Model predictive traffic control: A. A mixed-logical dynamic approach based on the link transmission model,

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1 Delft Unversty of Technology Delft Center for Systems and Control Techncal report Model predctve traffc control: A mxed-logcal dynamc approach based on the lnk transmsson model M. Hajahmad, B. De Schutter, and H. Hellendoorn If you want to cte ths report, please use the followng reference nstead: M. Hajahmad, B. De Schutter, and H. Hellendoorn, Model predctve traffc control: A mxed-logcal dynamc approach based on the lnk transmsson model, Proceedngs of the 13th IFAC Symposum on Control n Transportaton Systems (CTS 2012), Sofa, Bulgara, pp , Sept Delft Center for Systems and Control Delft Unversty of Technology Mekelweg 2, 2628 CD Delft The Netherlands phone: (secretary) fax: URL: Ths report can also be downloaded vahttp://pub.deschutter.nfo/abs/12_033.html

2 Model Predctve Traffc Control: A Mxed-Logcal Dynamc Approach Based on the Lnk Transmsson Model Mohammad Hajahmad Bart De Schutter Hans Hellendoorn Delft Center for Systems and Control, Delft Unversty of Technology, Delft, The Netherlands, (e-mal: m.hajahmad,b.deschutter,j.hellendoorn@tudelft.nl) Abstract: In ths paper, model predctve control of traffc networks usng frst-order macroscopc lnk transmsson model (LTM) s consdered. The LTM model provdes fast yet accurate predctons for traffc networks compared to other models. In order to use ths model for traffc control, t s extended to nclude ramp meterng. Usng the extended LTM model as predcton model n a model predctve control framework, one can determne optmal control sgnals for metered on-ramps. However, the optmzaton problem s stll nonlnear and nonconvex, and n general t s not tractable to fnd ts global optmum, as global or mult-start local optmzaton technques take consderable tme. Therefore, n ths paper the extended LTM model s transformed nto a mxed logcal dynamc model. The resultng optmzaton problem can be recast as a mxed nteger lnear program (MILP) that can be solved much more effcently than the nonlnear optmzaton problem, and t allows to determne a global optmum effcently. A smple case study s selected, frst to test the modelng performance of the extended LTM and next to compare the control performance of the MILP approach and the orgnal nonlnear formulaton n terms of computatonal effcency and total cost. Keywords: Traffc control, lnk transmsson model, predctve control, mxed logcal dynamcs, mxed nteger lnear programmng. 1. INTRODUCTION Wth the ncreasng number of vehcles, hghways are becomng more and more congested. Ths along wth ncreasngly strngent traffc requrements necesstates the use of effcent large-scale traffc management and control methods. One partcular soluton to ths problem s based on Model Predctve Control (MPC), where a fnte-horzon constraned optmal control problem s solved n a recedng horzon fashon (Rawlngs and Mayne, 2009; Macejowsk, 2002). In the MPC framework a model of the process s requred to predct ts behavor over a predcton wndow. For traffc networks, a wde range of traffc flow models have been developed (Hoogendoorn and Bovy, 2001). MPC requres a traffc model that can provde accurate predctons of the traffc states whle t has low computatonal complexty. The METANET model (Messmer and Papageorgou, 1990) s a second-order model that s able to model the traffc network wth good accuracy. However, ths s a nonlnear model and when t s used as predcton model n the MPC framework, a nonlnear nonconvex optmzaton problem results. Ths approach has been consdered n Kostsalos et al. (2002), Hegy et al. (2005), and Bellemans et al. (2006); for a smple case study, t has been shown that solvng the nonlnear optmzaton problem based on the METANET model takes consderable tme and n fact there s no guarantee to have a unque global optmal soluton. Groot et al. (2011) proposed a method to transform the orgnal nonlnear problem nto a mxed nteger lnear optmzaton problem. Ths has been done by approxmatng the METANET