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Compartmental Networks Approach on Urban Traffic Control

Catalin Dimon, Andreea Ioana Udrea Department of Automatic Control and Systems Engineering

Faculty of Automatic Control and Computers, University Politehnica of Bucharest Bucharest, Romania

catalin.dimon@acse.pub.ro

Abstract—Urban traffic networks play an important role in the modern society. Modeling and control of metropolitan traffic is a complex problem and a lot of research efforts are made towards solving this problem. We consider a macroscopic approach of the traffic and we propose a new solution for modeling and control based on mechanism offered by compartmental networks and positive systems theory. The traffic congestion is avoided by use of a nonlinear compartmental controller. A number of case studies and their results are presented

Keywords—urban traffic; compartmental network; nonlinear control

I. INTRODUCTION In recent decades, traffic has undergone significant

developments, the authors trying to improve existing models or propose new models, appropriate to the new traffic conditions. Depending on the level of detail considered, three types of models are usually encountered [1]: microscopic, mesoscopic and macroscopic. The approach adopted in this paper is based on the macroscopic approach of road traffic.

In a decentralized macroscopic approach [2], road traffic is considered in the following configuration: road network-road section-road element. Describing a road network is best achieved by using a compartmental network representation, emphasizing the links between the sections that make up the road network, as well as how to cross the traffic junctions.

Road junctions are critical points of a road network, their role being to direct traffic and this is done with the help of traffic lights. In this regard, we were interested in the ratio between the green time and red time, called signaling cycle. If this ratio is fixed, does not change according to the needs of traffic, traffic dynamics can be slowed down, which can cause congestion phenomenon.

In the proposed representation, the junction is not considered as a node of a network (usually seen as a storage compartment), but the directions of movement within represent arcs connecting road sections. The segments become nodes of the network, allowing the accumulation of vehicles and the travel directions in the intersection are the arcs that facilitate the transfer of vehicles in the network.

The most important property of compartmental networks is the conservation of mass. For what matters to us, a network compartment is conservative in terms of the number of vehicles moving in the network. The property is more evident when the network is closed, without input and output streams, where the volume of vehicles will be the same.

In order to give at any time a good command that reflects the reality of traffic, accurate measurements of the number of vehicles within each section and intersection are required. The length of a road segment may adversely affect the estimation of its dynamics. So instead of a road section we employ a road element, more precisely the last element of the section which will be the source node of the network. Where the road is short the road element is the same with the road section.

II. ROAD NETWORK MODEL In order to describe the behavior of traffic in an area, for

each road section the queues of stopped vehicles (if they exist) are associated with the nodes of the network. Thus, each node represents the section of the road where vehicles accumulate until an event such as the changing of traffic lights occurs. The arcs represent the directions of movement of vehicles leaving a section through an intersection and have associated values corresponding to the weight of each output direction, based on estimates of the desired trajectories of vehicles.

For describing a compartmental network we use a dynamic model, where the balance of flows between compartments is defined by equations of the form:

( ) ( ) , 1,...,i ji ik i i

j i i kx f x f x e s i n

≠ ≠

= − + − =∑ ∑ (1)

where:

ix represent the state variables (the number of vehicles associated with each section);

( )jif x and ( )ikf x are non-negative and continuous functions (transfer flows for the intersections within the traffic area);

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ie represent the input flows, injected from the outside into specific network nodes (the flow of vehicles that are entering the traffic area);

is represent the output flows of the network (flows of vehicles that come out of the traffic area).

In these equations there are only terms that correspond to links between network nodes. It is obvious that in an empty compartment there is no non-zero positive flow:

0 ( ) 0i ijx f x= ⇒ = and ( ) 0is x = (2) α + β = χ. (1) (1)

From the above condition, if ( )ijf x and ( )is x are differentiable, we can write:

( ) ( )

( ) ( )ij ij i

i i i

f x r x x

s x q x x

=

= (3) α + β = χ. (1) (1)

where ( ) : nijr x R R+ +→ and ( ) : n

iq x R R+ +→ are continuous functions, differentiable and strictly positive, called specific rates.

