#### 3.2. Traffic Control

A within-day framework combines a TC model based on the online scheduled synchronization approach (i.e., the procedure is able to optimize green timings, offsets, and stage sequences, also called NTC—Network Traffic Control) and a link metering method.

For the link metering, the input variables required for traffic control decision variable optimizations are based on a microscopic traffic flow model, while the input variables required for Network Traffic Control decision variable optimizations are based on a macroscopic traffic flow model (i.e., a Cell Transmission model; [

50]). Furthermore, in order to guarantee the consistency between flows and decision variable optimizations, a traffic flow prediction model, based on a Kalman Filter (see [

51] for a more detailed description) and a rolling horizon approach, have been adopted here. Finally, the Network Traffic Control simultaneously operates with the Link Metering method.

The Network Traffic Control may be classified as a mixed discrete–continuous linear optimization. In terms of the optimization procedure, a bi-level approach is pursued, where, at an upper level, the total delay minimization is considered, and, at the lower level, the minimization of the queue equidistribution criterion is adopted (see [

33]). Here, we list the parameters and constraints used in the model:

Δ: Approach-stage incidence matrix, with entries ${\mathsf{\delta}}_{\mathrm{kj}}=1$ if approach k is green during stage j and 0 otherwise;

c > 0: The cycle length;

${\mathrm{t}}_{\mathrm{j}}$ ∊ [0, c]: The length of stage j as an optimization variable;

${\mathrm{t}}_{\mathrm{ar}}$ ∊ [0, c]: The so-called all red period at the end of each stage;

${\mathrm{l}}_{\mathrm{k}}$ ∊ [0, c]: The lost time for approach k, which is assumed to be known;

${\mathrm{g}}_{\mathrm{k}}$: The effective green for approach k;

${\mathrm{g}}_{\mathrm{min}}$ The minimum value of the effective green;

${\mathrm{q}}_{\mathrm{k}}$ > 0: The arrival flow for approach k, which is assumed to be known;

${\mathrm{s}}_{\mathrm{k}}$ > 0: The saturation flow for approach k, which is assumed to be known;

b ∊ [0,1] and eventually t ∊ [1–3]: The discrete variables for stage sequence definition as decision variables;

i: The generic links;

r: The turning rates;

α_{l,r}: The split ratio of the traffic demand in the lth link and rth movement;

γ_{l,r}: The number of lanes assigned to the r movements in the lth link;

${\mathrm{Q}}_{\mathrm{l}}^{\mathrm{in}}$: The total inflow;

${\mathrm{q}}_{\mathrm{l},\mathrm{r}}^{\mathrm{out}}$: The discharging capacity, expressed as vehicles/hour/lane;

${\mathrm{t}}_{\mathrm{l},\mathrm{r}}$: The sum of the signal phase ratios for the rth movement in the lt^{h} link;

For each junction i, let ${\mathsf{\varphi}}_{\mathrm{i}}$ ∊ [0, c] be the node offset between the start of a reference stage of junction i and the start of the reference stage of the first junction, which is used as a reference for the clock;

For each pair of (adjacent) junctions (i, h) in the network, let ${\mathsf{\varphi}}_{\mathrm{ih}}={\mathsf{\varphi}}_{\mathrm{h}}-{\mathsf{\varphi}}_{\mathrm{i}}=-{\mathsf{\varphi}}_{\mathrm{hi}}$ be the link offset between junctions i and h, which is needed for computing the total delay through a traffic flow model;

For the computation of delay, the method for interacting junctions has been adopted, and, in particular, the total network delay has been computed by combining each junction j and each approach k, the deterministic delay (

${{\mathit{DTD}}_{k}}^{j}$), and the stochastic and oversaturation term

${{(\mathit{SOTD}}_{k}}^{j})$. Finally, the total delay is equal to the following formula:

Regarding queue equidistribution, the following objective function has been considered:

The proposed criterion aims to minimize the difference between the longitudinal capacity and the demand flow at each link.

With reference to the solution algorithm in this paper, meta-heuristic Simulated Annealing (see [

17]), working on a single objective problem, has been adopted.

Concerning the adopted methodology for the link control implementation, the approaches proposed by [

19,

20,

52] have been considered, which are based on occupancy as a control variable.

Here, we list the parameters used in the model:

k: The time step;

s: The section;

$\hat{\mathrm{o}}$: The desired occupancy;

q_{s}: The metered flow;

o_{s}: The observed occupancy;

K_{p}: The proportional gain;

K_{l}: The integral gain.

Regarding the control function, a proportional-integral-type (PI) feedback controller has been implemented, aiming to ensure the consistency between observed and desired occupancy.

#### 3.4. Day-to-Day Behavioral Modelling

In this section, the steps of the iterative procedure for anticipatory route guidance able to guarantee consistency between user behavior and traffic signal decision variables, guarantee consistency between information (EsTT_{t}) and user behavior (ETT_{t − 1}), and focus on the information reliability (ER_{t − 1}) is described.

From an analytical point of view, it is a fixed-point problem and the method of successive averages (MSA), which is based on a recursive approach, is the solution algorithm that has been adopted here.

In particular, the following parameters are set accordingly:

k: The iteration;

**f**^{k}_{t}: Network flows at iteration k, at day t;

EsTT_{t}: The estimated travel time at day t;

EsTT^{k}_{t}*****: The estimated travel times consistent with flow on day t;

ETT_{t − 1}: The experienced travel times at day t − 1;

ER_{t − 1}: The experienced reliability at day t − 1.

The main steps of the solution algorithm are listed below:

k = 0

**f**^{0}_{t} = Ω (**ER**_{t − 1} = 0; **ETT**_{t − 1}= 0; **EsTT**^{k − 1}_{t} = 0)

do

k = k + 1

//step 1 — flow updating*/

**f**^{k}_{t} = Ω (**ER**_{t − 1;} **ETT**_{t − 1;} **EsTT**^{k − 1}_{t}) //Ω is a generic function

//step 2 – consistent estimated travel times computation*/

**EsTT**^{k}_{t}***** = Ψ (**f**^{k}_{t}) //Ψ is a generic function

//step 3 – new estimated travel times estimation*/

**EsTT**^{k}_{t} = **EsTT**^{k −1} _{t} + 1/α [**EsTT**^{k}_{t}***** - **EsTT**^{k −1} _{t}]

Loop until

max │**EsTT**^{k}_{t}***** - **EsTT**^{k −1} _{t}│/**EsTT**^{k −1} _{t} > δ //δ = 0.02