model by pecewse affne (PWA) functons. Although ths can solve the aforementoned computatonal complexty problem, usng ths approach for larger networks s mpractcal and stll takes large amount of computaton tme. One way to overcome ths problem for large-scale traffc networks s to use frst-order models lke the cell transmsson model (Daganzo, 1994) and the lnk transmsson model (Yperman, 2007). These models are mostly used for dynamc traffc assgnment problems (Peeta and Zlaskopoulos, 2001). However, ths paper consders usng them for other control purposes. The Cell Transmsson Model (CTM) has been wdely used to model traffc evoluton. However, usng the CTM as predcton model n the MPC framework wll result n a nonlnear optmzaton problem. One way to tackle ths problem s usng the relaxed formulatons proposed by Zlaskopoulos (2000), Ukkusur and Waller (2008) to obtan lnear problems. Ths s however an approxmaton for the orgnal optmzaton problem. Moreover, accordng to Lo(2001) ths approxmaton leads to a phenomenon called vehcle holdng. Ths means that vehcles are unnecessarly held n cells for some tme perod despte there beng spare capacty at the next downstream

3 cell. Therefore, Ln and Wang (2004) and Lo (2001) have proposed a new formulaton based on the CTM that leads to a mxed nteger lnear optmzaton problem. Besdes not usng approxmaton, reachng a global optmum s also guaranteed. Recently the lnk transmsson model (LTM) has been developed by Yperman (2007). The LTM has a lower computatonal complexty than the CTM and METANET. Ths s due to the fact that the LTM calculates the traffc varables for only the boundares of the lnks. Moreover, to reduce computatonal efforts n CTM, one could enlarge the length of the tme step, but such an operaton leads to reducton n accuracy. In the LTM, t can be proved (Yperman et al., 2005) that one can get acceptable accuracy wth less computatonal effort. For the frst tme, we am at usng the LTM for predcton n the MPC framework. However, the LTM s stll a nonlnear model. In ths work, we am at usng a procedure for transformng the nonlnear LTM nto a system of lnear equatons and nequaltes wth real and nteger varables. Based on ths new formulaton, one can buld a mxed nteger lnear problem, whch can be solved more effcently than the orgnal nonlnear optmzaton problem based on the orgnal LTM. The remander of ths paper s organzed as follows. In Secton 2, frst the orgnal LTM s brefly ntroduced and next the model s extended to nclude ramp meterng. Next, n Secton 3 MPC for traffc networks s presented and traffc performance functons are brefly revewed. In Secton 4, rules for transformng the LTM nto a system of lnear equaltes and nequaltes are presented. At the end, the fnal mxed nteger lnear optmzaton problem based on the new formulaton of the LTM s establshed. A case study n Secton 5 s presented to test the nonlnear MPC based on the orgnal LTM and the new approach based on the new formulaton of the LTM. The performance of the two approaches s compared n terms of computatonal effcency and total cost. Conclusons and topcs for further research are gven n Secton THE LINK TRANSMISSION MODEL The Lnk Transmsson Model (LTM) s a model orgnally developed for dynamc traffc assgnment (Yperman, 2007). The role of dynamc traffc assgnment (DTA) (Peeta and Zlaskopoulos, 2001) s to frst assgn optmal routes to the travelers usng a route choce model and then smulate and evaluate the assgned routes usng a traffc model. The traffc model should be capable of descrbng the traffc evoluton of a transportaton network (e.g. freeway or urban networks). In the followng, the LTM s presented brefly. The reader s referred to Yperman(2007) and Yperman et al. (2005) for an n-depth descrpton of theltm.rghtafterthspart,anextensontotheorgnal model for ncludng ramp meterng sgnals s proposed. In the LTM framework, traffc networks consst of homogeneous lnks, that start at an upstream boundary denoted by x 0 and end at a downstream boundary denoted by xl. The lnks have a length L and they are connected to each other va