A simplified macroscopic urban traffic network model for model-based predictive control

Transcription

1 Delft University of Technology Delft Center for Systems an Control Technical report 9-28 A simplifie macroscopic urban traffic network moel for moel-base preictive control S. Lin, B. De Schutter, Y. Xi, an J. Hellenoorn If you want to cite this report, please use the following reference instea: S. Lin, B. De Schutter, Y. Xi, an J. Hellenoorn, A simplifie macroscopic urban traffic network moel for moel-base preictive control, Proceeings of the 12th IFAC Symposium on Transportation Systems, Reono Beach, California, pp , Sept. 29. Delft Center for Systems an Control Delft University of Technology Mekelweg 2, 2628 CD Delft The Netherlans phone: (secretary fax: URL: This report can also be ownloae viahttp://pub.eschutter.info/abs/9_28.html

2 A simplifie macroscopic urban traffic network moel for moel-base preictive control S. Lin B. De Schutter Y. Xi J. Hellenoorn Department of Automation, Shanghai Jiao Tong University No. 8 Dongchuan Roa, Minhang District, Shanghai, P. R. China ( lisashulin@gmail.com, ygxi@sjtu.eu.cn. Delft Center for Systems an Control, Delft University of Technology Mekelweg 2, 2628 CD Delft, The Netherlans ( b.eschutter@csc.tuelft.nl, j.hellenoorn@tuelft.nl. Abstract: A moel preictive control (MPC approach offers several avantages for controlling an coorinating urban traffic networks. To apply MPC in large urban traffic networks, a fast moel that has a low on-line computational complexity is neee. In this paper, a simplifie macroscopic urban traffic network moel is propose an teste. Compare with a previous moel, the moel reuces the computing time by enlarging its upating time intervals, an preserves the computational accuracy as much as possible. Simulation results show that the simplifie moel reuces the computing time significantly, compare with the previous moel that provie a goo trae-off between accuracy an computational complexity. We also illustrate that the simplifications introuce in the simplifie moel have a limite impact on the accuracy of the simulation results. As a consequence, the simplifie moel can be use as preiction moel for MPC for larger urban traffic network. Keywors: Macroscopic traffic moeling; Urban traffic control; Moel preictive control; Urban traffic network. 1. INTRODUCTION In recent years, the number of icles has grown larger an larger, an the requirements for traveling by icles are getting more an more stringent. To reuce traffic jams an to promote efficiency in traveling, effective traffic control algorithms are necessary. Many control theories have been applie to control traffic (Kachroo an Özbay, 1999; Papageorgiou, 1983, like fuzzy control, PID control, moel preictive control, an multiagent control, in combination with ifferent control structures like centralize, istribute, an hierarchical control. We focus on moel-base control methos, an on MPC in particular. Consiering on-line computational complexity, macroscopic traffic moels are usually use in MPC. However, for ifferent moel-base control approaches, there still exist ifferent levels of requirements for the macroscopic moel. Some moels just nee to express the relation between the input values an the performance inicators, but some are more etaile so as to escribe the ynamics of the traffic evolution; some moels are more precise in moeling the ynamics, while some are simpler so as to be fast for on-line computing. As a result, there exists a wie variety of macroscopic traffic moels with ifferent levels of etail. For ifferent control methos, appropriate traffic moels with the require moeling power nee to be selecte. In the past few years, various macroscopic urban traffic moels were evelope an use for traffic control. The store-an- S. Lin is a visiting researcher at the Delft Center for Systems an Control, Delft University of Technology. B. De Schutter is also with the Marine & Transport Technology epartment of Delft University of Technology. forwar moel, propose by Gazis an Potts (1963 an later use by Diakaki et al. (22, is a simple moel with low computational complexity, but it can only be use for saturate traffic, i.e., if the icle queues resulting from the re phase cannot be issolve completely at the en of the following green phase. The moel propose by Barisone et al. (22 an extene by Dotoli et al. (26 is computationally more intensive an it can escribe ifferent scenarios, but it is also more complicate. The moel propose by Kashani an Sariis (1983 has lower moeling power, but can not epict scenarios other than saturate traffic either. The moel of van en Berg