BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES • Volume 15, No 5 Special Issue on Control in Transportation Systems Sofia • 2015
Print ISSN: 1311-9702; Online ISSN: 1314-4081 DOI: 10.1515/cait-2015-0013
Macroscopic Traffic Flow Control via State Estimation H. Abouaïssa1, H. Majid2 1
Université Lille Nord France, F-59000 Lille, France. U-Artois, LGI2A, (EA. 3926) Technoparc Futura, F-62400 Béthune, France 2 University of Sulaimani, Faculty of Engineering Sciences, Department of Civil Engineering, Sulaimani, Iraq Emails:
[email protected] [email protected]
Abstract: The studies presented in this paper deal with traffic control in case of missing data and/or when the loop detectors are faulty. We show that the traffic state estimation plays an important role in traffic prediction and control. Two approaches are presented for the estimation of the main traffic variables (traffic density and mean speed). The state constructors obtained are then used for traffic flow control. Several numerical simulations show very promising results for both traffic state estimation and control. Keywords: Macroscopic model, estimation, freeway traffic control, sliding mode, algebraic method, differential flatness.
1. Introduction The freeway traffic control and the advent of the Intelligent Transportation Systems (ITS), which are able to provide continuous forecasting of the traffic and allow alleviating the daily congestion phenomena, are closely related to the ability to have the whole needed information about the traffic behaviour and then the traffic state evolution. This information is usually provided by a set of loop detectors or any other sensors, such as video camera and others, already installed along the freeway infrastructure. Nevertheless, like any other device, such sensors face several problems, such as failure, usury and default, as well as the high cost of installation and maintenance. In addition, the loop detectors could also undermine the road paved and accelerate the deterioration of the infrastructure. The state estimation and reconstruction of the missing data for a faulty detector play an important role for both traffic forecasting and control. 5
The amount of attention devoted to the development of an estimation strategy using a traffic flow model has been relatively small compared to the modeling stage. Among these works, N a h i and T r i v e d i [18] shows an interesting method for data processing and estimation of the traffic density closely in homogeneous situations. Notice that the most widely used tools for traffic state estimation are Kalman filters and their extensions. Thus, G a z i s and K n a p p [10], K n a p p [12] have proposed a Kalman filtering techniques for data processing. Such a method is based on time series of the mean speed and flow data from each detector and then it generates crude estimates of the vehicle counts. In [21] an extended Kalman filter method was utilized for traffic state estimation, using second order models. K o h a n [13] has designed a first order sliding mode state estimator for the estimation of both speed and density. Although the proposed method is robust against bounded modeling errors and various disturbances, it suffers from chattering phenomena. S u n et al. [20] have derived a density estimator at unmonitored locations along a freeway, from the so-called Switching Mode Model (SMM). Sun et al. have proposed an estimator based on a Mixture Kalman Filter (MKF) algorithm using also the SMM. M i h a y l o v a and B o e l [17] have used a particle filter for density estimation/prediction. Most of these works have dealt with state estimation for traffic prediction and forecasting. In this paper we propose a combination of the state estimation for traffic flow control purpose. Two approaches are provided. The first one aims at solving the problem of the chattering phenomena using high order sliding mode techniques. The second method is based on new setting of numerical differentiation. The obtained estimators are then used for freeway traffic flow control. Our paper is organized as follows. Section 2 recalls the used macroscopic model. The sliding mode and algebraic methods are presented in Section 3 and then applied to the state estimation. Section 4 details the developed traffic flow control algorithm using the state estimation. Numerical simulations and results are provided in Section 5. Finally, the last section summarizes some conclusions and further works.
2. A macroscopic traffic flow model 2.1. A modified second order continuum model As an illustration consider the following simple freeway stretch divided into three segments. The studied system is governed by the following space discretized equation, called a conservation law 1 (1) [qi−1 (t ) − qi (t ) + ai ri (t ) − bi pi (t )], ρ& i (t ) = Li λi where: ρ& i (t ) in (veh/km per lane) is the traffic density in segment I; qi (t ) denotes the traffic volume in (veh/h); Li and λi, are the segment length and the number of
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lanes in seegment i, resspectively; ai and bi are a binary variables v which indicate respectivelyy the presencce or the abseence of an on n-ramp ri (t ) and an off-rramp pi (t ) .
