Application of Stochastic Control Concepts to a Freeway Traffic Control Problem1

Copn-ight

©

IFAC Co ntrol. Computers.

COllllllllllicali())ls in ·I·ransportation.

Paris. France. I 'JH'J

APPLICATION OF STOCHASTIC CONTROL CONCEPTS TO A FREEWAY TRAFFIC CONTROL PROBLEM 1 S. A. Smulders ,\l(l/hnl/(I/in (lnd COII/fm/I'I Sril'llu'. P.O. Box -1079. 1009 AB AII/s/l'ld(lll/. Till' N ('/hl'll(ll/(Is~ DII/rh Minis/n of Tran spor/ , Trafl/r Engin/'l'Iing IJi"ision. l'.(). B ox 2()906 , 2500 EX . Th/' Hag ll/,. Th/' ,\'l'Ihl'lllllll/., :1 CI'II/n' f (JI'

In the Netherlands a freeway control and signalling system has been installed on several freeways some years ago. The system consists of induction loops embedded in the road surface and matrix signal boards mounted on gantries. The matrix boards may be used to display speed signals to the drivers. In this paper we investigate the control problem of choosing suitable speed values, with the aim of reducing the probability of congestion. A model for freeway traffic is presented that performs realistically in several basic traffic situations. An estimation algorithm or mter is derived that estimates the state of traffic at any time moment, using the measurements available from the signalling system. The filter is tested against simulated and against real traffic data and shown to behave satisfactorily . Finally, a control policy is designed for the speed signals. The design criterion is the expected total throughput of the freeway until congestion occurs. The optimal policy turns out to consist of switching on or off the control as the traffic density crosses specific switching levels. Keywords: Modelling; nonlinear filtering; stochastic control ; road traffic; traffic control. INTRODUCTION

tested against simulated data, and against traffic data obtained through the signalling system. Some preliminary results were presented by Smulders (1987). In the fourth section a control policy is designed for the homogenisation measure mentioned above. A simplified traffic model is considered and shown to represent the congestion phenomenon in a realistic fashion . The policy design criterion is the expected total throughput of the freeway until congestion occurs. Conclusions and suggestions for further research are given in the final section.

In the Netherlands a freeway control and signalling system has been installed on several freeways some years ago (Remeijn, 1982). The system consists of induction loops embedded in the road surface and matrix signal boards mounted on gantries. The paired induction loops are spaced approximately 500 meters apart and allow the detection of passing cars and measurement of the passing speed. The measurements are sent to the control centre associated with the signalling system. The gantries are spaced 500 to 1000 meters apart. The matrix boards on these gantries may be used to display advisory speed signals, lane arrows and road clear signals. These variable message signs are operated from the control centre.

THE FREEWAY TRAFFIC MODEL The model we present is based on a well-known macroscopic freeway traffic model described by Pay ne (1971) . This model was used by several other researchers: Cremer (1979), Van Maarseveen (1982,1983), Papageorgiou (1983), and Payne himself (1979) .

In this paper we investigate the control problem of choosing suitable speed signals, with the aim of reducing the probability of congestion. Experiments in practice have shown that a feasible and effective control measure exists, which consists of displaying one and the same speed value at a series of consecutive gantries, for all lanes. This is called the homogenisalion measure. The experiments have shown that this control measure leads to a considerable reduction of the instabilities that occur in dense freeway traffic (Van Toorenburg, 1983). The paper is divided in five sections. In the next section a model for freeway traffic is presented which displays realistic performance in several basic traffic situations. The model is based upon the model described by Van Maarseveen (1983). The modelling results were previously published by Smulders (1987). In the third section an estimation algorithm or filler is formulated that estimates the state of traffic at any time moment, from the available measurements (count data and passing speed measurements). The filter was extensively

The freeway is divided into sections of approximately 500 meters long with the measuring loops at the boundaries of each section. For section i we define the variables: p;(t) : the density in section i in vehicles per km per lane

v;(/) : the mean speed of the vehicles in section i at time I

The model consists of the following stochastic differential equations:

dPI(t)=_I_{~j_IPj _ IVj_ 1 -~jpjvjLt+ /jL; r -j,LI, [dm j _ I(I) -

dm,(t)]

(2. 1)

1. 1bi.s research is supported by the Technology Foundation. 2. Till December 31,1988. 3. From January I, 1989.

