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Modelling and optimal control of oversaturated transportation networks
Modelling and optimal control of oversaturated transportation networks Hisaaki
Nagase
Department of Control Engineering, Toyonaka, Osaka, 560, Japan (Received October 1979)
Faculty
of Engineering
Science,
Osaka University,
This paper treats an optimal control problem of automobile traffic flows in an oversaturated transportation network with a rectangular grid of intersections. All O-D (origin-destination) traffic flows are divided into portions at each origin and one route for travelling to the destination is assigned to the each portion. The traffic flows are controlled by the traffic signals placed at each intersection. Examples of numerical solutions, which are obtained by applying the linear-programming technique, are presented. The effect of route assignment for decreasing the traffic congestion is shown by the examples.
Introduction Nowadays, the roads in and around many cities are often congested by the heavy demands of automobile traffic, especially in the rush hours. These daily traffic demands often cause vehicular queues at some intersections. This daily congestion seems to be inevitable, since the extension of roads tends to induce additional demand, and since a forcible reduction in the current demand is also difficult. Therefore, control schemes are required that are applicable to oversaturated road networks. In this paper we consider a transportation network which becomes oversaturated during a period of heavy traffic demand. If the incoming vehicular flows into the network exceed some saturation limit, the vehicles may form queues. The queues may be formed along the roads incident to the interesections in which the combined arrival rates in two competing directions exceed the combined throughputs. In this case, it is important to minimize the total waiting time of all the vehicles in the queues over the entire period of oversaturation. The simplest case of oversaturated traffic systems is that of a single oversaturated intersection. This problem has been solved by Gazis’ using a graphical method and Pontryagin’s maximum principle. The optimal control of the traffic signal placed at the intersection was found to correspond to a succession from maximum to minimum service in one direction, and from minimum to maximum service for the other. In the same paper, the case of two consecutive oversaturated intersections was also treated. A natural extension of this control problem is the problem of an oversaturated network. In the case of a network, each vehicle is assumed to enter and leave the network at specified points in the network, 0307-904X/80/020101-08/$02.00 0 1980 IPC Business Press Ltd
namely origin and destination, 0 and D, respectively. Usually, several routes are available for the vehicle. One particular route among them is selected by the vehicle to minimize the cost. D’Ans and Gazis2 have studied oversaturated networks. In their study, however, the assignment of a route to each vehicle was left out of account. In this paper, it is assumed that each O-D traffic flow is divided into sections at the origin and one of the available routes is assigned to each section. We shall treat an oversaturated network with a rectangular grid of, no-turn, traffic intersections. Two competing O-D traffic flows, whose directions cross, are served by the network. The time-varying demands of the two O-D traffic flows to the network are assumed to be known during a sufficiently long interval including the rush period. For a network, the problem is formulated as an optimal control problem with multiple constraints and multiple transportation delays. In this paper, the problem is formulated as a discrete-time system. An optimal solution of this problem can be obtained by applying linear programming.374 Some numerical examples of the optimal solution are given for the case where each O-D traffic flow has two routes.
Modelling
of transportation
networks
Consider a transportation network serving two O-D traffic flows, OD1 and 002. Let the origin of ODl be 01, the destination of ODl be D,, the origin of 002 be O,, and the destination of 002 be D,. The numbers of routes used by OD1 and 002 are m and n, respectively. The routes taken by ODl will be called route 1, route 2, . , and route m, while the routes taken by 002 will be called route m + 1,
Appt. Math. Modelling, 1980, Vol. 4, April
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Control of oversaturated transportation networks:
H. Nagase
route m + 2, , and route m t n, as illustrated in Figure 1. Throughout this paper it is assumed that the symbol i covers 1 tom,whilejcoversm+l tom+n. We shall define the O-D traffic flows Q1@ .A t) and Q*(p.At), where the symbol p is an integer between 0 and N ~ 1 and the symbol At represents a constant time interval. They represent the numbers of vehicles per unit time generated at the origins O1 and 02, respectively. It is assumed that they are numbered at each origin at the interval of At. The number of vehicles will be treated as if it were continuous though it is actually discrete. Since a route is assigned to each vehicle at its origin:
Ql(@t)
= f YAkAt)
p=l,...,N-l
(1)
i=l Wl+?I
Qz(pAt)
=
c
p=l,...
q;@At)
,N-1
(2)
1
j=m+l
where [0, N. At] is the period of oversaturation and qk@At), k = 1, . . . , m + ~1,are the numbers of vehicles to which the route k is assigned. It is reasonable to assume that q&At) has the upper and lower bounds: qf’“@At)
7 Schematic
diagram of a network
gij@At), respectively. Equation (4) and the inequality imply that h,(pAt) is constrained by: 1 - lij - bij < h,@At)
< qpax@At)
Figure
k=l,...,mtn
(3)
The intersection of the route i (i = 1, . . . , m) and j (j = m + n) is denoted by (i, j). A section of a route, m+l,..., connecting two intersections in Figure I, will be termed a link. To traverse a link a vehicle spends some time, namely transportation delay. It is assumed that the delay time is a multiple of At at each link. The delay time for each link is as follows: t;i, m+1 . At: from 0, to (i, m + 1)
At:
Xii{@ + l)At}
Xii(o) + 1)At)
from02
from(i-
l,j)to(i,j),i=2,
t lij = 1
_Yij{(p t 1)AtI
(4)
G bit
(5)
where aij and bij are the lower and upper bounds for
Appl. Math. Modelling,
1980,
Vol. 4, April
(7a)
+ At{r;,j ~ lgi,j_l
- siihii(pA t)}
. ,m
where Zij is the rate of total lost time in one cycle used by acceleration and clearing and is assumed to be constant. Since the green light period cannot be too small nor too large, gii(pAt) is constrained by:
102
tl
- rijgij@At)J (7b)
y,{(ptl)At}=y,(pAt>tAt{qj((P_77ij)At)
