**3.2 Centralised traffic control strategy**

The Network Traffic Control (NTC) may be classified as a continuous linear optimisation problem and a multi - objective approach is pursued combining the total delay minimisation and the minimisation of the queue equidistribution criterion [13].

The parameters and constraints used in the model are listed below


Finally, for each junction the node offset is needed; it represents the period between the start of a reference stage of junction i and the start of the reference stage of the first junction used as master for clock.

**153**

*Centralised Traffic Control and Green Light Optimal Speed Advisory Procedure in Mixed…*

Regarding the queue equidistribution the following objective function (*of*, see

( , ) (max ( , , ,, , ))

Summing up the on-line synchronisation [14] is obtained by combining together:

The procedure is able to simultaneously optimise the green timings and the offsets. Regarding the solution algorithm in this paper the meta-heuristic Multi - objective Simulated Annealing [15, 16] has been applied. As a matter of fact, meta-heuristic algorithms can effectively address even optimisation problems with objective function not expressed in closed form, so that derivatives are not easily available, as

In particular the basic Simulated Annealing algorithm is a neighbourhood based meta-heuristic, which is inspired by the statistical mechanics to find solutions for

Regarding the link metering (LM), is a feedback method implemented in accordance with the proportional integral type proposed by [6, 17–19] and it is based on

Regarding the control function, a proportional-integral-type (PI) feedback controller (2) aiming to maintain the observed occupancy around the desired value

( ) ( ) ( ) ( ) ( ) é ù = -- - - + - é ù ë û ê ú ë û **1 1** *q k q k K o k o k K oo k s s ps s l s* (2)

The criterion aims to balance the rates of queue growth (or equalise them in an ideal case) in a network and then minimises the spill-over risk; it is based on traffic control decision variables design able to minimise the difference between the

• the continuous variables needed to completely define the signal plan, that is are: (i) the stage lengths, constrained by the consistency among the stage

*<sup>n</sup> in out k i i tr i l r i tr i tr*

= - åå**<sup>3</sup> <sup>2</sup>**

bx

**0**

*m ms* (1)

= =

*i tr*

discharging capacity and the traffic demand at each link.

lengths and the cycle length, (ii) the node offsets;

• the objective functions defined by the total delay and the queue

**1 1**

*DOI: http://dx.doi.org/10.5772/intechopen.95247*

*of g*

j

Eq. 1) has been considered:

equidistribution.

it occurs for the scheduled synchronisation.

occupancy as a control variable.

• k be the time step

• qs be the gated flow

• Kp be the proportional gain

as in following displayed has been applied:

• Kl be the integral gain

• s be the section

• **o**

both discrete and continuous optimisation problems.

We list here the parameters used in the model

be the desired downstream occupancy

• os be the observed occupancy at downstream

*Centralised Traffic Control and Green Light Optimal Speed Advisory Procedure in Mixed… DOI: http://dx.doi.org/10.5772/intechopen.95247*

Regarding the queue equidistribution the following objective function (*of*, see Eq. 1) has been considered:

$$\text{off}\left(\mathbf{g}\_k, \mathbf{g}\_l\right) = \sum\_{l=1}^{u} \sum\_{tr=1}^{3} \left( \max\left(\mathbf{g}\_{l,tr} m\_i^{iu} - \mathbf{g}\_{l,tr} m\_{i,tr}^{out} s\_{i,tr}, \mathbf{0}\right)\right)^2 \tag{1}$$

The criterion aims to balance the rates of queue growth (or equalise them in an ideal case) in a network and then minimises the spill-over risk; it is based on traffic control decision variables design able to minimise the difference between the discharging capacity and the traffic demand at each link.

Summing up the on-line synchronisation [14] is obtained by combining together:


The procedure is able to simultaneously optimise the green timings and the offsets. Regarding the solution algorithm in this paper the meta-heuristic Multi - objective Simulated Annealing [15, 16] has been applied. As a matter of fact, meta-heuristic algorithms can effectively address even optimisation problems with objective function not expressed in closed form, so that derivatives are not easily available, as it occurs for the scheduled synchronisation.

In particular the basic Simulated Annealing algorithm is a neighbourhood based meta-heuristic, which is inspired by the statistical mechanics to find solutions for both discrete and continuous optimisation problems.

Regarding the link metering (LM), is a feedback method implemented in accordance with the proportional integral type proposed by [6, 17–19] and it is based on occupancy as a control variable.

We list here the parameters used in the model


Regarding the control function, a proportional-integral-type (PI) feedback controller (2) aiming to maintain the observed occupancy around the desired value as in following displayed has been applied:

$$\mathbf{q}\_s(\mathbf{k}) = \mathbf{q}\_s(\mathbf{k} - \mathbf{1}) - \mathbf{K}\_p \left[ \mathbf{o}\_s(\mathbf{k}) - \mathbf{o}\_s(\mathbf{k} - \mathbf{1}) \right] + \mathbf{K}\_l \left[ \dot{\mathbf{o}} - \mathbf{o}\_s(\mathbf{k}) \right] \tag{2}$$
