Time and space discretized nonlinear optimal control of freeway traffic flows

Copyright © IFAC Automatic Systems for Building the Infrastructure in Developing Countries, Istanbul, Republic of Turkey, 2003

ELSEVIER

IFAC PUBLICATIONS www.elsevier.comllocatelifac

TIME AND SPACE DISCRETIZED NONLINEAR OPTIMAL CONTROL OF FREEWAY TRAFFIC FLOWS

AhmetAKB~

Marmara University, Vocational High School ofTechnical Sciences G6ztepe-Kadlk6y, 81040 Istanbul, Republic of Turkey Fax:0-21 6-4182505; E-mail: [email protected]

Abstract: Various linear and nonlinear control techniques available for lumped parameter systems can be adapted to the freeway traffic flow process by space discretized models. In any case, it is important to analyze the control laws designed by using different paradigms so that they can be compared for their effectiveness. Such a control model has been proposed in this study. It is based on the feedback linearization technique. In the proposed model, the macroscopic densities of the main-link segments (discrete space units) used in the feedback linearization technique are modified due to the direction of the shock waves between the contiguous segments. To reach this aim, the macroscopic flow and density parameters of the discrete space units are calculated at the end of each discrete time period., by using the cumulative occupancy and vehicle count measurements obtained from the traffic detectors. Simulation based comparisons studied in VlSSIM are shown that the effectiveness of the feedback linearization technique can be increased by this way. The test results are satisfied. Copyright © 2003 IFAC Keywords: Freeway traffic control; nonlinear optimal control; ramp metering.

1. INTRODUCTION

(Wattleworth and Berry, 1965; Yuan and Kreer, 1971; Wang and May, 1973; Chen and Cruz, 1974) or automatic control theory (Isaksen and Payne, 1973; Goldstein and Kumar, 1982; Papageorgiou et.al., 1983) have been proposed. Some of proposed algorithms have already been utilizing for local ramp metering controls at many sites of the world.

The steadily increasing traffic congestions on urban and inter-urban freeways have led to the use of several control mechanisms all over the world. Basically, these are formed by using ramp metering and variable speed control actions. With the ramp metering control action the freeway flow control mechanism is maintained by controlling the number of vehicles entering the freeway, while it is maintained by limiting the free flow speed of the vehicles between the specified freeway sections with the variable speed control; so that the volumes are being kept below the capacity through the main-link.

However, in the last years, many researchers have recognized that considering the effect of the coordinated controls on the network level has many advantages compared to local control. For global results, nonlinear control techniques become necessary. In this context, new studies must be planned for advanced controls. In any case, it is important to analyze control laws designed using different paradigms so that they can be compared for their effectiveness by using either field experiments or simulation studies.

Ramp metering is the most popular of these. It has been recognized as an effective way for releasing freeway congestion, typically the result of either a surge of demand during peak commuting hours or a temporary reduction of freeway capacity. The studies on ramp metering have been extended over the last 35 years. In this period a variety of algorithms which are based on either optimization techniques

In this study, it is proposed such a paradigm for coordinated control of freeway traffic flows along the freeway sections which are composed due to road

127

geometric conditions, so that each may has one onramp only. It is a time and space discretized nonlinear optimal control algorithm based on the feedback linearization technique which is modified depending on the shock wave evaluations (Zbang H., et.al.,1996; Kachroo P. and Krishen K., 2000). In the following sections; the macroscopic flow characteristics, ramp metering objective and the optimal control problem have been studied first. Then, discrete system dynamics, proposed control model, and the simulation based test studies which are realized in VISSIM simulation environment have been introduced. Finally, effectiveness of the feedback linearization model and modified type of it (proposed model), with respect to the uncontrolled conditions, have been evaluated.

critical value of its density, Pc, at any time and any position of the way. The flow which corresponds to these speed and density hae its maximum value, qmax, which is equal the flow capacity, qcap=qmtJ)!.