Replacing the relations (3) in (1), it results the model of the network:

( ) ( ) ( ) , 1,...,i ji j ik i i i i

j i i kx r x x r x x q x x e i n

≠ ≠

= − − + =∑ ∑(4) α + β = χ. (1) (1)

Although this representation applies to road traffic, it can also be used to model various industrial processes, highlighting the conservation of mass and allowing for the description of the process in terms of compartmental networks. These networks have important structural properties presented in [3].

III. CONTROL DESIGN The congestion of a compartmental network occurs when

demand inflow exceeds the capacity of the network. The effect of congestion is the endless accumulation of vehicles in the compartments of the network. To avoid congestion we propose a nonlinear controller with a structure similar to compartmental systems.

Let us consider a compartmental network, with n compartments, m input flows and p output flows, which has the following properties:

- it is fully connected to the input and completely connected to the output flow rates;

- transfer rates between compartments are bounded;

- the capacities of the network compartments are bounded;

- there is a demand for an input flow, referenced to each input node of the network, which represents the input flow that can enter the system.

Preventing the congestion involves checking the flow input ( )ie t which is actually injected into the system, by reducing it

relative to the demand ( )id t :

( ) ( ) ( )i i ie t u t d t= (5)

where 0 ( ) 1iu t< < represents a fraction of the requested

flow ( )id t .

Assuming that the output flows of the system ( )( ) ( )i is x t y t= are measurable, the state space

representation of the system becomes:

( ) ( )( )

x A x x B d uy C x x

= += (6)

with [ ]1 2T

nx x x x= , [ ]1 2T

mu u u u= ,

[ ]1 2T

ny s s s= and the matrices are of the form:

1 1 21 11

12 2 2 22

1 2

11

22

1

2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )

0 00 0

( )

0 0

( ) 0 00 ( ) 0

( )

i ni

i ni

n n n nii n

nm

q x r x r x r x

r x q x r x r xA x

r x r x q x r x

dd

B d

d

q xq x

C x

≠

≠

≠

⎡ ⎤⎛ ⎞− +⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥

⎢ ⎥⎛ ⎞− +⎢ ⎥⎜ ⎟= ⎝ ⎠⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎛ ⎞− +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

=

∑

∑

∑

0 0 ( )nq x

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ (7)

The control structure is illustrated in Fig. 1. The controller has a network structure with a number of compartments equal to the number of outputs of the controlled network. Each compartment of the controller is provided with a copy of the output rates of the controlled network. The flows coming out of the controller are distributed to the control variables, representing inputs for the controlled network, so that there is only one way, through the controller, from an output node to a input node, for each pair of nodes.

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Fig. 1. Closed loop structure of the compartmental system

In matrix form, the control law is [4]:

( ) ( )( )

z G d F z z yu K z z

= += (8) α + β = χ. (1) (1)

where:

( ) ,

( ) ( ),

i

i

ki k sk Q

i sk Q

G d diag d i I

F z diag z i I

α∈

∈

⎛ ⎞= − ∈⎜ ⎟

⎝ ⎠⎛ ⎞

= Φ ∈⎜ ⎟⎝ ⎠

∑

∑ (9) α + β = χ. (1) (1)

Relations (6) and (8) give the set of equations describing the closed-loop system:

( ) ( ) ( )( , )

( ) ( ) ( )x A x B d K z x x

L x zz C x G d F z z z

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ (10) α + β = χ. (1) (1)

where ( , )L x z is a compartmental matrix dependent on the demand.

The storage function of the system:

1 1

( , )pn

i ji j

M x z x z= =

= +∑ ∑ (11) α + β = χ. (1) (1)

is invariant, and the trajectories of the system states belong to the set:

( ){ }( , ) | ( , ) (0), (0) 0n nH x z R R M x z M x z σ+ += ∈ × = = >(12) α + β = χ. (1) (1)

Therefore, the state variables are bounded:

0 ( ) , 1,

0 ( ) , 1,i

j

x t i n

z t i p

σσ

≤ ≤ =

≤ ≤ = (13)

Thus, the congestion control is achieved by the structure of the proposed controller. If σ is less than the maximum capacity of the compartments, the saturation avoidance is guaranteed. In addition, it can be observed that σ depends on the initial conditions. In general, the system is considered with initial conditions equal to zero ( (0) 0x = ). Thus, σ is chosen only by the initial conditions of the controller, expressed as:

1

(0)p

jj

zσ=

=∑.