- y(Lj/j)2[fJpj + (I - fJ)pj

221

+

d[pj _ I

-

p;]dl

+ dw;(t)

(2 .2)

222

S. A . Sll1uldel's

for i-I, ... ,n, where

Pi = [api+(I-a)pi+Il Vi

120

= [avi+(I-a)vi + Il

100 v7

and

(kmIh)

80 60 40

(2.3) P7

20

(vch/km/\ane)O

The parameter values are given in Table 2.1. Suitable boundary conditions are obtained by prescribing the entrance intensity and assuming stationarity at the exit of the freeway stretch.

0 .0

0.1

0.2

0.3

0.4

0.5

time (h)

Fig. 2.1. Simulation of unstable traffic. TABLE 2.1 The Model Parameters parameter a

T Pjam

Vir.. Peril

a d

;

value 0.85 0.01 110.0 110.0 27.0 1.0 2970.0 6.5 0.5

120

unit

100

h veh / km / lane km / h veh / km / lane km 2 / h I/h km / h2

80

VI (km/h)

60 40 20 0

In (2.1) and (2.2) the parameters have the following meaning: li

ei Li

a T Vim Pjam Peril

d

r

f1

the number of lanes of section i the minimum of /j and /j - 1 the length of section i in km weighting factor E [O, IJ relaxation time in hours the equilibrium speed at zero density in km / h the jam density in veh / km / lane the critical density in veh / km / lane a constant of dimension I / h anticipation factor km / h2 anticipation weight factor.

The density equation represents the principle of conservation of vehicles. The stochastic process mj(t) that appears is a martingale and accounts for the stochastic character of the jumps in Pj(t), whereas the first term on the right-hand side of (2.1) represents the behaviour in the mean. The latter term is the difference of the intensities Ai _ 1 and Aj, where Aj=ejpjVj.

Note that we assume that there are no on- and off-ramps. These may easily be accounted for if necessary. The speed equation (2.2) is based on Payne's model, but was modified (Smulders,1987) to obtain realistic behaviour of the model. The Brownian motion Wj represents stochastic fluctuations. We will not explain how we arrived at the model (2.1), (2.2), but refer to Smulders (1987) for more details. The results of simulation experiments show the model to behave realistically in case of low, moderate and high density traffic. When density becomes 25-30 veh / km / lane instabilities occur, as the realisation plotted in Fig. 2.1 shows. When congestion has occurred it extends itself in the upstream direction. Moreover, slop-start ,mves are generated in the congested region, see Fig. 2.2.

0.0

0.05

O. JO

0. 15

0.20

0.25

0.30

0.35

time (h)

Fig. 2.2. Simulation of congested traffic.

The freeway control and signalling system does not provide the exact values of density and mean speed. The only information we receive are the passing times of vehicles at measuring sites and their speed. These measurements clearly contain information on the prevailing densities and mean speeds, but these cannot be computed exactly. We have to estimate the density and mean speed from the measurements. An algorithm for this is developed in the next section. Here we give a mathema tica l model of the relation between the measurements and the variables density and mean speed. Apart from passing times the system also provides passing speeds. It turns out that to obtain a practically useful algorithm the speed space has to be discretised in a finite number of classes. We therefore introduce the speed classes V) for j = I, .'" m where m is the number of speed classes. If we define

'111 : the number of vehicles that pass site i from 10 with speed in class V j then we assume the following relation

to

hold :

d'l11(t) = r1(Pj,vj )'Ajdl + dm1(t)·

Here r1 is the fraction of vehicles that pass location i with speed in class V} . This fraction is supposed to depend on traffic density and mean speed of the sections neighbouring location i. For a precise description of the model for y{ see Smulders (1987). The process m1 is a martingale. The observation equation of our traffic model then follows by accounting for measurement errors:

223 dn-/(t) = [1 +f{(j) -ff'(j»)·yj(pj. vj)'Ajdt + dim{ + drf,(j) - dr7'(j»)(t)

where

£{(/): the fraction of false counts at location i in vj

(3.6)

f,!,(j): the fraction of missed counts at location i in Vi

rf(j), r'!'(j): error process martingales. For i =O, ...• n and j = I •...• m the equations (2.4) present the measurement equations and will be used in the development of the filter in the next section.

