HIERARCHICAL NONLINEAR MODEL-PREDICTIVE RAMP METERING CONTROL FOR FREEWAY NETWORKS

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Abstract: A nonlinear rolling-horizon model-predictiv e hierarchical coordinated ramp metering scheme is presented. The hierarchical co ntrol structure consists of three layers: the estimation/prediction layer, the opti mization layer and the direct control layer. The second layer incorporates the prev iously designed optimal control tool AMOC while the local feedback strategy ALINEA is used in the third layer. Simulation results are presented for the Amsterdam ring -road. It is shown that control of all on-ramps including freeway intersection s leads to the optimal utilization of the available infrastructure. &RS\ULJKW‹,)$& Keywords: ramp metering; model-predictive control; multi ALINEA; freeway traffic control. 1. INTRODUCTION Ramp metering aims at improving the traffic conditions by appropriately regulating the inflow from the on-ramps to the freeway mainstream and is deemed as one of the most effective control measures for freeway network traffic. One of the most efficie nt local ramp metering strategies is the ALINEA feedback strategy and its variations (Papageorgiou, HW DO, 1991, 1998; Smaragdis and Papageorgiou, 2003; Smaragdis, HW DO, 2004). A number of design approaches have been proposed for coordinated ramp metering. These include multivariable control (Diakaki and Papageorgiou, 1994) and optimal control (Bellemans, HW DO, 2002; Hegyi, HW DO, 2003; Gomes and Horowitz, 2004). Kotsialos HWDO. (2002b) presented AMOC, an open-loop control tool which combines a nonlinear formulation with a powerful numerical optimization algorithm and is able to consider coordinated ramp metering, route guidance as well as integrated control combining both control measures. In (Kotsialos and Papageorgiou, 2001, 2004) the results from AMOC’s application to the problem of coordinated ramp metering at the Amsterdam ring-road are presented in detail. A more detailed overview of ramp metering algorithms may be found in (Papageorgiou and Kotsialos, 2002).

© 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006

layer control; AMOC;

Due to various inherent uncertainties, the open-loop optimal solution becomes suboptimal when directly applied to the freeway traffic process. In this paper, the optimal results are cast in a model-predictive frame and are viewed as targets for local feedback regulators which leads to a hierarchical control scheme. The rest of this paper is structured as follows. In section 2 the freeway network traffic flow model used for both simulation and control design purposes is briefly described. Section 3 introduces the formulation of the optimal control problem for ramp metering. The hierarchical control structure is described in section 4 while the results of applying ALINEA, as a stand-alone strategy, and the proposed hierarchical strategy are presented in section 5. Finally, conclusions and directions for future work are outlined in section 6. 2. TRAFFIC FLOW MODELLING A validated second-order traffic flow model is used for the description of traffic flow on freeway networks and provides the modeling part of the optimal control problem formulation. In fact, the

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same model is used in this paper for the traffic flow simulator (METANET) (Messmer and Papageorgiou, 1990) and for the control strategy (AMOC) albeit with different external disturbances. The network is represented by a directed graph whereby the links of the graph represent freeway stretches. Each freeway stretch has uniform characteristics, i.e., no on-/off-ramps and no major changes in geometry. The nodes of the graph are placed at locations where a major change in road geometry occurs, as well as at junctions, on-ramps, and off-ramps. The time and space arguments are discretized. The discrete time step is denoted by 7 (typically 7 10V ). A freeway link P is divided into 1 P segments of equal length /P (typically /P 500P ), such that the stability condition / ≥ 7 ⋅ υ , holds, where υ , is the free-flow speed of link P . This condition ensures that no vehicle traveling with free speed will pass a segment during one simulation time step. Each segment L of link P at time W = N7 , N = 0,..., . , where . is the time horizon, is macroscopically characterized via the following variables: the WUDIILFGHQVLW\ ρ , (N ) (veh/lane/km) is the number of vehicles in segment L of link P at time W = N7 divided by /P and by the number of lanes Λ P ; the PHDQ VSHHG υ , (N ) (km/h) is the mean speed of the vehicles included in segment L of link P at time W = N7 ; and the WUDIILFYROXPH or IORZ T , (N ) (veh/h) is the number of vehicles leaving segment L of link P during the time period  N7 , (N + 1)7  , divided by 7 . The evolution of traffic state in each segment is described by use of the interconnected state equations for the density and mean speed respectively (Kotsialos and Papageorgiou, 2001, 2004). Roughly speaking, the flow increases with increasing density until a density critical value is reached, at which flow becomes maximum ( Tµ ,FDS ). After this critical density, congestion sets on and the flow decreases reaching virtually zero at a jam density value. P

