## 2 Methodology
### Simulated Dwellings
* Since the insulation was assumed to be consistent with the type of climate, a poorly insulated building in a very cold climate was deemed to be better insulated than a poorly insulated building in a climate with milder winters. * The reference values assigned to the three levels of building insulation in each climate were drawn from the existing legislation in the different areas (see Table 1). The established law for the assignment of U-values was based on the degree days of the coldest month of each city. The general outcome was that existing buildings were the least insulated, while the highly insulated ones presented values above the minimum legal requirements. The decision was taken to establish a simple, climate-consistent rule to determine the three levels of insulation in each city since wall and roof thermal resistance (1/U-value) would vary linearly with the degree days in the coldest month in that city, as shown in Equation (1). The relationship obtained by establishing linearity between degree days and thermal resistance was obviously inverse with the U-values, as depicted in Equation (1) and Figure 2.  
\[\frac{1}{U}=k\cdot DD_{CM} \tag{1}\]  
where \(U\) is the U-value of the walls or roof, \(k\) a constant whose value depends on the type of element and insulation level (Table 2), and \(DD_{CM}\) the heating degree-days using a base of 20 \({}^{\circ}\)C in the coldest month in each city. In Figure 2, the triangular and rhomboid data points are the values listed in Table 1 further to the applicable legislation in each city, and the dotted lines are the values calculated with Equation (1) for the three degrees of insulation entered in the model for the simulations. The lower line plots the values for highly-insulated dwelling, the middle one for intermediately-insulated ones, and the upper line corresponds with poorly insulated dwellings. \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Country** & **Area** & \begin{tabular}{c} **Köppen–Geiger** \\ **Class** \\ \end{tabular} & \(\text{DD}_{\text{CM}}\) & \begin{tabular}{c} **U-Value** \\ **(Walls)** \\ \end{tabular} & \
\begin{tabular}{c} **U-Value** \\ **(Roof)** \\ \end{tabular} &
\begin{tabular}{c} **U-Value** \\ **(Windows)** \\ \end{tabular} & **Ref.** \\ \hline  & & & \({}^{\circ}\)C & \(\text{W/(m^{2}\text{ K})}\) & \(\text{W/(m^{2}\text{ K})}\) & \(\text{W/(m^{2}\text{ K})}\) & \\ \hline Italy & Lazio (Rome) & Csa & 372 & 0.50 & 0.46 & & [34] \\ Japan & Kanto (Tokyo) & Cfa & 423 & 0.53 & 0.37 & 4.65 & [35] \\ Turkey & Istanbul & Csa & 444 & 0.60 & 0.40 & 2.40 & [36] \\ Spain & Madrid & BSk & 452 & 0.66 & 0.38 & 3.00 & [37] \\ United Kingdom & London & Cfb & 467 & 0.28 & 0.16 & 1.6 & [38] \\  & British Columbia & & 526 & 0.36 & 0.21 & 1.70 & [39] \\ Canada & (Vancouver) & Cfb & 532 & 0.36 & 0.20 & & [34] \\ France & Ile-de-France (Paris) & Cfb & 532 & 0.36 & 0.20 & & [34] \\ South Korea & Kwangju-jikhalsi & Dfa & 617 & 0.45 & 0.25 & 1.80 & [40] \\ Germany & Berlin & Dfb & 625 & 0.30 & 0.20 & & [34] \\ Turkey & Yozgat & Dsb & 671 & 0.40 & 0.25 & 2.40 & [36] \\ United States & New York & Dfa & 674 & 0.46 & 0.15 & 1.81 & [41] \\ Hungary & Budapest & Dfa & 675 & 0.45 & 0.25 & 1.80 & [34] \\ South Korea & Seoul-jikhalsi & Dwa & 722 & 0.47 & 0.29 & 3.84 & [40] \\ United States & Alaska (Anchorage) & Dsc & 895 & 0.32 & 0.16 & 1.81 & [41] \\ Canada & Ontario (Ottawa) & Dfb & 954 & 0.32 & 0.20 & 1.70 & [39] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Some of the cities studied, Köppen–Geiger classification, and U-values for representative construction elements. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Values in (m^2/W)** & \begin{tabular}{c} **Poorly Insulated** \\ **Building** \\ \end{tabular} & \
\begin{tabular}{c} **Intermediately** \\ **Insulated Building** \\ \end{tabular} &
\begin{tabular}{c} **Highly Insulated** \\ **Building** \\ \end{tabular} \\ \hline Walls & 0.0028 & 0.0042 & 0.0075 \\ Roof & 0.0047 & 0.0072 & 0.0138 \\ \hline \hline \end{tabular}
\end{table}