*2.2. Simulation Model*

A dynamic simulation of the case-study building has been performed by the authors by means of the energy tool TRNSYS-17 [21], which allows to simulate the dynamic behavior multi-zone buildings and their systems. The software carries out the simulation basing on a thermal balance (typically on an hourly basis) taking into account the e ffects of the accumulation and of the thermal release of the opaque building envelope [22,23].

Both the building envelope and the heating system were modeled and simulated and the results were validated according to the real thermal energy measured at both the building level (through the main thermal energy meter) and at the dwelling level (by means of the data collected by the HCAs installed in the dwellings). The heating season 2017–2018 (approximately from November 1st to April 15th) has been chosen as reference period for the validation.

The 3-D building-type model was developed through the Google SketchUp plug-in "Trnsys3d" [24], as shown in Figure 3, in which the thermo-physical properties associated to the building envelope are the ones collected in Table 1.

Due to technical reasons (in Trnsys3d thermal zones must be convex), 24 thermal zones have been used to model the apartments, namely: 3 for type B and type C dwellings and 2 for type A dwellings. For the attic, 8 thermal zones with pitched roof have been implemented. The stairwell has been represented with one thermal zone for each floor.

**Figure 3.** 3-D model of the building case-study.

The infiltration rate (i.e., the number of air volumes in the unit of time entering the building from the outside through the windows due to their inadequate closure) has been set according to the conventional values for residential buildings [25], which depend on air tightness but also on wind speed.

The only considered internal heat gain contribution was the one coming from the inhabitants, since most of the apartments are not equipped with highly energy-consuming electrical devices and only a few of the rooms are actually used by the inhabitants.

The heating system has been modeled as the summation of four components by means of specific system-types, accounting for: (*i*) the emission system, whose heating power represents an input and is given in Table 2; (*ii*) the generation system, whose efficiency is an input parameter and has been calculated as the average generation efficiency of two consecutive years measured during the experimental campaign (i.e., the ratio between the thermal energy consumption measured by the direct heat meter installed downstream to the boiler and the natural gas consumption expressed in kWh and measured by the building natural gas meter); (*iii*) the system circulators; (*iv*) distribution and mixing systems.

Given the presence of TRVs on the emission terminals, with which the inhabitants can adjust the indoor temperature of the rooms, a fluctuant set-point temperature has been associated to each thermal zone. This is represented by the average weekly value assumed by the mean temperature measured through the T-logger sensors installed in the dwellings. The operation of the generation system is controlled by a daily and a seasonal control schedule, simulated, respectively, according to the actual on-off schedule of the heating system (from 6 to 8 a.m. and from 15 to 22 p.m.) and to the heating period established according to the Italian law [26] for the reference climatic zone (climatic zone: D, heating degree-days (HDD): 1911, heating period: 1st November–15th April).

## **3. Results and Discussion**

For each thermal zone, the heating demand was obtained for the base scenario, i.e., the simulation of the full operation of the building, where all apartments are heated with a fluctuating set-point temperature determined as the mean weekly temperature measured by the temperature sensors installed in the dwellings.

Figures 4–6 show the hourly distribution of the heating load for the analyzed dwellings obtained by the dynamic simulation, each representing a macro thermal-zone resulting from the summation of **Figure 4.** Simulated heating load of apartments 1C, 2A and 3B.

the thermal zones simulating the dwellings. For simplicity reasons, the graphs show only the heating season (i.e., from 1st January to 15th April, and from 1st November to 31st December).

**Figure 6.** Simulated heating load of apartments 7C, 8A and 9B.

The annual heating consumption was calculated as the area below the power curve obtained by the simulation. Both the estimated and the measured energy consumptions were normalized with respect to the reference HDDs of the building location (1911) as reported by Equations (1) and (2).

$$\text{(normalized consumption)}\_{\text{estimated}} = \frac{\text{estimated consumption}}{\text{HDD}\_{\text{(weather life)}}} \cdot 1911 \tag{1}$$

$$(\text{normalized consumption})\_{\text{measured}} = \frac{\text{measured consumption}}{\text{HDD}\_{2017-18}} \cdot 1911 \tag{2}$$

where *HDD*(*weather file*) are the HDDs calculated for the weather file used for the simulation and *HDD*2017−<sup>18</sup> are the HDDs of the heating season of 2017–2018 of the building location obtained by the local weather database.

A calibration of the model has been performed, aimed at decreasing the magnitude of the simulation error in some apartments. In fact, after the first simulation, apartments 5A, 6B, 7C and 8A showed the highest simulation errors (see Table 3). In this regard, information was retrieved by the authors in the field, during meetings specifically organized to collect data about end-user's behavior. In particular, it has been found that users of those apartments used to open their windows during the heating hours, instead of employing the TRVs, to decrease the indoor temperature. For this reason, for the above-mentioned apartments, a calibration has been performed by the authors by increasing the ventilation rate until the error decreased to a range of ±15%.

