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A Linear Formulation for Model Predictive Perimeter Traffic Control in Cities*
Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of 20th Congress Proceedings of the the 20th World World Congress The International Federation Automatic Proceedings of the 20th World Toulouse, France, July 9-14, 2017 The International International Federation Federation of of Congress Automatic Control Control The of Automatic Control Available online at www.sciencedirect.com Toulouse, France, July The International of Automatic Control Toulouse, France,Federation July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 8543–8548 Linear Linear Formulation Formulation for for Model Model Predictive Predictive Linear Formulation for Model Predictive Perimeter Traffic Control in Cities Linear Formulation for Model Predictive Perimeter Traffic Control in Cities Perimeter Traffic Control in Cities Perimeter Traffic Control in Cities Anastasios Kouvelas, Mohammadreza Saeedmanesh,
Anastasios Kouvelas, Mohammadreza Anastasios Mohammadreza Saeedmanesh, Nikolas Geroliminis Saeedmanesh, Anastasios Kouvelas, Kouvelas, Mohammadreza Saeedmanesh, Nikolas Geroliminis Saeedmanesh, Anastasios Kouvelas, Mohammadreza Nikolas Nikolas Geroliminis Geroliminis Nikolas Geroliminis Urban Transport Systems Laboratory Urban Transport Laboratory Urban Transport Systems LaboratoryEngineering School of Architecture, Civil Systems & Environmental Urban Transport Systems Laboratory School of Architecture, Civil & Environmental Engineering Urban Transport Systems ´ School of Architecture, Civil & Environmental Engineering School ofEcole Architecture, Civil F´ &ed´ Environmental Engineering Polytechnique eraleLaboratory de Lausanne ´ Ecole Polytechnique F´ e d´ e rale de Lausanne School of Architecture, Civil & Environmental Engineering ´ ´ Ecole Polytechnique F´ e d´ e rale de Lausanne CH-1015Ecole Lausanne, Switzerland {tasos.kouvelas, Polytechnique F´ed´e(e-mails: rale de Lausanne ´ Lausanne, CH-1015 Switzerland (e-mails: {tasos.kouvelas, Ecole Polytechnique F´ e d´ e rale de Lausanne CH-1015 Lausanne, Switzerland (e-mails: {tasos.kouvelas, mohammadreza.saeedmanesh, nikolas.geroliminis}@epfl.ch) CH-1015 Lausanne, Switzerland (e-mails: {tasos.kouvelas, mohammadreza.saeedmanesh, nikolas.geroliminis}@epfl.ch) CH-1015 Lausanne, Switzerland (e-mails: {tasos.kouvelas, mohammadreza.saeedmanesh, nikolas.geroliminis}@epfl.ch) mohammadreza.saeedmanesh, nikolas.geroliminis}@epfl.ch) mohammadreza.saeedmanesh, nikolas.geroliminis}@epfl.ch) Abstract: An alternative approach for real-time network-wide traffic control in cities that Abstract: An alternative approach forperimeter real-time flow network-wide traffic control in of cities that Abstract: An alternative approach for real-time traffic in has recentlyAn gained a lot ofapproach interest is control. The basiccontrol concept suchthat an Abstract: alternative for real-time network-wide network-wide traffic control in cities cities that has recently gained a lot of interest is perimeter flow control. The basic concept of such an Abstract: An alternative approach for real-time network-wide traffic control in cities that has recently gained a lot of interest is perimeter flow control. The basic concept of such an approach is togained partition heterogeneous cities into a small homogeneous regions (zones) has recently a lot of interest is perimeter flow number control. of The basic concept of such an approach isperimeter togained partition heterogeneous cities into aa small small number of homogeneous regions (zones) has recently a lot of interest is perimeter flow control. The basic concept of such an approach is to partition heterogeneous cities into number of homogeneous regions (zones) and apply control to the inter-regional flows along the boundaries between regions. approach is to partition heterogeneous cities into a small number of homogeneous regions (zones) and apply perimeter control to the the inter-regional inter-regional flows along the the boundaries between regions. approach isperimeter to partition heterogeneous a small number of homogeneous regionsbetween (zones) and apply perimeter control to flows along boundaries between regions. The apply transferring flowscontrol are controlled atcities the into traffic intersections located at the between borders and to the inter-regional flows along the boundaries regions. The transferring flows are controlled at the traffic intersections located at the borders between and apply perimeter control to the inter-regional flows along the boundaries between regions. The transferring are at traffic intersections located at borders regions, so as to flows distribute the congestion an optimal way and minimize total between delay of The transferring flows are controlled controlled at the the in traffic intersections located at the thethe borders between regions, so as as to focus distribute the congestion in anstudy optimal way and minimize the total between delay of The transferring flows arethe controlled at theisin traffic intersections located thenot borders regions, so to distribute the congestion in an optimal way and minimize the total delay of the system. The of current work toan three aspects thatatare covered in the regions, so as to distribute the congestion optimal way and minimize the total delay of the system. The focus of the thethe current work isinthe toanstudy study three aspects that are not covered in the regions, so as to distribute congestion optimal way and minimize the total delay of the system. The focus of current work is to three aspects that are not covered in the perimeter control literature, which are: (a) treatment of some model parameters that are the system. The focus of the current work is to study three aspects that are not covered in the perimeter control literature, which are: (a)isthe the treatment ofthe some model parameters that are the The in focus theimplementations currentare: work to study that are not in the perimeter control literature, which are: (a) treatment some model parameters that are not system. measurable realoflife and can three affectof performance of covered the controller perimeter control literature, which (a) the treatment ofaspects some model parameters that are not measurable in real life implementations and can affect the performance of the controller perimeter control literature, which are: (a) the treatment of some model parameters that are not measurable in real realestimation life implementations implementations and can affect the the performance of(b) theintegration controller (e.g.measurable advanced online schemes canand be can developed for this purpose),of not in life affect performance the controller (e.g. advanced external online estimation schemes can canthat be can developed for this this purpose), (b) integration not measurable in realestimation life implementations performance theintegration controller (e.g. advanced online estimation schemes be developed for purpose), (b) integration of appropriate demand information hasaffect been the considered systemof(b) disturbance in (e.g. advanced online schemes canand be developed for this purpose), of appropriate external demand information that has been and considered system (b) disturbance in (e.g. advanced online estimation schemes can be developed for this purpose), integration of appropriate external demand information that has been considered system disturbance in the derivation of feedback control laws in previous works, (c) mathematical formulation of appropriate external demand information that has been considered system disturbance in the derivation of feedback control laws in previous works, and (c) mathematical formulation of appropriate external demand information that has been considered system disturbance the derivation of feedback feedback controlinlaws laws in previous previous and mathematical formulation of the original nonlinear problem a linear form, soworks, that optimal can be applied inina the derivation of control in works, and (c) (c)control mathematical formulation of the original nonlinear problem inlaws a linear form, soworks, that optimal optimal control can be be applied in the derivation ofmodel feedback controlconcept. in This previous and mathematical formulation of the original nonlinear problem in form, so that control can applied aaa (rolling horizon) predictive work the(c) mathematical analysis ofin the of the original nonlinear problem in aa linear linear form, so presents that optimal control can be applied in (rolling horizon) model predictive concept. This work presents the mathematical analysis of the of the original nonlinear problem in a linear form, so that optimal control can be applied in (rolling horizon) model predictive concept. This work presents the mathematical analysis of the optimal horizon) control problem, as well asconcept. the approximations and simplifications that are assumed ina (rolling model predictive This work presents the mathematical analysis of the optimal control problem, as well wellofas asaconcept. the approximations and simplifications simplifications that are assumed in (rolling horizon) model predictive This work presents the mathematical analysis of the optimal control problem, as the approximations and that are assumed in order to derive the formulation linear optimization problem. Preliminary simulation results optimal control problem, as well as the approximations and simplifications that are assumed in order to derive the formulation of a linear optimization problem. Preliminary simulation results optimal control problem, as well as the approximations and simplifications that are assumed in order tocase derive the formulationenvironment of aa linear linear optimization optimization problem. results for theto of the a macroscopic (plant) are problem. presented,Preliminary in order tosimulation demonstrate the order derive formulation of Preliminary simulation results for thetocase case of the a macroscopic macroscopic environment (plant) are problem. presented, in order to demonstrate demonstrate the order derive formulation of a linear optimization Preliminary results for the of environment (plant) are presented, in to the efficiency of the approach. Results for the closed-loop model predictive control scheme for the case of aaproposed macroscopic environment (plant) are presented, in order order tosimulation demonstrate the efficiency of the proposed approach. Results for the closed-loop model predictive control scheme for the case of a macroscopic environment (plant) are presented, in order to demonstrate the efficiency of the proposed approach. Results for the closed-loop model predictive control scheme are presented forproposed the nonlinear case,Results which isforused “benchmark”, well as the linearscheme case. efficiency of the approach. the as closed-loop modelaspredictive control are presented presented forproposed the nonlinear nonlinear case,Results which is is used used as “benchmark”, aspredictive well as as the the linearscheme case. efficiency of the approach. the as closed-loop modelas control are for the case, which as “benchmark”, as well linear case. are presented for the nonlinear case, which isforused “benchmark”, well as the linear case. © 2017, IFAC (International Federation Automatic Control) Hosting by Elsevier Ltd.asAll reserved. are presented for the nonlinear case,ofwhich is used as “benchmark”, as well therights linear case. Keywords: Model predictive control; nonlinear optimisation; linear approximation; urban Keywords: Model predictive control; nonlinear optimisation; linear approximation; urban Keywords: Model predictive predictive control; control; nonlinear nonlinear optimisation; optimisation; linear linear approximation; approximation; urban urban perimeter control. Keywords: Model perimeter control. Keywords: Model predictive control; nonlinear optimisation; linear approximation; urban perimeter control. perimeter control. perimeter control. 