , i.e. er is the slope of the tangent of the fundamental diagram at 6 , and hence its value is positive on the left-hand side of the fundamental diagram. Because the control input r is updated every T (typically T = C = 40 ... 80s), time discretisation of (5) with sample T yields
r(k) = r(k-I) - KR[6 -ooutCk») - Kp[oout(k)-ooutCk-I») (11)
AOoulk+ I) = ~Aooulk) + [(I-In! Q' ][A'Ln(k)+ Ar(k») (6)
where Kp>O is a further regulator parameter. Deadbeat regulation is achieved with KR=Q' , Kp = ~Q' l(l-a) .
where k=O, I, 2, .. .. Thus 0out(k) is the occupancy at time kT, and A'Ln(k), ruCk) are assumed constant over the interval [(k-I)T, kT) . The parameter ~ results from the discretization to be ~ = exp(-Q ' . TI
3.3 Elimination of constant disturbances Regulator (8) is capable of elinifuating constant disturbances qin when applied to (7). In fact we have
(a·o)). ~ may be neglected, if the ratio ofT is sufficiently small. This will be the case if traffic volume entering the motorway reaches site 2 during the time interval T. In this case, the effect of entering traffic r(k) will be visible at site 2 before the end of the corresponding time interval. Setting ~=O in (6)
0out(Z) = qin(z)
(z-I)/Q'
(12)
z(z-I+KR /en
which is zero in the steady-state, i.e. for z=l. 51
What happens if the actual ramp volume is biased (i.e. equal to r+d) compared with the ramp volume r ordered by the regulator (e.g. due to inaccurate rsat in (1) or due to red light violations)? Such a bias d acts in (7) as a disturbance, in the same way as ctn does. Hence, the bias is automatically eliminated by the control system. This statement does not hold true, if the actual (measured) ramp volume is used as a retarded value r(k-l) in the feedback law (8) (as will be recommended in the next section for anti-wind up purposes). In this case, r(k-l)+d(k-l) takes the place of r(k-l) in (11), and we obtain the z-transfer function 00ut ( z )
d(z)
l/Q ~'
= -----''----
(13)
volumes, be visible in the measurements. A distance of 40m in Paris (Haj-Salem, et al., 1990) was found to be adequate as well as a distance of 400m in Amsterdam (Middelbam and Smulders, 1991). Measurement of r(k) may also be necessary, e.g. for real-time estimation of rsat, or for use in (8). However, ALINEA is also applicable directly to the green or red phase duration, which circumvents the need of estimating rsat and measuring r. In fact, combining (1) and (8), we obtain instead of (8) the feedback law g(k) = g(k-l) - K~ [6 -oout(k)]
(14)
where K~ =KRC/rsat• Regulator (14) has not been tested extensively in the field as yet.
z-I+(KR/Q')
Thus, for constant d, (13) yields a steady-state error of d/KR.
4. DISCUSSION OF ALINEA FEATURES The simple regulator (8) has been proposed by Papageorgiou, et al. (1991) for local, traffic responsive ramp metering under the name ALINEA (Asservissement LIneaire d' Entree Autoroutiere). Note that both the demand-capacity and the occupancy strategies react to excessive occupancies oout only after a threshold value is reached, and in a rather crude way, whereas ALINEA reacts smoothly even to slight differences 6 -oout(k) and may thus prevent congestion in an elegant way, stabilizing traffic flow at a high throughput level. It is interesting to note that the feedback structure of ALINEA appears to some members of the Traffic Engineering community, with little or no knowledge of Automatic Control theory, to be peculiar or even counter-intuitive. In order to avoid wind-up phenomena of the 1regulator, the measured actual ramp volume r(k-I) should be used in (8), any time one of the constraints mentioned in section 2.2 becomes active. In all other cases, r(k-I) should be equal to the last calculated ramp volume in order for the regulator to be able to eliminate possible deviations between the ordered and actual ramp volumes, as discussed in section 3.3. In field experiments, it was found that ALINEA is not very sensitive w.r.t. the choice of the regulator parameter KR. A value of K R=70 veh/h was found to yield excellent results in many different sites. ALINEA requires only one mainstream detector station for 00ut downstream of the ramp entrance. The measurement location should be such that a congestion, originating from excessive on-ramp 52
It was demonstrated in section 3.3 that, if 'lin is constant, then ALINEA leads to zero offset, whereas for time varying ctn ALINEA acts according to (12) as a smoothing filter.