nodes. A node can represent a change n the characterstcs of a road such as capacty, speed lmts, etc. (nhomogeneous nodes), mergng lanes and/or onramps (merge nodes), or dvergng lanes and/or off-ramps (dverge nodes). There are also orgn and destnaton nodes, whch can be ncluded n the nhomogeneous nodes category. Moreover, cross nodes are defned for modelng urban ntersectons. The LTM s capable of determnng tme-dependent lnk volumes, lnk travel tmes, and route travel tmes n traffc networks. To ths am, the LTM uses the cumulatve number of vehcles as a representaton for the traffc evoluton. The cumulatve number of vehcles N(x, t) s defned for the upstream and downstream boundares of the lnks. The values of N(x 0,t) and N(xL,t) are updated usng flow functons of lnks and nodes defned n the followng. The sendng number of vehcles 1 S (t) of lnk at tme t s defned as the maxmum amount of vehcles that could leave the downstream end of ths lnk durng the tme nterval [t,t + t], where t s the smulaton tme step. It s constraned by the lnk s maxmum flow q M, and s formulated as [ ( S (t) = mn N x 0,t+ t L ) ] N(x L,t), q M, t υ free, (1) where υ free, and L are the free-flow speed and the length of lnk. Smlarly, the recevng number of vehcles R (t) of lnk at tme t s defned as the maxmum amount of vehcles that could enter the upstream end of ths lnk durng the tme nterval [t,t+ t], and t s also lmted by the lnk s maxmum flow. [ It ( s formulated as follows R (t) = mn N x L,t+ t+ L ) +ρ max L N(x 0 w,t), ] q M, t (2) where w and ρ max are the congeston speed and the jam densty, respectvely. 2.1 Node models For each of the nodes, a transton number of vehcles G j (t) s defned and determned by usng the sendng and recevng numbers of vehcles of the connected lnks. In fact, the transmsson flow determnes the maxmum number of vehcles that can be transferred from ncomng lnks to outgong lnks of a node durng the tme nterval [t,t+ t]. For the nhomogeneous nodes, the transton number G j (t) s formulated as G j (t) = mn [ S (t),r j (t) ] (3) where s the ncomng lnk and j s the outgong lnk. For orgn nodes, the transton number of vehcles s determned as follows: G j (t) = mn [ N o (t+ t) N(x 0 j,t), R j (t) ] (4) 1 Yperman (2007) uses the term sendng flow for ths purpose, but snce t s not a flow, we prefer to use term number of vehcles. The same holds for other model varables that wll be defned later.

4 where j s the ndex of the frst lnk connected to the orgn and N o denotes the traffc demand n orgns n terms of the cumulatve number of vehcles. A smple queue model for orgns s defned as: ω o (t) = N o (t) N(x 0 j,t) (5) where ω o (t) and N(x 0 j,t) denote the number of vehcles standng n the queue and the cumulatve number of vehcles that already entered the network at tme t, respectvely. It should be noted that ths a Pont-Queue(P-Q) model. The transton number of vehcles for destnaton nodes s equal to the sendng number of vehcles of the last lnk connected to the destnaton node: G j (t) = S (t) (6) wth j and the destnaton and the last lnk. Merge nodes can represent mergng of lnks and/or on-ramps n traffc networks. To model a mergng node n wth predefned prortes for the ncomng lnks, Lebacque (1996) has proposed the followng equaton for the transton number of vehcles of the ncomng lnks of the merge node G j (t) = mn [ S (t),p j R j (t) ] for all I n (7) where p j s the prorty parameter assocated wth ncomng lnk connected to the only outgong lnk j va the merge node, and I n s the set of ncomng lnks to node n. The prorty parameter s determned for each lnk based on the characterstcs of the lnk (e.g. capacty, number of lanes,...) and t should be noted that p j = 1. Jn and Zhang (2003) proposed another model that does not have fxed prorty parameters. The prorty proportons are equal to the proportons of S (t) of the ncomng lnks. The transton number of vehcles s formulated as [ ] R j (t)s (t) G j (t) = mn S (t),s (t) for all I n (8) I n where j s the only outgong lnk of merge node n. Fnally, the thrd model for a merge node proposed