1. INTRODUCTION perimeter flow control (or gating). The basic concept of 1. INTRODUCTION INTRODUCTION perimeter flow control control (or gating). gating). The basic basic concept of 1. perimeter flow (or The of such an approach is to partition heterogeneous cities into 1. INTRODUCTION perimeter flow control (or gating). The basic concept concept of such an approach is to partition heterogeneous cities into 1. INTRODUCTION perimeter flow control (or gating). The basic concept of such an approach is to partition heterogeneous cities into a small number of homogeneous regions (zones) and apply Traffic congestion is a major problem for urban environ- such an approach is to partition heterogeneous cities into asuch small number of homogeneous regions (zones) and apply Traffic congestion ismetropolitan a major major problem problem for urban environan approach is to partition heterogeneous cities into a small number of homogeneous regions (zones) and apply Traffic congestion is a for urban environperimeter control to the inter-regional flows along the ments and modern areas. Most cities around a small number of homogeneous regions (zones) and apply Traffic congestion is a major problem for urban environ- perimeter control to the inter-regional flows along the ments and modern metropolitan areas. Most cities around small number of homogeneous regions (zones) Traffic congestion ismetropolitan a persistently major problem for urban environcontrol to the flows along the ments and modern metropolitan areas. Most cities around boundaries between The inter-transferring flows the world have been becoming denser and aperimeter perimeter control toregions. the inter-regional inter-regional flows and alongapply the ments and modern areas. Most cities around boundaries between regions. The inter-transferring the world have been persistently becoming denser and perimeter control to the inter-regional flows along the ments and modern metropolitan areas. Most cities around boundaries between regions. The inter-transferring flows the world have beendecades persistently becoming denser and are controlled at theregions. traffic intersections located atflows the widerworld over have the last and the problem of urban boundaries between The inter-transferring flows the been persistently becoming denser and are controlled at the traffic intersections located at the wider over the last decades and the problem of urban boundaries between regions. The inter-transferring flows the world have been persistently becoming denser and are controlled at the traffic intersections located at the wider over the last decades and the problem of urban borders between regions, so as to distribute the congestion traffic over management steadily momentum due are controlled at the traffic intersections located at the wider the last is decades andgaining the problem of urban between regions, so asintersections to distribute the congestion traffic management is steadily gaining momentum due borders are controlled the traffic at wider over the last decades andgaining the problem of urban borders between regions, so the congestion traffic management is steadily gaining momentum due in an optimal way and minimize the totallocated delay of the the to its economic, social and environmental impact. Many borders betweenat regions, so as as to to distribute distribute the congestion traffic management is steadily momentum due in an optimal way and minimize the total delay the to its economic, social and environmental impact. Many borders between regions, so as to distribute the congestion traffic management is steadily gaining momentum due way minimize the total of the to its economic, economic, social andout environmental impact.settings Many in system (an alternative could theof efforts have beensocial carried to optimize signal in an an optimal optimal way and andobjective minimize the maximize total delay delay oftotal the to its and environmental impact. Many system (an alternative objective could maximize the total efforts have been carried out to optimize signal settings in an optimal way and minimize the total delay of the to its economic, social and environmental impact. Many system (an alternative objective could maximize the total efforts have been carried out to optimize signal settings during have the peak networks face serious con- throughput). system (an alternative objective could maximize the total efforts beenhours, carriedwhere out to optimize signal settings during the peak hours, where networks face serious con- throughput). system (an alternative objective could maximize the total efforts have been carried out to optimize signal settings throughput). during the peak hours, where networks face serious congestion problems and the performance of the infrastructure during the peak hours, where networks face serious con- throughput). Perimeter flow control can be viewed as a high-level gestion problems and the performance of the the infrastructure during the peak and hours, where networks faceinfrastructure serious congestion problems and the performance of infrastructure degrades significantly. The state-of-practice strategies fail throughput). Perimeter flow control viewed aa high-level gestion problems the performance of the Perimeter flow can be viewed as scheme can and be be as combined with degrades significantly. The state-of-practice strategies fail Perimetercontrol flow control control can bemight viewed as a high-level high-level gestion problems and the performance of the infrastructure degrades significantly. The state-of-practice state-of-practice strategies fail regional to deal efficiently with oversaturated conditions (i.e. queue regional control scheme and might be combined with degrades significantly. The strategies fail Perimeter flow control can be viewed as a high-level regional control scheme and might be combined with other strategies (e.g. local, distributed or coordinated conto deal efficiently with oversaturated conditions (i.e. queue regional control scheme and might be combined with degrades significantly. The state-of-practice strategies fail to deal efficiently with oversaturated conditions (i.e. queue spillbacks and partial gridlocks), asconditions they are (i.e. either de- other strategies (e.g. local, distributed or coordinated conto deal efficiently with oversaturated queue regional control scheme and might be combined with other strategies (e.g. local, distributed or coordinated controllers) in a hierarchical control framework; this topic has spillbacks and partial gridlocks), as they are either deother strategies (e.g. local, distributed or coordinated conto deal by efficiently with oversaturated conditions (i.e. queue spillbacks andofpartial partial gridlocks), as they are accurately either de- trollers) signed use simplified models as that do not in a hierarchical control framework; this topic has spillbacks and gridlocks), they are either deother strategies (e.g. local, distributed or coordinated controllers) in aaofhierarchical control framework; this topic has gained a lot attraction in the research community lately. signed by use of simplified models that do not accurately trollers) in hierarchical control framework; this topic has spillbacks and partial gridlocks), as they are either designed by some use of oftraffic simplified models that do do notpropagation accurately gained replicate flow models phenomena (e.g. a lot of attraction in the research community lately. signed by use simplified that not accurately trollers) in a hierarchical control framework; this topic has gained a lot of attraction in the research community lately. For a recent review on this research direction the reader is replicate some traffic flow phenomena (e.g. propagation gained a lot of attraction in the research community lately. signed by use of simplified models that do not accurately replicate some or traffic flow phenomena (e.g. (e.g. propagation propagation of congestion), basedflow on application-specific heuristics. For aa recent review on this research direction the reader is replicate some traffic phenomena gained a lot of attraction in the research community lately. For recent review on this research direction the reader is referred to Keyvan-Ekbatani et al. (2012); Yildirimoglu of congestion), or based on application-specific heuristics. For a recent review on this research direction the reader is replicate some traffic flow phenomena (e.g. propagation of congestion), or based on application-specific heuristics. Ancongestion), alternative or approach forapplication-specific real-time network-wide traf- referred to Keyvan-Ekbatani et al. (2012); Yildirimoglu of based on heuristics. For a recent review on this research direction the reader is referred to Keyvan-Ekbatani et al. (2012); Yildirimoglu et al. (2015). The original model for the dynamics of An alternative approach for real-time network-wide trafreferred to Keyvan-Ekbatani et al. (2012); Yildirimoglu of congestion), or based on application-specific heuristics. An alternative approach for real-time network-wide traffic control that has recently gained a lot of interest is et al. (2015). The original model for the dynamics of An alternative approach for real-time network-wide traf- et referred to Keyvan-Ekbatani et al. (2012); Yildirimoglu al. (2015). The original model for the dynamics of multi-region process (plant) is highly fic control control thatapproach has recently recently gained aanetwork-wide lot of of interest interest is the et al. (2015). The original model for thenonlinear dynamicsand of An alternative for real-time traffic that has gained lot is the multi-region process is highly and fic control that has recently gained a lot of interest is et (2015). The original model for dynamics of the multi-region process (plant) is nonlinear and modelling tool that (plant) is utilized is the thenonlinear Macroscopic This research has been supported by the ERC (European Research theal. multi-region process (plant) is highly highly nonlinear and fic control that has recently gained a lot of interest is the modelling tool that is utilized is the Macroscopic research has been supported by the ERC (European Research multi-region process (plant) is highly nonlinear and This the modelling tool that is utilized is the Macroscopic Council) Starting Grant “METAFERW: Modelling and controlling This research has been supported by the ERC (European Research Fundamental Diagram (MFD) (see e.g. Ramezani et al. the modelling tool that is utilized is the Macroscopic This research has been supported by the ERC (European Research Council) Starting Grant “METAFERW: Modelling andmulti-modal controlling Thiscongestion Fundamental (see e.g. Ramezani et al. the modelling tool that is utilized is the Macroscopic research has been supported bylarge-scale theModelling ERC (European Research traffic and propagation in urban Council) Starting Grant “METAFERW: and Fundamental Diagram (MFD) (see e.g. Ramezani et (2015)). MFD Diagram provides a(MFD) concave, low-scatter relationship Council) Starting Grant “METAFERW: Modelling and controlling controlling Fundamental Diagram (MFD) (see e.g. Ramezani et al. al. traffic congestion propagation in large-scale urban (2015)). MFD Diagram provides aa(MFD) concave, low-scatter relationship Council) Starting Grant “METAFERW: Modelling andmulti-modal controlling networks” (Grant and #338205). traffic congestion and propagation in urban multi-modal Fundamental (see e.g. Ramezani et al. (2015)). MFD provides concave, low-scatter relationship traffic congestion and propagation in large-scale large-scale urban multi-modal (2015)). MFD provides a concave, low-scatter relationship networks” (Grant and #338205). traffic congestion propagation in large-scale urban multi-modal networks” (Grant (2015)). MFD provides a concave, low-scatter relationship networks” (Grant #338205). #338205).
