## 4 Conclusions  
In this work the issue of heat thefts between adjacent apartments has been investigated by performing a dynamic simulation on a case-study building. To this end, a dynamic model of a real social housing building supplied by a centralized heating plant has been implemented using TRNSYS \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**from\(\mathbf{to}\)** & **IC** & **2A** & **3B** & **4C** & **5A** & **6B** & **7C** & **SA** & **9B** & **Total** \\ \hline \multirow{2}{*}{**1C**} & \multirow{2}{*}{\(-193\)} & \multicolumn{3}{c}{315} & \multirow{2}{*}{\(-122\)} & \multirow{2}{*}{\(kWh\)} \\  & & & & (6.6\%) & & & & & & (2.6\%) \\ \hline \multirow{2}{*}{**2A**} & 193 & & 228 & 739 & & & & & 1160 & kWh \\  & (n.a.) & & (n.a.) & & (n.a.) & & & & & (n.a.) \\ \hline \multirow{2}{*}{**3B**} & \multirow{2}{*}{\(-228\)} & \multirow{2}{*}{\(-3.1\%)} & \multirow{2}{*}{\(-21\)} & \multirow{2}{*}{\(-0.3\%)} & \multirow{2}{*}{\(-24\)} & \multirow{2}{*}{\(-24\)} & \multirow{2}{*}{\(kWh\)} \\  & & & & & & (\(-0.3\%)) & & & (3.4\%) \\ \hline \multirow{2}{*}{**4C**} & \(-315\) & & & \(-54\) & \(-317\) & & & \(-68\) & kWh \\  & (\(-5.0\%\)) & & & (\(-0.9\%\)) & & (\(-5.0\%\)) & & (\(-10.9\%\)) \\ \hline \multirow{2}{*}{**5A**} & \multirow{2}{*}{\(-739\)} & \multirow{2}{*}{\(54\)} & \multirow{2}{*}{\(10\)} & \multirow{2}{*}{\(-107\)} & \multirow{2}{*}{\(-107\)} & \multirow{2}{*}{\(-782\)} & kWh \\  & & & & & & (\(-2.0\%\)) & & (\(-14.5\%\)) \\ \hline \multirow{2}{*}{**6B**} & \multirow{2}{*}{\((0.3\%)\)} & \multirow{2}{*}{\((-0.2\%)\)} & \multirow{2}{*}{\((-0.2\%)\)} & \multirow{2}{*}{\((2.5\%)\)} & \multirow{2}{*}{\(153\)} & \multirow{2}{*}{\(164\)} & kWh \\  & & & & & & & (2.5\%) & (\(2.7\%\)) \\ \hline \multirow{2}{*}{**7C**} & \multirow{2}{*}{\((4.1\%)\)} & \multirow{2}{*}{\((4.1\%)\)} & \multirow{2}{*}{\((0.0\%)\)} & \multirow{2}{*}{\(2\)} & \multirow{2}{*}{\(319\)} & kWh \\  & & & & & & (0.0\%) & & (4.1\%) \\ \hline \multirow{2}{*}{**8A**} & \multirow{2}{*}{\((1.6\%)\)} & \multirow{2}{*}{\((0.0\%)\)} & \multirow{2}{*}{\(-22\)} & \multirow{2}{*}{\(87\)} & \multirow{2}{*}{\(193\)} & kWh \\  & & & & & & (1.3\%) & (2.8\%) \\ \hline \multirow{2}{*}{**9B**} & \multirow{2}{*}{\((-153\)} & \multirow{2}{*}{\(-87\)} & \multirow{2}{*}{\(-240\)} & \multirow{2}{*}{\(kWh\)} \\  & & & & & (\(-18.8\%\)) & & (\(-1.0\%\)) & & (\(-2.8\%\)) \\ \hline \hline \end{tabular}  
* to be read row by column, (-) signs meaning that the row apartment is giving energy to the column apartment; (+) signs meaning that the row apartment is stealing energy to the column apartment.  
\end{table} 
Table 7: Share * of heat given or stolen between adjacent apartments, scenario b, kWh and (%). Figure 10: Indoor temperature variation of apartments 2A, 5A and 8A in a representative week of the heating season, scenario b.  
simulation software, validated with real energy consumption data and calibrated in order to obtain a maximum deviation from the measured data in a range of +6/-12% per apartment. The final estimation error of the model on the total building energy consumption was \(-\)2%. Three scenarios were simulated, in order to highlight the differences between the possible operation of some apartments: a base scenario considering a full operation of the heating plant (i.e., all apartments heated at different set-point temperatures) and two additional scenarios in which two apartments chosen as representative of favorable and unfavorable positions were considered, individually, unheated. The analysis highlighted that the amount of the heat thefts due to different set-point temperatures in case of full operation of the building is almost negligible in low-insulated buildings, such as the analyzed one. In particular, the share of heat exchanges calculated with respect to the total annual energy need for space heating of single apartments varies between 0 and 6.2%.