## 2 Mathematical Model  
### Degree-Day Method Incorporating Solar Radiation Effect  
The degree-day method is a fundamental and practical tool for estimating heating and cooling energy demand in buildings. Its simplicity, coupled with its ability to provide valuable insights into energy consumption patterns and climate change impacts, makes it a valuable resource for energy analysis and building design. For these reasons, the degree-day method has been preferred in many studies in the literature [16; 33; 34; 35; 36]. This method involves calculating heating degree days (_HDD_) and cooling degree days (_CDD_) to assess the energy required for heating and cooling purposes. _HDD_ quantifies the energy needed to heat a building to a comfortable level above a base temperature (\(T_{b}\)), while _CDD_ represents the energy required to cool a building to maintain comfort below a specific threshold temperature [37]. _HDD_ and _CDD_ are calculated by summing the differences between the average temperature of a specific day (\(T_{0}\)) and the base temperature (\(T_{b}\)). A positive difference signifies the need for energy to cool the building, while a negative difference indicates the need for energy to heat the building. A difference of zero means no heating or cooling is required. Equations (1) and (2) can be used to determine the total annual _HDD_ and _CDD_ values during the warmer periods and colder months, respectively. \[HDD=\sum_{d=1}^{365}(T_{b}-T_{0})\hskip 56.905512 ptfor\ T_{b}>T_{0} \tag{1}\]  
\[CDD=\sum_{d=1}^{365}(T_{0}-T_{b})\hskip 56.905512 ptfor\ T_{0}>T_{b} \tag{2}\]  
Solar radiation has not been mentioned in this section so far. In order to obtain more accurate results in practical applications, this effect should also be included in the calculations. Therefore, \(T_{sol}\), which is called solar air temperature, should be used instead of average daily temperature (\(T_{0}\)) in the above equations. Solar air temperature can be determined using the following equation [38]. \[T_{sol}=T_{0}+\frac{\dot{q}_{s}\alpha_{s}}{h_{0}}+\frac{\varepsilon\sigma\Big{(}T_{ sky}{}^{4}-T_{0}{}^{4}\Big{)}}{h_{0}} \tag{3}\]  
In this equation, \(T_{0}\) represents the daily average temperature value, as mentioned before. Additionally, \(\dot{q}_{s}\) is the incident solar radiation, \(\alpha_{s}\) is the solar absorptance of the surface, \(h_{0}\) is the combined radiation and convection coefficient, \(\varepsilon\) is the radiative emission coefficient of the surface, \(\sigma\) is the Stefan-Boltzmann constant, and \(T_{sky}\) is the sky and surrounding surface temperature [33]. In Equation (3), the second term represents the heat impact from solar exposure on the opaque surface. Here, the incident solar radiation (\(\dot{q}_{s}\)) needs to be determined, comprising reflected, direct, and diffuse solar radiations. This radiation value, which includes reflected, direct, and diffuse solar radiation components, is computed using Equation (4). \[\dot{q}_{s}=\dot{q}_{h}\Bigg{(}1-\frac{\dot{q}_{h,d}}{\dot{q}_{h}}\Bigg{)}R_{b }+\dot{q}_{h}\rho\bigg{(}\frac{1-\cos\beta}{2}\bigg{)}+\dot{q}_{h,d}\bigg{(} \frac{1+\cos\beta}{2}\bigg{)} \tag{4}\]  
Average daily diffuse radiation for the horizontal surface, denoted by the symbol (\(\dot{q}_{h,d}\)) in Equation (4), is calculated with the help of Equation (5) [39]. In this formula, \(\dot{q}_{h}\) represents the monthly average daily global solar radiation, while \(\dot{q}_{0,h}\) denotes the monthly average daily extraterrestrial radiation. In addition, the \(R_{b}\) value, which appears in Equation (4), can be calculated by dividing the azimuth angle by the zenith angle as shown in Equation (6). Azimuth angle and zenith angle are crucial parameters in the context of solar energy systems. The azimuth angle refers to the horizontal angle measured clockwise from true north to the direction of the sun. In contrast, zenith angle is the angle between the sun and the vertical, representing the angle of incidence of sunlight on a surface perpendicular to the sun's rays [40].