networks” (Grant #338205). Copyright © 2017 IFAC 8877 Copyright 2017 IFAC 8877 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 8877 Copyright © 2017 IFAC 8877Control. Peer review©under of International Federation of Automatic Copyright 2017 responsibility IFAC 8877 10.1016/j.ifacol.2017.08.1411
Proceedings of the 20th IFAC World Congress 8544 Anastasios Kouvelas et al. / IFAC PapersOnLine 50-1 (2017) 8543–8548 Toulouse, France, July 9-14, 2017
between network vehicle accumulations [veh] or density [veh/km] and network circulating flow [veh/h] or production [veh·km] for every region of the system which is homogeneously congested. Clustering techniques can help partition a network into regions with small variance of link densities (see e.g. Saeedmanesh and Geroliminis (2016)). Recently, there has been some effort to simplify this model; the authors in Hajiahmadi et al. (2015) proposed a hybrid model for formulating this problem and applied model predictive control (MPC) in conjunction with switching plans control. A pre-defined set of MFDs is introduced (which correspond to regional switching plans), and some affine approximations are made in order to reduce the complexity of the derived hybrid MPC approach. In the current work we devise a linear program (LP) formulation to solve the finite-time optimal perimeter flow control problem. Moreover, we investigate two more aspects of the problem that have not been covered in the literature: (a) the treatment of some model parameters that are not easily measurable with loop detectors data in real life implementations, and can affect the performance of the controller; especially the regional route choice parameters and the origin-destination information, and (b) integration of realistic external demand information that has been considered system disturbance in the derivation of multivariable feedback control laws in previous works (Kouvelas et al. (2015)). Finally, the derived LP is solved in a rolling time horizon using a feedback MPC framework, and the control decisions are applied to the nonlinear plant for evaluation. The efficiency of the control decisions is compared to a “benchmark” case, in which the nonlinear MPC problem is solved using state-of-the-art nonlinear numerical solvers. Simulation results for the case of a macroscopic model (plant) are presented. Note that this benchmark approach is more challenging for application in real life due to computational requirements, and most importantly, the lack of detailed data (i.e. high resolution vehicle trajectories data is needed). Essentially, real-world data availability can restrain the methodologies that can be applied, and, consequently, the real-time applicability of the benchmark approach is deemed cumbersome. 2. AGGREGATED DYNAMICS FOR A PARTITIONED CITY Consider an urban network partitioned in N homogeneous regions with well-defined MFDs (Figure 1). The index i ∈ N = {1, 2, . . . , N } denotes the region of the system, ni (t) the total accumulation (number of vehicles) in region i, and nij (t) the number of vehicles in region i with final destination region j ∈ N , at a given time t. Let Ni be the set of all regions that are directly reachable from the borders of region i, i.e. adjacent regions to region i. The discrete time MFD dynamics of the N -region system can be described by the following first order difference equations nii (kp + 1) = nii (kp ) + Tp −
h∈Ni
qii (kp ) − Mii (kp )
Miih (kp ) +
h∈Ni
i Mhi (kp )
(1)
𝑢𝑢𝑗𝑗𝑗𝑗
𝑃𝑃𝑖𝑖
𝑖𝑖
𝑛𝑛𝑖𝑖
𝑢𝑢𝑖𝑖𝑖𝑖
𝑗𝑗
𝑃𝑃𝑗𝑗
𝑛𝑛𝑗𝑗
Fig. 1. The network of a city partitioned into a multi-region MFD system.
nij (kp + 1) = nij (kp ) + Tp −
qij (kp )
h (kp ) + Mij
h∈Ni
i Mhj (kp )
h∈Ni
(2)
where i, j ∈ N , i = j; kp = 0, 1, . . . , Kp − 1 is the model discrete time index, Tp [sec] the sample time period of the model (i.e. time t = kp Tp ). The exogenous variables qij (kp ) [veh/sec] denote the (uncontrollable) traffic flow demand that is generated in region i, at time step kp , with final destination in region j (i.e. qii (kp ) is the demand generated in region i that has final destination in region i). h The variables Mij (kp ) [veh/sec] denote the transfer flows from region i to region h, that have final destination region j, while Mii (kp ) [veh/sec] is the internal trip completion rate of region i (without going through another region). Consequently, the total accumulation ni (kp ) for region i can be computed by ni (kp ) = j∈N nij (kp ).
We assume that for each region i there exists a production MFD between accumulation ni (kp ) and total production, Pi (ni (kp )) [veh·m/sec], which describes the performance of the system in an aggregated way. This MFD can be estimated using measurements from loop detectors and/or h GPS trajectories. Transfer flows Mij (kp ) and internal trip completion rates Mii (kp ) are calculated according to the corresponding production MFD of the region, and proportionally to the accumulations nij (kp ) as follows nii (kp ) Pi (ni (kp )) Mii (kp ) = θii (kp ) (3) ni (kp ) Li h Mij (kp ) = min Cih (nh (kp )), h (kp ) uih (kc )θij
nij (kp ) Pi (ni (kp )) ni (kp ) Li
(4)
where i, j ∈ N , h ∈ Ni ; kc = kp /Nc , with Nc some positive integer, is the control discrete time index (i.e. the control cycle is always a multiple of the plant sample time Tp ), and Li [m] is the average trip length for region i, which is assumed to be independent of time and destination,
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internal (inside region i) or external (to some other region h j). The parameters θii (kp ), θij (kp ) reflect the route choice of drivers and are assumed to be exogenous (i.e. can be constant or time varying and they are provided by h another methodology). The transfer flows Mij (kp ) are the minimum between the sending flow from region i (which only depends on the accumulation of the region), and the receiving capacity Cih (nh (kp )) [veh/sec] of region h. This flow capacity is a piecewise function of the accumulation nh (kp ) (usually modelled with two pieces, one constant value and a decreasing curve), and is introduced to prevent overflow phenomena within the regions, i.e. each region i has a maximum accumulation ni,max such that 0 ≤ ni (kp ) ≤ ni,max , ∀ i ∈ N . (5) If ni (kp ) = ni,max , the region reaches gridlock and all the inflows along the periphery are restricted. Finally, the control variables uih (kc ), ∀ i ∈ N , h ∈ Ni denote the fraction of the flow that is allowed to transfer from region i to region h for the time interval [(Nc kc )Tp , (Nc kc + Nc − 1)Tp ]. Physical constraints are applied to the values of the control variables as follows 0 ≤ uih (kc ) ≤ 1, ∀ i ∈ N , h ∈ Ni (6) but also – depending on the implementation – operational constraints of the following form might apply, i.e., |uih (kc ) − uih (kc − 1)| ≤ uR (7) ih , ∀ i ∈ N , h ∈ Ni . Equations (1)–(4) are a discretized version of equations presented in Ramezani et al. (2015) and represent the traffic dynamics of an N -region urban network considering the heterogeneity effect and integrating an aggregated routing model. Note, that these equations allow the drivers to choose any arbitrary sequence of regions as their route and their path can cross region boundaries multiple times.
subject to equations (1), (2), (3), (4), (5), (6), (7) nij (k) Pi (ni (k)) h h (k) = uih (κ)θij (k) Mij ni (k) Li ni (k) = nij (k)
2.2 Linearising the problem In the current work we derive a linear approximation of the model described in the previous section, and we formulate a linear MPC problem that can be utilized for real-time control purposes. In order to linearise the dynamic equations we assume the following simplifications and approximations: • We introduce the model parameters for the accumulation proportions, i.e. αii (k) = nii (k)/ni (k) and αij (k) = nij (k)/ni (k), i ∈ N , j ∈ Ni . One approach, that is implemented here, is to get feedback for the values of these parameters every time that we roll the horizon. They can be estimated in real-time from measurements (e.g. using extended Kalman filter or maximum likelihood approximation) and then be kept constant for all the optimisation horizon. Note that the parameters can be time varying but they need to be exogenous signals for the MPC framework. • New “dummy” control variables uii (k) are introduced ∀ k = kp , kp + 1, . . . , kp + Np − 1, that restrict the trip completion rates at every region i. Although these variables are not reasonable from a physical point of view, they are required in order for the problem to be linear. A conjecture is that the solution of MPC will always result in uii (k) = 1, ∀ i ∈ N , ∀ k = kp , kp + 1, . . . , kp + Np − 1, but this needs to be validated through results. • Most importantly, we introduce new decision variables fii (k) = uii (k)Gi (ni (k))θii (k)αii (k) (12) h fih (k) = uih (k)Gi (ni (k)) θij (k)αij (k) (13)
h • the quantities qii (k), qij (k) and θii (k), θij (k) are considered exogenous variables that can be measured or given by another algorithm beforehand, • as in many similar works, the capacity constraint Cih (nh (kp )) in (4) is dropped, since from a control viewpoint it is not necessary; the control actions will not allow the system to operate in states close to gridlock, and this constraint is never activated inside the NMPC.
k=kp
i∈N
(10)
k = kp , kp + 1, . . . , kp + Np − 1, κ = k/Nc (11) ∀ i, j ∈ N , h ∈ Ni The objective function (8) tries to maximize the total production of the system for a horizon of Np model steps (or Np Tp seconds). This problem can be solved in reasonable time by use of advanced nonlinear optimization toolboxes (e.g. ipopt 1 ), and in our case serves as a benchmark for the results reported later. Note that if we want to compare the results with the LP approach described later, the objective function should be the same in both cases in order to have a fair comparison.
The MFD dynamics described in the previous section derive a nonlinear model that has been used in other works (Geroliminis et al. (2013); Ramezani et al. (2015)) to apply nonlinear model predictive control (NMPC). Here, we solve the same problem again to obtain a benchmark, which is then used as a comparison to the LP formulation presented later. In order to have a well defined problem and without loss of generality – since this is a nonlinear MPC problem – the following assumptions are made for formulating the problem:
nii (k), nij (k), uih (κ)
(9)
j∈N
2.1 Nonlinear model predictive control (NMPC)
Given these two reasonable assumptions the nonlinear optimal control problem for a horizon of Np model steps is defined as follows kp +Np −1 h Li Mii (k) + uih (κ)Mij (k) (8) maximize
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j∈N
∀ i, j ∈ N , h ∈ Ni that help linearise the equations. In (12)–(13) the h variables θii (k), θij (k), αii (k), αij (k) are considered time varying exogenous signals and as a result the nonlinearity of the problem comes from the product of the control inputs uih (k) with the MFD functions.
1 http://www.i2c2.aut.ac.nz/Wiki/OPTI/index.php/Solvers/ IPOPT
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• To overcome this, we approximate the MFDs of the regions with piecewise affine (PWA) functions Gi (ni (k)) that form a convex set (see e.g. Figure 3 for a case study with 4 regions). Each MFD can be approximated with l = 1, 2, . . . , Ni affine functions, and we denote as Gli (ni (k)) each affine function l. In conclusion, the control variables have the property of being bounded (i.e. uih (k) ∈ [0, 1], ∀ i ∈ N , h ∈ Ni , k = kp , kp + 1, . . . , kp + Np − 1) and the MFDs that can be approximated by PWA functions. As a result, we are looking for an optimal solution within a convex set, and in this particular case the product can be linearised by introducing the new variables fii (k), fih (k) (see Gomes and Horowitz (2006) for some theoretical analysis of a similar convexification in a ramp metering control problem). Moreover, PWA approximations is a popular technique to reduce complexity and attempt to linearise nonlinear systems (see e.g. Xu et al. (2016)); alternatively someone could use robust control to solve another version of this problem, by introducing uncertain parameters (see e.g. Haddad (2015)). Finally, once the optimal solution is computed, there is a unique transformation between the new variables and the original control variables uii (k), uih (k). This is a modelling trick that allows us to simplify the problem without loosing any accuracy in the dynamics.