=

v, 1<0-------+--_

OL--j_~--+O'---=:::::....-~--

q

o

o

0

P

(voMunIlane)

donsIy

p.

o

PJ-

When the density of traffic flow exceeds the critical density, the volume and average speed of the flow begin to decrease, so that, at the jam density, P={}jam, both flow and average speed become zero. If the density of traffic flow is greater or smaller than this critical value; the flow can not achive the maximum value, so, the capacity usage and the transportation performance of the freeway become worst.

Traffic flow process on a freeway is characterized by three basic macroscopic parameters: flow (volume), q; density, P; and velocity, v. Flow is defmed as the number of vehicles passing a specific point in an hourly rate (veh/h). Density is defined as the number of vehicles occupying per kilometer of the way in a lane basis (veh/kmllane). Speed is defmed as the average rate of motion and is expressed in kilometers per hour (km/h). All of these macroscopic flow parameters are the function of time and space, q(x,t), p(x,t), v(x,t», as shown in Figure-I.

~lfq(·,,)

p

Figure 2. The static relations between the macroscopic flow parameters.

2. MACROSCOPIC PROPORTIES OF TRAFFIC FLOWS

q:F:::::=::

v

v ...

However, the importancy of performance decreasing are occured when density exceeds the critical density, P>Pc. Because of that, control action must be taken in consideration for these conditions. 3. FREEWAY TRAFFIC FLOW CONTROL

:::=::::?,t}

As indicated above, the flow-density relationship, q=f(p), is referred to as the fundamental diagram of traffic flow, for designing a control process. Due to this, the control aim is to ensure that:

- - - - -.....:------~

Figure 1. Distributed parameter form of the freeway traffic system with its parameters.

p(X,t) < Pmax

According to these definitions, the basic flow model is expressed as following (May, 90): q(X,t) = p(x,t)

* v(x,t)

(1 - P /Pjom )

E

L, t

E

R+

where Pmax is a maximum density between the Pc and So, the control aim can be achieved by having a critical density value close to the maximum flow density as the target density of the controller. In this context, ramp metering supports this aim by the traffic light (signal) activation which indicates wheather the vehicles can go into the freeway or not, Figure-3.

{}jam.

(1)

There are various static models which have been used to represent the relationship between velocity and density parameters. The most simple one of these models proposes a linear relationship between the two parameters (Greenshield, RD., 1934): v = vf

(3)

Pc 5.Pmax
q.

(2)

I

!qQtA ..

~

I

where VI is the free flow speed, {}jam is the jam density. By substituing (2) to (1), we can easily show that relations q=f(p) and q=f(v) have the parabolic characteristics. So, the static relations between the macroscopic flow parameters v=f(p}, q=f(p), q=f(v} can be shown as in Figure-2. From these relations it can be clearly stated that, each freeway flow has its optimum average speed, Vo , which corresponds a

i

/P

;:: r

{

I

L

G

i~ !

,_.

Figure 3. Freeway flow control activation mechanizm by ramp metering. In Figure-3, C, G and R indicate the cycle time, green time and red time of the signal, in seconds,

128

respectively; so that, G=c-R. The relation between the green time and the ramp flow with their constraints are as following, G

=[(r(t) / rm",). c]

Due to the above explanations, space discretization has been performed by dividing the considered freeway link into the segments as shown in Figure-4. The flow parameters have also been shown in time discretized form in Figure-4. Here, T is the discrete time interval in seconds; k indicates the time interval between kT and (k+ 1)T; i indicates the segment number. Thus, the time and space discretized form of the freeway traffic system can be considered as a lumped parameter system for each segment. So, the dynamic of the system can easily be derived in difference equations form.

