# NZEB Renovation Definition in a Heating Dominated Climate: Case Study of Poland
## 2 Calculation Methodology
### Stage I--Cost-Optimal U-Values for Renovated Elements of the Building Envelope  
The aim of the calculation was to determine the cost-optimal heat transfer coefficients for the elements of the building's fabric (external walls, roofs, floors, windows, etc.) for single-family houses to be renovated. The results depend on the type of building element (external wall, roof, etc. ), its initial U-value and the cost of energy. As an optimizing criterion, the minimal cumulative cost (investment + energy losses cost) was used, calculated over a 30-year period. The calculations were based on the methodology framework established by the Commission with the Delegated Act No. 244/2012 [34; 35]. The study by [PERSON] and [PERSON] [36] also shows a literature review of applied methodologies of the optimal insulation thickness calculation for the various building elements and in different climatic conditions. The cumulative cost for the purpose of the study was calculated on basis of the following formula:  
\[k_{Rd,j}=k_{M,j}+\sum_{i=1}^{30}\bigl{(}k_{E,j}\cdot R_{di}(i)\bigr{)} \tag{1}\]  
where:  
* \(k_{Rd,j}\)--unit cumulative cost for the renovation variant \(j\), EUR/m\({}^{2}\);
* \(k_{M,j}\)--renovation cost of the external building element for variant \(j\), EUR/m\({}^{2}\);
* \(k_{E,j}\)--annual cost of energy losses transferred through 1 m\({}^{2}\) of the element for variant \(j\) in the base year (depends on the thermal insulation of the external element, climate conditions and the cost of heat), EUR; and
* \(R_{di(i)}\)--the discount factor for the year \(i\), depending on the rate of inflation, the increase in the prices of energy and the discount rate. The cost optimal U-value of the renovated building element is the one for which the cumulative cost reaches the minimum value. The unit renovation cost of external building elements for different insulation thickness and techniques was calculated on the basis of the following formula:  
\[k_{M,j}=R_{j}+M_{j}+P_{V} \tag{2}\]  
where:\(P_{v}\)--tax on goods and services (8%), EUR/m\({}^{2}\);
* \(R_{j}\)--labour cost depending on building element type, insulation technique and thickness, EUR/m\({}^{2}\); and
* \(M_{j}\)--the cost of direct and auxiliary materials along with purchase costs (5.9%), EUR/m\({}^{2}\). The renovation cost was higher than the cost of new construction because it includes additional works like strengthening external walls or removal of old windows. The maximum analysed insulation thickness or minimum U-value was limited to the standard market available one. For example, in the case of ETICS (External Thermal Insulation Composite System) systems for external wall insulation it was 30 cm. The annual energy cost was calculated with the simplified method derived from the Polish energy audit regulation [37] with use of the following equation:  
\[k_{E,j}=0.0000864\cdot U_{j}\cdot Sd\cdot E \tag{3}\]  
where:  
* \(U_{j}\)--heat transfer coefficient for the variant j, W/m\({}^{2}\) K;
* \(Sd\)--heating degree days (HDD), Kd; and
* \(E\)--energy cost, EUR/GJ. The numbers of HDD for the main Polish cities is presented in Table 3. These values vary from 3488 (Opole) to 4117 (Olsztyn) [38] and are typical for a heating dominated climate. For comparison, the number of HDD in the European cities ranges from 500 [Kd] (Limassol, Cyprus) to 7000 (Kd) (Tromso, Norway) according to Eurostat [39]. Table 3 shows that the highest density of Polish cities falls at the value of around 3700 HDD. This value was used as a basis for further analysis. Another aspect, key because of the significant influence on the annual cost of energy losses through the building element, affecting strongly the results of the optimization was the type of energy carrier used for heating of the building. The price of 1 GJ of energy was significantly diversified and depending on the energy carrier type. Table 4 shows current energy prices in Poland for chosen energy carriers, ignoring the efficiency of the heat source.