## 3 Materials and Methods
The data concerning the heating season degree days (from the years 2001-2019 and the multi annual average) based on which the calculations were carried out were taken from the climate database Institute of Meteorology and Water Management-National Research Institute (IMWM-NRI) for the Lomaz region. The amount of final energy consumption was calculated using the formula:  
\[Q_{K,H}=\sum\
olimits_{i=1}^{n}\frac{HDD(t_{b})_{i}}{HDD(t_{b})_{0}}Q_{K,H_{i} }.\frac{1}{n}, \tag{1}\]  
where: \(Q_{K,H}\)--the final energy demand for the heating season, [kWh]; \(HDD(t_{b})_{0}\)--the number of degree days in a standard heating season, [\({}^{\circ}\)Cd]; \(HDD(t_{b})_{i}\)--the number of degree days for the \"i\" of this year, [\({}^{\circ}\)Cd]; \(Q_{K,H_{i}}\)--final energy consumption for heating in a measurement period for the \"i\" of this year, [kWh]. n-number of years of the measurement period. Due to the different periods of thermal improvement of buildings and the introduction of a weather-controlled central heating control system. Table 3 shows the values of measuring years n for energy consumption before and after modernization. The index of final energy demand for heating before and after the implementation of the improvement was calculated according to the formula:  
\[FE=\frac{Q_{K,H}}{A_{H}}, \tag{2}\]  
where: \(FE\)--index of final energy demand for heating, [kWh\(\cdot\)m\({}^{-2}\cdot\)year\({}^{-1}\)]; \(Q_{K,H}\)--the final energy demand for the heating season, [kWh]; \(A_{H}\)--calculated area of temperature-controlled rooms (heated surface), [m\({}^{2}\)]. For individual building states the \(FE\) indicator has been defined as follows: \(FE_{0}\)--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\({}^{-1}\)]; \(FE_{2}\)--index of final energy demand for heating converted to the conditions of the standard heating season, after the introduction weather controlled central heating system, [kWh\(\cdot\)m\({}^{-2}\cdot\)year\({}^{-1}\)]. These buildings had energy audits prepared, on the basis of which the optimum variants of thermal modernization were selected, the partitions that should be modernized were indicated and the appropriate thicknesses of layers of thermal insulation materials were selected. Some of them are measured and others calculated, as pointed out in Table 4. The table contains, among others, such information as: the year of the building's construction, the year in which the thermal improvement was performed and the year in which the modernization of the central heating system was introduced. Moreover, the table includes the following data characterizing the buildings: area and cubic capacity, area of walls through which heat losses occur, time constant of the building, power demand for heating and index of final energy demand for heating before and after the improvement consisting in insulation of external partitions. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Building**} \\ \cline{2-4}
**Building Condition** & **A** & **B** & **C** \\ \hline before modernization, [year] & 10 & 12 & 9 \\ \hline after thermal improvement, [year] & 2 & 1 & 2 \\ \hline after the introduction weather controlled central heating system, [year] & 6 & 5 & 7 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Number of years of the measurement period—n. The buildings subjected to the analysis were characterized by a similar value of the unit energy demand indicator for heating, which ranged from about 95 to 104 [kWh\(\cdot\)m\({}^{-2}\)-year\({}^{-1}\)]. The demand for heating power after thermal improvement is in the range from 125 kW (building B) to 236 kW (building C). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**No.