We now develop the algorithm for the estimation of the section densities and mean speeds from the measured passing times and speeds. To simplify notation introduce the state vector _

L

=lh6.h5, ... ,h;;,]T.

where h{ = y{ ·hj. PI x is an approximation of the matrix of conditional error covariances

THE FILTER

I PI(t) . VI(t) • ...•

XI -

In (3.5) A is a constant matrix of a special structure which accounts for the + I or -I jumps of the estimated densities when vehicles enter or leave a section. In (3.6) L is the vector of intensities:

Pn(t) . vn(t»)

E[(XI - XI)(XI - XI)T I g/].

l: is the covariance matrix of the, Brownian motion processes Wj. In the equations above F is given by the following

T

,

and the measurement vector NI =

I nO(t).

n5(t) • ... • ng'(t) {.

+ dZ, = H(X,)dt + dM,

dX, = F(X,)dt

(3.1 )

dN,

(3 .2)

where F and H and Z and M follow from (2.1). (2.2) and (2.4). For the estimation of the state X from the measurements N techniques have been developed in the area of system and control theory . Well-known is the Kalman filter for the case where F and H are linear and Z and M are Brownian motion processes. For the case of counting type measurements theory has been developed (Bremaud. 1981). The estimator of XI that minimizes the estimation error variance is given by

I g,N ].

the mean of XI conditioned upon the measurements. ~},N is the a-algebra generated by {Ns • s ~ t} and represents the information containeq in the measurements up to and including time t. For XI the following differential equation can be derived: dXI=EIF(X, ) I G]/jdt +4>,{dN, - EIH(Xt ) I ~T/ ]dl }

(3.3)

Here 4>1 is called the gain nwtrix which satisfies a complicated equation that we will not present here. In (3.3) the first term on the right-hand side shows that the filter follows the model. The second term is a correction. based on the measurements. Equation (3.3) ,gives the exact evolution of Xt in time. Unfortunately XI cannot be computed from (3 .3) in general. For the evaluation of EIF(X,) I ~i/ ) the computation of the entire conditional probability distribution of X t is needed, which is practically impossiple. The same holds for other terms in the equations for X t and 4>t. We therefore have to resort to approximations. The usual approach is to develop Taylor series of the nonlipear functions like F and H around the estimated state XI and neglect high order terms. Depending on whether one only takes first or second order terms into account one speaks of a first or second order filter. The equations of the second order filter are: dXI =F(XI)dt +4>,[dN, - H(Xt)dt]

4>1 ={p;X H'(XI { diag - I[H(X, )] +

+ 2}:}:-a-aj

Model and measurement equations (2.1) to (2.4) can then be summarised as

XI = El XI

a2 F j (.)

1

Fj(.) = Fj(.)

(3.4)

[~] }I -

(3.5)

k

,

(P/ )jk

Xj xk

H and L are given by analogous expressions. The first order filter may be obtained by setting F j equal to F j • etc.

Using equations (3.4). (3.5) and (3.6) we can compute estimates of the state of traffic from the measured passing times and speeds. A FORTRAN computer program was written for the numerical integration of the differential equations and was extensively tested against simulated and against real traffic data. Equation (3 .6) was not solved directly but by means of a square root method to improve numerical stability (Morf. Levy and Kailath, 1978). For a detailed account of the results of these tests we refer to Smulders (1988a). We only present a few examples and the main conclusions here. The proposed filter was tested against artificial and against real traffic data. The artificial data was generated by simulating the model equations (2.1 ).(2.2) and (2.4), the real data was obtained from the signalling system. Performance of the filter was evaluated using three criteria: an integrated squared error criterion. a local estimate criterion and an innovation increments criterion. To reduce the sensitivity to modelling errors in the observation equation (2.4) a structural modification of the filter was proposed and was shown to reduce the bias. The modification consists of replacing the gain 4> in (3.4).(3 .5) and (3.6) by cP. defined by