3.1 LMPC without OD information Moving one step forward with our approximation, the new model does not need to keep track of the OD information (aggregated information about each region can be sufficient for control purposes). Hence, by adding all the states nij and nii for each region i, we get a linear model that does not consider OD data, but only aggregated demands in the region level. In that case, the derived LP that approximates the original system and can be solved online is as follows
ni (k), fii (k), fih (k)
k=kp
i∈N
Li [fii (k) + fih (k)]
− fii (k) −
qi (k) fih (k) +
h∈Ni
fhi (k)
h∈Ni
0 ≤ fii (k) ≤ θii (k)αii (k)Gli (ni (k)) h θij (k)αij (k) 0 ≤ fih (k) ≤ Gli (ni (k))
(15)
(16) (17)
j∈N
0 ≤ ni (k) ≤ ni,max (18) (19) k = kp , kp + 1, . . . , kp + Np − 1 ∀ i, j ∈ N , h ∈ Ni , l = 1, 2, . . . , Ni where the objective function (14) again represents the total production of the system. All the constraints of this problem are linear, and, as a consequence, the computational requirements are quite low, even for a network with many regions and large prediction horizons. Note that constraint (7) can also be applied for the first step of the LMPC without breaking the linearity, by using the following linear inequalities P R h fih (kp ) ≤ uPR (kp )αij (kp ) θij ih + uih Gi (ni (kp )) j∈N
j∈N
The assumptions outlined above are reasonable approximations/simplifications of the nonlinear model in order to derive a linear formulation that can be used for online MPC. In this section we formulate a linear model predictive control (LMPC) problem, which does not keep track of the origin-destination information of vehicles. This dynamic model has less online data requirements (as it carries lower level of information, i.e. only state and demand trajectories di , instead of dij ), but under certain optimization horizons can provide similar optimal solutions for the control variables (see Section 4).
maximize
ni (k + 1) = ni (k) + Tp
(20) P h R fih (kp ) ≥ uPR (n (k )) θ (k )α (k − u G i p ij p ) ij p ih ih i
3. LINEAR MODEL PREDICTIVE CONTROL (LMPC)
kp +Np −1
subject to
(14)
(21) ∀ i, j ∈ N , h ∈ Ni PR where uR ij are user defined bounds, uij are the control commands applied to the plant in the previous control cycle, and GP i (ni (kp )) are the affine functions that the accumulations ni (kp ) of each region i belong to (this can be easily found as ni (kp ) are known). Besides, only the first control decision is applied to the nonlinear plant and then we roll the horizon. 4. PROOF OF CONCEPT This section presents some simulation results obtained for the described methodology. The simulation model (plant) is the nonlinear model presented (1)–(5). The test case network is a replica of the network used in Kouvelas et al. (2017) and corresponds to a part of the CBD of Barcelona in Spain (Figure 2(a)). The network is partitioned into 4 regions in order to apply perimeter control (Figure 2(b)). Figure 3 presents the MFDs of the 4 regions from data obtained from a microsimulation model in Kouvelas et al. (2017). The red lines present the PWA approximation of the MFDs with the affine functions utilised by the LP approximation. The approximation is pretty accurate and this relaxation should not cause any problems in the optimization procedure. In that respect, the nonlinear model and PWA approximation are almost identical; note also, that this approximation can be done with many more lines than presented here without hardening significantly the computations of the LP solver (6 pieces are shown for each MFD in Figure 3, while in the LMPC 30 pieces have been used for better accuracy).
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Fig. 3. MFDs for the 4 regions of the case study network (blue); piecewise affine approximation of the MFDs to be used in linear MPC (red).
Finally, Figure 6 displays the results for the regional accumulations when the two approaches are applied to control the transferring flows between regions. The objective of the controller is to maximize the production in the network and the parameters used for this simulation are Tp = 20sec, Nc = 3, Np = 21, uR ij = 0.2, Ni = 30, ∀ i, j ∈ N . It is clear that the controller improves the traffic states of the system, and the area between the blue and the dashed black lines (or dashed green lines) corresponds to the total delay improvement in the system. These results are quite promising, as it is demonstrated that we can
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First, we present a comparison of the plant (1)–(5) and the linear model presented earlier (12)–(13), (15), (18) for the no control (NC) case (i.e. uih (kc ) = 1, ∀ i ∈ N , h ∈ Ni , kc = kp /Nc ), which will also be our base scenario. Figure 4 presents a demand scenario (i.e. generated vehicles per time unit for all the simulation horizon) for the case study with 4 regions (e.g. 4×4 OD matrix). For this demand, Figure 5 displays the trajectories of accumulations nij for the plant and for the linear model, which actually correspond to the estimation of nij used within the LMPC framework (using ni and αij ). The prediction horizon for the linear model is 21 times higher than the sample time of the plant (e.g. Np = 21). The trajectories of the accumulations nij demonstrate that the linear model can be used to approximate the original nonlinear plant (given small prediction horizons and feedback of model parameters αij ). This model is a quite accurate representation of the original system, thus is appropriate to be used for the LMPC framework.
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Fig. 5. Vehicle accumulations nij for the plant (solid lines) and the linear model (dashed lines) when applied for 21 steps of prediction (Np = 21) and the NC case. achieve the same level of improvement by using the linear approximation of the model described in section 3.1, which also does not require OD information. Table 1 presents the performance evaluation of the two approaches. All simulated scenarios start and finish with an empty network and serve all the demand (79143 vehicles) 2 . The total travelled distance (TTD) in all scenarios is equal to 169.55 × 103 veh·km and the total travelled time (TTT) by all vehicles in the network is computed by TTT =
Kp −1
ni (k)
(22)
k=0 i∈N
and presented in Table 1 for each case. Both NMPC and LMPC achieve a substantial improvement in terms of network mean speed (NMS=TTD/TTS) and TTT (around 20–30%).Their performance is quite similar, however the LMPC approach has less data requirements and it is a LP formulation that guarantees convergence and optimality. 2 Note that for very small accumulation values at the beginning and end of simulation NC is applied in all cases.
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ACKNOWLEDGEMENTS
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and demand profiles. Finally, another research direction is to use perimeter control as a first-level controller in cities (as it deals with zone interactions) and develop a secondlevel of distributed control (e.g. Kouvelas et al. (2014)) for optimizing locally. The combination of the two provides a hierarchical control scheme that could potentially be more efficient in alleviating traffic congestion in cities, but this needs also to be further investigated.
4000 n 1 NC
10000
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Accumulation [veh]
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10000
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6000
˙ The authors are grateful to I¸sik Ilber Sırmatel for the valuable comments during the development of this work.
n 4 NMPC n 4 LMPC
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REFERENCES 0
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Fig. 6. Vehicle accumulations ni for the 4 regions for NC case (blue lines), NMPC (dashed black lines), and LMPC (dashed green lines) when applied for 21 steps of prediction (Np = 21). Table 1. Performance of different approaches. Criteria TTS (veh·h)×103 NMS (km/h)
NC
NMPC
(%)
LMPC
(%)
17.73
13.76
-22.39
13.58
-23.41
9.57
12.32
28.74
12.49
30.51
5. CONCLUSION A LP formulation is derived for solving the perimeter control problem in multi-region cities. The originally nonlinear system is relaxed and approximated by a simplified linear model that under certain assumptions can track the behaviour of the multi-region system. The new model requires less information in terms of real-time measurements (e.g. traffic states, OD demands), and, because it is formulated as a LP, it guarantees optimality and fast convergence of the solver. The simulation results need to be further investigated in order to assess the goodness of the solution obtained through the linear MPC to the benchmark, which consists of the nonlinear MPC problem with full information about demands and state feedback measurements. Note that a field implementation of the benchmark approach would require real-time measurements (or estimates) of nij and dij , while the LMPC version requires only region-based measurements (i.e. ni and di ) and not detailed origin-destination data. Future work will deal with the development of a more solid methodology for estimating the model parameters αij (e.g. online extended Kalman filter), as the values of these parameters are crucial for the optimization horizon. These parameters can be estimated at every control cycle and then considered constant for all the optimization horizon of the MPC and this could not be sufficient for the improvement of the system. Simple estimation/prediction techniques can be used to enhance the knowledge for this parameters and help the convex problem to track the nonlinear dynamics in a better way. Investigations about different convex objective functions for the MPC is also another research topic. The proposed methodology needs to be evaluated for different realistic objective functions
Geroliminis, N., Haddad, J., and Ramezani, M. (2013). Optimal perimeter control for two urban regions with macroscopic fundamental diagrams: A model predictive approach. IEEE Transactions on Intelligent Transportation Systems, 14(1), 348–359. Gomes, G. and Horowitz, R. (2006). Optimal freeway ramp metering using the asymmetric cell transmission model. Transportation Research Part C, 14(4), 244–262. Haddad, J. (2015). Robust constrained control of uncertain macroscopic fundamental diagram networks. Transportation Research Part C, 59, 323–339. Hajiahmadi, M., Haddad, J., De Schutter, B., and Geroliminis, N. (2015). Optimal hybrid perimeter and switching plans control for urban traffic networks. IEEE Transactions on Control Systems Technology, 23(2), 464–478. Keyvan-Ekbatani, M., Kouvelas, A., Papamichail, I., and Papageorgiou, M. (2012). Exploiting the fundamental diagram of urban networks for feedback-based gating. Transportation Research Part B, 46(10), 1393–1403. Kouvelas, A., Lioris, J., Fayazi, S.A., and Varaiya, P. (2014). Maximum pressure controller for stabilizing queues in signalized arterial networks. Transportation Research Record, (2421), 133–141. Kouvelas, A., Saeedmanesh, M., and Geroliminis, N. (2015). Feedback perimeter control for heterogeneous urban networks using adaptive optimization. In 18th IEEE International Conference on Intelligent Transportation Systems, 882–887. Kouvelas, A., Saeedmanesh, M., and Geroliminis, N. (2017). Enhancing model-based feedback perimeter control with data-driven online adaptive optimization. Transportation Research Part B, 96, 26–45. Ramezani, M., Haddad, J., and Geroliminis, N. (2015). Dynamics of heterogeneity in urban networks: aggregated traffic modeling and hierarchical control. Transportation Research Part B, 74, 1–19. Saeedmanesh, M. and Geroliminis, N. (2016). Clustering of heterogeneous networks with directional flows based on “Snake” similarities. Transportation Research Part B, 91, 250–269. Xu, J., van den Boom, T., De Schutter, B., et al. (2016). Model predictive control for continuous piecewise affine systems using optimistic optimization. In American Control Conference, 2016, 4482–4487. Yildirimoglu, M., Ramezani, M., and Geroliminis, N. (2015). Equilibrium analysis and route guidance in large-scale networks with MFD dynamics. Transportation Research Part C, 59, 404–420.
