**1. Introduction**

As well known, in recent years targeting energy efficiency in residential buildings has become a key issue for the European Union (EU) which has committed itself to cut its greenhouse gas emissions to 80–95% below 1990 levels by 2050 [1]. Indeed, buildings are responsible for 36% of global final energy consumption in Europe and for nearly 40% of total direct and indirect CO2 emissions [2], of which about 70% is due to heating the existing building stock. For this reason, in 2012 the EU issued the so-called "Energy Efficiency Directive" (EED) [3] amended in 2018 as part of the "Clean energy for all Europeans package", establishing a set of binding measures to help the EU reach its energy efficiency target. In particular, Article 9 highlights the need to ensure that final customers are provided with meters that accurately reflect their actual energy consumption and set as mandatory the installation of individual meters in buildings supplied from a central source, whether sub-metering is cost-efficient or the related costs are proportionate in relation to the potential energy savings [4]. In this way, end-users should be aware, and at the same time responsible, of their energy consumption.

This obligation has led to a number of issues related to a fair billing service [5,6]. Indeed, fair and transparent heat cost allocation for heating/cooling/hot water costs is a difficult subject. In fact, especially in old and existing buildings, the lack of insulation of walls can generate heat transfers between adjacent apartments, when these are heated at different set-point temperatures (the so-called "stolen heat" or "heat thefts"), being the cause of involuntary over-consumptions for dwelling surrounded by numerous unheated or occasionally heated apartments. In this sense, a number of authors addressed the problem, with some [7–9] from the perspective of heat cost allocation, others [10–13] analyzing the phenomenon in a quantitative manner.

Within the first group, Liu et al. [7] presented an alternative method for allocating heat costs in multi-apartment buildings based on measuring the on-off ratio of the thermostatic valves on the radiators. A cost-efficient method was developed for reallocation of heating costs based on heat transfers between the adjacent apartments by Siggelsten [14] which was then further developed by Michnikowski [9] basing on the data of the average indoor temperature provided by special heat cost allocators.

Among the second group, Gafsi and Lefebvre [10] demonstrated that an apartment can gain up to 90% of heat from adjacent apartments and highlighted the complexity of the phenomenon and the large number of influence factors. Pakanen and Karjalainen [11] proposed a method for estimating static heat flows between adjacent rooms in a hypothetical environment simulated using TRNSYS software. Luki´c et al. [12] showed that, in fully insulated buildings, an unheated dwelling can steal about 80% of its total energy need from the surrounding apartments, by simulating a real building using EnergyPlus simulation software. In [13], an energy simulation is adopted to calculate the heat transfer proportion with the validation of on-site measurement, showing an unexpected amount of adjacent room heat transfer up 70% of the total heating load.

That said, it is clear that apartment location and operation can play a significant role on its energy consumption. In fact, depending on the type of building, dwellings with favorable positions, such as those with a high interior to total wall surface ratio, the ones south oriented, etc., can have significant benefit by gaining heat from surrounding dwellings, if compared to apartments located in more disadvantaged positions.

In this paper, the issue of heat thefts has been investigated by performing a dynamic simulation on a real case-study represented by a social housing building supplied by a centralized natural gas boiler, using TRNSYS simulation software. To this end, the real operation of the building (base scenario) has been simulated and the model has been validated against real energy consumption data collected by means of direct and indirect heat metering systems installed. Thanks to the validated model, it was possible to simulate two additional scenarios (scenarios a and b) in which two apartments representative of the most and the least favored positions were considered, individually, unheated.

The main aim of this work is to analyze different scenarios of building operation, with particular reference to the effects of heat transfers between adjacent dwellings in terms of energy consumption and heat cost allocation under uneven use of the heating system (in terms of occupancy of the dwellings and different set-point temperatures). The dynamic of heat transfers has been simulated with reference to an Italian building with low thermal energy performances and in a Mediterranean climate, to highlight the differences with other existing studies. Thus, the novelty of this work relies on the fact that the authors focused on the issue of heat thefts in buildings from the point of view of the allocation of heating costs among the dwelling units. This allowed for greater understanding of the dynamic behavior of the heat fluxes in relation to the single apartments rather than the entire building, especially considering that the Mediterranean climatic conditions may enhance the dynamics of the heat flows due to temperature excursions and solar heat gains during the day.