**Table 3.** Results of validation before and after the calibration process (respectively BC and AC).


Results are reported in Table 3, namely: the normalized measured energy consumption of each dwelling for the heating season 2017–2018 (estimated basing on the reading of the HCAs); the normalized estimated energy consumption (i.e., obtained by means of the dynamic simulation implemented using TRNSYS17), respectively before and after the calibration process (BC and AC); the error between the estimated and the measured normalized thermal energy consumption before and after the calibration process (BC and AC), calculated as per Equation (3).

$$error\,\left[\%\right] = \frac{\text{estimated consumption} - \text{measured consumption}}{\text{measured consumption}} \cdot 100\tag{3}$$

In Table 3, only the energy consumptions normalized with respect to the reference HDDs of the building location are reported.

As shown in Table 3, following the calibration process on dwellings 5A, 6B, 7C and 8A, the error calculated with respect to the total energy consumption of the building decreased to −2%, while the one of single dwellings decreased to a range of <sup>+</sup>6/−12%. Thus, the validation results have been considered to be acceptable for the purpose of the present analysis.

Indeed, these small deviations may be caused by di fferent causes, as following:


*Appl. Sci.* **2020**, *10*, 2436

The heat flowing through the adjacent wall (*Qh* [Wh]) of two dwellings in one hour, *h,* has been calculated as per Equation (4):

$$Q\_h = \mathbb{U} \cdot A \cdot (T\_{in} - T\_{out}) \cdot h \tag{4}$$

where U, [W m<sup>−</sup><sup>2</sup> <sup>K</sup>−1] is the thermal transmittance of the wall, A [m2] is the area of the wall and *Tin* − *Tout*, [*K*] is the temperature difference between the inner and the outer layer of the wall (with *Tin* indicating the temperature measured at the inner surface—i.e., the surface of the analyzed dwelling—while *Tout* the outer surface temperature—i.e., the temperature of the adjacent dwelling).

The total heat stolen/given through each wall separating adjacent dwellings was calculated as per Equation (5).

$$Q\_{\text{stolen}/\text{griven}} = \sum\_{h=1}^{8760} Q\_h \tag{5}$$

In this way, the heat calculated as per Equations (4) and (5) is positive when the reference environment is gaining heat; on the contrary, it is negative when the heat is transferred from the reference environment to the adjacent one.

For each *i*-th apartment, the heat flowing through the walls adjacent with other apartments was calculated and it was assessed how much these heat flows would impact on the total energy consumption of the apartment itself as per Equation (6).

$$\frac{Q\_{stolen/given,i}}{Q\_{tot,i}} \cdot 100\tag{6}$$

The results are reported for each apartment in Table 4 for the base scenario first.


**Table 4.** Percentage \* of heat given or stolen between adjacent apartments, base scenario.

\* 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.

It is worth to observe that results obtained by this calculation depend both on the set-point temperature and the 'position' of the dwelling with respect to the others. For sake of completeness, a summary of the average seasonal value of the set-point temperature used to simulate the base scenario is given in Table 5.

Generally speaking, results show that in case of full operation of the building, the phenomenon of heat-thefts is almost negligible, representing on average 0.5% of the annual energy consumption for heating purposes of a single dwelling with the maximum absolute value of 6.2%. Observing the apartment with the lowest average set-point temperature (1C), it is possible to notice that this "steals heat" from both apartments 2A and 4C (respectively 1.0% and 6.2% of its total energy need for space heating).


**Table 5.** Average seasonal values of set-point temperature used for the simulation of the base scenario.

On the other hand, even a central apartment that could benefit from its position (as for example apartment 4C) can nullify this advantage when using set-point temperatures higher than the one of adjacent apartments, losing up to 10% of its energy need.

However, an unexpected behavior can be highlighted for apartments 2A, 5A and 8A. Although, in fact, apartment 5A has set the lowest average set point temperature, it still gives heat to both adjacent apartments on the same vertical (2A and 8A). This is because set-point temperature is reached in the apartments only during the operating hours of the heating plant, while, for the rest of the day, the temperature inside the apartment 5A is always higher than that in 8A and 2A, due to its favorable position. This is evident in Figure 7, which reports the indoor temperature for the above-mentioned three apartments in a representative week of the heating season.

**Figure 7.** Simulated indoor temperature variation of apartments 2A, 5A and 8A in a representative week of the heating season, base scenario.

Referring to Figure 7, it is highlighted that in a day there are two "heating periods", according to the schedule of the heating plant (from 6 to 8 a.m. and from 15 to 22 p.m.). From the figure, it is evident that the installed heating system quickly manages to bring the apartment up to the set-point temperature, even in situations of daily temperature excursions. For the sake of completeness, Figure 8a,b show the trends of the daily indoor temperature in the above-mentioned apartments during the warm-up times simulated for January the 5th, which is representative of a cold winter day. From the figure, it can be highlighted that, even under high load conditions (i.e., low outdoor temperature), the heating system is very e ffective and set-point temperatures are reached in less than about 15 min. Generally speaking, this limited warm-up time mainly depends on the time constant of the heating system and, to a lesser extent, on that of the building. Obviously, this heating transient is not representative of the indoor comfort conditions of the room, that are determined by both the indoor air temperature and the mean radiant temperature, which depends on the thermal mass of the building.