Fig. 1. Example of freeway f stretch h with three segm ments
The trraffic volume qi (t ) is reelated to the traffic denssity and the mean speed ks to the folloowing expression borrow wed by H i l l i g e s and W e i d l i c h vi (t ) thank [11] (see allso [13]): (2) qi(t) = avi+1(t))ρi(t) + (1 – α)ρi+1(t)vi+2(tt), for i = 1, …, … N – 1 and qN(t) = ρN(t))vN(t), wheree 0 ≤ α ≤ 0 reepresents ann appropriate weighting factor, whicch generallyy needs to be b identifiedd. Since the number off vehicles leaaving sectionn i is mainlyy defined as a function of o segment sstates rather than the traaffic conditioons further doownstream, this t parameteer is assumedd to be close to 1. The mean m speed iss defined thannks to the following diffeerential equaation ν 1 ξ (3) [ρ i+1 (t ) − ρ i (t )], v&i (t ) = [Ve (ρ i (t )) − vi (t )] + vi (t )[vi −1 (t ) − vi (t )] − Li τLi τ ⎡ 1 ⎛ ρ ⎞a ⎤ where Ve (ρ i (t ) ) = v f ,i exp e ⎢− ⎜ i ⎟ ⎥ is thee so-called “fundamentaal diagram” ⎢ a ⎜⎝ ρ c ,i ⎟⎠ ⎥ ⎣ ⎦ (see, e.g., [15]); vf,i is thhe free-flow speed, a is a model param meter and ρc,i represents the critical density of thhe segment i.
2.2. Initial conditions c From the exxample in Fig. 1 we assuume that thee traffic voluume and the mean speed ( qin (t ), vin (t ) ) are usually measured from the loo op detector located l at thee uppermost boundary of o the freewaay section. We W assume also a stationarry conditions at the exit segment. Thhen, for the last l downstreeam segmentt N, ρ N (t ) = ρ N +1 (t ). We coonsider then the t followingg initial cond ditions: • q0 = qin, v0 = vinn, q • ρ 0 = 0 , v0 ≠ 0, v0 • ρN = ρN+1. 7
3. Nonlinear traffic state estim mation The main objective o off the traffic state estimaation is to deesign an obsserver for a freeway strretch describeed by Equatiions (1), (2) and a (3). For a freeway streetch divided into N segm ments, the seet of state vaariables that govern the studied systtem are: [ρ1v1, …, ρNvN] ∈ R2N. We can easily oobserve that 2 equationss with 2N segment variabbles. Assumee, following the system consists of 2N [21], that a set of traaffic measurrement devicces, such ass loop detecctors, video y stretch at a separationn of several cameras, ettc., are instaalled along the freeway kilometers, i.e., at the end of eveery m segmeent and at on-ramps. o Suuch devices o the trafficc flow, space mean speedd and occupanncy at every provide meeasurements of T seconds. (For simulattion purpose the occupan ncy rate in % is convertedd into traffic T measurrements proovided by the sensorss allow thhe designed density.) The reconstructor to estimatte the main traffic state variables for each segm ment. Having s the freeway sysstem can be the freewayy stretch wiith N segments and m sensors, subdivided into approxximately M = N/m subsy ystems. Suchh systems suubdivision is achieved inn such a wayy that the subbsystem l+1 has little inffluence on thhe dynamics of the mainnstream subssystem l, l = 1, 2, ..., M − 1 [13]. Too reach this rrequirement, the length of o the last seegment of thhe sub- systeem l must bee small comppared to the first segmeent of the doownstream subsystem l + 1. This coondition means that the effect of the t anticipattion term inn expression n (3) for thhe last segm ment of the subsystem l is negligibble and one can consideer the decenntralized probblem of the state estimaation for eachh subsystem at a time. 3.1. A slidinng mode obsserver for traaffic state esttimation The slidingg mode techhnique is rellated to the Variable Sttructure Systems (VSS) theory. It is essentiallly based onn the resoluttion of diffferential equuations with ntroduced byy [4]. Histoorically, the discontinuoous right haand side, whhich was in twisting moode algorithm m is the firstt 2nd order sliding mode algorithm kknown [5]. It features tw wisting arounnd the origin of the 2nd order-slidingg plane. Thee trajectories perform ann infinite num mber of rotaations whilee convergingg in a finite time to the origin [9, 19].