The main findings of the analysis revealed dynamic effects of "inversion" of the heat transfers between adjacent apartments, that would not have been evident in static conditions, and that should be carefully taken into account for heat cost allocation in social housing context. The results may be particularly useful because the case-study is representative of the Italian building stock in terms of constructive characteristics and heating plant and also because the influence of solar heat gains

in Mediterranean climate conditions on the heat theft phenomenon is highlighted through the dynamic simulation.

## **2. Materials and Methods**

## *2.1. The Building Case-Study*

The object of this study is represented by a low-insulated building located in a Mediterranean climate. To this end, a social housing building located in the province of Frosinone, Italy has been chosen as the subject of this study and simulated under both the hypothesis of uneven operational temperatures and occasional heating. In particular, the building has been studied under actual operational conditions, meaning that real set-point temperatures and operation hours of the heating system e ffectively set by the users were employed for the simulation. Specifically, the set-point temperatures of the fully-occupied building simulation varied between 18.6 ◦C and 22.1 ◦C as a function of the given end-user, while the operation hours were equal for all the dwellings (i.e., from 6 to 8 a.m. and from 15 to 22 p.m. according to the real settings of the centralized heating plant). The outdoor climatic conditions are those typical of the climatic zone in which the building is located and vary between (23.2/−2.1 ◦C) in the simulation period.

The building was built in 1979 by ATER, the Italian Territorial Agency for Social Housing Buildings. The hydronic heating system consists on a central natural gas boiler with a maximum power of 152 kW, located in the ground floor, with uninsulated distribution pipes running mainly in the outer walls in vertical configuration [15] and emission terminals consisting of traditional cast iron radiators.

Nine dwellings are arranged in two blocks each of which with three floors. The first, consisting of three dwellings, is located above the front porch (type C); the second, located above the garages, is made up of six dwellings (two on each floor), three north-west oriented (type B) and three south-west oriented (type A). Figure 1a,b shows, respectively, the cross section of the building and the plant scheme of a representative floor, while Figure 2 shows a picture of the investigated building.

**Figure 1.** (**a**) Cross-section of the building case-study; (**b**) plant scheme of a representative floor.

Dwellings of types A and B have a net floor area of about 79 m2, while type C ones have a net floor area of about 86 m2. The net interstorey height is about 2.7 m.

It is underlined that almost all the walking surface of the first floor, which is located above the garages and the porch, is exposed to the external environment (the garages have open doors), as well as almost all the ceiling of the last floor faces the unheated attic.

**Figure 2.** Picture of the building case-study.

The main thermal and physical characteristics of the investigated building, collected through on-site inspections and document review, are listed in Table 1. A distinction has been made between two different types of floors, since the outer floor of the dwelling located on the porch has undergone a thermal insulation with the addition of an external layer of expanded polystyrene 0.05 m thick, while the outer floor of the other apartments located at the first floor did not.


**Table 1.** Main constructive characteristics of the investigated building.

In September 2017, two-sensors—Heat Cost Allocators (HCAs) and Thermostatic Radiator Valves (TRVs)—were installed on each radiator to comply with the EED obligation [16], transposed in Italy with the Legislative Decree 102/2014 and subsequent amendments and integrations [17]. To allow an optimal system operation, a variable speed circulation pump was installed in the central boiler room. Together with the HCAs and the TRVs, two temperature sensors were installed in the bedroom and in the living room of each dwelling, which are able to measure the daily average temperature of the room as an average of 24-h measurements with a 6-min acquisition frequency. Additionally, a main thermal energy meter was installed on the main distribution pipes downstream of the centralized boiler in order to measure the total energy consumption of the building.

The installed heating powers have been estimated through the dimensional method [18,19], after mapping the dimensions of all radiators, which was also required as preliminary action for the purposes of programming the HCAs. In the context of a specific project, survey questionnaires have been administrated to the inhabitants of the investigated building and on-field informative meetings have been organized, aimed at designing a suitable feedback strategy to enhance end-user's awareness. In particular, the analysis of surveys allowed authors to collect specific information about behavioral features which have been also meaningful for the present analysis, such as the ability to use thermostatic valves and chrono thermostat, and their feeling about indoor temperature in the apartment [20].

Table 2 contains the data related to the calculated installed heating power and the number of inhabitants of each apartment of the building case study.


**Table 2.** Installed heating power and number of inhabitants of each dwelling.