by Daganzo (1995), s formulated as follows: [ ( ( ) ) G j (t) = medan S (t),r j (t) S (t) S (t), I ] n p j R j (t) for all I n (9) G j s determned for each lnk from the set of ncomng lnks I n connected to outgong lnk j va the merge node n. For dverge nodes that are used to model dvergng lnks and/or off-ramps n traffc networks, two types of equatons have been proposed n the lterature. The frst one was proposed by Daganzo (1995): G j (t) = q j mn [ S (t), mn j J n ( )] Rj (t) for all j J n q j (10) where J n s the set of outgong lnks, q j s the splt factor and s the unque ncomng lnk of node n. The second type of equaton for a dverge node has been proposed by Lebacque (1996): G j (t) = mn [ p j S (t),r j (t) ] for all j J n (11) In general, for ntersectons wth two or more upstream and downstream lnks, we can combne the merge and dverge models. As n Yperman et al. (2005), we combne (8) and (10). The resultant equaton becomes: [ G j (t) = p j mn mn j J n ( R j (t)s (t) p j S (t) I n ) ],S (t) for all I n and for all j J n (12) Havng determned the transton number of vehcles of all nodes, the cumulatve number of vehcles for the upstream and downstream boundares of lnks can be updated usng the followng equatons: N(x L,t+ t) = N(x L )+ j J n G j (t) for all I n (13) N(x 0 j,t+ t) = N(x 0 j)+ I n G j (t) for all j J n (14) For each node n N where N s the set of all nodes n the traffc network. 2.2 Extenson of the LTM model for ramp meterng In ths secton the LTM model s extended to nclude control sgnals. In order to mplement the acton of ramp meterng n the LTM framework, one can add a constrant on the transton number of vehcles from the on-ramp to the manstream. Recall from (7), the modfed equaton s as follows: G j (t) = mn [ S j (t),p j R j (t),c r (t) ] (15) where s the ncomng lnk (the on-ramp), j s the outgong lnk, C s the capacty of the on-ramp (veh/h), and r (t) [0,1] s the meterng sgnal. A smlar modfcaton can be appled to ether (8) or (9). 3. MODEL PREDICTIVE CONTROL FOR TRAFFIC NETWORKS Model Predctve Control (MPC) (Macejowsk (2002), Rawlngs and Mayne (2009)) s an advanced control method for ndustral processes and traffc networks. The man dea s to use a predcton model of the process (n our case: the traffc network) and an objectve functon assessng the desred performance of the process, and to fnd the optmal control nputs by means of an optmzaton algorthm. In our case, the LTM s used to predct the behavor of a traffc network over a predcton horzon. The optmzaton algorthm mnmzes the objectve functon and fnds a sequence of optmal control nputs for the whole predcton horzon,but only the frst control nput sample s appled to the traffc network and the procedure s repeated n the next control step but n a rollng horzon style. In other words, the predcton horzon s shfted one step forward, and the predcton and optmzaton procedure over the shfted horzon are repeated usng new measurements. For a traffc network, one can defne dfferent objectve functons based on travel tme, fuel consumpton of vehcles, emssons, etc. The objectve functon we chose s the total tme spent n the traffc network, consstng of the tme vehcles spend n queues at manstream orgns and on-ramps and the travel tme on the freeway. The

5 Total Tme Spent (TTS) objectve functon for the MPC controller s formulated as follows M(k c+n p) 1( ) J TTS (k c ) = T ρ (k)l λ + ω o (k) I all o O all k=mk c (16) where T s the smulaton tme step length, k c s the controller tme step counter, and k s the model tme step counter. In fact, we assume that the controller tme step length s an nteger multple of the smulaton tme step length: T c = MT. Moreover, N p s the control horzon, ρ s the densty of lnk, ω o s the queue length at orgn o, and I all and O all are the set of all lnks and the set of all orgns, respectvely. In order to apply the objectve functon (16) to our contnuous-tme LTM model, we have to dscretze the model and take care of the delay n the sendng/recevng number of vehcles. Further, we have to estmate the densty of lnks usng the followng: