## 4 Calculation of Monthly Total Energy on an Inclined Surface  
The total solar energy which is received by a horizontal surface depends on the prevailing meteorological conditions and astronomical factors. For practical purposes, we use Tables with the monthly mean daily solar radiation values obtained from measurements on horizontal surfaces as shown in Table 13. The data refers to the mean daily values on the 15 th day of each month. The table shows the day number (\(d_{n}\)), the solar declination (\(\delta\)), and the sunset hour angle (\(\omega_{s}\)) in degrees, the measured shortwave radiation in kWh/m\({}^{2}\)/d, the radiation indices, the daily totals at the top of the atmosphere in kWh/m\({}^{2}\)/d, and the monthly totals of the measured shortwave variables in kWh/m\({}^{2}\)/month. Details of the calculation procedure are given in [24; 26]. In practice, photovoltaics are installed with a slope with respect to a horizontal surface and in southern direction. For the estimation of solar radiation on an inclined surface,  
Figure 29: Relationships between various daily radiation indices. there are three types of solar radiation, namely, (i) beam; (ii) diffuse; and (iii) reflected solar radiation from the surfaces surrounding it. Hence, there are three conversion factors, i.e., for beam (\(R_{B}\)), diffuse (\(R_{D}\)) and reflected (\(R_{R}\)) solar radiation factors. These conversion factors convert the beam and diffuse solar radiation of a horizontal surface to those of an inclined surface. The beam radiation which is received by an inclined surface depends on the incident angles on the horizontal and inclined surfaces. When the solar panel is installed on a horizontal surface with respect to south the incident angle is the solar zenith angle which is given in Equation (2). When the solar panel is installed with a slope \(\beta\) with respect to the horizontal level and in the south direction, the incident angle (\(\theta\)) has the following form:  
\[\cos(\theta)=\cos(\phi-\beta)*\cos(\delta)*\cos(\omega)+\sin(\phi-\beta)*\sin (\delta) \tag{52}\]  
where \(\phi\) is the latitude, \(\delta\) is the solar declination and \(\omega\) is the hour angle. Then the factor \(R_{B}\) can be estimated from the ratio of the incident angles:  
\[R_{B}=\frac{\cos(\theta)}{\cos(\theta_{2})}=\frac{\cos(\phi-\beta)*\cos( \delta)*\cos(\omega)+\sin(\phi-\beta)*\sin(\delta)}{\cos(\phi)*\cos(\delta)* \cos(\omega)+\sin(\phi)*\sin(\delta)} \tag{53}\]  
As a result, \(R_{B}\) depends on the latitude (\(\varphi\)), day of the year (\(\delta\)), hour of day (\(\omega\)) and the slope of the panel (\(\beta\)). Therefore, it changes during the day and the time of the year. The daily calculation of \(R_{B}\) can be done through the integration of the nominator of the solar hour angle from sunrise to sunset of Equation (53) for the inclined surface:  
\[R_{B}=\frac{\cos(\phi-\beta)*\cos(\delta)*\sin(\omega^{\prime}_{s})+(\pi/180)* \omega^{\prime}_{s}*\sin(\phi-\beta)*\sin(\delta)}{\cos(\phi)*\cos(\delta)* \sin(\omega_{s})+(\pi/180)*\omega_{s}*\sin(\phi)*\sin(\delta)} \tag{54}\]  
where \(\omega_{s}\) is the sunset hour angle when the panel is horizontal, and \({\omega_{s}}^{\prime}\) is the sunset hour angle when the panel has a slope \(\beta\) with respect to the horizontal surface. Both equations have the following form:  
\[\omega_{s}=\cos^{-1}(-\tan(\phi)*\tan(\delta)) \tag{55}\]  
\[\omega^{\prime}_{s}=\min(\omega_{s},\cos^{-1}(-\tan(\phi-\beta)*\tan(\delta))) \tag{56}\]  
The conversion factor for the diffuse radiation (\(R_{D}\)) is defined as the ratio of diffuse radiation incident on an inclined surface to that in a horizontal surface. The diffuse factor can be estimated by considering the sky as an isotropic source of diffuse radiation. Then \(R_{D}\) is given as follows:  
\[R_{D}=(1+\cos(\beta))/2 \tag{57}\]  
The reflected solar radiation is the reflected radiation from the ground and other objects near surface of interest.