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A A A A
IFAC PapersOnLine 50-1 (2017) 8543–8548 Linear Linear Formulation Formulation for for Model Model Predictive Predictive Linear Formulation for Model Predictive Perimeter Traffic Control in Cities Linear Formulation for Model Predictive Perimeter Traffic Control in Cities Perimeter Traffic Control in Cities Perimeter Traffic Control in Cities Anastasios Kouvelas, Mohammadreza Saeedmanesh,
Anastasios Kouvelas, Mohammadreza Anastasios Mohammadreza Saeedmanesh, Nikolas Geroliminis Saeedmanesh, Anastasios Kouvelas, Kouvelas, Mohammadreza Saeedmanesh, Nikolas Geroliminis Saeedmanesh, Anastasios Kouvelas, Mohammadreza Nikolas Nikolas Geroliminis Geroliminis Nikolas Geroliminis Urban Transport Systems Laboratory Urban Transport Laboratory Urban Transport Systems LaboratoryEngineering School of Architecture, Civil Systems & Environmental Urban Transport Systems Laboratory School of Architecture, Civil & Environmental Engineering Urban Transport Systems ´ School of Architecture, Civil & Environmental Engineering School ofEcole Architecture, Civil F´ &ed´ Environmental Engineering Polytechnique eraleLaboratory de Lausanne ´ Ecole Polytechnique F´ e d´ e rale de Lausanne School of Architecture, Civil & Environmental Engineering ´ ´ Ecole Polytechnique F´ e d´ e rale de Lausanne CH-1015Ecole Lausanne, Switzerland {tasos.kouvelas, Polytechnique F´ed´e(e-mails: rale de Lausanne ´ Lausanne, CH-1015 Switzerland (e-mails: {tasos.kouvelas, Ecole Polytechnique F´ e d´ e rale de Lausanne CH-1015 Lausanne, Switzerland (e-mails: {tasos.kouvelas, mohammadreza.saeedmanesh, nikolas.geroliminis}@epfl.ch) CH-1015 Lausanne, Switzerland (e-mails: {tasos.kouvelas, mohammadreza.saeedmanesh, nikolas.geroliminis}@epfl.ch) CH-1015 Lausanne, Switzerland (e-mails: {tasos.kouvelas, mohammadreza.saeedmanesh, nikolas.geroliminis}@epfl.ch) mohammadreza.saeedmanesh, nikolas.geroliminis}@epfl.ch) mohammadreza.saeedmanesh, nikolas.geroliminis}@epfl.ch) Abstract: An alternative approach for real-time network-wide traffic control in cities that Abstract: An alternative approach forperimeter real-time flow network-wide traffic control in of cities that Abstract: An alternative approach for real-time traffic in has recentlyAn gained a lot ofapproach interest is control. The basiccontrol concept suchthat an Abstract: alternative for real-time network-wide network-wide traffic control in cities cities that has recently gained a lot of interest is perimeter flow control. The basic concept of such an Abstract: An alternative approach for real-time network-wide traffic control in cities that has recently gained a lot of interest is perimeter flow control. The basic concept of such an approach is togained partition heterogeneous cities into a small homogeneous regions (zones) has recently a lot of interest is perimeter flow number control. of The basic concept of such an approach isperimeter togained partition heterogeneous cities into aa small small number of homogeneous regions (zones) has recently a lot of interest is perimeter flow control. The basic concept of such an approach is to partition heterogeneous cities into number of homogeneous regions (zones) and apply control to the inter-regional flows along the boundaries between regions. approach is to partition heterogeneous cities into a small number of homogeneous regions (zones) and apply perimeter control to the the inter-regional inter-regional flows along the the boundaries between regions. approach isperimeter to partition heterogeneous a small number of homogeneous regionsbetween (zones) and apply perimeter control to flows along boundaries between regions. The apply transferring flowscontrol are controlled atcities the into traffic intersections located at the between borders and to the inter-regional flows along the boundaries regions. The transferring flows are controlled at the traffic intersections located at the borders between and apply perimeter control to the inter-regional flows along the boundaries between regions. The transferring are at traffic intersections located at borders regions, so as to flows distribute the congestion an optimal way and minimize total between delay of The transferring flows are controlled controlled at the the in traffic intersections located at the thethe borders between regions, so as as to focus distribute the congestion in anstudy optimal way and minimize the total between delay of The transferring flows arethe controlled at theisin traffic intersections located thenot borders regions, so to distribute the congestion in an optimal way and minimize the total delay of the system. The of current work toan three aspects thatatare covered in the regions, so as to distribute the congestion optimal way and minimize the total delay of the system. The focus of the thethe current work isinthe toanstudy study three aspects that are not covered in the regions, so as to distribute congestion optimal way and minimize the total delay of the system. The focus of current work is to three aspects that are not covered in the perimeter control literature, which are: (a) treatment of some model parameters that are the system. The focus of the current work is to study three aspects that are not covered in the perimeter control literature, which are: (a)isthe the treatment ofthe some model parameters that are the The in focus theimplementations currentare: work to study that are not in the perimeter control literature, which are: (a) treatment some model parameters that are not system. measurable realoflife and can three affectof performance of covered the controller perimeter control literature, which (a) the treatment ofaspects some model parameters that are not measurable in real life implementations and can affect the performance of the controller perimeter control literature, which are: (a) the treatment of some model parameters that are not measurable in real realestimation life implementations implementations and can affect the the performance of(b) theintegration controller (e.g.measurable advanced online schemes canand be can developed for this purpose),of not in life affect performance the controller (e.g. advanced external online estimation schemes can canthat be can developed for this this purpose), (b) integration not measurable in realestimation life implementations performance theintegration controller (e.g. advanced online estimation schemes be developed for purpose), (b) integration of appropriate demand information hasaffect been the considered systemof(b) disturbance in (e.g. advanced online schemes canand be developed for this purpose), of appropriate external demand information that has been and considered system (b) disturbance in (e.g. advanced online estimation schemes can be developed for this purpose), integration of appropriate external demand information that has been considered system disturbance in the derivation of feedback control laws in previous works, (c) mathematical formulation of appropriate external demand information that has been considered system disturbance in the derivation of feedback control laws in previous works, and (c) mathematical formulation of appropriate external demand information that has been considered system disturbance the derivation of feedback feedback controlinlaws laws in previous previous and mathematical formulation of the original nonlinear problem a linear form, soworks, that optimal can be applied inina the derivation of control in works, and (c) (c)control mathematical formulation of the original nonlinear problem inlaws a linear form, soworks, that optimal optimal control can be be applied in the derivation ofmodel feedback controlconcept. in This previous and mathematical formulation of the original nonlinear problem in form, so that control can applied aaa (rolling horizon) predictive work the(c) mathematical analysis ofin the of the original nonlinear problem in aa linear linear form, so presents that optimal control can be applied in (rolling horizon) model predictive concept. This work presents the mathematical analysis of the of the original nonlinear problem in a linear form, so that optimal control can be applied in (rolling horizon) model predictive concept. This work presents the mathematical analysis of the optimal horizon) control problem, as well asconcept. the approximations and simplifications that are assumed ina (rolling model predictive This work presents the mathematical analysis of the optimal control problem, as well wellofas asaconcept. the approximations and simplifications simplifications that are assumed in (rolling horizon) model predictive This work presents the mathematical analysis of the optimal control problem, as the approximations and that are assumed in order to derive the formulation linear optimization problem. Preliminary simulation results optimal control problem, as well as the approximations and simplifications that are assumed in order to derive the formulation of a linear optimization problem. Preliminary simulation results optimal control problem, as well as the approximations and simplifications that are assumed in order tocase derive the formulationenvironment of aa linear linear optimization optimization problem. results for theto of the a macroscopic (plant) are problem. presented,Preliminary in order tosimulation demonstrate the order derive formulation of Preliminary simulation results for thetocase case of the a macroscopic macroscopic environment (plant) are problem. presented, in order to demonstrate demonstrate the order derive formulation of a linear optimization Preliminary results for the of environment (plant) are presented, in to the efficiency of the approach. Results for the closed-loop model predictive control scheme for the case of aaproposed macroscopic environment (plant) are presented, in order order tosimulation demonstrate the efficiency of the proposed approach. Results for the closed-loop model predictive control scheme for the case of a macroscopic environment (plant) are presented, in order to demonstrate the efficiency of the proposed approach. Results for the closed-loop model predictive control scheme are presented forproposed the nonlinear case,Results which isforused “benchmark”, well as the linearscheme case. efficiency of the approach. the as closed-loop modelaspredictive control are presented presented forproposed the nonlinear nonlinear case,Results which is is used used as “benchmark”, aspredictive well as as the the linearscheme case. efficiency of the approach. the as closed-loop modelas control are for the case, which as “benchmark”, as well linear case. are presented for the nonlinear case, which isforused “benchmark”, well as the linear case. © 2017, IFAC (International Federation Automatic Control) Hosting by Elsevier Ltd.asAll reserved. are presented for the nonlinear case,ofwhich is used as “benchmark”, as well therights linear case. Keywords: Model predictive control; nonlinear optimisation; linear approximation; urban Keywords: Model predictive control; nonlinear optimisation; linear approximation; urban Keywords: Model predictive predictive control; control; nonlinear nonlinear optimisation; optimisation; linear linear approximation; approximation; urban urban perimeter control. Keywords: Model perimeter control. Keywords: Model predictive control; nonlinear optimisation; linear approximation; urban perimeter control. perimeter control. perimeter control. 