F 2. Super-tw Fig. wisting algorithm m phase trajectory where e reppresents the erroor
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In the super-twistiing algorithm m, the trajecttories of the 2nd order-sliding plane b twisting around the origin o (Fig. 2; e.g., [14]] for a short are also chharacterized by review on the Super-tw wisting slidinng mode ob bserver and its i applicatioon to traffic flow estimaation). Fig. 3 illustrates thhe structure of o a super-tw wisting observver algorithm m.
F 3. Structuree of the super-tw Fig. wisting observeer
In the following liines we conssider the exam mple of Fig. 1 to show thhe design off the state esstimation alggorithm. In thhis example we assume that the outpput vector is F = [F1F2]T = [ρ3v3]T. In term ms of macrooscopic moddeling the sttudied freew way stretch oof Fig. 1 is described as a follows: ⎧ 1 ⎪ ρ&1 = [qin − αρ1v2 − (1 − α )ρ 2 v3 + r1 ], L 1 ⎪ ⎪ 1 (4) ⎨ ρ& 2 = [αρ1v2 + (1 − 2α )ρ 2 v3 − (1 − α )ρ 3v3 ], L2 ⎪ ⎪ 1 ⎪ ρ& 3 = [αρ 2 v3 + (1 − α )ρ 3v3 − qout ]; L3 ⎩
(5)
⎧ 1 ν ξ δ ⎪v&1 = [Ve (ρ1 ) − v1 ] + v1 [vin − v1 ] − [ρ 2 (t ) − ρ1 (t )] − r1v1 , L1 L1 τ L1 ⎪ ⎪ 1 ξ ν v2 [v1 − v2 ] − [ρ 3 (t ) − ρ 2 (t )], ⎨v&2 = [Ve (ρ 2 ) − v2 ] + L2 τ L2 ⎪ ⎪ 1 ν ξ ⎪v&3 = [Ve (ρ 3 ) − v3 ] + v3 [v2 − v3 ] − [ρ out (t ) − ρ 3 (t )] = F&v . τ L3 L3 ⎩
As above mentionned, in expreessions (4) an nd (5), ρ3 annd v3, are asssumed to be s be unnderlined thaat qout = ρouttvout forms thhe output meeasurements known. It should variables. If ρout0 is the estimation period. (See [8] and [16], for more details and the principle of resetting when the denominator becomes close to 0.) Using the above formulas allows obtaining the derivative estimations of the traffic density and mean speed.
4. Traffic flow control The traffic flow control represents the only efficient way to overcome the congestion problems. This section shows the efficiency of the state estimation in order to alleviate these phenomena by implementing the isolated ramp metering control. Such control measurement consists of acting on the traffic demand at the on ramp in order to maintain the traffic density in the merge freeway section close to the critical one. The proposed algorithm is based on the joint use of the sliding mode control and differential flatness. From the first term of Equation (4), the control variable r1 can be parameterized in term of the flat output F1 = ρ3 and its time-derivatives up to three: r1 (t ) = θ1 F1(3 ) + θ 2 F&&1 + θ 3 F&1 + θ 4 q&&out + θ 5 q& out + qout − qin , (17) where L3 L2 ⎡ 1 2 − 3α ⎤ θ1 = 2 , θ2 = ⎢ − ⎥, α ⎣ v3 v3 ⎦ α v 2 v3 2 ⎤ L2 L ⎡ 1 1 − 2α ⎤ L ⎡ v3 (1 − α ) θ3 = ⎢ , θ5 = ⎢ − + 2α − 1⎥ , θ 4 = 2 ⎥. α ⎢⎣ αv2 α ⎣ v3 αv 2 ⎦ α v 2 v3 ⎥⎦ Given a desired reference trajectory F1* (usually, the reference trajectory is taken around the critical density), for the controlled traffic density at the merge segment, a linearized control law achieving an exponential asymptotic tracking of the trajectory is given by the following expression: (18) r1 (t ) = θ1 w + θ 2 F&&1 + θ 3 F&1 + θ 4 q&&out + θ 5 q& out + qout − qin ,
where w = F1(3 ) is an auxiliary control input. Notice that the control variable in that equation needs a derivative estimation of the ouput flow. Such estimation is obtained in the same way as for F1 and F2. Using the high order sliding mode 12