where tij and vij are integers. Note that the symbol $ is used for the links utilized by ODl, while the symbol q by 002. Next, consider the traffic lights at the intersection (i, j). One cycle of the traffic lights is divided into three portions, that is, green periods in both directions and lost time. Let gijCPAt) denote the rate of effective green light period in one cycle time provided for ODl. The rate provided for 002 will be denoted by hi;(pAt). These variables have the following relationship:
0
=xij@At)
to (1,j)
j=mtl,...,mtn
gijCpAt) + hij0?At)
j=m
j=mt2,...,mtn j=mtl,...,mtn
77ij’ At:
+ At{qi(@ - gij)At)
((P - tij)At)
j=mt2,...,mtn ul,j . At:
=xij@At)
- rijgi;@A t)I
from (i, j ~ 1) to (i, j), i = 1, . . , m
(6)
For each intersection (i, j) two queues of vehicles may grow. Let x&At) be the number of vehicles belonging to the queue along the route i which connects O1 with D1. And let vij(pAt) be the size of the queue along the route j which connects O2 with D2.The maximum throughput rate for ODl passing (i, j) will be denoted by rij, while that for 002 by Sij. The state variables x,@At) andyii@At) satisfy the following difference equations:
i=l,...,m gij
< 1 - lij - aij
(5)
=Yi#At)
- At{si-I,$-l,j((P - sijhij@A t)}
i=l
(7c)
+ - Vij)At) i=2,...,m
(74
The first terms on the right-hand-sides of equations (7a)(7d) represent the arriving flows, while the second terms represent the flows served. Note that equations (7a-d) are based on the assumption that each intersection is oversaturated or just saturated. An intersection is said to be oversaturated when the queue is nonempty, i.e., Xij(pAt) > 0. An intersection is termed just saturated when the queue becomes empty only at the end moment of the green light period; in other words, all incoming vehicles to the intersection are served by consuming the effective green light period gij@At). The state variable in this case is zero. An intersection is termed undersaturated when the queue becomes empty before the end moment of the green light period. The state variable in this case becomes negative. But, in fact the flow rate becomes smaller than the maximum throughput rate rii, since the queue length cannot
Control of oversaturated
be negative. Consequently, equations (7a-d) do not represent the undersaturated intersections. Therefore, in our problem which will be formulated below, the state variables are constrained to be nonnegative. In conclusion, equations (7a-d) are valid as far as each intersection is oversaturated or just saturated. It should also be noted that if the link before a certain queue is completely jammed by the preceding vehicles, the queue cannot be served in the green light period. In this case the second term of the corresponding state equation does not represent the flow served. Such a case, however, can be avoided by constraints: ,Vij@A t) < Nii
Xij(pA t) < Mij
transportation
Xij(0)= Xt
0
>
Yij(O) =
q;@At) p’=
+,nl+i>.
q@At)
pnr,j>.
p’=
gij(bAt)
p’=
In this section we formulate the problem. Next, some additional constraints will be stated. Since we treat the oversaturated network, a reasonable choice of an objective function for control would be the one which minimize the total waiting time of all vehicles in the queues. This goal is equivalent to the minimization of the sum of queue sizes over the entire period of oversaturation. Now, our problem is stated as follows: : mtn
j-m+1
x{
xii (PAt) + WiiYij @At) 1
(9)
subject to the system equations: Xij{(p t l)At}
= X,@At)
+ At {qi((p - tii)At) j=m
~ rijgijCPAt)l
i-1
(lOa)
Xii ((p ~ 1)At) =X,(pAt) +At{ri,;-
rgi,j~-r(@ ~ 4ij)At))
~ rijgij(PA t)l l’=mt2,...,mtn Yij{(P + l)At>
(lob)
=Yij(PAt) + At {qj((p ~ n,)At) - sijhij(pA t)}
i=l
UOC)
Yij I@ t I)AtIl =Yij(pAt)} t Ar{si-I,jhi-l,j((P
~ Qij)At)l
-sijhij@At)},
i=2,...,m(lOd)
with constraints: 0 G Xij(pA t) < Mij
Q1@At>= f
0 G_Yi&JAt) G Nij
q&At)
(11) (12a)
i=l lTl+Yl
Qz(PAt)
=
C qj@At)
(12b)
j=m+l qp’“(pAt)
G qkm”“(pAt) k=l
gij(pAt)
+ hi#At)
0 < aij Ggi@At)
+ Zij = 1 G bij
. >b,.
, -1
(17a)
1 -1
(17b)
,b,. . . ,-l.j=m
_,
t2,.
hij($At)p’=
pQi,j,
, --l.i=
(18a) 2,.
,m (18b)
(8)
formulation
m
(16)
j=mt2....,m+n
Problem
1 c p=O i-l
. . ,r;,
--[i,j,.
and given O-D flow data:
N-1
YG> 0
initial flow data:
final conditions:
J=At.
H. Nagase
initial queue lengths:
where ~ij and Nij are the capacities of the links along which Xij@At) andy&At) may grow, respectively.
minimize
networks:
,...,mtn
(13)
Xij(NA t) = 0
(15)
k=1,2
!&(PAt)
(19)
(20)
wherep=O ,...,, /\r-l,i=l,..., m,andj=m+l...., mtn. This problem is an optimal control problem with multiple constraints and multiple transportation delays. The positive constants Vii and Wij in equation (9) are the weights for Xii and yij, respectively. We can choose their values taking various conditions on the network into account. Their values represent the weights of corresponding queues. If a queue Xii is particularly undesirable, for example, we can give a large value to Vii. If the weights of all queues are the same, they are Uij = Wij = 1 for all i and i. On account of the transportation delays &, . At and nij ’ At included in the equations (1 Oa- d), the data (17a18b) are required to specify the initial flow pattern in the network. That is, the symbols qi@A t), qj@A t), gij(pA t), hijO?A t), P < - 1, in equations (1 Oaad) represent given data, while they are unknown control variables when p > 0. Since the state variables cannot represent undersaturated intersections, they are restricted to be nonnegative by the constraints (1 1). The initial queue lengths (16) are assumed to be given, while the final time N. At and the final queue lengths, that will be zero in most cases as shown in equations (19), should be specified. The other equations and inequalities have been stated previously. It has been assumed that the O-D flow demands are measured during a sufficiently long interval including the rush period. By using these data, first, the initial time 0 is fixed. In the beginning of the rush period every intersection becomes oversaturated one after another. As the initial time 0, we should take the moment when the last intersection just becomes saturated. We cannot take an ealier moment for the constraints (11). Next, the final time N. At, or equivalently the integer N, is fixed. According to the experiential knowledge, one of the moderate values of N should be chosen. Next, we shall consider some further constraints that might be posed in treating real networks. Two types of such constraints will be briefly described in the following: (i) Upper bound of the total waiting time of vehicles which utilize a route i: N-l
(14)
_Vij(NAt) = 0
AtI
m+n
X,@A
Ix p=O
j=m
t)
<
Mi
(21)
+I
Appl. Math. Modelling, 1980, Vol. 4, April
103
Control
of oversaturated
transportation
networks:
H. Nagase
(ii) Upper bound of the waiting time of an individual vehicle at an intersection (i, j), at which the vehicle arrives at the time @ -$)At, where $ is a positive integer smaller than p:
xi&PAt)
G At.