ρ (k) = N(xL,k) N(x0,k) L (17) However, we are not able to apply control nputs that have hgh fluctuatons. Ths s due to the fact that n realty traffc sgnals cannot vary wth hgh frequency over tme. Further, hgh fluctuatons n control nputs may cause nstablty n some cases. Therefore, a penalty term on control nput devatons s usually added to the objectve functon. In our case, the control nputs are the meterng sgnals of on-ramps. The penalty term s formulated as k c+n p 1 ζ l=k c o O ramp ro (l) r o (l 1) (18) where r o s the meterng sgnal and O ramp s the set of ndces of metered ramps 2. The ζ s a weghtng coeffcent. Also, to reduce the complexty, control varables are sometmes taken constant after passng a predefned control horzon N c. Takng ths nto account, N p n (18) should be replaced by N c. Fnally, The penalty term s added to the TTS objectve functon (16). However, the total objectve functon s nonlnear 3. Ths along wth usng the LTM model for predcton leads to a nonlnear nonconvex optmzaton problem that has to be solved n the MPC framework to fnd the optmal control sgnals. At every control tme step, there s no guarantee to be able to fnd a unque global soluton. Furthermore, the nonlnear optmzaton may take consderable tme to fnd a (local) optmum. In the next secton, a soluton to ths problem s proposed. In fact, we am to transform the nonlnear nonconvex optmzaton problem nto a Mxed Integer Lnear Problem (MILP). 4. MLD-MPC FORMULATION FOR THE LTM Usng the methods proposed by Wllams (1993) and adopted n Bemporad and Morar (1999), one can transform the model and the objectve functon nto a system of 2 It should be noted that manstream orgns outflows can also be controlled n some cases, so n that case they can also be ncluded n the set O ramp. 3 But n fact t s pecewse-affne (PWA), a property that wll be used later on n the next secton. lnear equaltes and nequaltes consstng of real and nteger varables and end up wth an MILP. The MILP can be effcently solved usng exstng MILP solvers lke CPLEX, GLPK, or lp solve(see Atamturk and Savelsbergh(2005)). The MILP solvers can fnd the global optmum and ths s a sgnfcant advantage over the nonlnear optmzaton problem solvers. 4.1 Mxed Logcal Dynamc Models In order to get an MILP, we frst have to transform the model of the system nto a Mxed Logcal Dynamc (MLD) form. An MLD model s descrbed by the followng system of equatons (Bemporad and Morar (1999)) x(k +1) =Ax(k)+B 1 u(k)+b 2 δ(k)+b 3 z(k)+f y(k) =Cx(k)+D 1 u(k)+d 2 δ(k)+d 3 z(k)+g E 1 x(k)+e 2 u(k)+e 3 δ(k)+e 4 z(k) h, where δ(k) {0,1} n b denotes the vector of bnary varables used to ndcate whch regon of operaton the system s n, and z(k) R nz s the vector of auxlary varables. In the followng, we wll elaborate more on how to get the MLD form by ntroducng some basc transformaton rules. Consder the statement f(x) 0, where f s an affne functon over a bounded set X of the nput varable x.moreover,assumethattheconstantsmandm arelower and upper bounds of the functon f over X. By defnng δ {0,1}, the followng holds { f(x) M(1 δ) [f(x) 0] [δ = 1], ff (19) f(x) ǫ+(m ǫ)δ where ǫ s a small tolerance, typcally the machne precson 4. Moreover, δf(x) can be replaced by the auxlary real varable z = δf(x). In fact, z = δf(x) s equvalent to z Mδ z mδ (20) z f(x) m(1 δ) z f(x) M(1 δ) Now, usng the basc rules, we can transform the LTM model nto an MLD form. 4.2 MLD transformaton of the LTM The LTM model s contnuous-tme n nature. Thus, n order to apply the MLD transformatons, frst the equatons ( should ) be dscretzed. To ths am, we assume that L υ free, s a multple nteger of the sample tme T. Further, tme ndex t s replaced by k. Recallng the sendng number of vehcles equaton from (1), t can be rewrtten as follows S (k) = f,1 (k)+[f,2 (k) f,1 (k)]δ (k), (21) }{{} z (k) wth ( f,1 (k) = N x 0,k +1 L ) N(x L,k) (22) υ free, 4 The reason for ntroducng ǫ s that an equaton lke f(x) > 0 does not ft the MLD framework, n whch only nonstrct nequaltes are allowed. Therefore, the equaton f(x) > 0 s replaced by the equaton f(x) ǫ.