IS~I~{4>j'km_j)/~k

_

for

I ~i ~ n

4>j,km+j =

(3.7) 4>j,km+j

for n T

l~i ~ 2n

for k=O, ...,n. Further conclusions of the filter tests were: -to obtain a slightly better performance the parameters a and y had to be modified. The weighting factor a was set to 0.5 and the anticipation strength factor y was reduced from 6.5 to 1.0 km / h 2 : -the filter turned out to be insensitive to most other parameters. This is explained by the fact that density largely explains traffic behaviour. Accurate parameter estimates are therefore not required to achieve good estimates of traffic density and mean speed: -the first order fliter proved to be as accurate as the second order one and is therefore to be preferred: -two speed classes (m= 2) were shown to be sufficient to obtain satisfactory estimates.

s.

224

A. Sl11uldcrs signals. Several control methods have been considered in the literature and have been applied in practice. Control by means of variable speed signals is not considered very often in literature, however. Some experiments with control by means of variable speed limitations were used by Cremer (1979) to show (through simulation studies) the possible advantages of this type of control. One of the main difficulties associated with control by means of speed signs is to assess the effect on driver behaviour. In contrast to what apparently was found by Zackor (1972) speed limitations are not necessarily followed by drivers in the Netherlands. Moreover, the speed signs of the Dutch Motorway Control and Signalling System are not compulsory but advisory. The effects of speed signal control turn out to be far less dramatic than proposed by Cremer (1979). There an increase of capacity of 21 % is assumed to be achievable. According to our data capacity hardly increased, but control nevertheless had a significant positive effect on the stability of the traffic stream.

The filter that resulted from the tests was applied to large data sets to check the validity of the results. As an example consider the stretch consisting of four sections of the A12 freeway near Utrecht (km 65.55 to 63.50) and data from May 9, 1983. The weather was dry and clear. The period considered)s 7:15 - 8:15 am. The first order filter is applied with ~, using three speed classes: [0,80) , [SO,I(0) and [100,00). The filter is started with a large initial uncertainty. A result is presented in Fig. 3.1, where the estimated evolution of traffic density and mean speed in the third section are shown. Note that during the period density increases and as a result, the speed shows more and more instabilities. Running the filter on this data set required almost 6 minutes of cpu-time on a CDC Cyber 750 mainframe. 120 100

v3

(km/h)

80 60

40 P3

20

(veb/km/lane)

o

L-____

7:15

~

____

7:27

~

______

~

____

~

____

~

8:03

7:39 7:51 time (h :min)

Fig. 3.1. Estimated state trajectories. Another example of a result obtained with the filter. on a different data set, is shown in Fig. 3.2. Each point corresponds to a (.0, v) - pair, there is a point every 2 to 5 seconds. The equilibrium relation (2 .3) is plotted also. The model seems to agree with the real data very well. A large excursion occurs, in which density increases to 60 veh/lrnl/lane and the speed decreases to 40 km/ h. The cycle that is apparent in the plot is travelled clockwise. This in accordance with results obtained by Ceder (1979). 120 100 80

v

t

60

(km/h)

40 20

0

0

20

40 p

....

60 80 100 (veh / km / lane)

120

Fig. 3.2. Phase plot of state estimates.