I T=F_a gi, ;- 1((7 ~ &;)At>
Yi, ;-
max Ql
(22)
-,
Ql
--
O-
This constraint can be interpreted as follows. Considering the vehicles on the route i; especially, consider the vehicles which arrive at the intersection (i, j) during the interval [@ ~ i)At, pat]. Since the transportation delay of the preceding link is ,$ii. At, those vehicles have passed the intersections (i, j - 1) during the interval [(p - fi -&), (p gii>At]. If the intersection (i, j) were cumulative, i.e., yii = 0, those vehicles form a queue at (i, j). Its length at time pAt can be represented by the right-hand-side of the constraint (22). Therefore, the constraint (22) means that the first one of those vehicles have passed or is just passing the intersection (i, j) at the time pat. Our problem can be regarded as a problem of linear. programming. The equations and inequalities (IOa-d), (1 l), (12a and b), (13), (14), and (15) are regarded as the constraints of the LP problem. Both the state variables xij(pAt), P = 1, . . . , N - 1 and the control variablesgi@At), ,..., N-l,k=l,..., mtnare regarded as the unknown variables of the LP problem. The objective function of our problem (9) are regarded as that of the LP problem. That is, the weighting coefficients in the objective function of the LP problem are zero for gi@At), kii(pAt), and qk@At), Vii for xii(pAt), and Wii foryi@At). The LP problem has N(3mn + 2) equality constraints, N(4mn t m + n) inequality constraints, and N(4mn + m + n) unknown variables excluding the constraints (2 1) and (22). The further constraints (21) and (22) only increase the number of inequality constraints without introducing any conceptual difference in the formulation of the LP problem.
92
Table 1 List of parameter values in
examples l-3*
rlj
r2i
Si3
Si4
Vlj
V2j
wi3
Wi4 aii
bij
lij
2.0
1.1
1.3
2.1
2.0
1.0
1.0
2.0
0.6
0.1
*i
104
0.3
513
El4
523
E24
r)13
1114
‘723
r124
0
8
8
8
0
8
8
4
min 9k
max 41
max 42
max 43
max 44
Af
0
0.7rqj
O.Jr2j
O.Jsi3
O.JSi4
4
3,4
k=1,....4
= 1.2
Appl.
j=
Math.
Modelling,
1980,
Vol.
4, April
1’
s.c
20 1 y-P
:
Ll
-a”
7 ‘0
mar
___A
--
Q3
20’
’
73
-----
I
kii(pAt),qk@At),p=O
As an example of the transportation networks, we shall consider the case where each O-D traffic has two routes, i.e., m = II = 2. In this case there are eight state variables and six control variables. Three examples of the numerical solutions for this network will be given. The parameter values taken in these examples are given in Table I. The O-D flow data are given in Figure 2. The initial time of the oversaturated period and the initial flow pattern in the network are determined from the O-D flow
y2
f
yij@At),
Numerical examples
mar
_-----__” 10
0’
’
)
I
I
1
20
1
--_
L
I
’
?(
Time, imId Figure 2 0-D flow data and optimal in example 1
sequences of route assignment
data. As a first example we shall give an optimal solution of the problem, assuming that the further constraints (21) and (22) are absent and that the link capacitiesMii and Nij are sufficiently large. In the second example we assume that (i), the queue size yr4(pAt) is limited for the small capacity of the link or some other reasons and that (ii) the time lost by the vehicles in the queue x&pat) is also limited for some reasons. These constraints are expressed as follows: (i)
~t&At)
(ii)
At
< 250 (veh.)
N-l 1 x&@At)
p=l,...,N-1
< 100 (veh. .h)
(23) (24)
p=1
In our third example we consider the case where dynamic control is not applied to the traffic flows, that is, each timevarying O-D traffic flow data is divided into two portions with a suitable constant ratio and each route is assigned to each portion and furthermore the control variables of all traffic signals are fixed and unchanged. In these examples, the initial queue lengths (16) are adjusted to be zero for ease of computation. The initial flow pattern (17a-lSb), that cause the initial queue lengths to be zero, has been found by repeating the following procedure. First, the controls about the initial time of the rush period are arbitrarily specified. Where they satisfy the constraints (12aa15). Second, by using these data, the flows are simulated. Since every intersection in the network does not become oversaturated simultaneously, in general, this procedure provides us with an example of the initial conditions (I 6-l 8b). By varying the values of the controls, the controls by which all intersections in the network become saturated simultaneously have been found. In examples 1 and 2, optimal solutions were obtained by using a simplex procedure4 on the ACOS77NEAC/700 at
Control Table 2 Controls
in example
of oversaturated
transportation
networks:
H. Nagase
the control variables as well as the state variables. From to the first intersections for the flows. Incident to the second intersections for the flows, queues hardly develop. The queue size sizeYr4(PAt) in Figure 7 clearly shows the effect of constraint (23). Comparing the queue sizes xas@At) in Figures 4 and 7, we see that the constraint (24) decreases the time lost by the vehicles in the queue xas($At).