I P

I P

P L

P L

P L

Fig. 1. The origin-link queue model. For origin links, i.e., links that receive traffic dema nd GR and forward it into the freeway network, a simple queue model is used (Fig. 1). The outflow TR of an origin link R depends on the traffic conditions of the corresponding mainstream segment ( µ ,1 ), the ramp’s queue length ZR (veh) and the existence of ramp

© 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006

metering control measures. If ramp metering is applied, then the outflow TR (N ) that is allowed to leave origin R during period N, is a portion UR (N ) of the maximum outflow that would leave in absence of ramp metering. Thus, UR (N ) ∈  Umin, R ,1 is the metering rate for the origin link R , i.e., a control variable, where Umin,R is a minimum admissible value; typically, Umin,R > 0 is chosen in order to avoid ramp closure. If UR (N ) = 1 , no ramp metering is applied. A similar approach applies to freeway-to-freeway (ftf) interchanges. The evolution of the origin queue ZR is described by an additional state equation (conservation of vehicles). Note that the freeway f low Tµ ,1 in merge segments is maximized if the corresponding density ρ µ ,1 takes values near the critical density ρ µ ,FU . Freeway bifurcations and junctions (including onramps and off-ramps) are represented by nodes. Traffic enters a node Q through a number of input links and is distributed to the output links. The percentage of the total inflow at a bifurcation node Q that leaves via the outlink P is the turning rate β QP . 3. FORMULATION OF THE OPTIMAL CONTROL PROBLEM The overall network model has the state-space form

[ (N + 1) = I  [ (N ), X (N ), G (N )

(1)

where the state of the traffic flow process is describ by the state vector [ ∈ 1 and its evolution depends on the system dynamics and the input variables. Input variables are distinguished into control variables X ∈ 0 and external disturbances G ∈ ' . In this case, vector [ consists of the densities ρ , and mean speeds υ , of every segment L of every link P as well as the queues ZR of every origin R . The control vector X consists of the ramp metering rates UR of R every on-ramp under control, with UR,min ≤ UR (N ) ≤ 1 . Finally, the disturbance vector consists of the demands GR at every origin of the network and the turning rates β QP at the network’s bifurcations. The disturbance trajectories G (N ) must be known over the time horizon . 3 for optimal control. For practical applications, these values may be predicted more or less accurately based on historical data or on real-time estimations (Wang HW DO., 2003). P L

P L

The coordinated ramp metering control problem is formulated as a discrete-time dynamic optimal control problem with constrained control variables and can be solved numerically over a given optimization horizon . 3 (Papageorgiou and Marinaki, 1995). The chosen cost criterion is the Total Time Spent (TTS) of all vehicles in the network (including the waiting time experienced in the ramp queues) which is a natural objective for the traffic systems considered. Penalty terms are added appropriately to the cost criterion in

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ed

Direct Control Layer that has the task of realizing the suggested policy. For each on-ramp R with merging segment (µ ,1) (Fig. 1) a local regulator is applied with control sample time 7F = ]F7 , ]F ∈ , in order to calculate the on-ramp outflow TRU (NF ) , where NF = ]F N . The set-points for the local regulators are the corresponding optimal merge values of either density ρ µ ,1 or flow Tµ ,1 delivered by AMOC (and averaged over the ]F time intervals).

order for the solution to comply with the maximum ramp queue constraints. The solution determined by AMOC consists of the optimal ramp metering rate trajectories and the corresponding optimal state trajectory. 4. HIERARCHICAL CONTROL The solution provided by AMOC is of an open loop nature. As a consequence, its direct application may lead to traffic states different than the calculated optimal ones due to errors associated with the initial state estimate, the prediction of the disturbances and the model parameters used. A receding-horizon (model-predictive) approach can be employed to address any mismatch between the predicted and the actual system behavior. This approach is extended to the hierarchical control system shown in Fig.~2 which consists of three layers (Kotsialos 2004).

In the Direct Control Layer, regulators ALINEA (Papageorgiou, HW DO, 1991) or flow-based FLALINEA (Smaragdis and Papageorgiou, 2003) are employed as local regulators, with AMOC-optimal set-points for each controlled on-ramp. The ALINEA local regulator with set-point ρ µ ,1 reads

T (N ) = T (N − 1) + .  ρ µ ,1 − ρµ ,1 (N ) (2) where . is the feedback gain factor. The flow-based U R

F

U R

F

U

F

U

ALINEA with set-point Tµ ,1 reads

The Estimation/Prediction Layer receives as input historical data, information about incidents and realtime measurements from sensors installed in the freeway network. This information is processed in order to provide the current state estimate and the future predictions of the disturbances to the next layer.

