## 6 Illustrative Results
Indeed, the actual temperature inside a specific room is never more than 0.5 [\({}^{\circ}\)C] less than the required value in a given time instant. When radiators are present, the use of ramp signals for the set-point variation (which are more general than step ones) allows a significant decrement of the energy consumption with respect to step signals, as it is shown, for the considered day, in Table 4. Indeed, the total energy in the case of ramps is approximately 13% less than the case of steps. The energy saving obtained by using ramps instead of steps is less significant in case of the use of fan-coil units, as it is shown in Table 5. This is actually an expected result as the fan-coil units allow the temperature to increase more rapidly in a room. Figure 4: Four days simulation, from 15 November to 18 November, and optimal set-point signal planning, in case the optimized set point can change as a step signal. Red line: user defined set-point signal. Blue line: optimal set-point signal obtained by the genetic algorithm. Figure 5: Four days simulation, from 15 November to 18 November, and optimal set-point signal planning, in case the optimized set point can change as a ramp signal. Red line: user defined set-point signal. Blue line: optimal set-point signal obtained by the genetic algorithm. Results obtained by applying the determined set-point signals to the building model developed with TRNSYS (that is, with the model of the system that takes into account all the thermodynamics aspects) are shown in Figure 8 for the case of radiators and in Figure 9 for the case of fan-coil units, again for room 3. It appears that results are comparable to those obtained with the ARMAX model, which further proves the effectiveness of the linear modelling technique. The total energy consumption in case of TRNSYS model is summarized in Tables 6 and 7. Therein, two cases are considered: (i) the case where the weather forecasts are not available and therefore, the weather conditions of the day before are applied directly in the optimization procedure; (ii) the case where the weather conditions are supposed to be known. To analyze better the results, the external temperature and the solar radiation in the two days considered (the one used for the optimization and the one for the day where the optimized set-point is applied) are plotted in Figure 10.  
Also, these results confirm that, with fan-coil actuators, the use of ramp set-point signals does not provide a very significant improvement and, to save computation time, it is sufficient to calculate the time instants where the set-point steps must be applied to the control system to meet the user requirements. Figure 6: Results obtained by simulating the ARMAX model of room 3 with radiators for one day, 5 December. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Room** & **Step Case [kg/h]** & **Ramp Case [kg/h]** & **Saving [\%]** \\ \hline
1 & 3635 & 3216 & 13.03 \\
2 & 1801 & 1657 & 8.73 \\
3 & 2276 & 1781 & 27.8 \\
4 & 2950 & 2133 & 38.3 \\
5 & 2015 & 2054 & \(-\)1.93 \\
6 & 12,208 & 11,486 & 6.29 \\
7 & 6692 & 5547 & 20.64 \\ Total (J) & 31,577 & 27,874 & 13.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Radiators performance index value (4) for both step and ramp set-point signals determined by the genetic algorithm (5 December). Figure 7: Results obtained by simulating the ARMAX model of room 3 with fan-coil units for one day, 5 December. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Room** & **Step Case [K]** & **Ramp Case [K]** & **Saving [\%]** \\ \hline
1 & 8549 & 8554 & \(-\)0.059 \\
2 & 7907 & 7782 & 1.61 \\
3 & 7920 & 7548 & 4.93 \\
4 & 8495 & 8881 & \(-\)4.35 \\
5 & 8869 & 7894 & 12.4 \\
6 & 7232 & 7312 & \(-\)1.09 \\
7 & 6777 & 6822 & \(-\)0.658 \\ Total (J) & 55,750 & 54,794 & 1.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Fan-coil performance index value (4) for both step and ramp set-point signals determined by the genetic algorithm (5 December).