et al. (23; Hegyi (24; van en Berg et al. (24 is capable of simulating the evolution of traffic ynamics in all traffic scenarios (unsaturate, saturate, an over-saturate traffic conitions by upating the iscrete-time moel in small simulation steps. This moel provies a goo trae-off between accuracy an computational complexity compare with the microscopic moel, which is teste an further extene in Lin an Xi (28 an Lin et al. (28. In principle, a centralize MPC metho guarantees globally optimal control actions for traffic networks. It can maximize the throughput of the whole network, an provie network-wie coorination of the traffic control measures. However, the problem is that the on-line computational complexity for centralize MPC grows significantly, when the network scale gets larger, the preiction time span gets longer, an the traffic moel becomes more complex or gets a higher moeling power. There are two main approaches to aress this problem: (1 simplifying the traffic moel in orer to reuce the computational complexity, an (2 cutting the traffic network into small subnetworks or even intersections, which are then controlle using istribute or multi-agent control. In this paper we consier

3 the first approach, i.e., we evelop simplifie, yet sufficiently accurate, traffic moels, in particular, for urban traffic networks. We start with an urban traffic moel base on previous work of Kashani an Sariis (1983; van en Berg et al. (23; Lin an Xi (28. To reuce the computational buren, the simplifie moel enlarges the simulation time interval to one cycle time. During each simulation time interval, traffic states are upate once in each link accoring to the average input an output traffic flow rates in the current cycle. To a flexibility, every intersection in the network can have a ifferent cycle time, an the intersections share the same control time interval. This control time interval is the least common multiple of all the cycle times of the intersections in the network. It is necessary to efine this common control time interval to keep the moel running an communicating synchronously uner both centralize control an istribute control. For a given link the average input traffic flow rates are provie by the upstream links, which transform their own output traffic flow rates into input flow rates for the given link taking the ifferent simulation time intervals into account. We will emonstrate with examples that this simplifie traffic moel reuces the simulation time significantly, compare with the moel in van en Berg et al. (23 an Lin an Xi (28, with only a limite reuction in accuracy. This makes it possible to apply centralize MPC to larger urban traffic networks. 2. TWO MACROSCOPIC URBAN TRAFFIC NETWORK MODELS In this section we present the original moel of van en Berg et al. (23 an Lin an Xi (28 (inicate as the BLX moel as well as a new simplifie moel (calle the. But first we introuce some common notation for both moels. Define J the set of noes (intersections, an L the set of links (roas in the urban traffic network. Link (u, is marke by its upstream noe u (u J an ownstream noe ( J. The sets of input an output links for link (u, are I u, L an O u, L (e.g., for the situation of Fig. 1 we have I u, = {i 1,i 2,i 3 } an O u, ={o 1,o 2,o 3 }. In orer to escribe the evolution of the moels, we first efine some variables (see also Fig. 1: I u, : set of input links of link(u,, O u, : set of output links of link(u,, k : simulation step counter for the urban traffic moel, n u, (k : number of icles in link(u, at step k, q u, (k : queue length at step k in link(u,, q u,,om is the queue length of the sub-stream turning to link o m, m l u,,o m (k : number of cars leaving link (u, an turning to o m, m a u, (k : number of cars arriving at the (en of the queue in link (u, at step k, m a u,,o m (k is the number of arriving cars in the sub-stream towars o m, S u, (k : available storage space of link (u, at step k expresse in number of icles, αu, l (k : flow rate leaving link (u, at step k, αl u,,o m (k is the leaving flow rate of the sub-stream towars o m, αu, a (k : flow rate arriving at the en of the queue in link (u, at step k, αu,,o a m (k is the arriving flow rate of the sub-stream towars o m, αu, e (k : flow rate entering link(u, at step k, β u,,om (k : relative fraction of the traffic turning to o m at step k, µ u, : saturate flow rate leaving link(u,, g u,,om (k : green time length uring step k for the traffic stream towars o m in link(u,, b u,,om (k : boolean value inicating whether the traffic signal at intersection for the traffic stream in link (u, turning to o m is green (1 or re ( at