T (N ) = T (N − 1) + .  Tµ ,1 − Tµ ,1 (N ) (3) where . is the feedback gain factor. TRU is bounded U R

F

U R

F

I

F

by the maximum ramp flow 4R and a minimum admissible ramp flow TRU,min . In order to avoid windup, the term TRU (NF − 1) used in both (2) and (3) is bounded accordingly. I

Creation of over-long ramp queues can be avoided with the application of a queue control policy (Smaragdis and Papageorgiou, 2003) in conjunction with both local metering strategies (2) and (3). The queue control law takes the form

TRZ (NF ) = −

1

 ZR,max − ZR (NF ) + G R (NF − 1) (4)

7F 

where ZR,max is the maximum admissible ramp queue. The final on-ramp outflow is then

{

}

TR (NF ) = max TRU (NF ), TRZ (NF ) .

Fig. 2. Hierarchical control structure. The Optimization Layer (AMOC) considers the current time as the initial point and the state estimat as the initial condition. Given the predictions G (N ), N = 0,..., . 3 − 1 , the optimal control problem is solved delivering the optimal control trajectory (translated into optimal on-ramp outflows) and the corresponding optimal state trajectory. These trajectories are forwarded as input to the decentralize

e

d

© 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006

(5)

The flows Tµ ,1 are preferable as set-points for local regulation because they are directly measurable without the uncertainty caused by modeling. However, flows do not uniquely characterize the traffic state, as the same flow may be encountered under non-congested and congested traffic conditions. Hence a flow set-point Tµ ,1 is used (in conjunction with FL-ALINEA), only if ρ µ ,1 ≤ ρ µ ,FU and Tµ ,1 ≤ 0.9Tµ ,FDS , i.e. only if the optimal flows are well below the critical and congested traffic conditions. If ρ µ ,1 ≥ ρ µ ,FU and Tµ ,1 ≤ 0.9Tµ ,FDS then ALINEA is applied with ρ µ ,1 delivered by AMOC; in all other cases ALINEA with ρ µ ,1 = ρ µ , FU is applied in order to guarantee maximum flow even in presence of various mismatches. The update period or application horizon of the model predictive control is . $ ≤ . 3 , after which the optimal control problem is solved again with updated

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state estimation and disturbance predictions, thereby closing the control loop of AMOC as in modelpredictive control. The control actions will be generally more efficient with increasing . 3 and decreasing . $ . 5. APPLICATION RESULTS

The ring-road has been divided into 76 segments with average length 421m. This means that the state vector is 173-dimensional (including the 21 on-ramp queues). The disturbance vector is 41-dimensional while the dimension of the control vector depends on the number of controlled on-ramps.  7KHQRFRQWUROFDVH

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For the purposes of this study, the counter-clockwise direction of the A10 freeway, which is about 32km long is considered. There are 21 on-ramps on this freeway, including the junctions with the A8, A4, A2 and A1 freeways, and 20 off-ramps, including the connections with A8, A4, A2 and A1. The topological network model may be seen in Fig.~3. The model parameters for this network were determined from validation of the network traffic flow model against real data (Kotsialos, HWDO, 2002a).

Using real (measured) time-dependent demand and turning rate trajectories as input to the METANET simulator without any control measures, heavy congestion appears in the freeway and large queues are built in some on-ramps. The density evolution is displayed in Fig.~4 and the corresponding queue profile in Fig.~5. The excessive demand coupled with the uncontrolled entrance of the drivers into the main stream causes congestion (Fig. 4). This congestion originates at the junction of A1 with A10 and propagates upstream blocking A4 and a large part of the A10-West. As a result many vehicles are accumulated in the ftf on-ramp of A4, with a queue that exceeds 1,200 veh, and the surrounding on-ramps (Fig. 5). The TTS for this scenario is equal to 14,167 vehÂK  $SSOLFDWLRQRI$/,1($

Fig. 3. The Amsterdam ring-road.