6 f,2 (k) = q M, T (23) and δ (k) = 1 [f,2 (k) f,1 (k)] 0. Followng the MLD rules, we reach the followng lnear equaton S (k) = f,1 (k)+z (k), (24) subject to the followng constrants, {f,2 (k) f,1 (k)} M(1 δ (k)) {f,2 (k) f,1 (k)} ǫ+(m ǫ)δ (k) z (k) Mδ (k) z (k) mδ (k) z (k) [f,2 (k) f,1 (k)] m(1 δ (k)) z (k) [f,2 (k) f,1 (k)] M(1 δ (k)) where M and m are upper and lower bounds for {f,2 (k) f,1 (k)}. These constrants along wth (22), (23), and (24) are equvalent to (1). For the recevng number of vehcles (2), the transformaton procedure s smlar. The transton number of vehcles for mergng nodes can also be transformed nto the MLD form. For (7), one can assume that the prorty parameter p j s constant, whch s not far from realty (snce ths parameter s mostly related to physcal propertes of the lnks). By ths assumpton, the transformaton wll be smlar to the sendng number case. For (8), one can use smple approxmatons for the multplcaton of the sendng numbers lke assumng that they can be taken as constant over a certan perod of tme, or tryng to approxmate the functon wth pecewseaffne (PWA) functons Groot et al. (2011). Next, the PWA approxmaton can be transformed nto the MLD form (Bemporad and Morar, 1999). For (9), recall [ G j (k) =medan S (k),r j (k) (( S (k)) I n ) ] S (k),p j R j (k) for all I n (25) The medan functon s equal to the followng condtons: G =medan(a,b,c) = a f [(a b and c a) or (a c and b a)] b f [(a b and b c) or (c b and b a)] c f [(a c and c b) or (b c and c a)] Thus, followng the MLD rules, these condtons can also be transformed nto the MLD form, yeldng an expresson of the form G = aδ }{{} 1 + bδ 2 + cδ }{{} 3 (26) }{{} z 1 z 2 z 3 along wth a set of 34 nequaltes. Transformaton of other types of node models s straghtforward, snce they contan the mn functon and the transformaton procedure for that was explaned n the S (k) case. 4.3 Fnal MILP problem After transformng the LTM model nto the MLD form, one can recast the orgnal nonlnear optmzaton problem nto an MILP. However, to ths am a lnear objectve functon s needed. Recall from Secton 3, the penalty term (18) that has been added to the objectve functon s pecewse affne. Thus t can also be transformed nto a mxed-nteger lnear form by defnng addtonal bnary and auxlary varables. However, there s another approach to recast (18) as a lnear problem that does not need any bnary varable. It can be easly proved that the followng optmzaton problems have the same optmal soluton: mn β θ,β mn θ θ β θ β θ Hence, nstead of reformulatng (18) nto an MLD form too, one can use the above lnear problem. Fnally, usng the total lnear objectve functon and the MLD-LTM model, the fnal MILP problem can be constructed. The MILP can be solved usng effcent solvers lke CPLEX. 5. CASE STUDY In order to test the proposed approach, a benchmark traffc network example has been selected from Hegy et al. (2005). As shown n Fg. 1 the network conssts of a two-lane freeway wth an on-ramp. Both the manstream orgn and the on-ramp are controlled. The network s modeled usng the modfed verson of the LTM that ncludes the ramp meterng sgnals. We take the standard parameter settngs used by Hegy et al. (2005): υ free = 102 km/h, T s = 12 s, ρ max = 180 veh/km/lane, ρ crt = 33.5 veh/km/lane, L 1 = 4 km, L 2 = 2 km and smulate over a tme horzon of 2.5 hours. Smulaton results for the no-control case are shown n Fg. 2 The flows of the vehcles from the manstream orgn and the on-ramp are controlled n order to mnmze the sum of the TTS objectve functon (16) and the penalty term (18) wth ζ = 0.4. The control sgnals are obtaned frst by usng an MPC controller based on the nonlnear LTM and next by usng the MLD-MPC approach. The performance of the two approaches s compared n terms of computatonal effcency and total cost. The results are shown n Table 1. Based on the predcton horzon and the control horzon, dfferent scenaros have been defned. The smulaton tme step T and the control tme step T c are both 12 s. For each scenaro, the total tme spent over the full 2.5 hours smulaton perod s compared for both approaches. Also the computaton tme for one run of the optmzaton step s presented, averaged over the number of smulaton steps. As can be seen n the table, the MLD-MPC approach returns values that are close to the orgnal TTS, whle needng a shorter computaton tme. It should be noted that for the nonlnear optmzaton algorthm, a mult-start optmzaton approach wth several ntal ponts should be used (In our case we called the nonlnear optmzaton algorthm 6 tmes wthn each MPC optmzaton step). It means that the CPU tmes for nonlnear MPC can n fact be a multple of the current values presented n Table 1. Wth the ncrease n the predcton horzon and the control horzon, the mean computaton tme of one optmzaton step over the smulaton horzon for the MLD-MPC approach ncreases only a lttle whle n the