The proposed controller is also robust; in order to obtain the command we only need the structure of the network. The command is independent of the flows ( )ijr x and ( )is x . This is important because in many applications the precise knowledge of these flows is impossible.

IV. ROAD JUNCTION ANALYSIS We analyze the case of a cross junction, shown in Fig. 2. In

Fig. 3 the equivalent compartmental network is shown. We can identify a source road element (shown in red) and three destination road elements (shown in blue), the first representing the last element of a section of road and the other the first elements of the corresponding road sections. To be mentioned that the output flow of the source element is controlled by a traffic light, assumed for the moment with a fixed signaling cycle. While the configuration is maintained the same, the traffic lights signaling cycle can also be calculated based on traffic parameters.

Fig. 2. Structure of a cross junction

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Fig. 3. Compartmental network representation of a cross junction

The flows represented in the figures are:

- ( )d t input flow in the source node;

- 2 3 4, ,v v v output flows for the destination nodes;

- 12 13 14, ,f f f output flows from the source node (transfer flows between the source node and the destination nodes).

The input flow in the source node is variable in time and depends on the characteristics of the road section to which the node (as road element) belongs. In addition, this flow is seasonal; its variations depend on the time of day (traffic in the morning, afternoon and evening), day of the week (weekday traffic and weekend traffic) and the month of the year (the traffic during the months of summer and winter).

The output flows 2 3 4, ,v v v depend on the state of the node (road element) and the state of the section (specifically the state of the next road element in the composition of the section). Therefore, the output rates are defined by the relations:

( ) ( )

1i

i i i i ii

xv x g xx

μ μ= =+ (14) α + β = χ. (1) (1)

where iμ is the flow of vehicles associated with node i, and

function ( )i ig x models the behavior of vehicles leaving the node. The function is useful because the vehicles coming into the destination node need a time interval to arrive at the speed value of the respective section. Flow increases monotonically towards the value dictated by the steady state.

The output flows of the first node can be decomposed in a similar way to the transfer flows of a compartmental network:

12 12 1 1

13 13 1 1

14 14 1 1

( )( )( )

f v xf v xf v x

βββ

=== (15)

where the function 1 1( )v x is defined by (14) and

12 13 14, ,β β β represent the weights of the total flow coming from the source node 1 and distributed to the destination nodes 2, 3 and 4. The sum of 12 13 14, ,β β β is equal to 1. If we have a number of vehicles waiting at the traffic light, by dividing this number to the time unit, considered as the duration of a signaling cycle, we obtain the total output rate of the source node. In the same way, we can calculate the flow of vehicles that will go in front, right and left. By dividing each of these flows with the total output flow we can obtain the values of the weights 12 13 14, ,β β β .

For the cross-junction in Fig.2 we consider the following conditions:

- large input flow;

- small exit flows for destination sections;

- crowded destination road elements.

The objective is to avoid congestion at the junction.

The state space model of the network is as follows:

1 1 1

2 12 1 1 2 2

3 13 1 1 3 3

4 14 1 1 4 4

( )( ) ( )( ) ( )( ) ( )

x du v xx v x v xx v x v xx v x v x

βββ

= −= −= −= − (16)

where 1x is the state of the source node 1, 2x is the state of

the destination node 2 (right), 3x is the state of the destination

node 3 (in front) and 4x is the state of the destination node 4 (left).