ALINEA may be used at each ramp as a stand-alone strategy without any kind of coordination. The setpoint for each controlled on-ramp R is set to the critical density of the corresponding link µ , in order to maximize the local freeway throughput. Different scenarios will be examined. Urban on-ramps are always controlled and the maximum queue constraint (4) and (5) is active with an admissible ramp queue ZR,max equal to 30 veh as the ramp storage capacity is limited. If no control measure is applied to the ftf o ramps, TTS is reduced to 13,402 vehÂK EXW ODUJH queues are formed on A4 reaching 1,200 veh. When ALINEA is applied to the ftf on-ramps as well with an admissible ramp queue there equal to 200 veh, then TTS is reduced even more to 11,515 vehÂK However, queues can not be avoided on A4 reaching 935 veh. This queue profile is displayed in Fig. 6.

Fig. 4. No-control scenario: Density profile.

Fig. 6. ALINEA: Queue profile with active queue constraints (30 veh for the urban on-ramps and 200 veh for the ftf on-ramps). Fig. 5. No-control scenario: Ramp queue profile.

© 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006

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Fig. 7. TTS values for different admissible ramp queues for the ftf on-ramps and different control strategies. TTS values are plotted in Fig. 7 for different admissible ramp queues. These values converge towards the (dotted) TTS value that would have been achieved by ALINEA if the storage capacity of ftf onramps were infinite.  $SSOLFDWLRQRI$02&

The optimal open-loop solution under the assumption of perfect information with respect to the future disturbances for the entire simulation time serves as an "upper bound" for the efficiency of the hierarchical control strategy as it relies on ideal conditions. TTS values are plotted in Fig. 7 for all the scenarios.

Results obtained by the optimal open-loop solution are not realistic because the assumption of perfect knowledge of the future disturbances cannot hold in practice. The hierarchical control proposed is able to cope with this problem by employing the rolling horizon technique, e.g. with . 3 = 360 (1 hour) and . $ = 60 (10 min). It is assumed that the state of the system is known exactly when AMOC is applied every 10 min, which is a fairly realistic assumption. With respect to the prediction of the on-ramp demands and the turning rates it is assumed that a fairly good predictor is available so that the smoothed real trajectories are used as the predicted ones. Obtained TTS values are plotted in Fig. 7 for all scenarios. These values are lower than the ALINEA ones and reach faster the corresponding ideal case without maximum queue constraints for the ftf onramps. For admissible ramp queues equal to 30 veh for the urban on-ramps and 200 veh for the ftf onramps TTS is equal to 7,399 vehÂK ZKLFKLV D  improvement over the no-control case and only 4.6% worse than the open loop case; see the corresponding density evolution in Fig.~8 and queue profile in Fig.~9. Queues are built early in the simulation in anticipation of the future congestion and maximum queue constraints are taken into consideration without serious degradation of the strategy’s efficiency. Queues on the ftf on-ramps do not exceed the value of 200 veh. 6. CONCLUSIONS AND FUTURE WORK

Fig. 8. Hierarchical control: Density profile with active queue constraints (30 veh for the urban onramps and 200 veh for the ftf on-ramps).

Fig. 9. Hierarchical control: Ramp queue profile with active queue constraints (30 veh for the urban onramps and 200 veh for the ftf on-ramps).  $SSOLFDWLRQRIKLHUDUFKLFDOFRQWURO

© 11th IFAC Symposium on Control in Transportation Systems Delft, The Netherlands, August 29-30-31, 2006

The results of applying local feedback control, ideal open-loop control and rolling-horizon hierarchical coordinated control to the Amsterdam ring-road have been presented. Uncoordinated local control with ALINEA is quite successful in reducing TTS and lifting congestion up to a certain degree. However, large queues on A4 are unavoidable. As expected, hierarchical control outperforms the uncoordinated local ramp metering approach. In the network studied and for the specific disturbance profiles used, the introduction of ramp metering at some particular ramps within the network reduced some local traffic problems. However, a significant amelioration of the global traffic conditions in the network calls for comprehensive control of all on-ramps including ftf on-ramps in the aim of optimal utilization of the available infrastructure. By building queues that do not exceed 200 veh on the ftf on-ramps hierarchical control leads to a 47.8% improvement over the nocontrol case. The total available storage capacity in all on-ramps and ftf intersections is sufficient to effectively and ultimately combat freeway congestion for the network studied. Achievement of good results with the hierarchical control strategy requires reasonably accurate model, state estimates and disturbances prediction. Moreover, code complexity, relative intensive computations and