3
Figure 4, we see that the queues develop incident 41142
43144
913
914
923
924
0.71
0.5
0.44
0.45
0.49
0.47
Table 3 Values of objective Example J (veh..
function
in examples
1 h)
820.4
2
3
904.5
1045.0
-___
o6-------
l-3
Discussion
----03 I
2
06
03,
I o-
’
I
I
I
<’
09
c
In this section we discuss some aspects of the problem. First, let us consider the formulation of our problem, especially the system-optimization assumption. In automobile traffic systems, presence or absence of a trip as well as its route are determined individually by the driver, not collectively by the controller of the system. Such systems are usually assumed to operate on a user-optimization assumption. For example, it is assumed that the travel times of a vehicle on all available routes from its origin to its destination are equalized when each driver takes the route on which their travel time is not longer than on any other route. But we have assumed a system-optimization in this
_---------_-0.6
Time, Figure 3 Optimal
20 hn)
sequences of signal setting in example
36
%
log------
400 1
Osaka University. Computation time was 227 s for example 1 and 252 s for example 2. In example 3, we successively integrated the system equations and the computation time was less than 1 s. The results of example 1 are illustrated in Figures 2-4, those of example 2 in Figures 5- 7, and those of example 3 in Figure 8 and given in Table 2. The values which the objective function took in these examples are given in Table 3. Examining the results, a few remarks are in order, It seems that there are a few switching times in the optimal sequences of route assignment. It is not clear whether there are switching times in the optimal sequences of signal setting. The optimal controls in Figures 2 and 3 are not bang-bang.* It seems that they cannot take the maximum or the minimum on account of the various constraints on * The phrase ‘bang-bang control’ means that each control variable takes the maximum or the minimum in its admissible reasion with a finite number of switching time from the maximum to the minimum or reversely. The optimal control in case of m = M = 1 has been proved’ to be bang-bang.
3002
200. IOO0,; 500
-
400 ; k
300200
-
IOO-
Figure 4 Optimal
Appl.
state trajectories
in example
Math. Modelling,
1980,
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Vol. 4, April
105
Control of oversaturated
transportation
networks:
H. Nagase
--------qma~ O-
I
Ql
o--
2
-_
0
J
92 max Q2
Om
J
I-
400 1
_Jr-n-
20
1
I
-1
Time, imln)
Figure 5 Optimal
sequences of route assignment
in example
2
I
I 20
I
I
I
I
8 40
1
Time, (mln)
Figure 7 Optimal state trajectories
in example
2
0.3
06.
v
-z G
c 06
03----
0.
L
I
1
I
I
I
,09
I
paper; minimize the total time lost by the vehicles in queues by the route assignment and the traffic signals. Therefore, it will necessary to consider (i), the route assignment and (ii), the system-optimization assumption. Route assignment
0
I
I
Figure 6 Optimal
/
I
I
I
1
sequences of signal setting in example 2
106 Appl. Math. Modelling,
1980,
I 3609
20 Tlme,(mln)
Vol. 4, April
Nowadays, the recommendation of a better route by supplying information on congestion, construction, the weather, etc., operates at branching points of highways and urban roads. Each driver determines his route taking this information into account. The optimal sequences of route assignment obtained by solving our problem will be realized by supplying information to drivers which recommends a particular route. Since the assignment is not enforcible, the maximum and minimum possible values of the assignment, 9?‘“@At) and 9p’“(pAt), should be set suitable, using experiential knowledge. In our examples, these are assumed to be constants, i.e., &“@At) = 0 and 9Tax(9At) = 0.7sk, ,4, independent of the other conditions. k=l,...
Control of oversaturated transportation networks:
Tlme,(mmi Figure 8 State trajectories
System-optimization
in example
3
assumption
Let us consider the control policies which are desirable to treat the congested networks. Especially, when the congestion is inevitable, some policies will be required in addition to those for noncongested networks. When the network is oversaturated, the travel time of a vehicle is considerably affected by the queue lengths on the selected route. In such a case, a driver usually cannot know the best route for him, since the sufficient information about the current state of the network is not supplied to him. Therefore, the flow pattern in the oversaturated networks will be less useroptimized than that in the undersaturated networks. Taking this circumstances into account, the control schemes for congested networks should more or less satisfy the following conditions. (a) The oversaturated intersections disappear as soon as possible. (b) The total amount of congestion is minimized. (c) The congestion is not concentrated about some particular intersections. (d) Undersaturated intersections do not appear during the period of oversaturation, since they decrease the capacity of the network. (e) Travel times of all available routes for a vehicle are equalized. Of course, no control can satisfy all these conditions, since they are interdependent. Both conditions (a) and (b) represent system-optimization assumptions, while condition (e) represents a user-optimization assumption. Both conditions (c) and (d) represent the situations of each intersection and they are usuaIIy helpful in satisfying condition (e). In our approach these conditions are treated in the following. As shown in the previous sections, first the initial time and the initial flow pattern are decided from the O-D flow data, then the final time T = N. At is specified, and finally
H. Nagase
the objective function is minimized by a LP procedure. Solving the problem repeatedly with various values of N, we obtain an interval of N, Nr
Appl.
Math.
Modelling,
1980,
Vol. 4, April
IO7
Control
of oversaturated
transportation
networks:
H. Nagase
and (ii), the computation time required to execute a renewal procedure of the tableau. (i), is roughly proportional to the number of the unknown variables. (ii), is proportional to the column number of the tableau and also to the row number of the tableau. Multiplying these factors and neglecting lower-order terms, we can conclude that the corn utation P time of our problem is roughly proportional to N3m n3.
Conclusions The minimization problem of the traffic congestion in an oversaturated transportation network has been formulated as a dynamic optimization problem. The network treated here is that with a rectangular grid of intersections. The flow demand of each O-D traffic has been assumed to be given as a time-varying function. Traffic flows have been regulated simultaneously both by the dynamic route assignment and by the traffic light control. A method of linear programming is directly applicable to our problem and some numerical examples have been given in the case where each O-D traffic has two routes; consequently the network has
108
Appl.
Math.
Modelling,
1980,
Vol. 4, April
four intersections. These examples show that our approach is an effective method for controlling oversaturated networks.
Acknowledgements The author would like to express his thanks to Professor Y. Sakawa of the Faculty of Engineering Science, Osaka University for his continual guidance and discussion throughout this work. He is also indebted to Dr H. Kimura, Y. Inouye, and N. Fujii for their valuable discussions and encouragement.