1. INTRODUCTION perimeter flow control (or gating). The basic concept of 1. INTRODUCTION INTRODUCTION perimeter flow control control (or gating). gating). The basic basic concept of 1. perimeter flow (or The of such an approach is to partition heterogeneous cities into 1. INTRODUCTION perimeter flow control (or gating). The basic concept concept of such an approach is to partition heterogeneous cities into 1. INTRODUCTION perimeter flow control (or gating). The basic concept of such an approach is to partition heterogeneous cities into a small number of homogeneous regions (zones) and apply Traffic congestion is a major problem for urban environ- such an approach is to partition heterogeneous cities into asuch small number of homogeneous regions (zones) and apply Traffic congestion ismetropolitan a major major problem problem for urban environan approach is to partition heterogeneous cities into a small number of homogeneous regions (zones) and apply Traffic congestion is a for urban environperimeter control to the inter-regional flows along the ments and modern areas. Most cities around a small number of homogeneous regions (zones) and apply Traffic congestion is a major problem for urban environ- perimeter control to the inter-regional flows along the ments and modern metropolitan areas. Most cities around small number of homogeneous regions (zones) Traffic congestion ismetropolitan a persistently major problem for urban environcontrol to the flows along the ments and modern metropolitan areas. Most cities around boundaries between The inter-transferring flows the world have been becoming denser and aperimeter perimeter control toregions. the inter-regional inter-regional flows and alongapply the ments and modern areas. Most cities around boundaries between regions. The inter-transferring the world have been persistently becoming denser and perimeter control to the inter-regional flows along the ments and modern metropolitan areas. Most cities around boundaries between regions. The inter-transferring flows the world have beendecades persistently becoming denser and are controlled at theregions. traffic intersections located atflows the widerworld over have the last and the problem of urban boundaries between The inter-transferring flows the been persistently becoming denser and are controlled at the traffic intersections located at the wider over the last decades and the problem of urban boundaries between regions. The inter-transferring flows the world have been persistently becoming denser and are controlled at the traffic intersections located at the wider over the last decades and the problem of urban borders between regions, so as to distribute the congestion traffic over management steadily momentum due are controlled at the traffic intersections located at the wider the last is decades andgaining the problem of urban between regions, so asintersections to distribute the congestion traffic management is steadily gaining momentum due borders are controlled the traffic at wider over the last decades andgaining the problem of urban borders between regions, so the congestion traffic management is steadily gaining momentum due in an optimal way and minimize the totallocated delay of the the to its economic, social and environmental impact. Many borders betweenat regions, so as as to to distribute distribute the congestion traffic management is steadily momentum due in an optimal way and minimize the total delay the to its economic, social and environmental impact. Many borders between regions, so as to distribute the congestion traffic management is steadily gaining momentum due way minimize the total of the to its economic, economic, social andout environmental impact.settings Many in system (an alternative could theof efforts have beensocial carried to optimize signal in an an optimal optimal way and andobjective minimize the maximize total delay delay oftotal the to its and environmental impact. Many system (an alternative objective could maximize the total efforts have been carried out to optimize signal settings in an optimal way and minimize the total delay of the to its economic, social and environmental impact. Many system (an alternative objective could maximize the total efforts have been carried out to optimize signal settings during have the peak networks face serious con- throughput). system (an alternative objective could maximize the total efforts beenhours, carriedwhere out to optimize signal settings during the peak hours, where networks face serious con- throughput). system (an alternative objective could maximize the total efforts have been carried out to optimize signal settings throughput). during the peak hours, where networks face serious congestion problems and the performance of the infrastructure during the peak hours, where networks face serious con- throughput). Perimeter flow control can be viewed as a high-level gestion problems and the performance of the the infrastructure during the peak and hours, where networks faceinfrastructure serious congestion problems and the performance of infrastructure degrades significantly. The state-of-practice strategies fail throughput). Perimeter flow control viewed aa high-level gestion problems the performance of the Perimeter flow can be viewed as scheme can and be be as combined with degrades significantly. The state-of-practice strategies fail Perimetercontrol flow control control can bemight viewed as a high-level high-level gestion problems and the performance of the infrastructure degrades significantly. The state-of-practice state-of-practice strategies fail regional to deal efficiently with oversaturated conditions (i.e. queue regional control scheme and might be combined with degrades significantly. The strategies fail Perimeter flow control can be viewed as a high-level regional control scheme and might be combined with other strategies (e.g. local, distributed or coordinated conto deal efficiently with oversaturated conditions (i.e. queue regional control scheme and might be combined with degrades significantly. The state-of-practice strategies fail to deal efficiently with oversaturated conditions (i.e. queue spillbacks and partial gridlocks), asconditions they are (i.e. either de- other strategies (e.g. local, distributed or coordinated conto deal efficiently with oversaturated queue regional control scheme and might be combined with other strategies (e.g. local, distributed or coordinated controllers) in a hierarchical control framework; this topic has spillbacks and partial gridlocks), as they are either deother strategies (e.g. local, distributed or coordinated conto deal by efficiently with oversaturated conditions (i.e. queue spillbacks andofpartial partial gridlocks), as they are accurately either de- trollers) signed use simplified models as that do not in a hierarchical control framework; this topic has spillbacks and gridlocks), they are either deother strategies (e.g. local, distributed or coordinated controllers) in aaofhierarchical control framework; this topic has gained a lot attraction in the research community lately. signed by use of simplified models that do not accurately trollers) in hierarchical control framework; this topic has spillbacks and partial gridlocks), as they are either designed by some use of oftraffic simplified models that do do notpropagation accurately gained replicate flow models phenomena (e.g. a lot of attraction in the research community lately. signed by use simplified that not accurately trollers) in a hierarchical control framework; this topic has gained a lot of attraction in the research community lately. For a recent review on this research direction the reader is replicate some traffic flow phenomena (e.g. propagation gained a lot of attraction in the research community lately. signed by use of simplified models that do not accurately replicate some or traffic flow phenomena (e.g. (e.g. propagation propagation of congestion), basedflow on application-specific heuristics. For aa recent review on this research direction the reader is replicate some traffic phenomena gained a lot of attraction in the research community lately. For recent review on this research direction the reader is referred to Keyvan-Ekbatani et al. (2012); Yildirimoglu of congestion), or based on application-specific heuristics. For a recent review on this research direction the reader is replicate some traffic flow phenomena (e.g. propagation of congestion), or based on application-specific heuristics. Ancongestion), alternative or approach forapplication-specific real-time network-wide traf- referred to Keyvan-Ekbatani et al. (2012); Yildirimoglu of based on heuristics. For a recent review on this research direction the reader is referred to Keyvan-Ekbatani et al. (2012); Yildirimoglu et al. (2015). The original model for the dynamics of An alternative approach for real-time network-wide trafreferred to Keyvan-Ekbatani et al. (2012); Yildirimoglu of congestion), or based on application-specific heuristics. An alternative approach for real-time network-wide traffic control that has recently gained a lot of interest is et al. (2015). The original model for the dynamics of An alternative approach for real-time network-wide traf- et referred to Keyvan-Ekbatani et al. (2012); Yildirimoglu al. (2015). The original model for the dynamics of multi-region process (plant) is highly fic control control thatapproach has recently recently gained aanetwork-wide lot of of interest interest is the et al. (2015). The original model for thenonlinear dynamicsand of An alternative for real-time traffic that has gained lot is the multi-region process is highly and fic control that has recently gained a lot of interest is et (2015). The original model for dynamics of the multi-region process (plant) is nonlinear and modelling tool that (plant) is utilized is the thenonlinear Macroscopic This research has been supported by the ERC (European Research theal. multi-region process (plant) is highly highly nonlinear and fic control that has recently gained a lot of interest is the modelling tool that is utilized is the Macroscopic research has been supported by the ERC (European Research multi-region process (plant) is highly nonlinear and This the modelling tool that is utilized is the Macroscopic Council) Starting Grant “METAFERW: Modelling and controlling This research has been supported by the ERC (European Research Fundamental Diagram (MFD) (see e.g. Ramezani et al. the modelling tool that is utilized is the Macroscopic This research has been supported by the ERC (European Research Council) Starting Grant “METAFERW: Modelling andmulti-modal controlling Thiscongestion Fundamental (see e.g. Ramezani et al. the modelling tool that is utilized is the Macroscopic research has been supported bylarge-scale theModelling ERC (European Research traffic and propagation in urban Council) Starting Grant “METAFERW: and Fundamental Diagram (MFD) (see e.g. Ramezani et (2015)). MFD Diagram provides a(MFD) concave, low-scatter relationship Council) Starting Grant “METAFERW: Modelling and controlling controlling Fundamental Diagram (MFD) (see e.g. Ramezani et al. al. traffic congestion propagation in large-scale urban (2015)). MFD Diagram provides aa(MFD) concave, low-scatter relationship Council) Starting Grant “METAFERW: Modelling andmulti-modal controlling networks” (Grant and #338205). traffic congestion and propagation in urban multi-modal Fundamental (see e.g. Ramezani et al. (2015)). MFD provides concave, low-scatter relationship traffic congestion and propagation in large-scale large-scale urban multi-modal (2015)). MFD provides a concave, low-scatter relationship networks” (Grant and #338205). traffic congestion propagation in large-scale urban multi-modal networks” (Grant (2015)). MFD provides a concave, low-scatter relationship networks” (Grant #338205). #338205).
