the prodyn real time traffic algorithm

Copyright (C) IFAC Control in Transportation Systems Baden-Baden. Federal Republic of Germany 1983

THE PRODYN REAL TIME TRAFFIC ALGORITHM J. J.

Henry,

J.

L. Farges and

J.

Tuffal

Departement d'Etudes et de Recherches en Automatique (DERA) du Centre d'Etudes et de Recherches de Toulouse, BP 40 25, 31055, Toulouse Cedex, France

ABSTRACT PRODYN is an algorithm able to compute in real time the best signal settings with respect to the delay criterion for any flow demand in traffic networks. The hierarchical algorithm usesForward Dynamic Programming (FDP) to compute controls at the lower level (intersections) and decomposition coordination technics at the upper level. The implementation on the field relies on the use of network structure of microprocessors the tasks of which are optimization and state estimation. The major goal to reach w ~ ~ to solve for such an application the dimensionality problem of FDP. Results show suhtential gains on delay to give an idea of about 16% with respect to fixed time policies.

Keywords. Optimal control, Dynamic Programming, Large scale systems, Traffic control, Hierarchical systems, Nonlinear control systems, Computer control. 1. INTRODUCTION Most of the traffic control technics work upon the "fixed time" philosophy for instance with TRANSYT (Robertson, 1969) which is the most famous program. For intersection control "intervalle vehicule" or the algorithm of Miller (1963) can be considered as real time algorithms. For network control, we can mention the work of Gartner (1982), Singh (1973) Perrot (1970) and the SATURN method (Paquin, Henry and Leglise 1974). SCOOT: (Hunt and co-workers, 1981), appeared recently, seems to be the most sophisticated and operational real time method -but it is still based upon cyclic settings. PRODYN 1; an attempt to find acyclic settings in the most optimal sense. A traffic network is a large-scale system which can only be optimized by a hierarchical algorithm. PRODYN uses the decomposition coordination technics to convert the large initial optimization problem into several smaller problems which are solved by Dynamic Programming ,and then solves the global problem using a two level iterative calculation structure. The decomposition is spatial each smaller problem is assigned to an intersection. This paper will present the chosen modelization, the decomposition ··coordination method, the subsystem resolution, the practical aspects of implementation and the simulation results.

2.

FOR~1ULATION

For a given network we can set the optimization problem :

Control variables : with i : intersection index (two phases) k : time index (t=kT, T = 5s) we define the control as Vi I if there is a commutation k from one phase to the other o if not State variables

with

j : link index for intersection i we define at k xkji .. queue lenght in vehicles (vertical queue) i 0 green phase for links U and k 3 green phase for links 2 and 4 i W : green already e lap s ed k I j i .. = number of veh~cles ~n movement in the lth part of the link

Ak

(I) (2)

if j = I or 3 ji j Xk+l = Max (X i +1{ k + dO j i Vi) 0) I

if j

2 or 4

'

lJ~) (dV j i (I -V~)

(3)

J.J. Henry, J.1. Farges and J. Tuffal

306

saturation flow rate in veh/5s maximum saturation flow rate during the yellow (3$) and integral red (2s) in veh/5s

1 ji Ak + 1 -

-

':k.

2 ji

!

I

r10 ,1,0

.

1

' :!.

I

0,0, "

l

Ak+l-,

0,0,0

.•• •

0,0,0

~

N.j 1 j1 Ak+1 1 10 ,0,0 • ..• Nji+l jl I

: j:

~r 1Al(jl

.. ..

~

1

t

U j1 j1

0,0 , 1

~.o,o .. ..

I

"k

!! :

II

1+1:

I

JI 1

NH+l j l "'k

0,0,0

I , 0

11

L

Principle At the lower level for the intersection i the general criterion can be written under the form : J - J 1.. + J 1..

jl '

-_ aji

)1

1

where i' is the upstream intelrection of link j i j'i' turning % from j'i' to j i Pj i ji N respectively the integer and fractional ji part of the free travel time between R i' and i in parts of 5s. and Min[xti ' + I Ati',

sti'

+ dO

yr

r ]]

V

3. DECOMPOSITION COORDINATION TIle just present here one method among these already tested.

1

l~ Q

I)

'ji ~k

opt1.m1.zation algorithm for the subsyste~and a decomposition_coordination scheme.

(i-U~)[dvj'i' (l-V~

Ji concerns i itself, Ji is the contribution of the outputs of i to the rest of the network and will be approximated at the first order with respect to the outputs of i (inputs of the downstream intersection), The coefficients of this approximation (A~) are given by the upper level. The optimization of J follows an iterative process that can be described below with the help of figure I Supervisor

if j' - I or 3

k

1

upper level

~-.r--------------,-,--

1.Ji ;d r "'k' k

.,

V1. ]] if j'

2 or 4

k

Criterion The chosen criterion is the total delay approximated by :

N~AR[ ~L ,L~ L

J

i-I

NCAR K

F.1.

xji + F (Xli x 2i X3i x4i

k-I J- I

i

k

K

'-l(

,

i)

K '--K ,UK

_I 1/

2/

(5)

number of intersections tested and chosen here equal to 15 terminal criterion which is an approximation of the futur~ see Robertson

k

W~

0,

k

.::. VERMAX

Vi boolean k

i

(Wi - VERMINi)V i > 0 k

ji Xkji _< XMAX

k-

. ' queue constra1.nt

lower level

Fig. I i Definition of an initial control (Vk ) 1' 1-0 Simulation by the supervisor of the network with (V 1 computation a~1 dispa~~ to the local controners of A~1. and A~

k)

Optimization at the lower level (intersection i) of J, determination of (v~)) dispatch to the upper level

4/

Research by the supervisor of a new con tr?l (V~)lTI.as the best combination of 1. ..... 1. (V ) 1 and (Vk ) k

(6)

Contraints on the green time (7)

I_I

3/

(1974)

CoIEtraints vi(i _ Vi)

local contro:n.er

5/

~f (V~)~+I 1.S

(8 )

- (vt\

then the solution

(V~) 1

else co 1=1+1

and go to 2.

(9 )

Determination of the sensitivity factors Optimization problem We now have to optimize the criterion with respect to the control variables taking into account the state equations and constraints for a given initial state. The resolution of that problem gives an optimal control sequence for K intervals of 5s in open loop. If we repeat the process at each k using the measured state at k and the anticipated state at k+1 we get a closed loop control i. e. a real time .control. The main problem now lies in the resolution that is made in the example given below (4 intersection~for 35 states. That will be ~he purpose of PRODYN which uses a proper

~ e re

we define and compute the sensitivity factors A~1.