step k, v free u, : free-flow icle spee in link(u,, C u, : capacity of link (u, expresse in number of icles, : number of lanes in link(u,, c u, : offset between noe u an noe, l : average icle length. lane 2.1 In the a queue is moele as follows. For the sake of simplicity, the assumption is mae that at an intersection the cars going to the same estination move into the correct lane, so that they o not block the traffic flows going to other estinations. For each lane (or estination, a separate queue is constructe (with queue lengths enote by q. Furthermore, the simulation time step T s is typically set to 1 s an cars arriving at the en of a queue in simulation perio [kt s,(k+ 1T s are allowe to cross the intersection in that same perio (provie that they have green, that there is enough space in the estination link, an that there are no other restrictions. Consier link(u, (see Fig. 1. For each o m O u, the number of cars leaving link (u, for estination o m in the perio [kt s,(k+ 1T s is given by m l u,,o m (k= if b u,,om (k= max (,min(q u,,om (k+m a u,,o m (k, S om (k, β u,,om (k µ u, T s if b u,,om (k=1. The traffic arriving at the en of the queue in link(u, is given by the traffic entering the link via the upstream intersection elaye by the time τ(k T s + γ(k neee to rive from the upstream intersection to the en of the queue in the link; to this extent m a u, is upate as follows: where m a u, (k=(1 γ(k m l i m,u,(k τ(k+ i m I u, γ(k m l i m,u,(k τ(k 1, i m I u, Cu, q u, (k l } τ(k=floor vfree u, T, s Cu, q u, (k l } γ(k=rem vfree u, T, s

4 i 1 o 1 i 2 m l i 2,u, (αl i 2,u, m l i 1,u, (αl i 1,u, link(,u u q u,,o1 αu, e link(u, m a u, (αa u, q u,,o2 q u,,o3 m l i m l u,,o 3 (αu,,o l 3,u, (αl i 3,u, 3 m l u,,o 1 (αu,,o l 1 m l u,,o 2 (αu,,o l 2 o 2 i 3 o 3 Fig. 1. A link connecting two traffic-signal-controlle intersections with floor(x referring to the largest integer smaller than or equal to x, an rem(x is the remainer. The fraction of the arriving traffic in link (u, turning to o m O u, is m a u,,o m (k=β u,,om (k m a u, (k. The new queue lengths are given by the ol queue lengths plus the arriving traffic minus the leaving traffic q u,,om (k+ 1=q u,,om (k+m a u,,o m (k m l u,,o m (k for each o m O u,, an q u, (k= o m O u, q u,,om (k. The new available storage stage epens on the number of cars that enter an leave the link in the perio [kt s,(k+ 1T s : S u, (k+ 1=S u, (k i m I u, m l i m,u, (k+ o m O u, m l u,,o m (k. 2.2 Simplifie Moel (S Moel In the simplifie moel, every intersection takes the cycle time as its simulation time interval. The cycle times for intersection u an, which are enote by c u an c respectively, can be ifferent from each other, as Fig. 2 illustrates. Moreover, the S moel works with (average flow rates rather than with number of cars for escribing flows leaving or entering links. Taking the cycle time c as the length of the simulation time interval for link (u, an k as the corresponing time step counter, the number of the icles in link (u, is upate accoring to the input an output average flow rate over c at every time step k by ( n u, (k + 1=n u, (k + αu, e (k αu, l (k c. (1 The leaving average flow rate is the sum of the leaving flow rates turning to each output link: αu, l (k = αu,,o l m (k, o m O u,. (2 o m O u, ( αu,,o l m (k =min β u,,om (k µ u, g u,,om (k /c, q u,,om (k /c + αu,,o a m (k, β u,,om (k (C om n om (k /c. The number of icles waiting in the queue turning to link o m is upate as (3 q u,,om (k + 1=q u,,om (k + ( αu,,o a m (k αu,,o l m (k c. (4 Then, the number of waiting icles in link(u, is q u, (k = q u,,om (k. (5 o m O u, The flow rate entere link (u, will arrive at the en of the queues after a time elay τ(k c + γ(k, i.e., α a u, (k =(1 γ(k α e u, (k τ(k + γ(k α e u, (k τ(k 1, (6 Cu, q u, (k l } τ(k =floor vfree u, c, Cu, q u, (k l } γ(k =rem vfree u, c. (7 Before reaching the tail of the waiting queues in link(u,, the flow rate of arriving icles nee be ivie by multiplying the turning rates: α a u,,o m (k =β u,,om (k α a u, (k. (8 The flow rate entering link(u, is mae up from the flow rates from all the input links: αu, e (k = αi l m,u, (k. (9 i m I u, The leaving average flow rate over c is etermine by the capacity of the intersection, the number of cars waiting an/or arriving, an the available space in the ownstream link: In this formula, we see that the flow rate entering link (u, is provie by the combination of the flow rates leaving the upstream links. Recall that we have ifferent cycle times between