THE CONTROL POLICY The steady increase of traffic demand on freeways during the past decades has led to a high rate of congestion in the Netherlands. Increasing the capacity of the freeway by increasing the number of lanes is a solution that is not always acceptable or even preferable to alternative approaches. An alternative approach consists of exerting some kind of control over the flow of vehicles by means of

8: 15

A specific type of control policy is considered here, called homogenising control (Van Toorenburg.1983). One and the same advisory speed signal is displayed for all lanes at a given set of consecutive gantries at the same time. The speed value is chosen to be in correspondence with the actual speed of the traffic stream. This will in almost all cases lead to a value of 90 km/ h in practice. The only parameters left to optimize in this type of control are the switch on and switch off times. The motivation for this type of control is that congestion is caused by severe inhomogeneities of the traffic stream when the intensity approaches the capacity (Van Toorenburg, 1980). Shock waves occur, originating in a chain of vehicles that closely follow each other at high speed. Small disturbances are amplified and may finally lead to a standstill and congestion. The inhomogeneities in the traffic stream generate the small disturbances needed for congestion to set in. Examples of these inhomogeneities are speed differences between consecutive vehicles in one lane, speed differences between the lanes, flow differences between the lanes. An illustration of the large differences that exist between lanes may be found in Fig. 4.1. The difference between the probability density of time headways of left and right lane is apparent. The previous discussion suggests two ways to avoid congestion: remove the sources of disturbances (increase homogeneity) or reduce the number of short headways (increase stability). Homogenising control is aimed to do the former. It was applied in practice during experiments with the Dutch Motorway Control and Signalling System in 1983. A two-lane freeway stretch near Utrecht and a three-lane stretch near Rotterdam were considered, each about 6 km in length . Data obtained before, after and during the experiments were investigated and the results were reported by Van Toorenburg (1983). Further investigation of the data of the two-lane freeway allowed us to draw more detailed conclusions (Smulders, 1988b). The main conclusions are: -when control is applied a significant reduction occurs in the percentage of small time headways (.;;;; 1 sec), on the left lane. This effect does not occur on the right lane; -there is a reduction in the variance of the time headways on the left lane; -there is a slight but significant reduction in the mean speed, of equal size on both lanes; -the mean space and time headway on the right lane decrease significantly; -the instability of traffic flow, measured as the number of serious speed drops decreases significantly during homogenising control. The decrease that was measured amounted to about 50% ;

FrccII'a\' Traffic COlltrol Proble m

225

-during control capacity did not decrease. There is some indication for a small increase (1-2%).

The number of lanes is denoted by I, the length of the stretch by L (in km). In (4.1),(4.2) the variable i represents the control action: ;=0 means that no control is applied, i= 1 means that it is. The effect of control is modelled as 1 . 0 , - - - - - - - - - - --, 1.0 , - - - - - - - - - - - - , follows: RlGHT LANE

no control

0.8

0.8

LEFT LANE

,

i=O

{ vJree - Av ,

i =1

vJree

=

vfiiree

0.6

0.6

0.4 0.2

0.0 0'!--;O-=47""'''''''~6---;;-8- - ; ->

8

10

Ab.

time (5)

Fig. 4.1. Estimated probability densities of time headways. The general conclusion is that homogenising control is advantageous in that it increases safety and reduces the probability of congestion. The effect is not so much in the homogenisation however, as in the stabilization of the traffic stream. The most important result is the decrease of the fraction of small headways on the left lane. These small headways, corresponding to vehicles driving at close distance and at a high speed, are a major reason for the instability of the traffic stream. Note that the main effect of homogenising control occurs on the left lane and comes down to a regrouping of the available space and is local: only a fraction of the vehicles is involved. On the right lane the effect is global and leads to an increased intensity. Before coming to the design of a control policy for the homogenisation measure we now first discuss a simplified traffic model. A One-Dimensional Model

Although the model developed in Section 2 showed satisfactory behaviour and was able to represent the instabilities that occur in practice, we will not use it here. Instead, we will consider a model for one section of freeway only, for which the control policy design procedure is still feasible. To incorporate the effect of the homogenisation measure we note the following. On the left lane the percentage of short time headways decreased significantly, leading to a smaller variance of these headways. This may be shown to lead to a reduction of the variance of density increments over short time intervals. On the right lane the mean time headway decreased, implying a larger intensity. On both lanes the mean speed decreased slightly. The reduction in speed under control may immediately be expressed in v e by a reduction of vJree . The conclusion that capacity does not decrease and possibly increases slightly, together with the speed reduction of the previous paragraph implies that Peril will have to increase. For more details we refer to Smulders (1988b). The slight increase in total flow due to the effects on the right lane may be modelled by a small increase in ho. We now present the one-dimensional model with control:

The model given by (4.1) and (4.2) describes the evolution in time of the density of a freeway section when the entrance intensity Ao is given (in veh / h). This evolution is subject to considerable noise, modelled by a Brownian motion process with variance (oi)2. The mean speed is assumed to equal the equilibrium value v e defined by (2 .3).