References 1 2 3 4
Gazis, D.C. Oper. Rex 1964, 12,815 D’Ans, G. C. and Gazis, D. C. Tramp. Sci. 1976,10,1 Sakawa, Y. and Hayashi, C. Proc. IFAC To&o Symp. 1965, pp. 53-60 Kuester, J. L. and Mize, J. H. ‘Optimization techniques with fortran,’ McGraw-Hill, London, 1973
Nagase
Department of Control Engineering, Toyonaka, Osaka, 560, Japan (Received October 1979)
Faculty
of Engineering
Science,
Osaka University,
This paper treats an optimal control problem of automobile traffic flows in an oversaturated transportation network with a rectangular grid of intersections. All O-D (origin-destination) traffic flows are divided into portions at each origin and one route for travelling to the destination is assigned to the each portion. The traffic flows are controlled by the traffic signals placed at each intersection. Examples of numerical solutions, which are obtained by applying the linear-programming technique, are presented. The effect of route assignment for decreasing the traffic congestion is shown by the examples.
Introduction Nowadays, the roads in and around many cities are often congested by the heavy demands of automobile traffic, especially in the rush hours. These daily traffic demands often cause vehicular queues at some intersections. This daily congestion seems to be inevitable, since the extension of roads tends to induce additional demand, and since a forcible reduction in the current demand is also difficult. Therefore, control schemes are required that are applicable to oversaturated road networks. In this paper we consider a transportation network which becomes oversaturated during a period of heavy traffic demand. If the incoming vehicular flows into the network exceed some saturation limit, the vehicles may form queues. The queues may be formed along the roads incident to the interesections in which the combined arrival rates in two competing directions exceed the combined throughputs. In this case, it is important to minimize the total waiting time of all the vehicles in the queues over the entire period of oversaturation. The simplest case of oversaturated traffic systems is that of a single oversaturated intersection. This problem has been solved by Gazis’ using a graphical method and Pontryagin’s maximum principle. The optimal control of the traffic signal placed at the intersection was found to correspond to a succession from maximum to minimum service in one direction, and from minimum to maximum service for the other. In the same paper, the case of two consecutive oversaturated intersections was also treated. A natural extension of this control problem is the problem of an oversaturated network. In the case of a network, each vehicle is assumed to enter and leave the network at specified points in the network, 0307-904X/80/020101-08/$02.00 0 1980 IPC Business Press Ltd
namely origin and destination, 0 and D, respectively. Usually, several routes are available for the vehicle. One particular route among them is selected by the vehicle to minimize the cost. D’Ans and Gazis2 have studied oversaturated networks. In their study, however, the assignment of a route to each vehicle was left out of account. In this paper, it is assumed that each O-D traffic flow is divided into sections at the origin and one of the available routes is assigned to each section. We shall treat an oversaturated network with a rectangular grid of, no-turn, traffic intersections. Two competing O-D traffic flows, whose directions cross, are served by the network. The time-varying demands of the two O-D traffic flows to the network are assumed to be known during a sufficiently long interval including the rush period. For a network, the problem is formulated as an optimal control problem with multiple constraints and multiple transportation delays. In this paper, the problem is formulated as a discrete-time system. An optimal solution of this problem can be obtained by applying linear programming.374 Some numerical examples of the optimal solution are given for the case where each O-D traffic flow has two routes.
Modelling
of transportation
networks
Consider a transportation network serving two O-D traffic flows, OD1 and 002. Let the origin of ODl be 01, the destination of ODl be D,, the origin of 002 be O,, and the destination of 002 be D,. The numbers of routes used by OD1 and 002 are m and n, respectively. The routes taken by ODl will be called route 1, route 2, . , and route m, while the routes taken by 002 will be called route m + 1,
Appt. Math. Modelling, 1980, Vol. 4, April
101
Control of oversaturated transportation networks:
H. Nagase
route m + 2, , and route m t n, as illustrated in Figure 1. Throughout this paper it is assumed that the symbol i covers 1 tom,whilejcoversm+l tom+n. We shall define the O-D traffic flows Q1@ .A t) and Q*(p.At), where the symbol p is an integer between 0 and N ~ 1 and the symbol At represents a constant time interval. They represent the numbers of vehicles per unit time generated at the origins O1 and 02, respectively. It is assumed that they are numbered at each origin at the interval of At. The number of vehicles will be treated as if it were continuous though it is actually discrete. Since a route is assigned to each vehicle at its origin:
Ql(@t)
= f YAkAt)
p=l,...,N-l
(1)
i=l Wl+?I
Qz(pAt)
=
c
p=l,...
q;@At)
,N-1
(2)
1
j=m+l
where [0, N. At] is the period of oversaturation and qk@At), k = 1, . . . , m + ~1,are the numbers of vehicles to which the route k is assigned. It is reasonable to assume that q&At) has the upper and lower bounds: qf’“@At)
7 Schematic
diagram of a network
gij@At), respectively. Equation (4) and the inequality imply that h,(pAt) is constrained by: 1 - lij - bij < h,@At)
< qpax@At)
Figure
k=l,...,mtn
(3)
The intersection of the route i (i = 1, . . . , m) and j (j = m + n) is denoted by (i, j). A section of a route, m+l,..., connecting two intersections in Figure I, will be termed a link. To traverse a link a vehicle spends some time, namely transportation delay. It is assumed that the delay time is a multiple of At at each link. The delay time for each link is as follows: t;i, m+1 . At: from 0, to (i, m + 1)
At:
Xii{@ + l)At}
Xii(o) + 1)At)
from02
from(i-
l,j)to(i,j),i=2,
t lij = 1
_Yij{(p t 1)AtI
(4)
G bit
(5)
where aij and bij are the lower and upper bounds for
Appl. Math. Modelling,
1980,
Vol. 4, April
(7a)
+ At{r;,j ~ lgi,j_l
- siihii(pA t)}
. ,m
where Zij is the rate of total lost time in one cycle used by acceleration and clearing and is assumed to be constant. Since the green light period cannot be too small nor too large, gii(pAt) is constrained by:
102
tl
- rijgij@At)J (7b)
y,{(ptl)At}=y,(pAt>tAt{qj((P_77ij)At)
