**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

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 <sup>+</sup>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 di fferences between the possible operation of some apartments: a base scenario considering a full operation of the heating plant (i.e., all apartments heated at di fferent 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 di fferent 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%. Dynamic e ffects of inversion of the heat transfers between adjacent apartments have also been highlighted in the same scenario when an intermediate floor apartment, although having a lower set-point temperature during the heating hours, registered a higher indoor temperature compared to when the heating system was not working.

It has also been highlighted that, for the same end-user behavior (i.e., switching o ff the heating system), the benefit obtained from the favorably positioned apartment is higher to that of the apartment located in a disadvantaged position in terms of indoor temperature, although the amount of heat thefts of the apartment with a disadvantaged position is higher.

In fact, depending on apartment location, the thermal mass of the apartments in a favorable position would prevent their indoor temperatures from falling below acceptable threshold values if the users of those dwellings decide to turn down their radiators through the TRVs. On the contrary, this would not be true for apartments in unfavorable positions.

The results obtained show that greater attention should be paid to heat thefts for the purpose of heat cost allocation, especially in buildings with occasional and uneven occupation and/or operation and in buildings with low thermal performances (in particular those with poorly insulated partition surfaces between apartments).

More attention should be given to the heat theft phenomenon both in terms of: (*i*) design of new buildings or of retrofit interventions. The building envelope and the heating system should be designed keeping in mind the real operating conditions while normally, in design phase, the apartments are considered to be always heated, even in buildings (such as mountain residences) normally employed for occasional occupation; (*ii*) development of specific compensation techniques for heat cost allocation in buildings with particular reference to the estimation of the "involuntary" part of energy consumption (i.e., the one which is not determined by the end-user behavior). Moreover, this study highlights the need of an extensive analysis regarding compensation due to heat transfers between adjacent apartments in mild climatic conditions, which is still required to obtain a heat cost allocation methodology easily replicable for designers and building managers.

**Author Contributions:** Conceptualization M.D., G.P.; methodology, M.D., G.F., L.C.; formal analysis, L.C., V.B.; investigation, L.C., V.B.; resources, M.D., G.P.; data curation, L.C., V.B.; writing—original draft preparation, L.C.; writing—review and editing, L.C., G.F., M.D., V.B., G.P.; supervision, G.F., M.D., G.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work has been developed under the projects "Ricerca di Sistema Elettrico PAR 2016" funded by ENEA (grant number I12F16000180001) and "PRIN Riqualificazione del parco edilizio esistente in ottica NZEB" funded by MIUR (grant number 2015S7E247\_002). The authors wish to thank ATER of Frosinone, the Territorial Agency for Social Housing, for the technical support during the on field experimental campaign.

**Conflicts of Interest:** The authors declare no conflict of interest