The structure of the nonlinear controller corresponding to the previous model is the following:

2 2 2 2 2 12

3 3 3 3 3 13

4 4 4 4 4 14

( ) ( )( ) ( )( ) ( )

z v x z dz v x z dz v x z d

ααα

= − Φ= − Φ= − Φ (17)

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where 1 2 3, ,z z z are the states of the controller. The control law is given by the relation:

12 2 2 13 3 3 14 4 4( ) ( ) ( )u z z zα α α= Φ + Φ + Φ (18) α + β = χ. (1) (1)

The exit flows of compartmental network nodes are determined using expression (14). For the virtual output flows of controller nodes ( ), 1, 4i iz iΦ = , the following relationship is used:

( ) , 0.1i

i ii

zzz

εε

Φ = =+ (19) α + β = χ. (1) (1)

The percentages of the total flow exiting node 1 are as follows:

12 12

13 13

14 14

0.20.70.1

α βα βα β

= == == = (20) α + β = χ. (1) (1)

The maximum output flows of the nodes are chosen to reflect the considered situation:

1

2

3

4

300 / 6010 / 6025 / 605 / 60

μμμμ

==== (21) α + β = χ. (1) (1)

For simulation we considered two cases: constant input flow and variable input flow.

For the first case, with constant input flow, the evolution of the system states (the number of vehicles in each node) is shown in Fig. 4 for the open loop system and in Fig. 5 for the closed loop system. In the first situation, we note that vehicles are starting to accumulate, so the congestion of the road junction occurs. In the second situation, congestion is avoided; the number of vehicles does not exceed the maximum allowable (in this case equal to 5). In Fig. 6 is shown the input flow injected into the intersection, limited by the control law.

In the second case, the input flow is assumed to vary in time. The variation is shown in Fig. 7. The evolution of the system states is shown in Fig. 8 for the open loop system and in Fig. 9 for the closed loop system. The controlled input flow is illustrated in Fig. 10.

Fig. 4. Nodes state variation of open-loop system for constant input flow

Fig. 5. Nodes state variation of closed-loop system for constant input flow

Fig. 6. Input flow injected into the road junction for constant input flow

Fig. 7. Variable input flow

Fig. 8. Nodes state variation of open-loop system for variable input flow

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Fig. 9. Nodes state variation of closed-loop system for variable input flow

Fig. 10. Input flow injected into the road junction for variable input flow

It can be noted that the proposed nonlinear controller succeeds in each case to avoid congestion. It calculates a weighting control for the large input flow of the road junction in order to obtain a lower flow, which does not block the junction. The obtained command is always between 0 and 1. When you get a very small order, which tends to zero, it is best not to give the green light to vehicles, because the allotted time is very short and cars, trying to cross the junction, can block it. To avoid this, a minimum time interval for green phase can be

introduced, and if the controller calculates a lower time, it will choose to maintain the red phase during that time.

V. CONCLUSIONS Compartmental networks offer a solution for modeling a

road region. This type of modeling allows a connection in the model between the flow of vehicles on the source segments which intersect at a road junction and the free capacities of the destination segments after the junction. Based on this information, a nonlinear controller was designed to determine the optimal input flow in order to avoid a possible congested state in the junction. The obtained control allows the changing of the signaling cycle of a traffic light depending on existing traffic conditions.

The proposed solution offers the possibility of simulating the traffic flow for an urban road network, highlighting the intersections with risk of congestion, and can calculate a command for traffic lights signaling cycles, due to avoid congestion.

REFERENCES

[1] S.P. Hoogendoorn, P.H.L. Bovy, “State-of-the-art of vehicular traffic flow modelling”, Proceedings of the I MECH E Part I Journal of Systems & Control Engineering, Volume 215, Number 4, 19 August 2001, pp. 283-303.

[2] C. Dimon, G. Dauphin-Tanguy, D. Popescu, “Macroscopic modeling of road traffic by using hydrodynamic flow models”, 20th Mediterranean Conference on Control and Automation, Barcelona, Spain, 3-6 July 2012, page(s) 42-47.

[3] G. Bastin, “Sur la modélisation et le contrôle des réseaux dynamiques conservatifs”, Revue E-STA, Special CIFA 2006, Vol. 3(2), 2007.

[4] G. Bastin, V. Guffens, “Congestion control in compartmental network systems”, Systems and Control Letters, Vol. 55(8), 2006, pp. 689-696.