where tij and vij are integers. Note that the symbol $ is used for the links utilized by ODl, while the symbol q by 002. Next, consider the traffic lights at the intersection (i, j). One cycle of the traffic lights is divided into three portions, that is, green periods in both directions and lost time. Let gijCPAt) denote the rate of effective green light period in one cycle time provided for ODl. The rate provided for 002 will be denoted by hi;(pAt). These variables have the following relationship:
0
=xij@At)
to (1,j)
j=mtl,...,mtn
gijCpAt) + hij0?At)
j=m
j=mt2,...,mtn j=mtl,...,mtn
77ij’ At:
+ At{qi(@ - gij)At)
((P - tij)At)
j=mt2,...,mtn ul,j . At:
=xij@At)
- rijgi;@A t)I
from (i, j ~ 1) to (i, j), i = 1, . . , m
(6)
For each intersection (i, j) two queues of vehicles may grow. Let x&At) be the number of vehicles belonging to the queue along the route i which connects O1 with D1. And let vij(pAt) be the size of the queue along the route j which connects O2 with D2.The maximum throughput rate for ODl passing (i, j) will be denoted by rij, while that for 002 by Sij. The state variables x,@At) andyii@At) satisfy the following difference equations:
i=l,...,m gij
< 1 - lij - aij
(5)
=Yi#At)
- At{si-I,$-l,j((P - sijhij@A t)}
i=l
(7c)
+ - Vij)At) i=2,...,m
(74
The first terms on the right-hand-sides of equations (7a)(7d) represent the arriving flows, while the second terms represent the flows served. Note that equations (7a-d) are based on the assumption that each intersection is oversaturated or just saturated. An intersection is said to be oversaturated when the queue is nonempty, i.e., Xij(pAt) > 0. An intersection is termed just saturated when the queue becomes empty only at the end moment of the green light period; in other words, all incoming vehicles to the intersection are served by consuming the effective green light period gij@At). The state variable in this case is zero. An intersection is termed undersaturated when the queue becomes empty before the end moment of the green light period. The state variable in this case becomes negative. But, in fact the flow rate becomes smaller than the maximum throughput rate rii, since the queue length cannot
Control of oversaturated
be negative. Consequently, equations (7a-d) do not represent the undersaturated intersections. Therefore, in our problem which will be formulated below, the state variables are constrained to be nonnegative. In conclusion, equations (7a-d) are valid as far as each intersection is oversaturated or just saturated. It should also be noted that if the link before a certain queue is completely jammed by the preceding vehicles, the queue cannot be served in the green light period. In this case the second term of the corresponding state equation does not represent the flow served. Such a case, however, can be avoided by constraints: ,Vij@A t) < Nii
Xij(pA t) < Mij
transportation
Xij(0)= Xt
0
>
Yij(O) =
q;@At) p’=
+,nl+i>.
q@At)
pnr,j>.
p’=
gij(bAt)
p’=
In this section we formulate the problem. Next, some additional constraints will be stated. Since we treat the oversaturated network, a reasonable choice of an objective function for control would be the one which minimize the total waiting time of all vehicles in the queues. This goal is equivalent to the minimization of the sum of queue sizes over the entire period of oversaturation. Now, our problem is stated as follows: : mtn
j-m+1
x{
xii (PAt) + WiiYij @At) 1
(9)
subject to the system equations: Xij{(p t l)At}
= X,@At)
+ At {qi((p - tii)At) j=m
~ rijgijCPAt)l
i-1
(lOa)
Xii ((p ~ 1)At) =X,(pAt) +At{ri,;-
rgi,j~-r(@ ~ 4ij)At))
~ rijgij(PA t)l l’=mt2,...,mtn Yij{(P + l)At>
(lob)
=Yij(PAt) + At {qj((p ~ n,)At) - sijhij(pA t)}
i=l
UOC)
Yij I@ t I)AtIl =Yij(pAt)} t Ar{si-I,jhi-l,j((P
~ Qij)At)l
-sijhij@At)},
i=2,...,m(lOd)
with constraints: 0 G Xij(pA t) < Mij
Q1@At>= f
0 G_Yi&JAt) G Nij
q&At)
(11) (12a)
i=l lTl+Yl
Qz(PAt)
=
C qj@At)
(12b)
j=m+l qp’“(pAt)
G qkm”“(pAt) k=l
gij(pAt)
+ hi#At)
0 < aij Ggi@At)
+ Zij = 1 G bij
. >b,.
, -1
(17a)
1 -1
(17b)
,b,. . . ,-l.j=m
_,
t2,.
hij($At)p’=
pQi,j,
, --l.i=
(18a) 2,.
,m (18b)
(8)
formulation
m
(16)
j=mt2....,m+n
Problem
1 c p=O i-l
. . ,r;,
--[i,j,.
and given O-D flow data:
N-1
YG> 0
initial flow data:
final conditions:
J=At.
H. Nagase
initial queue lengths:
where ~ij and Nij are the capacities of the links along which Xij@At) andy&At) may grow, respectively.
minimize
networks:
,...,mtn
(13)
Xij(NA t) = 0
(15)
k=1,2
!&(PAt)
(19)
(20)
wherep=O ,...,, /\r-l,i=l,..., m,andj=m+l...., mtn. This problem is an optimal control problem with multiple constraints and multiple transportation delays. The positive constants Vii and Wij in equation (9) are the weights for Xii and yij, respectively. We can choose their values taking various conditions on the network into account. Their values represent the weights of corresponding queues. If a queue Xii is particularly undesirable, for example, we can give a large value to Vii. If the weights of all queues are the same, they are Uij = Wij = 1 for all i and i. On account of the transportation delays &, . At and nij ’ At included in the equations (1 Oa- d), the data (17a18b) are required to specify the initial flow pattern in the network. That is, the symbols qi@A t), qj@A t), gij(pA t), hijO?A t), P < - 1, in equations (1 Oaad) represent given data, while they are unknown control variables when p > 0. Since the state variables cannot represent undersaturated intersections, they are restricted to be nonnegative by the constraints (1 1). The initial queue lengths (16) are assumed to be given, while the final time N. At and the final queue lengths, that will be zero in most cases as shown in equations (19), should be specified. The other equations and inequalities have been stated previously. It has been assumed that the O-D flow demands are measured during a sufficiently long interval including the rush period. By using these data, first, the initial time 0 is fixed. In the beginning of the rush period every intersection becomes oversaturated one after another. As the initial time 0, we should take the moment when the last intersection just becomes saturated. We cannot take an ealier moment for the constraints (11). Next, the final time N. At, or equivalently the integer N, is fixed. According to the experiential knowledge, one of the moderate values of N should be chosen. Next, we shall consider some further constraints that might be posed in treating real networks. Two types of such constraints will be briefly described in the following: (i) Upper bound of the total waiting time of vehicles which utilize a route i: N-l
(14)
_Vij(NAt) = 0
AtI
m+n
X,@A
Ix p=O
j=m
t)
<
Mi
(21)
+I
Appl. Math. Modelling, 1980, Vol. 4, April
103
Control
of oversaturated
transportation
networks:
H. Nagase
(ii) Upper bound of the waiting time of an individual vehicle at an intersection (i, j), at which the vehicle arrives at the time @ -$)At, where $ is a positive integer smaller than p:
xi&PAt)
G At.
