## 2 Mathematical Model  
### Degree-Day Approach and Solar Radiation Effect  
Various methods are available for calculating the annual energy consumption. The degree-day approach is the most reasonable and essential approach to obtain a building's annual energy consumption [6; 20; 22; 23]. According to this approach, a building's energy requirement is correlated with the deviation from the base temperature (\(T_{b}\)) and the daily mean outdoor temperature. The base temperature represents the outdoor temperature, below or above which heating or cooling is required [24]. By accounting for the impact  
Figure 1: Natural gas price in recent years. of solar irradiance on the building exterior, the annual heating degree days (HDDs) is formulated as [24]  
\[HDD=\sum_{d=1}^{365}(T_{b}-T_{sol-air})^{+} \tag{1}\]  
where \(T_{b}\) and \(T_{sol-air}\) are the base temperature and solar air temperature, respectively. The \({}^{+}\) symbol above the parenthesis states that values exceeding zero should be included in the count; therefore, in the case of \(T_{sol-air}>T_{b}\), the temperature variance should be considered zero. Similarly, cooling degree days (CDDs) is calculated using Equation (2):  
\[CDD=\sum_{d=1}^{365}(T_{sol-air}-T_{b})^{+} \tag{2}\]  
This time, the variance in temperature should be considered zero in the case of \(T_{b}>T_{sol-air}\) in Equation (2). In place of the average outdoor temperature for the day, solar air temperature (\(T_{sol-air}\)) is used to account for the thermal load generated by solar irradiance when calculating the cooling and heating degree days. The term \"solar air temperature\" relates to both the solar radiative flux and the external ambient air temperature [25]. \[T_{sol-air}=T_{0}+\frac{\alpha_{s}\dot{q}_{s}}{h_{0}}-\frac{\varepsilon\sigma \left(T_{0}{}^{4}-T_{surr}{}^{4}\right)}{h_{0}} \tag{3}\]  
In this equation, \(\alpha_{s}\) represents the surface's solar absorption rate, \(T_{0}\) is the outdoor air temperature, \(h_{0}\) is the combined radiation and convection coefficient of the exterior surface, \(\dot{q}_{s}\) is the solar irradiance reaching the surface, \(\sigma\) is the [PERSON] constant, \(\varepsilon\) is radiative emission coefficient of surface, and \(T_{surr}\) is the sky and surrounding surface temperature. The second term denotes the heat from solar exposure impact on the opaque surface in Equation (3). If \(T_{surr}\) is not equal to \(T_{0}\), a correction factor for heat transfer via radiation between the surface and its surroundings is added to the equation, which is the last term. The solar irradiance received by a surface is contingent upon its orientation and slope. Solar irradiance received by vertical exterior walls can be determined using the following equations [26]. Firstly, the clearness index (\(K_{T}\)) is determined through Equation (4):  
\[K_{T}=\frac{\dot{q}_{h}}{\dot{q}_{0,h}}=\left(a+b\frac{S}{S_{0}}\right) \tag{4}\]  
Here, \(\dot{q}_{0,h}\) is the monthly mean daily extraterrestrial radiation, \(\dot{q}_{h}\) is the monthly mean daily global solar radiation, \(a\) and \(b\) are the empirical parameters relating to the zone, \(S/S_{0}\) is the relative sunshine duration. The \(a\) and \(b\) parameters are determined depending on the declination angle (\(\delta\)), altitude (\(Z\)), and latitude angle (\(\varnothing\)) with the following equations [27]:  
\[a=0.103+0.000017Z+0.198\text{cos}(\varnothing-\delta) \tag{5}\]  
\[b=0.533-0.165\text{cos}(\varnothing-\delta) \tag{6}\]  
The monthly mean daily extraterrestrial radiation can be computed using the following equation [28]:  
\[\dot{q}_{0,h}=\frac{G_{s,c}}{\pi}\left[1+0.033\text{cos}\left(n\frac{360}{36 5}\right)\right]\left[\text{cos}\varnothing\text{cos}\delta\text{sin} \omega_{s}+\frac{2\pi\omega_{s}}{360}\text{sin}\varnothing\text{sin}\delta\right] \tag{7}\]  
Here, \(n\) is the day of the year, \(\omega_{s}\) is the month's sunset hour angle, and \(G_{s,c}\) is the solar constant taken as 1367 W/m\({}^{2}\)[28].