control prinnciple and assuming a thaat the sliding g surface s = F1 – F1*, thhe auxiliary control inpuut is 1 ⎞ ⎛ w = −ςsign ⎜⎜ s& + ζ s 2 sign (s )⎟⎟. (19) ⎠ ⎝ Substiituting (19) in (18) proovides the control algorrithm for isoolated ramp metering. Simullation resultts. For the simulation pu urpose, consider the freeeway stretch depicted in Fig. 1 and assume a that thhe two loop--detectors alllowing the m measurement fic densities in i segment 1 and 2 are faulty. f Usingg either the sliding mode of the traffi observer orr the state reconstructor r r based on numerical n diifferentiationn allows the obtaining of o the missingg data of (ρ1, v1) and (ρ2, v2). We havve chosen thee trapezoidal traffic demaand, which is more or lesss comparablle to real-datta (Fig. 4).
Fig. 4. Exampple of the used traffic t demand
The siimulation tim me is about 1h 25 min. Th he model parrameters are provided in Table 1. Table 1. 1 Model param meters Vallue
Unitt
Criticaal density
35.86
veh/km peer lane
Free-fflow speed
10 05
km//h
Maxim mal density
18 80
veh/km peer lane
Segmeent Length
0.5
km
α
0.7 77
–
Parameter
a (Mayy fundamental diagram) 0.80 014
–
ς (Coontroller parameeter)
0.1
–
ζ
0.0 05
–
(Coontroller param meter)
c Figs 5aa and b show w the densitiies and the m mean speeds In the no-control case, f the measuured and estiimated state variables. evolutions for 13
b a F 5. Densitiees (veh/km) andd speed (km/h) time Fig. t evolution: no-control casse
These figures show w that after about a 12 min nutes, the trafffic reaches aand exceeds t freeway stretch s is in a congestion the critical density (Fig. 5a), which means that the ution, whichh decreases consequently mode. Thiss fact is illusttrated by thee speed evolu (Fig. 5b). ur control algorithm sshows very Remarrk that the implementtation of ou encouraging results (Fig. 6), where the densitiees are maintaained aroundd the desired t speed vallues. value. This fact is confiirmed by the increase in the
a b Fig. 6. Densitties (veh/km) annd speed (km/h h) time evolutionn: control case
0 ≤ r1 ≤ 1. Fig. 7 illustrates thhe control vaariable evoluttion, where 0.2
Fig. 7. Control variable evolution
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Obvioously, the im mplementatioon of the ram mp-meteringg algorithm lleads to the queue form mation depicteed in Fig. 8.
Fig. 8. Quueue length timee evolution
5. Concluusion The studiess presented in this papeer provide a new way too deal with traffic flow control in case c of missing data or when w a loop p detector is faulty. Two approaches for state esttimation werre proposed to t tackle thiss important problem. p Thee first one is based on the t so-calledd super-twistting sliding mode technnique. The ssecond one, which provvides also roobust estimaation, is baseed on the use u of newlyy introduced algebraic methods. m Havving this neeeded inform mation, we appplied a ram mp-metering algorithm, which combbines the higgh order slid ding mode coontrol and ddifferentially flat systemss concept. Furtheer on the research will deeal with com mparative stuudies of the m most widely used techniiques in the traffic t area, such s as Kalm man filters annd their extennsions. This fact leads to t the propoosal of a cooordinated ram mp-metering algorithm inn the future works, whiich takes measurements of o all the stu udied freewaay networks. In addition, another stuudy is condducted in order to deveelop integratted traffic fl flow control combining several traffi fic flow manaagement actiions. Acknowledgeement: This paaper is partly supported by FP7 project 316087 ACOM MIN: “Advance computing annd innovation”.
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