I T=F_a gi, ;- 1((7 ~ &;)At>
Yi, ;-
max Ql
(22)
-,
Ql
--
O-
This constraint can be interpreted as follows. Considering the vehicles on the route i; especially, consider the vehicles which arrive at the intersection (i, j) during the interval [@ ~ i)At, pat]. Since the transportation delay of the preceding link is ,$ii. At, those vehicles have passed the intersections (i, j - 1) during the interval [(p - fi -&), (p gii>At]. If the intersection (i, j) were cumulative, i.e., yii = 0, those vehicles form a queue at (i, j). Its length at time pAt can be represented by the right-hand-side of the constraint (22). Therefore, the constraint (22) means that the first one of those vehicles have passed or is just passing the intersection (i, j) at the time pat. Our problem can be regarded as a problem of linear. programming. The equations and inequalities (IOa-d), (1 l), (12a and b), (13), (14), and (15) are regarded as the constraints of the LP problem. Both the state variables xij(pAt), P = 1, . . . , N - 1 and the control variablesgi@At), ,..., N-l,k=l,..., mtnare regarded as the unknown variables of the LP problem. The objective function of our problem (9) are regarded as that of the LP problem. That is, the weighting coefficients in the objective function of the LP problem are zero for gi@At), kii(pAt), and qk@At), Vii for xii(pAt), and Wii foryi@At). The LP problem has N(3mn + 2) equality constraints, N(4mn t m + n) inequality constraints, and N(4mn + m + n) unknown variables excluding the constraints (2 1) and (22). The further constraints (21) and (22) only increase the number of inequality constraints without introducing any conceptual difference in the formulation of the LP problem.
92
Table 1 List of parameter values in
examples l-3*
rlj
r2i
Si3
Si4
Vlj
V2j
wi3
Wi4 aii
bij
lij
2.0
1.1
1.3
2.1
2.0
1.0
1.0
2.0
0.6
0.1
*i
104
0.3
513
El4
523
E24
r)13
1114
‘723
r124
0
8
8
8
0
8
8
4
min 9k
max 41
max 42
max 43
max 44
Af
0
0.7rqj
O.Jr2j
O.Jsi3
O.JSi4
4
3,4
k=1,....4
= 1.2
Appl.
j=
Math.
Modelling,
1980,
Vol.
4, April
1’
s.c
20 1 y-P
:
Ll
-a”
7 ‘0
mar
___A
--
Q3
20’
’
73
-----
I
kii(pAt),qk@At),p=O
As an example of the transportation networks, we shall consider the case where each O-D traffic has two routes, i.e., m = II = 2. In this case there are eight state variables and six control variables. Three examples of the numerical solutions for this network will be given. The parameter values taken in these examples are given in Table I. The O-D flow data are given in Figure 2. The initial time of the oversaturated period and the initial flow pattern in the network are determined from the O-D flow
y2
f
yij@At),
Numerical examples
mar
_-----__” 10
0’
’
)
I
I
1
20
1
--_
L
I
’
?(
Time, imId Figure 2 0-D flow data and optimal in example 1
sequences of route assignment
data. As a first example we shall give an optimal solution of the problem, assuming that the further constraints (21) and (22) are absent and that the link capacitiesMii and Nij are sufficiently large. In the second example we assume that (i), the queue size yr4(pAt) is limited for the small capacity of the link or some other reasons and that (ii) the time lost by the vehicles in the queue x&pat) is also limited for some reasons. These constraints are expressed as follows: (i)
~t&At)
(ii)
At
< 250 (veh.)
N-l 1 x&@At)
p=l,...,N-1
< 100 (veh. .h)
(23) (24)
p=1
In our third example we consider the case where dynamic control is not applied to the traffic flows, that is, each timevarying O-D traffic flow data is divided into two portions with a suitable constant ratio and each route is assigned to each portion and furthermore the control variables of all traffic signals are fixed and unchanged. In these examples, the initial queue lengths (16) are adjusted to be zero for ease of computation. The initial flow pattern (17a-lSb), that cause the initial queue lengths to be zero, has been found by repeating the following procedure. First, the controls about the initial time of the rush period are arbitrarily specified. Where they satisfy the constraints (12aa15). Second, by using these data, the flows are simulated. Since every intersection in the network does not become oversaturated simultaneously, in general, this procedure provides us with an example of the initial conditions (I 6-l 8b). By varying the values of the controls, the controls by which all intersections in the network become saturated simultaneously have been found. In examples 1 and 2, optimal solutions were obtained by using a simplex procedure4 on the ACOS77NEAC/700 at
Control Table 2 Controls
in example
of oversaturated
transportation
networks:
H. Nagase
the control variables as well as the state variables. From to the first intersections for the flows. Incident to the second intersections for the flows, queues hardly develop. The queue size sizeYr4(PAt) in Figure 7 clearly shows the effect of constraint (23). Comparing the queue sizes xas@At) in Figures 4 and 7, we see that the constraint (24) decreases the time lost by the vehicles in the queue xas($At).
