## 3 Results  
### Annual Energy Sumss  
Annual solar energy sums were derived from the appropriate data base of each site for every tilt angle in the range \(5^{\circ}\)-\(55^{\circ}\) (in \(5^{\circ}\) increments) by using both ground albedos, \(\rho_{\rm g0}\) and \(\rho_{\rm g}\) in Equations (1)-(7). From the calculated annual energy sums, the maximum sum and its corresponding (optimum) tilt angle were then obtained. Table 2 shows these maximum annual H\({}_{\rm g,tot.~{}\beta}\) sums for all sites with the corresponding optimum tilts, opt. \(\beta\) (columns 2 and 3 for \(\rho_{\rm g0}\) and \(\rho_{\rm g}\), respectively). From this table, it is seen that the average H\({}_{\rm g,tot.~{}\beta}\)/\(\rho_{\rm g0}\) (column 2) is 2745.5 kWhm\({}^{-2}\), while the average H\({}_{\rm g,tot.~{}\beta}\)/\(\rho_{\rm g}\) (column 3) is 2769.7 kWhm\({}^{-2}\), i.e., a 0.9% increase. Because of this small difference, the annual maximum solar energy sums from calculations with \(\rho_{\rm g}\) were only considered in this study, as done in [1]. The reason for using both ground-albedo values was to show the difference in the derived maximum annual energy sums (columns 2, and 3 in Table 2). Therefore, it is interesting to see how the annual energy sums for the opt. \(\beta\)-derived values under the \(\rho_{\rm g}\) calculations (H\({}_{\rm g,tot.~{}\beta}\)/\(\rho_{\rm g}\)) are distributed across the 82 sites in Saudi Arabia. Figure 2 shows this spatial coverage of the country. Surprisingly no consistent pattern in terms of opt. \(\beta\) exists; to the contrary, there is a great mix of opt. \(\beta\) in the country, which does not seem to follow any particular logic. This may be attributed to intrinsic errors when deriving the solar horizontal radiation values in the PV-GIS tool. Therefore, such a distribution has no practical value to the solar energy industry as no clear application zones are formed as energy-application guidelines (as in the case of southward-oriented solar collectors in [1]) and may simply create confusion about the selection of the most appropriate tilt angle for the installation of a solar system at any particular location in Saudi Arabia. This unexpected outcome led to the idea of an 'innocent manipulation' of the opt. \(\beta\) values. This manipulation was based on the adoption of the three solar energy zones (SEZ) defined in [1]. This means that at a site for which the methodology has selected a 'wrong' opt. \(\beta\), a manual selection of the 'correct' tilt (according to the SEZ where the site itself belongs to) was made. Then, the corresponding annual solar energy sum was obtained. This is shown in the fourth column of Table 2, which provides the correct distribution of the 82 sites along the 3 SEZs (SEZ-A with selected opt. \(\beta=40^{\circ}\), SEZ-B with selected opt. \(\beta=45^{\circ}\) and SEZ-C with selected opt. \(\beta=50^{\circ}\)). The corrected distribution is shown in Figure 3, which coincides with that of Figure 4 in [1]. After this correction, one would like to see what the created solar energy-sum differences are; Figure 4 shows these differences across all 82 sites. These energy differences, \(\Delta\)H\({}_{\rm g,\rm k,\beta/\rho_{\rm g}}\), are defined as: \(\Delta\)H\({}_{\rm g,\rm k,\rm B/\rho_{\rm g}}\) = H\({}_{\rm g,\rm tot.~{}\beta}\)/\(\rho_{\rm g}\) = H\({}_{\rm g,\rm tot.~{}\beta}\)/\(\rho_{\rm g}\) = H\({}_{\rm g,\rm tot,selected~{}\rm opt.~{}\beta}\)/\(\rho_{\rm g}\). From Figure 4a, it is seen that most of the sites 1-43 do not produce large differences, while the sites 44-82 result in annual energy deficits as low as \(-55\) kWhm\({}^{-2}\)year\({}^{-1}\). The mean annual solar energy sums for all sites 1-43 and 44-82 are 3026.3 kWhm\({}^{-2}\)year\({}^{-1}\) and 2954.7 kWhm\({}^{-2}\)year\({}^{-1}\), respectively, while their corresponding mean differences are \(-4.4\) kWhm\({}^{-2}\)year\({}^{-1}\) and \(-5.7\) kWhm\({}^{-2}\)year\({}^{-1}\).