7 Table 1. Comparson of TTS (veh.h) and CPU Tme (s) for two approaches Scenaro TTS (nonlnear MPC) TTS (MLD-MPC) CPU tme (nonlnear MPC) CPU tme (MLD-MPC) N p = 7,N c = veh.h veh.h s s N p = 7,N c = veh.h veh.h s s N p = 10,N c = veh.h veh.h s s manstream meterng Fg. 1. Set-up of the case study demand (veh/h) O 1 O 2 neffcent, a reformulaton of the LTM was proposed n order to eventually obtan an mxed nteger lnear problem. For the gven case study, ths new approach gves results close to the ones obtaned by the nonlnear MPC whle the CPU tme goes down sgnfcantly. Moreover, t s expected that for larger networks the benefts of the new approach over the nonlnear MPC wll become even more clear. Asanextensontothswork,theLTMcouldbemodfedn order to nclude the effects of varable speed lmts. Once the new modfcatons are evaluated and approved on a case study wth real data, one can use the basc rules to transform the new model nto an MLD form too. Wth ths, full control of traffc networks usng ramp meterng and varable speed lmts wll become possble usng a fast traffc model (extended LTM) and an effcent control approach (MLD-MPC). densty (veh/km) ρ 1 ρ 2 ACKNOWLEDGEMENTS Research supported by the BSIK project Next Generaton Infrastructures (NGI), the European 7th Framework Network of Excellence Hghly-complex and networked control systems (HYCON2), the Delft Research Center Next Generaton Infrastructures, and the European COST Actons TU0702 and TU1102. queue length (veh) Tme (h) Fg. 2. Smulaton results for the no control case nonlnear MPC, the amount of change s consderable. Therefore, from the ncreasng values n computaton tme for larger predcton and control horzons, the MLD-MPC approach s expected to even perform sgnfcantly better, when t s appled to larger traffc networks. 6. CONCLUSIONS AND FURTHER RESEARCH Modelng and control of traffc networks usng the Lnk Transmsson Model has been presented n ths research. The LTM model was extended to nclude ramp meterng and then was used to model a secton of a freeway. Smulaton results showed fast yet accurate modelng usng the LTM. For the frst tme, the LTM was used as predcton modelnthempcframeworknordertomnmzeatraffc objectve functon. Snce a drect MPC mplementaton based on the nonlnear LTM was stll computatonally ω 1 ω 2 REFERENCES Atamturk, A. and Savelsbergh, M.W.P. (2005). Integerprogrammng software systems. Annals of Operaton Research, 140(1), Bellemans, T., De Schutter, B., and De Moor, B. (2006). Model predctve control for ramp meterng of motorway traffc: A case study. Control Engneerng Practce, 14(7), Bemporad, A. and Morar, M. (1999). Control of systems ntegratng logc, dynamcs, and constrants. Automatca, 35(3), Daganzo, C.(1994). The cell transmsson model: A smple dynamc representaton of freeway traffc. Transportaton Research B, 28(4), Daganzo, C. (1995). The cell transmsson model, part : network traffc. Transportaton Research B, 29(1), Groot, N., De Schutter, B., Zegeye, S., and Hellendoorn, H. (2011). Model-based traffc and emsson control usng PWA models A mxed-logcal dynamc approach. In Proceedngs of the 14th Internatonal IEEE Conference on Intellgent Transportaton Systems (ITSC 2011), Washngton, DC. Hegy, A., De Schutter, B., and Hellendoorn, H. (2005). Model predctve control for optmal coordnaton of ramp meterng and varable speed lmts. Transportaton Research Part C, 13(3), Hoogendoorn, S. and Bovy, P. (2001). State-of-the-art of vehcular traffc flow modellng. Proceedngs of the Insttuton of Mechancal Engneers, Part I: Journal of Systems and Control Engneerng, 215(4),

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