3
Figure 4, we see that the queues develop incident 41142
43144
913
914
923
924
0.71
0.5
0.44
0.45
0.49
0.47
Table 3 Values of objective Example J (veh..
function
in examples
1 h)
820.4
2
3
904.5
1045.0
-___
o6-------
l-3
Discussion
----03 I
2
06
03,
I o-
’
I
I
I
<’
09
c
In this section we discuss some aspects of the problem. First, let us consider the formulation of our problem, especially the system-optimization assumption. In automobile traffic systems, presence or absence of a trip as well as its route are determined individually by the driver, not collectively by the controller of the system. Such systems are usually assumed to operate on a user-optimization assumption. For example, it is assumed that the travel times of a vehicle on all available routes from its origin to its destination are equalized when each driver takes the route on which their travel time is not longer than on any other route. But we have assumed a system-optimization in this
_---------_-0.6
Time, Figure 3 Optimal
20 hn)
sequences of signal setting in example
36
%
log------
400 1
Osaka University. Computation time was 227 s for example 1 and 252 s for example 2. In example 3, we successively integrated the system equations and the computation time was less than 1 s. The results of example 1 are illustrated in Figures 2-4, those of example 2 in Figures 5- 7, and those of example 3 in Figure 8 and given in Table 2. The values which the objective function took in these examples are given in Table 3. Examining the results, a few remarks are in order, It seems that there are a few switching times in the optimal sequences of route assignment. It is not clear whether there are switching times in the optimal sequences of signal setting. The optimal controls in Figures 2 and 3 are not bang-bang.* It seems that they cannot take the maximum or the minimum on account of the various constraints on * The phrase ‘bang-bang control’ means that each control variable takes the maximum or the minimum in its admissible reasion with a finite number of switching time from the maximum to the minimum or reversely. The optimal control in case of m = M = 1 has been proved’ to be bang-bang.
3002
200. IOO0,; 500
-
400 ; k
300200
-
IOO-
Figure 4 Optimal
Appl.
state trajectories
in example
Math. Modelling,
1980,
1
Vol. 4, April
105
Control of oversaturated
transportation
networks:
H. Nagase
--------qma~ O-
I
Ql
o--
2
-_
0
J
92 max Q2
Om
J
I-
400 1
_Jr-n-
20
1
I
-1
Time, imln)
Figure 5 Optimal
sequences of route assignment
in example
2
I
I 20
I
I
I
I
8 40
1
Time, (mln)
Figure 7 Optimal state trajectories
in example
2
0.3
06.
v
-z G
c 06
03----
0.
L
I
1
I
I
I
,09
I
paper; minimize the total time lost by the vehicles in queues by the route assignment and the traffic signals. Therefore, it will necessary to consider (i), the route assignment and (ii), the system-optimization assumption. Route assignment
0
I
I
Figure 6 Optimal
/
I
I
I
1
sequences of signal setting in example 2
106 Appl. Math. Modelling,
1980,
I 3609
20 Tlme,(mln)
Vol. 4, April
Nowadays, the recommendation of a better route by supplying information on congestion, construction, the weather, etc., operates at branching points of highways and urban roads. Each driver determines his route taking this information into account. The optimal sequences of route assignment obtained by solving our problem will be realized by supplying information to drivers which recommends a particular route. Since the assignment is not enforcible, the maximum and minimum possible values of the assignment, 9?‘“@At) and 9p’“(pAt), should be set suitable, using experiential knowledge. In our examples, these are assumed to be constants, i.e., &“@At) = 0 and 9Tax(9At) = 0.7sk, ,4, independent of the other conditions. k=l,...
Control of oversaturated transportation networks:
Tlme,(mmi Figure 8 State trajectories
System-optimization
in example
3
assumption
Let us consider the control policies which are desirable to treat the congested networks. Especially, when the congestion is inevitable, some policies will be required in addition to those for noncongested networks. When the network is oversaturated, the travel time of a vehicle is considerably affected by the queue lengths on the selected route. In such a case, a driver usually cannot know the best route for him, since the sufficient information about the current state of the network is not supplied to him. Therefore, the flow pattern in the oversaturated networks will be less useroptimized than that in the undersaturated networks. Taking this circumstances into account, the control schemes for congested networks should more or less satisfy the following conditions. (a) The oversaturated intersections disappear as soon as possible. (b) The total amount of congestion is minimized. (c) The congestion is not concentrated about some particular intersections. (d) Undersaturated intersections do not appear during the period of oversaturation, since they decrease the capacity of the network. (e) Travel times of all available routes for a vehicle are equalized. Of course, no control can satisfy all these conditions, since they are interdependent. Both conditions (a) and (b) represent system-optimization assumptions, while condition (e) represents a user-optimization assumption. Both conditions (c) and (d) represent the situations of each intersection and they are usuaIIy helpful in satisfying condition (e). In our approach these conditions are treated in the following. As shown in the previous sections, first the initial time and the initial flow pattern are decided from the O-D flow data, then the final time T = N. At is specified, and finally
H. Nagase
the objective function is minimized by a LP procedure. Solving the problem repeatedly with various values of N, we obtain an interval of N, Nr
Appl.
Math.
Modelling,
1980,
Vol. 4, April
IO7
Control
of oversaturated
transportation
networks:
H. Nagase
and (ii), the computation time required to execute a renewal procedure of the tableau. (i), is roughly proportional to the number of the unknown variables. (ii), is proportional to the column number of the tableau and also to the row number of the tableau. Multiplying these factors and neglecting lower-order terms, we can conclude that the corn utation P time of our problem is roughly proportional to N3m n3.
Conclusions The minimization problem of the traffic congestion in an oversaturated transportation network has been formulated as a dynamic optimization problem. The network treated here is that with a rectangular grid of intersections. The flow demand of each O-D traffic has been assumed to be given as a time-varying function. Traffic flows have been regulated simultaneously both by the dynamic route assignment and by the traffic light control. A method of linear programming is directly applicable to our problem and some numerical examples have been given in the case where each O-D traffic has two routes; consequently the network has
108
Appl.
Math.
Modelling,
1980,
Vol. 4, April
four intersections. These examples show that our approach is an effective method for controlling oversaturated networks.
Acknowledgements The author would like to express his thanks to Professor Y. Sakawa of the Faculty of Engineering Science, Osaka University for his continual guidance and discussion throughout this work. He is also indebted to Dr H. Kimura, Y. Inouye, and N. Fujii for their valuable discussions and encouragement.
References 1 2 3 4
Gazis, D.C. Oper. Rex 1964, 12,815 D’Ans, G. C. and Gazis, D. C. Tramp. Sci. 1976,10,1 Sakawa, Y. and Hayashi, C. Proc. IFAC To&o Symp. 1965, pp. 53-60 Kuester, J. L. and Mize, J. H. ‘Optimization techniques with fortran,’ McGraw-Hill, London, 1973