5 j u N u = 2 α l i m,u, (k u c u c u, T c N = 3 k c N u αi l m,u, (k 1 2 k u c t T c Fig. 2. Relationship between cycle times an control time interval the upstream an ownstream intersections, so the simulation time steps are not the same. Some operations nee to be carrie out to synchronize the leaving an entering flow rates. In orer to control the urban traffic network, a common control time interval nee to be efine for the network moel, so that intersections can communicate with each other an be synchronous. T c = N j c j, for j J (1 with N j an integer. So T c is the least common multiple of all the intersection cycle times in the traffic network. As Fig. 2 shows, we have T c = N u c u = N c. (11 For a given k c the simulation time step counters for both intersections can range as follows: k u = N u k c + p u, p u =,1,...,N u 1 (12 k = N k c + p, p =,1,...,N 1. Now we show how the flow rates expresse in the timing of intersection u can be recast into the timing of intersection. First, we smooth the leaving flow rates from the upstream links as α l i m,u, (t=αl i m,u, (k u, k u c u t <(k u + 1 c u, (13 an then sample them again to obtain the average flow rates in time step k so as to be able use by the ownstream link, as Fig. 3 shows: α l i m,u, (k = (k +1 c + c u, k c + c u, α l i m,u, (t c t. (14 3. SIMULATION EXPERIMENTS In centralize MPC, a fast running traffic network moel is neee to satisfy the on-line optimization requirements. So, simulations are esigne an carrie out to verify whether the new simplifie moel ( can save time compare with the more etaile moel ( while retaining a sufficiently high level of accuracy. The two moels are compare for ifferent network input flow rates, ifferent preiction horizons, an ifferent traffic network scales. During the experiment, the simulation time interval of the is set to 1 s, while the simulation time intervals of the are cycle times k c N Fig. 3. Illustration for synchronizing flow rates link 1 Fig. 4. The layout of a urban traffic network Noe k Two way link Structure: (3,3 # noes: 9 which are 12 s, the same for all intersections in the network. The preiction horizons an traffic network scales are liste in Table 1. Table 1. Traffic network characteristics an preiction horizon for each of the 5 simulation cases Case number Structure (1,2 (3,3 (8,8 (13,13 (18,18 Network # noes N p Each network consiere is a gri-like network, where the Structure of the network is expresse as the number of noes in each row an each column, an # noes inicates the number of noes. For example, Fig. 4 shows the layout of a (3,3 network containing 9 noes. N p is the number of the control time intervals the moel will run (i.e., simulation or preiction horizon expresse in steps of length T c. When using network 3, an N p = 1, the computing times of the two moels uner ifferent network input flow rates are shown in Fig. 5. The figure shows that the computing times are almost inepenent of the network input flow rates for both moels. This means the traffic scenarios almost o not have any influence on the running time. Moreover, we can see from the figure that the require a much shorter computation time, aroun.5 s, while the took about 7 s, which is 14 times longer.

6 Computing time (s Computing time (s Input traffic flow rate (/h Fig. 5. The computing time consume for ifferent input flow rates of the traffic network Number of noes Fig. 7. The computing time consume for ifferent traffic network scales 2 Computing time (s N p Fig. 6. The computing time consume for ifferent preiction horizons In each step of MPC for traffic control, a numerical optimization problem nees to be solve to obtain the optimal input value for the next step (using, e.g., a multi-start Sequence Quaratic Programming (SQP algorithm. During the optimization, the moel may nee run hunres to thousans of times. Therefore, by ecreasing the computing time of the moel, the on-line optimization time in MPC can be ramatically reuce. Fig. 6 shows the changing of the running time with N p, when the traffic network is set to network 5. Fig. 7 shows the changing of the running time with network scale, when N p = 4. From the two figures, we can see that the longer the moel is preicting, the larger scale the network is set to, the more time that the S moel will save. The same conclusions can also be rawn from Fig. 8. The is much faster than the, especially for longer preiction horizons N p an larger network scales, but this extra spee is obtaine by ignoring some etails when moeling. Therefore, we nee to verify whether the can still satisfy the requirements of control. The number of leaving icles can reflect the control effect of traffic lights Fig. 8. The computing time consume for both ifferent preiction horizons an ifferent traffic network scales TTS (*h Time (s Fig. 9. The TTS for two moels