Peril

,

{ Peril

+ Ap ,

i=O

i=l , i=O

{ho ho + AA

=

i=l.

The model represents the main feature we are interested in, the occurrence of congestion when ho is near the capacity level. There are two equilibrium points, a stable one denoted by pb and an unstable one denoted by p~. If density exceeds the unstable value, congestion results. For a detailed analysis of the stability of the model we refer to Smulders (1988b). The capacity >..cap of the freeway section is defined as Aeap =

max

fpve(p).

o"-P'-P,_

If we define T

= inj{(;;;'O : PI=Pjam Ip(O) }

then it is known (Mandl,1968) that E[Tlp(O) ]< oo , which with T;;;'O, leads to P(T
0

AA Pjam

2 0.5 14000.0 11000.0 O.OI·ho 110.0

P cril

Ap vJree

Av a d

27.0 2.0 105.0 3.0 0.58 3197.0

TABLE 4.2 Mean Time to Congestion

ho

~veh/ill

1000 2000 3000 3500 4000 4400 4600 4800

no control E[ Tlp5] (min) 9.6 • 1010 2.2 • 106 1044 8l.l5 15.28 6.68 4.94 3.83

control E[ Tlpb] ~mi~

2.3 * 10 14 2.0 • 108 8344 263 .3 25.82 8.40 5.78 4.25

It is clear that applying control leads to improved stability because of the following facts : -reduced variance of PI leads to a smaller probability of exceeding p~ ; -the distance between Pb and p~ increases because of the effects on speed and capaci ty.

226

S . .-\ . Smuldcl"s

lbis is confirmed by the results in Table 4.2. We conclude that for high intensity values congestion may only be postponed somewhat and that for intensities about 10% below capacity the effect has the most significance.

Control Policy Design

Now control policies may be designed of the homogenising type, based on the traffic model described above. It is the aim of control to reduce the instabilities present in traffic flow when intensity approaches capacity. The design is formulated as a problem of optimal control. The criterion to be maximised is defmed as min(T, T)

Vfi"(Po,t) = E[

f

IPsvsds I Po

1

(4.3)

where T : time horizon (h).

We are interested in Vfin(p,O) . Maximisation takes place over all stationary, piecewise continuous controls: i : [O'Pjaml-+ {O, I}.

Here i(P)= 1 means that control is applied when the density equals p. Maximising (4.3) comes down to maximising the total number of vehicles that pass through the section until congestion occurs or the time horizon is reached. lbis may be achieved by maximising the time to congestion and the intensity. We may use a result from Pliska (1973) to show that an optimal control, maximising (4.3) exists and is piecewise constant. The optimal policy therefore consists of switching the control on or off at several switching points. The optimal control policy and the associated criterion function may be shown to satisfy the Hamilton.Jacobi-Bellman or dynamiC programming equal ion

avfin { Tor--2I a vfin + --+max 1

at

ap

, ; 0, I

1 · aV/in } -I/[Xb - lpv ; (p)J-,- +Ipv; (p) = 0 vp J

(4.4)