networks” (Grant #338205). Copyright © 2017 IFAC 8877 Copyright 2017 IFAC 8877 2405-8963 © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 8877 Copyright © 2017 IFAC 8877Control. Peer review©under of International Federation of Automatic Copyright 2017 responsibility IFAC 8877 10.1016/j.ifacol.2017.08.1411
Proceedings of the 20th IFAC World Congress 8544 Anastasios Kouvelas et al. / IFAC PapersOnLine 50-1 (2017) 8543–8548 Toulouse, France, July 9-14, 2017
between network vehicle accumulations [veh] or density [veh/km] and network circulating flow [veh/h] or production [veh·km] for every region of the system which is homogeneously congested. Clustering techniques can help partition a network into regions with small variance of link densities (see e.g. Saeedmanesh and Geroliminis (2016)). Recently, there has been some effort to simplify this model; the authors in Hajiahmadi et al. (2015) proposed a hybrid model for formulating this problem and applied model predictive control (MPC) in conjunction with switching plans control. A pre-defined set of MFDs is introduced (which correspond to regional switching plans), and some affine approximations are made in order to reduce the complexity of the derived hybrid MPC approach. In the current work we devise a linear program (LP) formulation to solve the finite-time optimal perimeter flow control problem. Moreover, we investigate two more aspects of the problem that have not been covered in the literature: (a) the treatment of some model parameters that are not easily measurable with loop detectors data in real life implementations, and can affect the performance of the controller; especially the regional route choice parameters and the origin-destination information, and (b) integration of realistic external demand information that has been considered system disturbance in the derivation of multivariable feedback control laws in previous works (Kouvelas et al. (2015)). Finally, the derived LP is solved in a rolling time horizon using a feedback MPC framework, and the control decisions are applied to the nonlinear plant for evaluation. The efficiency of the control decisions is compared to a “benchmark” case, in which the nonlinear MPC problem is solved using state-of-the-art nonlinear numerical solvers. Simulation results for the case of a macroscopic model (plant) are presented. Note that this benchmark approach is more challenging for application in real life due to computational requirements, and most importantly, the lack of detailed data (i.e. high resolution vehicle trajectories data is needed). Essentially, real-world data availability can restrain the methodologies that can be applied, and, consequently, the real-time applicability of the benchmark approach is deemed cumbersome. 2. AGGREGATED DYNAMICS FOR A PARTITIONED CITY Consider an urban network partitioned in N homogeneous regions with well-defined MFDs (Figure 1). The index i ∈ N = {1, 2, . . . , N } denotes the region of the system, ni (t) the total accumulation (number of vehicles) in region i, and nij (t) the number of vehicles in region i with final destination region j ∈ N , at a given time t. Let Ni be the set of all regions that are directly reachable from the borders of region i, i.e. adjacent regions to region i. The discrete time MFD dynamics of the N -region system can be described by the following first order difference equations nii (kp + 1) = nii (kp ) + Tp −
h∈Ni
qii (kp ) − Mii (kp )
Miih (kp ) +
h∈Ni
i Mhi (kp )
(1)
𝑢𝑢𝑗𝑗𝑗𝑗
𝑃𝑃𝑖𝑖
𝑖𝑖
𝑛𝑛𝑖𝑖
𝑢𝑢𝑖𝑖𝑖𝑖
𝑗𝑗
𝑃𝑃𝑗𝑗
𝑛𝑛𝑗𝑗
Fig. 1. The network of a city partitioned into a multi-region MFD system.
nij (kp + 1) = nij (kp ) + Tp −
qij (kp )
h (kp ) + Mij
h∈Ni
i Mhj (kp )
h∈Ni
(2)
where i, j ∈ N , i = j; kp = 0, 1, . . . , Kp − 1 is the model discrete time index, Tp [sec] the sample time period of the model (i.e. time t = kp Tp ). The exogenous variables qij (kp ) [veh/sec] denote the (uncontrollable) traffic flow demand that is generated in region i, at time step kp , with final destination in region j (i.e. qii (kp ) is the demand generated in region i that has final destination in region i). h The variables Mij (kp ) [veh/sec] denote the transfer flows from region i to region h, that have final destination region j, while Mii (kp ) [veh/sec] is the internal trip completion rate of region i (without going through another region). Consequently, the total accumulation ni (kp ) for region i can be computed by ni (kp ) = j∈N nij (kp ).
We assume that for each region i there exists a production MFD between accumulation ni (kp ) and total production, Pi (ni (kp )) [veh·m/sec], which describes the performance of the system in an aggregated way. This MFD can be estimated using measurements from loop detectors and/or h GPS trajectories. Transfer flows Mij (kp ) and internal trip completion rates Mii (kp ) are calculated according to the corresponding production MFD of the region, and proportionally to the accumulations nij (kp ) as follows nii (kp ) Pi (ni (kp )) Mii (kp ) = θii (kp ) (3) ni (kp ) Li h Mij (kp ) = min Cih (nh (kp )), h (kp ) uih (kc )θij
nij (kp ) Pi (ni (kp )) ni (kp ) Li
(4)
where i, j ∈ N , h ∈ Ni ; kc = kp /Nc , with Nc some positive integer, is the control discrete time index (i.e. the control cycle is always a multiple of the plant sample time Tp ), and Li [m] is the average trip length for region i, which is assumed to be independent of time and destination,
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internal (inside region i) or external (to some other region h j). The parameters θii (kp ), θij (kp ) reflect the route choice of drivers and are assumed to be exogenous (i.e. can be constant or time varying and they are provided by h another methodology). The transfer flows Mij (kp ) are the minimum between the sending flow from region i (which only depends on the accumulation of the region), and the receiving capacity Cih (nh (kp )) [veh/sec] of region h. This flow capacity is a piecewise function of the accumulation nh (kp ) (usually modelled with two pieces, one constant value and a decreasing curve), and is introduced to prevent overflow phenomena within the regions, i.e. each region i has a maximum accumulation ni,max such that 0 ≤ ni (kp ) ≤ ni,max , ∀ i ∈ N . (5) If ni (kp ) = ni,max , the region reaches gridlock and all the inflows along the periphery are restricted. Finally, the control variables uih (kc ), ∀ i ∈ N , h ∈ Ni denote the fraction of the flow that is allowed to transfer from region i to region h for the time interval [(Nc kc )Tp , (Nc kc + Nc − 1)Tp ]. Physical constraints are applied to the values of the control variables as follows 0 ≤ uih (kc ) ≤ 1, ∀ i ∈ N , h ∈ Ni (6) but also – depending on the implementation – operational constraints of the following form might apply, i.e., |uih (kc ) − uih (kc − 1)| ≤ uR (7) ih , ∀ i ∈ N , h ∈ Ni . Equations (1)–(4) are a discretized version of equations presented in Ramezani et al. (2015) and represent the traffic dynamics of an N -region urban network considering the heterogeneity effect and integrating an aggregated routing model. Note, that these equations allow the drivers to choose any arbitrary sequence of regions as their route and their path can cross region boundaries multiple times.
subject to equations (1), (2), (3), (4), (5), (6), (7) nij (k) Pi (ni (k)) h h (k) = uih (κ)θij (k) Mij ni (k) Li ni (k) = nij (k)
2.2 Linearising the problem In the current work we derive a linear approximation of the model described in the previous section, and we formulate a linear MPC problem that can be utilized for real-time control purposes. In order to linearise the dynamic equations we assume the following simplifications and approximations: • We introduce the model parameters for the accumulation proportions, i.e. αii (k) = nii (k)/ni (k) and αij (k) = nij (k)/ni (k), i ∈ N , j ∈ Ni . One approach, that is implemented here, is to get feedback for the values of these parameters every time that we roll the horizon. They can be estimated in real-time from measurements (e.g. using extended Kalman filter or maximum likelihood approximation) and then be kept constant for all the optimisation horizon. Note that the parameters can be time varying but they need to be exogenous signals for the MPC framework. • New “dummy” control variables uii (k) are introduced ∀ k = kp , kp + 1, . . . , kp + Np − 1, that restrict the trip completion rates at every region i. Although these variables are not reasonable from a physical point of view, they are required in order for the problem to be linear. A conjecture is that the solution of MPC will always result in uii (k) = 1, ∀ i ∈ N , ∀ k = kp , kp + 1, . . . , kp + Np − 1, but this needs to be validated through results. • Most importantly, we introduce new decision variables fii (k) = uii (k)Gi (ni (k))θii (k)αii (k) (12) h fih (k) = uih (k)Gi (ni (k)) θij (k)αij (k) (13)
h • the quantities qii (k), qij (k) and θii (k), θij (k) are considered exogenous variables that can be measured or given by another algorithm beforehand, • as in many similar works, the capacity constraint Cih (nh (kp )) in (4) is dropped, since from a control viewpoint it is not necessary; the control actions will not allow the system to operate in states close to gridlock, and this constraint is never activated inside the NMPC.
k=kp
i∈N
(10)
k = kp , kp + 1, . . . , kp + Np − 1, κ = k/Nc (11) ∀ i, j ∈ N , h ∈ Ni The objective function (8) tries to maximize the total production of the system for a horizon of Np model steps (or Np Tp seconds). This problem can be solved in reasonable time by use of advanced nonlinear optimization toolboxes (e.g. ipopt 1 ), and in our case serves as a benchmark for the results reported later. Note that if we want to compare the results with the LP approach described later, the objective function should be the same in both cases in order to have a fair comparison.
The MFD dynamics described in the previous section derive a nonlinear model that has been used in other works (Geroliminis et al. (2013); Ramezani et al. (2015)) to apply nonlinear model predictive control (NMPC). Here, we solve the same problem again to obtain a benchmark, which is then used as a comparison to the LP formulation presented later. In order to have a well defined problem and without loss of generality – since this is a nonlinear MPC problem – the following assumptions are made for formulating the problem:
nii (k), nij (k), uih (κ)
(9)
j∈N
2.1 Nonlinear model predictive control (NMPC)
Given these two reasonable assumptions the nonlinear optimal control problem for a horizon of Np model steps is defined as follows kp +Np −1 h Li Mii (k) + uih (κ)Mij (k) (8) maximize
8545
j∈N
∀ i, j ∈ N , h ∈ Ni that help linearise the equations. In (12)–(13) the h variables θii (k), θij (k), αii (k), αij (k) are considered time varying exogenous signals and as a result the nonlinearity of the problem comes from the product of the control inputs uih (k) with the MFD functions.
1 http://www.i2c2.aut.ac.nz/Wiki/OPTI/index.php/Solvers/ IPOPT
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• To overcome this, we approximate the MFDs of the regions with piecewise affine (PWA) functions Gi (ni (k)) that form a convex set (see e.g. Figure 3 for a case study with 4 regions). Each MFD can be approximated with l = 1, 2, . . . , Ni affine functions, and we denote as Gli (ni (k)) each affine function l. In conclusion, the control variables have the property of being bounded (i.e. uih (k) ∈ [0, 1], ∀ i ∈ N , h ∈ Ni , k = kp , kp + 1, . . . , kp + Np − 1) and the MFDs that can be approximated by PWA functions. As a result, we are looking for an optimal solution within a convex set, and in this particular case the product can be linearised by introducing the new variables fii (k), fih (k) (see Gomes and Horowitz (2006) for some theoretical analysis of a similar convexification in a ramp metering control problem). Moreover, PWA approximations is a popular technique to reduce complexity and attempt to linearise nonlinear systems (see e.g. Xu et al. (2016)); alternatively someone could use robust control to solve another version of this problem, by introducing uncertain parameters (see e.g. Haddad (2015)). Finally, once the optimal solution is computed, there is a unique transformation between the new variables and the original control variables uii (k), uih (k). This is a modelling trick that allows us to simplify the problem without loosing any accuracy in the dynamics.