7 Accumulate number of leaving icles Time (s Fig. 1. The accumulate number of leaving icles for two moels on urban traffic, an Total Time Spent (TTS is usually use as the control performance. If the shows behavior that is similar to that of the for these two inexes, then it can be use as urban traffic control moel guaranteeing similar control effects but with less control efforts. Fig. 9 an 1 are rawn for link 1 of network 2 (see Fig. 4, an N p = 1. The figures show that the simplifie moel is accurate enough as a control moel for urban traffic network. 4. CONCLUSIONS A simplifie macroscopic moel has been establishe for controlling urban traffic network using moel preictive control (MPC. This moel takes the cycle times of the intersections as simulation time steps, where every intersection can have a ifferent simulation time step. A control time interval, which is the least common multiply of all the cycle times, is efine to guarantee the communication an synchronization in the urban traffic network. The simplifie moel also escribes how to ensure communication an synchronization between intersections with ifferent simulation time steps. The simplifie moel can take all typical traffic scenarios (saturate, unsaturate, an over-saturate traffic into consieration, an is more flexible by having ifferent cycle times. Moreover, it significantly reuces the computing time, which make it possible to be use for controlling larger urban traffic network. However, the increasing of computing spee is obtaine by enlarging the simulation time interval, which makes it lose some etails an sacrifice some accuracy at the same time. But simulation results show that it guarantees enough accuracy to be use as the control moel for urban traffic network. Further research will focus on eveloping MPC algorithm to control urban traffic network base on this moel, as well as an extensive assessment an comparison of the simplifie moel with a wie range of other traffic moels for various network layouts an traffic emans when use for MPC-base traffic control. ACKNOWLEDGEMENTS This research is supporte by a Chinese Scholarship Council (CSC grant, the National Science Founation of China (Grant No , the Specialize Research Fun for the Doctoral Program of Higher Eucation (Grant No , the European COST Action TU72, the BSIK projects Transition to Sustainable Mobility (TRANSUMO an Next Generation Infrastructures (NGI, the Delft Research Center Next Generation Infrastructures, an the Transport Research Centre Delft. REFERENCES Barisone, A., Giglio, D., Minciari, R., an Poggi, R. (22. A macroscopic traffic moel for real-time optimization of signalize urban areas. In Proc. of the 41st IEEE Conference on Decision an Control, Las Vegas, USA. Diakaki, C., Papageorgiou, M., an Abouolas, K. (22. A multivariable regulator approach to traffic-responsive network-wie signal control. Control Engineering Practice, 1(2, Dotoli, M., Fanti, M.P., an Meloni, C. (26. A signal timing plan formulation for urban traffic control. Control Engineering Practice, 14(11, Gazis, D.C. an Potts, R.B. (1963. The oversaturate intersection. In Proc. of the 2n International Symposium on Traffic Theory. Hegyi, A. (24. Moel Preictive Control for Integrating Traffic Control Measures. Ph.D. thesis, Delft University of Technology. Kachroo, P. an Özbay, K. (1999. Feeback Control Theory for Dynamic Traffic Assignment. Avances in Inustrial Control. Springer. Kashani, H. an Sariis, G. (1983. Intelligent control for urban traffic systems. Automatica, 19(2, Lin, S. an Xi, Y. (28. An efficient moel for urban traffic network control. In Proc. of the 17th Worl Congress The International Feeration of Automatic Control, Seoul, Korea. Lin, S., Xi, Y., an Yang, Y. (28. Short-term traffic flow forecasting using macroscopic urban traffic network moel. In Proc. of the 11th International IEEE Conference on Intelligent Transportation Systems, Beijing, China. Papageorgiou, M. (1983. Applications of Automatic Control Concepts to Traffic Flow Moeling an Control. Lecture Notes in Control an Information Sciences. Springer Verlag, Berlin, Germany. van en Berg, M., De Schutter, B., Hegyi, A., an Hellenoorn, J. (24. Moel preictive control for mixe urban an freeway networks. In Proc. of the 83r Annual Meeting of the Transportation Research Boar, 19. Washington, D.C. van en Berg, M., Hegyi, A., De Schutter, B., an Hellenoorn, J. (23. A macroscopic traffic flow moel for integrate control of freeway an urban traffic networks. In Proc. of the 42n IEEE Conference on Decision an Control. Maui, Hawaii USA.