## 4 A Case Study of a Low-Energy House in Bialystok (NE Poland)
The assumptions used for the calculations are summarized in Table 4.  
Table 5 contains the designed energy performance of the building [135] and energy indicators determined on the basis of the actual electrical energy consumption in the building over a period of 6 years (with the use of a remote heat pump manager). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Assumption** & **Value** & **Units** \\ \hline Design indoor air temperature & 20 & \({}^{\circ}\)C \\ Air exchange rate & 0.5 & 1/h \\ Envelope airtightness in n\({}_{50}\) (based on the air tightness test) & 0.48 & 1/h \\ Value of internal gains & 3.0 & Wm\({}^{2}\) \\ Seasonal space heating energy efficiency & 3.55 & - \\ Seasonal average efficiency of regulation and & & \\ heat use in the heated space/heating system; & & \\ heat transfer of the heating system and & & \\ energy storage efficiency & & \\ Designed hot water temperature & 55 & \({}^{\circ}\)C \\ Designed cold water temperature & 10 & \({}^{\circ}\)C \\ Units of hot water consumption & 6.8 & W/m\({}^{2}\) \\ Water heating energy efficiency & 3.10 & - \\ Seasonal average efficiency of DHW & 0.51 & - \\ preparation system & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Assumptions used for the energy need calculations [135]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Location of Thermal Bridges in the Building** & **\begin{tabular}{c} **\(\Psi\)-Value** \\ **(W/mK)** \\ \end{tabular} \\ \hline Wall/Wall & \(-\)0.084 \\ \hline Wall/Roof & \(-\)0.055 \\ \hline Wall/Floor on the ground & \(-\)0.078 \\ \hline \multirow{3}{*}{Windows} & \begin{tabular}{c} lintel \\ \end{tabular} & 0.007 \\ \cline{2-3}  & \
\begin{tabular}{c} reveal \\ \end{tabular} & \(-\)0.019 \\ \cline{2-3}  &
\begin{tabular}{c} sill \\ \end{tabular} & 0.007 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: A list of linear heat loss coefficients (\(\Psi\)-value) of the thermal bridges in the building in question [135]. The technical solutions used in the building made it possible to meet the requirements of the NF15 standard (which concentrates on the demand for usable energy for heating and ventilation), namely \(EU_{H}\) = 10.4 kWh/(m\({}^{2}\)year) \(<\) 15 kWh/(m\({}^{2}\)year). Since newly constructed buildings in Poland must meet the requirements of the Technical Conditions regarding the non-renewable primary energy demand indicator (\(EP\)), it was designated for the analyzed case study at the design stage. The amount of non-renewable primary energy depends on the amount of final energy and the coefficient of yield of non-renewable primary energy to produce and supply the energy carrier or energy (\(w_{i}\)). It is worth noting that in Poland this coefficient for electricity \(w_{i}\) = 3.0 [130] and is the highest among all heat sources. \(EP\) = 70.88 kWh/(m\({}^{2}\)year) for the analyzed building met the requirements of \(EP_{max}\) = 120 kWh/(m\({}^{2}\)year) in force at the time of its design but does not meet the current requirements (\(EP_{max}\) = 70 kWh/(m\({}^{2}\)year) according to WT2021 standard). Therefore, in order to meet the current requirements of thermal protection of buildings in Poland, it is necessary to use renewable energy. However, to reach ZEB or even the plus energy standard, the amount of energy produced on-site must be equal or higher than total final energy consumption over the course of a year [142]. Therefore, it was decided to analyze the use of photovoltaic solar panels for electrical energy production in the analyzed case study. The surface of the roof of the house on which the panels can be installed was adequate. Moncrystalline panels (inko-Cheetah JKM400M 72H-V with maximum power 400 Wp, efficiency STC = 19,88% and dimensions: 2008 \(\times\) 1002 \(\times\) 40 mm [143]) were selected.