(4)

GE[Gmin ,Gm",],Gmin >O,Gm", ~c

Here, r(t), indicates the ramp flow and r max its maximum value. Gmin and Gmax are the minimum and maximum of the green times, respectively. Theoretically, ramp metering is effective if the inflow, qin, on the main-link at the section immediately upstream of the ramp is less than outflow capacity, qOU1=qcap. (Zhang H., et.al., 1996) Depending on the above considerations, to attempt a controller design, first it must be decided to the macroscopic model of traffic flow which represents the dynamics of the flow. For purpose of this development, we assume that, the traffic flow process is governed by the macroscopic model which based on fluid dynamics (Lighthill and Whitham, 1955; Richards, 1956):

a ox

a at

-q(x,t)+-p(x,t) =

0

Figure 4. Time and space discretized form of the freeway traffic system. The following considerations have been considered for our control aims: Each segment can include one on-ramp, maximum; number of lanes must not be changed through the segment; the cycle time of the ramp control signals must be same and equal to the discrete time step, T=c, for coordination of local ramp metering actions; the green light starting time must be synchronized with the cycle starting time for all the signals; the green light durations calculated by the controller for every time index must be limited with its constraints, Gmin< Gi < Gmax., for all signals.

(5)

Equation (5) is the well known hyperbolic conservation law in PDE form. The nonlinear state equations can be derived from this law either in ODE form or in difference equations fonn, for optimal control of traffic process.

Due to above explanations, each freeway segment shown in Figure-4 can be represented with a generalized geometric structure as in Figure-5.

4. DISCRETE SYSTEM DYNAMICS Equation (5) is a steady-state representation of freeway traffic flow dynamics. Due to this equation, freeway traffic system is considered as a distributed parameter system which have infinite number of flow parameters along the way. But, practically it is not possible to measure the flow parameters at infinite number of discrete points. In reality, sensor measurements are made at discrete points with 500+1000 meters intervals, depending on the geometric structure of the way. This means that, the space base of the system must be descritized depending on the practical considerations.

d;lkJ

Figure 5. Generalized freeway segment structure. In this figure q,{k) represents the segment output flow, qout; qi-J(k) represents the the segment input flow, qin, which is the previous segment output flow in the same time. Here p,{k) is the segment density; r,{k) is the ramp flow; d,{k) is the demand; e,{k) is the exit flow from the main-link, respectively at time index k. Then, by using conservation principle stated with equation (5) the discrete system dynamics of an n segment freeway traffic system can be represented with following difference equations:

The time base of the control system must also be descritized. Because, the macroscopic flow parameters must be determined by using the microscopic flow parameters which can be measured by means of the traffic detectors within the very short time intervals, for real time control actions in practice. By this way, the macroscopic flow parameters can easily be calculated for every discrete time interval, so that the discrete time resolution of the system can be increased in satisfied manner, depending on the practical considerations.

Pi (hi) =Pi (k)+ 36~L. .[ qi-t(k)-qi(k) + ri(k)+ e;(k)] I

i=l, ..... j7

(6) All the flows represented in equations (6) are in veh/h., and densities are in vehlkm/lane dimensions. So, the macroscopic flow values must be determined

129

by multiplying the lane based flow values with the lane number of corresponding main-link segment.

N/k}

(9)

q;(k)= 3600. -r-·L1;

5. THE OPTIMAL CONTROL PROBLEM

Here, Ni(k) represents the number of vehicles passed from the detector, through the time index le; A is the number of segment lanes.

The objective of the optimal control of the dynamic process described by ordinary difference equations (6) is to optimize a performance index that consist of state variables, p,{k) q,{k), and control variables, r,{k), under the some constraints imposed on the control. Here, performance index can be stated due to many social and economic benefits such as the minimization of the total transportation time, total delay or fuel consumption. However, all of these objectives can be achieved by having a critical value close to maximum flow density, pmJJJ: defmed in equation (3), as the target density for the controller. So, the nonlinear optimal control problem can be stated as: find ro(k), the optimal r(k) which IDlDlIDlZes, J(r) =

As above considerations, the values of the macroscopic densities of the segment output positions, Piou/(k), are calculated due to the following equation at the end of each time index: N,(t)

~>o

p:",(k) = DS;(k).pj
Here, j represents the vehicle number and Oj is its occupancy time obtained from the traffic detectors; DSi(k) is the average degree of saturation at the segment output, for the time index k. As shown in equations (9) and (10), the macroscopic flow parameters are calculated by using cumulative occupancy times and number of vehicles obtained through the discrete time period. So, the macroscopic parameters obtained in this way can reflect the real life traffic conditions in precision.