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the "black box" character of the optimization procedure may be perceived as obstacles for ready and broad application of the method. In view of this, current work is focused on the development of a generic rule-based coordination scheme of local ramp metering actions that would reach similar efficiency as the presented hierarchical control albeit via a muc simpler code and without the need for predictions. 7. ACKNOWLEDGMENTS This work was partly funded by the European Commission in the framework of the project EURAMP (IST-2002-23110). The content of this paper is under the sole responsibility of the authors and in no way represents views of the European Commission. The authors would like to thank Mr. F. Middelham from the AVV-Rijkswaterstaat, The Netherlands, for providing the necessary data for the Amsterdam ring road. REFERENCES Bellemans, T., B. De Schutter and B. De Moor (2002). Model predictive control with repeated model fitting for ramp metering. In: 3URF2IWKH WK  ,(((,QWHO7UDQVS6\VWHPV&RQI, Singapore. Diakaki, C. and M. Papageorgiou (1994). Design and Simulation Test of Coordinated Ramp Metering Control (METALINE) for A10-West in Amsterdam. Internal report 1994-2, Dynamic Systems and Simulation Laboratory, Technical University of Crete, Chania, Greece. Gomes, G. and R. Horowitz (2004). Globally optimal solutions to the on-ramp metering problem-Part I & II. In: 3URF RI WKH WK ,((( ,QWHO 7UDQVS 6\VWHPV&RQI, Washington D.C. Hegyi. A., B. De Schutter and J. Hellendoorn (2003). MPC-based optimal coordination of variable speed limits to suppress shock waves in freeway traffic. In: 3URF RI WKH  $PHULFDQ &RQWURO &RQIHUHQFH, Denver, Colorado, USA. Kotsialos, A. (2004) Modeling and optimal control of traffic in large scale freeway networks (in greek). PhD Dissertation, Technical University of Crete, Chania, Greece Kotsialos, A. and M. Papageorgiou (2001). Optimal coordinated ramp metering with Advanced Motorway Optimal Control. 7UDQVSRUWDWLRQ 5HVHDUFK5HFRUG, , 55-71. Kotsialos, A. and M. Papageorgiou (2004). Efficiency and equity properties of freeway network-wide ramp metering with AMOC. 7UDQVSRUWDWLRQ 5HVHDUFK&, , 401-420. Kotsialos, A., M. Papageorgiou, C. Diakaki, Y. Pavlis and F. Middelham (2002a). Traffic flow modeling of large-scale motorway networks using the macroscopic modeling tool METANET. ,((( 7UDQV RQ ,QWHOOLJHQW 7UDQVSRUWDWLRQ6\VWHPV, , 282-292.

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Kotsialos, A., M. Papageorgiou, M. Mangeas and H. Haj-Salem (2002b). Coordinated and integrated control of motorway networks via nonlinear optimal control. 7UDQVSRUWDWLRQ 5HVHDUFK &, , 65-84. Messmer, A. and M. Papageorgiou (1990). METANET: A macroscopic simulation program for motorway networks. 7UDIILF(QJLQHHULQJ DQG &RQWURO, , 466-470. Papageorgiou, M., H. Haj-Salem and J.M. Blosseville (1991). ALINEA: A local feedback control law for on-ramp metering. 7UDQVSRUWDWLRQ 5HVHDUFK 5HFRUG, , 58-64. Papageorgiou, M., H. Haj-Salem and F. Middelham (1998). ALINEA local ramp metering: Summary of field results. 7UDQVSRUWDWLRQ5HVHDUFK5HFRUG, , 90-98. Papageorgiou, M. and A. Kotsialos (2002). Freeway ramp metering: an overview. ,((( 7UDQV RQ ,QWHOOLJHQW7UDQVSRUWDWLRQ6\VWHPV, , 271-281. Papageorgiou, M. and M. Marinaki (1995). A Feasible Direction Algorithm for the Numerical Solution of Optimal Control Problems. Dynamic Systems and Simulation Laboratory, Technical University of Crete, Chania, Greece. Smaragdis, E. and M. Papageorgiou (2003). A series of new local ramp metering strategies. 7UDQVSRUWDWLRQ5HVHDUFK5HFRUG, , 74-86. Smaragdis, E., M. Papageorgiou and E. Kosmatopoulos (2004). A flow-maximizing adaptive local ramp metering strategy. 7UDQVSRUWDWLRQ5HVHDUFK%, , 251-270. Wang Y., M. Papageorgiou and A. Messmer (2003). RENAISSANCE: a real-time motorway network traffic surveillance tool. In: 3UHSULQWV RI WK ,)$& 6\PSRVLXP RQ &RQWURO LQ 7UDQVSRUWDWLRQ

, pp.235-240. Tokyo, Japan.

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