**Figure 8.** Simulated indoor temperature trends of apartments 2A, 5A and 8A during the warm-up time of a representative day: (**a**) from 5:00 a.m. to 7:00 a.m.; (**b**) from 14:00 p.m.to 16:00 p.m.

On the other hand, the low thermal mass of the building is the cause of high temperature drops when the generator is turned <sup>o</sup>ff, especially in apartments in disadvantaged positions, despite their heat thefts from adjacent dwellings. Finally, it is worth to observe the behavior of the building during non-heating hours with the related heat exchanges among apartments.

For the purpose of evaluating the heat gains and losses due to a change of operation, the authors simulated two different scenarios in addition to the base one: (i) scenario a, in which only apartment 5A was assumed to be unheated; (ii) scenario b, in which only apartment 2A was assumed to be unheated. In both cases, the set-point temperatures of all other apartments remained unchanged with respect to the base scenario. These were chosen as representative of the most and the least favored positions of block A (but similar considerations can be made for block B except for minor changes due to the different orientation): in fact, apartment 5A is located in the second floor and surrounded by heated spaces (apartments 2A, 6B, 4C and 8A) while apartment 2A is located at the first floor and its uninsulated floor is completely exposed to outdoor temperature.

Table 6 shows the results obtained from switching off the heating system of the dwelling 5A (scenario a).

Obviously, in this case all the thermal zones adjacent to apartment 5A give energy to it, with apartment 2A and 8A giving, respectively, 8.5% and 5.2% of their total energy need for space heating (in fact these apartments have the largest heat exchange surfaces). Overall, apartment 5A is able to steal 1249 kWh from its neighbors, which would have accounted for 25% of its energy consumption in the base scenario.

Referring to Figure 9, it is important to highlight that when the heating system is not working, in some hours of the week, apartment 5A still has its indoor air temperature higher than the one of both adjacent dwellings, confirming again the existence of an inversion in the direction of the heat flows already highlighted in the base scenario. In this scenario, the average seasonal difference of indoor temperature between apartment 5A and the other heated apartment is about 2.6 ◦C. Table 7 shows the results obtained from switching off the heating system of the dwelling 2A (scenario b) while Figure 10 shows the trend of the indoor temperature of the investigated apartments in a representative week of the heating season.


**Table 6.** Percentage \* of heat given or stolen between adjacent apartments, scenario a, kWh and (%).

\* 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.

**Figure 9.** Indoor temperature variation of apartments 2A, 5A and 8A in a representative week of the heating season, scenario a.

The results show that apartment 2A is always able to gain heat from its neighbors (apartments 5A, 1C, 3B,) since its indoor temperature is always lower than that of the other ones, stealing up to 13.6% of the total energy need for space heating of apartment 5A. Overall, apartment 2A is able to steal 1160 kWh from its neighbors, which would have accounted for 20% of its energy consumption in the base scenario. However, in this scenario, the average seasonal difference of indoor temperature between apartment 2A and the other heated apartment is about 3.6 ◦C, which is 1.0 ◦C lower than the one of apartment 5A in the similar scenario.

From the analysis of the results it can be observed that heat transfer between a heated dwelling toward an adjacent unheated, ranges 20–25% of its theoretical energy need, which is much lower than the estimated 70–90% in available literature [10,12,13]. This is essentially due to the fact that the investigated building is poorly insulated (i.e., low thermal insulation both in the internal partitioning walls and towards the external environment); thus, the energy need of dwellings is mainly determined by their thermal dispersion towards the outdoor environment (compared to which the heat transfers toward other apartments are certainly lower). On the contrary, in [8–10,12,13] the investigated case-study buildings presented highly insulated external facades and they were mainly located in cold continental climates. The milder climatic conditions under which the present simulation has been carried out also affect the extent of heat stolen, thanks to not-negligible solar heat gains, which in scenario a determine an inversion of heat exchanges during the central hours of sunny days.


**Table 7.** Share \* of heat given or stolen between adjacent apartments, scenario b, kWh and (%).

\* 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.

**Figure 10.** Indoor temperature variation of apartments 2A, 5A and 8A in a representative week of the heating season, scenario b.

As a matter of fact, in a Mediterranean climate, the compensation of heat costs based on average indoor temperature [7–9] or on static heat flows [11], would tend to overestimate the stolen heat between apartments, not taking into consideration the dynamic behavior of this phenomenon. It is highlighted that the results obtained within the present research could be useful to analyze similar buildings in terms of climatic conditions and orientation. However, in order to allow a greater and general applicability of the results, it would be necessary to carry out a dedicated sensitivity analysis to the variation of the abovementioned parameters.