~[(r(k)- r....'y (r(k)- r..,..) + w ~(P;(k) - Pc (k)y ]

(7) Here, K represents the number of discrete time steps corresponding to time horizon; Pc is the critical density of the traffic flow (target density for the controller); and w is a weighting coefficient.

Depending on the above explanations, by using feedback linearization technique, our discrete time and discrete space type controller proposal can be stated as following:

We can state the control objective for a standard feedback control problem, such as steady state asymptotic stability criterion (Kachroo P. and Knshen K., 2000): IimK __

[~(P;(k)-Pc (k)y ]~O

Ti(k + 1) = K

3600L;

(8)

.

(11)

i=l, ..... ,n

(11) equations are obtained from the discrete system dynamics represented with (6) equations. These satisfy the steady state asymptotic stability condition stated with equation (8), for all segments (Kachroo P. and Knshen K., 2000). Here, the target density of controller has been chosen as the critical density of the freeway flow, p,{k+ J)=pc; s, represents a constant value which is determined due to field experiments for determining the exit flow (e,{k) = S.qi_l(k), 0< S < 1).

According to the proposed control model, the macroscopic flow parameters are determined by using the microscopic flow parameters (vehicle occupancy time, number of vehicles) obtained from the traffic detectors placed in a place around the segment output, for each segment, as shown in Figure-6. -

[Pc - p;(k)]+ [q;(k) - (1- s) qi-1(k)]

K=-T--'

6. THE CONTROL MODEL

[-----

DS;(k)~l (10)

-L i

Due to the proposed control model, the macroscopic segment densities, p,{k), used in the equations (11) are modified by making a simple decision analysis based on the shock wave phenomenon between the two contiguous section of the freeway. The segment density is assigned either the actual segment output density or previous segment output density, due to existing shock wave direction, as stated in following expressions (Zhang H., et.al., 1996):

DS.,(k)

-"------

Figure 6. Placement of the traffic detectors for measuring the microscopic flow parameters. The values of the macroscopic flow parameters, q,{k), are calculated due to the following equation at the end of each time index:

v (k)= q,_/(k)-q;(k) c P:'-~ (k) _ p;"1 (k)

130

Of

,1

OW(k) P,'_I

* P,'WI(k)

(12)

p,.(k) ,

={

p:(k) , if vc> 0 00' Pi (k), if Vc < 0 or p'(;'(k) = p'("'(k)

lanes; the off-ramp flow ratios SJ , S2 ,S3 are 0.20, 0.15, 0.17 respectively; the free flow speed distribution are 80-100 km/h for cars, 50-80 km/h for heavy good vehicles (HGY); vehicle length distributions are between 8-18 meters; HGV ratio is 0.20. Initial green times for signal groups are 70, 75, 80 seconds, respectively. The constant parameter assignments are; signal cycle c=IOO s; G min=5 s and Gmax=90 s for all signal groups; rmax =2000 vehlh for all ramp flows; Pc=40 and PJam=90 vehlkm!lane for all main-link segments. One of the on-ramp approach view of the freeway traffic system in VISSIM simulation screen has been shown in Figure-8.

(13)

The block diagram of the proposed control model has been shown in Figure-7.

Pc

Figure 7. Proposed nonlinear optimal control model of the freeway traffic system. For each time index, the green times of each on-ramp can be calculated by using the determined ramp flow values, rlk). Here the constraints are the maximum values of the ramp flows, rmax.i' the maximum of the ~een times, Gmax,i' and the minimum of the green tunes, Gmin,i' Due to these assignments, the green time calculation algorithm is stated as following: . =j(!j(k)/rmvc),c, if Gmin,;sG/kh;,Gmvci G,(k) Gmui ,lfG/k»G""xi Gmini

(14) Figure 8. An on-ramp approach view of the chosen freeway traffic system in VISSIM simulation screen.

