## 6 Illustrative Results  
As an illustrative example, we consider the apartment described in Section 5, in this case the model has been previously identified by applying a methodology described in [28]. To obtain a simulation as close as possible to the real case, a user defined set-point signal \(\hat{T}_{i}\), one for each room, has been applied to the building. In Table 3 the set-point temperature values for each room and for each time interval of the day are shown. The genetic algorithm implemented in the MATLAB Global Optimization Toolbox (The Mathworks Inc., Natick, MA, USA) has been used for the optimization procedure, with an initial population size of 100 and a maximum number of generations of 300. To practically ensure that a global minimum is achieved, the function tolerance, which determines the termination of the algorithm if the difference between the value of the fitness function from one iteration and the previous one is less than this threshold, has been selected equal to \(10^{-6}\). As a first illustrative result, Figures 4 and 5 show the set-point signal, for room 1 of the considered apartment, resulting from the optimization for a simulation of four days, starting from 15 November to 18 November. In particular, Figure 4 shows the results when the set-point value can change as a step signal, while Figure 5 shows the same example where the optimized set-point signal can change as a ramp. In both cases fan-coil units have been used. The optimization for each day is performed during the night, starting at midnight. In case the weather forecasts of the subsequent day are not available, the external temperature and solar radiation data of the previous day are used as exogenous inputs, by assuming that the environmental conditions are similar from one day to the following one (for this reason, results are different from one day to another one). Please note that, in case of step signals, the computational time for the optimization is about 700 [s] while in case of ramp signals it is 1200 [s], as in this case more parameters must be optimized. The developed approach can be considered as a centralized optimization, in fact it takes into account the building from a global point of view instead of a single subsystem. It is important to stress that a set of optimal set-point signals determined by decentralized algorithms can be different from a set determined by a global optimization, due to the thermal coupling between adjacent areas. As it can be expected, the set-point step (or ramp) signals are applied before the temperature change selected by the user to obtain the required temperature at the required time. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Room**} & \multicolumn{10}{c}{**Time Intervals (Hours of the Day)**} \\ \cline{2-13}  & **0-8** & **8-9** & **9-11** & **11–12** & **12–14** & **14–17** & **17–18** & **18–19** & **19–20** & **20–21** & **21–22** & **22–24** \\ \hline
1 & 17 & 20 & 20 & 20 & 21 & 21 & 17 & 17 & 17 & 17 & 17 & 17 \\
2 & 17 & 17 & 20 & 20 & 20 & 20 & 17 & 17 & 17 & 17 & 17 & 17 \\
3 & 17 & 18 & 18 & 20 & 20 & 20 & 20 & 17 & 17 & 17 & 17 & 17 \\
4 & 17 & 17 & 21 & 21 & 21 & 21 & 21 & 17 & 17 & 17 & 17 \\
5 & 17 & 19 & 19 & 19 & 19 & 21 & 21 & 21 & 17 & 17 & 17 & 17 \\
6 & 17 & 20 & 20 & 20 & 20 & 20 & 20 & 20 & 17 & 17 & 17 \\
7 & 17 & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 19 & 17 & 17 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Set-point values selected by the user for each room for one day. A detail of the results related to a single day (5 December) for room 3 (considered as illustrative example) is shown in Figure 6, for radiators. Analogously, results related to the fan-coil system is shown in Figure 7 for the same room 3. Please note that these results have been obtained by applying the obtained set-point signals directly to the ARMAX model. As it is possible to see, both step and ramp set-point signals can fulfill the user comfort requirements.