## 3 Materials and Methods
**} & \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Abbreviation**} & \multicolumn{4}{c}{**Building**} \\ \cline{3-6}  & & **A** & **B** & **C** \\ \hline
**1** & construction year of a building, [year] & \(C_{A}\) & 1984 & 1992 & 1994 \\ \hline
**2** & year in which the thermal improvement was performed, [year] & \(C_{I}\) & 2012 & 2014 & 2011 \\ \hline
**3** & year in which the installing weather controlled central heating system, [year] & \(C_{\text{L}_{\text{R}}}\) & 2013 & 2015 & 2012 \\ \hline
**4** & calculated from exterior measurements the heated volume of building, [m\({}^{3}\)] & \(V_{e}\) & 9876 & 10294 & 18376 \\ \hline
**5** & calculated from interior measurements total (net internal area), [m\({}^{2}\)] & \(A_{\text{L}_{\text{R}}}\) & 2231 & 1646 & 3572 \\ \hline
**6** & calculated surface of heated floors from interior measurements, [m\({}^{2}\)] & \(A_{f}\) & 1576 & 1051 & 2235 \\ \hline
**7** & calculated from exterior measurements surface of roof projection area & \(A_{e}\) & 602 & 869 & 1141 \\ \hline
**8** & calculated from exterior measurements total walls’ surface (net) area, [m\({}^{2}\)] & \(A_{uv}\) & 1560,1 & 1417,3 & 2762,3 \\ \hline
**9** & calculated surface of floor from interior measurements (floor over basement or floor on the ground), [m\({}^{2}\)] & \(A_{\text{G}}\) & 446 & 549 & 714 \\ \hline
**10** & calculated from exterior measurements total windows area, [m\({}^{2}\)] & \(A_{uv}\) & 383,5 & 272,5 & 742,2 \\ \hline
**11** & number of storeys, [pc\({}_{\text{J}}\)] & \(N_{\text{O}_{\text{S}}}\) & 5 & 3 & 5 \\ \hline
**12** & number of residential flats, premises [pc\({}_{\text{J}}\)] & \(N_{\text{O}_{\text{S}}}\) & 45 & 30 & 75 \\ \hline
**13** & number of living persons per building [N\({}_{\text{J}}\)] & \(N_{\text{O}_{\text{S}}}\) & 145 & 99 & 195 \\ \hline
**14** & shape coefficient of buildings (the ratio surface to volume, [m\({}^{2}\)-m\({}^{3}\)], [m\({}^{-1}\)] & \(S\)V\({}_{e}\) & 0,37 & 0,36 & 0,38 \\ \hline
**15** & calculated thermal transmittance of walls components before modernization, [W\(\cdot\)m\({}^{-2}\)-K\({}^{-1}\)] & \(U_{\text{L}_{\text{S}}}\) & 0,57 & 0,44 & 0,36 \\ \hline
**16** & calculated thermal transmittance of walls components before & \(U_{\text{L}_{\text{P}}  
The calculated building time constants \(\tau\) (according to EN 13790 [38]) range from 62 h for building B to 78 and 84 h for buildings A and C respectively. They can be classified as \"medium\" buildings [34], with buildings A and C having a higher thermal inertia compared to building B. For the analyzed buildings, the amount of energy savings was calculated for two states:  
(a) after thermal improvement, where the energy saving in percent is calculated according to equation:  
\[FE_{\%,ES,a}=\left(1-\frac{FE_{1}}{FE_{0}}\right)100, \tag{3}\]  
where: \(FE_{\%,ES,a}\)--the energy savings achieved through thermal improvement of buildings in percent; \(FE_{0}\)--(average) index of final energy demand for heating converted to the conditions of the standard heating season, before modernization, [kWh\(\cdot\)m\({}^{-2}\cdot\)year\({}^{-1}\)]; \(FE_{1}\)--index of final energy demand for heating converted to the conditions of the standard heating season, after thermal improved, [kWh\(\cdot\)m\({}^{-2}\cdot\)year\({}^{-1}\)]. (b) after the introduction weather controlled central heating system. The energy saving in percent is calculated as:  
\[FE_{\%,ES,b,i}=\left(1-\frac{FE_{2,i}}{FE_{1,r}}\right)100, \tag{4}\]  
where: \(FE_{\%,ES,b,i}\)--the annual energy savings achieved through the introduction weather controlled central heating system in percent; \(FE_{2,r}\)--index of final energy demand for heating for the \"i\" of this year, converted to the conditions of the standard heating season, after the introduction weather controlled central heating system, [kWh\(\cdot\)m\({}^{-2}\cdot\)year\({}^{-1}\)]; \(FE_{1,r}\)--index of final energy demand for heating converted to the conditions of the standard heating season, in the reference year, [kWh\(\cdot\)m\({}^{-2}\cdot\)year\({}^{-1}\)].