The set value 6 in (8) is provided by the user. This set value may be changed any time, and hence ALINEA may be directly embedded into a hierarchical control system with set values of the individual ramps being specified in real time by a superior coordination level or by an operator. The main reason for regulating occupancy, rather than volume, is due to the fact that traffic volume may have the same values both for light and congested traffic, see Fig. 2. An additional advantage in the case 6 =oc.. i.e. regulation to capacity flow, is that the critical occupancy ocr seems to be less sensitive w.r.t. weather conditions and other influences compared with the capacity qcap of a motorway stretch, see Keen, et aI., 1986. As a consequence, considering the set value 6 =ocr is a more robust and reliable way of achieving capacity flow than considering Cl. =qcap, because variations of qcap, caused by environmental or other conditions, are stronger compared with variations of ocr. Two possible extensions may be suggested if b!T»KR/a. The first extension is to use the PI regulator (11) with Kp=(ablT)-KR' as discussed in section 3.2. The second extension is to add to the feedback laws (8) or (11) a term Y['lin(k)-ctn(k-I)], if the value ctn(k) can be predicted accurately by use of further upstream measurements. This second extension corresponds to a feedforward disturbance rejection, with a positive smoothing parameter y5, I. These extensions were not tested in the field. Finally, it should be noted that, if the output site 2 is located far downstream of the on-ramp (at a downstream bottleneck), there is a risk of congestion, due to incidents or other disturbances, in the interior of the stretch (1 ,2), that will not be visible at the
output measurement site 2. In such cases, the feedback (or any other) control system may be of little help for eliminating the congestion, if no additional measurement stations are provided.
5. A LOW COST IMPLEMENTATION CONCEPT For the implementation of the ramp metering strategy, we can distinguish three main processes: - Real data collection - Calculation of the metering rate - Activation of the signal light. In order to minimize the implementation cost, the used hardware corresponds to a well-known IBM-PC or Compatible including a simple and specific hardware board which was developed at INRETS and commercialised since 1988 by a private company BECR. The cost is around 250$ US. This interface is called MASTER-PC2 (see Haj-Salem, 1986).
Fig. 4. Occupancies and constraints activation. Signs (VMS). MASTER-PC2 has been used for the implementation of ramp metering strategies on the Corridor Peripherique of Paris. The overall architecture can be summarised as follows: The signal loop data are transmitted to the central room though phone lines. The receiver, located at the central room, identifies each signal loop, which are considered as an input of MASTERPC2 board. The developed software performs data collection, data screening, application of the selected strategy, and command of the signal light according to the output of the ramp metering strategy (metering rate). For the Corridor Peripherique, the ramp metering system covers around 12 km of the motorway axis including 60 loop detectors and three signal lights (for three on-ramps).
In general the traffic data collection process using
loop detectors is based on the software elementary signal loop scrutinising. The scrutinise method is simple but it needs a powerful Central Processor Unit (CPU) and optimised software. On the other hand, the accuracy of the measurement is very much related to the used scrutinise time slice (e.g. 1 msec). The approach for data collection used in MASTERPC2 board is based on the hardware sampling loop detector signal with a hardware aggregation of the collected traffic variables. The start of loop detector sampling is done automatically using a very simple hardware logic. The time slice sampling is fixed by the user using a MASTER-PC2 internal and programmable clock. Three programmable internal clocks are available, one for the signal sampling (see Fig. 3) and two others for programming the data collection time slice (1 sec or more) and the signal light duration. All the counters are mapped in the central memory. This means that the data process is limited to one memory instruction read from the overall counters at each time slice in order to keep the aggregated information (traffic volume, occupancy) issued from MASTER-PC2 board. MASTER-PC2 is able of generating 16 binary control signals (0-12 Volts), that can be used for the control of several signal lights or Variable Message
6. FIELD RESULTS ALINEA was first implemented and tested at the onramp Branc:;:ion of the Boulevard Peripherique in Paris. On-ramp volumes are realised on the basis of a traffic cycle C=40sec with a minimum green phase of 10 sec. The set value is 6 =29%, and the regulation parameter KR=70 vehlh. Override tactics are applied to avoid interference of the on-ramp queue with the surface street traffic. The conversion. of r(k), ordered by ALINEA, into green phase durations was done using (1) with real-time estimates for rsat . An arising average bias of -2 veh/cycle or -180 vehlh (which is independent of ALINEA) between ordered and actual ramp volumes was automatically eliminated by the regulator. Typical results of a 30-min period are shown in Figs. 4, 5. Figure 4 shows the upstream and downstream occupancies and the periods of active constraints. Figure 5 shows the ordered and actual on-ramp volumes, and the green phase dQIation. Results of Fig. 4 indicate a proper functioning of ALINEA that keeps 00ut near 6 in the average, whenever constraints are not activated.
