## 3 Results
### Annual Solar Energy  
Annual solar energy sums were derived from the appropriate data base of each site by utilising MODEL = HAY in Equation (3). The annual solar energy sum (or yield) at each location was estimated by aggregating (summing up) all hourly solar radiation values within the period 2005-2016. These aggregations were done for the three configuration modes of solar systems mentioned in Section 2. Similar annual solar energy sums were derived in previous works [30; 31; 32] at the 82 sites by utilising MODEL = ISO. It should be noted here that for the system types (i) and (ii), certain optimal tilt angles were found in those works, and the country was divided into three solar energy zones (SEZ), which are depicted in Figure 2. The same optimum tilt angles were adopted here for the HAY model. Figure 3 shows the variation in the annual solar energy sums across the 82 sites; these sums were calculated by using both the ISO and the HAY models (see Equation (3)) applied to the three configuration modes of solar systems (see Equations (1)-(4)). It must be said here that all ISO and HAY calculations were performed by taking into account an annually averaged near-real ground-albedo value, \(\rho_{\rm g}\), specific for each site applied to Equation (4). The annual mean solar irradiation values found are: for a configuration-type-(i) solar system, 2414.16 kWhm\({}^{-2}\) year\({}^{-1}\) (ISO) and 245236 kWhm\({}^{-2}\) year\({}^{-1}\) (HAY); for a configuration-type-(ii) solar system, 2973.59 kWhm\({}^{-2}\) year\({}^{-1}\) (ISO) and 3170.38 kWhm\({}^{-2}\) year\({}^{-1}\) (HAY); and for a  
Figure 2: Distribution of the 82 sites in Saudi Arabia into the 3 identified SEZs. For configuration-type-(i) solar systems, the optimum tilt angle is: \(\beta=20^{\circ}\) S (SEZ-A), \(\beta=25^{\circ}\) S (SEZ-B), \(\beta=30^{\circ}\) S (SEZ-C). For configuration-type-(ii) solar systems, the optimum tilt angle is: \(\beta=40^{\circ}\) (SEZ-A), \(\beta=45^{\circ}\) (SEZ-B), \(\beta=50^{\circ}\) (SEZ-C). For configuration-type-(iii) solar systems, the tilt angle is always varying, and therefore, this mode is independent from the location of the site. The numbers in the circles coincide with those in Figure 1. This figure is a reproduction of Figure 4 in [30]. configuration-type-(iii) solar system, 3093.25 kWhm\({}^{-2}\) year\({}^{-1}\) (ISO) and 3319.00 kWhm\({}^{-2}\) year\({}^{-1}\) (HAY). From these figures, one can draw the following conclusions:  
* The configuration-(i) solar systems produce almost identical annual solar energy yields in both ISO and HAY cases (see blue symbols in Figure 3). * The annual mean solar energy yield increases from configuration-(i) to configuration-(iii) under the same calculation model (ISO or HAY). * The annual mean solar energy yields in each solar system configuration are greater under the HAY model than those under the ISO. * The annual mean solar energy yield generally increases from configuration-(i) ISO to configuration-(iii) HAY. * There is only one exception that regards the annual mean solar energy yield of the configuration-(ii) HAY being greater than that of the configuration-(iii) ISO, i.e., 3170.38 kWhm\({}^{-2}\) year\({}^{-1}\) (ii-HAY) and 3093.25 kWhm\({}^{-2}\) year\({}^{-1}\) (iii-ISO); the explanation is not obvious but it can be attributed to the effect of the variation in the ground albedo across Saudi Arabia on the solar radiation estimation on the inclined surface of mode-(ii) HAY and mode-(iii) ISO; this discrepancy is depicted in Figure 4 and is investigated further in Section 3.3. * The differences in the annual mean solar energy sums, \(\rm H_{g,HAY}-H_{g,ISO}\), are as little as 38.20 kWhm\({}^{-2}\) year\({}^{-1}\) for configuration mode (i), 196.79 kWhm\({}^{-2}\) year\({}^{-1}\) for configuration mode (ii), and 225.75 kWhm\({}^{-2}\) year\({}^{-1}\) for the configuration mode (iii); this outcome may justify the option of using the L-J transposition model to estimate solar radiation on inclined surfaces, especially those which face south.