,if G/k} < Gmini

In the control model, cycle times of the signals have been chosen equal to the discrete time duration, c=T, for all of the controlled on-ramps. All the calculated green times start at the begining of the next discrete ~e index simultaneously and continue up to green tune end. Remaining duration up to the end of the cycle time, the signals are converted to the red which indicate ramp flow is stopping.

The simulation package VISSIM consist of two different programs: traffic simulator and signal state generator. The traffic simulator is a microscopic traffic simulation program based on Weidmann statistical model including the car following and lane change logic. The signal state generator is a signal control software polling detector which processes information from the traffic simulator on a second by second basis. It then determines the actual signal status for the following second and gives this information back to the traffic simulator. The result of the simulation on-line is the animation of traffic operations and off-line the generation of output files gathering statistical data such as travel time, delay and number of stops. (Vissim, 2000)

7. SIMULATION BASED TEST STUDIES Simulation based test studies have been realized in VISSIM simulation environment to compare the effectivenes of the proposed control model and the non-modified type of it, both are based on the feedback linearization technique, with respect to the uncontrolled conditions. 3 different tests have been completed under the same traffic conditions: testl, for uncontrolled case; test-2, for modified (proposed) control case; test-3, for non-modified control case. For this aim, the traffic detectors, signal groups and other hardware elements of the freeway traffic system which consist of 1.8 km length of main-link and its ramps have been configured in VISSIM simulation environment. The main-link has been divided to 3 segments, each has one on-ramp and one off-ramp. Assignments about the the chosen freeway traffic system have been done as following:

Traffic control program can be transferred to the VISSIM test environment by using an additional program modul VAP (vehicle activated programme). For this aim, control model is written as a text file by using the functions and commands of VAP. The proposed control model and the non-modified type of it have been programmed in this way. The performance index have been stated as the total travel times along the main-link. However, in addition to the travel times, the total delay and the total number of stops for the main-link have also been evaluated by using VISSIM data gathered through the simulations. The traffic scenario has been chosen as shown in Table-I, for all the 10800 sec (3 hours) of simulation based tests.

The segment lengths L], L 2 ,L3 are 440m., 565m. and 595m. respectively; all the main-link segments have 3 lanes; all the on-ramps and off-ramps have 2

131

Table 1. The traffic scenario chosen for the tests. _lion time

moirHink Input qo (veh./h)

rampl
rwnp2 demand

(oeconds)

d' (Wlh.Ih)

rampJ demand d' (veh./h)

0·3600

6000

500

600

700

3601· 7200

6000

1200

1300

1400

nOl-l0800

6000

700

500

500

8. CONCLUSION The control model proposed in this study have been encouraged me for new studies about coordinated ramp metering and other network wide traffic controls. So, the new studies will be planned about network wide traffic control actions which deal with whole system amelioration including ramp queues.

For the perfonnance evaluations, 1793 meter of the main-link length which include all the 3 main-link segments is marked in VISSIM. For this length, the results obtained about the travel times and number of stops through the 3 hours of simulations have been shown in Figure-9 and Figure-lO, respectively.