3.1 LMPC without OD information Moving one step forward with our approximation, the new model does not need to keep track of the OD information (aggregated information about each region can be sufficient for control purposes). Hence, by adding all the states nij and nii for each region i, we get a linear model that does not consider OD data, but only aggregated demands in the region level. In that case, the derived LP that approximates the original system and can be solved online is as follows
ni (k), fii (k), fih (k)
k=kp
i∈N
Li [fii (k) + fih (k)]
− fii (k) −
qi (k) fih (k) +
h∈Ni
fhi (k)
h∈Ni
0 ≤ fii (k) ≤ θii (k)αii (k)Gli (ni (k)) h θij (k)αij (k) 0 ≤ fih (k) ≤ Gli (ni (k))
(15)
(16) (17)
j∈N
0 ≤ ni (k) ≤ ni,max (18) (19) k = kp , kp + 1, . . . , kp + Np − 1 ∀ i, j ∈ N , h ∈ Ni , l = 1, 2, . . . , Ni where the objective function (14) again represents the total production of the system. All the constraints of this problem are linear, and, as a consequence, the computational requirements are quite low, even for a network with many regions and large prediction horizons. Note that constraint (7) can also be applied for the first step of the LMPC without breaking the linearity, by using the following linear inequalities P R h fih (kp ) ≤ uPR (kp )αij (kp ) θij ih + uih Gi (ni (kp )) j∈N
j∈N
The assumptions outlined above are reasonable approximations/simplifications of the nonlinear model in order to derive a linear formulation that can be used for online MPC. In this section we formulate a linear model predictive control (LMPC) problem, which does not keep track of the origin-destination information of vehicles. This dynamic model has less online data requirements (as it carries lower level of information, i.e. only state and demand trajectories di , instead of dij ), but under certain optimization horizons can provide similar optimal solutions for the control variables (see Section 4).
maximize
ni (k + 1) = ni (k) + Tp
(20) P h R fih (kp ) ≥ uPR (n (k )) θ (k )α (k − u G i p ij p ) ij p ih ih i
3. LINEAR MODEL PREDICTIVE CONTROL (LMPC)
kp +Np −1
subject to
(14)
(21) ∀ i, j ∈ N , h ∈ Ni PR where uR ij are user defined bounds, uij are the control commands applied to the plant in the previous control cycle, and GP i (ni (kp )) are the affine functions that the accumulations ni (kp ) of each region i belong to (this can be easily found as ni (kp ) are known). Besides, only the first control decision is applied to the nonlinear plant and then we roll the horizon. 4. PROOF OF CONCEPT This section presents some simulation results obtained for the described methodology. The simulation model (plant) is the nonlinear model presented (1)–(5). The test case network is a replica of the network used in Kouvelas et al. (2017) and corresponds to a part of the CBD of Barcelona in Spain (Figure 2(a)). The network is partitioned into 4 regions in order to apply perimeter control (Figure 2(b)). Figure 3 presents the MFDs of the 4 regions from data obtained from a microsimulation model in Kouvelas et al. (2017). The red lines present the PWA approximation of the MFDs with the affine functions utilised by the LP approximation. The approximation is pretty accurate and this relaxation should not cause any problems in the optimization procedure. In that respect, the nonlinear model and PWA approximation are almost identical; note also, that this approximation can be done with many more lines than presented here without hardening significantly the computations of the LP solver (6 pieces are shown for each MFD in Figure 3, while in the LMPC 30 pieces have been used for better accuracy).
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8547
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u41 u14
4
Demand [veh/sec]
u43
3
u34
u24
u42
2
d
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d
2
d
(b)
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d
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d
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d
d41
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×10
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Fig. 4. Traffic demand for the four regions and all simulation horizon (4×4 OD matrix, i refers to origin and j to destination).
14
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4
n4,cr
Np 11
4
1.5
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d31
MFD4 PWA
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n2,cr
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2 ×10
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4000
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PWA
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1,cr
MFD2
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0.5
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PWA
0.5
6000
Demand [veh/sec]
22 MFD1
0
4000
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Demand [veh/sec]
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d
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Fig. 2. Test case network: (a) map of Barcelona CBD; (b) partitioning into 4 homogeneous regions and control variables.
n
14
d
2.5
Time [sec]
(a)
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0.5
500 m
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1.5
0
d
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Demand [veh/sec]
d
3
Nm 14
0
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8000
Nm 21 Np 22
1500
Nm 22 Np 23
1000
Nm 23 Np 24
500 0
10000
Np 21
2000
Nm 24
0
2000
Time [sec])
Fig. 3. MFDs for the 4 regions of the case study network (blue); piecewise affine approximation of the MFDs to be used in linear MPC (red).
Finally, Figure 6 displays the results for the regional accumulations when the two approaches are applied to control the transferring flows between regions. The objective of the controller is to maximize the production in the network and the parameters used for this simulation are Tp = 20sec, Nc = 3, Np = 21, uR ij = 0.2, Ni = 30, ∀ i, j ∈ N . It is clear that the controller improves the traffic states of the system, and the area between the blue and the dashed black lines (or dashed green lines) corresponds to the total delay improvement in the system. These results are quite promising, as it is demonstrated that we can
8000
10000
2000 Np 41
Accumulation [veh]
Np 31
2500
Nm 31 Np 32
2000
Nm 32
1500
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1000
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500 0
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Time [sec])
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Accumulation [veh]
First, we present a comparison of the plant (1)–(5) and the linear model presented earlier (12)–(13), (15), (18) for the no control (NC) case (i.e. uih (kc ) = 1, ∀ i ∈ N , h ∈ Ni , kc = kp /Nc ), which will also be our base scenario. Figure 4 presents a demand scenario (i.e. generated vehicles per time unit for all the simulation horizon) for the case study with 4 regions (e.g. 4×4 OD matrix). For this demand, Figure 5 displays the trajectories of accumulations nij for the plant and for the linear model, which actually correspond to the estimation of nij used within the LMPC framework (using ni and αij ). The prediction horizon for the linear model is 21 times higher than the sample time of the plant (e.g. Np = 21). The trajectories of the accumulations nij demonstrate that the linear model can be used to approximate the original nonlinear plant (given small prediction horizons and feedback of model parameters αij ). This model is a quite accurate representation of the original system, thus is appropriate to be used for the LMPC framework.
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1000
Np 43 Nm 43 Np 44
500
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6000
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Time [sec])
Fig. 5. Vehicle accumulations nij for the plant (solid lines) and the linear model (dashed lines) when applied for 21 steps of prediction (Np = 21) and the NC case. achieve the same level of improvement by using the linear approximation of the model described in section 3.1, which also does not require OD information. Table 1 presents the performance evaluation of the two approaches. All simulated scenarios start and finish with an empty network and serve all the demand (79143 vehicles) 2 . The total travelled distance (TTD) in all scenarios is equal to 169.55 × 103 veh·km and the total travelled time (TTT) by all vehicles in the network is computed by TTT =
Kp −1
ni (k)
(22)
k=0 i∈N
and presented in Table 1 for each case. Both NMPC and LMPC achieve a substantial improvement in terms of network mean speed (NMS=TTD/TTS) and TTT (around 20–30%).Their performance is quite similar, however the LMPC approach has less data requirements and it is a LP formulation that guarantees convergence and optimality. 2 Note that for very small accumulation values at the beginning and end of simulation NC is applied in all cases.
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Proceedings of the 20th IFAC World Congress 8548 Anastasios Kouvelas et al. / IFAC PapersOnLine 50-1 (2017) 8543–8548 Toulouse, France, July 9-14, 2017
n 2 NC
n 1,cr n 1 NMPC
8000
n 1 LMPC
6000 4000 2000 0
0
2000
4000
6000
8000
3000
n 2 NMPC n 2 LMPC
2000
1000
0
10000
n 2,cr
0
2000
Time [sec]
4000
6000
10000
8000
3000
n 3 NMPC n 3 LMPC
2000
1000
0
2000
4000
6000
8000
ACKNOWLEDGEMENTS
n 4 NC
n 3,cr
Accumulation [veh]
n 3 NC
Accumulation [veh]
8000
Time [sec]
4000
0
and demand profiles. Finally, another research direction is to use perimeter control as a first-level controller in cities (as it deals with zone interactions) and develop a secondlevel of distributed control (e.g. Kouvelas et al. (2014)) for optimizing locally. The combination of the two provides a hierarchical control scheme that could potentially be more efficient in alleviating traffic congestion in cities, but this needs also to be further investigated.
4000 n 1 NC
10000
Accumulation [veh]
Accumulation [veh]
12000
10000
n 4,cr
6000
˙ The authors are grateful to I¸sik Ilber Sırmatel for the valuable comments during the development of this work.
n 4 NMPC n 4 LMPC
4000
2000
0
REFERENCES 0
2000
Time [sec]
4000
6000
8000
10000
Time [sec]
Fig. 6. Vehicle accumulations ni for the 4 regions for NC case (blue lines), NMPC (dashed black lines), and LMPC (dashed green lines) when applied for 21 steps of prediction (Np = 21). Table 1. Performance of different approaches. Criteria TTS (veh·h)×103 NMS (km/h)
NC
NMPC
(%)
LMPC
(%)
17.73
13.76
-22.39
13.58
-23.41
9.57
12.32
28.74
12.49
30.51
5. CONCLUSION A LP formulation is derived for solving the perimeter control problem in multi-region cities. The originally nonlinear system is relaxed and approximated by a simplified linear model that under certain assumptions can track the behaviour of the multi-region system. The new model requires less information in terms of real-time measurements (e.g. traffic states, OD demands), and, because it is formulated as a LP, it guarantees optimality and fast convergence of the solver. The simulation results need to be further investigated in order to assess the goodness of the solution obtained through the linear MPC to the benchmark, which consists of the nonlinear MPC problem with full information about demands and state feedback measurements. Note that a field implementation of the benchmark approach would require real-time measurements (or estimates) of nij and dij , while the LMPC version requires only region-based measurements (i.e. ni and di ) and not detailed origin-destination data. Future work will deal with the development of a more solid methodology for estimating the model parameters αij (e.g. online extended Kalman filter), as the values of these parameters are crucial for the optimization horizon. These parameters can be estimated at every control cycle and then considered constant for all the optimization horizon of the MPC and this could not be sufficient for the improvement of the system. Simple estimation/prediction techniques can be used to enhance the knowledge for this parameters and help the convex problem to track the nonlinear dynamics in a better way. Investigations about different convex objective functions for the MPC is also another research topic. The proposed methodology needs to be evaluated for different realistic objective functions
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