-
PC IIuI
ALINEA was compared with other ramp metering strategies at the on-ramp Branc:;:ion over a period of one year, see Haj-Salem, et al., 1990, for detailed results. Evaluation criteria include:
Fig. 3. MASTER-PC2 data collection principle 53
On-....,_tw-nt4000c) 40 35
30
25
!r-\\
\::\,
\
f
•.•
L/\/:I'-~v'
15
'0
~~~-r-r~~~,~,~,~,-.,-.,-.,-.,-.,-.-.~~(~~'
'0
30
Fig. 5. On-ramp volume and green-phase duration. Total time spent (TTS), in veh ' h, equals total travel time on the mainstream stretch (1,2), plus total waiting time on the ramp. - Total travel distance (TTD), in veh' km, corresponds to the total served demand. - Mean speed (MS), in krn/h, equals TTD/TTS. Congestion duration (CD), in min, is the accumulated time with 0ou?'0cr during a peak period from 7:00 a.m. to 10:00 a.m.
- presented along with a low cost implementation concept. ALINEA is well funded on classical regulator theory and proved in extended field tests to be very efficient in ameliorating traffic conditions on motorways. Comparative evaluation with popular naive approaches at different sites demonstrated its clear superiority. ALINEA is recommended for any motorway site experiencing recurrent and/or nonrecurrent congestion. Particularly the numerous sites currently applying naive strategies would probably have much to gain through switching to the ALINEA concept.
-
Table 1 depicts the evaluation results for the no control, ALINEA, demand-capacity (DC), and occupancy (0) strategies. These results demonstrate the high efficiency and clear superiority of ALINEA. ALINEA is currently operational at three on-ramps of the Boulevard Peripherique in Paris. More recent evaluations considering a 6 km long motorway stretch and the parallel arterial have shown that it has a very beneficial impact also on the adjacent street network traffic, see Haj-Salem and Papageorgiou, 1995. Moreover, ALINEA was tested at the on-ramp Coen Tunnel on Al 0 near Amsterdam and was compared to a demand-capacity-type strategy with similar results as reported here, see Middelham and Smulders, 1991. More recently, ALINEA was implemented on four consecutive ramps of Al 0 near Amsterdam and was compared to a demandcapacity-type strategy. Once more, results have demonstrated the ability of ALINEA to reduce ITS in the order of 15 ... 20%, as well as its clear superiority compared with naive strategies, see Middelham, et al., 1995. Implementation of ALINEA is currently in preparation in various sites around Europe.
7. CONCLUSIONS ALINEA, a recently proposed local trafficresponsive strategy for ramp metering has been Table 1 Comparative evaluation results for the peak period 7:00 a.m to 10:00 a.m.
REFERENCES Haj-Salem, (1986). Interface d'acquisition des donnees de trafic et de commande de processus en temps reel. INRETS Research Report No. 11, Paris, France. Haj-Salem, H., J.M. Blosseville and M. Papageorgiou (1990). ALINEA: A local feedback control law for on-ramp metering-A real life study. In: Proc. 3rd lEE Intern. Con! on Road Traffic Control, London, U.K., pp. 194-198. Haj-Salem, H. and M. Papageorgiou (1995). Ramp metering impact on urban corridor traffic: Field results. Transportation Research A, 29, to appear. Keen, K.G., M.J. Schofield and G.C. Hay (1986). Ramp metering access control on M6 motorway. In: Proc. 2nd lEE Intern. Con! on Road Traffic Control, London, U.K. Koble, H.M., T.A. Adams and V.S. Samant (1980). Control strategies in response to freeway incidents. Report FHWA/RD-801005, Federal Highway Administration, Washington, DC. Masher, D.P., D.W. Ross, P.J. Wong, P.L. Tuan, H.M. Zeidler and S. Petracek (1975). Guidelines for design and operation of ramp control systems. Stanford Research Institute, Menid Park, Calif. Middelham, F. and S. Smulders (1991). Isolated ramp metering: Real-life study in The Netherlands. Del. No. 7a, DRIVE Project CHRlSTIANE (V 1035), Brussels, Belgium. Middelham, F., H. Taale and B.V. Velzen (1995). An assessment of multiple ramp metering on the Amsterdam ring-road. Ministry of Transport, Public Works and Water Management, Rotterdam, The Netherlands. Papageorgiou, M. (Ed.)(l99 I). Concise Encyclopedia of Traffic and Transportation Systems. Pergamon Press, London, U.K Papageorgiou, M., H. Haj-Salem and J.M. Blosseville (1991). ALINEA: A local feedback control law for on-ramp metering. Transportation Research Record, 1320, pp. 58-64.