*3.2. Heating Analysis*

The results show that the refurbishment had a significant impact in the heating demands of the building. First, observing the box plots from the vectorial analysis, as shown in the first row of Figure 7, it is clear that the heating demand was reduced on all floors. This fact is supported by vectorial tests

that reject similarity between samples on all floors (see Table 4). After that, with the functional graphs, as shown in the second row of Figure 7, it is also appreciated that the heating demand curves after refurbishment are below the initial curves. This is also proved by the functional tests that reject all the sample similarity hypothesis (see Table 4). The results shown in Table 4 change from vectorial to functional analysis. Both analyses come to the same conclusions, but the magnitude of change is different for each one. With vectorial approach, the reduction per floor ranges from 3667 W as the highest reduction on the third floor to 1057 W as the lowest on the first floor. In contrast, with the functional analysis, although the highest and lowest reduction were on the same floors, the values of the reduction are not the same. The heating demand, each minute, was reduced on average 3918 W on third floor and 1456 W on first floor. Therefore, the relative heating savings were significant (see Table 4). The floor most benefited was the third floor; both analysis obtained savings higher than a 30% on this floor. Again, the ground floor had to be analysed individually because, in addition to the installation of shading in 2017 that prevents solar gains, a false ceiling was already insulated on this floor in 2015. Thus, it was possible to obtain a saving of about 12% of the initial heating demands from vectorial results and about 17% from functional results (see Table 4).

**Figure 7.** Analysis of the heating demands of each floor measured in W. In the first row, the vectorial results (in form of box plots) are presented. In the second row, instead, the functional data are represented with the respective mean functions. The data are divided into winter days before and after the refurbishment.

**Table 4.** Numerical results on each floor results for heating demands. The vectorial results are presented with Dvec measuring the difference between medians. The functional results, accompanied with the average minute difference (Dfunc), the L2(*l*) distance between the mean functions (Ddist) and the smoothing adjustment (R2), are also presented. For both analyses, the change in the variability of the data ( Var) and the heating savings are displayed. Lastly, the *p*-values for the tests, both vectorial and functional, are calculated.


In the case of the vectorial analysis, Table 4 shows that, in general, the measurements are less variable in general (between 35% and 60% less). On the first floor, instead, this method detects an increase in data dispersion after the retrofitting. However, this fact is not supported by the functional approach (see Table 4). Figure 7 shows in its second row that the heating curves on this floor are similar or even less variable. In the case of functional analysis results, as shown in Table 4, the variation of the measurements is lower on all floors (between 23% and 51% less). The functional method is demonstrated to be more accurate and provides more information. For instance, functional analysis detects the demand peaks on ground floor in the morning when the heating starts, as it can be seen in second row of Figure 7. With the vector analysis, this information is lost as shown in first row of Figure 7.

The possible consequences of the retrofitting on the building temperatures are also studied. Both the vectorial approach and the functional approach conclude that the temperatures increased on every floor except on the third floor (see Figure 8 and Table 5). The tests do not detect any change in the average indoor temperature on this floor. This is probably because this floor is a large space with low occupancy where the temperatures were stable before and after refurbishment. This is supported, on the one hand, by the functional analysis in Figure 8 where the curves are almost overlapping. On the other hand, Table 5 shows that the temperature on the third floor increased very slightly and the FANOVA test does not detect a significant change (*p*-value = 0.17). On the contrary, the ground, first and second floors had higher temperatures after retrofitting, as shown in Figure 8 and Table 5. The increase, depending on the floor and method, was between 0.5 and 2 ◦C. The reason is that after the refurbishment the building is more insulated, heat losses are reduced and it is easier to keep it warmer. Moreover, the temperature set point have been increased in the common zones. Only with FDA it is detected that the retrofitting succeeds to reduce the influence of natural light on the ground floor temperatures. In the second row of Figure 8, it is observed that temperatures on the ground floor after refurbishment do not have peaks at the end of the day due to solar radiation. Finally, both analyses show that the homogeneity of monitored data is significantly improved. Table 5 shows that, on each floor and from both approaches, there is more homogeneity in the measurements related to the building's indoor temperatures.

**Figure 8.** Analysis of the temperatures on each floor measured in ◦C. In the first row, the vectorial results (in form of box plots) are presented. In the second row, instead, the functional data are represented with the respective mean functions. The data are divided into winter days before and after the refurbishment.

**Table 5.** Numerical results on each floor indoor temperatures. The vectorial results are presented with Dvec measuring the difference between medians. The functional results, accompanied with the average minute difference (Dfunc), the L2(*l*) distance between the functional mean (Ddist) and the smoothing adjustment (R2), are also presented. For both analyses, the change in the variability of the data is displayed ( Var). Lastly, the *p*-values for the tests, both vectorial and functional, are calculated.


As in the electrical analysis, the relative savings obtained in the heating analysis were calculated, as shown in Table 4. These savings reached values of more than 30% (in particular on third floor), and, in this case, with the functional approach are higher than with the vectorial approach. The floor with the smallest relative saving, observing the results of both methods, was the first floor, but with the vectorial method the reduction was almost 8% and with the functional method almost 17%. In this case, the results are more homogeneous among floors; there is not much difference from floor to floor. As expected, the form of the heating demands curves, before and after retrofitting, is the same although the values after retrofitting decreased (see Figure 7). Similar behaviour is appreciated in the temperature curves (see Figure 8).

After retrofitting the indoors temperature increased on all floors, a ventilation system with exterior air was installed and the internal gains were reduced with LED lighting. These changes should contribute to an increase of the heating demand. However, Figure 7 and Table 4 show that the heating demand on each floor has been reduced, demonstrating the effectiveness of façade insulation. Besides the decrease on the heating demands, indoor temperatures in the building are maintained or even increased, as demonstrated in Figure 8 and Table 5.