REFERENCES Chen, I.C., Cruz, J.B. and Pauget J.G. (1974), Entrance Ramp Control for Travel Rate Maxirnization in Expressways. Trans.Res.. Vol.8, pp.503-508. Goldstein N.B. and Kurnar K.S.P. (1982), A Decentralized Control Strategy for Freeway Regulation. Trans.Res., B.I6. pp_ 279-290. Greenshields, B.D. (1934), A Study of Traffic Capacity. Proc. US Highway Res.S.'vol 14, pp. 448-477. Isaksen L. and Payne HJ. (1973), Suboptimal Control of Linear Systems by Augmentation with Application to Freeway Traffic Regulation IEEE Trans.on Automatic Control, Vol.J8, pp.210-219. Kachroo P. and Knshen K. (2000), System Dynamics and Feedback Control Design Problem Fonnulations for Real Time Ramp Metering. Society for Design and Process Science, VolA, No.I, pp. 37-54. Lighthill M. J. and Whitham G.B. (1955), On Kinematic Waves: IT. A Theory of Traffic Flow on Long Crowded Roads. Proc. R. Soc. Lond_Ser. A. 229, pp. 317-345. May, AD. (1990), Traffic Flow Fundamentals. Prentice-Ha/l, New Jersey. Papageorgiou M. (1983), Applications of Automatic Control Concepts to Traffic Flow Modelling and Control. Lecture Notes in Control and Information Sciences, (Balaskrishnan AV. and Thoma M., Eds.) Spring, Berlin, 94-100. Richards, P. I. (1956), Shockwaves on the Highway. Operations Research. Vol. 4, pp. 42-51. VISSIM (2000), User Manual. PTV system Software and Consulting GmbH; StumpfstraJ3e 1 D-76131 Karlsruhe, Gennany. Wang J.1. and May AD. (1973), Computer Model for Optimal Freeway on-Ramp Control. Highway Res. Record, Vo1.469, pp. 16-25. Wattleworth J.A. and Berry D.S. (1965), Peak Period Control of a Freeway System -Some Theoretical Investigations. Highway Res.Rec.89, pp. 1-25. Yuan L.S. and Kreer J.B. (1971), Adjustment of Freeway Ramp Metering Ratesto Balance Entrance Ramp Queues. Trans.Res., Vol. 5, pp. 127-133. Zhang H., Rithchie S.G. and Recker W.W. (1996), Some General Results On The Optimal Ramp Control Problem, Trans. Res., C. Vol. 4, No. 2, pp 51-69.

According to the obtained results, average delays are 78.8, 64.2, 71.2 sec.lveh for testl, test2 and test3, respectively. Also, average number of stops are 1.93, 0.73, 0.83 stops/veh for test}, test2 and test3, respectively. Due to these results, average delay is decreased %18.5 by the proposed model, while it is decreased %9.6 by the non-modified model, with respect to uncontrolled conditions. Also, average number of stops is decreased %62.2 by the proposed model, while it is decreased %57 by the nonmodified model, with respect to uncontrolled conditions. These comparisons have shown that the effectiveness of the feedback linearization technique have been increased by using the proposed model.

I· -... _. :2

~ 350

~U) 300 41 250

E :: 200 Q)

iU

test1 - - test2 - - test31

400

.'

...

_._ •

• __ ....

,11':.

~

_

. . .--. - -. ---- ---- ---- --. ---;t

..

'

.. ---- ---. --- ------ --- - ---------; --- - - ---_.- -----~:-

-_.. -- - -- - ----- ----

--.~

\

';:.;;~-.: ~.-

150

~100 c:

'7

c:

'(ij

E

50

-.--------- ----.--.------------_.----------._---------

0 ~-.......- -......- _ - -......-_~ 0 2000 4000 6000 8000 10000 silTlJlation tirre (seconds)

Figure 9. Test results for main-link travel times.

1- - .. - test1 - - - test2 - - - test3! 16~:;;;;;;;;:;;;;;;;;:;;;;;;;;:;;;;;;;;;;;;;;;;;;;:;;;;;;;;:;;;;;;;;:;;;;;;;;:;;;;;;;;~ ~

14 ---------. -- --. ----------- -- -;.; -.------------- ---------

~

g

12 10

.9

en "0

8 6 4

--.--------- .... -.. -.. ;-~--j-~... ----------..--------- .. ---- ---- .. ------. ;---\ -:---} :~-·II-------- •• -- -- --. . .-- -- -"1:1 ---- ---- --. ··-0-· .:---0- ....... - - - - ---

E

2

-- .. ------.--r...--.-;--~-J\.-----~-;o('\---~

~

0

~ I/)

---_._--------------- -- ..-----~~ .. --.-----------.---------- .-.---- _.. _. ;_~

.:.1

a

1

__ -- _. ---------------_.-

:

I,.

~-------.-

o

2000

4000

6000

8000

10000

simulation time (seconds)

Figure 10. Test results for main-link number of stops.

132