in Vfi"(P) . lbis may be interpreted as postponing control as long as no significant benefit can be expected from it. Results of computations of the optimal p for several intensity levels are presented in Table 4.3 (The row indicated by p",,). Note that for small Ao control will hardly ever be applied, unless something exceptional occurs, like an accident, leading to extreme density values. Note also that for intensities above 3500 veh/h one and the same control policy suffices. The increase in p for Ao >4000 may be explained by the fact that when intensity is high the criterion is very "flat" and insensitive to p. Little benefit may be expected from the homogenisation measure when intensity is too high. To illustrate the effect of optimal control we present the result of a simulation for the case Ao =4000 veh / h. The optimal control appears to be I {p ;;' 29 } . Figure 4.2 shows a realisation of PI for the case with and without control. The positive effect is clear, without control congestion occurs at 0.35 h and with control congestion does not occur during the period considered. The respective criterion values are 1320 and 1997 veh. The simulation result shown here is typical in the sense that in most realisations when density is critical, congestion may occur, even when control is applied, though the probability is smaller when control is applied. Once congestion is avoided, however, it will take some time before another critical situation turns up. This means that control leads to a relatively small probability of incurring a relatively large benefit. A negative point in the implementation is the frequent switching that takes place: over 600 times in 30 minutes! It is clear that this is not acceptable from a practical point of view, as frequent switching would confuse drivers and might even be dangerous. This problem is addressed by Smulders (l988b). In the literature optirnisation problems with costs for switching from one control action to another have been addressed. Introducing switching costs leads to a hysteresis type of control : a switching point as it occurs in absence of switching costs splits up into two points when these costs are introduced.

with boundary conditions 120 r - - - - - - - - - - - - - - - - - ,

fi aV " ---ap(O,t)=O, Vfi"(Pjam , t) =0, Vfin(p, T) = O.

100

In our problem we expect one switching point to be important: when the density approaches the instability point of the model control is expected to become beneficial. This has been investigated and led to restrict attention to oneswitch controls: A one-switch control policy is defined policy of the form DEFINITION

10

be a

80

P

60

(veh/km/lane)

t

40 20

0.1

i(p) = I {p;;'p }

0.2

0.3

0.4

0.5

time (h)

which means that control is applied on~)' for p;" p.

Fig, 4.2. Simulation of traffic density. We will restrict attention to computation of the criterion value for several elements of the class of one-switch controls. lbis avoids the maxirnisation problem in (4.4) and leaves us with the differential equation: fin

aVfi" 1 a Vfi" 1 · Cl V -a+ T o f--2 - + -[Ab - Ipvf(p)l-~- + Ipv ;(p) = 0 t ap 1.1 up 2

where i(P)=I{p;;'p} and p is given. This parabolic equation is solved by means of the routine D03PAF of the NAG-library. Starting with the one-switch control corresponding to p=O we will increase p as long as no deterioration of the criterion function occurs. To be precise: p will be maximised under the condition that for all pe[O'Pjaml no deterioration of more than 5% should occur

The optimal control then consists of switching to one decision when the larger of the two points is crossed in upward direction, and returning to the original decision only after crossing the smaller point in the downward direction . The hysteresis clearly enables a reduction of the frequency of switching. A specific design procedure was followed by Smulders (1988b), resulting in the hysteresis control policy of Table 4.3. In the simulation experiment presented earlier PI exceeded 29 (= Pon) at t = 0.0 I h and only become less than 5 (= Poff) for a period of 36 seconds near t = 0.02 h. Implementing the hysteresis reduces the number of switchings from over 600 to 3. The criterion value is hardly different: 2005 veh.

FrcC\\'al' Traffic Control Problcm

TABLE 4.3 Optimal Hysteresis Control

Ao

(veh/h) 2000 3000 3500 4000 4400 4800 (veh/km/lane) 36 2 5 8 11 56 p"" (veh/km/lane) 75 51 29 29 30 31 Pne

CONCLUSIONS In this paper a freeway traffic control problem has been addressed and solved. Based upon a model which shows realistic performance in several traffic situations an estimation algorithm or filter has been derived. Using the local measurements obtained from the Dutch Motorway Control and Signalling System this filter produces estimates of traffic density and mean speed of a freeway stretch of several sections. This information about the state of traffic is a necessary prerequisite for the implementation of the control policy that is designed next. This policy consists of switching on or off a homogenisation measure when traffic density crosses certain switching levels and results in a smaller probability of congestion. Further research may consist of using a more detailed model in the design phase. A first example of this is presented by Smulders (1988b).

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